# Kerodon

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### 1.1.1 Simplicial and Cosimplicial Objects

We begin with some preliminaries.

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

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 $\operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}$. 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 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 1.1.1.4:

Definition 1.1.1.6. Let $\operatorname{Set}$ denote the category of sets. A simplicial set is a simplicial object of $\operatorname{Set}$: that is, a functor $\operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{Set}$. 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 1.1.1.7. 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

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

Our goal for the rest of this section is describe simplicial sets (and simplicial objects in general) in more concrete terms.

Variant 1.1.1.8. Let $\operatorname{{\bf \Delta }}_{\operatorname{inj}}$ denote the category whose objects are linearly ordered sets of the form $[n] = \{ 0 < 1 < \cdots < n \}$ (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 1.1.1.9. The category $\operatorname{{\bf \Delta }}_{\operatorname{inj}}$ of Variant 1.1.1.8 can be regarded as a (non-full) subcategory of the simplex category $\operatorname{{\bf \Delta }}$ of Definition 1.1.1.2. Consequently, any simplicial object $C_{\bullet }$ of a category $\operatorname{\mathcal{C}}$ determines an underlying semisimplicial object of $\operatorname{\mathcal{C}}$, given by the composition

$\operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}} \hookrightarrow \operatorname{{\bf \Delta }}^{\operatorname{op}} \xrightarrow { C_{\bullet } } \operatorname{\mathcal{C}}.$

We will often abuse terminology by identifying a simplicial object of $\operatorname{\mathcal{C}}$ with its underlying semisimplicial object.

To a first degree of approximation, a simplicial object of a category $\operatorname{\mathcal{C}}$ can be regarded as a sequence of objects $\{ C_ n \} _{n \geq 0}$. However, these sets are equipped with additional structure, arising from the morphisms in the simplex category $\operatorname{{\bf \Delta }}$. We now spell this out more concretely.

Notation 1.1.1.10 (Face Operators). Let $n$ be a positive integer. For $0 \leq i \leq n$, we let $\delta ^{i}_{n}: [n-1] \rightarrow [n]$ denote the unique strictly increasing function whose image does not contain the element $i$, given concretely by the formula

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

If $C_{\bullet }$ is a (semi)simplicial object of a category $\operatorname{\mathcal{C}}$, then we can evaluate $C_{\bullet }$ on the morphism $\delta ^{i}_{n}$ to obtain a morphism from $C_{n}$ to $C_{n-1}$. We will denote this map by $d^{n}_{i}: C_{n} \rightarrow C_{n-1}$ and refer to it as the $i$th face operator.

Remark 1.1.1.11 (Relations Among Face Operators). Let $n \geq 2$ be an integer. For every pair of integers $0 \leq i < j \leq n$, the diagram of linearly ordered sets

$\xymatrix@R =50pt@C=50pt{ [n-2] \ar [r]^-{\delta ^{i}_{n-1}} \ar [d]^{ \delta ^{j-1}_{n-1} } & [n-1] \ar [d]^{ \delta ^{j}_{n}} \\[n-1] \ar [r]^-{ \delta ^{i}_{n} } & [n] }$

is commutative. It follows that, if $C_{\bullet }$ is a semisimplicial object of a category $\operatorname{\mathcal{C}}$, then the face operators of $C_{\bullet }$ satisfy the following condition:

$(\ast )$

For $0 \leq i < j \leq n$, we have $d^{n-1}_{i} \circ d^{n}_{j} = d^{n-1}_{j-1} \circ d^{n}_{i}$ (as morphisms from $C_{n}$ to $C_{n-2}$).

Remark 1.1.1.11 admits the following converse:

Proposition 1.1.1.12. Let $\operatorname{\mathcal{C}}$ be a category and let $\{ C_ n \} _{n \geq 0}$ be a sequence of objects of $\operatorname{\mathcal{C}}$. Then a system of morphisms $\{ d^{n}_{i}: C_{n} \rightarrow C_{n-1} \} _{0 \leq i \leq n, n > 0}$ arise as the face operators of a semisimplicial object $C_{\bullet }$ of $\operatorname{\mathcal{C}}$ if and only if they satisfy condition $(\ast )$ of Remark 1.1.1.11. Moreover, if this condition is satisfied, then $C_{\bullet }$ is uniquely determined.

