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2.2.6 The Duskin Nerve of a Strict $2$-Category

Let $\operatorname{\mathcal{C}}$ be a strict $2$-category (Definition 2.1.0.1). Then we can regard $\operatorname{\mathcal{C}}$ as a $2$-category (in which the associativity and unit constraints are identity morphisms), and form the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ by applying Construction 2.2.1.1. However, the Duskin nerve of a strict $2$-category admits a more direct description, which can be formulated entirely in terms of strict $2$-categories (and strict functors between them). The proof is based on a construction which will play an important role in §.

Construction 2.2.6.1 (The $2$-Category of Strings). Let $(Q, \leq )$ be a partially ordered set. We define a strict $2$-category $\operatorname{String}(Q)$ as follows:

  • The objects of $\operatorname{String}(Q)$ are the elements of $Q$.

  • Given elements $x,y \in Q$, we let $\underline{\operatorname{Hom}}_{ \operatorname{String}(Q) }(x, y)$ denote the partially ordered set of all finite linearly ordered subsets $S \subseteq Q$ such that $x$ is a least element of $S$ and $y$ is the greatest element of $S$ (where the partial order is given by inclusion of linearly ordered subsets of $Q$). We regard the partially ordered set $\underline{\operatorname{Hom}}_{\operatorname{String}(Q)}(x,y)$ as a category, having a unique morphism $S \Rightarrow S'$ when $S$ is contained in $S'$.

  • For every element $x \in Q$, the identity $1$-morphism $\operatorname{id}_{x} \in \underline{\operatorname{Hom}}_{\operatorname{String}(Q)}( x, x)$ is given by the singleton $\{ x\} $ (regarded as a linearly ordered subset of $Q$, having greatest and least element $x$).

  • For every triple of objects $x,y,z \in Q$, the composition functor

    \[ \circ : \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(y,z) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(x,y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(x,z) \]

    is given (on objects) by $(S, T) \mapsto S \cup T$.

Remark 2.2.6.2. Let $Q$ be a partially ordered set, which we regard as a category (having a unique morphism from $x$ to $y$ when $x \leq y$). Note that, for every pair of elements $x,y \in Q$, the category $\underline{\operatorname{Hom}}_{\operatorname{String}(Q)}(x,y)$ is empty unless $x \leq y$. It follows that there is a unique (strict) functor $\operatorname{String}(Q) \rightarrow Q$ which is the identity on objects.

Construction 2.2.6.3. Let $Q$ be a partially ordered set, which we regard as a category having a unique morphism $e_{y,x}$ for every pair of elements $x,y \in Q$ with $x \leq y$. We define a strictly unitary lax functor $T_{Q}: Q \rightarrow \operatorname{String}(Q)^{\operatorname{c}}$ as follows:

  • On objects, the lax functor $T_{Q}$ is given by $T_{Q}(x) = x$.

  • On $1$-morphisms, the lax functor $T_{Q}$ is given by $T_{Q}( e_{y,x} ) = \{ x,y \} \in \underline{\operatorname{Hom}}_{ \operatorname{String}(Q)}(x,y)$ whenever $x \leq y$ in $Q$.

  • For every triple of elements $x,y,z \in Q$ satisfying $x \leq y \leq z$, the composition constraint $\mu _{z,y,x}: T_{Q}( e_{z,y} ) \circ T_{Q}( e_{y,z} ) \Rightarrow T_{Q}( e_{z,x} )$ is the $2$-morphism of $\operatorname{String}(Q)^{\operatorname{c}}$ corresponding to the inclusion of linearly ordered sets

    \[ T_ Q( e_{z,x} ) = \{ x, z \} \subseteq \{ x, y, z \} = \{ y, z \} \cup \{ x, y \} = T_{Q}( e_{y,x} ) \circ T_{Q}( e_{z,y} ). \]

