cell complex in nLab
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For more details see also at CW-complex.
Context
Model category theory
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
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
Contents
Idea
A cell complex is an object in a category which is obtained by successively “gluing cells” via pushouts.
Definition
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.
Examples
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A CW-complex is a cell complex in Top with respect to the generating cofibrations in the standard model structure on topological spaces.
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Every simplicial set is a cell complex with respect to the generating cofibrations in the standard model structure on simplicial sets.
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A Sullivan model is a cell complex with respect to the generating cofibrations in the standard model structure on dg-algebras.
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A cell spectrum is a cell complex in the category of topological sequential spectra.
examples of universal constructions of topological spaces:
References
Textbook account:
- Rudolf Fritsch, Renzo Piccinini, Cellular structures in topology, Cambridge University Press (1990) (doi:10.1017/CBO9780511983948, pdf)
A discussion in the context of algebraic model categories is in
- Emily Riehl, Cellularity in algebraic model structures (blog post)
Last revised on August 17, 2022 at 13:53:32. See the history of this page for a list of all contributions to it.