Michael Shulman
groupoidal object

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

More explicitly, AA is groupoidal iff any 2-cell α:fg:XA\alpha: f \to g: X \;\rightrightarrows\; A is an isomorphism. If KK has finite limits, this is equivalent to saying that AA 2A\to A^{\mathbf{2}} is an equivalence, where 2\mathbf{2} is the “walking arrow.”

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

Revised on January 30, 2009 20:09:52 by Mike Shulman (75.3.140.11)