Theory and Applications of Categories, Vol. 18, No. 22, 2007, pp. 665–732.

FAMILIAL 2-FUNCTORS AND PARAMETRIC RIGHT ADJOINTS

MARK WEBER

Abstract. We define and study familial 2-functors primarily with a view to the development of the 2-categorical approach to operads of [Weber, 2005]. Also included in this paper is a result in which the well-known characterisation of a category as a simplicial set via the Segal condition, is generalised to a result about nice monads on cocomplete categories. Instances of this general result can be found in [Leinster, 2004], [Berger, 2002] and [Moerdijk-Weiss, 2007b]. Aspects of this general theory are then used to show that the composite 2-monads of [Weber, 2005] that describe symmetric and braided analogues of the ω-operads of [Batanin, 1998], are cartesian 2-monads and their underlying endo-2-functor is familial. Intricately linked to the notion of familial 2-functor is the theory of fibrations in a finitely complete 2-category [Street, 1974] [Street, 1980a], and those aspects of that theory that we require, that weren’t discussed in [Weber, 2007], are reviewed here.

1. Introduction

The category Fam(X) of families of objects of a given category X is one of the most basic constructions in category theory. Indeed the fibration Fam(X)→Set which takes a family of objects to its indexing set is one of the guiding examples for the use of fibrations to organise logic and type theory [Bénabou, 1985] [Jacobs, 1999].

We shall view the assignment X ↦ Fam(X) as the object part of an endo-2-functor of CAT. The endo-2-functor Fam has many pleasant properties. It is parametrically representable^{1} in the sense of [Street, 2000]. Moreover, it preserves all types of fibrations one can define in a finitely complete 2-category [Street, 1974] [Street, 1980a]. It happens that all of Fam’s pleasant properties can be derived from two axioms that can be imposed on 2-functors between finitely complete 2-categories. The general concept is that of familial 2-functor, and is the main subject of this paper.

This work was completed while the author was a postdoctoral fellow at UQAM and Macquarie University, and I am indebted to these mathematics departments, André Joyal and Michael Batanin for their support. During the production of this work I have benefited greatly from discussions with Michael Batanin, Clemens Berger, Steve Lack, Nicola Gambino and André Joyal, as well as some insightful remarks of an anonymous referee.

Received by the editors 2007-02-27 and, in revised form, 2007-12-20. Transmitted by Stephen Lack. Published on 2007-12-31. 2000 Mathematics Subject Classification: 18D50, 55P48. Key words and phrases: gerbes, motives, tannakian categories. \begin{matrix}c \\ ◯ \end{matrix} Mark Weber, 2007. Permission to copy for private use granted. 1Set-valued parametrically representable functors were first identified by Diers in [Diers, 1977], and

called familially representable in [Carboni-Johnstone, 1995]. The general notion makes sense for functors between arbitrary categories [Weber, 2004].

The main motivation for this work is to facilitate the development of the 2-categorical approach to operads initiated in [Weber, 2005]. According to [Weber, 2005] the environment for a notion of operad is a triple (K, T, A) where K is a 2-category with products, T is a 2-monad on K and A is an object of K which has the structure of a monoidal pseudo-T -algebra. To obtain the classical examples K is CAT the 2-category of categories, T is the symmetric monoidal category monad, to say that A has a monoidal pseudo-T -algebra structure is to say that A is a symmetric monoidal category, and unwinding the general definition of operad in this case gives a sequence of objects of the category A together with symmetric group actions and substitution maps as with the usual definition of operad. To obtain the operads of [Batanin, 1998] as part of this setting, one takes K to be the 2-category of globular categories, and T to be D_{s} as defined in [Batanin, 1998].

The classical theory of operads in a good symmetric monoidal category A has four basic formal aspects. First the category of symmetric sequences in A (these are sequences of objects of A with symmetric group actions) has a monoidal structure, and monoids for this monoidal structure are operads. Second is the related fact that an operad in A determines a monad on A with the same algebras as the original operad. Third is the construction of the free operad on a symmetric sequence, and fourth is the process of freely adding symmetric group actions to an operad. One may consider this last aspect as relating different operad notions (symmetric and non-symmetric). Each of these formal aspects have more complicated analogues in the theory of ω-operads [Batanin, 1998] [Batanin, 2002]. In addition to this, one has the idea of operads internal to other operads of [Batanin, 2002]. The construction of operads which are universal among those which have a certain type of internal operad structure, has been shown in [Batanin, 2002] and [Batanin, 2006] to be fundamental to the theory of loop spaces.

To give a conceptual account of these formal aspects in the general setting of [Weber, 2005] requires an understanding of how one should specialise K, T and A. In the classical theory the formal aspects require that A have colimits that interact well with the monoidal structure. For the general setting then, it is desirable that one can discuss cocomplete objects in K in an efficient way. Thus K should be a 2-topos in the sense of [Weber, 2007]. In order for the 2-monads T in this general setting to interact well with this theory of internal colimits, they must have a certain combinatorial form. A complete discussion of this is deferred to [Weber], in which the notion of an analytic 2-monad on a 2-topos is defined and identified as the appropriate setting. In this paper we shall focus on those properties that an analytic 2-monad (T, η, μ) enjoys that don’t involve the size issues that are encoded by the notion of 2-topos. In particular T is a familial 2-functor, T 1 is a groupoid^{2} and η and μ are cartesian. By definition such 2-monads are cartesian monads in the usual sense, but interestingly, they are also cartesian monads in a bicategorical sense (see remark(7.13) below). The general theory of operads, using the notions developed in [Weber, 2007] and this paper, will be presented in [Weber].

This paper is organised as follows. In section(2) the notions of polynomial functor and parametric right adjoint (p.r.a) are recalled. While polynomial functors are interesting and important, p.r.a functors are more general, and there are examples fundamental to higher category theory which are p.r.a but not polynomial. Then in section(3) the theory of split fibrations internal to a finitely complete 2-category [Street, 1974] is recalled and in some cases reformulated in a way more convenient for us. Moreover that section includes any of the background in 2-category theory that we require, and that was not discussed already in [Weber, 2007]. Section(4) is devoted to the nerve characterisation theorem, which generalises the characterisation of categories as simplicial sets via the Segal condition to a result about a monad with arities on a cocomplete category. Moreover in that section applications to the theory of Dendroidal sets [Moerdijk-Weiss, 2007a] [Moerdijk-Weiss, 2007b], unpublished work of Tom Leinster [Leinster, 2004] and to the work of Clemens Berger [Berger, 2002] are provided.

2An object B in a 2-category K is a groupoid when for all X the hom-category K(X, B) is a groupoid, or in other words, when every 2-cell with 0-target B is invertible.

A self-contained discussion of the Fam construction and the definition of familial 2functor is presented in section(5.1). The basic properties of familial 2-functors are developed in section(6), and results enabling us to exhibit many examples of familial 2-functors are presented in section(7). Aspects of the nerve characterisation theorem are then used in section(8) to show that the unit and multiplication of the composite 2-monads of [Weber, 2005], whose corresponding operad notions include symmetric and braided versions of the higher operads of [Batanin, 1998], as cartesian.

The definitions and notations of [Weber, 2007] are used freely throughout this one. In particular, for a category ℂ we denote by ℂ̂ the category of presheaves on ℂ, that is the functor category [ℂ^{op}, Set]. When working with presheaves we adopt the standard practises of writing C for the representable ℂ(−, C) ∈ ℂ̂ and of not differentiating between an element x ∈ X(C) and the corresponding map x : C→X in ℂ̂. We denote by CAT(ℂ̂) the functor 2-category [ℂ^{op}, CAT] which consists of functors ℂ^{op} → CAT, natural transformations between them and modifications between those. We adopt the standard notations for the various duals of a 2-category K: K^{op} is obtained from K by reversing just the 1-cells, Kco is obtained by reversing just the 2-cells, and Kcoop is obtained by reversing both the 1-cells and the 2-cells.

When doing basic category theory, one can get a lot of mileage out of expressing one’s work in terms of the paradigmatic good yoneda structure [Street-Walters, 1978] [Weber, 2007] on CAT. In particular these “yoneda methods” always seem to lead to the most efficient proofs of basic categorical facts, and we shall adopt this approach and associated notation throughout this work. The basic construct of this structure is that given a functor f:A→B such that A is locally small, and for all a and b the hom B(fa, b) is small, one can organise the arrow maps of f into a 2-cell

[figure: A; f; y_{A}; B; B(f,1); Â; χ^{f}]

which is both an absolute left lifting and a pointwise left extension. See [Weber, 2007] (definition(3.1) and example(3.3)) for a fuller discussion of good yoneda structures and this example in particular. If B is itself locally small then y_{B}f enjoys the same size condition that f did above^{3}, and then B̂(y_{B}f, 1) is the functor B̂→Â obtained by precomposing with f^{op}, and so we denote it by res_{f}. If in addition A is small then res_{f} has both adjoints. Of course this is very basic category theory, but these adjoints have useful canonical yoneda structure descriptions. The right adjoint is obtained as Â(B(f, 1), 1) (see [Weber, 2007] proposition(3.7)), and the left adjoint lan_{f} is obtained as a pointwise left extension

[figure: A; y_{A}; f; Â; _{lan_{f}}; B; y_{B}; B̂; ϕ_{f}]

of y_{B}f along y_{A} (see [Weber, 2007] theorem(3.20)), and since y_{A} is fully faithful ϕ_{f} is an isomorphism.

2. Polynomial functors and parametric right adjoints

In this section we recall and develop the notion of parametric right adjoint functor. Such functors have been considered by several authors at various levels of generality [Carboni-Johnstone, 1995] [Street, 2000] [Weber, 2004]. Their importance to higher dimensional algebra was first noticed by Mike Johnson [Johnson, 1987]. Parametric right adjoints are a generalisation of the more commonly studied polynomial functors [Kock] which we recall first. Recently polynomial functors have been applied in categorical study of dependent type theories [Moerdijk-Palmgren, 2000] [Gambino-Hyland, 2004] and in a conceptual description of opetopes [Batanin-Joyal-Kock-Mascari]. However, as we argue in example(2.5) below, this class of functors is not quite general enough for higher dimensional algebra, hence the need for parametric right adjoints.

Let A be a finitely complete category. For f : X→Y in A we recall that the functor f_{!} : A/X→A/Y given by composition with f, has a right adjoint denoted by f^{∗} which is given by pulling back along f. When f^{∗} has a further right adjoint, denoted f_{∗}, f is said to be exponentiable. The category A is said to be locally cartesian closed when f_{∗} exists for every f, and from [Freyd, 1972] we know that this is equivalent to asking that the slices of A are cartesian closed. Thus every elementary topos is locally cartesian closed, but CAT, the category of categories and functors is not. However in this case the exponential maps have been characterised as the Giraud-Conduché fibrations [Giraud, 1972] [Conduché, 1972], and this class of functors includes Grothendieck fibrations and opfibrations. For X ∈ A we denote by t_{X} : X→1 the unique map into the terminal object of A. Obviously the functor t_{X}_{!} : A/X→A has a simpler description: it takes the domain of a map into X. One may verify directly that t_{X}_{!} creates connected limits, and so for any f, f_{!} preserves connected limits since f_{!}t_{Y} _{!} = t_{X}_{!}.

3This size condition is called admissibility.

It is instructive to unpack these definitions in the case A = Set because then objects of A/X can be regarded either as functions into X or as X-indexed families of sets via the equivalence Set/X≃Set^{X} which sends a function to its fibres. Thus one can describe f_{!} as the process of taking coproducts over the fibres of f, and f_{∗} takes cartesian products over the fibres of f.

Now from maps

\xymatrix{W & X \ar[l]_{f} \ar[r]^{g} & Y \ar[r]^{h} & Z}

in a finitely complete category A, where g is exponentiable, one can consider the composite functor

\xymatrix{A/W \ar[r]^{f^{∗}} & A/X \ar[r]^{g_{∗}} & A/Y \ar[r]^{h_{!}} & A/Z} .

Functors which arise in this way are called polynomial functors^{4}. In particular when W = Z = 1 we write P_{g} for the corresponding polynomial functor.

2.1. Example. This example illustrates how polynomials with natural number coefficients can be regarded as polynomial functors. The polynomial p(X) = X^{3} + 2X + 2 can be regarded as an endofunctor of Set: interpret X as a set, product as cartesian product of sets and sums as disjoint unions. In fact p is the following polynomial endofunctor of Set:

\xymatrix{Set \ar[r]^{t_{5}^{∗}} & Set/5 \ar[r]^{f_{∗}} & Set/5 \ar[r]^{(t_{5})_{!}} & Set}

where 5 denotes the set {0, 1, 2, 3, 4} and f : 5→5 is given by f(0) = f(1) = f(2) = 0, f(3) = 1 and f(4) = 2. To see this we trace X ∈ Set through this composite, representing objects of Set/5 as 5-tuples of sets:

\xymatrix{X \ar@{|->}[r]^{t_{5}^{∗}} & (X, X, X, X, X) \ar@{|->}[r]^{f_{∗}} & (X^{3}, X, X, 1, 1) \ar@{|->}[r]^{(t_{5})_{!}} & p(X)}

Obtaining f : A→B from p is easy. Note that p(X) = X^{3} +X +X +1+1 which is a 5-fold sum and so B = 5, and the exponents of the terms of this sum tell us what the cardinality of the fibres of f are. In this way all polynomials with natural number coefficients can be regarded as polynomial endofunctors of the form P_{f} for f a function between finite sets, and vice versa, although different f’s can give the same polynomial. For example preor post-composing f by a bijection doesn’t change the polynomial corresponding to the induced endofunctor.

In much the same way, multivariable polynomials with natural number coefficients correspond to polynomial functors of the form h_{!}g_{∗}f^{∗} where

\xymatrix{W & X \ar[l]_{f} \ar[r]^{g} & Y \ar[r]^{h} & Z}

4In the literature authors often use the term “polynomial functor” for a special case of what we have described here. For example our polynomial functors correspond to what [Gambino-Hyland, 2004] call dependent polynomial functors, and [Kock] restricts most of the discussion to the case A = Set.

are functions between finite sets. The cardinality of W corresponds to the number of input variables and the cardinality of Z corresponds to the number of coordinates of output. Much of the elementary mathematics of polynomials can be categorified by re-expressing in terms of polynomial functors [Kock].

2.2. Example. For X ∈ Set we denote by MX the free monoid on X. The elements of MX may be regarded as finite sequences of elements of X and so one has

MX ≅ ∐_{n∈ℕ} X^{n}

and so M ≅ P_{f} where f : ℕ_{•}→ℕ,

ℕ_{•} = {(n, m) : n, m ∈ ℕ and n < m}

and f(n, m) = m. In other words the underlying endofunctor of the free monoid monad on Set is a polynomial functor. Bénabou showed that this construction works when Set is replaced by an arbitrary elementary topos with a natural numbers object [Bénabou, 1991].

For more on polynomial functors see [Kock]. We now begin our discussion of parametric right adjoints. Given a functor T : A→B and X ∈ A the effect of T on arrows into X may be viewed as a functor

T_{X} : A/X → B/T X.

When A has a terminal object 1, we identify A/1 = A and then the original T can be factored as

\xymatrix{A \ar[r]^{T_{1}} & B/T 1 \ar[r]^{^{t_{T 1}^{!}}} & B} .

2.3. Definition. [Street, 2000] Let A be a category with a terminal object. A functor T : A→B is a parametric right adjoint (henceforth p.r.a) when T_{1} has a left adjoint. We shall denote such a left adjoint as L_{T}. A monad (T, η, μ) on A is p.r.a when its functor part is p.r.a and its unit and multiplication are cartesian^{5}.

Notice in particular that p.r.a functors preserve pullbacks, and in fact all connected limits, since a p.r.a is a composite of a right adjoint and a functor of the form f_{!} [Carboni-Johnstone, 1995].

2.4. Example. Polynomial functors are p.r.a: if T is of the form h_{!}f_{∗}g^{∗} for f, g and h as above, then T_{1} may be identified with f_{∗}g^{∗}, and so L_{T} is g_{!}f^{∗}. In fact L_{T} is also p.r.a because (L_{T})_{1} may be identified with f^{∗}. In particular, note that for a polynomial functor T, L_{T} preserves monomorphisms.

5meaning that the naturality squares for μ and η are pullbacks.

2.5. Example. We shall now give an important example of a p.r.a functor which is not polynomial. For us a graph consists of a set V of vertices, a set E of edges, and source and target functions \xymatrix{s, t : E \ar[r] \ar[r] & V}. Thus the category Graph of graphs is the category of presheaves on the category that looks like this: \xymatrix{• \ar[r] \ar[r] & •}. Let T be the endofunctor of Graph which sends X ∈ Graph to the free category on X. That is, the vertices of T X are the same as those of X, and the edges of T X are paths in X. In particular T 1 has one vertex, and an edge for each natural number. Thus an object of Graph/T 1 is a graph whose edges are labelled by natural numbers, and a morphism of this category is a morphism of graphs which preserves the labellings. Now applying L_{T} to such a labelled graph involves replacing it each edge labelled by n ∈ ℕ by a path of length n. So when n = 0, one identifies the source and target of the given edge, when n = 1 one leaves the edge as it is, and for n > 1 one must add new intermediate vertices for each new path. To see that L_{T} really is described this way, it helps to describe it more formally as a left kan extension

[figure: ℕ; i; E_{T}; Graph/T 1; L_{T}; Graph]

of E_{T} along i, where i(n) consists of a single edge labelled by n between distinct vertices, and E_{T} (n) = [n], the graph whose vertices are natural numbers k such that 0≤k≤n and edges are k→(k+1) for 0≤k<n. Given this formal description it is easy to see that L_{T} ⊣ T_{1}.

Let f be the inclusion of the vertices of the labelled graph

\xymatrix{x \ar[r]^{0} & y} .

This is a monomorphism in Graph/T 1, but applying L_{T} to f gives the unique function {x, y}→1 viewed as a graph morphism between graphs with no edges. Thus L_{T} (f) is not a monomorphism, and so by example(2.4) T is not polynomial.

It is easy to adapt example(2.5) to show that the free strict n-category endofunctor on the category of n-globular sets, and the free strict ω-category functor on the category of globular sets are p.r.a but not polynomial. The endofunctors just mentioned are fundamental to the combinatorics of higher dimensional algebra.

We shall now recall some alternative characterisations of p.r.a functors. Recall from [Weber, 2004] that f : B→T A is T -generic when for any α, β, and γ making the outside of

[figure: B; α; f; T X; T γ; T A; T β; T δ; T Z]

commute, there is a unique^{6} δ for which γ ∘ δ = β and T (δ) ∘ f = α. Such a δ is called a T -filler for the given square.

2.6. Proposition. Let A be a category with a terminal object and T : A→B. Then the following statements are equivalent:

1. T is a parametric right adjoint.

2. For all A ∈ A, T_{A} is a right adjoint.

3. Every f : B→T A factors as

\xymatrix{B \ar[r]^{g} & T D \ar[r]^{T h} & T A}

where g is T -generic.

