An object in a 2-category is groupoidal if the category is a groupoid for all objects of . Groupoidal objects are also called (1,0)-truncated objects since is a (1,0)-category (a groupoid).
More explicitly, is groupoidal iff any 2-cell is an isomorphism. If has finite limits, this is equivalent to saying that is an equivalence, where is the “walking arrow.”
We write for the full sub-2-category of on the groupoidal objects; it is a (2,1)-category and is closed under limits in .