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Let $X$ be a connected topological space. Then its based loops space $\Omega X$ is an examples of $A_{\infty}$ space. Let $PX$ be the free paths space. Which type of homotopical structures are naturally defined on $PX$?

Edit: The multiplication of loops on the based loop spaces is defined via the ordinary composition followed by a reparametrization, this construction is associative only up to homotopy (the resulting objects in the associative diagram differ by a parametrization) and give to $\Omega X$ the structure of an $A_{\infty}$ spaces. Why this structure is not well defined on $PX$? It seems to me that the composition of two paths is well defined on $PX$ and it is again associative up to homotopy...

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  • $\begingroup$ The pathspace $PX$ is homotopy equivalent to $X$, so it has exactly the same amount of structure than $X$ $\endgroup$ Mar 11, 2016 at 18:32
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    $\begingroup$ $PX$, together with the pair of projections $PX \to X$ sending a path to its left and right endpoint, is (loosely speaking) a topological groupoid. $\endgroup$ Mar 11, 2016 at 22:41

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