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Davi Sales Barreira (Jun 17 2022 at 17:37):So, in "Category Theory The Sciences", David Spivak tries to give an intuition regarding adjunctions by talking about how to translate sounds from "babies" to "adult words". I confess I was a bit lost in the analogy, and I was trying to come up with other examples of how to intuitively understand what's going on with adjunctions. At first, I thought adjunctions were like the Moore-Penrose inverse, where it's sort of a weaker version of how to invert a matrix... But it seems that this is not actually the case.
Hence, my question is what are some good analogies/intuitions in order to understand adjuntcions?
John Baez (Jun 17 2022 at 17:45):To answer a question with a question: do you know a bunch of examples of adjunctions? In my opinion, examples are more crucial in developing intuitions than examples. Different examples suggest different analogies.
Analogies come and go: you may decide one is great, then change your mind and decide it's misleading. But examples last forever.
Davi Sales Barreira (Jun 18 2022 at 11:50):Haven't seen many examples. I'll try to read more examples before trying to grasp the definition. I was trying to construct examples to check my understanding, but it seems that I have a wrong intuition.
Morgan Rogers (he/him) (Jun 18 2022 at 13:24):If you have some maths background, Emily Riehl's book Category Theory in Context is great for getting intuition from examples!
John Baez (Jun 18 2022 at 23:22):It's hard to know what examples to give you since I don't know what math you know, but adjunctions are literally everywhere so any category theorist should be able to quickly list at least 20 examples.
John Baez (Jun 19 2022 at 00:34):For example, whenever you have a pair of mathematical concepts, one with "more structure" and one with "less structure", you should expect an adjunction between them. For example: groups and sets.
There may not always be an adjunction, but there often is.
Matteo Capucci (he/him) (Jun 19 2022 at 09:37):If you want an analogy from linear algebra, think of 'adjoint operator'. But this is an analogy, not an example unfortunately.