The most basic motivation for introducing higher categories is the observation that CatU, the category of U-small categories, naturally has a structure of 2-category: the objects are categories, the morphisms are functors, and the 2-morphisms are natural transformations between functors. If we denote this 2-category by CAT2cat then its truncation τ≤1CAT2cat to a 1-category would have, as morphisms, the equivalence classes of functors up to natural equivalence. While it is often necessary to consider two naturally equivalent functors as being “the same,” identifying them formally leads to a loss of information.

Topologists are confronted with a similar situation when looking at the category of spaces. In homotopy theory one thinks of two homotopicmaps between spaces as being “the same” however, the homotopy category ho(Top) obtained after dividing by this equivalence relation doesn't retain enough information. This loss of information is illustrated by the question of diagrams. Suppose Ψ is a small category. A diagram of spaces is a functor T : Ψ → Top, that is a space T, (x) for each object x ∈ Ψ and a map T (a) : T (x) → T (y) for each arrow a ∈ Ψ(x, y), satisfying strict compatibility with identities and compositions. The category of diagrams Func(Ψ, Top) has a natural subclass of morphisms: a morphism f : S → T of diagrams is a levelwise weak equivalence if each f (x) : S(x) → T (x) is a weak equivalence.