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

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

Each such tree may naturally be regarded as specifying an (colored) 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 operads.,

Dendroidal sets

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

References

See the references at dendroidal set.