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Stream: theory: category theory

Topic: Finite dimensional objects in a symmetric monoidal category

Jean-Baptiste Vienney (Dec 21 2021 at 17:40):

I'm wondering how to define "finite dimensional objects" in a symmetric monoidal category. I would want some purely categorical definition which give me what I want in the obvious examples. The most obvious ones are $FSet \subseteq Set$ and $FVec_{\mathbb{K}} \subseteq Vec_{\mathbb{K}}$. I know two solutions, one by defining exterior powers and another by restricting to compact closed categories and by using the trace. I would prefer not using duality in my context and so I prefer the first one. I would want to restrict myself to the categories where exterior powers (as a coequalizer) exist. The problem is that you cannot define exterior powers in $Set$ because you can't multiply the morphisms by $-1$. When you have the exterior powers and say an initial object 0, you have just to say $E \in \mathcal{C}$ is finite dimensional iff there exists some $n \ge 1$ such that $E^{\wedge n} = 0$ and the dimension of the object is the smallest such $n$ minus $1$.

I know it makes no sense in the category $Set$, but it seems for me that the object $\mathcal{P}_{n}E$ for a set $E$ is so similar to $\Lambda^{n}E$ for a $\mathbb{K}$-vector space $E$. For a finite set $E$ you have $|\mathcal{P}_{k}E| = \binom{|E|}{k}$ and for a finite dimensional $\mathbb{K}$-vector space $E$, you have $dim(E^{\wedge k}) = \binom{dim(E)}{k}$. In this sense, it would be beautiful if these two things could be defined by a same categorical notion because it is obvious that a set $E$ is finite if and only if $\mathcal{P}_{n}E=\emptyset$ for some $n \ge 1$ and that the cardinality of the set is the smallest such $n$ minus $1$.

I was thinking that maybe you can use the fact that $\{x,..,x\} = \{x\}$ for as many copies $x$ as you want, in the same manner that $x \wedge ... \wedge x = 0$ for as many copies of $x$ as you want but it's not exactly the same thing, and you always have the problem that the applicaton $x \mapsto x \otimes ... \otimes x$ is not a morphism in $Vec_{\mathbb{K}}$.

In the case of symmetric powers, all work well. You have the symmetric powers in the category $Set$, and $E^{\otimes_{s}n}$ is the set of all multisets of $n$ elements with value in $E$. It seems that it gives you a structure of $\lambda$-ring on (the isomorphism classes of) $Set$ where the addition is $\bigsqcup$ and the multiplication is $\times$ (with the operators for symmetric powers).

With the operators $\mathcal{P}_{n}$, it seems also to give you a $\lambda$-ring (with the operators $\lambda_{n}$, usually for the exterior powers). But $\mathcal{P}_{n}$ is not an exterior power, in the category $Set$, it really annoys me ! Can we obtain the exterior powers or $\mathcal{P}_{n}$ in the same manner with a categorical definition ?

Nathanael Arkor (Dec 21 2021 at 17:51):

Do you know about [[locally presentable categories]]? They capture a notion of "finite object", namely the [[presentable objects]].

Jean-Baptiste Vienney (Dec 21 2021 at 17:54):

Not really ! It seems to be nice ! But it's sad that it doesn't capture the facts about $\lambda$-rings.

John Baez (Dec 21 2021 at 23:05):

It's worth noting that there's a symmetric monoidal category of [[super vector spaces]], and given a finite-dimensional odd super vector space, it's not true that its nth exterior power vanishes for large n.

John Baez (Dec 21 2021 at 23:07):

But both vector spaces and super vector spaces are finite-dimensional if and only if they are [[dualizable objects]].

Fawzi Hreiki (Dec 22 2021 at 00:26):

Also, a module is dualisable just when it is finite projective (or equivalently finite locally free) and hence a vector bundle

Patrick Nicodemus (Dec 22 2021 at 01:38):

https://arxiv.org/abs/1107.6032 this paper comes to mind, the authors address appropriate notions of "finiteness"