Chapter 1: The Language of $\infty $-Categories
Section 1.1: Simplicial Sets (cite)
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: Discrete Simplicial Sets
Subsection 1.1.5: Directed Graphs as Simplicial Sets
Subsection 1.1.6: Connected Components of Simplicial Sets
Subsection 1.1.7: The Singular Simplicial Set of a Topological Space
Subsection 1.1.8: The Geometric Realization of a Simplicial Set
Subsection 1.1.9: Kan Complexes
1.1 Simplicial Sets
For each integer $n \geq 0$, we let
\[ | \Delta ^{n} | = \{ (t_0, t_1, \ldots , t_ n) \in [0,1]^{n+1} : t_0 + t_1 + \cdots + t_ n = 1 \} \]
denote the topological simplex of dimension $n$. For any topological space $X$, we will refer to a continuous map $\sigma : | \Delta ^ n | \rightarrow X$ as a singular $n$-simplex in $X$. Every singular $n$-simplex $\sigma $ determines a finite collection of singular $(n-1)$-simplices $\{ d_ i \sigma \} _{0 \leq i \leq n}$, called the faces of $\sigma $, which are given explicitly by the formula
\[ (d_ i \sigma )( t_0, \ldots , t_{n-1} ) = \sigma ( t_0, t_1, \ldots , t_{i-1}, 0, t_{i}, \ldots , t_{n-1} ). \]
Let $\operatorname{Sing}_{n}(X) = \operatorname{Hom}_{\operatorname{Top}}( | \Delta ^ n |, X)$ denote the set of singular $n$-simplices of $X$. Many important algebraic invariants of $X$ can be directly extracted from the sets $\{ \operatorname{Sing}_ n(X) \} _{n \geq 0}$ and the face maps $\{ d_ i: \operatorname{Sing}_{n}(X) \rightarrow \operatorname{Sing}_{n-1}(X) \} _{0 \leq i \leq n}$.
Example 1.1.0.1 (Singular Homology). For any topological space $X$, the singular homology groups $\operatorname{ \mathrm{H} }_{\ast }(X; \operatorname{\mathbf{Z}})$ are defined as the homology groups of a chain complex
\[ \cdots \xrightarrow {\partial } \operatorname{\mathbf{Z}}[ \operatorname{Sing}_{2}(X) ] \xrightarrow { \partial } \operatorname{\mathbf{Z}}[ \operatorname{Sing}_1(X) ] \xrightarrow {\partial } \operatorname{\mathbf{Z}}[ \operatorname{Sing}_0(X) ], \]
where $\operatorname{\mathbf{Z}}[ \operatorname{Sing}_ n(X) ]$ denotes the free abelian group generated by the set $\operatorname{Sing}_ n(X)$ and the differential is given on generators by the formula
\[ \partial (\sigma ) = \sum _{i = 0}^{n} (-1)^{i} d_ i \sigma . \]
For some other algebraic invariants, it is convenient to keep track of a bit more structure. A singular $n$-simplex $\sigma : | \Delta ^ n | \rightarrow X$ also determines a collection of singular $(n+1)$-simplices $\{ s_ i \sigma \} _{0 \leq i \leq n}$, given by the formula
\[ (s_ i \sigma )( t_0, \ldots , t_{n+1} ) = \sigma ( t_0, t_1, \ldots , t_{i-1}, t_{i} + t_{i+1}, t_{i+2}, \ldots , t_{n+1}). \]
The resulting constructions $s_ i: \operatorname{Sing}_{n}(X) \rightarrow \operatorname{Sing}_{n+1}(X)$ are called degeneracy maps, because singular $(n+1)$-simplices of the form $s_ i \sigma $ factor through the linear projection $| \Delta ^{n+1} | \rightarrow | \Delta ^ n |$. For example, the map $s_0: \operatorname{Sing}_0(X) \rightarrow \operatorname{Sing}_{1}(X)$ carries each point $x \in X \simeq \operatorname{Sing}_0(X)$ to the constant map $\underline{x}: | \Delta ^1 | \rightarrow X$ taking the value $x$.
Example 1.1.0.2 (The Fundamental Group). Let $X$ be a topological space equipped with a base point $x \in X \simeq \operatorname{Sing}_0(X)$. Then continuous paths $p: [0,1] \rightarrow X$ satisfying $p(0) = x = p(1)$ can be identified with elements of the set $\{ \sigma \in \operatorname{Sing}_1(X): d_0(\sigma ) = x = d_1(\sigma ) \} $. The fundamental group $\pi _1(X,x)$ can then be described as the quotient
\[ \{ \sigma \in \operatorname{Sing}_1(X): d_0(\sigma ) = x = d_1(\sigma ) \} / \simeq , \]
where $\simeq $ is the equivalence relation on $\operatorname{Sing}_1(X)$ described by
\[ ( \sigma \simeq \sigma ' ) \Leftrightarrow ( \exists \tau \in \operatorname{Sing}_2(X) ) [ d_0(\tau ) = s_0(x) \text{ and } d_1(\tau ) = \sigma \text{ and } d_2(\tau ) = \sigma ' ]. \]
The datum of a $2$-simplex $\tau $ satisfying these conditions is equivalent to the datum of a continuous map $| \Delta ^2 | \rightarrow X$ with boundary behavior as indicated in the diagram
\[ \xymatrix { & x \ar [dr]^{ \underline{x} } \\ x \ar [ur]^{\sigma '} \ar [rr]^{\sigma } & & x; } \]
such a map can be identified with a homotopy between the paths determined by $\sigma $ and $\sigma '$.
