Proposition 4.3.6.1 (Joyal [MR1935979]). Let $K$ be a simplicial set, let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Then the projection map $\operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{\mathcal{C}}$ is a left fibration of simplicial sets, and the projection map $\operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}$ is a right fibration of simplicial sets. In particular, the simplicial sets $\operatorname{\mathcal{C}}_{f/}$ and $\operatorname{\mathcal{C}}_{/f}$ are $\infty $-categories (see Remark 4.2.1.4).
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
4.3.6 Slices of $\infty $-Categories
Recall that, if $\operatorname{\mathcal{C}}$ is a category containing an object $S$, then the forgetful functors
\[ \operatorname{\mathcal{C}}_{S/} \rightarrow \operatorname{\mathcal{C}}\quad \quad \operatorname{\mathcal{C}}_{/S} \rightarrow \operatorname{\mathcal{C}} \]
are left and right covering maps, respectively (Remark 4.3.1.6). In this section, we will prove an $\infty $-categorical counterpart of this assertion:
Remark 4.3.6.2. In the special case where $\operatorname{\mathcal{C}}$ is (the nerve of) an ordinary category, Proposition 4.3.6.1 follows from Corollary 4.3.5.17; in fact, both of the simplicial sets $\operatorname{\mathcal{C}}_{f/}$ and $\operatorname{\mathcal{C}}_{/f}$ are (the nerves of) ordinary categories.
We begin with some elementary remarks.
Construction 4.3.6.3. Let $f: A \hookrightarrow A'$ and $g: B \hookrightarrow B'$ be monomorphisms of simplicial sets. Using Remark 4.3.3.16, we see that the induced maps are also monomorphisms. Moreover, the intersection of their images is the image of the monomorphism $(f \star g): A \star B \hookrightarrow A' \star B'$. We therefore obtain a monomorphism of simplicial sets which we will refer to as the pushout-join of $f$ and $g$.
\[ A \star B' \xrightarrow { f \star \operatorname{id}_{B'} } A' \star B' \xleftarrow { \operatorname{id}_{A'} \star g } A' \star B \]
\[ (A \star B') \coprod _{ (A \star B) } (A' \star B) \hookrightarrow A' \star B', \]
We will deduce Proposition 4.3.6.1 from the following property of Construction 4.3.6.3:
Proposition 4.3.6.4 (Joyal [MR1935979]).] Let $f: A \hookrightarrow A'$ and $g: B \hookrightarrow B'$ be monomorphisms of simplicial sets. If $f$ is right anodyne or $g$ is left anodyne, then the pushout-join is an inner anodyne morphism of simplicial sets.
\[ (A \star B') \coprod _{ (A \star B) } (A' \star B) \hookrightarrow A' \star B' \]
Example 4.3.6.5. Let $f: A \hookrightarrow A'$ be a right anodyne morphism of simplicial sets. Applying Proposition 4.3.6.4 to the inclusion $\emptyset \hookrightarrow \Delta ^0$, we deduce that the natural map $A^{\triangleright } \coprod _{A} A' \hookrightarrow A'^{\triangleright }$ is inner anodyne. Similarly, if $g: B \hookrightarrow B'$ is left anodyne, the induced map $ B' \coprod _{B} B^{\triangleleft } \rightarrow B'^{\triangleleft }$ is inner anodyne.
Corollary 4.3.6.6. Let $f: A \hookrightarrow B$ be an inner anodyne morphism of simplicial sets. Then, for every simplicial set $K$, the induced map $g: A \star K \hookrightarrow B \star K$ is also inner anodyne.
Proof. The morphism $g$ factors as a composition
\[ A \star K \xrightarrow {g'} B \coprod _{ A } (A \star K) \xrightarrow {g''} B \star K. \]
The morphism $g'$ is inner anodyne since it is a pushout of $f$, and the morphism $g''$ is inner anodyne by virtue of Proposition 4.3.6.4. It follows that $g = g'' \circ g'$ is also inner anodyne. $\square$
Example 4.3.6.7. Let $f: A \hookrightarrow B$ be an inner anodyne morphism of simplicial sets. Then the inclusion maps $f^{\triangleright }: A^{\triangleright } \hookrightarrow B^{\triangleright }$ and $i^{\triangleleft }: A^{\triangleleft } \hookrightarrow B^{\triangleleft }$ are inner anodyne.
