For more see also at CW-complex.
model category, model -category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of -categories
Model structures
for -groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant -groupoids
for rational -groupoids
for rational equivariant -groupoids
for -groupoids
for -groups
for -algebras
general -algebras
specific -algebras
for stable/spectrum objects
for -categories
for stable -categories
for -operads
for -categories
for -sheaves / -stacks
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 cell complex is an object in a category which is obtained by successively “gluing cells” via pushouts.
Let be a category with colimits and equipped with a set of morphisms.
In practice is usually a cofibrantly generated model category with set of generating cofibrations and set of acyclic generating cofibrations.
An -cell complex in is an object which is connected to the initial object by a transfinite composition of pushouts of the generating cofibrations in .
A relative -cell complex (relative to any object ) is any morphism obtained like this starting from .
A finite cell complex or countable cell complex is a cell complex with a finite set or a countable set of cells, respectively.
A CW-complex is a cell complex in Top with respect to the generating cofibrations in the standard model structure on topological spaces.
Every simplicial set is a cell complex with respect to the generating cofibrations in the standard model structure on simplicial sets.
A Sullivan model is a cell complex with respect to the generating cofibrations in the standard model structure on dg-algebras.
A cell spectrum is a cell complex in the category of topological sequential spectra.
examples of universal constructions of topological spaces:
Textbook account:
A discussion in the context of algebraic model categories is in
Last revised on October 5, 2024 at 08:19:17. See the history of this page for a list of all contributions to it.