# Example of a functor which preserves all small limits but has no left adjoint

The General Adjoint Functor Theorem (Category Theory) states that for a locally small and complete category $D$, a functor $G\colon D \to C$ has a left adjoint if and only if $G$ preserves all small limits and for each object $A$ in $C$, $A \downarrow G$) has a weakly initial set.

Could someone help by giving an example of a functor $G$ that preserves all small limits but has no left adjoint?

• Let $\emptyset$ be the empty category. Then $G : \emptyset \to \mathcal{C}$ preserves all limits (vacuously), but $G$ has no left adjoint if $\mathcal{C}$ is non-empty. Apr 27 '12 at 7:39
• @Zhen: but the empty category is not complete. Jan 14 '18 at 23:33

## 2 Answers

A nontrivial example is mentioned in MacLane's Categories for the Working Mathematician , on page 123: consider the category $\mathbf{CompBool}$ of complete boolean algebras. The forgetful functor $\mathbf{CompBool} \to \mathbf{Set}$ has no left adjoint, but preserves all limits ($\mathbf{CompBool}$ is also small-complete). The reason is that, given a denumerable set $D$, one can construct an arbitrarily large complete Boolean algebra generated by $D$ (a fact that was apparently proved by Solvay in 1966), and so the solution set condition in the General AFT fails.

Martin's answer is probably the one you want, but here's another marginally trivial example.

Let $\mathcal{C}$ be a category and let $\mathbf{1}$ be the terminal category with only one object and one morphism. The unique functor $G : \mathcal{C} \to \mathbf{1}$ obviously preserves all limits, but $G$ has a left adjoint if and only if $\mathcal{C}$ has an initial object. Dually, $G$ preserves all colimits and has a right adjoint if and only if $\mathcal{C}$ has a terminal object.

• Thanks Zhen Lin - your example(s) are very helpful too. May 4 '12 at 10:44
• This example is incomplete; you need to exhibit an example of a category $C$ which is cocomplete but has no terminal object (or dually). Jan 14 '18 at 23:32
• Berci gives examples here: math.stackexchange.com/questions/3980824/… The large posets of sets or ordinals are both cocomplete with no terminal object! Jan 11 '21 at 8:38