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1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeDec 21st 2014
   • (edited Dec 21st 2014)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexHere's something I'd never noticed before yesterday: in the category of posets, letting $Down(P)$ denote the internal hom $[P^{op}, \mathbf{2}]$, there is at most one retraction $\pi$ of the yoneda = principal down-set embedding $y: P \to Down(P)$, and this occurs precisely when $\pi \dashv y$ (so that $P$ is a sup-lattice). A "lowbrow" proof might go like this: if $D \subseteq P$ is a down-set, then for all $d \in D$ we have $y(d) \subseteq D$, whence $d = \pi y(d) \leq \pi(D)$, showing that $\pi(D)$ is an upper bound of $D$. On the other hand, if $e$ is any upper bound of $D$, then $D \subseteq y(e)$, whence $\pi(D) \leq \pi y(e) = e$, so that $\pi(D)$ is a least upper bound of $D$. I don't know why exactly, but I found this slightly disconcerting. Offhand, I would have expected *many* retractions are possible. But there's at most one! Slightly more generally: suppose $V$ is a commutative quantale (I'm gearing up to general enriched categories) which we can think of as a small [[cosmos]], and suppose $P$ is a $V$-enriched category. If $f: P^{op} \to V$ is a $V$-functor, then we can use the enriched so-called "co-yoneda" lemma to write $$f = \int^p f(p) \cdot y(p)$$ and now if we suppose $\pi: V^{P^{op}} \to P$ is a $V$-functor retraction of $y$, then we may put $e = y \circ \pi: V^{P^{op}} \to V^{P^{op}}$. The enrichment of $e$ yields a canonical transformation $$f = \int^p f(p) \cdot e y(p) \to e(f)$$ (where the first equation uses $e y = y \pi y = y$) which gives the unit $1 \to y\pi$ of an adjunction $\pi \dashv y$, with counit the retraction isomorphism $\pi y \stackrel{\sim}{\to} 1_P$. So again, there's only one possible retraction. For a while I thought this type of calculation might generalize to a general cosmos $V$. If there is $\pi: V^{C^{op}} \to C$ together with a $V$-natural isomorphism $\pi y_C \to 1_C$, one can indeed manufacture a $V$-natural candidate for a unit $1 \to y_C \pi$ along the above lines, but I wasn't able to see the triangular equations (which come for free in posetal cases like the above). (Not even for the case $V = Set$.) Either this is because I'm blind or stupid here, or in fact in general there is no such adjunction. Which is it? So I'm putting this question to readers here: is there an example of a locally small category $C$ and a functor $F: Set^{C^{op}} \to C$ and an invertible transformation $F y_C \to 1_C$ but where $F$ is *not* left adjoint to $y_C$?

   Here’s something I’d never noticed before yesterday: in the category of posets, letting denote the internal hom , there is at most one retraction of the yoneda = principal down-set embedding , and this occurs precisely when (so that is a sup-lattice).

   A “lowbrow” proof might go like this: if is a down-set, then for all we have , whence , showing that is an upper bound of . On the other hand, if is any upper bound of , then , whence , so that is a least upper bound of .

   I don’t know why exactly, but I found this slightly disconcerting. Offhand, I would have expected many retractions are possible. But there’s at most one!

   Slightly more generally: suppose is a commutative quantale (I’m gearing up to general enriched categories) which we can think of as a small cosmos, and suppose is a -enriched category. If is a -functor, then we can use the enriched so-called “co-yoneda” lemma to write

   and now if we suppose is a -functor retraction of , then we may put . The enrichment of yields a canonical transformation

   (where the first equation uses ) which gives the unit of an adjunction , with counit the retraction isomorphism . So again, there’s only one possible retraction.

   For a while I thought this type of calculation might generalize to a general cosmos . If there is together with a -natural isomorphism , one can indeed manufacture a -natural candidate for a unit along the above lines, but I wasn’t able to see the triangular equations (which come for free in posetal cases like the above). (Not even for the case .) Either this is because I’m blind or stupid here, or in fact in general there is no such adjunction. Which is it?

   So I’m putting this question to readers here: is there an example of a locally small category and a functor and an invertible transformation but where is not left adjoint to ?

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeDec 22nd 2014
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   Author: Mike Shulman
   Format: MarkdownItexInteresting question! At first I thought I could also generalize your calculation, but now I think I was wrong. I do think I can prove that if such an $F$ exists, then $C$ is "weakly total", in that it has weak colimits (existence but not uniqueness of factorizations) of all the diagrams that a total category would have colimits of, and $F$ assigns those weak colimits. Conversely, if $C$ is weakly total and we can choose those weak colimits functorially in a way that assigns the actual colimit of each representable, then that functorial choice ought to be such an $F$. Can we think of any weakly total category that is not total?

   Interesting question! At first I thought I could also generalize your calculation, but now I think I was wrong.

   I do think I can prove that if such an exists, then is “weakly total”, in that it has weak colimits (existence but not uniqueness of factorizations) of all the diagrams that a total category would have colimits of, and assigns those weak colimits. Conversely, if is weakly total and we can choose those weak colimits functorially in a way that assigns the actual colimit of each representable, then that functorial choice ought to be such an . Can we think of any weakly total category that is not total?