Re: question on terminology

[David Roberts <david.roberts@adelaide.edu.au>]

The 'groupoidal quotient' is also the fundamental groupoid of the category
considered as a homotopy type via the nerve. I would be tempted to call
what you have a covering 'space' of the category, as the category of all
these functors is equivalent to the category of
covering spaces of the geometric realisation of the nerve.

David

On 25 August 2012 23:28, claudio pisani <pisclau@yahoo.it> wrote:
>
>
> --- 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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