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Some folks over at nLab want to use categories as a foundation for all of mathematics, I'm guessing as an alternative to sets. Sets work fine, and so do categories, so I have started wondering what other kinds of objects everything could be founded on. The Wikipedia page for categories has a nice table of different group-like structures. In addition to the group-like structures, there are also a lot of other structures that come up in abstract algebra, like rings, vector spaces and modules. Could there be an v-Lab where everyone gets together to phrase everything in math in terms of vector spaces, or is there something special about categories? Is there some kind of property or requirement that an abstract object must have in order to work as a foundational concept?

There is a lot of theory about how powerful each different kind of logic is (for example, some theorems can be proven in ZFC but not in constructive mathematics), so I am curious if there is a similar multiverse for foundational definitions.

Here is a closely related question, but for groups. It is encouraging to see that for groups the answer is "yes." This makes it even more interesting to ask, what are the requirements for foundational objects in general?

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    $\begingroup$ Is there any evidence that “the folks at nLab want to use categories as a foundation for all of mathematics”? I have the impression that most of them (if they think about these issues) like (homotopy) types. $\endgroup$ Jun 26 '20 at 19:39
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    $\begingroup$ The title and first sentence are both bad. Obviously categories are not special since you say sets also work fine. And who are the people at nlab who “want to use categories as a foundation”? Many nlab-ers would probably say something more guarded instead, perhaps that they “find categories an appropriate foundation if foundations are appropriate at all”. $\endgroup$
    – Matt F.
    Jun 26 '20 at 20:46
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    $\begingroup$ This must be referring to ETCC, but I do not know anybody on the nLab who wants to take categories with noninvertible morphisms as a foundation of mathematics. Both sets (e.g., ZFC, elementary toposes, and ETCS) and ∞-groupoids (e.g., HoTT) can be successfully used as a foundation, but I don't think much has been done with ETCC, and we don't even have a reasonably precise working definition of ETCC. $\endgroup$ Jun 26 '20 at 21:20
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    $\begingroup$ By the way, using ETCS as foundation is not "using categories as foundation" $\endgroup$
    – Qfwfq
    Jun 26 '20 at 23:08
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    $\begingroup$ Yes, any more than using ZFC is "using first-order logic as foundations". $\endgroup$ Jun 27 '20 at 5:16
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The term "foundations of mathematics" is all well and good when one has fixed a foundation to work with. But when you start trying to compare different foundations of mathematics, you quickly realize that the term "foundations of mathematics" requires a great deal of unpacking. I recall once reading a perspicuous article by Penelope Maddy [1] identifying on the order of a dozen different roles we implicitly expect a "foundations of mathematics" to play. This is a testament to the resounding successes of the most widely-used foundations of math, namely ZFC layered on top of (classical, finitary) first-order logic, and variations thereof.

Anyway, if the question is "What are the requirements for foundational objects in general?", then I'm not sure I can do better than pointing you to Maddy. I don't think the question has a simple answer.

Maybe I'll also point out that there are many different ways one might "use category theory as a foundations of mathematics".

  1. One way would be to use Lawvere's ETCC (plus additional axioms as needed) in a similar way to the way ZFC (plus additional axioms as needed) is used by most mathematicians. That is, we write down a particular first-order theory (either ZFC or ETCC) and then just encode everything we want to do inside of it. This is perfectly doable, but I think most would agree it is awkward and doesn't really have any tangible benefits.

  2. Another way would be to use ETCS, or perhaps the theory of an elementary topos with natural number object, or something like this, instead of ZFC. Again, this is perfectly doable. I think there are some who feel that this approach has merit.

  3. Another way would be to back up, and decide that one isn't interested in just one foundational theory, but in comparing many different ones (note that even the usual foundations are open-ended in this way -- we always allow ourselves to extend ZFC by large cardinal hypotheses and whatnot when necessary). We might find a privileged role for category theory in deciding what is and isn't a "foundations of math", or in comparing different foundations of math. For instance, we might want to compare two different foundations by comparing what each of them says about "the category of sets".

