Braided monoidal categories and Hopf algebras have applications for invariants in knot theory and 3-dimensional manifolds. The classical results involving the relationship between k-bialgebras (quasi-triangular k-bialgebras) and monoidal categories (braided monoidal categories) have been known for some time. Motivated by problems in the deformation of Witt algebras Jonas T. Hartwig, Daniel Larsson, and Sergei D. Silvestrov introduced hom-Lie algebras in 2006. Over the last decade, many hom-associative algebraic structures and their properties were established.

This dissertation addresses the categorical settings of hom-associative algebras analogous to the aforementioned classical results. To facilitate this objective we first introduce a new type of category called a hom-tensor category (4.1.1) and show that it provides the appropriate categorical framework for modules over a hom-bialgebra (4.1.16). Next we introduce the notion of a hom-braided category (4.2.3) and show that this is the right categorical setting for modules over quasitriangular hom-bialgebras (4.2.5). We also prove that, under certain conditions, one can obtain a pre-tensor category (respectively a quasi-braided category) from a hom-tensor category (respectively a hom-braided category) and explain how the hom-Yang-Baxter equation fits into the framework of hom-braided categories. Finally, we show how the category of Yetter-Drinfeld modules over a hom-bialgebra with a bijective structural map can be organized as a hom-braided category and discuss some open questions.