The category of $\pi$-finite spaces

Abstract

We show that the category of truncated spaces with finite homotopy invariants ($\pi$\=/finite spaces) has many of the features expected of an elementary \oo topos. It should be thought of as the natural higher analogue of the elementary 1-topos of finite sets, with which it shares several initiality properties. The paper has also an appendix about univalent families in \oo pretopoi.