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5.1.6 Pinched Morphism Spaces

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. In ยง5.1.1, we associated to every pair of objects $X,Y \in \operatorname{\mathcal{C}}$ a Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$, which we refer to as the space of morphisms from $X$ to $Y$ (Construction 5.1.1.1). In this section, we discuss a variant of this construction which is often more technically convenient to work with.

Construction 5.1.6.1. Let $\operatorname{\mathcal{C}}$ be a simplicial set containing vertices $X$ and $Y$. We let $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Y)$ denote the fiber product $\operatorname{\mathcal{C}}_{X/} \times _{\operatorname{\mathcal{C}}} \{ Y\} $, and we let $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y)$ denote the fiber product $\{ X\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/Y}$. We will be primarily interested in these constructions in the situation where $\operatorname{\mathcal{C}}$ is an $\infty $-category. In this case, we refer to $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Y)$ as the left-pinched space of morphisms from $X$ to $Y$ and to $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y)$ as the right-pinched space of morphisms from $X$ to $Y$.

Remark 5.1.6.2. Let $\operatorname{\mathcal{C}}$ be a simplicial set containing vertices $X$ and $Y$. For every integer $n \geq 0$, one can identify $n$-simplices of the left-pinched morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Y)$ with $(n+1)$-simplices $\sigma : \Delta ^{n+1} \rightarrow \operatorname{\mathcal{C}}$ for which $\sigma (0) = X$ and the face $d_0(\sigma )$ is the constant map $\Delta ^{n} \rightarrow \{ Y\} \hookrightarrow \operatorname{\mathcal{C}}$. Similarly, one can identify $n$-simplices of the right-pinched morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y)$ with $(n+1)$-simplices $\sigma ': \Delta ^{n+1} \rightarrow \operatorname{\mathcal{C}}$ for which $\sigma (n+1) = Y$ and the face $d_{n+1}(\sigma )$ is the constant map $\Delta ^{n} \rightarrow \{ X\} \hookrightarrow \operatorname{\mathcal{C}}$. In particular, we have canonical bijections

\[ \{ \textnormal{Vertices of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Y)$} \} \simeq \{ \textnormal{Edges $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$} \} \simeq \{ \textnormal{Vertices of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y)$} \} . \]

Remark 5.1.6.3. Let $\operatorname{\mathcal{C}}$ be a simplicial set containing vertices $X$ and $Y$, which we also regard as vertices of the opposite simplicial set $\operatorname{\mathcal{C}}^{\operatorname{op}}$. Then we have canonical isomorphisms of simplicial sets

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}^{\operatorname{op}}}^{\mathrm{L}}(X,Y) \simeq \operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(Y,X)^{\operatorname{op}} \quad \quad \operatorname{Hom}_{\operatorname{\mathcal{C}}^{\operatorname{op}}}^{\mathrm{R}}(X,Y) \simeq \operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(Y,X)^{\operatorname{op}}. \]

Proposition 5.1.6.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the pinched morphism spaces $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Y)$ and $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y)$ are Kan complexes.

Proof. By virtue of Proposition 4.3.6.1, the projection map $\operatorname{\mathcal{C}}_{X/} \rightarrow \operatorname{\mathcal{C}}$ is a left fibration. Applying Corollary 4.4.2.2, we deduce that the fiber $\operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}(X,Y) = \operatorname{\mathcal{C}}_{X/} \times _{\operatorname{\mathcal{C}}} \{ Y\} $ is a Kan complex. A similar argument shows that $\operatorname{Hom}^{\mathrm{R}}_{\operatorname{\mathcal{C}}}(X,Y)$ is a Kan complex. $\square$

Remark 5.1.6.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a pair of morphisms $f,g: X \rightarrow Y$ having the same source and target. Then the datum of an edge $e: f \rightarrow g$ in the left-pinched morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Y)$ is equivalent to the datum of a homotopy from $f$ to $g$, in the sense of Definition 1.3.3.1. In particular, $f$ and $g$ are homotopic if and only if they belong to the same connected component of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Y)$. We therefore have a canonical bijection $\operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y) \simeq \pi _0( \operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}(X,Y) )$.

