inftycategory

An infinity-category, also known as an ∞-category, is a higher categorical structure that generalizes the notion of a category. In a standard category, objects are related by morphisms, and compositions of these morphisms are associative. An infinity-category extends this by allowing for higher-dimensional morphisms. Instead of just 0-morphisms (objects) and 1-morphisms (morphisms), an infinity-category has k-morphisms for all natural numbers k. These k-morphisms can be thought of as paths between (k-1)-morphisms.

The key property of an infinity-category is that compositions of these higher-dimensional morphisms are associative only up to a higher coherent equivalence. This means that composing three morphisms A to B, B to C, and C to D in different ways might not yield the exact same result, but rather objects that are equivalent in a specific, well-defined sense. This "higher homotopy coherence" is what distinguishes infinity-categories from ordinary categories and their usual generalizations like n-categories.

There are several equivalent definitions of infinity-categories, each highlighting different aspects of their structure. One common approach is through the language of simplicial sets, where an infinity-category is represented by a simplicial set satisfying certain Segal conditions. Another perspective uses opetopes, which are combinatorial objects that generalize n-cubes and can be used to define the structure of higher compositions.

Infinity-categories are a fundamental concept in modern abstract mathematics, particularly in areas like algebraic topology, homological algebra, and theoretical physics. They provide a framework for understanding and unifying diverse mathematical theories that exhibit hierarchical structures and equivalences. Their development has led to significant advancements in fields such as stable homotopy theory and categorical abstract nonsense.