Most general context where a "disjoint sum" definition of a direct limit is applicable and always exist

Question

I am a bit out of my element here so I'm hopefully not saying something stupid.

Anyways, wikipedia gives two ways to define direct limits, one for "algebraic structures" and one for general categories.

The definition for "algebraic structures" is as follows:

Suppose that $(\mathbb A,\le)$ is a directed set, and $X_\alpha$ is an object in some category of algebraic structures (eg. groups, rings, modules, etc.) for each $\alpha\in\mathbb A$ and for each $\alpha\le\beta$ in $\mathbb A$ there is a morphism $\varphi_{\beta\alpha}:X_\alpha\rightarrow X_\beta$ such that $\varphi_{\alpha\alpha}=\mathrm{Id}_{X_\alpha}$ and $\varphi_{\gamma\alpha}=\varphi_{\gamma\beta}\circ\varphi_{\beta\alpha}$ for $\alpha\le\beta\le\gamma$.

Then the direct limit is $$\lim_{\longrightarrow}X_\alpha=\bigsqcup_{\alpha\in\mathbb A}X_\alpha/\sim,$$ where the equivalence relation $\sim$ is defined such that $x_\alpha\in X_\alpha$ and $x_\beta\in X_\beta$ are equivalent iff there is a $\gamma\in\mathbb A$ with $\alpha\le\gamma$ and $\beta\le\gamma$ such that $\varphi_{\gamma\alpha}(x_\alpha)=\varphi_{\gamma\beta}(x_\beta)$.

The category-theoretical definition is way more abstract.

The way the wikipedia article is worded seems to imply that direct limits always exist for "algebraic structures" but not necessarily for general categories.

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I mainly wish to have a working definition of a direct limit that I can understand stalks of sheaves with, and I don't like the category-theoretical definition right now, so I thought about defining the direct limit for a direct system consisting of objects of a category $\mathcal C$ which is a subcategory of $\mathrm{Set}$ (these are called concrete categories right?) the exact same way I have defined it above for "algebraic structures".

However I have some suspicions of issues, namely that

• I find it likely that direct limits of objects in a concrete category $\mathcal C$ exist over $\mathrm{Set}$ but not necessairly over $\mathcal C$, right? In particular, I have a feeling that the category of smooth manifolds is not stable under direct limits, since smooth manifolds are not stable under either uncountable disjoint sums or quotients.
• If so, what exactly are those concrete categories which are stable under direct limits?
• Mostly unrelated, but I mainly want to deal with sheaves of sections of smooth fibre bundles here, so I kinda wanna ask, if $\pi:E\rightarrow M$ is a smooth fibre bundle without any additional assumption on it then the sheaf of sections $\Gamma(\pi):U\mapsto\Gamma_U(\pi)$ is a $\mathrm{Set}$-valued functor, but can we - in general - restrict its target category further? (Like, the sheaf of sections of a vector bundle is a sheaf of modules)

Answer 1

There’s a standard property that closely matches what you ask for. For a category $C$ with a functor $U : C \to \newcommand{\Set}{\mathrm{Set}}\Set$ (think of the “forgetful” functor from a category of algebraic structures), we say $U$ creates direct limits over directed sets if (roughly) all such direct limits exist in $C$ and are computed the same way as they are in $\Set$, i.e. by a disjoint sum + quotient construction as you describe.

This holds for lots of concrete categories, i.e. categories with a faithful functor $U : C \to \Set$ (not just subcategories of $\Set$, by the way). In particular, it does hold for all categories of finitary algebraic structures. (But not infinitary algebraic structures: e.g. for Banach spaces, which can be presented algebraically with “completeness” as an infinitary algebraic operation, the forgetful functor doesn’t create directed limits of directed sets.)

Regarding the category-theoretic definition being “way more abstract”, though: I really encourage pushing back against the initial fear of “abstractness”. The only aspect that’s really more “abstract” is that it’s a characterisation of the direct limit, not an explicit construction/description of it. (The other thing which can seem more abstract, but really isn’t, is that the general definition is phrased in terms of arbitrary categories, objects, and morphisms; but in any specific case, these become become perfectly concrete, as statements about e.g. groups and group homomorphisms, or whatever.) So I quite agree it’s reassuring to have the concrete construction too; but for working with the direct limit in practice, it turns out remarkably often that the category-theoretic universal property is more useful than the concrete construction.