Aspects of algebraic exponentiation

Abstract

We analyse some aspects of the notion of algebraic exponentiation introduced by the second author [16] and satisfied by the category Gp of groups. We show how this notion provides a new approach to the categorical-algebraic question of the centralization. We explore, in the category Gp, the unusual universal properties and constructions determined by this notion, and we show how it is the origin of various properties of this category.

A characterisation of Lie algebras amongst anti-commutative algebras

“…In comparison, (LACC) is very strong: as mentioned above, we have groups, Lie algebras, crossed modules, and cocommutative Hopf algebras over a field of characteristic zero as "natural" semi-abelian examples, next to all abelian categories. An example of a slightly different kind-because it is non-pointed-is any category of groupoids with a fixed object of objects [8].…”

“…The aim of this article is to prove that, if a variety of anti-commutative algebrasnot necessarily associative, where xx " 0 is an identity-over an infinite field admits algebraic exponents in the sense of James Gray's Ph.D. thesis [17], so when it is locally algebraically cartesian closed (or (LACC) for short, see [19,8]), then it must necessarily be a variety of Lie algebras. Since, as shown in [18], the category Lie K of Lie algebras over a commutative unitary ring K is always (LACC), this condition may be used to characterise Lie algebras amongst anti-commutative algebras.…”

Central reflections and nilpotency in exact Mal’tsev categories

“…The existence of centralisers has far-reaching consequences, as shown by James Gray and the second author, cf. [13,35,36]. Since they are useful for our study of nilpotency, we discuss some of them here.…”

“…A category with the property that for each object Z, the functor (ω Z ) * has a right adjoint is called algebraically cartesian closed [13]. Algebraic cartesian closedness implies canonical isomorphisms (X × Z) + Z (Y × Z) ∼ = (X + Y ) × Z for all objects X, Y, Z, a property we shall call algebraic distributivity, cf.…”

A characterisation of Lie algebras via algebraic exponentiation

“…Recently, in the work of Gray [20,22] and Bourn-Gray [4], a different kind of closedness was considered, which is meant to be more appropriate to non-abelian algebraic contexts [24,13]. Here the cartesian product functor Bˆp´q is replaced by a functor B5p´q which happens not to be induced by a monoidal product.…”