abelian category and coimage isomorphic to image

1. Proposition

Let 20240121-abelian_category_and_coimage_isomorphic_to_image_3038b4682e1f6d64ae501574b4029ad43f8f615e.svg be a category. TFAE:

  1. 20240121-abelian_category_and_coimage_isomorphic_to_image_3038b4682e1f6d64ae501574b4029ad43f8f615e.svg is abelian
  2. for the image and coimage, the induced morphism
20240121-abelian_category_and_coimage_isomorphic_to_image_d0855968fa8bc9c9cf5a1ccf2be557a80a684e48.svg

is an isomorphism

2. Proof

Let

20240121-abelian_category_and_coimage_isomorphic_to_image_a6f44cf07e8f3f555fe7e9dacb204296d98e9975.svg

Then we get

20240121-abelian_category_and_coimage_isomorphic_to_image_4459eebd755680eedbe14ae8e73f4d451719d73d.svg

Note that since

20240121-abelian_category_and_coimage_isomorphic_to_image_6af86ad101cb3aa56adde543211b7b3d97cc175b.svg

is a cofork, we get

20240121-abelian_category_and_coimage_isomorphic_to_image_3aba3d4526938aed47e104a5f941184aac0866c1.svg

as (mostly) commuting diagram

Therefore, by universal property of the kernel, we get a unique map

20240121-abelian_category_and_coimage_isomorphic_to_image_fd3dff59cdc74c10d5699fb2f5b0b5cc6f3e6600.svg

TODO

Date: nil

Author: Anton Zakrewski

Created: 2024-10-20 So 05:33