[Paper Review] Algebraic theories, span diagrams and commutative monoids in homotopy theory

James Cranch|arXiv (Cornell University)|Sep 7, 2011

Homotopy and Cohomology in Algebraic Topology6 references3 citations

TL;DR

This paper develops an algebraic theory of commutative monoids in the context of quasicategories, using span diagrams to model natural operations on monoids. It establishes that the resulting notion of a commutative monoid is equivalent to both Lurie's concept of a commutative algebra object and the classical $E_∞$-monoid, providing a combinatorially explicit and geometrically intuitive framework for homotopical commutative algebra.

ABSTRACT

We adapt the notion of an algebraic theory to work in the setting of quasicategories developed recently by Joyal and Lurie. We develop the general theory at some length. We study one extended example in detail: the theory of commutative monoids (which turns out to be essentially just a 2-category). This gives a straightforward, combinatorially explicit, and instructive notion of a commutative monoid. We prove that this definition is equivalent (in appropriate senses) both to the classical concept of an E-infinity monoid and to Lurie's concept of a commutative algebra object.

Motivation & Objective

• To extend the classical notion of algebraic theories to the setting of quasicategories, enabling a homotopical generalization of algebraic structures.
• To provide a combinatorially explicit and geometrically intuitive definition of a commutative monoid in homotopy theory using span diagrams.
• To establish the equivalence between this new definition and established notions: $E_∞$-monoids and Lurie's commutative algebra objects.
• To develop a foundational theory of algebraic theories in quasicategories, including free models, fibrations, and limits.

Proposed method

• Adapts the concept of algebraic theories to quasicategories using the language of $(∞,1)$-categories and Joyal-Lurie's quasicategory theory.
• Introduces the $“\mathrm{Span}\u201d$ quasicategory as a model for span diagrams between finite sets, capturing copying and addition operations.
• Uses Lawvere-style duality to interpret commutative monoids as algebras over a specific quasicategorical theory derived from the span category.
• Applies higher categorical tools such as $q$-colimits, $q$-Kan extensions, and left/right Kan extension criteria from Lurie's *Higher Topos Theory*.
• Employs the category $\mathrm{Span}^\times$ and the inclusion $\mathrm{FinSet}_* \to \mathrm{Span}$ to model monoidal structures and test algebra object conditions.
• Proves equivalence via a two-step Kan extension argument: showing existence of $q$-Kan extensions and verifying the right Kan extension condition via subcategory restriction.

Experimental results

Research questions

• RQ1 How can the classical notion of an algebraic theory be generalized to the homotopical setting of quasicategories?
• RQ2 Can span diagrams over finite sets provide a complete and combinatorially explicit presentation of natural operations on commutative monoids in homotopy theory?
• RQ3 Is the resulting notion of a commutative monoid in this framework equivalent to Lurie's definition of a commutative algebra object in a symmetric monoidal quasicategory?
• RQ4 Does this construction recover the classical $E_\infty$-space structure in the homotopical setting?
• RQ5 What is the role of $q$-colimits and $q$-Kan extensions in verifying the universal properties of these algebraic structures?

Key findings

• The category $\mathrm{Span}$ of span diagrams between finite sets forms a quasicategory that models the natural operations on commutative monoids via copying and addition.
• The theory of commutative monoids in this framework is equivalent to the classical $E_\infty$-monoid structure, as both are classified by the same quasicategorical algebraic theory.
• The construction establishes that a commutative monoid in the span quasicategory is equivalent to a commutative algebra object in the sense of Lurie, via a $q$-Kan extension argument.
• The restriction functor from the quasicategory of algebras over the span theory to the quasicategory of $\mathcal{C}^\otimes$-algebras is acyclic Kan, ensuring the existence and uniqueness of lifts.
• The key technical result is that a functor $f: \mathcal{J} \to \mathcal{C}^\times$ is a $q$-right Kan extension of its restriction to $\mathrm{FinSet}_*$ if and only if it satisfies the $q$-colimit condition on the subcategory of pointed finite sets.
• The proof relies on showing that the $q$-colimit of the restriction to $\mathcal{K}^{\prime\prime}_K$ (the subcategory of spans with isomorphisms and injections) exists and is preserved, thereby implying the existence of $q$-Kan extensions.