Simplex Category

Definitions

The simplex category $\Delta$ is the category of finite non-empty ordinals and monotone maps.

$$[n]=\{0<1<\cdots <n\}(n\ge 0),$$

An augmented simplex category (written ${\Delta }_{+}$ or ${\Delta }_a$) is a simplex category that also includes the empty ordinal $[-1]=\varnothing$.

Write ${\Delta }_{\le n}$ for the full subcategory of $\Delta$ restricted to objects.

Remarks

• Some authors use simplex category as including $[-1]=\varnothing$. Conventions vary.
• It is often helpful to think of the simplex category as a full subcategory of Cat on the finite linear orders, regarded as categories. Its object categories are the finite ordinal categories and its morphisms are all functors between objects. This construction is an instance of the general poset→category.