Proposition 4.3.6.4 implies the following stronger version of Proposition 4.3.6.1:
Proposition 4.3.6.8. Let $q: X \rightarrow S$ be an inner fibration of simplicial sets, let $f: K \rightarrow X$ be any morphism of simplicial sets, let $K_0$ be a simplicial subset of $K$, and set $f_0 = f|_{ K_0}$. Then the restriction map is a right fibration, and the restriction map is a left fibration.
\[ X_{/f} \rightarrow X_{/f_0} \times _{ S_{ / (q \circ f_0)}} S_{ / (q \circ f)} \]
\[ X_{f/} \rightarrow X_{f_0/} \times _{ S_{(q \circ f_0)/}} S_{(q \circ f)/} \]
Proof. We will prove the first assertion; the second follows by a similar argument. By virtue of Proposition 4.2.4.5, it will suffice to show that for every right anodyne morphism $i: A \hookrightarrow A'$, every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ A \ar [d]^{i} \ar [r] & X_{/f} \ar [d] \\ A' \ar [r] \ar@ {-->}[ur] \ar [r] & X_{/f_0} \times _{ S_{ / (q \circ f_0)}} S_{ / (q \circ f)} } \]
admits a solution. Unwinding the definitions, this is equivalent to solving an associated lifting problem
\[ \xymatrix@R =50pt@C=50pt{ (A \star K) \coprod _{ A \star K_0 } (A' \star K_0) \ar [r] \ar [d] & X \ar [d]^{q} \\ A' \star K \ar@ {-->}[ur] \ar [r] & S, } \]
where the left vertical morphism is the pushout-join of Construction 4.3.6.3. Proposition 4.3.6.4 guarantees that this morphism is inner anodyne, so that the desired extension exists by virtue of our assumption that $q$ is an inner fibration (Proposition 4.1.3.1). $\square$
Corollary 4.3.6.9. Let $q: X \rightarrow S$ be an inner fibration of simplicial sets and let $f: K \rightarrow X$ be any morphism of simplicial sets. Then the restriction map is a right fibration, and the restriction map is a left fibration.
\[ X_{/f} \rightarrow X \times _{S} S_{ / (q \circ f)} \]
\[ X_{f/} \rightarrow X \times _{ S } S_{(q \circ f)/} \]
Proof. Apply Proposition 4.3.6.8 in the special case $K_0 = \emptyset $. $\square$
Corollary 4.3.6.10. Let $q: X \rightarrow S$ be an inner fibration of simplicial sets and let $f: K \rightarrow X$ be any morphism of simplicial sets. Then the induced maps are inner fibrations.
\[ X_{/f} \rightarrow S_{ / (q \circ f)} \quad \quad X_{f/} \rightarrow S_{(q \circ f)/} \]
Corollary 4.3.6.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets, and let $f_0 = f|_{K_0}$ be the restriction of $f$ to a simplicial subset $K_0 \subseteq K$. Then the restriction map $\operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}_{/f_0}$ is a right fibration, and the restriction map $\operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{\mathcal{C}}_{f_0/}$ is a left fibration.
Proof. Apply Proposition 4.3.6.8 to the inner fibration $q: \operatorname{\mathcal{C}}\rightarrow \Delta ^{0}$. $\square$
Proof of Proposition 4.3.6.1. Apply Corollary 4.3.6.11 in the special case $K_0 = \emptyset $. $\square$
Proposition 4.3.6.4 also yields the following:
Proposition 4.3.6.12. Let $q: X \rightarrow S$ be an inner fibration of simplicial sets, let $f: K \rightarrow X$ be any morphism of simplicial sets, let $K_0$ be a simplicial subset of $K$, and set $f_0 = f|_{K_0}$. If the inclusion $K_0 \hookrightarrow K$ is left anodyne, then the restriction map $X_{/f} \rightarrow X_{/f_0} \times _{ S_{ / (q \circ f_0)}} S_{ / (q \circ f)}$ is a trivial Kan fibration. If the inclusion $K_0 \hookrightarrow K$ is right anodyne, then the restriction map $X_{f/} \rightarrow X_{f_0/} \times _{ S_{(q \circ f_0)/}} S_{(q \circ f)/}$ is a trivial Kan fibration.