We now compare the pinched morphism spaces of Construction 5.1.6.1 with the morphism spaces of Construction 5.1.1.1.

Construction 5.1.6.6. Let $\operatorname{\mathcal{C}}$ be a simplicial set containing vertices $X$ and $Y$, and let

\[ \iota ^{ \mathrm{L} }_{X}: \operatorname{\mathcal{C}}_{X/} \hookrightarrow (X \downarrow \operatorname{\mathcal{C}}) \quad \quad \iota ^{\mathrm{R}}_{Y}: \operatorname{\mathcal{C}}_{/Y} \hookrightarrow (\operatorname{\mathcal{C}}\downarrow Y) \]

be the pinch inclusion morphisms of Construction 5.1.5.12. Restricting to the fibers over the objects $Y,X \in \operatorname{\mathcal{C}}$, we obtain morphisms of Kan complexes

\[ \operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}(X,Y) = \operatorname{\mathcal{C}}_{X/} \times _{\operatorname{\mathcal{C}}} \{ Y\} \rightarrow (X \downarrow \operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} Y = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \]
\[ \operatorname{Hom}^{\mathrm{R}}_{\operatorname{\mathcal{C}}}(X,Y) = \{ X\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/Y} \rightarrow \{ X\} \times _{\operatorname{\mathcal{C}}} (\operatorname{\mathcal{C}}\downarrow Y) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y), \]

which we will denote by $\iota ^{\mathrm{L}}_{X,Y}$ and $\iota ^{\mathrm{R}}_{X,Y}$, respectively. We will refer to $\iota ^{\mathrm{L}}_{X,Y}$ as the left-pinch inclusion map and to $\iota ^{\mathrm{R}}_{X,Y}$ as the right-pinch inclusion map.

Remark 5.1.6.7. Let $\operatorname{\mathcal{C}}$ be a simplicial set containing vertices $X$ and $Y$. Then the pinch inclusion maps

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Y) \xrightarrow { \iota ^{\mathrm{L}}_{X,Y} } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \xleftarrow { \iota ^{\mathrm{R}}_{X,Y} } \operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y) \]

are monomorphisms (see Remark 5.1.5.13).

Remark 5.1.6.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing objects $X$ and $Y$. Then the pinch inclusion maps

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Y) \xrightarrow { \iota ^{\mathrm{L}}_{X,Y} } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \xleftarrow { \iota ^{\mathrm{R}}_{X,Y} } \operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y) \]

are bijective on vertices: vertices of each simplicial set can be identified with morphisms from $X$ to $Y$ in the $\infty $-category $\operatorname{\mathcal{C}}$ (Remarks 5.1.1.2 and 5.1.6.2). However, they are generally not bijective on edges. Note that edges of the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ can be identified with diagrams

\[ \xymatrix@R =30pt@C=30pt{ X \ar [rrrr]^{f} \ar [dddd]^{\operatorname{id}_{X}} \ar [ddddrrrr]^{g} & & & & Y \ar [dddd]^{\operatorname{id}_{Y}} \\ & & & \sigma & \\ & & & & \\ & \tau & & & \\ X \ar [rrrr]^{f'} & & & & Y } \]

in the $\infty $-category $\operatorname{\mathcal{C}}$. Such a diagram belongs to the image of the left-pinch inclusion map $\iota ^{\mathrm{L}}_{X,Y}$ if and only $\tau = s_0( g )$ (so that the simplex $\tau $ is degenerate, $f' = g$, and the entire diagram is determined by $\sigma $). Similarly, the diagram belongs to the image of the right-pinch inclusion map $\iota ^{\mathrm{R}}_{X,Y}$ if and only if $\sigma = s_1(g)$ (so that the simplex $\sigma $ is degenerate, $f = g$, and the entire diagram is determined by $\tau $).

