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In 2013 Solomon Feferman in Foundations of unlimited category theory: what remains to be done (The Review of Symbolic Logic, 6 (2013) pp 6-15, link) laid out three desirable axioms for "unlimited category theory":

  1. There exists a category containing all objects of a given type (set, group, topological space, etc). It should literally include all such objects with no size limitations.
  2. For any two categories $A$, $B$, there should exist a functor category $B^A$.
  3. It should enable all standard constructions such as $\mathbb{N}$ and sums and products, etc.

Then in 2014 in The prospects of unlimited category theory: doing what remains to be done, 2014 (The Review of Symbolic Logic, 8 (2015) pp 306-327, link), Ernst discusses Feferman's program. Ernst shows that these axioms are inconsistent, by proving a version of Cantor's theorem for the category of reflexive graphs, thereby demonstrating a version of Cantor's paradox, showing that desired axioms are inconsistent.

He concludes that the search for such foundations is misguided, and so are objections to a ZFC-based set theoretic foundation for category theory.

To me this seems analogous to the situation with formal set theory with unrestricted comprehension in the days of Russell's paradox. Or the situation Girard's paradox for Martin-Löf type theory. Those formalisms were able to be saved. Is there any hope of salvaging Feferman's unlimited category theory?

What is the state of the art in category-theoretic foundations for category theory in 2018?

