I practice identifying adjoint functors on simple categories.

Now I came across a case, where it seems, I have a counit-unit adjunction, but not a hom-set adjunction. Is this possible?

For the concrete example:

$$ F: \mathcal{C} \to \mathcal{D},~~ G: \mathcal{C} \leftarrow \mathcal{D} \\ a, b, b', ~h: a \to b, ~h': b' \to b \in \mathcal{C} \\ Fa, Fb, ~Fh: Fa \to Fb \in \mathcal{D} \\ Fb' = Fb, ~Fh' = 1_{Fb} $$

I can work out $\eta$ and $\epsilon$:

$$ \eta_a = 1_a, ~~\eta_b = 1_b, ~~\eta_{b'} = h' \\ \varepsilon_{Fd} = 1_{Fd}, ~~\varepsilon_{Fe} = 1_{Fe} $$ And they fulfill triangle identities.

But I can't work out the bijections for $\Phi_{Fa,b'}: Hom(GFa,b') \cong Hom(Fa,Fb')$.

$Hom(Fa,Fb')$ contains $Fh$, but $Hom(GFa,b')$ seems to be the empty set.


In your adjunction, $G$ is the right adjoint and $F$ is the left adjoint, so the bijection of Hom-sets you ask for is incorrect. There would instead be a bijection $\mathrm{Hom}(a,GFb')\cong\mathrm{Hom}(Fa,Fb')$, and indeed there is.


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.