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Stream: learning: questions

Topic: Linear transformation category

David Egolf (Apr 08 2020 at 02:47):

Linear transformations are central to my engineering work in medical imaging research, as many imaging schemes can be modeled as linear transformations from imaging targets to observations. To understand linear transformations (and imaging systems) better - and in particular how they can be stuck together - I want to learn more about what kind of constructions on linear transformations are induced by common categorical constructions.

To do this, I would like to learn more about the category $\mathbf{LinTran}$ (is there a standard name?) where the objects are linear transformations between finite dimensional vector spaces, the morphisms are commutative squares from one linear transformation to another, and composition is done component-wise. In particular, I would like to figure out what limits and colimits exist, and what linear transformations (and matrices) they correspond to. To begin with, it might make sense to think about initial or terminal objects, but I'm especially interested in limits over diagrams of shape $\cdot \to \cdot$, as this corresponds to two interlinked linear transformations, and so to two interlinked imaging systems. (Interlinked imaging systems are relevant to thinking about imaging schemes involving multiple related observers / observations).

I don't yet know a good way to begin learning about a new (to me!) category of interest, being new to category theory. I suspect this category is well-studied, but am not sure what it might be called.

Any hints (general or specific) are appreciated!

Thibaut Benjamin (Apr 08 2020 at 08:21):

It looks to me like this category lacks something about linear transformations : there is no operation of composition of the linear transformations! I mean you could try to add a monoidal product (ie a way to combine objects together), but it is not defined for all pairs of objects. That is not to say that there is nothing interesting about the category you are considering : you might be able to see many cool things with it, but keep in mind that's not the whole picture.

I believe, what you want to define here is rather a "double category" : that's like a category, but with vertical morphisms and horizontal morphisms, plus 2-cells (ie morphisms between morphisms), that have shape of a square. Here the objects would be vector spaces, the horizontal and vertical morphsism would be the linear transformations, and the 2-cells, the commutative squares. This kind of objects might be scary if you are new to category, so probably best not to go too deep into this too quickly, but I think it's good to keep this in mind as well

Jules Hedges (Apr 08 2020 at 09:19):

There's a standard construction called an arrow category $\mathcal C^\to$ that you can make from any category $\mathcal C$, where objects of $\mathcal C^\to$ are morphisms in $\mathcal C$ and morphisms in $\mathcal C^\to$ are commutative squares in $\mathcal C$. It sounds like you're talking about $\mathbf{FVect}^\to$

sarahzrf (Apr 08 2020 at 14:44):

and good news: the questions you are asking probably have well-established answers if this is what you have in mind :-)

sarahzrf (Apr 08 2020 at 14:45):

for example, if $\mathcal{C}$ is complete or cocomplete, then so is $\mathcal{C}^\to$

sarahzrf (Apr 08 2020 at 14:46):

or more generally, if it has limits or colimits of a given shape, then so does $\mathcal{C}^\to$

sarahzrf (Apr 08 2020 at 14:48):

(this follows from the fact that $\mathcal{C}^\to$ is equivalently the category of functors from the "single-nontrivial-arrow category" $0 \to 1$ to $\mathcal{C}$, and the fact that if a category has all limits or colimits of a given shape, then so does the category of functors from a given domain into that codomain)
(also, in such a scenario, the limits/colimits are computed pointwise)

John Baez (Apr 08 2020 at 15:46):

By the way, algebraists call the category $\mathsf{Vect}^{\to}$ "the category of representations of the $A_2$ quiver", and they know shitloads about that.

A quiver $Q$ is a graph, and we can freely turn any quiver into a category $F(Q)$, and then $\mathsf{Vect}^{F(Q)}$ is called the category of representations of the quiver $Q$.

They call the graph with $n$ objects and a chain of $n-1$ composable arrows $A_n$.

John Baez (Apr 08 2020 at 15:50):

A quiver has finite type if it has finitely many isomorphism classes of indecomposable representations: ones that aren't direct sums (=coproducts) of others in a nontrivial way. Gabriel's theorem classifies the quivers of finite type, and $A_n$ has finite type.

John Baez (Apr 08 2020 at 15:50):

This is an amazing theorem, because it sets up connections between quivers and other nice things.

John Baez (Apr 08 2020 at 16:00):

Anyway, $\mathsf{Vect}^{\to}$ has 3 indecomposable objects. If $k$ is our field these are:

$k \stackrel{1}{\longrightarrow} k$
$k \stackrel{0}{\longrightarrow} 0$
$0 \stackrel{0}{\longrightarrow} k$

Every object of $\mathsf{Vect}^{\to}$ is isomorphic to a direct sum of copies of these.

sarahzrf (Apr 08 2020 at 16:53):

o:

sarahzrf (Apr 08 2020 at 16:53):

what about the function $\mathbb{R} \to \mathbb{R}$ given by $x \mapsto 1.5x$?

Morgan Rogers (he/him) (Apr 08 2020 at 16:58):

It's an isomorphism, so in particular isomorphic to the identity on $\mathbb{R}$ :tada:

David Egolf (Apr 08 2020 at 17:16):

Thanks, everyone. This gives me some good direction to do some reading.

sarahzrf (Apr 08 2020 at 20:04):

hmm, hold on...

sarahzrf (Apr 08 2020 at 20:06):

oooh, i was glossing "object of Vect^→ [...] isomorphic to" in my head as "morphism of Vect [...] is" in my head w/o thinking about it

sarahzrf (Apr 08 2020 at 20:06):

but i suppose "is" for morphisms of Vect is equality, whereas "isomorphism" for objects of Vect^→ is considerably weaker

sarahzrf (Apr 08 2020 at 20:06):

neat!