Kerodon

Variant 2.2.2.2 (Spans). Let $\operatorname{\mathcal{C}}$ be a category. If $X$ and $Y$ are objects of $\operatorname{\mathcal{C}}$, we define a span from $X$ to $Y$ to be a diagram $X \leftarrow M \rightarrow Y$ in the category $\operatorname{\mathcal{C}}$. If $\operatorname{\mathcal{C}}$ admits fiber products, then we can dualize Example 2.2.2.1 to produce a $2$-category $\operatorname{Span}(\operatorname{\mathcal{C}})$ having the same objects, where $1$-morphisms from $X$ to $Y$ in $\operatorname{Span}(\operatorname{\mathcal{C}})$ are given by spans from $X$ to $Y$ in $\operatorname{\mathcal{C}}$. More precisely, we define $\operatorname{Span}(\operatorname{\mathcal{C}})$ to be the conjugate of the $2$-category $\operatorname{Cospan}( \operatorname{\mathcal{C}}^{\operatorname{op}} )$.