Re: question on terminology

[claudio pisani <pisclau@yahoo.it>]

--- Sab 25/8/12, Fred E.J. Linton <fejlinton@usa.net> ha scritto:

> Da: Fred E.J. Linton <fejlinton@usa.net>
> Oggetto: Re: categories: question on terminology
> A: "claudio pisani" <pisclau@yahoo.it>, categories@mta.ca
> Data: Sabato 25 agosto 2012, 05:35
> 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
> 

Dear Fred,
thanks for the suggestion.
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.
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.

Claudio









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