Theory and Applications of Categories, Vol. 27, No. 4, 2012, pp. 47–64.

THE CORE OF ADJOINT FUNCTORS

ROSS STREET

Abstract. There is a lot of redundancy in the usual definition of adjoint functors. We define and prove the core of what is required. First we do this in the hom-enriched context. Then we do it in the cocompletion of a bicategory with respect to Kleisli objects, which we then apply to internal categories. Finally, we describe a doctrinal setting.

1. Introduction

Kan [7] introduced the notion of adjoint functors. By defining the unit and counit natural transformations, he paved the way for the notion to be internalized to any 2-category. This was done by Kelly [8] whose interest at the time was particularly in the 2-category of V-categories for a monoidal category V (in the sense of Eilenberg-Kelly [4]).

During my Topology lectures at Macquarie University in the 1970s, the students and I realized, in proving that a function f between posets was order preserving when there was a function u in the reverse direction such that f(x) ≤ a if and only if x ≤ u(a), did not require u to be order preserving. I realized then that knowing functors in the two directions only on objects and the usual hom adjointness isomorphism implied the effect of the functors on homs was uniquely determined. Writing this down properly led to the present paper.

Section 2 merely reviews adjunctions between enriched categories. Section 3 introduces the notion of core of an enriched adjunction: it only involves the object assignments of the two functors and a hom isomorphism with no naturality requirement. The main result characterizes when such a core is an adjunction.

The material becomes increasingly for mature audiences; that is, for those with knowledge of bicategories. Sections 4 and 5 present results about adjunctions in the Kleisli object cocompletion of a bicategory in the sense of [13]. In particular, this is applied in Section 6 to adjunctions for categories internal to a finitely complete category. By a different choice of bicategory, where enriched categories can be seen as monads (see [2]), we could rediscover the work of Section 3; however, we leave this to the interested reader. In Section 7 we describe a general setting involving a pseudomonad (doctrine) on a bicategory.

The author gratefully acknowledges the support of an Australian Research Council Discovery Grant DP1094883.

Received by the editors 2012-01-01 and, in revised form, 2012-05-29.

Published on 2012-06-12 in the volume of articles from CT2011.

2000 Mathematics Subject Classification: 18A40; 18D10; 18D05.

Key words and phrases: adjoint functor; enriched category; bicategory; Kleisli cocompletion.

\begin{matrix}c \\ ◯ \end{matrix} Ross Street, 2012. Permission to copy for private use granted.

2. Adjunctions

For V-categories A and X, an adjunction consists of

1. V-functors U : A ⟶ X and F : X ⟶ A;

2. a V-natural family of isomorphisms π : A(F X, A) ≅ X (X, UA) in V indexed by A ∈ A, X ∈ X.

We write π : F ⊣ U : A ⟶ X.

The following result is well known; for example see Section 1.11 of [10].

2.1. Proposition. Suppose U : A ⟶ X is a V-functor, F : obX ⟶ obA is a function, and, for each X ∈ X, π : A(F X, A) ≅ X (X, UA) is a family of isomorphisms V-natural in A ∈ A. Then there exists a unique adjunction π : F ⊣ U : A ⟶ X for which F : obX ⟶ obA is the effect of the V-functor F : X ⟶ A on objects.

3. Cores

3.1. Definition. For V-categories A and X, an adjunction core consists of

1. functions U : obA ⟶ obX and F : obX ⟶ obA;

2. a family of isomorphisms π : A(F X, A) ≅ X (X, UA) in V indexed by A ∈ A, X ∈ X.

Given such a core, we make the following definitions:

(a) β_{X} : X ⟶ UF X is the composite

I ⟶^{j} A(F X, F X) ⟶^{π} X (X, UF X);

(b) α_{A} : F UA ⟶ A is the composite

I ⟶^{j} X (UA, UA) ⟶^{π^{−1}} A(F UA, X);

(c) U_{AB} : A(A, B) ⟶ X (UA, UB) is the composite

A(A, B) ⟶^{A(α_{A},1)} A(F UA, B) ⟶^{π} X (UA, UB);

(d) F_{XY} : X (X, Y) ⟶ A(F X, F Y) is the composite

X (X, Y) ⟶^{X (1,β_{Y} )} X (X, UF Y) ⟶^{π^{−1}} A(F X, F Y).

Clearly each adjunction π : F ⊣ U : A ⟶ X includes an adjunction core as part of its data. Then it follows directly from the Yoneda lemma and the definitions (a) and (b) that the effect of U and F on homs are as in (c) and (d).

3.2. Theorem. An adjunction core extends to an adjunction if and only if one of the diagrams (3.1) or (3.2) below commutes. The adjunction is unique when it exists.

\xymatrix{A(A, B) ⊗ A(F X, A) \ar[r]^{U_{A,B}⊗π} \ar[d]_{comp} & X (UA, UB) ⊗ X (X, UA) \ar[d]^{comp} \\ A(F X, B) \ar[r]_{π} & X (X, UB)}  (3.1)

\xymatrix{X (Y, UA) ⊗ X (X, Y) \ar[r]^{π^{−1}⊗F_{X,Y}} \ar[d]_{comp} & A(F Y, A) ⊗ A(F X, F Y) \ar[d]^{comp} \\ X (X, UA) \ar[r]_{π^{−1}} & A(F X, A)}  (3.2)

Proof. We deal first with the version involving diagram (3.1). For an adjunction, (3.1) expresses the V-naturality of π in A ∈ A. Conversely, given an adjunction core satisfying (3.1), we paste to the left of (3.1) with X = UC, the diagram

\xymatrix{A(A, B) ⊗ A(C, A) \ar[r]^{1⊗A(α_{C},1)} \ar[d]_{comp} & A(A, B) ⊗ A(F UC, A) \ar[d]^{comp} \\ X (X, UA) \ar[r]_{π^{−1}} & A(F X, A)}  (3.3)

which commutes by naturality of composition. This leads to the following commutative square.

