Algebra and Number Theory, Simplicial sets; simplicial objects in a category, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), locally finite simplicial complexes, Hochschild cohomology, category of diagrams, simplicial cohomology, (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.), unital algebras, presheaves, cup product, Preorders, orders, domains and lattices (viewed as categories)