topoidal

Towards a Topos for Persistence

Joao Pita Costa * and Mikael Vejdemo-Johasson * ** and Primož Škraba *

A topos theoretic approach to set theory permits ideas like time variable sets and provides tools for unification of techniques for mathematics having had a great importance in the recent developments of Quantum Theory. Persistent homology is a central tool in topological data analysis, which examines the structure of data through topological structure. The basic technique is extended in many different directions, permuting the encoding of topological features by barcodes and correspondent persistence diagrams. The set of points of all such diagrams determines a complete Heyting algebra that can explain aspects of the relations between correspondent persistence bars and provide a global perspective over this approach. We are fundamentally interested in the algebraic foundations of applied and computational algebraic topology, in particular in a unifying theory for the various flavors of persistent homology that have emerged so far. In this poster we shall look at the topos of sheaves over such algebra, discuss its construction and potential for a generalized simplicial homology over it.

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* Institute Jozef Stefan, Slovenia

** KTH, Sweden