nLab (infinity,1)-profunctor

Idea

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

Definition

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

$$F \colon D^{op}\times C \to \infty Grpd \,.$$

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

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

Every (∞,1)-functor $f\colon C\to D$ induces two (∞,1)-profunctors $D(1,f)\colon C ⇸ D$ and $D(f,1)\colon D ⇸ C$, defined by $D(1,f)(d,c) = D(d,f(c))$ and $D(f,1)(c,d) = D(f(c),d)$. (Here $D(-,-)$ denotes the hom functor of $D$ and $1$ 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 $(\infty, 1)$-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

$$\mathcal{C} ⇸ \mathcal{D}$$

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

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

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.

Related pages

• cograph of an (∞,1)-profunctor

• (∞,1)Prof

References

• Emily Riehl, Dominic Verity, Kan extensions and the calculus of modules for ∞-categories, (arXiv:1507.01460)

• Rune Haugseng, Bimodules and natural transformations for enriched ∞-categories, (arXiv:1506.07341)