Proof. We will prove the first assertion; the second follows by a similar argument. Assume that the inclusion $K_0 \hookrightarrow K$ is left anodyne. We wish to show that, for every monomorphism of simplicial sets $i: A \hookrightarrow A'$, every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ A \ar [d]^{i} \ar [r] & X_{/f} \ar [d] \\ A' \ar [r] \ar@ {-->}[ur] \ar [r] & X_{/f_0} \times _{ S_{ / (q \circ f_0)}} S_{ / (q \circ f)} } \]
admits a solution. Unwinding the definitions, this is equivalent to solving an associated lifting problem
\[ \xymatrix@R =50pt@C=50pt{ (A \star K) \coprod _{ A \star K_0 } (A' \star K_0) \ar [r] \ar [d] & X \ar [d]^{q} \\ A' \star K \ar@ {-->}[ur] \ar [r] & S, } \]
where the left vertical morphism is the pushout-join of Construction 4.3.6.3. Since the left vertical map is inner anodyne (Proposition 4.3.6.4), the desired solution exists by virtue of our assumption that $q$ is an inner fibration (Proposition 4.1.3.1). $\square$
Corollary 4.3.6.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets, and let $f_0 = f|_{K_0}$ be the restriction of $f$ to a simplicial subset $K_0 \subseteq K$. If the inclusion $K_0 \hookrightarrow K$ is left anodyne, then the restriction map $\operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}_{/f_0}$ is a trivial Kan fibration. If the inclusion $K_0 \hookrightarrow K$ is right anodyne, then the restriction map $\operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{\mathcal{C}}_{f_0/}$ is a trivial Kan fibration.
Proof. Apply Proposition 4.3.6.12 to the inner fibration $q: \operatorname{\mathcal{C}}\rightarrow \Delta ^{0}$. $\square$
Example 4.3.6.14 (Composition Functors). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$, which we identify with a diagram $\Delta ^1 \rightarrow \operatorname{\mathcal{C}}$. The inclusions $\{ 0\} \hookrightarrow \Delta ^1 \hookleftarrow \{ 1\} $ then induce restriction functors $\operatorname{\mathcal{C}}_{X/} \xleftarrow { e_0 } \operatorname{\mathcal{C}}_{f/} \xrightarrow { e_1 } \operatorname{\mathcal{C}}_{Y/}$. It follows from Corollary 4.3.6.13 that $e_1$ is a trivial Kan fibration, and therefore admits a section $s: \operatorname{\mathcal{C}}_{Y/} \rightarrow \operatorname{\mathcal{C}}_{f/}$ (which is unique up to isomorphism). The composition $e_0 \circ s$ can then be viewed as a functor from $\operatorname{\mathcal{C}}_{Y/}$ to $\operatorname{\mathcal{C}}_{X/}$, which we will refer to as precomposition with $f$. Concretely, this functor takes an object $g: Y \rightarrow Z$ of the $\infty $-category $\operatorname{\mathcal{C}}_{Y/}$ to an object $h: X \rightarrow Z$ of the $\infty $-category $\operatorname{\mathcal{C}}_{X/}$, which is characterized (up to isomorphism) by the requirement that there exists a $2$-simplex so that $h$ is a composition of $f$ with $g$ in the sense of Definition 1.4.4.1. Applying the same construction in the opposite $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$, we obtain a functor $\operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}_{/Y}$ which we will refer to as postcomposition with $f$; concretely, it carries an object $e: W \rightarrow X$ of the $\infty $-category $\operatorname{\mathcal{C}}_{/X}$ to an object $W \rightarrow Y$ of $\operatorname{\mathcal{C}}_{/Y}$ which is a composition of $e$ with $f$.
\[ \xymatrix { & Y \ar [dr]^{g} & \\ X \ar [ur]^{f} \ar [rr]^{h} & & Z, } \]
We now turn to the proof of Proposition 4.3.6.4.
