Is there an functor without an adjoint?

Question

So I'm doing some research into category theory, and I don't know whether this is a trivial question or not so I'll ask it anyway. Which functors don't have left adjoints?

I know there must be some, as otherwise there would be no point for the adjoint functor theorem or knowing that right adjoints preserve limits etc. But I just can't think of any. A couple of examples would be great, and even better if you could explain how they defy the adjoint functor theorem, but just to know some to look into would be really helpful. Many thanks.

Answer 1

Special Adjoint Functor Theorem (SAFT) states that a functor $T\colon\mathcal{A}\to\mathcal{B}$ from a good category $\mathcal{A}$ (category which is locally small, complete, well-powered, has a small cogenerating family) to a locally small category $\mathcal{B}$ has a left adjoint if and only if it is continuous (preserves small limits). There are many other formulations, see nlab article.

For example, take $\mathbf{Set}$. It satisfies SAFT, hence the functor $T\colon\mathbf{Set}\to\mathcal{B}$ has a left adjoint iff it preserves limits. Take a non-empty set $X$, then the functor $$(-\times X)\colon\mathbf{Set}\to\mathbf{Set}$$ hasn't a left adjoint (because it doesnt't preserve products), but it actually has a right adjoint.

Вesides SAFT, there are plenty of cases when a functor has no left adjoints. For instance, take a functor $\mathbf{0}\to\mathcal{A}$ from the empty category to a non-empty category. Then it of course has no left (and right) adjoints by the trivial reason.

Note, that if you find a functor without a right adjoint, then you automatically get a functor without a left adjoint (you can simply take its dual). For example, the functor $$\text{Mor}\colon\mathbf{Cat}\to\mathbf{Set},$$ which maps a small category to its set of morphisms, has no right adjoints (because it doesn't preserve coequalizers). Hence, the dual functor $\text{Mor}^{\text{op}}$ has no left adjoints.