范畴论

In this paper we study equivariant crossed modules in its link with strict graded categorical groups. The resulting Schreier theory for equivariant group extensions of the type of an equivariant crossed module generalizes both the theory of…

In this paper we consider the categorical Isbell completion construction for generalized metric spaces in the sense of Lawvere. We show that this is an analogue of the tight span construction of classical metric spaces, and that the Isbell…

Starting from any unital colored PROP $P$, we define a category $P(P)$ of shapes called $P$-propertopes. Presheaves on $P(P)$ are called $P$-propertopic sets. For $0 \leq n \leq \infty$ we define and study $n$-time categorified $P$-algebras…

We develop the theory of distributive inverse semigroups as the analogue of distributive lattices without top element and prove that they are in a duality with those etale groupoids having a spectral space of identities, where our spectral…

We develop a homotopy theoretical version of classical Morita theory using the notion of a strong monad. It was Anders Kock who proved that a monad T in a monoidal category E is strong if and only if T is enriched in E. We prove that this…

As shown in a previous paper by the same authors, the theory of Galois functors provides a categorical framework for the characterisation of bimonads on any category as Hopf monads and also for the characterisation of opmonoidal monads on…

Extending previous work, we define monoidal algebraic model structures and give examples. The main structural component is what we call an algebraic Quillen two-variable adjunction; the principal technical work is to develop the category…

We study the monoidal structure of the standard strictification functor $\textrm{st}:\mathbf{Bicat} \rightarrow \mathbf{2Cat}$. In doing so, we construct monoidal structures on the 2-category whose objects are bicategories and on the…

The category $_{A}\mathbb{S}_{A}$ of bisemimodules over a semialgebra $A,$ with the so called Takahashi's tensor product $-\boxtimes_{A}-,$ is semimonoidal but not monoidal. Although not a unit in $_{A}\mathbb{S}%_{A},$ the base semialgebra…

This dissertation has two main parts. The first part deals with questions relating to Haghverdi and Scott's notion of partially traced categories. The main result is a representation theorem for such categories: we prove that every…

We show that the functor from bialgebras to vector spaces sending a bialgebra to its subspace of primitives has monadic length at most 2.

We introduce a new topos in order to give a semantic account of the nonstandard functional interpretation introduced by Eyvind Briseid, Pavol Safarik and the author.

A full subcategory of a Grothendieck category is called deconstructible if it consists of all transfinite extensions of some set of objects. This concept provides a handy framework for structure theory and construction of approximations for…

Dual monoidal category $\mathcal C^\ast$ of a monoidal functor $F:\mathcal C\to \mathcal V$ has been constructed by S. Majid. In this paper, we extend the construction of dual structures for an Ann-functor $F:\mathcal B\to \mathcal A$. In…

Any $\Gamma$-graded categorical group is determined by a factor set of a categorical group. This paper studies the factor set of the group $\Gamma$ with coefficients in the categorical group of the type $(\Pi,A).$ Then, an interpretation of…

A braided Ann-category $\A$ is an Ann-category $\A$ together with the braiding $c$ such that $(\A, \otimes, a, c, (I,l,r))$ is a braided tensor category, and $c$ is compatible with the distributivity constraints. The paper shows the…

Regular Ann-functor classification problem has been solved with Shukla cohomology. In this paper, we would like to present a solution to the above problem in the general case and in the case of strong Ann-functors with, respectively, Mac…

There are different categorizations of the definition of a {\it ring} such as {\it Ann-category} (see N. T. Quang [6]), {\it ring category} (see M. Kapranov and V.Voevodsky [2]),... The main result of this paper is to prove that every axiom…

In this paper, we have studied the axiomatics of {\it Ann-categories} and {\it categorical rings.} These are the categories with distributivity constraints whose axiomatics are similar with those of ring structures. The main result we have…

In this paper we present some applications of Ann-category theory to classification of crossed bimodules over rings, classification of ring extensions of the type of a crossed bimodule.