cokernel and epimorphism in an ab-enriched, pointed category

1. Proposition

Let 20240125-cokernel_and_epimorphism_in_an_abelian_category_3038b4682e1f6d64ae501574b4029ad43f8f615e.svg be a pointed and 20240125-cokernel_and_epimorphism_in_an_abelian_category_e08d810fed313dafa6119deb03285ae027678f08.svg a morphism

TFAE:

  1. 20240125-cokernel_and_epimorphism_in_an_abelian_category_ad6a03aaa8d3a8bd22f0e7659be6109d78aaf52a.svg is an epimorphism
  2. the cokernel is the zero object: 20240125-cokernel_and_epimorphism_in_an_abelian_category_eca1bfb468b7300554cbade65ef925c171660af0.svg

2. Proof

2.1. 1) 20240125-cokernel_and_epimorphism_in_an_abelian_category_5667b5be7592236ab833642a1d2a85ce8a5490a6.svg 2)

corollary of epimorphism with zero as cokernel

2.2. 2) 20240125-cokernel_and_epimorphism_in_an_abelian_category_5667b5be7592236ab833642a1d2a85ce8a5490a6.svg 1)

Suppose there exists 20240125-cokernel_and_epimorphism_in_an_abelian_category_1b7dc6a0ca66cb9435eee7ceecd6c5d5a5d161f2.svg such that

20240125-cokernel_and_epimorphism_in_an_abelian_category_7754c3add816d73f9979a441b7058d6b0ab6efd0.svg

Then

20240125-cokernel_and_epimorphism_in_an_abelian_category_d89e7cb5125beae67fd749ad03438216a052f89a.svg

and hence by universal property we get

20240125-cokernel_and_epimorphism_in_an_abelian_category_3dafe070b6be9c86dc86ac1ac8905cc611747680.svg

Thus it follows, that

20240125-cokernel_and_epimorphism_in_an_abelian_category_f92ac5cdfe864dd01835aaba6db26539295d79dc.svg

Date: nil

Author: Anton Zakrewski

Created: 2024-10-20 So 09:14