There are seemingly 4 different levels of linearity that a category can satisfy:(adsbygoogle = window.adsbygoogle || []).push({});

preadditivity: hom-sets are abelian groups and composition is bilinear

additivity: all finite biproducts exist

pre-Abelian: all morphisms have kernels and cokernels

Abelian: every monic is a kernel and every epic is a cokernel.

and these are all nice properties, but I wonder how useful these classifications are, since every preadditive category I've ever met is actually Abelian. OK, well I saw an additive category which wasn't Abelian in a homework exercise, but it was a rather contrived example. So does anyone have a bucket of fun examples of categories that are preadditive but not additive, additive but not pre-Abelian, or pre-Abelian but not Abelian? Do such categories occur in mathematics, or are they only for use in proving abstract nonsense theorems in category theory?

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Preadditive categories are all Abelian

Loading...

Similar Threads for Preadditive categories Abelian | Date |
---|---|

I What is the equivalent of a group in category theory? | Nov 7, 2017 |

I Categories of Pointed Sets - Aluffi, Example 3.8 | May 3, 2016 |

Trying To Learn Category Theory | Dec 31, 2015 |

Category-theoretic proof of free groups | Mar 20, 2014 |

**Physics Forums - The Fusion of Science and Community**