Notation 1.1.1.1. For every nonnegative integer $n$, we let $[n]$ denote the linearly ordered set $\{ 0 < 1 < 2 < \cdots < n-1 < n \} $.
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1.1.1 Simplicial and Cosimplicial Objects
We begin with some preliminaries.
Definition 1.1.1.2 (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 1.1.1.3. 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 even better: for every nonempty finite linearly ordered set $I$, there is a unique order-preserving bijection $I \simeq [n]$, for some $n \geq 0$.
Definition 1.1.1.4. Let $\operatorname{\mathcal{C}}$ be any category. A simplicial object of $\operatorname{\mathcal{C}}$ is a functor $\operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}$. Dually, a cosimplicial object of $\operatorname{\mathcal{C}}$ is a functor $\operatorname{{\bf \Delta }}\rightarrow \operatorname{\mathcal{C}}$.
Notation 1.1.1.5. We will often use the expression $C_{\bullet }$ 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 use the notation $C^{\bullet }$ to indicate a cosimplicial object of $\operatorname{\mathcal{C}}$, and $C^{n}$ for its value on $[n] \in \operatorname{{\bf \Delta }}$.
Variant 1.1.1.6. Let $\operatorname{{\bf \Delta }}_{\operatorname{inj}}$ denote the category whose objects are sets of the form $[n]$ (where $n$ is a nonnegative integer) and whose morphisms are strictly increasing functions $\alpha : [m] \hookrightarrow [n]$. If $\operatorname{\mathcal{C}}$ is any category, we will refer to a functor $\operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}$ as a semisimplicial object of $\operatorname{\mathcal{C}}$. We typically use the notation $C_{\bullet }$ to indicate a semisimplicial object of $\operatorname{\mathcal{C}}$, whose value on an object $[n] \in \operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}}$ we denote by $C_{n}$.
Remark 1.1.1.7. The category $\operatorname{{\bf \Delta }}_{\operatorname{inj}}$ of Variant 1.1.1.6 can be regarded as a (non-full) subcategory of the category $\operatorname{{\bf \Delta }}$ of Definition 1.1.1.2. Consequently, any simplicial object $C_{\bullet }$ of a category $\operatorname{\mathcal{C}}$ determines a semisimplicial object of $\operatorname{\mathcal{C}}$, given by the composition We will often abuse notation by identifying a simplicial object $C_{\bullet }$ of $\operatorname{\mathcal{C}}$ with the underlying semisimplicial object of $\operatorname{\mathcal{C}}$.
\[ \operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}} \hookrightarrow \operatorname{{\bf \Delta }}^{\operatorname{op}} \xrightarrow { C_{\bullet } } \operatorname{\mathcal{C}}. \]
To a first degree of approximation, a simplicial object $C_{\bullet }$ of a category $\operatorname{\mathcal{C}}$ can be identified with the collection of objects $\{ C_ n \} _{n \geq 0}$. However, these objects are equipped with additional structure, arising from the morphisms in the simplex category $\operatorname{{\bf \Delta }}$. We now spell this out more concretely.
Notation 1.1.1.8. Let $n$ be a positive integer. For $0 \leq i \leq n$, we let $\delta ^{i}: [n-1] \rightarrow [n]$ denote the unique strictly increasing function whose image does not contain the element $i$, given concretely by the formula If $C_{\bullet }$ is a (semi)simplicial object of a category $\operatorname{\mathcal{C}}$, then we can evaluate $C_{\bullet }$ on the morphism $\delta ^{i}$ to obtain a morphism from $C_{n}$ to $C_{n-1}$. We will denote this map by $d_{i}: C_{n} \rightarrow C_{n-1}$ and refer to it as the $i$th face map. Dually, if $C^{\bullet }$ is a cosimplicial object of a category $\operatorname{\mathcal{C}}$, then the evaluation of $C^{\bullet }$ on the morphism $\delta ^{i}$ determines a map $d^{i}: C^{n-1} \rightarrow C^{n}$, which we refer to as the $i$th coface map.
