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One way to think about what the monoidal structure on vector spaces is doing is that it is telling us that vector spaces do not really form a category, or not "just" a category: they form a multicateg …

I find it less confusing to work directly with $A^{op}$ so let me do that; I'll rename it $C$. We have a complete abelian category $C$ (completeness is equivalent to being closed under small products) …

Schur's lemma has the same proof in a $k$-linear abelian category $C$ as usual: if $T : M \to M$ is a nonzero endomorphism of a simple object, by simplicity it must have trivial kernel and cokernel, s …

$k$-linear cocomplete categories admit a "tensor product over $\text{Mod}(k)$" (thinking of them as module categories over $\text{Mod}(k)$) and the only thing you need to know about it to answer this …

Sure, we can define such things. Let's work in the Morita 2-category $\text{Mor}(k)$ over a commutative ring $k$, which has objects $k$-algebras $A$, morphisms $k$-bimodules (with composition given …

The category $[C^{op}, \text{Ab}]$ of $\text{Ab}$-valued presheaves on any (small, for simplicity) $\text{Ab}$-enriched category is about as nice as it gets - locally finitely presentable, Grothendiec …

No. Let $C$ be a category constructed as follows. Let $G$ be a group and let $BG$ be the one-object category with automorphisms $G$. Then adjoin to $BG$ a zero object. This means that $C$ has two obje …

You want to look up Morita theory for enriched categories. By a "linear category" I will mean an $\text{Ab}$-enriched category. Write $\widehat{A}$ for the category of presheaves of abelian groups o …

Suppose that $V, W$ are vector spaces and $f, g : V \to W$ are two parallel morphisms such that $f \le g$, for some preorder $\le$ satisfying your conditions. Claim: $f$ is a scalar multiple of $ …

This is not an answer. Below "finite" means "finite-dimensional over $k$," so "profinite" means "pro-finite-dimensional" and so forth. The category of coalgebras is the ind-category of the category o …

Regarding schemes: on a scheme $X$, the functor of global sections is exact iff no quasicoherent sheaves have higher cohomology. By Serre's criterion for affineness, for reasonable schemes this is equ …

I'll denote the abelian category by $A$ and an object in it by $a$. Any $a \in A$ gives rise to a ring $\text{End}(a)$. The inclusion $a \to A$ induces a functor from the category of finitely generate …

No. What follows appears to be a counterexample for $C = \text{Vect}$ (I don't understand where in your argument you prove fullness). Let $M = \text{Vect}^{op}, C = \text{Vect}$, and let $h : \text{V …

(Crossposted from math.SE.) Call a category acyclic if only the identity morphisms are invertible and the endomorphism monoid of every object is trivial. Let $C, D$ be two finite acyclic categories. …