\xymatrix{A(A, B) ⊗ A(C, A) \ar[r]^{U⊗U} \ar[d]_{comp} & X (UA, UB) ⊗ X (UC, UA) \ar[d]^{comp} \\ A(C, B) \ar[r]_{U} & X (UC, UB)}  (3.4)

We also have the equality

(I ⟶^{j} A(A, A) ⟶^{U} X (UA, UB)) = (I ⟶^{j} X (UA, UB))  (3.5)

straight from the definitions (b) and (c). Together (3.4) and (3.5) tell us that U is a V-functor. Now the general diagram (3.1) expresses the V-naturality of π in A. By Proposition 2.1, we have an adjunction determined uniquely by the core.

Writing V^{rev} for V with the reversed monoidal structure A⊗^{rev}B = B⊗A, and applying the first part of this proof to the V^{rev}-enriched adjunction π^{−1} : U^{op} ⊣ F^{op} : X^{op} ⟶ A^{op}, which is the same as an adjunction π : F ⊣ U : A ⟶ X, we see that it is equivalent to an adjunction core satisfying (3.2).

3.3. Corollary. If V is a poset then adjunction cores are adjunctions.

Proof. All diagrams, including (3.1), commute in such a V.

An object X of a category is often called preterminal when, for all objects K, there is at most one morphism K ⟶ X. In the category Set, both the initial and terminal objects are preterminal. A poset is a category in which every object is preterminal. A V-category A is locally preterminal when each hom A(A, B) is preterminal in V. Using this terminology, we can state a more general corollary.

3.4. Corollary. If in Definition 3.1 either A or X is locally preterminal then the adjunction core is an adjunction.

4. Adjunctions between monads

This section will discuss adjunctions in a particular bicategory KL(K) of monads in a bicategory K. The results will apply to adjunctions between categories internal to a category C with pullbacks.

As well as defining bicategories Bénabou [1] defined, for each pair of bicategories A and K, a bicategory Bicat(A, K) whose objects are morphisms A ⟶ K of bicategories (also called lax functors), whose morphisms are transformations (also called lax natural transformations), and whose 2-cells are modifications. In particular, Bicat(1, K) is one bicategory whose objects are monads in K; it was called Mnd(K) in [17] for the case of a 2-category K, where it was used to discuss Eilenberg–Moore objects in K. We shall also use the notation Mnd(K) when K is a bicategory.

We write K^{op} for the dual of K obtained by reversing morphisms (not 2-cells). Monads in K^{op} are the same as monads in K. So we also have the bicategory

Mnd^{op}(K) = Bicat(1, K^{op})^{op}

whose objects are monads in K. This was used in [17] to discuss Kleisli objects in K.

Two more bicategories EM(K) and KL(K), with objects monads in K, were defined in [13]. The first freely adjoins Eilenberg–Moore objects and the second freely adjoins Kleisli objects to K. In fact, EM(K) has the same objects and morphisms as Mnd(K) but different 2-cells while KL(K) has the same objects and morphisms as Mnd^{op}(K) but different 2-cells.

A monad in a bicategory K is an object A equipped with a morphism s : A ⟶ A and 2-cells η : 1_{A} ⟶ s and μ : ss ⟶ s such that

\xymatrix{ & s(ss) \ar[dr]^{sμ} \\ (ss)s \ar@{-->}[ur]^{≅} \ar[d]_{μs} &  & ss \ar[d]^{μ} \\ ss \ar[rr]_{μ} &  & w}  (4.1)

and the composites

s1 ⟶^{sη} ss ⟶^{μ} s and 1s ⟶^{ηs} ss ⟶^{μ} s  (4.2)

should be the canonical isomorphisms. We shall use the same symbols η and μ for the unit and multiplication of all monads; so we simply write (A, s) for the monad.

For monads (A, s) and (A′, s′) in K, a monad opmorphism (f, ϕ) : (A, s) ⟶ (A′, s′) consists of a morphism f : A ⟶ A′ and a 2-cell ϕ : fs ⟶ s′f in K such that

((fs) s ⟶^{ϕs} (s′f) s ⟶^{≅} s′(fs) ⟶^{s′ϕ} s′(s′f) ⟶^{≅} s′(s′f) ⟶^{μf} s′f)

= ((fs) s ⟶^{≅} f(ss) ⟶^{fμ} fs ⟶^{ϕ} s′f) (4.3)

and

(f1 ⟶^{fη} fs ⟶^{ϕ} s′f) = (f1 ⟶^{≅} 1f ⟶^{ηf} s′f) .  (4.4)

The composite of monad opmorphisms (f, ϕ) : (A, s) ⟶ (A′, s′) and (f′, ϕ′) : (A′, s′) ⟶ (A′′, s′′) is defined to be (f′f, ϕ′ ⋆ ϕ) : (A, s) ⟶ (A′′, s′′) where ϕ′ ⋆ ϕ is the composite

(f′f) s ⟶^{≅} f′(fs) ⟶^{f′ϕ} f′(s′f) ⟶^{≅} (f′s′) f ⟶^{ϕ′f} (s′′f′) f ⟶^{≅} s′′(f′f) .  (4.5)

The objects of both Mnd^{op}(K) and KL(K) are monads (A, s) in K. The morphisms in both are the opmorphisms (f, ϕ). The 2-cells σ : (f, ϕ) ⟶ (g, ψ) : (A, s) ⟶ (A′, s′) in Mnd^{op}(K) are 2-cells σ : f ⟶ g in K such that the following square commutes.

