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Ring theory considers things with 2 operations and category theory 101 talks about products and coproducts.

I maybe understand why binary operations are more common to look at that trinary, quaternary, …, but why look at the interactions between only 2 of them? Surely the extension from 2 to many interlocking operations is not an obvious one…

isomorphismes
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    Have you ever heard of operads? I recommend this article. – Najib Idrissi Oct 19 '14 at 19:23
  • @NajibIdrissi Yes. But I'd forgotten about them. Thanks. – isomorphismes Oct 19 '14 at 19:25
  • I'm not saying it's the answer, but it's at least one answer. Another possible answer is "the majority of interesting algebraic structures are binary". – Najib Idrissi Oct 19 '14 at 19:26
  • @NajibIdrissi OK, so for the second possible answer: (1) do we really know that? (2) how do we know that? – isomorphismes Oct 19 '14 at 19:38
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    A structure with several operations certainly is a non-obvious extension of the case of two operations, and there are concrete examples. For instance, consider $C(\mathbb{R},\mathbb{R})$, the set of all real valued functions of one real variable: there are three interesting binary operations on it: sum, product, and composition, giving rise to a very rich structure. But I'm not sure that a general theoretic approach here would be a useful approach to understand the structure of $C(\mathbb{R},\mathbb{R})$. – Pietro Majer Oct 19 '14 at 19:42
  • @NajibIdrissi Think about the amount of effort that goes into showing that the Cartesian product ("boring") satisfies a universal property. Is there some similar way of showing that it's "easy" to extend 2 to many ops? – isomorphismes Oct 19 '14 at 19:45
  • @PietroMajer How do we know +,×,∘ are the only three interesting binary operations on $C(\mathbb{R},\mathbb{R})$? – isomorphismes Oct 19 '14 at 19:47
  • (I'm not claiming that --say at least three...) – Pietro Majer Oct 19 '14 at 19:55
  • @PietroMajer Ah ok. Gotcha. – isomorphismes Oct 19 '14 at 19:56
  • https://www.ias.edu/articles/2-theories ←feels like Voevodsky trying to break with this kind of pattern – isomorphismes May 23 '15 at 19:01
  • Could this be related: http://math.stackexchange.com/questions/116771 ? – Watson Jan 23 '17 at 12:42

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As usual, if you're confused about some general fact about categories, you should first see what it says about posets. In the context of posets products and coproducts correspond to two very natural operations, namely min and max. These are the two fundamental ways we might try to combine two elements of a poset to get another one, and roughly speaking the fact that there are two of them comes from the fact that posets have two "directions," namely "increasing" and "decreasing."

More generally, a category has two "directions," namely "morphisms in" and "morphisms out." That's why, for example, a functor has two adjoints (when they exist), a left adjoint and a right adjoint, and that's it. (The relationship to limits and colimits is that limits and colimits can be described as adjoints.)

When the poset is in particular a poset of subsets of a set, thought of as Boolean propositions, we can furthermore think of our two natural operations as and and or.

To the extent that there's any connection to ring theory, it's because e.g. you can think of the primordial (semi)ring as being $\mathbb{N}$ with addition and multiplication, which in turn is a decategorification of the category of finite sets equipped with coproduct and product. But there are other ways to think about rings, e.g. as monoid objects in $(\text{Ab}, \otimes)$. From this perspective "two binary operations" is missing the point; we could replace $(\text{Ab}, \otimes)$ with an arbitrarily complicated monoidal category here.

In any case, I dispute your claim that we only look at two binary operations. For example, a Poisson algebra has three. On the categorical side, a (bi)cartesian closed category also has three. The ring of symmetric functions has five.

Qiaochu Yuan
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  • Thanks Qiaochu. This is exactly the kind of answer I was hoping for. One question though, it seems like you're taking a 1-category as natural (in the first part). And maybe some other sort of object that doesn't naturally split into in/out is just as natural? (I only meant to give rings and cat 101 as examples.) – isomorphismes Oct 19 '14 at 19:51
  • @isomorphismes: well, who knows? We could spend all day arguing about precisely what is and is not natural. What I'm willing to say is that categories have been a very fruitful point of view historically and I expect they'll continue to be in the future. – Qiaochu Yuan Oct 19 '14 at 20:11
  • OK. That comment helps me contextualise your answer; thanks. – isomorphismes Oct 19 '14 at 20:18
  • Qiaochu, I was reviewing your answer and remembered Voevodsky's 2-theories from his 2014 IAS talk. Relating to your second paragraph—this "3-D diagram" has more than two directions. – isomorphismes Apr 30 '15 at 16:16