Terminology of locally small categories without replacement

Terminology of locally small categories without replacement

[Colin McLarty]

Locally small categories are always defined as categories such that:

LS) for any objects A,B there is a set of all arrows A-->B.

When the base set theory includes the axiom scheme of replacement that
is equivalent to a prima facie stronger property:

??) for any set of objects there is a set of all arrows between them.

These two are not equivalent in the absence of the axiom scheme of
replacement.  There the second is much stronger, but it remains
important.  Is there a good term for it?

thanks, Colin


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Re: Terminology of locally small categories without replacement

[Vaughan Pratt]

On 12/1/2010 2:00 PM, Colin McLarty wrote:
> These two [weak and strong notions of locally small] are not
  > equivalent in the absence of the axiom scheme of
> replacement.  There the second is much stronger, but it remains
> important.  Is there a good term for it?

Sure: "Locally small."  In the absence of Replacement it would make more
sense to call the weaker concept "weakly locally small" than the
stronger one "strongly locally small" since it is presumably the strong
one that is more often intended.  As you say, Replacement identifies the
concepts, and one then defines the common concept with whichever
definition is shorter or simpler, namely the weak one.

A downside of allowing multiple set theories is the proliferation of a
menagerie of definitions.  Considerations like the above can help manage
the menagerie, though the benefit of the menagerie in the first place
would seem to accrue more to logic than to mathematics.  The role of
logic in mathematics should be to understand the latter, not to
complicate it.

Vaughan


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RE: Terminology of locally small categories without replacement

[F. William Lawvere]

Dear Colin,I think your stronger definition is the correct one, by analogy withother categories (where properness and other manifestations ofintensive and extensive objective quantities come up).  If your definition of 'category' itself is equivalent to 'internal category  C3=>C2->C1 x C1 in the category of classes ', then your notion seems to be a case of the condition (on E ->B) that the pullback of any small S ->B is small.The AXIOM of  replacement would perhaps be the extra condition on the universe that the pullbacks of S = 1 suffice to test the above, a condition that is perhaps appropriate for abstract constant discrete sets but not for cohesive variable ones.It would not be the 'scheme' of replacement that is relevant here since the category of objective classes (not their sometimes representing  subjective formulas) is directly under consideration.I presume that you are here trying to extend the Bernays-Mac Lane framework.It is not clear what  would result if we alternatively considered that a category C itself is just a formula, i.e objectively, a subset naturally defined in every model. Bill
> Date: Wed, 1 Dec 2010 17:00:51 -0500
> Subject: categories: Terminology of locally small categories without replacement
> From: colin.mclarty@case.edu
> To: categories@mta.ca
> 
> Locally small categories are always defined as categories such that:
> 
> LS) for any objects A,B there is a set of all arrows A-->B.
> 
> When the base set theory includes the axiom scheme of replacement that
> is equivalent to a prima facie stronger property:
> 
> ??) for any set of objects there is a set of all arrows between them.
> 
> These two are not equivalent in the absence of the axiom scheme of
> replacement.  There the second is much stronger, but it remains
> important.  Is there a good term for it?
> 
> thanks, Colin
> 

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RE: Terminology of locally small categories without replacement

[Michael Shulman]

A slightly different way of formulating this answer is in terms of
indexed categories / fibrations.  The standard definition of "locally
small indexed category" is, for a "naively Set-indexed category,"
precisely the stronger definition referring to all families (or sets)
of objects.

On Thu, Dec 2, 2010 at 7:18 AM, F. William Lawvere <wlawvere@hotmail.com> wrote:
>
> Dear Colin,I think your stronger definition is the correct one, by analogy withother categories (where properness and other manifestations ofintensive and extensive objective quantities come up).  If your definition of 'category' itself is equivalent to 'internal category  C3=>C2->C1 x C1 in  the category of classes ', then your notion seems to be a case of the condition (on E ->B) that the pullback of any small S ->B is small.The AXIOM of   replacement would perhaps be the extra condition on the universe that the pullbacks of S = 1 suffice to test the above, a condition that is perhaps appropriate for abstract constant discrete sets but not for cohesive variable ones.It would not be the 'scheme' of replacement that is relevant here since the category of objective classes (not their sometimes representing  subjective formulas) is directly under consideration.I presume that you are here trying to extend the Bernays-Mac Lane framework.It is not clear what  would result if we alternatively considered that a category C itself is just a formula, i.e objectively, a subset naturally defined in every model. Bill
>> Date: Wed, 1 Dec 2010 17:00:51 -0500
>> Subject: categories: Terminology of locally small categories without replacement
>> From: colin.mclarty@case.edu
>> To: categories@mta.ca
>>
>> Locally small categories are always defined as categories such that:
>>
>> LS) for any objects A,B there is a set of all arrows A-->B.
>>
>> When the base set theory includes the axiom scheme of replacement that
>> is equivalent to a prima facie stronger property:
>>
>> ??) for any set of objects there is a set of all arrows between them.
>>
>> These two are not equivalent in the absence of the axiom scheme of
>> replacement.  There the second is much stronger, but it remains
>> important.  Is there a good term for it?
>>
>> thanks, Colin
>>


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Re: Terminology of locally small categories without replacement

[Richard Garner]

Dear Jean,

> Could you please tell me:
> (i) Given a category S how does one chose the finitely complete 2-
> category K and the class C of small objects so that the locally small
> objects of K in your sense, are the locally small fibrations over S ?

