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In gluing categories from localizations (zoranskoda) the main section
From a family of localizations to a comonad
is fully rewritten in improved notation. In other way, it is explained better how to get a comonad from a cover of a category by not necessarily compatible flat localizations. This generalizes the Sweedler's coring to relative situations. Now from such data one can make a two category, which I will explain in few days.
This is a preliminary to something I am writing at the moment namely to explain in such terms actions of comonads and monoidal categories on such descent categories. This part will be analogous to description of equivariant maps among G-manifolds in pairs of local charts, but because of the distributive laws with coherences, the thing complicates.
In fact, because the whole text is part of a long nlab entry in change let me copy here the new inserted section. The very construction in fact does not need flat localization, but a cover of a cartesian category by any family of flat functors (if it is infinite one requires in addition commuting with products of the same size, which in turn are also supposed to exist). Here is the text (for some reason the formula for $\mathbf{\delta}^\lambda$ did not render, something with $*$-symbols (until I added this remark!), even after cleaning it, while it does in nlab entry!):
By a flat localization (functor) we will mean an exact additive functor $Q^*$ having a fully faithful right adjoint $Q_*$. The composition $Q_* Q^*$ is then often denoted by $Q$. A family $F_\lambda^*: \mathcal{A}\to \mathcal{B}_\lambda$, $\lambda\in\Lambda$ of functors with the same domain is conservative if $Q^*_\lambda(f)$ is invertible for all $\lambda$ only if $f$ is invertible. A flat cover of an abelian category $\mathcal{A}$ is by definition a conservative family of flat functors $F_{\lambda}:\mathcal{A}\to\mathcal{B}_{\lambda}$.
Localization functors typically do not mutually commute. Namely, given a family of flat localizations $Q_\lambda^*: \mathcal{A}\to\mathcal{A}_\lambda$, $\lambda\in\Lambda$, the functors $Q_{\lambda} Q_\mu = Q_{\lambda*}Q_\lambda^{*}Q_{\mu*}Q_\mu^*:\mathcal{A}\to\mathcal{A}$ and $Q_\mu Q_\lambda$ for $\lambda\neq\mu$ are in general not isomorphic. If the family is a cover then define the product category $\mathcal{A}_\Lambda = \prod_{\lambda\in\Lambda}\mathcal{A}_\lambda$ and the functor $\mathbf{Q}^*:\mathcal{A}\to\mathcal{A}_\Lambda$, $\mathbf{Q}^*:M\to(Q_\lambda^* M)_{\lambda}$ where the notation $(N_\lambda)_\lambda = (N_\lambda)_{\lambda\in\Lambda}$ denotes the ordered $\Lambda$-tuple in $\mathcal{A}_\Lambda$. If $\mathcal{A}$ has products of families of $\mathrm{card}\,\Lambda$ objects, then $\mathbf{Q}^*$ has a right adjoint $\mathbf{Q}_* :\mathcal{A}_\Lambda\to\mathcal{A}$ given by $(M^\lambda)_\lambda\mapsto \prod_\lambda Q_{\lambda *} M^\lambda$. Indeed,
$\array{ Hom_{\mathcal{A}_\Lambda}((M^\lambda)_\lambda,(Q^*_\lambda N)_\lambda) &:=& \prod_{\lambda\in\Lambda} Hom_{\mathcal{A}_\lambda}(M^\lambda,Q^*_\lambda N) \\ &=& \prod_{\lambda\in\Lambda} Hom_{\mathcal{A}}(Q_{\lambda*}M^\lambda,N) \\ &=& Hom_{\mathcal{A}}(\prod_{\lambda*}Q_{\lambda *}M^\lambda,N)\,\, }$The unit $\mathbf{\eta}:Id_{\mathcal{A}}\to\mathbf{Q}_*\mathbf{Q}^*:\mathcal{A}\to\mathcal{A}$ of the adjunction $\mathbf{Q}^*\dashv\mathbf{Q}_*$ is the map induced from the units $\eta^\lambda$, by the universality of Cartesian product in $\mathcal{A}$, namely $\mathbf{\eta} = (\eta^\lambda)_{\lambda\in\Lambda} : M\to\prod_\lambda Q_{\lambda*}Q^*_\lambda M$. The counit $\mathbf{\epsilon}:\mathbf{Q}^*\mathbf{Q}_*\to Id_{\mathcal{A}_\Lambda}$ has the components given by the compositions
$\mathbf{\epsilon}_{(N^\lambda)_\lambda} : (Q^*_\lambda\prod_\mu Q_{\mu*} N^\mu)_\lambda \stackrel{(Q^*_\lambda (\mathrm{pr}_\lambda))_\lambda}\to (Q^*_\lambda Q_{\lambda*}N^\lambda)_\lambda \stackrel{(\epsilon^\lambda)_\lambda}\to (N^\lambda)_\lambda$where in the first functor the projections for the Cartesian product are used.
Denote $\Omega:= \mathbf{Q}^* \mathbf{Q}_{*} :\mathcal{A}_\Lambda\to\mathcal{A}_\Lambda$; then $\mathbf{\Omega} = (\Omega,\mathbf{\delta},\mathbf{\epsilon})$ is the comonad on $\mathcal{A}_\Lambda$ induced by the adjunction $\mathbf{Q}^*\dashv\mathbf{Q}_*$, where for each $(N^\lambda)_\lambda\in\mathcal{A}_\Lambda$ the component $\mathbf{\delta}^\lambda_{(N^\lambda)_\lambda}$ of the comultiplication $\mathbf{\delta} = \mathbf{Q}^*\mathbf{\eta}\mathbf{Q}_*$ is more explicitly the map
$\mathbf{\delta}^\lambda_{(N^\lambda)_\lambda} = (Q_\mu^* (\eta^\rho_{\prod_\lambda Q_{\lambda*} N^\lambda})_\rho)_\mu : (Q^*_\mu\prod_\lambda Q_{\lambda*} N^\lambda)_\mu \to (Q_\mu^* \prod_\rho Q_{\rho*} Q^*_\rho\prod_\lambda Q_{\lambda*} N^\lambda)_\mu$Again, if each $Q_\mu^*$ commutes with $\Lambda$-products then the products can be placed in front: $$(\prod_\lambda Q^*_\mu Q_{\lambda *} N^\lambda)_\mu \to (\prod_{\lambda\rho}Q_\mu^*Q_{\rho*}Q^*_\rho Q_{\lambda *} N^\lambda)_\mu$$
There is a comparison functor
$K_{\mathbf{\Omega}}:\mathcal{A}\to(\mathcal{A}_\Lambda)_{\mathbf{\Omega}}, \,\,\,M\mapsto (\mathbf{Q}^*M,\mathbf{Q}^*(\mathbf{\eta}_M)) =((Q_{\lambda}^* M)_\lambda,(Q^*_\lambda(\eta^\mu_M)_\mu)_\lambda);$under the appropriate (Beck comonadicity criteria) conditions $K_{\mathbf{\Omega}}$ is an equivalence, with the (quasi)inverse mapping sending an $\mathbf{\Omega}$-comodule $(N,\nu)\in(\mathcal{A}_\Lambda)_{\mathbf{\Omega}}$, into the equalizer of morphisms $\mathbf{\eta}_{\mathbf{Q}_*N}$ and $\mathbf{Q}_*(\nu) : \mathbf{Q}_* N\to \mathbf{Q}_*\mathbf{Q}^*\mathbf{Q}_* N$ in $\mathcal{A}$, thus identifying $\mathcal{A}$ with the Eilenberg-Moore category of comodules for the comonad $\mathbf{\Omega}$.
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