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I have heard a joke like this:

The Yoda embedding, contravariant it is.

And a joke concerning "How to put an elephant into a refrigerator", a comment from "Category Theorist" says

Isn’t this just a special case of Yoneda’s lemma?

Since I'm not familiar with category theory, is there any way for me to understand these punch lines? I'm not quite sure if I could post this here, if not, please tell me and I'll delete it. Thanks!

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Golbez
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1 Answers1

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I could be really wrong, but I think it's just a general joke about the way category theory makes almost everything a special case of something more abstract.

That's in the spirit of the next quip by the theoretical computer scientist: "I can't decide!"

(That said, before I wrote this I was googling to see if there was some sort of Japanese refrigerator company with a "Yoneda model," or an elephant species with that name since either would have been awesome if true...)

The "Yoda embedding" joke is a pun based on Yoda's name and the way he speaks. Roughly speaking, "contravariance" in category theory is a quality of a functor that "reverses arrows in diagrams." An old arrow is replaced with a new arrow pointing in the opposite direction. So not only do "Yoda" and "Yoneda" sound alike, but Yoda also has a peculiar habit of reversing words in his sentences, which reminds you of contravariance.

(Here are a bunch of examples of category theory making special cases of stuff. It will not be exhaustive, and possibly not 100% accurate, given the current state of my category theoretic knowledge.)

Firstly, category theory makes entire disciplines of mathematics special cases of categories. There is the category of groups, the category of rings, the category of manifolds, the category of sets etc. etc. ad nauseum. Then you study the properties of these categories abstractly.

But if that's not your cup of tea, there is a way to describe a (single) group as a category. Monoids too. Actually, if you make a monoid inside the category of groups, you get a ring, so rings are special cases of monoids. You can view a poset as a category also. It turns out that groups, equivalence relations and the fundamental groupoid of a topological space all are special cases of something called a groupoid in category theory.

In another direction, you're probably aware of products, coproducts, limits, colimits, pullbacks, pushouts etc in the various categories. The abstract definitions for each of these is the same for each category, but the actual construction of each can vary wildly by category.

These constructions are all generalized by the idea of "universal objects," of which they are all examples.

But universal objects aren't so special, they're just special cases of initial and terminal objects in categories built up from old categories.

But terminal objects aren't so special, they are just initial objects in the opposite category of the category we're in.

But categories aren't so special, they're just special cases of $n$-categories.

But...

I didn't even mention adjoints. The famous motto of adjoints is "Adjoints arise everywhere." I think I'll have to leave it up to the wiki page on adjoint functors to list examples, but glancing over it, I think I have heard many more examples that appear even there.

rschwieb
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  • Thanks! Seems that's really a general interpretation of these jokes! But can you explain it further? Such as the "contravariant" and its relation with "Yoda embedding". – Golbez Sep 25 '13 at 17:09
  • @Golbez Sure, I'll add a bit. – rschwieb Sep 25 '13 at 17:25
  • @Golbez I also see in the "related links" to this question: "Yoneda lemma as generalization of Cayley's theorem?" http://math.stackexchange.com/questions/1701/yoneda-lemma-as-generalization-of-cayleys-theorem?rq=1 – rschwieb Sep 25 '13 at 18:05
  • This is a detailed answer that give me some motivation to learn some category theory, thank you very much!! – Golbez Sep 26 '13 at 01:31
  • This link also provides me some other useful topics here and in MO! Such as http://math.stackexchange.com/questions/37165/can-someone-explain-the-yoneda-lemma-to-an-applied-mathematician and http://mathoverflow.net/questions/3184/philosophical-meaning-of-the-yoneda-lemma – Golbez Sep 26 '13 at 13:16
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    "Firstly, category theory makes entire disciplines of mathematics special cases of categories." This is simply false. – goblin GONE Jun 30 '14 at 00:17
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    Dear @user18921 : I guess that discourteous blurt translates to: "what did you mean when you said that?" To this I would say "For example, groups and their morphisms form a particular category, as do topological spaces and posets." This apparent fact is all I meant, not much more. – rschwieb Jun 30 '14 at 02:18
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    @rschwieb, well if that's all you meant, I suggest you perform an edit. What you have actually written is that entire disciplines are subsumed by category theory. You seem to be suggesting that: "Don't worry learning probability theory, general topology, commutative algebra, or number theory, because hey! they are all just special cases of categories." Frankly, I'm tired of people overplaying the importance of category theory. Yes, it is important. No, it is not a substitute for actually learning a new discipline from people who have worked in that discipline their whole lives. – goblin GONE Jul 01 '14 at 00:51
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    Dear @user18921 : thanks for the suggestion. I see what you mean, but at the moment I think it's an exaggeration to say that my sentence means "disciplines are subsumed in category theory." It is no more than "there's a category for most every discipline." If I see someday votes accumulating on your comment, I might change my mind though. Until this point there has been no confusion. – rschwieb Jul 01 '14 at 10:11