Generalized tensor products

Thistle, W. Wayne (1970) Generalized tensor products. Masters thesis, Memorial University of Newfoundland.

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Abstract

If R is a commutative ring and A and B are R-modules then hom(A,B), Hom(A,B) and A ⓧ B will denote the set of morphisms A → B, the set of morphisms A → B regarded as a R-module and the usual algebraic tensor product of A and B, respectively. The R-module, A ⓧ B can be defined by any of the following results: -- (i) t: A x B → A ⓧ B; (a,b) |→ a ⓧ b is a universal bilinear function in the sense that any other bilinear function A x B → C factors uniquely through t. -- (ii) there is a natural isomorphism hom(A ⓧ B, C) ≌ hom(A, Hom(B,C)). -- (iii) the functor - ⓧ B is a left adjoint to the functor Hom(B,-), i.e. the isomorphism of (ii) is natural in the variables A and C. -- (iv) there is a natural isomorphism Hom(A ⓧ B, C) ≌ Hom(A, Hom(B,C)). ⓧ also has the property that: -- (v) there exists natural isomorphisms: -- a: A ⓧ (B ⓧ C) ≌ (A ⓧ B) ⓧ C -- r: A ⓧ R ≌ A where R is regarded as an R-module -- e: R ⓧ A ≌ A -- c: A ⓧ B ≌ B ⓧ A. -- The existence of these isomorphisms does not constitute a definition of ⓧ since analogous isomorphisms exist for the direct sum A ⓧ B of R-modules A and B. -- In this thesis we abstract the definitions (i), (ii), (iii) and (iv) from the category of R-modules to a general category C calling the tensor products so defined the (i) Bimorphism Product (ii) the Exponential Product (iii) the Adjoint Product (iv) the Strong Exponential Product, respectively. The relation between (ii), (iv) and (v) is discussed in (17); (i) is related to these ideas in (24); (iii) does not seem to have been discussed elsewhere. -- Our main purpose is to examine the conditions under which the different products coincide and the extent to which the products satisfy the conditions (v). The value of this theory lies in the number and diversity of the examples. -- Chapter I gives the necessary details about category theory and defines many of the terms which occur in the main discussion. In Chapter II the Adjoint Product and the properties of associativity, commutativity and left and right identities are introduced. The "coherence" of the above isomorphisms forms the content of the third chapter which is a survey of the works of MacLane (21) and Kelly (17). Chapter IV gives the definition of the Bimorphism Product as explained in Pumplun (24). In Chapter V it is proved that any Bimorphism Product is an Exponential Product and conversely any commutative Exponential Product is a Bimorphism Product. In Chapter VI (sections 1 to 3) the relationships between the natural isomorphisms a, r, e and c and the Exponential and Strong Exponential Products are discussed; in section 6.4 these natural isomorphisms and the Strong Exponential Product are related to the Bimorphism Product. In Chapter VII the Exponential Product and the Adjoint Product are shown to be distinct, in general; conditions are given for their equivalence. Chapter VIII is devoted to a selection of examples from many branches of mathematics. The Appendix gives necessary and sufficient conditions for the concepts of a morphism and bimorphism to coincide. The lengthy bibliography lists papers related to the thesis. -- Our results are proved in full detail (apart from Chapter III which is a survey of results from (18) and (21), detailed proofs being given in these papers, and two proofs involving trimorphisms in section 6.4). Chapter VII and the Appendix are basically original work. The proofs of the theorems in Chapter V are given in greater detail than by the original author.

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