An object of a 2-category is discrete if the category is equivalent to a discrete set for all objects of . Discrete objects are also called 0-truncated objects since they are characterized by being a 0-category (a set).
More explicitly, an object is discrete if and only if every pair of parallel 2-cells are equal and invertible. If has finite limits, this can be expressed equivalently by saying that is an equivalence, where is the “walking parallel pair of arrows.”
We write for the full sub-2-category of on the discrete objects; it is equivalent to a 1-category, and is closed under limits in .
A morphism is called discrete if it is discrete as an object of the slice 2-category .