Proposition 5.1.6.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the pinch inclusion morphisms

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Y) \xrightarrow { \iota ^{\mathrm{L}}_{X,Y} } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \xleftarrow { \iota ^{\mathrm{R}}_{X,Y} } \operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y) \]

are homotopy equivalences of Kan complexes.

Proof. We will prove that the left-pinch inclusion morphism $\iota ^{\mathrm{L}}_{X,Y}$ is a homotopy equivalence; the proof for the right-pinch inclusion morphism $\iota ^{\mathrm{R}}_{X,Y}$ is similar. Note that we have a commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{X/} \ar [r]^-{ \iota ^{\mathrm{L}}_{X} } \ar [d] & (X \downarrow \operatorname{\mathcal{C}}) \ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^-{\operatorname{id}} & \operatorname{\mathcal{C}}, } \]

where the horizontal maps are equivalences of $\infty $-categories (Theorem 5.1.5.14) and the vertical maps are left fibrations (Propositions 4.3.6.1 and 5.1.5.10), hence isofibrations (Example 4.4.1.10). Applying Corollary 4.5.4.4, we deduce that the induced map of fibers

\[ \iota ^{\mathrm{L}}_{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Y) = (\operatorname{\mathcal{C}}_{X/} ) \times _{\operatorname{\mathcal{C}}} \{ Y\} \rightarrow (X \downarrow \operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{C}}} \{ Y\} = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \]

is an equivalence of $\infty $-categories, hence a homotopy equivalence of Kan complexes (Remark 4.5.1.4). $\square$

Corollary 5.1.6.10. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between $\infty $-categories. The following conditions are equivalent:

  • The functor $F$ is fully faithful. That is, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the functor $F$ induces a homotopy equivalence of Kan complexes $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}(F(X), F(Y) )$.

  • For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the functor $F$ induces a homotopy equivalence of left-pinched morphism spaces $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}^{\mathrm{L}}( F(X), F(Y) )$.

  • For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the functor $F$ induces a homotopy equivalence of right-pinched morphism spaces $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}^{\mathrm{R}}( F(X), F(Y) )$.

Example 5.1.6.11. Let $\operatorname{\mathcal{C}}$ be an ordinary category containing objects $X$ and $Y$. Then the pinch inclusion morphisms

\[ \iota ^{\mathrm{L}}_{X}: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{X/} \rightarrow (X \downarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})) \quad \quad \iota ^{\mathrm{R}}_{Y}: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{/Y} \rightarrow (\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \downarrow Y) \]

are isomorphisms (see Remark 4.3.1.6). In particular, we can identify the pinched morphism spaces $\operatorname{Hom}_{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})}^{\mathrm{L}}(X,Y)$ and $\operatorname{Hom}_{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})}^{\mathrm{R}}(X,Y)$ with the constant simplicial set $\operatorname{Hom}_{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})}(X,Y)$ associated to the usual morphism set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$.

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a pair of objects $X$ and $Y$. By virtue of Proposition 5.1.6.9, the pinched morphism spaces $\operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}(X,Y)$ and $\operatorname{Hom}^{\mathrm{R}}_{\operatorname{\mathcal{C}}}(X,Y)$ of Construction 5.1.6.1 contain the same homotopy-theoretic information as the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ of Construction 5.1.1.1. However, they package this information in a more efficient way: an $n$-simplex of the Kan complex $\operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}(X,Y)$ can be identified with a single $(n+1)$-simplex of the $\infty $-category $\operatorname{\mathcal{C}}$ (see Remark 5.1.6.2), but to specify an $n$-simplex of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ one must supply $n+1$ different $(n+1)$-simplices of $\operatorname{\mathcal{C}}$ (see Remark 5.1.6.8 for the case $n=1$).

