Notation 1.1.0.1. For every nonnegative integer $n$, we let $[n]$ denote the linearly ordered set $\{ 0 < 1 < 2 < \cdots < n-1 < n \} $.
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1.1 Simplicial Sets
In this section we provide an introduction to the theory of simplicial sets, which will play an essential role throughout this book. We begin with some preliminaries.
Definition 1.1.0.2 (The Simplex Category). We define a category $\operatorname{{\bf \Delta }}$ as follows:
The objects of $\operatorname{{\bf \Delta }}$ are linearly ordered sets of the form $[n]$ for $n \geq 0$.
A morphism from $[m]$ to $[n]$ in the category $\operatorname{{\bf \Delta }}$ is a function $\alpha : [m] \rightarrow [n]$ which is nondecreasing: that is, for each $0 \leq i \leq j \leq m$, we have $0 \leq \alpha (i) \leq \alpha (j) \leq n$.
We will refer to $\operatorname{{\bf \Delta }}$ as the simplex category.
Remark 1.1.0.3. The category $\operatorname{{\bf \Delta }}$ is equivalent to the category of all nonempty finite linearly ordered sets, with morphisms given by nondecreasing maps. In fact, we can say something better: for every nonempty finite linearly ordered set $I$, there is a unique nondecreasing bijection $I \simeq [n]$, for some $n \geq 0$.
Definition 1.1.0.4. Let $\operatorname{\mathcal{C}}$ be a category. A simplicial object of $\operatorname{\mathcal{C}}$ is a functor $\operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}$. A cosimplicial object of $\operatorname{\mathcal{C}}$ is a functor $\operatorname{{\bf \Delta }}\rightarrow \operatorname{\mathcal{C}}$.
Notation 1.1.0.5. We will often use an expression like $C_{\bullet }$ to denote a simplicial object of a category $\operatorname{\mathcal{C}}$. In this case, we write $C_{n}$ for the value of the functor $C_{\bullet }$ on the object $[n] \in \operatorname{{\bf \Delta }}$. Similarly, we often use an expression like $C^{\bullet }$ to indicate a cosimplicial object of $\operatorname{\mathcal{C}}$, and $C^{n}$ for its value on $[n] \in \operatorname{{\bf \Delta }}$.
We will be primarily interested in the following special case of Definition 1.1.0.4:
Definition 1.1.0.6. Let $\operatorname{Set}$ denote the category of sets. A simplicial set is a simplicial object of $\operatorname{Set}$: that is, a functor $\operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{Set}$.
Notation 1.1.0.7. We let $\operatorname{Set_{\Delta }}= \operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{Set})$ denote the category of functors from $\operatorname{{\bf \Delta }}^{\operatorname{op}}$ to $\operatorname{Set}$. We refer to $\operatorname{Set_{\Delta }}$ as the category of simplicial sets.
Remark 1.1.0.8. Since the category of sets has all (small) limits and colimits, the category of simplicial sets also has all (small) limits and colimits. Moreover, these limits and colimits are computed levelwise: for any functor and any nonnegative integer $n$, we have canonical bijections
\[ S_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}\quad \quad (C \in \operatorname{\mathcal{C}}) \mapsto S_{\bullet }(C), \]
\[ (\varinjlim _{C \in \operatorname{\mathcal{C}}} S(C))_{n} \simeq \varinjlim _{C \in \operatorname{\mathcal{C}}} ( S_ n(C) ) \quad \quad (\varprojlim _{C \in \operatorname{\mathcal{C}}} S(C))_{n}\simeq \varprojlim _{C \in \operatorname{\mathcal{C}}} ( S_ n(C) ). \]
Example 1.1.0.9 (The Standard Simplex). Let $n \geq 0$ be an integer. We let $\Delta ^{n}$ denote the functor Then $\Delta ^ n$ is a simplicial set, which we will refer to as the standard $n$-simplex. By convention, we extend this construction to the case $n = -1$ by setting $\Delta ^{-1} = \emptyset $.
\[ \operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{Set}\quad \quad [m] \mapsto \operatorname{Hom}_{\operatorname{{\bf \Delta }}}( [m], [n] ). \]
Example 1.1.0.10. The standard $0$-simplex $\Delta ^{0}$ is a final object of the category of simplicial sets: that is, it carries each $[n] \in \operatorname{{\bf \Delta }}^{\operatorname{op}}$ to a set having a single element.