Proof. Let $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{inj}}$ denote the category which is freely generated by a collection of objects $\{ [n] \} _{n \geq 0}$ and a collection of morphisms $\{ \widetilde{\delta }_{n}^{i}: [n-1] \rightarrow [n] \} _{n > 0, 0 \leq i \leq n}$. Let $\overline{ \operatorname{{\bf \Delta }}}_{\operatorname{inj}}$ denote the quotient of $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{inj}}$ obtained by imposing the relation

1.3
\begin{eqnarray} \label{equation:relation-semisimplicial-identity} \widetilde{\delta }^{n}_{j} \circ \widetilde{\delta }^{n-1}_{i} & = & \widetilde{\delta }^{n}_{i} \circ \widetilde{\delta }^{n-1}_{j-1} \end{eqnarray}

for every integer $n \geq 2$ and every pair $0 \leq i < j \leq n$. Using Remark 1.1.1.11, we see that there is a unique functor $F_{\operatorname{inj}}: \overline{\operatorname{{\bf \Delta }}}_{\operatorname{inj}} \rightarrow \operatorname{{\bf \Delta }}_{\operatorname{inj}}$ which carries each object $[n] \in \overline{\operatorname{{\bf \Delta }}}_{\operatorname{inj}}$ to itself, and each generating morphism $\widetilde{\delta }_{n}^{i}$ to the monomorphism $\delta _{n}^{i}: [n-1] \hookrightarrow [n]$ of Notation 1.1.1.10. To prove Proposition 1.1.1.12, it will suffice to show that the functor $F_{\operatorname{inj}}$ is an isomorphism of categories.

Fix integers $0 \leq m \leq n$, and set $b = n-m+1$. In the category $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{inj}}$, every morphism $\beta : [m] \rightarrow [n]$ admits a unique factorization $\beta = \widetilde{\delta }_{n}^{i_0} \circ \widetilde{\delta }_{n-1}^{i_1} \circ \cdots \circ \widetilde{\delta }_{n-b}^{i_{b}}$, where the superscripts are nonnegative integers satisfying $0 \leq i_ a \leq n - a$ for $0 \leq a \leq b$. Let us say that $\beta$ is in standard form if, in addition, the integers $i_ a$ satisfy the inequalities $i_0 < i_1 < i_2 < \cdots < i_ b$. Note that, by repeatedly applying the relation (1.3), we can convert any morphism of $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{inj}}$ to a morphism which is in standard form. More precisely, every morphism $\overline{\beta }: [m] \rightarrow [n]$ in $\overline{\operatorname{{\bf \Delta }}}_{\operatorname{inj}}$ can be lifted to a morphism $\beta : [m] \rightarrow [n]$ which is in standard form.

By construction, the functor $F_{\operatorname{inj}}$ is bijective on objects. To complete the proof, it will suffice to show that for every morphism $\alpha : [m] \hookrightarrow [n]$, there is a unique morphism $\overline{\beta }: [m] \rightarrow [n]$ in $\overline{\operatorname{{\bf \Delta }}}_{\operatorname{inj}}$ satisfying $F_{\operatorname{inj}}( \overline{\beta } ) = \alpha$. By virtue of the preceding discussion, it will suffice to show that $\alpha$ can be lifted uniquely to a morphism $\beta : [m] \rightarrow [n]$ in the category $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{inj}}$ which is in standard form. We now observe that $\beta = \widetilde{\delta }_{n}^{i_0} \circ \widetilde{\delta }_{n-1}^{i_1} \circ \cdots \circ \widetilde{\delta }_{n-b}^{i_{b}}$ is characterized by the requirement that $\{ i_0 < i_1 < \cdots < i_ b \} \subseteq [n]$ is the complement of the image of $\alpha$. $\square$

To classify simplicial objects of a category $\operatorname{\mathcal{C}}$, we will need to record more information, given by surjective morphisms in the simplex category $\operatorname{{\bf \Delta }}$.

Notation 1.1.1.13 (Degeneracy Operators). Let $\operatorname{{\bf \Delta }}_{\operatorname{surj}}$ denote the category whose objects are the linearly ordered sets $[n] = \{ 0 < 1 < \cdots < n \}$ for $n \geq 0$, and whose morphisms are nondecreasing surjective functions $[m] \twoheadrightarrow [n]$. For every pair of integers $0 \leq i \leq n$ we let $\sigma ^{i}_{n}: [n+1] \twoheadrightarrow [n]$ denote the morphism of $\operatorname{{\bf \Delta }}_{\operatorname{surj}}$ given by the formula

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

If $C_{\bullet }$ is a simplicial object of a category $\operatorname{\mathcal{C}}$, then we can evaluate $C_{\bullet }$ on the morphism $\sigma ^{i}_{n}$ to obtain a morphism from $C_{n}$ to $C_{n+1}$. We will denote this map by $s^{n}_{i}: C_{n} \rightarrow C_{n+1}$ and refer to it as the $i$th degeneracy operator.