Remark 2.2.6.4. Let $Q$ be a partially ordered set, let $T_{Q}: Q \rightarrow \operatorname{String}(Q)^{\operatorname{c}}$ be the lax functor of Construction 2.2.6.3, and let $F: \operatorname{String}(Q)^{\operatorname{c}} \rightarrow Q^{\operatorname{c}} = Q$ be (the conjugate of) the functor of Remark 2.2.6.2 (so that $F$ is the identity on objects). Then the composition

\[ Q \xrightarrow {T_ Q} \operatorname{String}(Q)^{\operatorname{c}} \xrightarrow {F} Q \]

is the identity functor from $Q$ to itself. Beware that the composition

\[ \operatorname{String}(Q)^{\operatorname{c}} \xrightarrow {F} Q \xrightarrow {T_ Q} \operatorname{String}(Q)^{\operatorname{c}} \]

is not the identity (as a lax functor from $\operatorname{String}(Q)^{\operatorname{c}}$ to itself). This composition carries each object of $\operatorname{String}(Q)^{\operatorname{c}}$ to itself, but is given on $1$-morphism by the construction $\{ x_0 < x_1 < \cdots < x_ n \} \mapsto \{ x_0 < x_ n \} $.

The $2$-category $\operatorname{String}(Q)$ of Construction 2.2.6.1 is characterized by the following universal property:

Theorem 2.2.6.5. Let $Q$ be a partially ordered set and let $T_{Q}: Q \rightarrow \operatorname{String}(Q)^{\operatorname{c}}$ be the lax functor of Construction 2.2.6.3. For every strict $2$-category $\operatorname{\mathcal{C}}$, composition with $T_{Q}$ induces a bijection

\[ \{ \text{\em Strict functors $F: \operatorname{String}(Q)^{\operatorname{c}} \rightarrow \operatorname{\mathcal{C}}$} \} \rightarrow \{ \text{\em Strictly unitary lax functors $F^{+}: Q \rightarrow \operatorname{\mathcal{C}}$} \} . \]

Before giving the proof of Theorem 2.2.6.5, let us note . For each integer $n \geq 0$, we let $\operatorname{String}[n]$ denote the strict $2$-category obtained by applying Construction 2.2.6.1 to the linearly ordered set $[n] = \{ 0 < 1 < \cdots < n-1 < n \} $. The construction $[n] \mapsto \operatorname{String}[n]^{\operatorname{c}}$ determines a functor from the simplex category $\operatorname{{\bf \Delta }}$ of Definition 1.1.1.2 to the (ordinary) category $\operatorname{2Cat}_{\operatorname{Str}}$ of strict $2$-categories (Definition 2.1.5.5). We will view this functor as a cosimplicial object of $\operatorname{2Cat}_{\operatorname{Str}}$ which we denote by $\operatorname{String}[\bullet ]$. Applying the construction of Variant 1.1.5.3, we obtain a functor $\operatorname{Sing}_{\bullet }^{ \operatorname{String}[\bullet ]^{\operatorname{c}} }: \operatorname{2Cat}_{\operatorname{Str}} \rightarrow \operatorname{Set_{\Delta }}$, which carries each strict $2$-category $\operatorname{\mathcal{C}}$ to the simplicial set $[n] \mapsto \operatorname{Hom}_{ \operatorname{2Cat}_{\operatorname{Str}} }( \operatorname{String}[n]^{\operatorname{c}}, \operatorname{\mathcal{C}})$. Using Theorem 2.2.6.5, we can identify this construction with the Duskin nerve functor

\[ \operatorname{2Cat}_{\operatorname{Str}} \hookrightarrow \operatorname{2Cat}_{\operatorname{ULax}} \xrightarrow { \operatorname{N}_{\bullet }^{\operatorname{D}} } \operatorname{Set_{\Delta }}. \]

In particular, we have the following:

