They're called toposes, and you already asked a very similar question. The idea is that you can model topoi axiomatically, which is what Lawvere's ETCS and another set theory called SEAR do. To even talk about anything at all formally, you first have to have some underlying logic powerful enough to even express these concepts in the first place. That is, you need to be able to interpret definitions in some sort of complete and compact theory of first order logic or slightly weaker equivalent. You need to understand how things work before you start using them. You need rules in place before you can interpret axiom schema, or else all you have is a bunch of symbols on a page, strictly speaking.

Also, please read Categories for the Working Mathematician and Sheaves in Geometry and Logic by S. Mac Lane at least part of the way through before you post another topic on category theory. You're making too many mistakes that make your questions meaningless, and you've now done it twice over the course of a few days.

mod|link|edit|undelete deleted by Harry 12 mins ago

answered 5 hours ago Harry 618●11

1 [this is a great comment] [flag this comment as offensive, spam, or hate speech] If you don't like my comment, and you want to downvote me, then leave a reason. – Harry 4 hours ago [delete this comment]

1 [this is a great comment] [flag this comment as offensive, spam, or hate speech] Toposes give provability semantics for higher-order logic. This is way more than needed for just first-order logic. There are two kinds of categorical model for first-order logic I'm aware of. The first are hyperdoctrines, which consist of a functor from some category with products to Posets, plus stuff to make everything work. (This is a general way of phrasing the usual model construction.) The second is via locally cartesian closed categories. This gives you a semantics of proofs, and can actually model a bit more than first-order logic -- you can interpret dependent type theory in it. – Neel Krishnaswami 1 hour ago [delete this comment]

1 [this is a great comment] [flag this comment as offensive, spam, or hate speech] I didn't downvote you, but telling people who are asking fairly reasonable questions, on the whole, to "read... Mac Lane at least part of the way through before you post another topic on category theory" because of some minor mistakes and a somewhat obtuse style does not invite a lot of sympathy for when you say something that's wrong. (Even if it's wrong for entirely understandable reasons -- I thought of topoi too, but this didn't look like what topoi are about.) – Harrison Brown 47 mins ago [delete this comment]

[this is a great comment] [flag this comment as offensive, spam, or hate speech] You can also take a logical theory and turn it into a category. The objects are the types and the arrows are the terms. This is done for toposes in Elementary Categories, Elementary Toposes, by Colin McLarty (and in many other places, but this one is especially clear). The technique has been known for something like thirty years and has been applied in many settings, for example linear logic. – SixWingedSeraph 41 mins ago [delete this comment]

[this is a great comment] [flag this comment as offensive, spam, or hate speech] -1: If you don't like the question, either don't answer it, or explain why (politely -- there is, frankly, some room for improvement here) in the comments. An answer should not consist primarily of criticism. Also, I disagree that the question is the same: his previous question concerned set theory and this one concerns logic. – Pete L. Clark 26 mins ago [delete this comment]

[this is a great comment] [flag this comment as offensive, spam, or hate speech] He asked the same question this week. The fact is, you can't talk about modeling things without first having a meta-logic to work within. This meta-logic needs to have rules for interpreting axioms and axiom schema, cf. Bourbaki Ch.1 [Theory of Sets]. That is, you have to introduce a symbolic logic to begin to construct a theory. His question is also stupid because he doesn't know what he's talking about. I've seen this before. "I don't know anything about category theory, but it's really general, and I want to know how general we can get." – Harry 24 mins ago [delete this comment]

@Harry: also, I'll warn you that "His question is also stupid because he doesn't know what he's talking about." almost certainly contravenes our recently instituted (but perhaps not yet announced!) rule that people can not make inflammatory remarks unless they're using their full real name! :-)

@Harry: Is being foreign some kind of disability (or something)?

@Harry: Look, this is the last time I even comment on anything related to you and your behavior. Many times you are impolite and downright offensive, and I have tried telling you several times in a civil manner to no avail, as made clear by the answer that is the topic of this thread. Other times you try to be funny, and even in those cases you still manage to annoy people. Since it seems you won't stop, I'll shut up now. MOers know what to expect from you.

I'm absolutely in favor of letting controversial or argumentative answers be deleted. I could be wrong, but the impression I have is that Harry deleted his answer essentially because he got the message that a lot of people thought it was rude and unhelpful, and the comments section was threatening to degenerate into further unhelpfulness. (Harry, is that about right?) Which is a totally reasonable reason to delete an answer!

