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This thesis proposes a framework based on a notion of combinatorial cellcomplex (cc) whose cells are defined simply as finite sets of vertices. Thecells of a cc are subject to four axioms involving a rank function that assignsa rank (or a dimension) to each cell. Our framework focuses on classes of ccadmitting an inclusion-reversing duality map. We introduce a combinatorialnotion of cobordism that allows us to single out a category whose morphisms arecobordisms having a causal structure. Our aim is to offer an approach to lookfor a combinatorial notion of quantum field theory having a built-in dualityoperation acting on the underlying space and not relying on any manifoldstructure. The introduction includes links with fields in Theoretical andMathematical Physics related to Quantum Gravity and motivating our framework.We start by introducing cc and the duality map on a class of cc with emptyboundary called closed cc. We then focus on the problem of reconstructing acertain class of cc from their cells of rank 2 and lower. Such cc are inparticular duals to simplicial complexes with no boundary and theirreconstruction is realized using a discrete notion of connection. Our next mainresult extends the duality map we defined on closed cc to a class of cc withboundary. An important by-product of the study of this extended duality map isthe combinatorial notion of cobordism used in this work. We also introduce ageneral notion of subdivision of a cc via a map called reduction, as well asthe dual notion of reduction called collapse. These two types of mapcharacterize the structure of certain cc called slices, using sequences of mapscalled slice sequences. Slices are the basic building blocs of our definitionof causal cobordisms and the dual of a slice sequence defines the compositionof cobordisms, providing us with a category whose morphisms are causalcobordisms.

Combinatorial Cell Complexes: Duality, reconstruction and causal cobordisms