[Paper Review] A note on coring extensions

Tomasz Brzeziński|ArXiv.org|Oct 1, 2004

Algebraic structures and combinatorial models9 references3 citations

TL;DR

This paper introduces the concept of a coring extension—defined as a right extension of an A-coring by a B-coring—linking it to k-additive functors between comodule categories that factor through forgetful functors. The key contribution is establishing a correspondence between coring extensions and factorisable functors, illustrated via descent data and enriched by a category of corings with morphisms as coring extensions.

ABSTRACT

A notion of a coring extension is defined and it is related to the existence of an additive functor between comodule categories that factorises through forgetful functors. This correspondence between coring extensions and factorisable functors is illustrated by functors between categories of descent data. A category in which objects are corings and morphisms are coring extensions is also introduced.

Motivation & Objective

• To define and formalize the notion of a coring extension as a generalization of algebra morphisms in the context of comodules.
• To establish a correspondence between coring extensions and k-additive functors between comodule categories that factor through forgetful functors.
• To demonstrate this correspondence via categories of descent data arising from entwining structures.
• To construct a category whose objects are corings and morphisms are coring extensions, enriching the categorical framework of corings.
• To clarify the asymmetry between left and right coring extensions, contrasting with symmetric behavior in algebras or coalgebras.

Proposed method

• Defining a coring extension as a B-coring D that makes an A-coring C a (C,D)-bicomodule with the left regular coaction ΔC.
• Using the Sweedler sigma notation to express coactions and compatibility conditions in the bicomodule structure.
• Establishing a factorisation property for functors between comodule categories via forgetful functors to Mk.
• Applying the notion of a measuring map ν:C⊗B→A to define how an A-coring measures a B-algebra.
• Constructing isomorphisms such as C□D(D⊗D E) ≅ C⊗D E to verify the structure of comodule functors.
• Utilizing the left dual ring *C = HomA(C,A) to analyze dual structures and compatibility in the comodule framework.

Experimental results

Research questions

• RQ1 How can the notion of an algebra morphism be generalized to corings using categorical factorisation?
• RQ2 What conditions ensure that a functor between comodule categories factors through forgetful functors?
• RQ3 How do coring extensions relate to descent data and entwining structures in noncommutative geometry?
• RQ4 Can a category of corings be defined where morphisms are coring extensions rather than standard coring morphisms?
• RQ5 Why does the left-right symmetry of algebra extensions not extend to coring extensions?

Key findings

• A coring extension is defined as a B-coring D such that an A-coring C is a (C,D)-bicomodule with the left regular coaction ΔC.
• Any A-coring morphism γ:C→D induces a coring extension via the right coaction (C⊗γ)∘ΔC.
• The existence of a coring extension implies that C is a right B-module and ΔC is right B-linear.
• A k-additive functor F:MC→MB factors through forgetful functors if and only if it arises from a coring extension under suitable purity conditions.
• The category of corings with morphisms as coring extensions is well-defined and provides a new categorical framework for studying comodule functors.
• The correspondence between coring extensions and factorisable functors is illustrated explicitly in the context of descent data from entwining structures.