[Paper Review] Group operads as crossed interval groups

Jun Yoshida|arXiv (Cornell University)|Jun 8, 2018

Homotopy and Cohomology in Algebraic Topology13 references4 citations

TL;DR

This paper establishes a fully faithful functor from the category of group operads to the category of crossed interval groups, proving that group operads are precisely the tame and operadic crossed interval groups over the symmetric group operad. The key contribution is a categorical equivalence that embeds group operads as a reflective subcategory within crossed interval groups, unifying two frameworks for encoding symmetries in algebraic topology and homotopy theory.

ABSTRACT

The goal of the paper is to establish and to investigate a fully faithful embedding of the category of group operads into that of crossed interval groups. For this, we introduce a monoidal structure on the slice of the category of operads over the operad of symmetric groups. Comparing with the monoidal structure on the category of interval sets discussed in the author's previous work, we obtain a monoidal functor connecting these two categories. It will be shown that this actually induces a fully faithful functor on monoid objects and does not change the underlying sets, so we obtain a required embedding. The conditions for crossed interval groups to belong to the essential image will be proposed; namely in terms of commutativity of certain elements. As a result, it will turn out that the group operads form a reflective subcategory of the category of crossed interval groups. Finally, we will discuss monoid objects in symmetric monoidal category and Hochschild homologies on them.

Motivation & Objective

• To unify two frameworks for encoding symmetries in operads: group operads and crossed interval groups.
• To establish a categorical equivalence between group operads and a specific class of crossed interval groups.
• To show that group operads correspond exactly to crossed interval groups that are both operadic and tame over the symmetric group operad.
• To construct a fully faithful functor from group operads to crossed interval groups, enabling a reflective subcategory structure.
• To provide an explicit description of the essential image of this embedding, clarifying the structural conditions that define group operads within the broader context of crossed interval groups.

Proposed method

• Construct a monoidal structure $\rtimes$ on the slice category $\mathbf{Op}^{/\mathfrak{S}}$ of operads over the symmetric group operad $\mathfrak{S}$, enabling the encoding of group structures as monoid objects.
• Define a functor $\Psi: (\mathbf{Op}^{/\mathfrak{S}})^{*/} \to \mathbf{Set}^{/\mathfrak{S}}_{\nabla}$ that is strictly monoidal, inducing a functor between categories of monoid objects.
• Use the strict monoidality of $\Psi$ to lift the monoid structure of group operads to the category of crossed interval groups, yielding a fully faithful functor $\widehat{\Psi}: \mathbf{GrpOp} \to \mathbf{CrsGrp}^{/\mathfrak{S}}_{\nabla}$.
• Characterize the essential image of $\widehat{\Psi}$ by identifying two conditions: operadicity (commutativity of certain elements) and tameness (a finiteness and compatibility condition).
• Apply the adjoint functor theorem to show $\widehat{\Psi}$ admits a left adjoint, confirming $\mathbf{GrpOp}$ is a reflective subcategory of $\mathbf{CrsGrp}^{/\mathfrak{S}}_{\nabla}$.
• Leverage base change along the functor $\mathfrak{J}: \widetilde{\Delta} \to \nabla$ to relate Hochschild chain complexes to the constructed crossed interval groups, particularly in the case of braid group operads.

Experimental results

Research questions

• RQ1 How can group operads be embedded into the category of crossed interval groups in a fully faithful and structure-preserving way?
• RQ2 What conditions on a crossed interval group over $\mathfrak{S}$ are necessary and sufficient for it to arise from a group operad?
• RQ3 Is there a categorical equivalence between group operads and a specific subclass of crossed interval groups, and if so, what is the nature of this equivalence?
• RQ4 Can the Hochschild chain complex of a monoid in a $\mathcal{G}$-symmetric monoidal category be reconstructed from the associated crossed interval group structure?
• RQ5 What is the role of the augmented crossed simplicial group $\mathbfcal Z$ in connecting group operads to Hochschild homology via the functor $\mathfrak{J}^\natural\mathcal{G}$?

Key findings

• The category of group operads $\mathbf{GrpOp}$ fully faithfully embeds into the category of crossed interval groups over $\mathfrak{S}$, denoted $\mathbf{CrsGrp}^{/\mathfrak{S}}_{\nabla}$, via a functor $\widehat{\Psi}$.
• Group operads are characterized as precisely those crossed interval groups that are both operadic and tame over $\mathfrak{S}$, providing a structural classification.
• The embedding $\widehat{\Psi}$ is fully faithful and admits a left adjoint, making $\mathbf{GrpOp}$ a reflective subcategory of $\mathbf{CrsGrp}^{/\mathfrak{S}}_{\nabla}$.
• The Hochschild chain complex of a monoid in a $\mathcal{G}$-symmetric monoidal abelian category is isomorphic to the chain complex associated to the restriction of the paracyclic object $A^{\circledcirc}_\bullet$ to $\Delta^{\mathrm{op}}$, when $\mathcal{G}$ is a group operad.
• For the braid group operad $\mathcal{B}$, there exists a canonical pullback square involving $\mathbfcal Z$, $\mathfrak{J}^\natural\mathcal{B}$, and $\mathfrak{J}^\natural\mathfrak{S}$, which ensures a canonical Hochschild chain construction.
• Maps from the augmented crossed simplicial group $\mathbfcal Z$ to $\mathfrak{J}^\natural\mathcal{G}$ correspond bijectively to maps of augmented crossed simplicial groups, ensuring the Hochschild complex is well-defined and independent of choices in the connected case.