Proof. (1)⇒(3): Let f : B→T A and then we denote D_{f} = L_{T} (T (t_{A})f) and g_{f} : T (t_{A})f→T_{1}(D_{f}) as the component of the unit of L_{T} ⊣ T_{1} at T (t_{A})f. Thus we have a commutative square

[figure: B; f; g_{f}; T A; T t_{A}; T D_{f}; T t_{D_{f}}; T h_{f}; T 1]

in B, and by the universal property of g_{f} as the unit, a unique h_{f} making the two triangles commute. It suffices to show that g_{f} is generic. Suppose that we have α, β and γ as in the left square

\xymatrix{B \ar[rr]^{α} \ar[dd]_{g_{f}} &  & T X \ar[dd]^{T γ} & f \ar[rr]^{α} \ar[dd]_{g_{f}} &  & T_{1}X \ar[dd]^{T_{1}γ} \\  & = &  &  & = \\ T D_{f} \ar[rr]_{T β} &  & T Z & T_{1}D_{f} \ar[rr]_{T_{1}β} &  & T_{1}Z}

which is in B. Composing this with T t_{Z} and regarding the data in B/T 1 we obtain the square on the right. The unique filler δ : D_{f}→X is now obtained because of the universal property of g_{f} as the unit of the adjunction L_{T} ⊣ T_{1}. (3)⇒(2): For f : B→T A choose a factorisation

\xymatrix{B \ar[r]^{g_{f}} & T D_{f} \ar[r]^{T h_{f}} & T A}

of f with g_{f} generic. Then the genericness of g_{f} implies its universal property as the unit of an adjunction with right adjoint T_{A}, where the object map of the left adjoint is given by f ↦ h_{f}. (2)⇒(1): by definition.

6We are adopting a slight change of terminology from [Weber, 2004]. The T -generics defined here were called strict T -generic in [Weber, 2004], and a weaker notion in which the uniqueness of δ is dropped, are the T -generics of [Weber, 2004]. We only consider the strict notion in this paper.

A factorisation of f as in (3) is called a generic factorisation. From the proof of proposition(2.6) we have:

2.7. Scholium. A choice of L_{T} in proposition(2.6) amounts to a choice, for all f : B→T A of generic factorisation

\xymatrix{B \ar[r]^{g_{f}} & T D_{f} \ar[r]^{T h_{f}} & T A}

of f, such that for all α : A→A′, g_{f} = g_{T (α)f} and αh_{f} = h_{T (α)f}. In terms of this notation, the left adjoint to T_{A} has object map f ↦ h_{f}.

When working with a p.r.a functor T, we shall usually assume a choice of L_{T} has been made, and the notation of scholium(2.7) for the corresponding choice of generic factorisations will be used freely. In the case where T is the functor part of a p.r.a monad, these generic factorisations are in fact part of a factorisation system on Kl(T) [Berger, 2002], the Kleisli category of T, with every map of Kl(T) factoring as a generic map followed by a free map. The free maps are those which are in the image of the identity on objects left adjoint functor A→Kl(T). We shall reflect these circumstances in the notation we use: writing the chosen generic factorisation of a map f : B→A in Kl(T) as

\xymatrix{B \ar[r]^{g_{f}} & D_{f} \ar[r]^{h_{f}} & A}

where g_{f} is generic and h_{f} is free. Moreover we shall assume that our choice of generic factorisations is normalised: namely that when f is itself a free map D_{f} = B, g_{f} = id and h_{f} = f. This is clearly always possible and is a useful simplification. Indeed, another way to express the condition that our choice of generic factorisations is normalised, is to say that all the components of the unit η are chosen generics.

2.8. Example. Let T be the free category endofunctor of Graph and recall the graphs [n] from example(2.5). To give a graph morphism f : [1]→T X is to give a path in X, that is, a graph morphism p : [n]→X. Writing g : [1]→T [n] for the path in [n] starting at 0 and finishing at n (ie the maximal path), we have a factorisation

\xymatrix{[1] \ar[r]^{g} & T [n] \ar[r]^{T p} & T X}

of f which the reader may easily verify is the generic factorisation of f.

2.9. Example. Let p : E→B be a discrete fibration between small categories. Also denote by p the presheaf on B corresponding to p : E→B via the Grothendieck construction. Then the functor lan_{p} : Ê→B̂ obtained by left kan extension along p^{op}, factors as the canonical equivalence Ê≃B̂/p followed by the domain functor B̂/p→B̂. We may identify this canonical equivalence with (lan_{p})_{1} and so lan_{p} is p.r.a.

Parametric right adjoints between presheaf categories are particularly easy to understand. In particular, to verify p.r.a’ness of a functor T one need only generically factor maps with whose domain is representable and codomain is T 1.

2.10. Proposition. For a functor T : 𝔹̂→ℂ̂ the following statements are equivalent:

1. Any f : C→T 1 has a generic factorisation where C ∈ ℂ (is regarded as a representable presheaf).

2. T is p.r.a.

Proof. (2)⇒(1) by proposition(2.6). On the other hand given (1) one can define the functor

E_{T} : y/T 1 → 𝔹̂ x : C→T 1 ↦ D_{x}

and by the definition of “generic” the assignment

z : C→T Z ↦ h_{z} : D_{z}→X

and scholium(2.7) gives bijections

T (Z)(C) ≅ ∐_{x∈T 1(C)} ℂ̂(E_{T} (x), Z).  (1)

Thus T_{1} ≅ ℂ̂(E_{T} , 1) and so L_{T} is obtained as the left kan extension of E_{T} along the composite of the yoneda embedding and the canonical equivalence { \atop y/T 1}≃ℂ̂/T 1.

2.11. Remark. From the definition of E_{T} : y/T 1 → 𝔹̂ in the above proof and lemma(5.7) of [Weber, 2004], an object X ∈ 𝔹̂ is isomorphic to an object in the image of E_{T} iff there exists C ∈ ℂ a generic morphism C→T X.

2.12. Remark. In the proof of proposition(2.10) we reduced the data for a p.r.a T : B̂→Ĉ to the functor E_{T} : y/T 1 → B̂. Denoting by p : E→C the discrete fibration corresponding to T 1 by the Grothendieck construction (so E = y/T 1), T amounts to a span

\xymatrix{B̂ & E \ar[l]_{e} \ar[r]^{p} & C}

(ie e = E_{T}), and one can recapture T from such a span as the composite lan_{p}B̂(e, 1): lan_{p} is p.r.a by example(2.9) and B̂(e, 1) has left adjoint given by left kan extension as in the proof of proposition(2.10). Now a functor e : E→B̂ can in turn be regarded as a 2-sided discrete fibration with small fibres [Street, 1974] [Street, 1980a] [Weber, 2007], and writing

\xymatrix{B & D \ar[l]_{d} \ar[r]^{c} & E}

for the underlying span in Cat of this discrete fibration we see that the data for a p.r.a functor between presheaf categories consists of functors

\xymatrix{B & D \ar[l]_{d} \ar[r]^{c} & E \ar[r]^{p} & C}

between small categories such that the pair (d, c) form a discrete fibration from B to E, and p is a (one-sided) discrete fibration. In the case where the categories in question are all discrete, T is just the polynomial functor p_{!}c_{∗}d^{∗}. Hence if T : B̂→Ĉ and B and C are discrete, then T is p.r.a iff T is polynomial. Thus we have exhibited p.r.a functors between presheaf categories as a natural generalisation of polynomial functors defined from Set.

Moreover p.r.a functors between presheaf categories have been characterised

2.13. Theorem. [Weber, 2004] Let B and C be small categories. A functor T : B̂→Ĉ with rank is p.r.a iff it preserves connected limits.

although we shall not make much use of this characterisation in this paper. Instead we use the explicit descriptions provided by proposition(2.10) and remark(2.12). When giving a self-contained discussion of any particular example of a p.r.a monad T on a presheaf category ℂ̂, guided by these results one orders the discussion in the following way. First one describes T 1 and constructs the functor E_{T}. Then T is obtained as in the above remark. With these descriptions in hand one then is in a position to describe explicitly the unit and multiplication of the monad. Implicitly we followed this approach already in example(2.5) for the category monad on Graph, and in [Batanin, 1998] this was the way in which the strict ω-category monad on globular sets was described. To illustrate further we will now discuss in detail an example which is central to [Moerdijk-Weiss, 2007a] and [Moerdijk-Weiss, 2007b]. In particular one should compare the following example to the discussion in section(3) of [Moerdijk-Weiss, 2007a].

2.14. Example. A multigraph consists of a set of vertices and a set of multi-edges. A multi-edge goes from an n-tuple of vertices to a single vertex, and a typical such is depicted as

f : (a_{1}, ..., a_{n}) → b

where a_{i} and b are vertices. When n = 1 f is called an edge. A coloured operad^{7} is a multigraph X admitting:

1. units: for each vertex a, one has an edge 1_{a} : (a)→a.

2. symmetric group actions: for all n ∈ ℕ and vertices a_{i} and b for 1≤i≤n, one has an action of the n-th symmetric group on the set X(a_{1}, ..., a_{n}; b) of multi-edges from (a_{1}, ..., a_{n}) to b.

3. substitution of multi-edges: as in a multicategory satisfying associativity and unit laws, and equivariant with respect to the symmetric group actions.

Informally the free coloured operad T X on a multigraph X can be described as follows. The vertices are the same as those of X, and a multi-edge of T X may be pictured as a tree-with-permutations labelled in a compatible way by the vertices and multi-edges of X. To illustrate, here is a picture of a multi-edge (a, b, c, d, e)→z in T X:

7Coloured operads are also called symmetric multicategories, or as in [Moerdijk-Weiss, 2007a] [Moerdijk-Weiss, 2007b], are sometimes just referred to as operads.

[figure: a; b; c; d; e; f; g; h; k; z; w; x; y]

where

f : (b, c)→x g : (a, d, e)→w h : ()→y k : (w, y, x)→z

are multi-edges in X. We will now formalise this, and along the way exhibit T as a p.r.a monad on the presheaf category of multigraphs.

Denote by 𝕄 the following category. There is an object 0 and for n ∈ ℕ an object (n, 1). For each n ∈ ℕ there is an arrow τ_{n} : 0→(n, 1), and for 1≤i≤n there are arrows σ_{n,i} : 0→(n, 1). There are no relations. Clearly an object X of 𝕄̂ is a multigraph: the elements of X(0) are the vertices of X and the elements X(n, 1) are the multi-edges with source of length n.

The multigraph T 1 has one object which E_{T} sends to the representable multi-graph 0. Informally the multi-edges of T1 are trees-with-permutations. We now give an inductive definition of the multi-edges of T 1 and their images by the functor E_{T}. For each p ∈ T 1(n, 1) we must also define objects (σ_{1}, ..., σ_{n}, τ) of E_{T} (p) in order that the arrow map of the functor E_{T} be defined. We refer to σ_{i} as the i-th source of p and τ as the target of p. The initial step of the definition is that T 1(1, 1) has an element which gets sent to the representable 0 ∈ 𝕄̂ by E_{T}, and so σ_{1} and τ for this element are both equal to the unique object of 0. By induction p ∈ T 1(n, 1) is a triple (q, f, ρ), where q ∈ T 1(k),

f : {1, ..., n} → {1, ..., k}

is an order preserving function, and ρ is a permutation of n symbols. The multi-graph E_{T} (p) contains E_{T} (q) and has the same target, and in addition there are n new objects (a_{1}, ..., a_{n}), and k new multi-edges

(a_{j} : α(i)≤j≤β(i)) → σ′_{i}

where σ′_{i} is the i-th source of q and α(i) (resp. β(i)) is the least (resp. greatest) element of the fibre f^{−1}(i). The i-th source of p is given by σ_{i} = a_{ρi}. Having described T 1 and E_{T} one now obtains a description of T X for any X ∈ 𝕄̂ from equation(1).

Here are some examples to enable the reader to reconcile the formal description of T 1 just given with the intuitive idea of trees-with-permutations. The representables 0 and (2, 1) = (0, t_{2}, id), (0, t_{2}, (12)), and q = (0, t_{3}, (132)) are pictured as

[figure]

respectively, and (q, f, (132)) where f : {1, 2, 3, 4, 5}→{1, 2, 3} is given by f(1) = f(2) = 1 and f(3) = f(4) = f(5) = 2, is drawn as

and is an element of T 1(5, 1). The multigraph E_{T} p has one object for each edge of the tree and a multi-edge of arity k for each node with k edges coming in from above. So in this last example E_{T} p has 9 objects, a multi-edge of arity 0, a multi-edge of arity 2, and two multi-edges of arity 3. Thus equation(1) for T X expresses formally the intuition of trees-with-permutations labelled by X: by the equation a multi-edge of T X consists of p ∈ T 1(n, 1) together with a morphism E_{T} p→X in 𝕄̂, and such a morphism amounts to a labelling of the edges of the tree p by the objects of X. By definition T is a p.r.a endofunctor of 𝕄̂.

One reconciles the above discussion of trees with that of [Moerdijk-Weiss, 2007a], in which non-planar trees are used, by observing that in an obvious way one can associate a non-planar tree to any multi-edge of T 1, and p and q have the same associated non-planar tree iff E_{T} p≅E_{T} q. Thus one has a bijection between non-planar trees and isomorphism classes of multigraphs in the image of E_{T}.

There is an inclusion η_{X} : X→T X of multigraphs which is the identity on objects, and on arrows is given by

f : (a_{1}, ..., a_{n})→b ↦ [figure: a_{1}; a_{n}; f; b; ...]

Grafting of labelled trees provides substitution for T X, and permuting input edges of labelled trees provides symmetric group actions for the T X(n, 1) with respect to which multi-composition is equivariant. That is, T X is a coloured operad in an obvious way. Moreover the reader will easily verify^{8} directly that composition with η_{X} gives a bijection between morphisms T X→Z of coloured operads and morphisms X→Z of multigraphs, and so T X really is the free coloured operad on X. The monad multiplication of T encodes substitution of trees: given a tree whose nodes are labelled by labelled trees, that is a multi-edge of T^{2}(X), applying μ_{X} amounts to substituting the labels and removing the outer nodes. One may verify directly that μ_{X} and η_{X} are natural in X, and that for each X one has

8Indeed the definition of T X described here was selected so that this universal property follows pretty much by definition.

[figure: X; t_{X}; η_{X}; T X; T t_{X}; 1; η_{1}; T 1; T^{2}X; T^{2}t_{X}; μ_{X}; T X; T t_{X}; T^{2}1; μ_{1}; T 1]

and so (T, η, μ) is indeed a p.r.a monad on 𝕄̂.

The remainder of this section is devoted to two lemmas, which are useful technical consequences of parametric right adjointness. In fact the second of these results only requires that the functor in question preserves pullbacks. Let T : A→B be a functor between categories which have products, and

(A_{i} : i ∈ I)

be a family of objects of A. Writing p_{A_{i}} for the i-th projection of the product, one has the comparison map

\xymatrix{T (∏ A_{i}) \ar[r]^{k_{T,A_{i}}} & ∏ T (A_{i})}

defined by p_{T (A_{i})}k_{T,A_{i}} = T (p_{A_{i}}). In general this comparison map is natural in the A_{i}, however when T is p.r.a slightly more than this is true.

2.15. Lemma. Let T : A→B be a p.r.a functor between categories which have products. Then the comparison maps k_{T,A_{i}} just defined are cartesian natural^{9} in the A_{i}.

Proof. Given functors

\xymatrix{A \ar[r]^{T} & B \ar[r]^{S} & C}

between categories with products, if both S and T have cartesian natural product comparison maps and S preserves pullbacks, then the composite ST will have cartesian natural product comparison maps because these maps are given by the composites

\xymatrix{ST (∏ A_{i}) \ar[r]^{Sk_{T}} & S∏T (A_{i}) \ar[r]^{k_{S}} & ST ∏A_{i}}

for each family (A_{i} : i ∈ I) of objects of A. Thus since any p.r.a functor is a composite of a right adjoint and a functor of the form t_{X,!} : A/X→A, it suffices to show that the lemma holds for this last special case, and this is straight-forward to verify directly.

9meaning that the commuting squares that witness the naturality in the given variables are in fact pullbacks.

The final result of this section is the analogue of lemma(2.15) for coproducts. Recall that a category E is small-extensive [Carboni-Lack-Walters, 1993] when it has coproducts and when for each family of objects (X_{i} : i ∈ I) of E the functor

∏_{i∈I} E/X_{i} → E/(∑_{i∈I} X_{i})

which sends a family of maps (h_{i} : Z_{i}→X_{i}) to their coproduct is an equivalence. There are many examples of small-extensive categories: for instance every Grothendieck topos is small extensive, as is CAT. For a family of objects of E as above we shall denote the i-th coproduct inclusion as c_{X_{i}}. Let (f_{i} : X_{i}→Y_{i}) be a family of maps and recall that small-extensivity implies that each of the squares

\xymatrix{X_{i} \ar[d]_{f_{i}} \ar[r]^{c_{X_{i}}} & ∐ X_{i} \ar[d]^{∐ f_{i}} \\ Y_{i} \ar[r]_{c_{i}} & ∐ Y_{i}}

are pullbacks, and that given an arrow g : A→B and a family of pullback squares as shown on the left,

[figure: X_{i}; f_{i}; A; g; Y_{i}; B; ∐ X_{i}; ∐ f_{i}; A; g; ∐ Y_{i}; B]

then the induced square shown on the right in the previous display is also a pullback. Let T : A→B be a functor between small-extensive categories, and (A_{i} : i ∈ I) be a family of objects of A. One has the comparison map

\xymatrix{∐ T A_{i} \ar[r]^{k_{T,A_{i}}} & T (∐ A_{i})}

defined by k_{T,A_{i}}c_{T (A_{i})} = T (c_{A_{i}}). As an immediate consequence of the definitions and the basic consequences of small extensivity that we have just recalled, one has

2.16. Lemma. Let T : A→B be a pullback preserving functor between small-extensive categories. Then the comparison maps k_{T,A_{i}} just defined are cartesian natural in the A_{i}.

3. Some 2-categorical background

In this section we recall the descriptions of split fibrations, iso-fibrations and bicategorical fibrations internal to a finitely complete 2-category K. This theory is due to Ross Street [Street, 1974] [Street, 1980b]. We shall also provide a mild reformulation of Street’s theory of split fibrations that is useful for this work.

3.1. Split fibrations. One has a notion of (Grothendieck) fibration in any finitely complete 2-category [Street, 1974]. The definition is based on the idea of a cartesian 2-cell. For a functor f : A→B, recall that a morphism α : a_{1}→a_{2} in A is f-cartesian when for all α_{1} and β as shown:

[figure: fa_{1}; fα; fa_{2}; fa_{3}; β; fα_{1}; =]

there is a unique γ : a_{3}→a_{1} such that fγ = β and αγ = α_{1}. Then for f : A→B in K one says that a 2-cell

[figure: X; a_{1}; a_{2}; A; α]

is f-cartesian when for all g : Y →X in K, αg ∈ K(Y, A) is K(Y, f)-cartesian. A fibration in K is then defined to be a map f : A→B such that for all

[figure: X; a; b; A; f; B; β]

there exists f-cartesian \overline{β} : c⇒a so that fc = b and f\overline{β} = β. That is every β as shown here has a “cartesian lift” \overline{β}.

There is a nice characterisation of fibrations: f is a fibration iff the canonical map η_{f} : A→B/f has a right adjoint over B. In general, this adjunction η_{f} ⊣ a should be regarded as a choice of cartesian liftings, just as in the familiar case K = CAT, and is referred as a cleavage for f. It is convenient to denote cleavages as ordered pairs (a, ε), where ε is counit of η_{f} ⊣ a.

