[Paper Review] Building the bicategory Span$_2(\mathcal {C})$

Franciscus Rebro|arXiv (Cornell University)|Jan 5, 2015

Homotopy and Cohomology in Algebraic Topology8 references3 citations

TL;DR

This paper constructs a bicategory Span₂(𝒞) from any category 𝒞 with pullbacks and a terminal object, where objects are 𝒞-objects, morphisms are spans in 𝒞, and 2-morphisms are isomorphism classes of spans of spans. The key contribution is proving that this structure satisfies the axioms of a bicategory, generalizing earlier constructions in cobordism and groupoid categories, and applying it to $n$-manifolds and collared cobordisms to yield a symmetric monoidal bicategory structure.

ABSTRACT

Given any category $\mathcal{C}$ with pullbacks and a terminal object, we show that the data consisting of the objects of $\mathcal{C}$, the spans of $\mathcal{C}$, and the isomorphism classes of spans of spans of $\mathcal{C}$, forms a bicategory. We denote this bicategory Span$_2(\mathcal{C})$ and show that this construction can be applied to give a bicategory of $n$-manifolds, cobordisms of those manifolds, and cobordisms of cobordisms.

Motivation & Objective

• To generalize the construction of bicategories of spans to include spans of spans, forming a higher-dimensional structure.
• To prove that Span₂(𝒞), defined using isomorphism classes of spans of spans, forms a bicategory when 𝒞 has pullbacks and a terminal object.
• To apply the construction to $n$-manifolds and cobordisms, yielding a bicategory of collared cobordisms of cobordisms.
• To lay the foundation for extending Span₂(𝒞) to a symmetric monoidal bicategory using the product structure in 𝒞.

Proposed method

• Define morphisms in Span₂(𝒞) as spans in 𝒞, with 2-morphisms as isomorphism classes of spans of spans.
• Use pullbacks to define composition of 2-morphisms, ensuring associativity via canonical isomorphisms.
• Establish identity 2-morphisms using identity spans and verify unitality via pullback properties.
• Prove coherence conditions (associativity and unit constraints) using universal properties of pullbacks and isomorphisms of spans of spans.
• Construct a functor from the category $K^n$ to the category of $n$-manifolds with corners to model cubical $σ$-manifolds and their collared cobordisms.
• Apply the construction to $n$-manifolds, showing that collared cobordisms of cobordisms form a well-defined bicategory under the Span₂(𝒞) framework.

Experimental results

Research questions

• RQ1 Can a bicategory be constructed from any category 𝒞 with pullbacks and a terminal object, using spans and spans of spans as 1- and 2-morphisms?
• RQ2 Does the composition of 2-morphisms defined via pullbacks of spans of spans satisfy the associativity and unitality constraints of a bicategory?
• RQ3 Can this construction be applied to the category of $n$-manifolds with corners to yield a bicategory of collared cobordisms of cobordisms?
• RQ4 Is the resulting bicategory Span₂(𝒞) amenable to further enrichment, such as a symmetric monoidal structure?

Key findings

• Span₂(𝒞) is a well-defined bicategory when 𝒞 has pullbacks and a terminal object, with composition of 2-morphisms given by pullbacks of spans of spans.
• Isomorphism classes of spans of spans serve as 2-morphisms, and the construction respects the universal properties of pullbacks, ensuring coherence.
• The bicategory structure is stable under isomorphism, as isomorphisms of spans of spans are identified as equal 2-morphisms.
• The construction applies to $n$-manifolds, yielding a bicategory of $(n-2)$-manifolds, $(n-1)$-dimensional collared cobordisms, and $n$-dimensional collared cobordisms of cobordisms.
• Cubical $σ$-manifolds naturally give rise to functors $K^n o σ$-Man, enabling a systematic construction of collared $n$-tuple cobordisms.
• The framework provides a foundation for extending Span₂(𝒞) to a symmetric monoidal bicategory using the product structure in 𝒞, as outlined in future work.