  In previous work, we showed that there are appropriate model category
structures on the category of simplicial categories and on the category of
Segal precategories, and that they are Quillen equivalent to one another and to
Rezk's complete Segal space model structure on the category of simplicial
spaces. Here, we show that these results still hold if we instead use groupoid
or "invertible" cases. Namely, we show that model structures on the categories
of simplicial groupoids, Segal pregroupoids, and invertible simplicial spaces
are all Quillen equivalent to one another and to the standard model structure
on the category of spaces. We prove this result using two different approaches
to invertible complete Segal spaces and Segal groupoids.
