Exploring the Homotopy Category: Weak Equivalences and Isomorphisms
Table of Contents:
- Introduction
- Isolating the Homotopy-Related Parts of a Model Category
- The Construction of Host C
3.1 Definition of Host C
3.2 Homotopic Classes and Equivalence
3.3 Composition of Homotopic Classes
- The Whitehead Theorem
4.1 Upgrading the Result to Model Categories
4.2 Homotopy Equivalences
- The Localization Functor: Gamma PQ
5.1 Object Level Function
5.2 Morphism Level Function
5.3 Well-definedness and Functoriality
- Weak Equivalences and Isomorphisms in the Homotopic Category
- The Localization Theorem
7.1 Definition of Localization
7.2 Universality of the Localization Functor
7.3 Unique Factorization
- Vibration and Co-fibration Categories
- Homotopy Fiber Sequences
- Hom Sets and Bijection
- Commuting Squares in the Homotopic Category
11.1 Existence of Corresponding Commuting Squares in the Original Category
Introduction
In the field of mathematics, particularly in the study of model categories, there exists a concept called the homotopic category. This category is derived from the original model category by isolating the homotopy-related parts. In this article, we will explore the construction of the homotopic category, the properties of its objects and morphisms, and its significance in the study of weak equivalences and isomorphisms. We will also discuss the notion of vibration and co-fibration categories within the homotopic category and examine the existence of commuting squares. So, let's dive into the fascinating world of the homotopic category.
Isolating the Homotopy-Related Parts of a Model Category
To fully understand the construction of the homotopic category, we first need to grasp the idea of isolating the homotopy-related parts of a model category. In this context, a model category is denoted as C. We introduce a construction called "Host C," which is a category that contains the objects of C that are both fibrant and cofibrant. The morphisms in Host C are homotopy classes of morphisms in C. It is important to note that these classes are equivalence classes, hence their name. The composition of two classes in Host C is defined as the class of the composition of the representatives. We refer to Host C as the homotopic category of the model category C. This construction allows us to focus solely on the homotopy aspects of the original model category.
The Construction of Host C
3.1 Definition of Host C
Let's delve deeper into the construction of Host C. As mentioned earlier, Host C consists of the objects of C that are both fibrant and cofibrant. This means that the objects in Host C exhibit properties related to both fibrations and cofibrations. These objects play a crucial role in the study of homotopy in the model category.
3.2 Homotopic Classes and Equivalence
In Host C, the morphisms are represented by homotopy classes. These classes capture the essence of homotopy and allow us to study the relationship between morphisms in a more abstract manner. It is important to distinguish these classes from true homotopy classes, as the former represents the equivalence classes of morphisms instead of the usual notion of homotopic classes.
3.3 Composition of Homotopic Classes
One of the key operations in Host C is the composition of homotopic classes. In this category, the composition of two classes is defined as the class of the composition of their representatives. This composition operation ensures that the homotopic category retains the essential properties of the original model category.
The Whitehead Theorem
The Whitehead Theorem, known in the context of topology, also applies to model categories. The topological Whitehead Theorem states that a weak equivalence between CW complexes implies a homotopy equivalence. In the realm of model categories, we can upgrade this theorem to hold for any model category C. This theorem, known as the Whitehead Theorem for model categories, further solidifies the connection between weak equivalences and homotopy equivalences.
4.1 Upgrading the Result to Model Categories
In model categories, the Whitehead Theorem establishes that a weak equivalence in C can be factored as an acyclic cofibration followed by an acyclic fibration. This factorization provides insights into the behavior of weak equivalences in the model category, shedding light on how they relate to homotopy equivalences.
4.2 Homotopy Equivalences
Within the homotopic category, the notion of isomorphism is defined as homotopy equivalence. This means that in the homotopic category, isomorphisms are represented by homotopy equivalences. The Whitehead Theorem plays a crucial role in this definition as it bridges the gap between weak equivalences and homotopy equivalences.
