5
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ What functors do you know? $\endgroup$
    – Oskar
    Mar 24, 2016 at 18:05
  • 1
    $\begingroup$ I mean that the best way to find such examples is to take all the functors you familiar with and check them for existence of left/right adjoints. $\endgroup$
    – Oskar
    Mar 24, 2016 at 18:49

1 Answer 1

6
$\begingroup$

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.

$\endgroup$
3
  • $\begingroup$ Is it an equivalent condition? I know it's a sufficient condition. The General Adjoint Functor Theorem gives an equivalent condition. $\endgroup$ Mar 24, 2016 at 19:57
  • 1
    $\begingroup$ @PatrickStevens Every right-adjoint functor preserves all limits. $\endgroup$
    – Oskar
    Mar 24, 2016 at 20:00
  • $\begingroup$ Oh, that was embarrassing. Thanks. $\endgroup$ Mar 24, 2016 at 20:02

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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