To make the choice of liftings from a given cleavage explicit write

[figure: B/f; q; p; A; f; B; 1_{B}; B; λ]

for the defining lax pullback square of B/f and let (a, ε) be a cleavage for f. Given a 2-cell β as above, one defines its cartesian lift \overline{β} to be qεβ′ where β′ is unique such that

β = [figure: X; b; a; β′; B/f; A; f; B; 1_{B}; B; =; =; λ]

One may show that the 2-cell qε is f-cartesian and so qεβ′ is f-cartesian for any β. This choice of cartesian liftings has the following property: for any x : Z→X, \overline{ϕx} = \overline{ϕ}x. Conversely given such a choice of cartesian liftings, one obtains a cleavage by lifting the defining pullback square for B/f and using its universal property to induce the counit ε : η_{f}a→1 and so one has:

3.2. Lemma. To give a cleavage for f : A→B is the same as giving for all 2-cells

[figure: X; a; b; A; f; B; ϕ]

a choice \overline{ϕ} of f-cartesian lift of ϕ such that \overline{ϕx} = \overline{ϕ}x for all 1-cells x : Z→X.

Thus amongst those 2-cells whose 0-target is A, there are those f-cartesian 2-cells which are chosen by the given cleavage for f. In the special case K = CAT, it is enough to consider X = 1 and then one speaks of chosen-cartesian arrows of A.

3.3. Definition. Let f : A→B be a fibration in K and be a cleavage (a, ε) for f. A 2-cell

[figure: X; a_{1}; a_{2}; A; ϕ]

is chosen-(a, ε)-cartesian when the cartesian lift of fϕ provided by the cleavage (a, ε) is ϕ. When a cleavage (a, ε) for f is given, we will often abuse terminology and say that ϕ is chosen-f-cartesian when there is little risk of confusion.

Thus one can define split fibrations in K in an elementary way by analogy with K = CAT. Recall that in this special case a cleavage is split when: (1) identity arrows are chosencartesian, and (2) chosen-cartesian arrows are closed under composition.

3.4. Definition. A cleavage (a, ε) for f : A→B is split when for the resulting chosencartesian 2-cells we have:

1. For any k : X→A, the identity 2-cell 1_{k} is chosen-f-cartesian.

2. A vertical composite of chosen-f-cartesian 2-cells is chosen-f-cartesian.

A split fibration in K is a fibration f : A→B which has a split cleavage for f.

One can define the 2-category Spl(B) of split fibrations over B as follows:

• objects consist of a map f : A→B and a split cleavage (a, ε) for f.

• an arrow of Spl(B) is an arrow in K/B which preserves chosen-cartesian 2-cells.

• a 2-cell of Spl(B) is just a 2-cell in K/B. and denote the forgetful 2-functor by U_{B} : Spl(B)→K/B. In Street [Street, 1974] split fibrations over B are defined a little differently to definition(3.4): as the strict algebras for the 2-monad on K/B whose underlying endofunctor

Φ_{B} : K/B → K/B

obtained by lax pullback along 1_{B}. The reader will easily supply the formal definitions of the unit and multiplication of this monad, and verify that it is in fact a cartesian monad. More importantly from [Street, 1974], we know that Φ_{B} is colax idempotent (or in older terminology, Kock-Zöberlein of limit-like variance). We omit the straight-forward proof of the following result, which reconciles the two definitions of split fibrations we have been discussing.

3.5. Theorem. Let K be a finitely complete 2-category and B ∈ K. There is an isomorphism Spl(B) ≅ Φ_{B}-Alg_{s} commuting with the forgetful functors into K/B. ^{10}

The basic, elementary, well-known and easy to prove facts about split fibrations at this level of generality are summarised in

3.6. Theorem.

1. A composite of split fibrations is a split fibration.

2. The pullback of a split fibration along any map is a split fibration.

3. If \xymatrix{A & E \ar[l]_{d} \ar[r]^{c} & B} is a discrete fibration from A to B then d is a split fibration.

and in light of the fact that split fibrations over a given base form a 2-category, with little more effort one can verify the following more comprehensive expression the pullback stability.

3.7. Theorem. Let K be a finitely complete 2-category and f : A→B be a morphism of f. The 2-functor f^{∗} given by pulling back along f lifts as in the square on the left,

\xymatrix{Spl(B) \ar@{.>}[rr] \ar[dd]_{U_{B}} &  & Spl(A) \ar[dd]^{U_{A}} & Spl(A) \ar@{.>}[rr] \ar[dd]_{U_{A}} &  & Spl(B) \ar[dd]^{U_{B}} \\  & = &  &  & = \\ K/B \ar[rr]_{f^{∗}} &  & K/A & K/A \ar[rr]_{f_{!}} &  & K/B}

and when f is a split fibration, the 2-functor f_{!} given by composition with f lifts as in the square on the right.

Of course for a locally discrete 2-category Spl(B) and K/B coincide and everything discussed in this subsection becomes trivial. For instance Φ_{B} is the identity monad. Thus one may regard Spl(B) as the “real” 2-categorical slice over B, with K/B its “naive” cousin. This point of view will become important in section(5).

10Recall that for a 2-monad T, T -Alg_{s} denotes the 2-category of strict algebras, strict algebra morphisms and algebra 2-cells (see [Blackwell-Kelly-Power, 1989]).

3.8. Iso-fibrations. An iso-fibration in a finitely complete 2-category K is a map f : A→B such that for all invertible 2-cells

[figure: X; a; b; A; f; B; β]

there exists an invertible \overline{β} : c⇒a so that fc = b and f\overline{β} = β. For example, any fibration and any opfibration is an iso-fibration since the (op)cartesian lift of an invertible 2-cell is invertible. Another important example is provided by

3.9. Example. Let K be a finitely complete 2-category, A ∈ K and (t, η, μ) a monad on A in K. Then the underlying forgetful arrow u : A^{t}→A of the Eilenberg-Moore object of t is a discrete iso-fibration: given b : X→A^{t}, a : X→A and β : a≅ub there exists a unique \overline{β} : c≅a so that u\overline{β} = β. To see this note that by the representability of the notions involved it suffices to consider the case K = CAT in which case u is just the forgetful functor from the category of t-algebras. In this case the assertion amounts to the fact that for any isomorphism between an object of a ∈ A and the underlying object of a t-algebra, one has a unique t-algebra structure on a with respect to which the given isomorphism lives in t-Alg.

One has the following analogue of the adjoint characterisation of a fibration, which one can prove by adapting the proof of its analogue by assuming that all 2-cells of that proof are invertible.

3.10. Proposition. Let K be a finitely complete 2-category and f : A→B in K. Then the following statements are equivalent:

1. f is an iso-fibration.

2. For all g : X→B, the map i : g/_{=}f→g/≅f has a pseudo inverse in K/X.

3. The map i : A→B/≅f has a pseudo inverse in K/B.

Moreover when B is a groupoid the following are equivalent for f : A→B:

• f is a fibration.

• f is an opfibration.

• f is an iso-fibration.

because a 2-cell ϕ with 0-target A is f-cartesian iff ϕ is invertible iff ϕ is f-opcartesian. Thus a cleavage for the fibration f, thought of as a choice of f-cartesian 2-cells as in lemma(3.2), is also a cleavage for f seen as an opfibration. Thus we have an isomorphism Spl(B)≅SplOp(B) commuting with the forgetful 2-functors into K/B.

3.11. Bipullbacks. Taking the representable definition of pseudo pullback one can ask instead that the induced functor be an equivalence. In a finitely complete 2-category this is equivalent to the following definition. A bipullback of

\xymatrix{A \ar[r]^{f} & B & C \ar[l]_{g}}

in a finitely complete 2-category K consists of an invertible 2-cell

\xymatrix{P \ar[rr]^{q} \ar[dd]_{p} &  & C \ar[dd]^{g} \\  & ≅ \\ A \ar[rr]_{f} &  & B}

in K such that the arrow P →f/≅g it induces is an equivalence.

3.12. Example. While it is not true in general that a pullback square

[figure: P; p; q; C; g; A; f; B]

in K is a bipullback, it is true whenever g is an isofibration by proposition(3.10).

It is well-known and straight-forward to prove that bipullbacks satisfy an analogous composition/cancellation result to those enjoyed by lax and pseudo pullbacks.

3.13. Proposition. Let K be a finitely complete 2-category and

[figure: X; x; y; A; v; u; C; g; Y; h; B; f; D; ϕ; ψ]

be invertible 2-cells in K such that ψ exhibits A as a bipullback of f and g. Then ϕ exhibits X as a bipullback of u and h iff the composite isomorphism exhibits X as a bipullback of fh and g.

3.14. Bifibrations. While isomorphisms of categories are obviously fibrations, equivalences of categories may not be. For a simple example recall the functor ch : SET→CAT which sends any set X to the category whose objects are the elements of X, and for which there is a unique morphism between any two objects of ch(X). Clearly if X is non-empty then ch(X)≃1, and ch sends functions between non-empty sets to equivalences of categories. Apply ch to a function from a one element set to a two element set for an example of an equivalence which is not a fibration^{11}. Thus the concept of fibration described so far is 2-categorical in nature. However there is an analogous bicategorical concept [Street, 1980b], which we refer to as bifibration^{12}, that we shall now discuss.

We content ourselves with simply providing the elementary definition of bifibration in a finitely complete 2-category, proving that bifibrations compose and are stable under bipullback, and providing an adjoint characterisation. In [Street, 1980b] much more than this is achieved: in particular a pseudo-monadic description of bifibrations is obtained analogous to the description of fibrations as pseudo algebras.

Let K be a finitely complete 2-category. A map f : A→B in K is a bifibration when for all

[figure: X; a; b; A; f; B; β]

there exists f-cartesian \overline{β} : c⇒b and invertible ι : b⇒fc such that

β = [figure: X; c; a; b; A; f; B; \overline{β}; ι]

3.15. Examples.

1. By definition fibrations are bifibrations.

2. Equivalences are bifibrations. Let f : A→B be an equivalence. Thus there is an adjunction i⊣f whose unit η and counit ε are invertible. Moreover f is representably fully faithful. One may check easily that for any fully faithful functor g : X→Y, all the morphisms of X are g-cartesian. Thus by definition all 2-cells between arrows with codomain A are f-cartesian. Now observe that the equations

[figure: X; a; b; A; f; B; β; X; β; b; A; f; 1_{A}; A; f; B; i; 1_{B}; B; β; η; ε; =]

for all β, exhibit f as a bifibration since η is invertible.

3. An object B of K is said to be a groupoid when for all X ∈ K, the hom-category K(X, B) is a groupoid in CAT. Any map f : A→B where B is a groupoid is a bifibration, because any 2-cell β as in the above definition of bifibration is invertible.

The basic facts on bifibrations are 3.16. Proposition. [Street, 1980b]

11In fact this functor is not even a Giraud-Conduché fibration, so we have here an example of an equivalence of categories which is not even an exponentiable functor.

12Although in the literature the word “bifibration” is sometimes taken to mean a functor which is both a fibration and a opfibration, we prefer to stick to the convention of using the prefix “bi” for bicategorical notions.

1. The composite of bifibrations in a finitely complete 2-category is a bifibration.

2. Bifibrations are stable under bipullback.

Finally one can also give an adjoint characterisation of bifibrations by adapting the proof of the usual adjoint characterisation of fibrations in the obvious way. In this case this criterion establishes a nice relationship between forming lax pullbacks and forming pseudo pullbacks along a bifibration. Thus for f : A→B and g : X→B, we denote by i : g/≅f→g/f the arrow induced by the defining isomorphism

[figure: g/≅f; A; f; X; g; B]

for g/≅f, and the universal property of g/f.

3.17. Theorem. [Street, 1980b] Let K be a finitely complete 2-category and f : A→B in K. Then the following statements are equivalent:

1. f is a bifibration.

2. For all g : X→B, the map i : g/≅f→g/f has a right adjoint in K/X with invertible unit.

3. The map i : B/≅f→B/f has a right adjoint in K/B with invertible unit.

4. Abstract nerves

In this section we analyse the monad theory which provides a far-reaching generalisation of the characterisation of categories as simplicial sets satisfying the Segal condition. A general theorem in this vein is an unpublished result of Tom Leinster [Leinster, 2004], and applies to a p.r.a monad T on a presheaf category. Here we describe a more general result and I am grateful to Steve Lack for suggesting the general setting expressed in definition(4.1). Throughout this section we make serious use of the yoneda structures notation discussed in the introduction.

4.1. Definition. A monad with arities consists of a monad (T, η, μ) on a cocomplete category A together with a fully faithful and dense functor

i_{0} : Θ_{0} → A

such that Θ_{0} is small, and the functor A(i_{0}, T) preserves the left extension

[figure: Θ_{0}; i_{0}; i_{0}; A; 1; A; id]

We shall denote such a monad with arities as a pair (T, Θ_{0}).

4.2. Remark. Since A(i_{0}, 1) is fully faithful and fully faithful maps reflect left extensions, T will automatically preserve the above left extension since A(i_{0}, T) does.

4.3. Examples. One can exhibit any accessible monad T on a locally presentable category A as a monad with arities. In this situation there is a regular cardinal α such that A is locally α-presentable and T preserves α-filtered colimits, and then one takes i_{0} to be the inclusion of the α-presentable objects. The functor A(i_{0}, T) preserves the required left extension in this case since this left extension amounts to the description of each object of A as an α-filtered colimit of α-presentable objects, and T and A(i_{0}, 1) both preserve α-filtered colimits [Adamek-Rosický, 1994].

Writing U : T -Alg→A for the forgetful functor and F for its left adjoint, factor the composite F i_{0} as

\xymatrix{Θ_{0} \ar[r]^{i_{0}} \ar[ddr]_{j} & A \ar[r]^{F} & T −Alg \\  & = \\  & Θ_{T} \ar[uur]_{i}}

an identity on objects functor j followed by a fully faithful functor i to define the category Θ_{T}.

4.4. Definition. Let (T, Θ_{0}) be a monad with arities on A. Then the nerve functor for (T, Θ_{0}) is

T -Alg(i, 1) : T -Alg → Θ̂_{T}

and so for a T -algebra X we call T -Alg(i, X) ∈ Θ̂_{T} the nerve of X. For Z ∈ Θ̂_{T} satisfies the Segal condition for (T, Θ_{0}) when res_{j}Z is in the image of

A(i_{0}, 1) : A→Θ̂_{0}.

The main purpose of this section is to characterise the nerves of T -algebras as those presheaves satisfying the Segal condition, and to explain the important instances of this result. For a given choice of unit \overline{η} and counit \overline{ε} of lan_{j} ⊣ res_{j}, we denote by (\overline{T} , \overline{η}, \overline{μ}) the monad on Θ̂_{0} which arises from this adjunction. First we note that \overline{T} -Alg may be identified with Θ̂_{T}.

4.5. Lemma. res_{j} : Θ̂→Θ̂_{0} is monadic.

Proof. Let f_{1}, f_{2} : \xymatrix{X_{1} \ar[r] \ar[r] & X_{2}} be maps in Θ̂_{T}. Since coequalisers are formed componentwise in presheaf categories, any coequaliser of f_{1}j^{op} and f_{2}j^{op} lifts to a unique coequaliser of f_{1} and f_{2}. Thus by the absolute coequaliser form of the Beck theorem [Paré, 1969] the comparison functor Θ̂_{T} →\overline{T} -Alg is an isomorphism.

We shall now induce canonical isomorphisms

[figure: T -Alg; T −Alg(i,1); U; Θ̂_{T}; ^{res^{j}}; A; A(i_{0},1); Θ̂_{0}; κ; A; A(i_{0},1); F; Θ̂_{0}; lan^{j}; T -Alg; T −Alg(i,1); Θ̂_{T}; ψ]

which when pasted together provide a monad functor from T to \overline{T}. We define κ as the unique 2-cell such that

[figure: A; F; 1; T -Alg; U; T −Alg(i,1); Θ̂_{T}; ^{res^{j}}; Θ_{0}; i_{0}; i_{0}; y; A; A(i_{0},1); Θ̂_{0}; χ^{i_{0}}; id; η; κ; Θ_{0}; j; y; Θ_{T}; y; i; T -Alg; T −Alg(i,1); Θ̂_{0}; Θ̂_{T}; _{res_{j}}; χ^{yj}; χ^{i}; =]  (2)

Note that χ^{i_{0}} exhibits A(i_{0}, 1) as a left extension of y along i_{0}, and since η as the unit of an adjunction is an absolute left extension, we have by the composability of left extensions that A(i_{0}, U) is exhibited as a left extension of y along F i_{0} on the left hand side of (2). Thus the above definition of κ makes sense since F i_{0} = ij. In addition to this we have

4.6. Lemma. The 2-cell κ defined by (2) is an isomorphism and exhibits res_{j} as a left extension of A(i_{0}, U) along T -Alg(i, 1).

Proof. The right hand side of (2) exhibits res_{j}T -Alg(i, 1) as a left extension of y along ij by proposition(3.4) of [Weber, 2007], and so κ is an isomorphism. Since i is fully faithful χ^{i} is invertible. Since χ^{yj} exhibits res_{j} as a left extension, it follows that the right hand side of (2) exhibits res_{j} as a left extension of y along T -Alg(i, ij) and so by the composability of left extensions, κ exhibits res_{j} as a left extension of A(i_{0}, U) along T -Alg(i, 1).

Now we define ψ to be the unique 2-cell such that

[figure: Θ_{0}; j; i_{0}; A; F; A(i_{0},1); Θ̂_{0}; ^{lan^{j}}; Θ_{T}; i; y; T -Alg; T −Alg(i,1); Θ̂_{T}; χ^{i}; id; ψ; Θ_{0}; i_{0}; y; j; A; A(i_{0},1); Θ_{T}; y; Θ̂_{0}; _{lan_{j}}; Θ̂_{T}; χ^{i_{0}}; ϕ_{j}; =]  (3)

Since F is a left adjoint and i_{0} is dense the identity cell on the left hand side of (3) exhibits F as a left extension of ij along i_{0}. By the isomorphism κ and from the condition that i_{0} endows T with arities, this left extension is preserved by res_{j}T -Alg(i, 1). However res_{j} creates colimits, and so this left extension is in fact preserved by T -Alg(i, 1). Thus since χ^{i} is an isomorphism, T -Alg(i, F) is exhibited as a left extension of y_{Θ_{T}} j along i_{0} on the left hand side of (3), and so the above definition of ψ does indeed make sense. In addition to this we have

4.7. Lemma. The 2-cell ψ defined by (3) is an isomorphism and exhibits lan_{j} as a left extension of T -Alg(i, F) along A(i_{0}, 1).

Proof. Since χ^{i_{0}} is an isomorphism the right hand side of (3) exhibits lan_{j} as a left extension of y_{Θ_{T}} j along A(i_{0}, i_{0}), and so by the composability of left extensions ψ exhibits lan_{j} as a left extension of T -Alg(i, F) along A(i_{0}, 1). The 2-cell ϕ_{j} is invertible and lan_{j} is a left adjoint and so the composite cell of (3) exhibits lan_{j}A(i_{0}, 1) as a left extension of y_{Θ_{T}} j along i_{0}, and so ψ is an isomorphism.