\xymatrix{fs \ar[r]^{ϕ} \ar[d]_{σs} & s′f \ar[d]^{s′σ} \\ gs \ar[r]_{ψ} & s′g}  (4.6)

Vertical and horizontal composition in Mnd^{op}(K) are performed in the obvious way so that the projection Und : Mnd^{op}(K) ⟶ K, taking (A, s) to A, (f, ϕ) to f, and σ to σ, preserves them. The associativity and unit isomorphisms in Mnd^{op}(K) are also such that Und preserves them, making Und a strict morphism of bicategories.

A 2-cell ρ : (f, ϕ) ⟶ (g, ψ) : (A, s) ⟶ (A′, s′) in KL(K) is a 2-cell ρ : f ⟶ s′g in K such that

(fs ⟶^{ϕ} s′f ⟶^{s′ρ} s′(s′g) ⟶^{≅} (s′s′) g ⟶^{μg} s′g)

= (fs ⟶^{ρs} (s′g) s ⟶^{≅} s′ (gs) ⟶^{s′ψ} s′ (s′g) ⟶^{≅} (s′s′) g ⟶^{μg} s′g) . (4.7)

The vertical composite of the 2-cells ρ : (f, ϕ) ⟶ (g, ψ) and τ : (g, ψ) ⟶ (h, θ) is the 2-cell

f ⟶^{ρ} s′g ⟶^{s′τ} s′(s′h) ⟶^{≅} (s′s′) h ⟶^{μh} s′h .  (4.8)

The horizontal composite of 2-cells ρ : (f, ϕ) ⟶ (g, ψ) : (A, s) ⟶ (A′, s′) and ρ′ : (f′, ϕ′) ⟶ (g′, ψ′) : (A′, s′) ⟶ (A′′, s′′) is the 2-cell

f′f ⟶^{f′ρ} f′(s′g) ⟶^{≅} (f′s′) g ⟶^{ϕ′g} (s′′f′) g

⟶^{(s′′ρ′)g} (s′′(s′′g′)) g ⟶^{≅} (s′′s′′) (g′g) ⟶^{μ(g′g)} s′′(g′g) . (4.9)

Each 2-cell σ : (f, ϕ) ⟶ (g, ψ) in Mnd^{op}(K) defines a 2-cell ρ : (f, ϕ) ⟶ (g, ψ) in KL(K) via ρ = ηg · σ. The associativity and unit isomorphisms for KL(K) are determined by the condition that we have a strict morphism of bicategories

K : Mnd^{op}(K) ⟶ KL(K)

which is the identity on objects and morphisms and takes each 2-cell σ to ηg · σ.

Henceforth we shall invoke the coherence theorem (see [15] and [5]) that every bicategory is biequivalent to a 2-category to write as if we were working in a 2-category KL(K). We also recommend reworking the proofs below using the string diagrams of [6] as adapted for bicategories in [19] and [20].

Now we are in a position to examine what is involved in an adjunction

(f, ϕ) ⊣ (u, υ) : (A, s) ⟶ (X, t)  (4.10)

with counit α : (f, ϕ) · (u, υ) ⟶ 1_{(A,s)} and unit β : 1_{(X,t)} ⟶ (u, υ) · (f, ϕ) in KL(K).

We have morphisms u : A ⟶ X and f : X ⟶ A in K. We have 2-cells υ : uf ⟶ tu and ϕ : ft ⟶ sf both satisfying (4.3) and (4.4) with the variables appropriately substituted.

We have a 2-cell α : fu ⟶ s satisfying

(fus ⟶^{αs} ss ⟶^{μ} s) = (fus ⟶^{fυ} ftu ⟶^{ϕu} sfu ⟶^{sα} ss ⟶^{μ} s)  (4.11)

which is (4.7) for α.

We have a 2-cell β : 1_{X} ⟶ tuf satisfying

(t ⟶^{tβ} ttuf ⟶^{μuf} tuf) = (t ⟶^{βt} tuft ⟶^{tuϕ} tusf ⟶^{tυf} ttuf ⟶^{μuf} tuf)  (4.12)

which is (4.7) for β.

Using the rules for compositions in KL(K), we see that the two triangle conditions for the counit and unit of an adjunction become, in this case, the identities

(f ⟶^{ηf} sf) = (f ⟶^{fβ} ftuf ⟶^{ϕuf} sfuf ⟶^{sαf} ssf ⟶^{μf} sf)  (4.13)

and

(u ⟶^{ηu} tu) = (u ⟶^{βu} tufu ⟶^{tuα} tus ⟶^{tυ} ttu ⟶^{μu} tu) .  (4.14)

It is common to call a morphism f : X ⟶ A in a bicategory K a map when it has a right adjoint. We write f^{⋆} : A ⟶ X for a selected right adjoint, η_{f} : 1_{X} ⟶ f^{⋆}f for the unit, and ε_{f} : ff^{⋆} ⟶ 1_{A} for the counit.

