Talk:Loop space

Quasigroup?

In what sense does loop space form a quasigroup? If I have two loops f and g there need not be a loop h such that fh = g. Loop space forms a nonassocitive, noncommutative magma, but surely not a quasigroup. -- Fropuff

Yes, you're right. Got my definitions mixed up. -- Staecker 01:30, 20 October 2005 (UTC)

adjointness to suspension

In the unbased case, the left adjoint to the loop functor is given by ordinary product with the circle (since the unbased loop functor is by definition the mapping space of maps from the circle). So IMHO the article cannot be quite correct in saying that the loop space functor is right adjoint to the unreduced suspension. What do you guys think? - Saibot2 21:36, 13 February 2007 (UTC)

adjointness and terminology

Saibot2 is right, the free loop functor (I think that's a more common name than unbased loop, but less descriptive) is right adjoint to the cartesian product. I also think that ΩX is a notation used only for the based loop space; the free loops are usually denoted L(X) or $\mathcal{L}(X)$. But maybe the usage outside of topology differs. -- Tilmanbauer (talk) 19:25, 19 February 2008 (UTC)