Definition of Colimit

Question

In our course of Algebraic Topology we defined the colimit, or direct limit, of a direct system as follows:

Direct Set: A partially ordered set $(\Lambda,\leq)$ such that $\forall \alpha,\beta\in\Lambda \hspace{0.2 cm}\exists \gamma \mid \alpha\leq \gamma, \beta \leq \gamma$

Direct system: (of abelian groups with indices in a direct set $\Lambda$) is a family of abelian groups $\{A_\alpha\}_{\alpha \in \Lambda}$ with homomorphisms $\{\varphi_{\alpha \beta}:A_\alpha\rightarrow A_\beta\}_{\alpha \leq \beta}$ such that $\varphi_{\alpha \alpha}=1_{A_\alpha}$ and if $\alpha \leq \beta \leq \gamma$ then $\varphi_{\alpha\gamma}=\varphi_{\beta\gamma}\circ\varphi_{\alpha\beta}$.

Direct Limit of a direct system is $\varinjlim A_\alpha = \bigoplus_\limits{\alpha \in \Lambda}A_\alpha /<a_\alpha - \varphi_{\alpha \beta}(a_\alpha)>_{\alpha \leq \beta}$

I don't really understand the definition. Here are some questions i am thinking about these days:

1. Can somebody explain with some examples how this quotient works? The relationship is on a specific entry of a string with finite non-zero elements (an element of the sum) and not on the strings themselves.

2. Wikipedia defines it on the disjoint union and sounds more intuitive, how should i think this direct sum? As a set is it a quotient of the disjoint union?

3. I also thought about a nested direct system with inclusion, the direct limit should be the union? It makes sense to me only if i use the wikipedia definition (and only heuristically).

Answer 1

Q1:

The definition of $\varinjlim A_\alpha$ in your question is not correct. What is meant is this:

For each $a_\alpha \in A_\alpha$ and each $\beta \ge \alpha$ let us denote by $[a_{\alpha,\beta}]$ the element of $\bigoplus_{\alpha} A_\alpha$ having $a_\alpha$ as the $\alpha$-th coordinate, $-\varphi_{\alpha \beta}(a_\alpha)$ as the $\beta$-th coordinate and all other coordinates $0$. Let $N$ be the subgroup of $\bigoplus_{\alpha} A_\alpha$ generated by all elements of the form $[a_{\alpha,\beta}]$ (where $\alpha$, $a_\alpha \in A_\alpha$ and $\beta \ge \alpha$ are arbitrary). $N$ is that what you denote by $<a_\alpha - \varphi_{\alpha \beta}>_{\alpha \leq \beta}$. BUT: Although you probably have a typo and should have $<a_\alpha - \varphi_{\alpha \beta}(a_\alpha)>_{\alpha \leq \beta}$, it is not really clear what the interpretation of this expression should be. I hope the above definition of $N$ clarifies it.

Then define $$\varinjlim A_\alpha = \left(\bigoplus_{\alpha} A_\alpha \right) / N .$$

Q2:

The Wikipedia definition differs from the above and you may regard it as more intuitive. As a set it is in fact a quotient of the disjoint union. The benefit of the above definition is that it is obvious that the direct limit is an abelian group, whereas the Wikipedia definition requires a definition of a multiplication and a proof that we obtain an abelian group. See Direct (inductive) limit of groups.

One can show that both definitions produce isomorphic groups. To do so, you may show that both constructions have the universal property of the direct limit. See Wikipedia.

Q3:

The direct limit of a direct system whose bondings $\varphi_{\alpha \beta}$ are inclusions is in fact the union of all $A_\alpha$; the group multiplication of elements $x \in A_\alpha,y \in A_\beta$ is simply performed by choosing $\gamma \ge \alpha, \beta$ and multiplying in $A_\gamma$. It is now an easy exercise to show that the canonical mapping $$\coprod_\alpha A_\alpha \to \bigcup_\alpha A_\alpha$$ induces a group isomorphism $\varinjlim A_\alpha \to \bigcup_\alpha A_\alpha$.