infinity1categories

infinity1categories, also known as ∞-categories orinfty-categories, are a generalization of the notion of a category. In a standard category, morphisms are just arrows, and composition is associative. In an ∞-category, the "morphisms" are higher-dimensional structures, often visualized as cells or n-cells, and the composition of these higher-dimensional structures is associative up to a higher coherent homotopy. This means that composing three or more morphisms in different ways might not yield the exact same result, but rather results that are equivalent through a chain of higher-dimensional equivalences.

The formal definition of an ∞-category can be approached in several ways, including through the use of simplicial sets, quasicategories, or Segal spaces. These different definitions aim to capture the same underlying categorical structure. The motivation for studying ∞-categories comes from various areas of mathematics, particularly algebraic topology and theoretical physics. They provide a framework for understanding structures that exhibit a high degree of symmetry or degeneracy, which often appear in these fields.

One key concept related to ∞-categories is the idea of "homotopy coherence." In a standard category, associativity of composition is a strict identity: (f o g) o h = f o (g o h). In an ∞-category, this becomes an equivalence: (f o g) o h ≃ f o (g o h). This equivalence itself can be composed with other equivalences, leading to a complex web of higher-dimensional relationships. The coherence conditions ensure that all such compositions are consistently related, preventing paradoxes. Infinity-categories are a fundamental tool in higher category theory and have significant applications in areas like stable ∞-categories and their connections to K-theory and topological field theories.