Remark 1.1.1.14 (Relations Among Degeneracy Operators). For every triple of integers $0 \leq i \leq j \leq n$, the diagram of linearly ordered sets

$\xymatrix@R =50pt@C=50pt{ [n+2] \ar [r]^-{\sigma ^{i}_{n+1}} \ar [d]^{ \sigma ^{j+1}_{n+1} } & [n+1] \ar [d]^{ \sigma ^{j}_{n}} \\[n+1] \ar [r]^-{ \sigma ^{i}_{n} } & [n] }$

is commutative. It follows that, if $C_{\bullet }$ is a simplicial object of a category $\operatorname{\mathcal{C}}$, then the degeneracy operators of $C_{\bullet }$ satisfy the following condition:

$(\ast ')$

For $0 \leq i \leq j \leq n$, we have an equality $s^{n+1}_{i} \circ s^{n}_ j = s^{n+1}_{j+1} \circ s^{n}_ i$ (as morphisms from $C_{n}$ to $C_{n+2}$).

Proposition 1.1.1.12 has a counterpart for the category $\operatorname{{\bf \Delta }}_{\operatorname{surj}}$:

Proposition 1.1.1.15. Let $\operatorname{\mathcal{C}}$ be a category and let $\{ C_ n \} _{n \geq 0}$ be a sequence of objects of $\operatorname{\mathcal{C}}$. Then a system of morphisms $\{ s^{n}_{i}: C_{n} \rightarrow C_{n+1} \} _{0 \leq i \leq n}$ can be obtained from a functor $C_{\bullet }: \operatorname{{\bf \Delta }}_{\operatorname{surj}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}$ if and only if they satisfy condition $(\ast ')$ of Remark 1.1.1.14. In this case, the functor $C_{\bullet }$ is uniquely determined.

Proof. We proceed as in the proof of Proposition 1.1.1.12. Let $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{surj}}$ denote the category which is freely generated by a collection of objects $\{ [n] \} _{n \geq 0}$ and a collection of morphisms $\{ \widetilde{\sigma }_{n}^{i}: [n+1] \rightarrow [n] \} _{0 \leq i \leq n}$. Let $\overline{ \operatorname{{\bf \Delta }}}_{\operatorname{surj}}$ denote the quotient of $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{surj}}$ obtained by imposing the relation

1.6
\begin{eqnarray} \label{equation:relation-semisimplicial-identity2} \widetilde{\sigma }^{j}_{n} \circ \widetilde{\sigma }^{i}_{n+1} & = & \widetilde{\sigma }^{i}_{n} \circ \widetilde{\sigma }^{j+1}_{n+1} \end{eqnarray}

for every triple of integers $0 \leq i \leq j \leq n$. Using Remark 1.1.1.14, we see that there is a unique functor $F_{\operatorname{surj}}: \overline{\operatorname{{\bf \Delta }}}_{\operatorname{surj}} \rightarrow \operatorname{{\bf \Delta }}_{\operatorname{surj}}$ which carries each object $[n] \in \overline{\operatorname{{\bf \Delta }}}_{\operatorname{surj}}$ to itself, and each generating morphism $\widetilde{\sigma }_{n}^{i}$ to the monomorphism $\sigma _{n}^{i}: [n-1] \hookrightarrow [n]$ of Notation 1.1.1.13. To prove Proposition 1.1.1.15, it will suffice to show that the functor $F_{\operatorname{surj}}$ is an isomorphism of categories.

Fix integers $0 \leq m \leq n$, and set $b = n-m+1$. In the category $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{surj}}$, every morphism $\beta : [n] \rightarrow [m]$ admits a unique factorization $\beta = \widetilde{\sigma }_{m}^{i_0} \circ \widetilde{\sigma }_{m+1}^{i_1} \circ \cdots \circ \widetilde{\sigma }_{m+b}^{i_{b}}$, where the superscripts are nonnegative integers satisfying $0 \leq i_ a \leq m+b$ for $0 \leq a \leq b$. Let us say that $\beta$ is in standard form if, in addition, the integers $i_ a$ satisfy the inequalities $i_0 < i_1 < i_2 < \cdots < i_ b$. Note that, by repeatedly applying the relation (1.6), we can convert any morphism of $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{surj}}$ to a morphism which is in standard form. More precisely, every morphism $\overline{\beta }: [n] \rightarrow [m]$ in $\overline{\operatorname{{\bf \Delta }}}_{\operatorname{surj}}$ can be lifted to a morphism $\beta : [m] \rightarrow [n]$ which is in standard form.

