nLab
(infinity,1)-profunctor

Contents

Contents

Idea

The concept of (,1)(\infty, 1)-profunctor is the categorification of that ofprofunctors from category theory to (∞,1)-category theory.

Definition

If CC and DD are (∞,1)-categories, then a profunctor from CC to DD is a (∞,1)-functor of the form

F:D op×CGrpd. F \colon D^{op}\times C \to \infty Grpd \,.

Such a profunctor is usually written as F:CDF \,\colon\, C ⇸ D. Composition of (∞,1)-profunctors in (∞,1)Prof is by the “tensor product of (∞,1)-functors” homotopy coend construction: if H:ABH\colon A ⇸ B and K:BCK\colon B ⇸ C, their composite is given as a functor C op×AGrpdC^{op}\times A \to \infty Grpd by

(c,a) bBH(b,a)×K(c,b).(c,a)\mapsto \int^{b\in B} H(b,a)\times K(c,b).

Every (∞,1)-functor f:CDf\colon C\to D induces two (∞,1)-profunctors D(1,f):CDD(1,f)\colon C ⇸ D and D(f,1):DCD(f,1)\colon D ⇸ C, defined by D(1,f)(d,c)=D(d,f(c))D(1,f)(d,c) = D(d,f(c)) and D(f,1)(c,d)=D(f(c),d)D(f,1)(c,d) = D(f(c),d). (Here D(,)D(-,-) denotes the hom functor of DD and 11 denotes the identity (∞,1)-functor on the respective category.)

Since a profunctor is also known as a (bi)module or a distributor or a correspondence, we should expect other names to be used for (,1)(\infty, 1)-profunctor. In Higher Topos Theory (Definition 2.3.1.3), Lurie speaks of a correspondence.

Properties

Relation to presentable \infty-categories

In analogy to the situation for profunctors between 1-categories (see there), \infty-profunctors

𝒞𝒟 \mathcal{C} ⇸ \mathcal{D}

are equivalently plain but \infty -colimit-preserving \infty -functors

PSh (𝒞)PSh (𝒟) PSh_\infty(\mathcal{C}) \xrightarrow{\;} PSh_\infty(\mathcal{D})

between the corresponding \infty -categories of \infty -presheaves.

In this way small \infty -categories with \infty-profunctors between them is a full sub- \infty -category of of the \infty-category Pr(∞,1)Cat that of presentable \infty -categories with cocontinuous \infty -functors between them.

References

Last revised on October 15, 2021 at 10:48:18. See the history of this page for a list of all contributions to it.