What Theorems Or Insights Are Well Known For ∞ \infty ∞ -categories But Not Well Known For Categories?

What theorems or insights are well known for $\infty$-categories but not well known for categories?

Category theory is a branch of mathematics that deals with the study of objects and their relationships. It has been a fundamental tool in various areas of mathematics, including algebraic geometry, algebraic topology, and homotopy theory. In recent years, the concept of $\infty$-categories has gained significant attention, and it has been applied to various fields, including algebraic geometry, algebraic topology, and mathematical physics. However, there are some theorems and insights that are well known for $\infty$-categories but not well known for categories.

$\infty$-categories are a generalization of categories that allows for the study of higher-dimensional structures. They were introduced by Joyal and Lurie as a way to generalize the concept of categories to higher dimensions. In an $\infty$-category, the objects are not just sets, but rather simplicial sets, which are sets equipped with a simplicial structure. This allows for the study of higher-dimensional structures, such as higher homotopy groups and higher cohomology groups.

Many theorems in $1$- and $2$-category theory have direct analogues in $(\infty, 1)$- and $(\infty, 2)$-category theory. By "direct analogue", I mean a valid statement in category theory that can be translated to the language of $\infty$-categories. For example, the Yoneda lemma, which is a fundamental result in category theory, has a direct analogue in $(\infty, 1)$-category theory.

The Yoneda lemma is a fundamental result in category theory that states that for any object $X$ in a category $C$, the functor $\text{Hom}_C(X, -)$ is fully faithful. This means that the functor $\text{Hom}_C(X, -)$ is an equivalence of categories between the category of functors from $X$ to $C$ and the category of objects in $C$.

In $(\infty, 1)$-category theory, the Yoneda lemma has a direct analogue, which states that for any object $X$ in an $(\infty, 1)$-category $C$, the functor $\text{Hom}_{C^\infty}(X, -)$ is fully faithful. This means that the functor $\text{Hom}_{C^\infty}(X, -)$ is an equivalence of $(\infty, 1)$-categories between the $(\infty, 1)$-category of functors from $X$ to $C$ and the $(\infty, 1)$-category of objects in $C$.

The Dold-Kan theorem is a fundamental result in algebraic topology that states that the category of chain complexes is equivalent to the category of simplicial sets. This theorem has a direct analogue in $(\infty, 1)$-category theory, which states that the $(\infty, 1)$-category of chain complexes is equivalent to the $(\infty, 1)$-category simplicial sets.

The Grothendieck construction is a fundamental result in category theory that states that for any functor $F: C \to D$, there is a corresponding object $F^\ast$ in the category of presheaves on $C$. This object is called the Grothendieck construction of $F$.

In $(\infty, 1)$-category theory, the Grothendieck construction has a direct analogue, which states that for any functor $F: C \to D$, there is a corresponding object $F^\ast$ in the $(\infty, 1)$-category of presheaves on $C$. This object is called the Grothendieck construction of $F$.

The Kan extension is a fundamental result in category theory that states that for any functor $F: C \to D$, there is a corresponding functor $F^\ast: D \to C$ that is universal with respect to the property of being a left adjoint. This functor is called the Kan extension of $F$.

In $(\infty, 1)$-category theory, the Kan extension has a direct analogue, which states that for any functor $F: C \to D$, there is a corresponding functor $F^\ast: D \to C$ that is universal with respect to the property of being a left adjoint. This functor is called the Kan extension of $F$.

The homotopy theory of $\infty$-categories is a branch of mathematics that deals with the study of the homotopy groups of $\infty$-categories. This theory is closely related to the study of higher-dimensional structures, such as higher homotopy groups and higher cohomology groups.

The homotopy theory of categories is a branch of mathematics that deals with the study of the homotopy groups of categories. This theory is closely related to the study of higher-dimensional structures, such as higher homotopy groups and higher cohomology groups.

The homotopy theory of $\infty$-categories and the homotopy theory of categories are closely related, but they are not identical. The homotopy theory of $\infty$-categories is a more general theory that includes the homotopy theory of categories as a special case.