Following [Lack-Street, 2002] we shall refer to a monad in a 2-category K as a pair (A, s) where s : A→A is the underlying endoarrow of the monad, and the unit and multiplication are left unmentioned. Recall from [Street, 1972] [Lack-Street, 2002] that a monad morphism (A, s)→(B, t) consists of a pair (f, ϕ), where f : A→B and ϕ : tf→fs, which are compatible with the monad structures in the sense that

[figure: s; f; 1; f; ϕ; η; =; s; f; 1; f; =; η; s; f; t; t; f; ϕ; μ; =; s; f; t; t; f; ϕ; ϕ; μ]

Defining φ as the inverse of the composite

[figure: A; F; A(i_{0},1); T -Alg; U; T −Alg(i,1); A; A(i_{0},1); Θ̂_{0}; _{lan_{j}}; Θ̂_{T}; _{res_{j}}; Θ̂_{0}; κ; ψ]

and recalling that the monad structure of \overline{T} is determined by a choice of unit \overline{η} and counit \overline{ε} of lan_{j} ⊣ res_{j}, we have

4.8. Proposition. One can choose the unit \overline{η} and counit \overline{ε} of lan_{j} ⊣ res_{j} in a unique way so that (A(i_{0}, 1), φ) is a monad morphism from T to \overline{T}.

Proof. For any 2-category K one may associate another 2-category C(K) as follows. The objects of C(K) are the arrows of K. Given arrows f and g of K, an arrow f→g consists of (h, k, ϕ) as in

[figure: f; h; k; g; ϕ]

where ϕ is invertible and exhibits k as a left extension of gh along f, and k is a left adjoint. Composition of arrows is given by vertical pasting and is well-defined since left adjoints preserve left extensions. Given arrows (h_{1}, k_{1}, ϕ_{1}) and (h_{2}, k_{2}, ϕ_{2}) from f to g in C(K), a 2-cell consists of α : h_{1}→h_{2} and β : k_{1}→k_{2} such that

[figure: α; ϕ_{2}; β; ϕ_{1}; =]

By the elementary properties of left extensions the 2-functor dom : C(K)→K whose object map takes the domain of an arrow is locally fully faithful^{13}. Thus given arrows (l, \overline{l}, ϕ_{1}) and (r, \overline{r}, ϕ_{2}) in C(K), and an adjunction l ⊣ r in K, one obtains a unique adjunction \overline{l} ⊣ \overline{r} compatible with ϕ_{1} and ϕ_{2}. By lemmas(4.6) and (4.7) one can apply this last observation to κ and ψ from which the result follows by the definition of φ.

Before moving on to the proof of our most general nerve characterisation, we describe a general lemma on monad morphisms that applies to (A(i_{0}, 1), φ). First recall that given a monad (A, s) in a 2-category K, that an algebra for s consists of x : X→A and a : sx→x such that a(ηx) = id and a(μx) = a(sa). For example the Eilenberg-Moore object of s consists of u : A^{s}→A and an s-algebra σ : su→u which satisfies a universal property. Now for any monad morphism (f, ϕ) : (A, s)→(B, t) in K the assignment

\xymatrix{sx \ar[r]^{a} & x & ↦ & tfx \ar[r]^{ϕx} & fsx \ar[r]^{fa} & fx}

sends s-algebra structures on x to t-algebra structures on fx, and the following observation is immediate.

4.9. Lemma. Let (f, ϕ) be a monad morphism (A, s)→(B, t) in a 2-category K such that f is fully-faithful and ϕ is an isomorphism. Then the above assignment gives a bijection between s-algebra structures on x and t-algebra structures on fx.

4.10. Theorem. Let (T, Θ_{0}) be a monad with arities on A.

1. The nerve functor T -Alg(i, 1) is fully faithful.

2. Z ∈ Θ̂_{T} is the nerve of a T -algebra iff it satisfies the Segal condition.

13Meaning that dom’s hom-functors are fully faithful.

Proof. We denote the pullback and pseudo pullback of A(i_{0}, 1) and res_{j} as

[figure: P_{1}; p_{1}; q_{1}; Θ̂_{T}; ^{res^{j}}; A; A(i_{0},1); Θ̂_{0}; P_{2}; q_{2}; p_{2}; Θ̂_{T}; ^{res^{j}}; A; A(i_{0},1); Θ̂_{0}; λ]

and write k_{1} : P_{1}→P_{2} and k_{2} : T -Alg→P_{2} for the canonical maps induced by the universal property of P_{2}: k_{1}λ = id and k_{2}λ = κ. We have

[figure: T -Alg; T −Alg(i,1); k_{2}; P_{2}; q_{2}; Θ̂_{T}; P_{1}; k_{1}; q_{1}; =; =]

and k_{1} is an equivalence by examples(3.9) and (3.12). Since fully faithful maps are pullback stable q_{1} is fully faithful since A(i_{0}, 1) is. Thus by the above diagram q_{2} is fully faithful. If k_{2} is an equivalence then T -Alg(i, 1) is also fully faithful by the above diagram, and the essential images^{14} of q_{1} and T -Alg(i, 1) are the same. Since the essential image of q_{2} consists of those Z ∈ Θ̂_{T} satisfying the Segal condition by definition, to finish the proof it suffices to show that k_{2} is an equivalence.

Now res_{j}q_{2} is a \overline{T} -algebra with structure map res_{j}\overline{ε}q_{2}, and so by example(3.9) we can define a_{1} : \overline{T} A(i_{0}, p_{2})→A(i_{0}, p_{2}) as the unique \overline{T} -algebra structure on A(i_{0}, p_{2}) making λ an isomorphism of \overline{T} -algebras. By lemma(4.9) we can define a_{2} : T p_{2}→p_{2} as the unique T -algebra structure on p_{2} such that A(i_{0}, a_{2})∘φp_{2} = a_{1}. By the universal property of Uε : T U→U as the Eilenberg-Moore object of T, there is a unique k_{3} : P_{2}→T -Alg such that Uk_{3} = p_{2} and Uεk_{3} = a_{2}. We will now verify that k_{3} is a pseudo inverse for k_{2}. We have p_{2}k_{2}k_{3} = p_{2} and so to obtain k_{2}k_{3}≅1 it suffices, by the 2-dimensional of the universal property of P_{2}, to give an isomorphism ϕ : q_{2}→q_{2}k_{2}k_{3} such that res_{j}ϕ∘λ = λk_{2}k_{3}. But this is the same as giving a \overline{T} -algebra isomorphism ϕ_{2} : res_{j}q_{2}→res_{j}q_{2}k_{2}k_{3} such that ϕ_{2}∘λ = λk_{2}k_{3}. There is one such isomorphism because λ and λk_{2}k_{3} are themselves \overline{T} -algebra isomorphisms. To give an isomorphism 1≅k_{3}k_{2} is the same as giving an isomorphism U≅Uk_{3}k_{2} of T -algebras. We have Uk_{3}k_{2} = U and so it suffices to see that this is an identity of T -algebras, that is, that Uε = Uεk_{3}k_{2}. Since A(i_{0}, 1) is fully faithful and by the definition of k_{3}, this is the same as showing that A(i_{0}, Uε) = A(i_{0}, a_{2}k_{2}). From the definitions of the various 2-cells involved it is straight forward to verify that

λk_{2} ∘ A(i_{0}, Uε) ∘ φU = res_{j}(T -Alg(i, 1)ε ∘ ψ^{−1}U)
λk_{2} ∘ A(i_{0}, a_{2}k_{2}) ∘ φU = res_{j}(\overline{ε}T -Alg(i, 1) ∘ lan_{j}κ)

and so it suffices to show that T -Alg(i, 1)ε ∘ ψ^{−1}U = \overline{ε}T -Alg(i, 1) ∘ lan_{j}κ, and this follows from the definition of \overline{ε} via proposition(4.8).

14The essential image of a fully faithful functor f : A→B is the full subcategory of B consisting of the b ∈ B such that fa≅b for some a in A.

We will now begin to specialise this result and discuss examples. First we assume that A is a presheaf category: A = ℂ̂ for some small category ℂ, and that i_{0} is the inclusion of a full subcategory which includes the representables. Thus we have

[figure: ℂ; j_{0}; y_{ℂ}; Θ_{0}; i_{0}; ℂ̂; id]  (4)

where i_{0} and j_{0} are the inclusions. For any p ∈ Θ_{0} we write

[figure: y/p; π_{p}; 1; p; ℂ; y; ℂ̂; \overline{p}]

for the canonical lax pullback square, and recall that this cocone exhibits p as a colimit of representables. Explicitly, the component of \overline{p} corresponding to x : C→p in y/p is just x as an arrow of ℂ̂. By definition the cocone \overline{p} lives in Θ_{0}, and so one can consider the cocone j\overline{p} in Θ_{T}.

We shall now show that Z ∈ Θ̂_{T} satisfies the Segal condition iff it sends each cocone j\overline{p} to a limit cone, by characterising the image of ℂ̂(i_{0}, 1). Since i_{0} is fully faithful (4) exhibits j_{0} as an absolute left lifting of y_{ℂ} along i_{0}, and so (4) exhibits i_{0} as a pointwise left extension of y along j_{0} by axiom(2) of the good yoneda structure for CAT. This implies i_{0}≅Θ_{0}(j_{0}, 1) so that res_{j_{0}} ⊣ ℂ̂(i_{0}, 1). Thus we have

4.11. Lemma. ℂ̂(i_{0}, 1) is fully-faithful and has left adjoint res_{j_{0}}.

Write η̃ for the unit of res_{j_{0}} ⊣ ℂ̂(i_{0}, 1). We want to characterise the image of ℂ̂(i_{0}, 1), and by lemma(4.11), X ∈ Θ̂_{0} is in this image iff η̃_{X} is invertible. Thus we want a useful explicit description of η̃_{X}. For p ∈ Θ_{0} we must describe a function

η̃_{X,p} : Xp → ℂ̂(p, Xj_{0}^{op})

and this definition must be natural in X and p. Since \overline{p} is a colimit cocone, the functions

ℂ̂(x, Xj_{o}^{op}) : ℂ̂(p, Xj_{o}^{op}) → ℂ̂(C, Xj_{o}^{op})

for all x : C→p in y/p, form a limit cone in Set. Thus we can define η̃_{X,p} to be the unique function such that for all x : C→p the square

\xymatrix{Xp \ar[r]^{X(x)} \ar[d]_{η̃_{X,p}} & XC \ar[d]^{ι_{C,X}} \\ ℂ̂(p, Xj_{o}^{op}) \ar[r]_{ℂ̂(x,Xj_{o}^{op})} & ℂ̂(C, Xj_{o}^{op})}

commutes, where ι is the induced isomorphism in

[figure: ℂ; y; y; ℂ̂; ℂ̂(y,1); ℂ̂; χ^{y}; ℂ; y; y; ℂ̂; 1; ℂ̂(y,1); ℂ̂; id; ι; =]

4.12. Lemma. The 2-cell η̃ just defined is a unit for res_{j_{0}} ⊣ ℂ̂(i_{0}, 1).

Proof. We already know there exists such an adjunction and that ℂ̂(i_{0}, 1) is fully faithful. Thus we have res_{j_{0}} ℂ̂(i_{0}, 1)≅1ℂ̂ and so by [Weber, 2007] lemma(2.6) it suffices to show that res_{j_{0}} η̃ and η̃ℂ̂(i_{0}, 1) are invertible. This in turn amounts to saying that η̃_{X,p} is invertible when p is a representable (ie in ℂ) or when X is of the form ℂ̂(i_{0}, Z) for some Z ∈ ℂ̂. In each case η̃_{X,p} turns out, by definition, to be a component of ι.

From lemmas(4.11) and (4.12) the following characterisation of the image of ℂ̂(i_{0}, 1) is now immediate.

4.13. Proposition. X ∈ Θ̂_{0} is in the image of ℂ̂(i_{0}, 1) iff for each p ∈ Θ_{0}, X sends the colimit cocone \overline{p} to a limit cone.

4.14. Corollary. [Leinster, 2004] Let (T, Θ_{0}) be a monad with arities on ℂ̂ and i_{0} be the inclusion of a full subcategory that contains the representables. Then Z ∈ Θ̂_{T} satisfies the Segal condition iff for all p ∈ Θ_{0} it sends the cocone j\overline{p} to a limit cone.

4.15. Example. One can apply theorem(4.10) and corollary(4.14) in the case where ℂ = 1 (so ℂ̂ = Set) and T is a finitary monad on Set seen as having arities via the inclusion of a skeleton of the category of finite sets. In this case Θ_{T}^{op} is precisely the Lawvere theory associated to T. Recall that a model of the theory Θ_{T}^{op} in a category E with products is a finite product preserving functor Θ_{T}^{op}→E. Theorem(4.10) gives the wellknown identification between the algebras of T and the models of its associated Lawvere theory in Set: to say that Z ∈ Θ̂_{T} satisfies the Segal condition is the same as saying that Z : Θ_{T}^{op}→Set preserves finite products.

We now consider the case of a p.r.a monad (T, η, μ) on a presheaf category ℂ̂, and explain how such a monad always comes equipped with arities. So we shall identify the objects of Θ_{0} in the next definition, and then proceed to explain how they provide arities for T.

4.16. Definition. Let ℂ be a small category and (T, η, μ) be a p.r.a monad on ℂ̂. We define Θ_{0} to be the full subcategory of ℂ̂ consisting of those objects in the image of E_{T} : y_{ℂ}/T 1→ℂ̂ (see proposition(2.10) and remark(2.12)). An object p ∈ ℂ̂ is a T -cardinal when it is isomorphic to an object of Θ_{0}.

4.17. Example. Let T be the category monad on Graph. The representables in Graph are the graphs [0] and [1]. Trivially [0]→T X is T -generic iff X≅[0]. Thus by example(2.8), T -cardinals are those graphs of the form [n]. Moreover the category Θ_{T} in this case is the category Δ of non-empty ordinals and order-preserving maps.

4.18. Example. Let T be the strict ω-category monad on the category of globular sets. By [Weber, 2004] proposition(9.1) T -cardinals are exactly the globular cardinals of [Batanin, 1998] [Street, 2000]. Moreover the category Θ_{T} in this case is Joyal’s category Θ [Joyal, 1997].

4.19. Example. Let T be the symmetric multicategory monad on 𝕄̂, the category of multigraphs, described in example(2.14). Then the category Ω as defined in [Moerdijk-Weiss, 2007a] section(3) is equivalent to Θ_{T}. As explained in example(2.14) Θ_{0} is not skeletal, although its isomorphism classes are in bijection with planar trees. In [Moerdijk-Weiss, 2007a], the category Ω is defined to have non-planar trees as objects, and the objects of Ω̂ are called dendroidal sets.

Note that given a p.r.a monad T on ℂ̂, an object X ∈ ℂ̂ is a T -cardinal iff there exists C ∈ ℂ a generic morphism C→T X, by remark(2.11).

4.20. Proposition. Let T be a p.r.a monad on ℂ̂.

1. Representables are T -cardinals.

2. If p is a T -cardinal and g : p→T q is T -generic then q is a T -cardinal.

Proof. (1): for C ∈ ℂ we shall see that η_{C} : C→T C is T -generic. From

\xymatrix{C \ar[rr]^{α} \ar[dd]_{η_{C}} &  & T X \ar[dd]^{T γ} \\  & = \\ T C \ar[rr]_{T β} &  & T Z}

one obtains the desired T -filler δ as follows:

[figure: C; η; β; δ; α; T X; T γ; X; η; Z; η; T C; T β; T Z; =; =; =]

(2): since p is a T -cardinal we have a T -generic g_{1} : C→T p, thus the composite T (g)g_{1} is T^{2}-generic by [Weber, 2004] lemma 5.14. Since μ is cartesian μ_{q}T (g)g_{1} is T -generic, by [Weber, 2004] proposition 5.10, and so exhibits q as a T -cardinal.

4.21. Examples. By proposition(4.20) one obtains a “generic-free” factorisation system on the full sub-category of Kl(T) given by the T -cardinals. In the case where T is the free category monad, this sub-category is the category of non-empty ordinals and order preserving maps. The factorisation system is given by the factorisation of monotone maps [n]→[m] into a map which preserves top and bottom elements (the T -generics), followed by a monotone injection which preserves consecutive elements (the free maps). The analogous factorisation for the strict ω-category monad was described in [Berger, 2002].

4.22. Proposition. Let ℂ be a small category and (T, η, μ) be a p.r.a monad on ℂ̂. Then the inclusion i_{0} : Θ_{0}→ℂ̂ of the T -cardinals provides T with arities.

Proof. We have

[figure: ℂ; j_{0}; y_{ℂ}; Θ_{0}; i_{0}; i_{0}; ℂ̂; 1; ℂ̂; id; id]

and we saw above that the left most 2-cell here is a left extension. The composite 2-cell is also a left extension since y_{ℂ} is dense, and so the right most 2-cell is a left extension also, and so i_{0} is dense. To finish the proof we must show that the functor ℂ̂(i_{0}, T) preserves the left extension

[figure: Θ_{0}; i_{0}; i_{0}; ℂ̂; 1; ℂ̂; id]

For X ∈ ℂ̂ the lax pullback

[figure: i_{0}/X; π; 1; X; Θ_{0}; i_{0}; ℂ̂; λ]

exhibits X as a colimit of T -cardinals by the density of i_{0}. We must show that this colimit is preserved by ℂ̂(i_{0}, T). Given

[figure: i_{0}/X; π; 1; Z; Θ_{0}; ℂ̂(i_{0},T i_{0}); Θ̂_{0}; ϕ]

we define \overline{ϕ} : ℂ̂(i_{0}, T X)→W by \overline{ϕ}_{p}(f) = ϕ_{h_{f} ,p}(g_{f}) for f : p→T X. To see that this is natural in p let k : q→p be in Θ_{0}. Inducing δ as the unique T -fill in the commutative diagram on the left we obtain the commutative diagram on the right

[figure: q; k; g_{fk}; p; f; g_{f}; T X; T D_{fk}; T δ; T h_{fk}; T D_{f}; T h_{f}; ℂ̂(q, T D_{f}); ϕ_{h_{f} ,q}; ℂ̂(q, T D_{fk}); ℂ̂(q,T δ); ϕ_{h_{fk},q}; ℂ̂(p, T D_{f}); ℂ̂(k,T D_{f}); ϕ_{h_{f} ,p}; Zq; Zp; Zk]

and so by

W (k)\overline{ϕ}_{p}(f) = ϕ_{h_{f} ,p}(g_{f}) = W (k)ϕ_{h_{f} ,p}(g_{f}) = ϕ_{h_{f} ,q}(g_{f}k)
= ϕ_{h_{f} ,q}(T (δ)g_{fk}) = ϕ_{h_{fk},q}(g_{fk}) = \overline{ϕ}_{q}(fk)

\overline{ϕ}_{p} is indeed natural in p. The equation

[figure: i_{0}/X; π; 1; Z; Θ_{0}; ℂ̂(i_{0},T i_{0}); Θ̂_{0}; ϕ; i_{0}/X; π; 1; Z; Θ_{0}; ℂ̂(i_{0},T i_{0}); Θ̂_{0}; λ; \overline{ϕ}; =]  (5)

says that for each h : p→X and q ∈ Θ_{0}, that ϕ_{h,q}(f) = \overline{ϕ}_{q}ℂ̂(q, T h), and this follows by the definition of \overline{ϕ} and the naturality of ϕ. On the other hand for f : p→T X, taking h = h_{f} we obtain \overline{ϕ}_{p}(f) = ϕ_{h_{f} ,p}(g_{f}) from this equation and so \overline{ϕ} is unique satisfying (5).

