# 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 ${\underline{B}}_{1}=\left({B}_{2}⇉{B}_{1}\right)$ equipped with a functor

$\left(S,T\right):{\underline{B}}_{1}\to {B}_{0}×{B}_{0},$(S,T):\underline{B}_1 \to B_0\times B_0,
• A (horizontal) composition functor

${\underline{B}}_{1}{×}_{S,{B}_{0},T}{\underline{B}}_{1}\to {\underline{B}}_{1}$\underline{B}_1 \times_{S,B_0,T} \underline{B}_1 \to \underline{B}_1

over ${B}_{0}×{B}_{0}$

• A unit functor

${B}_{0}\to {\underline{B}}_{1}$B_0 \to \underline{B}_1

over ${B}_{0}×{B}_{0}$

• A (horizontal) inverse functor

${\underline{B}}_{1}\to {\underline{B}}_{1}$\underline{B}_1 \to \underline{B}_1

covering the swap map from ${B}_{0}×{B}_{0}$ 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)