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1. Constructing smooth manifolds of loop spaces. canonical page. The point of this is to figure out exactly when the "standard method" (alluded to by Tim) works. The distinction between "loop" and "path" is irrelevant.

2. The Smooth Structure of the Space of Piecewise-Smooth Loops, Glasgow Mathematical Journal, 59(1) (2017) pp27-59. (arXiv:0803.0611, doi:10.1017/S0017089516000033). Why you should be very, very nervous whenever anyone says "consider piecewise-smooth maps"; and take as a cautionary tale as to the inadvisability of going beyond smooth maps in general.

3. Work of David Roberts on the nLab. This is where I got the 2-idea that I mentioned above.

4. Other relevant nLab pages: http://ncatlab.org/nlab/show/generalized+smooth+space, http://ncatlab.org/nlab/show/smooth+loop+space and further.

5. Of course, the magnificent book by Kriegl and Michor. (I'm going to create a separate MO account for that book; its role will be to post an answer on relevant questions simply saying "Read Me".)

1. Constructing smooth manifolds of loop spaces, Proc. London Math. Soc. 99 (2009) pp195–216 (doi:10.1112/plms/pdn058, arXiv:math/0612096). The point of this is to figure out exactly when the "standard method" (alluded to by Tim) works. The distinction between "loop" and "path" is irrelevant.

2. The Smooth Structure of the Space of Piecewise-Smooth Loops, Glasgow Mathematical Journal, 59 (2017) pp27-59. (arXiv:0803.0611, doi:10.1017/S0017089516000033). Why you should be very, very nervous whenever anyone says "consider piecewise-smooth maps"; and take as a cautionary tale as to the inadvisability of going beyond smooth maps in general.

3. Work of David Roberts on the nLab. This is where I got the 2-idea that I mentioned above.

4. Other relevant nLab pages: http://ncatlab.org/nlab/show/generalized+smooth+space, http://ncatlab.org/nlab/show/smooth+loop+space and further.

5. Of course, the magnificent book by Kriegl and Michor. (I'm going to create a separate MO account for that book; its role will be to post an answer on relevant questions simply saying "Read Me".)

1. Constructing smooth manifolds of loop spaces. canonical page. The point of this is to figure out exactly when the "standard method" (alluded to by Tim) works. The distinction between "loop" and "path" is irrelevant.

2. The Smooth Structure of the Space of Piecewise-Smooth Loops. canonical page. Why you should be very, very nervous whenever anyone says "consider piecewise-smooth maps"; and take as a cautionary tale as to the inadvisability of going beyond smooth maps in general.

3. Work of David Roberts on the nLab. This is where I got the 2-idea that I mentioned above.

4. Other relevant nLab pages: http://ncatlab.org/nlab/show/generalized+smooth+space, http://ncatlab.org/nlab/show/smooth+loop+space and further.

5. Of course, the magnificent book by Kriegl and Michor. (I'm going to create a separate MO account for that book; its role will be to post an answer on relevant questions simply saying "Read Me".)

1. Constructing smooth manifolds of loop spaces. canonical page. The point of this is to figure out exactly when the "standard method" (alluded to by Tim) works. The distinction between "loop" and "path" is irrelevant.

2. The Smooth Structure of the Space of Piecewise-Smooth Loops, Glasgow Mathematical Journal, 59(1) (2017) pp27-59. (arXiv:0803.0611, doi:10.1017/S0017089516000033). Why you should be very, very nervous whenever anyone says "consider piecewise-smooth maps"; and take as a cautionary tale as to the inadvisability of going beyond smooth maps in general.

3. Work of David Roberts on the nLab. This is where I got the 2-idea that I mentioned above.

4. Other relevant nLab pages: http://ncatlab.org/nlab/show/generalized+smooth+space, http://ncatlab.org/nlab/show/smooth+loop+space and further.

5. Of course, the magnificent book by Kriegl and Michor. (I'm going to create a separate MO account for that book; its role will be to post an answer on relevant questions simply saying "Read Me".)

Update: The biggest failing is actually that the obvious model space is not a vector space. The space of paths mod thin homotopy in $\mathbb{R}^n$ does not inherit a well-defined addition from the space of paths in $\mathbb{R}^n$. Full details at the nLab page http://ncatlab.org/nlab/show/smooth+structure+of+the+path+groupoid.

(Update added here, rather than at the end, as it's the most direct answer to the specific question; the rest should be viewed as extra for those interested in more than just whether or not this space is a smooth manifold.)