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    • CommentRowNumber1.
    • CommentAuthorHarry Gindi
    • CommentTimeSep 3rd 2011
    • (edited Sep 3rd 2011)

    Let CC be a presheaf category, and let F:C m+1+nCF : C^{m+1+n} \to C be a functor such that for any family of m+nm+n objects of CC, A 1,,A miA_1,\dots,A_{m-i} and B 1,,B n+iB_1,\dots,B_{n+i} (letting ii vary between 00 and mm), F(A 1,,A mi,,B 1,,B n+i)F(A_1,\dots,A_{m-i},-,B_1,\dots,B_{n+i}) is a parametric left adjoint, that is, the induced functor

    L:CF(A 1,,A mi,,B 1,,B n+i)CL:C\to F(A_1,\dots,A_{m-i},\emptyset,B_1,\dots,B_{n+i})\downarrow C

    admits a right adjoint.

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

    L Δ:CG()CL_\Delta:C\to G(\emptyset)\downarrow C

    admits a right adjoint?