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1. • CommentRowNumber1.
   • CommentAuthorHarry Gindi
   • CommentTimeSep 3rd 2011
   • (edited Sep 3rd 2011)
   • PermaLink
   Author: Harry Gindi
   Format: MarkdownItexLet $C$ be a presheaf category, and let $F : C^{m+1+n} \to C$ be a functor such that for any family of $m+n$ objects of $C$, $A_1,\dots,A_{m-i}$ and $B_1,\dots,B_{n+i}$ (letting $i$ vary between $0$ and $m$), $F(A_1,\dots,A_{m-i},-,B_1,\dots,B_{n+i})$ is a parametric left adjoint, that is, the induced functor $$L:C\to F(A_1,\dots,A_{m-i},\emptyset,B_1,\dots,B_{n+i})\downarrow C$$ admits a right adjoint. Let $\Delta^{m+1+n}:C\to C^{m+1+n}$ denote the $m+1+n$-fold diagonal functor. Let $G=F\circ \Delta^{m+1+n}$. Then is it the case that the induced functor $L_\Delta:C\to G(\emptyset)\downarrow C$ admits a right adjoint?

   Let $C$ be a presheaf category, and let $F : C^{m+1+n} \to C$ be a functor such that for any family of $m+n$ objects of $C$, $A_1,\dots,A_{m-i}$ and $B_1,\dots,B_{n+i}$ (letting $i$ vary between $0$ and $m$), $F(A_1,\dots,A_{m-i},-,B_1,\dots,B_{n+i})$ is a parametric left adjoint, that is, the induced functor

   $$L:C\to F(A_1,\dots,A_{m-i},\emptyset,B_1,\dots,B_{n+i})\downarrow C$$

   admits a right adjoint.

   Let $\Delta^{m+1+n}:C\to C^{m+1+n}$ denote the $m+1+n$-fold diagonal functor. Let $G=F\circ \Delta^{m+1+n}$. Then is it the case that the induced functor

   $L_\Delta:C\to G(\emptyset)\downarrow C$

   admits a right adjoint?