The Injective category number on continuous maps

We present the notion of injective category number of a map f X→ 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→ X/τ, where X is a normal space and τ:X→ X is a fixed-point free involution, we show that the injective category of q is equal to the index of (X,τ) plus $2$.