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1.1.1 Simplicial and Cosimplicial Objects

We begin with some preliminaries.

Notation 1.1.1.1. For every positive integer $n$, we let $[n]$ denote the linearly ordered set $\{ 0 < 1 < 2 < \cdots < n-1 < n \} $.

Definition 1.1.1.2 (The Simplex Category). Let $\operatorname{{\bf \Delta }}$ denote the category whose objects are sets of the form $[n]$, where $n$ is a nonnegative integer, where a morphism $\alpha : [m] \rightarrow [n]$ is a nondecreasing map of linearly ordered sets (that is, a map which satisfies the condition $\alpha (i) \leq \alpha (j)$ whenever $i \leq j$). We 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 $[n] \in \operatorname{{\bf \Delta }}$.

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 1.1.1.6. 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

\[ \delta ^{i}( j) = \begin{cases} j & \text{ if } j < i \\ j+1 & \text{ if } j \geq i. \end{cases} \]

If $C_{\bullet }$ is a 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 on $C^{\bullet }$ 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.

Notation 1.1.1.7. For every pair of nonnegative integers $i$ and $j$, we let $\sigma ^{i}: [n+1] \rightarrow [n]$ denote the function given by the formula

\[ \sigma ^{i}( j) = \begin{cases} j & \text{ if } j \leq i \\ j-1 & \text{ if } j > i. \end{cases} \]

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.

Exercise 1.1.1.8. Let $C_{\bullet }$ be a simplicial object of a category $\operatorname{\mathcal{C}}$. Show that the face and degeneracy maps of Notations 1.1.1.6 and 1.1.1.7 satisfy the simplicial identities

$(1)$

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 $C_{n-2}$).

$(2)$

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 $C_{n+2}$).

$(3)$

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 maps $\{ d_ i: C_ n \rightarrow C_{n-1} \} _{0 \leq i \leq n}$, $\{ s_ i: C_ n \rightarrow C_{n+1} \} _{0 \leq i \leq n}$ 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 1.1.1.9. Let $\operatorname{Set}$ denote the category of sets. A simplicial set is a simplicial object of $\operatorname{Set}$: that is, a functor $\operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{Set}$. If $S_{\bullet }$ is a 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 1.1.1.10. 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

\[ S_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}\quad \quad (C \in \operatorname{\mathcal{C}}) \mapsto S_{\bullet }(C), \]

and any nonnegative integer $n$, we have canonical bijections

\[ (\varinjlim _{C \in \operatorname{\mathcal{C}}} S)_{n}(C) \simeq \varinjlim _{C \in \operatorname{\mathcal{C}}} ( S_ n(C) ) \quad \quad (\varprojlim _{C \in \operatorname{\mathcal{C}}} S)_{n})(C) \simeq \varprojlim _{C \in \operatorname{\mathcal{C}}} ( S_ n(C) ). \]