abelian category as balanced category

1. Proposition

Let 20231214-abelian_category_as_balanced_category_eb04be82533afa6bcab2b0d2106fd2828974c0d1.svg be an abelian category. Then is balanced

2. Proof

Suppose 20231214-abelian_category_as_balanced_category_97a87bf3386d9d22e8b4e5c831f624db9871c29b.svg is a bimorphism. Then by monomorphism as kernel of the cokernel and cokernel and epimorphism in an abelian category we conclude, that

20231214-abelian_category_as_balanced_category_7f18cdb4b5598111427bf1375c24af4bcba86403.svg

is an equalizer

Thus using

20231214-abelian_category_as_balanced_category_201d4ce3dcbb3faf321212eebfd5360ec8c1892b.svg

we get a morphism 20231214-abelian_category_as_balanced_category_5f965d0a56374955f8a17da7129fa95cde5a6603.svg such that

20231214-abelian_category_as_balanced_category_e240f49830a363a466ea74312750ff3ef26222ff.svg

hence by monic split epi as isomorphism 20231214-abelian_category_as_balanced_category_5461866a7289ba4c56a10a82a5622bfbdcb4ae22.svg is an isomorphism

Alternatively:

direct consequence of the image coimage factorization

2.1. a)

Suppose 20231214-abelian_category_as_balanced_category_97a87bf3386d9d22e8b4e5c831f624db9871c29b.svg is a bimorphism. By equalizer and equal maps it suffices to show, that is an equalizer of some map.

By regular monomorphism it follows that each monomorphism, hence also 20231214-abelian_category_as_balanced_category_5461866a7289ba4c56a10a82a5622bfbdcb4ae22.svg is an equalizer.