Just exactly what the title says; often, in mathematics, particularly in the vicinity of Grothendieck, I see reference to "the yoga of...". What exactly does the term "yoga" mean in these contexts?

I've taken "yoga" to mean a part of the body of mathematics which does not consist of many actual theorems or results  or in fact could not be formalized as just a few theorems  but rather a collection of principles and techniques that one needs to wrap one's head around completely, after which one will be able to use them almost effortlessly. As an example, I would say that there is a yoga of generating functions in combinatorics. (Perhaps this is the simplest example of a yoga.) 


"Yoga" and "yoke" (as in of oxen) are derived from the same IndoEuropean root, meaning a linkage. Of course "linkage", and "relation", and "connection", and "join", all have mathematical meanings already, so one must go further afield to talk about two mathematical concepts being yoked to one another. When I have seen the word used by mathematicians (esp. Bott), it is usually in exactly this way  the yoga of X and Y, not of X. (As such I must disagree with the comment `of course it has nothing to do with any proper meaning of the word "yoga".') 


I sometimes have used the word myself, without ever having sat down and asked myself what do I mean by that exactly. I've used it roughly to mean a coherent body of techniques; I'm not sure if I can amplify much further. "The yoga of adjunctions and mates", "the yoga of the Yoneda lemma and its correlates", and you will find a bunch more scattered around the nLab ("the yoga of 'generalized the' ") if you use the search function. To me, a "yoga" is not quite as formalized (or pretentious) as a "calculus", but it's somewhat in that vein. 

