fibration of points

Given a category C and an object IOb(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


This can be unpacked the following way:

  • An object of Pt I(C) is a pair (p:XI: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:YI:t)(p:XI:s) is any morphism f:YX such that pf=q and ft=s.

By the construction, the category Pt I(C) of points over I is pointed and finitely complete; moreover the inverse image functor v *:Pt I(C)Pt J(C) induced by v:JI is a left exact functor.

The category Pt(C) of points of C has objects Ob(Pt(C))= IOb(Pt I(C)). In other words, the objects are the split epimorphisms p:XI 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:YJ:t)(p:XI:s) is a pair of morphisms u:YX, v:JI in C such that ut=sv and vq=pu.


The fibration of points is the codomain-assigning functor π:Pt(C)C, π:(p:XI:s)I, (u,v)u. It is a fibered category in the sense of Grothendieck. Its fibers are 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 vv * above.

