homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
symmetric monoidal (∞,1)-category of spectra
A simplicial ring is a simplicial object in the category Ring of rings.
This may be understood conceptually as follows:
as ordinary rings are algebras over the ordinary algebraic theory of rings, if we regard this as an (∞,1)-algebraic theory then simplicial rings model the -algebras over that;
the category Ring is naturally equipped with the structure of a pregeometry. The corresponding geometry (for structured (∞,1)-toposes) is , the opposite of the category of simplicial rings.
A simplicial ring is a simplicial object in the category Ring of rings.
Given a simplicial ring , its connected components (the 0th “homotopy group”)) is an ordinary ring
Forming connected components is a functor from simplicial rings to plain rings, which is left adjoint to the inclusion of ordinary rings as simplicially constant simplicial rings, exhibiting a reflective subcategory inclusion
So on formal duals of commutative (simplicial) rings this is a coreflection of affine schemes in affine derived schemes
Notice that coreflective embeddings are also given for instance by the inclusion of manifolds into formal manifolds. This is one way in which formal duals of simplicial rings manifest themselves as infinitesimal neighbourhoods of formal duals of plain rings.
The higher homotopy groups of a simplicial ring are naturally modules over the ring of connected components, so is weakly contractible, as a simplicial set, iff .
(Again this is a manifestation of the simplicial ring being just an infinitesimal thickening of its connected components.)
Notice that is equivalently the cokernel of . Accordingly, any chain of face maps compose with the projection to is independent of the choices. These maps
give a surjective map from to the constant simplicial ring .
(This is just the simplest piece of the Postnikov tower.) If we can then compose with a surjective map to a constant simplicial field.
There is a model category structure on simplicial rings that presents -rings. See model structure on simplicial T-algebras for more.
We describe here the model category presentation of the (∞,1)-category of modules over simplicial rings.
Let be a simplicial commutative algebra. Write for the category which, with regarded as a monoid in the category of abelian simplicial groups is just the category of -modules in . This means that
Equip with the structure of a model category by setting:
Proposition This defines a model category structure which is
All simplicial fields are simplicially constant. This is because the composite is the identity, so is surjective, but all field homomorphisms are injective, so is an isomorphism.
Introduction and survey includes
Bertrand Toën, chapter 4 of Simplicial presheaves and derived algebraic geometry , lecture at Simplicial methods in higher categories (pdf)
Bertrand Toën, Derived Algebraic Geometry (arXiv:1401.1044)
See model structure on simplicial algebras for references on the model structure discussed above.
Some of the above material is taken from this MO entry.
Discussion in the context of homotopy theory, hence for simplicial ring spectra includes
Last revised on April 19, 2024 at 04:01:56. See the history of this page for a list of all contributions to it.