Covariant & Contravariant Homotopy Theories (1908.06879v1)

Abstract: Given a locally presentable category together with a suitable functorial cylinder object, we construct model structures which are sensitive to the `direction' of the cylinder. We show that the Covariant and Contravariant model structures on simplicial sets as well as the coCartesian and Cartesian model structures on marked simplicial sets are examples of our formalism. In this setting, notions of final and initial maps and smooth and proper maps arise very naturally and we will identify these maps in the examples.

• The paper develops covariant and contravariant model structures through the use of functorial cylinder objects in locally presentable categories.
• It employs anodyne extensions and weak factorization systems to formalize right and left homotopical structures, ensuring properness under pullbacks.
• Concrete examples with simplicial and marked simplicial sets illustrate the framework's applicability in higher category theory.

Overview of "Covariant Content Contravariant Homotopy Theories" by Hoang Kim Nguyen

This paper introduces a formal framework for constructing covariant and contravariant model structures within the context of locally presentable categories equipped with functorial cylinder objects. The author's objective is to extend the theoretical foundation provided by Cisinski and others to develop model structures sensitive to the "direction" of a cylinder, thereby leading to distinct covariant and contravariant model structures. These constructions harness the underlying topological intuitions of homotopy theory while being applicable to a wide range of mathematical contexts.

Key Contributions and Methodology

1. Framework for Model Structures: The author delineates a method for generating covariant and contravariant model structures from an exact cylinder object within a locally presentable category. The paper underscores the significance of directionality in these cylinders, introducing homotopical data that enables a bifurcation into two distinct model structures.
2. Anodyne Classes and Right/Left Homotopical Structures: Utilizing the interplay of anodyne extensions and weak factorization systems, the research defines right and left homotopical structures that lead to well-formed model categories. These structures are pivotal in formally advancing the covariant and contravariant model structures.
3. Abstract Finality and Properness: Implicit within these structures are the notions of finality and initiality, correlated with proper and smooth map properties. These concepts are explored in relation to their capacity to hold stable under model category operations such as pullbacks, giving rise to a particularly functional understanding of fibrations within these contexts.
4. Concrete Examples: Two principal examples are explored in depth: the covariant and contravariant model structures for simplicial sets, based on pioneering work by Joyal, and the Cartesian model structures for marked simplicial sets introduced by Lurie. Through these examples, the paper showcases the applicability of the developed theory to prominent structures in higher category theory and homotopy theory.
5. Generative Process and Proof of Properness: The framework delineated in the work includes a constructive approach to identifying generating sets for right and left anodyne extensions, pivotal in defining weak equivalences within the model structures. This leads to a comprehensive proof of properness for coCartesian fibrations, offering new insights into the stability of these mappings under homotopical perturbations.

Theoretical and Practical Implications

The methodological contributions presented in this paper provide a versatile toolset for homotopy theorists seeking to explore directional behavior in model categories. Practically, the extension of Cisinski’s work to consider directional homotopy not only broadens the applicability of model categorical constructs but also deepens the potential for theoretical exploration in areas such as higher category theory.

The paper aligns well with existing frameworks in abstract homotopy theory while bridging connections to tangible structures within simplicial and marked simplicial sets, reinforcing its relevance across varying levels of theoretical mathematics.

Future Directions

Further research might explore the implications of these model structures in more diverse categories, potentially extending applicability to complex categorical structures in algebraic topology and geometry. Additionally, establishing connections between these local developments and broader categorical homotopy-theoretic frameworks could provide a more unified understanding of model-theoretic phenomena in mathematics.

Hoang Kim Nguyen’s exploration of covariant and contravariant homotopy theories thus opens new avenues for inquiry into the manipulation and understanding of categorical structures through the lens of model theory, offering robust theoretical tools for researchers in the field.