nLab
Theta category

Contents

Idea

For n the category Θ nJoyal’s disk category – may be thought of as the full subcategory of the category StrnCat of strict n-categories on n-categories that are something like free n-categories on pasting diagrams of n-globes.

For instance Θ 2 contains an object that is depicted

a b c b \array{ & \nearrow &\Downarrow& \searrow && && \nearrow && \searrow \\ a && \to && b &\to & c && \Downarrow && b \\ & \searrow &\Downarrow& \nearrow && && \searrow && \nearrow && }

Such pasting diagrams may be encoded in planar trees, the above one corresponds to the tree:

2 1 *.\array{ \nwarrow \nearrow & & & \uparrow &&& 2 \\ & \nwarrow & \uparrow & \nearrow &&& 1 \\ && {*} } \,.

Accordingly, Θ n is the category of planar rooted trees of level n.

In low degree we have

  • Θ 0=* is the point.

  • Θ 1=Δ is the simplex category: the n-simplex [n] is thought of as a linear quiver and as such the pasing diagram of n 1-morphisms

    01n.0 \to 1 \to \cdots \to n \,.

    Dually, this is the planar rooted tree of the form

    *\array{ \nwarrow &\uparrow & \cdots \nearrow \\ &{*} }

    with n-branches.

Definition

Examples

In Θ 0 write O 0 for the unique object. Then write in Θ n

O n:=[1](O n1).O_n := [1](O_{n-1}) \,.

This is the strict n-category free on a single n-globe.

Theta-spaces

A local model structure on simplicial presheaves on the Theta categories is called Theta spaces and models (n,r)-categories.

References

The Θ-categories were introduced in

An discussion with lots off pictures is in chapter 7 of

A useful discussion is in

  • Clemens Berger, Iterated wreath product of the simplex category and iterated loop spaces (arXiv)

and in section 3 of

there leading over to the notion of Theta space.