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A long time ago I studied mathematics at the University of Stockholm. I had a romantic view of modern algebra and manage to make the first two algebra courses by self studies in order to immediately study homological algebra, Galois theory and such topics. That is not the best way to study. Later as a graduate student I did rather well - until the gaps in my basic knowledge and abilities began to affect too much. Then I stopped focusing on mathematics about 35 years ago.

I did self studies in category theory because we were supposed to do that and because it was a good idea. Category theory worked fine with the mathematics evolved at 1950 or so. The universal definitions and duality simplified a lot of mathematics as tensor products and injective/projective modules etc and the functors opened new possibilities.

The last 40 years or so the interest in and the development of category theory has exploded and seems nowaday be very abstract but also very consistent.

My question is, what modern category theory could be interesting for a person mainly interested in the mathematics concerning structured sets?

The bounty will soon expire and there is 50+ in reputation to earn - aren't there anything to express on this topic?

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    $\begingroup$ @TobiasKildetoft To my knowledge nobody uses the word multicategory for anything other than a colored operad... And if the question is really about higher categories then I'm even more baffled because very many people are already interested in them, even more so than multicategories. $\endgroup$
    – Najib Idrissi
    Jul 19, 2016 at 11:47
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    $\begingroup$ People are eager to share their knowledge, but you have to ask a clear and precise question first... As we can now see, @Tobias was right and you weren't talking about multicategories at all (why use that word without looking it up?). Now, this question is too broad, it's not reasonable to ask for motivation for category and higher category theory! If you want category theory see this, if you want higher category see this. This is another problem with your question: it doesn't appear you did any research... $\endgroup$
    – Najib Idrissi
    Jul 19, 2016 at 14:39
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    $\begingroup$ The question was already clear and precise enough, even if the english phrasing isn't the natural/native. With a minimum of charity, the (original) question reads as "what are reasons for someone interested in the mathematics of structured sets to study category theory or multicategories?" This is not broad; it's about the relevance of category theory to a particular (I think Bourbakian) way of looking at and thinking about mathematics. $\endgroup$
    – Vladimir Sotirov
    Jul 19, 2016 at 16:11
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    $\begingroup$ @NajibIdrissi you have your views on what's "acceptable", I have mine. I wasn't trying to twist your words, I am disagreeing that multicategories came out of nowhere in the post, but that's not the point (although I am irked by requiring beginners, and especially possibly amateur beginners, to navigate the category theoretic literature on their own, and to consider the lack of that effort as reason to dismiss their question). The point I want to make is that the question was never about modern category theory in general, but about its specific relation to the structured sets point of view. $\endgroup$
    – Vladimir Sotirov
    Jul 19, 2016 at 17:45
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    $\begingroup$ For a start I'd recommend to browse something on the subject, say, the Intro to Tom Leinster's book That's could be done quite quickly. It has a motivation section. Then you'd have a more clear idea what you might want to ask. $\endgroup$
    – quid ♦
    Jul 20, 2016 at 16:10

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Perhaps the most compelling answer to the question:

My question is, what modern category theory could be interesting for a person mainly interested in the mathematics concerning structured sets?

is just that a great deal of modern algebraic geometry, as a result of Grothendieck and others, makes heavy use of category theory. Here, I'm basically replacing "structured sets" with "rings, fields and modules." However, the study of groups and monoids has also been immensely impacted by category theory. In particular, category theory has played a really large role in representation theory.

If "structured sets" also includes topological spaces then there's even more category theory that ends up being relevant. And of course, topological spaces, and the study of their invariants, are important in things mentioned above like algebraic geometry and representation theory.

One problem with your question is that it's immensely broad. You're basically asking "How is category theory used in algebra?" And well, the answer at this point is almost "In what cases is it not used?"

My suggestion, if you want to have some sense of what category theory is all about, is pick up Saunders Mac Lane's book and force yourself to learn the foundations (e.g. categories, functors, natural transformations) from a purely formal point of view, and then read about the examples. Then pick some topic you like (e.g. ring theory, or representation theory) and ask a more specific question about category theory in, say, representation theory. Again, for any of this to make any sense at all, you'll have to have a pretty good grip on whatever topic it is you're interested in. Category theory tends to produce "large scale" structural theorems, and so if you're not familiar enough with a topic to be interested in how all of its pieces fit together, it (in my opinion) will be very hard to motivate category theory.

However, after writing all of this, and then reading through the comments above a bit more, I see that someone has really already provided you with an answer, which is this MSE question, so that's probably a pretty good place to start.

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    $\begingroup$ I disagree with your characterization of the question as, "How is category theory used in algebra?". That's probably an oversimplification of the question which I thought to be along the lines expressed here. $\endgroup$
    – user170039
    Jun 24, 2018 at 13:47
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    $\begingroup$ The wide applications of Category Theory doesn't make it more significant than that of "structured sets" (which I think should be made more precise @Lehs) if there is not some advantages of using categories over that of "structured sets". $\endgroup$
    – user170039
    Jun 24, 2018 at 13:48
  • $\begingroup$ Thanks for your nice answer, but I used Saunders-McLane during my self studies and with modern category theory I rather ment the developement the last 30 years. $\endgroup$
    – Lehs
    Jun 24, 2018 at 20:04
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    $\begingroup$ Ok well, I think that I really don't understand what the question is asking. Is it asking for a defense of the use of category theory? I think that's hardly necessary. Is it asking for an encyclopedic description of the uses of category theory in modern mathematics? I'm pretty sure that is the definition of being "too broad." $\endgroup$
    – Jonathan Beardsley
    Jun 24, 2018 at 20:09
  • $\begingroup$ @Jonathan I had no big problem with the original category theory, but it seems difficult to overview the modern topics and undetstand it's connection to classical mathematics. Maybe someone should make a map and a manual. I'm sure there are certain developement for the sake of it's own, but some might be useful also in the case of sets with structures. $\endgroup$
    – Lehs
    Jun 25, 2018 at 8:49

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