In a recent paper, motivated by the study of central extensions of associative algebras, George Janelidze introduces the notion of weakly action representable category. In this paper, we show that the category of Leibniz algebras is weakly action representable and we characterize the class of acting morphisms. Moreover, we study the representability of actions of the category of Poisson algebras and we prove that the subvariety of commutative Poisson algebras is not weakly action representable.

C-systems were defined by Cartmell as models of generalized algebraic theories. B-systems were defined by Voevodsky in his quest to formulate and prove an initiality conjecture for type theories. They play a crucial role in Voevodsky’s construction of a syntactic C-system from a term monad. In this work, we construct an equivalence between the category of C-systems and the category of B-systems, thus proving a conjecture by Voevodsky.

Just like group actions are represented by group automorphisms, Lie algebra actions are represented by derivations: up to isomorphism, a split extension of a Lie algebra $B$ by a Lie algebra $X$ corresponds to a Lie algebra morphism $B\to {\mathit {Der}}(X)$ from $B$ to the Lie algebra ${\mathit {Der}}(X)$ of derivations on $X$. In this article, we study the question whether the concept of a derivation can be extended to other types of non-associative algebras over a field ${\mathbb {K}}$, in such a way that these generalized derivations characterize the ${\mathbb {K}}$-algebra actions. We prove that the answer is no, as soon as the field ${\mathbb {K}}$ is infinite. In fact, we prove a stronger result: already the representability of all abelian actions – which are usually called representations or Beck modules – suffices for this to be true. Thus, we characterize the variety of Lie algebras over an infinite field of characteristic different from $2$ as the only variety of non-associative algebras which is a non-abelian category with representable representations. This emphasizes the unique role played by the Lie algebra of linear endomorphisms $\mathfrak {gl}(V)$ as a representing object for the representations on a vector space $V$.

Given a partial action $\unicode[STIX]{x1D703}$ of a group on a set with an algebraic structure, we construct a reflector of $\unicode[STIX]{x1D703}$ in the corresponding subcategory of global actions and study the question when this reflector is a globalization. In particular, if $\unicode[STIX]{x1D703}$ is a partial action on an algebra from a variety $\mathsf{V}$, then we show that the problem reduces to the embeddability of a certain generalized amalgam of $\mathsf{V}$-algebras associated with $\unicode[STIX]{x1D703}$. As an application, we describe globalizable partial actions on semigroups, whose domains are ideals.

Our main result implies that for any choice 1 ≤ m ≤ n ≤ p of integers there exist finitary algebras A1 and A2 that generate the same variety, and such that the initial k-segments of their centralizer clones coincide exactly when k ≤ m, are isomorphic exactly when k ≤ n and are elementarily equivalent exactly when k ≤ p. The proof uses the existence and properties of disciplined topological spaces which we introduce and investigate here.

We give a revised and updated exposition of the theory of full dualities initiated by Clark, Davey, Krauss and Werner, introducing the (stronger) notion of a strong duality. All known full dualities turn out to be strong. A series of theorems which provide necessary and sufficient conditions for a strong duality to exist is proved. All full dualities in the literature can be obtained from these results and many new strong dualities can be derived. In particular, we show that within congruence distributive varieties every duality can be upgraded to a strong duality. Amongst the new strong dualities are the dualities of Davey, Priestley and Werner for the varieties of pseudocomplemented distributive lattices which are either strong as they stand or can easily be made strong by the addition of partial operations to the dual structures.