I've recently been led to believe some version of the following statement:

Weak homotopy types, or equivalently $\infty$-groupoids (let me not commit myself to a particular model of these), are freely generated under homotopy colimits by a point in the sense that, if $C$ is an $(\infty, 1)$-category with small homotopy colimits, then the $(\infty, 1)$-category of homotopy colimit-preserving $(\infty, 1)$-functors $\infty\text{-Gpd} \to C$ should be equivalent to $C$, with the equivalence on objects being given by evaluating the functor at a point.

This is the $(\infty, 1)$-categorical version of the $1$-categorical statement that $\text{Set}$ is freely generated under colimits by a point.

So we should be able to construct weak homotopy invariants by finding nice $(\infty, 1)$-categories and nice objects in them and seeing what the corresponding functor above produces. I think the following are examples, although the functors will be contravariant and I could have the details wrong:

- Take $C$ to be the ($(\infty, 1)$-category presented by) the dg-category of complexes of abelian groups, and take the object assigned to the point to be $\mathbb{Z}$. The corresponding weak homotopy invariant should be singular cochains on a space. This should be some version of Eilenberg-Steenrod.
- Take $C$ to be the $(\infty, 1)$-category of dg-categories, and take the object assigned to the point to be the dg-category of complexes of abelian groups. The corresponding weak homotopy invariant should be the dg-category of $\infty$-local systems (with values in complexes of abelian groups) on a space.

What other interesting weak homotopy invariants can be described this way?