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Rigid symmetric monoidal abelian categories over a field $k$ with a fiber functor (i.e., a strong monoidal exact functor) to $K$-vector spaces for some extension $K$ of $k$ (the functor is assumed to …

These categories are known as skeletons, or skeletal categories. Unfortunately, given a category $\mathcal C$, one can not define in general the structure of a category on the class of isomorphic clas …

This is not true; for example, take $X = \mathbb R^2$, $U = \mathbb R^2 \smallsetminus \{(0,0)\}$. Then your $\mathbb Z_U$ coincides with $\mathbb Z_X$, and $Hom(\mathbb Z_U, F)$ is $F(X)$, not $F(U)$ …

The dual of the category of abelian group is the category of compact topological abelian groups, by Pontryagin duality.

In the case of a quasi-separated scheme, the center of the category of quasi-coherent sheaves is $\mathcal O(X)$. Suppose that $f$ is in the center. Let $a \in \mathcal O(X)$ be the scalar that descri …

About your first question: the relation is transitive. If $a$ is equivalent to $b$ using a covering, and $b$ is equivalent to $c$ using another covering, it is easy to see that $a$ is equivalent to $c …

Categorical ideas should be certainly introduced early, as they are quite useful. On the other hand, as Andreas and Terry say, studying category theory at the beginning of your mathematical education …

The split Grothendieck group for vector bundles on a complete variety appears in Nori's PhD thesis on the fundamental group scheme; this was published in the Proceedings of the Indian Academy of Scien …

I don't think this is true. Let $X$ be the quotient of the action of $\mathbb Q$ on $\mathbb R$ by translation. This is a sheaf, and its automorphism groups are trivial. Suppose that there exist a loc …

I would think so. Two pretopologies are equivalent when they generate the same topology, that is, when the have the same sieves. If $A$ is an object of $C$, a sieve on $A$ is a subfunctor of the fun …

This is far from being a technical issue, there are many examples when it fails. Suppose that A is the category of $\mathbb F_p$-vector spaces, $B = C$ the category of abelian groups, $F$ the embeddin …