David Roberts
bigroupoid

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

• A space $B_0$,
• An internal groupoid $\underline{B}_1 = (B_2 \rightrightarrows B_1)$ equipped with a functor
  $$(S,T):\underline{B}_1 \to B_0\times B_0,$$
• A (horizontal) composition functor
  $$\underline{B}_1 \times_{S,B_0,T} \underline{B}_1 \to \underline{B}_1$$

  over $B_0\times B_0$

• A unit functor
  $$B_0 \to \underline{B}_1$$

  over $B_0\times B_0$

• A (horizontal) inverse functor
  $$\underline{B}_1 \to \underline{B}_1$$

  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)