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I was pondering the Functor again, this time prompted by John Baez's recent series of posts. The concrete definition, as a Functor from to is
This leads me to ponder: given categories and , what conditions on them would mean that there would be a functor whose name could reasonably be .
A naive, too literal generalization, would be:
But that seems too literal. Is there a semantically more faithful generalization to as-general-as-possible and ?
@Jacques Carette that doesn't look like a functor to me. Should it be and in the conditions instead?
Mine or the original? The original is exactly as Joyal defined it, and reproduced faithfully from Baez's post.
The domain of the functor should be the category of finite sets and bijections.
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.
All "species-like" category, i.e. one of the categories appearing in §6 here https://arxiv.org/pdf/2401.04242 support a functor like ""; it's the representable on the object "1" (which has slightly different meanings in those categories, but it's mostly always the singleton).
I should have figured that out from the topic heading, thanks for clarifying!
@fosco could you unwind that a bit for me please? My representable-foo has always been weak. And doesn't that assume that ? 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 ?
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 "" 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
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.
Just in reference to your first message, is not contractible (nor is its object set). I think you perhaps meant that should be a groupoid?
I'm quite aware that is not contractible. I guess I phrased what I meant badly: I wanted to generalize to non-sets. So consider how to do that as part of my question!