Ordinal category theory? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T22:53:37Z http://mathoverflow.net/feeds/question/75610 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/75610/ordinal-category-theory Ordinal category theory? Martin Brandenburg 2011-09-16T15:26:04Z 2012-05-13T11:36:25Z <p>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.</p> http://mathoverflow.net/questions/75610/ordinal-category-theory/96829#96829 Answer by Buschi Sergio for Ordinal category theory? Buschi Sergio 2012-05-13T11:36:25Z 2012-05-13T11:36:25Z <p>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 $n$. </p>