A comparison theorem for simplicial resolutions

It is well known that Barr and Beck's definition of comonadic homology makes sense also with a functor of coefficients taking values in a semi-abelian category instead of an abelian one. The question arises whether such a homology theory has the same convenient properties as in the abelian case. Here we focus on independence of the chosen comonad: conditions for homology to depend on the induced class of projectives only.Comment: 16 page

Approximate Hagemann-Mitschke co-operations

We show that varietal techniques based on the existence of operations of a certain arity can be extended to n-permutable categories with binary coproducts. This is achieved via what we call approximate Hagemann-Mitschke co-operations, a generalisation of the notion of approximate Mal'tsev co-operation. In particular, we extend characterisation theorems for n-permutable varieties due to J. Hagemann and A. Mitschke to regular categories with binary coproducts.Comment: 11 pages. Dedicated to George Janelidze on the occasion of his sixtieth birthda

The Yoneda isomorphism commutes with homology

We show that, for a right exact functor from an abelian category to abelian groups, Yoneda's isomorphism commutes with homology and, hence, with functor derivation. Then we extend this result to semiabelian domains. An interpretation in terms of satellites and higher central extensions follows. As an application, we develop semiabelian (higher) torsion theories and the associated theory of (higher) universal (central) extensions.Comment: Fixed an inaccuracy in (3.6

Categories vs. groupoids via generalised Mal'tsev properties

We study the difference between internal categories and internal groupoids in terms of generalised Mal'tsev properties---the weak Mal'tsev property on the one hand, and $n$-permutability on the other. In the first part of the article we give conditions on internal categorical structures which detect whether the surrounding category is naturally Mal'tsev, Mal'tsev or weakly Mal'tsev. We show that these do not depend on the existence of binary products. In the second part we focus on varieties of algebras.Comment: 30 pages; final published versio

A characterisation of Lie algebras amongst anti-commutative algebras

Let $\mathbb{K}$ be an infinite field. We prove that if a variety of anti-commutative $\mathbb{K}$-algebras - not necessarily associative, where $xx=0$ is an identity - is locally algebraically cartesian closed, then it must be a variety of Lie algebras over $\mathbb{K}$. In particular, $\mathsf{Lie}_{\mathbb{K}}$ is the largest such. Thus, for a given variety of anti-commutative $\mathbb{K}$-algebras, the Jacobi identity becomes equivalent to a categorical condition: it is an identity in~$\mathcal{V}$ if and only if $\mathcal{V}$ is a subvariety of a locally algebraically cartesian closed variety of anti-commutative $\mathbb{K}$-algebras. This is based on a result saying that an algebraically coherent variety of anti-commutative $\mathbb{K}$-algebras is either a variety of Lie algebras or a variety of anti-associative algebras over $\mathbb{K}$.Comment: Final version to appear in Journal of Pure and Applied Algebr

Further remarks on the "Smith is Huq" condition

We compare the 'Smith is Huq' condition (SH) with three commutator conditions in semi-abelian categories: first an apparently weaker condition which arose in joint work with Bourn and turns out to be equivalent with (SH), then an apparently equivalent condition which takes commutation of non-normal subobjects into account and turns out to be stronger than (SH). This leads to the even stronger condition that weighted commutators (in the sense of Gran, Janelidze and Ursini) are independent of the chosen weight, which is known to be false for groups but turns out to be true in any two-nilpotent semi-abelian category.Comment: 13 page

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 $\mathbb{K}$ an infinite field of characteristic different from $2$, we prove that the variety of Lie algebras over $\mathbb{K}$ is the only variety of non-associative $\mathbb{K}$-algebras which is a non-abelian locally algebraically cartesian closed (LACC) category. More generally, a variety of $n$-algebras $\mathcal{V}$ is a non-abelian (LACC) category if and only if $n=2$ and $\mathcal{V}=\mathsf{Lie}_\mathbb{K}$. In characteristic $2$ the situation is similar, but here we have to treat the identities $xx=0$ and $xy=-yx$ separately, since each of them gives rise to a variety of non-associative $\mathbb{K}$-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

A decomposition formula for the weighted commutator

We decompose the weighted subobject commutator of M. Gran, G. Janelidze and A. Ursini as a join of a binary and a ternary commutator.Comment: 7 pages. Dedicated to George Janelidze on the occasion of his sixtieth birthda

A note on split extensions of bialgebras

We prove a universal characterization of Hopf algebras among cocommutative bialgebras over a field: a cocommutative bialgebra is a Hopf algebra precisely when every split extension over it admits a join decomposition. We also explain why this result cannot be extended to a non-cocommutative setting.Comment: Reduced the context to algebraically closed field

On the "three subobjects lemma" and its higher-order generalisations

We solve a problem mentioned in an article of Berger and Bourn: we prove that in the context of an algebraically coherent semi-abelian category, two natural definitions of the lower central series coincide. In a first, "standard" approach, nilpotency is defined as in group theory via nested binary commutators of the form $[[X,X],X]$. In a second approach, higher Higgins commutators of the form $[X,X,X]$ are used to define nilpotent objects. The two are known to be different in general; for instance, in the context of loops, the definition of Bruck is of the former kind, while the commutator-associator filtration of Mostovoy and his co-authors is of the latter type. Another example, in the context of Moufang loops, is given in Berger and Bourn's paper. In this article, we show that the two streams of development agree in any algebraically coherent semi-abelian category. Such are, for instance, all Orzech categories of interest. Our proof of this result is based on a higher-order version of the Three Subobjects Lemma of Cigoli-Gray-Van der Linden, which extends the classical Three Subgroups Lemma from group theory to categorical algebra. It says that any $n$-fold Higgins commutator $[K_1, \dots,K_n]$ of normal subobjects $K_i$ of an object $X$ may be decomposed into a join of nested binary commutators.Comment: 20 pages; revised version, with some simplified proof