2,112 research outputs found
Algebraic exponentiation and internal homology in general categories
Includes bibliographical references (p. 101-102).We study two categorical-algebraic concepts of exponentiation:(i) Representing objects for the so-called split extension functors in semi-abelian and more general categories, whose familiar examples are automorphism groups of groups and derivation algebras of Lie algebras. We prove that such objects exist in categories of generalized Lie algebras defined with respect to an internal commutative monoid in symmetric monoidal closed abelian category. (ii) Right adjoints for the pullback functors between D. Bourns categories of points. We introduce and study them in the situations where the ordinary pullback functors between bundles do not admit right adjoints in particular for semi-abelian, protomodular, (weakly) Maltsev, (weakly) unital, and more general categories. We present a number of examples and counterexamples for the existence of such right adjoints. We use the left and right adjoints of the pullback functors between categories of points to introduce internal homology and cohomology of objects in abstract categories
A characterisation of Lie algebras amongst anti-commutative algebras
Let be an infinite field. We prove that if a variety of
anti-commutative -algebras - not necessarily associative, where
is an identity - is locally algebraically cartesian closed, then it must
be a variety of Lie algebras over . In particular,
is the largest such. Thus, for a given variety of
anti-commutative -algebras, the Jacobi identity becomes equivalent
to a categorical condition: it is an identity in~ if and only if
is a subvariety of a locally algebraically cartesian closed
variety of anti-commutative -algebras. This is based on a result
saying that an algebraically coherent variety of anti-commutative
-algebras is either a variety of Lie algebras or a variety of
anti-associative algebras over .Comment: Final version to appear in Journal of Pure and Applied Algebr
A characterisation of Lie algebras via algebraic exponentiation
In this article we describe varieties of Lie algebras via algebraic
exponentiation, a concept introduced by Gray in his Ph.D. thesis. For
an infinite field of characteristic different from , we prove
that the variety of Lie algebras over is the only variety of
non-associative -algebras which is a non-abelian locally
algebraically cartesian closed (LACC) category. More generally, a variety of
-algebras is a non-abelian (LACC) category if and only if
and . In characteristic the
situation is similar, but here we have to treat the identities and
separately, since each of them gives rise to a variety of
non-associative -algebras which is a non-abelian (LACC) category.Comment: The ancillary files contain the code used in the proofs. Final
version to appear in Advances in Mathematic
On some categorical-algebraic conditions in S-protomodular categories
In the context of protomodular categories, several additional conditions have
been considered in order to obtain a closer group-like behavior. Among them are
locally algebraic cartesian closedness and algebraic coherence. The recent
notion of S-protomodular category, whose main examples are the category of
monoids and, more generally, categories of monoids with operations and
Jo\'{o}nsson-Tarski varieties, raises a similar question: how to get a
description of S-protomodular categories with a strong monoid-like behavior. In
this paper we consider relative versions of the conditions mentioned above, in
order to exhibit the parallelism with the "absolute" protomodular context and
to obtain a hierarchy among S-protomodular categories
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