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The simplex category $\Delta$ is defined as the category of (non-empty) finite ordinals and order preserving maps. Furthermore, a simplicial set $X$ is defined as a contravariant functor $X: \Delta \rightarrow \text{Set}$.

I am interested in the possibility of generalizing the notion of a simplicial set by considering the category of infinite ordinals $\mathcal{O}$, if such a thing exists, and then defining an $\infty$-simplicial set as $X^{\infty}: \mathcal{O} \rightarrow \text{Set*}$, where $\text{Set*}$ is the appropriate adjustment of the category of small sets $\text{Set}$ such that mapping from objects in the category of infinite ordinals will still satisfy contravariance.

Assuming such a generalization exists, how does this affect the geometric definition of an $(\infty,1)$-category? Is there no longer a capturing of the geometric model desired from simplicial sets or can something more general than an $(\infty,1)$-category be defined? It would also be interesting to see how this would change $(\infty,1)$-functors between simplicial to sets to some other type of functor between $X^{\infty}$ sets. What would natural transformations look like, if they could still be defined properly?

EDIT: I asked this question to a graduate student who is doing work in $\infty$-categories, and he said that you would not get the same geometric model you want by quasicategories if you allowed infinite ordinals. Does this make sense to anyone? He said, that by taking the geometric realization "you probably wouldn't get anything back", but I don't really know what to make of that.

EDIT 2: This question has really been interesting me, and I can't find anything on it in Lurie's Higher Topos Theory or any other literature I have looked through. It seems like any time an author introduces the idea of a simplicial set to aid in defining quasicategories they don't think about possible variations on the simplex category that might change the entire construction they are making into something completely different. Let me know if you want a more specific question to answer!

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I've asked a very similar question here: – Martin Brandenburg Mar 14 '12 at 20:27
One refinement: the simplex category is not usually defined as the category of finite ordinals. Although usage varies, it is far more frequently the category of nonempty finite ordinals. – Tom Leinster Mar 14 '12 at 20:36
You could also restrict to the category of nonempty countable ordinals. If you take all ordinals, size considerations become even more important than they already are when dealing with simplicial sets and (oo,1)-categories. – David Roberts Mar 15 '12 at 0:29
No idea! I think there was something once about defining infinity categories as a set with operations, in a way which included infinity-arrows, but it was an isolated observation. The big question is why? – David Roberts Mar 22 '12 at 3:29
Maybe I'm being dense, but is there any reason to ask this question? It sounds to me like saying "I'm tired of multiplying only finitely many elements in a group, can I have a structure that's like a group but lets me multiply an arbitrary ordinal's worth of elements together?". Would you mind sharing some reasons you think this might be of some interest? (I doubt you'll get many answers unless you do.) – Omar Antolín-Camarena Mar 26 '12 at 2:24
up vote 11 down vote accepted

It seems to me that there should be some not too nebulous motivation for such a question. Rather than answer the question as stated, I'll give an open problem along the same lines. It starts with an old and much neglected paper: Daniel M. Kan Semisimplicial spectra Illinois J. Math. Volume 7 (1963), 463-478. That gives an analogue of based simplicial sets that allows for simplices of negative dimension and is designed to give an alternative definition of spectra. There are problems with the smash product and there are several later papers that flesh out the theory (Kan and Whitehead, Burghelea and coauthors). I would be curious to see how this definition fits into the modern world of spectra. There is a notion of Kan semisimplicial spectrum (opus cit) and it seems very natural to wonder if there is an interesting version of stable quasicategories sitting as a subcategory of the category of Kan semisimplicial spectra.

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