  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. We prove several initiality results for this \oo
category. The paper has also an appendix about univalent families in \oo
pretopoi.
