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8.1.5 Morphisms in the Duskin Nerve

Let $S$ be a simplicial set. Recall that, for every pair of vertices $X,Y \in S$, the morphism space $\operatorname{Hom}_{S}(X,Y)$ is defined by the formula

\[ \operatorname{Hom}_{S}(X,Y) = \{ X\} \operatorname{\vec{\times }}_{S} \{ Y\} = \{ X\} \times _{ \operatorname{Fun}(\{ 0\} , S) } \operatorname{Fun}(\Delta ^1,S) \times _{ \operatorname{Fun}(\{ 1\} , S) } \{ Y\} . \]

In this section, we specialize to the case where $S = \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ is the Duskin nerve of a $2$-category $\operatorname{\mathcal{C}}$. In this case, we will see that there is close relationship between the simplicial set $\operatorname{Hom}_{S}(X,Y)$ and the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$ of $1$-morphisms from $X$ to $Y$. More precisely, we will construct a comparison map

\[ \operatorname{Cospan}( \operatorname{N}_{\bullet } \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) ) \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}) }( X, Y), \]

and show that it is an isomorphism of simplicial sets (Corollary 8.1.5.6).

Warning 8.1.5.1. Let $\operatorname{\mathcal{C}}$ be a $2$-category. If every $2$-morphism in $\operatorname{\mathcal{C}}$ is invertible, then the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ is an $\infty $-category (Theorem 2.3.2.1). It follows that, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the simplicial set $\operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}) }( X,Y )$ is a Kan complex. Beware that, in the case where $\operatorname{\mathcal{C}}$ contains non-invertible $2$-morphisms, then the simplicial set $\operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}) }( X,Y )$ is generally not an $\infty $-category (in fact, it is not even an $(\infty ,2)$-category unless the category $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ admits pushouts: see Proposition 8.1.6.1). In such cases, it may be more useful to consider the pinched morphism spaces of $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$: see Example 4.6.5.12 and Remark 8.1.5.8.

Construction 8.1.5.2. Let $\operatorname{\mathcal{A}}$ be a category, let $\operatorname{\mathcal{C}}$ be a $2$-category containing objects $X$ and $Y$, and let $F: \operatorname{Tw}(\operatorname{\mathcal{A}}) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$ be a functor. We define a strictly unitary lax functor $U_ F: [1] \times \operatorname{\mathcal{A}}\rightarrow \operatorname{\mathcal{C}}$ as follows:

$(1)$

The lax functor $U_{F}$ is given on objects by $U_ F(0,A) = X$ and $U_ F(1,A) = Y$ for each object $A \in \operatorname{\mathcal{A}}$.

$(2)$

Let $f: A \rightarrow B$ be a morphism in the category $\operatorname{\mathcal{A}}$, which we also regard as an object of the twisted arrow category $\operatorname{Tw}(\operatorname{\mathcal{A}})$. For $0 \leq i \leq j \leq 1$, we let $f_{ji}$ denote the corresponding morphism from $(i,A)$ to $(j,B)$ in the product category $[1] \times \operatorname{\mathcal{A}}$. Then the lax functor $U_{F}$ is given on $1$-morphisms by the formula

\[ U_{F}( f_{ji} ) = \begin{cases} \operatorname{id}_{X} & \text{ if } i = j = 0 \\ \operatorname{id}_{Y} & \text{ if } i = j = 1 \\ F(f) & \text{ if } 0 = i < j = 1. \end{cases} \]
$(3)$

Let $f: A \rightarrow B$ and $v: B \rightarrow C$ be composable morphisms in the category $\operatorname{\mathcal{A}}$, and let $0 \leq i \leq j \leq k \leq 1$. Then the composition constraint $\mu _{g_{kj}, f_{ji} }$ for the lax functor $U_ F$ is given as follows:

  • If $i=j=k=0$, then $\mu _{g_{kj},f_{ji} }$ is the unit constraint $\upsilon _{X}: \operatorname{id}_{X} \circ \operatorname{id}_{X} \xRightarrow {\sim } \operatorname{id}_{X}$ of the $2$-category $\operatorname{\mathcal{C}}$.

  • If $i=0$ and $j=k=1$, then $\mu _{g_{kj},f_{ji} }$ is given by the composition

    \[ \operatorname{id}_{Y} \circ F(f) \xRightarrow { \lambda _{F(f)} } F(f) \xRightarrow { F(\operatorname{id}_ A, g) } F( g \circ f), \]

    where $\lambda _{F(f)}$ is the left unit constraint of Construction 2.2.1.11 and we regard the pair $(\operatorname{id}_ A,g)$ as an element of $\operatorname{Hom}_{\operatorname{Tw}(\operatorname{\mathcal{A}})}( f, g \circ f)$.

