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d  

d  
e f

C
9.4 Free Triples 315
commutes.
Proof. Begin with f0 = f . Let f1 : C1 ’ B be de¬ned using the de¬ning property

of a sum by f1 —¦ e0 = f and f1 —¦ c1 = b —¦ Rf . We will de¬ne fn for n ≥ 1 inductively;
notice that after f1 we are using a pushout rather than a sum, so our induction
hypothesis will have to carry with it a commutativity condition.
So assume that f0 , · · · , fn have been de¬ned in such a way that
(a) fi : Ci ’ B;

(b) fi —¦ ci = b —¦ Rfi’1 ; and
(c) fi —¦ ei’1 = fi’1 .

Then b —¦ Rfi —¦ Rei’l = b —¦ Rfi’1 , so we can legitimately de¬ne fi+1 by requiring
that fi+1 —¦ ci+1 = b —¦ Rfi and fi+1 —¦ ei = fi . The induced map f then clearly
satis¬es the required identities.
By Lemma 8, the objects (C , c : RC ’ C ) form a solution set for the un-

derlying functor U : (R: C ) ’ C , which therefore has a left adjoint as required.


Exercises 9.4

1. Prove that if C is a category with countable sums and R an endofunctor which
preserves countable sums, then R generates a free triple.

2. (Lambek)
(i) Let R be an endofunctor of a category C . Show that if a: RA ’ A is an

initial object in (R: C ), then a is an isomorphism.
(ii) Prove that an endofunctor which has no ¬xed points does not generate a
free triple.
(iii) Prove that the covariant power set functor which takes a map to its direct
image does not generate a free triple.

3. Formulate and prove a trans¬nite generalization of Theorem 7 analogous to
the way in which Exercise 2 of Section 9.3 generalizes Theorem 8 of that section.
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