Michael Shulman
groupoidal object

An object A in a 2-category K is groupoidal if the category K(X,A) is a groupoid for all objects X of K. Groupoidal objects are also called (1,0)-truncated objects since K(X,A) is a (1,0)-category (a groupoid).

More explicitly, A is groupoidal iff any 2-cell α:fg:XA is an isomorphism. If K has finite limits, this is equivalent to saying that AA 2 is an equivalence, where 2 is the “walking arrow.”

We write gpd(K) for the full sub-2-category of K on the groupoidal objects; it is a (2,1)-category and is closed under limits in K.

Revised on January 30, 2009 20:09:52 by Mike Shulman (