By construction, the functor $F_{\operatorname{surj}}$ is bijective on objects. To complete the proof, it will suffice to show that for every morphism $\alpha : [n] \twoheadrightarrow [m]$ in $\operatorname{{\bf \Delta }}_{\operatorname{surj}}$, there is a unique morphism $\overline{\beta }: [n] \rightarrow [m]$ in $\overline{\operatorname{{\bf \Delta }}}_{\operatorname{surj}}$ satisfying $F_{\operatorname{surj}}( \overline{\beta } ) = \alpha$. By virtue of the preceding discussion, it will suffice to show that $\alpha$ can be lifted uniquely to a morphism $\beta : [n] \rightarrow [m]$ in the category $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{surj}}$ which is in standard form. We now observe that $\beta =\widetilde{\sigma }_{m}^{i_0} \circ \widetilde{\sigma }_{m+1}^{i_1} \circ \cdots \circ \widetilde{\sigma }_{m+b}^{i_{b}}$ is characterized by the requirement that $\{ i_0 < i_1 < \cdots < i_ b \}$ is the collection of integers $0 \leq j < n$ satisfying $\alpha (j) = \alpha (j+1)$. $\square$

Exercise 1.1.1.16 (Relations Between Face and Degeneracy Operators). Let $C_{\bullet }$ be a simplicial object of a category $\operatorname{\mathcal{C}}$. Show that the face and degeneracy operators of $\operatorname{\mathcal{C}}$ satisfy the following compatibility condition:

$(\ast '')$

For $0 \leq i,j \leq n$, we have an equality

$d^{n+1}_{i} \circ s^{n}_ j = \begin{cases} s^{n-1}_{j-1} \circ d^{n}_ i & \text{ if } i < j \\ \operatorname{id}_{ C_ n } & \text{ if } i = j \text{ or } i = j + 1 \\ s^{n-1}_{j} \circ d^{n}_{i-1} & \text{ if } i > j+1 \end{cases}$

(as morphisms from $C_{n}$ to $C_{n}$).

The structure of a simplicial object can be recovered from its face and degeneracy operators:

Proposition 1.1.1.17. Let $\operatorname{\mathcal{C}}$ be a category containing a sequence of objects $\{ C_{n} \} _{n \geq 0}$. Then morphisms

$\{ d^{n}_{i}: C_{n} \rightarrow C_{n-1} \} _{ 0 \leq i \leq n, n > 0} \quad \quad \{ s^{n}_{i}: C_{n} \rightarrow C_{n+1} \} _{0 \leq i \leq n}$

are the face and degeneracy operators for a simplicial object $C_{\bullet }$ of $\operatorname{\mathcal{C}}$ if and only if they satisfy condition $(\ast )$ of Remark 1.1.1.14, condition $(\ast ')$ of Remark 1.1.1.14, and condition $(\ast '')$ of Exercise 1.1.1.16.

Proof. We proceed as in the proofs of Propositions 1.1.1.12 and 1.1.1.15. Let $\widetilde{\operatorname{{\bf \Delta }}}$ denote the category which is freely generated by a collection of objects $\{ [n] \} _{n \geq 0}$ together with morphisms $\{ \widetilde{\delta }_{n}^{i}: [n-1] \rightarrow [n] \} _{n > 0, 0 \leq i \leq n}$ and $\{ \widetilde{\sigma }_{n}^{i}: [n+1] \rightarrow [n] \} _{0 \leq i \leq n}$. Let $\overline{ \operatorname{{\bf \Delta }}}$ denote the quotient of $\widetilde{\operatorname{{\bf \Delta }}}$ obtained by imposing the relations (1.3) and (1.6), together with the following:

1.10
\begin{eqnarray} \label{equation:relation-semisimplicial-identity3} \widetilde{\sigma }^{j}_{n} \circ \widetilde{\delta }_{n+1}^{i} & = & \begin{cases} \widetilde{\delta }_{n}^{i} \circ \widetilde{\sigma }^{j-1}_{n-1} & \text{ if } i < j \\ \operatorname{id}_{ [n] } & \text{ if } i = j \text{ or } i = j + 1 \\ \widetilde{\delta }^{i-1}_{n} \circ \widetilde{\sigma }^{j}_{n-1} & \text{ if } i > j+1. \end{cases}\end{eqnarray}

for every triple of integers $0 \leq i \leq j \leq n$. There is a unique functor $F: \overline{\operatorname{{\bf \Delta }}} \rightarrow \operatorname{{\bf \Delta }}$ which carries each object $[n] \in \overline{\operatorname{{\bf \Delta }}}$ to itself and satisfies $F( \widetilde{\delta }_{n}^{i} ) = \delta _{n}^{i}$ and $F( \widetilde{\sigma }_{n}^{i} ) = \sigma _{n}^{i}$. To prove Proposition 1.1.1.17, it will suffice to show that the functor $F$ is an isomorphism of categories.

