nLab
fibration of points

Given a category $C$ and an object $I\in \mathrm{Ob}(C)$, the category of points over $I$ is the category ${\mathrm{Pt}}_I(C)$ of pointed objects of the slice category $C/I$; or alternatively the coslice-slice category

$${\mathrm{Pt}}_I(C)=(I,{\mathrm{id}}_I)\backslash (C/I)$$Pt_I(C) = (I,id_I)\backslash(C/I)

This can be unpacked the following way:

• An object of ${\mathrm{Pt}}_I(C)$ is a pair $(p:X\leftrightarrow I:s)$ of morphisms of $C$ such that $s$ is a section of $p$; in other words it is a split epimorphism $p$ to $I$ with a fixed choice of splitting $s$.

• A morphism $f:(q:Y\leftrightarrow I:t)\to (p:X\leftrightarrow I:s)$ is any morphism $f:Y\to X$ such that $p\circ f=q$ and $f\circ t=s$.

By the construction, the category ${\mathrm{Pt}}_I(C)$ of points over $I$ is pointed and finitely complete; moreover the inverse image functor $v^{*}:{\mathrm{Pt}}_I(C)\to {\mathrm{Pt}}_J(C)$ induced by $v:J\to I$ is a left exact functor.

The category $\mathrm{Pt}(C)$ of points of $C$ has objects $\mathrm{Ob}(\mathrm{Pt}(C))={\coprod }_I\mathrm{Ob}({\mathrm{Pt}}_I(C))$. In other words, the objects are the split epimorphisms $p:X\to I$ of $C$ with a choice of a splitting, or equivalently the retracts in $C$. At the level of morphisms it is just a bit more complex than the morphisms in each ${\mathrm{Pt}}_I(C)$, namely the slice and coslice triangles become squares. More precisely, a morphism $(u,v):(q:Y\leftrightarrow J:t)\to (p:X\leftrightarrow I:s)$ is a pair of morphisms $u:Y\to X$, $v:J\to I$ in $C$ such that $u\circ t=s\circ v$ and $v\circ q=p\circ u$.

$$\begin{array}{ccc}Y& \stackrel{u}{\to }& X\\ q\downarrow \uparrow t& & p\downarrow \uparrow s\\ J& \stackrel{v}{\to }& I\end{array}$$\array{ Y &\stackrel{u}\to & X \\ q\downarrow\uparrow t & & p\downarrow\uparrow s\\ J&\stackrel{v}\to & I }

The fibration of points is the codomain-assigning functor $\pi :\mathrm{Pt}(C)\to C$, $\pi :(p:X\leftrightarrow I:s)\to I$, $(u,v)\mapsto u$. It is a fibered category in the sense of Grothendieck. Its fibers are ${\mathrm{Pt}}_I(C)$ which are (as mentioned above) pointed and finitely complete. A morphism $(u,v)$ (in notation as above) is cartesian] iff $q,u,p,v$ are the sides of a pullback square in $C$ (i.e. $q$ is a pullback of $p$ along $v$ and $u$ a pullback of $v$ along $p$). The inverse image functor for this fibration is exactly described by the rule $v\mapsto v^{*}$ above.

References

• Borceux-Bourn