Combinatorial Cell Complexes: Duality, reconstruction and causal cobordisms

Abstract: This thesis proposes a framework based on a notion of combinatorial cell complex (cc) whose cells are defined simply as finite sets of vertices. The cells of a cc are subject to four axioms involving a rank function that assigns a rank (or a dimension) to each cell. Our framework focuses on classes of cc admitting an inclusion-reversing duality map. We introduce a combinatorial notion of cobordism that allows us to single out a category whose morphisms are cobordisms having a causal structure. Our aim is to offer an approach to look for a combinatorial notion of quantum field theory having a built-in duality operation acting on the underlying space and not relying on any manifold structure. The introduction includes links with fields in Theoretical and Mathematical Physics related to Quantum Gravity and motivating our framework. We start by introducing cc and the duality map on a class of cc with empty boundary called closed cc. We then focus on the problem of reconstructing a certain class of cc from their cells of rank 2 and lower. Such cc are in particular duals to simplicial complexes with no boundary and their reconstruction is realized using a discrete notion of connection. Our next main result extends the duality map we defined on closed cc to a class of cc with boundary. An important by-product of the study of this extended duality map is the combinatorial notion of cobordism used in this work. We also introduce a general notion of subdivision of a cc via a map called reduction, as well as the dual notion of reduction called collapse. These two types of map characterize the structure of certain cc called slices, using sequences of maps called slice sequences. Slices are the basic building blocs of our definition of causal cobordisms and the dual of a slice sequence defines the composition of cobordisms, providing us with a category whose morphisms are causal cobordisms.