4.1. Theorem. Suppose (4.10) is an adjunction in KL(K) with counit α and unit β, and suppose f : X ⟶ A is a map in K. Then the composite 2-cell

π : f^{⋆}s ⟶^{βf^{⋆}s} tuff^{⋆}s ⟶^{tuε_{f} s} tus ⟶^{tυ} ttu ⟶^{μu} tu  (4.15)

in K is invertible with inverse defined by the composite 2-cell

π^{−1} : tu ⟶^{η_{f} tu} f^{⋆}ftu ⟶^{f^{⋆}ϕu} f^{⋆}sfu ⟶^{f^{⋆}sα} f^{⋆}ss ⟶^{f^{⋆}μ} f^{⋆}s.  (4.16)

Proof. Without yet knowing that π^{−1} as given by (4.16) is inverse to π, we calculate ππ^{−1}:

μu · tυ · tuε_{f}s · βf^{⋆}s · f^{⋆}μ · f^{⋆}sα · f^{⋆}ϕu · η_{f}tu
= μu · tυ · tuμ · tusα · tuϕu · tuε_{f}ftu · tufη_{f}tu · βtu
= μu · tυ · tuμ · tusα · tuϕu · βtu
= μu · μtu · ttυ · tυs · tusα · tuϕu · βtu
= μu · tυ · tuα · μufu · tυfu · tuϕu · βtu
= μu · tυ · tuα · μufu · tβu
= μu · tυ · μus · ttuα · tβu
= μu · tμu · ttυ · ttuα · tβu
= μu · tηu
= 1_{tu}.

The first, fourth, sixth and seventh equalities above follow purely from properties of composition in K. The second equality uses the triangular equation appropriate to the unit and counit for f and its right adjoint. The third equality uses the opmorphism property of (u, υ) and associativity of μ. The fifth equality uses (4.12). The eighth equality uses (4.14).

Now we calculate π^{−1}π:

f^{⋆}μ · f^{⋆}sα · f^{⋆}ϕu · η_{f}tu · μu · tυ · tuε_{f}s · βf^{⋆}s
= f^{⋆}μ · f^{⋆}sμ · f^{⋆}ssα · f^{⋆}sϕu · f^{⋆}ϕtu · η_{f}ttu · tυ · tuε_{f}s · βf^{⋆}s
= f^{⋆}μ · f^{⋆}μs · f^{⋆}sαs · f^{⋆}ϕus · η_{f}tus · tuη_{f}sβf^{⋆}s
= f^{⋆}μ · f^{⋆}sε_{f}s · f^{⋆}ηff^{⋆}s · η_{f}f^{⋆}s
= 1_{f^{⋆}s},

using the associativity and unit conditions for the monads, the opmorphism property of (f, ϕ), equation (4.11), and equation (4.14).

As expected by general principles of doctrinal adjunction [9], a monad opmorphism (f, ϕ) : (X, t) ⟶ (A, s) for which f is a map in K gives rise to a monad morphism (f^{⋆}, ϕ̂) : (A, s) ⟶ (X, t) where ϕ̂ : tf^{⋆} ⟶ f^{⋆}s is the mate of ϕ under the adjunction f ⊣ f^{⋆} in the sense of [11].

4.2. Remark. The Kleisli construction for K is a left biadjoint for the pseudofunctor K ⟶ KL(K) taking A to (A, 1_{A}). The value of the biadjoint at (A, t) is denoted by A_{t}. When the Kleisli construction exists, it takes the adjunction (4.10) in KL(K) to an adjunction f_{ϕ} ⊣ u_{υ} : A_{s} ⟶ X_{t} in K.

5. Cores between monads

5.1. Definition. An adjunction core (u, g, π) between monads (A, s) and (X, t) in a bicategory K consists of the following data in K:

1. morphisms u : A ⟶ X and g : A ⟶ X;

2. an invertible 2-cell π : gs ⟶ tu.

Given such a core, we make the following definitions:

(a) β̄ : g ⟶ tu is the composite

g ⟶^{gη} gs ⟶^{π} tu;

(b) ᾱ : u ⟶ gs is the composite

u ⟶^{ηu} tu ⟶^{π^{−1}} gs;

(c) υ : us ⟶ tu is the composite

us ⟶^{ᾱs} gss ⟶^{gμ} gs ⟶^{π} tu;

(d) ψ : tg ⟶ gs is the composite

tg ⟶^{tβ̄} ttu ⟶^{μu} tu ⟶^{π^{−1}} gs.

5.2. Proposition. An adjunction core between monads (A, s) and (X, t) is obtained from the data of Theorem 4.1 by putting g = f^{⋆}. Moreover, the β̄ of (a) and the ᾱ of (b) are the mates of the unit β and counit α, respectively, the composite in (c) recovers υ, and the ψ of (d) is the mate ϕ̂ of ϕ.

Proof. That we have an adjunction core follows from the invertibility of π according to Theorem 4.1. Next we look at the composite in (a):

π · f^{⋆}η
= μu · tυ · tuε_{f}s · βf^{⋆}s · f^{⋆}η
= μu · tυ · tuη · tuε_{f} · βf^{⋆}
= μu · tηu · tuε_{f} · βf^{⋆}
= tuε_{f} · βf^{⋆}

which is the mate β̄ of β. That the composite in (b) gives ᾱ is a similar calculation. Next we calculate:

π^{−1}s · ηus
= f^{⋆}μs · f^{⋆}sαs · f^{⋆}ϕus · η_{f}tus · ηus
= f^{⋆}μs · f^{⋆}sαs · f^{⋆}ϕus · f^{⋆}fη_{f}us · η_{f}us
= f^{⋆}μs · f^{⋆}sαs · f^{⋆}ηfus · η_{f}us
= f^{⋆}μs · f^{⋆}ηss · f^{⋆}αs · η_{f}us
= f^{⋆}αs · η_{f}us,

so the composite in (c) is:

