nLab singular simplicial complex



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



Paths and cylinders

Homotopy groups

Basic facts




This construction ‘probes’ a space XX by mapping geometric simplices into it. It is one of the classical approaches to determining invariants of the homotopy type of the space.


The singular simplicial complex S (X)S_\bullet(X) of a topological space XX is the nerve of XX with respect to the standard cosimplicial topological space Δ Top:ΔTop\Delta_{Top} : \Delta \to Top. It is thus the simplicial set, S (X)S_\bullet(X), having

S n(X)=Hom Top(Δ Top n,X). S_n(X) = Hom_{Top}(\Delta_{Top}^n, X) \,.

as its set of nn-simplices, and fairly obvious faces and degeneracy mappings obtains by restriction along the structural maps of Δ Top:ΔTop\Delta_{Top} : \Delta \to Top. This is always a Kan complex and as such has the interpretation of the fundamental ∞-groupoid Π(X)\Pi(X) of XX.

The nn-simplices of this are just singular n-simplices generalising paths in XX. (The term singular is used because there is no restriction that the continuous function used should be an embedding, as would be the case in, for instance, a triangulation where a simplex in the underlying simplicial complex corresponds to an embedding of a simplex.)


Preservation of model structure

The singular complex functor preserves all five classes of maps in a model category: weak equivalences, cofibrations, acyclic cofibrations, fibrations, and acyclic fibrations.

Relation to geometric realization

Together with its adjointgeometric realization ||:sSetTop|-| : sSet \to Top—the functor Sing:TopsSetSing : Top \to sSet is part of the Quillen equivalence between the model structure on topological spaces and the model structure on simplicial sets that is sometimes called the homotopy hypothesis-theorem.

Relation to ordinary (co)homology

Forming from the singular simplicial complex Sing(X)Sing(X) first the free simplicial abelian group [Sing(X)]\mathbb{Z}[Sing(X)] and then under the Dold-Kan correspondence the corresponding normalized chain complex yields the chain complex of (normalized!) singular chains, which computes the singular homology of XX.

Last revised on June 6, 2020 at 14:21:38. See the history of this page for a list of all contributions to it.