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Example 2.2.2.1 (Correspondences). Let $\operatorname{\mathcal{C}}$ be a category containing a pair of objects $X$ and $Y$. A correspondence from $X$ to $Y$ is an object $M \in \operatorname{\mathcal{C}}$ together with a pair of morphisms $X \xleftarrow {f} M \xrightarrow {g} Y$ in $\operatorname{\mathcal{C}}$. The correspondences from $X$ to $Y$ can be regarded as the objects of a category $\operatorname{\mathcal{M}}_{X,Y}$, where a morphism from $(M,f,g)$ to $(M',f',g')$ in $\operatorname{\mathcal{M}}_{X,Y}$ is given by a morphism $u: M \rightarrow M'$ for which the diagram

\[ \xymatrix { & M \ar [dd]^{u} \ar [dr]^{ g } \ar [dl]_{f} & \\ X & & Y \\ & M' \ar [ul]^{f'} \ar [ur]_{g'} & } \]

commutes.

Assume now that the category $\operatorname{\mathcal{C}}$ admits fiber products. We can then construct a $2$-category $\operatorname{Corr}(\operatorname{\mathcal{C}})$ as follows:

  • The objects of $\operatorname{Corr}(\operatorname{\mathcal{C}})$ are the objects of $\operatorname{\mathcal{C}}$.

  • For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, we take $\underline{\operatorname{Hom}}_{ \operatorname{Corr}(\operatorname{\mathcal{C}})}(X,Y)$ to be the category $\operatorname{\mathcal{M}}_{X,Y}$; in particular, $1$-morphisms from $X$ to $Y$ in the category $\operatorname{Corr}(\operatorname{\mathcal{C}})$ can be identified with correspondences from $X$ to $Y$.

  • For every triple of objects $X,Y,Z \in \operatorname{\mathcal{C}}$, the composition law

    \[ \circ : \underline{\operatorname{Hom}}_{\operatorname{Corr}(\operatorname{\mathcal{C}})}( Y, Z) \times \underline{\operatorname{Hom}}_{\operatorname{Corr}(\operatorname{\mathcal{C}})}( X, Y) \rightarrow \underline{\operatorname{Hom}}_{ \operatorname{Corr}(\operatorname{\mathcal{C}})}( X, Z ) \]

    is given on objects by the construction $(N, M) \mapsto M \times _{Y} N$.

  • For every object $X \in \operatorname{\mathcal{C}}$, the identity $1$-morphism from $X$ to itself in $\operatorname{\mathcal{C}}$ is given by the correspondence $X \xleftarrow { \operatorname{id}_ X } X \xrightarrow { \operatorname{id}_ X} X$, and the unit constraint $\upsilon _{X}: X \times _{X} X \rightarrow X$ is the map given by projection onto either factor.

  • For every triple of composable $1$-morphisms

    \[ W \xrightarrow {M} X \xrightarrow {N} Y \xrightarrow {P} Z \]

    in $\operatorname{Corr}(\operatorname{\mathcal{C}})$, the associativity constraint $\alpha _{P,N,M}: P \circ (N \circ M) \Rightarrow (P \circ N) \circ M$ is the canonical isomorphism of iterated fiber products $(M \times _{X} N) \times _{Y} P \simeq M \times _{X} (N \times _{Y} P)$.

We will refer to $\operatorname{Corr}(\operatorname{\mathcal{C}})$ as the $2$-category of correspondences in $\operatorname{\mathcal{C}}$.