μu · tυ · tuε_{f}s · βf^{⋆}sa · f^{⋆}μ · π^{−1}s · ηus
= μu · tυ · tuε_{f}s · βf^{⋆}s · f^{⋆}μ · f^{⋆}αs · η_{f}us
= μu · tυ · tuε_{f}s · tuff^{⋆}μ · βf^{⋆}ss · f^{⋆}αs · η_{f}us
= μu · tυ · tuε_{f}s · tuff^{⋆}μ · tuff^{⋆}αs · βf^{⋆}fus · η_{f}us
= μu · tυ · tuμ · tuε_{f}ss · tuff^{⋆}αs · βf^{⋆}fus · η_{f}us
= μu · tμu · ttυ · tυs · tuε_{f}ss · tuff^{⋆}αs · βf^{⋆}fus · η_{f}us
= μu · tμu · ttυ · tυs · tuαs · tuε_{f}fus · βf^{⋆}fus · η_{f}us
= μu · tμu · ttυ · tυs · tuαs · tuε_{f}fus · tufη_{f}s · βus
= μu · tμu · ttυ · tυs · tuαs · βus
= μu · tυ · μus · tυs · tuαs · βus
= μu · tυ · ηus
= μu · ηtu · υ
= υ,

as required. The calculation for the composite in (d) is similar.

The Corollary of the following result should be compared with Theorem 3.2.

5.3. Theorem. For an adjunction core (u, g, π) between monads (A, s) and (X, t) in a bicategory K, the following two commutativity conditions (5.1) and (5.2) are equivalent.

\xymatrix{ & tus \ar[dr]^{tυ} \\ gss \ar[ur]^{πs} \ar[d]_{gμ} &  & ttu \ar[d]^{μu} \\ gs \ar[rr]_{π} &  & tu}  (5.1)

\xymatrix{ & tgs \ar[dr]^{ψs} \\ ttu \ar[ur]^{tπ^{−1}} \ar[d]_{μu} &  & gss \ar[d]^{gμ} \\ tu \ar[rr]_{π^{−1}} &  & gs}  (5.2)

Moreover, under these conditions, using definitions (a), (b), (c) and (d),

(i) (u, υ) : (A, s) ⟶ (X, t) is a monad opmorphism; (ii) (g, ψ) : (A, s) ⟶ (X, t) is a monad morphism; (iii) π is equal to the composite

gs ⟶^{β̄s} tus ⟶^{tυ} ttu ⟶^{μu} tu;

(iv) π^{−1} is equal to the composite

tu ⟶^{tᾱ} tgs ⟶^{ψs} gss ⟶^{gμ} gs;

(v) the following identity holds

(us ⟶^{ᾱs} gss ⟶^{gμ} gs) = (us ⟶^{υ} tu ⟶^{tᾱ} tgs ⟶^{ψs} gss ⟶^{gμ} gs);

(vi) the following identity holds

(tg ⟶^{tβ̄} ttu ⟶^{μu} tu) = (tg ⟶^{ψ} gs ⟶^{β̄s} tus ⟶^{tυ} ttu ⟶^{μu} tu);

(vii) the following identity holds

(g ⟶^{gη} gs) = (g ⟶^{β̄} tu ⟶^{tᾱ} tgs ⟶^{ψs} gss ⟶^{gμ} gs);

(viii) the following identity holds

(u ⟶^{ηu} tu) = (u ⟶^{ᾱ} gs ⟶^{β̄s} tus ⟶^{tυ} ttu ⟶^{μu} tu).

Proof. Assuming (5.1) at the first step, we have the calculation:

π · gμ · ψs
= μu · tυ · πs · ψs
= μu · tυ · πs · π^{−1}s · μus · tβ̄s
= μu · tυ · μus · tβ̄s
= μu · μtu · ttυ · tβ̄s
= μu · tμu · ttυ · tβ̄s
= μu · tπ,

proving (5.2). The converse is dual.

(i) Using (5.1) at the second step, we have the calculation:

μu · tυ · υs
= μu · tυ · πs · gμs · ᾱss
= π · gμ · gμs · ᾱss
= π · gμ · gsμ · ᾱss
= π · gμ · ᾱs · uμ
= υ · uμ.

We also have:

υ · uη
= π · gμ · ᾱs · uη
= π · gμ · π^{−1}s · ηus · uη
= π · gμ · π^{−1}s · tuη · ηu
= π · gμ · gsη · π^{−1} · ηu
= π · π^{−1} · ηu
= ηu.

Hence (u, υ) is a monad opmorphism.

(ii) This is dual to (i) using (5.2) instead of (5.1).

(iii) Using (5.1), we have:

π
= π · gμ · gηs
= μu · tυ · πs · gηs
= μu · tυ · β̄s.

(iv) This is dual to (iii) using (5.2) instead of (5.1).

(v) Using (iv), we immediately have:

gμ · ψs · tᾱ · υ
= π^{−1} · υ
= π^{−1} · π · gμ · ᾱs
= gμ · ᾱs.

(vi) This is dual to (v) using (iii) instead of (iv).

(vii) Using (iv), we immediately have:

gμ · ψs · tᾱ · β̄
= π^{−1} · β̄
= π^{−1} · π · gη
= gη.

(viii) This is dual to (vii) using (iii) instead of (iv).

5.4. Corollary. An adjunction core of the form (u, f^{⋆}, π) between monads (A, s) and (X, t) in a bicategory K extends to an adjunction (4.10) in KL(K) if and only if one of the diagrams (5.1) or (5.2) commutes. The adjunction is unique when it exists.

Proof. Properties (i)–(viii) of Theorem 5.3, when re-expressed with the mates α, β and ϕ replacing ᾱ, β̄ and ψ, give precisely what is required for an adjunction (4.10). The converse is Proposition 5.2.

5.5. Remark. Recall from [13] that the free completion EM(K) of K with respect to the Eilenberg–Moore construction is obtained as

EM(K) = KL(K^{op})^{op} .

