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$\newcommand{\la}[1]{\kern-1.5ex\leftarrow\phantom{}\kern-1.5ex}\newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex}\newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\phantom{}\kern-1.5ex} \newcommand{\ua}[1]{\bigg\uparrow\raise.5ex\rlap{\scriptstyle#1}}\newcommand{\da}[1]{\bigg\downarrow\raise.5ex\rlap{\scriptstyle#1}} $There's a commutative diagram $$ \begin{array}{c} \mathbb{N}_{\mathsf{disc}} & \ra{} & \mathbb{Z}_{\mathsf{disc}}\\ \da{}& & \da{}\\ \mathbb{B} & \ra{} & \tau_{\leq1}\Omega^2S^2\\ \da{}& & \da{}\\ \mathbb{F} & \ra{} & \tau_{\leq1}\mathbb{S}\\\end{array} $$ of monoidal categories and braided strong monoidal functors where going down a row increases commutativity (nothing $\to$ braided $\to$ symmetric) and going right adds inverses (monoidal categories $\to$ $2$-groups). It involves the following categories:

  • $\mathbb{N}_{\mathsf{disc}}$, the free monoidal category on $\mathsf{pt}$, which is the discrete monoidal category on $\mathbb{N}$;
  • $\mathbb{Z}_{\mathsf{disc}}$, the free $2$-group on $\mathsf{pt}$, which is the discrete monoidal category on $\mathbb{Z}$;
  • $\mathbb{B}$, the free braided monoidal category on $\mathsf{pt}$, which is the braid category;
  • $\tau_{\leq1}\Omega^2S^2$, the free braided $2$-group on $\mathsf{pt}$, is described below$^\dagger$;
  • $\mathbb{F}$, the free symmetric monoidal category on $\mathsf{pt}$, which is the groupoid of finite sets and permutations;
  • $\tau_{\leq1}\mathbb{S}$, the free symmetric $2$-group on $\mathsf{pt}$, which is the $1$-truncation of the sphere spectrum.

The functors involved are the following ones:

  • The functors involving $\mathbb{N}_{\mathsf{disc}}$ or $\mathbb{Z}_{\mathsf{disc}}$ are either the identity on objects or the inclusion $\mathbb{N}\hookrightarrow\mathbb{Z}$; on morphisms they are the inclusions of the trivial group or the identity;
  • The functors $\mathbb{B}\to\tau_{\leq1}\Omega^2S^2$ and $\mathbb{F}\to\tau_{\leq1}\mathbb{S}$ are the inclusions $\mathbb{N}\hookrightarrow\mathbb{Z}$ on objects and the abelianisation map on $\mathrm{Hom}$-sets (i.e. $\mathrm{B}_{n}\to\mathbb{Z}$ and $\mathrm{sgn}\colon\Sigma_n\to\mathbb{Z}_2$);
  • The functors $\mathbb{B}\to\mathbb{F}$ and $\tau_{\leq1}\Omega^2S^2\to\tau_{\leq1}\mathbb{S}$ are the identity on objects and (respectively) the underlying permutation on a braid map $\mathrm{B}_n\to\Sigma_n$ and the mod 2 map $\mathbb{Z}\to\mathbb{Z}_2$.

To each of these categories, one has a notion of graded algebra, defined as a lax monoidal functor from them to $(\mathsf{Mod}_R,\otimes_{R},R)$. Moreover, precomposition and (the $1$-categorical analogue of) operadic left Kan extensions give us a number of change of grading functors, giving us a diagram of adjunctions

Here:

  • $\mathbb{N}$- and $\mathbb{Z}$-graded algebras are already famous, not requiring an explanation;
  • $\mathbb{F}$-graded algebras and commutative $\mathbb{F}$-graded algebras are known in the literature as "twisted associative algebras" and "twisted commutative algebras" respectively. They are roughly $\mathbb{N}$-graded algebras whose $n$th graded piece carries an action of the symmetric group playing well with block permutation and multiplication; see here, here, or here.
  • $\tau_{\leq1}\mathbb{S}$-graded commutative algebras include things like $\mathbb{Z}$-graded commutative algebras (i.e. those satisfying $ab=(-1)^{\deg(a)\deg(b)}ba$), and in general are commutative in even degrees and otherwise are so up to an automorphism. I have asked if there's a name for them (along with some mathematical questions) here.
  • A $\mathbb{B}$-graded algebra is defined as follows:

