Definition of direct limit in Bredon In Bredons book Geometry and Topology he introduces the direct limit of groups in the following way. Let $D$ be a directed set and let $G_{\alpha}$ be an abelian group defined for each $\alpha \in D$. Suppose we are given homomorphisms $f_{\beta,\alpha}:G_{\alpha} \to G_{\beta}$ for each $\alpha<\beta$ in $D$. Assume that for all $\gamma > \beta > \alpha$ in $D$ we have $f_{\gamma,\beta}f_{\beta,\alpha}=f_{\gamma,\alpha}$. Then $\varinjlim G_{\alpha}$ is defined to be the quotient of $G=\bigoplus_{\alpha \in D} G_{\alpha}$ by the relations $f_{\beta,\alpha}(g) \sim g$ for all $g \in G_{\alpha}$ and $\beta>\alpha$.
I don't understand this definition at all. $(1)$ $G$ is defined to be the group that has families $(g_{\alpha})_{\alpha \in D}$ with finitely many nonzero values as its elements. How does $\sim$ define an equivalence relation on this set?
$(2)$ What is the resulting group? Assuming $\sim$ defines an equivalence relation, does one then define $\varinjlim G_{\alpha}$ to be $G/\sim$ with an appropriate group multiplication? What is the multiplication on this set of equivalence classes?
 A: Note that he doesn't say it is a quotient by a subgroup: he says it is a quotient by a set of relations. The author is being a bit sloppy in identifying the groups $G_{\alpha}$ with their images in the direct sum, so perhaps that is what is contributing to the confusion.
To make this a bit more precise, let $G=\bigoplus_{\alpha\in D}G_{\alpha}$, and let $\iota_{\alpha}\colon G_{\alpha}\to G$ be the canonical embedding.
The author instructs us to mod out $G$ by the smallest subgroup that "codes" all relations of the form $\iota_{\beta}(f_{\beta,\alpha}(g)) \sim \iota_{\alpha}(g)$, where $\alpha$ and $\beta$ range over all pairs of elements of $D$ with $\alpha\lt \beta$, and $g$ ranges over all elements of the corresponding $G_{\alpha}$.
This is the subgroup of $G$ generated by all elements of the form
$$\iota_{\beta}(f_{\beta,\alpha}(g))-\iota_{\alpha}(g).$$
As this is a subgroup, by definition, the quotient is just the quotient group with its usual addition and multiplication.
These generators are tuples with at most two nonzero entries:  one in entry $\alpha$, namely $-g$, and the second in entry $\beta$ with $\beta\gt \alpha$, with entry $f_{\beta,\alpha}(g)$, the image of $g$ in $G_{\beta}$. This identifies $g$ with all its "forward images". Once we have identified all of these images by taking their differences and putting them in a subgroup $N$, moding out by $N$ "makes them $0$" in the quotient, so that $\iota_{\alpha}(g)N = \iota_{\beta}(f_{\beta,\alpha}(g))N$ in the quotient.
A: The relation $f_{\beta \alpha}(g) \sim g$ is, of course, not an equivalence relation. What is meant is the following: take the (normal, since all groups are involved) subgroup $N$ of $G$ that is generated by the elements $f_{\beta \alpha}(g)-g$ (as $\alpha$ and $\beta$ run over $D$ and $g$ runs over $G_\alpha$; see the last paragraph for an explanation of the notation) and define the direct limit to be the quotient group $H=G/N$. This group $H$ admits---by construction---homomorphisms $\phi_\alpha: G_\alpha \to H$, defined to be the composition of the inclusion of $G_\alpha$ into the direct sum followed by the quotient morphism.
These satisfy $\phi_\beta f_{\beta \alpha}=\phi_\alpha$, and so one is left to check that the resulting group is actually initial (in the appropriate category) with respect to such data. But this follows since we have modded out by precisely the relations needed to obtain the appropriate commutativity relations for the $\phi_\alpha$'s.
It's worth noting that we have not used the fact that $D$ is directed at all here. This condition is sometimes useful for the following reason: it allows you to give an alternative construction of the direct limit, as the union of the groups $G_\alpha$ modulo the equivalence relation for which $g \in G_\alpha$ and $h \in G_\beta$ are equivalent if there is some $\gamma \geq \alpha, \beta$ with $f_{\gamma \alpha}(g)=f_{\gamma \beta}(h)$.
A word about notation: the direct sum $G=\bigoplus_{i \in I} G_i$ of a family of groups admits inclusions $\iota_i$ of each $G_i$ in $G$ (and in fact is univeral among groups admitting such homomorphisms). So for an element $g \in G_i$, when we wish to work with its image in $G$ we might write $\iota_i(g)$ (as A.M. does in the other answer, which does not differ substantively from the first two paragraphs of this one). However, since these inclusions are canonical it usually results in less cluttered formulas, and no possible confusion, to omit them from the notation as I have done in the first paragraph of this answer. Thus in the expression $f_{\beta \alpha}(g)-g$ above the elements $f_{\beta \alpha}(g) \in G_\beta$ and $g \in G_\alpha$ are both regarded as elements of $G$, where it makes sense to compute their difference. In practice, once one is used to this convention it makes proof writing and reading much easier, and I therefore strongly recommend it to students.
