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

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

Category theory is a branch of mathematics that studies the commonalities and patterns between different mathematical structures. It has been a fundamental tool in various areas of mathematics, including algebraic geometry, homotopy theory, and algebraic topology. In recent years, the concept of ∞\infty-categories has gained significant attention, and many theorems and insights have been developed specifically for these structures. However, there is a lack of awareness about the differences between ∞\infty-categories and traditional categories. In this article, we will explore some well-known theorems and insights in ∞\infty-category theory that are not well known in traditional category theory.

Before we dive into the theorems and insights, let's briefly discuss what ∞\infty-categories are. An ∞\infty-category is a higher category that is defined using a simplicial set. A simplicial set is a collection of sets, one for each non-negative integer, together with face and degeneracy maps between them. The face maps are used to define the boundary of a simplex, while the degeneracy maps are used to define the interior of a simplex. An ∞\infty-category is a simplicial set that satisfies certain properties, such as being a Kan complex.

Many theorems in 11- and 22-category theory have direct analogues in (∞,1)(\infty, 1)- and (∞,2)(\infty, 2)-category theory. By "direct analogue", we mean a valid statement in category theory that can be translated directly to the ∞\infty-category setting. For example, the Yoneda lemma, which states that a functor from a category to the category of sets is fully faithful if and only if it is representable, has a direct analogue in the ∞\infty-category setting.

While many theorems in traditional category theory have direct analogues in ∞\infty-category theory, there are some theorems and insights that are unique to the ∞\infty-category setting. Here are a few examples:

Homotopy Theory

Homotopy theory is a branch of mathematics that studies the properties of topological spaces that are preserved under continuous deformations. In the context of ∞\infty-categories, homotopy theory is used to study the properties of ∞\infty-categories that are preserved under weak equivalences. One of the key theorems in homotopy theory is the Whitehead theorem, which states that a map between connected spaces is a weak equivalence if and only if it induces an isomorphism on all homotopy groups. This theorem has a direct analogue in the ∞\infty-category setting, where it is used to study the properties of ∞\infty-categories that are preserved under weak equivalences.

Higher Algebra

Higher algebra is a branch of mathematics that studies the properties of algebraic structures that are preserved under higher morphisms. In the context of ∞\infty-categories, higher algebra is used to study the properties of ∞\infty-categories that are preserved under higher morphisms. One of the key theorems in higher algebra is the Lurie theorem, which states that a ∞\infty-category is a higher algebra if and only if it is a simplicial set that satisfies certain properties. This theorem has a direct analogue in the traditional category setting, where it is used to study the properties of categories that are preserved under higher morphisms.

Higher Topos Theory

Higher topos theory is a branch of mathematics that studies the properties of higher categories that are preserved under geometric morphisms. In the context of ∞\infty-categories, higher topos theory is used to study the properties of ∞\infty-categories that are preserved under geometric morphisms. One of the key theorems in higher topos theory is the Giraud theorem, which states that a higher topos is a higher category that satisfies certain properties. This theorem has a direct analogue in the ∞\infty-category setting, where it is used to study the properties of ∞\infty-categories that are preserved under geometric morphisms.

Cohomology Theories

Cohomology theories are a branch of mathematics that studies the properties of topological spaces that are preserved under continuous deformations. In the context of ∞\infty-categories, cohomology theories are used to study the properties of ∞\infty-categories that are preserved under weak equivalences. One of the key theorems in cohomology theory is the Eilenberg-Steenrod axioms, which state that a cohomology theory is a functor from the category of topological spaces to the category of abelian groups that satisfies certain properties. This theorem has a direct analogue in the ∞\infty-category setting, where it is used to study the properties of ∞\infty-categories that are preserved under weak equivalences.

Derived Algebraic Geometry

Derived algebraic geometry is a branch of mathematics that studies the properties of algebraic varieties that are preserved under derived functors. In the context of ∞\infty-categories, derived algebraic geometry is used to study the properties of ∞\infty-categories that are preserved under derived functors. One of the key theorems in derived algebraic geometry is the Artin-Lurie theorem, which states that a derived algebraic variety is a higher category that satisfies certain properties. This theorem has a direct analogue in the ∞\infty-category setting, where it is used to study the properties of ∞\infty-categories that are preserved under derived functors.

