Linear algebra from the categorical point of view

Question

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.

Answer 1

As pointed out by Martin my masters thesis touches on this topic, indeed, so may be of interest. In particular, as it utilises only the (nowadays) basics of Category Theory as can be found in MacLane's Category for the Working Mathematician it might serve as a good collection of examples of categorical reasoning particular to linear algebra. (Note that most of the basic categorical results derived at length in the thesis can be obtained directly from familiar general results of Categorical Algebra.)

Link to thesis on ResearchGate

In said thesis my focus has been on affine spaces though, and the first main observation was that the the slice category $\mathit{Vct}_K/K$ of the category of vector spaces over a field $K$ is a symmetric monoidal category that is the "join" of the symmetric monoidal categories of vector spaces and of affine spaces. (The situation becomes more interesting when you do this for $R$-modules over rings with nil-potents, but that is not discussed in the thesis.)

Vector spaces are embedded via zero maps, whereas every linear form that is an epimorphism represents an affine space. You can think of this linear form as a weight and the vector space as a vector space of weighted points. (We identify the affine space with its barycentric calculus of points; a construction going back to A. F. Möbius. Alternatively, one might consider it as adjoining an external zero vector.) This fact re-appears when euclidean transformations of 3-space are encoded as $4\times 4$ matrices, for example. The usual explanation is that we are looking at projective space, but this is geometrically incorrect: we are looking at the barycentric calculus of an affine space.

The second main observation is the construction of a Clifford algebra of an affine space. The product of two points $PQ$ is an invariant representing uniform motion with velocity $\overrightarrow{PQ}$ and the Spin group acts by translations. As noted by Lawvere this invariant can be understood from the ‘internal dynamics’ of the affine space; i.e. one looks as the monoidal action of the bi-pointed affine line $K$ on an affine space. (Note that the pointed affine line $0:1\to K$ together with the familiar addition in $K$ is an internal monoid here.) It turns out that there are many such algebras. For example, Lawvere himself considered the Grassmann algebra of an affine space , which is what Grassmann's long forgotten original construction of the exterior algebra was about, but which is itself not an affine space.

Note that the thesis is a bit of a mathematical meditation and exploration. Although it has a set goal, it goes into exploring almost every nook and cranny along the way (with sometimes unnecessary pedantic proofs). Its aim is to provide a modern mathematical framework for what Leibniz coined ‘Geometric Analysis’, and nowadays would be called ‘Geometric Algebra’. In particular, it explores the somewhat forgotten ideas of H. G. Grassmann in his first edition of his book "Lineale Ausdehnungslehre" (Linear theory of extension), which is a marvellous piece of work that was way ahead of its time.

It should also be noted that the thesis fails to deliver a clear answer to two of its goals:

1. What is the geometric algebra of affine space? One could potentially argue it is the Clifford algebra of affine space; but the thesis fails to present a distinct feature to fully justify this.
2. Is there a geometric algebra of Euclidean space? The thesis fails to investigate the various proposals of Clifford algebra constructions, which utilise embeddings of Euclidean space into non-Euclidean spaces from the categorical viewpoint. It also fails to study the category of Euclidean spaces and its monoidal structures in a more serious attempt to answer this question.

Answer 2

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.).

Answer 3

Professor F. William Lawvere's Introduction to Linear Categories and Applications along with his Categories of Space and of Quantity, where contrasting linear and distributive categories are compared to those of quantities and spaces, respectively, provide an in-depth discussion of the categories encountered in linear algebra. Linear categories and linear algebra are also discussed in Lawvere & Schanuel Conceptual Mathematics and in Lawvere & Rosebrugh Sets for Mathematics. Categories of linear algebra are also discussed in Arbib and Manes Arrows, Structures, and Functors textbook. It is interesting to note that the categories of linear structures quantifying spaces are within the categories of spaces.