derived smooth geometry
structures in a cohesive (∞,1)-topos
A geometry is an (∞,1)-category equipped in a compatible way with
These two structures gives rise to
for how to map test spaces into them.
that send each obect to a -valued structure sheaf.
Using the additional structure of a site on allows to identify those structure sheaves that are local in that they respect coverings. This constitutes a generalized notion of locally ringed toposes called -structured (∞,1)-toposes. Equivalently these local structure sheaves are given by (∞,1)-geometric morphisms to the big topos over .
the extra structure given the information of which covering morphisms are to be thought of as local homeomorphisms
the extra property that it has all finite limits.
If only all finite products exist we speak of a pre-geometry. Every pregeometry extends uniquely to an enveloping geometry .
When the objects of the geometry are thought of as test spaces (affine schemes), the objects of the pregeometry are to be thought of as the affine spaces. This distinction is used to encode smoothness of maps between test spaces: a morphism in is smooth if it locally factors through admissible maps between objects in .
An admissibility structure on an (∞,1)-category is a Grothendieck topology on that is generated from its intersection with a subcategory whose morphisms – called the admissible morphisms have the following properties
admissible morphisms are stable under (∞,1)-pullback;
admissible morphisms satisfy “left cancellability”, meaning that whenever in
and are admissible, then so is .
admissible morphisms are closed under retracts.
Equivalently, this is a Grothendieck topology on which is generated from admissible morphisms.
As will become clear when looking at examples, the notion of admissible morphisms models the idea of maps between test spaces that behave like open injections or, more generally, as local homeomorphisms .
A geometry (for -toposes) is
The discrete geometry on is given by
the admissible morphisms in are precisely the equivalences
Every small (∞,1)-category becomes a geometry by regarding it as a discrete geometry in the above way.
A pregeometry (for structured (∞,1)-toposes) is
an (∞,1)-category ;
equipped with an admissibility structure (homotopical topology)
Various concepts for geometries have immediate analogues for pregeometries.
A morphism in a pregeometry is called smooth if it is locally stably admissible in that there exists a cover (meaning: generators of a covering sieve) of by admissible morphisms, such that on the morphism factors admissibly through some in that there is a commuting diagram
To interpret this, recall that we think of admissible morphisms as injections of open subsets.
Smooth morphisms are stable under pullback.
pregeometric -structures preserve pullbacks of smooth morphisms.
Let be a pregeometry and an (∞,1)-topos.
A -structure on is an (∞,1)-functor such that
The first clause says that is in particular an -algebra over the (multi-sorted) (∞,1)-algebraic theory .
The other two clauses encode that this -algebra indeed behaves like a function algebra .
…the universal geometry extending a pregeometry…
Let be a pregeometry and a morphism that exhibits the geometry as a geometric envelope of . Then for every (∞,1)-topos precomposition with induces an equivalence of (∞,1)-categories of - and -structures on :
See Deligne-Mumford stack for details.
The general theory is developed in
The definition of a geometry is def. 1.2.5.
A -structure on an (∞,1)-topos is in def. 1.2.8.
The notion of -spectrum – which are (∞,1)-toposes – is the subject of section 2.1 .
is definition 2.1.2.
The definition of -generalized scheme is definition 2.3.9, page 51.
is the topic of section 2.4, theorem 2.4.1
The special case of “smoothly structured spaces” called derived smooth manifold is discussed in
Apart from looking at the special case this article also contains useful introduction and details on the general case.
In the version of this that is available on the arXiv (arXiv) the point of view is more on bi-presheaves, a useful discussion to the relation to structured morphisms here is in section 10.1 there.