ziggurism
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    Paraconsistency is one solution. – user40276 Jun 07 '18 at 21:05
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    Are the titles in all caps really necessary? – David Roberts Jun 08 '18 at 02:33
  • I don't understand why the problems you are talking about are objections to a ZFC-based set-theoretic foundation for category theory. The only problem is the acceptance or not of the notion of Grothendieck universe and it seems that this answer explains how to get rid of this notion (but to be sure, it is required to reread carefully every use of Grothendieck universes in the mathematical literature and I did not do that). – Philippe Gaucher Jun 08 '18 at 07:09
  • @PhilippeGaucher I'm not a set theorist, but in some parts of mathematics (I'm thinking algebraic K-theory, but also topos theory) you might want to consider the large category of small categories and the huge category of (some flavour of) large categories [e.g. presentable categories, topoi,...]. I don't know how to do it rigorously without universes, and while one can imagine using the techniques in the answer you link to address that, I feel this is going to be impossibly cumbersome for anything nontrivial. – Denis Nardin Jun 08 '18 at 08:44
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    @DenisNardin And the link I give also says : "But should they do this? [i.e. using this approach] For most purposes, I don't think so. The main purpose of universes is as a simplifying device of convenience to stratify the full universe by levels, which can be fruitfully compared by local notions of large and small. This makes for a very convenient theory, having numerous local concepts of large and small." – Philippe Gaucher Jun 08 '18 at 08:51
  • And also for the OP, the answer below the link I gave : it talks about Feferman's axiom which is a conservative extension of ZFC. Note: I am not a set-theorist either, my knowledge in this field is therefore limited. – Philippe Gaucher Jun 08 '18 at 09:00
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    @PhilippeGaucher Perhaps I was unclear, but Ernst does not frame this inconsistency as an object to ZFC based foundation of category theory. No, quite the opposite: objections are misguided, since Feferman's axioms are inconsistent, it is unreasonable to expect ZFC to encode them. It is a refutation of objections to ZFC based category theory. – ziggurism Jun 08 '18 at 12:04
  • But I think Mike Shulman's Set theory for category theory https://arxiv.org/abs/0810.1279 describes some of the issues of ZFC based category theory, that makes me want to hold out hope for Feferman. – ziggurism Jun 08 '18 at 12:06
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    The formalisms were "able to be saved" by removing the problematic "object of all objects". In Feferman's case that means dropping his first desideratum, which makes it no longer "unlimited". (This is unrelated to the question of category-theoretic foundations for category theory.) – Mike Shulman Jun 08 '18 at 13:56
  • By the way, even if one has to iterate a countable number times (or even more) the use of classes (that is classes of classes of clases ...), you don't usually need the existence of a Grothendieck universe. One can usually find a conservative extension of ZFC by adding a constant symbol satisfying some conditions (about transitivity). This symbol will behave like a Grothendieck universe (except for second-order substitution and maybe a little more). – user40276 Jun 09 '18 at 01:00
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    And expanding in my previous comment, I'm not aware of any important inconsistency that can be derived from assuming the existence of an absolute sets of all sets if one excludes the ex falso quodlibet ($0 \rightarrow A$). – user40276 Jun 09 '18 at 01:05
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    It seems to me the reasons that size considerations are necessary are legitimate mathematical reasons, not foundational annoyances. Even if there is a formalism where one has a category of all sets, that category will still not have limits or colimits indexed by any category, or else it would be a poset. Since you will have to introduce a notion of smallness to correctly formulate completeness properties anyway, why not just take the notion seriously to begin with? – Tim Campion Jun 09 '18 at 05:35
  • @user40276 I have one, hopefully not too far-fetched :-). If you assume that the set of all sets is actually a set, all categories have a set of generators and cogenerators: all objects. Then e.g. a lot of miraculous left and right adjoints start "existing". – Philippe Gaucher Jun 09 '18 at 08:25
  • @PhilippeGaucher Yes, every limit preserving functor will have a right adjoint. But that actually seems make things easier. For instance, presentability that's a big issue in homotopy theory wouldn't be that much problematic anymore. Do you have a specific example in mind? – user40276 Jun 09 '18 at 17:26
  • @TimCampion To conclude that a small complete category is a poset you need the excluded middle. So stating this as a problem might be a little controversial I think. – user40276 Jun 09 '18 at 17:32
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    @user40276 Excluding EFQ doesn't save you, because of Curry's paradox. Assuming you have the axiom of separation, if there is a set of all sets then you can form the set $C = { x \mid (x\in x) \to P }$ for a arbitrary proposition $P$. Now reproduce Russell's paradox with $P$ in place of $\bot$. If $C\in C$, then by definition of $C$, $(C\in C)\to P$, whence $P$. Therefore, $(C\in C) \to P$; hence $C\in C$ by definition of $C$, and therefore $P$. So every statement is provable, making for a rather uninteresting theory. – Mike Shulman Jun 12 '18 at 23:24
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    You can also phrase this more categorically: if every limit-preserving functor between complete categories has a left adjoint, then every endofunctor of Set has an initial algebra and therefore (by Lambek's lemma) a fixed point. In particular, this is true for the covariant powerset functor $\Omega^{(-)}$, so there is a set $X$ such that $r : A\cong \Omega^A$. Now let $C = r^{-1}({x\in A \mid (x\in r(x))\to P})$ for an arbitrary statement $P$ and argue as before. – Mike Shulman Jun 12 '18 at 23:28
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    So mere paraconsistency isn't enough to make a nontrivial theory with a set of all sets; you have to drop or restrict the contraction rule $P \to P\wedge P$ as well. This is known by philosophers who argue for dialetheism/paraconsistency (search for "contraction" at https://plato.stanford.edu/entries/dialetheism/). It's also known that this is enough: nontrivial naive set theories with a set of all sets can be formulated in linear logic (and affine logic). – Mike Shulman Jun 12 '18 at 23:32
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    @MikeShulman "The formalisms were 'able to be saved' by removing the problematic 'object of all objects'". -- Yes, that's how ZFC and ML type theory solved it. But according to Ernst, Feferman works with NFU, a set theory which still does retain a "set of all sets". Instead he loses desideratum 3. Perhaps a deep understanding of how NFU resolves Cantor's paradox could lead to a good notion of unlimited categories, with #1 or #3 weakened but in a natural or nice way? – ziggurism Jun 13 '18 at 13:08
  • @MikeShulman (This is unrelated to the question of category-theoretic foundations for category theory.) -- what do you mean here? Would not a formulation of Feferman's axioms that did not have inconsistencies be a category-theoretic foundation? – ziggurism Jun 13 '18 at 13:10
  • @MikeShulman nontrivial naive set theories with a set of all sets can be formulated in linear logic -- Do you think linear logic may offer a lifeline to unlimited category theory? That would be very interesting. – ziggurism Jun 13 '18 at 13:12
  • @ziggurism I am not hopeful that there is any way to weaken #1 or #3 to obtain something that I would call "unlimited category theory", but since I don't have a definition of the latter I don't have a proof. However, consider the categorical version of Curry's paradox that I sketched above using an initial algebra for the covariant powerset functor. I can't see anywhere to break that argument that wouldn't either eviscerate the meaning of "unlimited" or of "category theory". – Mike Shulman Jun 13 '18 at 15:28
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    @ziggurism Re: linear logic, until recently I would have said no, that linear set theories are technically interesting but the prospect of re-doing significant amounts of mathematics in linear logic is unrealistic. However, https://arxiv.org/abs/1805.07518 is leading me to re-think that (and in particular makes me think I would at least know how to start, by giving some intuition for when to use additive or multiplicative connectives). – Mike Shulman Jun 13 '18 at 15:31
  • @ziggurism Aside from the sociological issues of convincing mathematicians that doing math in linear logic is possible, I think the main problem is the exponential modalities. Without some kind of exponential modalities, linear logic is too constrained to be able to express very much; but with unrestricted exponentials, naive comprehension becomes inconsistent again. One possibility are "light logics" such as "light linear logic" and "soft linear logic" that have weaker exponentials; but I'm not sure whether they are strong enough for substantial mathematics. – Mike Shulman Jun 13 '18 at 15:34
  • @ziggurism I'm not sure what you mean by a "category-theoretic foundation". I would call Feferman's NFU-like foundation also a "set-theoretic foundation" for category theory, because the foundational theory uses sets as its basic objects and basic intuitions. The sets of NFU are different from the sets of ZFC, but they're still sets. By a "category-theoretic foundation" for category theory I would mean a foundation whose basic objects are either categories or at least uses some categorical intuition: ETCS is somewhat more in this direction, ETCC even more so. – Mike Shulman Jun 13 '18 at 15:35
  • @ziggurism To me this is an orthogonal dichotomy to unlimitedness: ETCS and ETCC contain size distinctions just like ZFC. – Mike Shulman Jun 13 '18 at 15:36
  • @MikeShulman Thanks for the example. I don't know much about paraconsistent stuff besides what other people have told me, so I haven't noticed this problem. But I was aware that EFQ alone was usually not enough whenever one wants to do paraconsistent stuff. By the way, instead of weakening contraction or weakening modus ponens, it's also possible to weak full comprehension by allowing only positive formulae. People usually call this positive set theory. It has some non trivial interesting topological models. – user40276 Jun 14 '18 at 06:18
  • By the way, a survey of some (but not all known) possible alternatives http://apcz.umk.pl/czasopisma/index.php/LLP/article/view/LLP.2003.004 . – user40276 Jun 14 '18 at 06:52
  • @TimCampion what would it mean to "take the notion [of smallness] seriously to begin with"? – ziggurism Jan 27 '23 at 13:40
  • It would mean to work in more standard foundations, with an explicit notion of smallness (as opposed to Feferman's foundations, which seems to be set up with the goal of "not having to ever talk about size" -- I'm saying this goal seems misguided to me). – Tim Campion Jan 27 '23 at 16:17
  • @TimCampion so Feferman's axioms should have included an axiomatization of category size. Does such an axiomatization exist today? – ziggurism Jan 28 '23 at 17:13

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