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.:
- homTop(|X|, Y) ≅ homS(X, SY)
for any simplicial set X and any topological space Y.
Read more about this topic: Simplicial Set
Famous quotes containing the words singular, set and/or space:
“And that singular anomaly, the lady novelist—
I don’t think she’d be missed—I’m sure she’d not be
missed!”
—Sir William Schwenck Gilbert (1836–1911)
“It’s the set of the soul that decides the goal,
And not the storms or the strife.”
—Ella Wheeler Wilcox (1850–1919)
“No being exists or can exist which is not related to space in some way. God is everywhere, created minds are somewhere, and body is in the space that it occupies; and whatever is neither everywhere nor anywhere does not exist. And hence it follows that space is an effect arising from the first existence of being, because when any being is postulated, space is postulated.”
—Isaac Newton (1642–1727)