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One version of the Axiom of Collection says that any surjection $A\to B$ from a class $A$ to a set $B$ is factored through by some surjection $C\to B$ where $C$ is a set.

Note that assuming $B$ is a set, the axiom of replacement ensures that $C$ is a set if and only if each fiber of $C\to B$ is a set (i.e. the map $C\to B$ is "small" in the sense of algebraic set theory). Thus it is natural to consider the following "class-collection axiom": any surjection $A\to B$ of classes is factored through by some surjection $C\to B$ whose fibers are all sets.

Has this "class-collection axiom" been studied at all? Assuming classical logic, it seems to follow from the axiom of foundation in the same way that collection follows from replacement and foundation: let the fiber of $C$ over $b\in B$ consist of all those $a\in A$ lying over $b$ and of minimal rank. Can it be proven in any intuitionistic context?

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I am pretty sure I do not understand the question. However, I feel that "Scott's trick" (pick a representative of minimal rank out of each equivalence class) means that the answer is no for set theories with Foundation. You might look at those texts (Aczel, maybe?) whose authors work without Foundation. Your question may be of some interest there. Gerhard "Ask Me About System Design" Paseman, 2011.10.11 –  Gerhard Paseman Oct 12 '11 at 5:34
    
I haven't found any reference to such a thing in the anti-foundation literature that I've looked at. –  Mike Shulman Oct 12 '11 at 16:28
    
Also, can you say what you don't understand about the question, so that I can attempt to clarify it? –  Mike Shulman Oct 12 '11 at 16:28
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Mike, I wonder if you have tried to prove that this holds in any known models of intuitionistic set theory? E.g., I would be tempted to look at these models in categories of ideals that have been considered in the algebraic set theory literature (such models certainly satisfy collection). –  Michael A Warren Oct 13 '11 at 13:26
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Hmm, assuming my interpretation is right, then what happens for the specific case $\chi(x) :\leftrightarrow \exists y \phi(x,y)$ ? The existence hypothesis is trivial and we have no useful witnessing information on the LHS to work with, unlike in the set-collection case. –  Daniel Mehkeri Oct 15 '11 at 1:55
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1 Answer

Maybe in

A. Joyal, I. Moerdijk: A completeness theorem for open maps, Annals of Pure and Applied Logic 70 (1994) 51-86

an axiom related to this occurs - but its study is not the main purpose of this paper. So I'm looking for a more extensive treatment of it myself.

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