Unified Functorial Signal Representation II: Category action, Base Hierarchy, Geometries as Base structured categories · arXivDesk
1611.02437Nov 8, 20161.Notations revised to reflect the difference in abstract and concrete category actions. 2.Both Category and set-theoretic versions of the definition of groupoid geometries made explicit
Unified Functorial Signal Representation II: Category action, Base Hierarchy, Geometries as Base structured categories
In this paper we propose and study few applications of the base structured categories X⋊FC, ∫CFˉ
Nearby in the stack
,
X⋊FC
and
∫CFˉ
. First we show classic transformation groupoid
X//G
simply being a base-structured category
∫GFˉ
. Then using permutation action on a finite set, we introduce the notion of a hierarchy of base structured categories
[(X2a⋊F2aB2a)⨿(X2b⋊F2bB2b)⨿...]⋊F1B1
that models local and global structures as a special case of composite Grothendieck fibration. Further utilizing the existing notion of transformation double category
(X1⋊F1B1)//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⋊FH
) of
F:HFMan∞USet
. 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