  We investigate the universal cover of a topological group that is not
necessarily connected. Its existence as a topological group is governed by a
Taylor cocycle, an obstruction in 3-cohomology. Alternatively, it always exists
as a topological 2-group. The splitness of this 2-group is also governed by an
obstruction in 3-cohomology, a Sinh cocycle. We give explicit formulas for both
obstructions and show that they are inverse of each other.
