From Groups to Categorial Algebra by Dominique Bourn

Author:Dominique Bourn
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

makes the monomorphism u normal to R.

Proof

We get . Consider now the following diagram:

The right hand side upper squares are pullbacks by Proposition 1.​6.​10, while the left hand side ones are so since the lower square is a pullback. Accordingly the upper morphism of equivalent relations is a fibrant morphism and is normal to R.  □

Proposition 4.3.10

Let be a protomodular category. Then a monomorphism u is normal to at most one internal equivalence relation in .

Proof

Suppose is a monomorphism normal to two equivalence relations R and S on X in . We are going to show that , which will imply . For that let us consider the following diagram where denotes the inclusion:

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