As a topological space it is equivalently a mapping space from the circle to some pointed topological space : a space of loops in . Here and the loops might be equipped with further geometric structure such as smooth structure and then one may consider a smooth loop space, etc.
Strictly speaking, and as considered here, a loop space consists only of loops that start and end at a fixed base point in . Without this restriction one speaks of a free loop space.
Let Top be a nice category of topological spaces, in particular one which is complete, cocomplete, and cartesian closed. Let be the circle, i.e., 1-dimensional sphere, with chosen basepoint, and let be a space with a chosen basepoint. Then the loop space of (at ) is an internal hom
in the category of based spaces. Explicitly, it is given by the pullback in
with basepoint provided by the right vertical arrow.
A loop space is an example of a A-∞ space, in particular it is an H-space. Loop spaces admit this rich algebraic structure which arises from the fact that the based space carries a correspondingly rich co-algebraic structure, starting from the fact that the based space is an H-cogroup.
An important theoretical consideration is when an H-space, and particularly one having the type of a CW-complex, has the homotopy type of a loop space of another CW-complex: . In this circumstance, one calls a delooping of ; an important example is where carries a topological group structure , and is the classifying space of .
The most basic fact about deloopings is the shift in homotopy groups:
which follows straight from the adjunction plus the fact that the suspension of is . (This isomorphism needs to be developed at greater length.)
An H-space admits a delooping if and only if the monoid induced from the H-space structure is a group, and the H-space structure can be extended to a structure of algebra over an operad over Stasheff’s A-∞ operad .
This is due to (Stasheff). The analogous statement holds true in every (∞,1)-topos other than Top. Details on this more general statement are at loop space object and at groupoid object in an (∞,1)-category.
Let the space be locally 0-connected and semi-locally 1-connected? (i.e. it admits a universal covering space). The loop space for any basepoint is locally path connected, as is the free loop space . If is locally 1-connected and admits a basis of open sets such that is the zero map, then is locally 0-connected and semi-locally 1-connected, and so admits a universal covering space.
In general, if is locally -connected, is locally -connected. This process can obviously be iterated up to times, so that is locally 0-connected. This can be weakened to locally -connected and semi-locally -connected: this is just like the case but replacing with (as was done in the previous paragraph with ). We will actually define a space to be semi-locally -connected to include the condition that it is locally -connected. This result was proved for more general mapping spaces and various subspaces when is Hausdorff and a finite polyhedron in (Wada) but a much simpler and direct proof for general and or is possible.
The fundamental -groupoid of a space (Trimblean for choice) can be topologised to be an internal -groupoid in when is semi-locally -connected. Furthermore, the homotopy groups of the -groupoid, a priori topological groups, are discrete.
There is a Quillen equivalence
between the model structure on simplicial groups and the model structure on reduced simplicial sets, thus exhibiting both of these as models for infinity-groups (Kan 58). Its left adjoint , the simplicial loop space construction, is a concrete model for the loop space construction with values in simplicial groups.
|(∞,1)-operad||∞-algebra||grouplike version||in Top||generally|
|A-∞ operad||A-∞ algebra||∞-group||A-∞ space, e.g. loop space||loop space object|
|E-k operad||E-k algebra||k-monoidal ∞-group||iterated loop space||iterated loop space object|
|E-∞ operad||E-∞ algebra||abelian ∞-group||E-∞ space, if grouplike: infinite loop space Γ-space||infinite loop space object|
|connective spectrum||connective spectrum object|
Daniel Kan, On homotopy theory and c.s.s. groups, Ann. of Math. 68 (1958), 38-53
Jim Stasheff, Homotopy associative -spaces I, II, Trans. Amer. Math. Soc. 108, 1963, 275-312
H. Wada, Local connectivity of mapping spaces, Duke Mathematical Journal, vol ? (1955) pp 419-425
The simplicial loop group functor is discussed in chapter V, section 5 of
See also the references at looping and delooping.