5
$\begingroup$

Say that $\mathcal{X}$ is a conglomerate if $\mathcal{X} = \{X_i: i \in I\}$, where each $X_i$ and $I$ are classes. The Axiom of Choice for Conglomerates is the statement: Whenever $\mathcal{X}$ and $\mathcal{Y}$ are conglomerates and $f: \mathcal{X} \rightarrow \mathcal{Y}$ is a surjective map, then $f$ has a right inverse.

The "Axiom of Choice for Classes" and the (usual) "Axiom of Choice for Sets" may be defined in the obvious, analogous way (i.e., "Whenever $\mathcal{X}$ and $\mathcal{Y}$ are classes $\ldots$ etc." in the first case, and "Whenever $X$ and $Y$ are sets $\ldots$ etc." in the second).The Axiom of Choice for Classes implies the existence of a global choice function (meaning, a class function which is a choice function for the class of all non-empty sets), and the latter is, in turn, equivalent to the existence of a well-ordering of the universe.

Well, it is well-known that the Axiom of Choice for Conglomerates implies that "Every category has a skeleton" (see, e.g., Adamék/Herrlich/Strecker book). On the other hand, Isbell and Wright have proved in the 60's that the statement "Every category has a skeleton" implies the existence of a well-ordering of the universe.

My question is: considering the following assertions,

"The Axiom of Choice for Conglomerates"

and

"Every category has a skeleton"

are they equivalent statements ? At first glance, my conjecture is that the answer is "Yes", but I didn't find any reference for that.

Added (1): In Freyd/Scedrov book it is shown that the Axiom of Choice for Sets is equivalent to the statement "Every small category has a skeleton". So I also wonder whether there is some ladder of equivalences between forms of the Axiom of Choice and assertions regarding the existence of skeletons for certain categories. I mean, considering statements as

"Every small category has a skeleton",

"Every locally small category has a skeleton", and

"Every category has a skeleton", and maybe other similar statements of this kind,

could we put each one of them in correspondence with some equivalent form of the Axiom of Choice ? The first one is equivalent to the Axiom of Choice for Sets, as I have just commented.

Note: I have posted, some few hours later, the very same question at MathOverflow, but after that I was told that this was not the right procedure (I should wait a few days and then contact the moderators and ask the question to be migrated). Sorry about that, I won't do this again.

Added (2): After getting some comments it seems that the set-theoretic background should be more specified. The question was posed assuming the usual foundations of category theory - but, of course, this is a very debatable issue. In a first moment, we could think of ZFC + 2 inaccessibles, and then in such environment code the notions of: set, class and conglomerate. However, whether we are in first or second order seems also to have influence on this matter.

I was also told that, indeed, the Axiom of Choice for Classes is equivalent to the Axiom of Choice for Conglomerates: Eric Wofsey's argument (see his response) seems correct to me.

So, I guess that, indeed, my real question is: how can we define precisely a background (both categorically and set-theoretically) where we could analyze statements as the ones listed at the end of the question,

"Every small category has a skeleton",

"Every locally small category has a skeleton",

"Every category has a skeleton", etc.,

and what forms of the Axiom of Choice would these statements be equivalent to ?

$\endgroup$
1
$\begingroup$

I don't know what kind of background set theory you're working with, but it seems that for any reasonable choice, the "axiom of choice for conglomerates" is equivalent to the "axiom of choice for classes". A surjection $f:\{X_i\}_{i\in I}\to\{Y_j\}_{j\in J}$ of conglomerates can be interpreted as a surjective relation between the index classes $I$ and $J$ (namely, $i$ is related to $j$ if $f(X_i)=Y_j$), and so if you can well-order $I$, you can describe a right inverse (send $Y_j$ to $X_i$ for the $<$-least $i$ such that $f(X_i)=Y_j$, for some well-ordering $<$ of $I$).

$\endgroup$
  • $\begingroup$ Thanks Eric. The background of set theory I was thinking is the one who serves as foundation for category theory - but, of course, this is a very debatable issue. We could think of ZFC + 2 inaccessible cardinals, and then code in such environment the notions of set, class and conglomerate. But indeed, your argument looks fine, thanks a lot ! The possibility of well-ordering any class seems to go a long way. $\endgroup$ – Samuel G. Silva Oct 15 '15 at 17:58
  • $\begingroup$ Two inaccesibles is way too much. The intended interpretation of the set/class/conglomerate distinction needs much less – one inaccessible would be enough. $\endgroup$ – Zhen Lin Oct 15 '15 at 22:26
  • $\begingroup$ ... You are suggesting something as considering $V_{\kappa + 2}$ for $\kappa$ inacessible, that's it ? $\endgroup$ – Samuel G. Silva Oct 15 '15 at 22:40
  • $\begingroup$ If $\kappa$ is inaccessible then you can think of elements of $V_\kappa$ as sets, $V_{\kappa + 1}$ as classes, and elements of $V_{\kappa + 2}$ as conglomerates. $\endgroup$ – Zhen Lin Oct 16 '15 at 11:52
  • $\begingroup$ ... And probably, in order to discuss equivalences as above, consider that the Axiom of Choice holds only for sets and then investigate equivalences of the Axiom of Choice for classes. That's an idea. Thank you. $\endgroup$ – Samuel G. Silva Oct 16 '15 at 13:57

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.