Is there an operad homotopifying the Koszul rule?

In homotopy theory one has the idea of a homotopy-commutative multiplication, in which one replaces the relation $$ab=ba$$ in a commutative monoid/group/ring/etc. for an unspecified homotopy. One then notices that with such a notion there is no way to relate the multiple homotopies involving $$2$$-fold multiplications, with terms such as $$abc$$ and $$cba$$. This provides the motivation for passing to the setting of homotopy-coherent structures, where one considers these homotopies as part of the data.

Formalising this story, one obtains the notion of an $$\mathbb{E}_{k}$$-operad, forming an infinite sequence of operads $$\mathbb{E}_1$$, $$\mathbb{E}_2$$, $$\ldots$$, $$\mathbb{E}_{\infty}$$. The algebras over these, called $$\mathbb{E}_{k}$$-algebras, then provide a notion of a multiplicative structure which is "$$k$$-times commutative". For example, the $$\mathbb{E}_{k}$$-algebras in sets are monoids for $$k=1$$, and then commutative monoids for $$k=2$$, at which step the process stabilises, and we get commutative monoids again for any $$k\geq2$$. Similarly, $$\mathbb{E}_1$$-, $$\mathbb{E}_2$$-, and $$\mathbb{E}_{\geq 3}$$-algebras in categories are given by monoidal categories, braided monoidal categories, and symmetric monoidal categories, after which we have stabilisation again.

However, some of the algebraic structures one finds in practice are often noncommutative, but still satisfy some special conditions that are close to ordinary commutativity. One of these structures is given by $$\mathbb{Z}$$-graded-commutative algebras, in which one instead considers the relation $$ab=(-1)^{\deg(a)\deg(b)}ba,$$ often called the Koszul sign rule. A primordial example of such an algebraic structure is given by the exterior algebra $$\bigwedge^\bullet_R(M)$$ of an $$R$$-module $$M$$.

Question. Are there analogues of $$\mathbb{E}_{k}$$-operads that are to these as the relation $$ab=(-1)^{\deg(a)\deg(b)}ba$$ is to $$ab=ba$$?

• typically $E_k$ operads in a linear context would already be in the ground category of graded vector spaces with the graded tensor product (i.e. the Koszul rule). Sep 9, 2021 at 1:51
• If $\mathrm{Mod}_R$ means ungraded modules then I don't think you will find an operad whose algebras come with any direct sum decomposition. Sep 10, 2021 at 7:31
• If $\mathrm{Mod}_R$ is $\mathbb{Z}$-graded modules with the naive symmetrizer then you will not be able to encode that coefficient operadically. Sep 10, 2021 at 7:35
• One more restricted category that can be encoded operadically in ungraded $R$-modules is the category of $R$-algebras with $\mathbb{Z}$-grading constrained to be between fixed finite $p$ and $q$. Give your operad unary operators "project to degree n" which are idempotent, mutually annihilatory and sum to the identity. Then you can encode your commutativity conditions degree by degree. This doesn't work for unrestricted $R$-algebras in ordinary operad theory because the "sum to the identity" relation is an infinite sum. Sep 11, 2021 at 1:26
• @GabrielC.Drummond-Cole Oh, this is really cool! So the $\mathbb{Z}$-graded and $\mathbb{Z}/2$-graded cases are fundamentally different (at least in this sense)!
– Théo
Sep 15, 2021 at 3:57

These kinds of things come up naturally.

For example, think about operads acting on spaces that themselves have the action of a (Lie) group, i.e. topological operads that act on spaces with $$G$$-actions. For example, the symmetry axiom of such an operad would involve not just an action of the symmetric group but a wreath product of the symmetric group with the group $$G$$.

There is a natural generalization of the cubes operads to this context as well. I don't have any particular name for them. I suppose semi-direct (or wreath product) of the cubes operad with the group $$G$$ would be appropriate.

My splicing operad is an example of an operad of this form, and it naturally contains the semi-direct product of the cubes operad with an orthogonal group. You can get any orthogonal group (and cubes operad), depending on which type of embedding space you want this operad to act on.

https://arxiv.org/abs/1004.3908

• Thank you, this looks really interesting!
– Théo
Sep 9, 2021 at 19:51