= Higher category theory
<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.<Higher Topos Theory by Jacob Lurie>[sections 1.1–1.2, motivation and language of higher categories]{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.
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