I am struggling with the following exercise from Emily Riehl's Category Theory in Context, regarding adjunction morphisms: paraphrasing, let $F : C \to D, G: D \to C$ and $F' : C' \to D', G' : D' \to C'$ be functors such that $F \dashv G$ and $F' \dashv G'$. We now consider functors $H : C \to C'$ and $K : D \to D'$ such that the squares of adjunctions commute, that is so that $HG = G'K$ and $KF = F'H$. The exercise then asks to prove that the following are equivalent:

  1. if $\eta, \eta'$ are the units of the adjunctions, then $H \eta = \eta' H$, i.e. $H\eta_c = \eta'_{Hc}$.
  2. if $\epsilon, \epsilon'$ are the counits of the adjunctions, then $K\epsilon = \epsilon'K$.
  3. the arrow compositions $$ D(Fc,d) \xrightarrow{\simeq} C(c,Gd) \xrightarrow{H} C(Hc,HGd) = C(Hc,G'Kd) $$ and $$ D(Fc,d) \xrightarrow{K} D(KFc,Kd) = D(F'Hc,Kd) \xrightarrow{\simeq} C(Hc, G'Kd) $$ are equal.

I've though about this for a fair amount of time, trying to use the adjunction relations between transposes/adjunts and the commutativity relations, with no luck: the only implication I have managed to prove is $(3) \Rightarrow (1)$, via tracking $1_{Fc}$ along both arrows, which give $H\eta_c$ and $\eta'_{Hc}$ respectively.

Any hints?

  • 2
    $\begingroup$ Write out the isomorphisms of homsets explicitly in terms of units and counits. $\endgroup$ – Derek Elkins Jan 29 at 13:23
  • $\begingroup$ @DerekElkins Thanks for the idea, I think I was using (co) units as black boxes a bit too much. I believe I have managed to conclude the result via your hint, and would really appreciate a sanity check. Cheers. $\endgroup$ – Guido A. Jan 29 at 18:44

Hint with another perspective:

An adjunction $F\dashv G$ with $F:C\to D$ can be represented as a category $\mathcal F$ containing (disjoint isomorphic copies of) $C$ and $D$ with a further arrow $\tilde\delta:c\to d$ for each $\delta:Fc\to d$.
Due to the adjunction, $C$ is a coreflective full subcategory, while $D$ is a reflective full subcategory of $\mathcal F$.

Now the conditions imply that $H$ and $K$ induces a functor $\mathcal H:\mathcal F\to\mathcal F'$, and both maps in 3. describe the action of this functor on the newly added homsets.

  • $\begingroup$ This is a bit different from what I was thinking, and also more advanced than my current abilities on the subject, as I have not studied (co) reflective subcategories yet. I will give this another read later on. Thanks (once again) for taking the time to answer :) $\endgroup$ – Guido A. Jan 29 at 18:41

Following Derek Elkins' hint, I think I have managed to solve the problem: first off, by noting that the commutative diagram of $(3)$ is equivalent to the commutative diagram with the isomorphism legs reversed, from the technique used to solve $(1) \iff (3)$ one can similarly prove $(2) \iff (3)$.

Having said that, $(1) \iff (3)$ follows from the fact that the transpose of an arrow $f : Fc \to d$ can be defined (or characterized) as the composition $\eta_cGf$. Suppose that $(1)$ holds and let $f : Fc \to d$ be an arrow. Now, commutativity of the diagram amounts to showing $$ H(\eta_cGf) = \eta'_{Hc}G'Kf \tag{$\star$} $$ since the transpose of $Kf : F'Hc = KFc \to d$ is, via the same remark, $\eta'_{Hc}G'Kf$. In effect, $$ H(\eta_cGf) = H\eta_c HGf \stackrel{(1)}{=} \eta'_{Hc}HGf = \eta'_{Hc}G'Kf $$ as desired.


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.