David Roberts
bigroupoid
An internal bigroupoid (in Top Top , for argument’s sake, but any finitely complete category will do, or at least one with some pullbacks, like Diff Diff ) consists of the following data:
A space B 0 B_0 ,
An internal groupoid B ̲ 1 = ( B 2 ⇉ B 1 ) \underline{B}_1 = (B_2 \rightrightarrows B_1) equipped with a functor( S , T ) : B ̲ 1 → B 0 × B 0 ,
(S,T):\underline{B}_1 \to B_0\times B_0,
A (horizontal) composition functorB ̲ 1 × S , B 0 , T B ̲ 1 → B ̲ 1
\underline{B}_1 \times_{S,B_0,T} \underline{B}_1 \to \underline{B}_1
over B 0 × B 0 B_0\times B_0
A unit functorB 0 → B ̲ 1
B_0 \to \underline{B}_1
over B 0 × B 0 B_0\times B_0
A (horizontal) inverse functorB ̲ 1 → B ̲ 1
\underline{B}_1 \to \underline{B}_1
covering the swap map from B 0 × B 0 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)
Last revised on October 24, 2012 at 14:46:09.
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