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3.3. Schreier internal groupoids 41
which is the unique possible multiplication, is a morphism. We have that m((a′1, a1) · (a′2, a2)) = m(a′1 · a′2, a1 · a2) =
=q(a′1 ·a′2)·a1 ·a2 =q(a′1)·q(s0d0(a′1)·q(a′2))·a1 ·a2 = = q(a′1) · q(s0d1(a1) · q(a′2)) · t(a1) · s0d1(a1) · a2 =
= q(a′1) · t(a1) · q(s0d1(a1) · q(a′2)) · s0d1(a1) · a2 =
= q(a′1) · t(a1) · s0d1(a1) · q(a′2) · a2 =
= q(a′1) · a1 · q(a′2) · a2 = m(a′1, a1) · m(a′2, a2). Moreover, m respects domain and codomain, indeed:
and
d0m(a′, a) = d0q(a′) · d0(a) = 1 · d0(a) = d0(a),
d1m(a′, a) = d1q(a′) · d1(a) = d1q(a′) · d0(a′) =
= d1q(a′) · d1s0d0(a′) = d1(q(a′) · s0d0(a′)) = d1(a′).
It is associative, because
m(m(a′′, a′), a) = m(q(a′′) · a′, a) = q(q(a′′) · a′) · a =
= qq(a′′) · q(s0d0q(a′′) · q(a′)) · a = q(a′′) · q(1 · q(a′)) · a = = q(a′′) · q(a′) · a = m(a′′, m(a′, a)).
Finally, m preserves identities, since
m(s0d1(a), a) = qs0d1(a) · a = 1 · a = a,
and
m(a, s0d0(a)) = q(a) · s0d0(a) = a. This proves that (a) implies (b).
The equivalence between (b) and (c) comes from Proposition 3.3.2.
Gran proved in [20] that, in a Mal’tsev category, internal groupoids are closed under quotients inside the category of internal reflexive graphs. Here we have a similar result:
