Michael Shulman
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 $A^2\to A^{\mathrm{ppr}}$ is an equivalence, where $2$ is the “walking arrow” and $\mathrm{ppr}$ is the “walking parallel pair of arrows.”

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