An orbifold datum is a collection A of algebraic data in modular fusion category C. It allows one to define new CA construction that generalisation taking the Drinfeld centre category. Under certain simplifying assumptions we characterise terms scalars satisfying polynomial equations and give an explicit expression which computes number isomorphism classes simple objects CA. In Ising-type categ...

We compare two possible ways of defining a category 1-combs, the first intensionally as coend optics and second extensionally quotient by operational behaviour 1-combs on lower-order maps. show that there is full bijective objects functor quotienting intensional definition to extensional one give some sufficient conditions for this be an isomorphism categories. also how constructions can extend...

‎We firstly prove the completeness of the category of crossed modules in a modified category of interest‎. ‎Afterwards‎, ‎we define pullback crossed modules and pullback cat objects that are both obtained by pullback diagrams with extra structures on certain arrows‎. ‎These constructions unify many corresponding results for the cases of groups‎, ‎commutative algebras and can also be adapted to ...

‎In this paper we prove that a simplicial complex is determined‎ ‎uniquely up to isomorphism by its barycentric subdivision as well as‎ ‎its comparability graph‎. ‎We also put together several algebraic‎, ‎combinatorial and topological invariants of simplicial complexes‎.

Martindale proved that under some conditions every multiplicative isomorphism between two rings is additive. In this paper, we extend this theorem to a larger class of mappings and conclude that every multiplicative $(alpha, beta)-$derivation is additive.

In this paper the concurrent semantics of double-pushout (DPO) graph rewriting, which is classically deened in terms of shift-equivalence classes of graph derivations, is axiomatised via the construction of a free monoidal bi-category. In contrast to a previous attempt based on 2-categories, the use of bi-categories allows to deene rewriting on concrete graphs. Thus, the problem of composition ...