Just out of curiosity: Is there a notion of $\alpha$-category for an ordinal number $\alpha$, extending the given notions for $\alpha \leq \omega$? If there is none, which one would you propose? Feel free to draw images.
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asked Sep 16, 2011 at 15:26
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3$\begingroup$ One can also dream of a notion of x-category for any real (or whatever) number x. $\endgroup$– Jonathan ChicheCommented Sep 16, 2011 at 16:56
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$\begingroup$ I imagine that the difficulty would be in what you must do for limit ordinals. $\endgroup$– Spice the BirdCommented Sep 17, 2011 at 5:40
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1$\begingroup$ Here is an idea: giving a higher category is equivalent so giving its nerve, which is a simplicial that satisfies some horn filling condition. Now one could define $\alpha$-categories by taking "huge simplical sets" which are indexed not only by the finite ordinals but by ordinals smaller than $\alpha$ and again impose some horn filling condition. $\endgroup$– Jan WeidnerCommented Sep 18, 2011 at 11:28
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$\begingroup$ @Jan: I don't think that would work. That would give you a notion of an $(\alpha,1)$-category. The simplicial models of Verity and Street for (even strict) higher categories with noninvertible higher cells make use of thin cells, which are "marked" cells in a stratified simplicial set. $\endgroup$– Harry GindiCommented Sep 20, 2011 at 10:53
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$\begingroup$ This question seems similar to the question I asked here: mathoverflow.net/questions/91213/… $\endgroup$– Samuel ReidCommented May 14, 2012 at 0:59
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1 Answer
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Let $\Delta_2$ the 2-tronked simplicial category (objects are the finite orders $0=$(0), $1=$(0,1), $2=$(0, 1, 2) and order-preserving functions). Then $Cat$ (category of small categories)is isomorphically to $Fun_c(\Delta_2^{op}, Set)$ (finite limits preserving functors and natural transformations). More in general the (hyper)category of n-double small categories $n$-$Double$ is isomorphically to $Fun_c((\Delta_2^{op}\times\ldots\times\Delta_2^{op}), Set)$ (n fold products), and $n$-$Cat$ is a subcategory of $n$-$Double$ (elements of $Fun_c((\Delta_2^{op}\times\ldots\times\Delta_2^{op}), Set)$ that send some morphisms to identities). Of course you can generalize this for a infinite cardinal $n$.
answered May 13, 2012 at 11:36
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$\begingroup$ Thanks! Can you give a reference for the isomorphism $\mathrm{Cat} \cong \mathrm{Fun}_c(\Delta^{\mathrm{op}},\mathrm{Set})$? The functor is probably the nerve, which is fully faithful, but I haven't come across this description of the image. $\endgroup$ Commented May 13, 2012 at 17:04
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2$\begingroup$ Its a old folklore property, for example P.Johnstone "Topos theory" p.48, remark. 2.13. That double categories are Cat-object (internal categories) inside Cat (this last is the category of Cat object come from JW Gray, Formal category theory Lnm 391, or R. Brown article "Double categorie, 2.categories, thin-structures, connections" (is mentioned as a result, no more hard than a nice exercise). $\endgroup$ Commented May 13, 2012 at 17:52
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$\begingroup$ YOu can find a more simple algebraic description of $\omega$-category (stright generalizable to infinite cardinals) in R. Street "The algebra of oriented simplexes" J. Pure Appl. Alg.49 (math.mq.edu.au/~street/aos.pdf) $\endgroup$ Commented May 13, 2012 at 17:53