Example 5.1.6.12 (Pinched Morphism Spaces in the Duskin Nerve). Let $\operatorname{\mathcal{C}}$ be a $2$-category (Definition 2.2.1.1) and let $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ denote its Duskin nerve (Construction 2.3.1.1). Let $X$ and $Y$ be objects of $\operatorname{\mathcal{C}}$, which we identify with vertices of the simplicial set $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$. Using Example 5.1.1.12, we can identify $n$-simplices $\sigma $ of the simplicial set $\operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})}( X, Y)$ with commutative diagrams

\[ \xymatrix@R =20pt@C=20pt{ & & & & f_{0,n} & & & & \\ & & & f_{0,n-1} \ar@ {=>}[ur] & & f_{1,n} \ar@ {=>}[ul] & & & \\ & & f_{0,n-2} \ar@ {=>}[ur] & & f_{1,n-1} \ar@ {=>}[ul] \ar@ {=>}[ur] & & f_{2,n} \ar@ {=>}[ul] & & \\ & \cdots \ar@ {=>}[ur] & & \cdots \ar@ {=>}[ur] \ar@ {=>}[ul] & & \cdots \ar@ {=>}[ul] \ar@ {=>}[ur] & & \cdots \ar@ {=>}[ul] & \\ f_{0,0} \ar@ {=>}[ur] & & f_{1,1} \ar@ {=>}[ul] \ar@ {=>}[ur] & & \cdots \ar@ {=>}[ul] \ar@ {=>}[ur] & & f_{n-1,n-1} \ar@ {=>}[ul] \ar@ {=>}[ur] & & f_{n,n} \ar@ {=>}[ul] } \]

in the category of $1$-morphisms $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$. The image of the left-pinch inclusion morphism

\[ \iota ^{\mathrm{L}}_{X,Y}: \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})}^{\mathrm{L}}( X, Y) \hookrightarrow \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})}( X, Y) \]

consists of those simplices which correspond (under this identification) to commutative diagrams in which each of the leftward pointing $2$-morphisms $f_{i,j} \Rightarrow f_{i-1,j}$ is an identity map. In this case, the entire diagram is determined by the sequence of composable morphisms $f_{0,0} \Rightarrow f_{0,1} \Rightarrow f_{0,2} \Rightarrow \cdots \Rightarrow f_{0,n}$ in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$. Similarly, the image of the right-pinch inclusion morphism

\[ \iota ^{\mathrm{R}}_{X,Y}: \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})}^{\mathrm{R}}( X, Y) \hookrightarrow \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})}( X, Y) \]

consists of those simplices which correspond to commutative diagrams in which the rightward pointing $2$-morphisms $f_{i,j} \Rightarrow f_{i,j+1}$ is an identity map, which ensures that the entire diagram is determined by the sequence of composable morphisms $f_{n,n} \Rightarrow f_{n-1,n} \Rightarrow f_{n-2,n} \Rightarrow \cdots \Rightarrow f_{0,n}$ in $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$. Allowing $[n] \in \operatorname{{\bf \Delta }}$ to vary, we obtain canonical isomorphisms of simplicial sets

\[ \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})}^{\mathrm{L}}( X, Y) \simeq \operatorname{N}_{\bullet }( \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) ) \quad \quad \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})}^{\mathrm{R}}( X, Y) \simeq \operatorname{N}_{\bullet }( \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) )^{\operatorname{op}}. \]

Example 5.1.6.13. Let $X$ be a topological space containing a pair of points $x$ and $y$, which we regard as objects of the $\infty $-category $\operatorname{Sing}_{\bullet }(X)$. Using Example 4.3.5.9, we obtain canonical isomorphisms of Kan complexes

\[ \operatorname{Hom}^{\mathrm{L}}_{\operatorname{Sing}_{\bullet }(X)}(x,y) \simeq \operatorname{Sing}_{\bullet }( P_{x,y} ) \simeq \operatorname{Hom}_{\operatorname{Sing}_{\bullet }(X)}^{\mathrm{R}}(x,y), \]

where $P_{x,y}$ denotes the topological space of continuous paths $p: [0,1] \rightarrow X$ satisfying $p(0) = x$ and $p(1) = y$ (equipped with the compact-open topology). Combining this observation with Example 5.1.1.4, we can identify the pinch inclusion maps $\iota ^{\mathrm{L}}_{x,y}$ and $\iota ^{\mathrm{R}}_{x,y}$ with monomorphisms from the simplicial set $\operatorname{Sing}_{\bullet }( P_{x,y} )$ to itself. Beware that these maps are not the identity (though one can show that they are homotopic to the identity).

