Examples

Remarks

The term `non-Abelian'-instead of noncommutative- is often the preferred qualifier for theories, with the possible exeception of `noncommutative geometry', which is a non-Abelian theory. On the other hand, the term `anabelian' (in French, ``anabelienne'') was employed by Alexander Grothendieck to describe a new field of research called ``Anabelian Geometry'' in his `Esquisse d'un Programme''

The commutativity property in all Abelian structures, such as Abelian groups and Abelian categories is a global rather than partial, or local, property. Thus, many categories or toposes/topoi may exhibit local but not global commutativity properties; for example, a category is still non-Abelian if $Hom_{Ab}(-, -)$ does not have the structure of a commutative (or Abelian) group; alternatively, a category that does not satisfy one of the four $Ab$ -axioms of Freyd, or one of the $Ab1$ to $Ab6$ axioms in the current abelian category definition is non-Abelian.

Many-valued, algebraic logic examples

Examples of Abelian categories

1. The category of Abelian (or commutative) groups is Abelian;
2. The Grothendieck category is a special case of $\mathcal{\A}b5$ category;
3. Local Grothendieck categories are Abelian categories;
4. The Grassmann category is Abelian;
5. The category of (commutative) semi-noetherian rings is Abelian. ([1]);
6. The category of Heyting logic algebras, $Hy$ and the category of Boolean logic algebras, Boole are both categories of commutative lattices, and are Abelian categories;
7. If $\grp$ is a topological groupoid the category of sheaves, $S_{hA}$ of Abelian groups over $\grp$ is an Abelian category.

Related results If $\mathcal{\G}$ is a Grothendieck catgeory and $\mathcal{\A}$ is a localizing subcategory of $\mathcal{\G}$ , then $\mathcal{\G} \slash \mathcal{\A}$ is Abelian, as it is also a Grothendieck category (COROLLARY 6.2 on p. 186 in [1]) .

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