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    • CommentRowNumber1.
    • CommentAuthorJohn Baez
    • CommentTimeMay 18th 2023

    I added a remark after “Rafael’s theorem”. By the way, the equations obeyed by ν\nu and ζ\zeta don’t seem to match the verbal descriptions of them as right and left inverses in the statement of Rafael’s theorem, so I hope someone can sort this out and fix this:

    Rafael’s theorem. Let FGF\dashv G be a pair of adjoint functors. Then FF is separable iff the unit η:1GF\eta:1\to G F has a section (= a natural transformation ν\nu which is its right inverse, νη=1\nu\circ\eta = 1). GG is separable iff the counit ε:FG1\epsilon:F G \to 1 has a retraction (i.e., a natural transformation ζ\zeta that is its left inverse, ηζ=1\eta\circ\zeta =1).

    diff, v9, current

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 19th 2023

    The compositions in the equations for η\eta and ε\epsilon (where the latter was misnamed η\eta!) needed to be reversed, so I did that.

    diff, v11, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMay 19th 2023

    touched formatting and wording, such as of the “Definition” (which still remains incomplete, but I added a pointer to where it is given).

    diff, v12, current