When the definition of a set starts to matter in category theory

Question

In most introductory courses to category theory, the precise definition of a set is more-or-less ignored. The idea being that all basic results in the subject hold for any reasonable definition of a set.

At some point one feels that this must cease to be the case. So what is the simplest (a simple) example of a categorical concept or result that actually relies on the details of the definition of a set that one is using?

As you said, "The idea being that all basic results in the subject hold for any reasonable definition of a set." — Paul Taylor, Feb 12 '21 at 13:23
@MaximeRamzi: A set is an object in a category of sets. A category of sets is a category with finite limits, powerobjects, natural numbers objects, and possibly also well-pointed, Boolean, and satisfying the axiom of choice. — Dmitri Pavlov, Feb 12 '21 at 15:26
@DmitriPavlov : but then what's a category (hom-sets), a finite limit, etc. ? My point was that no one defines the word "set", or at least the primitive in their theory (e.g. a type if you like type theory) because it doesn't really make sense to do so, they're closer to syntactic objects. Maybe the OP actually wanted to ask about the set theory we use, rather than the definition of set ( I guess if I had to answer I would answer with which set theory we use) — Maxime Ramzi, Feb 12 '21 at 15:31
@MaximeRamzi: You do have hom-sets, but as a defined notion: using the listed properties, you can prove that the category is cartesian closed. To define a category, one does not need hom-sets, only individual morphisms. Finite limits reduce to pullbacks and terminal objects, and neither one requires sets or finite sets. — Dmitri Pavlov, Feb 12 '21 at 15:34
The excellent survey by Mike Shulman at https://arxiv.org/abs/0810.1279 presents a couple of options for set theories to use for category theory. — Ingo Blechschmidt, Feb 12 '21 at 15:56
@MaximeRamzi: See also the nLab article fully formal ETCS for a presentation that does not make any use of sets whatsoever. — Dmitri Pavlov, Feb 12 '21 at 16:03
@DmitriPavlov that doesn't define what a set is more than saying "it's something that satisfies ZF(C)" does.Ultimately, this is still only defining what set theory we're using. (note that I'm not saying it's a bad thing, or that we should try to define what a set is, but it is what the question asks). I do think that it is the more reasonable question to try to answer — Maxime Ramzi, Feb 12 '21 at 16:46
@MaximeRamzi: Your objection is equally applicable to the definition of a vector as an element of a vector space. The latter also does not define what a vector "is", only that it is an element of some ambient structure, which is defined. I fail to see how sets are any different from vectors. — Dmitri Pavlov, Feb 12 '21 at 17:15
@DmitriPavlov : as pointed out in the comments below Tim's answer, "a vector" is not a well-defined thing, anything is a vector (anything is an element of a vector space if you let me pick the vector space). "A vector in $V$" is completely well-defined if you defined $V$. As suggested again in those comments, in the most common math foundation, everything is defined except precisely what a set is (or a type, etc.). My view is that it does not matter and that it is not a sensible question, but still. — Maxime Ramzi, Feb 12 '21 at 17:28
@MaximeRamzi: A vector is an element of a vector space, i.e., a map of sets u: 1→U(V), where 1 is the terminal object in a category of sets and U(V) is the underlying set of a vector space V, or equivalently (by adjunction), a morphism of vector spaces v: k→V. Thus, any vector v knows what vector space it lives in: it's precisely codom(v). — Dmitri Pavlov, Feb 13 '21 at 00:15
A set is a collection of things. That’s what it is. It comes from ordinary language and human intuition. The real question is what are the rules and axioms for forming sets. — Monroe Eskew, Feb 13 '21 at 15:09

score 17 · Accepted Answer · edited May 06 '22 at 08:39

I didn't have time to write up a proper answer on initially seeing your question and much of what I have to say has been said in the comments and Tim's answer, but I'll still offer some specifics on things mentioned in the comments.

The definition of a set is, as Tim Campion pointed out, the axioms of whatever set theory you're working in. They do not so much define a set as describe a primitive notion that behaves like what we think sets should behave like, with different axiomatizations giving rise to differing notions of set.

