Timeline for Independence and Category Theory
Current License: CC BY-SA 2.5
25 events
| when toggle format | what | by | license | comment | |
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| Oct 18, 2010 at 22:52 | comment | added | Andrej Bauer | @burned: Ok, last attempt: are you worried about the existence of the triangle which I defined at the end of my answer? If not, why are you worried about limits of small diagrams? If yes, well then we are closer to discovering what bothers you. | |
| Oct 18, 2010 at 22:51 | comment | added | Andrej Bauer | @burned: Excuse me, but you asked explicitly "How can you define the limit of $F$ inside $\mathrm{Set}$?" This means you explicitly worried that somehow independence phenomena prevent us from constructing limits of small diagrams. They do not. All I can say is that you are confused in some trivial issue, but unfortunately I cannot see what the issue is (and certainly it is unrelated to Category theory). I thought I answered your problem. Anyhow, I am not going to prolong this discussion anymore. Talk to me privately if you will. | |
| Oct 18, 2010 at 19:34 | comment | added | Todd Trimble | This discussion should probably be brought to a close (if you want you can write me at topological dot musings at gmail dot com), but in which statement of mine does it seem to you that I am missing the well-known fact that independence phenomena are embedded in theories? The question as to "how these issues are dealt with in category theory" has been partially addressed by Francois, but you seem to be advancing the idea that "category theory" gets it wrong somehow. I guess there are inexperienced category theorists who get it wrong sometimes, but I don't see that's happened here. Bye! | |
| Oct 18, 2010 at 19:31 | comment | added | Harry Gindi | The key point here is that $D$ is provably small in ZFC. I know basically no model theory, but the tiny tiny bit of it I've picked up is this: a statement $R$ is provable in a theory if and only if it is true in every model of that theory. | |
| Oct 18, 2010 at 19:26 | comment | added | Harry Gindi | In particular, we can define the forgetful functor $D\to Set$ independently of $D$ by restriction of the forgetful functor from the category of all (topological spaces, groups). In particular, as we showed above, this functor now has a limit since its domain is small. But then we're done by the earlier observations. | |
| Oct 18, 2010 at 19:24 | comment | added | Harry Gindi | Since the functoriality of this new operation is not in question (I'm not about to check compatibility conditions for you. They're straightforward), we have given a characterization of the limit that depends only on the smallness of $D$. By my (and Andrej's) assumptions, if we put an upper bound on the size of the diagram $D$ of (suslin objects, whitehead groups), no matter what set theory we pick, $D$ is provably small within $ZFC$. This means that given any functor (including the forgetful functor) from $D\to Set$, we can take its limit. | |
| Oct 18, 2010 at 19:19 | comment | added | Harry Gindi | $G(d)^F(d)$ is also a set with cardinality $\lambda_d$, the set of all $\gamma$-tuples of functions as above has cardinality $\prod_{d\in Ob(D) \lambda_d$ which is well-defined since $D$ is small. The set of natural transformations is a subset of this set (satisfying the compatibility condition for natural transformations), so our resulting category is locally small. Now, given a small category $D$, we define a functor $lim^D:Set^D\to Set$ defined by sending a functor $F$ to the homset $Hom_{Set^D}(*,F)$. | |
| Oct 18, 2010 at 19:14 | comment | added | Harry Gindi | @burned: I will give a list of provable statements: Fix ZFC as the background theory of sets. Let $Set$ be the category of ZFC sets. Let $D$ be any small category. We may form the functor category $Set^D$ where the objects of this category are functors $D\to Set$ and the morphisms are natural transformations. To see that the hom-sets form a legitimate set, notice that since $Ob(D)$ is a set ((with cardinality $\gamma$), and given two functors $F,G\in Ob(Set^D)$, we can look at the set of all $\gamma$-tuples of functions $\{F(d)\to G(d)\}_{d\in D}$. Since each homset | |
| Oct 18, 2010 at 19:11 | comment | added | Not Mike | And my question pertains to how these issues are dealt with in category theory. | |
