Could not include topos theory - contents
A geometric morphism $f : E \to F$ between toposes is a functor of the underlying categories that is consistent with the interpretation of $E$ and $F$ as generalized topological spaces.
If $F = Set = Sh(*)$ is the terminal sheaf topos, then $E \to Set$ is essential if $E$ is a locally connected topos . In general, $f$ being essential is a necessary (but not sufficient) condition to ensure that $f$ behaves like a map of topological spaces whose fibers are locally connected: that it is a locally connected geometric morphism.
Given a geometric morphism $(f^* \dashv f_*) : E \to F$, it is an essential geometric morphism if the inverse image functor $f^*$ has not only the right adjoint $f_*$, but also a left adjoint $f_!$:
A point of a topos $x : Set \to E$ which is given by an essential geometric morphism is called an essential point of $E$.
There are various further conditions that can be imposed on a geometric morphism:
If $f_!$ can be made into an $E$-indexed functor and $f^*$ satisfies some extra conditions, the geometric morphism $f$ is a locally connected geometric morphism (see there for details).
If $f_!$ preserves finite products then $f$ is called connected surjective.
If in addition to the above $f$ is a local geometric morphism in that there is a further functor $f^! : F \to E$ which is right adjoint $(f_* \dashv f^!)$ and full and faithful then the geometric morphism $f$ is called cohesive.
For $C$ and $D$ small categories write $[C,Set]$ and $[D,Set]$ for the corresponding copresheaf toposes. (If we think of the opposite categories $C^{op}$ and $D^{op}$ as sites equipped with the trivial coverage, then these are the corresponding sheaf toposes.)
This construction extends to a 2-functor
from the 2-category Cat${}_{small}$ with 2-morphisms reversed) to the sub-2-category of Topos on essential geometric morphisms, where a functor $f : C \to D$ is sent to the essential geometric morphism
where $Lan_f$ and $Ran_f$ denote the left and right Kan extension along $f$, respectively.
This 2-functor is a full and faithful 2-functor when restricted to Cauchy complete categories:
For all small categories $C,D$ we have an equivalence of categories
between the opposite category of the functor category between the Cauchy completions of $C$ and $D$ and the the category of essential geometric morphisms between the copresheaf toposes and geometric transformations between them.
In particular, since every poset – when regarded as a category – is Cauchy complete, we have
Sometimes it is useful to decompose this statement as follows.
There is a functor
which assigns to each poset a locale called its Alexandroff locale. By a theorem discussed there, a morphisms of locales $f : X \to Y$ is in the image of this functor precisely if its inverse image morphism $f^* Op(Y) \to Op(X)$ of frames has a left adjoint in the 2-category Locale.
Moreover, for any poset $P$ the sheaf topos over $Alex P$ is naturally equivalent to $[P,Set]$
With this, the fact that $[-,Set] : Poset \to Topos$ hits precisely the essential geometric morphisms follows with the basic fact about localic reflection, which says that $Sh : Locale \to Topos$ is a full and faithful 2-functor.
Let $f : E \to F$ be an essential geometric morphism.
For every $\phi : X \to f^* f_* A$ in $E$ the diagram
commutes, where the vertical morphisms are unit and counit, respectively, and where the bottom horizontal morphism is the adjunct of $\phi$ under the composite adjunction $(f^* f_! \dashv f^* f_*)$.
The morphism $\phi : X \to f^* f_* A$ is the component of a natural transformation
The composite $X \stackrel{\phi}{\to} f^* f_* A \to A$ is the component of this composed with the counit $f^* f_* \Rightarrow Id$.
We may insert the 2-identity given by the zig-zag law
Composing this with the counit $f^* f_* \Rightarrow Id$ produces the transformation whose component is manifestly the morphism $X \to f^* f_! X \to A$.
For any morphism $f\colon A\to B$ in a topos $E$, the induced geometric morphism $f\colon E/A \to E/B$ of overcategory toposes is essential.
For the case $B = *$ the terminal object, the geometric morphism
is also called an etale geometric morphism.
A locally connected topos $E$ is one where the global section geometric morphism $\Gamma : E \to Set$ is essential.
In this case, the functor $\Gamma_! = \Pi_0 : E \to Set$ sends each object to its set of connected components. More on this situation is at homotopy groups in an (∞,1)-topos.
Note, though that if $p\colon E\to S$ is an arbitrary geometric morphism through which we regard $E$ as an $S$-topos, i.e. a topos “in the world of $S$,” the condition for $E$ to be locally connected as an $S$-topos is not just that $p$ is essential, but that the left adjoint $p_!$ can be made into an $S$-indexed functor (which is automatically true for $p^*$ and $p_*$). This is automatically the case for $Set$-toposes (at least, when our foundation is material set theory—and if our foundation is structural set theory, then our large categories and functors all need to be assumed to be $Set$-indexing anyway). For more see locally connected geometric morphism.
The tiny objects of a presheaf topos $[C,Set]$ are precisely the essential points $Set \to [C,Set]$. See tiny object for details.
essential geometric morphism
The definition of geometric morphism appears before Lemma A.4.1.5 in
Connected surjective and local geometric morphisms are discussed in
Further refinements are in