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.
[Math] Reference request: Book of Linear algebra from categorical point of view
ct.category-theorylinear algebrareference-request
Related Solutions
Let's make a list here. Everyone is invited to add and complete the list and the proofs.
List
0) locally compact Hausdorff spaces $\longleftrightarrow$ commutative C*-algebras
0') proper continuous maps $\longleftrightarrow$ non-degenerate C*-homomorphisms $A\to B$
0'') Continuous maps $\longleftrightarrow$ non-generate C*-homomorphisms $A\to M(B)$
1) compact $\longleftrightarrow$ unital
1') $\sigma$-compact $\longleftrightarrow$ has a countable approx. unit ($\iff$ has a strictly positive element)
2) point $\longleftrightarrow$ maximal ideal
3) closed embedding $\longleftrightarrow$ closed ideal
4) surjection/injection $\longleftrightarrow$ injection/surjection
5) homeomorphism $\longleftrightarrow$ automorphism
6) clopen subset $\longleftrightarrow$ projection
7) totally disconnected $\longleftrightarrow$ AF-algebra (AF = approximately finite dimensional)
8) One-point compactification $\longleftrightarrow$ unitalization
9) Stone-Cech compactification $\longleftrightarrow$ multiplier algebra
10) Borel measure $\longleftrightarrow$ positive functional
11) probability measure $\longleftrightarrow$ state
12) disjoint union $\longleftrightarrow$ product
13) product $\longleftrightarrow$ completed tensor product
14) topological K-Theory $K^0$ $\longleftrightarrow$ algebraic K-theory $K_0$
15) second countable $\longleftrightarrow$ separable w.r.t. the C*-norm
Proofs
0),1),2),3),5) follow directly from Gelfand duality - details can be found, for example, in Murphey's book about C*-algebras. For 0'), see here (I wrote this up because I didn't know any reference). A C*-homomorphism $A \to B$ is nondegenerate if the ideal generated by the image is dense. For 4) see here. 6) is given by characteristic functions. A reference for 7) is Kenneth R. Davidson, C*-Algebras by Example, Theorem III.2.5. It is related to 6) because a commutative C*-algebra is AF iff it is separable and topologically generated by the projections. 8) follows from abstract nonsense and 1). 9) ?. 10) is the Riesz representation Theorem. 11) follows from 10). 12) asserts $C_0(X \coprod Y) = C_0(X) \times C_0(Y)$, which is trivial. 13) asserts that the canonical map $C_0(X) \hat{\otimes} C_0(Y) \to C_0(X \times Y)$ is an isomorphism - this follows from the Theorem of Stone-Weierstraß. 14) is the Theorem of Serre-Swan.
The category of Lie algebras is equivalent to a certain category of cocommutative Hopf algebras, with the equivalence given by sending a Lie algebra $\mathfrak{g}$ to its universal enveloping algebra $U(\mathfrak{g})$. These cocommutative Hopf algebras can in turn be thought of as group objects in a certain category of cocommutative coalgebras, and hence can potentially pop up as automorphism objects in any category enriched over cocommutative coalgebras.
You might object that you don't know any interesting examples of such categories, but in fact you do: the category of commutative algebras admits such an enrichment (see the nLab), and this is one abstract way to see why Lie algebras can act on commutative algebras (by derivations).
Speaking more philosophically, you should expect to be able to extract Lie algebras from any situation where you can cook up a sensible notion of infinitesimal automorphism or more generally an infinitesimal element of some group. Enriching over cocommutative coalgebras gives you one fairly general way to do this; if $X$ is an object in your category and $\text{End}(X)$ is the cocommutative bialgebra of endomorphisms of $X$, then the primitive elements of $\text{End}(X)$ (the ones satisfying $\Delta X = 1 \otimes X + X \otimes 1$, where $1 = \text{id}_X \in \text{End}(X)$) should be regarded as the infinitesimal endomorphisms of $X$, and indeed these naturally form a Lie algebra under the commutator bracket.
Best Answer
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.).