Understanding the Homotopy Category: A Vital Aspect of Homotopy Theory

Table of Contents

1. Introduction
2. Isolation of Homotopy-Related Parts of a Model Category
3. Construction of the Homotopic Category
4. Whitehead Theorem in Model Categories
5. Localization Functor and the Homotopic Category
6. Vibration and Co-Fibration Categories
7. Homotopy Fiber Sequences
8. Bijection between Hom Sets in the Homotopic Category
9. Surjectivity of the Localization Functor
10. Commuting Squares in the Homotopic Category

Introduction

In the field of mathematics, particularly in algebraic topology, the concept of model categories plays a vital role in the study of homotopy theory. Model categories provide a framework to study homotopy-related objects and maps between them. One important construction in model categories is the creation of the homotopic category, which isolates the homotopy-related parts of a given model category.

Isolation of Homotopy-Related Parts of a Model Category

To understand the construction of the homotopic category, let's start with a model category C. In this model category, there are objects that are both fibrant and cofibrant, and morphisms between them that are homotopy classes of morphisms. These homotopy classes represent equivalence classes of morphisms up to homotopy. Hence, we refer to this category as Host(C).

It is essential to note that the composition of two classes in Host(C) is defined as the class of the composition of their representatives. This composition respects the homotopy equivalence relations, making Host(C) a well-defined category.

Construction of the Homotopic Category

The homotopic category, denoted as Homotopy(C), is the category that forces weak equivalences in C to become isomorphisms. Intuitively, an isomorphism in Homotopy(C) corresponds to a homotopy equivalence. In other words, a map that induces an isomorphism on homotopy groups.

To ensure this, there exists a lemma known as the Whitehead Theorem in model categories. This theorem states that a weak equivalence between CW complexes implies a homotopy equivalence. CW complexes are both fibrant and cofibrant objects in model categories. Thus, the Whitehead Theorem extends the result to all model categories.

Whitehead Theorem in Model Categories

The Whitehead Theorem in model categories establishes the equivalence between weak equivalences and homotopy equivalences. It is a powerful tool in studying homotopy theory and allows us to understand the behavior of weak equivalences in model categories.

Localization Functor and the Homotopic Category

To move from the original model category C to its homotopic category Homotopy(C), we employ a functor called the localization functor. This functor maps objects and morphisms from C to Homotopy(C).

For instance, considering an object X in C, the localization functor maps it to PQ(X), where P and Q are factorization functors. These factorization functors ensure that the resulting object is both fibrant and cofibrant.

Similarly, the localization functor maps morphisms between objects in C to morphisms in Homotopy(C). This functor exhibits universality in the sense that it sends weak equivalences in C to isomorphisms in Homotopy(C). Any other functor from C that sends weak equivalences to isomorphisms factors through the localization functor.

Vibration and Co-Fibration Categories

The full subcategories of fibrant objects and cofibrant objects in a model category inherit additional structure. The category of fibrant objects is known as a vibration category, while the category of cofibrant objects is called a co-fibration category. These subcategories capture specific properties of the original model category.

Homotopy Fiber Sequences

Homotopy fiber sequences play a crucial role in understanding homotopy theory. In a model category, a homotopy fiber sequence is a sequence of maps that represents a specific type of mapping. The vibration and co-fibration categories inherit these sequences and allow for further investigation and analysis.

Bijection between Hom Sets in the Homotopic Category

In the homotopic category Homotopy(C), there exists a bijection between hom sets. Specifically, the hom set from the object PQ(X) to the object Q(Y) in Homotopy(C) is in bijection with the hom set from the object X to the object Y in the original model category C.

This bijection is a result of the properties of the localization functor, which preserves the underlying morphisms between objects. It allows us to establish a correspondence between the hom sets and analyze the relationship between objects in both categories.

Surjectivity of the Localization Functor

The localization functor exhibits surjectivity when considering maps between cofibrant and fibrant objects. If X is a cofibrant object and Y is a fibrant object, there exists a surjection from Homotopy(C)(PQ(X), Y) to C(X, Y), where C(X, Y) represents the morphisms between X and Y in the original category.

This surjectivity arises due to the properties of the localization functor, which yields various morphisms in Homotopy(C) representing a single map in C. Hence, the localization functor captures the essence of the original morphism while introducing additional elements from the homotopic category.

Commuting Squares in the Homotopic Category

In the homotopic category Homotopy(C), every commuting square can be represented by a corresponding square in the original model category C. Although the objects may not be the same, the existence of a corresponding commuting square ensures that the properties of morphisms are preserved.

By constructing the localization functor and mapping objects and morphisms from C to Homotopy(C), we establish a relationship that allows us to analyze and understand commuting squares in both categories.

Conclusion

The creation of the homotopic category in a model category provides an avenue to study homotopy-related objects and maps. By isolating the homotopy-related parts of the model category, we gain valuable insights into the behavior of weak equivalences and the properties of morphisms. The localization functor plays a crucial role in this process, allowing for the translation of objects and morphisms between categories. Additionally, the presence of vibration and co-fibration categories expands our understanding of the underlying structure within a model category. Through the bijection between hom sets and the representation of commuting squares, we can further explore the relationships and properties present in both the homotopic category and the original model category.

Note: The content provided here is a simplified overview of the topic. For a more comprehensive understanding, further study and research in the field of model categories and homotopy theory are recommended.

🌐 Resources:

• Model Categories: An Introduction
• Homotopy Theory and Model Categories