Combinatorial Cobordism Theory

TL;DR

This paper develops a combinatorial framework for cobordism theory that allows for a functorial approach to field theories without relying on manifold structures, introducing a new categorical structure with causal and duality features.

Contribution

It introduces a novel combinatorial formalism for cobordism, enabling a functorial definition of field theories in a purely discrete setting without manifolds.

Findings

Defines a combinatorial notion of cobordism with composition operation.

Establishes a bijective correspondence between sequences of maps and higher-dimensional cell complexes.

Creates a category of cobordisms with causal structure and dualities.

Abstract

We introduce a formalism based on a combinatorial notion of cell complex subject to an inclusion-reversing duality operation. Our main goal is to open the way for a functorial definition of field theories in a context where no manifold or topological structure is assumed. This is achieved via a discrete notion of cobordism for which a composition operation is defined. Our main theorem enables the composition of cobordisms by showing that certain sequences of maps between cell complexes are in bijective correspondence with a cell complex of dimension one higher. As a result we obtain a category whose morphisms are cobordisms having a causal structure generalizing that of Causal Dynamical Triangulations as well as dualities inherited from the duality map defined on cell complexes.