Inftycategories

An infinity-category, often called a ∞-category or (lowercase) infinity-category, is a higher categorical structure that generalizes the notion of a category. In a standard category, the objects are sets and the morphisms are functions between these sets. In an infinity-category, the concept is extended to allow for "morphisms between morphisms" and so on, to any level of iterated composition.

More formally, an infinity-category can be thought of as a simplicial set satisfying certain properties, or as a category enriched over the category of small categories. This means that instead of just having sets of morphisms between objects, we have categories of morphisms. The internal structure of these categories of morphisms captures the higher-dimensional relationships.

The theory of infinity-categories is a cornerstone of modern algebraic topology, abstract homotopy theory, and theoretical physics, particularly in areas like string theory and quantum field theory. They provide a framework for understanding and manipulating complex structures that arise in these fields, such as spectra, topological spaces with additional structure, and various forms of non-abelian duality. Research in this area aims to develop a coherent and powerful theory that can unify diverse mathematical concepts.