Introduction to ∞-categories

This chapter provides the \(\infty \)-categorical foundations that underlie the rest of the book. Our goal is to develop a usable language of \(\infty \)-categories that matches intuition from classical category theory and aligns with the way it is used in stable homotopy theory and higher algebra.

One peculiar aspect of the theory of \(\infty \)-categories is that, while its foundations can be set up in traditional set-based mathematics, the actual practice of the theory involves some significant conceptual shifts regarding several basic mathematical notions. The premise of this book is that the most effective way to get into the field is by learning about these conceptual shifts first, and starting to do some basic mathematics with them, before discussing how these concepts can be modeled within traditional mathematics. We will achieve this by taking an axiomatic approach to \(\infty \)-categories, focusing on just those basic properties of the theory that are needed in practice.

Section 1.1 explains the heuristics behind the theory, and Section 1.2 turns them into working syntax by setting out the axioms we assume. Section 1.3 records the constructions that build new \(\infty \)-categories out of old ones: products and coproducts, pullbacks and pushouts, and functor categories. Section 1.4 then turns to the internal structure of a single \(\infty \)-category: it axiomatizes which commutative diagrams are determined by their visible edges, deduces the Segal axiom and with it the composition of morphisms, and introduces animae and hom animae, so that the heuristics of Section 1.1 become part of the formal setup. The section closes with the Rezk axiom. Afterwards, Section 1.5 returns to constructions with opposite categories, subcategories and localizations.

Section 1.6 discusses quasicategories, the standard set-based model. It is the one place in this chapter where we commit to a specific model, and the specifically quasicategorical parts may be skipped. Finally, Section 1.7 treats limits and colimits, and Section 1.8 collects the foundational results that later chapters use without further proof, among them the \(\infty \)-categories \(\An \) and \(\Cat _{\infty }\) themselves and the Yoneda lemma.

The exercises in this chapter vary considerably in difficulty. Some are routine checks, while others invite the reader to work through technical consequences of the axioms in detail. The latter may instead be read together with their solutions, and are not checkpoints that must be cleared on a first reading. We use a separate proof guide when we state a result for later use but omit its proof from the main exposition. These guides indicate the structure of the omitted argument and may likewise be skipped on a first reading. Complete proofs and further solutions may be found on the author’s webpage.

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