Infinity-Category

An infinity-category (or ∞-category) is a higher category in which the tower of morphisms continues through all finite dimensions and stabilizes in the limit, encoding a coherent notion of equivalence at every level. Unlike an n-category, which truncates after n levels of structure, an ∞-category treats homotopy as primitive: morphisms are paths, 2-morphisms are homotopies between paths, 3-morphisms are homotopies between homotopies, and the tower never ends. This makes ∞-categories the natural setting for homotopy theory, algebraic topology, and any domain where the

Models and Realizations

An infinity-category is not a single structure but a family of them, realized through different model categories that present the same homotopy theory. The most common models include:

• Quasi-categories (or weak Kan complexes): simplicial sets in which every inner horn has a filler. Developed by Joyal and refined by Lurie, this is the most widely used model in practice.
• Topological categories': categories enriched over topological spaces, where the hom-sets carry a topology that encodes the higher morphisms.
• Simplicial categories: categories enriched over simplicial sets, closely related to topological categories but more combinatorial.
• Complete Segal spaces: simplicial spaces satisfying conditions that ensure composition is well-defined up to coherent homotopy.

The equivalence of these models is itself a theorem in higher category theory: they all present the same underlying homotopy theory, confirming that the notion of infinity-category is robust against changes in presentation.

The Univalent Perspective

In homotopy type theory, the concept of an infinity-category arises naturally from the structure of identity types. The identity type Id_A(a,b) is not merely a proposition (a is equal to b) but a space of paths from a to b. The identity type of the identity type, Id_{Id_A(a,b)}(p,q), is the space of homotopies between paths, and this tower continues indefinitely. This means that every type in HoTT is automatically an infinity-groupoid — a special case of an infinity-category in which all morphisms are invertible.

This is not a coincidence but a deep structural fact: the rules of type theory generate exactly the structure of infinity-categories. The syntax and the geometry are the same object viewed from different angles.

The infinity-category is the correct notion of category, and ordinary categories are merely approximations. We teach 1-categories first not because they are fundamental, but because they are the truncation — the low-dimensional projection — of a structure that is inherently infinite-dimensional. The pedagogical order is backward. The history of mathematics is the history of discovering that the structures we invented for convenience are actually shadows of deeper structures we did not know we needed.