In conclusion, while many theorems in traditional category theory have direct analogues in ∞\infty-category theory, there are some theorems and insights that are unique to the ∞\infty-category setting. These theorems and insights have far-reaching implications for various areas of mathematics, including homotopy theory, higher algebra, higher topos theory, cohomology theories, and derived algebraic geometry. As the field of ∞\infty-category theory continues to evolve, it is likely that we will see even more theorems and insights emerge that are specific to this area of mathematics.

For further reading on ∞\infty-category theory, we recommend the following resources:

  • The book "Higher Topos Theory" by Jacob Lurie is a comprehensive introduction to the subject.
  • The notes on quasicategories by André Joyal provide a detailed overview of the subject.
  • The book "Les préfaisceaux comme modèles des types d'homotopie" by Denis-Charles Cisinski provides a detailed introduction to the subject.
  • The book "Derived Algebraic Geometry" by Jacob Lurie provides a comprehensive introduction to the subject.

In our previous article, we explored some well-known theorems and insights in ∞\infty-category theory that are not well known in traditional category theory. In this article, we will answer some frequently asked questions about ∞\infty-category theory.

A: A category is a mathematical structure that consists of objects and morphisms between them. An ∞\infty-category, on the other hand, is a higher category that is defined using a simplicial set. This means that an ∞\infty-category has not only objects and morphisms, but also higher morphisms between morphisms, and so on.

A: A simplicial set is a collection of sets, one for each non-negative integer, together with face and degeneracy maps between them. The face maps are used to define the boundary of a simplex, while the degeneracy maps are used to define the interior of a simplex.

A: A Kan complex is a simplicial set that satisfies certain properties, such as being a Kan complex. This means that the face and degeneracy maps satisfy certain conditions, such as being injective and surjective.

A: The Yoneda lemma is a theorem in category theory that states that a functor from a category to the category of sets is fully faithful if and only if it is representable. This means that the functor is isomorphic to the functor that sends an object to the set of morphisms from that object to a fixed object.

A: The Whitehead theorem is a theorem in homotopy theory that states that a map between connected spaces is a weak equivalence if and only if it induces an isomorphism on all homotopy groups. This theorem has a direct analogue in the ∞\infty-category setting, where it is used to study the properties of ∞\infty-categories that are preserved under weak equivalences.

A: The Lurie theorem is a theorem in higher algebra that states that a ∞\infty-category is a higher algebra if and only if it is a simplicial set that satisfies certain properties. This theorem has a direct analogue in the traditional category setting, where it is used to study the properties of categories that are preserved under higher morphisms.

A: The Giraud theorem is a theorem in higher topos theory that states that a higher topos is a higher category that satisfies certain properties. This theorem has a direct analogue in the ∞\infty-category setting, where it is used to study the properties of ∞\infty-categories that are preserved under geometric morphisms.

A: The Eilenberg-Steenrod axioms are a set of axioms define a cohomology theory. A cohomology theory is a functor from the category of topological spaces to the category of abelian groups that satisfies certain properties. This theorem has a direct analogue in the ∞\infty-category setting, where it is used to study the properties of ∞\infty-categories that are preserved under weak equivalences.

A: The Artin-Lurie theorem is a theorem in derived algebraic geometry that states that a derived algebraic variety is a higher category that satisfies certain properties. This theorem has a direct analogue in the ∞\infty-category setting, where it is used to study the properties of ∞\infty-categories that are preserved under derived functors.

In conclusion, ∞\infty-category theory is a rich and fascinating area of mathematics that has far-reaching implications for various areas of mathematics, including homotopy theory, higher algebra, higher topos theory, cohomology theories, and derived algebraic geometry. We hope that this Q&A article has provided a useful overview of some of the key concepts and theorems in ∞\infty-category theory.

For further reading on ∞\infty-category theory, we recommend the following resources:

  • The book "Higher Topos Theory" by Jacob Lurie is a comprehensive introduction to the subject.
  • The notes on quasicategories by André Joyal provide a detailed overview of the subject.
  • The book "Les préfaisceaux comme modèles des types d'homotopie" by Denis-Charles Cisinski provides a detailed introduction to the subject.
  • The book "Derived Algebraic Geometry" by Jacob Lurie provides a comprehensive introduction to the subject.