Date: Wed, 18 Nov 1992 10:15:16 -0500 (EST) From: D_FELDMAN@UNHH.UNH.EDU

Have categories that are equipped with a faithful functor to some group (or perhaps monoid) ever been singled out as objects of study?

One reference is: A. Froehlich & C.T.C. Wall "Graded monoidal categories" Compositio Matematica Vol 28 Fasc 3 (1974) 229-285 (graded here is wrt a group; and the applications they had in mind were equivariant Brauer group, representations of groups by automorphisms of algebraic structures, theory of quadratic & Hermitian forms). Fibrations into groupoids are very important. The only reference that comes to mind (but I hope other people can remember more) is the recent Memoir of the AMS by R. (Bob) Gordon, and here is my review: Review of: "G-Categories" by Gordon The literature of algebraic topology and representation theory is sparsely sprinkled with ad hoc results about categories on which a group G acts; here called G-categories. Category theory has a lot to say about G-categories because they are special cases of many structures: for example, categories internal to a presheaf topos, G-graded categories, categories fibred over G (or G-indexed categories), categories with homs enriched in a (non-symmetric) biclosed monoidal category, algebras for a doctrine. Yet, it is precisely this diversity of views that makes it worthwhile beginning a systematic theory as done in the present memoir. All the peculiarities of G-categories are explained by no single view, and some by none. The author is motivated, it seems, by the prospect of applications to modules over G-graded algebras. There is particular value in having this work done by someone whose speciality is outside category theory. Limits, representability, adjointness and cotripleability are all adapted to the G-context. The author is interested in the control the stable subcategory has over its ambient G-category. However, the reviewer believes stability considerations entered too strongly into the choice to consider only conical limits (= Kan extensions along the unique functor into the trivial G-set), and that a more versatile notion would have been Kan extension along functors into the G-set G (acting on itself by multiplication); this would have been discovered had a formula for Kan extensions been included. For the variant of the Yoneda Lemma, a special G-category Par(G-Set) is introduced to receive the hom functors of G-categories. It is interesting to observe that this does actually come out of the viewpoint of G-sets as presheaves on G (as a one-object category). The gross internal full subcategory of the category of presheaves on C in the sense of the reviewer ["Cosmoi of internal categories", Transactions of the American Mathematical Society 258 (1980) 271-318] was shown to be the category-valued presheaf on C whose value at u C is the category of presheaves on C/u; it can be checked that this is equivalent to Par(G-Set) when C = G. Indeed, the G-groupoid C/u is used by the author to discuss systems of isomorphisms and stably closed G-categories. There is a section on "transversals". Transversaled functors between G-categories arise, for a category theorist, from the view that G-categories are categories fibred over G; they are then the appropriate morphisms (more general than G-functors). Alternatively, they are pseudonatural transformations between category-valued functors on G. Because of the inverses in G, the author is able to prove that, if one of the functors in an adjunction is a G-functor, then the other admits a transversal, but is generally not a G-functor. The memoir is written for algebraists with a basic knowledge of category theory, yet there is much to stimulate the expert. ---------------------- Regards, Ross ==============================================================================