Zoran Skoda
domain globalization of functors

Let $\{Q_{\lambda }^{*}:A\to A_{\lambda }{\}}_{\lambda \in \Lambda }$ be a cover of $A$ by functors having a right adjoint, $Q_{\lambda }^{*}⊣Q_{\lambda *}$ with unit ${\eta }^{\lambda }$ and counit ${ϵ}^{\lambda }$. Each pair defines a monad with underlying endofunctor $Q_{\lambda *}{Q}_{\lambda }^{*}$ and if $A_{\lambda }$ admit products of $\mathrm{card}\Lambda$ objects, the whole cover defines a comonad $\Omega$ on ${\prod }_{\lambda }{A}_{\lambda }$, induced by the adjunction $A\leftrightarrow {\prod }_{\lambda }{A}_{\lambda }$. (Cf. gluing categories from localizations).

Denote ${\Omega }_{\mu }^{\lambda }=Q_{\mu }^{*}{Q}_{\lambda *}$ the projections ${\pi }_{\lambda }:A_{\Lambda }\to A_{\lambda }$, ${\omega }^{\lambda }:\Omega \Rightarrow {\Omega }^{\lambda }{\pi }_{\lambda }$, then $\Omega ={\prod }_{\lambda }{\Omega }^{\lambda }{\pi }_{\lambda }$. The unit defines the coaction $\Omega \to \Omega \Omega$ which locally gives rise to ${\Omega }_{\lambda }^{\nu }\stackrel{{\delta }_{\lambda }^{\nu }(\mu )}{⟶}{\Omega }_{\lambda }^{\mu }{\Omega }_{\mu }^{\nu }$ and the counit locally gives ${ϵ}^{\lambda }:{\Omega }_{\lambda }^{\lambda }\to 1_{A_{\lambda }}$.

Domain localization for a cover

and $G:A\to B$ a functor. (I will present here a sketch following mainly Gabi’s improvements and notation.) Then $G$ defines the following local data

$$G^{\lambda }:=\left(A_{\lambda }\stackrel{Q_{\lambda *}}{⟶}A\stackrel{G}{⟶}B\right)$$G^\lambda := \left( A_\lambda\stackrel{Q_{\lambda *}}\longrightarrow A\stackrel{G}\longrightarrow B\right)

$${\psi }^{\lambda }(\nu ):=(G^{\lambda }=G{Q}_{\lambda *}\stackrel{G{\eta }_{\mu }{Q}_{\lambda *}}{⟶}G{Q}_{\nu *}{Q}_{\nu }^{*}{Q}_{\lambda *}=G^{\nu }{\Omega }_{\nu }^{\lambda })$$\psi^\lambda(\nu) := \left( G^\lambda = G Q_{\lambda*}\stackrel{G\eta_\mu Q_{\lambda*}}\longrightarrow G Q_{\nu*} Q^*_\nu Q_{\lambda *} = G^\nu \Omega^\lambda_\nu \right)

where $\lambda ,\nu \in \Lambda$. Then the composition

$$G^{\lambda }\stackrel{{\psi }^{\lambda }(\lambda )}{⟶}{G}^{\lambda }{\Omega }_{\lambda }^{\lambda }\stackrel{G^{\lambda }{ϵ}^{\lambda }}{⟶}{G}^{\lambda }$$G^\lambda \stackrel{\psi^\lambda(\lambda)}\longrightarrow G^\lambda \Omega^\lambda_\lambda \stackrel{G^\lambda \epsilon^\lambda}\longrightarrow G^\lambda

