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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Colin When a class C of small objects has been distinguished in a finitely complete 2-category K, an object x of K is locally small when each cospan a : u --> x <-- v : b with u and v small has a small comma object a/b. It is only seldom that R has an object w such that this gives a logically equivalent definition by restricting to u = v = w. Best wishes, Ross On 02/12/2010, at 9:00 AM, Colin McLarty wrote:
??) for any set of objects there is a set of all arrows between them.
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Ross, I'm sure you know what a locally small fibration P: X --> S is. You also know that this is a very strong notion from which many many results can be derived, especially if you make very mild assumptions on S, e.g. that S has pullbacks. In your mail you give a the following definition:
When a class C of small objects has been distinguished in a finitely complete 2-category K, an object x of K is locally small when each cospan a : u --> x <-- v : b with u and v small has a small comma object a/b. It is only seldom that R has an object w such that this gives a logically equivalent definition by restricting to u = v = w.
Best wishes, Ross
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 ? (ii) What can you prove about the locally small objects of K, especially since you assume nothing on C ? (iii) What significant mathematical examples can you give of your notion ? Best wishes, Jean [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (7)
-
Colin McLarty -
F. William Lawvere -
JeanBenabou -
Michael Shulman -
Richard Garner -
Ross Street -
Vaughan Pratt