| Oct 18, 2010 at 19:11 | comment | added | Not Mike | I'm sorry for any confusion of the matter at hand. My question was a poorly stated attempt to ask: how things which are independent of the theories of ZFC+I, or any other extension of ZFC (NOTE: I'm not talking about any particular model of set theory, again a point of confusion I should have cleared up) are handled within category theory. I was not asking for the particular definition of the limit and colimit, or that they are always defined. The very subtle point you and Harry seem to be missing is that: any system of axioms worth anything will always have statements independent of it. | |
| Oct 18, 2010 at 18:17 | comment | added | Todd Trimble | ("Penultimate" now referring to the one beginning "no, if the set you have defined...") | |
| Oct 18, 2010 at 18:15 | comment | added | Todd Trimble | @burned: your penultimate comment seems very confused. I'll reply to two things though. (1) "if the set you have defined exists or doesn't (in any form) then you have just made an assertion that extends beyond ZFC". What set? Say, the set of subsets of the reals whose cardinality is uncountable but less than the continuum? That set of subsets exists (using separation), even though such subsets may not. (2) "my problem is with logical consequences of being able to actually produce the value of such a limit". Whoever said that you can (always)? | |
| Oct 18, 2010 at 17:51 | comment | added | Not Mike | An even better example of what I'm talking about, I guess would be whether or not the subcategory of Top, which consist of topological spaces with the countable chain condition is closed under topological products. | |
| Oct 18, 2010 at 17:46 | comment | added | Not Mike | This is clear when you are within ZFC. The limits we have talked about thus far would be labeled as independent of ZFC, and any discussion of them would have to take place in an extension of ZFC which decides the elements. | |
| Oct 18, 2010 at 17:42 | comment | added | Not Mike | The answer I was hoping for was one that actually dealt with the existence a of notion of independence from the axioms, and that would allow for independence results from ZFC to transfer to CT (as they in some sense should). | |
| Oct 18, 2010 at 17:39 | comment | added | Not Mike | No, if the set you have defined exists or doesn't (in any form) then you have just made an assertion that extends beyond ZFC. Moreover, I understand that you can compute small limits or colimits within ZFC, my problem is with logical consequences of being able to actually produce the value of such a limit. By asserting you know the value, you have asserted that the elements take a particular form. (And in fact have left ZFC) | |
| Oct 18, 2010 at 17:30 | comment | added | Todd Trimble | @burned: in ZFC it is provable that small diagrams have limits and colimits. (The completely different question of whether the set of Suslin lines is empty or nonempty is different to the existence of that set. Are you confusing existence of the set with existence of elements of the set?) | |
| Oct 18, 2010 at 17:20 | comment | added | Not Mike | of such objects. But once you have done either, you are no longer working inside of just ZFC. | |
| Oct 18, 2010 at 17:17 | comment | added | Not Mike | You two don't seem to understand what I'm looking for: Namely whether category theory can correctly handle independence results. That is to say, without actually trying to compute or define objects that depend on them. That being said the examples you have given me are fine, and your answer is a nice attempt. The only problem is they rely on the assumption that you can make a choice about the existence of such objects. That is to say they only make sense when you have asserted the existence or non-existence. | |
| Oct 18, 2010 at 14:39 | comment | added | Andrej Bauer | Yes, I know I just rewrote Todd's anwser and your comments. But I think in this case the answer should be as specific and down-to-earth as possible, so I thought it was worth doing. | |
| Oct 18, 2010 at 13:36 | comment | added | Harry Gindi | In fact, it's almost identical to the comment I made on Todd's post. +1 again (this time in imaginary votes). | |
| Oct 18, 2010 at 13:34 | comment | added | Harry Gindi | This is the correct answer to the question (more aptly put than mine, as well). It should be voted up. | |
| Oct 18, 2010 at 13:33 | comment | added | Andrej Bauer | Now I fill dizzy for having used the law of excluded middle in such a horrible way. | |
| Oct 18, 2010 at 13:29 | history | edited | Andrej Bauer | CC BY-SA 2.5 |
better explanation; added 28 characters in body
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| Oct 18, 2010 at 13:24 | history | answered | Andrej Bauer | CC BY-SA 2.5 |