The Localization Functor: Gamma PQ
The localization functor, denoted as gamma pq, serves as a fundamental tool in moving from the original model category to the corresponding homotopic category. This functor allows us to project objects and morphisms onto the homotopic category and reveals the inherent structure preserved within this category. Let's explore the details of gamma pq.
5.1 Object Level Function
At the object level, the gamma pq functor considers an object x in the original category C. To obtain an object in the homotopic category, gamma pq performs a two-step process. Firstly, it applies the initial morphism to x, factoring it as a co-vibration followed by an acyclic vibration. If x is vibrant, an additional requirement is imposed to ensure that the second map is the identity on x. This step is intended to capture the essence of x in the homotopic category. Secondly, gamma pq applies the terminal morphism to x, factoring it as an acyclic co-fibration followed by a vibration. Similar to the previous step, an additional requirement is imposed for fibrant objects, ensuring that the first map is the identity on x. With this process, gamma pq generates an object in the homotopic category corresponding to x.
5.2 Morphism Level Function
The morphism level function of the gamma pq functor deals with the mapping between morphisms in the original category C and the corresponding morphisms in the homotopic category. For a morphism f from x to y in C, gamma pq first applies the initial morphism to x and the terminal morphism to y. This step allows us to generate intermediate objects that facilitate the lifting of f to the homotopic category. By utilizing the properties of co-vibrations and acyclic vibrations, gamma pq ensures the existence of a lift from x to y. This lift guarantees that the morphism f in the original category corresponds to a morphism in the homotopic category.
5.3 Well-definition and Functoriality
To ensure the well-definition and functoriality of the gamma pq functor, we need to establish that the choices made during its construction do not impact its properties. Although we initially introduced the construction of pq, which might appear as a choice, we have shown that it doesn't affect the resulting functor. Therefore, we can refer to gamma pq as the unique functor that captures the essence of the homotopic category.
Weak Equivalences and Isomorphisms in the Homotopic Category
In the homotopic category, weak equivalences are transformed into isomorphisms. This property, which sets the homotopic category apart, is a consequence of the localization process. By localizing at the weak equivalences, the homotopic category ensures that weak equivalences become true isomorphisms.
The Localization Theorem
The localization theorem formalizes the properties of the homotopic category. It states that the homotopic category is the localization of the original category, considering the weak equivalences as the subclass for localization. This theorem guarantees the existence of a functor, gamma pq, that maps the original category to the homotopic category. Furthermore, this functor possesses the universality property, making it the unique functor that sends weak equivalences to isomorphisms.
7.1 Definition of Localization
The term "localization" refers to the process of creating a new category from an existing category by considering a specific subclass of morphisms and turning them into isomorphisms. In the case of the homotopic category, the localization is applied to the weak equivalences.
7.2 Universality of the Localization Functor
The universality of the localization functor, gamma pq, ensures that any other functor from the original category to another category, which sends weak equivalences to isomorphisms, factors through gamma up to natural isomorphism. This property establishes an essential connection between the homotopic category and other categories that share similar properties.
7.3 Unique Factorization
Another significant aspect of the localization theorem is the unique factorization property. It guarantees that the factorization of a functor, f, through the localization functor, gamma pq, is unique up to natural isomorphism. This means that despite the potential existence of multiple factorizations, they are all isomorphic in a natural way, eliminating any ambiguity in the construction of the homotopic category.
Vibration and Co-fibration Categories
In the homotopic category, we can analyze the subcategories of fibrant objects and co-fibrant objects separately. These subcategories inherit additional structure from the original model category. The category of fibrant objects is referred to as a vibration category, while the category of co-fibrant objects is known as a co-fibration category. These names reflect the fact that each subcategory retains half of the factorization axioms of the overall model category.
Homotopy Fiber Sequences
Homotopy fiber sequences represent an important concept within the homotopic category. These sequences capture the relationship between fibrations, co-fibrations, and isomorphisms in the homotopic category. By examining homotopy fiber sequences, we can gain insights into the properties and behavior of objects and morphisms in the model category.
