Spahn
Clemens Berger, A Cellular Nerve for Higher Categories (Rev #10)

1. Globular theories and cellular nerves

Contents:

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 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.)

Definition

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.

Lemma

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. (…)

Definition

(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.

Lemma

The forgetful functor

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

is an equivalence of categories.

Proof

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.

Revision on November 19, 2012 at 00:44:52 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.