Page 60 - Textos de Matemática Vol. 41

46 Chapter 7. Images and preimages
so that f∗(x→y) = f∗(x)→f∗(y).
On the other hand, from the last equation we immediately conclude that
f∗(x) ∧ a ≤ f∗(y) iff a ≤ f∗(x→y) iff φ(a) ∧ x ≤ y.
(ii)⇔(iii): Condition (7.3.1) means that for every b ∈ M the diagram
L f! //M
f ∗ (b)∧(−) b∧(−) //
L f! M
is commutative. This is equivalent to saying that the corresponding square of
right adjoints
oo f∗ LM
OO
OO
f ∗ (b)→(−)
b→(−)
Loof∗ M commutes, which means precisely that
f∗(b → y) = f∗(b) → f∗(y) for all y ∈ M.
(iii)⇔(iv): Condition (7.3.1) means also that for every a ∈ L the square
oo f∗ LM
a∧(−) f! (a)∧(−) //
L f! M
commutes. Again this is so iff the corresponding square of right adjoints
L f //M
OO
OO
f! (a)→(−) Loof∗ M
commutes, which is precisely condition (7.3.2).
a→(−)