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Consider category theory as one theory among many others: with a simple signature and some simple axioms.

Compare it with - e.g. - group theory as another theory with a simple signature and some simple axioms.

Compare it with set theory as still another theory with a simple signature and some (not so simple) axioms.

How could you tell in advance that (especially and somehow exclusively) category theory gives rise to (and makes definable) such a fundamental concept like universal property?

Consider the way universal properties are defined: Why isn't one able to define comparable abstract and useful concepts on top of groups, sets, and so on? Or is one?

What makes categories special in this respect - from an abstract point of view?

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    Why do you think one should be able to tell in advance what a theory gives rise to? In mathematics, when you know a theory can do something, the theory did.2012-10-11
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    Given that the 'traditional' foundation for all of mathematics is set theory, it seems strange to say that one can't define useful concepts on top of sets...2012-10-11
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    I don't know historical facts, but it seems likely that universal properties are important because it is a nice property expressible in categories. This is similar to being a transitive set having a nice defining property in set theory, and playing important roles.2012-10-11
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    @StevenStadnicki, that's debatable. Connes wrote that sets don't explain anything and that we get most of the information about nature through spectral methods. Granted that's an informal assertion, but being an engineer, one I agree with: whether it's spectral methods in signal processing or in databases, functions and relations set the agenda, not sets.2012-10-11
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    @alancalvitti Ahhh, but what is a relation, if not a set of pairs? :P2012-10-11
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    @StevenStadnicki, that's what Prof Jan Mycielsky asked me as well, so I pointed him to the paragraph in Goldblatt's *Topoi* where he points out a limitation of the function-as-pairs concept. I don't know that I convinced him, but I can tell you in computer science, you can form tables of pairs "lookup tables" but that's only one way (and a minority) in which functions (or relations) are actually computed.2012-10-11
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    @alancalvitti: computing something and defining it are fairly different things.2012-10-12
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    @MartinArgerami, that's true, but even so, in software, functions can be both defined and computed without defining or computing lookup tables.2012-10-12
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    @alancalvitti: but we are talking math here, not applications. And in particular category theory, which is a fairly abstract branch of mathematics. Whatever intuitive or practical concept of function you have, it does not really bear on what a function is from a mathematical point of view. And the same applies to the Connes' comment that you mentioned: it talks about "gathering information about nature", not about developing pure abstract math.2012-10-12
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    @MartinArgerami, constructive mathematics, ie, without Choice and excluded middle, is essentially about computation. Computer Science is about algorithms not just data structures, and algorithms are part and parcel of constructive math.2012-10-12
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    @alancalvitti: But constructive mathematics in that sense is a very limited part of mathematics, and for many of us a not particularly interesting part.2012-10-12
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    @BrianM.Scott, I'm aware of that perception - and it's a complex subject - but it doesn't seem accurate. For example, Herrlich compiled a list of "disasters" with choice, disasters without choice, and disasters either way. Would you say that non-Euclidean geometry is more limited part of math than Euclidean? It's just a different direction.2012-10-12
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    @alancalvitti: I’m quite familiar with HH’s nice little book; I don’t, however, agree that there are actually any disasters with choice. I’ve never seen a consequence of AC that bothered me, and I’m a set-theoretic topologist, so I’m reasonably familiar with the subject. I don’t think that your geometric analogy is apt. First, I would say instead that Euclidean geometry is a only a part of geometry, and not the most interesting, either. Also, when I said that constructive mathematics is a very limited part of mathematics, I was talking about human endeavor as well as the subject itself; ...2012-10-12
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    @alancalvitti: ... most mathematicians don’t take that approach. Indeed, for many of us it amounts to tying one hand behind one’s back (or worse).2012-10-12
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    @BrianM.Scott, Schafarevich said: math gets implemented, otherwise it's just marks on paper. The world runs on computers, which are finite state machines. Kolmogorov also remarked that at the end of the day there's no reason to believe reality is described by ODEs rather than difference equations or state machines. Keep in mind the relativity of approaches. Maybe you're tying your own hands?2012-10-12
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    @AsafKaragila, universal properties were important long before categories where defined or even identified as a concept.2012-10-12
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    I do not think this is a real question, and this is turning into a discussion... If my vote were not binding, I would vote to close and suggest starting a blog with comments.2012-10-12
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    @alancalvitti: Why should I pay any more attention to the opinions of Shafarevich (who is obviously wrong about this) and Kolmogorov than to my own? And why should I care whether the world runs on computers (which in fact it does not), or $-$ when I’d doing mathematics and not attempting to describe physical reality $-$ what mathematical model best describes that reality? That last is your bias. Not yours alone, of course, but it’s far from universal, and in particular I don’t share it. Enough. This is off the subject and not a mathematical question anyway.2012-10-12
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    @BrianM.Scott, fair enough, let me explain the computer comment: obviously it's people who program machines (thus I'm skeptical on "machine learning"), but the degree of automation is increasing: eg there's no need for pilots on most commercial jet flights (Airbus avionics often veto human pilot response)2012-10-12
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    @MarianoSuárez-Alvarez I do not think there is something wrong with the question. There is something wrong with (some of) the comments. They are just off-topic.2012-10-12
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    @Mariano: Yes, I am aware that universality property was important *before* categories. However it is the fact that it is so easy to express this property with arrows made it so important, in my opinion, as a construction or a defining property.2012-10-12
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    @alancalvitti: You have completely lost all mathematical context in your last comment.2012-10-12
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    @AsafKaragila, categories were defined precisely in order to be able to express the notion of universality and of naturality, in fact. Doing this well is their *raison d'être*!2012-10-12
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    I'd spontaneously argue it is because the CT axioms have (set) function composition "$\circ$" as a model.2014-06-20