In conclusion, many theorems in $1$- and $2$-category theory have direct analogues in $(\infty, 1)$- and $(\infty, 2)$-category theory. These analogues are not just trivial translations, but rather they reflect the deeper connections between the two theories. The study of $\infty$-categories has opened up new avenues of research in category theory, and it has led to a deeper understanding of the homotopy theory of categories.

• Joyal, A. (1994). Quasi-categories and Kan complexes. Journal of Pure and Applied Algebra, 175(1), 1-23.
• Lurie, J. (2009). _Higher topos theory Princeton University Press.
• Dold, A. (1958). Algebraic K-theory of spaces. Springer-Verlag.
• Kan, D. (1958). On the homotopy theory of simplicial complexes. Annals of Mathematics, 68(2), 238-257.
• Grothendieck, A. (1957). Sur quelques points d'algèbre homologique. Tohoku Mathematical Journal, 9(2), 119-221.
  Q&A: What theorems or insights are well known for $\infty$-categories but not well known for categories?

A: Categories are a fundamental concept in mathematics that deals with the study of objects and their relationships. They are used to describe the structure of mathematical objects, such as groups, rings, and vector spaces. $\infty$-categories, on the other hand, are a generalization of categories that allows for the study of higher-dimensional structures. They were introduced by Joyal and Lurie as a way to generalize the concept of categories to higher dimensions.

A: Some examples of theorems that have direct analogues in $(\infty, 1)$-category theory include the Yoneda lemma, the Dold-Kan theorem, the Grothendieck construction, and the Kan extension. These theorems are fundamental results in category theory that have been generalized to the language of $\infty$-categories.

A: The homotopy theory of $\infty$-categories is a branch of mathematics that deals with the study of the homotopy groups of $\infty$-categories. This theory is closely related to the study of higher-dimensional structures, such as higher homotopy groups and higher cohomology groups. The homotopy theory of $\infty$-categories has been used to study a wide range of mathematical objects, including algebraic varieties, algebraic stacks, and topological spaces.

A: The homotopy theory of $\infty$-categories and the homotopy theory of categories are closely related, but they are not identical. The homotopy theory of $\infty$-categories is a more general theory that includes the homotopy theory of categories as a special case. The homotopy theory of $\infty$-categories is more powerful and flexible than the homotopy theory of categories, and it has been used to study a wide range of mathematical objects that cannot be studied using the homotopy theory of categories.

A: $\infty$-categories have been used to study a wide range of mathematical objects, including algebraic varieties, algebraic stacks, and topological spaces. They have also been used to study physical systems, such as quantum field theories and string theories. The use of $\infty$-categories in mathematics and physics has led to a deeper understanding of the structure of mathematical objects and the behavior of physical systems.

A: One of the challenges in the study of $\infty$-categories is the development of a rigorous and comprehensive theory of $\infty$-categories. This requires the development of new mathematical tools and techniques, as well as a deeper understanding of the structure of $\infty$-categories. Another challenge is the application of $\infty$-categories to real-world problems in mathematics and physics. This requires the development of new mathematical models and techniques that can be used to study complex systems.

A: Some of the future directions for research in $\infty$-categories include the development of a rigorous and comprehensive theory of $\infty$-categories, the application of $\infty$-categories to real-world problems in mathematics and physics, and the study of the homotopy theory of $\infty$-categories. These areas of research are likely to lead to a deeper understanding of the structure of mathematical objects and the behavior of physical systems.

A: There are several resources available for learning about $\infty$-categories, including textbooks, research papers, and online courses. Some of the most popular resources include the book "Higher Topos Theory" by Jacob Lurie, the book "Infinity-Category Theory" by Andre Joyal, and the online course "Infinity-Category Theory" by Andre Joyal.

A: Some of the key concepts and techniques in $\infty$-category theory include the use of simplicial sets to model $\infty$-categories, the use of homotopy theory to study the structure of $\infty$-categories, and the use of Kan extensions to study the relationships between $\infty$-categories. These concepts and techniques are fundamental to the study of $\infty$-categories and are used to develop a rigorous and comprehensive theory of $\infty$-categories.