In mathematics, well-behaved topological spaces can be localized at primes, in a similar way to the localization of a ring at a prime. This construction was described by Dennis Sullivan in 1970 lecture notes that were finally published in .
The reason to do this was in line with an idea of making topology, more precisely algebraic topology, more geometric. Localization of a space X is a geometric form of the algebraic device of choosing 'coefficients' in order to simplify the algebra, in a given problem. Instead of that, the localization can be applied to the space X, directly, giving a second space Y.
We let A be a subring of the rational numbers, and let X be a simply connected CW complex. Then there is a simply connected CW complex Y together with a map from X to Y such that
This space Y is unique up to homotopy equivalence, and is called the localization of X at A.
If A is the localization of Z at a prime p, then the space Y is called the localization of X at p.
The map from X to Y induces isomorphisms from the A-localizations of the homology and homotopy groups of X to the homology and homotopy groups of Y.