Zoran Skoda
domain globalization of functors

Let {Q λ *:AA λ} λΛ be a cover of A by functors having a right adjoint, Q λ *Q λ* with unit η λ and counit ϵ λ. Each pair defines a monad with underlying endofunctor Q λ*Q λ * and if A λ admit products of cardΛ objects, the whole cover defines a comonad Ω on λA λ, induced by the adjunction A λA λ. (Cf. gluing categories from localizations).

Denote Ω μ λ=Q μ *Q λ* the projections π λ:A ΛA λ, ω λ:ΩΩ λπ λ, then Ω= λΩ λπ λ. The unit defines the coaction ΩΩΩ which locally gives rise to Ω λ νδ λ ν(μ)Ω λ μΩ μ ν and the counit locally gives ϵ λ:Ω λ λ1 A λ.

Domain localization for a cover

and G:AB a functor. (I will present here a sketch following mainly Gabi’s improvements and notation.) Then G 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. Then the composition

G λψ λ(λ)G λΩ λ λG λϵ λG λG^\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 Λ which depends on the cover of A and on B.

Given a natural transformation θ:GG, 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 form a morphism {G λ,ψ λ(ν)} λ,ν{G λ,ψ λ(ν)} λ,ν in DomLoc Λ. 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) and DomLoc pp of product preserving functors. The above correspondences do not require that Q λ * 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) in the case of comonadicity of the cover ( λA λ) ΩA.

Starting with a datum {G λ,ψ λ(ν)} λ,ν one defines G Λ:A λB by G Λ= λG λπ λ and ψ Λ:G ΛG ΛΩ as the unique map simultaneously factorizing for all λ the composition

G Λω G λG λπ λψ λ(μ)π λG μΩ ν λπ λG^\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 λ.

Similarly, given a morphism {θ λ:G λG λ} λ:{G λ,ψ λ(ν)} λ,ν{G λ,ψ λ(ν)} λ,ν, one defines θ Λ:G ΛG Λ as the unique natural transformation such that for each λΛ 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) with another functor Fun pp( λA λ,B) Fun(Ω,B)Fun pp(( λA λ) Ω,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) one defines the value of GFun pp(( λA λ) Ω,B) on the Ω-comodule (N,ν) in ( λA λ) Ω, by the equalizer of

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