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. Cheers, -- Fred ---
I have sometimes called them "biactions" since any such X (considered as, say, a left action of C) is paired with the obvious presheaf X' on C^op (a right action of C): X'f = (Xf)^-1. Of course, they correspond, as categories over C, to discrete bifibrations. I also know that the separable or decidable presheaves are those for which every Xf is injective.
Claudio
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