A $\mathbb{B}$-graded $R$-algebra is a pair $(R_{\bullet},\{\sigma_{n}\}_{n\in\mathbb{N}})$ with

  • $R_{\bullet}$ an $\mathbb{N}$-graded algebra;
  • $\sigma_n\colon\mathrm{B}_{n}\to\mathrm{End}(R_n)$ an action of the $n$th braid group $\mathrm{B}_n$ on $R_n$;

such that multiplication plays well with braid addition (the morphisms $\oplus\colon\mathrm{B}_n\times\mathrm{B}_m\to\mathrm{B}_{n+m}$) in that $$\sigma_{n+m,\tau_1\oplus\tau_2}(ab)=\sigma_{n,\tau_1}(a)\sigma_{m,\tau_2}(b)$$ for each $n,m\in\mathbb{N}$, each $a\in R_n$, each $b\in R_m$, each $\tau_1\in\mathrm{B}_{n}$ and each $\tau_2\in\mathrm{B}_{m}$.

Moreover, such a $\mathbb{B}$-graded $R$-algebra is $\mathbb{B}$-graded commutative if, for each $a\in R_n$ and each $b\in R_m$, we have $$ab=\sigma_{n+m,\tau}(ba),$$ where $\tau$ is the braid move pictured (when $(n,m)=(5,4)$) as in the image

taken from Joyal–Street, p. 7.

  • A $\tau_{\leq1}\Omega^2S^2$-graded algebra is defined as follows:

A $\tau_{\leq1}\Omega^2S^2$-graded ring is a pair $(R_{\bullet},\{\sigma_{k}\}_{k\in\mathbb{Z}})$ with

  • $R_{\bullet}$ a $\mathbb{Z}$-graded algebra (corresponding to $\pi_0(\Omega^2 S^2)\cong\pi_2(S^2)\cong\mathbb{Z}$);
  • $\sigma_k\colon\mathbb{Z}\to\mathrm{End}(R_k)$ a $\mathbb{Z}$-action on $R_k$ (corresponding to $\pi_1(\Omega^2 S^2)\cong\pi_3(S^2)\cong\mathbb{Z}$);

such that multiplication plays well with integer addition in that $$\sigma_{k+\ell,k_1+k_2}(ab)=\sigma_{k,k_1}(a)\sigma_{\ell,k_2}(b)$$ for each $k,\ell\in\mathbb{Z}$, each $a\in R_k$, each $b\in R_\ell$, and each $k_1,k_2\in\mathbb{B}$.

Moreover, such a $\tau_{\leq1}\Omega^2S^2$-graded $R$-algebra is $\tau_{\leq1}\Omega^2S^2$-graded commutative if, for each $a\in R_k$ and each $b\in R_\ell$, we have $$ab=\sigma_{k+\ell,k\ell}(ba).$$

Question. Have the notions of $\mathbb{B}$- and $\tau_{\leq1}\Omega^2S^2$-graded (commutative or not) algebras been studied before, as is the case of $\mathbb{F}$-graded algebras (commutative or not)?


$^\dagger$The braided $2$-group $\tau_{\leq1}\Omega^2S^2$ is the fundamental groupoid of the double loop space of the $2$-sphere (given equivalently by the double de-delooping $\mathbf{B}^{-2}\Pi_{\leq3}(S^2)$ of the fundamental trigroupoid of the $2$-sphere), and explicitly it is equivalent to the category where

