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Stream: learning: questions

Topic: "Infinite dimensional" simplicial sets

André Beuckelmann (Jan 08 2021 at 12:45):

Simplicial sets are defined as presheaves on the category of finite ordinals with increasing maps. Now, there is nothing to stop us from, instead, considering all ordinals less than some cardinal $\kappa$ - let us call this category $\Delta_\kappa$ - or, if we don't worry about set theoretic complications, all ordinals (let's call it $\Delta_\infty$) and consider the functors

$\Delta_\kappa^{\text{op}}\to \text{Set}$

which we might imagine as simplicial sets which also have infinite dimensional simplices. Now, if we understand the faces not as a single simplex of dimension one lower, but, instead, as a suitable collection of simplices in all lower dimensions (by including all their faces, and the faces of those, and so on), it seems like we should also be able to define an analogue of the Kan condition for limit ordinals. Further, it seems like we should also be able to define a geometric realisation functor for those objects (unless we are considering $\Delta_\infty$ - then we get set theoretic problems), and I'd suspect many other constructions for ordinary simplicial sets also carry over to this setting.

I was wondering, whether those objects have been studied and whether anyone knows any references to read about them.

John Baez (Jan 08 2021 at 18:37):

I've never read about them, but they sound interesting. The usual $\Delta_\omega$ is the free strict monoidal category on a monoid object, and this accounts for a lot of its importance in cohomology theory: e.g. any comonad gives a simplicial object for this reason. So, it would be interesting to ask: what's the corresponding universal property of $\Delta_\kappa$ for other ordinals $\kappa$?

André Beuckelmann (Jan 08 2021 at 18:53):

If $\kappa$ is a cardinal, you probably get $\Delta_\kappa$ by also requiring your category to be closed under transfinite composition of length at most $\kappa$, I assume, but I'm not sure how useful this property would be. I don't suspect there to be any nice characterisation, if $\kappa$ is not a cardinal, and even less so, if it is not even a limit ordinal.

John Baez (Jan 08 2021 at 19:00):

I don't mind restricting to cardinals! I suspect $\Delta_\kappa$ will only be interesting to the extent we can find interesting universal properties for it. The ability to do transfinite composition of length up to $\kappa$ probably is interesting - or in other words, if someone finds that "too infinite to be interesting", they probably won't be interested in $\Delta_\kappa$.

To get interested in $\Delta_\kappa$, it's probably best to imagine oneself as a person with a transfinite lifetime, for whom transfinite composition seems just as easy as composing a chain of 32 morphisms.

André Beuckelmann (Jan 08 2021 at 19:05):

Whoops, I think I answered a bit too soon: We also have to assume $\kappa$ to be the regular and even then, we get transfinite compositions of length strictly less than $\kappa$ and not at most $\kappa$. I'd suspect that $\Delta_\kappa$ gets more interesting the more nice properties we require $\kappa$ to have - I'd imagine the case with $\kappa$ being some large cardinal - i.e. where $\Delta_\kappa$ behaves almost like the category of all ordinals - to be the most interesting (where we promote $\aleph_0$ to being an honorary large cardinal).

John Baez (Jan 08 2021 at 19:06):

$\aleph_0$ would be a strongly inaccessible cardinal if people hadn't modified the definition of strongly inaccessible cardinal specifically to prevent this.

John Baez (Jan 08 2021 at 19:07):

0 and 1 would also count as strongly inaccessible, I believe.