Is there any book of Linear algebra in the modern language of Category theory?

I refer to the (systematic, formalist) study of the category whose objects are vector spaces and whose morphisms are linear maps and its consequences.

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    $\begingroup$ I suspect the answer is no, but find the question intriguing. At least in the US market, the typical linear algebra customer wants only matrix algorithms and specific problems that can be solved that way. Books tending toward abstraction have become almost extinct, even at the level of a second or third course. Bourbaki on the other hand stopped before venturing into category language. There are of course books on "universal algebra" and "category theory" but not what you are looking for. Good luck. $\endgroup$ – Jim Humphreys Aug 15 '14 at 23:05
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    $\begingroup$ reddit.com/r/math/comments/1eowe8/… $\endgroup$ – Carlo Beenakker Aug 16 '14 at 0:38
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    $\begingroup$ I don't remember right out of my hat whether Paolo Aluffi's "Algebra 0" does much linear algebra (and I'm not at the computer which has it as PDF), but it certainly looks like a step in the right direction. Also, Kostrikin/Manin "Linear Algebra and Geometry", while not using the categorical approach right away, does introduce categories at some point (as well as tons of other interesting things). But now I'm seeing that these are exactly the first two suggestions on the Reddit thread... $\endgroup$ – darij grinberg Aug 16 '14 at 7:24
  • $\begingroup$ At least over symmetric monoidal model categories, Bertrand Toën, Gabriele Vezzosi, Homotopical algebraic geometry II: geometric stacks and applications, 2004, arXiv:math/0404373, is perhaps what you're looking for. $\endgroup$ – user62675 Aug 18 '14 at 23:32
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    $\begingroup$ See also at ncatlab.org/nlab/show/integral+transforms+on+sheaves the analogy between locally presentable $\infty$-categories and vector spaces. "For $C,D\in\mathscr{Pr}\infty\mathscr{Cat}$, ..., we may think of $C,D$ as analogous to vector spaces". The nlab doesn't provide much literature on this correspondence, though. $\endgroup$ – user62675 Aug 18 '14 at 23:38

Filip Bár's master thesis, "On the Foundations of Geometric Algebra" might be a beginning (I don't know if this is online, but perhaps you can ask the author). This thesis develops some ideas by Grassmann in modern language, especially concerning affine spaces and affine algebras, but Chapter 2 deals with vector spaces from a basic categorical point of view.

Meanwhile there are some accounts on commutative algebra from a category-theoretic point of view (Toën-Vezzosi, Lurie, B.).

  • $\begingroup$ I searched the web but could not find any contact details for Filip Bár. Could I email you a request for them? $\endgroup$ – Arrow Dec 9 '16 at 11:57
  • $\begingroup$ @Arrow: He is/was at the University of Cambridge. Sorry for not telling more. $\endgroup$ – Martin Brandenburg Apr 12 '17 at 11:57

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