  • If $i=j=0$ and $k=1$, then $\mu _{g_{kj},f_{ji} }$ is given by the composition

    \[ F(g) \circ \operatorname{id}_{X} \xRightarrow { \rho _{F(g)} } F(g) \xRightarrow { F(f, \operatorname{id}_ C) } F( g \circ f), \]

    where $\rho _{F(g)}$ is the right unit constraint of Construction 2.2.1.11 and we regard the pair $(f, \operatorname{id}_ C)$ as an element of $\operatorname{Hom}_{\operatorname{Tw}(\operatorname{\mathcal{A}})}(g, g \circ f)$.

  • If $i=j=k=1$, then $\mu _{g_{kj},f_{ji} }$ is equal to the unit constraint $\upsilon _{Y}: \operatorname{id}_{Y} \circ \operatorname{id}_{Y} \xRightarrow {\sim } \operatorname{id}_{Y}$ of the $2$-category $\operatorname{\mathcal{C}}$.

Exercise 8.1.5.3. Show that Construction 8.1.5.2 is well-defined. That is, given a functor $F: \operatorname{Tw}(\operatorname{\mathcal{A}}) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$ as in Construction 8.1.5.2, show that there is a unique strictly unitary lax functor $U_{F}$ satisfying properties $(1)$, $(2)$, and $(3)$ of Construction 8.1.5.2.

We can now formulate the main result of this section.

Theorem 8.1.5.4. Let $\operatorname{\mathcal{A}}$ be a category and let $\operatorname{\mathcal{C}}$ be a $2$-category containing objects $X$ and $Y$. Then the assignment $F \mapsto U_{F}$ of Construction 8.1.5.2 induces a monomorphism of sets

\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Functors $F: \operatorname{Tw}(\operatorname{\mathcal{A}}) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$} \} \ar [d] \\ \{ \textnormal{Strictly unitary lax functors $U: [1] \times \operatorname{\mathcal{A}}\rightarrow \operatorname{\mathcal{C}}$} \} . } \]

The image of this monomorphism consists of those strictly unitary lax functors $U: [1] \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ having the property that $U|_{ \{ 0\} \times \operatorname{\mathcal{A}}}$ and $U|_{ \{ 1\} \times \operatorname{\mathcal{A}}}$ are the constant functors taking the values $X$ and $Y$, respectively.

Remark 8.1.5.5. Let $\operatorname{\mathcal{C}}$ be a $2$-category containing objects $X$ and $Y$. For every category $\operatorname{\mathcal{A}}$, we can use Theorem 2.3.4.1 to identify strictly unitary lax functors $U: [1] \times \operatorname{\mathcal{A}}\rightarrow \operatorname{\mathcal{C}}$ with morphisms of simplicial sets $G: \Delta ^1 \times \operatorname{N}_{\bullet }(\operatorname{\mathcal{A}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$. Consequently, Theorem 8.1.5.4 supplies a bijection

\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Functors $F: \operatorname{Tw}(\operatorname{\mathcal{A}}) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$} \} \ar [d]^{\sim } \\ \{ \textnormal{Morphisms of simplicial sets $\operatorname{N}_{\bullet }(\operatorname{\mathcal{A}}) \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})}(X,Y)$} \} . } \]

Note that the bijection of Remark 8.1.5.5 depends functorially on the simplicial set $\operatorname{\mathcal{A}}$. Specializing to categories of the form $\operatorname{\mathcal{A}}= [n]$, we obtain the following:

Corollary 8.1.5.6. Let $\operatorname{\mathcal{C}}$ be a $2$-category containing objects $X$ and $Y$. Then Construction 8.1.5.2 induces an isomorphism of simplicial sets

\[ \operatorname{Cospan}( \operatorname{N}_{\bullet } \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) ) \xrightarrow {\sim } \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}) }( X, Y). \]

Exercise 8.1.5.7. Show that Theorem 8.1.5.4 follows from Corollary 8.1.5.6. In other words, to prove Theorem 8.1.5.4, there is no loss of generality in assuming that $\operatorname{\mathcal{A}}$ has the form $\{ 0 < 1 < \cdots < n \} $ for some integer $n \geq 0$.

