About definition of direct limit

Question

In the definition of direct limit of abelian groups (or modules or ...), one takes abelian groups $G_i$ ($i\in I)$ with a morphism $f_{ij}:G_i\rightarrow G_j$ with following conditions:

• for every $i\in I$, $f_{ii}:G_i\rightarrow G_i$ is identity

• if $f_{ij}:G_i\rightarrow G_j$ and $f_{jk}:G_j\rightarrow G_k$ is a homomorphism, then their composition is $f_{ik}$.

• for each $G_i$ and $G_j$, there is a $G_k$ such that one can pass from $G_i$ and $G_j$ into $G_k$ with homomorphisms $f_{ik}$ and $f_{jk}$.

Q. In most of the standard books (Jacobson, Dummit-Foote, Atiyah-MacDonald or Wikipedia), the direct limit is defined in above way. My question is, why one imposes third condition above to define direct limit?

In terms of the indexing set $I$, the references say that for any $i,j\in I$, there is $k\in I$ such that $i\le k$ and $j\le k$. Why this condition is posed?

If we have just two abelian groups $G_1$ and $G_2$ with no homomorphism from one to other, can't we define their direct limit to be direct sum? This could be generalized even we have a family of groups $\{G_i\}_{i\in I}$ with no homomorphism from one to other, and define their direct limit to be the direct sum?

Answer 1

You can do it in a more general way, but let me first make precise what you wrote in your question.

Although you do not mention it, you work with a directed set $(I,\le)$. You associate to each $i \in I$ an object $G_i$ and to each pair $(i,j) \in I \times I$ with $i \le j$ a morphisms $f_{ij} : G_i \to G_j$, and require that the three conditions in your questions are satisfied.

Let us adopt a more abstract perspective. We can regard $(I,\le)$ as a small category $\mathcal I$ such that

• The set of objects of $\mathcal I$ is $I$.
• The set of morphisms from $i$ to $j$ is empty if $i \not\le j$ and consists of the pair $(i,j)$ if $i \le j$.

With this interpretation, a direct system is nothing else than a functor $G : \mathcal I \to \mathcal C$ into the desired category $\mathcal C$ (you consider the category of abelian groups).

But for each functor $F : \mathcal D \to \mathcal C$ living on any small category $\mathcal D$ we can define the concept of a colimit (see for example here). By a co-cone for $F$ we mean a system consisting of an object $C$ of $\mathcal C$ and a collection of morphisms $l_d : C \to F(d)$, $d \in Ob(\mathcal D)$, such that $F(\mu) \circ l_d = l_{d'}$ for all morphisms $\mu : d \to d'$. A colimit of $F$ is a co-cone $(C^*,l^*_d)$ for $F$ with the following universal property:

For each co-cone $(C,l_d)$ for $F$ there exists a unique morphism $\phi : C \to C^*$ such that $l^*_d \circ \phi = l_d$ for all $d$.

The existence of colimits has to be proved; it depends on $\mathcal D$, $\mathcal C$ and $F$.

If $\mathcal D = \mathcal I$ as above and $\mathcal C$ is the category of abelian groups (or more generally of $R$-modules), then each functor $F$ has a colimit and one calls it direct limit in this case.

What happens if we omit the third condition in your question? In that case we essentially consider partially ordered sets $(I,\le)$ instead of directed sets, and in fact all functors on such $I$ have a colimit. More generally, all functors living on small categories have colimits. See the above wiki-link. But recall that they are not called direct limits.

If you consider a category $\mathcal D$ having no morphisms except identities, then a functor $F: \mathcal D \to \mathcal C$ is nothing else than a collection of objects $F_d = F(d)$ indexed by the $d \in Ob(\mathcal D)$. The colimit of $F$ then is the coproduct of the $F_d$ (if it exists). In the category of abelian groups we get the direct sum $S = \bigoplus_d F_d$ with "canonical embeddings" $\iota_d : F_d \to S$.

Answer 2

To your first question:
If I recall correctly, the condition you mention is needed in the proof of existence of direct limits of (abelian) groups (which I don't want to give here as it is a bit lengthy and can be found in suitable literature) in the general setting, e.g. when the index set $I$ is not totally ordered.

To your second question:
Indeed, the direct limit is the direct sum in this case, though one ought to be careful with notation: A direct limit is defined for a direct system $\langle X_i,f_{ij}\rangle$ of objects AND morphisms in a category, where for all $i\le j$ in the direct set $I$ we require the existence of a morphism $X_i\to X_j$.
Hence writing $G_1,G_2$ yields - under the assumption that you use the common order inherited from the natural numbers $\mathbb N\supset I=\{1,2\}$ - that we need to be given a morphism $G_1\to G_2$. Less confusion might arise for the direct set $I=\{∗,+\}$ when requiring the two elements to be non-comparable to each other.