# infinity categories – Using the universal property of spaces

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 6.1.3.14)?
• possibly even that $$Cat_infty$$ is compactly generated by $$*$$ and $$Delta^1$$?