nLab fibration of points

Idea

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

$$Pt_I(C) \;=\; (I,id_I)\backslash(C/I)$$

This can be unpacked the following way:

• An object of $Pt_I(C)$ 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 $Pt_I(C)$ of points over $I$ is pointed and if the category $C$ has finite limits, then finitely complete; moreover the inverse image functor $v^*:Pt_I(C)\to Pt_J(C)$ induced by $v:J\to I$ is a left exact functor.

The category $Pt(C)$ of points of $C$ has objects $Ob(Pt(C))=\coprod_I Ob(Pt_I(C))$. 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 $Pt_I(C)$, 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 $v\circ q =p\circ u$.

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

The fibration of points is the codomain-assigning functor $\pi \,\colon\, Pt(C)\to C$, $\pi:(p:X\leftrightarrow I:s)\to I$, $(u,v)\mapsto v$. It is a fibered category in the sense of Grothendieck (cf. codomain fibration). Its fibers are the $Pt_I(C)$ 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.

Related concepts

• codomain fibration

References

The notion appears in Section 3 of:

• Dominique Bourn, Normalization equivalence, kernel equivalence and affine categories, Lecture Notes in Mathematics 1488 (1991) [doi:10.1007/BFb0084212]