Kerodon

Remark 10.2.6.2. By construction, the category $\operatorname{{\bf \Delta }}_{ \mathrm{min}}$ is a (non-full) subcategory of the simplex category $\operatorname{{\bf \Delta }}$ of Definition 1.1.0.2. It can therefore also be regarded as a subcategory of the augmented simplex category $\operatorname{{\bf \Delta }}_{+}$ of Definition 10.2.1.10. The inclusion functor $\operatorname{{\bf \Delta }}_{\mathrm{min}} \hookrightarrow \operatorname{{\bf \Delta }}_{+}$ admits a left adjoint $C_{+}: \operatorname{{\bf \Delta }}_{+} \rightarrow \operatorname{{\bf \Delta }}_{\mathrm{min}}$, given concretely by the construction $C_{+}( [n] ) = [0] \star [n] \simeq [n+1]$. We will refer to $C_{+}$ as the concatenation functor. We let $C: \operatorname{{\bf \Delta }}\rightarrow \operatorname{{\bf \Delta }}_{\mathrm{min}}$ denote the restriction of $C_{+}$ to the simplex category $\operatorname{{\bf \Delta }}$, which we will also refer to as the concatenation functor.