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 $A\to A^{ppr}$ is an equivalence, where $ppr$ is the “walking parallel pair of arrows.”

We write $disc(K)$ 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 $K/B$.

Revised on June 12, 2012 11:10:00
by Andrew Stacey?
(129.241.15.200)