Corollary 2.2.6.6. For every strict $2$-category $\operatorname{\mathcal{C}}$, there is a canonical isomorphism of simplicial sets

\[ \operatorname{Sing}_{\bullet }^{\operatorname{String}[\bullet ]^{\operatorname{c}} }(\operatorname{\mathcal{C}}) \simeq \operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}}), \]

given on $n$-simplices by composition with the lax functor $T_{[n]}: [n] \rightarrow \operatorname{String}[n]^{\operatorname{c}}$ of Construction 2.2.6.3. In other words, the Duskin nerve $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ is given by

\[ \operatorname{N}^{\operatorname{D}}_{n}(\operatorname{\mathcal{C}}) \simeq \{ \text{\em Strict functors $\operatorname{String}[n]^{\operatorname{c}} \rightarrow \operatorname{\mathcal{C}}$} \} . \]

Remark 2.2.6.7. It is not difficult to see that the category $\operatorname{2Cat}_{\operatorname{Str}}$ of strict $2$-categories admits small colimits (beware that this is not true for the larger category $\operatorname{2Cat}$). Combining Corollary 2.2.6.6 with Proposition 1.1.6.18, we deduce that the Duskin nerve functor $\operatorname{N}^{\operatorname{D}}_{\bullet }: \operatorname{2Cat}_{\operatorname{Str}} \rightarrow \operatorname{Set_{\Delta }}$ admits a left adjoint $\operatorname{Set_{\Delta }}\rightarrow \operatorname{2Cat}_{\operatorname{Str}}$, which carries a simplicial set $S_{\bullet }$ to the generalized geometric realization $| S_{\bullet } |^{\operatorname{String}[\bullet ]^{\operatorname{c}} }$. Composing this left adjoint with the fully faithful embedding $\operatorname{N}^{\operatorname{D}}_{\bullet }: \operatorname{2Cat}_{\operatorname{ULax}} \rightarrow \operatorname{Set_{\Delta }}$ (Theorem 2.2.4.1), we deduce that the inclusion functor $\operatorname{2Cat}_{\operatorname{Str}} \hookrightarrow \operatorname{2Cat}_{\operatorname{ULax}}$ has a left adjoint, given by the construction $\operatorname{\mathcal{C}}\mapsto | \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}) |^{ \operatorname{String}[\bullet ]^{\operatorname{c}} }$. We can regard Theorem 2.2.6.5 as providing an explicit description of this left adjoint in a special case: it carries each partially ordered set $Q$ to the strict $2$-category $\operatorname{String}(Q)^{\operatorname{c}}$ given by Construction 2.2.6.1.

Proof of Theorem 2.2.6.5. Let $\operatorname{\mathcal{C}}$ be a strict $2$-category, let $Q$ be a partially ordered set, and let $F: Q \rightarrow \operatorname{\mathcal{C}}$ be a strictly unitary lax functor. We wish to show that $F$ factors uniquely as a composition

\[ Q \xrightarrow {T_ Q} \operatorname{String}(Q)^{\operatorname{c}} \xrightarrow {F^{+} } \operatorname{\mathcal{C}}, \]

where $T_{Q}$ is the strictly unitary lax functor of Construction 2.2.6.3 and $F^{+}$ is a strict functor from $\operatorname{String}(Q)^{\operatorname{c}}$ to $\operatorname{\mathcal{C}}$.

For every pair of elements $x,y \in Q$ satisfying $x \leq y$, we let $e_{y,x}: x \rightarrow y$ denote the unique morphism from $x$ to $y$ in the category $Q$, and for every triple $x,y,z \in Q$ satisfying $x \leq y \leq z$, we let $\mu _{z,y,x}: F( e_{z,y} ) \circ F( e_{y,x} ) \Rightarrow F( e_{z,x} )$ denote the composition constraint for the lax monoidal functor $F$. Unwinding the definitions, we see that a strict functor $F^{+}: \operatorname{String}(Q)^{\operatorname{c}} \rightarrow \operatorname{\mathcal{C}}$ satisfies $F^{+} \circ T_{Q} = F$ if and only if the following conditions are satisfied:

$(0)$

For every element $x \in Q$, we have $F^{+}(x) = F(x)$ (as objects of the $2$-category $\operatorname{\mathcal{C}}$).

