4
$\begingroup$

Does the composition of adjoint functors again form an adjunction?

Say $\langle F_1,G^1\rangle$ is an adjunct pair between two categories A and B and $\langle F_2,G^2\rangle$ is also an adjoint pair between the categories B and C, then would $\langle F_2\circ F_1,G^1\circ G^2\rangle$ be an adjoint pair between the categories A and C?

If not is there some way of obtain an adjunction given the assumptions made on the 4 functors, or by adding extra properties to the categories?

Particularly, here I am assuming A is the category $_RMod$ for some (apriori) arbitrary ring.

$\endgroup$
  • $\begingroup$ Woops.. I recopied the wrong thing from my note pad :0 $\endgroup$ – AIM_BLB Feb 19 '14 at 14:29
  • 2
    $\begingroup$ Ok. What have you tried? The proof is one line long. $\endgroup$ – Martin Brandenburg Feb 19 '14 at 14:37
3
$\begingroup$

Let a be an object in A, c be an object in C and b ,\tilde{b} be objects in B.

The adjunctions hypothesized give: $C(F_2(b),c)\cong B(b,G^2(c))$ and $B(F_1(a),\tilde{b}) \cong A(a,G^1(\tilde{b}))$.
Now setting b:=$F_1(a)$ and $\tilde{b}:=G^2(c)$ it may be concluded: $A(a,G^1(G^2(c))) \cong B(F_1(a),G^2(c)) \cong C(F_2(F_1(a)),c)$.

So $F_2\circ F_1$ is left adjoint to $G^1\circ G^2$.

$\endgroup$
  • 1
    $\begingroup$ I guess I should stop being afraid of my own solutions :P Thanks Martin :) $\endgroup$ – AIM_BLB Feb 19 '14 at 14:52
  • $\begingroup$ Its not letting me.. it says I have to wait 2 days :( $\endgroup$ – AIM_BLB Feb 19 '14 at 18:06

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.