**Title**: On the structure of geometrically injective modules indexed by partially ordered sets

**Speaker**: Crichton Ogle (Ohio State University)

**Abstract**: If $C$ is the categorical representation of a finite partially ordered set, then a $C$-module refers to a covariant functor from $C$ to the category $(\mbox{vect}/k)$ of finite-dimensional vector spaces over a field $k$. When $C$ represents a totally ordered set, one simply recovers the notion of finite persistence module, so this framework represents a natural generalization of persistence modules, and includes all finite multi-dimensional persistence modules.

For a fixed $C$ the moduli space of isomorphism classes of $C$-modules is in general non-discrete, and a fundamental problem - for multi-dimensional persistence modules in particular - has been to find an appropriate generalization of the classification theorem for finite 1-dimensional persistence modules which is nevertheless consistent with the consequences of Gabriel's Theorem.

In these two talks we present such a generalization in the case the $C$-module admits an inner product (is an IPC-module). This is additional structure which occurs naturally in applications; specifically when the module is geometrically injective - it arises as the homology of a $C$-diagram of spaces where the morphisms are all closed cofibrations.

We show that any such IPC-module $M$ admits a weakly tame cover $T(M) \twoheadrightarrow M$, functorial in $M$, from which one can recover the block codes of the module even when the module $M$ itself is not weakly tame. This cover $T(M)$ is the closest approximation to $M$ by a weakly tame $C$-module, in the sense that $M$ is weakly tame iff the projection $T(M) \twoheadrightarrow M$ is an isomorphism. Finally, in the event the indexing category $C$ is holonomy-free, the block summands of $T(M)$ may be further decomposed as a direct sum of generalized bar codes (GBCs). In particular, this is the case when $C$ indexes a finite multi-dimensional persistence diagram.

The paper forming the basis for this talk is linked here: http://arxiv.org/abs/1803.08108

**Seminar URL**: https://research.math.osu.edu/topology/