Clemens Berger, A Cellular Nerve for Higher Categories (changes)

Showing changes from revision #12 to #13: Added | Removed | Changed

This entry draws from

  • Clemens Berger, A cellular nerve for higher categories, Advances in Mathematics 169, 118-175 (2002) (pdf)

0. Notation and terminology


0.2 Higher graphs and higher categories


The globe category GG is defined to be the category with one object in each degree and the globular operators s,ts,t are defined by the identities

ss=st s\circ s=s\circ t
tt=tst\circ t= t\circ s

A presheaf on GG is called a globular set or omega graph or ω\omega-graph.

ω\omega-graphs with natural transformations as morphisms form a category denoted by omegaGraph\omegaGraph.

Definition (Godement´s interchange rules)

Let CC be 22-category with underlying reflexive 22-graph {(C_i)_{i=0,1,2} with globular operators given by source, target, and identity.

Then (C i) i(C_i)_i comes with three composition laws

i j:C j× iC jC j \circ_i^j: C_j\times_i C_j\to C_j

for 0i<j20\le i\lt j\le 2. Spelled out this means:

i=0,j=1i=0, j=1: composition of 11-morphism along 00-morphisms (i.e.objects)

i=0,j=2i=0,j=2: composition of 22-morphisms along 00-morphisms (i.e.objects), also called horizontal composition.

i=1,j=2i=1,j=2: composition of 22-morphisms along 11-morphisms, also called vertical composition.

Then Godement´s interchange rule or Godement´s interchange law or just interchange law is the assertion that the immediate diagrams commute.

Note that there is on more type of composition of a 11-morphism with a 22-morphism called whiskering.

Definition (ω\omega-category)

An ω\omega-category is defined to be a reflexive graph (C i) i(C_i)_i such that for every triple i<j<ki\lt j\lt k, the family (C i,C j,C k; i j, i k, j )(C_i,C_j,C_k;\circ_i^j,\circ_i^k, \circ_j^) has the structure of a 22-category.

1. Globular theories and cellular nerves


Batanin’s ω\omega-operads are described by their operator categories which are called globular theories.

Definition (finite planar level tree)

A finite planar level tree ( or for short just a tree) is a graded set (T(n)) n 0(T(n))_{n\in \mathbb{N}_0} endowed with a map i T:T >0i_T: T_{\gt 0} decreasing the degree by one and such that all fibers i T 1(x)i_T^{-1}(x) are linearly ordered.

The collection of trees with maps of graded sets commuting with ii defines a category 𝒯\mathcal{T}, called the category of trees.


The finite ordinal [n]Δ[n]\in \Delta we can regard as the 1-level tree with nn input edges. Hence the simplex category embeds in the tree category Δ𝒯\Delta\hookrightarrow\mathcal{T}.

The following *{}_*-construction is due to Batanin.

Lemma and Definition (ω\omega-graph of sectors of a tree)

Let TT be a tree.

A TT-sector of height kk is defined to be a cospan

y y y\array{ y^\prime&&y^{\prime\prime} \\ \searrow&&\swarrow \\ &y }

denoted by (y;y ,y )(y;y^\prime,y^{\prime\prime}) where yT(k)y\in T(k) and y<y y\lt y^{\prime\prime} are consecutive vertices in the linear order i T 1(y)i_T^{-1}(y).

The set GTGT of TT-sector is graded by the height of sectors.

The source of a sector (y;y ,y )(y;y^\prime,y^{\prime\prime}) is defined to be (i(y);x,y)(i(y);x,y) where x,yx,y are consecutive vertices.

The target of a sector (y;y ,y )(y;y^\prime,y^{\prime\prime}) is defined to be (i(y);y,z)(i(y);y,z) where y,zy,z are consecutive vertices.

y y x y z i i(y)\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 TT 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)i^{-1}(x). We denote this new tree by T¯\overline T and the set of its sectors by T *:=GT¯(k)T_*:=G\overline T(k) and obtain source- and target operators s,t:T *T *s,t:T_*\to T_*. This operators satisfy

ss=sts\circ s=s\circ t
tt=tst\circ t =t\circ s

as one sees in the following diagram depicting an “augmented” tree of height 33

T(3) y y T(2) x y z i T(1) u v w i T(0) r\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 *T_* is an ω\omega-graph (also called globular set).

Now let GG denote the globe category whose unique object in degree nn\in \mathbb{N} is n Gn_G, and let n\mathbf{n} denotes the linear nn-level tree.

Then we have n *G(,n G)\mathbf{n}_*\simeq G(-,n_G) is the standard nn-globe. (Note that the previous diagram corresponds to the standard 33 globe.)


Let f:S *T *f:S_*\to T_* be a monomorphism.

ff is called to be cartesian if

(S *) n f n (T *) n s t (S *) n1 f n1 (T *) n1\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 nn.


Let S,TS,T be level trees.

(1) Any map S *T *S_*\to T_* is injective.

(2) The inclusions S *T *S_*\hookrightarrow T_* correspond bijectively to cartesian subobjects of T *T_*.

(3) The inclusions S *T *S_*\hookrightarrow T_* correspond bijectively to plain subtrees of TT with a specific choice of TT-sector for each input vertex of SS. (…)


(1) The category Θ 0\Theta_0 defined by having as objects the level trees and as morphisms the maps between the associated ω\omega-graphs. These morphisms are called immersions. This category shall be equipped with the structure of a 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 Θ A\Theta_A such that

Θ 0Θ A\Theta_0\hookrightarrow \Theta_A

is an inclusion of a wide subcategory such that representable presheaves on Θ A\Theta_A restrict to sheaves on Θ 0\Theta_0.

(3) Presheaves on Θ A\Theta_A which restrict to sheaves on Θ 0\Theta_0 are called Θ A\Theta_A-models.


The forgetful functor

Sh(Θ 0)ωGraph:=Psh(G)Sh (\Theta_0)\to \omega Graph:=Psh (G)

is an equivalence of categories.


Let XPsh(Θ 0)X\in Psh(\Theta_0) and show that XSh(Θ 0)X\in Sh(\Theta_0) iff X(T)hom Psh(G)(T *,X)X(T)\simeq hom_{Psh(G)}(T_*,X) by writing XX as a colimit of representables.


There is a monad (w,η,μ)(w,\eta,\mu) on ωGraph\omega Graph defined by

w(X) n:= T:ht(T)nhom ωGraph(T *,X)w(X)_n:=\coprod_{T:ht(T)\le n}hom_{\omega Graph}(T_*,X)

η:id Psh(G)w\eta:id_{Psh(G)}\to w is induced by Yoneda: X nhom ωGraph(n *,X)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 ω\omega-categories

Last revised on November 20, 2012 at 21:13:27. See the history of this page for a list of all contributions to it.