[Paper Review] Delooping totalization of a multiplicative operad

Victor Tourtchine|arXiv (Cornell University)|Dec 29, 2010

Advanced Topics in Algebra10 references3 citations

TL;DR

This paper establishes a homotopy equivalence between the totalization of a cosimplicial space derived from a multiplicative operad and the double loop space of derived morphisms from the associative operad to the operad itself. Using explicit cofibrant models in categories of bimodules and operads, the author constructs a geometric, homotopy-theoretic proof that deloops the totalization space twice, yielding a double loop space structure.

ABSTRACT

The paper shows that under some conditions the totalization of a cosimplicial space obtained from a multiplicative operad is a double loop space of the space of derived morphisms from the associative operad to the operad itself.

Motivation & Objective

• To establish a homotopy equivalence between the totalization of a cosimplicial space from a multiplicative operad and a double loop space of derived morphisms.
• To provide a geometric, homotopy-theoretic proof of delooping the totalization space, avoiding heavy reliance on abstract homotopy techniques.
• To construct explicit cofibrant models for the associative operad in nested categories: weak bimodules, bimodules, and operads.
• To demonstrate that the totalization space is homotopy equivalent to the double loop space of derived morphisms in the category of operads.

Proposed method

• Construct a cofibrant model $\widetilde{\triangle}$ for the associative operad in the category of weak bimodules over $\mathcal{A}_{\text{ssoc}}$.
• Build a cofibrant model $\widetilde{\square}$ for $\mathcal{A}_{\text{ssoc}}$ in the category of bimodules over $\mathcal{A}_{\text{ssoc}}$.
• Construct a cofibrant model $\widetilde{\text{\textasciigrave}}$ for $\mathcal{A}_{\text{ssoc}}$ in the category of non-$\Sigma$ operads.
• Establish a homotopy equivalence $\Omega^2 \operatorname{Operads}(\widetilde{\text{\textasciigrave}}, \mathcal{O}) \simeq \operatorname{Tot} \mathcal{O}(\bullet)$ via successive delooping steps.
• Use fiber-wise analysis of maps between totalization spaces to prove that induced maps on loop spaces are homotopy equivalences.
• Leverage the equivalence between semicosimplicial totalization and full totalization to simplify the construction by ignoring degeneracies initially.

Experimental results

Research questions

• RQ1 Is the totalization of a cosimplicial space from a multiplicative operad homotopy equivalent to a double loop space of derived morphisms from the associative operad to the operad?
• RQ2 Can the double delooping of the totalization space be constructed geometrically, without relying on advanced homotopy-theoretic machinery?
• RQ3 What explicit cofibrant models exist for the associative operad in the categories of weak bimodules, bimodules, and operads?
• RQ4 How do the inclusion of cells in the bimodule structures affect the homotopy type of the totalization space?
• RQ5 What is the precise relationship between the totalization of a multiplicative operad and the derived morphism space in the category of operads?

Key findings

• The totalization $\operatorname{Tot} \mathcal{O}(\bullet)$ of a multiplicative operad is homotopy equivalent to $\Omega^2 \widetilde{\operatorname{Operads}}(\mathcal{A}_{\text{ssoc}}, \mathcal{O})$.
• A homotopy equivalence $\Omega^2 \operatorname{Operads}(\widetilde{\text{\textasciigrave}}, \mathcal{O}) \simeq \operatorname{Tot} \mathcal{O}(\bullet)$ is constructed explicitly using cofibrant models.
• The space $\widetilde{\triangle}$, a cofibrant model of $\mathcal{A}_{\text{ssoc}}$ in $\underset{\mathcal{A}_{\text{ssoc}}}{\operatorname{Wbimod}}$, is a contractible, cofibrant cosimplicial space with infinite-dimensional $CW$-complex components.
• The map $\tilde{\zeta}_{n,i}$ induces a homotopy equivalence on fibers between loop spaces of totalization and bimodule spaces, proving the delooping step-by-step.
• The construction relies on free cell attachments in $B\widetilde{\text{\textasciigrave}}_{n-1,i}$, and the inclusion of punctured discs in the bimodule fiber analysis.
• The result is consistent with an earlier result by Dwyer and Hess, but the proof is more geometric and less reliant on homotopy-theoretic tools.