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I am going to rephrase your questions, the way I understand them. I hope this is a reasonably correct interpretation.

1- question: how can you tell by just looking that category theory (CT) (a theory with a simple signature and some simple axioms, in your words) is more powerful than say group theory or..... Answer: take CT axioms (for ex. Here) and take monoid theory (MT) axioms (for ex. Here). You see almost immediately that MT is a special case of CT where all identities are the same. You also see that groupoid theory (GdT) is a special case of CT when you assume that all arrows are invertible. Since group theory (GT) is a special case of MT or GdT you already see that:

CT is more general (thus presumably more powerful) than MT, GdT or GT. That's not bad, considering that all this can be seen almost immediately.

If you specialize CT is more complicated ways you can prove that the resulting theory is a model for (a flavour of) Set theory. And so on with Rings, topological spaces, ect.

So we can say that the simple axioms of CT can be augmented to obtain many other previously known mathematical structures.

CT - as defined above - has been generalized further with higher categories so I would not say that it is unique in any (permanent) sense. Perhaps in the future we will abstract even more with some other theory, who knows. At the moment Category theorIES are at the forefront of generality and abstraction. That's all we can say.

2- question: Universal properties (UP) are logical statements expressible in the language of CT. Can we guess by looking at their formal structure that they are going to be a fundamental concept in CT?

Answer: I do not think this is possible at the moment. We can feed a computer with a theory and a statement and ask whether the statement is true or not (a Theorem). But we cannot decide - by just looking at the structure of the statement - whether it is going to be very useful or just moderately useful in the future development of the theory. This can only be decided ex-post. In the case of UP, they are not even theorems, just properties (definitions basically), which may or may not apply to a specific category/functor. They turn out to be fundamental concepts by the fact that they appear to be satisfied by many important categories/functors. Ex-post unfortunately.

Samuel defined UP in 1948 and Kan went on with adjoints in 1958. CT was founded in 1942. So UP and adjoints were not obvious things.

3- question:Why isn't one able to define comparable abstract and useful concepts on top of groups, sets, and so on?

Answer: even the most abstract construction in group theory is just something that applies to groups (and some derived set or ring, or...) only. It will never be automatically applicable in a non-group setting (topological spaces which are not groups, for example).

Conclusion. It seems that much axiomatic mathematics has been developed by "reverse-engineering". You take some nice theorem (Pythagoras' theorem for ex. ) and you work backward to find axioms such that the theorem can be deduced from them. This s apparently what Euclid did. Perhaps UP were invented that way. Basic CT axioms certainly were developed that way.

hth

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    Thank you for taking my question serious. I would say you rephrased the question very well - and I'm quite satified with your answer.2012-10-12
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    You are welcome @HansStricker .2012-10-13
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A theory that people actually study is rarely the result a a randomly chosen collection of axioms. If I recall correctly Saunders Mac Lane said categories were defined so that functors could be defined and functors were defined so that natural transformations could be defined. As Asaf Karagila points out transitive sets pop out of set theory. So to universal properties pop out of category theory. Situations like these make for interesting mathematics.

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    Is it transitive sets that pop up or transitivity in general?2012-10-12