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Is there a reason we consider $\infty$-categories to be the $\omega^{th}$ step in the 2-internalization inside Cat (or enrichment over Cat if you prefer)* process made invertible above some finite ordinal, and don't continue on to higher steps in the recursion? Is there nothing to be gained, or is the $\omega^{th}$ step already mysterious enough that going further is foolhardy?

For example, it seems (very naively) that something like a $(\omega_1,\omega)$-category or higher categories defined up to large cardinals that become invertible at smaller large cardinals might be interesting, or in a $\neg CH$ universe we could ask about $(\omega_1,\mathfrak{c})$-categories and the like. My apologies if this question is trivial, but I couldn't find a discussion/explanation in the literature.


*This is an incorrect characterization of how to arrive at a 'fully weak' $\infty$-category (thanks Mike for catching the error), and it appears as though it's an open question wether we can give an algebraic definition of a fully weak $\infty$-category. For details on how to correctly iterate internalization to arrive at a correct definition for weak $n$-categories for all $n$, see this excellent paper by Simona Paoli.

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    $\begingroup$ At least for me the reason I care about $(\omega,n)$-categories is that they are the one which show up in practice. But I'd love to see examples of more exotic things around! $\endgroup$ – Denis Nardin Feb 28 at 22:19
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    $\begingroup$ I was thinking about this at some point, and I had some thoughts (although I might be completely wrong). The numbers appearing can be seen as dimensions of cells, and infinite-dimensional sphere is contractible. So it seems, at least from that naive perspective, that infinite ordinals will contribute no homotopical information. $\endgroup$ – Shay Ben Moshe Feb 28 at 22:59
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    $\begingroup$ A different generalization that I think is interesting is the notion of "$\mathbb{Z}$-category", in which negative (but finite) dimensional cells are also allowed. In particular, a "$\mathbb{Z}$-groupoid" is, or should be, the same as a spectrum in the sense of homotopy theory. $\endgroup$ – Mike Shulman Mar 1 at 0:23
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    $\begingroup$ @MikeShulman Very interesting! My intuition in response to Harry was that the Grothendieck ring of a $\delta$-number ordinal (or all of $O_n$) has predecessors for all limit ordinals and so might admit a better definition but I thought it was far fetched, but $\mathbb{Z}$ is the Grothendieck ring of $\omega$, so maybe this generalization could also make sense? $\endgroup$ – Alec Rhea Mar 1 at 0:30
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    $\begingroup$ @AlecRhea I think you should probably start by trying to think of examples, and then, if you find any, formulate the definition so as to describe them. $\endgroup$ – Mike Shulman Mar 1 at 1:03
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Using Street's "one type" definition of strict $\infty$-category one can see that the concept of "strict $P$-category" makes sense not justs for any ordinals $P$ but in fact for any posets $P$ (though I expect the definition below is not right when $P$ is not totally ordered, see the remark at the end)

Definition : a $P$-category is a set $X$ endowed with:

  • (source and target) For each $p \in P$, two functions $\pi_p^+$ and $\pi_p^{-}$ from $X$ to $X$.

  • (composition) For each $p \in P$ a partially defined composition operation $\#_p : X \times X \rightarrow X$.

Satisfying the following conditions:

  • (globularity relations) For any $q \leqslant p$ one has $ \pi_p^{\mu} \pi_q^{\epsilon} = \pi_q^{\epsilon}$ and if $q>p$ one has $ \pi_p^{\mu} \pi_q^{\epsilon} = \pi_p^{\mu}$.

  • (dimension axiom) For any $x \in X$, there exists $p \in P$ such that $\pi^+_p(x)=x$.

  • (domain of composition) $x \#_p y$ is defined if and only if $\pi^+_p x= \pi^-_p y$.

  • (boundaries of compositions) $\pi_p^+(x \#_p y)=\pi_p^+ y$ ; $\pi_p^-(x \#_p y)=\pi_p^- x$ and $\pi^{\epsilon}_q(x \#_p y) = \pi^{\epsilon}_q(x) \#_p \pi^{\epsilon}_q(y)$ if $q>p$.

  • (unit law) For any $x \in X$ and any $n$ one has $x \#_n \pi_n^+ x= x = \pi^-_n x \#_n x$.

  • (associativity of compositions) $(x \#_p y )\#_p z = x \#_p (y \#_p z)$ when either side is defined.

  • (exchange law) for $p< q$, $(w \#_q x) \#_p (y \#_q z) = (w \#_p y) \#_q (x \#_p z)$ when the left hand side is defined.

(where $\epsilon$ and $\mu$ denotes arbitrary signs)

Saying that a cell is ``invertible'' makes sense exactly as in strict $\infty$-category, so you can talk about $(\alpha,\beta)$ categories for any ordinals $\alpha$ and $\beta$.

Now:

  • So far we are lacking motivation (interesting examples) to develop such a theory.

  • weak version of this notion have not been defined. Moving from strict $\infty$-categories to weak $\infty$-categories is a difficult jump, so there might be a lot of work to came up with such a notion. And as I said we don't really have a motivation to do so.

  • Even in the strict case, the "homotopy theory" of such object has not been developed (I'm talking about an analogue of the Folk model structure). This is probably easier than the point above, but still would need to be done.

