Functoriality of infinite suspension spectrum functor on infinity groupoids!

Question

Consider the functor $F: C \rightarrow D $ of $\infty$-groupoids. Is there any explicit proof somewhere in the literature that $\Sigma^{\infty}$ construction is functorial? I mean how do we define $\Sigma^{\infty} F$ and how does it respect the composition of maps between spaces?

Answer 1

I'm not sure if this counts as a direct way, but a more practical way to show functoriality at the level of $\infty$-categories is to not construct $\Sigma^\infty$ as a functor directly, but instead show that it is left adjoint to the functor $\Omega^\infty : \mathrm{Sp} \to \mathcal{S}$ which is must easier to define (it is given by projection). One can extract an explicit description of $\Sigma^\infty f$ from this, but this method also handles showing functoriality on higher morphisms for free.

That is, one proves that for every space $X$ there is some spectrum $\Sigma^\infty X$ along with a natural equivalence $\hom_{\mathcal{S}}(X, \Omega^\infty -) \cong \hom_{\mathrm{Sp}}(\Sigma^\infty X, -) : \mathrm{Sp} \to \mathcal{S}$. (Using the Yoneda lemma, this is equivalent to producing a map $X \to \Omega^\infty\Sigma^\infty X$ which induces this equivalence.) It then follows from general results that (1) the assignment $X \mapsto \Sigma^\infty X$ is a functor $\mathcal{S} \to \mathrm{Sp}$ (2) this functor is left adjoint to $\Omega^\infty$, which we'll surely want anyways.

For references, one can find the general fact that a collection of objects representing the appropriate functors induce a functor in e.g., Kerodon or Higher Categories and Homotopical Algebra Proposition 6.1.11. For the proof that $\Sigma^\infty$ actually satisfies this condition, one can consult Lemma 2.9 of Denis Nardin's notes.

If one doesn't mind sacrificing an explicit description of $\Sigma^\infty X$ and is content with a characterization as just the left adjoint to $\Omega^\infty$, one can do better still: one can prove that the category of spectra is presentable as the limit of a collection of presentable categories and show that $\Omega^\infty$ satisfies the conditions of the adjoint functor theorem to produce a fully functorial $\Sigma^\infty$ for free. This is done in e.g., Higher Algebra Proposition 1.4.3.4.