2016/11/08 by Samant, Salil, Joshi, Shiv Dutt
#Category Theory (math.CT) #FOS: Mathematics
paper · doi:10.48550/arxiv.1611.02437
In this paper we propose and study few applications of the base structured categories X \rtimesF C, ∫C F, X \rtimes_\mathbbF C and ∫C \mathbbF. First we show classic transformation groupoid X / / G simply being a base-structured category ∫G F. Then using permutation action on a finite set, we introduce the notion of a hierarchy of base structured categories [(X2a \rtimes_\mathbfF2a B2a) \amalg (X2b \rtimes_\mathbfF2b B2b) \amalg ...] \rtimes_\mathbfF1 B1 that models local and global structures as a special case of composite Grothendieck fibration. Further utilizing the existing notion of transformation double category (X1 \rtimes_\mathbfF1 B1) / / 2G, we demonstrate that a hierarchy of bases naturally leads one from 2-groups to n-category theory. Finally we prove that every classic Klein geometry is the Grothendieck completion (G = X \rtimes_\mathbbF H) of \mathbbF: H \xrightarrowF Man∞ \xrightarrowU Set. This is generalized to propose a set-theoretic definition of a groupoid geometry (G,B) (originally conceived by Ehresmann through transport and later by Leyton using transfer) with a principal groupoid G = X \rtimes B and geometry space X = G/B; which is essentially same as G = X \rtimes_\mathbbF B or precisely the completion of \mathbbF: B \xrightarrowF Man∞ \xrightarrowU Set.