exact functor and exact sequence

1. Proposition

Let 20240122-covariant_exact_functor_and_exact_sequence_8d761a8e0310248d2b232dd1cfa3d15c4ce348ba.svg be pre-abelian categories categories and 20240122-covariant_exact_functor_and_exact_sequence_4834e0933441585e455612f47acbea83f96321b7.svg be an enriched functor. TFAE:

  1. 20240122-covariant_exact_functor_and_exact_sequence_f9d4fff07675f599df559c5e656b52a1fd42a7f3.svg is an covariant exaÖact functor
  2. 20240122-covariant_exact_functor_and_exact_sequence_f9d4fff07675f599df559c5e656b52a1fd42a7f3.svg preserves exact sequences

2. Proof

corollary of:

2.1. 1) 20240122-covariant_exact_functor_and_exact_sequence_5667b5be7592236ab833642a1d2a85ce8a5490a6.svg 2)

Alternatively, let

20240122-covariant_exact_functor_and_exact_sequence_ec21a486d1efb365b3cb142e531eb31afc2d91f5.svg

be exact

Then there exists an isomorphism between the image and the kernel

20240122-covariant_exact_functor_and_exact_sequence_9038535598f3d2dc8f1e4063d1baedb09d626b80.svg

and applying 20240122-covariant_exact_functor_and_exact_sequence_f9d4fff07675f599df559c5e656b52a1fd42a7f3.svg, which by definition presreves

2.2. 2) 20240122-covariant_exact_functor_and_exact_sequence_5667b5be7592236ab833642a1d2a85ce8a5490a6.svg 1)

Date: nil

Author: Anton Zakrewski

Created: 2024-10-20 So 09:02