Spahn
Clemens Berger, A Cellular Nerve for Higher Categories (Rev #7, changes)

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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 n n n\in \mathbb{N} is n Gn_G , for and letnn n\in \mathbf{n} \mathbb{N} let denotes the linearnn \mathbf{n} 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 . The This category shall be equipped with the structure of asite by defining the covering sieves by epimorphic families (of immersions). This site is called the globular site.

(2)

Revision on November 18, 2012 at 22:40:32 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.