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 $\alpha: f \to g: X \;\rightrightarrows\; A$ is an isomorphism. If $K$ has finite limits, this is equivalent to saying that $A\to A^{\mathbf{2}}$ is an equivalence, where $\mathbf{2}$ 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
(75.3.140.11)