How can category theory help my research in set theory?

Question

How can category theory help my research in set theory?

I rarely use category theory as such in my current work, and one almost never sees any category theory in set-theoretic research papers or at conferences (except of course when the issue is to apply set theory to category theory rather than conversely). Why is this, when category theoretic language and thinking has proved so successful in other parts of mathematics?

Since there seems to be a relatively sizable community of category theorists on this site, many of whom appear to know a lot of set theory or at least have opinions about it, I hope that I might gain some insight.

Note that I am not looking for a reason to do category theory instead of set theory. I am already inspired by a collection of topics, questions and results within set theory, which I find compelling and sometimes profound. What I want to know is whether category theory can provide me with techniques to use to attack those problems.

Answer 1

I believe this is a question that has not been adequately explored.

I view topos-like set theory and ZF-like set theory as exposing two faces of the same subject. In ZF-like theory, sets come equipped with a "membership" relation $\in$, while in topos-like theory, they do not. The former, which I call "material set theory," is the standard viewpoint of set theorists, but the second, which I call "structural set theory," is much closer to the way sets are used by most mathematicians.

However, the two viewpoints really contain exactly the same information. Of course, any material set theory gives rise to a category of sets, but conversely, as J Williams pointed out, from the topos of sets one can reconstruct the class of well-founded relations. With suitable "axioms of foundation" and/or "transitive-containment" imposed on either side, these two constructions set up an equivalence between "topoi of sets, up to equivalence of categories" and "models of (material) set theory, up to isomorphism."

Of course, it happens quite frequently in mathematics that we have two different viewpoints on one underlying notion, and in such a case it is often very useful to compare the meaning of particular statements from both viewpoints. Usually both viewpoints have advantages and disadvantages and each can easily solve problems that seem difficult to the other. Thus, I see a tremendous and (mostly) untapped potential here, if the ZF-theorists and topos theorists would talk to each other more. How much of the structure studied by ZF-theorists can be naturally seen in categorical language? Does this language provide new insights? Does it suggest new structure that hasn't yet been noticed?

One example is the construction of new models. Many of the constructions used by set theorists, such as forcing, Boolean-valued models, ultrapowers, etc. can be seen very naturally in a topos-theoretic context, where category theory gives us many powerful techniques. I personally never understood set-theoretic forcing until I was told that it was just the construction of the category of sheaves on a site. From this perspective the "generic" objects in forcing models can be seen to actually have a universal property, so that for instance one "freely adjoins" to a model of set theory a particular sort of set (say, for instance, a set with cardinality strictly between $\aleph_0$ and $2^{\aleph_0}$), with exactly the same universal property as when one "freely adjoins" a variable $x$ to a ring $R$ to produce the polynomial ring $R[x]$.

On the other hand, some constructions seem more natural in the world of material set theory, such as Gödel's constructible universe. I don't know what the category-theoretic interpretation of that is. So both viewpoints are important.

Another example is the study of large cardinals. Many or most large cardinal axioms have a natural expression in structural terms. For example, there exists a measurable cardinal if and only if there exists a nontrivial exact endofunctor of $Set$. And there exists a proper class of measurable cardinals if and only if $Set^{op}$ does not have a small dense subcategory. Some people at least would argue that Vopenka's principle is much more naturally formulated in category-theoretic terms. I have asked where there are nontrivial logical endofunctors of $Set$; this seems to be a sort of large-cardinal axiom, but it's unclear how strong it is. It seems possible to me that categorial thinking may suggest new axioms of this sort and new relationships between old ones.

Answer 2

This is not a direct answer to the question but more a small note on the difference between set theory as usually done and set theory viewed through the lenses of category theory.

For me, the most striking aspect of topos theory was the unearthing of vast families of categories that appear in "ordinary" mathematics and that behave very much like the category of sets (denoted by Set).

One can reverse the point of view and ask what is special from a categorial point of view about the category of sets? Well, one partial answer is that Set is well-pointed, that is, the terminal is a separator (or generator). This means that every arrow $f:a\to b$ is determined uniquely by the family of points $fx:t\to b$, with $t$ the terminal and $x:t\to a$ a (global) point of $a$. This is a precise categorial version of our view of sets as "discrete, structureless piles of sand".

If we did set theory this way, what would be different? There is a (at least one) major philosophical difference. There is no global $\in$. Since the equality predicate of sets is defined by

$X=Y \iff (\forall x, x\in X \iff x\in Y)$

it follows that there is no global equality predicate for sets. This may be unintuitive from our conception of sets, but it is the Right Thing in category land, because equality between objects is not preserved by equivalences. The Freyd / Scedrov book Categories, allegories contains a formal treatment of a language invariant under equivalences and a considerable part of M. Makkai's work on n-categories is building this insight right into the foundation of higher categories.

Hope it helps a little, regards.

Answer 3

One reason why category theory has not been too successful when dealing with set theory is due to the fact that set functions do not preserve most of the structure of a set. By this I mean that functions do not preserve $\in$, i.e. we do not have $S \in T \Rightarrow f(S) \in f(T)$. In most other fields in which category theory has been successful the morphisms preserve a lot of the structure of objects.

Because set functions do not preserve $\in$ most category theorists consider sets in quite a different way to set theorists - for a category theorist a set is a "bag of dots", in which the internal structure of such a bag is irrelevant, only the morphisms between them are relevant. This is the view taken by Lawvere and Rosebrugh in Sets for Mathematics.

Topos theory has often been considered as a generalization of sets, but I think of it more as a generalization of set functions, not sets themselves. To get sets in the standard $\in$ sense you have to take objects in a topos along with a rigid, well-founded tree structures. See MacLane and Moerdijk - Sheaves in Geometry and Logic, VI.10 for this view.

There is also the book Joyal and Moerdijk called Algebraic Set Theory, which I have not read, but I believe applies topos and category theoretic methods to the study of models of set theory.

Answer 4

The statement has been made in several places that forcing in logic is similar to or the same as sheafification. Also MacLane, Categories for the Working Mathematician, has an appendix entitled "Foundations", with the following statement: "However, categories can be described directly---and they can then be used as a possible foundation for all of mathematics, thus replacing the use in such a foundation of the usual Zermelo-Fraenkel axioms for set theory."

The summary in this appendix is (for me at least) just an informal sketch of the point. When I read this before, it was never clear to me to what extent you really get a full alternative to set-theoretic axioms. (Or to first-order logic? Or to propositional calculus?)

Or to what extent some other category can be addressed by axioms like the category of sets. In fact in standard set theory you do work with the category of ordinals as well as the category of sets. Could there be useful axioms expressed in terms of the category of groups or rings or something like that?

It seems strange that a lot of mathematics involves two orthogonal formalisms thrown together, category theory and set theory. For instance, for me personally the distinction between small and large categories has only been useful for negative reasons. How much is there to what MacLane promises? (Maybe there is a lot and I just don't know about it.)

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Well, there is a book that could help with my head-scratching, Topoi, the categorical analysis of logic, by Robert Goldblatt.

Answer 5

This might have already been mentioned but algebraic set theory is a current program in progress that explores various questions about sets from the categorical perspective. Here's the link: https://www.phil.cmu.edu/projects/ast/

Answer 6

I'm much more interested in categories than in sets, so take what I say with a healthy dose of salt. Solomon Feferman gave a talk at the Berkeley Logic Colloquium a while back about trying to come up with a good "set-theoretic" (at least in style) description of what category theorists actually do. So category theory helps his set theory research by suggesting new questions.