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Vect is the category with objects as vectors and arrows as linear transformations between them. Then these arrows have quite a bit of structure. We can take the transpose, trace, determinant, eigenvectors, etc of them. Do these operations make sense in other categories? Is there a more general treatment of this I can read?

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    $\begingroup$ Trace, detrminant, eigenvectors make sense only for endmmorphisms (and not always for infinte dimensions), not general linear transformations. The transpose is related to the notion of dual space. $\endgroup$ Apr 22, 2014 at 4:13
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    $\begingroup$ The operations you talk about aren't necessarily the things you want to generalize. I think googling about "Abelian Categories" and, once you're comfortable with abelian categories, "Tensor Categories" would be useful to you. I think, however, that the first generalization of Vect that you should read about (if you haven't already) is the category of modules over a ring. This is basically the concept of a vector space except now, instead of a base field, we just have a base ring. Surprisingly, this changes a great deal about the structure of the category. $\endgroup$ Apr 22, 2014 at 4:53

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Traces can be defined in monoidal categories: arXiv:1010.4527

The transpose generalizes to morphisms between dualizable objects in monoidal categories. They are known as mates.

The determinant is the top exterior power, and the latter can be defined (for example) in Cauchy complete $\mathbb{Q}$-linear symmetric monoidal categories. A famous use is Deligne's Catégories Tannakiennes.

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