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Adjoint functors and triples

1965; Duke University Press; Volume: 9; Issue: 3 Linguagem: Inglês

10.1215/ijm/1256068141

1945-6581

Samuel Eilenberg, John C. Moore,

Advanced Topics in Algebra

A riple F (F, , ) in ctegory a consists of functor F a nd morphisms la F, F F stisfying some identities (see 2, (T.1)- (T.3)) nlogous to those stisfied in monoid.Cotriples re defined dually.It has been recognized by Huber [4] that whenever one hs pir of adoint functors T a , S a (see 1), then the functor TS (with appro- priate morphisms resulting from the adjointness relation) constitutes a triple in nd similarly ST yields cotriple in a.The main objective of this pper is to show that this relation between d- jointness nd triples is in some sense reversible.Given triple Y in a we de- fine new ctegory a nd adoint functors T a a, S a a such that the triple given by TS coincides with .There my be mny adoint pirs which in this wy generate the triple Y, but among those there is a uni- versal one (which therefore is in a sense the "best possible one") nd for this one the functor T is faithful (Theorem 2.2).This construction cn best be illustrated by n example.Let a be the ctegory of modules over a commu- tative ring K nd let A be K-lgebm.The functor F A@ together with morphisms nd resulting from the morphisms K A, h @ A A given by the K-algebra structure of A, yield then a triple Y a.The ctegory a is then precisely the ctegory of A-modules.The general construction of a closely resembles this example.As another example, let a be the category of sets nd let F be the functor which to ech set A ssigns the underlying set of the free group generated by A. There results triple Y in a nd a is the category of groups. Let G(, e, G) be cotriple in category A. It has been recognized by Godement [3] and Huber [4], that the iterates G of G together with face and degeneracy morphisms G + G , G G + defined using e and yield a simplicial structure which can be used to define homology and cohomology.Now if Y is a triple in a, then one has an adjoint pair T" aa, S and therefore one has an associated cotriple G in .This in turn yields a simplicial complex for every object in a , thus paving the way for homology and cohomology in ar.In 4 we show that under suitable