ScalingStacks

Say a map $f\colon X\rightarrow Y$ of spaces is a homotopy monomorphism if

1. (1)

   it is injective on $\pi_{0}$, and

2. (2)

   it is a weak equivalence of each component of $X$ to the corresponding component of $Y$.

Equivalently, $f$ is a homotopy monomorphism if the square

is a homotopy pullback square. Since homotopy limits commute, the homotopy limit functor applied to a homotopy monomorphism between two diagrams yields a homotopy monomorphism.