Page 117 - Textos de Matemática Vol. 45

9.6. Internal groupoids, equivalence relations, groups 109
such that the following part is an iterated kernel relation:
oo X3
oo d2 d3
d0 d0
s0 oo s0 d0 //
d1// //oos
// X2
LL oo
d1 X1 0 X0 LL //
s1 d1 d2
which implies that any commutative square is a pullback. An internal groupoid is an equivalence relation as soon as the morphisms d0 and d1 (from X1 to X0) are jointly monomorphic. It is then sufficient to have a 2-truncated simplicial object, the 3-truncation part coming for free:
R[d0 ]
p0
oo s0 d0 //
p1 // X1 oo s0
oo LL //
s1 d1 d2
X0
If E is a variety of universal algebra, then an internal equivalence is just a con- gruence, i.e. an equivalence relation which is compatible with all the operations.
According to the previous observation an internal monoid X is an internal group as soon as the following square is a pullback:
X×X d1 //X d0 τX
// XτX 1
In set theoretical terms, this means that the map from X × X to itself sending (m, y) to (m, m · y) is a bijection. This implies that any element has a right inverse, and thus an inverse.
Given a left exact functor U : E → E′, it preserves internal categories, groupoids, monoids and groups, preorders and equivalence relations. Suppose moreover it reflects pullbacks, i.e. suppose that a commutative square in E is a pullback as soon as its image by U is a pullback; it is clear that, in those cir- cumstances, the functor U reflects the internal categories, groupoids, monoids and groups, preorders and equivalence relations.