I'm using notation close to Street-Walters "Yoneda structures".

For any locally small category $\textbf{A}$ there are, of course, $\hat{\textbf{A}}:=\textbf{set}^{\textbf{A}^{op}}$ and $\check{\textbf{A}}:=(\textbf{set}^{\textbf{A}})^{op}$ as well as the corresponding Yoneda embeddings $Y(\textbf{A}):\textbf{A}\rightarrow\hat{\textbf{A}}$ and $Z(\textbf{A}):\textbf{A}\rightarrow\check{\textbf{A}}$.

For any locally small functor $F:\textbf{A}\rightarrow\textbf{B}$, there is a covariant $F$-weighted-hom functor $\textbf{B}(F,1):\textbf{B}\rightarrow\hat{\textbf{A}}$ as well as a contravariant one $\textbf{B}\langle1,F\rangle:\textbf{B}\rightarrow\check{\textbf{A}}$ and evaluation of $F$ on arrows can be encoded via a natural transformation from the covariant Yoneda $\chi^F:Y(\textbf{A})\Rightarrow\textbf{B}(F,1)F$ or via one from the contravariant one $\psi^F:Z(\textbf{A})\Rightarrow\textbf{B}\langle 1,F \rangle F$.

It is a fact that, in $\textbf{Cat}$, the covariant $F$-weighted-hom is a left Kan extension along $F$ of the covariant Yoneda of its domain (SW Axiom 1 for the Yoneda structure of $\textbf{Cat}$) and that $F$ is the absolute left lifting of Yoneda along the $F$-weighted-hom (SW Axiom 2), namely:
$(1) \hspace{12pt} (\textbf{B}(F,1),\chi^F)=lan_FY(\textbf{A})$
$(2) \hspace{12pt} (F,\chi^F)=LIF_{\textbf{B}(F,1)} Y(\textbf{A})$
By pedantically adapting the proof of $(1)$ I can see its contravariant version:
$(1*) \hspace{8pt}(\textbf{B} \langle 1,F \rangle,\psi^F)=lan_FZ(\textbf{A})$

Is $(1*)$ a consequence of $(1)$ in the sense that there is a direct way, that I guess should pass through the underlying profunctors, to prove $(1*)$ assuming $(1)$ ?

Is there a contravariant version of $(2)$ ?

  • $\begingroup$ I seem that from $(1)$, replacing $A$ and $F$ with its duals $A^{op}$ and $F^{op}$ and from $(\check{A})^{op}$-$lan_{F^{op}}Y(A^{op})=(\check{A})$-$ran_{F^{op}}Y(A^{op})=lan_{F}Z(A)$ follow $(1_\ast)$ $\endgroup$ Dec 19 '11 at 13:27

(I tried to put this as a comment, but likely space limitations were exceeded. I apologise if this is not the usual practice)
Thanks to Sergio Buschi for the suggestion.
Indeed (1*) is nothing but the opposite of another instance of SW-axiom 1, namely
$(1_a) \hspace{24pt}(\textbf{B}^{op}(F^{op},1) ,\chi^{F^{op}})=lan_{F^{op}}Y(\textbf{A}^{op})$
In facts, it turns out that:
$Y(\textbf{A}^{op})=Z(\textbf{A})^{op} \hspace{24pt} \textbf{B}^{op}(F^{op},1)=\textbf{B} \langle1, F \rangle^{op} \hspace{24pt} \chi^{F^{op}}=\psi^{F}$
(where $\textbf{B} \langle1, F \rangle$ denotes the functor $\textbf{B} \rightarrow \check{\textbf{A}}:B \mapsto \textbf{B}(B,F-)$)
Then, the duality involution $(-)^{op}: \textbf{Cat} \rightarrow \textbf{Cat}^{co}$ turns $(1_a)$ into a right Kan extension in $\textbf{Cat}^{co}$ which is the left Kan extension $(1*)$ in $\textbf{Cat}$.
A similar application of SW-axiom 2 yields:
$(2_a) \hspace{24pt}(F^{op},\psi^{F})=LIF_{\textbf{B}\langle 1,F \rangle^{op}}Z(\textbf{A})^{op}$
(where $LIF$ means absolute left lifting)

  • $\begingroup$ A second question then arises. <br> Is $(2_a)$ equivalent to the following ? <br> $(F,\psi^{F})=LIF_{\textbf{B}\langle 1,F \rangle}Z(\textbf{A})$ $\endgroup$ Dec 20 '11 at 15:15
  • $\begingroup$ If I remeber well, the R. Street article is about a representable 2-category (i.e. it has comma objects, see the article). If you get the double dualization (on 1-morphisms, and cells) and apply the $(2)$ to the dougle dual of (meta)2-category $CAT$ of the categories ($CAT$ as cocomma construction), then its double dual is representable) , you get the wanted assert about point $(2_a)$. $\endgroup$ Dec 20 '11 at 15:42

I realize that the question was not precise, but I now hope to understand.
For any locally small functor $F:\textbf{A}\rightarrow\textbf{B}$, evaluation of $F$ on arrows can be encoded via $\chi^F:Y(\textbf{A})\Rightarrow\textbf{B}(F,1)F$ or $\psi^F:\textbf{B}\langle 1,F \rangle F \Rightarrow Z(\textbf{A})$.
SW Axioms 1 and 2 are
$(1) \hspace{12pt} (\textbf{B}(F,1),\chi^F)=lan_FY(\textbf{A})$
$(2) \hspace{12pt} (F,\chi^F)=LIF_{\textbf{B}(F,1)} Y(\textbf{A})$
The same axioms applied to $F^{op}$ directly yield
$(1*) \hspace{8pt}(\textbf{B} \langle 1,F \rangle,\psi^F)=ran_FZ(\textbf{A})$
$(2*) \hspace{12pt} (F,\psi^F)=RIF_{\textbf{B}\langle 1,F \rangle} Z(\textbf{A})$
(where $RIF$ means absolute right lifting). So the contravariant side of a Yoneda structure can be exploited in a 2-category with a duality involution.


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