The construction of a universal arrow ud: G(cd)→d from G: C→D to d usually depends on d. If this construction can always be performed, the function d |_ cd can be extended to a functor F: D→C. We shall see in the next section that such G and F relate in an important way called adjunction; for the moment we concentrate on the construction of the functor F.
In this and in the following section, we assume that we are dealing with locally small categories. 5.2.1 Theorem. Let G: C→D be a functor such that ∀d∈ObD ∃<ud,cd> universal from d to
G Then there exists a functor F: D→C such that
i. F(d) = cd
ii. C[F_,_] ≅D[_,G_].
(Note that C[F_,_], D[_,G_] : D op×C→Set). Moreover, the functor F is unique, up to
isomorphism.
Proof: By assumption we know that, for all f∈D[d,d'] ,
Set then F(f) = g, that is, ud'°f = G(Ff))°ud . By the uniqueness property, G(id) = id. Moreover, by twice the definition of F,
And again by unicity of F(h ° f), one then has F(f°h) = F(f) ° F(h).
We need now to define a natural isomorphism ϕ: D[_,G_] ≅ C[F_,_]. Thus we first need to check, for a suitable ϕ, that for all g∈D[d',d] and h∈C[c,c'] the following diagram commutes:
Equivalently,
1. ∀f∈D[d,Fc] ϕ(G(h)°f°g) = h°ϕ(f)°Fg .
Now write u (u') for ud (ud', respectively). We know then that ∀f∈D[d,G(c)] ∃!f'∈C[F(d),c] f = G(f') ° u . Define ϕ(f) = f', that is, f = G(ϕ(f))°u (compare with the definition of F ). ϕ is clearly a set-theoretic isomorphism; thus, we have only to prove the naturality (1).
That is, G(ϕ(f°g))°u' = G(ϕ(f)°F(g) )°u' , since G is a functor. By unicity, 2. ϕ(f ° g) = ϕ(f) ° F(g) .
Moreover, for all f∈D[d,G(c)],
by the definition of G . Therefore,
(G(h) ° f ° g) = h °ϕ(f ° g) by the diagram and unicity = h °ϕ(f) ° F(g) by (2).
This proves (1), i.e. the naturality of ϕ , and by proposition 3.2.3 the proof is completed. ♦
Dually, we have the following:
5.2.2 Theorem. Let F: D→C be a functor such that ∀c∈ObC ∃<uc,dc> universal from F to
c. Then there exists a (unique) functor G: C→D such that i. G(c) = dc
Proof The result follows by duality; anyway we explicitly reprove it, but by using a different technique from the one used above. As the reader will see, the difference is essentially notational, but she or he is invited to study both proofs since they are good examples of two common proof styles in Category Theory.
Let GOb: ObC→ObD be the function defined by GOb(c) = dc, where uc: F(dc)→c is the universal arrow. We have
∀f∈C[F(d),c] ∃!g∈D[d,GOb(c)] f = uc ° F(g) Now define, ∀g∈D[d,GOb(c)] τd,c(g) = uc ° F(g) .
For every d∈ObD and c∈ObC, τd,c:D[d,GOb(c)]→C[F(d),c] is clearly a set-theoretic isomorphism. Note that, ∀h∈D[d',d],
1. τd',c(g ° h) = uc ° F( g ° h ) = uc ° F(g) ° F(h) = τd,c(g) ° F(h)
By taking τd,c-1(f) for g in (1), we have τd',c( τd,c-1(f) ° h ) = f ° F(h), or equivalently, 2. τd,c-1(f) ° h = τd,c-1( f ° F(h) )
For simplicity, we now omit the indexes of τ and τ-1 .
Let GMor: MorC →MorD be the function defined by
∀k∈C[c,c'] GMor(k) = τ-1(k ° uc) ∈D[GOb(c), GOb(c')] We want to prove that G = (GOb, GMor) is a functor:
G(idc) = τ-1(uc)
= τ-1(uc ° idF(G(c)) )
= τ-1(uc ° F(idG(c)) )
= τ-1( τ(idG(c) ) )
= idG(c) and, for every f: c'→c", k: c→c',
G(f ° k) = τ-1( f ° k ° uc)
= τ-1(f ° uc' ° F( τ-1(k ° uc) ) )
= τ-1(f ° uc') °τ-1(k ° uc) by (2) = G(f) ° G(k)
(1) proves the naturality of τd,c in the component d . We have still to prove the naturality in c, that
is, ∀g∈D[d,GOb(c)], ∀k∈C[c,c'] 3. τd,c'(G(k) ° g ) = k °τd,c(g) We have: τ(G(k) ° g ) = τ( τ-1(k ° uc) ° g ) = τ( τ-1(k ° uc ° F(g) ) ) by (2) = k ° uc ° F(g) = k °τ(g)
By taking τd,c-1(f) for g in (3) we obtain τd,c'(G(k) °τd,c-1(f) ) = k ° f , or equivalently:
(2) and (4) state the naturality of τ-1.
Note that since G must satisfies (4) , then
G(k) = G(k) ° id = G(k) °τ-1(uc) = τ-1(k ° uc)
which shows that the adopted definition for G was actually forced. This proves the unicity of the functor G. ♦
5.2.3 Example An interesting example of application of theorem 5.2.2 refers to Cartesian closed categories. By the previous section, we know that if C is a CCC, then for all a,b in ObC, (p: a×b→a, p2: a×b→b) is universal from ∆ to (a,b), and evala,b: ba×a→b is universal from _×a to b. Then the functions _×_: ObC××××C→ObC and _a : ObC→ObC which respectively take (a,b) to a×b and b to ba, can be extended to two functors _×_: C××××C→C and _a : C→C. The explicit definition is the following: for every f: a→c , g: b→d
(_×_)(f,g) = f×g = < f ° p1, g ° p2 > : a×b→c×d (_a)(g) = Λ(evala,b ° g) : ba→da
For every object c in C, even the unique arrow !c: c→t may be seen as universal arrow from the unique functor !C: C→1 to t. In this case, the extension of the function that takes 1∈Ob1 to t to a functor T: 1→C is trivial, but it is interesting that the existence of the terminal object t in C may be expressed by the natural isomorphism 1[!C(c)=1 ,1] ≅ C[c,T(1)=t].
5.2.4 Example Consider C, Ct and Inc as in example 5.1.4, and assume that for each object a∈ObC there exists the lifting a°. By example 5.1.4 we know that exa: a°→a is an universal arrow from the embedding functor Inc: Ct→Cp to a. By theorem 5.2.2 the function _°: ObC→ObC which takes every object a to its lifting a°, may be extended to a functor _°: Cp→Ct. The explicit definition of the functor _° on a partial arrow f: b→c is the following
(_°)(f) = f° = τ(f ° exb)∈Ct[b°,c°] where τ(f°exa) is the only arrow such that exc °τ(f°exb) = f°exb.
Note that nearly all the facts about partiality and extendability we proved depend directly on properties of natural transformations and adjunctions. That is, it was not possible to derive the properties of the lifting of b by assuming just a set-theoretic isomorphism between Cp[a,b] and Ct[a,b°] for all a, as one may be tempted at first thought. The expressive categorical notion of natural transformation turns out to be essential for these purposes.