Group-like structures in general categories I multiplications and comultiplications

Abstract

I n t r o d u c t i o nThe starting point of the investigation whose results are to be presented in a series of three papers, of which this is the first, was an a t t e m p t to give precise meaning to and to answer the question how group structures arise in homotopy theory and why these group structures satisfy the familiar conditions of naturality. This problem, set in the restricted context of homotopy theory, was discussed in [2], but it was already clear that, by restricting the category of study in this way, we were disguising the generality of the approach and of the results. A more general treatment was indicated in Chapter 14 of [6] but not taken very far.A group is a set A with a multiplication m : A × A -> A satisfying certain axioms. The basic idea of this paper is to consider categories ~ rich enough in objects and maps to enable us to formulate a set of axioms which, in the case where ~ is the category ~ of (based) sets, are equivalent to the group axioms. Of course these axioms are formulated entirely in 'terms of the maps of the category ~. I t is then a basic observation that if (A, m) is a "group" in ~ or, as we shall prefer to say, a G-object, and if H(X, A) is the set of maps from X to A, then H(X, A) acquires a group-structure, in the familiar sense, from the structure m a p m; and the group-structure is commutative if m is "commutative" in a sense applicable to the category ~. Moreover the group-structure is natural with respect to maps X -> Y in ~ in the sense that, for such a m a p ], the

The Hilton-Heckmann argument for the anti-commutativity of cup products

“…In their paper [1] on group objects in general categories, B. Eckmann and P. J. Hilton show that two monoid structures on a set which are such that one of them is a homomorphism for the other actually coincide and are commutative; this is stated-for groups-as Theorem 5.4.2 in that paper.…”

“…The result just quoted implies then that these two structures are equal, and that [X, Y ] is commutative. Instances of this are the well-known facts that both the fundamental group π 1 (G, e) of a topological group G and the higher homotopy groups π p (X, x 0 ) for p > 1 of any space X are abelian; see section 5.3 in [1].…”

Generalized entropy in expanded semigroups and in algebras with neutral element

“…Eckmann and Hilton [5] established the following strong result about commuting binary operations with a common neutral element. By Theorem 2.2, every entropic groupoid (A; ·) with a neutral element is a commutative monoid.…”

Cohomology of Groups in Arbitrary Categories

“…Since we do not assume that the category 6 is a category with zeros in the sense of [3 ] we have to use another axiom for the neutral element for group-like structures. The complete set of axioms which we shall use is: (I) There exists a morphism p: XXX-*X. p is called a multiplication on X.…”