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Stream: theory: category theory

Topic: Endofunctors of FVec_{K}

Jean-Baptiste Vienney (Jun 04 2022 at 17:12):

Let $\mathbb{K}$ be a field of characteristic $0$. Let $FVec_{\mathbb{K}}$ be the category of finite-dimensional $\mathbb{K}$-vector spaces. Is there a description of all the endofunctors $FVec_{\mathbb{K}} \rightarrow FVec_{\mathbb{K}}$ ? I don't want just a restriction like polynomial or Schur functors but everything possible.

John Baez (Jun 06 2022 at 01:23):

I don't know such a characterization. My feeling is that there are too many crazy ones. But let me try to come up with some.

Morgan Rogers (he/him) (Jun 06 2022 at 09:34):

Observe that any such must induce a non-decreasing function on dimension, since if $\mathrm{dim}(V) \leq \mathrm{dim}(W)$ then $V$ is a retract of $W$, and this must remain true after applying the endofunctor. Since you are considering all functors, the 0 morphisms and additive structure on morphisms won't necessarily be preserved (if they were, 0 and direct sums would have to be preserved too).

There will be many many such functors. Even if the functor preserves one-dimensional spaces, the induced map $\mathbb{K} \to \mathbb{K}$ doesn't have to preserve addition of morphisms so you have a monoid homomorphism from the multiplicative monoid of $\mathbb{K}$ to itself; for $\mathbb{K}=\mathbb{Q}$ this is determined by a function $\mathbb{P} \to \mathbb{Q}$, where $\mathbb{P}$ is the set of prime numbers.

Morgan Rogers (he/him) (Jun 06 2022 at 09:35):

So, like John, I'm pessimistic about the possibility of neatly classifying these things.

Oscar Cunningham (Jun 06 2022 at 10:03):

Previously, we talked about endofunctors of FinSet: https://categorytheory.zulipchat.com/#narrow/stream/229136-theory.3A-category-theory/topic/classification.20End.20.28FinSet.29.2C.20End.20.28FinCat.29

Oscar Cunningham (Jun 06 2022 at 10:04):

On objects they have to not just be increasing but have positive differences of all orders. I expect the same thing is true for vector spaces

Morgan Rogers (he/him) (Jun 06 2022 at 12:44):

@Oscar Cunningham isn't that only true for monads? We can always take constant functors on any object we like.

Oscar Cunningham (Jun 06 2022 at 12:47):

Hmmm, maybe I misunderstood the answers in the page I linked

Oscar Cunningham (Jun 06 2022 at 12:48):

I meant 'nonnegative', so the constant map isn't a contradiction (and for sets this claim only works for the nonempty sets).

Morgan Rogers (he/him) (Jun 07 2022 at 06:15):

If you meant non-negative then you were agreeing with me ;)

Also note that the zero vector space isn't a strict initial object in Vect, so we can have a functor which is 0 for as long as we like and then goes up in dimension. Or at least this is possible with the axiom of choice where we can pick a basis for each vector space and do something like "project onto a subspace of codimension k".