Re: module for a category

[Stefan Forcey <sforcey@math.vt.edu>]

 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.

 > 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:
 >
 > 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
 >
 >
 >
 >
 >