# Kerodon

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### 4.6.5 Pinched Morphism Spaces

Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. In §4.6.1, 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 4.6.1.1). In this section, we discuss a variant of this construction which is often more technically convenient to work with.

Construction 4.6.5.1. 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 4.6.5.2. 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 4.6.5.3. 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 4.6.5.4. 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 4.4.2.3, 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 4.6.5.5. 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 4.6.5.1 with the morphism spaces of Construction 4.6.1.1.

Construction 4.6.5.6. Let $\operatorname{\mathcal{C}}$ be a simplicial set containing vertices $X$ and $Y$, and let

$\delta _{X/}: \operatorname{\mathcal{C}}_{X/} \hookrightarrow \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\quad \quad \delta _{/Y}: \operatorname{\mathcal{C}}_{/Y} \hookrightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ Y\}$

be the coslice and slice diagonal morphisms of Construction 4.6.4.13. Restricting to the fibers over the objects $Y,X \in \operatorname{\mathcal{C}}$, we obtain morphisms of Kan complexes

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

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 4.6.5.7. 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 4.6.4.15).

Remark 4.6.5.8. 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 4.6.1.2 and 4.6.5.2). 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 4.6.5.9. 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

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{X/} \ar [r] \ar [d] & \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^-{\operatorname{id}} & \operatorname{\mathcal{C}}, }$

where the horizontal maps are equivalences of $\infty$-categories (Corollary 4.6.4.18) and the vertical maps are left fibrations (Propositions 4.3.6.1 and 4.6.4.11), hence isofibrations (Example 4.4.1.10). Applying Corollary 4.5.2.26, we deduce that the induced map of fibers

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

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

Corollary 4.6.5.10. 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 4.6.5.11. Let $\operatorname{\mathcal{C}}$ be an ordinary category containing objects $X$ and $Y$. Then the slice and coslice diagonal morphisms

$\delta _{X/}: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{X/} \rightarrow \{ X\} \operatorname{\vec{\times }}_{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \quad \quad \delta _{/Y}: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{/Y} \rightarrow (\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \operatorname{\vec{\times }}_{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})} \{ Y\}$

are isomorphisms (see Remark 4.3.1.7). 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 4.6.5.9, 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 4.6.5.1 contain the same homotopy-theoretic information as the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ of Construction 4.6.1.1. 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 4.6.5.2), 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 4.6.5.8 for the case $n=1$).

Example 4.6.5.12 (Pinched Morphism Spaces in the Duskin Nerve). Let $\operatorname{\mathcal{C}}$ be a $2$-category (Definition 2.2.1.1). For each integer $n \geq 0$, we can use Remark 2.3.1.8 to identify $(n+1)$-simplices $\sigma$ of the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{\mathcal{C}})$ with the following data:

$(0)$

A collection of objects $\{ Z_ i \} _{ 0 \leq i \leq n+1}$ of the $2$-category $\operatorname{\mathcal{C}}$.

$(1)$

A collection of $1$-morphisms $\{ f_{j,i}: Z_ i \rightarrow Z_ j \} _{0 \leq i \leq j \leq n+1 }$ in the $2$-category $\operatorname{\mathcal{C}}$, satisfying $f_{j,i} = \operatorname{id}_{ Z_ i }$ when $i = j$.

$(2)$

A collection of $2$-morphisms $\{ \mu _{k,j,i}: f_{k,j} \circ f_{j,i} \Rightarrow f_{k,i} \} _{0 \leq i \leq j \leq k \leq n+1}$ in the $2$-category $\operatorname{\mathcal{C}}$, satisfying some additional constraints (see $(b)$ and $(c)$ of Proposition 2.3.1.9).

Fix a pair of objects $X$ and $Y$. Then $\sigma$ represents an $n$-simplex of the right pinched morphism space $\operatorname{Hom}^{\mathrm{R}}_{ \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}) }( X, Y)$ if and only if the above data satisfies the following additional conditions:

• For $0 \leq i \leq n$, the object $Z_ i$ is equal to $X$. For $i = n+1$, the object $Z_{i}$ is equal to $Y$.

• For $0 \leq i \leq j \leq n$, the $1$-morphism $f_{j,i}$ is equal to the identity $1$-morphism $\operatorname{id}_{X}$.

• For $0 \leq i \leq \leq j \leq k \leq n$, the $2$-morphism $\mu _{k,j,i}$ is equal to the unit constraint $\upsilon : \operatorname{id}_{X} \circ \operatorname{id}_{X} \Rightarrow \operatorname{id}_{X}$.

In this case, we can identify $(1)$ with a collection of $1$-morphisms $\{ g_ i: X \rightarrow Y \} _{0 \leq i \leq n}$ given by $g_ i = f_{n+1,i}$, and $(2)$ with a collection of $2$-morphisms $\{ \nu _{j,i}: g_ j \Rightarrow g_ i \} _{ 0 \leq i \leq j \leq n}$, where $\nu _{j,i}$ is given by the composition

$g_{j} \xRightarrow {\sim } g_{j} \circ \operatorname{id}_{X} = f_{n+1,j} \circ f_{j,i} \xRightarrow {\mu _{n+1,j,i} } f_{n+1,i} = g_ i.$

Unwinding the definitions, condition $(b)$ translates to the requirement that $\nu _{j,i}$ is an identity $2$-morphism when $i = j$, and condition $(c)$ translates to the identity $\nu _{k,j} \circ \nu _{j,i} = \nu _{k,i}$ for $0 \leq i \leq j \leq k \leq n$. In this case, we can identify the pair $( \{ g_ i \} _{0 \leq i \leq n}, \{ \nu _{j,i} \} _{0 \leq i \leq j \leq n} )$ with a functor $[n] \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)^{\operatorname{op}}$. These identifications depends functorially on $[n] \in \operatorname{{\bf \Delta }}$, and therefore determine a canonical isomorphism of simplicial sets

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

Using similar reasoning, we obtain an isomorphism of simplicial sets

$\operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})}^{\mathrm{L}}( X, Y) \simeq \operatorname{N}_{\bullet }( \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) ).$

Example 4.6.5.13. 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 4.6.1.4, 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 4.6.5.14 (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 4.6.5.2). 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

4.55
$$\label{equation:pinched-morphism-in-dg-nerve} \partial f_{I} = \begin{cases} 0 & \textnormal{ if } k=0 \\ \sum _{a=0}^{k} (-1)^{a+1} f_{I \setminus \{ i_ a\} } & \textnormal{ if } k > 0. \end{cases}$$

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 (4.55) 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.