Clemens Berger, A Cellular Nerve for Higher Categories (Rev #3, changes)

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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)) nmathhbN 0(T(n))_{n\in \mathhb{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} y^\prime&&y^{\prime\prime} \\ \searrow&\swarrow \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 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) }

Revision on November 18, 2012 at 15:26:14 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.