By George M. Bergman

This ebook experiences representable functors between recognized types of algebras. All such functors from associative jewelry over a hard and fast ring $R$ to every of the types of abelian teams, associative earrings, Lie jewelry, and to a number of others are made up our minds. effects also are acquired on representable functors on kinds of teams, semigroups, commutative jewelry, and Lie algebras. The booklet contains a ``Symbol index'', which serves as a thesaurus of symbols used and an inventory of the pages the place the themes so symbolized are handled, and a ``Word and word index''. The authors have strived--and succeeded--in making a quantity that's very straightforward.

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**Sample text**

15) p : \R\ i i ... i i \R\ —> S for some object S. ) Such an epimorphism will correspond to a certain «-tuple of elements JCJ, ... ,a ) of elements of a set C(\R\, A) is defined to satisfy the relation if and only if there exists a (necessarily unique) morphism / : S —> A such that a> = fx^. For example, let C be the category whose objects are pairs (X, r) where X is a set and r a symmetric binary relation on X, and where morphisms are setmaps respecting these relations. Consider the functor V from Group to C taking a group G to the pair (IGI, {(g, h) elGI x IGI | gh = hg}).

And indeed, the construction of tensor products of bimodules is a special case of this concept: For any rings A and B, a representable functor from 28 II. -module structure on the sets (AMod)(AMB, AN). The left adjoint of this functor is j^MB ®£ ~ : #Mod —> ^Mod. ) So, as claimed, Freyd's ''tensor product" operation on representing coalgebras is in this case the usual tensor product operation on bimodules. One might generalize the notation used for bimodules, and write a V-coalgebra object R in a variety U as " j j / ?

Qn). (iii) For all objects A of C, the algebra C(R, A) lies in V. (iv) Interpreted as an algebra object of C o p , R is a V-object. 11. 7(H). The full subcategory of V consisting of the representable covariant functors will be denoted Rep(C, V). Note that algebra objects represent contra variant functors, while covariant functors are represented by coalgebra objects. 12) C ( - , - ) : C o p x C -* Set is covariant in one variable and contravariant in the other. Thus, if we put a "structured" object R in one position, obtaining a functor in the other variable, we will get contravariance either in the relation between the structure on R and the induced structure on the output sets, or in the relation between input object and output object, but not both.