Open topological field theories and 2-segal objects

In this thesis we analyze 2-dimensional open topological field theories in both 1-categorical and ∞-categorical contexts. Making use of the formalism, introduced by Dyckerhoff and Kapranov, of graphs structured over a crossed simplicial group ∆G, we give combinatorial models for 2-dimensional open cobordism categories with additional structure — orientations, N-spin structures, etc. We then use this model to effect a classification of the corresponding classes of 1-categorical topological field theories. This classification retrieves, in special cases, a number of results known in the literature, as well as providing new results. We then turn to 2-dimensional open oriented topological field theories valued in an ∞-category Span(C) of spans in an ∞-category C. Applying a theorem stated by Lurie in [33], such topological field theories are classified by Calabi-Yau algebras in Span(C). We define two 1-categories whose functors to C parameterize, respectively, associative algebras and Calabi-Yau algebras in Span(C). We prove that there is an equivalence of ∞-categories between associative algebras in Span(C) and 2-Segal simplicial objects in C; and we prove an equivalence of ∞-categories between Calabi-Yau algebras in Span(C) and 2-Segal cyclic objects in C. We discuss the invariants the resultant topological field theories assign to surfaces, and develop the example provided by cyclic structures on Čech nerves