Simplicial Set - Singular Set For A Space

Singular Set For A Space

The singular set of a topological space Y is the simplicial set defined by S(Y): n → hom(|Δn|, Y) for each object n ∈ Δ, with the obvious functoriality condition on the morphisms. This definition is analogous to a standard idea in singular homology of "probing" a target topological space with standard topological n-simplices. Furthermore, the singular functor S is right adjoint to the geometric realization functor described above, i.e.:

hom_Top(|X|, Y) ≅ hom_S(X, SY)

for any simplicial set X and any topological space Y.