Let $X$ be a topological space. Consider the category $\mathrm{Sh}(X)$ of sheaves on $X$ and natural transformations between sheaves.
It is well-known that colimits in $\mathrm{Sh}(X)$ are computed as follows: firstly, compute the colimit in the category $\mathrm{PSh}(X)$ of presheaves on $X$. Secondly, take the sheafification of that colimit in the category of presheaves.
In particular, if $A$ and $B$ are sheaves, then their coproduct is $(iA + iB)^\#$, where
$$i\colon \mathrm{Sh}(X) \to \mathrm{PSh}(X)$$
is the inclusion functor and
$$(-)^\#\colon \mathrm{PSh}(X) \to \mathrm{Sh}(X)$$
is it's left adjoint – the sheafification functor.
Does that fact that $(-)^\#$ is left adjoint to $i$ have something to do with the fact that colimits are computed using sheafification? I'm curious whether there's a general categorical fact about computing colimits using adjoint functors (or something like that) I'm missing. Also: is sheafification on morphisms, i.e., the function $$(-)^\#\colon \mathrm{Mor}(\mathrm{PSh}(X)) \to \mathrm{Mor}(\mathrm{Sh}(X)),$$
used in the construction of colimits in $\mathrm{Sh}(X)$? That would be an indication that really the functor $(-)^\#$ is closely connected to colimits in $\mathrm{Sh}(X)$ (and not only the "object function"!).
Are there other examples of situations in which one constructs colimits using an adjoint functor (to an inclusion functor)?
Best Answer
The general result is as follows (which can be found in almost every category theory textbook): Let $\mathcal{D}$ be a reflective subcategory of a category $\mathcal{C}$, i.e. the inclusion has a left adjoint $L : \mathcal{C} \to \mathcal{D}$. Then, if a diagram $(X_i)$ in $\mathcal{D}$ has a colimit $\mathrm{colim}_i X_i$ in $\mathcal{C}$, then $L(\mathrm{colim}_i X_i)$ is its colimit in $\mathcal{D}$. The proof is just one line: For $T \in \mathcal{D}$ we have natural bijections $$\hom(L(\mathrm{colim}_i X_i),T) \cong \hom(\mathrm{colim}_i X_i,T) \cong \lim_i \hom(X_i,T). ~~ \square$$ Also, it is well-known that left adjoints only have to be defined on objects; the action on morphisms follows from the universal property. Specifically, for $X \in \mathcal{C}$ we have a universal morphism $X \to L(X)$ with $L(X) \in \mathcal{D}$, and a morphism $f : X \to X'$ is mapped to the unique(!) morphism $L(X) \to L(X')$ such that the evident square commutes.