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1 The Language of $\infty $-Categories

A principal goal of algebraic topology is to understand topological spaces by means of algebraic and combinatorial invariants. Let us consider some elementary examples.

  • To any topological space $X$, one can associate the set $\pi _0(X)$ of path components of $X$. This is the quotient of $X$ by an equivalence relation $\simeq $, where $x \simeq y$ if there exists a continuous path $p: [0,1] \rightarrow X$ satisfying $p(0) = x$ and $p(1) = y$.

  • To any topological space $X$ equipped with a base point $x \in X$, one can associate the fundamental group $\pi _{1}(X,x)$. This is a group whose elements are homotopy classes of continuous paths $p: [0,1] \rightarrow X$ satisfying $p(0) = x = p(1)$.

For many purposes, it is useful to combine the set $\pi _0(X)$ and the fundamental groups $\{ \pi _1(X,x) \} _{x \in X}$ into a single mathematical object. To any topological space $X$, one can associate an invariant $\pi _{\leq 1}(X)$ called the fundamental groupoid of $X$. The fundamental groupoid $\pi _{\leq 1}(X)$ is a category whose objects are the points of $X$, where a morphism from a point $x \in X$ to a point $y \in X$ is given by a homotopy class of continuous paths $p: [0,1] \rightarrow X$ satisfying $p(0) = x$ and $p(1) = y$. The set of path components $\pi _0(X)$ can then be recovered as the set of isomorphism classes of objects of the category $\pi _{\leq 1}(X)$, and each fundamental group $\pi _{1}(X,x)$ can be identified with the automorphism group of the point $x$ as an object of the category $\pi _{\leq 1}(X)$. The formalism of category theory allows us to assemble information about path components and fundamental groups into a single convenient package.

The fundamental groupoid $\pi _{\leq 1}(X)$ is a very important invariant of a topological space $X$, but is far from being a complete invariant. In particular, it does not contain any information about the higher homotopy groups $\{ \pi _{n}(X,x) \} _{n \geq 2}$. We therefore ask the following:

Question 1.0.0.1. Let $X$ be a topological space. Can one devise a “category-theoretic” invariant of $X$, in the spirit of the fundamental groupoid $\pi _{\leq 1}(X)$, which contains information about all the homotopy groups of $X$?

We begin to address Question 1.0.0.1 in §1.1 by introducing the theory of simplicial sets. A simplicial set $S_{\bullet }$ is a collection of sets $\{ S_ n \} _{n \geq 0}$, which are related by face maps $\{ d_ i: S_ n \rightarrow S_{n-1} \} _{0 \leq i \leq n}$ and degeneracy maps $\{ s_ i: S_ n \rightarrow S_{n+1} \} _{0 \leq i \leq n}$ satisfying suitable identities (see Definition 1.1.1.9 and Exercise 1.1.1.8). Every topological space $X$ determines a simplicial set $\operatorname{Sing}_{\bullet }(X)$, called the singular simplicial set of $X$, with the property that each $\operatorname{Sing}_{n}(X)$ is the collection of continuous maps from the topological $n$-simplex into $X$ (Construction 1.1.5.1). Moreover, the homotopy groups of $X$ can be reconstructed from the simplicial set $\operatorname{Sing}_{\bullet }(X)$ by a simple combinatorial procedure (see §). Kan observed that this procedure can be applied more generally to any simplicial set $S_{\bullet }$ satisfying the following Kan extension condition:

$(\ast )$

For $0 \leq i \leq n$, every map $\sigma _0: \Lambda ^{n}_{i} \rightarrow S_{\bullet }$ admits an extension $\sigma : \Delta ^{n} \rightarrow S_{\bullet }$.

Here $\Delta ^ n$ denotes a certain simplicial set called the standard $n$-simplex (Construction 1.1.2.1), and $\Lambda ^{n}_{i}$ denotes a certain simplicial subset of $\Delta ^ n$ called the $i$th horn (Construction 1.1.2.9). Simplicial sets satisfying condition $(\ast )$ are called Kan complexes. Every simplicial set of the form $\operatorname{Sing}_{\bullet }(X)$ is a Kan complex (Proposition 1.1.7.3), and the converse is true up to homotopy. More precisely, Milnor proved in [MR0084138] that the construction $X \mapsto \operatorname{Sing}_{\bullet }(X)$ induces an equivalence from the (geometrically defined) homotopy theory of CW complexes to the (combinatorially defined) homotopy theory of Kan complexes; we will discuss this point in Chapter .

