Toward categories and higher structures

Take two continuous paths that have the same endpoints. They may be different as functions but deformable into one another while their endpoints stay fixed. We can identify them by passing to homotopy classes, or retain the deformation as additional data. Once we retain it, we can ask about deformations between deformations. Higher category theory supplies ways to organize this information while still composing maps. The motivation is concrete: replacing everything by equivalence classes can discard relationships that later constructions need. Learning the subject means understanding how those relationships are encoded and made coherent.

Have ordinary examples first

Begin with ordinary category theory in Emily Riehl's Category Theory in Context. Chapters 1–4 provide the main spine: categories and functors, natural transformations, representability and Yoneda, limits, and adjunctions. Keep a familiar construction beside each definition. For example, the product of two vector spaces has projection maps through which every pair of linear maps from a third space factors uniquely. Verify that assertion before calling the product a limit. The point of a universal property is that it tells you how to use a construction and compare it with another candidate.

The useful preparation is a stock of examples from linear algebra, groups, rings, or topology, together with confidence proving statements about maps. You can start with one or two of these subjects and expand your examples as needed. To skip the introductory categorical material, construct a natural transformation, prove a simple universal property, and explain an adjunction through its hom-set bijection, including naturality. Merely recognizing the diagrams is a weaker test. If Yoneda feels detached from everything else, calculate a representable functor on a small category and work out what a natural transformation from it actually determines.

Why keep homotopical information?

The next bridge should involve topology or homological algebra with computations. Paths and homotopies make higher morphisms visible; chain complexes make equivalences and derived constructions useful. Depending on your destination, review fundamental groups and basic homotopy, or work through chain maps, chain homotopies, and homology. These are alternative sources of motivation, not instructions to finish every course before looking ahead. What matters is being able to explain one example where preserving extra homotopical information changes the mathematical question.

Riehl's A Leisurely Introduction to Simplicial Sets offers a short, concrete entry into one language for that information. Study the simplex category and face and degeneracy maps, then the nerve of a category and the singular complex of a space. Draw the simplices in low dimensions. In the nerve, a pair of composable arrows determines a triangle expressing composition. Section 5 reaches Kan complexes and quasi-categories; the difference between filling every horn and filling inner horns distinguishes two important kinds of structure. Short length does not remove the need to check the definitions on examples.

A further change of language

For deeper study, Emily Riehl and Dominic Verity's Elements of ∞-Category Theory develops an approach using infinity-cosmoi and their homotopy 2-categories. Read the preface to understand the aim, then Section 1.1 on quasi-categories before proceeding to the axiomatic framework. This is a later text, with substantial categorical and homotopical demands. A concrete simplicial route and this more axiomatic route emphasize different work: the former makes the encoding visible, while the latter develops arguments that apply across several models. Choose according to the examples and constructions you want to study.

Keep equality, isomorphism, equivalence, and homotopy separate. A diagram that commutes up to a chosen homotopy contains data beyond an assertion that its composites represent the same homotopy class; compatibility among such choices creates further questions. Also specify which higher categories you mean: in an $(\infty,1)$-category, morphisms above dimension one are invertible in the appropriate higher sense. After the foundations, algebraic examples can lead toward stable categories and derived algebra, while geometric examples can lead toward spaces, descent, and higher stacks. Carry an explicit example through each change of language, and check what the new framework preserves before adopting its general vocabulary.

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