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Posts Tagged ‘enriched categories’

The goal of today’s post is to introduce and discuss semiadditive categories. Roughly speaking, these are categories in which one can add both objects and morphisms. Prominent examples include the abelian categories appearing in homological algebra, such as categories of sheaves and modules and categories of chain complexes.

Semiadditive categories display some interesting categorical features, such as the prominence of pairs of universal properties and the surprising ways in which commutative monoid structures arise, which seem to be underemphasized in textbook treatments and which I would like to emphasize here. I would also like to emphasize that their most important properties are unrelated to the ability to subtract morphisms which is provided in an additive category.

In this post, for convenience all categories will be locally small (that is, \text{Set}-enriched).

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In the previous post we learned that it is possible to recover the center Z(R) of a ring R from its category R\text{-Mod} of left modules (as an \text{Ab}-enriched category). For commutative rings, this justifies the idea that it is sensible to study a ring by studying its modules (since the modules know everything about the ring).

For noncommutative rings, the situation is more interesting. Two rings R, S are said to be Morita equivalent if the categories R\text{-Mod}, S\text{-Mod} are equivalent as \text{Ab}-enriched categories. As it turns out, there exist examples of rings which are non-isomorphic but which are Morita equivalent, so Morita equivalence is a strictly coarser equivalence relation on rings than isomorphism. However, many important properties of a ring are invariant under Morita equivalence, and studying Morita equivalence offers an interesting perspective on rings on general.

Moreover, Morita equivalence can be thought of in the context of a fascinating larger structure, the bicategory of bimodules, which we briefly describe.

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The center Z(G) of a group is an interesting construction: it associates to every group G an abelian group Z(G) in what is certainly a canonical way, but not a functorial way: that is, it doesn’t extend (at least in any obvious way) to a functor \text{Grp} \to \text{Ab} (unlike the abelianization G/[G, G]). We might wonder, then, exactly what kind of construction the center is.

Of course, it is actually not hard to come up with a rather general example of a canonical but not functorial construction: in any category C we may associate to an object c \in C its automorphism group \text{Aut}(c) or endomorphism monoid \text{End}(c)), and this is a canonical construction which again doesn’t extend in an obvious way to a functor C \to \text{Grp} or C \to \text{Mon}. (It merely reflects some special part of the bifunctor \text{Hom}(-, -).)

It turns out that the center can actually be thought of in terms of automorphisms (or endomorphisms), not of a group G, but of the identity functor G \to G, where G is regarded as a category with one object. This definition generalizes, and the resulting general definition has some interesting specializations. Moreover, an important general property is that the center is always abelian, and this has a very elegant proof.

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In this post, I’d like to record a few basic definitions and results regarding noncommutative rings. This is a subject clearly of great importance and generality, but I haven’t had much exposure to it, and I’m trying to fix that. I am working mostly from Lam’s A first course in noncommutative rings.

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