We consider algebras and Frobenius algebras, internal to a monoidal category, that are graded over a finite abelian group. For the case that A is a twisted group algebra in a linear abelian monoidal category we obtain a graded generalization of the Schur Lemma for the category of induced A-modules. We further show that if the monoidal category is braided and A is commutative up to a bicharacter of the grading group, then the category of induced A-modules can be endowed with a graded-monoidal structure that is twisted by the bicharacter. In the particular case that the grading group is ℤ/2⁢ℤ, these findings reproduce known results about superalgebras and super-monoidal structures.

The purpose of a proof is to reduce the complexity of a statement until it becomes a sequence of trivialities. To this end, the choice of notation, diagrams and overall paradigm can aid in conveying large amounts of information in a simple manner. This compilation thesis focuses on the choice of visual tools to convey algebraic results in the context of mathematical physics, using a categorical paradigm with various topological semantics. The topics range from covering known results in knot theory, abstract diagram categories and low-dimensional topological quantum field theory, to novel results such as the topological rack exclusion principle, tetrahedral symmetry of framed associators and new diagrammatics for graded-monoidal categories based on the Kleisli presentation.We demonstrate how these diagrammatic methods can be used to simplify algebraic proofs and communicate across disciplines.

We establish tetrahedral symmetries of 6j-symbols for arbitrary fusion categories under minimal assumptions. As a convenient tool for our calculations we introduce the notion of a veined fusion category, which is generated by a finite set of simple objects but is larger than its skeleton. Every fusion category C contains veined fusion subcategories that are monoidally equivalent to C and which suffice to compute many categorical properties for C. The notion of a veined fusion category does not assume the presence of a pivotal structure, and thus in particular does not assume unitarity. We also exhibit the geometric origin of the algebraic statements for the 6j-symbols. 

We define and explore rack objects internal to categories with products. In demonstration, we classify the group-racks, and use homotopy to prove both existence and exclusion theorems for path-connected topological racks.

In this paper we study the distribution of level crossings for the spectra of linear families A + lambda B, where A and B are square matrices independently chosen from some given Gaussian ensemble and lambda is a complex-valued parameter. We formulate a number of theoretical and numerical results for the classical Gaussian ensembles and some of their generalisations. Besides, we present intriguing numerical information about the monodromy distribution in case of linear families for the classical Gaussian ensembles of 3 x 3-matrices.