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2.3.3 Thin $2$-Simplices of a Duskin Nerve

Let $\operatorname{\mathcal{C}}$ be a $2$-category and let $\sigma $ be a $2$-simplex of the Duskin nerve $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$, corresponding to a diagram

\[ \xymatrix { & Y \ar [dr]^{g} \ar@ {=>}[]+<0pt,-15pt>;+<0pt,-30pt>^-{\gamma } & \\ X \ar [ur]^{f} \ar [rr]_{h} & & Z. } \]

Our goal is to prove Theorem 2.3.2.5, which asserts that $\sigma $ is thin (in the sense of Definition 2.3.2.3) if and only if the $2$-morphism $\gamma : g \circ f \Rightarrow h$ is invertible. This follows from Propositions 2.3.3.1 and Proposition 2.3.3.2 below.

Proposition 2.3.3.1. Let $\operatorname{\mathcal{C}}$ be a $2$-category, let $n \geq 3$, and let $u: \Lambda ^{n}_{\ell } \rightarrow \operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ be a map of simplicial sets for some $0 < \ell < n$. Let $\sigma $ denote the $2$-simplex of $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ obtained by composing $u$ with the map $\Delta ^2 \rightarrow \Lambda ^{n}_{\ell }$ given by the map of linearly ordered sets

\[ [2] \simeq \{ \ell -1, \ell , \ell +1 \} \subseteq [n], \]

corresponding to a diagram

\[ \xymatrix { & X_{\ell } \ar [dr] \ar@ {=>}[]+<0pt,-15pt>;+<0pt,-30pt>^-{\gamma } & \\ X_{\ell -1} \ar [ur] \ar [rr] & & X_{\ell +1} } \]

in the $2$-category $\operatorname{\mathcal{C}}$. If $\gamma $ is invertible, then $u$ extends uniquely to an $n$-simplex of $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$.

Proof. Using Examples 2.3.1.13 and 2.3.1.14, we see that the restriction of $u$ to the $1$-skeleton of $\Lambda ^ n_{\ell }$ is given by a collection of objects $\{ X_ i \} _{0 \leq i \leq n}$ of $\operatorname{\mathcal{C}}$, together with $1$-morphisms $\{ f_{ji}: X_ i \rightarrow X_ j \} _{0 \leq i < j \leq n}$. For $n \geq 5$, the horn $\Lambda ^{n}_{\ell }$ contains the $3$-skeleton of $\Delta ^ n$, so the existence and uniqueness of the desired extension is automatic by virtue of Corollary 2.3.1.10 (in particular, we do not need to assume that $0 < \ell < n$ or that $\gamma $ is invertible). We now treat the case $n = 3$. We will assume that $\ell = 1$ (the case $\ell = 2$ follows by symmetry), so that we can use Example 2.3.1.15 to identify $u$ with a triple of $2$-morphisms

\[ \mu _{210}: f_{21} \circ f_{10} \Rightarrow f_{20} \quad \mu _{310}: f_{31} \circ f_{10} \Rightarrow f_{30} \quad \mu _{321}: f_{32} \circ f_{21} \Rightarrow f_{31}. \]

Using the description of $3$-simplices of $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ supplied by Example 2.3.1.16, we see an extension of $u$ to a $3$-simplex of the Duskin nerve $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ can be identified with a $2$-morphism $\mu _{320}: f_{32} \circ f_{20} \Rightarrow f_{30}$ satisfying the equation

\[ \mu _{320} (\operatorname{id}_{ f_{32} } \circ \mu _{210}) = \mu _{310} (\mu _{321} \circ \operatorname{id}_{ f_{10}} ) \alpha _{ f_{32}, f_{21}, f_{10} }. \]

Our assumption guarantees that $\gamma = \mu _{210}$ is an isomorphism; it follows that the preceding equation has a unique solution, given by

\[ \mu _{320} = \mu _{310} (\mu _{321} \circ \operatorname{id}_{ f_{10}} ) \alpha _{ f_{32}, f_{21}, f_{10} } (\operatorname{id}_{ f_{32} } \circ \mu _{210}^{-1} ). \]

We now treat the case $n=4$. For simplicity, we will assume that $\ell = 2$ (the cases $\ell = 1$ and $\ell = 3$ follow by a similar argument). To simplify the notation in what follows, we will denote the composition of a pair of $1$-morphisms of $\operatorname{\mathcal{C}}$ by $hg$, rather than $h \circ g$. Note that the horn $\Lambda ^{n}_{\ell }$ contains the $2$-skeleton of $\Delta ^ n$, so the morphism $u$ can be identified with a collection of $2$-morphisms $\mu _{kji}: f_{kj} f_{ji} \Rightarrow f_{ki}$. Using Example 2.3.1.16, we note that the extension of $u$ to a $4$-simplex of $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ is automatically unique, and exists if and only if the outer cycle commutes in the diagram