Definition 1.1.0.11. Let $S_{\bullet }$ be a simplicial set and let $n$ be a nonnegative integer. An $n$-simplex of $S_{\bullet }$ is an element of the set $S_{n}$. We will also refer to elements of $S_{0}$ as vertices of $S_{\bullet }$, and to elements of $S_{1}$ as edges of $S_{\bullet }$. We often write $v \in S_{\bullet }$ to indicate that $v$ is a vertex of $S_{\bullet }$.
Proposition 1.1.0.12. Let $n$ be a nonnegative integer and regard the identity map $\operatorname{id}_{[n]}: [n] \rightarrow [n]$ as an $n$-simplex of $\Delta ^ n$. For every simplicial set $S_{\bullet }$, evaluation on $\operatorname{id}_{[n]}$ induces a bijection
\[ \operatorname{Hom}_{ \operatorname{Set_{\Delta }}}( \Delta ^ n, S_{\bullet } ) \rightarrow S_{n} \quad \quad f \mapsto f( \operatorname{id}_{[n] } ). \]
Proof. This is a special case of Yoneda's lemma. $\square$
Notation 1.1.0.13. Let $S_{\bullet }$ be a simplicial set and let $\sigma \in S_{n}$ be an $n$-simplex of $\operatorname{\mathcal{C}}$. By virtue of Proposition 1.1.0.12, there is a unique morphism $f_{\sigma }: \Delta ^ n \rightarrow S_{\bullet }$ in the category of simplicial sets which satisfies $f_{\sigma }( \operatorname{id}_{[n]} ) = \sigma $. In practice, we will often abuse notation by identifying the $n$-simplex $\sigma $ with the morphism $f_{\sigma }$.
Remark 1.1.0.14 (Simplicial Subsets). Let $S_{\bullet }$ be a simplicial set. Suppose that:
For every integer $n \geq 0$, we are given a subset $T_{n} \subseteq S_{n}$,
For every morphism $\alpha : [m] \rightarrow [n]$ in the simplex category $\operatorname{{\bf \Delta }}$, the associated map $S_{n} \rightarrow S_{m}$ carries $T_{n}$ into $T_{m}$.
Then we the construction $[n] \mapsto T_{n}$ determines another simplicial set $T_{\bullet }$. In this case, we will say that $T_{\bullet }$ is a simplicial subset of $S_{\bullet }$ and write $T_{\bullet } \subseteq S_{\bullet }$.
Example 1.1.0.15. Let $S_{\bullet }$ be a simplicial set and let $v$ be a vertex of $S_{\bullet }$. Then $v$ can be identified with a map of simplicial sets $\Delta ^0 \rightarrow S_{\bullet }$. This map is automatically a monomorphism (note that $\Delta ^0$ has only a single $n$-simplex for every $n \geq 0$), whose image is a simplicial subset of $S_{\bullet }$. It will often be convenient to denote this simplicial subset by $\{ v \} $. For example, we can identify vertices of the standard $n$-simplex $\Delta ^ n$ with integers $i$ satisfying $0 \leq i \leq n$; every such integer $i$ determines a simplicial subset $\{ i \} \subseteq \Delta ^ n$ (whose $k$-simplices are the constant maps $[k] \rightarrow [n]$ taking the value $i$).
Our first goal in this section is to make Definition 1.1.0.6 more concrete. To a first degree of approximation, a simplicial set $S_{\bullet }$ can be viewed as a collection of sets $\{ S_{n} \} _{n \geq 0}$. However, this collection is endowed with additional structure, arising from morphisms in the simplex category $\operatorname{{\bf \Delta }}$. For example, let $n$ be a positive integer. For each $0 \leq i \leq n$, there is a unique order-preserving bijection $[n-1] \simeq [n] \setminus \{ i\} \subset [n]$. This induces a function $d^{n}_{i}: S_{n} \rightarrow S_{n-1}$ which we will refer to as a face operator for the simplicial set $S_{\bullet }$ (Construction 1.1.1.4). For $n \geq 2$ and $0 \leq i < j \leq n$, it is not difficult to show that these face operators satisfy the identity
1.1
\begin{eqnarray} \label{equation:semisimplicial-identity} d^{n-1}_{i}( d^{n}_{j}(\sigma ) ) & = & d^{n-1}_{j-1}( d^{n}_{i} (\sigma ) ) \end{eqnarray}
(see Remark 1.1.1.7). In §1.1.1, we prove a partial converse: a collection of sets $\{ S_{n} \} $ and face operators $\{ d^{n}_{i}: S_{n} \rightarrow S_{n-1} \} $ which satisfy (1.1), we can uniquely reconstruct the data of a semisimplicial set: that is, a (contravariant) set-valued functor on the subcategory $\operatorname{{\bf \Delta }}_{\operatorname{inj}} \subset \operatorname{{\bf \Delta }}$ whose morphisms are strictly increasing functions (see Proposition 1.1.1.9).
