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?
2026-04-08 20:45:46.1775681146
categorification and linear algebra
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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.