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I would like to ask for some reference textbooks or articles, or any information that you know about developing topological concepts and differential geometric concepts using as a foundation ETCS which is elementary theory of category of sets by Lawvere.

My motivation for it is that recently I have seen a reasonable discussion that ETCS could be a valid alternative foundation for set theory and being young I would like to take the risk and develop my machinery for differential geometry using this foundation with a hope that it could give me some useful intuition and an interesting point of view that those who used conventional set theory like ZFC/NBG don't have (interesting question is, is it really possible? I am pretty sure that both foundational systems can be shown to be equivalent if you equip ETCS with some additional axioms, but I would argue that the method of developing set theory differently makes you think about set theory differently as well).

As a sidenote, I would also be very interested in your opinions about developing topology and differential geometry using ETCS. Maybe the set theoretic equivalents are not possible, but it is possible to construct something equivalent or more general than topological spaces/manifolds etc.?

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    $\begingroup$ Do differential geometers really "use" ZFC/NBG? Don't they, like most working mathematicians, just use set theory in an informal way without caring too much about foundational issues? $\endgroup$ – Lord Shark the Unknown Jan 15 at 6:15
  • $\begingroup$ @LordSharktheUnknown in principle set theory is first order logic and topology and differential geometry is developed using this theory. Of course, after mathematicians train their intuition they are using informal shortcuts for developing theorems, but there are still rigorous foundations. I am interested in formulating these rigorous foundations and then I can proceed as informal as I want once I am sure I understand what I am assuming and what not, and once I understand how everything is defined. $\endgroup$ – Daniels Krimans Jan 15 at 6:17
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    $\begingroup$ I'm sure you can use whatever foundations you want, and I'm sure they will give you intuition for neither topology nor geometry. $\endgroup$ – user98602 Jan 15 at 17:20
  • $\begingroup$ I agree completely with the other responses you have received. I highly doubt there is a book on differential geometry that spells out its foundations in ETCS... I’m not even sure if there would be one that explicitly uses ZFC. I would be surprised, but less surprised in the case of topology since there are topology books out there that make set theoretical foundations explicit. On a more positive note, if you haven’t seen Lawvere’s “Sets for Mathematics,” you may want to have a look. $\endgroup$ – spaceisdarkgreen Jan 15 at 19:50
  • $\begingroup$ @spaceisdarkgreen Not sure I agree. For example, most constructions in differential geometry are topological spaces. So you need to know what topological space is. Topological spaces are then defined using unions and intersections. So, you really need to know what unions and intersections are. In ZFC and ETCS these are pretty different. In ZFC they are pretty obvious but in ETCS you have to know how to make sense of them. So, there are observable differences between foundations you use in the most basic level. $\endgroup$ – Daniels Krimans Jan 15 at 20:27
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I agree with Mike and Lord Shark. As you say, ETCS is not really a different theory than ZFC, up to the more abstruse axioms. It's not clear what it would even mean to develop topology in ETCS as opposed to ZFC. Indeed, I think most enthusiasts of ETCS would agree that most mathematicians effectively already work in ETCS moreso than in ZFC, insofar as nobody ever asks whether $3$ is an element of $\pi$; in fact, this observation was a core motivation for ETCS, in my understanding.

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  • $\begingroup$ thanks for your answer, but I am not sure I agree. For example, most constructions in differential geometry are topological spaces. So you need to know what topological space is. Topological spaces are then defined using unions and intersections. So, you really need to know what unions and intersections are. In ZFC and ETCS these are pretty different. In ZFC they are pretty obvious but in ETCS you have to know how to make sense of them. So, there are observable differences between foundations you use in the most basic level. $\endgroup$ – Daniels Krimans Jan 15 at 20:28
  • $\begingroup$ @DanielsKrimans And none of that distinction has to do with topology or differential geometry. You just now have "Step 0: get comfortable with pushouts and pullbacks in the category of sets" in your list of requirements before you do any actual topology. $\endgroup$ – user98602 Jan 15 at 20:39
  • $\begingroup$ @MikeMiller how is this distinction has nothing to do with topology? In order to know definitions in topology I have to know this distinction. Anyways, if you want, I can rephrase my question as "Can you give some references about ETCS such that while studying topology and differential geometry one is comfortable with all of the constructions that are used?". $\endgroup$ – Daniels Krimans Jan 15 at 20:41
  • $\begingroup$ @DanielsKrimans This would be a list of references about ETCS. All of your questions are about set-theoretic constructions, and that's all you need to know. $\endgroup$ – user98602 Jan 15 at 20:42
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    $\begingroup$ @DanielsKrimans The category of sets is not very complicated. It would be strange to try to learn any mathematics without understanding it up to essentially constructing all limits and colimits (if not necessarily in that language.) Insofar as there are multiple fully faithful functors from sets to topological spaces, I'm not sure what constructions you could be hoping to avoid. I'd suggest just reading some ETCS, if you want to learn a very nice foundational system. If you want to learn topology, read a topology book. $\endgroup$ – Kevin Carlson Jan 15 at 21:58

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