As I said we are lacking examples, so it is not clear this notion has any interest at all, but I don't think the notion is trivial in any sense, and actually we do have a few notable examples beyond $\infty$-categories:

For example a $2 \omega$-category with only one cell of dimension $<\omega$ is exactly the same as a strict $\infty$-category with a commutative monoide structure, and I expect that the weak version should be a $E_{\infty}$-monoidal $\infty$-category in the same way that a weak $\infty$-category with only one cell of dimension $<k$ is the same as a $E_{k}$-monoidal $\infty$-category

As mentioned by Mike above, $\mathbb{Z}$-groupoids are closely connected to spectra. I believe using a similar argument to the Dold-Kan equivalence for strict $\infty$-category one can show that a strict $\mathbb{Z}$-groupoids is the same as an unbounded chain complexes (not quite sure about this... maybe there are problems in $-\infty$ and only a special type of $\mathbb{Z}$-groupoids needs to be considered). If true this is definitely a good indicator that a weak $\mathbb{Z}$-groupoids should be a spectra.

(In fact I remember seeing an arxiv preprint using $\mathbb{Z}$-categories to model spectra or something in this spirit... but I did not really read it and I havn't been able to find it back.)

To me the observation about infinite dimensional sphere is not really relevant here: to me what it says is that spaces don't have information past $\omega$, but this is just the fact that spaces up to homotopy are $\omega$-groupoids and not $\omega^+$-groupoids or anything like this.

Also if I'm correct the $\omega^+$-category freely generated by an arrows $\theta$ of dimension $\omega$ has for arrows $\theta$ and all the $\pi^{\epsilon}_n \theta$ for $ n < \omega$ and $\epsilon=+/-$ and that they are all different, so the type of collapse Harry mentioned in his answer do not seem to happen, at least with this definition.

Finally, I believe the definition given here is only right when $P$ is totally ordered, but I suspect that there is a modification of this definition, which is equivalent for totally ordered set, and such that for example a $\mathcal{P}(\{1,\dots,n\})$-category is the same as an $n$-fold category.

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    $\begingroup$ Fascinating, much appreciated Simon! Should the functions $\pi^+_p$ and $\pi^-_p$ be from $X$ to $X$ though? $\endgroup$ – Alec Rhea Mar 2 at 16:03
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    $\begingroup$ Oh yes absolutely, and the composition are on X as well ! $\endgroup$ – Simon Henry Mar 2 at 16:36
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Because what we today call $(\infty,n)$-categories are really homotopy-coherent strict $n$-categories. It is an open question if they are equivalent to some of the algebraic definitions of weak higher categories where you could try to formalize this.

The only case I've ever looked into was Street's strict $\omega^+$-categories, and the additional layer ends up forcing some things to be trivial, so it was concluded that this is not a natural or useful model.

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  • $\begingroup$ Thank you, but if I understand the nlab article linked above correctly the notion of strict $\omega$-categories is what you get if you internalize at the $1$-categorical level repeatedly, but if you internalize inside a $2$-category with prior knowledge of $2$-cells you can loosen the strict horizontal equalities internalization usually gives to isos as desired and correctly iterate the recursion all the way to $\omega$ to get the same thing as the homotopy-based definition (if you also add 'completeness' to guarantee that horizontal and vertical composition 'play well'). What am I missing? $\endgroup$ – Alec Rhea Feb 28 at 22:40
  • $\begingroup$ @AlecRhea An infinite dimensional cell must have the same n-source and n-target for all n. $\endgroup$ – Harry Gindi Feb 28 at 22:45
  • $\begingroup$ Ah, so the issue is that $n$-cells usually map between $n-1$ cells but $\omega$ has no predecessor so the algebraic definition goes wonky at the first limit ordinal stage? Are you taking an $\omega$-cell to be a collection of $n$-cells where $n$ ranges over $\omega$? $\endgroup$ – Alec Rhea Feb 28 at 22:54
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    $\begingroup$ @MikeShulman At least part of the discussion was at the end of this long thread nforum.ncatlab.org/discussion/3185/… Unfortunately I think another part was on the cat theory mailing list. $\endgroup$ – Harry Gindi Mar 1 at 1:17
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    $\begingroup$ @HarryGindi I don't see anything in that discussion that leads me to believe that the source and target of an $\omega$-cell must be the same. $\endgroup$ – Mike Shulman Mar 2 at 5:33
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The numbers that appear in terms such as $n$-categories, are the dimensions of cells. However, infinite-dimensional spheres are contractible. Therefore, at least from that perspective, it seems that infinite ordinals will contribute no homotopical information.

This of course assumes that we are interested in homotopy-theoretic information. It is plausible that one may alter the definitions, or be interested in something different altogether, and then infinite ordinals will give new information.

(this is a rewrite of my comment above)

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  • $\begingroup$ Thank you for posting this as an answer Shay. If nobody (myself included) comes up with an interesting definition using infinite ordinals in the next few days I'll accept this answer since it explains why things become trivial as mentioned in Harry's answer, and the homotopy reason is compelling. I would however like to leave open the possibility of someone coming along and posting an interesting definition here, so I reserve the right to move the acceptance check to a new answer of that type should one arrive. (even if you post it, but then you could edit this post and save me 15 points ;) $\endgroup$ – Alec Rhea Mar 2 at 7:45

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