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A concretization of a category is a faithful functor to the category of sets. A category is concretizable if there exists such a functor.

An evident necessary condition for concretizability is local smallness. Of course, for many local smallness is part of the definition of category. In the 1960s, Isbell exhibited a further necessary condition and conjectured its sufficiency.

The following theorems were established in the early 1970s:

1) (Freyd) Isbell's necessary condition for concretizability is sufficient.

2) (Kučera) Every category is a quotient of a concretizable category by a congruence.

Precise definitions, statements, and proofs may be found in Chapter 6 of "Theory of Mathematical Structures" by Adámek. These results are formulated in the language of class-set theory and proved in GB+Global Choice. The arguments rely on having an appropriate well-ordering of the class of all sets.

Another result from the same period is:

3) (Freyd) The Homotopy Category is not concretizable.

By definition, this category is a quotient of a concretizable category. Here the evident weakening to a schema (asserting that no formula yields a concretization) is formalizable in the language of set theory, and the argument appears to carry over as well.

Are the evident weakenings of assertions 1) and 2) to schemata (for definable categories, concretizations and congruences) provable in ZFC?

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    $\begingroup$ Could you add precise statements or precise references? $\endgroup$ Commented Sep 10, 2013 at 0:33
  • $\begingroup$ @FrançoisG.Dorais, the Isbell condition was formulated in "Two set-theoretical theorems in categories" by Isbell (matwbn.icm.edu.pl/ksiazki/fm/fm53/fm5315.pdf), and the Freyd's theorem is stated in "CONCRETENESS" by Freyd (sciencedirect.com/science/article/pii/0022404973900315), though I don't think the question is interesting... $\endgroup$ Commented Sep 10, 2013 at 10:42
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    $\begingroup$ @Adam not sure how I missed this. Yes, 1) and 2) are theorems in ZF (say relativised, or for definable locally small categories) or NBG minus choice; 2) is definitely a theorem even in a slightly weaker version of Algebraic Set Theory assuming the category of classes is boolean. I believe so is 1), if we add in an axiom about a cumulative hierarchy that always holds for ZF. $\endgroup$ Commented Aug 16, 2023 at 5:35
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    $\begingroup$ Set-like linear orders ("small" in AST parlance) and well-ordered classes are kind of orthogonal generalisations of well-ordered sets, mind. $ORD+ORD$ is well-ordered, but not set-like, and any linearly-ordered set is clearly set-like (eg the rationals, with the usual order). $\endgroup$ Commented Aug 18, 2023 at 12:32
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    $\begingroup$ @Adam I think I meant relativised to a given model of ZF. Getting a function constant on equivalence classes is easy: take a constant function! Getting on that takes distinct values on different equivalence classes is trickier. I think you need the ranks to be well-ordered, or some other mechanism that avoids choices to find sets of representatives of all equivalence classes. $\endgroup$ Commented Aug 20, 2023 at 2:49

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It's over a year after I said I would write it, but here's a summary of my results with Martti Karvonen:

tl;dr No choice is needed

For the following, if working in ZF then a 'large, locally small category' should be taken to mean a definable category: the objects and the morphisms are each defined by a formula representing a class, and the source, target, composition and identity arrow functions are class functions also so encoded (because one is really working with the syntactic category of ZF). With that caveat:

  1. Freyd's theorem holds in both ZF, in NBG sans AC, and even in a weak version of set theory whenever Scott's trick is available (for instance assuming an axiom of hierarchy, so there is a small rank function $V\to ORD$).
  2. Kučera's theorem holds in ZF, in NBG sans AC, and even in slightly weaker assumptions than Freyd's theorem, namely in a set theory where the universe $V$ has a preorder such that for each set $x \in V$, the class $\{y\in V \mid y \leq x\}$ is a set.

These results are achieved by working in a category of classes à la algebraic set theory + LEM, and so they aren't specific to the foundational systems mentioned above, but whatever foundations whose models are categories with the structural properties of AST+LEM. In particular they will also apply in any foundation based ultimately on boolean toposes that can talk about large objects, so for instance if one is working relative to universes and with a model of ETCS coming from the small sets relative to one universe. One can also get finitary versions where 'small' means finite and 'proper class' means 'infinite set', or even just countably infinite.

For a few more details see my talk at Un colloque en hommage à Jean Bénabou, How to be concrete when you don't have a choice. The paper is on the long to-write list.

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