Zoran Skoda
compatible idempotent monads

Given two idempotent monads in a category AA, we say that they are mutually (strongly) compatible if there is an invertible distributive law between them. Such a distributive law is automatically unique and can be given by a formula.

Motivated by the localization theory, denote Q μ=Q μ*Q μ *Q_\mu = Q_{\mu *}Q_\mu^* and Q λ=Q λ*Q λ *Q_\lambda = Q_{\lambda*}Q_\lambda^* the underlying endofunctors of the monads, where Q μ *Q_\mu^* and Q λ *Q_\lambda^* are the free functors and Q μ*,Q λ*Q_{\mu*},Q_{\lambda*} the forgetful functors between the Eilenberg-Moore categories A λ=A Q λA_\lambda = A^{Q_\lambda}, A μ=A Q μA_\mu = A^{Q_\mu} and AA.

Regardless the compatibility, in this situation define the category A μλA_{\mu\lambda} as the equalizer

Suppose there is an invertible distributive law Q λQ μQ μQ λQ_\lambda Q_\mu \cong Q_\mu Q_\lambda, then one has the lift Q¯ λ:A Q μA Q μ\bar{Q}_\lambda : A^{Q_\mu}\to A^{Q_\mu} and the composed monad Q μQ λQ_\mu Q_\lambda with A Q μQ λ(A Q μ) Q λA^{Q_\mu Q_\lambda}\cong (A^{Q_\mu})^{Q_\lambda} and the decomposition of Q¯ λ\bar{Q}_\lambda as

A Q μQ¯ λ *(A Q μ) Q λQ¯ λ*A Q μ A^{Q_\mu}\stackrel{\bar{Q}_\lambda^*}\longrightarrow (A^{Q_\mu})^{Q_\lambda} \stackrel{\bar{Q}_{\lambda*}}\longrightarrow A^{Q_\mu}

The free functor Q¯ μ *\bar{Q}_\mu^* above is a localization, hence in particular essentially surjective on objects, and Q¯ λ*\bar{Q}_{\lambda*} is fully faithful, thus A Q μQ λA^{Q_\mu Q_\lambda} is the essential image of Q¯ λ\bar{Q}_\lambda.

Claim (ZŠ, GB) Under the invertible compatibility, the equalizer above, the essential image of Q¯ λ\bar{Q}_\lambda, and the consecutive EM category are equivalent:

A μλ(A Q μ) Q λEssImQ¯ λ.A_{\mu\lambda} \cong (A^{Q_\mu})^{Q_\lambda} \cong EssIm \bar{Q}_\lambda.

Furthermore, under these conditions, A λμA μλA_{\lambda\mu}\cong A_{\mu\lambda} and the latter equivalence commutes with the forgetful functor to AA. See also compatible localization.

Created on June 26, 2011 at 13:29:31. See the history of this page for a list of all contributions to it.