A space is called formally unramified if every morphism into it has for every infinitesimal thickening of at most one infinitesimal extension.
(If all thickenings exist it is called a formally smooth morphism. If the thickening exist uniquely, it is called a formally etale morphism.)
Traditionally this has been considered in the context of geometry over formal duals of rings and associative algebras. This we discuss in the section (Concrete notion). But generally the notion makes sense in any context of infinitesimal cohesion. This we discuss in the section General abstract notion.
General abstract notion
be a triple of adjoint functors with a full and faithful functor that preserves the terminal object.
We may think of this as exhibiting infinitesimal cohesion (see there for details, but notice that in the notation used there we have , and ).
We think of the objects of as cohesive spaces and of the objects of as such cohesive spaces possibly equipped with infinitesimal extension.
As a class of examples that is useful to keep in mind consider a Q-category of infinitesimal thickening of rings and let
be the corresponding Q-category of copresheaves.
For any such setup there is a canonical natural transformation
Details of this are in the section Adjoint quadruples at cohesive topos.
From this we get for every morphism in a canonical morphism
A morphism in is called formally unramified if (1) is a monomorphism.
This appears as (KontsevichRosenberg, def. 5.1, prop. 188.8.131.52).
The dual notion, where the morphism is required to be an epimorphism is that of formally smooth morphisms. If both conditions hold, hence if the morphism is in fact an isomorphism, one speaks of formally etale morphisms.
An object is called formally unramified if the morphism to the terminal object is formally unramified.
The object is formally unramified precisely if
is a monomorphism.
This appears as (KontsevichRosenberg, def. 5.3.2).
Formally unramified morphisms are closed under composition.
This appears as (KontsevichRosenberg, prop. 5.4).
For the moment see the discussion at unramified morphism.