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
A simplicial set is (sometimes) called reduced if it has a single vertex, .
More generally, for a simplicial set is -reduced if its -skeleton is the point, .
Write sSet for the full subcategory inclusion of the reduced simplicial sets into all of them.
This is a reflective subcategory. The reflector
identifies all vertices of a simplicial set.
Write for the category of pointed simplicial sets. There is also a full inclusion . This has a right adjoint which sends a pointed simplicial set to the subobject all whose -cells have as 0-faces the given point.
The inclusion into pointed simplicial sets is coreflective. The coreflector is the Eilenberg subcomplex construction in degree 1.
There is a model structure on reduced simplicial sets (see there) which serves as a presentation of the (∞,1)-category of pointed connected ∞-groupoids.
There is a Quillen equivalence
between the model structure on simplicial groups and the model structure on reduced simplicial sets (thus exhibiting both of these as models for infinity-groups). Its left adjoint , the simplicial loop space construction, is a concrete model for the loop space construction with values in simplicial groups.
Last revised on June 11, 2022 at 16:46:26. See the history of this page for a list of all contributions to it.