Lemma 4.3.6.15 (Joyal [MR1935979]). Let $p,q \geq 0$ be nonnegative integers. Then:
Assume $p > 0$. Then, for $0 \leq i \leq p$, the pushout-join monomorphism
of Construction 4.3.6.3 is isomorphic to the horn inclusion $\Lambda ^{p+1+q}_{i} \hookrightarrow \Delta ^{p+1+q}$.
Assume $q> 0$. Then, for $0 \leq j \leq q$, the pushout-join monomorphism
of Construction 4.3.6.3 is isomorphic to the horn inclusion $\Lambda ^{p+1+q}_{p+1+j} \hookrightarrow \Delta ^{p+1+q}$.
\[ (\Lambda ^{p}_{i} \star \Delta ^ q) \coprod _{ ( \Lambda ^{p}_{i} \star \operatorname{\partial \Delta }^ q) } ( \Delta ^ p \star \operatorname{\partial \Delta }^ q) \hookrightarrow \Delta ^{p} \star \Delta ^ q \]
\[ (\operatorname{\partial \Delta }^ p \star \Delta ^ q) \coprod _{ ( \operatorname{\partial \Delta }^ p \star \Lambda ^{q}_{j} ) } ( \Delta ^ p \star \Lambda ^ q_ j) \hookrightarrow \Delta ^{p} \star \Delta ^ q \]
Proof. We will prove the first assertion; the second follows by symmetry. We begin by observing that there is a unique isomorphism of simplicial sets $u: \Delta ^{p} \star \Delta ^{q} \simeq \Delta ^{p+1+q}$ (Example 4.3.3.23). Let $\sigma $ be an $n$-simplex of the join $\Delta ^{p} \star \Delta ^{q}$; we wish to show that $u(\sigma )$ belongs to the horn $\Lambda ^{p+1+q}_{i}$ if and only if $\sigma $ belongs to the union of the simplicial subsets
\[ \Lambda ^{p}_{i} \star \Delta ^{q} \subseteq \Delta ^{p} \star \Delta ^{q} \supseteq \Delta ^{p} \star \operatorname{\partial \Delta }^{q}. \]
We consider three cases (see Remark 4.3.3.15):
The simplex $\sigma $ belongs to the simplicial subset $\Delta ^{p} \subseteq \Delta ^{p} \star \Delta ^{q}$. In this case, $\sigma $ is contained in $\Delta ^{p} \star \operatorname{\partial \Delta }^{q}$ and $u(\sigma )$ is contained in $\Lambda ^{p+1+q}_{i}$.
The simplex $\sigma $ belongs to the simplicial subset $\Delta ^{q} \subseteq \Delta ^{p} \star \Delta ^{q}$. In this case, $\sigma $ is contained in $\Lambda ^{p}_{i} \star \Delta ^{q}$ and $u(\sigma )$ is contained in $\Lambda ^{p+1+q}_{i}$ (since $p > 0$).
The simplex $\sigma $ factors as a composition
\[ \Delta ^{n} = \Delta ^{p' + 1 + q'} \simeq \Delta ^{p'} \star \Delta ^{q'} \xrightarrow { \sigma _{-} \star \sigma _{+} } \Delta ^{p} \star \Delta ^{q}. \]Let us abuse notation by identifying $\sigma _{-}$ and $\sigma _{+}$ with nondecreasing functions $[p'] \rightarrow [p]$ and $[q'] \rightarrow [q]$, and $u(\sigma )$ with the nondecreasing function $[n] \rightarrow [p+1+q]$ given by their join. In this case, $\sigma $ fails to belong to the union $(\Lambda ^{p}_{i} \star \Delta ^{q}) \cup ( \Delta ^{p} \star \operatorname{\partial \Delta }^{q} )$ if and only if both of the following conditions are satisfied:
The image of the nondecreasing function $\sigma _{-}: [p'] \rightarrow [p]$ contains $[p] \setminus \{ i\} $.
