Before setting up the axiomatics of \(\infty \)-categories used in this book, we take a moment to explain the heuristics behind the theory, which will help motivate our choice of setup.
The fundamental principle of homotopy theory
The first concept we need to discuss is the notion of equality. In traditional mathematics, equality is typically viewed as a mere property: two mathematical objects are either equal or they are not. In homotopy theory and related fields, however, we encounter a more sophisticated notion of ‘sameness’ where it is not just important that two objects are ‘the same’, but precisely how they are ‘the same’. In particular, the sameness relationship between objects becomes a mathematical object in its own right: think of a homotopy between two continuous maps, or of an isomorphism between two abelian groups. This perspective is so central to understanding \(\infty \)-categories that it deserves to be elevated to a guiding principle:
Fundamental Principle of Homotopy Theory: When expressing that two entities are equal, we must always specify how they are equal by providing a homotopy/isomorphism between them.
To get a feeling for this principle, let us look at three familiar settings in which it already appears:
Example 1.1.1 (Algebraic topology). Algebraic topology provides the most natural example of the Fundamental Principle:
- Two continuous maps \(f,g\colon X \to Y\) are considered ‘equal up to homotopy’ if we provide a homotopy \(H\colon X \times [0,1] \to Y\) between \(f\) and \(g\);
- Two spaces \(X\) and \(Y\) are then considered ‘homotopically the same’ when equipped with a homotopy equivalence: continuous maps \(f\colon X \to Y\) and \(g\colon Y \to X\) together with chosen homotopies \(gf \sim \id _X\) and \(fg \sim \id _Y\).
- When studying diagrams of spaces, we similarly replace strict commutativity by homotopy
commutativity. For instance, given a square we do not ask for a strict equality \(h \circ f = k \circ g\), but instead require a specified homotopy \(H\colon h \circ f \sim k \circ g\) witnessing how the two compositions are ‘the same’. This perspective naturally leads to homotopy coherent diagrams, where we must also specify compatibilities between different choices of homotopies.
Example 1.1.2 (Homological algebra). The phenomenon of ‘equality as structure’ also naturally appears in homological algebra, where it leads us to think about chain complexes not as rigid algebraic objects but as representatives of more flexible homotopical entities:
- Instead of asking whether two chain maps \(f,g\colon C_{\bullet } \to D_{\bullet }\) between chain complexes are equal, we ask whether they are chain homotopic, meaning that there exists a family of maps \(\phi _n\colon C_n \to D_{n+1}\) satisfying \(f_n - g_n = \partial ^D \circ \phi _n + \phi _{n-1} \circ \partial ^C\). Likewise, a chain map \(C_{\bullet } \to D_{\bullet }\) is regarded as an equivalence if it admits an inverse up to chain homotopy.
- In practice, one often wants to go further and even identify two chain complexes whenever they are connected by a quasi-isomorphism, that is, a chain map inducing isomorphisms on all homology groups. Forcing the quasi-isomorphisms to become actual isomorphisms results in the derived category of chain complexes.
Example 1.1.3 (Category theory). A third example comes from category theory:
- For two objects \(X\) and \(Y\) in a 1-category \(C\), the natural notion of equality between them is often not the strict set-theoretic equality, but rather the existence of an isomorphism \(f\colon X \iso Y\). Likewise, universal constructions in category theory are typically unique only ‘up to unique isomorphism’ rather than strictly equal.
- This pattern extends to functors: we wish to think of two functors \(F,G\colon C \to D\) as ‘the same’ whenever we are given a natural isomorphism \(\eta \colon F \iso G\). If we think of the individual isomorphisms \(\eta _X \colon F(X) \iso G(X)\) as ‘equalities’, then the naturality condition becomes a statement about structural properties of these equalities.
