# Why are left adjoints often given by weighted colimits?

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?

• Left adjoints and colimits are described via morphisms out of them. – HeinrichD Apr 17 '17 at 20:59
• Yeah that's why I assumed that left adjoints should be given by weighted colimits (and they often seem to be in my experience). I'm getting confused after reading that left adjoints are right Kan extensions, and hence the pointwise formula is a weighted limit rather than a weighted colimit. I'm trying to figure out how to reconcile this apparent contradiction. – ಠ_ಠ Apr 17 '17 at 23:11
• If $F\dashv U$ then $-\circ U \dashv -\circ F$ but this says that $\mathsf{Ran}_U(G) = G \circ F$. Now simply consider $\mathsf{Ran}_U(Id)$. – Derek Elkins left SE Apr 18 '17 at 0:49
• Thanks! This is helpful to know. So I guess the fact that $F \dashv U$ iff $F \cong \operatorname{Ran}_U \mathrm{id}$ (and the extension is absolute) is due to some sort of argument involving the Yoneda embedding? – ಠ_ಠ Apr 18 '17 at 3:05
• Yeah, there are size issues with the above that severely limit it's range of applicability. (Basically, to ensure the functor categories exist without further argument, $F$ and $G$ would need to be functors between small categories.) The argument, not going through an adjunction but instead using your end characterization, is: $\mathsf{Ran}_U(G)\cong\int_{c\in\mathsf{C}}(Gc)^{\mathsf{D}(-,Uc)}\cong\int_{c\in\mathsf{C}}(Gc)^{\mathsf{D}(F-,c)}\cong G\circ F$ using Yoneda reduction. See also 4.28 in Basic Concepts of Enriched Category Theory. – Derek Elkins left SE Apr 18 '17 at 17:18

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 $\coprod_{x\in X} -$.

• Oops that was typo (not including the continuity condition). Is there some general idea that explains how to turn the end (weighted limit) formula for the left adjoint into a coend (weighted colimit) formula for the left adjoint? For instance, say that I was trying to develop module theory from scratch and I discovered that the internal hom satisfies the hypotheses of the adjoint functor theorem, and is given by an end. How could I turn this end formula into the usual coend formula for the tensor product? – ಠ_ಠ Apr 17 '17 at 23:18

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!

• Thanks! I am just confused because the nLab states that a left adjoint $L$ to a functor $R: \mathsf{C} \to \mathsf{D}$ is a right Kan extension $\operatorname{Ran}_R 1_{\mathsf{C}}$, whose pointwise formula is given by a weighted limit rather than a weighted colimit. I expected that the formula for $L$ would be a weighted colimit instead, since it seems that many left adjoints I'm familiar with are given by (weighted) colimits (tensor products, inverse image etc.). Am I missing something, or is the nLab in error? – ಠ_ಠ Apr 17 '17 at 23:05
• It is presumably an error. Left adjoints preserve all colimits, and thus morally should be computed by colimits. – Joe Berner Jan 6 '18 at 14:08