Motivated by the preceding examples, we can ask the following:
Question 1.1.0.3. Given a topological space $X$, what can we say about the collection of sets $\{ \operatorname{Sing}_{n}(X) \} _{n \geq 0}$, together with the face and degeneracy maps
\[ d_{i}: \operatorname{Sing}_{n}(X) \rightarrow \operatorname{Sing}_{n-1}(X) \quad \quad s_ i: \operatorname{Sing}_{n}(X) \rightarrow \operatorname{Sing}_{n+1}(X)? \]
What sort of mathematical structure do they form?
In [MR0035434], Eilenberg and Zilber supplied an answer to Question 1.1.0.3 by introducing what they called complete semi-simplicial complexes, which are now more commonly known as simplicial sets. Roughly speaking, a simplicial set $S_{\bullet }$ is a collection of sets $\{ S_ n \} _{n \geq 0}$ indexed by the nonnegative integers, equipped with face and degeneracy operators $\{ d_{i}: S_{n} \rightarrow S_{n-1}, s_ i: S_ n \rightarrow S_{n+1} \} _{0 \leq i \leq n}$ satisfying a short list of identities. These identities can be summarized conveniently by saying that a simplicial set is a presheaf on the simplex category $\operatorname{{\bf \Delta }}$, whose definition we review in §1.1.1.
Simplicial sets are connected to algebraic topology by two closely related constructions:
For every topological space $X$, the face and degeneracy operators defined above endow the collection $\{ \operatorname{Sing}_{n}(X) \} _{n \geq 0}$ with the structure of a simplicial set. We denote this simplicial set by $\operatorname{Sing}_{\bullet }(X)$ and refer to it as the singular simplicial set of $X$ (see Construction 1.1.7.1). These simplicial sets tend to be quite large: in any nontrivial example, the sets $\operatorname{Sing}_{n}(X)$ will be uncountable for every nonnegative integer $n$.
Any simplicial set $S_{\bullet }$ can be regarded as a "blueprint" for constructing a topological space $| S_{\bullet } |$ called the geometric realization of $S_{\bullet }$, which can be obtained as a quotient of the disjoint union $\coprod _{n \geq 0} S_ n \times | \Delta ^ n |$ by an equivalence relation determined by the face and degeneracy operators on $S_{\bullet }$. Many topological spaces of interest (for example, any space which admits a finite triangulation) can be realized as a geometric realization of a simplicial set $S_{\bullet }$ having only finitely many nondegenerate simplices; we will discuss some elementary examples in §1.1.2.
These constructions determine adjoint functors
\[ \xymatrix@1{ \operatorname{Set_{\Delta }} \ar@ <.4ex>[r]^-{ | \, | } & \operatorname{Top} \ar@ <.4ex>[l]^-{ \operatorname{Sing}_{\bullet } }} \]
relating the category $\operatorname{Set_{\Delta }}$ of simplicial sets to the category $\operatorname{Top}$ of topological spaces. We review the constructions of these functors in §1.1.7 and §1.1.8, viewing them as instances of a general paradigm (Variant 1.1.7.7 and Proposition 1.1.8.22) which will appear repeatedly in Chapter 2.
For any (pointed) topological space $X$, Examples 1.1.0.1 and 1.1.0.2 show that the singular homology and fundamental group of $X$ can be recovered from the simplicial set $\operatorname{Sing}_{\bullet }(X)$. In fact, one can say more: under mild assumptions, the entire homotopy type of $X$ can be recovered from $\operatorname{Sing}_{\bullet }(X)$. More precisely, there is always a canonical map $| \operatorname{Sing}_{\bullet }(X) | \rightarrow X$ (given by the counit of the adjunction described above), and Giever proved that it is always a weak homotopy equivalence (hence a homotopy equivalence when $X$ has the homotopy type of a CW complex; see Proposition 3.5.3.8). Consequently, for the purpose of studying homotopy theory, nothing is lost by replacing $X$ by $\operatorname{Sing}_{\bullet }(X)$ and working in the setting of simplicial sets, rather than topological spaces. In fact, it is possible to develop the theory of algebraic topology in entirely combinatorial terms, using simplicial sets as surrogates for topological spaces. However, not every simplicial set $S_{\bullet }$ behaves like the singular complex of a space; it is therefore necessary to single out a class of "good" simplicial sets to work with. In §1.1.9 we introduce a special class of simplicial sets, called Kan complexes (Definition 1.1.9.1). By a theorem of Milnor ([MR0084138]), the homotopy theory of Kan complexes is equivalent to the classical homotopy theory of CW complexes; we will return to this point in Chapter 3.