posetal object

An object $A$ in a 2-category $K$ is **posetal** if the category $K(X,A)$ is a preorder (equivalent to a poset) for all objects $X$ of $K$. Posetal objects are also called **(0,1)-truncated objects** since $K(X,A)$ is a (0,1)-category (a poset).

More explicitly, $A$ is posetal iff any parallel 2-cells $\alpha,\beta:f \;\rightrightarrows\; g: X\;\rightrightarrows\; A$ are equal. If $K$ has finite limits, this is equivalent to saying that $A^{\mathbf{2}} \to A^{ppr}$ is an equivalence, where $\mathbf{2}$ is the “walking arrow” and $ppr$ is the “walking parallel pair of arrows.”

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

Revised on January 30, 2009 20:10:20
by Mike Shulman
(75.3.140.11)