[Paper Review] Note on the construction of globular weak omega-groupoids from types, topological spaces etc

John Bourke|arXiv (Cornell University)|Feb 25, 2016

Homotopy and Cohomology in Algebraic Topology9 references3 citations

TL;DR

This paper reinterprets van den Berg and Garner's construction of a Batanin weak ω-groupoid from a type using Grothendieck's simpler, more transparent definition of weak ω-groupoids. By recasting the construction in the language of globular theories and endomorphism structures, it demonstrates that Grothendieck weak ω-groupoids provide a more intuitive and workable framework for modeling higher-dimensional homotopical structures from types, topological spaces, and Kan complexes.

ABSTRACT

A short introduction to Grothendieck weak omega-groupoids is given. Our aim is to give evidence that, in certain contexts, this simple language is a convenient one for constructing globular weak omega-groupoids. To this end, we give a short reworking of van den Berg and Garner's construction of a Batanin weak omega-groupoid from a type using the language of Grothendieck weak omega-groupoids.

Motivation & Objective

• To demonstrate that Grothendieck's definition of weak ω-groupoids offers a more transparent and accessible framework than Batanin's original definition for constructing higher groupoids.
• To rework van den Berg and Garner's construction of a weak ω-groupoid from a type using the language of Grothendieck weak ω-groupoids instead of Batanin's weak ω-categories.
• To show that the same construction applies not only to types in intensional type theory but also to topological spaces and Kan complexes, broadening its applicability.
• To argue that endomorphism globular theories and the factorization system (L,R) provide a cleaner and more intuitive technical foundation than globular operads for such constructions.
• To provide a self-contained, expository account that makes the core ideas of the original construction more accessible through a simplified formalism.

Proposed method

• Uses Grothendieck's definition of weak ω-groupoids via globular sets equipped with contractible endomorphism theories and a factorization system (L,R) on the category of globular objects.
• Applies the path object construction iteratively in an identity type category to generate a globular object encoding higher homotopies.
• Employs the factorization system (L,R) to prove that the inclusion maps $ i_{ar{n}}: A(0) o A(ar{n}) $ are L-maps, using induction and pullback stability.
• Demonstrates that the endomorphism theory $ ext{End}(i/A) $ is contractible by showing that every parallel pair of m-cells admits a lifting via the (L,R)-factorization.
• Uses the universal property of globular sums and pullbacks to construct diagonal fillers in diagrams involving $ i_{ar{n}} $, ensuring the contractibility of the endomorphism structure.
• Relies on the fact that in identity type categories (such as those of topological spaces and Kan complexes), the path object construction yields a well-behaved globular object with the required structure.

Experimental results

Research questions

• RQ1 Can Grothendieck's definition of weak ω-groupoids provide a simpler and more transparent framework for constructing higher groupoids from types and spaces?
• RQ2 How does the construction of a weak ω-groupoid from a type via the path object iteration compare when formulated in Grothendieck's language versus Batanin's original framework?
• RQ3 What structural properties of the resulting globular object ensure that it satisfies the axioms of a Grothendieck weak ω-groupoid?
• RQ4 To what extent can the same construction be applied to topological spaces and Kan complexes, and how do they fit into the identity type category framework?
• RQ5 Why are endomorphism globular theories and the (L,R)-factorization system better suited for this construction than globular operads or Batanin's approach?

Key findings

• The construction of a weak ω-groupoid from a type, originally achieved via Batanin's weak ω-categories, can be re-expressed more transparently using Grothendieck's definition of weak ω-groupoids.
• The inclusion maps $ i_{ar{n}}: A(0) o A(ar{n}) $ are shown to be L-maps via induction and pullback stability, a key step in verifying the (L,R)-factorization structure.
• The endomorphism theory $ ext{End}(i/A) $ is contractible, which is the central condition ensuring that the resulting globular object is a Grothendieck weak ω-groupoid.
• The path object construction iteratively applied in an identity type category yields a globular object whose structure satisfies the axioms of a Grothendieck weak ω-groupoid.
• Topological spaces and Kan complexes are shown to be identity type categories, so the construction applies directly to them, extending the scope beyond type theory.
• The use of globular sums and the (L,R)-factorization system simplifies the verification of coherence conditions compared to Batanin’s operadic approach.