adjoint

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  • 2021-10-08

    Set up the adjoint \begin{align*} \mathsf{D} \coloneqq\adjunction{p^*}{p_*}{{\mathsf{QCoh}}({\mathbf{B}}G)}{{\mathsf{QCoh}}(\operatorname{Spec}R)} \coloneqq\mathsf{C} .\end{align*} Then \(LR \coloneqq p^*p_*\), and Barr-Beck yields \begin{align*} {\mathsf{QCoh}}({\mathbf{B}}G)\underset{\tilde{p^*}}{{ { \, \xrightarrow{\sim}\, }}} {\mathsf{(p^*p_*)}{\hbox{-}}\mathsf{coMod}}({\mathsf{QCoh}}(\operatorname{Spec}R)) .\end{align*}

  • 2021-06-05

    Theorem: limit of \(F\) in \({\mathsf{sSet}}_{{ \mathsf{quasiCat} } }\) is a homotopy limit of its adjoint in \({\mathsf{sSet}}{\hbox{-}}\mathsf{Cat}_{{\mathsf{Kan}}}\), and the limit of its adjoint in ${\mathsf{sSet}}^{\Delta^{\operatorname{op}}}_{ \mathsf{CSS} } $.

  • 2021-04-26

    Finding adjoint is usually easy, because checking isomorphisms on hom sets is concrete.

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