$(1)$

For every pair of elements $x,y \in Q$ satisfying $x \leq y$, we have $F^{+}( \{ x, y\} ) = F( e_{y,x} )$ (as $1$-morphisms from $F(x)$ to $F(y)$ in the strict $2$-category $\operatorname{\mathcal{C}}$).

$(2)$

For every triple of elements $x,y,z \in Q$ satisfying $x \leq y \leq z$, the functor $F^{+}$ carries the inclusion $\{ x, z \} \subseteq \{ x,y,z \} $ (regarded as a $2$-morphism from $\{ y, z\} \circ \{ x,y\} $ to $\{ x, z\} $ in the strict $2$-category $\operatorname{String}(Q)^{\operatorname{c}}$) to $F( \mu _{z,y,x} )$ (regarded as a $2$-morphism from $F( e_{z,y} ) \circ F( e_{ y,x} )$ to $F( e_{z,x} )$ in the strict $2$-category $\operatorname{\mathcal{C}}$).

Note that, since we are requiring $F^{+}$ to be a strict functor, we can replace $(1)$ by the following stronger condition:

$(1')$

For every nonempty finite linearly ordered subset $S = \{ x_0 < x_1 < \cdots < x_ n \} \subseteq Q$, the functor $F^{+}$ carries $S$ (regarded as a $1$-morphism from $x_0$ to $x_ n$ in the strict $2$-category $\operatorname{String}(Q)^{\operatorname{c}}$) to the composition $F( e_{ x_{n}, x_{n-1} } ) \circ \cdots \circ F( e_{ x_{1}, x_0} )$ (regarded as a $1$-morphism from $F( x_0)$ to $F(x_ n)$ in the strict $2$-category $\operatorname{\mathcal{C}}$). In what follows, we will denote this composition by $F(S)$.

Let $S = \{ x_0 < x_1 < \cdots < x_ n \} $ be a nonempty finite linearly ordered subset of $Q$, and let $x_ i$ be an element of $S$ which is neither the largest nor the smallest (so that $0 < i < n$). In this case, we let $\gamma _{S, x_ i}: F( S ) \Rightarrow F(S \setminus \{ x_ i \} )$ denote the $2$-morphism of $\operatorname{\mathcal{C}}$ given by the horizontal composition

\[ \gamma _{S,x_ i} = \operatorname{id}_{ e_{x_ n, x_{n-1} } } \circ \cdots \circ \operatorname{id}_{ e_{x_{i+2}, x_{i+1} } } \circ F(\mu _{ x_{i+1}, x_{i}, x_{i-1} }) \circ \operatorname{id}_{ e_{ x_{i-1}, x_{i-2} } } \circ \cdots \circ \operatorname{id}_{ e_{x_1, x_0} }. \]

More generally, given a sequence of distinct elements $s_1, s_2, \cdots , s_ m \in S \setminus \{ x_0, x_ n \} $, we let $\gamma _{ S, s_1, \ldots , s_ m }: F(S) \Rightarrow F( S \setminus \{ s_1, \ldots , s_ m \} )$ denote the $2$-morphism of $\operatorname{\mathcal{C}}$ given by the vertical composition

\[ F(S) \xRightarrow { \gamma _{S, s_1} } F(S \setminus \{ s_1\} ) \xRightarrow { \gamma _{ S \setminus \{ s_1\} , s_2} } F( S \setminus \{ s_1, s_2\} ) \Rightarrow \cdots \Rightarrow F( S \setminus \{ s_1, \ldots , s_ m \} ). \]

Since the strict functor $F^{+}$ is required to be compatible with vertical and horizontal composition, we can replace $(2)$ by the following stronger condition:

$(2')$

Let $S = \{ x_0 < x_1 < \cdots < x_ n \} $ be be a nonempty finite linearly ordered subset of $Q$. Then, for every sequence of distinct elements $s_1, \ldots , s_ m \in S \setminus \{ x_0, x_ n \} $, the functor $F^{+}$ carries the inclusion $S \setminus \{ s_1, \ldots , s_ m \} \subseteq S$ (regarded as a $2$-morphism from $S$ to $S \setminus \{ s_1, \ldots , s_ m \} $ in the strict $2$-category $\operatorname{String}(Q)^{\operatorname{c}}$) to the $2$-morphism $\gamma _{S, s_1, \ldots , s_ m }$ (regarded as a $2$-morphism from $F(S)$ to $F( S \setminus \{ s_1, \ldots , s_ m \} )$ in the strict $2$-category $\operatorname{\mathcal{C}}$).

