Let $$(C, \otimes, 1)$$ be a additive symmetric monoidal category with internal hom (well, lets think about $$\mathrm{Mod}_R$$ with $$R$$ a commutative ring), I am trying to figure out what would be the anticommutative operad in $$C$$. By that I mean: an operad $$P$$ such that $$P-$$algebras are the skew-commutative monoids in $$C$$.

My try:

Take $$\mathcal{A}lt(n) = 1 \circlearrowright S_n$$ with the signal representation. This seens natural just because the Schur functor would be something like an antisymmetrization

\begin{align*} S_{\mathcal{A}lt}(A) = \coprod_{n \in \mathbb{N}} 1 \square_{S_n} A^{\square n} \simeq \coprod_{n \in \mathbb{N}} \mathrm{coeq}_{g \in S_n} \left( \mathrm{sng}(g) g: A^{\square n} \to A^{\square n} \right) \end{align*}

Now the unit map $$1 \to P(1)=1$$ seems to be just identity map, fine. But what about multiplication map $$\mu(m; i_1, \cdots, i_m): \mathcal{A}lt(m) \otimes \mathcal{A}lt(i_1) \otimes \cdots \otimes \mathcal{A}lt(i_m) \to \mathcal{A}lt(i_1 + \cdots + i_m)$$ ? Well, we have a natural isomorphism $$1^{m} \otimes 1^{i_1} \otimes \cdots \otimes 1^{i_m} \simeq 1$$, I would like to use this map probably with a sign that depends on $$m, i_1, \cdots, i_m$$. Operad axioms give me restrictions to that sign:

1. Equivariance axiom gives: \begin{align} \mathrm{sng}(\sigma) \mu(m; i_1, \cdots, i_m) = \mathrm{sgn}(\sigma_{i_1, \cdots, i_m}) \mu(m; i_{\sigma^{-1}(1)}, \cdots, i_{\sigma^{-1}(m)}) \end{align} where $$\sigma \in S_m$$ and $$\sigma_{i_1, \cdots, i_m} \in S_{i_1 + \cdots + i_m}$$ is a block permutation

2. Associativity gives: \begin{align} \mu(m; j_1, \cdots, j_m) \mu(i_1; j_{1,1}, \cdots, j_{1, i_1}) \cdots \mu(i_m; j_{m,1}, \cdots, j_{m, i_m}) = \mu(i; j_{1,1}, \cdots, j_{m, i_m}) \mu (m; i_1, \cdots, i_m) \end{align} where $$i = \sum_k i_k$$, $$j_l = \sum_k j_{l, k}$$ and $$j = \sum_l j_l$$.

3. Unity gives: \begin{align} \mu(1; n) = \mathrm{id} \quad \quad \quad \quad \mu(n; 1, \cdots, 1) = \mathrm{id} \end{align}

Maybe equation (2) gives me some recursion? I really don't know. It seems I can't just put $$\mu(\cdots)$$ to be the natural isomorphism because of equation (1).

Well, with all that, a $$\mathcal{A}lt-$$algebra would be $$A\in C$$ with morphisms $$\gamma(m): \mathcal{A}lt(m) \otimes A^{\otimes m} \simeq A^{\otimes m} \to A$$ such that

i. Equivariance: Let $$g \in S_m$$ action on $$A^{\otimes m}$$ by permutation of factors, then \begin{align} \gamma(m) g = \mathrm{sgn}(g) \gamma(m) \end{align}
Nice, it is what i want.

ii. Associativity: \begin{align} \gamma(m) (\gamma(i_1) \otimes \cdots \otimes \gamma(i_m) ) = \mu(m; i_1, \cdots, i_m) \gamma(i_1 + \cdots + i_m) \end{align} It seems really weird to be that sign over there.

iii. Unity: $$m(1): 1 \times A \to A$$ is just the natural isomorphism. Ok.

Problem:

Well, equation (ii) don't make any sense to me unless all those $$\mu(\cdots)$$ are the natural isomorphisms, but I can't do that. What is going on here? Maybe if I understand better this block permutation sign things will go easier.

• Have you tried $C=\mathbb{Z}$ or $M_n(\mathbb{Z})$? Commented Mar 1, 2023 at 14:07