David Roberts
bigroupoid

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

  • A space B 0B_0,
  • An internal groupoid B̲ 1=(B 2B 1)\underline{B}_1 = (B_2 \rightrightarrows B_1) equipped with a functor
    (S,T):B̲ 1B 0×B 0, (S,T):\underline{B}_1 \to B_0\times B_0,
  • A (horizontal) composition functor
    B̲ 1× S,B 0,TB̲ 1B̲ 1 \underline{B}_1 \times_{S,B_0,T} \underline{B}_1 \to \underline{B}_1

    over B 0×B 0B_0\times B_0

  • A unit functor
    B 0B̲ 1 B_0 \to \underline{B}_1

    over B 0×B 0B_0\times B_0

  • A (horizontal) inverse functor
    B̲ 1B̲ 1 \underline{B}_1 \to \underline{B}_1

    covering the swap map from B 0×B 0B_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)

Revised on October 24, 2012 14:46:09 by David Roberts (121.216.167.70)