It is now clear that the functor $F^{+}$ is unique if it exists: its values on objects, $1$-morphisms, and $2$-morphisms of $\operatorname{String}(Q)^{\operatorname{c}}$ are determined by conditions $(0)$, $(1')$, and $(2')$, respectively. To prove existence, it will suffice to show that this prescription is well-defined: namely, that the $2$-morphism $\gamma _{S, s_1, \ldots , s_ m}$ defined above depends only on the sets $S$ and $T = S \setminus \{ s_1, \ldots , s_ m \} $, and not on the order of the sequence $(s_1, \ldots , s_ m )$ (it then follows easily from the construction that the definition of $F^{+}$ on $2$-morphisms is compatible with vertical and horizontal composition). Since group of all permutations of the set $\{ s_1, \ldots , s_ m \} $ is generated by transpositions of adjacent elements, it will suffice to show that we have

\[ \gamma _{S, s_1, \cdots , s_{i-1}, s_{i}, s_{i+1}, s_{i+2}, \cdots , s_ m } = \gamma _{S, s_1, \cdots , s_{i-1}, s_{i+1}, s_{i}, s_{i+2}, \cdots , s_ m } \]

for each $1 \leq i < m$. Replacing $S$ by $S \setminus \{ s_1, \ldots , s_{i-1} \} $, we are reduced to proving that $\gamma _{S, s, t} = \gamma _{ S, t, s }$ whenever $s < t$ are elements of $S - \{ x_0, x_ n \} $. We now distinguish two cases:

  • Suppose that the elements $s$ and $t$ are non-consecutive elements of $S$: that is, we have $s = x_ i$ and $t = x_ j$ where $j > i + 1$. In this case, we can identify both $\gamma _{S, s,t}$ and $\gamma _{S, t, s}$ with the horizontal composition

    \[ \operatorname{id}_{ e_{x_ n, x_{n-1} } } \circ \cdots \circ F(\mu _{ x_{j+1}, x_{j}, x_{j-1} }) \circ \cdots \circ F( \mu _{ x_{i+1}, x_ i, x_{i-1} }) \circ \cdots \circ \operatorname{id}_{ e_{x_1, x_0} }. \]
  • Suppose that the elements $s$ and $t$ are consecutive: that is, we have $S = \{ x_0 < \cdots < r < s < t < u < \cdots < x_ n \} $. In this case, to verify the identity $\gamma _{S, s, t} = \gamma _{S, t, s}$, we can replace $S$ by the subset $\{ r < s < t < u \} $ and thereby reduce to checking the commutativity of the diagram

    \[ \xymatrix@C =80pt@R=50pt{ F( e_{ u, t} ) \circ F( e_{t,s} ) \circ F( e_{s,r} ) \ar@ {=>}[r]^-{ \operatorname{id}_{ F( e_{u,t} )} \circ \mu _{t,s,r} } \ar@ {=>}[d]^-{ \mu _{u,t,s} \circ \operatorname{id}_{ F( e_{s,r})} } & F( e_{u,t} ) \circ F( e_{ t, r} ) \ar@ {=>}[d]^{ \mu _{u,t,r} } \\ F( e_{u,s} ) \circ F( e_{s,r} ) \ar@ {=>}[r]^{ \mu _{ u,s,r} } & F( e_{u,r} ) } \]

    in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( F(r), F(u) )$, which is the coherence condition required by the composition contraints for the lax functor $F$ (axiom $(c)$ of Definition 2.1.4.3).

$\square$