nLab
small fibration

A fibered category is a small fibration if it is equivalent to a fibered category which is obtained via a specific construction: the input is an internal category $C=(C_0,C_1,s,t,c,i)$ in a category $E$ with pullbacks, and the output is a fibered category $P:F\to E$, whose fiber over an object $I$ (“index set”) in $C$ is the small category $C^I={\mathrm{hom}}_E(I,C)$ whose object part is $C_0^I$ and morphism part is $C_1^I$; and the structure maps of $C^I$ are obtained by postcomposing with structure maps of $C$.

Conventions for the internal category $C$: the composition $c$ takes arguments in Leibniz convention, and source map is the right leg of the span $C_0\leftarrow C_1\to C_0)$; the projection $p_1,p_2:C_1{\times }_{C_0}{C}_1$ have $t\circ p_1=s\circ p_2$. Thus $t\circ p_1=s\circ p_2$.

Consider objects $m:I\to C_0$ and $n:J\to C_0$ in fibers over $I$ and $J$. Morphisms in ${\mathrm{hom}}_F(m,n)$ are pairs $(\alpha ,f):m\to n$ where $\alpha :I\to J$ and $f:J\to C_1$ are morphisms in $E$ satisfying $s\circ f=m$, $t\circ f\circ \alpha =n$. The interesting part of the construction is the composition: given also $p:K\to C_0$ and $m\stackrel{(\alpha ,f)}{\to }n\stackrel{(\beta ,g)}{\to }p$, their composition is $(\beta \circ \alpha ,c\circ h)$ where $c:C_0{\times }_{C_1}{C}_0\to C_0$ is the composition and $h=(f,g\circ \alpha ):I\to C_0{\times }_{C_1}{C}_0$ is the map given by the universal property of the fibered product and $f=p_1\circ h$ and $g\circ \alpha =p_2\circ h$. One checks that so defined composition is associative.

Finally, the projection $P:F\to C$ is rather obvious: $P(m:I\to C_0)=I$, and in notation from above, $P(\alpha ,f)=\alpha$. One checks that $P$ is a fibered category with a canonical cleavage: the chosen cartesian morphism over $\alpha :I\to J$ with given target $n:J\to C_0$ is $(\alpha ,i\circ n\circ \alpha ):(I,n\circ \alpha )\to (J,n)$. Indeed, for any object $l:L\to C_0$ over $L$ and a morphism $(\lambda ,g):l\to n$ with $\lambda =\alpha \circ \beta$, one can decompose $(\lambda ,g)=(\alpha ,i\circ n\circ \alpha )\circ (\beta ,g)$ where $(\beta ,g)$ is a morphism in $F$ with $P(\beta ,g)=\beta$.

The basic example is the small fibration $\mathrm{Set}(C)\to \mathrm{Set}$ of set-indexed families of objects in $C$, for a small category $C$. The objects in the fiber $C^I$, that is the functions $m:I\to C_0$, $m:i\mapsto m_i\in C_0$ are the $I$-indexed families $(m_i{)}_{i\in I}$ of objects in $C$. A map $(\alpha ,f):m\to n$ is a map $\alpha :I\to J$ together with an $I$-indexed family $(f_i{)}_{i\in I}$ of morphisms $f_i:m_i\to n_{\alpha (i)}$ in $C$. The composition is $(\beta ,(g_j{)}_{j\in J})\circ (\alpha ,(f_i{)}_{i\in I})=(\beta \circ \alpha ,(g_{\alpha (i)}\circ f_i{)}_{i\in I})$.