4.23. Example. Applying proposition(4.22) and theorem(4.10) to the category monad on Graph gives the well-known characterisation of nerves of categories via the Segal condition.

4.24. Example. Applying proposition(4.22) and theorem(4.10) to the strict ω-category monad on the category of globular sets gives the characterisation of nerves of strict ωcategories given by [Berger, 2002] theorem(1.12).

4.25. Remark. In the above analysis of a p.r.a monad T on ℂ̂ one can replace Θ_{0} by any full subcategory Θ′_{0} of ℂ̂ satisfying the following conditions:

1. Θ′_{0} contains the representables,

2. if p ∈ Θ′_{0} and g : p→T q is T -generic then q ∈ Θ′_{0}, and

3. Θ′_{0} is equivalent to a small category.

Thus it follows that given a cartesian monad morphism ϕ : S→T, the inclusion i_{0} : Θ_{0}→ℂ̂ also endows S with arities. To see this one need only check (2) for (S, Θ_{0}), and this follows from the corresponding property for (T, Θ_{0}) because composition with components of ϕ preserves generics by [Weber, 2004] proposition(5.10).

4.26. Example. In [Berger, 2002] theorem(1.17) nerves of algebras of ω-operads are characterised. An ω-operad is a cartesian monad morphism S→T where T is the strict ωcategory monad on the category of globular sets. In view of remark(4.25) one recaptures this result by applying theorem(4.10) and corollary(4.14) to (S, Θ_{0}), where Θ_{0} is the full subcategory consisting of the T -cardinals. Remember that Θ_{S} is defined from Θ_{0} via the identity on objects fully faithful factorisation of F i_{0}, and this Θ_{S} is what in [Berger, 2002] is called the globular theory of the associated ω-operad. Note that one gets another nerve characterisation of S-algebras by using the S-cardinals instead of the T -cardinals.

4.27. Example. In [Moerdijk-Weiss, 2007b] nerves of operads are characterised in proposition(5.3) and theorem(6.1). In view of examples(2.14) and (4.19), one obtains these results by applying theorem(4.10), corollary(4.14) and proposition(4.20) to the symmetric multicategory monad on the category 𝕄̂ of multigraphs.

5. Familial 2-functors

In this section we introduce familial 2-functors and then discuss in detail the paradigmatic example Fam.

5.1. The definition of familial 2-functor. The notion of familial 2-functor is intended to be a 2-categorical analogue of p.r.a-ness. The notion of p.r.a has an obvious naive 2-categorical analogue because for a 2-category A and X∈A, one has the naive slice 2-category A/X, and for a 2-functor T : A→B, the effect of T on arrows into X is a 2-functor T_{X} : A/X→B/T X. Thus one can just imitate definition(2.3) at this level to provide the definitions of p.r.a 2-functor and p.r.a 2-monad. Generic morphisms work in much the same way, except that in order to obtain the 2-categorical analogue of proposition(2.6), one must describe a 2-dimensional aspect. That is, a one-cell f : B→T A is T -generic when it is T -generic in the 1-categorical sense as defined in section(2) and, given

[figure: B; f; α_{1}; α_{2}; T X; T γ; T A; T β; T Z; ϕ]

such that T (β)f = T (γ)α_{1} = T (γ)α_{2} and T (γ)ϕ = id, and writing δ_{1} (resp. δ_{2}) for the T -filler for T (β)f = T (γ)α_{1} (resp. T (β)f = T (γ)α_{2}), there exists unique ϕ′ : δ_{1}→δ_{2} such that T (ϕ′)f = ϕ and γϕ′ = id. With this 2-categorical definition of T -generic morphism at hand, the proof of proposition(2.6) is easily adapted so that it applies to 2-functors. With the naive 2-categorical analogue of p.r.a-ness understood, we now give the more sophisticated analogue which allows for the role of fibrations in our finitely complete 2-categories (see the remarks just before subsection(3.8)).

5.2. Definition. A 2-functor T : A→B between finitely complete 2-categories is familial when it is p.r.a and the 2-functor

T_{1} : A → B/T 1

factors through U_{T 1} : Spl(T 1)→B/T 1. T is opfamilial when the 2-functor T co : Aco→Bco is familial^{15}. A 2-monad is familial (resp. opfamilial) when its underlying 2-functor is familial (resp. opfamilial), and its unit and multiplication are cartesian.

As already discussed in section(3), for a finitely complete 2-category K and X ∈ K, one can consider various alternatives to the naive slice K/X because of the presence of the various concepts of “fibration” internal to K. As in the one dimensional case p.r.a-ness for T : A→B asks more of the assignment

A ↦ T t_{A} : T A→T 1,

namely, that it is the object map of a right adjoint. Familiality asks further that T t_{A} be a split fibration, or more precisely that this right adjoint factor through Spl(T 1). Thus familiality is the 2-categorical analogue of parametric right adjointness provided that one interprets Spl(X) as the 2-categorical analogue of the slice category of K over X. In the example of Fam discussed below, T t_{A} is one of the most well-known examples of a split fibration: it is the functor

Fam(A) → Set

which sends a family of objects to its indexing set.

Following the same philosophy one might have considered an alternative to definition(5.2) in which one asked that T_{1} factors as

\xymatrix{A \ar[r]^{^{\overline{T}^{1}}} & Spl(T 1) \ar[r]^{U_{T 1}} & B/T 1}

where \overline{T}_{1} is a right adjoint. This of course implies that T_{1} is a right adjoint since U_{T 1} is, but given a mild condition on A this is in fact a consequence of the original definition.

5.3. Lemma. If T : A→B is a familial 2-functor between finitely complete 2-categories and A has coequalisers, then the 2-functor \overline{T}_{1} has a left adjoint.

Proof. From the definition one has the following situation

[figure: Spl(T 1); U_{T 1}; A; T_{1}; _{\overline{T}_{1}}; B/T 1; =]

to which one may apply Dubuc’s adjoint triangle theorem [Dubuc, 1972] to construct the left adjoint to \overline{T}_{1}.

15In other words just reverse all the 2-cells in the definition of “familial” to define “opfamilial”. Note that this includes replacing split fibrations by split opfibrations.

Another observation which supports the idea that familial 2-functors are a 2-categorical analogue of p.r.a-ness is

5.4. Lemma. Let K be a finitely complete 2-category and B ∈ K. Then the composite

\xymatrix{Spl(B) \ar[r]^{U_{B}} & K/B \ar[r]^{t_{B!}} & K}

is familial.

Proof. Note that U_{B} has a left adjoint F_{B} and so t_{B!}U_{B} is p.r.a. Moreover (t_{B!}U_{B})_{1} is just U_{B}, which trivially factors through U_{B} and so t_{B!}U_{B} is familial.

Thus denoting the composite t_{B!}U_{B} by \overline{t}_{B}, one has

5.5. Corollary. Let T : A→B be a familial 2-functor between finitely complete 2categories and suppose that A has coequalisers. Then T factors as a right adjoint followed by a 2-functor of the form \overline{t}_{B}.

It is sometimes useful to note that the generic morphisms for a familial 2-functor satisfy a stronger 2-dimensional property. We make use of this below in describing Fam’s pseudo monad structure and in section(6).

5.6. Definition. Let T : A→B be a 2-functor between finitely complete 2-categories such that T_{1} factors through U_{T 1} : Spl(T 1)→B/T 1. Thus for all A∈A, T t_{A} is a split fibration. A map g : B→T A is lax-T -generic if for all α, β, γ and ϕ as on the left hand side of the equation below, there are unique δ, ϕ_{1} and ϕ_{2} so that

[figure: B; α; g; T X; T γ; T A; T β; T Z; ϕ; B; g; α; T X; T γ; T A; T δ; T β; T Z; ϕ_{1}; T ϕ_{2}; =]  (6)

and ϕ_{1} is chosen T t_{X}-cartesian. Reversing the direction of the 2-cells involved gives the definition of oplax-T -generic morphism.

We will now provide some basic useful properties lax-T -genericness, in particular that it implies ordinary T -genericness, and then we will show that the T -generics for familial T are in fact lax.

5.7. Lemma. Let T : A→B be a 2-functor between finitely complete 2-categories such that T_{1} factors through U_{T 1} : Spl(T 1)→B/T 1.

1. In the situation of equation(6) ϕ is an identity iff ϕ_{1} and ϕ_{2} are identities.

2. If g : B→T A lax-T -generic then g is T -generic.

3. In the situation of equation(6) ϕ is invertible iff ϕ_{1} and ϕ_{2} are invertible.

Proof. (1): the implication (⇐) is trivial, so let ϕ be an identity. Now T (t_{X})ϕ_{1} = T (t_{Z})ϕ = id and so since ϕ_{1} is chosen-T (t_{X})-cartesian, ϕ_{1} = id. We have

[figure: B; g; T (γ)α; T Z; T t_{Z}; T A; T (γδ); T t_{A}; T 1; =; =; B; g; T (γ)α; T Z; T t_{Z}; T A; T β; T t_{A}; T 1; =; =]

and so by uniqueness β = γδ, and applying uniqueness this time to equation(6), one must have ϕ_{2} = id. (2): it suffices to verify only the 2-dimensional aspect of genericness as the one-dimensional part is immediate from (1). Given

[figure: B; g; α_{1}; α_{2}; T X; T γ; T A; T β; T Z; ϕ]

where g satisfies the hypothesis of lemma(5.8), T (γ)α_{1} = T (β)g = T (γ)α_{2}, and T (γ)ϕ = id, we have unique δ_{1}, δ_{2} : A→X such that

γδ_{1} = β = γδ_{2} T (δ_{1})g = α_{1} T (δ_{2})g = α_{2}

We must provide ϕ′ : δ_{1}→δ_{2} unique such that γϕ′ = id and T (ϕ′)g = ϕ. Since g satisfies the hypothesis of lemma(5.8) we have unique δ_{3}, ϕ_{1} and ϕ_{2} so that

ϕ = [figure: B; g; α_{2}; T X; T 1; T A; T δ_{3}; T δ_{1}; T X; ϕ_{1}; T ϕ_{2}]

and ϕ_{1} is chosen-T t_{X}-cartesian. Now T (t_{X})ϕ_{1} = T (t_{X})ϕ = id and so ϕ_{1} = id. Composing the above diagram-equation with T γ and using uniqueness, one obtains γϕ_{2} = id since T (γ)α = id. Thus δ_{2} = δ_{3} and ϕ_{2} = ϕ′ the required unique 2-cell. (3): the implication (⇐) is trivial, so let ϕ be an isomorphism. This time ϕ_{1} is the cartesian lift of the invertible 2-cell T (t_{Z})ϕ, and so is invertible. Thus T (ϕ_{2})g is invertible, and so since g is T -generic, ϕ_{2} is invertible.

5.8. Lemma. If T : A→B is familial and g : B→T A is T -generic then g is in fact lax-T -generic.

Proof. Consider α, β, γ and ϕ as on the left hand side of equation(6). Since T t_{X} is a split fibration there are α′ : B→T X and ϕ_{1} : α′→α chosen T t_{X}-cartesian such that T (t_{X})ϕ_{1} = T (t_{Z})ϕ. This last equation is an equation of 2-cells, and taking their domain arrows one has T (t_{X})α′ = T (t_{A})g, but the genericness of g ensures that α′ = T (δ)g for a unique δ : A→X. Since ϕ_{1} is chosen T t_{X}-cartesian and T γ is a morphism of split fibrations T t_{X}→T t_{Z}, T (γ)ϕ_{1} is chosen T t_{Z} cartesian. Thus there is ϕ′_{2} : T (β)g→T (γδ)g unique such that ϕ = (T (γ)ϕ_{1})ϕ′_{2}, but the 2-dimensional aspect of g’s genericness ensures that there is a unique ϕ_{2} such that ϕ′_{2} = T (ϕ_{2})g.

5.9. Fam as a familial 2-functor. For X ∈ CAT we recall the explicit definition of the category Fam(X):

• objects: are pairs (I, f) where I ∈ Set and f : I→X is a functor.

• arrows: an arrow (I, f)→(J, g) is a pair (k, \overline{k}) where k : I→J is a function and \overline{k} is

a natural transformation

[figure: I; k; f; J; g; X; \overline{k}]

• identities: are those (k, \overline{k}) such that k is an identity function and \overline{k} is an identity natural transformation.

• composition: the composite of \xymatrix{(I, f) \ar[r]^{(k,\overline{k})} & (J, g) \ar[r]^{(m,\overline{m})} & (M, h)} has function part mk

and 2-cell part given by

[figure: I; k; f; J; m; g; M; h; X; \overline{k}; \overline{m}]

In the obvious way Fam is a 2-functor. A functor f : I→X is a family of objects of X, I is the indexing set of this family, and for i ∈ I we say that fi ∈ X is the label of i. Similarly for (k, \overline{k}) as above one may regard the components of \overline{k} as labeling the function k. In more detail, for i ∈ I the component \overline{k}_{i} : fi→gki in X is the label of the assignment i↦ki. In this way one regards Fam(X) as the category of X-labeled sets and X-labeled functions. As already recalled in the previous subsection it is well-known that the functor

Fam(t_{X}) : Fam(X) → Fam(1)=Set

given explicitly on objects by (I, f)↦f is a split fibration. The chosen cartesian arrows of the split cleavage for Fam(t_{X}) are those (k, \overline{k}) such that \overline{k} is an identity. More generally, (k, \overline{k}) is Fam(t_{X})-cartesian iff \overline{k} is invertible. The cleavage just described is 2-functorial in X, that is one has a factorisation

\xymatrix{CAT \ar[r] & Spl(Set) \ar[r]^{U_{Set}} & CAT/Set}

of Fam_{1} as required by definition(5.2).

In order to establish that Fam is p.r.a we must consider functors f : B→Fam(A). By way of notation we shall denote f’s object and arrow maps as follows:

b ↦ (fb : Ib→A) (β : b_{1}→b_{2})↦[figure: Ib_{1}; Iβ; fb_{1}; Ib_{2}; fb_{2}; A; fβ]

and so I is the functor Fam(t_{A})f : B→Set. Such a functor f will send each b ∈ B to an A-labeled set and each morphism of B to an A-labeled function. For any object or arrow of A, one can observe how often it is used as a label in the description of f.

5.10. Definition. A functor f : B→Fam(A) endows B with elements when:

1. For all a ∈ A, there is a unique b ∈ B and a unique i ∈ Ib such that fb(i) = a.

2. For all α : a_{1}→a_{2}, there is a unique β : b_{1}→b_{2} in B and i ∈ Ib_{1} such that (fβ)_{i} = α.

In other words, f endows B with elements when each object and each arrow of A is used exactly once as a label by f. Thus one may regard a ∈ A as an element of the unique b ∈ B such that fb(i) = a, and similarly for the arrows of A. We will now see that f is Fam-generic iff it endows B with elements.

5.11. Lemma. Any f : B→Fam(A) factors as

\xymatrix{B \ar[r]^{g} & Fam(C) \ar[r]^{Fam(h)} & Fam(B)}

where g endows B with elements. Moreover if B is small and discrete then so is C.

Proof. We will use the notation for f’s object and arrow maps described prior to definition(5.10). Let C = 1/I and define g : B→Fam(C) as follows. For b ∈ B, gb : Ib→C is defined by gb(i) = i : 1→Ib. For β : b_{1}→b_{2} we write its image by g as

[figure: Ib_{1}; Iβ; gb_{1}; Ib_{2}; gb_{2}; C; gβ]

where

(gβ)_{i} = \xymatrix{ & 1 \ar[ddl]_{i} \ar[ddr] \\  & = \\ Ib_{1} \ar[rr]_{Iβ} &  & Ib_{2}}

By definition g endows B with elements. Defining h : C→A as

[figure: (i : 1→Ib); ↦; fb(i); 1; i_{1}; i_{2}; Ib_{1}; Iβ; Ib_{2}; =; ↦; 1; i_{1}; i_{2}; Ib_{1}; Iβ; fb_{1}; Ib_{2}; fb_{2}; A; fβ; =]

it follows by definition that f = Fam(h)g. Now C fits into a pullback square

[figure: C; p; Set_{•}; τ; B; I; Set]

thus p : C→B is a discrete opfibration with small fibres, and so if B is small and discrete, then C is small and discrete also.

5.12. Lemma. If f : B→Fam(A) endows B with elements then f is Fam-generic.

Proof. Considering

\xymatrix{B \ar[dd]_{f} \ar[rr]^{p} &  & Fam(X) \ar[dd]^{Fam(r)} \\  & = \\ Fam(A) \ar[rr]_{Fam(q)} &  & Fam(Z)}  (7)

we use the notation for f’s object and arrow maps described prior to definition(5.10), similarly we use the following notation for p’s object and arrow maps,

b ↦ (pb : Jb→X) (β : b_{1}→b_{2})↦[figure: Jb_{1}; Jβ; pb_{1}; Jb_{2}; pb_{2}; X; pβ]

and so J is the functor Fam(t_{X})p : B→Set. However by the commutativity of (7) we have Ib = Jb and q(fb) = r(pb) for all b, and Iβ = Jβ and q(fβ) = r(pβ) for all β. Since f endows B with elements we may define δ : A→X as follows. For a ∈ A, let b be the unique object of B and i the unique element of Ib such that fb(i) = a, and then δ(a) = (pb)(i). For α : a_{1}→a_{2}, let β : b_{1}→b_{2} and i ∈ Ib_{1} be unique such that fβ_{i} = α, and then δ(α) = (pβ)_{i}. By definition δ is unique such that Fam(δ)f = p and rf = q.

5.13. Proposition. Fam is a familial 2-functor. Moreover f : B→Fam(A) is Famgeneric iff it endows B with elements.

Proof. If f endows B with elements then it is generic by lemma(5.12). Thus by lemma(5.11) Fam is p.r.a. Moreover by that lemma if f is generic, then you can factor it as Fam(h)g where g endows B with elements. But this implies that g is itself generic, and so h is an isomorphism, whence f also endows B with elements.

5.14. Corollary. Fam is a polynomial 2-functor: Fam ≅ P_{τ}, where τ : Set_{•}→Set is the forgetful functor from the category of pointed sets.

Proof. Since Fam is p.r.a it suffices to show that L_{Fam} is isomorphic to the composite

\xymatrix{CAT/Set \ar[r]^{τ^{∗}} & CAT/Set \ar[r]^{(t_{Set})_{!}} & CAT} .

By proposition(5.13) the generic factorisation of f : B→Fam(A) was being constructed in lemma(5.11). Applying this construction in the case A = 1, gives the result by proposition(2.6).

To see that familiality is stronger than p.r.a’ness, note that by theorem(6.2) below, familial 2-functors preserve discrete fibrations, and dually, opfamilial 2-functors preserve discrete opfibrations. However applying Fam to the discrete opfibration τ : Set_{•}→Set does not give a discrete opfibration. Consider for instance a set S with 2 distinct elements x and y. Then the 2-element family of pointed sets ((x, S), (y, S)) is sent by Fam(τ) to the 2element family (S, S). There is a unique chosen Fam(tSet)-cartesian map f : (S, S)→(S) in Fam(Set), where (S) denotes the singleton family consisting of the set S, and this map admits no lifting to a map ((x, S), (y, S))→((z, S)) in Fam(Set_{•}): if such a lifting existed we would have x = z = y, but x and y are different. Thus Fam is familial but not opfamilial, and dually the endo-2-functor of CAT whose object map is X ↦ Fam(X^{op})^{op} is opfamilial but not familial. An example of a p.r.a 2-functor which is neither familial nor opfamilial is Φ_{B}, the underlying endofunctor of the fibrations 2-monad of subsection(3.1). Even in the case K = CAT one can easily verify that Φ_{B} is neither familial nor opfamilial.