is the identity and the diagram

$$\begin{array}{ccc}{G}^{\lambda }& \stackrel{{\psi }^{\lambda }(\nu )}{⟶}& G^{\nu }{\Omega }_{\mu }^{\lambda }\\ {}_{{\psi }^{\lambda }(\mu )}\downarrow & & \downarrow {}_{G^{\mu }{\delta }_{\nu }^{\lambda }(\mu )}\\ G^{\nu }{\Omega }_{\mu }^{\lambda }& \underset{{\psi }^{\mu }(\nu ){\Omega }_{\mu }^{\lambda }}{⟶}& G^{\nu }{\Omega }_{\nu }^{\mu }{\Omega }_{\mu }^{\lambda }\end{array}$$\array{ G^\lambda & \stackrel{\psi^\lambda(\nu)}\longrightarrow & G^\nu \Omega^\lambda_\mu \\ {}_{\mathllap{\psi^\lambda(\mu)}}\downarrow && \downarrow {}_{\mathrlap{G^\mu \delta^\lambda_\nu(\mu)}}\\ G^\nu \Omega^\lambda_\mu & \underset{\psi^\mu(\nu)\Omega^\lambda_\mu}\longrightarrow & G^\nu \Omega^\mu_\nu \Omega^\lambda_\mu }

commutes. Such pairs are objects of a category ${\mathrm{DomLoc}}_{\Lambda }$ which depends on the cover of $A$ and on $B$.

Given a natural transformation $\theta :G\Rightarrow G\prime$, one defines the corresponding local data by whiskering:

$${\theta }^{\lambda }:=(G^{\lambda }=G{Q}_{\lambda *}\stackrel{\theta Q_{\lambda *}}{⟶}G{Q}_{\lambda *}=G{\prime }^{\lambda })$$\theta^\lambda := \left( G^\lambda = G Q_{\lambda*} \stackrel{\theta Q_{\lambda*}}\longrightarrow G Q_{\lambda*} = G'^\lambda\right)

The data, $\{{\theta }^{\lambda }{\}}_{\lambda \in \Lambda }$ form a morphism $\{G^{\lambda },{\psi }^{\lambda }(\nu ){\}}_{\lambda ,\nu }\to \{G{\prime }^{\lambda },\psi {\prime }^{\lambda }(\nu ){\}}_{\lambda ,\nu }$ in ${\mathrm{DomLoc}}_{\Lambda }$. By the definition that means that the diagram

$$\begin{array}{ccc}{G}^{\lambda }& \stackrel{{\psi }^{\lambda }(\nu )}{⟶}& G^{\nu }{\Omega }_{\nu }^{\lambda }\\ {}_{{\theta }^{\lambda }}\downarrow & & \downarrow {}_{{\theta }^{\nu }{\Omega }_{\nu }^{\lambda }}\\ G{\prime }^{\lambda }& \stackrel{\psi {\prime }^{\lambda }(\nu )}{⟶}& G{\prime }^{\nu }{\Omega }_{\nu }^{\lambda }\end{array}$$\array{ G^\lambda &\stackrel{\psi^\lambda(\nu)}\longrightarrow& G^\nu \Omega^\lambda_\nu \\ {}_{\mathllap{\theta^\lambda}}\downarrow &&\downarrow{}_{\mathrlap{\theta^\nu\Omega^\lambda_\nu}} \\ G'^\lambda &\stackrel{\psi'^\lambda(\nu)}\longrightarrow& G'^\nu \Omega^\lambda_\nu }

of natural transformations commutes. One also considers the subcategories ${\mathrm{Fun}}^{\mathrm{pp}}(A,B)$ and ${\mathrm{DomLoc}}^{\mathrm{pp}}$ of product preserving functors. The above correspondences do not require that $Q_{\lambda }^{*}$ are localizations.

Domain globalization

To get to the inverse of the localizarion of funtors for a cover, we first consider ${\mathrm{DomLoc}}_{\Lambda }\to \mathrm{Fun}({\prod }_{\lambda },B{)}^{\mathrm{Fun}(\Omega ,B)}\cong \mathrm{Fun}(A,B)$ in the case of comonadicity of the cover $({\prod }_{\lambda }{A}_{\lambda }{)}^{\Omega }\cong A$.

