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at.algebraic topology – Measure the failure of colimit to commute with taking free loops (or Hochschild homology)?

For an area $X$, let $mathcal{L}X = mathrm{Maps}(S^1, X)$ breathe the free loop house.

The functor from areas to areas $X mapsto mathcal{L}X$ doesn’t commute with (homotopy) colimits. Indeed, suppose $X$ is offered as a CW complicated, i.e. as a colimit of contractible areas: $X = mathrm{Colim}, D_alpha$. Then the unaffected 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 ceaseless loops.

The inclusion of ceaseless loops will not be an equivalence except $X$ is contractible, as a result of it splits the fibration $Omega X to mathcal{L} X to X$.

I’d love to know that it isn’t an isomorphism “as a result of” loops don’t commute with colimits. That is:

It is workable to behold that $X mapsto mathcal{L} X$ will not be an isomorphism by computing some invariant which, in common, measures the failure of loops to commute with colimits?

Here I’ve in intellect that if I wished to understand how a functor did not breathe require, I’d seek its derived functors.

When I search the web for loop areas and colimits, invariably I discover myself studying about “calculus of functors”.

Is the calculus of functors going to ameliorate me right here?

In reality for my functions I’m in the end within the analogous query for the functor “Hochschild homology” from dg classes to train complexes, so would breathe completely fortunate with an respond to the above query after taking chains.

(As for what these functions are, it has to do with an issue about dynamics involved manifolds. An clarification of how that’s associated is maybe too lengthy to comprise right here, so I wrote a brief point to about it to encourage this query.)

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