About the equivalence between monads and monadic functors
About the equivalence between monads and monadic functors
复制标题
关于 monad 和 monadic functor 之间的等价
DOI：
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
Hadrian Heine
中科院分区：
文献类型：
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作者：
Hadrian Heine
Given a $(\infty, 2)$-category $\mathcal{C}$ we call a morphism $ \mathrm{Y} \to \mathrm{X}$ of $ \mathcal{C}$ monadic if it induces for every $\mathrm{Z} \in \mathcal{C}$ a monadic functor $ [\mathrm{Z},\mathrm{Y}] \to [\mathrm{Z},\mathrm{X}] $ on mapping categories. Given a monad $\mathrm{T} $ on some object $\mathrm{X}$ of $\mathcal{C} $, i.e. an associative algebra in the monoidal category of endomorphisms of $\mathrm{X},$ we call a morphism $\psi : \mathrm{Y} \to \mathrm{X}$ of $ \mathcal{C}$ an Eilenberg-Moore object of $\mathrm{T}$ if $\psi$ admits a left action over $\mathrm{T}$ (where $[\mathrm{X},\mathrm{X}]$ acts on $[\mathrm{Y},\mathrm{X}] $ from the left by composition) that exhibits $\mathrm{T} $ as an endomorphism object of $\psi.$
For every $(\infty, 2)$-category $\mathcal{C},$ in which every monad admits an Eilenberg- Moore object, we construct a localization $\mathrm{End} : (\mathcal{C}_{/ \mathrm{X}})^{\mathrm{R}} \rightleftarrows \mathrm{Alg}([\mathrm{X}, \mathrm{X}])^{\mathrm{op}} : \mathrm{Alg} $ between the $\infty$-category of right adjoint morphisms with target $\mathrm{X}$ and the $\infty$-category of monads on $\mathrm{X},$ whose local objects are the monadic morphisms with target $\mathrm{X}.$ We show that for every $\infty$-category $\mathrm{S}$ the $(\infty, 2)$-category ${\mathrm{Cat}_\infty}_{/ \mathrm{S}} $ of $\infty$-categories over $\mathrm{S}$ admits both Eilenberg-Moore objects and coEilenberg- Moore objects.
From this result we deduce that for every categorical pattern $\mathfrak{P} $ on a $\infty$-category S the subcategory of ${\mathrm{Cat}_\infty}_{/ \mathrm{S}} $ with objects the $\mathfrak{P} $-fibered objects admits Eilenberg-Moore objects and coEilenberg-Moore objects that are preserved by the subcategory inclusion to ${\mathrm{Cat}_\infty}_{/ \mathrm{S}}. $
Given a $(\infty, 2)$-category $\mathcal{C}$ we call a morphism $ \mathrm{Y} \to \mathrm{X}$ of $ \mathcal{C}$ monadic if it induces for every $\mathrm{Z} \in \mathcal{C}$ a monadic functor $ [\mathrm{Z},\mathrm{Y}] \to [\mathrm{Z},\mathrm{X}] $ on mapping categories. Given a monad $\mathrm{T} $ on some object $\mathrm{X}$ of $\mathcal{C} $, i.e. an associative algebra in the monoidal category of endomorphisms of $\mathrm{X},$ we call a morphism $\psi : \mathrm{Y} \to \mathrm{X}$ of $ \mathcal{C}$ an Eilenberg-Moore object of $\mathrm{T}$ if $\psi$ admits a left action over $\mathrm{T}$ (where $[\mathrm{X},\mathrm{X}]$ acts on $[\mathrm{Y},\mathrm{X}] $ from the left by composition) that exhibits $\mathrm{T} $ as an endomorphism object of $\psi.$
For every $(\infty, 2)$-category $\mathcal{C},$ in which every monad admits an Eilenberg- Moore object, we construct a localization $\mathrm{End} : (\mathcal{C}_{/ \mathrm{X}})^{\mathrm{R}} \rightleftarrows \mathrm{Alg}([\mathrm{X}, \mathrm{X}])^{\mathrm{op}} : \mathrm{Alg} $ between the $\infty$-category of right adjoint morphisms with target $\mathrm{X}$ and the $\infty$-category of monads on $\mathrm{X},$ whose local objects are the monadic morphisms with target $\mathrm{X}.$ We show that for every $\infty$-category $\mathrm{S}$ the $(\infty, 2)$-category ${\mathrm{Cat}_\infty}_{/ \mathrm{S}} $ of $\infty$-categories over $\mathrm{S}$ admits both Eilenberg-Moore objects and coEilenberg- Moore objects.
From this result we deduce that for every categorical pattern $\mathfrak{P} $ on a $\infty$-category S the subcategory of ${\mathrm{Cat}_\infty}_{/ \mathrm{S}} $ with objects the $\mathfrak{P} $-fibered objects admits Eilenberg-Moore objects and coEilenberg-Moore objects that are preserved by the subcategory inclusion to ${\mathrm{Cat}_\infty}_{/ \mathrm{S}}. $