I was trying to understand Chen’s iterated integrals a bit more abstractly and ran into the following confusion about loop spaces.

Let $X$ be a smooth manifold, viewed as a simplicial presheaf over $\text{Mfld}$ or $\text{CartSp}$. It seems to me there are two different notions of the free loop space of $X$. The first is what’s described on free loop space object. This is as the powering $X^{S^1}$ or alternatively as the internal hom $[\text{LConst}(S^1), X]$. The second is the internal hom $[S^1, X]$ where here $S^1$ is the usual smooth manifold under Yoneda.

These two notions seem to be quite different — roughly, the first does not seem to remember the smooth structure on $S^1$. Is that true? (for instance the global sections seem to be different…) How are these two objects related?

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