Using the universal property of spaces The $\infty$-category of spaces has the following properties:

*

*It is 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.)

*(As Tim Campion points out in a comment, another characterization, also in HTT): Spaces are obtained by freely adding arbitrary colimits to the category $\{*\}$.

Can either of these characterizations 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 generated by (the compact objects) $*$ and $\Delta^1$?

 A: Here is a "model-independent" proof that colimits in $Spaces$ are universal. Of course, the ingredients going into the proof may have model-dependent proofs.
Fact: Colimits in $Spaces$ are universal.
Proof: We want to show that for any map of spaces $f: Y \to X$, the pullback functor $$f^\ast: Spaces_{/X} \to Spaces_{/Y}$$ preserves colimits. We may view $X,Y$ as $\infty$-categories which happen to be $\infty$-groupoids and $f: Y \to X$ as a functor between them. By straightening / unstraightening, the functor $f^\ast :Spaces_{/X} \to Spaces_{/Y}$ is identitified with the "precompose $f$" functor $$f^\ast: Psh(X) \to Psh(Y)$$ Now, $Psh(X),Psh(Y)$ are functor categories with values in the cocomplete category $Spaces$. So colimits are computed "objectwise" in these categories. That is, for $F: I \to Psh(X)$, we have $(\varinjlim_{i \in I} F(i))(x) = \varinjlim_{i \in I} (F(i)(x))$, and similarly in $Y$. From these formulas, it is immediate that precomposing $f$ preserves colimits. That is, we have $$(\varinjlim_{i \in I} f^\ast F(i))(y) = \varinjlim_{i \in I} F(i)(f(y)) = \varinjlim_{i \in I} (f^\ast F(i))(y)$$ as desired.

This proof doesn't precisely use the universal property of $Psh(X),Psh(Y)$ of freely adding colimits, but perhaps it could be tweaked to do so.