Remark 8.1.5.8. Let $\operatorname{\mathcal{C}}$ be a $2$-category containing a pair of objects $X$ and $Y$. Then we have a commutative diagram of simplicial sets

\[ \xymatrix@C =50pt@R=50pt{ \operatorname{N}_{\bullet } \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \ar [r]^-{\rho _{+} } \ar [d]^{\sim } & \operatorname{Cospan}( \operatorname{N}_{\bullet } \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) ) \ar [d]^{\sim } & \operatorname{N}_{\bullet } \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)^{\operatorname{op}} \ar [l]_-{ \rho _{-} } \ar [d]^{\sim } \\ \operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})}(X,Y) \ar [r]^-{ \iota ^{\mathrm{L}} } & \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}) }(X,Y) & \operatorname{Hom}^{\mathrm{R}}_{ \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})}(X,Y) \ar [l]_-{ \iota ^{\mathrm{R}} } } \]

where the upper horizontal maps are the inclusions of Example 8.1.4.5, the lower horizontal maps are the the pinch inclusion maps of Construction 4.6.5.6, the outer vertical maps are the isomorphisms of Example 4.6.5.12, and the inner vertical map is the isomorphism of Corollary 8.1.5.6.

Stated more concretely, Corollary 8.1.5.6 asserts that we can identify $n$-simplices of the simplicial set $\operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}) }( X, Y)$ with commutative diagrams

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

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

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

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

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

consists of those simplices which correspond to commutative diagrams in which the rightward pointing $2$-morphisms $f_{i,j} \Rightarrow f_{i,j+1}$ are identity maps, 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 $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$.

Proof of Theorem 8.1.5.4. Let $\operatorname{\mathcal{A}}$ be an ordinary category, let $\operatorname{\mathcal{C}}$ be a $2$-category containing objects $X$ and $Y$, and let $U: [1] \times \operatorname{\mathcal{A}}\rightarrow \operatorname{\mathcal{C}}$ be a strictly unitary lax functor having the property that $U|_{ \{ 0\} \times \operatorname{\mathcal{A}}}$ and $U|_{ \{ 1\} \times \operatorname{\mathcal{A}}}$ are the constant functors taking the values $X$ and $Y$, respectively. We wish to show that there exists a unique functor of ordinary categories $F: \operatorname{Tw}(\operatorname{\mathcal{A}}) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$ such that $U$ is equal to the strictly unitary lax functor $U_{F}$ given by Construction 8.1.5.2. To prove this, we may assume without loss of generality that the $2$-category $\operatorname{\mathcal{C}}$ is strictly unitary (Proposition 2.2.7.7). Given a morphism $f: A \rightarrow B$ in the category $\operatorname{\mathcal{A}}$ and a pair of integers $0 \leq i \leq j \leq 1$, we write $f_{ji}: (i,A) \rightarrow (j,B)$ for the corresponding morphism in the product category $[1] \times \operatorname{\mathcal{A}}$. Unwinding the definitions, we see that the identity $U = U_{F}$ imposes the following requirements on the functor $F$:

$(1)$

Let $f: A \rightarrow B$ be a morphism in the category $\operatorname{\mathcal{C}}$, which we identify with an object of the twisted arrow category $\operatorname{Tw}(\operatorname{\mathcal{A}})$. Then $F(f)$ is equal to $U( f_{10} ) \in \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$.

$(2)$

Let $f: A \rightarrow B$ and $g: B \rightarrow C$ be composable morphisms in the category $\operatorname{\mathcal{A}}$, and regard the pairs $(\operatorname{id}_ A, g)$ and $(f, \operatorname{id}_ C)$ as elements of $\operatorname{Hom}_{\operatorname{Tw}(\operatorname{\mathcal{A}})}(f, g \circ f)$ and $\operatorname{Hom}_{\operatorname{Tw}(\operatorname{\mathcal{A}})}( g, g \circ f)$, respectively. Then $F( \operatorname{id}_ A, g)$ and $F(f, \operatorname{id}_ C)$ are equal to the composition constraints $\mu _{ g_{11}, f_{10} }$ and $\mu _{ g_{10}, f_{00} }$ for the lax functor $U$, respectively.

We now establish the uniqueness of the functor $F$. The value of $F$ on objects is determined by condition $(1)$. If $f: A \rightarrow B$ and $f': A' \rightarrow B'$ are objects of the twisted arrow category $\operatorname{Tw}(\operatorname{\mathcal{A}})$, then an element of $\operatorname{Hom}_{\operatorname{Tw}(\operatorname{\mathcal{A}})}(f, f')$ can be identified with a pair $(u,v)$ where $u \in \operatorname{Hom}_{\operatorname{\mathcal{A}}}(A',A)$ and $v \in \operatorname{Hom}_{\operatorname{\mathcal{A}}}(B,B')$ satisfy $f' = v \circ f \circ u$. In this case, the morphism $(u,v)$ factors as a composition $(u, \operatorname{id}_{B'} ) \circ (\operatorname{id}_{A}, v)$, so condition $(2)$ guarantees the identity

\[ F(u,v) = F(u, \operatorname{id}_{B'} ) \circ F( \operatorname{id}_{A}, v) = \mu _{ (vf)_{10}, u_{00} } \circ \mu _{ v_{11}, f_{10} }. \]

This proves the uniqueness of $F$ on morphisms.