These differing axiomatizations do have an impact on category theory because they change the behavior of ${\bf Set}$, and consequently the behavior of presheaf categories, and they also impact our ability to manipulate large categories. As mentioned by Ingo Blechschmidt in the comments, Mike Shulman has an excellent paper surveying some of these consequences. I will summarize some of them here, but I highly recommend you check out his paper.

A striking result referenced in the Shulman paper is due to Colin McLarty, establishing that the NF axiomatization of what a set is yields a ${\bf Set}$ that isn't Cartesian closed.

In ZFC we really only run into issues if we want to manipulate large categories as a whole, for example ${\bf Set}$ or ${\bf Group}$, which are not actually objects in ZFC since they're proper classes. We can get around this with shenanigans about formulas the metalanguage, but anyone looking for an integrated and 'natural' treatment of large categories on level footing with small ones will be disappointed in this setting.

NBG is a conservative extension of ZFC (meaning it doesn't prove anything about sets that ZFC can't) which does allow proper classes to be real objects in the theory, but we still run into some discomfort when dealing with large categories. NBG manages to be conservative over ZFC by restricting it's comprehension axiom to only apply to sets, not proper classes -- in practical terms, as Mike points out in his paper linked above, this means (for example) that we can't prove by induction that a large category $\mathcal{C}$ has an $n$-fold Cartesian product $\mathcal{C}^n$. We can get around this by constructing it directly as the category of functions from $n$ into $\mathcal{C}$, but the unavailability of canonical proof methods like induction is troubling.

MK is a non-conservative extension of ZFC, essentially NBG but with full class comprehension allowed so we have access to all of the standard proof tools for large categories. This new theory can prove things ZFC can't, like the consistency of ZFC, and is thusly strictly stronger in a meaningful sense. MK also has its own serious issues when working with large categories -- we can't define the category of functors between two large categories, and this applies to NBG as well.

Using full MK further suggests that we try to look at the category of classes, since they're really the category of collections we want to work with right? And bam, once again we're back to a situation where we have to play games in the metalanguage, or conservatively extend/step up the consistency strength of our theory. This leads mathematicians to situations like Grothendieck universes, where it's always possible to step up to the next universe if we need to talk about 'all the somethings' in the current universe. This is equivalent to working in ZFC plus an axiom asserting the existence of an inaccessible cardinal.

All the extra baggage of universes or inaccessibles is still somewhat of a sledgehammer for the problem at hand, though; all we want is for large categories to 'be like small categories' in enough ways that we can carry out all the constructions we care about with large categories, but inaccessibles or Grothendieck universes also have a plethora of other consequences (like the need to juggle universes¹). A solution to these problems comes in the form of reflection principles, which are essentially axioms asserting that proper classes look enough like sets that we don't have to soil ourselves when they appear, but don't endow them with enough independence to give rise to a whole hierarchy of universes we need to ask questions about. All of this is discussed at length in the Shulman paper referenced above, with additional references therein.

First paper: Shulman, Mike. Set theory for category theory. arXiv:0810.1279v2 [math.CT]

Second paper: McLarty, Colin. Failure of Cartesian Closedness in NF. J. Symbolic Logic 57 (1992), no. 2, 555--556. https://doi.org/10.2307/2275291

¹As Tim points out in the comments, how many universes we have to juggle when taking this route is up to us. Skilled jugglers may use an infinite number, while those new to the approach may use only two.

Nice answer! Just to nitpick -- when working with Grothendieck universes, there is some freedom to decide just how many universes / inaccessibles we're assuming to exist. Your description in words seems to describe the assumption of a proper class of inaccessibles, or at least infinitely many, while your formal gloss only assumes one inaccessible. — Tim Campion, Feb 13 '21 at 15:07
The nitpick game is a fun way to learn stuff. I like to nitpick mostly just to make sure I'm not missing something that makes my nitpick incorrect :) — Tim Campion, Feb 13 '21 at 15:31
So would the tl;dr be that there's a technical solution to the issues that can arise, but when first learning about category theory you don't focus on this because it's somewhat ancillary to the purpose of the theory? — Sam Hopkins, Feb 13 '21 at 16:14
@SamHopkins Essentially yes; I definitely wouldn't recommend focusing on this stuff in an introductory class, as we aren't really gaining anything beyond assurances that we can manipulate large categories how we want to without running into Russell's paradox type issues. There are some technicalities involved, but for the purposes of an introductory class everything should work as expected, so you can just present 'the category of sets/groups/categories' without further mention of size issues as long as you specify that the category of categories isn't an object of itself ;). — Alec Rhea, Feb 13 '21 at 16:38

score 14 · Answer 2 · answered Feb 12 '21 at 16:52

14

To consolidate a clarification from the comments:

