nLab fibration of points



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

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

This can be unpacked the following way:

  • An object of Pt I(C)Pt_I(C) is a pair (p:XI:s)(p:X\leftrightarrow I:s) of morphisms of CC such that ss is a section of pp; in other words it is a split epimorphism pp to II with a fixed choice of splitting ss.

  • A morphism f:(q:YI:t)(p:XI:s)f:(q:Y\leftrightarrow I:t)\to(p:X\leftrightarrow I:s) is any morphism f:YXf:Y\to X such that pf=qp\circ f= q and ft=sf\circ t=s.

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

The category Pt(C)Pt(C) of points of CC has objects Ob(Pt(C))= IOb(Pt I(C))Ob(Pt(C))=\coprod_I Ob(Pt_I(C)). In other words, the objects are the split epimorphisms p:XIp:X\to I of CC with a choice of a splitting, or equivalently the retracts in CC. At the level of morphisms it is just a bit more complex than the morphisms in each Pt I(C)Pt_I(C), namely the slice and coslice triangles become squares. More precisely, a morphism (u,v):(q:YJ:t)(p:XI:s)(u,v): (q:Y\leftrightarrow J:t)\to (p:X\leftrightarrow I:s) is a pair of morphisms u:YXu:Y\to X, v:JIv:J\to I in CC such that ut=svu\circ t = s\circ v and vq=puv\circ q =p\circ u.

Y u X qt ps J v I\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 π:Pt(C)C\pi \,\colon\, Pt(C)\to C, π:(p:XI:s)I\pi:(p:X\leftrightarrow I:s)\to I, (u,v)v(u,v)\mapsto v. It is a fibered category in the sense of Grothendieck (cf. codomain fibration). Its fibers are the Pt I(C)Pt_I(C) which are (as mentioned above) pointed and finitely complete. A morphism (u,v)(u,v) (in notation as above) is cartesian iff q,u,p,vq,u,p,v are the sides of a pullback square in CC (i.e. qq is a pullback of pp along vv and uu a pullback of vv along pp). The inverse image functor for this fibration is exactly described by the rule vv *v\mapsto v^* above.


The notion appears in Section 3 of:

Last revised on February 8, 2023 at 15:08:51. See the history of this page for a list of all contributions to it.