Math 2.0

Algebraic topology Definition

Singular simplicial set

Builds on Topological spaceContinuous map of topological spaces

Definition

For n≥0n \ge 0 let [n]={0,1,…,n}[n] = \set{0, 1, \dots, n}, and let ∣Δn∣={(t0,…,tn)∈ℝn+1:ti≥0, t0+⋯+tn=1}\abs{\Delta^n} = \set{(t_0, \dots, t_n) \in \R^{n+1} : t_i \ge 0, \ t_0 + \cdots + t_n = 1} be the standard topological nn-simplex, with vertices e0,…,ene_0, \dots, e_n. An order-preserving map θ ⁣:[m]→[n]\theta \colon [m] \to [n] induces the affine map θ∗ ⁣:∣Δm∣→∣Δn∣\theta_* \colon \abs{\Delta^m} \to \abs{\Delta^n} with θ∗(ei)=eθ(i)\theta_*(e_i) = e_{\theta(i)}. Let XX be a topological space. The singular simplicial set Sing⁡X\operatorname{Sing} X is the simplicial set with:

  1. (Sing⁡X)n(\operatorname{Sing} X)_n the set of continuous maps σ ⁣:∣Δn∣→X\sigma \colon \abs{\Delta^n} \to X, called singular nn-simplices.
  2. For θ ⁣:[m]→[n]\theta \colon [m] \to [n] order-preserving, the map θ∗ ⁣:(Sing⁡X)n→(Sing⁡X)m\theta^* \colon (\operatorname{Sing} X)_n \to (\operatorname{Sing} X)_m given by θ∗(σ)=σ∘θ∗\theta^*(\sigma) = \sigma \circ \theta_*.

Remarks

The identities (θ∘ψ)∗=θ∗∘ψ∗(\theta \circ \psi)_* = \theta_* \circ \psi_* and id⁡∗=id⁡\id_* = \id show that Sing⁡X\operatorname{Sing} X is a functor Δop→Set\Delta^{\mathrm{op}} \to \mathrm{Set}, that is, a simplicial set. A continuous map f ⁣:X→Yf \colon X \to Y induces Sing⁡f\operatorname{Sing} f by σ↦f∘σ\sigma \mapsto f \circ \sigma, so Sing⁡\operatorname{Sing} is a functor from spaces to simplicial sets.

Example: if XX is a single point, each (Sing⁡X)n(\operatorname{Sing} X)_n has exactly one element, so Sing⁡X\operatorname{Sing} X is the terminal simplicial set Δ0\Delta^0.

Non-example: the simplicial set whose nn-simplices are all functions ∣Δn∣→X\abs{\Delta^n} \to X, continuous or not, is a simplicial set, but it is not Sing⁡X\operatorname{Sing} X for X=[0,1]X = [0, 1]. It contains discontinuous functions ∣Δ1∣→[0,1]\abs{\Delta^1} \to [0, 1], which clause 1 excludes.

Sing⁡X\operatorname{Sing} X is always a Kan complex.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectSet, definition, no evidence levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionEmpty set, object, no evidence levelEmpty setobjectFamily of sets, definition, no evidence levelFamily of setsdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionIntersection, definition, no evidence levelIntersectiondefinitionUnion, definition, no evidence levelUniondefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUnion of a family of sets, definition, no evidence levelUnion of a family of …definitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionTopological space, definition, no evidence levelTopological spacedefinitionFunction, definition, no evidence levelFunctiondefinitionPreimage of a set, definition, no evidence levelPreimage of a setdefinitionMetric space, definition, no evidence levelMetric spacedefinitionOpen ball, definition, no evidence levelOpen balldefinitionMetric topology, definition, no evidence levelMetric topologydefinitionOpen set, definition, no evidence levelOpen setdefinitionContinuous map of topological spaces, definition, no evidence levelContinuous map of top…definitionSingular simplicial set, definition, no evidence levelSingular simplicial s…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Element definition
  2. Proposition concept
  3. Real numbers object
  4. Set definition
  5. Conjunction definition
  6. Disjunction definition
  7. Implication definition
  8. Predicate definition
  9. Empty set object
  10. Family of sets definition
  11. Ordered pair definition
  12. Intersection definition
  13. Union definition
  14. Existential quantifier definition
  15. Universal quantifier definition
  16. Cartesian product definition
  17. Subset definition
  18. Union of a family of sets definition
  19. Uniqueness quantifier definition
  20. Binary relation definition
  21. Topological space definition
  22. Function definition
  23. Preimage of a set definition
  24. Metric space definition
  25. Open ball definition
  26. Metric topology definition
  27. Open set definition
  28. Continuous map of topological spaces definition
  29. Singular simplicial set definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Simplicial_set
About this page
TagDJM5H15WV5
URIm2://DJM5H15WV5
Kinddefinition
WritingW0 stub
MSC55U10, 18N50, 55N10
DefinesSingular simplicial set
UsesTopological space, Continuous map of topological spaces
Changed2 November 2026

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