- In monoidal categories, the associativity and unitality conditions are expressed not as strict equalities but as natural isomorphisms: for objects \(X\), \(Y\), and \(Z\), we have an associator \(\alpha _{X,Y,Z}\colon (X \otimes Y) \otimes Z \iso X \otimes (Y \otimes Z)\) and unitors \(\lambda _X\colon 1 \otimes X \iso X\) and \(\rho _X\colon X \otimes 1 \iso X\). The coherence conditions (like the Mac Lane pentagon) then govern how these isomorphisms relate to each other.
The common pattern in all these examples is that it is not enough to know that two things are the same: one must also remember the witness, and often also the compatibilities between different such witnesses.
Animae
Once equality is no longer a mere property, the usual notion of a collection also has to change. In traditional set-based mathematics, the primitive notion of a collection is that of a set. The behavior of sets is governed by axioms which strongly reflect the traditional perspective on equality: given a set \(X\), two of its elements \(x\) and \(y\) are either equal or not equal. The question of how \(x\) and \(y\) are equal is a meaningless one.
If we take the Fundamental Principle fully seriously, we are thus led to a different notion of collection, one in which equality between elements is a structure rather than a property. In recent years, the homotopy theory community has started to adopt the term anima for this concept. In terms of traditional mathematics, one may think of animae as homotopy types of topological spaces; we will discuss this viewpoint at length in Chapter 2. But the philosophy we adopt in this book is that animae are best thought of as primitive objects in their own right, whose role is similar to that of sets in traditional mathematics. In particular, the familiar constructions on sets still have analogues for animae:
- Given animae \(X\) and \(Y\), there is a product \(X \times Y\), whose elements are pairs \((x,y)\) consisting of an element \(x\) of \(X\) and an element \(y\) of \(Y\);
- Given animae \(X\) and \(Y\), there is a disjoint union \(X \sqcup Y\), an element of which is either an element of \(X\) or an element of \(Y\);
- There is an empty anima \(\emptyset \), which does not have any elements;
- There is a one-point anima \(*\), which is non-empty, and in which all elements are equal.
The main difference between animae and sets lies in what it means for two elements \(x\) and \(y\) of an anima \(X\) to be equal. Rather than equality being a yes/no question, an equality between \(x\) and \(y\) should be an actual mathematical object. The collection of all such equalities should then itself form an anima, which we will denote by \(X(x,y)\). In turn, if \(\alpha \) and \(\beta \) are two such equalities, we may form the anima \(X(x,y)(\alpha ,\beta )\) of equalities between \(\alpha \) and \(\beta \). This process may be repeated ad infinitum, resulting in equalities between equalities between (...) between equalities.
The following are some examples of animae:
Example 1.1.4. Every set \(X\) gives rise to an anima, which we will again denote by \(X\). Given two elements \(x,y \in X\), the anima \(X(x,y)\) of equalities between \(x\) and \(y\) is either the one-point anima (if \(x = y\)) or the empty anima (if \(x \neq y\)).
Example 1.1.5. Every 1-category \(C\) gives rise to an anima \(C^{\simeq }\), called its groupoid core. The elements of \(C^{\simeq }\) are the objects of \(C\). The anima \(C^{\simeq }(X,Y)\) between two objects \(X\) and \(Y\) is the set of isomorphisms between \(X\) and \(Y\) (regarded as an anima via the previous example).
Example 1.1.6. Every topological space \(X\) gives rise to an anima \(\Pi _{\infty }(X)\), called its underlying anima (or fundamental \(\infty \)-groupoid). Every point in \(X\) defines an element of \(\Pi _{\infty }(X)\). Given points \(x,y \in X\), every path \(p\colon [0,1] \to X\) from \(x\) to \(y\) defines an equality in \(\Pi _{\infty }(X)\) between \(x\) and \(y\). Given two such paths \(p\) and \(q\), every homotopy between \(p\) and \(q\) defines an equality between them in \(\Pi _{\infty }(X)(x,y)\). Every homotopy between homotopies defines an equality of equalities. This pattern repeats ad infinitum. We will study the assignment \(X \mapsto \Pi _{\infty }(X)\) in more detail in Chapter 2.
