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11 votes

The category of abelian group objects

$\DeclareMathOperator\Ab{Ab}\DeclareMathOperator\Hom{Hom}$If $C$ is any additive category then every object has a unique structure as an abelian group object so $\Ab(C)=C$; but typically $C$ is not ab …
Neil Strickland's user avatar
9 votes

Abelian category which is not well-powered

There is a construction of Peter Freyd that embeds any triangulated category $\mathcal{T}$ in an abelian category $\mathcal{A}(\mathcal{T})$. Explicitly, we start with the category of arrows in $\mat …
Neil Strickland's user avatar
18 votes

Abelian categories that are not monoidal

Let $\mathcal{A}$ be an additive category with a monoidal structure such that the maps $$ \otimes \colon \mathcal{A}(A,B)\times\mathcal{A}(C,D) \to \mathcal{A}(A\otimes C,B\otimes D) $$ are biaddi …
Neil Strickland's user avatar
13 votes

Examples of applications of the Freyd-Mitchell embedding theorem.

An interesting class of examples comes from the following construction, also due to Freyd. Let $\mathcal{T}$ be a small triangulated category. Form a new category $\mathcal{A}$, with one object $I(u …
Neil Strickland's user avatar
3 votes

Categories with canonical factorizations into products satisfying two particular properties

First, you should make clear whether LCA groups are assumed to be Hausdorff or not. Next, whichever way you answer the previous question, the category of LCA groups will not be an abelian category. …
Neil Strickland's user avatar
4 votes

Module category equivalent to graded module category?

This answer is just an elaboration of Moosbrugger's comment. For simplicity, assume that $R$ is just a field $k$ concentrated in degree zero. Put $S=\text{Map}(\mathbb{Z},k)$, and let $e_n\in S$ be …
Neil Strickland's user avatar