In mathematics as normally practiced, there is no such thing as a "definition" of the notion of set. There are various axiomatic theories one can use to reason about objects which we call "sets" (or think of as "set-like"), such as ZFC, ETCS, type theory, etc. But in these theories, "set" is typically a primitive, undefined notion. This is much as in classical Euclidean geometry, where the terms "point", "line", etc. are undefined, but we have well-defined rules for reasoning about them. The only place you would "define what a set is" is if you were constructing a model of one of these theories -- then you have to define what the model is going to say a "set" is, relative to whatever metatheory you're using.

But I think the larger question is -- "when does one have to worry about set-theoretic details when doing category theory?" The question is a bit ill-defined, but maybe there are a few things to say anyway:

The most prominent "set-theoretic consideration" in category theory is that of size. When doing category theory, it's important to develop a sense for which "set-like things" one considers are actually sets, and which ones are "too big to be sets" -- i.e. are actually proper classes [1]. Two theorems of basic category theory come immediately to mind -- both due to Freyd:

Any small category with all small products is a preorder. This theorem tells us that if we want to work with completeness conditions in category theory, then we really have to live with the fact that there's a difference between large and small.
The adjoint functor theorem crucially requires that certain collections appearing in the hypotheses be small (i.e. that they be sets rather than proper classes). The standard counterexample is: for each cardinality $\kappa$, let $G_\kappa$ be a simple group of cardinality $\kappa$, and define $F: Grp \to Set$ to be the functor $G \mapsto \prod_\kappa Hom(G_\kappa,G)$. This functor preserves limits, but has no left adjoint.

There are also "set-theoretic considerations" which are really just an annoyance. For instance, for most purposes it doesn't matter whether, say, the category $Ab$ of abelian groups is skeletal or not (because $Ab$ is equivalent to any of its skeleta, which are all isomorphic -- there is nothing special about $Ab$ here). But sometimes you might want to do something like take the product of all countable abelian groups. When you do this, it technically matters -- if $Ab$ is skeletal, this is a perfectly well-defined small abelian group, whereas if $Ab$ is defined as usual in ZFC say, there is a proper class of countable abelian groups (though just a small number of isomorphism classes). So to perform this construction in usual set theory, you should probably say something like "the product of one copy from each isomorphism class of countable abelian groups" and then proceed as you would have otherwise.

[1] In this post, I will use the language of a set / class distinction to talk about size. There are also other approaches, most notably the approach of using one or more Grothendieck universes. In the Grothendieck universe approach, there are no proper classes -- everything is a set. It's just that some sets are "small" while others are "large".

