Clemens Berger, A Cellular Nerve for Higher Categories (Rev #4)

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.

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 1i_T^{-1}.

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 *GT¯ *s,t:T_*\to G\overline 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).

This **-construction is due to Batanin.

Revision on November 18, 2012 at 16:58:49 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.