When we have a discrete groupoid, we have the concept of equivalence of groupoids as categories. Given two equivalent discrete groupoids, we obtain a simplicial homotopy between their respective nerves. Now, I'm curious about the topological case. How is the equivalence of topological groupoids defined? How do we define the homology of a topological groupoid? Is the homology of topological groupoids invariant with respect to such equivalence?
If someone can provide me with some references, I would be very thankful.