Starting with a datum $\{G^{\lambda },{\psi }^{\lambda }(\nu ){\}}_{\lambda ,\nu }$ one defines $G^{\Lambda }:\prod A_{\lambda }\to B$ by $G^{\Lambda }={\prod }_{\lambda }{G}^{\lambda }{\pi }_{\lambda }$ and ${\psi }^{\Lambda }:G^{\Lambda }\Rightarrow G^{\Lambda }\Omega$ as the unique map simultaneously factorizing for all $\lambda$ the composition

$$G^{\Lambda }\stackrel{{\omega }_G^{\lambda }}{⟶}{G}^{\lambda }{\pi }_{\lambda }\stackrel{{\psi }^{\lambda }(\mu ){\pi }_{\lambda }}{⟶}{G}^{\mu }{\Omega }_{\nu }^{\lambda }{\pi }_{\lambda }$$G^\Lambda \stackrel{\omega^\lambda_G}\longrightarrow G^\lambda \pi_\lambda \stackrel{\psi^\lambda(\mu)\pi_\lambda}\longrightarrow G^\mu\Omega^\lambda_\nu \pi_\lambda

through the composition $G^{\Lambda }\Omega \stackrel{{\omega }_G^{\mu }\Omega }{⟶}{G}^{\mu }{\pi }_{\mu }\Omega \stackrel{G^{\mu }{\pi }_{\mu }{\omega }^{\lambda }}{⟶}{G}^{\mu }{\Omega }_{\mu }^{\lambda }{\pi }_{\lambda }$.

Similarly, given a morphism $\{{\theta }^{\lambda }:G^{\lambda }\to G^{\lambda }{\}}_{\lambda }:\{G^{\lambda },{\psi }^{\lambda }(\nu ){\}}_{\lambda ,\nu }\to \{G{\prime }^{\lambda },\psi {\prime }^{\lambda }(\nu ){\}}_{\lambda ,\nu }$, one defines ${\theta }^{\Lambda }:G^{\Lambda }\to G^{\Lambda }$ as the unique natural transformation such that for each $\lambda \in \Lambda$ the diagram

$$\begin{array}{ccc}{G}^{\Lambda }& \stackrel{{\theta }^{\Lambda }}{⟶}& G{\prime }^{\Lambda }\\ {}_{{\omega }_G^{\lambda }}\downarrow & & \downarrow {}_{{\omega }_{G\prime }^{\lambda }}\\ G^{\lambda }{\pi }_{\lambda }& \stackrel{{\theta }^{\lambda }{\pi }_{\lambda }}{⟶}& G{\prime }^{\lambda }{\pi }_{\lambda }\end{array}$$\array{ G^\Lambda &\stackrel{\theta^\Lambda}\longrightarrow& G'^\Lambda \\ {}_{\mathllap{\omega^\lambda_G}}\downarrow && \downarrow {}_{\mathrlap{\omega^\lambda_{G'}}}\\ G^\lambda\pi_\lambda &\stackrel{\theta^\lambda\pi_\lambda}\longrightarrow & G'^\lambda\pi_\lambda }

Next step is to compose the restriction of the above functor to product preserving data and functors ${\mathrm{DomLoc}}^{\mathrm{pp}}\to {\mathrm{Fun}}^{\mathrm{pp}}({\prod }_{\lambda }{A}_{\lambda },B{)}^{\mathrm{Fun}(\Omega ,B)}$ with another functor ${\mathrm{Fun}}^{\mathrm{pp}}({\prod }_{\lambda }{A}_{\lambda },B{)}^{\mathrm{Fun}(\Omega ,B)}\cong {\mathrm{Fun}}^{\mathrm{pp}}(({\prod }_{\lambda }{A}_{\lambda }{)}^{\Omega },B)$. For the latter, we need the existence of the appropriate equalizers, as in the construction of the inverse of the comparison functor in the proof of the Beck comonadicity theorem. Namely, given $(G^{\Lambda },{\psi }^{\Lambda })$ one defines the value of $G\in {\mathrm{Fun}}^{\mathrm{pp}}(({\prod }_{\lambda }{A}_{\lambda }{)}^{\Omega },B)$ on the $\Omega$-comodule $(N,\nu )$ in $({\prod }_{\lambda }{A}_{\lambda }{)}^{\Omega }$, by the equalizer of

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