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Added the reference:
How is ω−Cat the directed limit of n-categories for finite n? It seems like the result of this process wouldn’t be cocomplete: an object would have to be an n-category for some arbitrarily large n, not an honestly infinite dimensional one. Any particular ω-category is the colimit of its n-skeleta for finite n, but on the level of all ω-categories it seems like we’d want to take the inverse limit of the skeleta-assigning functors (the functor from strict (n+1)-categories to strict n-categories where we discard all the (n+1)-cells). Although I guess since these are the right adjoints of the inclusions, we are technically taking the directed limit in PrL. But that seems too confusing to be the intent.
The claim seems correct to me, as far as I can see. If we take the ∞−Grpd, the category of strict globular ∞-groupoids, and try to use the inclusion functors n−Grpd→∞−Grpd to exhibit it as a colimit of the inclusions n−Grpd→(n+1)−Grpd, then given some category C such that the relevant diagram commutes, and given some strict ∞-groupoid G, to obtain what to do on an n-cell of G to construct a functor ∞−Grpd→C, truncate G to an n-groupoid, and do what the functor n−Grpd→C does on n-cells. Similarly with functors between strict ∞-groupoids.
There’s probably some general result from which this can be derived, e.g. one can probably use some general facts about monads to reduce it to the fact that the category of globular sets is the colimit of the categories of n-truncated globular sets with inclusions between them. And whilst I haven’t checked it carefully, this fact is completely general I think when one has a sequence of fully faithful right adjoints (arising by Kan extension from truncation of the category of globes) which assemble in the way they do here.
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The discussion is about the claim here.
I have touched the typesetting of that formula:
(0Cat≔Set)−↪(1Cat≔Set-Cat)−↪(2Cat≔Cat-Cat)−↪(3Cat≔(2Cat)-Cat≃(Cat-Cat)-Cat)−↪⋯.Because, when one types “$Set-Cat$” then the renderer interprets it as “Set minus Cat” and makes a big minus sign. To instead get the intended hyphen, better to use “\text” and type “$Set\text{-}Cat$”
Re the anonymous comment above:
How does this work if I take C to be the category of all finite dimensional ω-groupoids? For similar reasons I don’t think it’s true that the category of globular sets is the colimit of the n-globular set categories for finite n. Rather the globe category is the colimit of the truncated globe categories, but once we take presheaves (i.e. map out) this should turn into a limit.
I’d think the colimit of a sequence of inclusions of full subcategories would be the full subcategory on the union of their object sets, and in this case that does not give us all ω-categories. Take any chain complex which is nonzero in arbitrarily high degrees, construct an ω-category from this, and you’ll get something which is not an n-category for any finite n
Dear Shamrock, it is not enough to just give C, one needs a whole diagram ;-). If you try to do that, I believe you will either run into trouble or be able to construct the universal functor.
I sort of see what you are getting at, but I do think the claim is correct. In case authority weighs more than my remarks, I am pretty sure that it appears in the literature in numerous places, though I do not have a reference to hand.
I think perhaps your intuition about the passage from globes to presheaves is a little off: the inclusion functor from n-truncated globes to globes induces three functors on the level of presheaves, one of which (the ’skeleton’ functor) behaves essentially exactly the same way as the one on globes. It would be a bit messy to write out formally, but as I tried to suggest, one can see that since has a truncation functor with a fully faithful left (apologies for the typo in my previous comment) adjoint easily constructible from the universal property, an object of the colimit will amount exactly to a specification of an object of n−Grpd for every n compatible with n−Grpd→(n+1)−Grpd inclusions, which basically means the choice of some n-globes for every n, and the same for globular sets.
Similarly with your last paragraph, something is a touch off I think. Any strict ∞-groupoid certainly, for any finite n, is an n-groupoid if one throws away i-globes for iUnknown charactern. It is not isomorphic or equivalent to any such truncation, though. In the first sentence of that paragraph I think “union” is doing a bit too much work (one needs to be careful how to make sense of it outside of material set theory): in general, a rigorous statement is that the colimit will be a quotient of the disjoint union of all of the objects involved by the equivalence relation defined by the inclusions, and again if one thinks about that one will end up with the notion of a strict ∞-groupoid.
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