Tensor Products of Theories, Application to Infinite Loop Spaces*

In [4] Lawvere formalized the concept of an algebraic theory defined by operations and laws without existential quantifiers. Examples are the theories of monoids, groups, loops, rings, modules etc., whose axioms can be put into the required form; although not the theory of fields. The advantage of this approach lies in the unified treatment of a variety of universal constructions, especially the construction of free objects. Let .& and 93 be algebraic theories, possibly topologized (we recall definitions in Section 2). One frequently is interested in the algebraic structure of .%-objects in the category of d-spaces. This is given by the tensor product theory & 0 93 of & and 93. For example, group objects in the category of groups are exactly the abelian groups, so that Y? 0 3 is the theory of abelian groups if ‘3 denotes the theory of groups. In general, the structure of ti 0 3 is far from clear and few examples are known. We prove the following somewhat surprising result.