David Roberts
bigroupoid

An internal bigroupoid (in $\mathrm{Top}$, for argument’s sake, but any finitely complete category will do, or at least one with some pullbacks, like $\mathrm{Diff}$) consists of the following data:

• A space $B_0$,

• An internal groupoid ${\underline{B}}_1=(B_2\rightrightarrows B_1)$ equipped with a functor

  $$(S,T):{\underline{B}}_1\to B_0\times B_0,$$(S,T):\underline{B}_1 \to B_0\times B_0,

• A (horizontal) composition functor

  $${\underline{B}}_1{\times }_{S,B_0,T}{\underline{B}}_1\to {\underline{B}}_1$$\underline{B}_1 \times_{S,B_0,T} \underline{B}_1 \to \underline{B}_1

  over $B_0\times B_0$

• A unit functor

  $$B_0\to {\underline{B}}_1$$B_0 \to \underline{B}_1

  over $B_0\times B_0$

• A (horizontal) inverse functor

  $${\underline{B}}_1\to {\underline{B}}_1$$\underline{B}_1 \to \underline{B}_1

  covering the swap map from $B_0\times B_0$ to itself.

Together with natural transformations… (see for the time being Definition 5.21 in my thesis - I need to grok how to do diagrams here)