Page 112 - Textos de Matemática Vol. 45

104 Chapter 9. Appendix
The main example we are interested in is the so-called fibration of points. Let E be a finitely complete category. The category PtE is the category whose objects (called points) are the split epimorphisms in E (with a fixed splitting), and whose arrows are the commutative squares between them. It is not difficult to see that the functor
¶E : P tE → E,
associating with any split epimorphism its codomain, is a fibration. Indeed,
givenamorphismα:J→EinEandasplitepimorphism Xoo s //E,the f
morphism (α′,α), where α′ is given by the following pullback square P α′ //X
OO OO
f′s′ fs //
JαE
is cartesian over α.
Given any morphism p: E → B in E, we can define a functor, called the
change-of-base functor
p∗ : P tB E → P tE E
by pulling back along p any split epimorphism with codomain B. Thanks to the commutativity of limits, we get that any change-of-base functor is left exact, which means that it preserves finite limits.
9.4 Simplicial objects
We need a uniform notation for internal relations, categories and groupoids in E. For that we chose the simplicial one. A simplicial object in a category E is a graded set of objects Xn,n ∈ N, together with a family of (face) maps di:Xn+1 →Xn, 0≤i≤n+1,andof(degeneracy)mapssi:Xn →Xn+1, 0≤ i ≤ n:
d0d0 d0
oos0 oo s0 oos0 d0 //
Xn Xn−1 .... .... X2 d1 // X1 oo s0 X0
.... Xn+1
dn+1 dn d2
ooLLooKK ooLL sn sn−1 s1
// d1