ScalingStacks

Given a functor

let us describe the salient features of the resulting cocartesian fibration

which it classifies.

1. (1)

   Over an object $b\in\mathcal{B}$, the fiber $f^{-1}(b)\subset\mathcal{E}$ is canonically equivalent to $F(b)\in{{\mathcal{C}\textup{at}}_{\infty}}$.

2. (2)

   Given a morphism $b_{1}\xrightarrow{\varphi}b_{2}$ in $\mathcal{B}$ and an object $e\in f^{-1}(b_{1})$ of the fiber over its source, there is a canonical morphism

   $$e\rightarrow\varphi_{*}(e)$$

   in $\mathcal{E}$ which projects to $\varphi$, called an $f$-cocartesian lift (or simply a cocartesian lift) of $\varphi$ relative to $e$, such that the canonical equivalence $f^{-1}(b_{2})\simeq F(b_{2})$ identifies the object $\varphi_{*}(e)\in f^{-1}(b_{2})$ with the object $(F\varphi)(e)\in F(b_{2})$. This is illustrated in Figure 1.

3. (3)

   An arbitrary morphism $e_{1}\rightarrow e_{2}$ in $\mathcal{E}$ admits a unique factorization as a cocartesian morphism followed by a morphism lying in the fiber $f^{-1}(b_{2})$ – which we will therefore refer to as a fiber morphism –, as illustrated in Figure 2.

   Under the equivalence $f^{-1}(b_{2})\simeq F(b_{2})$, the morphism $\psi_{*}(e_{1})\rightarrow e_{2}$ in $f^{-1}(b_{2})$ corresponds to a morphism $(F\psi)(e_{1})\rightarrow e_{2}$ in $F(b_{2})$. Thus, we have canonical equivalences

   $$\hom_{\mathcal{E}}(e_{1},e_{2})\simeq\hom_{f^{-1}(b_{2})}(\psi_{*}(e_{1}),e_{2})\simeq\hom_{F(b_{2})}((F\psi)(e_{1}),e_{2})$$

   of hom-spaces.