Here is another twist on this circle of ideas which appeared in the introductory chapter of my 1976 thesis. Robin Cockett and I are working on a redevelopment of it. For monoidal (V,\ten, i), (promonoidal V will suffice) consider Brian Day's convolution (closed) monoidal structure on set^{V^op}. If A is a set^{V^op} category, it is helpful to think of A(-,-):A^op x A ---> set^{V^op} as A(-,-,-):A^op x V^op x A ---> set with the interpretation that A(a,v,b) provides a set of `v-indexed families' of arrows from a to b. The composite of a v-indexed family (v;f):a--->b with a w-indexed family (w;g):b--->c is a w\ten v family (w\ten v;gf):a--->c. Of course it may happen that for each a,b in A, A(a,-,b) is representable, by an object A[a,b] in V. In this case each (v;f):a--->b takes the form f:v--->A[a,b]. If for each a in A, and each v in V, A(a,v,-) is representable, by an object a.v in A, then the (v;f):a--->b take the form f:a.v--->b. Note that the identity a.v--->a.v considered as a v-indexed family (v,j):a--->a.v can be construed as a family of `sum-injections' for the `multiple' a.v. (Asking for a representing object {v,b} for A(-,v,b) leads to dual considerations.) Simultaneous representability in a,b and a,v is equivalent to the notion of `tensored V-category' mentioned below. In part this work was motivated by questions raised by Linton in `The multilinear Yoneda lemmas' SLN 195, 209--229, and also pursued by Reynolds in his 1973 Wesleyan thesis. For example, if A is a V-category and M is a V-actegory, in the nomenclature below, what is a V-functor A--->M, a V-functor M--->A? The familial approach, suggested by the 1970s work of Benabou, Pare/Schumacher, Rosebrugh and others provides a straightforward intuitive answer. For general set^{V^op}-categories A and M, the data for a set^{V^op}-functor F:A--->M sends, for each v in V, each v-indexed family (v;f):a--->b to a v-indexed family (v;Ff):Fa--->Fb. Each representability possibility for A and B allows for a compact presentation of the data. When A is a V-category then it suffices to know F on the generic families g:A[a,b]--->A[a,b]. In other words, one requires (A[a,b];Fg):Fa--->Fb. If M is also a V-category then Fg is what is usually denoted F_{a,b}:A[a,b]--->M[Fa,Fb], the effect of F on homs, but if M is a V-actegory it will take the form Fa.A[a,b]--->Fb. If A is a V-actegory then it suffices to know F on the generic (v,j):a--->a.v. For M a V-category we have Fj:v--->M[Fa,F(a.v)], while for M also a V-actegory we have Fj:Fa.v--->F(a.v), a form called `tensorial strength' by Anders Kock in a seeries of papers about mononoidal monads. In fact the 3x3 possibilities for `strengths' can be tabulated easily using these considerations: Write 1) for `powers' {v,b}, 2) for homs [a,b] and 3) for `multiples' a.v. Then the i,j th entry below provides the form of strength for a set^{V^op}-functor F:A--->M where A is of type i) and M is of type j) 1) 2) 3) 1) F{v,b}--->{v,Fb} v--->M[F{v,b},Fb] F{v,b}.v--->Fb 2) Fa--->{A[a,b],Fb} A[a,b]--->M[Fa,Fb] Fa.A[a,b]--->Fb 3) Fa--->{v,F(a.v)} v--->M[Fa,F(a.v)] Fa.v--->F(a.v) Best regards RJ Wood

What you are looking for may be similar to something I queried Ross Street in regard to earlier this summer. I'll save him some time by putting here the relevant part of his response.

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I think the one you first mention is what we have been calling V-actegories. Benabou looked at these rather than (as well as?) V-categories in the early days of monoidal categories. Pareigis also made use of them. More recently, publications of Paddy McCrudden involve them. There is a close connection with V-categories. A V-module V x A --> A in this sense for which we have a parametrized adjoint V(x,[a,b]) =~ A(x.a,b) makes A a V-category with V-valued hom [a,b].

Conversely, a tensored V-category becomes such a V-module.

I recommend the work of McCrudden, who has developed among other things a descent theoretic approach to the tensor product of V-actegories. There is also resource in the work of Harald Lindner. His paper, Enriched Categories and Enriched Modules, in Cahiers, Vol XXII-2 (1981) develops morphisms between enriched categories and actegories, which he calls modules. I'm curious about why it is that I have never seen his work referenced.

Paul B Levy writes:

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Hi

Is there a standard reference for the notion of "left module for a category"? (or right module, or bimodule)

Is there any reference in the setting of ordinary categories rather than (or as well as) enriched categories or bicategories?

Thanks Paul