Monads in K^{op} are the same as monads in K. An adjunction opcore between monads (X, t) and (A, s) in K is an adjunction core in K^{op}. This consists of morphisms f, v : X ⟶ A and an invertible 2-cell π : sv ≅ ft. Also recall from [13] that a wreath in K is a monad in EM(K). Many examples of wreaths were provided in that paper. A wreath on A consists of a monad t : A ⟶ A with unit η : 1 ⟹ t and multiplication μ : tt ⟹ t, an endomorphism s : A ⟶ A with 2-cells σ : 1 ⟹ st and ν : ss ⟹ st, and a 2-cell λ : ts ⟹ st satisfying conditions. This gives an adjunction cocore s, 1_{A} : A ⟶ A between the monad (A, s) and the wreath product monad (A, st) where π is the identity of st. Since 1_{A} has itself as adjoint and condition (5.1) can be checked, we obtain an example of an adjunction between monads from every wreath. To be explicit, we have the adjunction

(s, sμ.νt.sλ) ⊣ (1_{A}, sμ.σt) : (A, st) ⟶ (A, t)

associated with the Eilenberg–Moore construction for the monad (s, λ) on (A, t) in EM(K).

6. Cores between internal categories

This section will apply our results to categories internal to a category C which admits pullbacks. For this example, we take the bicategory K of the previous sections to be the bicategory Span(C) of spans in C as constructed by Bénabou in [1].

The objects of the bicategory Span(C) are those of C. A morphism S = (s_{0}, S, s_{1}) : U ⟶ V is a so-called span

U ⟵^{s_{0}} S ⟶^{s_{1}} V

from U to V in C. A 2-cell r : (s_{0}, S, s_{1}) ⟶ (t_{0}, T, t_{1}) : U ⟶ V is a morphism r : S ⟶ T in C such that t_{0}r = s_{0} and t_{1}r = s_{1}. Vertical composition of 2-cells is simply that of C. Horizontal composition uses pullback in C; more precisely,

(U ⟶^{(s_{0},S,s_{1})} V ⟶^{(t_{0},T,t_{1})} W) = (U ⟶^{(s_{0}p,P,t_{1}q)} W)

where

\xymatrix{P \ar[r]^{q} \ar[d]_{p} & T \ar[d]^{t_{0}} \\ S \ar[r]_{s_{1}} & V}  (6.1)

is a pullback square.

Each morphism f : U ⟶ V in C determines a span f_{⋆} = (1_{U}, U, f) : U ⟶ V. We write f^{⋆} : V ⟶ U for the span (f, V, 1_{V}) : V ⟶ U. It is well known that we have an adjunction f_{⋆} ⊣ f^{⋆} in Span(C); in fact, it is shown in [3] that the maps in Span(C) are all isomorphic to spans of the form f_{⋆} : U ⟶ V for some f : U ⟶ V in C.

One of the reasons for interest in the free Kleisli object cocompletion KL(K) in the paper [13] is that the 2-category Cat(C) of categories in C is equivalent to the sub-2-category of KL(Span(C)) obtained by restricting to the morphisms whose underlying morphisms in Span(C) are maps. We shall explain this in a bit more detail.

A category in C is a monad (A, S) in Span(C)). The object A of C is called the object of objects. The span S = (s_{0}, S, s_{1}) : A ⟶ A provides the object of morphisms S, the source operation s_{1} and the target operation s_{0}. The multiplication for the monad provides the composition operation and the unit for the monad provides the identities operation.

A functor between categories in C is a monad opmorphism of the form (f_{⋆}, ϕ) : (X, T) ⟶ (A, S) in Span(C)). The morphism f : X ⟶ A in C is called the effect on objects of the functor and the morphism ϕ : T ⟶ S in C is called the effect on morphisms of the functor.

A natural transformation between functors in C is precisely a 2-cell between them in KL(Span(C)).

As an immediate consequence of Corollary 5.4 we have:

6.1. Corollary. An adjunction core of the form (u_{⋆}, f^{⋆}, π) between categories (A, S) and (X, T) in a category C extends to an adjunction (f_{⋆}, ϕ) ⊣ (u_{⋆}, υ) in Cat(C) if and only if one of the diagrams (5.1) or (5.2) mutatis mutandis commutes. The adjunction is unique when it exists.

6.2. Remark. If in Corollary 6.1 we denote the categories (A, S) and (X, T) in C by 𝔸 and 𝕏 and the functors (f_{⋆}, ϕ) and (u_{⋆}, υ) by F : 𝕏 ⟶ 𝔸 and U : 𝔸 ⟶ 𝕏 then the π is an isomorphism of spans from A to X between the object of objects of the comma category F/𝔸 and the object of objects of the comma category 𝕏/U. We leave it to the reader to interpret diagram (5.1) in this notation.

7. A doctrinal setting

Let D be a pseudomonad (also called a doctrine in [14], [9], [21] and [18]) on a bicategory K. It means that we have a pseudofunctor D : K ⟶ K, a unit pseudonatural transformation denoted by n : 1_{K} ⟶ D, and a multiplication pseudonatural transformation denoted by m : DD ⟶ D. For example, see [16] or [12] for the axioms.

A lax D-algebra (A, s) consists of an object A, a morphism s : DA ⟶ A, and 2-cells μ : s · Ds ⟹ s · m_{A} and η : 1_{A} ⟹ s · n_{A}, satisfying coherence conditions.

