Spahn factorization of a monad (changes)

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Example

Let (i !⊣i *⊣i *⊣i !):H th→i !H(i_!\dashv i^*\dashv i_*\dashv i^!):H_{th}\stackrel{\i^!}{\to} H be an adjoint quadruple defining infinitesimal cohesion; i.e. i !i_! and i *i_* are full and faithful and i !i_! preserves the terminal object.

Then there are two factorizations of the terminal geometric morphism? on H thH_{th} through the geometric morphisms (i *⊣i *)(i^*\dashv i_*) and (i *⊣i !)(i_*\dashv i^!):

H →(i *⊣i *) H th Γ H↘ ↙ Γ th ∞Grpd \array{ \mathbf{H} && \stackrel{(i^* \dashv i_*)}{\to} && \mathbf{H}_{th} \\ & {}_{\mathllap{\Gamma_H}}\searrow && \swarrow_{\mathrlap{\Gamma_{th}}} \\ && \infty Grpd }

as is the induced geometric morphism (i *⊣i !):H th→H(i_* \dashv i^!) : \mathbf{H}_{th} \to \mathbf{H}

H th →(i *⊣i !) H Γ th↘ ↙ Γ H ∞Grpd. \array{ \mathbf{H}_{th} && \stackrel{(i_* \dashv i^!)}{\to} && \mathbf{H} \\ & {}_{\mathllap{\Gamma_{th}}}\searrow && \swarrow_{\mathrlap{\Gamma_H}} \\ && \infty Grpd } \,.

Created on December 17, 2012 at 02:15:09. See the history of this page for a list of all contributions to it.

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