In this section we study the relation between two seemingly distant concepts as adjunction and monad. As a matter of fact, every adjunction immediately defines a monad, and conversely every monad can be thought of as generated by an adjunction, called a resolution for the monad (see 5.4.2 below). Resolutions for a given monad can be build up in a category by introducing a natural notion of morphism between them; it then happens that the Eilenberg-Moore and the Kleisli Categories associated with the monad (see definitions 4.2.3 and 4.2.4) are respectively the terminal and initial object of the category.
The presentation is rather technical; at first reading, the reader may just look at the first theorem, which will be applied in the next section.
5.4.1 Proposition Let (F, G, η , ε) be an adjunction from Cto D; then (T = GF, η , µ = GεF)
Proof Note first that GF, η, and GεF have the correct types, i.e., T = GF: C→C, η: IdC→GF, and µ = GεF: GFGF→GF. We must prove the unity and associative laws for the monad.
As for the unity laws we have
µ° Tη = GεF ° GFη = G(εF ° Fη) = G(idF) = idGF µ°ηT = GεF °ηGF = (Gε°ηG)F = idG(F) = idGF For the associative law, note first that
ε°εFG = ε° FGε .
Indeed, for any d∈ObD , and letting f = εd: FG(d)→d, εd °εFG(d) = f °εFG(d)
= εd ° FG(f) by naturality of ε
= εd ° FG(εd)
Then one has the following:
µ°µT = GεF ° GεFGF = G(ε°εFG)(F) = G(ε° FGε)(F) = GεF ° GFGεF = µ° Tµ . The rest is an exercise in duality. ♦
5.4.2 Definition Let (T, η , µ) be a monad over a category C. A resolution for (T, η , µ) is a
category D and an adjunction (F, G, η , ε) from C to D such that T = GF and µ = GεF. A
morphism between two resolutions (F, G, η , ε): C→D and (F', G', η , ε'): C→D' (for the
same monad) is a functor H: D→D' such that F' = H ° F, G = G' ° H , and Hε = ε'H.
It is easily proved that resolutions with the associated morphisms form a category. Now we are going to prove that the Eilenberg-Moore and Kleisli categories associated with a monad (T, η , µ) both give rise to resolutions. In particular, they are respectively the terminal and initial objects in the category of all resolutions for that monad.
5.4.3 Proposition Let (T, η , µ) be a monad over a category C , and let CT be the Eilenberg-
Moore category associated with the monad. Then there exists a resolution for (T, η , µ) which is an
adjunction from CT to C.
Proof Let UT: CT→C be the forgetful functor that takes every algebra (c,α) to c, and every morphism of algebras h to the same h regarded as a morphism in C. Let FT: C→CT be the
functor which takes every object c to its free algebra (T(c),µc), and every morhism f: c→c' to FT(f) = T(f). Let εΤ: FTUT→idCT be the natural transformation defined by εΤ(c,α) = α (note that α: T(c)=FTUT(c,α)→c). We want to prove that (FT, UT, η , εT) is a resolution for (T, η , µ).
Obviously UT ° FT = T . UT εT FT = µ, since for any object c one has (UT εT FT)(c) = UT(εT FT(c))
= UT(εT(T(c), µc)) by def. of FT
= UT(µc) by def. of εT
= µc by def. of UT
We still haveto prove that (FT, UT, η , εT) is an adjunction from CT to C, that is,
(UTεT) ° (ηUT) = idUT
(εTFT) ° (FTη) = idFT
One has, for every T-algebra (c,α),
(UTεT °ηUT) (c,α) = UT(εT(c,α) ) °ηUT(c,α)
= α°ηc by def. of εT and UT
= idc by def. of T-algebra
And for every c∈ObC :
(εTFT ° FTη) (c) = εTFT(c) ° FT(ηc)
= µc ° T(ηc) by def. of εT and FT
= idT(c) by the unity law of the monad.♦
We say that the resolution (FT, UT, η , εT) from C to the Eilenberg-Moore category CT, and given
by proposition 5.4.3, is associated with CT.
5.4.4 Proposition Let (T, η , µ) be a monad over a category C .Then the resolution (FT, UT,
η, εT): C→CT, associated with the Eilenberg-Moore Category CT, is a terminal object in the
category of all the resolution for the monad (T, η , µ).
Proof Let (F, G, η , ε): C→D be another resolution for (T, η , µ). We must prove that there exists a unique arrow from (F, G, η , ε) to (FT, UT, η , εT). Remember (cf. definition 5.4.2) that
such an arrow is a functor H: D→CT , such that FT = H ° F, G = UT ° H , and Hε = εTH.
Define, for any object d, and any morphism f of D, H(d) = (G(d), G(ε(d)) )
H(f) = G(f).
Then one has, for any c∈ObC, any h∈MorC,
H(F(c)) = (G(F(c)), G(ε(F(c))) ) = (T(c), µc) = FT(c) H(F(h)) = G(F(h)) = T(h) = FT(h)
that proves the equality H ° F = FT.
Moreover, for any d∈ObD, and any f∈MorD, UT(H(d)) = UT(G(d), G(ε(d)) ) = G(d)
UT(H(f)) = UT(G(f)) = G(f), as UT is the identity on morphisms. That proves the equality G = UT ° H .