Take K = Fib(S) and take C to be the representable fibrations. For me
this is actually the easiest way to remember the definition of locally
small fibration.

> (iii) What significant mathematical examples can you give of your
> notion ?

See (i).

Best regards,

Richard


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Re: Terminology of locally small categories without replacement

[JeanBenabou]

Dear Richard,

Thank you for your quick answer, which unfortunately is both  
incorrect and incomplete.

It is incorrect because Fib(S) does not have pullbacks or equalizers  
hence it not finitely complete
It is incomplete for two reasons:
(i) When I asked for significant mathematical examples, I meant apart  
from locally small fibrations, because I do not believe in abstract  
nonsense "generalizations" which have no genuine examples except well  
known special cases.
(ii) In my mail there was another question which you seem to have  
forgotten namely: What can you prove about the locally small objects  
of K, especially since you assume nothing on C ?
To complete my remark (ii) I would mind a little less the lack of  
genuine examples of this generalized notion if at least under the  
mere assumptions of Street on could prove a few non totally trivial  
results.
I would like to point out for example that, with Street's definition,  
one cannot even prove that a small object of K is locally small.

I'm sure that Ross, who gave this definition, will very soon give a  
correct and complete answer to the three questions I asked him in my  
previous mail.

Best regards,

Jean


Le 7 déc. 10 à 23:22, Richard Garner a écrit :

> Dear Jean,
>
>> Could you please tell me:
>> (i) Given a category S how does one chose the finitely complete 2-
>> category K and the class C of small objects so that the locally small
>> objects of K in your sense, are the locally small fibrations over S ?
>
> Take K = Fib(S) and take C to be the representable fibrations. For me
> this is actually the easiest way to remember the definition of locally
> small fibration.
>
>> (iii) What significant mathematical examples can you give of your
>> notion ?
>
> See (i).
>
> Best regards,
>
> Richard



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Re: Terminology of locally small categories without replacement

[Richard Garner]

Dear Jean,

> It is incorrect because Fib(S) does not have pullbacks or equalizers hence
> it not finitely complete

Yes, that's true; however Fib(S) does have PIE limits (or even just
bilimits) which is enough for Ross's definition to make sense. In
fact, one need not even assume the existence of any limits at all: an
object may be defined to be locally small just when, for every cospan
of arrows with small domain, the comma object exists and is again
small.

> It is incomplete for two reasons:
> (i) When I asked for significant mathematical examples, I meant apart from
> locally small fibrations.

In fact there are no other examples; the two notions are essentially
equivalent. Given the 2-category K with a class of small objects
therein, we can consider the full sub-2-category of K spanned by those
objects, and the underlying ordinary category C of this. Now each
object x in K induces a fibration p: E -->  C whose total category E
has as objects, morphisms f: c --> x in K with small domain; and as
arrows (f, c) --> (f',c'), pairs of a morphism h: c --> c' and a
2-cell f => f'h in K. It's now not hard to show that this fibration
will be locally small just when the object x is locally small in the
sense described by Ross (under the additional, and reasonable
assumption that the class of small objects is dense in K---which in
particular is the case when K = Fib(S) and C are the representable
fibrations).

Richard


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Re: Terminology of locally small categories without replacement

[Ross Street]

Dear Jean

Thank you for your message. I have been in Canberra and am only now
catching up with emails, reference writing and the like.
In the meantime, I see that you provided an answer to the original
question under this "Subject" (involving locally small fibrations) and
that Richard Garner has responded well to your questions showing the
close relationship between the concepts. I think the comma object
observation (I would not claim it competitively as "my notion"
particularly) was a helpful viewpoint for some researchers. As Richard
points out that, to make the definition, the 2-category K does not
really need any limits however comma objects are useful (in the same
way that internal category can be defined in any category but having
pullbacks makes one feel better). (I now notice in my message that C
somehow was mistyped later as R. As indicated, more conditions on C
give stronger consequences.)

I would be pleased to hear what you have in mind as some of the
significant and numerous results derivable using locally small
fibrations.

Walters and I were interested at one time in developing category
theory in a 2-category with an analogue P of the presheaf construction
("Yoneda structures"). Mark Weber has recently been able to make use
of some of these ideas in developing foundations for recent advances
in Batanin's operad theory. I believe my paper [The petit topos of
globular sets, J. Pure Appl. Algebra 154 (2000) 299-315] was some help
in this respect.

In any case, one example of a 2-category K is provided by a finitely
complete cartesian closed category E with an internal full subcategory
S; we take K = Cat(E) and Pa = [a opposite,S] where [ , ] is cartesian
internal hom in Cat(E).

In particular, we can take E = Cat so that K is the 2-category Dbl of
double categories. Given any internal full subcategory set of Set,
there is an internal subcategory fun of Cat which is the double
category of squares based on categories in set. The objects of fun are
categories in set, the horizontal and vertical morphisms are functors,
and the squares are natural transformations in the squares. The small
objects of K = Dbl are defined to be the double categories in set.

Best wishes,
Ross


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