In an abelian category, a commutative quadrangle is called bicartesian if its diagonal sequence is short exact, i.e. if it is a pullback and a pushout. A commutative quadrangle is bicartesian if and only if we get induced isomophisms on the horizontal kernels and on the horizontal cokernels. In a triangulated category in the sense of Verdier [3, Def. 1-1], a commutative quadrangle is called hom...

We construct extended TQFTs associated to Rozansky--Witten models with target manifolds $T^*\mathbb{C}^n$. The starting point of the construction is 3-category whose objects are such models, and morphisms defects all codimensions. By truncation, we obtain a (non-semisimple) 2-category $\mathcal{C}$ bulk theories, surface defects, isomorphism classes line defects. Through systematic application ...

‎This paper develops a basic theory of $H$-groups‎. ‎We‎ ‎introduce a special quotient of $H$-groups and‎ ‎extend some algebraic constructions of topological groups to the category‎ ‎of H-groups and H-maps and then present a functor from this category to the category of quasitopological groups‎.

We consider presentations that were derived in [3] for the interval groups associated with proper quasi-Coxeter elements of Coxeter group W(Dn). use combinatorial methods to derive alternative groups, and these new show a element W(Dn) cannot be isomorphic Artin type Dn. While specific problems we solve arise from study their solution provides an illustration how techniques indicated by computa...

We establish an explicit comparison between two constructions in homotopy theory: the left adjoint of coherent nerve functor, also known as rigidification and Kan loop groupoid functor. This is achieved by considering localizations unraveling a construction Hinich, using sequence operators originally introduced Szczarba 1961. As result, we obtain several combinatorial models for path category s...

We investigate the relations of almost isometric embedding and of almost isometry between metric spaces. These relations have several appealing features. For example, all isomorphism types of countable dense subsets of R form exactly one almost-isometry class, and similarly with countable dense subsets of Uryson’s universal separable metric space U. We investigate geometric, set-theoretic and m...

The features of a usual conformal field theory were captured by the formalism of a tensor or fusion category. The notion of a Verlinde algebra which arises from a fusion category, the action of a modular group on the algebra, and the Verlinde formula gave us beautiful means to study these theories. Such they were that in [Ki] Kirillov put forth the notion of a G-equivariant fusion category so t...

Abstract We compare two associative algebras which encode the “quantum topology” of Legendrian curves in contact threefolds product type $$S\times {\mathbb {R}}$$ S × R . The first is skein algebra graded links and second Hall Fukaya category S construc...

Abstract Consider a connected manifold of dimension at least two and the group compactly supported diffeomorphisms that are isotopic to identity through isotopy. This acts n -transitively: any -tuple points can be moved other by an element this group. The orbifold is typically not -transitive: simple obstructions given isomorphism classes isotropy groups points. In paper we investigate transiti...