For lax D-algebras (A, s) and (A′, s′) in K, a lax opmorphism (f, ϕ) : (A, s) ⟶ (A′, s′) consists of a morphism f : A ⟶ A′ and a 2-cell ϕ : fs ⟹ s′Df in K such that

\xymatrix{ & s′Df · Ds \ar[r]^{s′Dϕ} & s′Ds′ · D^{2}f \ar@{-->}[dr]^{μD^{2}f} \\ fsDs \ar[ur]^{ϕDs} \ar[dr]_{fμ} &  &  & s′m_{A′}D^{2}f \\  & fsm_{A} \ar[r]_{ϕm_{A}} & s′Df · m_{A} \ar@{-->}[ur]_{s′m_{f}}}  (7.1)

\xymatrix{fsn_{A} \ar[r]^{ϕn_{A}} & s′Df · n_{A} \ar[d]^{s′n_{f}} \\ f \ar[r]_{ηf} \ar[u]_{fη} & s′n_{A′}}  (7.2)

The composite of lax opmorphisms (f, ϕ) : (A, s) ⟶ (A′, s′) and (f′, ϕ′) : (A′, s′) ⟶ (A′′, s′′) is defined to be (f′f, ϕ′ ⋆ ϕ) : (A, s) ⟶ (A′′, s′′) where ϕ′ ⋆ ϕ is the composite

f′fs ⟶^{f′ϕ} f′s′Df ⟶^{ϕ′Df} s′′Df′ · Df .  (7.3)

There is a bicategory KL(K, D) whose objects are lax D-algebras and whose morphisms are lax opmorphisms. A 2-cell ρ : (f, ϕ) ⟹ (g, ψ) : (A, s) ⟶ (A′, s′) in KL(K, D) (in non-reduced form, in the terminology of 13) is a 2-cell ρ̂: fs ⟹ s′Dg in K such that the diagrams (7.4) and (7.5) commute.

\xymatrix{ & s′Dg · Ds \ar[r]^{s′Dψ} & s′Ds′D^{2}g \ar[dr]^{μD^{2}g} \\ fsDs \ar[ur]^{ρ̂Ds} \ar[dr]_{fμ} &  &  & s′m_{A′}D^{2}g \\  & fsm_{A} \ar[r]_{ρ̂m_{A}} & s′Dg · m_{A} \ar[ur]_{s′m_{g}}}  (7.4)

\xymatrix{ & s′Df · Ds \ar[r]^{s′Dρ̂} & s′Ds′D^{2}g \ar[dr]^{μD^{2}g} \\ fsDs \ar[ur]^{ϕDs} \ar[dr]_{fμ} &  &  & s′m_{A′}D^{2}g \\  & fsm_{A} \ar[r]_{ρ̂m_{A}} & s′Dg · m_{A} \ar[ur]_{s′m_{g}}}  (7.5)

The reduced form of the 2-cell KL(K, D) is a 2-cell ρ : f ⟹ s′Dg · n_{A} in K such that equality (7.6) holds.

fs ⟶^{ϕ} s′Df ⟶^{s′Dρ} s′Ds′D^{2}gDn_{A} ⟶^{μD^{2}gDn_{A}} s′m_{A′}D^{2}gDn_{A}

⟶^{s′m_{g}^{−1}Dn_{A}} s′Dg · m_{A}Dn_{A} ⟶^{≅} s′Dg

=

fs ⟶^{ρs} s′Dg · n_{A}s ⟶^{s′n_{g}s} s′n_{A′}gs ⟶^{s′n_{A′}ψ} s′n_{A′}s′Dg

⟶^{s′n_{s′}^{−1}Dg} s′Ds′n_{DA′}Dg ⟶^{μn_{DA′}Dg} s′m_{A′}n_{DA′}Dg ⟶^{≅} s′Dg. (7.6)

The bijection between reduced and non-reduced forms takes ρ to the ρ̂ defined by either side of (7.6).

The vertical composite of the 2-cells ρ : (f, ϕ) ⟶ (g, ψ) and τ : (g, ψ) ⟶ (h, θ) is the 2-cell

f′fs ⟶^{f′ρ̂} f′s′Dg ⟶^{s′τ} s′(s′h) ⟶^{≅} (s′s′) h ⟶^{μh} s′h .  (7.7)

The horizontal composite of 2-cells ρ : (f, ϕ) ⟶ (g, ψ) : (A, s) ⟶ (A′, s′) and ρ′ : (f′, ϕ′) ⟶ (g′, ψ′) : (A′, s′) ⟶ (A′′, s′′) has non-reduced form the 2-cell

f ⟶^{ρ} s′Dg · n_{A} ⟶^{s′Dτ·n_{A}} s′Ds′D^{2}hDn_{A} · n_{A} ⟶^{μD^{2}hDn_{A}·n_{A}} s′m_{A′}D^{2}hDn_{A} · n_{A}

⟶^{s′m_{h}^{−1}Dn_{A}·n_{A}} s′Dh · m_{A}Dn_{A} · n_{A} ⟶^{≅} s′Dh · n_{A} . (7.8)

This completes the definition of the bicategory KL(K, D) with the exception of giving the coherent associativity and unit isomorphisms. As forewarned, we have been writing as if K were a 2-category in which case KL(K, D) would also be a 2-category. Putting in all the coherent isomorphisms as we did in the definition of KL(K), we can readily give them for KL(K, D) as we did for the special case where D was the identity pseudomonad.

In a future paper we shall explain the universal properties of the construction taking K to KL(K).

7.1. Definition. An adjunction core (u, g, π) between lax D-algebras (A, s) and (X, t) in a bicategory K consists of the following data in K:

1. morphisms u : A ⟶ X and g : A ⟶ X;

2. an invertible 2-cell π : gs ⟶ tDu.

