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1. • CommentRowNumber1.
   • CommentAuthorJohn Baez
   • CommentTimeMay 18th 2023
   • PermaLink
   Author: John Baez
   Format: MarkdownItexI added a remark after "Rafael's theorem". By the way, the equations obeyed by $\nu$ and $\zeta$ 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 $F\dashv G$ be a pair of adjoint functors. Then $F$ is separable iff the unit $\eta:1\to G F$ has a section (= a natural transformation $\nu$ which is its right inverse, $\nu\circ\eta = 1$). $G$ is separable iff the counit $\epsilon:F G \to 1$ has a retraction (i.e., a natural transformation $\zeta$ that is its left inverse, $\eta\circ\zeta =1$). <a href="https://ncatlab.org/nlab/revision/diff/separable+functor/9">diff</a>, <a href="https://ncatlab.org/nlab/revision/separable+functor/9">v9</a>, <a href="https://ncatlab.org/nlab/show/separable+functor">current</a>

   I added a remark after “Rafael’s theorem”. By the way, the equations obeyed by $\nu$ and $\zeta$ 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 $F\dashv G$ be a pair of adjoint functors. Then $F$ is separable iff the unit $\eta:1\to G F$ has a section (= a natural transformation $\nu$ which is its right inverse, $\nu\circ\eta = 1$). $G$ is separable iff the counit $\epsilon:F G \to 1$ has a retraction (i.e., a natural transformation $\zeta$ that is its left inverse, $\eta\circ\zeta =1$).

   diff, v9, current

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeMay 19th 2023
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexThe compositions in the equations for $\eta$ and $\epsilon$ (where the latter was misnamed $\eta$!) needed to be reversed, so I did that. <a href="https://ncatlab.org/nlab/revision/diff/separable+functor/11">diff</a>, <a href="https://ncatlab.org/nlab/revision/separable+functor/11">v11</a>, <a href="https://ncatlab.org/nlab/show/separable+functor">current</a>

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

   diff, v11, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 19th 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItextouched formatting and wording, such as of the "Definition" (which still remains incomplete, but I added a pointer to where it is given). <a href="https://ncatlab.org/nlab/revision/diff/separable+functor/12">diff</a>, <a href="https://ncatlab.org/nlab/revision/separable+functor/12">v12</a>, <a href="https://ncatlab.org/nlab/show/separable+functor">current</a>

   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