Let be a profunctor, i.e. a functor . Its cograph, also called its collage, is the category whose set of objects is the disjoint union of the sets of objects of and , and where
where composition is defined as in , , and according to the actions of and on .
The cograph of a functor is the special case when is a corepresentable profunctor of the form for some functor .
The cograph of a profunctor can be given a universal property: it is the lax colimit of that profunctor, considered as a single arrow in the bicategory of profunctors. (The word “collage” is also sometimes used more generally for any lax colimit, especially in a -like bicategory.) The cograph of a profunctor is also a cotabulator in the proarrow equipment of profunctors. Furthermore, the cospans which are cographs of profunctors can be characterized as the two-sided codiscrete cofibrations in the 2-category Cat.
Cographs of profunctors can also be characterized as categories equipped with a functor to the interval category , where is the fiber over and is the fiber over . See Distributors and barrels.
The notion of a cograph of a profunctor generalizes to (∞,1)-category theory.
For and two (∞,1)-categories a correspondence between them is an -category over the interval category with an equivalences and .
This appears as (Lurie, def 2.3.1.3).
There is a canonical bijection between equivalence classes of correspondences between and and equivalence classes of (∞,1)-profunctors, i.e., (∞,1)-functors
from the product of with the opposite-(∞,1)-category of to ∞Grpd.
This appears as (Lurie, remark 2.3.1.4).
Therefore the correspondence corresponding to a profunctor is its cograph/collage.
An -profunctor comes from an ordinary (∞,1)-functor precisely if its cograph is not just an inner fibration but a coCartesian fibration.
And it comes from a functor precisely if it is a Cartesian fibration. And precisely if both is the case is the right adjoint (∞,1)-functor to .
Because by the (∞,1)-Grothendieck construction
coCartesian fibrations correspond to (∞,1)-functors (∞,1)Cat;
Cartesian fibrations correspond to (∞,1)-functors (∞,1)Cat.
and as discussed at adjoint (∞,1)-functor, an -functor has an adjoint precisely if the coCartesian fibration corresponding to it is also Cartesian.
cograph of a functor, cograph of a profunctor
For ordinary and enriched categories, cographs were studied (and used to characterize profunctors) by:
Ross Street, “Fibrations in bicategories”
Carboni and Johnson and Street and Verity, “Modulated bicategories”
The -category theoretic notion (“correspondence”) is the topic of section 2.3.1 of
See Ross Street’s post in category-list 2009, Re: pasting along an adjunction.
Last revised on August 5, 2023 at 22:16:19. See the history of this page for a list of all contributions to it.