This entry draws from * [[nLab:Clemens Berger]], _A cellular nerve for higher categories_, Advances in Mathematics 169, 118-175 (2002) ([pdf](http://math1.unice.fr/~cberger/nerve.pdf)) ## 0. Notation and terminology ### 0.1 ### 0.2 Higher graphs and higher categories +-- {: .num_defn #globe category} ###### Definition The *globe category* $G$ is defined to be the category with one object in each degree and the *globular operators* $s,t$ are defined by the identities $$ s\circ s=s\circ t$$ $$t\circ t= t\circ s$$ A presheaf on $G$ is called a *globular set* or *omega graph* or *$\omega$-graph*. $\omega$-graphs with natural transformations as morphisms form a category denoted by $\omegaGraph$. =-- +-- {: .num_defn} ###### Definition (Godement´s interchange rules) Let $C$ be $2$-category with underlying reflexive $2$-graph $C_i$ for $i=0,1,2$ with globular operators given by source, target, and identity. Then $(C_i)_i$ comes with three composition laws $$ \circ_i^j: C_j\times_i C_j\to C_j$$ for $0\le i\lt j\le 2$. Spelled out this means: $i=0, j=1$: composition of $1$-morphism along $0$-morphisms (i.e.objects) $i=0,j=2$: composition of $2$-morphisms along $0$-morphisms (i.e.objects), also called *[[nLab:horizontal composition]]*. $i=1,j=2$: composition of $2$-morphisms along $1$-morphisms, also called *[[nLab:vertical composition]]*. Then *Godement´s interchange rule* or *Godement´s interchange law* or just *[[nLab:interchange law]]* is the assertion that the immediate diagrams commute. Note that there is on more type of composition of a $1$-morphism with a $2$-morphism called *[[nLab:whiskering]]*. =-- +-- {: .num_defn #omega-category} ###### Definition ($\omega$-category) An $\omega$-category is defined to be a reflexive graph $(C_i)_i$ such that for every triple $i\lt j\lt k$, the family $(C_i,C_j,C_k;\circ_i^j,\circ_i^k, \circ_j^)$ has the structure of a $2$-category. =-- ## 1. Globular theories and cellular nerves Contents: Batanin's $\omega$-operads are described by their operator categories which are called *globular theories*. +-- {: .num_definition} ###### Definition (finite planar level tree) A _finite planar level tree_ ( or for short just a _tree_) is a graded set $(T(n))_{n\in \mathbb{N}_0}$ endowed with a map $i_T: T_{\gt 0}$ decreasing the degree by one and such that all fibers $i_T^{-1}(x)$ are linearly ordered. The collection of trees with maps of graded sets commuting with $i$ defines a category $\mathcal{T}$, called the *[[nLab:tree category|category of trees]]*. =-- +-- {: .num_lemma} ###### Example The finite ordinal $[n]\in \Delta$ we can regard as the 1-level tree with $n$ input edges. Hence the simplex category embeds in the tree category $\Delta\hookrightarrow\mathcal{T}$. =-- The following ${}_*$-construction is due to Batanin. +-- {: .num_lemma} ###### Lemma and Definition ($\omega$-graph of sectors of a tree) Let $T$ be a tree. A _$T$-sector of height $k$_ is defined to be a cospan $$\array{ y^\prime&&y^{\prime\prime} \\ \searrow&&\swarrow \\ &y }$$ denoted by $(y;y^\prime,y^{\prime\prime})$ where $y\in T(k)$ and $y\lt y^{\prime\prime}$ are consecutive vertices in the linear order $i_T^{-1}(y)$. The set $GT$ of $T$-sector is graded by the height of sectors. The _source of a sector $(y;y^\prime,y^{\prime\prime})$_ is defined to be $(i(y);x,y)$ where $x,y$ are consecutive vertices. The _target of a sector $(y;y^\prime,y^{\prime\prime})$_ is defined to be $(i(y);y,z)$ where $y,z$ are consecutive vertices. $$\array{ &y^\prime&&y^{\prime\prime} \\ & \searrow&&\swarrow \\ x&&y&&z \\ \searrow&&\downarrow^i&&\swarrow \\ &&i(y) }$$ To have a source and a target for every sector of $T$ we adjoin in every but the highest degree a lest- and a greatest vertex serving as new minimum and maximum for the linear orders $i^{-1}(x)$. We denote this new tree by $\overline T$ and the set of its sectors by $T_*:=G\overline T(k)$ and obtain _source- and target operators_ $s,t:T_*\to T_*$. This operators satisfy $$s\circ s=s\circ t$$ $$t\circ t =t\circ s$$ as one sees in the following diagram depicting an "augmented" tree of height $3$ $$\array{ T(3)&&&y^\prime&&y^{\prime\prime} \\ &&& \searrow&&\swarrow \\ T(2)&&x&&y&&z \\ &&\searrow&&\downarrow^i&&\swarrow \\ T(1)&&u&&v&&w \\ &&\searrow&&\downarrow^i&&\swarrow \\ T(0)&&&&r }$$ which means that $T_*$ is an $\omega$-graph (also called [[nLab:globular set]]). Now let $G$ denote the [[nLab:globe category]] whose unique object in degree $n\in \mathbb{N}$ is $n_G$, and let $\mathbf{n}$ denotes the linear $n$-level tree. Then we have $\mathbf{n}_*\simeq G(-,n_G)$ is the standard $n$-globe. (Note that the previous diagram corresponds to the standard $3$ globe.) =-- +-- {: .num_defn} ###### Definition Let $f:S_*\to T_*$ be a monomorphism. $f$ is called to be *cartesian* if $$\array{ (S_*)_n&\stackrel{f_n}{\to}&(T_*)_n \\ \downarrow^s&&\downarrow^t \\ (S_*)_{n-1}&\stackrel{f_{n-1}}{\to}&(T_*)_{n-1} }$$ is a pullback for all $n$. =-- +-- {: .num_lemma} ###### Lemma Let $S,T$ be level trees. (1) Any map $S_*\to T_*$ is injective. (2) The inclusions $S_*\hookrightarrow T_*$ correspond bijectively to cartesian subobjects of $T_*$. (3) The inclusions $S_*\hookrightarrow T_*$ correspond bijectively to plain subtrees of $T$ with a specific choice of $T$-sector for each input vertex of $S$. (...) =-- +-- {: .num_definition} ###### Definition (1) The category $\Theta_0$ defined by having as objects the level trees and as morphisms the maps between the associated $\omega$-graphs. These morphisms are called *immersions*. This category shall be equipped with the structure of a [[nLab:site]] by defining the covering sieves by epimorphic families (of immersions). This site is called _the globular site_. (2) A *globular theory* is defined to be a category $\Theta_A$ such that $$\Theta_0\hookrightarrow \Theta_A$$ is an inclusion of a wide subcategory such that representable presheaves on $\Theta_A$ restrict to sheaves on $\Theta_0$. (3) Presheaves on $\Theta_A$ which restrict to sheaves on $\Theta_0$ are called $\Theta_A$-models. =-- +-- {: .num_lemma} ###### Lemma The forgetful functor $$Sh (\Theta_0)\to \omega Graph:=Psh (G)$$ is an equivalence of categories. =-- +-- {: .proof} ###### Proof Let $X\in Psh(\Theta_0)$ and show that $X\in Sh(\Theta_0)$ iff $X(T)\simeq hom_{Psh(G)}(T_*,X)$ by writing $X$ as a colimit of representables. =-- +-- {: .num_defn} ###### Definition There is a monad $(w,\eta,\mu)$ on $\omega Graph$ defined by $$w(X)_n:=\coprod_{T:ht(T)\le n}hom_{\omega Graph}(T_*,X)$$ $\eta:id_{Psh(G)}\to w$ is induced by Yoneda: $X_n\mapsto hom_{\omega Graph}(n_*,X)$ =-- ## 2. Cellular sets and their geometric realization ## 3. A closed model structure for cellular sets ## 4. Homotopy structure for weak $\omega$-categories