The $infty$-category of spaces is known to be the $infty$-category obtained from the (ordinary) category of finite sets by freely adding sifted colimits. (See e.g. Cesnavicius-Scholze https://arxiv.org/abs/1912.10932 §5.1 for a review of this notion and for pointers to Lurie’s HTT where this is proven.)
Can this characterization be used (ideally without referring to the model of quasi-categories) to show other properties, such as:
- that colimits in spaces are universal (proven by Lurie in HTT Lemma 220.127.116.11)?
- possibly even that $Cat_infty$ is compactly generated by $*$ and $Delta^1$?