Certainly there's a possibility that people won't be as professional if they can "remove the evidence." But -- and maybe this is naive -- I really don't think that it'll be a problem in practice, since we're hopefully all civil in our professional lives outside of MO, and have gotten into the habit. And if a user does abuse the system, someone else can always get in touch with a moderator (or post something on meta, as someone did a while ago after someone else went through and downvoted all their posts) and specific action could be taken regarding that person.

Also, plclark makes a good point. People my age to a few years older have to worry enough already about what might get on these here Interwebs. Why add MO to that list?

I do disagree with plclark on one thing, though. I'll come out and publicly state that "Why is a subgroup of a free abelian group always abelian?" is in fact a stupid question.

@Harry: the apology is appreciated (speaking for myself, but hopefully for others too).

To try and move this back towards structural as opposed to personal matters: I am inclined to go with what Mike Shulman said above. In an unthreaded system, deletion of comments just makes things confusing - it would be preferable to at least have some visible record of "Comment by X - deleted" so that a late arrival can get some idea of who was responding to what.

Also, and this is not meant to be a personal swipe at particular people: I think that dissuading deletion of comments would hopefully lead to people considering their comments more carefully. I've deleted one or two of my own due to typos, but perhaps also a few due to being boneheaded; and IMHO it'd be a salutary reminder to me to engage my brain or judgment, if comment deletion wasn't so easy.

Lastly, and this is meant as a personally directed comment - sad, but inevitable: I don't agree with Harry's metaphor or conclusion when he says

"I deleted my post because I put my hand in to test the waters, and I was bitten by pirahnas. If we don't allow deletion or voting locks on closed answers, nobody will be willing to make unorthodox statements."

There is a difference between putting your hand in the water and being bitten, and poking a beehive. From what I could see, the "piranhas" you encountered were no less civil than you were, and in the case of the SixWingedSeraph were informative at some level.

Also, on the thread linked to, one of your comments says "I deleted my post because too many people were downvoting it." That seems to me - though this is a matter of taste/perspective - to go against some of the point of MO. I've given some answers which I thought should be rated higher, and some which have been rated higher than I myself would have judged: but when participating in a community, that's what happens.

@Scott Morrison: Yeah, people are jerks on the Internet in general. (Actually, there's an amusing related comic and acronymic term -- GIFT -- whose expansion probably isn't appropriate for this forum, but I'll just say that three of the words are "Greater", "Internet", and "Theory". Your imagination and/or Google should be able to fill in the rest.) Part of that is anonymity, which is why I *am* in favor of the "no-rude-comments-unless-you're-using-your-name" thing. But even when you're not anonymous (some people are jerks even on places where they're registered and regularly contribute), part of what lets you get away with being, um, an F is the assumption that your behavior on one corner of the internet is not going to relate to how people think of you anywhere else. This is manifestly *not* the case with MO, since the set of people you interact with here and the set of people you interact with professionally are probably not disjoint.

So I think that smart people will pretty much know to self-censor on MO. And if they're habitually rude/argumentative/whatever, people who use the site will get the picture whether or not they delete their worse comments. But if someone makes a mistake a couple of times and then gets the picture, the system should allow them to get as much of a chance at forgiveness as possible. It's kind of the difference between, I don't know, Michelle Obama and Silvio Berlusconi. One of them has said and done a couple of regrettable things, and one of them has their foot surgically implanted in their mouth. Therefore, the recent news for Michelle was her favorability rating dipping to a stunning low of 68%, while Silvio got punched in the face. So, that's my point.

Unrelatedly, I'm tempted to offer an apologia on Harry's behalf, but it's probably not wise unless Harry's okay with it and not worth my time unless someone wants to hear it. Harry? Everyone else?

@Harrison- Do you really mean an apologia? I don't see what one there could be. I think many of us are annoyed from time to time with people who keep posting questions on the site when it seems clear to one that they should do some background reading, but that's no reason to write such an answer. At some point you just have to accept that you're not getting through and do something else.

The other problem with the internet is everything sounds more mean-spirited on the internet than it does in person. Even if you really don't mean to say anything inflammatory, it's far too easy to. Probably some of Harry's "witticisms" actually work when delivered in person, but deprived of context, body language and voice tone, everything is easy to take as an attack.

I'm a bit late to the party, but I just want to say that I agree with Harrison ("I'm absolutely in favor of letting controversial or argumentative answers be deleted ...")

@Harry: my bad, I was reading one of your later comments rather than your "controversial answer". Note how I am not editing my previous answer, so people can see me being hasty, and I can be reminded of same next time I jump in without engaging my brain.