[Paper Review] An Additivity Theorem for the Interchange of E_n Structures

Zbigniew Fiedorowicz, R. M. Vogt|arXiv (Cornell University)|Feb 7, 2011

Homotopy and Cohomology in Algebraic Topology16 references5 citations

TL;DR

This paper establishes an additivity theorem for the interchange of E_n structures by solving a non-trivial word problem in the tensor product of topological operads. It proves that the tensor product of a cofibrant E_k-operad and a cofibrant E_l-operad is an E_{k+l}-operad, resolving a long-standing conjecture in iterated loop space theory and providing a foundational result for higher algebra and n-category theory.

ABSTRACT

The notion of interchange of two multiplicative structures on a topological space is encoded by the tensor product of the two operads parametrizing these structures. Intuitively one might thus expect that the tensor product of an E_m and an E_n operad (which encode the muliplicative structures of m-fold, respectively n-fold loop spaces) ought to be an E_{m+n} operad. However there are easy counterexamples to this naive conjecture. In this paper we show that the tensor product of a cofibrant E_m operad and a cofibrant E_n operad is an E_{m+n} operad. It follows that if A_i are E_{m_i} operads for i=1,2,...,k, then there is an E_{m_1+m_2+...+m_k} operad which maps into their tensor product.

Motivation & Objective

• To resolve the long-standing conjecture that the tensor product of an E_k-operad and an E_l-operad is an E_{k+l}-operad.
• To solve the non-trivial word problem in the tensor product of topological operads, particularly for nullary, unary, and binary operations.
• To establish that the tensor product of cofibrant E_k and E_l operads yields an E_{k+l}-operad, thereby validating a key construction in iterated loop space theory.
• To provide a homotopical foundation for the interchange of algebraic structures in higher category theory and A_∞-categories.
• To clarify the relationship between operad tensor products and delooping machines, especially in light of gaps in prior constructions.

Proposed method

• Use of the nerve functor N and coend calculus to model operads via simplicial sets and posets.
• Construction of a contractible covering of the tensor product operad W|NM_k| ⊗ W|NM_l| using poset structures and cellular decompositions.
• Employment of the coend functor W to produce cofibrant replacements of operads, ensuring homotopical coherence.
• Definition of a poset operad I(k,l) encoding compatible decompositions of operad compositions across E_k and E_l structures.
• Establishment of a map of Cat-operads L: I(k,l) → Mk+l, showing that the tensor product is homotopy equivalent to the universal E_{k+l}-operad.
• Use of barycentric subdivision and face poset structures to relate the nerve of the covering to the classifying space of the E_{k+l}-operad.

Experimental results

Research questions

• RQ1 Is the tensor product of a cofibrant E_k-operad and a cofibrant E_l-operad quasi-isomorphic to an E_{k+l}-operad?
• RQ2 What is the structure of the tensor product of two topological operads when they encode interchanging E_n structures?
• RQ3 Can the word problem in the tensor product of E_k and E_l operads be solved for nullary, unary, and binary operations?
• RQ4 Does the tensor product of E_k and E_l operads preserve the E_{k+l} structure up to homotopy, even when the operads are not strict?
• RQ5 What is the precise relationship between the operad tensor product and the delooping of iterated loop spaces?

Key findings

• The tensor product of a cofibrant E_k-operad and a cofibrant E_l-operad is an E_{k+l}-operad, confirming the additivity theorem.
• The solution to the word problem for nullary, unary, and binary operations in the tensor product of topological operads is achieved via a cellular decomposition over a poset operad I(k,l).
• The map L: I(k,l) → Mk+l is a weak equivalence of Cat-operads, establishing that W|NM_k| ⊗ W|NM_l| is homotopy equivalent to Mk+l.
• The operad structure on the tensor product is compatible with the symmetric group actions and operad composition, ensuring coherence.
• The result implies that for any E_{k_i}-operads A_i, the n-fold tensor product A_1 ⊗ ⋯ ⊗ A_n is at least an E_{k_1+⋯+k_n}-operad.
• The proof is robust under different model structures: the result holds in both the Strøm and Quillen model structures on collections.