[Paper Review] Operads as polynomial 2-monads

Mark Weber|arXiv (Cornell University)|Dec 24, 2014

Advanced Topics in Algebra31 references14 citations

TL;DR

This paper introduces a new construction of a polynomial 2-monad from any operad, where the algebras of the 2-monad correspond to categorified algebras of the operad rather than strict algebras. Unlike the standard monad construction, this approach works without requiring Σ-freeness and establishes a canonical framework unifying operads, Cat-operads, and clubs via polynomial 2-monads, with a Quillen equivalence between strict algebras of the new and standard 2-monads when the operad is Σ-free.

ABSTRACT

In this article we give a construction of a polynomial 2-monad from an operad and describe the algebras of the 2-monads which then arise. This construction is different from the standard construction of a monad from an operad in that the algebras of our associated 2-monad are the categorified algebras of the original operad. Moreover it enables us to characterise operads as categorical polynomial monads in a canonical way. This point of view reveals categorical polynomial monads as a unifying environment for operads, Cat-operads and clubs. We recover the standard construction of a monad from an operad in a 2-categorical way from our associated 2-monad as a coidentifier of 2-monads, and understand the algebras of both as weak morphisms of operads into a Cat-operad of categories. Algebras of operads within general symmetric monoidal categories arise from our new associated 2-monad in a canonical way. When the operad is sigma-free, we establish a Quillen equivalence, with respect to the model structures on algebras of 2-monads found by Lack, between the strict algebras of our associated 2-monad, and those of the standard one.

Motivation & Objective

• To provide a new construction of a polynomial 2-monad from any operad, independent of Σ-freeness.
• To characterize the algebras of this 2-monad as categorified algebras of the original operad.
• To unify operads, Cat-operads, and clubs within the framework of categorical polynomial monads.
• To recover the standard monad construction from the new 2-monad via coidentification in a 2-categorical setting.
• To establish a Quillen equivalence between strict algebras of the new 2-monad and those of the standard 2-monad when the operad is Σ-free.

Proposed method

• Construct a polynomial 2-monad on Cat/I from an operad T using a 2-categorical universal property, leveraging polynomial functors in Cat.
• Define the 2-monad via a diagram of functors I ← E → B → I, where the middle functor is exponentiable, forming a polynomial in the bicategory Poly_Cat.
• Use coidentification of 2-natural transformations to define the monad structure, ensuring compatibility with fibrations and opfibrations via the familial and opfamilial properties of the structure functors.
• Show that the algebras of the new 2-monad correspond to weak morphisms of operads into a Cat-operad of categories, generalizing algebras in symmetric monoidal categories.
• Establish a Quillen equivalence between the model categories of strict algebras of the new 2-monad and the standard 2-monad when T is Σ-free, using Lack’s model structures on algebras of 2-monads.
• Apply the construction to recover the PROP of an operad via universal properties in 2-dimensional monad theory, as developed in companion works.

Experimental results

Research questions

• RQ1 How can a polynomial 2-monad be constructed from an operad in a way that captures categorified algebras rather than strict ones?
• RQ2 In what sense does this new 2-monad unify operads, Cat-operads, and clubs within the framework of polynomial monads?
• RQ3 How is the standard monad construction from an operad recovered from this new 2-monad via 2-categorical coidentification?
• RQ4 What is the relationship between the strict algebras of the new 2-monad and those of the standard 2-monad when the operad is Σ-free?
• RQ5 How can algebras of an operad in a symmetric monoidal category be canonically described using this new 2-monad?

Key findings

• The algebras of the new 2-monad associated to an operad T are precisely the categorified algebras of T, such as symmetric monoidal categories when T is the terminal operad Com.
• For the terminal operad Com, a strict algebra of the new 2-monad is a symmetric strict monoidal category, while a strict algebra of the standard 2-monad T/Σ is a commutative monoid in Cat.
• When the operad T is Σ-free, there exists a Quillen equivalence between the model categories of strict algebras of the new 2-monad and the standard 2-monad T/Σ.
• The new 2-monad construction recovers the standard monad T/Σ as a coidentifier of 2-monads in a 2-categorical setting, providing a new perspective on the classical construction.
• Algebras of an operad in any symmetric monoidal category arise canonically from the new 2-monad via weak morphisms into a Cat-operad of categories.
• The construction establishes that operads, Cat-operads, and clubs are all naturally embedded as categorical polynomial monads, revealing a unifying framework.