John Baez asked:

Does anyone know references that address the question of strictifying weak 2-groupoids? Here by a "weak 2-groupoid" I mean something like a bicategory in which every 2-morphism is invertible and every 1-morphism is an equivalence, and by "strictifying" such a thing I mean finding a biequivalent strict 2-category in which every 1-morphism and every 2-morphism is invertible.

Dear John and others This is a question that people ask from time to time. Below is an answer I sent to Prof. Dr. Klaus Heiner Kamps back on Fri, 21 Feb 1997. Regards, Ross Dear Heiner The fact that every monoidal category V is equivalent to a strict monoidal category st(V) follows from Mac Lane's coherence theorem (see "Cats for the working mathematician" Springer GTM 5). Pare' & Mac Lane adapted that proof to a coherence theorem for bicategories JPAA 37 (1985) 59-80 and then it follows that every bicategory W is biequivalent to a 2-category st(W). [In fact, st(W) is easy to describe: the arrows are paths in W and the 2-cells are just 2-cells in W between the composites of the paths bracketed from the right.) But there are much quicker proofs of these facts. For the fact that every monoidal category V is equiv to a strict monoidal category e(V) see the early part of (with A. Joyal) Braided tensor categories, Advances in Math 102 (1993) 20-78; MR94m:18008. For the generalisation to bicategories, see page 3 of (with R. Gordon and A.J. Power) Coherence for tricategories, Memoirs of the American Math. Society 117 (1995) Number 558 (ISSN 0065-9266); MR96j:18002. So now suppose we have a bigroupoid B. By the above, B is biequivalent to a 2-category K. Since every 2-cell in B is invertible and every arrow in B is an equivalence, the same will be true in K. So it remains to show that K is biequivalent to a 2-groupoid. Consider a 2-category K' whose objects are those of K and whose arrows (f,g,a,b) : A --> B consist of adjunctions g -| f with unit a, counit b in K. The 2-cells (u,v) : (f,g,a,b) --> (f',g',a',b') are 2-cells u : f --> f', v : g' --> g which determine each other under the adjunctions. Compositions are the obvious ones. The 2-functor K' --> K taking (f,g,a,b) : A --> B to f : A --> B is a biequivalence (it is the identity on objects and an equivalence on each hom category). Yet, in K', each arrow is invertible (not merely an equivalence). This last construction is the several-object version of a special case of a construction used in the early part of (with A. Joyal) The geometry of tensor calculus II (in preparation). The approach I give above is how I would do it as a categorical exercise. A proof using cohomology of groups was given in (with A. Joyal) Braided monoidal categories, Macquarie Math Reports # 860081 (Nov. 1986). Hope this helps. --Ross 11-Mar-2002 02:53:17 -0400,1858;000000000000-00000000