  4. We might also just use category theory as a tool in the foundations of mathematics. For instance, type theoretic foundations are often compared to classical foundations by building models of the former in the latter, and what a "model" is is often expressed in terms of some kind of category.

  5. One might stick with ZFC or whatever, but emphasize that category theory plays a privileged role in comparing different types of mathematical objects, based on the empirical fact that mathematical objects often organize themselves naturally into categories, and such categories can be compared via functors, etc.

  6. As an example of (3), different notions of constructivism can be fruitfully compared in terms of what properties they say the category of sets has (things like being an "exact category", a "pretopos", or whatnot). It's probably due to my ignorance, but I'm not aware of similarly enlightening concepts being more naturally formulated in terms of membership-based set theory. Moreover, new constructivist theories can be formulated and motivated starting from such categorical considerations.

  7. In another direction, support for certain large cardinal principles, such as measurable cardinals or Vopenka's principle, can be given based on categorical formulations of these principles, and categorical considerations about these formulations. Conceivably, one might imagine formulating new large cardinal principles based primarily on categorical considerations, although I'm not sure this has been done much in practice, unless you count Vopenka's Principle or Weak Vopenka's Principle. I actually find it very suggestive that the strongest large cardinal principles (Reinhardt and Berkeley cardinals) are inconsistent with ZFC, so even classically most naturally retreats to ZF to study them. Although I don't have any evidence for the following scenario, I wouldn't be surprised if even stronger large cardinal principles might one day be found which are inconsistent with ZF, and instead most naturally studied in the context of topos theory or something like that.

  8. And so forth.

[1] I'm not sure, but it may have been What do we want a Foundation to do? (non-paywalled link). Full citation: Maddy P. (2019) What Do We Want a Foundation to Do?. In: Centrone S., Kant D., Sarikaya D. (eds) Reflections on the Foundations of Mathematics. Synthese Library (Studies in Epistemology, Logic, Methodology, and Philosophy of Science), vol 407. Springer, Cham

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    $\begingroup$ Maddy’s answers to her title question are mostly bolded (and thus easily scanned) in that article: Elucidation, Epistemic Source, Essential Guidance, Generous Arena, Metamathematical Corral, Metaphysical Insight, Proof Checking, Risk Assessment, Shared Standard, etc $\endgroup$
    – Matt F.
    Jun 27 '20 at 0:48
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    $\begingroup$ Excellent answer! I’d add a suggestion that your 5 points fall into two main groups, and this distinction is worth emphasising. Points 1 & 2 use “foundation” in a comparable sense to how ZFC is a foundation: a basic logical formal system for encoding. Points 3–5 use “foundation” in a quite different sense: a set of particularly fundamental organising tools (with 3 & 4 emphasising the application of these tools to “foundations” in the first sense). $\endgroup$ Jun 27 '20 at 9:46
  • $\begingroup$ Regarding (1): it is common nowadays to assume that of course all the axioms we use in math are unproblematic, and of course all of math can be brought under a single umbrella. Given this, there is indeed "no tangible benefit" to showing these obvious facts. But such conclusions were not always so obvious, and foundational systems such as ZFC were in large part motivated by them. As Maddy says, it's important to be explicit about what you want a foundation to do. You'll get very different answers depending on whether or not you care about Metamathematical Corral or Risk Assessment. $\endgroup$ Jun 27 '20 at 15:45
  • $\begingroup$ @TimothyChow I see the confusion -- when I wrote "I think most would agree it is awkward and doesn't really have any tangible benefits", I should have clarified that "it" refers to encoding all of mathematics in terms of ETCC. By contrast, I think most would agree that encoding all of mathematics in terms of ZFC is less awkward than encoding all of mathematics in terms of ETCC, and does have the tangible benefits you allude to. What I'm driving at is that to the mathematician interested in "replacing ZFC", ETCS or a topos or type-theoretic approach is probably more attractive than ETCC. $\endgroup$
    – Tim Campion
    Jan 23 '21 at 21:28

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