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.

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)

nota special set, it isadditional structureon a set. It is coming from the fact that your sections map into a set with additional structure (namely vector spaces). $\endgroup$ – Andrej Bauer Oct 9 '19 at 12:40smoothsections?) just be the germs? Are germs concrete enough for your purposes? $\endgroup$ – Andrej Bauer Oct 9 '19 at 20:56