# Talk:Abelian category

WikiProject Mathematics (Rated B-class, Mid-importance)
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Mathematics rating:
 B Class
 Mid Importance
Field: Foundations, logic, and set theory

## Early redesign

I intend to redesign this article over the next few days, in light of the existence of the articles Preadditive_category, Additive_category, and Pre-Abelian_category. I'm going to tighten up the definition, since much of that repeats items that can be found in those articles, and add reference links to those articles. — Toby 11:34 Jul 21, 2002 (PDT)

## Module (category theory)

Hi there! Does the concept of Module (category theory) make any sense to you? If so, would anybody write an article about this? This article seems to be requested on Wikipedia:Requested articles/mathematics. I suspect the person requesting this article confused something, but I could be wrong. Thanks. Oleg Alexandrov 00:15, 29 Dec 2004 (UTC)

## Some suggestions for the page

I have never participated in Wikipedia before so I hope I do not make mistakes. The definition of abelian category given on this page is in a very category-theoretic language; I do not know offhand the definition of "normality" for monomorphisms and epimorphisms and I do not believe most working mathematicians who use abelian categories, and who need to prove that a given category is an abelian category, know that definition. The definition of abelian category given in this article is certainly in line with Freyd's approach to the topic, and it is a good definition, but perhaps also including a more elementary definition would be good, such as the one on Wolfram's site ( [1] ), which is basically identical to the one given in Lang's Algebra. The advantage of this definition is that it is more elementary and familiar to mathematicians who, like myself at this very moment, run across this page because they need to check that a certain category is abelian, and they do not have their standard reference book for abelian categories with them.

There are also some standard properties which you would like an abelian category to have, which they do not always have, but which guarantee different kinds of "nice" behavior in the category; but they are important in the theory of abelian categories and you typically want to know, for any abelian category, which of these axioms it satisfies. They are usually called (AB3), (AB3*), (AB4), (AB4*), (AB5), and (AB5*), and they are written up in most standard references on abelian categories, and appear in most papers and books in algebraic geometry which use any examples of abelian categories which are at all unusual, for instance, Milne's Etale Cohomology. The properties (AB3),...,(AB5*) are defined and discussed in Bucur-Deleanu's book on the subject and I would add the definitions to this page if I had a copy of the book with me right now. They would be a very appropriate addition to this article. --128.151.22.38 (talk) 21:04, 5 May 2008 (UTC)

I am sure that any contributions you make would be welcome additions to the article. If you feel the current definition can be improved, feel free to edit it as you see fit. If other editors disagree you can work together towards the best solution. If you plan to do a lot of editing around here its helpful to create a user account. Drop a note on my talk page if you need any help. Welcome to Wikipedia! -- Fropuff (talk) 05:05, 6 May 2008 (UTC)

## Locally small

In the context of the Wikipedia entry on categories one should add that an abelian category has to be locally small! A category having a zero object, having pullbacks/pushouts and which is normal does not fit into the "peacemeal" definition if the category is not locally small because then there exists a proper Hom-class which cannot be a group (as groups are sets!). —Preceding unsigned comment added by 131.246.83.5 (talk) 19:00, 23 March 2009 (UTC)

You actually already explained why that is not necessary: the fact that the hom-sets are groups already implies that they are sets. 71.227.119.236 (talk) 23:50, 30 September 2012 (UTC)

## Citation for the Theorem of Freyd

Does anyone know where one could find the theorem cited in this article on the equivalence between the more theoretic definition of an abelian category and the "piecemeal" definition? — Preceding unsigned comment added by 174.109.222.113 (talk) 06:42, 6 February 2012 (UTC)

I'm also interested in understanding that apparently strange definition: does Ab-enrichment follows from assuming a zero object, pullbacks, pushouts, and (co)normality? I mean: is there a unique way to *Ab*-enrich C, when I've got those three properties?
It seems to me the definition is missing the existence of kernels and cokernels (or something equivalent). At least in the cited book (Abelian Categories by Freyd), he uses the following list of axioms: zero object + binary sums/products + kernels and cokernels + monos are kernels, epis are cokernels — Preceding unsigned comment added by 212.201.77.20 (talk) 13:22, 24 August 2012 (UTC)

## Epimorphisms should be conormal?

In the article normal morphism the term normal is reserved for monomorphisms, while conormal is used for epimorphisms. The distinction is not made in this article. I think consistent terminology should be used throughout Wikipedia. Onrandom (talk) 13:35, 24 October 2012 (UTC)