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Higher category theory studies structures with morphisms between objects, morphisms between those morphisms, and further layers of comparison. In a two-dimensional example, categories are objects, functors are one-morphisms and natural transformations are two-morphisms. A two-morphism can be composed along a shared functor or alongside another natural transformation; the compatibility of these compositions is part of the structure.^[ref]

Homotopy theory gives the motivating picture: points are connected by paths, paths are compared by homotopies, and homotopies themselves can be compared by further homotopies. Replacing every path by only its homotopy class produces the fundamental groupoid and forgets these higher comparisons. Weak higher categories retain coherent comparison data when associativity holds only up to equivalence. One must specify a model and whether higher morphisms are required to be invertible; “higher category” does not impose one universal convention.