Augmented Simplex Category

The augmented simplex category $\Delta_{a}$ is the category where

• Objects are Natural Numbers
• Morphisms $m\to n$ are monotone maps $[m]\to[n]$ between Finite Ordinals.

Properties

• The augmented simplex category is a Monoidal Category with respect to Ordinal Sum. In fact, the augmented simplex category is the walking Monoid Object, where the unit is the unique face map $\eta:[0]\to[1]$ and join is the unique degeneracy $\sigma:[2]\to[1]$.

• The object $1$ is a Terminal Object in the augmented simplex category, and the unique map $[n]\to[1]$ is always a degeneracy.

• The object $0$ is a Initial Object in the augmented simplex category, and the unique map $[0]\to[n]$ is always a face map. As $0$ is the unit for ordinal sum, this makes the augmented semisimplex category Semicocartesian Monoidal Category.

• However, the ordinal sum $m\oplus n$ is not a coproduct. The problem boils down to the fact that the obvious coduplication map $[n+n]\to[n]$ is not order preserving. We could try to use the map that arises from dividing by $2$, but this will not commute with the inclusions.

Questions

• Does the augmented simplex category have pullbacks? Pullbacks + a terminal object give products, so I suppose that is the real question to ask.