free loop space
Stable homotopy theory
The free loop space of a topological space (based or not) is the space of all loops in . This is in contrast to the based loop space of a based space for which the loops are at the fixed base point .
For a topological space, the free loop space is the topological space of continuous maps in compact-open topology.
If we work in a category of based spaces, then still the topological space is in the non-based sense bit it itself has a distinguished point which is the constant map where is the base point of .
General abstract description
If is a topological space, the free loop space of is defined as the free loop space object of formed in the (∞,1)-category Top.
Revised on September 18, 2013 01:55:36
by Urs Schreiber