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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Let $\Delta_2$ the 2tronked simplicial category (objects are the finite orders $0=$(0), $1=$(0,1), $2=$(0, 1, 2) and orderpreserving 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 ndouble 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 $n$. 

