Infinity-Categories Unveiling Functors, Simplex Universality, And Higher Order Categories

Infinity-categories, a fascinating realm within higher category theory, often spark questions about their relationship to traditional categories and the role of simplex notions. The initial impression that infinity-categories are merely a generalization of higher-order categories – categories with arrows between arrows, and so on – is a natural starting point. This article aims to explore the nature of infinity-categories, their connection to functors, and the universality of simplex notions, offering a comprehensive understanding for those venturing into this abstract yet powerful domain.

Understanding Infinity-Categories

At their core, infinity-categories (also known as ∞-categories) are a sophisticated framework for extending the concepts of category theory to higher dimensions. In classical category theory, we deal with objects and morphisms (arrows) between them. Infinity-categories, however, introduce morphisms between morphisms, morphisms between morphisms between morphisms, and so on, ad infinitum. This hierarchical structure allows us to model more complex mathematical structures and relationships than traditional categories can handle.

The key idea is that the composition of morphisms, which is a fundamental operation in category theory, is no longer a strict operation in infinity-categories. Instead, it is defined up to a higher-level morphism, capturing the notion of homotopy. This flexibility is crucial for capturing the subtle nuances of many mathematical structures, particularly in topology and homotopy theory.

Imagine a map between two topological spaces. In classical category theory, we might consider two maps equivalent if they are strictly equal. However, in homotopy theory, we consider maps equivalent if they can be continuously deformed into each other. This notion of continuous deformation is precisely what infinity-categories capture through their higher-level morphisms. The morphisms between morphisms represent the homotopies between the original morphisms, and the morphisms between those morphisms represent homotopies between homotopies, and so on.

One way to think about infinity-categories is as a generalization of groupoids. A groupoid is a category where every morphism is invertible. Similarly, an infinity-groupoid is an infinity-category where every morphism, every morphism between morphisms, and so on, is invertible up to a higher-level morphism. Infinity-groupoids play a crucial role in homotopy theory, as they provide a powerful way to encode the homotopy type of a topological space.

To formalize the notion of an infinity-category, mathematicians have developed several equivalent models, each with its own strengths and weaknesses. Some of the most prominent models include:

• Simplicially enriched categories: These are categories where the hom-sets (sets of morphisms between two objects) are replaced by simplicial sets, which are combinatorial objects that encode topological information.
• Quasi-categories (or weak Kan complexes): These are simplicial sets that satisfy a weak version of the Kan condition, which ensures that certain horn-filling problems have solutions. This condition allows for a flexible notion of composition of morphisms.
• Complete Segal spaces: These are simplicial spaces (simplicial objects in the category of spaces) that satisfy certain completeness and Segal conditions, which ensure that they behave like infinity-categories.
• Model categories: These are categories equipped with a model structure, which provides a framework for doing homotopy theory within the category. Infinity-categories can be extracted from model categories using various techniques.

Each of these models provides a different perspective on what an infinity-category is, but they are all equivalent in the sense that they define the same underlying mathematical structure. Choosing the right model often depends on the specific problem at hand.

Infinity-Categories as Functors

The statement that an infinity-category is a functor might seem surprising at first, given the abstract nature of infinity-categories. However, this perspective offers a valuable insight into their structure and behavior. To understand this, we need to delve into the world of simplicial sets and their role in modeling infinity-categories.

As mentioned earlier, one popular model for infinity-categories is that of quasi-categories, which are simplicial sets satisfying a weak Kan condition. A simplicial set is a sequence of sets indexed by the natural numbers, along with face and degeneracy maps that satisfy certain compatibility conditions. These sets and maps encode the structure of a simplicial complex, a generalization of triangles and tetrahedra to higher dimensions.

A functor, in its most basic form, is a map between categories that preserves their structure. It maps objects to objects and morphisms to morphisms, while respecting composition and identities. In the context of infinity-categories, we can think of a functor as a map between simplicial sets that preserves the simplicial structure. Specifically, it maps the $n$-simplices of one simplicial set to the $n$-simplices of another, while respecting the face and degeneracy maps.