The singular simplicial set $\operatorname{Sing}_{\bullet }(X)$ is a natural candidate for sort of invariant requested in Question 1.0.0.1: it is a mathematical object of a purely combinatorial nature which contains complete information about the homotopy groups of $X$ and their interrelationship (from which we can even reconstruct $X$ up to homotopy equivalence, provided that $X$ has the homotopy type of a CW complex). But in order to see that it qualifies as a complete answer, we must address the following:

Question 1.0.0.2. Let $X$ be a topological space. To what extent does the simplicial set $\operatorname{Sing}_{\bullet }(X)$ behave like a category? What is the relationship between $\operatorname{Sing}_{\bullet }(X)$ with the fundamental groupoid of $X$?

Our answer to Question 1.0.0.2 begins with the observation that the theory of simplicial sets is closely related to category theory. To every category $\operatorname{\mathcal{C}}$, one can associate a simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, called the nerve of $\operatorname{\mathcal{C}}$ (we will review the construction of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ in §1.2; see Construction 1.2.1.1). The construction $\operatorname{\mathcal{C}}\mapsto \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is fully faithful (Proposition 1.2.2.1): in particular, a category $\operatorname{\mathcal{C}}$ is determined (up to canonical isomorphism) by the simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. Throughout much of this book, we will abuse notation by not distinguishing between a category $\operatorname{\mathcal{C}}$ and its nerve $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$: that is, we will view a category as a special kind of simplicial set. These simplicial sets have a simple characterization: according to Proposition 1.2.3.1, a simplicial set $S_{\bullet }$ has the form $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (for some category $\operatorname{\mathcal{C}}$) if and only it it satisfies the following variant of the Kan extension condition (Proposition 1.2.3.1):

$(\ast ')$

For $0 < i < n$, every map $\sigma _0: \Lambda ^{n}_{i} \rightarrow S_{\bullet }$ admits a unique extension $\sigma : \Delta ^{n} \rightarrow S_{\bullet }$.

The extension conditions $(\ast )$ and $(\ast ')$ are closely related, but differ in two important respects. The Kan extension condition requires that every map of simplicial sets $\sigma _0: \Lambda ^{n}_{i} \rightarrow S_{\bullet }$ admits an extension $\sigma : \Delta ^ n \rightarrow S_{\bullet }$. Condition $(\ast ')$ requires the existence of an extension only only in the case $0 < i < n$, but demands that the extension is unique. Neither of these conditions implies the other: a simplicial set of the form $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ satisfies condition $(\ast )$ if and only if the category $\operatorname{\mathcal{C}}$ is a groupoid (Proposition 1.2.4.2), and a simplicial set of the form $\operatorname{Sing}_{\bullet }(X)$ satisfies condition $(\ast ')$ if and only if every continuous path $[0,1] \rightarrow X$ is constant. However, conditions $(\ast )$ and $(\ast ')$ admit a common generalization. We will say that a simplicial set $S_{\bullet }$ is an $\infty $-category if it satisfies the following variant of $(\ast )$ and $(\ast ')$, known as the weak Kan extension condition:

$(\ast '')$

For $0 < i < n$, every map $\sigma _0: \Lambda ^{n}_{i} \rightarrow S_{\bullet }$ admits an extension $\sigma : \Delta ^ n \rightarrow S_{\bullet }$.

