I know that both the reduced suspension and loop space are covariant functors on the category of compactly generated, topological spaces with a basepoint and I think they are homotopy invariant. Now, they are supposed to be adjoint in that category, that is we have a natural bijection $map_\bullet(\Sigma X, Y)\cong map_\bullet(X, \Omega)$, right? But the only version that seems to appear is $[\Sigma X,Y]_\bullet\cong[X, \Omega Y]_\bullet$ which would follow from the first if both are homotopy invariant, right?In everything I have come across so far, it's is always the second version that is mentioned. Now I am not sure that I have understood everything right. Could somebody explain that to me?
Adjointness of reduced suspension and loop space
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general-topology
algebraic-topology
1 Answers
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You're right, $\mathsf{Map}_*(\Sigma X, Y) \cong \mathsf{Map}_*(X, \Omega Y).$
Let $\mathsf{Top}_*$ be the category of based topological spaces. Then $\Sigma$ is a functor from $\mathsf{Top}_*$ to $\mathsf{Top}_*$, which assigns to every object $X$, its suspension $\Sigma X$. Now let $\mathsf{Map}_* : \mathsf{Top}_* \times \mathsf{Top}_* \to \mathsf{Top}_*$ be the functor which assigns to every two objects $X$ and $Y$ the function space $\mathsf{Map}_*(X,Y)$ (... with the compact open topology!).
Note: I'm being very careless here with keeping track of base points, but hopefully this will give you and idea of the proof.
Now let $F : \mathsf{Top}_* \to Set$ which assigns to every object $X$ its underlying set. Then it is clear that $\mathsf{Map}_*$ is a hon-functor rel $F$. Let $\Omega = \mathsf{Map}_*(S^1, -)$ be the loop functor. Finally a homeomorphism $\beta : \mathsf{Map}_*(\Sigma X, Y) \cong \mathsf{Map}_*(X, \Omega Y)$ can be given which is natural.