Holmstrom Pointed category

A category (e.g. a model category) is pointed if the morphism from the initial to the terminal object is an isomorphism.

For any model category $C$, the corresponding category under the terminal object is a pointed model category $C_*$, and there is a faithful, but not full, embedding $C \to C_*$.

One could also consider similar constructions of model structures on the category under (or over) any object $A$.

nLab page on Pointed category