Remark 8.3.2.5 (0464): From Couplings to Profunctors

Remark 8.3.2.5 (From Couplings to Profunctors). Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be $\infty $-categories. By virtue of Corollary 5.6.0.6, the construction of Example 8.3.2.4 induces a monomorphism

\[ \xymatrix@C =50pt@R=50pt{ \{ \textnormal{Profunctors $\mathscr {K}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$} \} / \textnormal{Isomorphism} \ar [d] \\ \{ \textnormal{Couplings $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$} \} / \textnormal{Equivalence}, } \]

whose image consists of equivalence classes of couplings $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ having essentially small fibers.

In particular, to every coupling $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ with essentially small fibers, we can associate a profunctor $\mathscr {K}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$, which is characterized (up to isomorphism) by the requirement that it is a covariant transport representation for $\lambda $ (see Definition 5.6.5.1).