Re: question on terminology

Re: question on terminology

[claudio pisani]

--- 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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Re: question on terminology

[David Roberts]

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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Re: question on terminology

[Fred E.J. Linton]

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



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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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question on terminology

[claudio pisani]

Dear categorists,

Is there a standard name for those presheaves X on a  category C such that Xf is a bijection for any f in C?
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: Question on terminology

[Steve Lack]

I would suggest "cancellative categories" - I believe the corresponding term
"cancellative monoid" is standard.

Steve Lack.


On 5/02/10 4:39 AM, "lamarche" <lamarche@loria.fr> wrote:

> 
> Is there a standard accepted name for categories all whose morphisms
> are both epi and mono? This includes groupoids, posets and preorders,
> along with free categories (from graphs), so it's not a trivial class
> at all.
> 
> I am leaning towards calling them "integral categories", by analogy
> with integral domains, but Google searches have been frustrating.
> Another possiblility would be "cancellation categories" (they have
> both the left and right cancellation property), but I'm not sure I
> like this one.
> 
> Is there anything about these in the literature already?
> 
> Thanks in advance,
> 
> François
> 
> 
> 
> 
> 
> 
> 
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Re: Question on terminology

[Toby Bartels]

lamarche wrote:

>Is there a standard accepted name for categories all whose morphisms are
>both epi and mono?

...

>Another
>possiblility would be "cancellation categories" (they have both the left
>and right cancellation property), but I'm not sure I like this one.

I believe that the usual term in monoid theory
would be "cancellative monoid" (not "cancellation monoid").
Google gets relevant hits for "cancellative category",
although that might not appeal to you any better.


--Toby


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Question on terminology

[lamarche]

Is there a standard accepted name for categories all whose morphisms  
are both epi and mono? This includes groupoids, posets and preorders,  
along with free categories (from graphs), so it's not a trivial class  
at all.

I am leaning towards calling them "integral categories", by analogy  
with integral domains, but Google searches have been frustrating.  
Another possiblility would be "cancellation categories" (they have  
both the left and right cancellation property), but I'm not sure I  
like this one.

Is there anything about these in the literature already?

Thanks in advance,

François







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