5
$\begingroup$

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?

$\endgroup$

1 Answer 1

5
$\begingroup$

The answer is yes - more generally, "coherent invertibility" is just invertibility, which is what makes conditions of the form "such and such things are invertible" extremely practical.

A more precise statement could be something like: the forgetful functor $map(\mathbb S, C)\to C$ realizes an equivalence between the source and the full subgroupoid of $C^\simeq$ spanned by the objects whose class in $\pi_0(C^\simeq)$ is invertible.

In a sense, it all boils down to the following lemma:

Lemma: Let $M$ be an $\mathbb E_\infty$-monoid, and assume $\pi_0(M)$ is a group. Then $M$ is an $\mathbb E_\infty$-group.

Now, for some people, the hypothesis of the lemma is the definition of $\mathbb E_\infty$-groups, so this is only interesting if I give you another definition.

Possible ones, for which the proof of that lemma is more or less complicated are:

1- there exists a map $i: M\to M$ such that $M\xrightarrow{(i,id_M)} M\times M\xrightarrow{\mu} M$ is constant equal to the neutral element;

2- The shear map, $M\times M\xrightarrow{(\mu,pr_2)} M\times M$ ($(x,y)\mapsto (xy,y)$) is an equivalence;

3- There exists a functor $Proj_\mathbb S\to Spaces$ whose underlying $\mathbb E_\infty$-monoid is $M$;

4- There is a spectrum $m$ with $\Omega^\infty m \simeq M$, i.e. $M$ is an infinite loop space.

Because of these things, invertible elements are very well-behaved - e.g. the result from HA you mention becomes an immediate consequence of the adjoint functor theorem. The interesting part of that result is that (under Ore-type conditions, which are there in ordinary algebra too) one can actually describe the homotopy groups of $S^{-1}R$ ! (the existence of something with that universal property works for any $S$, not only nice ones)

$\endgroup$
3
  • $\begingroup$ Thanks, Maxime, this is great! $\endgroup$
    – Emily
    Apr 9, 2023 at 22:56
  • $\begingroup$ What is "a map" in the first item $i\colon M\to M$? A map of anima, $E_1$-monoids, or $E_\infty$-monoids? $\endgroup$
    – Z. M
    Apr 10, 2023 at 10:17
  • $\begingroup$ If $M$ is an $\mathbb E_{n+1}$-monoid (including $n=\infty$, where $\infty+1 = \infty$), you can make $i$ into a map of $\mathbb E_n$-monoids. In fact, you can always make it into a map of $\mathbb E_{n+1}$-monoids, but then the souce (or target) has to be tempered with, something like $M^{op}$ $\endgroup$ Apr 10, 2023 at 11:58

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.