The theory of $\infty $-categories can be viewed as a simultaneous generalization of homotopy theory and category theory. Every Kan complex is an $\infty $-category, and every category $\operatorname{\mathcal{C}}$ determines an $\infty $-category (given by the nerve $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$). In particular, the notion of $\infty $-category answers the first part of Question 1.0.0.2: simplicial sets of the form $\operatorname{Sing}_{\bullet }(X)$ are almost never (the nerves of) categories, but are always $\infty $-categories. At this point, the reader might reasonably object that this is terminological legerdemain: to address the spirit of Question 1.0.0.2, we must demonstrate that simplicial sets of the form $\operatorname{Sing}_{\bullet }(X)$ (or, more generally, all simplicial sets satisfying condition $(\ast '')$) really behave like categories. We begin in §1.3 by explaining how to extend various elementary category-theoretic ideas to the setting of $\infty $-categories. In particular, we can associate to each $\infty $-category $S_{\bullet }$ a collection of objects (these are the elements of $S_0$), a collection of morphisms (these are the elements of $S_1$), and a composition law on morphisms. In particular, we show that any $\infty $-category $S_{\bullet }$ determines an ordinary category $\mathrm{h} \mathit{S}_{\bullet }$, called the homotopy category of $S_{\bullet }$ (Proposition 1.3.5.2). The construction of the homotopy category allows us to answer the second part of Question 1.0.0.2: for every topological space $X$, the singular simplicial set $\operatorname{Sing}_{\bullet }(X)$ is an $\infty $-category, whose homotopy category $\mathrm{h} \mathit{\operatorname{Sing}}_{\bullet }(X)$ is the fundamental groupoid $\pi _{\leq 1}(X)$ (see Example 1.3.5.5).

Roughly speaking, the difference between an $\infty $-category $S_{\bullet }$ and its homotopy category $\mathrm{h} \mathit{S}_{\bullet }$ is that the former can contain nontrivial homotopy-theoretic information (encoded by simplices of dimension $n \geq 2$, which can be loosely understood as “$n$-morphisms”) which is lost upon passage to the homotopy category $\mathrm{h} \mathit{S}_{\bullet }$. We can summarize the situation informally with the heuristic equation

\[ \{ \text{Categories} \} + \{ \text{Homotopy Theory} \} = \{ \text{$\infty $-Categories} \} , \]

or more precisely with the diagram

\[ \xymatrix { \{ \text{Categories} \} \ar@ {}[r]|{\xhookrightarrow { \operatorname{N}_{\bullet } }} & \{ \text{$\infty $-Categories} \} \ar@ {}[d]|{\cap } & \{ \text{Kan Complexes} \} \ar@ {}[l]|{\supset } \\ & \{ \text{Simplicial Sets} \} & \{ \text{Topological Spaces} \} \ar [u]_{\operatorname{Sing}_{\bullet } } } \]

Structure

  • Section 1.1: Simplicial Sets
    • Subsection 1.1.1: Simplicial and Cosimplicial Objects
    • Subsection 1.1.2: Simplices and Horns
    • Subsection 1.1.3: The Skeletal Filtration
    • Subsection 1.1.4: Directed Graphs as Simplicial Sets
    • Subsection 1.1.5: The Singular Simplicial Set of a Topological Space
    • Subsection 1.1.6: The Geometric Realization of Simplicial Set
    • Subsection 1.1.7: Kan Complexes
  • Section 1.2: The Nerve of a Category
    • Subsection 1.2.1: Construction of the Nerve
    • Subsection 1.2.2: Recovering a Category from its Nerve
    • Subsection 1.2.3: Characterization of Nerves
    • Subsection 1.2.4: The Nerve of a Groupoid
  • Section 1.3: $\infty $-Categories
    • Subsection 1.3.1: Objects and Morphisms
    • Subsection 1.3.2: The Opposite of an $\infty $-Category
    • Subsection 1.3.3: Homotopies of Morphisms
    • Subsection 1.3.4: Composition of Morphisms
    • Subsection 1.3.5: The Homotopy Category
    • Subsection 1.3.6: The Universal Property of $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$
    • Subsection 1.3.7: Equivalences
  • Section 1.4: Functors of $\infty $-Categories
    • Subsection 1.4.1: Examples of Functors
    • Subsection 1.4.2: Commutative Diagrams
    • Subsection 1.4.3: The $\infty $-Category of Functors
    • Subsection 1.4.4: Digression: Lifting Properties
    • Subsection 1.4.5: Trivial Kan Fibrations
    • Subsection 1.4.6: Uniqueness of Composition