nLab cubical-simplicial set

Idea

A notion intermediate of those of bisimplicial set and bicubical set

Definition

A cubical-simplicial set is a functor

$$F : \Box^{op} \times \Delta^{op} \to \mathrm{Set}$$

where $\Box$ is a category of cubes and $\Delta$ is the simplex category.

This is the same as a cubical object in the category of simplicial sets.

Examples

• triangulated type theory has categorical semantics in cubical-simplicial sets where the category of cubes used is the category of finite flat distributive lattices.

Related concepts

• simplicial set

• bisimplicial set

• cubical set

• bicubical set

• cubical infinity-groupoid