To prove existence, we define the functor $F$ on objects $f \in \operatorname{Tw}(\operatorname{\mathcal{A}})$ by setting $F(f) = U( f_{10} )$, and on morphisms $(u,v) \in \operatorname{Hom}_{\operatorname{Tw}(\operatorname{\mathcal{A}})}(f,f')$ by the formula

\[ F(u,v) = \mu _{ (vf)_{10}, u_{00} } \circ \mu _{ v_{11}, f_{10} }. \]

Note that this prescription automatically satisfies condition $(1)$. Since $U$ is a strictly unitary functor between strictly unitary $2$-categories, its composition constraints $\mu _{g,h}$ are the identity whenever either $g$ or $h$ is an identity morphism (Remark 2.2.7.5), which shows that $F$ satisfies condition $(2)$ and that it carries identity morphisms to identity morphisms. We will complete the proof by showing that $F$ is compatible with composition. Let $f: A \rightarrow B$, $f': A' \rightarrow B'$, and $f'': A'' \rightarrow B''$ be objects of the twisted arrow category $\operatorname{Tw}(\operatorname{\mathcal{A}})$, and suppose we are given morphisms $(u,v) \in \operatorname{Hom}_{\operatorname{Tw}(\operatorname{\mathcal{A}})}( f, f')$ and $(u',v') \in \operatorname{Hom}_{\operatorname{Tw}(\operatorname{\mathcal{A}})}( f',f'')$. We wish to prove an equality $F( u \circ u', v' \circ v) = F(u',v') \circ F(u,v)$ of morphisms from $F(f)$ to $F(f'')$ in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$. Unwinding the definitions, this is equivalent to the commutativity of the outer cycle of the diagram

\[ \xymatrix@R =50pt@C=50pt{ & & F(vfu) \ar@ {=>}[dr]^{ \mu _{v'_{11}, (vfu)_{10} } } & & \\ & F(vf) \ar@ {=>}[ur]^{ \mu _{ (vf)_{10}, u_{00}}} \ar@ {=>}[dr]^{ \mu _{v'_{11}, (vf)_{10} } } & & F(v'vfu) \ar@ {=>}[dr]^{ \mu _{ (v'vfu)_{10}, u'_{00} } } & \\ F(f) \ar@ {=>}[ur]^{ \mu _{v_{11}, f_{10} } }\ar@ {=>}[rr]^{ \mu _{ (v'v)_{11}, f_{10} }} & & F(v'vf) \ar@ {=>}[ur]^{ \mu _{ (v'vf)_{10}, u_{00} }} \ar@ {=>}[rr]^{ \mu _{ (v'vf)_{10}, (uu')_{00} }} & & F( v'vfuu') } \]

in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$. In fact, the entire diagram commutes. The commutativity of the upper square follows by applying property $(c)$ of Definition 2.2.4.5 to the composable triple of morphisms

\[ (0,A') \xrightarrow {u_{00} } (0,A) \xrightarrow { (vf)_{10} } (1,B') \xrightarrow { v'_{11} } (1,B'') \]

in the product category $[1] \times \operatorname{\mathcal{A}}$. The commutativity of the lower left triangle follows by applying property $(c)$ to the composable triple of morphisms

\[ (0,A) \xrightarrow {f_{10}} (1,B) \xrightarrow { v_{11} } (1,B') \xrightarrow { v'_{11} } (1,B'') \]

and noting that the composition constraint $\mu _{ v'_{11}, v_{11} }$ is equal to the identity (by virtue of our assumption that the lax functor $U|_{ \{ 1\} \times \operatorname{\mathcal{A}}}$ is constant). Similarly, the commutativity of the lower right triangle follows by applying $(c)$ to the compostable triple of morphisms

\[ (0,A'') \xrightarrow { u'_{00} } (0,A') \xrightarrow {u_{00} } (0,A) \xrightarrow { (v'vf)_{10} } (1,B'') \]

and noting that the composition constraint $\mu _{ u_{00}, u'_{00} }$ is equal to the identity (by virtue of our assumption that the lax functor $U|_{ \{ 0\} \times \operatorname{\mathcal{A}}}$ is constant). $\square$