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Stream: learning: questions

Topic: Generalizing Species' X Functor


view this post on Zulip Jacques Carette (Jul 01 2025 at 16:07):

I was pondering the XX Functor again, this time prompted by John Baez's recent series of posts. The concrete definition, as a Functor from FinBij\mathsf{FinBij} to Set\mathsf{Set} is

X(S)={1if S=1if S1X(S) = \begin{cases} 1 & \text{if\ } |S|=1 \\ \emptyset & \text{if\ } |S| \neq 1 \\ \end{cases}

This leads me to ponder: given categories CC and DD, what conditions on them would mean that there would be a functor X:CDX : C \rightarrow D whose name could reasonably be XX.

A naive, too literal generalization, would be:

  1. DD needs to have both terminal and initial objects
  2. CC would require a \textsf{isContr} (is contractible) decidable predicate, i.e. [[contractible+type]] on objects.

But that seems too literal. Is there a semantically more faithful generalization to as-general-as-possible CC and DD ?

view this post on Zulip Morgan Rogers (he/him) (Jul 01 2025 at 16:15):

@Jacques Carette that XX doesn't look like a functor to me. Should it be 1\geq 1 and =0=0 in the conditions instead?

view this post on Zulip Jacques Carette (Jul 01 2025 at 16:19):

Mine or the original? The original is exactly as Joyal defined it, and reproduced faithfully from Baez's post.

view this post on Zulip Nathanael Arkor (Jul 01 2025 at 16:30):

The domain of the functor should be the category of finite sets and bijections.

view this post on Zulip Jacques Carette (Jul 01 2025 at 16:41):

Nathanael Arkor said:

The domain of the functor should be the category of finite sets and bijections.

Oops, absolutely right, thanks - I fixed my original message.

view this post on Zulip fosco (Jul 01 2025 at 17:04):

All "species-like" category, i.e. one of the categories appearing in §6 here https://arxiv.org/pdf/2401.04242 support a functor like "XX"; it's the representable on the object "1" (which has slightly different meanings in those categories, but it's mostly always the singleton).

view this post on Zulip Morgan Rogers (he/him) (Jul 01 2025 at 17:30):

I should have figured that out from the topic heading, thanks for clarifying!

view this post on Zulip Jacques Carette (Jul 01 2025 at 17:43):

@fosco could you unwind that a bit for me please? My representable-foo has always been weak. And doesn't that assume that D=SetD = \mathsf{Set}? That doesn't work for L-Species, as far as I understand.

By "representable on", I guess you mean that it's those functors where the representing object is "1" in CC ?

view this post on Zulip fosco (Jul 01 2025 at 19:53):

on second reading, I probably misunderstood what you were asking, and yes, for what I've been saying, D must be Set; I guess my point is that "XX" has a specific meaning for species, and it can be given many characterizations there:

generalizing different characterizations will surely give different answers -and it will impossible some times

view this post on Zulip Jacques Carette (Jul 02 2025 at 01:14):

This is part of a (quixotic!!) quest of mine to understand the 'fundamental ingredients' of species by understanding all the parts that need to align to make things be species-like. Your linked paper is definitely an important part of that puzzle.

I was approaching via trying to find the minimal requirements for each piece to behave in a recognizable manner, and then moving on to combination of pieces.

In this quest, I think that "monoida unit for substitution" and "singleton" are probably the more important properties that need to be preserved.

view this post on Zulip Nathan Corbyn (Jul 02 2025 at 01:20):

Just in reference to your first message, FinBij\mathrm{FinBij} is not contractible (nor is its object set). I think you perhaps meant that CC should be a groupoid?

view this post on Zulip Jacques Carette (Jul 02 2025 at 01:29):

I'm quite aware that FinBij\textsf{FinBij} is not contractible. I guess I phrased what I meant badly: I wanted to generalize S=1|S| = 1 to non-sets. So consider how to do that as part of my question!