\[ \xymatrix@C =0pt{ f_{43} (f_{31} f_{10} ) \ar@ {=>}[rrrr]^{\sim } \ar@ {=>}[ddddd]^{ \mu _{310} } & & & & (f_{43} f_{31}) f_{10} \ar@ {=>}[ddddd]^{\mu _{431} } \\ & f_{43}( (f_{32} f_{21} ) f_{10} ) \ar@ {=>}[ul]_{\mu _{321}}^{\sim } \ar@ {=>}[rr]^{\sim } & & (f_{43} (f_{32} f_{21})) f_{10} \ar@ {=>}[ur]^{\mu _{321}}_{\sim } \ar@ {=>}[d]^{\sim } & \\ & f_{43} ( f_{32} (f_{21} f_{10} ) ) \ar@ {=>}[u]^{\sim } \ar@ {=>}[dr]^{ \sim } \ar@ {=>}[d]^{\mu _{210}} & & (( f_{43} f_{32}) f_{21}) f_{10} \ar@ {=>}[d]^{\mu _{432} } & \\ & f_{43} ( f_{32} f_{20} ) \ar@ {=>}[d]^{\sim } \ar@ {=>}[ddl]_{ \mu _{320} } & (f_{43} f_{32} ) (f_{21} f_{10}) \ar@ {=>}[dl]_{\mu _{210} } \ar@ {=>}[dr]^{ \mu _{432} } \ar@ {=>}[ur]^{\sim } & ( f_{42} f_{21} ) f_{10} \ar@ {=>}[ddr]^{ \mu _{421} } \ar@ {=>}[d]_{\sim } & \\ & (f_{43} f_{32} ) f_{20} \ar@ {=>}[r]_-{\mu _{432}} & f_{42} f_{20} \ar@ {=>}[d]^{\mu _{420}} & f_{42} (f_{21} f_{10} ) \ar@ {=>}[l]^-{\mu _{210}} & \\ f_{43} f_{30} \ar@ {=>}[rr]^{\mu _{430}} & & f_{04} & & f_{41} f_{10}; \ar@ {=>}[ll]_{ \mu _{410} } } \]

here the unlabelled $2$-morphisms are induced by the associativity constraints of $\operatorname{\mathcal{C}}$. This follows from a diagram chase, since $\mu _{321} = \gamma $ is an isomorphism and each of the inner cycles of the diagram commutes (the $4$-cycles commute by functoriality, the central $5$-cycle commutes by the pentagon identity in $\operatorname{\mathcal{C}}$, and the remaining $5$-cycles commute by virtue of our assumption that $u$ is defined on the $0$th, $1$st, $3$rd, and $4$th face of the simplex $\Delta ^{4}$). $\square$

Proposition 2.3.3.2. Let $\operatorname{\mathcal{C}}$ be a $2$-category and let $\sigma $ be a $2$-simplex of the Duskin nerve $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$, corresponding to a diagram

\[ \xymatrix { & Y \ar [dr]^{ g } \ar@ {=>}[]+<0pt,-15pt>;+<0pt,-30pt>^-{\gamma } & \\ X \ar [ur]^{f } \ar [rr]_{ h } & & Z. } \]

in the $2$-category $\operatorname{\mathcal{C}}$. Assume that the following condition is satisfied:

$(\ast )$

Let $n \in \{ 3,4\} $ and let $u: \Lambda ^{n}_{1} \rightarrow \operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ be a map of simplicial sets such that $u|_{ \Delta ^2} = \sigma $; here we identify $\Delta ^2$ with a simplicial subset of $\Lambda ^{n}_{1} \subseteq \Delta ^{n}$ via the inclusion map $[2] \hookrightarrow [n]$. Then $u$ extends to an $n$-simplex of $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$.

Then $\gamma $ is invertible.

Proof. Without loss of generality, we may assume that $\operatorname{\mathcal{C}}$ is strictly unitary (Example A.0.0.7). Applying $(\ast )$ in the case $n = 3$, we can extend $\sigma $ to a $3$-simplex of $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ which is represented by the pair of diagrams

\[ \xymatrix@C =100pt@R=50pt{ & Y \ar [r]^{ g } \ar@ {=>}[]+<5pt,-10pt>;+<15pt,-25pt>^-{\gamma } & Z \ar [dr]^{ \operatorname{id}_ Z } \ar@ {=>}[]+<-20pt,-20pt>;+<-45pt,-45pt>^{\delta } & \\ X \ar [ur]^{ f} \ar [urr]_{ h } \ar [rrr]^{g \circ f} & & & Z \\ & Y \ar [r]^{ g } \ar [drr]_{ g } \ar@ {=>}[]+<20pt,-20pt>;+<45pt,-45pt>_{ \operatorname{id}_{g \circ f} } & Z \ar [dr]^{ \operatorname{id}_ Z } \ar@ {=>}[]+<-5pt,-10pt>;+<-15pt,-25pt>^{\operatorname{id}_ g } & \\ X \ar [ur]^{f} \ar [rrr]^{g \circ f} & & & Z. } \]

It follows that $\gamma $ admits a left inverse, given by the vertical composition $\delta : h \Rightarrow g \circ f$. To show that this composition is also a right inverse, we apply $(\ast )$ in the case $n = 4$ to construct a $4$-simplex $\tau $ of $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ whose two-dimensional faces correspond to the $2$-morphisms

\[ \mu _{2,1,0} = \mu _{4,1,0} = \gamma \quad \quad \mu _{3,1,0} = \operatorname{id}_{ g \circ f} \quad \mu _{3,2,0} = \delta \quad \mu _{4,2,0} = \operatorname{id}_ h \]

\[ \mu _{4,3,0} = \gamma \quad \quad \mu _{3,2,1} = \mu _{4,2,1} = \mu _{4,3,1} = \operatorname{id}_ g \quad \quad \mu _{4,3,2} = \operatorname{id}_{\operatorname{id}_ Z}. \]

The $3$-simplex $d_1(\tau )$ then witnesses the identity

\[ \mu _{4,2,0} (\mu _{4,3,2} \circ \operatorname{id}_ h ) = \mu _{4,3,0} (\operatorname{id}_{ \operatorname{id}_ Z} \circ \mu _{3,2,0}), \]

which shows that $\delta $ is also a right inverse to $\gamma $. $\square$