Let $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{inj}}$ and $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{surj}}$ be the categories appearing in the proofs of Proposition 1.1.1.12 and Proposition 1.1.1.15, respectively. Let us identify $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{inj}}$ and $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{surj}}$ with (non-full) subcategories of $\widetilde{\operatorname{{\bf \Delta }}}$. We will say that a morphism $\beta : [m] \rightarrow [n]$ of $\widetilde{\operatorname{{\bf \Delta }}}$ is weakly standard if it factors as a composition $[m] \xrightarrow { \beta _{\operatorname{surj}} } [k] \xrightarrow { \beta _{\operatorname{inj}} } [n]$, where $\beta _{\operatorname{inj}}$ belongs to $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{inj}}$ and $\beta _{\operatorname{surj}}$ belongs to $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{surj}}$. In this case, the morphisms $\beta _{\operatorname{inj}}$ and $\beta _{\operatorname{surj}}$ are uniquely determined. We will say that $\beta$ is in standard form if it is weakly standard and, in addition, the morphisms $\beta _{\operatorname{inj}}$ and $\beta _{\operatorname{surj}}$ are in standard form (as in the proofs of Propositions 1.1.1.12 and 1.1.1.15). Note that, by repeatedly applying the relation (1.10), we can convert any morphism of $\widetilde{\operatorname{{\bf \Delta }}}$ into a morphism $\beta$ which is weakly standard. Using the relations (1.3) and (1.6), we can further arrange that $\beta$ is in standard form. It follows that every morphism $\overline{\beta }: [m] \rightarrow [n]$ in $\overline{\operatorname{{\bf \Delta }}}$ can be lifted to a morphism $\beta : [m] \rightarrow [n]$ of $\widetilde{\operatorname{{\bf \Delta }}}$ which is in standard form.

By construction, the functor $F$ is bijective on objects. To complete the proof, it will suffice to show that for every morphism $\alpha : [m] \rightarrow [n]$ in $\operatorname{{\bf \Delta }}$, there is a unique morphism $\overline{\beta }: [m] \rightarrow [n]$ in $\overline{\operatorname{{\bf \Delta }}}$ satisfying $F( \overline{\beta } ) = \alpha$. Let $\widetilde{F}$ denote the composite functor $\widetilde{\operatorname{{\bf \Delta }}} \twoheadrightarrow \overline{\operatorname{{\bf \Delta }}} \xrightarrow {F} \operatorname{{\bf \Delta }}$. By virtue of the preceding discussion, it will suffice to show that there is a unique morphism $\beta : [m] \rightarrow [n]$ in $\widetilde{ \operatorname{{\bf \Delta }}}$ which is in standard form and satisfies $\widetilde{F}( \beta ) = \alpha$. In the simplex category $\operatorname{{\bf \Delta }}$, the morphism $\alpha$ factors uniquely as a composition $[m] \xrightarrow { \alpha _{\operatorname{surj}} } [k] \xrightarrow { \alpha _{\operatorname{inj}} } [n]$, where $\alpha _{\operatorname{inj}}$ is an injection and $\alpha _{\operatorname{surj}}$ is a surjection. If $\beta : [m] \rightarrow [n]$ is a weakly standard morphism of $\widetilde{\operatorname{{\bf \Delta }}}$, then the identity $\widetilde{F}( \beta ) = \alpha$ holds if and only if $\widetilde{F}( \beta _{\operatorname{inj}} ) = \alpha _{\operatorname{inj}}$ and $\widetilde{F}( \beta _{\operatorname{surj}} ) = \alpha _{\operatorname{surj}}$. We are therefore reduced to proving that $\alpha _{\operatorname{inj}}$ and $\alpha _{\operatorname{surj}}$ can be lifted uniquely to morphisms of $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{inj}}$ and $\widetilde{\operatorname{{\bf \Delta }}}_{\operatorname{surj}}$ which are in standard form, which was established in the proofs of Proposition 1.1.1.12 and Proposition 1.1.1.15. $\square$