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Results tagged with abelian-categories
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user 11640
7
votes
Accepted
Why does tensor product in Ab(V) require colimits in V?
It is easy to define the tensor product as being the object that represents the bilinear maps functor, but to prove that tensor products exist requires something extra. If you have free abelian groups …
- 15.9k
14
votes
Accepted
Projectives and Injectives in Functor Categories
For each object $c$ in $\mathcal{C}$, let $c^* : [\mathcal{C}, \mathcal{A}] \to \mathcal{A}$ be evaluation at $c$. It is an exact functor, so if a left adjoint $c_! : \mathcal{A} \to [\mathcal{C}, \ma …
- 15.9k
6
votes
Accepted
Is any abelian category a subcategory of $\mathrm{Ab}^I$?
Observe that $\bigoplus_I : \textbf{Ab}^I \to \textbf{Ab}$ is a conservative exact functor: it is right exact by general nonsense, it preserves monomorphisms (because e.g. $\bigoplus_{i \in I} A_i$ is …
- 15.9k