Page 109 - Textos de Matemática Vol. 45
P. 109
Chapter 9
Appendix
This appendix is mainly devoted to set notations and to recall some notions and results without proofs. Given a category E we shall denote by E(X, Y ) the set of morphisms between X and Y . We shall suppose here that the category E is finitely complete.
9.1 Pullbacks
Given any map f : X → Y in E, its kernel equivalence relation
p0 // R[f] oo s0 // X
p1
is given by the pullback of the map f along itself. In the set-theoretical context it is defined as the set {(x0, x1) ∈ X × X | f(x0) = f(x1)}. This means that xR[f]x′ if and only if we have f(x) = f(x′). One of the most useful results concerning pullbacks is the following one with its associated variations:
Proposition 9.1.1. Given two commutative squares where both the whole rect- angle and the right hand side square are pullbacks
• // • // • // //
then so is the left hand side square.
Whence the following corollary: 101
f
// Y.
•••
