HoTT Models: CCC Vs Quillen Model Categories Explained

Let's dive into the fascinating world of Homotopy Type Theory (HoTT) and its topological models, specifically focusing on the comparison between Cartesian Closed Categories (CCCs) and Quillen Model Categories. If you're anything like me, you've probably spent hours wrestling with the intricacies of HoTT, trying to wrap your head around how it all connects to our familiar topological spaces. I recently delved into the Awodey-Warren paper, Homotopy Theoretic Models of Identity Types, and I'm eager to share some insights and questions that arose during my exploration.

Understanding the Landscape

Before we get into the nitty-gritty, let's set the stage. HoTT, at its core, is a foundational system that treats types as spaces and terms as points within those spaces. Identity types, a crucial component of HoTT, capture the notion of equality between terms in a very refined way, allowing us to reason about paths and higher-dimensional structures within types. To make this more concrete, we turn to models, which provide interpretations of HoTT within established mathematical frameworks.

Two prominent contenders for modeling HoTT are CCCs and Quillen Model Categories. CCCs, with their function spaces and cartesian product structures, offer a natural setting for interpreting types and terms. On the other hand, Quillen Model Categories, equipped with notions of weak equivalences, fibrations, and cofibrations, provide a powerful framework for homotopy theory, allowing us to work with spaces up to homotopy. The central question is: how do these two approaches relate, and what are their respective strengths and weaknesses when it comes to modeling HoTT?

Cartesian Closed Categories (CCCs) and HoTT

Cartesian Closed Categories (CCCs) provide a very intuitive model for Homotopy Type Theory. In a CCC, we can interpret types as objects and terms as morphisms. The crucial aspect is the existence of function spaces, which allow us to represent functions between types as objects themselves. This is essential for interpreting dependent types and the identity types in HoTT. Specifically, the internal hom, denoted as [A, B], represents the type of functions from A to B. The evaluation map, eval: [A, B] × A → B, allows us to apply these functions to arguments.

In the context of HoTT, identity types can be modeled using path objects in CCCs. Given an object A, the path object PA represents the space of paths within A. The identity type between two terms x, y: A is then interpreted as the space of paths from x to y within A. This interpretation captures the intuition that equality in HoTT is not just a yes/no proposition but rather a structure with higher-dimensional information, reflecting the paths connecting the two terms. A significant advantage of using CCCs is their relative simplicity and the directness of the interpretation. Many constructions in HoTT translate very naturally into categorical constructions in CCCs, making it easier to reason about the semantics of type theory.

However, CCCs also have limitations. While they provide a good setting for interpreting the basic type-theoretic constructions, they may not fully capture the homotopy-theoretic aspects of HoTT, especially when dealing with more complex homotopy-theoretic structures. This is where Quillen Model Categories come into play.

Quillen Model Categories and HoTT

Quillen Model Categories offer a more sophisticated framework for modeling HoTT, emphasizing the homotopy-theoretic aspects. A Quillen Model Category is a category equipped with three classes of morphisms: weak equivalences, fibrations, and cofibrations, satisfying certain axioms. These structures allow us to do homotopy theory in a very general setting, abstracting away from specific topological spaces. The weak equivalences define which morphisms should be considered "homotopy equivalences," the fibrations capture the notion of "nice" projections, and the cofibrations are their duals.

In the context of HoTT, Quillen Model Categories allow us to model types as objects and terms as morphisms, but with a focus on homotopy invariance. The identity types are interpreted using the fibrations and path objects in the model category. A key advantage of using Quillen Model Categories is that they provide a rich set of tools for studying homotopy theory. We can use techniques like homotopy limits and colimits, derived functors, and spectral sequences to analyze the structure of types and their identity types. This is particularly useful when dealing with higher inductive types and other advanced features of HoTT.

However, working with Quillen Model Categories can be more challenging than working with CCCs. The axioms of a model category can be intricate, and constructing a model category structure on a given category often requires significant effort. Furthermore, the interpretation of HoTT in a Quillen Model Category may not be as direct as in a CCC, requiring a deeper understanding of homotopy theory.

CCC vs Quillen: A Detailed Comparison

Let's delve deeper into comparing these two models. CCCs are generally easier to work with for basic interpretations of HoTT. The direct correspondence between type-theoretic constructions and categorical constructions simplifies reasoning. For example, the function type A -> B is directly represented by the internal hom [A, B], and application is modeled by the evaluation morphism. However, CCCs might fall short when dealing with more advanced homotopy-theoretic concepts inherent in HoTT.

On the other hand, Quillen Model Categories provide a richer homotopy theory. The notions of weak equivalences, fibrations, and cofibrations allow for a more nuanced understanding of homotopy. They are better suited for modeling higher inductive types and other constructions that rely heavily on homotopy invariance. For example, the interval type, which is crucial for defining paths and homotopies, can be naturally modeled using the interval object in a Quillen Model Category. However, the increased power comes with increased complexity. Setting up a Quillen Model Category structure and interpreting HoTT within it can be significantly more challenging.

Another key difference lies in how identity types are treated. In a CCC, identity types are typically modeled using path objects, which represent the space of paths between two terms. In a Quillen Model Category, identity types are often modeled using fibrations. The fibrations capture the idea that the identity type is a "nice" projection, reflecting the homotopy-theoretic properties of equality. This difference in approach highlights the different perspectives offered by CCCs and Quillen Model Categories.

Specific Questions and Considerations

After reading the Awodey-Warren paper, I'm pondering a few specific points. One key question is about the relationship between univalence and model structures. Univalence, a central axiom in HoTT, states that the identity type between two types is equivalent to the type of equivalences between them. How does the choice of model category affect the interpretation and validation of the univalence axiom? Are there specific model categories that make it easier to prove univalence?

Another area of interest is the modeling of higher inductive types. Higher inductive types are types defined by constructors that introduce not only points but also paths and higher-dimensional structures. How do CCCs and Quillen Model Categories compare in their ability to model these types? Do Quillen Model Categories offer a more natural or powerful framework for dealing with higher inductive types?

Finally, I'm curious about the computational aspects of these models. Can we use CCCs or Quillen Model Categories to develop computational tools for working with HoTT? Are there implementations of HoTT based on these models, and how do they perform in practice? Understanding the computational implications of these models is crucial for making HoTT a practical tool for formalizing mathematics and developing software.

Conclusion

In summary, both CCCs and Quillen Model Categories offer valuable perspectives on modeling HoTT. CCCs provide a straightforward and intuitive interpretation, while Quillen Model Categories offer a richer homotopy theory. The choice between the two depends on the specific application and the desired level of sophistication. As we continue to explore the landscape of HoTT, understanding the strengths and weaknesses of these models will be crucial for unlocking the full potential of this exciting field. So, keep experimenting, keep questioning, and let's continue this journey together!