Why are left adjoints often given by weighted colimits?

Question

Freyd's adjoint functor theorem states that a continuous functor $R: \mathsf{C} \to \mathsf{D}$ has a left adjoint if its domain $\mathsf{C}$ is locally small and complete, and $R$ satisfies the solution set condition. Moreover, the proof constructs the left adjoint as a limit.

Also, we know that if $L$ exists then it is given by the Right Kan extension $$L \cong \operatorname{Ran}_R 1_{\mathsf{C}} \cong \int_{c \in \mathsf{C}} c^{\mathsf{D}(-,Rc)},$$ whose pointwise formula is an end, and hence a weighted limit.

However, it seems that plenty of left adjoints seem to be given by colimit formulas. For instance, the inverse image of (pre)sheaves is given by a colimit formula and also in the answer to another question of mine, Roland explains that a far left adjoint to inverse image of sheaves (when the map of spaces is etale) can be constructed as a colimit as well.

Also, in algebra, the left adjoint to the internal hom (i.e. tensor product) of (bi)modules is constructed as a coend, which is a colimit.

Is there a reason why so many left adjoints are given by weighted colimit formulas, even though they are right Kan extensions, and hence weighted limits? Is there a general weighted colimit or coend formula for left adjoints under certain circumstances?

Answer 1

Freyd's Adjoint Functor Theorem states that if a limit-preserving (aka continuous) functor $R : \mathsf{C \to D}$, where $\mathsf{C}$ is locally small and complete, satisfies the solution set condition, then it is a right adjoint. In my opinion, the main condition is the continuity one. This is underscored by e.g. posets, toposes, and (particularly for the dual statement regarding left adjoints) locally presentable categories.

A left adjoint has to preserve colimits. For, e.g. locally presentable categories, that is all it needs to do to be a left adjoint. In general, since colimits commute with each other, functors defined by colimits are a natural place to look for left adjoints. (Dually, for right adjoints.) Conversely, a reason why a left adjoint preserves all colimits may be because it is itself a colimit, so it's natural to wonder if a left adjoint can be reexpressed as a colimit. As a simple example, as $\mathbf{Set}$ is locally presentable, any left adjoint $\mathbf{Set}\to\mathbf{Set}$ is a colimit. Since $X\times - : \mathbf{Set}\to\mathbf{Set}$ is a left adjoint, it is also expressible as a colimit, namely $\coprod_{x\in X} -$.

Answer 2

As an overall comment, you tend to forget that many adjoints can be computed as co/limits because they are colimits :-) also, many functors are left or right Kan extension since they are defined between diagram categories, and among these many Kan extensions are pointwise. So, again, they are co/limits by the well known formula.

Now a minor comment on a detail you mention:

Is there a reason why so many left adjoints are given by colimit formulas, even though it seems that they should be given by weighted limits?

As soon as your functors are between locally small categories, there is no difference among weighted and conical co/limits: this is because there exists a "category of elements" construction $^{\cal C}\!\!\int_W\to {\cal C}$ for a functor $W : {\cal C}\to \bf Set$ [1] that lets you write (say) a weighted colimit $W\cdot F$ as a conical colimit over the category $^{\cal C}\!\!\int_W$. You appreciate this in the formula for pointwise Kan extensions: $$ \text{Lan}_GF\cong \underset{(G\downarrow x)}{\text{colim }} Fc, $$ and a dual formula holds on the right. Kan extensions are colimit weighted on functors $W=\hom_{\cal C}(G,\bullet)$, and comma categories are precisely the categories of elements of such functors. So, the circle is closed, as the right Kan extension giving the adjunction $F\dashv \text{Ran}_F1$ is expressed as a limit over (the opposite of) a comma category.

[1] As an aside, this is on its own right a weighted limit or a lax limit, think about it!