  Relying on sheaf theory, we introduce the notions of projected barcodes and
projected distances for multi-parameter persistence modules. Projected barcodes
are defined as derived pushforward of persistence modules onto $\mathbb{R}$.
Projected distances come in two flavors: the integral sheaf metrics (ISM) and
the sliced convolution distances (SCD). We conduct a systematic study of the
stability of projected barcodes and show that the fibered barcode is a
particular instance of projected barcodes. We prove that the ISM and the SCD
provide lower bounds for the convolution distance. Furthermore, we show that
the $\gamma$-linear ISM and the $\gamma$-linear SCD which are projected
distances tailored for $\gamma$-sheaves can be computed using TDA software
dedicated to one-parameter persistence modules. Moreover, the time and memory
complexity required to compute these two metrics are advantageous since our
approach does not require computing nor storing an entire $n$-persistence
module.
