inftybicategories

An infty-category, also known as an infinity-category or ∞-category, is a higher categorical structure that generalizes the notion of a category. In a standard category, objects are related by morphisms, and these morphisms can be composed. Infty-categories extend this by allowing for not just morphisms, but also "morphisms between morphisms" (2-morphisms), "morphisms between 2-morphisms" (3-morphisms), and so on, up to an infinite hierarchy of higher-dimensional cells.

These structures are fundamental in various areas of mathematics, particularly in algebraic topology, abstract homotopy theory, and theoretical physics, notably in string theory and quantum field theory. They provide a framework for understanding and organizing complex, multi-level relationships that arise in these fields.

The precise definition of an infty-category can be approached in several ways. One common approach is through the use of simplicial sets, where an infty-category is represented by a simplicial set satisfying certain properties, often related to the Kan condition. Another approach uses the language of operads, where infty-categories can be seen as models for the nerves of certain types of operads.

The concept of equivalence in infty-categories is also enriched. Two infty-categories are considered equivalent if there exists a sequence of invertible higher morphisms connecting them, reflecting a topological notion of homotopy equivalence. This means that structures that are indistinguishable up to these higher-dimensional deformations are considered the same.

Research in infty-categories aims to develop their theory rigorously and to explore their applications. This includes understanding their properties, constructing important examples, and establishing connections to other mathematical and physical theories. The development of stable infty-categories, for instance, has been particularly fruitful.