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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 …
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 …
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 …
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 …
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. …
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 …