Related papers: Action representability of the category of interna…

By providing a suitable generalization of Newman's bijective correspondence known for cocommutative Hopf algebras, we prove that the category of cocommutative Hopf monoids in any abelian symmetric monoidal category is semi-abelian, once…

Recently it was shown that the category of cocommutative Hopf algebras over an arbitrary field $\Bbbk$ is semi-abelian. We extend this result to the category of cocommutative color Hopf algebras, i.e. of cocommutative Hopf monoids in the…

We prove that the category 2-$ \mathrm{Grpd}(\mathscr{C}) $ of internal $2$-groupoids is a Birkhoff subcategory of the category $ \mathrm{Grpd}^2(\mathscr{C}) $ of double groupoids in a regular Mal'tsev category $\mathscr{C}$ with finite…

The aim of this paper is to characterize the notion of internal category (groupoid) in the category of Leibniz algebras and investigate the properties of well-known notions such as covering groupoid and groupoid operations (actions) in this…

In this paper, for given an algebraic theory $T$ whose category $C$ of models is semi-abelian, we consider the topological models of $T$ called topological $T$-algebras and obtain some results related to the fundamental groups of…

The purpose of this article is to prove that the category of cocommutative Hopf $K$-algebras, over a field $K$ of characteristic zero, is a semi-abelian category. Moreover, we show that this category is action representable, and that it…

We study the categorical-algebraic properties of the semi-abelian variety $\ell \mathbb{G}rp$ of lattice-ordered groups. In particular, we show that this category is fiber-wise algebraically cartesian closed, arithmetical, and strongly…

Let $G$ and $A$ be objects of a finitely cocomplete homological category $\mathbb C$. We define a notion of an (internal) action of $G$ of $A$ which is functorially equivalent with a point in $\mathbb C$ over $G$, i.e. a split extension in…

The notion of a group G acting on a group X is well-known. Fixing X, the corresponding functor Act(-,X) is representable by the group [X] of automorphisms of X. The notion of G-action on X has been generalized to the context of a…

The concept of a pair of compatible actions was introduced in the case of groups by Brown and Loday and in the case of Lie algebras by Ellis. In this article we extend it to the context of semi-abelian categories (that satisfy the…

We prove that the category of cocommutative Hopf algebras over a field is a semi-abelian category. This result extends a previous special case of it, based on the Milnor-Moore theorem, where the field was assumed to have zero…

We give an explicit description of internal actions in the semi-abelian categories of pro-groups and non-unital pro-rings in terms of actions of group objects and ring objects in $\mathrm{Pro}(\mathbf{Set})$, as well as in some related…

We prove that commutative algebras in braided tensor categories do not admit faithful Hopf algebra actions unless they come from group actions. We also show that a group action allows us to see the algebra as the regular algebra in the…

Derived actions in the category of groups with action on itself $\mathbf{Gr}^{\bullet}$ are defined and described. This category plays a crucial role in the solution of Loday's two problems stated in the literature. A full subcategory of…

We establish rigidity for partial transformation groupoids associated with algebraic actions of semigroups: If two such groupoids (satisfying appropriate conditions) are isomorphic, then the globalizations of the initial algebraic actions…

We describe the envelope C*-algebra associated to a partial action of a countable discrete group on a locally compact space as a groupoid C*-algebra (more precisely as a C*-algebra from an equivalence relation) and we use our approach to…

In this paper, we investigate *-homomorphisms between C*-algebras associated to \'etale groupoids. First, we prove that such a *-homomorphism can be described by closed invariant subsets, groupoid homomorphisms and cocycles under some…

We construct an abelian category C and exact functors in C which on the Grothendieck group descend to the action of a simply-laced quantum group in its adjoint representation. The braid group action in the adjoint representation lifts to an…

The aim of this paper is to prove the statement in the title. As a by-product, we obtain new globalization results in cases never considered before, such as partial corepresentations of Hopf algebras. Moreover, we show that for partial…

Suppose that $G$ is a groupoid acting on a small category $H$ in the sense of \cite[Definition 4]{NOT} and $H\times_\alpha G$ is the resulting semi-direct product category (as in \cite[Proposition 8]{NOT}). We show that there exists a…