It is well known that Fam is the underlying 2-functor of a lax-idempotent pseudomonad. The only thing that is pseudo about the monad we describe below, is that the associative law of the monad holds up to a canonical isomorphism. The other monad laws can be made to hold strictly. However when one considers a finite analogue of Fam, which we denote Fam_{f}, we do indeed obtain an honest example of a familial 2-monad. We shall now recover Fam’s pseudo monad structure, and also show that the unit and multiplication of this monad structure is cartesian. Interestingly we are able to obtain these facts from our understanding of Fam-generics.

The unit η_{X} : X→Fam(X) picks out singleton families, that is,

x ↦ (x : 1→X) (f : x→x′) ↦ [figure: 1; 1; x; 1; x′; X; f]

and η so defined is clearly a cartesian transformation. Now we shall fix a choice of Famgeneric factorisation and use the notation of scholium(2.7). Without loss of generality we assume that

1. the components of η are chosen generics.

2. if f : I→X in Fam(X), then

\xymatrix{1 \ar[r]^{1_{I}} & Fam(I) \ar[r]^{Fam(f)} & Fam(X)}

is the chosen generic factorisation of f : 1→Fam(X).

An object of Fam^{2}(X) consists of I ∈ Set and a functor f : I→Fam(X). We have our chosen generic factorisation

\xymatrix{I \ar[r]^{g_{f}} & Fam(D_{f}) \ar[r]^{Fam(h_{f})} & Fam(X)}

of f, and D_{f} ∈ Set by lemma(5.11). We define μ_{X}(f) = h_{f}. The arrow map of μ_{X} is given by

[figure: I_{1}; α; f_{1}; I_{2}; f_{2}; Fam(X); \overline{α}; D_{f_{1}}; δ; h_{f_{1}}; D_{f_{2}}; h_{f_{2}}; X; α_{2}; ↦]

where

\overline{α} = [figure: I_{1}; g_{f_{1}}; g_{f_{2}}α; Fam(D_{f_{2}}); Fam(h_{f_{2}}); Fam(D_{f_{2}}); Fam(δ); Fam(h_{f_{1}}); Fam(X); α_{1}; Fam(α_{2})]

where α_{1} is chosen-Fam(t_{D_{f_{2}}})-cartesian.

5.15. Proposition. (Fam, η, μ) is a lax idempotent pseudo-monad, whose underlying 2-functor is familial and whose unit and multiplication are cartesian.

Proof. The verification that μ is a cartesian 2-natural transformation follows easily from the definition of μ and lemma(5.8). The unit laws

μ_{X}Fam(η_{X}) = id = μ_{X}η_{Fam(X)}

follow from the assumptions we imposed on our chosen Fam-generic factorisations. To finish the proof we provide c : 1→ηFamμ such that μc = id and cηFam = id, because then c is the unit for an adjunction μ ⊣ ηFam with identity counit. For X ∈ CAT consider f : I→Fam(X) in Fam^{2}(X). We have h_{f} : D_{f}→X with D_{f} ∈ Set, and then η_{Fam(X)}μ_{X}(I, f) = h_{f} : 1→Fam(X). By the description of Fam-generic factorisations given in lemma(5.11) , we have for i ∈ I

\xymatrix{I(fi) \ar[rr]^{c_{i}} \ar[ddr]_{fi} &  & D_{f} \ar[ddl]^{h_{f}} \\  & = \\  & X}

where the c_{i} form a coproduct cocone. Regarding the above triangle as a morphism of Fam(X), in fact a chosen Fam(t_{X})-cartesian morphism since the 2-cell part is an identity, these morphisms provide the components of

[figure: I; t_{I}; f; 1; h_{f}; Fam(X); _{\overline{c}_{f}}]

which we regard as an arrow of Fam^{2}(X). These are the components of c : 1→ηFamμ. To see that μ_{X}c_{X} = id, note that \overline{c} is chosen cartesian, and so by the definition of μ, will be sent by μ_{X} to identities. To see that c_{X}η_{X} = id one must show that \overline{c}_{f} is an identity whenever I = 1, but this also follows from the assumptions we imposed on our chosen Fam-generic factorisations. Note that the isomorphism μFam(μ) ≅ μμFam is obtained by adjointness from ηFamη = Fam(η)η which holds by the naturality of η.

5.16. Example. We consider now a variation on Fam, where Fam_{f}(X) is defined as Fam is, except that the indexing sets are only allowed to be natural numbers n, regarded as sets:

n = {0, 1, ..., n − 1}.

All of the above analysis of Fam proceeds in the same way for Fam_{f}, except that Fam_{f}(1) is the category of natural numbers and functions between them (instead of Set), and the isomorphism μFam(μ) ≅ μμFam can be strictified. The reason for this is that coproducts in Fam(1) = Set, which are coherently associative and unital, are replaced by the strictly associative and unital ordinal sums in Fam_{f}(1). Thus (Fam_{f}, η, μ) is a lax-idempotent familial 2-monad.

6. Basic properties of familial 2-functors

In this section we establish the basic properties that familial 2-functors satisfy. In particular we study the senses in which these 2-functors preserve lax and pseudo pullbacks. Moreover we will see how familial 2-functors preserve all the different notions of fibration in a finitely complete 2-category which have been discussed so far.

As a parametric right 2-adjoint any familial 2-functor must preserve pullbacks. We shall now discuss how familial 2-functors are also well-behaved with respect to lax and pseudo pullbacks. Recall that a right adjoint retraction for an arrow f : A→B in a 2category K is an adjunction f ⊣ r in K such that the unit is an identity. Such a retraction for f is determined by the data: r : B→A and ε : fr→1_{B}, satisfying the equations:

rf = 1_{A} rε = id εf = id

When ε is invertible r is a pseudo-inverse retraction for f, and in this case f and r are part of an adjoint equivalence. Similarly a left adjoint retraction for f is an adjunction l ⊣ f whose counit is an identity. In [Weber, 2007] a 2-functor T : A→B between finitely complete 2-categories was said to preserve lax pullbacks up to a left adjoint section when for all

\xymatrix{A \ar[r]^{f} & C & B \ar[l]_{h}}

in A, the comparison map k : T (f/h)→T f/T h has a right adjoint retraction in B/T A. It is also interesting to consider the case when k has a pseudo-inverse in B/T A, in which case T is said to preserve lax pullbacks up to a left equivalence section. Note that this is different to asking that T preserve lax pullbacks up to a right equivalence section because the relevant equivalence lives in a different slice. One can of course make all of the analogous definitions for pseudo pullbacks as well, and we shall freely use the corresponding terminology.

6.1. Theorem. If T : A→B is a familial 2-functor between finitely complete 2-categories.

1. T preserves lax pullbacks up to a left adjoint section.

2. T preserves pseudo pullbacks up to a left equivalence section.

Proof. (1): For f : A→C and h : B→C in A we have lax pullbacks

[figure: f/h; q; p; B; h; A; f; C; λ; T f/T h; q_{2}; p_{2}; T B; T h; T A; T f; T C; λ_{2}]

and the comparison k : T (f/h)→T f/T h is the unique map such that λ_{2}k = T λ. Generically factorising p_{2} as p_{2} = T (s)g and by lemma(5.8) one obtains

λ_{2} = [figure: T f/T h; g; q_{2}; T B; T h; T D; T δ; T s; T A; T f; T B; ϕ_{1}; T ϕ_{2}]

where ϕ_{1} is chosen-T (t_{C})-cartesian. Letting t : D→f/h be the unique map such that λt = ϕ_{2}, we define r : T f/T h→T (f/h) as r = T (t)g. We have the 2-cells id : p_{2}kr→p_{2} and ϕ_{1} : q_{2}kr→q_{2}, and note that

\xymatrix{T (f)p_{2}kr \ar[r]^{id} \ar[d]_{λ_{2}kr} & T (f)p_{2} \ar[d]^{λ_{2}} \\ T (h)q_{2}kr \ar[r]_{T (h)ϕ_{1}} & T (h)q_{2}}

commutes since λ_{2}kr = T (λtg = T (ϕ_{2})g and by the above factorisation of λ_{2}. Thus we can define ε : kr→1 as the unique 2-cell such that p_{2}ε = id and q_{2}ε = ϕ_{1}. By definition r and ε can be regarded as living in B/T A, so it suffices to verify: rk = 1, εk = id and rε = id.

The key observation which enables us to perform these verifications is that since T is a parametric right 2-adjoint, the 2-cell T (λ) is a lax pullback in B/T 1, because it is the result of applying the right 2-adjoint T_{1} to the lax pullback λ.

To see that rk = 1 first note that we have

T (λ)rk = T (λt)gk = T (ϕ_{2})gk T (λ) = (T (λ)rk)(ϕ_{1}k)

Now ϕ_{1}k is chosen-T (t_{C})-cartesian and T (t_{C})ϕ_{1} = T (t_{B})T (λ) = id, whence ϕ_{1}k = id. Thus we have T (λ) = T (λ)rk. Now rk may be regarded as an arrow of B/T 1 because T (t_{f/h})rk = T (t_{D})gk = T (t_{A}p) = T (t_{f/h}), and so we may exploit the universal property of T (λ) as a lax pullback in B/T 1 to infer that rk = 1.

By the definition of ε, εk is the unique 2-cell such that p_{2}εk = id and q_{2}εk = ϕ_{1}k = id. Thus εk = id. To see that rε = id notice that

[figure: T f/T h; rkr; r; T (f/h); T q; T p; T B; T h; T A; T f; T C; rε; T λ] = T (λ)r

since T (p)rε = p_{2}ε = id. Regarding the object T f/T h as over T 1 via the map T (t_{d})g : T f/T h→T 1, r is in B/T 1 since T (t_{f/h})r = T (t_{D})g, and rkr and rε are also in B/T 1 since T (t_{f/h}rε = T (t_{D})gε = T (t_{A})p_{2}ε = id. Thus one may exploit the universal property of T (λ) as a lax pullback in B/T 1 to infer from the previous diagram that rε = id. (2): by lemma(5.7)(3) when one replaces lax by pseudo pullbacks in the proof of (1), all the induced 2-cells will be invertible, and so one obtains a proof of this result.

We shall explain how familial 2-functors preserve the various notions of fibration that one can consider. Since most of these notions have an adjoint characterisation, we can use the previous result to help establish this.

6.2. Theorem. Let T : A→B be a familial 2-functor between finitely complete 2-categories. Then T preserves fibrations, bifibrations, iso-fibrations and one-sided discrete fibrations.

Proof. Let f : E→A be in A. If f is a fibration then the map η_{f} : E→A/f has a right adjoint a in A/A. Since η_{T f} = kT (η_{f}) and

\xymatrix{T E \ar[rr]^{T η_{f}} & ⊥ & T (A/f) \ar[ll]^{T a} \ar[rr]^{k} & ⊥ & T A/T f \ar[ll]^{r}}

are adjoints in B/T A, where k is the comparison map, the composite adjunction exhibits T f as a fibration. The case of iso-fibrations is identical except for the replacement of lax pullbacks by pseudo pullbacks. If f is a bifibration then the canonical map i_{1} : A/≅f→A/f has a right adjoint a in A/A. Thus we have

\xymatrix{T (A/≅f) \ar[rr]^{T i_{1}} \ar[dd]_{k_{1}} & ⊥ & T (A/f) \ar@{.>}[ll]^{T a} \ar[dd]_{k_{2}} \\  &  & ⊣ \\ T A/≅T f) \ar[rr]_{i_{2}} \ar@{.>}[uu]_{r_{1}} &  & T (A/f) \ar@{.>}[uu]_{r_{2}}}

in B/T A, where the k_{j} are comparison maps, r_{1} is a pseudo inverse for k_{1} and the square involving just the solid arrows commutes. Thus i_{2} ⊣ k_{1}T (a)r_{2} exhibits T f as a bifibration. Now suppose that f is a discrete fibration and consider

[figure: X; α; β; T E; T f; T A; ϕ]

in B. Fix a generic factorisation β = gT (h) and by lemma(5.8) we obtain

ϕ = [figure: X; g; α; T E; T γ; T C; T δ; T h; T A; ϕ_{1}; T ϕ_{2}]

for unique ϕ_{1}, ϕ_{2} and δ with ϕ_{1} chosen-T t_{A}-cartesian. Since f is a discrete fibration we have ϕ_{3} : δ_{2}→δ unique such that fϕ_{3} = ϕ_{2}. Thus we have \overline{ϕ} = (T (ϕ_{3})g)(ϕ_{1}) with T (f)\overline{ϕ} = ϕ. To see that \overline{ϕ} is a unique lifting suppose that we have

ϕ = [figure: X; γ; α; β; T E; T f; T A; ϕ_{4}; =]

Then γ = T (δ_{3})g for a unique δ_{3} such that h = fδ_{3}, and we obtain ϕ_{5}, δ_{4} and ϕ_{6} unique such that

ϕ_{4} = [figure: X; g; α; T E; T 1; T C; T δ_{4}; T δ_{3}; T E; ϕ_{5}; T ϕ_{6}]

ϕ_{5} is chosen-T t_{A}-cartesian. Composing this last equation with T f and applying uniqueness we obtain ϕ_{5} = ϕ_{1} and fϕ_{6} = ϕ_{2}. Since f is a discrete fibration this last equation implies that ϕ_{6} = ϕ_{3} whence ϕ_{4} = \overline{ϕ}.

To see that a familial 2-functor T : A→B also preserves split fibrations, we look more closely at how a cleavage for T f was constructed from a cleavage for f. In the following lemma we shall characterise the chosen-T f-cartesian maps in terms of the chosenf-cartesian maps. To this end we let f : E→A be a fibration in A with cleavage (a, ε) and

[figure: X; α_{1}; α_{2}; T E; ϕ]

be in B. Take generic factorisations α_{1} = T (h_{1})g_{1} and α_{2} = T (h_{2})g_{2} so that we have

ϕ = [figure: X; g_{1}; g_{2}; T D_{2}; T h_{2}; T D_{1}; T δ; T h_{1}; T E; ϕ_{1}; T ϕ_{2}]

for unique δ, ϕ_{1} and ϕ_{2} with ϕ_{1} chosen-T -cartesian.

6.3. Lemma. In the situation just described ϕ is chosen-T f-cartesian iff ϕ_{2} is chosen-f-cartesian.

Proof. We have lax pullbacks

[figure: A/f; q; p; E; f; A; 1; A; λ; T A/T f; q_{2}; p_{2}; T E; T f; T A; 1; T A; λ_{2}]

and recall that qε = λ and that

[figure: Z; κ_{1}; κ_{2}; T E; ψ]

is chosen-f-cartesian iff ψ = qεψ′, where ψ′ : Z→A/f is defined by λψ′ = fψ. As in the proof of theorem(6.1) one can express λ_{2} as the composite

[figure: T A/T f; g; q_{2}; T E; T f; T C; T h; T s; T δ_{2}; T (A/f); T p; T q; T a; T A; 1; T A; =; =; =; λ_{3}; T (qε)]

where p_{2} = T (h)g is a generic factorisation and r, the right adjoint to the comparison map k : T (A/f)→T A/T f is defined as r = T (s)g. The counit ε_{2} of k ⊣ r is defined by p_{2}ε_{2} = id and q_{2}ε_{2} = λ_{3}. From the proof of theorem(6.2) the induced cleavage is given by (\overline{a}, \overline{ε}) where

\overline{a} = T (as)g  \overline{ε} = ε_{2}(kT (εs)g)

Notice how the previous diagram expresses λ_{2} = q_{2}\overline{ε}. Define ϕ′ : X→T A/T f as the unique map such that p_{2}ϕ′ = T (f)α_{1}, q_{2}ϕ′ = α_{2} and λ_{2}ϕ′ = T (f)ϕ. Define δ_{3} as in

[figure: X; g_{1}; ϕ′; T A/T f; g; T D_{1}; T h_{1}; T δ_{3}; T C; T h; T E; T f; T A]

such that gϕ′ = T (δ_{3})g_{1} and hδ_{3} = fh_{1}. The pasting composite of the previous two diagrams is T (f)ϕ, but T (f)ϕ is also the composite of

[figure: X; g_{1}; g_{2}; T D_{2}; T h_{2}; T h_{2}; T E; T f; T D_{1}; T δ; T h_{1}; T E; T f; T A; =; ϕ_{1}; T ϕ_{2}]

and so by the uniqueness part of lemma(5.8) we have

h_{2}δ = δ_{2}δ_{3} T (h_{2})ϕ_{1} = λ_{3}ϕ′ fϕ_{2} = fqεsδ_{3}

Now ϕ is chosen-T f-cartesian iff q_{2}\overline{ε}ϕ′=ϕ which is true iff

h_{2}δ = δ_{2}δ_{3} T (h_{2})ϕ_{1} = λ_{3}ϕ′ ϕ_{2} = qεsδ_{3}

by the uniqueness part of lemma(5.8). Since the first two equations hold automatically, this is equivalent to saying that ϕ_{2} = qεsδ_{3}, which exhibits ϕ_{2} as chosen-f-cartesian. Conversely if ϕ_{2} is chosen-f-cartesian then there is a map t : D→A/f such that qεt = ϕ_{2}, so that λsδ_{3} = fϕ_{2} = λt, whence sδ_{3} = t by the universal property of λ, and so ϕ_{2} = qεsδ_{3}, which we saw above is equivalent to saying that ϕ is chosen-T f-cartesian.

6.4. Remark. The characterisation of chosen-T f-cartesians given in lemma(6.3) is independent of the generic factorisations of α_{1} and α_{2} chosen at the outset.

Proof. Choose alternative generic factorisations α_{1} = T (h′_{1})g′_{1} and α′_{2} = T (h′_{2})g′_{2}, and then form δ′, ϕ′_{1} and ϕ′_{2} in the same way. Thus ϕ is the following composite

[figure: X; g_{1}; g′_{1}; g′_{2}; g_{2}; T D_{1}; T ι_{1}; T h_{1}; T D′_{1}; T h′_{1}; T δ′; T D′_{2}; T h′_{2}; T ι_{2}; T D_{2}; T h_{2}; T E; ϕ′_{1}; T ϕ′_{2}; =; =; =; =]

where ι_{1} and ι_{2} were induced uniquely so that the commutative regions of the above diagram are obtained. Since all the maps g_{1}, g′_{1}, g_{2} and g′_{2} are T -generic, ι_{1} and ι_{2} are in fact invertible. Since ϕ′_{1}ι_{1} is chosen-T t_{D_{2}}-cartesian, this factorisation of ϕ coincides with the original one described prior to lemma(6.3). By the uniqueness part of lemma(5.8), one obtains ϕ_{2} = ϕ′_{2}ι_{1}. Since ι_{1} is invertible this last equation implies that ϕ_{2} is chosenf-cartesian iff ϕ′_{2} is.

6.5. Corollary. If T : A→B is a familial 2-functor between finitely complete 2-categories then T preserves split fibrations.

