Homotopy coherent Invertibility.

Similarly to how $\mathbb{E}_k$-commutative spectra are a homotopy-coherent version of homotopy commutative spectra, encoding commutativity with higher homotopies, we have a similar phenomenon for invertibility: given a monoidal $\infty$-groupoid $\mathcal{C}$, we can speak of both:

• Objects $A$ of $\mathcal{C}$ for which $[A]\in\pi_0(\mathcal{C})$ is invertible (homotopy invertibility);

• Objects $A$ of $\mathcal{C}$ which are homotopy-coherently invertible, coming with:

  • A choice of another object $A^{-1}$ of $\mathcal{C}$;

  • Morphisms of the form $A^{-1}\otimes_{\mathcal{C}}A\to1_\mathcal{C}$ and $A\otimes_{\mathcal{C}}A^{-1}\to1_{\mathcal{C}}$;

  • Homotopies filling a diagram of the form

    

    (along with another homotopy filling a similar diagram for $A\otimes_{\mathcal{C}}A^{-1}\otimes_{\mathcal{C}}A$)

  • and so on.

  (More precisely, a handy way to encode all these higher homotopies is to define a homotopy-coherent invertible object $A$ of a symmetric monoidal $\infty$-category $\mathcal{C}$ to be a symmetric strong monoidal functor $F\colon\Omega^{\infty}\mathbb{S}\to\mathcal{C}$ with $F(1)=A$, where $\mathbb{S}$ is the sphere spectrum.)

Localisation of ring spectra.

Lurie's Higher Algebra, Section 7.2.3 defines the localisation $S^{-1}E$ of an $\mathbb{E}_1$-ring spectrum $E$ at a nice subset $S$ of homogeneous elements of $\pi_*(E)$. This localisation comes also with a natural map $\phi\colon R\to S^{-1}R$ and admits the following universal property (HA 7.2.3.27):

Given an $\mathbb{E}_1$-ring $A$, precomposition with $\phi$ defines a fully faithful map $$\phi^{*}\colon\mathrm{Map}_{\mathsf{Alg}^{(1)}}(S^{-1}R,A)\to\mathrm{Map}_{\mathsf{Alg}^{(1)}}(R,A)$$ with essential image given by those maps of $\mathbb{E}_1$-rings $\psi\colon R\to A$ such that $\psi(s)$ is invertible in $\pi_*A$ for each $s\in S$.

The question.

Is the localisation procedure in HA homotopy coherent in the above sense? I.e. if we take $E$ to be a connective spectrum, so that we may also view it as a symmetric monoidal $\infty$-groupoid via $\Omega^{\infty}$, and let $S$ be a subset of $\pi_0(E)$, is the object in $\Omega^{\infty}(S^{-1}E)$ corresponding to $s\in S$ homotopy-coherently invertible, for each $s\in S$?

If not, is there a suitable homotopy-coherent analogue of Lurie's localisation?

Homotopy coherent Invertibility.

Similarly to how $\mathbb{E}_k$-commutative spectra are a homotopy-coherent version of homotopy commutative spectra, encoding commutativity with higher homotopies, we have a similar phenomenon for invertibility: given a monoidal $\infty$-groupoid $\mathcal{C}$, we can speak of both:

• Objects $A$ of $\mathcal{C}$ for which $[A]\in\pi_0(\mathcal{C})$ is invertible (homotopy invertibility);

• Objects $A$ of $\mathcal{C}$ which are homotopy-coherently invertible, coming with:

  • A choice of another object $A^{-1}$ of $\mathcal{C}$;

  • Morphisms of the form $A^{-1}\otimes_{\mathcal{C}}A\to1_\mathcal{C}$ and $A\otimes_{\mathcal{C}}A^{-1}\to1_{\mathcal{C}}$;

  • Homotopies filling a diagram of the form

    

    (along with another homotopy filling a similar diagram for $A\otimes_{\mathcal{C}}A^{-1}\otimes_{\mathcal{C}}A$)

  • and so on.

  (More precisely, a handy way to encode all these higher homotopies is to define a homotopy-coherent invertible object $A$ of a symmetric monoidal $\infty$-category $\mathcal{C}$ to be a symmetric strong monoidal functor $F\colon\Omega^{\infty}\mathbb{S}\to\mathcal{C}$ with $F(1)=A$, where $\mathbb{S}$ is the sphere spectrum.)

Localisation of ring spectra.

Lurie's Higher Algebra, Section 7.2.3 defines the localisation $S^{-1}R$ of an $\mathbb{E}_1$-ring spectrum $R$ at a nice subset $S$ of homogeneous elements of $\pi_*(R)$. This localisation comes also with a natural map $\phi\colon R\to S^{-1}R$ and admits the following universal property (HA 7.2.3.27):

Given an $\mathbb{E}_1$-ring $A$, precomposition with $\phi$ defines a fully faithful map $$\phi^{*}\colon\mathrm{Map}_{\mathsf{Alg}^{(1)}}(S^{-1}R,A)\to\mathrm{Map}_{\mathsf{Alg}^{(1)}}(R,A)$$ with essential image given by those maps of $\mathbb{E}_1$-rings $\psi\colon R\to A$ such that $\psi(s)$ is invertible in $\pi_*A$ for each $s\in S$.

The question.

Is the localisation procedure in HA homotopy coherent in the above sense? I.e. if we take $R$ to be a connective spectrum, so that we may also view it as a symmetric monoidal $\infty$-groupoid via $\Omega^{\infty}$, and let $S$ be a subset of $\pi_0(R)$, is the object in $\Omega^{\infty}(S^{-1}R)$ corresponding to $s\in S$ homotopy-coherently invertible, for each $s\in S$?

If not, is there a suitable homotopy-coherent analogue of Lurie's localisation?