Cohomology and homotopy
In higher category theory
A geometric morphism between toposes is a functor of the underlying categories that is consistent with the interpretation of and as generalized topological spaces.
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.
Given a geometric morphism , it is an essential geometric morphism if the inverse image functor has not only the right adjoint , but also a left adjoint :
A point of a topos which is given by an essential geometric morphism is called an essential point of .
Relation to morphisms of (co)sites
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
from the 2-category Cat with 2-morphisms reversed) to the sub-2-category of Topos on essential geometric morphisms, where a functor is sent to the essential geometric morphism
where and denote the left and right Kan extension along , respectively.
This 2-functor is a full and faithful 2-functor when restricted to Cauchy complete categories:
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.
In particular, since every poset – when regarded as a category – is Cauchy complete, we have
Some morphism calculus
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 .
Etale geometric morphisms
For any morphism in a topos , the induced geometric morphism of overcategory toposes is essential.
For the case the terminal object, the geometric morphism
is also called an etale geometric morphism.
Locally connected toposes
A locally connected topos is one where the global section geometric morphism is essential.
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.
The tiny objects of a presheaf topos are precisely the essential points . See tiny object for details.
As many other things, it all started as an exercise in
Speaking of exercises, consider the results of Roos reported in exercise 7.3 of
- Peter Johnstone, Topos Theory , Academic Press New York 1977. (Dover reprint 2014, pp.254f)
The case of sheaves valued in FinSet is considered in
- J. Haigh, Essential geometric morphisms between toposes of finite sets , Math. Proc. Phil. Soc. 87 (1980) pp.21-24.
The standard reference for essential localizations 1, aka levels, is
- G. M. Kelly, F. W. Lawvere, On the Complete Lattice of Essential Localizations , Bull. Soc. Math. de Belgique XLI (1989) pp.261-299.
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
- Bill Lawvere, Axiomatic cohesion Theory and Applications of Categories, Vol. 19, No. 3, 2007, pp. 41–49. (pdf)