接微分と余接微分、射影関係式 - (新) 檜山正幸のキマイラ飼育記メモ編

多様体のあいだの写像に、その接微分〈tangent differential〉と余接微分〈cotangent differential〉を対応させる。 $\newcommand{\TD}{{\mathscr T}} \newcommand{\coTD}{{\mathscr T}^*} \newcommand{\hyph}{\mbox{-}} \newcommand{\incat}{\:\:\:\mbox{in}\:}$

接微分関手 $\TD = \TD_* : {\bf Man} \to \Phi\hyph{\bf Mod\hyph Sheaf}^{\dashv}$
余接微分関手 $\coTD : {\bf Man}^{op} \to \Phi\hyph{\bf Mod\hyph Sheaf}_{\vdash}$

圏の約束：

$\Phi\hyph{\bf Mod\hyph Sheaf}^{\dashv} := {\displaystyle \int_{\to\; {\bf Man}}} \Phi\hyph{\bf Mod\hyph Sh}^{\dashv}[\hyph]$
$\Phi\hyph{\bf Mod\hyph Sheaf}_{\vdash} := {\displaystyle \int^{\leftarrow\; {\bf Man}}} \Phi\hyph{\bf Mod\hyph Sh}_{\vdash}[\hyph]$

次は同値：

$\TD f : \TD M \to \TD N \incat \Phi\hyph{\bf Mod\hyph Sheaf}^{\dashv}$
$\TD f : \TD M \to f^{\dashv}(\TD N) \incat \Phi\hyph{\bf Mod\hyph Sh}^{\dashv}[M]$
$\TD f : \Phi M \to (\TD M)^* \otimes_{\Phi M} f^{\dashv}(\TD N) \incat \Phi\hyph{\bf Mod\hyph Sh}^{\dashv}[M]$
$\TD f \in (\TD M)^* \otimes_{\Phi M} f^{\dashv}(\TD N) \incat \Phi\hyph{\bf Mod\hyph Sh}^{\dashv}[M]$

次は同値：

$\coTD f : \coTD N \to \coTD M \incat \Phi\hyph{\bf Mod\hyph Sheaf}_{\vdash}$
$\coTD f : \coTD N \to f_{\vdash }(\coTD M) \incat \Phi\hyph{\bf Mod\hyph Sh}_{\vdash}[N]$
$\coTD f : \Phi M \to (\coTD N)^* \otimes_{\Phi N} f_{\vdash }(\coTD M) \incat \Phi\hyph{\bf Mod\hyph Sh}_{\vdash}[N]$
$\coTD f \in (\coTD N)^* \otimes_{\Phi N} f_{\vdash }(\coTD M) \incat \Phi\hyph{\bf Mod\hyph Sh}_{\vdash}[N]$
$\coTD f \in f_{\vdash }( (\TD M)^* ) \otimes_{\Phi N} \TD N \incat \Phi\hyph{\bf Mod\hyph Sh}_{\vdash}[N]$