Non-mechanical proof in direct limit (or colimit)

Question

Let $I$ be a directed set and for the sake of simplicity, let us work with a directed system of modules over some commutative ring $A$ (although I am looking for an answer which can be extended to a directed system of groups, rings, etc.).

Let $(M_i)_{i\in I}$ be a directed system and $M$ is its direct limit. Suppose $m_i\in M_i$ be an element of a module in the system such that its image in $M$ is zero. I want to show that there exists an $M_j$ where $i\leq j$ and the image of $m_i$ in $M_j$ is zero.

Now, using the construction of the direct limit and since we know that this is the only possible construction of the direct limit up to isomorphism, we can show the desired result. My issue is that this proof uses the construction of the direct limit rather than the UMP (Universal mapping property). Could someone please help me with a proof that only uses the UMP of direct limit and not the construction.

Thanks in advance.

Answer 1

I think there are good reasons to think that you won't easily be able to escape from a proof involving the concrete construction of the colimit. Well, first, of course in the end you want to prove something about the colimit, so it makes sense that you would have to use its description.

But more to the point, let us think about what it would mean to "only use the universal property". Usually this means we want a purely categorical proof, using only commutative diagrams. In this spirit, we could replace elements of a module $M$ by morphisms $A\to M$; the statement you want to prove becomes:

If a morphism $A\to M_i$ is such that the composition $A\to M_i\to M$ is zero, then there is some $j\geqslant i$ such that $A\to M_i\to M_j$ is zero.

Now the thing is: this statement is false if we replace $A$ by an arbitrary module $N$. It only works if we add some "smallness" assumption, such as $N$ being finitely generated (because then we can just look at what happens to a finite set of generators, and we can apply the result you want to prove). This should be a first clue that this is not a purely formal property of colimits, and that there is something a little special happening with elements.

Perhaps a more advanced point of view is to restate your problem as:

The forgetful functor from modules to sets preserves directed colimits.

This is true in most "algebraic" situations (groups, rings, etc.), but might be false in a general concrete category, even when directed colimits always exist.

This all goes to indicate that the property you're interested in is actually not completely formal, and thus it makes sense that you would have to use the explicit description of a directed colimit in your category, which reflects this special property.