Measure the failure of colimit to commute with taking free loops (or Hochschild homology)?

Question

For a space¹ $X$, let $\mathcal{L}X = \mathrm{Maps}(S^1, X)$ be the free loop space.

Inclusion of constant loops gives a natural map $X \to \mathcal{L}X$. This is not a homotopy equivalence unless $X$ is contractible, because it splits the fibration $\Omega X \to \mathcal{L} X \to X$.

I would like to know that it is not an equivalence ``because'' loops do not commute with (homotopy) colimits.

That is: suppose $X$ is presented as a CW complex, i.e. as a colimit of contractible spaces: $X = \mathrm{Colim}\, D_\alpha$. Then the natural map

$$X = \mathrm{Colim} \, D_\alpha = \mathrm{Colim} \, \mathcal{L} D_\alpha \to \mathcal{L} \, \mathrm{Colim} \, D_\alpha = \mathcal{L} X $$

is (homotopic to) the inclusion of constant loops.

It is possible to see that $X \mapsto \mathcal{L} X$ is not an equivalence by computing some invariant which, in general, measures the failure of loops to commute with colimits?

Here I have in mind that if I wanted to know how a functor failed to be exact, I would study its derived functors.

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When I search the internet for loop spaces and colimits, invariably I find myself reading about ``calculus of functors''.

Is the calculus of functors going to help me here?

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In fact for my purposes I am ultimately interested in the analogous question for the functor ``Hochschild homology'' from dg categories to chain complexes, so would be perfectly happy with an answer to the above question after taking chains.

(As for what these purposes are, it has to do with a problem about dynamics in contact manifolds. An explanation of how that is related is a bit afield and perhaps too long to include here, but you can see this short note.)

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¹ To avoid more complaints in the comments, let us say a connected CW complex.

Answer 1

If you consider Goodwillie calculus for functors ${\rm Top} \to Ch_{\mathbb Z}$, then a functor will be linear if and only if it preserves all small (homotopy) colimits (as in Maxime's comment). Indeed, in the stable setting, a square is a pushout if and only if it is a pullback, so excisive functors to a stable target preserve pushouts, and therefore all small colimits. So the functor $X \mapsto C_*(LX)$ will not be linear.

There may be issues when $X$ is not simply connected and I don't know a reference, but it seems to me that you should be able to reconstruct the Goodwillie tower of $X \mapsto C_*(LX)$, from the (co)-Hochschild model for $C_*(LX)$ by truncating the Hochschild complex. In particular it seems that the $n$th associated homogeneous functor is $C_*(X)^{\otimes n}$ (up to a shift).