Title: The local structure of modules indexed by small categories
Speaker: Crichton Ogle, OSU
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.
For C-modules with stable local structure, there exists a morphism of multi-flags p:F(T(M)) -> F(M); when the local structure F(M) of M is in general position w.r.t. the morphisms of M, the induced map p_* of associated graded objects is an isomorphism. If, in addition, F(M) is in general position at all of the objects, then p itself is an isomomorphism of multi-flags.
Finally, if the C-module M admits an inner product, then p is induced by a morphism of C-modules p':T(M) -> M (i.e., p = F(p')). In this case T(M) can be viewed as the closest approximation to M by a tame C-module. As before, when F(M) is in general position w.r.t. all morphisms, p' induces an isomorphism of associated graded local structures. In this case, p' induces an isomorphism of C-modules T(M)\cong M iff F(M) is in general position at all objects iff M itself is tame. Thus, in the presence of an inner product, the numerical general position vectors (for the set of morphisms and objects respectively) provide a complete set of discrete numerical invariants to M being tame.
Note: This is part of the Invitations to Mathematics lecture series given each year in Autumn Semester. Pre-candidacy PhD students can sign up for this lecture series by registering for one or two credit hours of Math 6193 with Professor Henri Moscovici.
