Homology of Small Categories and Its Applications
Open AccessMotivated by knot theory, we introduce a homology theory for small categories with functor coefficients. Under this general framework, different familiar homology theories such as group homology, chromatic homology, poset homology, and Khovanov homology can be realized as homology of small categories for which coefficients are specified functors. For the category of an abstract simplicial complex, we define chain groups via two different approaches and prove that these two definitions are equivalent in the sense that homology groups under these two definitions are isomorphic via an interpretation of barycentric subdivision. As one of its applications, we develop cohomology theory for quivers (multi-digraphs). We introduce quiver cohomology for noncommutative algebras motivated by Wagner and Turner's work. We analyze and speculate on properties of the quiver cohomology groups via some calculations.I will further investigate on how to generalize homology of distributive structures to Yang-Baxter homology. Motivated by Przytycki's one-term distributive homology and Carter's homology of set-theoretic Yang-Baxter operators, Yang-Baxter homology is introduced here as their generalizations. Some promising future research directions may include searching for connections between Yang-Baxter homology and Khovanov homology (Khovanov-Rozansky homology).
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