$\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 $R$-algebrais 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 commutativeif, 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 imagetaken from Joyal–Street, p. 7.

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

A

$\tau_{\leq1}\Omega^2S^2$-graded ringis 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 commutativeif, 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.)

braidedoperads 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$2more comments