Given such a core, we make the following definitions:

(a) β̄ : g ⟹ tDu · n_{A} is the composite

g ⟶^{gη} gsn_{A} ⟶^{πn_{A}} tDu · n_{A} ;

(b) ᾱ : u ⟹ gsn_{A} is the composite

u ⟶^{ηu} tn_{X}u ⟶^{≅} tDu · n_{A} ⟶^{π^{−1}n_{A}} gsn_{A} ;

(c) υ : us ⟹ tDu is the composite

us ⟶^{ᾱs} gsn_{A}s ⟶^{gsn_{s}} gsDs · n_{DA} ⟶^{gμn_{DA}} gsm_{A}n_{DA} ⟶^{≅} gs ⟶^{π} tDu ;

(d) ψ : tDg ⟹ gs is the composite

tDg ⟶^{tDβ̄} tDtD^{2}uDn_{A} ⟶^{μn_{u}^{−1}} tm_{X}Dn_{X}Du ⟶^{≅} tDu ⟶^{π^{−1}} gs .

If g = f^{⋆} for some map f : X ⟶ A in K then ᾱ and β̄ have mates α : fu ⟹ sn_{A} and β′ : 1_{X} ⟹ tDu · n_{A}f. We take β : 1_{X} ⟹ tDuDf · n_{X} to be the composite of β′ and tn_{f} : tDu · n_{A}f ≅ tDuDf · n_{X}.

7.2. Theorem. An adjunction core of the form (u, f^{⋆}, π) between lax D-algebras (A, s) and (X, t) in a bicategory K extends to an adjunction

(f, ϕ) ⊣ (u, υ) : (A, s) ⟶ (X, t)  (7.9)

with counit α and unit β in KL(K, D) if and only if one of the diagrams (7.10) or (7.11) commutes. The adjunction is unique when it exists.

\xymatrix{ & tDuDs \ar[r]^{tDυ} & tDtD^{2}u \ar[dr]^{μD^{2}u} \\ gsDs \ar[ur]^{πDs} \ar[dr]_{gμ} &  &  & tm_{X}D^{2}u \\  & gsm_{A} \ar[r]_{πm_{A}} & tDu · m_{A} \ar[ur]_{tm_{u}}}  (7.10)

\xymatrix{ & tDgDs \ar[r]^{ψDs} & gsDs \ar[dr]^{gμ} \\ tDtD^{2}u \ar@{-->}[ur]^{tDπ^{−1}} \ar[dr]_{μD^{2}u} &  &  & gsm_{A} \\  & tm_{X}D^{2}u \ar[r]_{tm_{u}^{−1}} & tDu · m_{A} \ar[ur]_{π^{−1}m_{A}}}  (7.11)

References

[1] Jean Bénabou, Introduction to bicategories, Lecture Notes in Mathematics 47 (Springer-Verlag, 1967) 1–77.

[2] Renato Betti, Aurelio Carboni, Ross Street and Robert Walters, Variation through enrichment, Journal of Pure and Applied Algebra 29 (1983) 109–127.

[3] Aurelio Carboni, Stefano Kasangian and Ross Street, Bicategories of spans and relations, Journal of Pure and Applied Algebra 33 (1984) 259–267.

[4] Samuel Eilenberg and G. Max Kelly, Closed categories, Proceedings of the Conference on Categorical Algebra (La Jolla, 1965), Springer (1966) 421–562.

[5] Robert Gordon, A. John Power and Ross Street, Coherence for tricategories, Memoirs of the American Math. Society 117 no. 558 (1995) vi+81 pp.

[6] André Joyal and Ross Street, The geometry of tensor calculus, Advances in Math. 88 (1991) 55–112.

[7] Daniel M. Kan, Adjoint functors, Transactions of the American Mathematical Society 87 (1958) 294–329.

[8] G. Max Kelly, Adjunction for enriched categories, Lecture Notes in Mathematics 106 (Springer-Verlag, 1969) 166–177.

[9] G. Max Kelly, Doctrinal adjunction, Lecture Notes in Mathematics 420 (Springer-Verlag, 1974) 257–280.

[10] G. Max Kelly, Basic concepts of enriched category theory, London Mathematical Society Lecture Note Series 64 (Cambridge University Press, Cambridge, 1982).

[11] G. Max Kelly and Ross Street, Review of the elements of 2-categories, Lecture Notes in Mathematics 420 (Springer-Verlag, 1974) 75–103.

[12] Stephen Lack, A coherent approach to pseudomonads, Advances in Math. 152 (2000) 179–202.

[13] Stephen Lack and Ross Street, The formal theory of monads II, Journal of Pure and Applied Algebra 175 (2002) 243–265.

[14] F. William Lawvere, Ordinal sums and equational doctrines, Lecture Notes in Mathematics 80 (Springer-Verlag, 1969) 141–155.

[15] Saunders Mac Lane and Robert Paré, Coherence for bicategories and indexed categories, Journal of Pure and Applied Algebra 37 (1985) 59–80.

[16] Francisco Marmolejo, Distributive laws for pseudomonads, Theory and Applications of Categories 5 (1999) 91–147.

[17] Ross Street, The formal theory of monads, Journal of Pure and Applied Algebra 2 (1972) 149–168.

[18] Ross Street, Fibrations in bicategories, Cahiers de topologie et géométrie différentielle 21 (1980) 111–160.

[19] Ross Street, Higher categories, strings, cubes and simplex equations, Applied Categorical Structures 3 (1995) 29–77.

[20] Ross Street, Categorical structures, Handbook of Algebra 1 (edited by M. Hazewinkel, Elsevier Science, 1996) 529–577.

[21] Volker Zöberlein, Doctrines on 2-categories, Math. Zeitschrift 148 (1976) 267–279.

Department of Mathematics, Macquarie University, NSW 2109, Australia

Email: ross.street@mq.edu.au

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