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102 Chapter 9. Corollary 9.1.2. Suppose the right hand side square is a pullback:
p1 R(x)
Appendix
p0
// f R[f]oo s0 // X // Y
//
then so is any of the left hand side commutative squares.
9.1.1 Left exact conservative functors
′oop′0
R [ f ] s ′0 // X′
// Y ′.
xy
p′ 1
f′
A functor U : C → D is said to be conservative when it reflects the isomorphims.
Proposition 9.1.3. Suppose that U : C → D is a left exact functor such that, for any monomorphism m in C, if Um is an isomorphism in D then m is an isomorphism. Then U is conservative.
Proof. Given any morphism f in C, consider the kernel equivalence relation of f:
p0 // f R[f]oo s0 //X //Y.
p1
Since U is left exact, we have that UR[f] is the kernel equivalence relation of
Uf:
ooUp0 // Uf // UR[f] = R[Uf] Us0 // UX
Up1
UY.
Suppose that Uf is an isomorphism. Then Us0 is an isomorphism. Since s0 is a monomorphism, our hypothesis implies that s0 is an isomorphism. But then f is a monomorphism, hence an isomorphism by our hypothesis.
9.2 Regular epimorphisms
A map f : X → Y in E is a regular epimorphism when it is the coequalizer of its kernel equivalence relation; when it is the case we denote this map by f : X Y . In any variety of universal algebra, as the categories M on of monoids, Gp of groups, SRng of semirings or Rng of rings, the regular epimorphisms are precisely the surjective homomorphisms. The regular epimorphisms are not stable under pullbacks in general; but they are in any variety of universal algebra. In this context the proposition of the previous section has a partial converse (see Theorem 3 in [7] and Proposition 2.7 in [21]):
