claudio pisani <pisclau@yahoo.it> commented as follows on my remarks to his initial question:
Claudio Pisani asked,
Is there a standard name for those presheaves X on a category C such that Xf is a bijection for any f in C?
Well, those presheaves are exactly the "restrictions to C" of the presheaves on the grouppoid reflection (the grouppoidal 'quotient') of C (by which I mean the category got by declaring invertible every C-morphism).
Does that suggest "grouppoidal action of C" might work? I think I'd tend to lobby against the use of the prefix "bi-" unless there were *really* compelling reasons in favor of it.
It seems to me that its disadvantage is that "groupoidal action of C" may suggest that C itself is a groupoid, but probably the ambiguity disappears in the right context.
Please don't misunderstand my question -- I wasn't trying to suggest that "grouppoidal action of C" was a _good_ name for the notion -- only that it might be better than "bi-...". Surely there are many other candidate names far better still :-) . I suggest one later on, but there are probably still others better yet. "Bi-" rarely is permanent -- think of the defunct terms "bi-compact", "bi-regular", "bi-morphism", "bi-normal", ... . Only where the motivation includes some irreducible sort of "two"-ness does "bi-" survive (e.g., "bicategory", "bilateral", "bisexual", "bipedal", "bicarbonate", "bilingual", ...). Well, there may be counterexamples to that extravagant claim, but I suspect not very many :-) . For presheaves X in general, there's no more reason for the restriction maps Xf to be invertible than, in physics, for a thermoodynamical event to be reversible -- physical processes are often simply irreversible, with no reason whatsoever to be expected to be reversible, just as the restrictions (or transitions) Xf have no reason (apart from when the maps f are invertible) to be invertible. If you want to come up with a term to indicate otherwise, i.e., to restrict attention to those presheaves X for which each Xf _is_ indeed invertible, it might pay, thinking of Xf as a *process* (and X as a compendium of such processes), to borrow the physics terminology and _think_ of the process Xf as _reversible_ if it's invertible, and then to speak of the whole presheaf as _reversible_ if each of its processes -- each of its transition maps Xf -- is. I won't go so all out as to claim that's the best term for these things, but I do feel it beats "bi-action" by a country mile, and is certainly clearly preferable to my silly "grouppoidal action" as well. Why do I not suggest "invertible" as adjective for the presheaf X when each Xf is invertible? Well, a presheaf is a functor, there's already a well established meaning for "invertible" as applied to a functor, and the present notion of "reversibility" is quite incompatible with that sort of invertibility. For, here one is concerned with the invertibility of each value Xf of the functor X, not on that of the functor X itself, qua functor. Anyway, it then becomes natural, given a reversible presheaf X on C, to call the presheaf Y on C^(op) defined by the formula Yf = (Xf)^(-1) the _reverse_ of the presheaf X, which helps give a better basis for the link you describe below:
By the way, I am actually interested in the (full and faithful, indexed) inclusion of presheaves on C' (where C' is the groupoid reflection of C) in presheaves on C and C^op (that is of groupoidal actions in left and in right actions). In fact it seems to provide a useful link between left and right actions.
I hope these remarks and their motivations prove useful. And if there's a still better term for the attribute you seek, by all means use it in preference to "reversible". Cheers, -- Fred [For admin and other information see: http://www.mta.ca/~cat-dist/ ]