Example 8.3.2.2 (03MC)

Example 8.3.2.2. Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be ordinary categories. Then every functor $K: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{Set}$ determines a morphism of simplicial sets

\[ \operatorname{N}_{\bullet }(K): \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{-} )^{\operatorname{op}} \times \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{+} ) \rightarrow \operatorname{N}_{\bullet }( \operatorname{Set}) \subset \operatorname{\mathcal{S}}. \]

This construction determines a monomorphism from the collection of profunctors from $\operatorname{\mathcal{C}}_{+}$ to $\operatorname{\mathcal{C}}_{-}$ (in the sense of classical category theory) to the collection of profunctors from $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_{+})$ to $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_{-})$ (in the sense of Definition 8.3.2.1). Beware that this map is (usually) not bijective: its image consists of those profunctors

\[ \mathscr {K}: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_{-})^{\operatorname{op}} \times \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_{+}) \rightarrow \operatorname{\mathcal{S}} \]

having the property that for every pair of objects $X \in \operatorname{\mathcal{C}}_{-}$ and $Y \in \operatorname{\mathcal{C}}_{+}$, the Kan complex $\mathscr {K}(X,Y)$ is a constant simplicial set (see Proposition 1.3.3.1 and Remark 5.5.1.7).