Isocategories and tensor functors
journal contributionposted on 01.01.1992 by Walter Noll
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Abstract: "In this paper, I show how the concepts of an isocategory (category all of whose morphisms are isomorphisms) and the corresponding concept of an isofunctor can be used to improve the conceptual infrastructure of many branches of mathematics. The crucial new idea is that of a natural assignment, a variant of the idea of a natural transformation introduced by Eilenberg and Mac Lane. Isofunctors that involve the isocategory LIS of all linear isomorphisms of finite-dimensional linear spaces are called tensor functors, because they can be used to clarify most uses of the word 'tensor' in the literature of mathematics and physics. Of particular importance are the 'analytic tensor functors', which can serve to simplify and generalize the treatment of tensor fields given in the standard textbooks on differentiable manifolds."