One of the fundamental invariants of algebraic topology arises when we regard a topological space as something like a category, or rather a groupoid. The points of the space become objects of the category. A path, here a continuous function from the standard unit interval, represents a morphism between its starting and ending points. More accurately, for there to be an associative composition law, we must revise this outline slightly and define a morphism to be an endpoint-preserving homotopy class of paths. This defines the fundamental groupoid of the space.

But from the topological perspective, it seems artificial to take homotopy classes of paths in pursuit of strict associativity. The more natural construction forms a (weak) ∞-groupoid with objects the points of X, 1-morphisms the paths in X, 2-morphisms the homotopies between paths, 3-morphisms the homotopies between these homotopies, and so on. With this example in mind, the homotopy hypothesis, a principle guiding these definitions, says that an ∞-groupoid should be the same thing as a topological space.

Continuing in this vein, mathematical structures admitting a topological enrichment assemble into (∞, 1)-categories, loosely defined to be categories with morphisms in each dimension such that every morphism above dimension 1 is invertible. One way to encode this definition is to say that an (∞, 1)-category is a category (weakly) enriched in ∞-groupoids, which are also called (∞, 0)-categories.