[Paper Review] Weights for Objects of Monoids

Łukasz Sienkiewicz, Marek Zawadowski|arXiv (Cornell University)|Jun 13, 2013

Homotopy and Cohomology in Algebraic Topology14 references3 citations

TL;DR

This paper introduces a unified framework for constructing monoid-related objects—such as bi-monoids and monoid actions—within 2-categories using weighted limits. By employing matrices of symmetric (possibly colored) operads to define auxiliary categories and 2-categories, it systematically encodes weights, enabling the description of complex algebraic structures as weighted limits in a structured and generalizable way.

ABSTRACT

The main objective of the paper is to dene the construction of the object of monoids, over a monoidal category object in any 2-category with nite products, as a weighted limit. To simplify the denition of the weight, we use matrices of symmetric (possibly colored) operads that dene some auxiliary categories and 2-categories. Systematic use of these matrices of operads allows us to dene several similar objects as weighted limits. We show, among others, that the constructions of the object of bi-monoids over a symmetric monoidal category object or the object of actions of monoids along an action of a monoidal category object can be also described as weighted limits.

Motivation & Objective

• To define the object of monoids over a monoidal category object in any 2-category with finite products using weighted limits.
• To simplify the construction of weights for such limits by introducing matrices of symmetric (possibly colored) operads.
• To generalize the framework to describe other algebraic structures, such as bi-monoids and monoid actions, as weighted limits.
• To provide a systematic method for encoding complex algebraic structures through operadic matrices in 2-categories.
• To unify seemingly distinct constructions in 2-category theory under the common formalism of weighted limits.

Proposed method

• The paper uses matrices of symmetric operads to define auxiliary categories and 2-categories that encode the structure of weights for limits.
• It constructs weights for weighted limits by leveraging the algebraic data encoded in these operadic matrices.
• The framework is applied to define the object of monoids as a weighted limit in a 2-category with finite products.
• The method extends to structures like bi-monoids and monoid actions by adapting the operadic matrices to capture the relevant algebraic axioms.
• The use of symmetric and possibly colored operads allows for a flexible and general encoding of algebraic operations and their coherence conditions.
• The entire construction is formalized within a 2-categorical setting, ensuring compatibility with higher-dimensional categorical reasoning.

Experimental results

Research questions

• RQ1 How can the object of monoids over a monoidal category object in a 2-category be systematically defined using weighted limits?
• RQ2 What role do matrices of symmetric operads play in simplifying the definition of weights for such limits?
• RQ3 Can the construction of bi-monoids over a symmetric monoidal category object also be expressed as a weighted limit?
• RQ4 How can monoid actions along a monoidal category action be captured using the same weighted limit framework?
• RQ5 What is the general mechanism by which operadic matrices can encode diverse algebraic structures as weighted limits in 2-categories?

Key findings

• The object of monoids over a monoidal category object in a 2-category with finite products can be rigorously defined as a weighted limit.
• Matrices of symmetric operads provide an effective and systematic way to construct the weights needed for such weighted limits.
• The framework successfully generalizes to include bi-monoids, demonstrating that their construction is also a weighted limit.
• Monoid actions along a monoidal category action are shown to admit a description as weighted limits using the proposed operadic matrix method.
• The use of operadic matrices enables a uniform treatment of diverse algebraic structures within the same categorical framework.
• The method establishes a coherent and reusable approach to defining complex monoidal and action-related structures in 2-categories.