The nondecreasing function $\sigma _{+}: [q'] \rightarrow [q]$ is surjective.
Together, these are equivalent to the assertion that the image of the nondecreasing function $u(\sigma ): [n] \rightarrow [p+1+q]$ contains $[p+1+q] \setminus \{ i\} $: that is, it fails to belong to the horn $\Lambda ^{p+1+q}_{i} \subseteq \Delta ^{p+1+q}$.
Proof of Proposition 4.3.6.4. For every pair of morphisms of simplicial sets $f: A \rightarrow A'$ and $g: B \rightarrow B'$, let
\[ \theta _{f,g}: (A \star B') \coprod _{ (A \star B) } (A' \star B) \rightarrow A' \star B' \]
denote their pushout join. We will show that, if $f$ is right anodyne and $g$ is a monomorphism, then $\theta _{f,g}$ is inner anodyne (the analogous assertion for the case where $g$ is left anodyne follows by a similar argument). Let us first regard $f$ as fixed, and let $T$ be the collection of all morphisms $g$ of simplicial sets for which $\theta _{f,g}$ is inner anodyne. Then $T$ is weakly saturated (in the sense of Definition 1.5.4.12). We wish to prove that $T$ contains every monomorphism of simplicial sets. By virtue of Proposition 1.5.5.14, we are reduced to proving that the morphism $\theta _{f,g}$ is inner anodyne in the special case where $g$ is the boundary inclusion $\operatorname{\partial \Delta }^{q} \hookrightarrow \Delta ^{q}$ for some $q \geq 0$.
Let us now regard $g: \operatorname{\partial \Delta }^ q \hookrightarrow \Delta ^ q$ as fixed, and let $S$ denote the collection of all morphisms of simplicial sets for which $\theta _{f,g}$ is inner anodyne. To complete the proof, we must show that $S$ contains every right anodyne morphism of simplicial sets. As before, we note that $S$ is weakly saturated. It will therefore suffice to show that $S$ contains every horn inclusion $\Lambda ^{p}_{i} \hookrightarrow \Delta ^ p$ for $0 < i \leq p$ (see Variant 4.2.4.2). In other words, we are reduced to checking that the pushout-join
\[ \theta _{f,g}: (\Lambda ^{p}_{i} \star \Delta ^ q) \coprod _{ ( \Lambda ^{p}_{i} \star \operatorname{\partial \Delta }^ q) } ( \Delta ^ p \star \operatorname{\partial \Delta }^ q) \hookrightarrow \Delta ^{p} \star \Delta ^ q \]
is inner anodyne. This is clear, since $\theta _{f,g}$ can be identified with the inner horn inclusion $\Lambda ^{p+1+q}_{i} \hookrightarrow \Delta ^{p+1+q}$ by virtue of Lemma 4.3.6.15. $\square$
Using Lemma 4.3.6.15, we can also establish a converse to Proposition 4.3.6.1:
Corollary 4.3.6.16. Let $\operatorname{\mathcal{C}}$ be a simplicial set. The following conditions are equivalent:
The simplicial set $\operatorname{\mathcal{C}}$ is an $\infty $-category.
For every vertex $X$ of $\operatorname{\mathcal{C}}$, the projection map $\operatorname{\mathcal{C}}_{X/} \rightarrow \operatorname{\mathcal{C}}$ is a left fibration of simplicial sets.
For every vertex $Y$ of $\operatorname{\mathcal{C}}$, the projection map $\operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}$ is a right fibration of simplicial sets.
