Which nice/deep elaborations on the (operators <-> sheaves) / (endomorphisms <-> objects) theme are there?

Question

A linear operator $T:V\to V$ on a (say) vector space over a field $k$ is just a $k[T]$-module, and may be viewed as the sheaf $\mathscr F_T$ over $\mathbb A^1_k$, with fibre over $\lambda\in k$ equal to $V/\langle T(v)-\lambda v\mid v\in V\rangle$; going backwards one may reconstruct $V$ as global sections of $\mathscr F_T$, with $T$ acting via $T(s)(\lambda)=\lambda s(\lambda)$.

This is of course a commonplace observation nowadays, but for some reason I still stay fascinated by this fact, and I learned about it many years ago.

I remember being particularly struck, having shortly before that studied some category theory, by the fact that this construction establishes a connection between endomorphisms in one category and objects in another category, which seemed somewhat unforeseen by category theory.

Later I learned about categories of bordisms, where also an endobordism of an $n$-manifold produces an $n+1$-manifold, but I still have no idea whether this is related in any way to the fact I started with.

[LATER - Re: comment by Will Sawin - this could be in principle formulated in terms of (category-theoretic) traces; however I only know traces of endomorphisms with values in an object of the same category, whereas here we seemingly should have traces with values themselves being objects of another category]

I've still got a feeling that there is something so deep hidden in this that its deepest roots still wait to be dug out.

In algebraic geometry, this is a simple particular case of the correspondence between modules over a ring and sheaves over the corresponding affine scheme;

The whole spectral theory may be viewed as built on it, things like direct integrals of linear spaces, etc. included;

Various dualities involving vector bundles and projective modules may be also viewed as generalizations of this...

Let me also mention certain "noncommutative" version of the spectrum of a $C^*$-algebra invented by Segal, which I briefly described in an answer to Why the Dold-Thom theorem?

But I will stop here and ask my questions.

Does this circle of ideas/facts have a common name? Which constructions/theories/theorems may be viewed as stemming from it? Is there a way to describe its place in mathematics? Is there really a connection between it and the endobordism thing I mentioned? Is there a category-theoretic or other abstract/axiomatic treatment of similar phenomena?

Answer 1

What you describe in the opening of your question is most naturally an equivalence of categories, not some dimension-shifting thing between endomorphisms and objects as you suggest. Indeed, let $\mathcal C$ be a category. The category of endomorphisms in $\mathcal C$ is the functor category $\mathcal C^{\circlearrowleft}$, where $\circlearrowleft = \mathrm B \mathbb N$ denotes the category with one object and endomorphism monoid $\mathbb N$, i.e. the category freely generated by an object with an endomorphism. In some corners of category theory, $\circlearrowleft$ would be called "the walking endomorphism". In particular, it is a category, being a category of functors.

If $\mathcal C$ is a nice enough monoidal category ($\mathrm{Vect}$ is nice enough), then $\mathcal C^{\circlearrowleft}$ can also be described as the category of $\mathcal C$-linear functors from the free $\mathcal C$-linearization of $\circlearrowleft$. This is the $\mathcal C$-enriched category with one object and endomorphism algebra $\mathbf 1_{\mathcal C}[x] \in \mathcal C$, the "polynomial algebra" in one variable with coefficients in the unit object $\mathbf 1_{\mathcal C} \in \mathcal C$.

But to move to your more geometric questions requires more than just this elementary category theory. For example, it happens that $\mathbf 1_{\mathcal C}[x]$ is a commutative algebra, and so its spectrum is an affine variety. This would not hold if $\circlearrowleft$ were replaced by some other more complicated walking object, say if you were asking about pairs of endomorphisms and not just endomorphisms. (On the other hand, $\circlearrowleft \times \circlearrowleft$ is the walking pair of commuting endomorphisms.)

I don't think there's anything deeper to your observation about endobordisms except to observe that $\mathcal C^{\circlearrowleft}$, being a functor category, is naturally a category.

Actually, there's something else to say about bordisms. The category of bordisms is naturally a double category, i.e. a category object internal to categories. Bordisms and their compositions form the "horizontal" morphisms but the "vertical" morphisms are smooth functions. (Important variations: use just embeddings or just local diffeomorphisms.) In any double category, if you fix an object, you get a category whose objects are the horizontal endomorphisms of that object. In the case of $\emptyset \in \mathrm{Bord}$, you get the category of $n$-manifolds and smooth maps (or embeddings or local diffeomorphisms).