Augmented Group

An augmented group is an algebraic structure corresponding to a set-indexed family of groups $G_{i}$. An augmented group consists of

• An indexing H-Set $I$
• A family of H-Sets $G:I\to\mathrm{Sets}$
• For every $i$, an identity element $0_{i}$
• A fibrewise multiplication operation $\cdot:G_{i}\to G_{i}\to G_{i}$
• Fibrewise inverses $(-)^{-1}:G_{i}\to G_{i}$

Such that the obvious group identities hold.

Note that this is not a Group Object in the Family Fibration; that honor goes to Displayed Groups. Rather, this is what you get by passing to Augmented Simplicial Sets.

Moreover, augmented groups behave quite differently from groups: as James Deikun pointed out, the Initial Augmented Group consists of an empty family of groups.