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Chapter 3
Schreier internal relations, categories and groupoids
We recall that an internal reflexive graph in a category E is a diagram of the form
d0 // X1oos0 //X0
d1
such that d0 s0 = 1X0 = d1 s0 . A reflexive relation is a reflexive graph such that the pair (d0,d1) is jointly monomorphic. If the category E has binary products,thismeansthattheinducedmorphism⟨d0,d1⟩:X1 →X0×X0 isa monomorphism.
Definition 3.0.10. An internal reflexive graph in the category Mon of monoids d0 //
X1oos0 //X0 d1
is a Schreier reflexive graph if the split epimorphism (d0 , s0 ) is a Schreier one. It is homogeneous if (d0, s0) is homogeneous.
As a consequence of Proposition 2.3.3, we have that Schreier (resp. homo- geneous) reflexive graphs are closed under finite products inside the category of internal reflexive graphs. The same is true for Schreier (resp. homogeneous) reflexive relations and for Schreier (resp. homogeneous) equivalence relations.
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