So, how does this relate to the idea of an infinity-category being a functor? The key is to realize that an infinity-category itself can be represented as a simplicial set, and a functor between infinity-categories can be represented as a map between these simplicial sets. In this sense, an infinity-category is not just a category; it is a structured object that can be manipulated and studied using the tools of functorial algebra.

This perspective is particularly useful when considering the nerve of a category. The nerve of a category $C$ is a simplicial set $N(C)$ whose $n$-simplices are sequences of $n$ composable morphisms in $C$. The face and degeneracy maps are defined in a way that reflects the composition and identity operations in the category. The nerve construction provides a way to embed ordinary categories into the world of simplicial sets, and it turns out that the nerve of a category is a quasi-category. This means that we can view ordinary categories as special cases of infinity-categories.

Furthermore, functors between categories induce maps between their nerves. If $F: C o D$ is a functor between categories, then it induces a map of simplicial sets $N(F): N(C) o N(D)$. This map preserves the simplicial structure, and it reflects the fact that $F$ preserves composition and identities. In this sense, we can think of a functor between categories as a special type of functor between infinity-categories.

Therefore, viewing infinity-categories as functors allows us to leverage the powerful tools of category theory and functorial algebra to study their structure and properties. It provides a bridge between the abstract world of infinity-categories and the more concrete world of simplicial sets and functors, making it a valuable perspective for anyone working in this area.

Simplex Notions: A Universal Language

The notion of a simplex plays a central role in the construction and understanding of infinity-categories. A simplex is a geometric object that generalizes the notion of a triangle and a tetrahedron to higher dimensions. An $n$-simplex is the convex hull of $n+1$ affinely independent points in Euclidean space. For example, a 0-simplex is a point, a 1-simplex is a line segment, a 2-simplex is a triangle, and a 3-simplex is a tetrahedron.

Simplexes are fundamental building blocks for constructing more complex geometric objects, such as simplicial complexes and simplicial sets. A simplicial complex is a collection of simplexes that are glued together along their faces. A simplicial set is a combinatorial object that encodes the structure of a simplicial complex. It consists of a sequence of sets indexed by the natural numbers, where the $n$-th set represents the set of $n$-simplexes, along with face and degeneracy maps that describe how the simplexes are glued together.

In the context of infinity-categories, simplexes provide a powerful way to represent higher-level morphisms. An $n$-simplex in an infinity-category can be thought of as an $n$-dimensional morphism, capturing the relationships between lower-dimensional morphisms. For example, a 0-simplex represents an object, a 1-simplex represents a morphism between objects, a 2-simplex represents a homotopy between morphisms, and so on.

The universality of simplex notions stems from their ability to encode a wide range of mathematical structures and relationships. Simplicial sets, in particular, provide a flexible and powerful framework for modeling homotopy types, which are the fundamental objects of study in homotopy theory. As mentioned earlier, infinity-groupoids, which are infinity-categories where all morphisms are invertible up to homotopy, provide a way to encode the homotopy type of a topological space. This means that we can use simplicial sets to study the topological properties of spaces by studying the corresponding infinity-groupoids.

The Kan condition, which is a key property of simplicial sets used to model infinity-categories, further underscores the universality of simplex notions. The Kan condition states that certain horn-filling problems have solutions. A horn is a simplicial complex obtained by removing one face from a simplex. The Kan condition ensures that we can always fill in a horn to obtain a complete simplex, which corresponds to composing morphisms in an infinity-category. This condition is crucial for capturing the flexible notion of composition that is characteristic of infinity-categories.

Moreover, the concept of simplicial approximation highlights the universality of simplexes in topology. The simplicial approximation theorem states that any continuous map between topological spaces can be approximated by a simplicial map between their triangulations. This theorem allows us to reduce many topological problems to combinatorial problems involving simplexes, making it a powerful tool for studying topological spaces.

In conclusion, simplex notions are universal in the sense that they provide a fundamental building block for constructing and understanding a wide range of mathematical structures, including infinity-categories, homotopy types, and topological spaces. Their ability to encode higher-level morphisms and their role in the Kan condition and the simplicial approximation theorem make them indispensable tools for mathematicians working in higher category theory and homotopy theory.

Is an Infinity-Category a Category?

The question of whether an infinity-category is