equivalences in/of $(\infty,1)$-categories
The concept of $(\infty, 1)$-profunctor is the categorification of that ofprofunctors from category theory to (∞,1)-category theory.
If $C$ and $D$ are (∞,1)-categories, then a profunctor from $C$ to $D$ is a (∞,1)-functor of the form
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
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.
In analogy to the situation for profunctors between 1-categories (see there), $\infty$-profunctors
are equivalently plain but $\infty$-colimit-preserving $\infty$-functors
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.
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)
Last revised on October 15, 2021 at 10:48:18. See the history of this page for a list of all contributions to it.