To fully recover the structure of a simplicial set $S_{\bullet }$, it is not enough to remember the face operators alone: one also needs to encode the data supplied by non-injective maps in the simplex category $\operatorname{{\bf \Delta }}$. For every pair of integers $0 \leq i \leq n$, there is a unique nondecreasing surjection $[n+1] \twoheadrightarrow [n]$ which is constant on the subset $\{ i, i+1\} $. This induces a function $s^{n}_{i}: S_{n} \rightarrow S_{n+1}$, which we refer to as the $i$th degeneracy operator (Construction 1.1.2.1). In §1.1.2, we show that a simplicial set $S_{\bullet }$ can be reconstructed from its face and degeneracy operators, which are required only to satisfy a handful of compatibility conditions (Proposition 1.1.2.14).
Let $S_{\bullet }$ be a simplicial set. We say that an $n$-simplex $\sigma \in S_{n}$ is degenerate if it belongs to the image of some degeneracy operator $s^{n-1}_{i}: S_{n-1} \rightarrow S_{n}$ (Definition 1.1.2.3). We say that $S_{\bullet }$ has dimension $\leq k$ if every $n$-simplex of $S_{\bullet }$ is degenerate for $n > k$ (Definition 1.1.3.1). Simplicial sets of low dimension are easy to describe:
A simplicial set of dimension $\leq 0$ is essentially just an ordinary set. More precisely, in §1.1.5 we show that a simplicial set $S_{\bullet }$ has dimension $\leq 0$ if and only if it is isomorphic to a constant functor $\operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{Set}$ (Proposition 1.1.5.14); in this case, we will say that $S_{\bullet }$ is discrete (Definition 1.1.5.10).
A simplicial set of dimension $\leq 1$ is essentially a directed graph. More precisely, in §1.1.6 we construct a functor from the category of simplicial sets to the category of directed graphs, and show that it is an equivalence when restricted to simplicial sets of dimension $\leq 1$ (Proposition 1.1.6.9).
Let $S$ be an arbitrary simplicial set. For every integer $k$, there is a largest simplicial subset of $S$ which has dimension $\leq k$. We will denote this simplicial subset by $\operatorname{sk}_{k}( S )$ and refer to it as the $k$-skeleton of $S$ (Construction 1.1.4.1). Allowing $k$ to vary, we can realize $S$ as the union of an increasing sequence
\[ \emptyset = \operatorname{sk}_{-1}(S ) \subseteq \operatorname{sk}_{0}( S ) \subseteq \operatorname{sk}_{1}( S ) \subseteq \operatorname{sk}_{2}( S ) \subseteq \cdots \]
which we refer to as the skeletal filtration. In §1.1.4, we analyze the transition maps which appear in the skeletal filtration. Our main result is that each of the inclusions $\operatorname{sk}_{k-1}( S ) \hookrightarrow \operatorname{sk}_{k}( S )$ is a pushout of coproducts of the inclusion map $\operatorname{\partial \Delta }^{k} \hookrightarrow \Delta ^{k}$ (Proposition 1.1.4.12). Here $\operatorname{\partial \Delta }^{k} = \operatorname{sk}_{k-1}( \Delta ^{k} )$ denotes the boundary of the standard simplex $\Delta ^{k}$ (Construction 1.1.4.10). Stated more informally, the $k$-skeleton $\operatorname{sk}_{k}( S )$ can be obtained from the $(k-1)$-skeleton $\operatorname{sk}_{k-1}(S)$ by attaching cells of dimension $k$.