When a reduced Segal space is group-like, it becomes a model for an infinity-group aka a loop space. Group-like reduced Segal spaces characterize loop spaces by means of finite products and weak equivalences and as such transparently show that the property of being a loop space is invariant under any product-preserving endofunctor of topological spaces.
If is a topological monoid, its classifying space has a model as a simplicial space in which the 0th level is a point and the nth level is n-times the product of the 1st level. The idea of a reduced Segal space is requiring the above two properties of the simplicial space to hold up to homotopy and capture in this way all -monoids.
The passage from reduced Segal spaces to group-like reduced Segal spaces is the step of adding “inverses up to coherent homotopy”. It turns out to be equivalent to simply requiring that the monoid of path components of the 1st level admits a group structure.
A simplicial space is called a reduced Segal space if:
(1) the space is weakly contractible;
(2) for each , the Segal map is a weak equivalence.
is called group-like reduced Segal space if in addition:
(3) the monoid structure on , induced from the -space structure on , admits inverses (i.e. it is a group).
Proposition(G. Segal): if is a group-like reduced Segal space, the map is a weak equivalence.
G. Segal, “Categories and Cohomology Theories”, Topology 13 (1974).
C. Balteanu, Z. Fiedorowicz, R. Schwanzl and R. Vogt, “Iterated Monoidal Categories”, Advances in Mathematics (2003).