Combinatorial Cobordism Theory

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.