\[ \delta ^{i}( j) = \begin{cases} j & \text{ if } j < i \\ j+1 & \text{ if } j \geq i. \end{cases} \]
Notation 1.1.1.9. For every pair of integers $0 \leq i \leq n$ we let $\sigma ^{i}: [n+1] \rightarrow [n]$ denote the function given by the formula If $C_{\bullet }$ is a simplicial object of a category $\operatorname{\mathcal{C}}$, then we can evaluate $C_{\bullet }$ on the morphism $\sigma ^{i}$ to obtain a morphism from $C_{n}$ to $C_{n+1}$. We will denote this map by $s_{i}: C_{n} \rightarrow C_{n+1}$ and refer to it as the $i$th degeneracy map. Dually, if $C^{\bullet }$ is a cosimplicial object of a category $\operatorname{\mathcal{C}}$, then the evaluation on $C^{\bullet }$ on the morphism $\sigma ^{i}$ determines a map $s^{i}: C^{n+1} \rightarrow C^{n}$, which we refer to as the $i$th codegeneracy map.
\[ \sigma ^{i}( j) = \begin{cases} j & \text{ if } j \leq i \\ j-1 & \text{ if } j > i. \end{cases} \]
Exercise 1.1.1.10. Let $C_{\bullet }$ be a semisimplicial object of a category $\operatorname{\mathcal{C}}$. Show that the face maps of Notation 1.1.1.8 satisfy the following condition:
For $n \geq 2$ and $0 \leq i < j \leq n$, we have $d_{i} \circ d_{j} = d_{j-1} \circ d_{i}$ (as a map from $C_{n}$ to $C_{n-2}$).
Conversely, show that any collection of objects $\{ C_ n \} _{n \geq 0}$ and morphisms $\{ d_ i: C_ n \rightarrow C_{n-1} \} _{0 \leq i \leq n}$, satisfying $(\ast )$ determines a unique semisimplicial object of $\operatorname{\mathcal{C}}$.
Exercise 1.1.1.11. Let $C_{\bullet }$ be a simplicial object of a category $\operatorname{\mathcal{C}}$. Show that the face and degeneracy maps of Notations 1.1.1.8 and 1.1.1.9 satisfy the simplicial identities
For $n \geq 2$ and $0 \leq i < j \leq n$, we have $d_{i} \circ d_{j} = d_{j-1} \circ d_{i}$ (as a map from $C_{n}$ to $C_{n-2}$).
For $0 \leq i \leq j \leq n$, we have $s_{i} \circ s_ j = s_{j+1} \circ s_ i$ (as a map from $C_{n}$ to $C_{n+2}$).
For $0 \leq i,j \leq n$, we have
\[ d_{i} \circ s_ j = \begin{cases} s_{j-1} \circ d_ i & \text{ if } i < j \\ \operatorname{id}_{ C_ n } & \text{ if } i = j \text{ or } i = j + 1 \\ s_{j} \circ d_{i-1} & \text{ if } i > j+1. \end{cases} \]
Conversely, show that any collection of objects $\{ C_ n \} _{n \geq 0}$ and morphisms $\{ d_ i: C_ n \rightarrow C_{n-1} \} _{0 \leq i \leq n}$, $\{ s_ i: C_ n \rightarrow C_{n+1} \} _{0 \leq i \leq n}$ satisfying $(1)$, $(2)$, and $(3)$ determines a (unique) simplicial object of $\operatorname{\mathcal{C}}$.
We will be primarily interested in the following special case of Definition 1.1.1.4:
Definition 1.1.1.12. Let $\operatorname{Set}$ denote the category of sets. A simplicial set is a simplicial object of $\operatorname{Set}$: that is, a functor $\operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{Set}$. A semisimplicial set is a semisimplicial object of $\operatorname{Set}$: that is, a functor $\operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}} \rightarrow \operatorname{Set}$. If $S_{\bullet }$ is a (semi)simplicial set, then we will refer to elements of $S_{n}$ as $n$-simplices of $S_{\bullet }$. We will also refer to the elements of $S_{0}$ as vertices of $S_{\bullet }$, and to the elements of $S_{1}$ as edges of $S_{\bullet }$. 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 1.1.1.13. Since the category of sets has all (small) limits and colimits, the category of (semi)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
\[ S_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}\quad \quad (C \in \operatorname{\mathcal{C}}) \mapsto S_{\bullet }(C), \]
\[ (\varinjlim _{C \in \operatorname{\mathcal{C}}} S(C))_{n} \simeq \varinjlim _{C \in \operatorname{\mathcal{C}}} ( S_ n(C) ) \quad \quad (\varprojlim _{C \in \operatorname{\mathcal{C}}} S(C))_{n}\simeq \varprojlim _{C \in \operatorname{\mathcal{C}}} ( S_ n(C) ). \]