nLab
cograph of a profunctor

Contents

In category theory

Definition

Let H:ABH\colon A ⇸B be a profunctor, i.e. a functor B op×ASetB^{op}\times A\to Set. Its cograph, also called its collage, is the category H¯\bar{H} whose set of objects is the disjoint union of the sets of objects of AA and BB, and where

H¯(a 1,a 2) =A(a 1,a 2) H¯(b 1,b 2) =B(b 1,b 2) H¯(b,a) =H(b,a) H¯(a,b) = \begin{aligned} \bar{H}(a_1,a_2) &= A(a_1,a_2)\\ \bar{H}(b_1,b_2) &= B(b_1,b_2)\\ \bar{H}(b,a) &= H(b,a)\\ \bar{H}(a,b) &= \emptyset \end{aligned}

where composition is defined as in AA, BB, and according to the actions of AA and BB on HH.

The cograph of a functor is the special case when HH is a “representable profunctor” of the form B(f,)B(f-,-) for some functor f:ABf\colon A\to B.

Properties

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 ProfProf-like bicategory.) The cograph of a profunctor is also a cotabulation? in the proarrow equipment of profunctors. Furthermore, the cospans AH¯BA\to \bar{H} \leftarrow B 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 (01)(0\to 1), where BB is the fiber over 00 and AA is the fiber over 11. See Distributors and barrels.

In (,1)(\infty,1)-category theory

The notion of a cograph of a profunctor generalizes to (∞,1)-category theory.

Definition

Definition

For CC and DD two (∞,1)-categories an correspondence between them is an (,1)(\infty,1)-category p:KΔ[1]p : K \to \Delta[1] over the interval category Δ[1]={01}\Delta[1] = \{0 \to 1\} with an equivalences K 0CK_0 \simeq C and K 1DK_1 \simeq D.

This appears as (Lurie, def 2.3.1.3).

Properties

Proposition

There is a canonical bijection between equivalence classes of correspondences between CC and DD and equivalence classes of (,1)(\infty,1)-profunctors (∞,1)-functors

C op×DGrpd C^{op} \times D \to \infty Grpd

from the product of DD with the opposite-(∞,1)-category of CC to ∞Grpd.

This appears as (Lurie, remark 2.3.1.4).

Therefore the correspondence corresponding to a profunctor is its cograph/collage.

Proposition

An (,1)(\infty,1)-profunctor comes from an ordinary (∞,1)-functor F:CDF : C \to D precisely if its cograph p:KΔ[1]p : K \to \Delta[1] is not just an inner fibration but a coCartesian fibration.

And it comes from a functor G:DCG : D \to C precisely if it is a Cartesian fibration. And precisely if both is the case is FF the right adjoint (∞,1)-functor to GG.

Because by the (∞,1)-Grothendieck construction

References

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 (,1)(\infty,1)-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.

Revised on October 31, 2011 22:05:48 by Mike Shulman (71.136.247.199)