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?

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    $\begingroup$ I'm not really sure what you expect to get out of this question--the answer is (tautologically) "all (homotopy) colimit-preserving invariants". When the target category is spectra (or their opposite), this gives all (co)homology theories, as in your first example. $\endgroup$ – Eric Wofsey Oct 2 '13 at 4:41
  • $\begingroup$ Indeed this question is already answered in the first statement paragraph. $\endgroup$ – Fernando Muro Oct 2 '13 at 9:46
  • $\begingroup$ I learned the point in your opening paragraph from Clark Barwick, and it's discussed in his and Chris Schommer-Pries's Unicity paper. That should at the very least help you to see the analog of what you're asking for $(\infty,n)$ categories and their paper might also contain information about other homotopy invariants that come from this idea. $\endgroup$ – David White Oct 2 '13 at 10:32
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    $\begingroup$ @Eric: hmm. Then maybe I'm asking a question which is too broad (something like "what are examples of objects in $(\infty, 1)$-categories")... I guess what I was hoping for was examples of invariants that people were already interested in and had already defined in more concrete ways but that can fit into this framework. $\endgroup$ – Qiaochu Yuan Oct 2 '13 at 15:07
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    $\begingroup$ This is already in Lurie's "Higher Topos Theory", if maybe a bit hidden. It's made explicit in this nLab entry here: ncatlab.org/nlab/show/limit+in+a+quasi-category#Tensoring $\endgroup$ – Urs Schreiber Oct 4 '13 at 19:16

Here is a slight generalization of what you are thinking about, related to Thom spectra (disclaimer, I am still learning this stuff, so, someone correct me if I make a mistake):

Let $R$ be some $A_\infty$-ring spectrum. Let $R\mbox{-line}$ denote the connected space (infinity group) of self (homotopy) automorphisms of $R$ in the $(\infty,1)$-category of $R$-modules. Note that there is a canonical functor $$\theta:R\mbox{-line} \to R\mbox{-Mod}.$$ Also note that there is a canonical equivalence of $(\infty,1)$-categories $$\mbox{Fun}(R\mbox{-line}^{op},\infty Gpd) \simeq \infty Gpd/R\mbox{-line}.$$ An object on the right may be thought of as a generalized spherical fibration, since if $R$ is the sphere spectrum, (stable) spherical fibrations are classified by maps into $BGl_1 R = R\mbox{-line}.$

The functor $\theta$ may be extended to a unique (homotopy) colimit preserving functor $${Th}_R:\mbox{Fun}(R\mbox{-line}^{op},\infty Gpd) \simeq \infty Gpd/R\mbox{-line} \to R\mbox{-Mod},$$ such that ${Th}_R$ restricted to the Yoneda embedded image of $R\mbox{-line}$ is just $\theta$. A simple calculation shows that in fact ${Th}_R$ sends a map (functor) $$f:X \to R\mbox{-line}$$ to the (homotopy) colimit of $\theta \circ f.$ This colimit is the generalized Thom spectrum of $f$. E.g., if $R$ is the sphere spectrum, and $M$ is a manifold with its canonical spherical fibration $$T_M:M \to R,$$ coming from the image of its tangent bundle under the $J$-homomorphism, then $Th_R\left(T_M\right)$ is the Thom spetrum of $M$. More generally, for the $J$-homomorphism itself $$J:BU \to BGL_1S,$$ $Th\left(J\right)$ is $MU.$

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