# categorification and linear algebra

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?

• 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. – Hagen von Eitzen Apr 22 '14 at 4:13
• 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. – Siddharth Venkatesh Apr 22 '14 at 4:53

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.