Zoran Skoda
domain globalization of functors

Let {Q λ *:AA λ} λΛ\{Q_\lambda^*:A\to A_\lambda\}_{\lambda\in\Lambda} be a cover of AA by functors having a right adjoint, Q λ *Q λ*Q^*_\lambda\dashv Q_{\lambda*} with unit η λ\eta^\lambda and counit ϵ λ\epsilon^\lambda. Each pair defines a monad with underlying endofunctor Q λ*Q λ *Q_{\lambda*} Q^*_\lambda and if A λA_\lambda admit products of cardΛcard \Lambda objects, the whole cover defines a comonad Ω\Omega on λA λ\prod_\lambda A_\lambda, induced by the adjunction A λA λA\leftrightarrow \prod_\lambda A_\lambda. (Cf. gluing categories from localizations).

Denote Ω μ λ=Q μ *Q λ*\Omega^\lambda_\mu = Q^*_\mu Q_{\lambda *} the projections π λ:A ΛA λ\pi_\lambda:A_\Lambda\to A_\lambda, ω λ:ΩΩ λπ λ\omega^\lambda:\Omega\Rightarrow\Omega^\lambda\pi_\lambda, then Ω= λΩ λπ λ\Omega = \prod_\lambda \Omega^\lambda \pi_\lambda. The unit defines the coaction ΩΩΩ\Omega\to \Omega\Omega which locally gives rise to Ω λ νδ λ ν(μ)Ω λ μΩ μ ν\Omega^\nu_\lambda \stackrel{\delta^\nu_\lambda(\mu)}\longrightarrow \Omega^\mu_\lambda \Omega^\nu_\mu and the counit locally gives ϵ λ:Ω λ λ1 A λ\epsilon^\lambda : \Omega^\lambda_\lambda\to 1_{A_\lambda}.

Domain localization for a cover

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

G λ:=(A λQ λ*AGB) G^\lambda := \left( A_\lambda\stackrel{Q_{\lambda *}}\longrightarrow A\stackrel{G}\longrightarrow B\right)
ψ λ(ν):=(G λ=GQ λ*Gη μQ λ*GQ ν*Q ν *Q λ*=G νΩ ν λ) \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 λ,νΛ\lambda,\nu\in \Lambda. Then the composition

G λψ λ(λ)G λΩ λ λG λϵ λG λ 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

G λ ψ λ(ν) G νΩ μ λ ψ λ(μ) G μδ ν λ(μ) G νΩ μ λ ψ μ(ν)Ω μ λ G νΩ ν μΩ μ λ\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 DomLoc ΛDomLoc_\Lambda which depends on the cover of AA and on BB.

Given a natural transformation θ:GG\theta : G\Rightarrow G', one defines the corresponding local data by whiskering:

θ λ:=(G λ=GQ λ*θQ λ*GQ λ*=G λ) \theta^\lambda := \left( G^\lambda = G Q_{\lambda*} \stackrel{\theta Q_{\lambda*}}\longrightarrow G Q_{\lambda*} = G'^\lambda\right)

The data, {θ λ} λΛ\{\theta^\lambda\}_{\lambda\in\Lambda} form a morphism {G λ,ψ λ(ν)} λ,ν{G λ,ψ λ(ν)} λ,ν\{G^\lambda, \psi^\lambda(\nu)\}_{\lambda,\nu}\to \{G'^\lambda,\psi'^\lambda(\nu)\}_{\lambda,\nu} in DomLoc ΛDomLoc_\Lambda. By the definition that means that the diagram

G λ ψ λ(ν) G νΩ ν λ θ λ θ νΩ ν λ G λ ψ λ(ν) G νΩ ν λ\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 Fun pp(A,B)Fun^{pp}(A,B) and DomLoc ppDomLoc^{pp} of product preserving functors. The above correspondences do not require that Q λ *Q^*_\lambda are localizations.

Domain globalization

To get to the inverse of the localizarion of funtors for a cover, we first consider DomLoc ΛFun( λ,B) Fun(Ω,B)Fun(A,B)DomLoc_\Lambda\to Fun(\prod_\lambda,B)^{Fun(\Omega,B)} \cong Fun(A,B) in the case of comonadicity of the cover ( λA λ) ΩA(\prod_\lambda A_\lambda)^\Omega \cong A.

Starting with a datum {G λ,ψ λ(ν)} λ,ν\{G^\lambda, \psi^\lambda(\nu)\}_{\lambda,\nu} one defines G Λ:A λBG^\Lambda : \prod A_\lambda \to B by G Λ= λG λπ λG^\Lambda = \prod_\lambda G^\lambda \pi_\lambda and ψ Λ:G ΛG ΛΩ\psi^\Lambda : G^\Lambda\Rightarrow G^\Lambda \Omega as the unique map simultaneously factorizing for all λ\lambda the composition

G Λω G λG λπ λψ λ(μ)π λG μΩ ν λπ λ 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 ΛΩω G μΩG μπ μΩG μπ μω λG μΩ μ λπ λG^\Lambda\Omega \stackrel{\omega^\mu_G\Omega}\longrightarrow G^\mu\pi_\mu\Omega \stackrel{G^\mu \pi_\mu \omega^\lambda}\longrightarrow G^\mu\Omega^\lambda_\mu \pi_\lambda.

Similarly, given a morphism {θ λ:G λG λ} λ:{G λ,ψ λ(ν)} λ,ν{G λ,ψ λ(ν)} λ,ν\{\theta^\lambda : G^\lambda\to G^\lambda\}_\lambda : \{G^\lambda,\psi^\lambda(\nu)\}_{\lambda,\nu}\to \{G'^\lambda,\psi'^\lambda(\nu)\}_{\lambda,\nu}, one defines θ Λ:G ΛG Λ\theta^\Lambda:G^\Lambda\to G^\Lambda as the unique natural transformation such that for each λΛ\lambda\in\Lambda the diagram

G Λ θ Λ G Λ ω G λ ω G λ G λπ λ θ λπ λ G λπ λ\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 DomLoc ppFun pp( λA λ,B) Fun(Ω,B)DomLoc^{pp}\to Fun^{pp}(\prod_\lambda A_\lambda,B)^{Fun(\Omega,B)} with another functor Fun pp( λA λ,B) Fun(Ω,B)Fun pp(( λA λ) Ω,B)Fun^{pp}(\prod_\lambda A_\lambda,B)^{Fun(\Omega,B)} \cong Fun^{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 Λ,ψ Λ)(G^\Lambda,\psi^\Lambda) one defines the value of GFun pp(( λA λ) Ω,B)G \in Fun^{pp}((\prod_\lambda A_\lambda)^\Omega,B) on the Ω\Omega-comodule (N,ν)(N,\nu) in ( λA λ) Ω(\prod_\lambda A_\lambda)^\Omega, by the equalizer of

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Revised on May 8, 2011 10:06:01 by Zoran Škoda (77.237.96.26)