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.
Since for a Grothendieck topos the inverse image of an essential point is of the form where is projective and connected, objects satisfying these three conditions are sometimes called essential objects (cf. Johnstone 1977, p.255).
The inverse image of an essential geometric morphism preserves small limits since it is a right adjoint. Hence, this provides a minimal requirement to satisfy for a general geometric morphism in order to qualify for being essential. In case, the toposes involved are Grothendieck toposes this condition is not only necessary but sufficient.
Let a geometric morphism between Grothendieck toposes. Then is essential iff preserves small limits iff preserves small products.
Grothendieck toposes are locally presentable and has rank i.e. it preserves -filtered colimits for some regular cardinal since it is a left adjoint. But by (Borceux vol. 2 Thm.5.5.7, p.275) a functor between two locally presentable categories has a left adjoint precisely if it has rank and preserves small limits.
Since limits can be constructed from products and equalizers and preserves the latter, it preserves small limits precisely when it preserves small products.
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 .
A particularly important instance of this situation is the following:
In the special case the terminal object, the essential geometric morphism
is also called an etale geometric morphism.
If the left adjoint of a logical functor furthermore preserves equalizers, then the corresponding essential geometric morphism is up to equivalence an etale geometric morphism for an object in that is determined up to isomorphism.
For a proof see e.g. Johnstone (1977, p.37).
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 -indexed anyway). For more see locally connected geometric morphism.
Like many other things, it all started as an exercise in
Speaking of exercises, consider the results of Roos on essential points reported in exercise 7.3 of
The case of sheaves valued in FinSet is considered in
For more on this see also
The definition of essential geometric morphisms appears before Lemma A.4.1.5 in
Connected surjective and local geometric morphisms are discussed in
Further refinements are in