answered Feb 12 '21 at 16:52

Tim Campion

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I'm deeply confused by this answer. I thought a very standard point of view was that ZFC does define what a set is, in much the same way that the axioms of Turing machines define what an algorithm is (i.e., it makes some vague but intuitive idea mathematically precise). – Sam Hopkins Feb 12 '21 at 17:00
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@SamHopkins I hope we can at least agree on the following. There is a narrow sense of "definition" in mathematics, where we define what one object is in terms of previously-defined objects. E.g. when we define $\mathbb R$ to be the set of Dedekind cuts in $\mathbb Q$, this is a definition. When we define a group to be a model of the theory of groups, this is a definition. In this narrow sense at least, we never define what a "set" is, because there are no previously-defined concepts in terms of which to define it. – Tim Campion Feb 12 '21 at 17:08
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@SamHopkins I agree with Tim. The $\mathsf{ZFC}$ axioms describe how sets relate to each other via $\in$, but they don't tell us what a set "is" in a good sense. The right analogy in my mind is: the triple (ZFC, ZFC-models, objects in ZFC-models) is similar to the triple (group axioms, groups, elements of groups). The group axioms define groups, not elements of groups, and in fact we never define what an element of a group is. Similarly, it's hard to pin down a sense in which we define what a set is. – Noah Schweber Feb 12 '21 at 17:12
This is all muddled of course by the foundational nature of the "set" idea, and the way sets infiltrate mathematical discourse at all levels. But I still ultimately take the stance above. – Noah Schweber Feb 12 '21 at 17:14
@NoahSchweber: so would you also say we have no definition of what a real number is? – Sam Hopkins Feb 12 '21 at 17:14
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@SamHopkins Sort of? It depends what it's allowed to build off of. If we're taking (say) $\mathbb{Q}$ for granted, then yes, we can define real numbers; but if we're working "ex nihilo," then no, I don't see any meaningful way to distinguish elements of one Archimedean real closed field from another. (And in fact even if we're allowed to "build off of $\mathbb{Q}$," we still have the problem that there are multiple different natural definitions - classes of Cauchy sequences, Dedekind cuts, etc.) – Noah Schweber Feb 12 '21 at 17:15
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Okay, well I feel like this attitude leads to the conclusion that we never define anything in mathematics. Which could be a defensible position, I'm not sure, but anyways I would imagine it is not a plurality view among working mathematicians. – Sam Hopkins Feb 12 '21 at 17:18
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@SamHopkins I don't think this "attitude" (or rather, this definition of "definition") leads to the conclusion that we never define anything in mathematics. In fact, it seems to me that the most "standard" understanding of what we do in mathematics is that "set" is the only formally undefined notion, and everything else is defined in terms of it. A group is defined to be a model of the group axioms. The term "element of the group $G$" is well-defined for any given $G$, but the term "element of a group" is not defined (it's a silly notion anyway -- anything is an element of some group). – Tim Campion Feb 12 '21 at 17:23
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The most "standard" understanding is that we really do define $\mathbb R$ to be the set of Dedekind cuts in $\mathbb Q$. In practice, we don't think about this definition too much, preferring to reason about real numbers using the fact that $\mathbb R$ is a complete Archimedean field. The notion of "complete archimedean field" is just as well-defined as "group", but the notion of "element of a complete archimedean field" is just as undefined (or else useless) as "element of a group". – Tim Campion Feb 12 '21 at 17:24
In type theory, on the other hand, one typically has logical primitives which allow one to do things like define $\mathbb N$ to be the universal structure with certain properties, so that we're not in this awkward position of defining $\mathbb N$ to be the ordinal $\omega$ and then later mostly forgetting this definition in favor of "$\mathbb N$ is (up to unique isomorphism) the initial well-ordered semiring on an element" or something. That is, I think type theory supports the informal notion of "definition" better than ZFC does. But even there, "type" is undefined. – Tim Campion Feb 12 '21 at 17:35
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@SamHopkins Actually, once the details are teased out I think this is a pretty common "structuralist" view. How many mathematicians would say something like "The Dedekind cut definition of reals is true and the equivalence classes of Cauchy sequences definition is false" with a straight face, rather than asserting relative usefulness in particular situations? In fact, isn't this a pretty common objection to ZFC-style foundations in the first place, on the grounds that they necessarily invoke "junk information" which isn't really contentful? – Noah Schweber Feb 12 '21 at 17:35
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Tim : I didn't know your "standard counterexample", but it's really nice ! – Maxime Ramzi Feb 12 '21 at 17:38
