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

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1. Globular theories and cellular nerves

Contents:

Batanin’s $\omega$ -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 $(T(n))_{n\in \mathhb{N}_0}$ endowed with a map $i_T: T_{\gt 0}$ decreasing the degree by one and such that all fibers $i_T^{-1}(x)$ are linearly ordered.

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

Let $T$ be a tree.

A $T$ -sector of height $k$ is defined to be a cospan

\begin{matrix} y^{'} & y^{''} \\ ↘ & ↙ \\ y \end{matrix}

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

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

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

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

\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.

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

1. Globular theories and cellular nerves

Definition (finite planar level tree)

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

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