  • We have $\mathrm{Obj}(\tau_{\leq1}\Omega^2S^2)=\mathbb{Z}$, coming from $\pi_{0}(\Omega^2 S^2)\cong\pi_2(S^2)\cong\mathbb{Z}$;
  • For each $k,\ell\in\mathrm{Obj}(\tau_{\leq1}\Omega^2S^2)$ we have $$\mathrm{Hom}_{\tau_{\leq1}\Omega^2S^2}(k,\ell)\overset{\mathrm{def}}{=}\begin{cases}\mathbb{Z}&\text{if $k=\ell$,}\\\emptyset&\text{otherwise,}\end{cases}$$ coming from $\pi_{1}(\Omega^2 S^2)\cong\pi_{3}(S^2)\cong\mathbb{Z}$;

together with the strict monoidal structure where

  • The tensor product of $\tau_{\leq1}\Omega^2S^2$ is given by integer addition on objects and hom-sets;
  • The monoidal unit is given by $0\in\mathrm{Obj}(\tau_{\leq1}\Omega^2S^2)$;
  • The braiding $$\beta^{\tau_{\leq1}\Omega^2S^2}_{k,\ell}\colon k+\ell \to \ell+k$$ of $\tau_{\leq1}\Omega^2S^2$ at $k,\ell\in\mathrm{Obj}(\mathbb{B'})$ is given by the element $k\ell$ of $\mathrm{Hom}_{\tau_{\leq1}\Omega^2S^2}(k+\ell,k+\ell)$.

(P.S. A general explanation for the appearance of homotopy groups of spheres here is the following: the free $\mathbb{E}_{\infty}$-group on the point in $\infty\text{-}\mathsf{Grpd}$ is the sphere spectrum $\mathbb{S}$, and considering instead the free $\mathbb{E}_{\infty}$-group on the punctual $k$-category in $k\text{-}\mathsf{Cats}$ gives $\tau_{\leq k}\mathbb{S}$. Rings graded by these carry a $\mathbb{Z}$-grading together with actions by the first $k$ stable homotopy groups of spheres, with the exception of $\pi_0(\mathbb{S})\cong\mathbb{Z}$. Similarly, we may consider the free $\mathbb{E}_{n}$-group on the point in $\infty\text{-}\mathsf{Grpd}$, given by $\Omega^n S^n$. Doing the same in $k$-categories again gives the $k$-truncation. This time however $\tau_{\leq k}\Omega^n S^n$-graded rings carry a $\mathbb{Z}$-grading corresponding to $\pi_{0}(\Omega^n S^n)\cong\pi_n(S^n)\cong\mathbb{Z}$ and actions by the groups $\pi_{1}(\Omega^n S^n)$, $\ldots$, $\pi_{k}(\Omega^n S^n)$. These are the unstable homotopy groups of the spheres in the range $\pi_{n+1}(S^n)$, $\ldots$, $\pi_{n+k}(S^n)$! This is in particular the case for $\tau_{\leq1}\Omega^2S^2$ above, which is the free $\mathbb{E}_2$-group on the punctual category, making it into the free braided $2$-group.)

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  • $\begingroup$ By the way, I really owe John Baez many, many thanks for explaining very paciently $\mathbb{B}'$ to me here! Thank you again, John! :) $\endgroup$
    – Théo
    Sep 8, 2021 at 3:35
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    $\begingroup$ I appreciate the Proustian title :) $\endgroup$ Sep 8, 2021 at 15:02
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    $\begingroup$ @SamHopkins It was additionally partly inspired by the first comment here. It seems from personal experience that reading the comment in context has the unexpected property of turning one's day into a better/happier one :) $\endgroup$
    – Théo
    Sep 8, 2021 at 15:24
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    $\begingroup$ @LSpice Thanks! The second case was actually intentional: I first wrote it pointing to Ivan's comment directly, but thought it had a chance of not being as funny without knowing first that varkor's question was a reference request :) $\endgroup$
    – Théo
    Sep 10, 2021 at 3:56
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    $\begingroup$ Tamarkin did consider braided operads in his well known paper on formality of $D_2$. Essentially, one replaces (as you did) the symmetric group action $S_n$ on $\mathcal P(n)$ by an action of $B_n$. One thus gets an operad by quotieting out pure braids, i.e if $\mathcal P$ is a braid operad, then $\mathcal P/PB$ is a usual symmetric one. $\endgroup$
    – Pedro
    Oct 19, 2021 at 13:46

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