Hom Sets and Bijection
In the homotopic category, the hom sets play a crucial role in understanding the relationships between objects and morphisms. For a specific case where x is co-fibrant and y is fibrant, we can establish a bijection between the hom set in the homotopic category and the hom set in the original category modulo weak equivalences. This bijection allows us to establish connections between objects and morphisms in both categories, providing a deeper understanding of their intrinsic properties.
Commuting Squares in the Homotopic Category
Within the homotopic category, every commuting square is part of the image of the localization functor, gamma pq. This means that for every commuting square in the homotopic category, there exists a corresponding square in the original category. However, it is important to note that the objects in these corresponding squares might not be the same. Nevertheless, the existence of a proper commuting square in the original category demonstrates the rich structure preserved within the homotopic category.
11.1 Existence of Corresponding Commuting Squares in the Original Category
To prove the existence of corresponding commuting squares in the original category, we analyze a specific commuting square in the homotopic category. By applying the pushout to the mapping cylinder, we obtain a factorization of the original square via universal properties. It is worth mentioning that the induced map from the codomain to the mapping cylinder is a homotopy equivalence, adding further insights to the behavior of commuting squares in the original category.
Conclusion
The homotopic category, constructed by isolating the homotopy-related parts of a model category, offers a deeper understanding of weak equivalences and isomorphisms within the context of model categories. The localization functor, gamma pq, plays a central role in connecting the original category to the homotopic category, ensuring that weak equivalences become true isomorphisms. The existence of vibration and co-fibration categories within the homotopic category further strengthens the understanding of morphisms and objects. Through the exploration of homotopy fiber sequences and the bijection between hom sets, we gain insights into the intricate relationships within the homotopic category. Finally, the existence of proper commuting squares in the original category based on the corresponding squares in the homotopic category highlights the interconnectedness of these two realms.
Pros:
- The construction of the homotopic category provides a powerful tool for studying weak equivalences and homotopy equivalences in model categories.
- The localization functor, gamma pq, serves as a bridge between the original category and the homotopic category, allowing for the analysis of objects and morphisms in both realms.
- The existence of vibration and co-fibration categories within the homotopic category showcases the underlying structure inherited from the original model category.
Cons:
- The construction of the homotopic category and its various properties might require a solid background in model categories and algebraic topology to fully grasp.
Highlights:
- The homotopic category is the localization of the original model category at weak equivalences.
- The Whitehead Theorem extends the result of weak equivalence to homotopy equivalence.
- The localization functor, gamma pq, maps the original category to the homotopic category.
- Weak equivalences become isomorphisms in the homotopic category.
- Vibration and co-fibration categories exist within the homotopic category.
- Homotopy fiber sequences capture the relationships between fibrations, co-fibrations, and isomorphisms.
- Bijection between hom sets in the homotopic category and the original category modulo weak equivalences.
- Commuting squares in the homotopic category have corresponding squares in the original category.
FAQs:
Q: What is the homotopic category?
A: The homotopic category is a construction derived from a model category by isolating the homotopy-related parts. It allows for the study of weak equivalences and isomorphisms.
Q: How are weak equivalences and isomorphisms related in the homotopic category?
A: In the homotopic category, weak equivalences are transformed into isomorphisms. This property highlights the significance of the homotopic category in capturing the essence of homotopy.
Q: What is the role of the localization functor, gamma pq?
A: The localization functor, gamma pq, serves as a bridge between the original category and the homotopic category. It maps objects and morphisms from the original category to their corresponding counterparts in the homotopic category.
Q: What are vibration and co-fibration categories?
A: Vibration and co-fibration categories are subcategories within the homotopic category that inherit additional structure from the original model category. Vibration categories pertain to fibrant objects, while co-fibration categories are associated with co-fibrant objects.
Q: What are homotopy fiber sequences?
A: Homotopy fiber sequences represent the relationship between fibrations, co-fibrations, and isomorphisms within the homotopic category. They provide insights into the behavior of objects and morphisms in the model category.
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