Spahn factorization of a monad

Example

Example

Let (i !i *i *i !):H thi !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 thH(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.