@NoahSchweber: Huh, now I am more confused. I had thought what you and Tim were describing was roughly the notion that a definition "creates" a certain class of objects. Whereas I would say a definition makes precise (or more precise) an intuitive, pre-definition conceptualization of a kind of object we already have. For instance even with the example of group, of course the 19th century group theorists did not work with our now standard set of axioms for groups, yet they worked with groups all the same. But maybe you agree with this second perspective. Anyways sorry for the long convo. – Sam Hopkins Feb 12 '21 at 17:40
@MaximeRamzi Yeah, I'm pretty sure it's in MacLane for example. I like that it doesn't use any exotic category -- just $Grp$ and $Set$ -- nor any exotic notions in such a category! I'm surprised we haven't gotten a "continue this discussion in chat" button yet, I'm feeling a bit constrained by the comment interface. I don't know how to create a chat room by hand. The top hit on meta doesn't actually explain how, just points out to the questioner that they didn't have enough rep at the time. – Tim Campion Feb 12 '21 at 17:45
Although this comment might be out of context, but David Lewis did show that it is possible to define what "set" and "set membership" means in Mereological terms! He did that in his Parts of Classes book, and in Mathematics is Megethology. So we can indeed define "set" in terms of more primitive notions of part-hood, and a singleton function, or simply as Holmes puts it a "labeling" function. – Zuhair Al-Johar Feb 12 '21 at 23:01
His approach explains all about sets and classes, Ur-elements, non-well founded sets, etc... Although it's not usually seen in textbooks on sets (though Randall Holmes did mention it in his online book on elementary set theory with a universal set), yet still it does provide a precise formal definition of what a set is. – Zuhair Al-Johar Feb 12 '21 at 23:01
@ZuhairAl-Johar Thanks for pointing that out -- I'm a big fan of David Lewis, so I'll have to check out his work on mereology. I certainly wouldn't want to claim that the "irreducibility" of the notion of set is necessary -- only that that's the way it is in fact formally treated in "standard" foundations of math. – Tim Campion Feb 12 '21 at 23:21
@TimCampion: In your description "There is a narrow sense of "definition" in mathematics, where we define what one object is in terms of previously-defined objects." nothing says that the collection of previously-defined objects must be nonempty. So it is perfectly legitimate (at least according to your description) to define a set simply as a term (if we work in ZFC) or as a term of the appropriate sort (if we work in a multisorted theory like ETCS). – Dmitri Pavlov Feb 13 '21 at 00:19
@TimCampion: "the term "element of a group" is not defined (it's a silly notion anyway -- anything is an element of some group).": an element of a group is a morphism of sets a:1→U(G), or, by adjunction, a morphism of groups b:Z→G. Thus, any element b of a group automatically knows what group it belongs to: it's precisely codom(b). – Dmitri Pavlov Feb 13 '21 at 00:21
The article mathematical object from the nLab is pertinent to this discussion. – Dmitri Pavlov Feb 13 '21 at 00:23
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@DmitriPavlov #1 Maybe I'm not taking into account radically formalist views. Personally, I feel that if one is going to define sets to be certain syntactic objects, then it's a bit unclear in what sense one is actually defining what a "set" is, and it would probably be more clear to say that one doesn't talk about mathematical objects at all -- one just shuffles around syntax. Certainly this sort of notion of "definition" is not something which sheds much light on how set theory affects category theory. – Tim Campion Feb 13 '21 at 00:29
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@DmitriPavlov #2 Sure, maybe I didn't phrase that so well. I think we agree that "element of a group" doesn't mean much unless it comes along with a group to be an element of. And I was mostly trying to discuss the situation with respect to the most "standard" foundation in ZFC. – Tim Campion Feb 13 '21 at 00:29
@TimCampion: I am not certain what meaning you assign to the adjective “radically”; what was described appears to be perfectly mainstream formalist definition. It is also possible to give a Platonic definition of a set; the original definition by Cantor qualifies, and the axioms of (say) ETCS or ZFC describe legitimate operations on such objects. So it appears to me that the "narrow sense" of "definition" is rather artificial and does not match any existing philosophy of mathematics or the actual practice of mathematicians. – Dmitri Pavlov Feb 13 '21 at 01:10
Let us continue this discussion in chat. – Tim Campion Feb 13 '21 at 01:27
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@TimCampion: On a second thought, the discussed differences seem to point in the direction of synthetic vs. analytic definitions. For instance, ∞-groupoids could be defined analytically as simplicial sets, or synthetically as a term in homotopy type theory. For sets essentially the only practical definition is synthetic. But I would not say that synthetic definitions are not definitions, just a different type of definitions. – Dmitri Pavlov Feb 14 '21 at 18:31
@DmitriPavlov I think that's fair. Regarding "radical" formalism earlier, I regret that wording. I'm a big fan of the principle that whatever our philosophical preferences, our use of terminology should not obstruct communication with others with different philosophical preferences. I think I jumped to the conclusion that "defining" a set to be a syntactic object, would complicate communication with, say, platonists, and this seemed "radical" to me. But if this "definition" is not meant to be in the narrow sense I had in mind, but in a broader sense, then I think it's not really an issue. – Tim Campion Feb 14 '21 at 21:22

When the definition of a set starts to matter in category theory

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