In finitely cocomplete homological categories, co-smash products give rise to (possibly higher-order) commutators of subobjects. We use binary and ternary co-smash products and the associated commutators to give characterisations of internal crossed modules and internal categories, respectively. The ternary terms are redundant if the category has the Smith is Huq property, which means that two equivalence relations on a given object commute precisely when their normalisations do. In fact, we show that the difference between the Smith commutator of such relations and the Huq commutator of their normalisations is measured by a ternary commutator, so that the Smith is Huq property itself can be characterised by the relation between the latter two commutators. This allows to show that the category of loops does not have the Smith is Huq property, which also implies that ternary commutators are generally not decomposable into nested binary ones. Thus, in contexts where Smith is Huq need not hold, we obtain a new description of internal categories, Beck modules and double central extensions, as well as a decomposition formula for the Smith commutator. The ternary commutator now also appears in the Hopf formula for the third homology with coefficients in the abelianisation functor.Comment: Revised version; 32 page

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Dedicated to the memory of Jean-Louis Loday for his generosity and benevolence.Abstract. In this paper, we give a description of polynomial functors from (finitely generated free) groups to abelian groups in terms of non-linear Mackey functors generalizing those given in a paper of Baues-Dreckmann-Franjou-Pirashvili published in 2001. This description is a consequence of our two main results: a description of functors from (finitely generated free) P-algebras (for P a set-operad) to abelian groups in terms of non-linear Mackey functors and the isomorphism between polynomial functors on (finitely generated free) monoids and those on (finitely generated free) groups. Polynomial functors from (finitely generated free) P-algebras to abelian groups and from (finitely generated free) groups to abelian groups are described explicitely by their cross-effects and maps relating them which satisfy a list of relations.Mathematics Subject Classification: 18D; 18A25; 55UKeywords: polynomial functors; non-linear Mackey functors; set-operads Polynomial functors play a prominent role in the representation theory of algebraic groups, algebraic K-theory as well as in the theory of modules over the Steenrod algebra (see, for instance, [11]). In particular it is a main computational tool for computing the stable cohomology of classical groups (considered as discrete groups) with twisted coefficients (see [4], [12] and [7]).The study of polynomial functors, in their own right, has a long history starting with the work of Schur in 1901 in [27], even before the notion of a functor was defined by Eilenberg and MacLane. In fact, without using the language of functors, Schur proved that over a field of characteristic zero any polynomial functor is a direct sum of homogeneous functors and the category of homogeneous functors of degree d is equivalent to the category of representations of the symmetric group on d letters. There were many attempts to generalize Schur's theorem for general rings. For polynomial functors from (finitely generated free) abelian groups to abelian groups a satisfactory answer is given in [2] where the authors obtained a description of those polynomial functors in terms of non linear Mackey functors.The principal aim of this paper is to generalize this result. The two main results of this paper are a description of functors from (finitely generated free) P-algebras (for P a set-operad) to abelian groups and a description of polynomial functors from (finitely generated free) groups to abelian groups, obtained from the first result and an isomorphism between polynomial functors on (finitely generated free) monoids and those on (finitely generated free) groups.More precisely, for P a set-operad, F ree(P) the category of (finitely generated free) Palgebras and Ab the category of abelian groups, we consider the category F unc(F ree(P), Ab) of functors F : F ree(P) → Ab. The first principal result of this paper is the following: where P Mack(Ω(P), Ab) is the category of pseudo-Mackey functors defined in Definition 1.28.T...