  We present the notion of injective category number of a map $f\colon X\to Y$
with basic results about this numerical invariant. For instance, we explore the
injective category number under pullbacks and compositions. In the case that
$f$ is the quotient map $q:X\to X/\tau$, where $X$ is a normal space and
$\tau:X\to X$ is a fixed-point free involution, we show that the injective
category of $q$ is equal to the index of $(X,\tau)$ plus $2$.
