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Stream: theory: category theory

Topic: classification End (FinSet), End (FinCat)

Arshak Aivazian (Dec 30 2021 at 21:07):

Is it possible to classify objects of End (FinSet) (up to isomorphism)? What about End(FinCat)?

Todd Trimble (Dec 30 2021 at 22:22):

Notation and abbreviations: it's probably a good idea to say what your notation/abbreviations refer to, the first time around . Are you seeking to understand endofunctors on the category of finite sets, for example? (Way too ambitious, methinks.)

Oscar Cunningham (Dec 31 2021 at 10:56):

Can we say which sequences of natural numbers are allowed as the object part of endofunctors on $\mathbf{FinSet}$?

Todd Trimble (Dec 31 2021 at 11:14):

One thing you could say is that past $0$, those sequences are nondecreasing, because (letting $[n]$ denote an $n$-element set) if $0 < m, n$, we have $m \leq n$ iff $[m]$ is a retract of $[n]$, and (endo)functors preserve retracts.

Another thing you could say is that is that if $F([n]) = [0]$ for any $n > 0$, then $F([m]) = [0]$ for all $m$. This is because for all $m$, there is a morphism $f: m \to n$, and thus a morphism $F(f): F([m]) \to [0]$, but this forces $F([m]) = [0]$ by strictness of $[0]$.

I don't know any other constraints.

Oscar Cunningham (Jan 05 2022 at 17:00):

This page has a partial solution: https://mathoverflow.net/questions/346290/when-is-an-integer-sequence-the-trace-of-a-monad-on-finset

Arshak Aivazian (Jan 06 2022 at 23:49):

Yes, I meant endofunctors. Isn't that standard notation?

John Baez (Jan 07 2022 at 00:53):

Yes, it's standard. But it's so hopeless to classify all endofunctors on FinSet up to isomorphism that while Todd correctly guessed that's what you were talking about, he had trouble believing that's what you meant.

John Baez (Jan 07 2022 at 00:55):

So, one quick answer to your question

Is it possible to classify objects of End (FinSet) (up to isomorphism)?

is "no".

John Baez (Jan 07 2022 at 00:58):

Of course, to turn this answer into a theorem we would have to define "classify" and "possible" more precisely.

John Baez (Jan 07 2022 at 00:59):

With some very limited sense of "classify" and some very ambitious definition of "possible" the answer could become "yes".

John Baez (Jan 07 2022 at 00:59):

Classifying endofunctors on FinCat up to isomorphism is even harder.

Joshua Meyers (Jan 07 2022 at 03:19):

There is actually some work on this. See

• A good starting point is Presentation of Set Functors: A Coalgebraic Perspective, J. Adámek, H. Gumm, V. Trnková
  https://www.semanticscholar.org/paper/Presentation-of-Set-Functors%3A-A-Coalgebraic-Ad%C3%A1mek-Gumm/0638f4cccc0fe20fe7868d8e2d84af2301863dcd

• There's also Trnkova, On descriptive classification of set functors, written in an older style

• And this book https://www.google.com/books/edition/Categorical_Perspectives/PjzTBwAAQBAJ is cool

Joshua Meyers (Jan 07 2022 at 03:19):

But I don't think anybody has a complete classification yet

Joshua Meyers (Jan 07 2022 at 03:19):

Not sure