nLab tree category

Idea

The simplex category may be regarded as the category of all linear directed graphs. The tree category generalizes this to directed rooted trees.

Definition

Finite planar level tree

(… ) see Berger

Finite symmetric rooted trees

We define the category $\Omega$ finite symmetric rooted trees.

The objects of $\Omega$ are non-empty non-planar trees with specified root.

Each such tree may naturally be regarded as specifying an (colored) symmetric operad with one color per edge of the tree. A morphism of trees in $\Omega$ is a morphism of the corresponding operads.

As such, $\Omega$ is by construction a full subcategory of that of symmetric operads enriched over Set.

Operadic structure

$\Omega$ can be turned into a (symmetric) operad by grafting trees.

Dendroidal sets

A presheaf on $\Omega$ is a dendroidal set, a generalization of a simplicial set.

References

• Clemens Berger, A cellular nerve for higher categories, Advances in Mathematics 169, 118-175 (2002) (pdf)

• See the references at dendroidal set.