Title: The local structure of modules indexed by small categories
Speaker: Crichton Ogle - The Ohio State University
Abstract: For a (small) category C, a C-module (over a field k) refers to a covariant functor C-> (fin. dim. v.s./k). Associated to any C-module is a bi-closed multi-flag F(M) (a concept we will introduce in the talk) referred to as its local structure. In most cases of interest (e.g., if C is any finite category and k a finite field), M has stable local structure. From this structure one is able to recover (in a basis-free manner) the "blocks" of M indexed on the set of admissible subcategories of C, whose direct sum comprises the tame cover T(M) of M, also a C-module. This tame cover exists regardless of whether or not M itself is tame, and equals M when it is. These blocks may be further decomposed as a sum of generalized bar-codes when the nerve N(C) is simply-connected. In the very special case C is the categorical representation of a finite totally ordered set, one recovers the interval submodule decomposition of a finite persistence module.