Re: question on terminology

Re: question on terminology

[Fred E.J. Linton]

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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Re: Re: question on terminology

[Colin McLarty]

I'm with Fred on this.

Colin



On Fri, Aug 24, 2012 at 11:35 PM, Fred E.J. Linton <fejlinton@usa.net> wrote:
> 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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