Proof. It suffices to show that the assignment (a, ε) ↦ (\overline{a}, \overline{ε}) described in lemma(6.3) sends split cleavages to split cleavages, so we suppose that (a, ε) is a split cleavage for f : E→A. If ϕ : α_{1}→α_{2} as above is an identity, then ϕ_{2} is an identity by lemma(5.7), which is chosen-f-cartesian since (a, ε) is split, and so by lemma(6.3) ϕ is chosen-T fcartesian. Suppose that ϕ : α_{1}→α_{2} as above and ψ : α_{2}→α_{3} are chosen-T f-cartesian. Factoring ψ as we did ϕ we have

ψϕ = [figure: X; g_{1}; g_{2}; g_{3}; T D_{1}; T δ; T h_{1}; T D_{2}; T δ′; T h_{2}; T D_{3}; T h_{3}; T E; ϕ_{1}; T ϕ_{2}; ψ_{1}; T ψ_{2}]

where ϕ_{2} and ψ_{2} are chosen-f-cartesian by lemma(6.3). Now (ψ_{2}δ)ϕ_{2} is chosen-f-cartesian since (a, ε) is split, and so ψϕ is chosen-T f-cartesian by lemma(6.3).

The preservation of split fibrations by familial 2-functors is expressed more comprehensively in the following result.

6.6. Theorem. If T : A→B is a familial 2-functor between finitely complete 2-categories then for all A ∈ A, the effect of T on morphisms

T_{A} : A/A → B/T A

can be lifted to 2-functors between 2-categories of split fibrations as in:

\xymatrix{Spl(A) \ar@{.>}[rr] \ar[dd]_{U_{A}} &  & Spl(T A) \ar[dd]^{U_{T A}} \\  & = \\ A/A \ar[rr]_{T_{A}} &  & B/T A}

Proof. The proofs of lemma(6.3) and corollary(6.5) provide the object mapping of the desired lifting, and since all 2-cells between morphisms of split fibrations are 2-cells of split fibrations, it suffices to provide the 1-cell mapping to finish defining the lifted 2-functor. Let f : E→A be a split fibration and ϕ be given as above. Suppose also that f′ : C→A is another split fibration with a given cleavage, and that k : f→f′ is a morphism of Spl(A). If ϕ is chosen-T f-cartesian, then ϕ_{2} is chosen-f-cartesian and so kϕ_{2} is chosen-f′-cartesian since k is a morphism of split fibrations. But kϕ_{2} = (T (k)ϕ)_{2} by the uniqueness part of lemma(5.8), and so T (k)ϕ is chosen-T (f′)-cartesian by lemma(6.3). That is, T k is indeed a map of split fibrations.

This lifting will be denoted as T_{A} : Spl(A)→Spl(T A).

7. Examples of familial 2-functors

So far we have exhibited a few fundamental examples of familial 2-functors. We shall now discuss results which enable us to exhibit more examples. In particular we exhibit the underlying endofunctors of the 2-monads which describe higher symmetric and braided operads within the framework of [Weber, 2005] as being familial and opfamilial.

7.1. Proposition. Suppose that the 2-categories A and B are finitely complete. If T : A→B is a parametric right 2-adjoint and T preserves lax pullbacks then T is familial and opfamilial.

Proof. It suffices to show that T is familial: reversing the 2-cells will give opfamiliality. First note that A∈A is discrete iff

[figure: A; 1; 1; A; 1; A; 1; A; id]

is a lax pullback, so T preserves discrete objects, and thus T 1 is discrete. Thus for all A ∈ A, T (t_{A}) is a split fibration for which the chosen cartesian cells are identities, and for all f : A_{1}→A_{2}, T f obviously preserves these chosen cartesian cells, and so T_{1} factors through U_{T 1} : Spl(T 1)→B/T 1 as required.

7.2. Corollary. Let T : ℂ̂→𝔻̂ be a parametric right adjoint. Then

Cat(T) : CAT(ℂ̂) → CAT(𝔻̂)

is familial and opfamilial.

Proof. Since T preserves pullbacks, and lax pullbacks in CAT(ℂ̂) and CAT(𝔻̂) can be constructed in terms of pullbacks in ℂ̂ and 𝔻̂ respectively, Cat(T) preserves lax pullbacks. By the previous proposition, it suffices to show that Cat(T) is a parametric right 2-adjoint. This follows by applying Cat(−) to the factorisation

\xymatrix{ℂ̂ \ar[r]^{T_{1}} & 𝔻̂/T 1 \ar[r]^{t_{T 1}} & 𝔻̂}

because Cat(−) applied to t_{T 1} is the equivalence Cat(𝔻̂/T 1)≃Spl(T 1) followed by \overline{t}_{T 1} which is p.r.a by lemma(5.4).

7.3. Examples. An ω-operad in Span the sense of [Batanin, 1998] gives rise to a p.r.a monad on the category 𝔾̂ of globular sets, and so applying it to internal categories, gives a 2-monad on the 2-category CAT(𝔾̂) of globular categories which is familial and opfamilial.

7.4. Theorem. If T : A→B is a familial 2-functor between finitely complete 2-categories and ϕ : S→T is a cartesian 2-natural transformation then S is a familial 2-functor.

Proof. By theorem(3.7) one has the diagram

\xymatrix{ &  &  & Spl(T 1) \ar[dd]^{U_{T 1}} \ar[rr]^{ϕ_{1}^{∗}} &  & Spl(S1) \ar[dd]^{U_{S1}} \\  &  & = \\ A \ar@{.>}[uurrr] \ar[rrr]_{T_{1}} \ar[rrrrr]_{S_{1}} & ⊥ &  & B/T 1 \ar[lll]_{L_{T}} \ar[rr]_{ϕ_{1}^{∗}} & ⊥ & B/S1 \ar[ll]_{_{ϕ_{1}_{!}}} \\  &  &  & ≅}

from which it is clear that S is familial.

7.5. Example. Recall the familial Fam_{f} from example(5.16). The monad S whose (strict) algebras are symmetric (strict) monoidal categories is a sub-monad of Fam_{f}. The category S(X) has the same objects as Fam_{f}(X), but a map

[figure: n; f; a; m; b; X; ϕ]

is in S(X) iff n = m and f is a bijection. One can easily check that Fam_{f}’s monad structure restricts to S, and so S is a cartesian 2-monad. The inclusion S→Fam_{f} is a cartesian 2-monad morphism and so by theorem(7.4) S is in fact a familial 2-monad.

7.6. Examples. Any club, as originally defined by Max Kelly in [Kelly, 1972], gives rise to a cartesian 2-monad morphism T →S where T is a cartesian 2-monad, and the algebras of T coincide with the algebras of the original club. By theorem(7.4) T is a familial 2-monad. As a particular example, T could be the 2-monad for braided monoidal categories. Its explicit description is the same as S’s above, except that f is a braid on n strings rather than a mere permutation. Yet another example, also obtainable from corollary(7.2), is the 2-monad for monoidal categories. As with this last case, we will see below that all of these examples are opfamilial also.

7.7. Theorem. If T : A→B is a familial 2-functor between finitely complete 2-categories and ℂ is a category then

[ℂ, T] : [ℂ, A] → [ℂ, B]

is a familial 2-functor.

Proof. Since [ℂ, T](1) is constant at T 1 and [ℂ, B]/[ℂ, T](1) ≅ [ℂ, B/T 1], one can regard the canonical factorisation of [ℂ, T] as given by the solid arrows in

\xymatrix{[ℂ, A] \ar[rr]_{[ℂ,T_{1}]} & ⊥ & [ℂ, B/T 1] \ar[r]^{[ℂ,t_{T 1}_{!}]} \ar@{.>}[ll]_{[ℂ,L_{T}]} & [ℂ, B]}

and thus [ℂ, T] is a parametric right 2-adjoint. Moreover since limits in [ℂ, B] are formed componentwise, the description of the fibrations monad can be made componentwise, and so Spl([ℂ, T](1)) ≅ [ℂ, Spl(T 1)], and so [ℂ, T]_{1} factors through U_{[ℂ,T](1)}.

7.8. Examples. For section(8) below and the general operad theory of [Weber, 2005]

[Weber] we observe that for any category ℂ, and any of the familial 2-monads T among example(5.16) and examples(7.6), composition with T gives a familial 2-monad on CAT(ℂ̂).

7.9. Theorem. If T : A→B and S : B→C are familial 2-functors between finitely complete 2-categories then ST is familial.

Proof. By theorem(6.6) we have

[figure: Spl(T 1); U_{T 1}; S_{T 1}; Spl(ST 1); U_{ST 1}; A; T_{1}; B/T 1; S_{T 1}; C/ST 1; =; =]

and so ST_{1} = S_{T 1}T_{1} factors through U_{ST 1}. Since T is p.r.a T_{1} has a left adjoint, and since S is p.r.a and by proposition(2.6), S_{T 1} also has a left adjoint, and so ST is p.r.a.

7.10. Examples. Important for the theory higher symmetric operads [Weber, 2005] [Weber] are the examples of familial 2-functors that one obtains on CAT(𝔾̂) by composing the familial 2-functors coming from examples(7.8) and examples(7.3).

In [Weber] analytic 2-monads on 2-toposes will be defined, and the operad theory of [Weber, 2005] will be developed further in this setting. The underlying 2-functor of an analytic 2-monad is in particular a familial 2-functor, and one of its additional properties is that evaluating it at 1 gives a groupoid. We now consider this extra condition. Let T : A→B be a 2-functor between finitely complete 2-categories, and suppose that T 1 is a groupoid. Recall that by proposition(3.10) we have Spl(T 1)≅SplOp(T 1) commuting with the forgetful functors into B/T 1, and so we immediately obtain

7.11. Proposition. Let T : A→B be a 2-functor between finitely complete 2-categories such that T 1 is a groupoid. Then T is familial iff T is opfamilial.

Since familial 2-functors T such that T 1 is a groupoid are also opfamilial, they enjoy all the properties that familials were shown to in sections(6) and (7), and the corresponding dual properties. For instance, analytic 2-functors preserve split fibrations and split opfibrations. In addition we have

7.12. Theorem. Let T : A→B be a familial 2-functor such that T 1 is a groupoid.

1. T preserves groupoids.

2. T preserves lax pullbacks up to a left equivalence section.

3. T preserve lax pullbacks up to a right equivalence section.

Proof. (1): let T : A→B be an analytic 2-functor between finitely complete 2-categories. If G ∈ A is a groupoid,

[figure: X; α_{1}; α_{2}; T G; ϕ]

and let

\xymatrix{X \ar[r]^{g} & T D \ar[r]^{T h} & T G}

be a T -generic factorisation for α_{1}. Then we have δ, ϕ_{1} and ϕ_{2} unique such that

ϕ = [figure: X; g; α_{2}; T G; T 1_{G}; T D; T δ; T h; T G; ϕ_{1}; T ϕ_{2}]

and ϕ_{1} is chosen-T t_{G}-cartesian. Since ϕ_{1} is T t_{G}-cartesian and T 1 is a groupoid, ϕ_{1} is invertible. Since G is a groupoid ϕ_{2} is also invertible. (2): we now refer to the proof of theorem(6.1)(1). The 2-cell ϕ_{1} is chosen-T (t_{C})-cartesian and thus invertible in this case since T 1 is a groupoid, and so ε : kr→1 the counit of k ⊣ r, is invertible. (3): apply (2) to T co.

Proposition(7.1), corollary(7.2), theorem(7.7), and theorem(7.9) remain true after replacing familial 2-functors by those familial T such that T 1 is a groupoid. With the exception of Fam and Fam_{f}, all the examples of familial 2-monads given so far all satisfy the groupoid condition. The only result of this section with no such refinement is theorem(7.4), which asserts that if ϕ : S⇒T is a cartesian 2-natural transformation and T is familial, then S is familial. The condition that T 1 is a groupoid has no effect on whether S1 is a groupoid. For assume that T 1 is a groupoid, and let f : B→T 1 be a morphism of B. Then one can define a 2-functor S and a cartesian 2-natural transformation ϕ : S⇒T from the pullbacks

[figure: SA; ϕ_{A}; T A; T t_{A}; B; f; T 1]

and by definition S1 = B.

7.13. Remark. We know that a familial 2-monad (T, η, μ) is cartesian: its functor part preserves pullbacks and the naturality squares of its unit and multiplication are pullbacks. If in addition T 1 is a groupoid then (T, η, μ) is also cartesian in the obvious bicategorical sense. By theorem(7.12) the underlying 2-functor preserves bipullbacks. Moreover, the naturality squares of its unit and multiplication are bipullbacks. To see this for the multiplication (the proof for the unit is identical), note that for all X ∈ K that t_{X} is a split fibration and thus an isofibration, and so T t_{X} is an isofibration by theorem(6.2). Thus the naturality square

[figure: T^{2}X; T^{2}t_{X}; μ_{X}; T X; T t_{X}; T^{2}1; μ_{1}; T 1]

is a bipullback by example(3.12). Thus by proposition(3.13) all of the naturality squares of μ are bipullbacks.

8. Cartesianness of composite 2-monads

In [Weber, 2005] the situation was considered in which one has a cartesian monad S on CAT and a p.r.a monad T on 𝔾̂ coming from a Batanin higher operad. One then regards both S and T as 2-monads on CAT(𝔾̂): S is viewed as acting componentwise and T is viewed as acting on category objects in 𝔾̂ because its functor part preserves pullbacks. Then theorem(7.12) of [Weber, 2005] says that there is a distributive law λ : T S→ST between S and T. The importance of this result is that for appropriate S and T, ST - operads are a symmetric analogue of the higher operads of [Batanin, 1998] and so are going to be appropriate for studying higher dimensional “weakly symmetric” monoidal structures and the stabilisation conjecture in the globular setting.

In this section we extend and in fact reprove theorem(7.12) of [Weber, 2005], so that the distributive laws obtained are seen to be cartesian, and so the composite monads that are obtained are familial. Moreover we work in an arbitrary presheaf category, replacing 𝔾 by an arbitrary small category ℂ. So we let (T, η, μ) be a p.r.a monad on ℂ̂ as in sections(2) and (4), and use all the associated notation and results developed in those sections. We let S be a p.r.a 2-monad on CAT. As in the globular case we also regard S and T as 2-monads on CAT(ℂ̂).

Recall from section(4) the monad (\overline{T} , \overline{η}, \overline{μ}) on Θ̂_{0} and the monad morphism (ℂ̂(i_{0}, 1), φ) from T to \overline{T} established in proposition(4.8). We would also like to interpret this monad morphism in our 2-categorical setting, and so we must verify that \overline{T} does indeed preserve pullbacks so that it can be applied to category objects. In fact we will now see that \overline{T} is another p.r.a monad. To do this we first spell out in detail the adjunction lan_{j} ⊣ res_{j}. For X ∈ Θ̂_{0} and p ∈ Θ_{T} we define

lan_{j}(X)(p) = ∑_{f∈Kl(T)(p,1)} XD_{f}  (8)

and we will justify this definition in lemma(8.1). Given a map f : p→1 in Kl(T) we denote by c_{f} : XD_{f}→lan_{j}(X)(p) the corresponding coproduct inclusion. For k : q→p in Θ_{T} we define lan_{j}(X)(k) as the unique map such that

\xymatrix{XD_{f} \ar[dd]_{Xh_{f,k}} \ar[rr]^{c_{f}} &  & lan_{j}(X)(p) \ar[dd]^{lan_{j}(X)(k)} \\  & = \\ XD_{fk} \ar[rr]_{c_{fk}} &  & lan_{j}(X)(q)}

where h_{f,k} is the unique free map so that

[figure: q; k; g_{fk}; p; g_{f}; f; D_{fk}; h_{f,k}; t_{D_{fk}}; D_{f}; t_{D_{f}}; 1; =; =; =]

in Kl(T). Thus lan_{j}(X) is well-defined as an object of Θ̂_{T}. Now for X ∈ Θ̂_{0} and p ∈ Θ_{0} we define \overline{η}_{X,p} : Xp→lan_{j}(X)(p) to be the coproduct inclusion c_{t_{p}}. This makes sense because our choice of generic factorisations is normalised, and is clearly natural with respect to maps in Θ_{0}.

8.1. Lemma. For X ∈ Θ̂_{0} the natural transformation

[figure: Θ_{0}^{op}; j^{op}; X; Θ_{T}^{op}; lan_{j}(X); Set; _{\overline{η}_{X}}]

described above exhibits lan_{j}(X) as a pointwise left extension of X along j^{op}.

Proof. Since Set is cocomplete and Θ_{0} and Θ_{T} are small it suffices to prove that \overline{η}_{X} is a left extension. Given Z and ϕ as follows

[figure: Θ_{0}^{op}; j^{op}; X; Θ_{T}^{op}; Z; Set; ϕ; Θ_{0}^{op}; j^{op}; X; Θ_{T}^{op}; Z; Set; ^{\overline{η}^{X}}; ϕ̂; =]  (9)

we must define ϕ̂ unique such that equation(9) holds. For p ∈ Θ_{T} we define ϕ̂_{p} as the unique function such that

\xymatrix{XD_{f} \ar[rr]^{c_{f}} \ar[dd]_{ϕ_{D_{f}}} &  & lan_{j}(X)(p) \ar[dd]^{ϕ̂_{p}} \\  & = \\ ZD_{f} \ar[rr]_{Zg_{f}} &  & Zp}

for all f : p→1 in Kl(T). To see that this definition of ϕ̂ is natural in p let k : q→p be in Θ_{T}. We must show that for all f : p→1 in Kl(T) that Z(k)ϕ̂_{p}c_{f} = ϕ̂_{q}lan_{j}(X)(k)c_{f}, and this follows from

[figure: XD_{f}; ϕ_{D_{f}}; c_{f}; Xh_{f,k}; XD_{fk}; c_{fk}; ϕ_{D_{fk}}; lan_{j}(X)(p); ϕ̂_{p}; lan_{j}(X)(k); lan_{j}(X)(q); ϕ̂_{q}; Zp; Zk; Zq; ZD_{f}; Zh_{f,k}; Zg_{f}; ZD_{fk}; Zg_{fk}; =; =; =; =]

the outside of which also commutes. By definition for p ∈ Θ_{0} we have ϕ̂_{p}c_{t_{p}} = ϕ_{p} and so ϕ̂ satisfies equation(9). Finally we verify the uniqueness of ϕ̂. Notice that for all p ∈ Θ_{T} and f : p→1 in Kl(T), we have

lan_{j}(X)(g_{f})c_{t_{D_{f}}} = c_{f}  (10)

by the definition of the arrow map of lan_{j}(X). Thus all the regions of

[figure: XD_{f}; c_{f}; c_{t_{D_{f}}}; ϕ_{D_{f}}; lan_{j}(X)(D_{f}); lan_{j}(X)(g_{f}); ϕ̂_{D_{f}}; ZD_{f}; Zg_{f}; lan_{j}(X)(p); ϕ̂_{p}; Zp]

commute, and so the definition of ϕ̂ is forced by equation(9) and the required naturality of ϕ̂.

