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• CommentRowNumber1.
• CommentAuthorThomas Holder
• CommentTimeOct 10th 2020

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeOct 10th 2020

I have hyperlinked combinatorial functors, since it seems wrong not to.

Since the entry does’t exist yet, I’ll create a stub. Best if you touch it afterwards.

• CommentRowNumber3.
• CommentAuthorThomas Holder
• CommentTimeOct 10th 2020

I must confess that I’ve picked up the terminology from Lawvere without seeing much connection to combinatorics whose enumeration problems seem more naturally connected to bijections à la Joyal than injections.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeOct 10th 2020

So the “combinatorial functors” are not combinatorial functors?

• CommentRowNumber5.
• CommentAuthorThomas Holder
• CommentTimeOct 10th 2020

Ah, interesting reference you dug up there! That might in fact be the concept that Lawvere had in mind. I’ll check it out when occasion arises.

• CommentRowNumber6.
• CommentAuthorThomas Holder
• CommentTimeOct 11th 2020
• (edited Oct 11th 2020)

I shuffled the link to combinatorial functor downwards and motivated the terminology with a quote from Lawvere. It looks to me that their strict combinatorial functors $Set_{mono}\to Set_{mono}$ apparently studied by Myhill might correpond to objects in the Schanuel topos but we have to wait for an energetic model theorist to sort out the connection to the Crossley-Nerode concept.

• CommentRowNumber7.
• CommentAuthorThomas Holder
• CommentTimeOct 13th 2020

Added a further description of the objects bringing them closer to what might rightfully be called combinatorial functor.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeOct 13th 2020
• (edited Oct 13th 2020)

Thanks for looking into it. Though I’ll say that I find there remains room to clarify the remark on combinatorial functors (I admit it remains unclear to me, without digging into the references).

Further in the vein of hyperlinking all technical terms (that’s what eventually constitutes the power of the wiki), I have added double square brackets to binomial coefficient and to name binding.

• CommentRowNumber9.
• CommentAuthorThomas Holder
• CommentTimeOct 13th 2020
• (edited Oct 13th 2020)

I replaced your link with a (hopefully in this context more suggestive) link to falling factorial where I terminologically highlighted the binomial coefficients.

To sort out the messy details of the Crossley-Nerode reference I count on the energetic model theorist.

• CommentRowNumber10.
• CommentAuthorDavidRoberts
• CommentTimeOct 13th 2020

Slightly off-topic The category $FinSet_{mono}$ appears under the name $FI$ in the work of algebraists/representation theorists. Apparently it does a lot of cool things. I do wonder how any of these relate to the sheaves on it, though not terribly seriously.

1. Re #10: Thanks for raising that! I think it would be good to make a page for this category, linked to from Schanuel topos, and mentioning this line of work in representation theory. No times myself just now though…!

• CommentRowNumber12.
• CommentAuthorThomas Holder
• CommentTimeOct 16th 2020
• (edited Oct 16th 2020)

Minor clarification concerning $[FinSet_mono,Set]$ added.

How about generalizing to $(\infty,1)$-toposes !?

I guess it still makes sense to define $Sh_\infty((\infty Grpd_{fin})_{mono}^{op},J_{at})$ where $(\infty Grpd_{fin})_{mono}$ is the $(\infty, 1)$-category of finite homotopy types with monomorphisms (aka (-1)-truncated morphisms) as morphisms and $J_{at}$ is generated by singletons.

• CommentRowNumber13.
• CommentAuthormaxsnew
• CommentTimeSep 13th 2022

x-ref the category of G-sets