Just as sets come with maps between them, animae come with maps between them. A map of animae \(f\colon X \to Y\) assigns to every element \(x\) of \(X\) an element \(f(x)\) of \(Y\), to every equality between \(x\) and \(x'\) an equality between \(f(x)\) and \(f(x')\), to every equality between two such equalities an equality between their images, and so on. The role played by bijections of sets is now played by equivalences of animae: those maps which admit an inverse, together with the equalities witnessing that both composites are the identity.
Uniqueness
The homotopical perspective on equality also forces us to reconsider uniqueness. Classically, saying that a set \(X\) contains a unique element with some given property means that the subset of all valid choices has exactly one element, or equivalently that the map from this subset to the one-point set \(*\) is a bijection. Replacing sets by animae and bijections by equivalences, the homotopical analogue reads: the anima of all valid choices is contractible, meaning that the map from it to the one-point anima \(*\) is an equivalence. In this sense, contractibility is the homotopical incarnation of ‘there is exactly one’: it says that a choice exists, that any two choices are equal, that any two equalities between choices are themselves equal, and so on.
The homotopical interpretation of uniqueness appears under various names in the classical literature. In category theory, the phrase ‘up to unique isomorphism’ is often used for objects satisfying a universal property. It expresses precisely that the anima (i.e., groupoid) of such objects, together with the isomorphisms compatible with their universal structures, is contractible. In homotopy theory, a common phrase is ‘up to contractible choice’. For example, to concatenate a finite sequence of loops in a pointed space, one may first choose a subdivision of the interval. Although no particular subdivision is preferred, the anima (i.e., space) of such choices is contractible. In this book, all uniqueness statements should be interpreted in the homotopical sense, and we simply say unique.
Categories
Having replaced sets by animae, we can now make the same replacement in the notion of a category. Instead of a set of objects and sets of morphisms between them, we now allow the objects to form an anima, and for each pair of objects the morphisms between them to form an anima as well. The resulting composition law is then no longer expected to be strictly associative and unital, but only associative and unital up to coherent higher isomorphism. This is the heuristic origin of the notion of an \((\infty ,1)\)-category, or just an \(\infty \)-category for short.1
One should not take this heuristic too literally as a definition: replacing sets by animae suggests the right shape of the theory, but turning this into a workable notion of \(\infty \)-category requires encoding highly non-trivial coherence data for composition. This is one of the reasons why, later in this chapter, we will take \(\infty \)-categories as primitive and describe their behavior axiomatically rather than trying to define them directly from this heuristic.
Let us now go through some basic families of examples of \(\infty \)-categories:
Example 1.1.7. Every classical category \(C\) defines an \(\infty \)-category, whose anima of objects is the groupoid core \(C^{\simeq }\) from Example 1.1.5, and whose hom animae are the hom sets of \(C\).
Example 1.1.8. Combining the previous example with Example 1.1.6, also every topologically enriched category defines an \(\infty \)-category, by applying \(\Pi _{\infty }\) to the hom spaces.
Example 1.1.9. Every anima \(X\) defines an \(\infty \)-category, again denoted \(X\), whose objects are the elements of \(X\) and whose animae of morphisms from \(x\) to \(y\) are the animae \(X(x,y)\) of equalities. The categorical composition corresponds to the fact that equality is reflexive and transitive. Symmetry of equality is encoded by the condition that every morphism in \(X\) admits an inverse.
In fact, in our axiomatic setup, this is how animae will be defined: as those \(\infty \)-categories in which every morphism is invertible. The animae \(X(x,y)\) from before are then nothing but the hom animae \(\Hom _X(x,y)\) of this \(\infty \)-category.
While the conceptual shifts involved in \(\infty \)-category theory may feel daunting at first, the theory broadly behaves just like classical category theory once one gets used to it. Standard constructions from category theory still make sense for \(\infty \)-categories, such as products, coproducts, pullbacks, pushouts, functor categories, groupoid cores, subcategories, and localizations.
Notes
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