Anima-parametrized objects | Higher Topos Theory notes

Definition 6.157.

Let $$C$$ be a category, not necessarily small. We define the functor $\pi\colon \int_{\An}C \to \An$ as the cartesian unstraightening of the functor

\Fun(-,C)\colon \An\catop \to \Cat, \qquad A \mapsto \Fun(A,C).

We refer to $\int_{\An}C$ as the category of $\An$ -parametrized objects of $$C$$ . Note that an object of $\int_{\An}C$ is a pair $$(A,X)$$ , where $A \in \An$ is an anima and $X\colon A \to C$ is an $$A$$ -indexed family of objects of $$C$$ . A morphism $(A,X) \to (B,Y)$ consists of a morphism of animae $f\colon A \to B$ and a map $X \to f^*Y$ in $\Fun(A,C)$ , where $f^*Y := Y \circ f\colon A \to C$ .

The fiber of $\int_{\An}C$ over $A \in \An$ is $\Fun(A,C)$ . By taking $$A = *$$ , we in particular obtain an inclusion $C \hookrightarrow \int_{\An}C$ . Note that this is fully faithful, since $\{*\} \hookrightarrow \An$ is fully faithful.