The multilinear yoneda lemmas: Toccata, fugue, and fantasia on themes by eilenberg-kelly and yoneda

Abstract

Although the notion of a covariant v-valued v-functor, exemplified by the v-valued hom functors a(A, -) on a v-category a, has been recognized by Eilenberg and Kelly, in their comprehensive foundational treatise [EK, esp. pp. 454, ff.] on closed and monoidal categories, for general closed categories v, those authors pointedly renounce consideration of that notion's contravariant counterpart until v is at least symmetric, and carefully refrain from even mentioning the two analogous possibilities for general (not necessarily closed) monoidal categories v.

The purpose of the present note is to provide these definitions, to formulate, somewhat after the fashion of Day and Kelly [DK, §§3, 4] or of Yoneda [Y, §§4.0, 4.1], the notions of the v-object of v-natural transformations between two such v-functors of similar variance and the v-object tensor product of a contravariant v-valued v-functor with a covariant one, and to establish the pertinent Yoneda Lemmas (extending [DK, (5.1)], the v-valued case of [DK, (3.5)], and [Y, (4.3.1), .2), .1*), and .2*)]). These will facilitate the description (elsewhere), for not necessarily symmetric v, of the algebras over a v-triple [L] in terms of v-functors on the associated Kleisli v-category, generalizing Dubuc's work [D₂] for closed symmetric monoidal v.

Preliminary speculations on these matters were aired in talks delivered at McGill University, Oct. 18, 1968, and at a meeting of the Midwest Category Seminar in San Antonio, Jan. 24, 1970.

During the preparation of the bulk of this paper, the author, on leave from his home university, was a Killam Senior Research Fellow at Dalhousie University, Halifax, Nova Scotia, and was supported in part by Canadian N.R.C. Grant # A 7565.