infinitycategories

Infinitycategories are a generalization of categories in category theory. In a standard category, the hom-sets (sets of morphisms between objects) are actual sets. In an infinitycategory, these hom-sets are replaced by higher categorical structures, typically spaces or more generally infinity-categories themselves. This allows for a richer structure where not only objects and morphisms exist, but also higher-dimensional "cells" representing compositions of morphisms, compositions of compositions, and so on.

The development of infinitycategories is motivated by the need to formalize certain constructions and concepts in algebraic topology, homotopy theory, and theoretical physics. For example, in homotopy theory, spaces are often studied up to homotopy equivalence, which suggests a categorical framework where objects are spaces and morphisms are homotopy classes of maps. However, simply considering homotopy classes of maps doesn't fully capture the structure of how these maps compose. Infinitycategories provide a way to retain this compositional information.

A key feature of infinitycategories is that they form a higher categorical analogue of the category of sets. Just as categories can be thought of as spaces with zero-dimensional cells (objects) and one-dimensional cells (morphisms), infinitycategories can be seen as spaces with cells of all non-negative dimensions. This perspective is often formalized using models such as simplicial sets, cubical sets, or operads, which serve as concrete representations of these abstract structures. The study of infinitycategories has led to significant advancements in understanding the relationship between algebra and topology, and they are a fundamental tool in modern stable homotopy theory and derived algebraic geometry.