Michael Shulman
discrete object

An object $A$ of a 2-category $K$ is discrete if the category $K(X,A)$ is equivalent to a discrete set for all objects $X$ of $K$. Discrete objects are also called 0-truncated objects since they are characterized by $K(X,A)$ being a 0-category (a set).

More explicitly, an object $A$ is discrete if and only if every pair of parallel 2-cells $\alpha ,\beta :f\rightrightarrows g:X\rightrightarrows A$ are equal and invertible. If $K$ has finite limits, this can be expressed equivalently by saying that $A\to A^{\mathrm{ppr}}$ is an equivalence, where $\mathrm{ppr}$ is the “walking parallel pair of arrows.”

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

A morphism $A\to B$ is called discrete if it is discrete as an object of the slice 2-category $K/B$.