Comultiplications in algebra and topology. (English) Zbl 0901.18004
The notion of a group, or more generally, a set with a multiplication, has been described in categorical terms. Thus for certain categories one can consider a multiplication on an object in the category or a group-object in the category. In this generality, it is possible to dualize these notions and obtain a comultiplication on an object (i.e., a co-\(H\)-object) in a category or a co-\(H\)-group-object in a category. In this paper we study these concepts. We show that they yield familiar mathematical objects for many specific categories. In addition, we investigate certain functors which preserve this structure and obtain several results about these functors. Thus our approach is to unify a number of diverse, known results by showing that they are specializations of general categorical considerations.
We briefly summarize the contents of this paper. We consider categories with coproducts and introduce co-\(H\)-objects, associative co-\(H\)-objects, co-\(H\)-group-objects and co-\(H\)-morphisms in these categories. We show that functors which preserve coproducts, carry co-\(H\)-objects, associative co-\(H\)-objects, etc. of the domain category into the corresponding objects of the range category. We examine these concepts in the following categories: the homotopy category of based spaces, the category of commutative, graded algebras, the category of associative graded algebras, the category of graded Lie algebras and the category of groups. We study functors whose images lie in these categories by means of the methods from above and put a number of classical results into this setting.
We have focussed our attention on some of those categories which appear in algebraic topology and on some of those functors which map the homotopy category of based spaces into the algebraic categories mentioned above. For more information on comultiplications and co-\(H\)-group-objects in many other algebraic categories, we refer to the recent book of G. M. Bergman and A. O. Hausknecht which contains a wealth of material [“Cogroups and co-rings in categories of associative rings”, AMS, Math. Surv. Mongr. 45 (1996; Zbl 0857.16001)]. For details on the categorical approach to multiplications and comultiplications, see the papers of B. Eckmann and P. J. Hilton [Math. Ann. 145, 227-255 (1962; Zbl 0099.02101); ibid. 150, 165-187 (1963); ibid. 151, 150-186 (1963; Zbl 0115.01403)].
We briefly summarize the contents of this paper. We consider categories with coproducts and introduce co-\(H\)-objects, associative co-\(H\)-objects, co-\(H\)-group-objects and co-\(H\)-morphisms in these categories. We show that functors which preserve coproducts, carry co-\(H\)-objects, associative co-\(H\)-objects, etc. of the domain category into the corresponding objects of the range category. We examine these concepts in the following categories: the homotopy category of based spaces, the category of commutative, graded algebras, the category of associative graded algebras, the category of graded Lie algebras and the category of groups. We study functors whose images lie in these categories by means of the methods from above and put a number of classical results into this setting.
We have focussed our attention on some of those categories which appear in algebraic topology and on some of those functors which map the homotopy category of based spaces into the algebraic categories mentioned above. For more information on comultiplications and co-\(H\)-group-objects in many other algebraic categories, we refer to the recent book of G. M. Bergman and A. O. Hausknecht which contains a wealth of material [“Cogroups and co-rings in categories of associative rings”, AMS, Math. Surv. Mongr. 45 (1996; Zbl 0857.16001)]. For details on the categorical approach to multiplications and comultiplications, see the papers of B. Eckmann and P. J. Hilton [Math. Ann. 145, 227-255 (1962; Zbl 0099.02101); ibid. 150, 165-187 (1963); ibid. 151, 150-186 (1963; Zbl 0115.01403)].
MSC:
| 18D35 | Structured objects in a category (MSC2010) |
| 16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |
| 16S10 | Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.) |
| 55P45 | \(H\)-spaces and duals |
| 57T05 | Hopf algebras (aspects of homology and homotopy of topological groups) |
Keywords:
co-group; co-\(H\)-space; Hopf algebra; tensor algebra; free group; comultiplication; co-\(H\)-object; coproducts; Lie algebrasReferences:
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