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The fundamental bigroupoid of a topological space is one way of capturing itshomotopy 2-type. When the space is semilocally 2-connected, one can lift theconstruction to a bigroupoid internal to the category of topological spaces, asBrown and Danesh-Naruie lifted the fundamental groupoid to a topologicalgroupoid. For locally relatively contractible spaces the resulting topologicalbigroupoid is locally trivial in a way analogous to the case of the topologisedfundamental groupoid.

A bigroupoid's topology (or, Topologising the homotopy bigroupoid of a space)