If is the terminal sheaf topos, then is essential if is a locally connected topos . In general, being essential is a necessary (but not sufficient) condition to ensure that behaves like a map of topological spaces whose fibers are locally connected: that it is a locally connected geometric morphism.
A point of a topos which is given by an essential geometric morphism is called an essential point of .
There are various further conditions that can be imposed on a geometric morphism:
If preserves finite products then is called connected surjective.
with full and faithful then the geometric morphism is called local geometric morphism.
In this case in particular
is a geometric embedding and hence makes a subtopos of .
For and small categories write and for the corresponding copresheaf toposes. (If we think of the opposite categories and as sites equipped with the trivial coverage, then these are the corresponding sheaf toposes.)
This construction extends to a 2-functor
where and denote the left and right Kan extension along , respectively.
For all small categories we have an equivalence of categories
between the opposite category of the functor category between the Cauchy completions of and and the the category of essential geometric morphisms between the copresheaf toposes and geometric transformations between them.
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 is in the image of this functor precisely if its inverse image morphism of frames has a left adjoint in the 2-category Locale.
Moreover, for any poset the sheaf topos over is naturally equivalent to
Let be an essential geometric morphism.
For every in the diagram
commutes, where the vertical morphisms are unit and counit, respectively, and where the bottom horizontal morphism is the adjunct of under the composite adjunction .
The morphism is the component of a natural transformation
The composite is the component of this composed with the counit .
We may insert the 2-identity given by the zig-zag law
Composing this with the counit produces the transformation whose component is manifestly the morphism .
For the case the terminal object, the geometric morphism
is also called an etale geometric morphism.
In this case, the functor 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 is an arbitrary geometric morphism through which we regard as an -topos, i.e. a topos “in the world of ,” the condition for to be locally connected as an -topos is not just that is essential, but that the left adjoint can be made into an -indexed functor (which is automatically true for and ). This is automatically the case for -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 -indexing anyway). For more see locally connected geometric morphism.
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