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Colin McLarty | 1 Dec 2010 23:00
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Terminology of locally small categories without replacement

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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Vaughan Pratt | 2 Dec 2010 15:01
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Re: Terminology of locally small categories without replacement


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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Ellis D. Cooper | 2 Dec 2010 15:55

Severe Strict Monoidal Category Naivete

(1) Is strict monoidal category the same as monoid in category of categories?
(2) Is it not true that in a strict monoidal category if
$X\xrightarrow{f}Y\xrightarrow{g}Z$ then $f\square g= g\circ f$?
(3) Is the pentagon axiom automatically satisfied in a strict
monoidal category?

Many thanks for your patience and pointers.

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F. William Lawvere | 2 Dec 2010 16:18
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RE: Terminology of locally small categories without replacement


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 <at> case.edu
> To: categories <at> 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?
>
(Continue reading)

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Steve Lack | 3 Dec 2010 03:54
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Re: Severe Strict Monoidal Category Naivete

Dear Ellis,

On 03/12/2010, at 1:55 AM, Ellis D. Cooper wrote:

> (1) Is strict monoidal category the same as monoid in category of categories?

Yes. 

> (2) Is it not true that in a strict monoidal category if
> $X\xrightarrow{f}Y\xrightarrow{g}Z$ then $f\square g= g\circ f$?

If I understand correctly, you have arrows f:X->Y and g:Y->Z and you 
are comparing the tensor products f <at> g:X <at> Y->Y <at> Z and g <at> f:Y <at> X->Z <at> Y.
They have different domain and codomain, so cannot be equal.

If you considered a commutative monoid in the category of categories, then these arrows would be equal. But
such commutative monoids are very rare. 

> (3) Is the pentagon axiom automatically satisfied in a strict
> monoidal category?
> 

Yes. In that case it asserts that two identity arrows with the same domain and codomain
are equal. 

Steve Lack.

> Many thanks for your patience and pointers.
> 
>
(Continue reading)

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Michael Shulman | 3 Dec 2010 05:15

RE: Terminology of locally small categories without replacement

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 <at> 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 <at> case.edu
>> To: categories <at> 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:
(Continue reading)

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JeanBenabou | 3 Dec 2010 09:07
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Locally small categories without replacement, or anything


Dear Colin,

I gave 36 years ago a definition of locally small fibration over an  
arbitrary base category S. Now look at the following very special  
case: S = Set, C is a category, and  P: Fam(C) --> S the canonical  
fibration where the fiber over the set I is the category C^I. To say  
that this P is locally small in my sense coincides exactly with the  
"more general" notion of C being locally small that you suggest.
Note that my definition of locally small fibration does not suppose  
that S has a terminal object 1, let alone that 1 is a strong  
generator in S. To show that this definition is equivalent to the  
"usual" one, you need not only a replacement scheme in Set, but also  
the fact that 1 is a strong generator in Set.
Thus I think that the correct general definition of "local smallness"  
is the one I gave for fibrations.
As a side important remark, the identity fibration Id(S): S --> S is  
always locally small without any assumption on S, in particular  S  
need not have a terminal object, pull-backs or any kind of limit.  
None of this is true with any of the "variants" of my definition you  
can find e.g. in the Elephant, where you have to assume that S has  
finite limits.
Thus "evil" fibrations can be interesting after all.

Best to all,

Le 1 déc. 10 à 23:00, Colin McLarty a écrit :

> Locally small categories are always defined as categories such that:
>
(Continue reading)

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Ellis D. Cooper | 3 Dec 2010 16:51

Severe Strict Monoidal Category Naivete

Dear Steve,

At 09:54 PM 12/2/2010, you wrote:
If I understand correctly, you have arrows f:X->Y and g:Y->Z and you
are comparing the tensor products f <at> g:X <at> Y->Y <at> Z and g <at> f:Y <at> X->Z <at> Y. They
have different domain and codomain, so cannot be equal.

I was thinking more about f:X->Y and g:Y->Z and tensoring  f with the
identity morphism of Y to get f <at> 1_Y:X <at> Y->Y <at> Y, and also tensoring 1_Y
with g to get 1_Y <at> g:Y <at> Y->Y <at> Z. So I get the composition f <at> 1_Y followed
by 1_Y <at> g is a morphism from
X <at> Y->Y <at> Z, and you made me realize that f followed by g as a morphism
X->Y cannot possibly equal f <at> 1_Y followed by 1_Y <at> g.

Then again, if the ambient strict monoidal category is symmetric, so
that the latter composition is a morphism X <at> Y->Z <at> Y, then to my mind
somehow this is pretty much the same as the composition X->Z of f
followed by g, basically because only the identity morphism of Y is involved.

The context of my inquiry is chemical reaction, as suggested by John
Baez a while ago. That is, if f and g are chemical reactions that
transform X to Y and Y to Z, respectively, then the net effect is
just transformation of X to Y, since the Y produced by f is
completely consumed by g. Bottom line: I would like a correct way to
say that tensoring f with an identity morphism is somehow no different from f.

Ellis

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(Continue reading)

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David Roberts | 5 Dec 2010 00:44
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Re: Severe Strict Monoidal Category Naivete

  Hi Ellis,

  to push the chemical reaction analogy further, tensoring with an identity
  morphism 1_Y is like having a chemical present that doesn't take part in the
  reaction: it's there are the beginning and end, but doesn't change. But you
  can't take it away (and no, it's not like a catalyst, in that your original
  arrow f was there to begin with), unless perhaps Y has some sort of dual or
  (weak) tensor inverse, and by now the analogy is stretched beyond breaking
  point.

  David

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Ellis D. Cooper | 4 Dec 2010 15:00

Severe Strict Monoidal Category Naivete

Hi David,

At 11:50 PM 12/3/2010, you wrote:
>to push the chemical reaction analogy further, tensoring with an
>identity morphism 1_Y is like having a chemical present that doesn't
>take part in the reaction: it's there are the beginning and end, but
>doesn't change. But you can't take it away (and no, it's not like a
>catalyst, in that your original arrow f was there to begin with).

I agree completely. All I am saying is that since there is no effect
on the reaction by the presence of a neutral chemical, it might just
as well not be mentioned. Perhaps what I am getting at is a quotient
category in which X <at> Y->Z <at> Y is identified with X->Z in this specific
situation. I do believe this is how chemists think of their
applications of Hess' Law, which is what my inquiry is really all about.

Ellis

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