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A 2-category consists of objects, morphisms (1-cells) between objects, and 2-morphisms (2-cells) between morphisms, with composition operations at both levels. Weak (or lax) functors between 2-categories preserve the 2-categorical structure up to invertible 2-morphisms, generalizing both ordinary functors and natural transformations. This framework encompasses categories, functors, and natural transformations as a single 2-categorical structure.
Study the 2-category Cat of all categories, functors, and natural transformations. Understand how ordinary categories and functors sit inside this structure. Explore other 2-categories: 2-categories arising from partial orders, from rings, and from algebraic structures.
In a 2-category, 2-morphisms need not have inverses; strict equality of compositions is replaced by isomorphism. Weak functors are less restrictive than strict functors and are often more natural, but this requires care in applications.
You already know three layers of categorical structure: objects (things), morphisms (maps between things, satisfying composition and identity laws), and natural transformations (maps between functors, which are themselves maps between categories). A 2-category takes these three layers and treats them as a unified structure. Objects are 0-cells, morphisms are 1-cells, and natural transformations — or their generalizations — are 2-cells. The defining example is Cat itself: objects are categories, 1-cells are functors, and 2-cells are natural transformations. The strictness of Cat (composition of functors is strictly associative on the nose) makes it the prototypical strict 2-category.
The power of the 2-categorical framework is that it unifies phenomena that otherwise require separate language. The statement "a functor F is an equivalence of categories" becomes, in 2-categorical terms, "F is a 1-cell with a quasi-inverse": there exists G and 2-cell isomorphisms FG ≅ Id and GF ≅ Id. Adjunctions, too, are 2-categorical data: the unit η: Id → GF and counit ε: FG → Id are 2-cells satisfying the triangle identities, so an adjunction is a structured pair of 1-cells with specified 2-cells. Once you recognize that functors, natural transformations, and adjunctions are all just 0-, 1-, and 2-dimensional cells in Cat, you can lift these concepts to any 2-category and reason about them uniformly.
Strict 2-functors between 2-categories preserve all structure exactly: composition of 1-cells is preserved on the nose, and 2-cells are sent to 2-cells respecting all compositions. But in practice, many natural constructions only preserve composition up to coherent isomorphism, not up to strict equality. A weak functor (also called a pseudofunctor or homomorphism of 2-categories) preserves composition of 1-cells only up to specified invertible 2-cells, together with coherence conditions ensuring these comparison 2-cells are compatible with associativity and identity. The prototypical example is the functor that assigns to each ring its category of modules: the "tensor product of modules" construction gives a functor that is only associative up to natural isomorphism, not strictly.
The distinction between strict and weak is the first instance of a deep pattern in higher category theory: as dimension increases, equality of composites is progressively weakened to "equivalence up to a cell one dimension higher," with coherence conditions ensuring the cells behave consistently. A strict 2-functor sends F(g ∘ f) = F(g) ∘ F(f) exactly. A weak 2-functor sends F(g ∘ f) ≅ F(g) ∘ F(f) via an invertible 2-cell φ_{g,f}, and this family of 2-cells must satisfy a pentagon-like coherence condition when composing three 1-cells. These coherence diagrams are the 2-categorical analogues of the associativity and unit axioms for monoidal categories — they ensure all ways of re-bracketing composites using the comparison 2-cells give the same result.
The builds-toward topic of higher category theory extends this pattern further: in a 3-category, 3-cells are maps between 2-cells, and weak functors between 3-categories preserve composition only up to invertible 3-cells, and so on up through the n-categorical hierarchy. The 2-categorical layer is where the essential new phenomena first appear — the distinction between strict and weak, the need for coherence data, and the recognition that "equality" of higher-dimensional structure is too strong a demand. Understanding 2-categories and weak functors gives you the conceptual vocabulary to engage with ∞-categories, homotopy type theory, and modern algebraic topology, all of which are grounded in the same hierarchical logic of cells and coherence conditions at increasing dimensions.