Direct limit

Remarks

The direct limit is written $\varinjlim A_i$. It is the colimit of the diagram given by the system, and it is unique up to a unique isomorphism. In sets, groups, rings and modules it always exists. There it is the disjoint union of the $A_i$ divided by a relation: $a_i \in A_i$ and $a_j \in A_j$ are identified when $f_{ik}(a_i) = f_{jk}(a_j)$ for some $k \ge i, j$.

Example: let $A_n = \Z$ for $n \in \N$, and let $f_{n,n+1}$ be multiplication by $2$. The maps $\phi_n \colon \Z \to \Z[1/2]$, $\phi_n(a) = a/2^n$, make $\Z[1/2]$ a direct limit. Clause 1 holds because $\phi_{n+1}(2a) = 2a/2^{n+1} = \phi_n(a)$.

Non-example: the maps $\phi_n(a) = a/2^n$ into $\Q$ satisfy clause 1 but not clause 2. For $B = \Z[1/2]$ with the maps above, there is no group homomorphism $u \colon \Q \to \Z[1/2]$ with $u(a/2^n) = a/2^n$ for all $a$ and $n$. Indeed $u(1/3)$ would be an element $x$ with $3x = 1$, and $\Z[1/2]$ has no such element.