I am a graduate student: I know the basics of Category Theory and Homological Algebra, but nothing too deep.
I have spent quite some time studying modules over the ring of real polynomials in one indeterminate.Some of you may already be acquainted with such objects, but I will refresh some of the ideas and make some observations. If you’re already familiar with these considerations, feel free to skip ahead.
Let $V$ be a (left) module over the ring $\mathbb{R}[X]$.It means that the elements of $V$ can be multiplied on the left by polynomials.In particular, we can multiply by polynomials of degree zero, namely real scalars.So if $V$ has a structure of $\mathbb{R}[X]$-module, it necessarily has a structure of real vector space too.
Moreover, consider the action of $X \in \mathbb{R}[X]$ on $V$, hereby denoted with a dot.\begin{align}V &\longrightarrow V \\v &\longmapsto X.v\end{align}
It is easy to check that this map $V \to V$ is a linear map.So the action of $X$ uniquely determines a linear endomorphism, call it $f \in \mathrm{End}_\mathbb{R}V$.
In general, to multiply a vector of $V$ by a polynomial is equivalent to letting the same polynomial, evaluated in $f \in \mathrm{End}_\mathbb{R}V$, act on the vector.For instance$$ (2X^3 - 2X + 1).v = (2f^3 - 2f + \mathrm{Id})(v)$$
Viceversa, let us start with an ordered pair $(V, f)$, where $V$ is a vector space and $f$ is any element of $\mathrm{End}_\mathbb{R}V$.This uniquely determines a structure of left $\mathbb{R}[X]$-module on $V$.It is enough to define the action of a polynomial just like in the equation above.
I have realized that one can also go further.With a little abuse, given $(V, f)$ and $(W, g),$ two left $\mathbb{R}[X]$-modules, a morphism of modules is precisely a linear function $T \colon V \to W$, such that the following diagram commutes.
$\require{AMScd}$\begin{CD}V @>{f}>> V\\@VVTV @VVTV\\W @>{g}>> W\end{CD}
I have shown that the category $\mathbb{R}[X]-\mathrm{Mod}$ is equivalent to the category $\mathcal{C}$, whose objects are pairs of the form $(V, f)$, where $V$ is an object of the category $\mathrm{Vect}_\mathbb{R}$ and $f \in \mathrm{End}_\mathbb{R}V$.The arrows of $\mathcal{C}$ are the arrows of $\mathrm{Vect}_\mathbb{R}$ with the additional property that they satisfy the diagram above.
My question is: there is a name for categories such as $\mathcal{C}$? Is this even a thing in category theory or is it just a fun piece of trivia?My knowledge of categorical jargon is very limited.
I am very tempted to call $\mathcal{C}$ a product category, since its objects are pairs, but I know that this is incorrect: in fact, the domain of the second entry depends on the first, so $\mathcal{C}$ cannot actually be written as a sort of Cartesian product.
I might be saying a ton of horsepoop right now, but is this maybe a topos? Or anything similar? Maybe Fibred Categories?The behaviour of $\mathcal{C}$ is reminiscent of sheaves or fibred manifolds to me.In fact, when we fix the first entry of the ordered pair (the vectorspace) and we let the other entry vary freely, we get a subcategory; these subcategories are indexed by $\mathrm{Vect}_\mathbb{R}$.Over each one of these categories we have a ring, or a real algebra, namely that of endomorphisms.It’s as if each of these subcategories had an algebra towering above.Is this meaningful in any way?
I really have no idea of what I am working with, but I would be very excited to find its place in the bigger picture, possibly even generalizations!Please excuse my newbiness in category theory.