Kerodon

Example 4.6.5.12. Let $\operatorname{\mathcal{C}}$ be an ordinary category containing objects $X$ and $Y$. Then the slice and coslice diagonal morphisms

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 $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$.

Example 4.6.5.13 (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 $0 \leq i \leq j \leq k \leq n$, 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

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

Using similar reasoning, we obtain an isomorphism of simplicial sets

Example 4.6.5.14. 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

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 4.6.1.5, 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 4.6.5.15 (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 $d^{n+1}_0(\sigma )$ 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

4.56

\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 (4.56) as

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

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.