[Paper Review] A Universal Characterisation of Codescent Objects

Alexander S. Corner|arXiv (Cornell University)|Sep 5, 2017

Computability, Logic, AI Algorithms13 references3 citations

TL;DR

This paper introduces extrapseudonatural transformations as a 2-dimensional generalization of extranatural transformations, using them to provide a universal characterization of codescent objects in bicategories. It establishes a Fubini theorem for codescent objects by proving that iterated codescent objects are equivalent under suitable conditions, generalizing coend Fubini theorems to the higher-dimensional setting.

ABSTRACT

In this work we define a 2-dimensional analogue of extranatural transformation and use these to characterise codescent objects. They will be seen as universal objects amongst extrapseudonatural transformations in a similar manner in which coends are universal objects amongst extranatural transformations. Some composition lemmas concerning these transformations are introduced and a Fubini theorem for codescent objects is proven using the universal characterisation description.

Motivation & Objective

• To extend the concept of extranatural transformations to a 2-dimensional setting using pseudofunctors and bicategories.
• To define and formalize extrapseudonatural transformations as a foundational tool for characterizing codescent objects.
• To establish a universal property for bicodescent objects analogous to the universal property of coends via extranatural transformations.
• To prove a Fubini-type theorem for codescent objects, showing that iterated codescent constructions are equivalent under appropriate conditions.

Proposed method

• Introduce extrapseudonatural transformations as a weak 2-dimensional generalization of extranatural transformations, involving pseudofunctors and invertible 2-cells.
• Define bicodescent objects as universal objects among extrapseudonatural transformations, capturing both 1-dimensional and 2-dimensional universal properties.
• Use coherence isomorphisms and pasting diagrams to verify the axioms of extrapseudonatural transformations, particularly EB2 for invertible 2-cells.
• Construct a composition of coherence cells to relate iterated codescent objects, proving that the order of coend-like constructions can be interchanged.
• Leverage the universal property of codescent objects to show that the composite transformations satisfy the required adjoint equivalence.
• Apply the Fubini theorem to both bicodescent and codescent objects, noting that codescent objects satisfy the stronger uniqueness condition.

Experimental results

Research questions

• RQ1 How can extranatural transformations be generalized to a 2-categorical setting to describe higher-dimensional colimits?
• RQ2 What universal property characterizes codescent objects in bicategories, analogous to the coend's universal property?
• RQ3 Can a Fubini theorem be established for codescent objects, allowing interchange of iterated codescent constructions?
• RQ4 What conditions ensure that iterated codescent objects are equivalent up to adjoint equivalence?
• RQ5 How do bicodescent objects relate to codescent objects in terms of universal properties and coherence?

Key findings

• Extrapseudonatural transformations are defined as a 2-dimensional analogue of extranatural transformations, providing a framework for higher-dimensional universal constructions.
• Bicodescent objects are characterized as universal objects among extrapseudonatural transformations, generalizing the role of coends in 1-category theory.
• A Fubini theorem for codescent objects is proven, showing that ∫^a∫^b P(a,b,a,b) ≃ ∫^b∫^a P(a,b,a,b) via an adjoint equivalence.
• The equivalence is established through a composition of coherence isomorphisms and pasting diagrams that satisfy the EB2 axiom for invertible 2-cells.
• The result applies to both bicodescent and codescent objects, with codescent objects satisfying the stronger uniqueness condition.
• The proof relies on the universal property of codescent objects and the coherence of pseudofunctors, ensuring that the interchange of iterated constructions is well-defined.