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No Pains, no Gains: Chapter 1

I had the idea to collect the struggles that we (or atleast a non-empty subset of us) had during learning chapter 1. Very likely future learners will struggle with those same concepts, so we can save them some time. Also, it is encouraging to see that even the best do not grasp everything instantly :)

I will create one such thread per chapter. Later on, the answers can be composed by topic if need be. Let me now what you think and especially where you struggled yourself! I am also open for change in terminology. (Is No Pains, no Gains suitable? are struggle and resolution as used below the correct terms?) My proposed format:

Struggle: < A struggle in the form of a question or a description >

Resolution: < Short informal solution, optionally with the some formalism >

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1.
edited August 2018

Struggle: So can a right adjoint exist without a right adjoint? Can we look for a right adjoint only? (and vice versa for left and right.)

Resolution: No, a left adjoint always comes with a right adjoint and vice versa. When the question is "Is there a right adjoint $$g(y)$$ to $$f(x)?$$" that more clearly means: "Is there a right adjoint $$g(y)$$ such that $$f(x)$$ is its left adjoint?"

Comment Source:**Struggle:** So can a right adjoint exist without a right adjoint? Can we look for a right adjoint *only*? (and vice versa for left and right.) **Resolution:** No, a left adjoint always comes with a right adjoint and vice versa. When the question is "Is there a right adjoint \$$g(y) \$$ to \$$f(x)? \$$" that more clearly means: "Is there a right adjoint \$$g(y) \$$ such that \$$f(x) \$$ is its left adjoint?"
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2.
edited August 2018

Struggle: Getting left and right adjoints straight, intensional vs. extensional equality of functions.

Resolution: Writing code to calculate left and right adjoints for simple posets helped me to clarify this quite a bit, though I think working through many more examples will help. Eventually, I'll post a link to some code here when it's in better shape.

Comment Source:**Struggle**: Getting left and right adjoints straight, intensional vs. extensional equality of functions. **Resolution**: Writing code to calculate left and right adjoints for simple posets helped me to clarify this quite a bit, though I think working through many more examples will help. Eventually, I'll post a link to some code here when it's in better shape. 
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3.

Eventually, I'll post a link to some code here when it's in better shape.

That would be great!

Comment Source:> Eventually, I'll post a link to some code here when it's in better shape. That would be great!