HA, Lurie, Characterization of suspension

Question

In HA, p24 Lurie wrote

The suspension functor $\Sigma_{C} :C \rightarrow C $ is essentially characterized by the existence of a natural homotopy equivalence $$ Map_C(\Sigma X,Y) \rightarrow \Omega Map_C(X,Y) $$

I am confused. How is this deduced?

Answer 1

The object $\Sigma X$ is defined as the pushout (in the $\infty$-categorical sense) of $0\leftarrow X\to 0$. By definition, this means that $Map_C(\Sigma X,Y)$ is the pullback of spaces (in the $\infty$-categorical sense) $Map_C(0,Y)\times_{Map_C(X,Y)}Map_C(0,Y)\equiv 0\times_{Map_C(X,Y)}0$, where $0$ is denoting the zero object both in $C$ and in pointed spaces. The latter pullback is $\Omega Map_C(X,Y)$, by definition.

Answer 2

If $[]$ denotes descending to the homotopy category, the equivalence implies $[\operatorname{Map} (\Sigma X, -)]$ is naturally isomorphic to $[\Omega \operatorname{Map}(X,-)]$. If we had another functor $F$ with the same property as in the question, then it would also be the case $[\operatorname{Map} (FX, -)]$ is naturally isomorphic to $[\Omega \operatorname{Map}(X,-)]$. Basic category theory says that two representing objects are isomorphic. The naturality of the homotopy equivalence implies that the isomorphisms can be chosen to assemble into a natural isomorphism $\Sigma \rightarrow F$.

If you are in a suitable category like compactly generated spaces, then this can also be deduced from the adjointness of smash and mapping space.