Definition:Direct Limit of Sequence of Groups/Definition 2

Definition
Let $\N$ be the order category on the natural numbers.

Let $\mathbf{Grp}$ be the category of groups.

Let $G: \N \to \mathbf{Grp}$ be an $\N$-diagram in $\mathbf{Grp}$.

A direct limit for $G$ is a colimit ${\varinjlim \,}_n \, G_n$, and is denoted $G_\infty$.

Also see

 * Equivalence of Definitions of Direct Limit of Sequence of Groups
 * Existence and Uniqueness of Direct Limit of Sequence of Groups