#
Lax Familial Representability and Lax Generic Factorizations

##
Charles Walker

A classical result due to Diers shows that a copresheaf F: A -> Set
on a category A is a coproduct of representables precisely
when each connected component of F's category of elements has an
initial object. Most often, this condition is imposed on a copresheaf
of the form B(X,T-) for a functor T: A -> B,
in which case this property says that T admits generic factorizations
at X, or equivalently that T is familial at X.

Here we generalize these results to the two-dimensional setting, replacing
A with an arbitrary bicategory A, and Set
with Cat. In this two-dimensional setting, simply asking
that a pseudofunctor A -> Cat be a coproduct
of representables is often too strong of a condition. Instead, we
will only ask that F be a lax conical colimit of representables.
This in turn allows for the weaker notion of lax generic factorizations
(and lax familial representability) for pseudofunctors of bicategories
T: A -> B.

We also compare our lax familial pseudofunctors to Weber's familial
2-functors, finding our description is more general (not requiring
a terminal object in A), though essentially equivalent
when a terminal object does exist. Moreover, our description of lax
generics allows for an equivalence between lax generic factorizations
and lax familial representability.

Finally, we characterize our lax familial pseudofunctors as right
lax F-adjoints followed by locally discrete fibrations
of bicategories, which in turn yields a simple definition of parametric
right adjoint pseudofunctors.

Keywords:
generic factorizations, lax conical colimit of representables

2020 MSC:
18N10, 18D30

*Theory and Applications of Categories,*
Vol. 35, 2020,
No. 37, pp 1424-1475.

Published 2020-08-20.

http://www.tac.mta.ca/tac/volumes/35/37/35-37.pdf

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