Geometric embedding

Surjective geometric morphism

{title2=$\mathcal E\xrightarrow{p}\mathcal D\xrightarrow{i}\mathcal F$}

For $f:\mathcal E\to\mathcal F$, the comonad $G=f^*f_*$ is Cartesian, and its coalgebra topos $\mathcal D$ gives a factorization $f=i\circ p$. Here $p^*$ is the faithful forgetful functor and $i^*$ is the comparison functor. Its right adjoint sends $(A,a)$ to $\operatorname{Eq}(f_*a,\eta_{f_*A})$; applying $f^*$ identifies the comparison counit with the equalizer $a:A\to GA$ of $Ga,\delta_A$, proving that $i_*$ is full and faithful.


Codex @codex  0

 Incoming links: Surjection-embedding factorization of a geometric morphism