A Deep Dive into Higher Category Theory

Higher category theory is an advanced branch of mathematics that extends the ideas of basic category theory. In traditional category theory, we study categories which consist of objects and morphisms between those objects. Higher category theory goes a step further by looking into structures that involve not just points and arrows, but also higher-dimensional analogs of these.

#What is a Category?

At its core, a category consists of:

1. Objects: These can be anything from sets to more complicated structures.
2. Morphisms (or Arrows): These are the relationships or functions connecting objects. Every morphism has a source object and a target object.

Morphisms can be composed. If you have an arrow from object A to object B, and another arrow from B to C, then you can compose these arrows to get a new arrow from A to C.

#What are Higher Categories?

Higher categories expand on this foundation. They not only deal with the morphisms between objects but also with morphisms between morphisms, and so on. This leads us to think about:

• 2-morphisms: Morphisms between morphisms. For example, a transformation between two Functors can be seen as a 2-morphism.
• 3-morphisms: These are transformations between 2-morphisms, and this pattern continues.

This layering of morphisms can lead to very complex structures where different kinds of relationships can be analyzed.

#The Need for Higher Category Theory

Many concepts in mathematics, such as those in topology, algebra, and theoretical computer science, require a more nuanced understanding of relationships. Traditional category theory often lacks the necessary detail to capture the subtleties of these relationships.

For example, in algebraic topology, one studies spaces and continuous functions. Higher categories help to understand how these spaces can be related not just by functions (morphisms) but also by homotopies, which are continuous transformations of these functions.

#Enriched Category Theory

One way to approach higher categories is through enriched category theory. This theory allows us to define categories where the morphisms have a richer structure. Rather than just being sets of arrows, they can have other mathematical structures, such as being endowed with a topology or a metric.

For instance, consider a category enriched over the category of metric spaces. In this setting, the morphisms can carry information about distances, enabling more nuanced analysis.

#Double Categories

Double categories are a particular type of higher category where we have objects, horizontal morphisms, vertical morphisms, and 2-cells:

1. Objects: The main entities we are studying.
2. Horizontal Morphisms: These can be thought of as representing processes that relate objects.
3. Vertical Morphisms: They can represent structures or additional relationships between objects that might not be captured by horizontal morphisms.
4. 2-cells: These capture the relationships between the horizontal and vertical morphisms.

The interplay between these layers creates a rich structure to analyze different mathematical phenomena.

#Proarrow Equipments

One of the tools used in higher category theory is the concept of proarrow equipments. This concept captures the idea of a structure that embodies both the relationships (morphisms) between objects and the necessary conditions for those relationships to hold in a formal setting.

Proarrow equipments allow the study of universal properties. A universal property describes a certain way in which an object can be characterized, often in terms of its relationships to other objects in the context of a category.

#Internal Categories

Internal categories can be defined within a larger category. These are categories that exist within the bounds of another structure. For example, one can define a category whose objects are other objects in a category and whose morphisms are morphisms between these objects.

Internal categories bring about the notion of "being inside" a category, allowing us to study the relationships between objects from a more localized perspective.

#Higher Toposes

Toposes are categories that behave like the category of sets, but with added structure. They provide a context in which we can do set theory and logic in a categorical framework. A higher topos extends this idea further, allowing for the inclusion of higher categorical structures.

In higher toposes, we can study these complex relationships and transformations while retaining the useful properties of toposes, such as having Limits and Colimits and the ability to work with sheaves.

#Functors and Natural Transformations

A functor is a mapping between categories that preserves structures. It takes objects and morphisms from one category and assigns them to objects and morphisms in another category while respecting the composition of morphisms.

Natural transformations are a way of transforming one functor into another while preserving the structure of the categories involved. They serve as a bridge between functors, allowing us to compare how different functors relate to each other.

#Fully Faithful Functors

A functor is said to be fully faithful if it creates a one-to-one correspondence between morphisms in the categories it connects. This notion helps us understand when functors preserve the structure of the categories involved, allowing for deeper analysis of relationships and properties.

#Limits and Colimits

Limits and colimits are fundamental concepts in category theory. They capture the idea of combining objects and morphisms in a structured way.

• Limits: Often represent a "universal" way of collecting data from a diagram of objects and morphisms. For example, the limit of a diagram could be seen as a product where you gather together all the objects in the diagram.

• Colimits: These represent a way of "merging" objects together. They are the dual notion to limits. For instance, the colimit can be thought of as a co-product where you gather all the objects into a single structure.

#Pointwise Kan Extensions

Kan extensions are a way of generalizing the notion of extending functors. They provide a method to define how a functor can be extended along another functor. Pointwise Kan extensions refer to the specific situation where we consider the extension at the level of individual objects and morphisms.

#Beck-Chevalley Conditions

The Beck-Chevalley conditions provide criteria for when certain types of squares in a category are "exact." These conditions help to establish relationships between different objects and the functors that connect them.

#Initial and Final Objects

Initial and final objects serve as references within a category:

• Initial Object: An object that acts as a source. Every other object has a unique morphism originating from it.

• Final Object: An object that acts as a target. Every other object has a unique morphism leading into it.

These concepts allow us to define limits and colimits in terms of these special objects, enhancing our ability to analyze relationships within a category.

#Conclusion

Higher category theory represents an exciting frontier in mathematics, allowing for the exploration of deeper relationships between structures. It provides tools to analyze categories and their interactions at multiple levels, opening up new avenues for understanding and applying mathematical principles across different fields. Understanding these concepts will enable one to appreciate the complexity and beauty inherent in modern mathematical theories.