[Paper Review] Concrete Categories in Homotopy Type Theory

James Cranch|arXiv (Cornell University)|Nov 8, 2013

Homotopy and Cohomology in Algebraic Topology2 references3 citations

TL;DR

This paper introduces a framework for constructing genuine $(\infty,1)$-categories within homotopy type theory by leveraging the category of types as a foundational, fully coherent example. It defines $n$-concrete $(\infty,1)$-categories and establishes key structures like cocartesian fibrations, proving their type-theoretic well-behavedness via univalence and truncation, thus enabling practical category theory in HoTT.

ABSTRACT

We introduce some classes of genuine higher categories in homotopy type theory, defined as well-behaved subcategories of the category of types. We give several examples, and some techniques for showing other things are not examples. While only a small part of what is needed, it is a natural construction, and may be instructive for people seeking to provide a fully general construction.

Motivation & Objective

• To develop a practical framework for $(\infty,1)$-categories in homotopy type theory, which currently lacks general definitions.
• To demonstrate that the category of types in HoTT naturally supports higher categorical structures due to built-in coherence.
• To define and analyze $n$-concrete $(\infty,1)$-categories as subcategories of the category of types, with increasing truncation levels.
• To investigate the behavior of cocartesian fibrations in this setting, proving their type-theoretic well-definedness.
• To lay groundwork for future formalization and extension to exotic homotopy type theories.

Proposed method

• Uses the category of types in homotopy type theory as a foundational, fully coherent $(\infty,1)$-category due to definitional equality of function composition.
• Defines $n$-concrete $(\infty,1)$-categories as subcategories of the category of types, with objects and morphisms satisfying $n$-truncation conditions.
• Introduces arrowlike categories as categories with two disjoint object types $A$ and $B$, and morphisms only from $A$ to $B$.
• Defines cocartesian morphisms in arrowlike categories as those inducing equivalences on hom-types, ensuring universal lifting properties.
• Applies the univalence axiom to prove that the type of cocartesian morphisms out of a given object is a proposition, ensuring uniqueness up to equivalence.
• Uses dependent type theory with path types and truncation levels to formalize coherence and equivalence conditions.

Experimental results

Research questions

• RQ1 Can meaningful $(\infty,1)$-categories be constructed in homotopy type theory despite the lack of a general definition?
• RQ2 How can coherence issues in higher category definitions be resolved using the built-in structure of the category of types?
• RQ3 What conditions on truncation levels allow for well-behaved constructions like cocartesian fibrations in HoTT?
• RQ4 Is the type of cocartesian morphisms out of a given object a proposition, ensuring uniqueness up to equivalence?
• RQ5 Can this framework be extended to other homotopy type theories beyond the standard one based on spaces?

Key findings

• The category of types in homotopy type theory is a fully coherent $(\infty,1)$-category due to definitional equality of function composition, eliminating higher coherence data.
• The type of proofs that a morphism is cocartesian is a proposition, ensuring that cocartesian morphisms are unique up to equivalence.
• In a univalent $n$-concrete $(\infty,1)$-category, the type of cocartesian morphisms out of any object is a proposition, due to univalence and the equivalence of objects.
• The type of proofs that an arrowlike category is a cocartesian fibration is a proposition, confirming the well-definedness of the concept.
• Cocartesian fibrations in arrowlike categories are well-behaved and can be characterized via equivalence-inducing composition maps.
• The framework is formalized in Agda using the HoTT library, demonstrating practical implementability despite limitations in forming functor categories.