By lemma(8.1) the following description of lan_{j}’s arrow maps is forced, and with respect to these arrow maps \overline{η}_{X} is natural in X. For ψ : X→X′ in Θ̂_{0} and p ∈ Θ_{0}, the function lan_{j}(ψ)_{p} is unique such that

\xymatrix{XD_{f} \ar[d]_{ψ_{D_{f}}} \ar[r]^{c_{f}} & lan_{j}(X)(p) \ar[d]^{lan_{j}(ψ)_{p}} \\ X′D_{f} \ar[r]_{c_{f}} & lan_{j}(X′)(p)}

commutes for all f : p→1 in Kl(T). Now for Z ∈ Θ̂_{T} and p ∈ Θ_{T} we define the function

\overline{ε}_{Z,p} : lan_{j}(Zj^{op})(p) → Zp

to be unique such that \overline{ε}_{X,p}c_{f} = Xg_{f} for all f : p→1 in Kl(T).

8.2. Lemma. The functions \overline{ε}_{Z,p} just defined are natural in Z and p, and the resulting \overline{ε} together with \overline{η} form the counit and unit of an adjunction lan_{j} ⊣ res_{j}.

Proof. To verify naturality in p let k : q→p be in Θ_{T}. For each f : p→1 in Kl(T) we have

[figure: ZD_{f}; c_{f}; Zh_{f,k}; Zg_{f}; lan_{j}(Zj^{op})(p); \overline{ε}^{Z,p}; lan_{j}(Zj^{op})(k); Zp; Zk; ZD_{fk}; c_{fk}; Zg_{fk}; lan_{j}(Zj^{op})(q); _{\overline{ε}_{Z,q}}; Zq; =; =; =]

in which the outside also commutes, and so Z(k)\overline{ε}_{Z,p}c_{f} = \overline{ε}_{Z,q}lan_{j}(Zj^{op})c_{f} as required. To verify naturality in Z let ψ : Z→Z′ in Θ̂_{T}, then for all f : p→1 in Kl(T) we have

[figure: ZD_{f}; c_{f}; ψ_{D_{f}}; Zg_{f}; lan_{j}(Zj^{op})(p); \overline{ε}^{Z,p}; lan_{j}(ψj^{op})_{p}; Zp; ψ_{p}; Z′D_{f}; c_{fk}; Z′g_{f}; lan_{j}(Z′j^{op})(p); _{\overline{ε}_{Z′,p}}; Z′p; =; =; =]

in which the outside also commutes, and so ψ_{p}\overline{ε}_{Z,p}c_{f} = \overline{ε}_{Z′,q}lan_{j}(ψj^{op})c_{f} as required. We already know from lemma(8.1) that \overline{η} is the unit of an adjunction lan_{j} ⊣ res_{j}, and so by example(2.17) of [Weber, 2007] for instance, it suffices to show that one of the triangular identities of an adjunction are satisfied by \overline{ε} and \overline{η}. To this end notice that for p ∈ Θ_{T} that \overline{ε}_{Z,p}c_{t_{p}} = Z(g_{t_{p}}) by definition, and so res_{j}\overline{ε} ∘ \overline{η}res_{j} = id as required.

8.3. Corollary. (\overline{T} , \overline{η}, \overline{μ}) is a p.r.a monad.

Proof. By theorem(2.13) and the explicit description of lan_{j} given above \overline{T} is p.r.a. Since presheaf categories are small-extensive \overline{η} is a cartesian transformation by definition. To see that \overline{μ} is cartesian it suffices, by elementary properties of pullback squares, to show that

\xymatrix{\overline{T}^{2}(X)(p) \ar[d]_{\overline{T}^{2}(t_{X})_{p}} \ar[r]^{^{\overline{μ}^{X.p}}} & \overline{T} (X)(p) \ar[d]^{\overline{T} (t_{X})_{p}} \\ \overline{T}^{2}(1)(p) \ar[r]_{_{\overline{μ}_{1,p}}} & \overline{T} (1)(p)}

is a pullback for all X and p. An element of \overline{T}^{2}(1)(p) is a pair (f : p→1, f_{2} : D_{f}→1) of maps in Kl(T). Given these maps we have

\xymatrix{\overline{T} (X)(D_{f}) \ar[rr]^{c_{f}} \ar[ddrr]_{\overline{T} (X)(g_{f})} &  & \overline{T}^{2}(X)(p) \ar[dd]^{\overline{μ}_{X,p}} & XD_{f_{2}} \ar[rr]^{Xh_{f_{2},g_{f}}} \ar[dd]_{c_{f_{2}}} &  & XD_{f_{2},g_{f}} \ar[dd]^{c_{f_{2},g_{f}}} \\  & = &  &  & = \\  &  & \overline{T} (X)(p) & \overline{T} (X)(D_{f}) \ar[rr]_{\overline{T} (X)(g_{f})} &  & \overline{T} (X)(p)}

where h_{f_{2},g_{f}} is the unique free map such that

[figure: p; g_{f}; g_{f_{2}g_{f}}; D_{f}; f_{2}; f_{2}; D_{f_{2}g_{f}}; h_{f_{2},g_{f}}; t_{D_{f_{2}g_{f}}}; D_{f_{2}}; t_{D_{f_{2}}}; 1; =; =; =]

and \overline{μ}_{1,p}(f, f_{2}) = f_{2}g_{f}. An element α of \overline{T} (X)(p) which is sent to f_{2}g_{f} by \overline{T}^{2}(t_{X})_{p} is by definition an element of XD_{f_{2},g_{f}}. Since g_{f_{2}}g_{f} is generic, h_{f_{2},g_{f}} is invertible, and so there is a unique element β such that Xh_{f_{2},g_{f}} (β) = α. Now β regarded as an element of \overline{T}^{2}(X)(p), is unique such that \overline{T}^{2}(t_{X})_{p}(β) = (f, f_{2}) and \overline{μ}_{X,p}(β) = α.

Thus as promised we have a monad morphism

(CAT(ℂ̂)(i_{0}, 1), φ) : (CAT(ℂ̂), T) → (CAT(Θ̂_{0}), \overline{T})

in the 2-category of 2-categories.

Let S be a 2-monad on CAT whose underlying 2-functor is p.r.a, and we regard S and \overline{T} as 2-monads on CAT(Θ̂_{0}). We now describe a cartesian distributive law \overline{λ} : \overline{T} S→S\overline{T}. For X ∈ CAT(Θ̂_{0}) and p ∈ Θ_{0} define the functor \overline{λ}_{X,p} as the unique functor such that

\xymatrix{S(XD_{f}) \ar[rr]^{c_{f}} \ar[ddrr]_{Sc_{f}} &  & \overline{T} (S∘X)(p) \ar[dd]^{_{\overline{λ}_{X,p}}} \\  & = \\  &  & S(\overline{T} (X)(p))}

8.4. Lemma. The functors \overline{λ}_{X.p} just defined are the components of a cartesian transformation \overline{λ} : \overline{T} S→S\overline{T}.

Proof. To verify that the above definition is natural in p let k : q→p be in Θ_{0}. Then the desired naturality follows since for all f : p→1 in Kl(T) we have

[figure: S(XD_{f}); c_{f}; S(Xh_{f,k}); Sc_{f}; \overline{T} (S∘X)(p); \overline{λ}_{X,p}; \overline{T} (S∘X)(k); S(\overline{T} (X)(p)); S(\overline{T} (X)(k)); S(XD_{fk}); c_{fk}; Sc_{fk}; \overline{T} (S∘X)(q); _{\overline{λ}_{X,q}}; S(\overline{T} (X)(p)); =; =; =]

where the outside also commutes. By definition \overline{λ}_{X,p} is 2-natural in X. It remains to see that it is in fact cartesian natural in X. Notice that \overline{λ}_{X,p} is a coproduct comparison map, and so by lemma(2.16) the result follows.

We must now verify that \overline{λ} satisfies the axioms of a distributive law. Instead of verifying the axioms directly we exploit instead the correspondence between distributive laws and monad liftings. First we recall this correspondence.

Given a 2-category K and monad (A, t) in K, an Eilenberg-Moore object consists of an object A^{t} of K together with a monad morphism (u, τ) : (A^{t}, 1)→(A, t) satisfying a universal property^{16}. For instance in CAT the object A^{t} is the category of algebras of the monad t, and u is the forgetful functor. As one would expect from this basic case, the forgetful arrow u has a left adjoint f and uf = t. Given monads (A, s) and (A, t) in K a distributive law of s over t consists of λ : ts→st such that (s, λ) is a monad morphism (A, t)→(A, t) and

[figure: 1; s; t; t; s; η; λ; =; 1; t; t; 1; s; =; η; s; s; t; t; s; λ; μ; =; s; s; t; t; s; λ; λ; μ]

In other words, λ satisfies 4 axioms which express its compatibility with the units and multiplications of s and t. Let K be a 2-category with Eilenberg-Moore objects. To give a distributive law λ : ts→st in K is the same as giving a lifting of the monad s to a monad \overline{s} on A^{t}, the object of t-algebras, in the sense that we have a monad (\overline{s}, \overline{η}, \overline{μ}) on A^{t} such that su = u\overline{s} and

[figure: A^{T}; 1; u; A^{T}; u; A; s; A; \overline{η}; η; A^{T}; ^{\overline{s}^{2}}; u; A^{T}; u; A; s; A; \overline{μ}; μ]

commute. Given a lifting \overline{s} as above, one obtains the corresponding distributive law as the composite

\xymatrix{ts \ar[r]^{tsη} & tst \ar[r]^{uε\overline{s}f} & st} .

In our situation K is the 2-category of 2-categories, 2-functors and 2-natural transformations. Monads in K are 2-monads and the object A^{t} of algebras of a 2-monad t is the 2-category t-Alg_{s}. Observe that the 2-monad on CAT(Θ̂_{0}) given by composition with S lifts to a monad on CAT(Θ̂_{T}) which by lemma(4.5) may be identified with \overline{T} -Alg_{s}, and so corresponds to a distributive law, and we shall now see that this distributive law is given by \overline{λ}. To see this note that for all f : p→1 in Kl(T) we have

16Eilenberg-Moore objects are a special kind of 2-categorical limit. In particular note that by [Street, 1976] any finitely complete 2-category has Eilenberg-Moore objects. See [Street, 1972] [Lack-Street, 2002] for further discussion.

[figure: \overline{T} (S∘X)(p); \overline{T} (S∘\overline{η}_{X})_{p}; \overline{T} (S∘\overline{T} X)(p); _{\overline{ε}_{S∘\overline{T} X,p}}; S(XD_{f}); Sc_{t_{D_{f}}}; c_{f}; c_{f}; S(\overline{T} (X)(D_{f})); c_{f}; S(\overline{T} X(g_{f})); S(\overline{T} (X)(p)); =; (I); =]

and we note that region (I) also commutes by by equation(10) in the proof of lemma(9). By the definition of \overline{λ} this yields \overline{λ}_{X,p} = \overline{ε}_{S∘\overline{T} X,p}\overline{T} (S∘\overline{η}_{X})_{p} and so \overline{λ} is indeed the distributive law corresponding to the canonical lifting of S to CAT(Θ̂) as claimed. We have proved:

8.5. Corollary. Let T be a p.r.a monad on ℂ̂ and S be a 2-monad on CAT whose underlying 2-functor is p.r.a. Then the functors \overline{λ}_{X.p} defined above are the components of a cartesian distributive law \overline{λ} : \overline{T} S→S\overline{T} between the 2-monads S and \overline{T} on CAT(Θ̂_{0}).

Now we assume that T -cardinals are connected. We shall now explain how \overline{λ} restricts to a cartesian distributive law λ : T S→ST. Let p ∈ Θ_{0} and recall its description as a canonical colimit of representables that we described prior to lemma(4.12). For x : C→p in y/p and X ∈ CAT(ℂ̂) we write π_{X,x} for the composite functor

\xymatrix{CAT(ℂ̂)(p, X) \ar[r]^{−∘x} & CAT(ℂ̂)(C, X) ≅ XC}

and note that the π_{X,x} for x : C→p, form the components of a limit cone. Define

κ_{X,p} : S(CAT(ℂ̂)(p, X)) → CAT(ℂ̂)(p, S∘X)

as the unique functor such that π_{S∘X,x}κ = S(π_{X,x}) for all x : C→p.

8.6. Lemma. The functors κ_{X,p} just described are the components of a natural isomorphism.

Proof. By definition κ_{X,p} is natural in p and 2-natural in X, and is the comparison map for a connected limit since T -cardinals are connected. It is an isomorphism since S as a p.r.a 2-functor preserves connected limits.

8.7. Lemma. κ is the 2-cell part of a monad morphism

(CAT(ℂ̂)(i_{0}, 1), κ) : (S, CAT(ℂ̂))→(S, CAT(Θ̂_{0}))

Proof. To see that κ is compatible with the unit note that for all x : C→p we have

\xymatrix{CAT(ℂ̂)(p, X) \ar[r]^{π} \ar[dd]_{η} & XC \ar[dd]^{η} \\  &  & CAT(ℂ̂)(p, X) \ar[r]^{π} \ar[dd]_{CAT(ℂ̂)(p,η)} & XC \ar[dd]^{η} \\ S(CAT(ℂ̂)(p, X)) \ar[r]^{Sπ} \ar[dd]_{κ} & SXC \\  &  & SCAT(ℂ̂)(p, X) \ar[r]_{π} & SXC \\ CAT(ℂ̂)(p, S∘X) \ar@{-->}[uur]_{π}}

in which all the regions are commutative by the naturality of η, the definition of κ and the naturality of π_{X,x} in X. Since the π_{S∘X,x} form limit cones the result follows. To see that κ is compatible with the multiplication note that for all x : C→p we have

\xymatrix{S^{2}(CAT(ℂ̂)(p, X)) \ar[d]_{Sκ} \ar[r]^{S^{2}π} & S^{2}X(C) \ar[d]^{μ} & S^{2}(CAT(ℂ̂)(p, X)) \ar[dr]^{S^{2}π} \ar[d]_{μ} \\ S(CAT(ℂ̂)(p, S∘X)) \ar@{-->}[ur]^{Sπ} \ar[d]_{κ} & SX(C) & S(CAT(ℂ̂)(p, X)) \ar[d]_{κ} \ar[dr]^{Sπ} & S^{2}X(C) \ar[d]^{μ} \\ CAT(ℂ̂)(p, S^{2}∘X) \ar[uur]_{π} \ar[r]_{CAT(ℂ̂)(p,μ)} & CAT(ℂ̂)(p, S∘X) \ar[u]_{π} & CAT(ℂ̂)(p, S∘X) \ar[r]_{π} & SX(C)}

in which all the regions are commutative by the naturality of μ, the definition of κ and the naturality of π_{X,x} in X. Since the π_{S∘X,x} form limit cones the result follows.

To describe λ, we shall use the string diagrams of [Joyal-Street, 1991] to depict arrows and 2-cells in a 2-category. As an illustration, the axioms for a monad (A, s) are depicted as

[figure: η; μ; s; s; =; \begin{matrix}s \\ s \end{matrix}; =; η; μ; s; s; μ; s; s; μ; s; s; =; μ; s; s; μ; s; s]

so our string diagrams go from top to bottom, and we always use η to denote the unit of a monad, and μ the multiplication. In subsequent diagrams we shall omit labels on the strings when no ambiguity results. The reader will easily translate the axioms for a monad morphism and for a distributive law into the language of string diagrams. Now λ is defined to be to be the unique 2-natural transformation such that

[figure: λ; =; φ^{−1}; \overline{λ}; κ; φ; κ^{−1}]

The left hand side of the above equation is CAT(ℂ̂)(i_{0}, 1)λ, and so this definition makes sense since CAT(ℂ̂)(i_{0}, 1) is fully faithful.

8.8. Theorem. Let T be a p.r.a monad on ℂ̂ such that T -cardinals are connected, and let S be a 2-monad on CAT whose underlying 2-functor is p.r.a. Then λ : T S→ST defined above is a cartesian distributive law.

Proof. It is cartesian since \overline{λ} is cartesian and CAT(ℂ̂)(i_{0}, 1), as a fully faithful right adjoint, reflects pullbacks. We now verify the 4 distributive law axioms. The axiom which expresses the compatibility of λ with T ’s unit is given on the left

[figure: η^{S}; =; λ; η; η^{S}^{CAT(ℂ̂)(i_{0},1)}; =; λ; η]

but it suffices to verify this axiom after composition with CAT(ℂ̂)(i_{0}, 1) because this 2-functor is fully faithful, in other words, it suffices to verify the axiom given on the right in the previous display. Similarly with the other three distributive law axioms: we verify them after composition with CAT(ℂ̂)(i_{0}, 1). Proceeding in this way we verify λ’s compatibility with T ’s unit,

[figure: η; λ; =; φ^{−1}; η; \overline{λ}; κ; φ; κ^{−1}; =; η; \overline{λ}; κ; φ; κ^{−1}]

[figure: =; κ^{−1}; φ; η; κ; =; κ^{−1}; κ; η; =; η]

λ’s compatibility with S’s unit,

[figure: η; λ; =; φ^{−1}; \overline{λ}; κ; φ; κ^{−1}; η; =; φ^{−1}; \overline{λ}; κ; φ; η]

= [figure: φ^{−1}; φ; κ; η] =  η

λ’s compatibility with T ’s multiplication,

[figure: λ; λ; μ; =; φ^{−1}; λ; φ; κ; μ; κ^{−1}; λ; =; φ^{−1}; φ^{−1}; λ; κ; φ; μ; κ^{−1}; φ; λ; κ; κ^{−1}]

728  MARK WEBER

[figure: =; φ^{−1}; φ^{−1}; κ^{−1}; \overline{λ}; \overline{λ}; φ; φ; κ; μ; =; φ^{−1}; φ^{−1}; κ^{−1}; \overline{λ}; \overline{λ}; μ; φ; κ; =; φ^{−1}; φ^{−1}; κ^{−1}; μ; \overline{λ}; φ; κ]

[figure: =; μ; φ^{−1}; κ^{−1}; λ; φ; κ; =; μ; λ]

and finally λ’s compatibility with S’s multiplication,

[figure: λ; λ; μ; =; φ^{−1}; λ; φ; κ; λ; μ; κ^{−1}; =; φ^{−1}; κ^{−1}; λ; φ; φ^{−1}; κ^{−1}; λ; φ; κ; κ; μ]

[figure: =; φ^{−1}; κ^{−1}; κ^{−1}; λ; λ; φ; κ; κ; μ; =; φ^{−1}; κ^{−1}; κ^{−1}; λ; λ; φ; μ; κ; =; φ^{−1}; κ^{−1}; κ^{−1}; μ; λ; φ; κ]

[figure: =; φ^{−1}; κ^{−1}; μ; \overline{λ}; φ; κ; =; μ; λ]

and so λ is indeed a distributive law.

Since the monad structure on ST is constructed using the monad structures of S and T and the distributive law established in theorem(8.8), by corollary(7.2) we immediately obtain the following result.

8.9. Corollary. Let T be a p.r.a monad on ℂ̂ such that T -cardinals are connected.

1. If S is a p.r.a 2-monad on CAT then the 2-functor ST on CAT(ℂ̂) has the structure of a p.r.a 2-monad.

2. If S is a familial (resp. opfamilial) 2-monad on CAT then the 2-functor ST on CAT(ℂ̂) has the structure of a familial (resp. opfamilial) 2-monad. Moreover if S1 is a groupoid then so is ST 1.

8.10. Examples. A monad T on 𝔾̂ arising from an ω-operad of [Batanin, 1998] is a p.r.a monad and T -cardinals, being globular cardinals, are connected. Thus the 2-monads of [Weber, 2005] giving rise to symmetric and braided analogues of higher operads are all familial.

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Email: weber@pps.jussieu.fr

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