Equivalence of Definitions of Direct Limit of Sequence of Groups

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The following definitions of the concept of Direct Limit of Sequence of Groups are equivalent:

Explicit Definition

Let $\sequence {G_n}_{n \mathop \in \N}$ be a sequence of groups.

For each $n \in \N$, let $g_n: G_n \to G_{n + 1}$ be a group homomorphism.

A direct limit for the sequences $\sequence {G_n}_{n \mathop \in \N}$ and $\sequence {g_n}_{n \mathop \in \N}$ comprises:

$(1): \quad$ a group $G_\infty$
$(2): \quad$ for each $n \in \N$, a group homomorphism $u_n: G_n \to G_\infty$

such that, for all $n \in \N$:

$u_{n + 1} \circ g_n = u_n$

and, for all groups $H$ together with group homomorphisms $h_n: G_n \to H$ satisfying $h_{n + 1} \circ g_n = h_n$, there exists a unique group homomorphism:

$h_\infty: G_\infty \to H$

such that for all $n \in \N$:

$h_n = h_\infty \circ u_n$

Definition by Category Theory

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 $G_\infty$.