Example 5.1.6.14 (Pinched Morphism Spaces in the Differential Graded Nerve). Let $\operatorname{\mathcal{C}}$ be a differential graded category (Definition 2.5.2.1), let $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ denote the differential graded nerve of $\operatorname{\mathcal{C}}$ (Definition 2.5.3.7), and let $X$ and $Y$ be objects of $\operatorname{\mathcal{C}}$ (which we also view as objects of the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$), and let $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }$ denote the chain complex of morphisms from $X$ to $Y$. For $n \geq 0$, we can identify $n$-simplices of the left-pinched morphism space $\operatorname{Hom}^{\mathrm{L}}_{\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})}(X,Y)$ with $(n+1)$-simplices $\sigma : \Delta ^{n+1} \rightarrow \operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$ for which $\sigma (0) = X$ and $d_0(\sigma )$ is the constant $n$-simplex with the value $Y$ (Remark 5.1.6.2). Concretely, such a simplex can be described as a datum $I \mapsto f_{I}$, defined for each subset $I = \{ i_0 > i_1 > i_2 > \cdots > i_ k > i_{k+1} \} \subseteq [n+1]$ having at least two elements, with the following properties:

$(1)$

If $i_{k+1} > 0$, then $f_{I}$ is an element of the abelian group $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Y)_{k}$, which is equal to $\operatorname{id}_ Y$ in the case $k=0$ and vanishes for $k > 0$.

$(2)$

If $i_{k+1} = 0$, then $f_{I}$ is an element of the abelian group $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{k}$ which satisfies the identity

\[ \partial f_{I} = \sum _{a=1}^{k} (-1)^{a} ( f_{ \{ i_0 > i_1 > \cdots > i_ a \} } \circ f_{ \{ i_ a > \cdots > i_{k+1} \} } - f_{I \setminus \{ i_ a \} } ). \]

Note that, by virtue of $(1)$, we can rewrite this identity as

5.12
\begin{equation} \label{equation:pinched-morphism-in-dg-nerve} \partial f_{I} = \begin{cases} 0 & \textnormal{ if } k=0 \\ \sum _{a=0}^{k} (-1)^{a+1} f_{I \setminus \{ i_ a\} } & \textnormal{ if } k > 0. \end{cases} \end{equation}

Let $J = \{ j_0 < j_1 < \cdots < j_ k \} $ be a nonempty subset of $[n]$. For $\{ f_ I \} $ as above, define $g_{J} \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{k}$ by the formula $g_ J = (-1)^{k(k-1)/2} f_{ \{ j_ k + 1 > j_{k-1} + 1 > \cdots > 0 \} }$. We can then rewrite the identity (5.12) as

\begin{eqnarray*} \partial g_{J} & = & \sum _{b = 0}^{k} (-1)^{b} g_{J \setminus \{ j_ b\} }. \end{eqnarray*}

The construction $J \mapsto g_{J}$ can then be identified with a morphism from the normalized chain complex $\mathrm{N}_{\ast }( \Delta ^ n)$ of Construction 2.5.5.9 to the chain complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$. This identification depends functorially on $n$, and therefore determines an isomorphism of simplicial sets

\[ \operatorname{Hom}^{\mathrm{L}}_{\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})}( X, Y) \simeq \mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast } ), \]

where $\mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast } )$ denotes the Eilenberg-MacLane space associated to the chain complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }$ (Construction 2.5.6.3). In particular, the left-pinched morphism space $\operatorname{Hom}^{\mathrm{L}}_{\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})}( X, Y)$ has the structure of a simplicial abelian group.