Proof. The implications $(1) \Rightarrow (2)$ and $(1) \Rightarrow (3)$ are special cases of Proposition 4.3.6.1. We will complete the proof by showing that $(3)$ implies $(1)$; the proof that $(2)$ implies $(1)$ is similar. Assume that $(3)$ is satisfied, and suppose that we are given a map $\sigma _0: \Lambda ^{n}_{i} \rightarrow \operatorname{\mathcal{C}}$, where $0 < i < n$; we wish to show that $\sigma _0$ can be extended to an $n$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$. Setting $Y = \sigma _0(n)$ and using the isomorphism $\Lambda ^{n}_{i} \simeq \Delta ^{n-1} \coprod _{ \Lambda ^{n-1}_{i} } (\Lambda ^{n-1}_{i})^{\triangleright }$ supplied by Lemma 4.3.6.15, we are reduced to solving a lifting problem of the form
\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n-1}_{i} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_{/Y} \ar [d] \\ \Delta ^{n-1} \ar@ {-->}[ur] \ar [r] & \operatorname{\mathcal{C}}. } \]
Since $0 < i \leq n-1$, the desired solution exists by virtue of our assumption that the projection map $\operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}$ is a right fibration. $\square$
For future use, let us record a variant of Lemma 4.3.6.15:
Variant 4.3.6.17. Let $p$ and $q$ be nonnegative integers. Then the pushout-join monomorphism of Construction 4.3.6.3 is isomorphic to the boundary inclusion $\operatorname{\partial \Delta }^{p+1+q} \hookrightarrow \Delta ^{p+1+q}$.
\[ (\operatorname{\partial \Delta }^ p \star \Delta ^ q) \coprod _{ ( \operatorname{\partial \Delta }^ p \star \operatorname{\partial \Delta }^ q) } ( \Delta ^ p \star \operatorname{\partial \Delta }^ q) \hookrightarrow \Delta ^{p} \star \Delta ^ q \]
Proof. We proceed as in the proof of Lemma 4.3.6.15. Let $u: \Delta ^{p} \star \Delta ^{q} \simeq \Delta ^{p+1+q}$ be the isomorphism supplied by Example 4.3.3.23, and let $\sigma $ be an $n$-simplex of the join $\Delta ^{p} \star \Delta ^{q}$. We wish to show that $u(\sigma )$ belongs to the boundary $\operatorname{\partial \Delta }^{p+1+q}$ if and only if $\sigma $ belongs to the union of the simplicial subsets
\[ \operatorname{\partial \Delta }^ p \star \Delta ^{q} \subseteq \Delta ^{p} \star \Delta ^{q} \supseteq \Delta ^{p} \star \operatorname{\partial \Delta }^{q}. \]
We consider three cases (see Remark 4.3.3.15):
The simplex $\sigma $ belongs to the simplicial subset $\Delta ^{p} \subseteq \Delta ^{p} \star \Delta ^{q}$. In this case, $\sigma $ is contained in $\Delta ^{p} \star \operatorname{\partial \Delta }^{q}$ and $u(\sigma )$ is contained in $\operatorname{\partial \Delta }^{p+1+q}$.
The simplex $\sigma $ belongs to the simplicial subset $\Delta ^{q} \subseteq \Delta ^{p} \star \Delta ^{q}$. In this case, $\sigma $ is contained in $\operatorname{\partial \Delta }^{p} \star \Delta ^{q}$ and $u(\sigma )$ is contained in $\operatorname{\partial \Delta }^{p+1+q}$.
The simplex $\sigma $ factors as a composition
\[ \Delta ^{n} = \Delta ^{p' + 1 + q'} \simeq \Delta ^{p'} \star \Delta ^{q'} \xrightarrow { \sigma _{-} \star \sigma _{+} } \Delta ^{p} \star \Delta {q}. \]In this case, $\sigma $ belongs to the union $(\operatorname{\partial \Delta }^{p} \star \Delta ^{q}) \cup ( \Delta ^{p} \star \operatorname{\partial \Delta }^{q} )$ if and only if either $\sigma _{-}$ or $\sigma _{+}$ fails to be surjective at the level of vertices. This is equivalent to the requirement that the map $u(\sigma ): \Delta ^{n} \rightarrow \Delta ^{p+1+q}$ fails to be surjective at the level of vertices: that is, it is a simplex of the boundary $\operatorname{\partial \Delta }^{p+1+q}$.