Explicit examples of (co)limit arguments in other fields

Over the past weeks, I have noticed that high level lecture notes in subjects like algebraic geometry, algebra, and algebraic topology often sketch proofs in the following form:

• Proof sketch for a "simple" case e.g Noetherian ring/scheme, CW complex, finitely generated subobject
• "Magically" get the general case by 'the usual (co)limit arguments'

Unless I am mistaken "(co)limit arguments" just mean exactness properties of all kinds of functors. The thing is, though, I can hardly think of any specific examples of proofs of relatively basic facts from these exactness properties.

So... I'd like to compile a big list of specific examples of proofs in which the general case can be obtained from a simple case along with explicit mention of the (co)limits involved.

I know any mathematician will be tempted to say that this method of proof pervades all of mathematics - and I'm sure it does - but I want to explicitly see the functors and (co)limits involved; too often I hear "this is really the continuity of the functor $F$" in disguise" without understanding the details.

Update: Let me broaden what I mean by 'the usual (co)limit arguments' in hopes this will inspire some answers. Apart from exactness, some categories have the property that arrows into certain (co)limits factor through one of the components. For instance, in Grothendieck abelian category, I'm pretty sure a map into a filtered colimit of monos factors through one of them. This can be used to construct functorial injective resolutions.

So what are some more interesting cases and applications of "an arrow $A\rightarrow \varinjlim _\alpha B_\alpha$ factors through some $A\rightarrow B_\alpha$" type results?

Solution 1:

I'm not sure if this is the sort of thing that you're looking for, but colimit (and limit) arguments prove really useful when dealing with $R$-modules. In particular we can think of direct limits (i.e. colimits over a directed poset) as a union (in some sense that can be made precise).

As an example, working in $\mathsf{Ab}=R\hbox{-}\mathsf{mod}$, we have the following isomorphisms: \begin{align*} \lim\nolimits_{n\in\mathbb{N}^\text{op}}\mathbb{Z}/p^n\mathbb{Z}&\cong\mathbb{Z}_p\\ \mathrm{colim}_{n\in\mathbb{N}}\mathbb{Z}/p^n\mathbb{Z}&\cong \mathbb{Z}[p^{-1}]/\mathbb{Z}\\[1em] \prod_{p\text{ prime}}\mathbb{Z}_p&\cong\hat{\mathbb{Z}}\\ \coprod_{p\text{ prime}}\mathbb{Z}[p^{-1}]/\mathbb{Z}&\cong\mathbb{Q}/\mathbb{Z} \end{align*} where $\mathbb{Z}_p$ is the $p$-adic integers and $\hat{\mathbb{Z}}$ is the profinite completion of the integers.

Now if we ask something like 'what is $\mathrm{Hom}_\mathsf{Ab}(\mathbb{Q}/\mathbb{Z},\mathbb{Q}/\mathbb{Z})$' we can find an answer comparatively easily: we know that $$\mathrm{Hom}_\mathsf{Ab}(\mathbb{Z}[p^{-1}]/\mathbb{Z},\mathbb{Q}/\mathbb{Z})\cong\mathbb{Z}[p^{-1}]/\mathbb{Z}$$ since the image of the morphism must land in the copy of $\mathbb{Z}[p^{-1}]/\mathbb{Z}\subset\mathbb{Q}/\mathbb{Z}$. Then we can just 'take a colimit': \begin{align*} \mathrm{Hom}_\mathsf{Ab}(\mathbb{Q}/\mathbb{Z},\mathbb{Q}/\mathbb{Z})&=\mathrm{Hom}_\mathsf{Ab}\left(\coprod_{p\text{ prime}}\mathbb{Z}[p^{-1}]/\mathbb{Z},\mathbb{Q}/\mathbb{Z}\right)\\&=\prod_{p\text{ prime}}\mathrm{Hom}_\mathsf{Ab}\left(\mathbb{Z}[p^{-1}]/\mathbb{Z},\mathbb{Q}/\mathbb{Z}\right)\\&=\prod_{p\text{ prime}}\mathbb{Z}[p^{-1}]/\mathbb{Z}\\&=\hat{\mathbb{Z}} \end{align*} More generally, for lots of the 'nasty' groups we can express them as a colimit of their finitely-generated direct summands and work with these well-behaved objects, as a simple case, and then take a colimit to obtain the more general case.

I know this isn't quite how your question was phrased, but I think that the idea is sort of still there: work with simple objects and then take a colimit to pass to big objects.