Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let C, E be small categories, let = SetCop, and let F:Ĉ → Ê be cocontinuous. I think F will always have a right adjoint when C, E are small, but not necessarily if they're large. Is that right?

share|improve this question
1  
please add your proof. –  Martin Brandenburg Apr 14 '10 at 17:23

2 Answers 2

up vote 4 down vote accepted

The part "F will always have a right adjoint when C, E are small" is definitely right. Using some mildly overkill machinery: in this case and are locally presentable categories, and the result is then the adjoint functor theorem for locally presentable categories:

Theorem: Let C and D be locally presentable categories and F : C → D a functor. Then

  1. F has a right adjoint iff F preserves small colimits.

  2. F has a left adjoint iff F is accessible (preserves κ-filtered colimits for some κ) and preserves small limits.

(Reference: Higher Topos Theory Corollary 5.5.2.9 for the (∞,1)-categorical version)

share|improve this answer
    
If you use universes, does that mean that it always holds? –  Harry Gindi Apr 14 '10 at 18:40
3  
Since the question has nothing to do with $(\infty,1)$-categories, a better reference would probably be Adamek+Rosicky, "Locally presentable and accessible categories." –  Mike Shulman Apr 18 '10 at 3:38

The answer to the first part is indeed true. In fact, something more general is true. Let $\mathcal{A}$ be a small category and let $\mathcal{C}$ be a cocomplete category (which is locally small, i.e., there is just a set of morphism between any two objects). Then any cocontinuous functor $L \colon \mathrm{Set}^{\mathcal{A}^\mathrm{op}} \rightarrow \mathcal{C}$ has a right adjoint, given by $C \mapsto \mathcal{C}(K-,C)$, where $K \colon \mathcal{A} \rightarrow \mathcal{C}$ is the composite of the Yoneda embedding and $L$.

This is for example proved in Kelly's "Basic concepts of enriched category theory", Theorem 4.51. He proves the enriched version of this result, where $\mathrm{Set}$ is replaced by any complete and cocomplete category $\mathcal{V}$. I must say I don't know of a reference that just treats the $\mathrm{Set}$-case.

If the target is the category of presheaves on some large category, then this might fail. Take for example $\mathcal{D}$ the large discrete category whose objects are sets, and let $F \colon \mathcal{D} \rightarrow \mathrm{Set}$ be the canonical inclusion functor. Then the functor $\mathrm{Set}\rightarrow \mathrm{Set}^{\mathcal{D}}$ which sends a set $X$ to the functor $F\times X$ (i.e., the functor which sends a set $A$ to $A\times X$) is cocontinuous, because $A\times -$ preserves colimits. However, there is a proper class of natural transformations $F \rightarrow F$ (a natural transformation just amounts to choosing an endomorphism of each set with no compatibility conditions), so if this functor had a right adjoint $R$, then we would have a bijection $\mathrm{Set}(\ast,RF) \cong \mathrm{Set}^{\mathcal{D}}(F,F)$, i.e., $RF$ would have to be a proper class. The reason for this failure is of course that $\mathrm{Set}^\mathcal{D}$ is not locally small. Note that this problem doesn't go away when we use universes: the above example would give you an isomorphism between a small set and a large set, i.e., a set outside of the universe.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.