A note on strong protomodularity, actions and quotients

2001

An action ∗ : V× A−→ Aof a monoidal category V on a category A corresponds to a strong monoidal functor F : V−→ (A, A)into the monoidal category of endofunctors of A. In many practical cases, the ordinary functor f : V−→ (A, A) underlying the monoidal F has a right adjoint g; and when this is so, F

Applied Categorical Structures

We characterize fibrations and *-fibrations in the 2-category of internal groupoids in terms of the comparison functor from certain pullbacks to the corresponding strong homotopy pullbacks. As an application, we deduce the internal version of the Brown exact sequence for *-fibrations from the internal version of the Gabriel-Zisman exact sequence. We also analyse fibrations and *-fibrations in the category of arrows and study when the normalization functor preserves and reflects them. This analysis allows us to give a characterization of protomodular categories using strong homotopy kernels and a generalization of the Snake Lemma. Financial support from FNRS grant 1.A741.14 and from NSERC is gratefully acknowledged by the first author. The second and the third author acknowledge the financial support of the I.N.D.A.M. Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni.

Arxiv preprint math/0607100, 2006

Arxiv preprint arXiv:1107.3850, 2011

Abstract: In this paper we introduce the definition of partial action on small $ k $-categories generalizing the similar well known notion of partial actions on algebras. The point of view of partial action which we use in this paper is the one which was introduced by Exel in his ...

2022

A notion of pentaction of any object in the category $\\mathbf{rGr}^{\\bullet}$ of reduced groups with action is introduced. The operations are defined in the set $\\mathsf{Pentact}(A)$ of pentactions of an object $A$ of $\\mathbf{rGr}^{\\bullet}$. It is proved that if an object $A$ is perfect with zero weak stabilizer in the sense defined in the paper, then $\\mathsf{Pentact}(A)$ is an object of $\\mathbf{rGr}^{\\bullet}$, it has a derived action on $A$, the object $A$ is action representable and $\\mathsf{Pentact}(A)$ represents all actions on $A$.

Theory and Applications of Categories

The existence of the split extension classifier of a crossed module in the category of associative algebras is investigated. According to the equivalence of categories XAss Cat 1 -Ass we consider this problem in Cat 1 -Ass. This category is not a category of interest, it satisfies its all axioms except one. The action theory developed in the category of interest is adapted to the new type of category, which will be called modified category of interest. Applying the results obtained in this direction and the equivalence of categories we find a condition under which there exists the split extension classifier of a crossed module and give the corresponding construction. Second author was supported by Ministerio de Economía y Competitividad (Spain), grant MTM2013-43687-P (European FEDER support included). Third author was partially supported by Shota Rustaveli Science Foundation grant DI/27/5-103/12, D-13/2 "Homological and categorical methods in topology, algebra and theory of stacks" and TUBITAK grant 2221 Konuk veya akademik izinli bilim insanı destekleme programi. The authors are grateful to the referee for his comments and suggestions.

2017

In order to deduce the internal version of the Brown exact sequence from the internal version of the Gabriel-Zisman exact sequence, we characterize fibrations and ∗-fibrations in the 2-category of internal groupoids in terms of the comparison functor from certain pullbacks to the corresponding strong homotopy pullbacks. A similar analysis in the category of arrows allows us to give a characterization of protomodular categories using strong homotopy kernels.

Applied Categorical Structures, 2016

A quasi-pointed category in the sense of D. Bourn is a finitely complete category C having an initial object such that the unique morphism from the initial object to the terminal object is a monomorphism. When instead this morphism is an isomorphism, we obtain a (finitely complete) pointed category, and as it is well known, the structure of zero morphisms in a pointed category determines an enrichment of the category in the category of pointed sets. In this note we examine quasi-pointed categories through the structure formed by the zero morphisms (i.e. the morphisms which factor through the initial object), with the aim to compare this structure with an enrichment in the category of pointed sets.

arXiv (Cornell University), 2008

We consider actions of groups on categories and bimodules, the related crossed group categories and bimodules, and prove for them analogues of the result know for representations of crossed group algebras and categories.

Turkish Journal of Mathematics

In this paper, we introduce the notion of lifting via a homomorphism of monoids for a crossed semimodule and give some properties. Further, we characterize actions and coverings of Schreier internal categories in the category Mon of monoids and prove the natural equivalence between their categories. Then, we prove that liftings of a certain crossed semimodule are naturally equivalent to the actions of Schreier internal category in Mon, where the Schreier internal category corresponds to the crossed semimodule. Finally, we give a relation between crossed semimodules and simplicial monoids.