Finally, for any d∈ObD,
εTH(d) = εT(G(d), G(ε(d)) ) = G(ε(d)) = H(ε(d)) that proves the equality Hε = εTH.
We have still to prove that H is the unique morphism from (F, G, η , ε) to (FT, UT, η , εT).
Let H' be another morphism; then, for any f∈MorD, H'(f) = UT(H'(f) = G(f) = UT(H(f) = H(f) and, for any d∈MorD,
H'(d) = ( UT(H'(d), εTH'(d)) by def. of UT and εT
= ( G(d), H'(εT(d)) ) as G = UT ° H' = ( UT(H(d), H(εT(d)) ) as H'(f) = H(f) = ( UT(H'(d), εTH(d))
= H(d). This completes the proof. ♦
The unique functor to (FT, UT, η , εT) in the category of all resolutions for a given monad (T, η , µ) is called comparison functor and it is usually denoted by KT.
The category of resolutions of a monad has also an initial object, which is based on the Kleisli category associated with the monad.
5.4.5 Proposition Let (T, η , µ) be a monad over a category C, and let CT be the Kleisli
category associated with the monad. Then there exists a resolution for (T, η, µ) that is an adjunction
from CT to C.
Proof: Let UT: CT→C be the functor defined by the following:
for any object c of CT (i.e., of C ), and any morphism h∈CT[c,c'] (and thus h∈C[c,Τ(c')]) UT(c) = T(c);
UT(h) = µc' ° T(h).
Let FT: C→CT be functor defined by the following: for any object c of C, and any morphism f∈C[c,c']
FT(c) = c;
FT(f) = ηc' ° f ( = T(f) °ηc ).
Let εΤ: FTUT→id be the natural transformation defined by the following: for any object c of CT
εΤ(c) = idT(c) (in C).
We want to prove that (FT, UT, η , εT): C→CT is a resolution for (T, η , µ). Obviously, UT ° FT = T .
Moreover, UT εT FT = µ, since for any object c one has
= UT(idT(c)) by def. of εT
= µc. by def. of UT
We have still to prove that (FT, UT, η , εT) is an adjunction from C to CT, that is, (UTεT) ° (ηUT) = id: UT→UT
(εTFT) ° (FTη) = id: FT→FT
One has, for every object c of CT:
(UTεT °ηUT) (c) = UT(idT(c)) °ηT(c) by def. of εT and UT
= µc ° T(idT(c)) °ηT(c) by def. UT on morphisms
= µc °ηT(c) as T is a functor
= idc. by the unity law of the monad
And, for every c∈ObC,
(εTFT ° FTη) (c) = εT(c) °(ηΤ(c) °ηc) by def. of FT
= idT(c) °(ηΤ(c) °ηc) by def. of εT
= µC ° T(idT(c)) °ηΤ(c) °ηc by def. of composition ° in CT = µC °ηΤ(c) °ηc
= ηc by the unity law of the monad
= idc by def. of the identity in CT. ♦
5.4.6 Proposition Let (T, η, µ) be a monad over a category C. The resolution (FT, UT, η , εT):
C→CT associated with the Kleisli Category CT is an initial object in the category of resolutions for
the monad (T, η , µ).
Proof Let (F, G, η , ε): C→D be another resolution for (T, η , µ). We must prove that there exists a unique arrow from (FT, UT, η , εΤ) to (F, G, η , ε), that is a unique functor K: CT→D, such that F = K ° FT, UT = G ° K , and KεΤ = εK.
Define, for any object cof CT, and any morphism f∈CT[c,c'], K(c) = F(c);
K(f) = εF(c') ° F(f).
where c and f are regarded as object and morphism of C. Then one has, for any c∈ObC, any h∈C[c,c'],
K(FT(c)) = K(c) = F(c)
K(FT(h)) = K(ηc' ° h) by def. of FT
= εF(c') ° F(ηc' ° h) by def. of K = εF(c') ° F(ηc') ° F(h) as F is a functor
= F(h) as (F, G, η , ε) is an adjunction
This proves the equality K ° FT = F .
Moreover, for any object cof CT, and any morphism f∈CT[c,c'], G(K(c)) = G(F(c)) = T(c) = UT(c)
G(K(f)) = G(εF(c') ° F(f)) by def. of K = G(εF(c')) ° G(F(f)) as G is a functor
= µc' ° T(f) as (F, G, η , ε) is a resolution
= UT(f) by def. of UT
that proves the equality UT = G ° K . Finally, for any d∈ObD,
K(εΤ(c)) = Κ(idT(c)) by def. of εΤ = εF(c') ° F(idT(c)) by def. of K
= εF(c') as F is a functor
= εK(c') by def. of K
that proves the equality KεΤ = εK.
We have still to prove that K is the unique morphism from (FT, UT, η , εT) to (F, G, η , ε). Let K': CT→D be another morphism; then, for every object c of CT,
K'(c) = K'(FT(c)) by def. of FT
= F(c) as K' ° FT = F
= K(FT(c)) as K ° FT = F
= K(c) by def. of FT
and, for any f∈CT[c,c'],
K'(f) = K'(idc' ° f )
= K'(µc' °ηT(c') ° f) by the unitary law of the monad = K'(µc' ° T(idT(c')) °ηT(c') ° f)
= K'(idT(c') ° (ηT(c') ° f) ) by def. of composition ° in CT = K'(idT(c')) ° K'(ηT(c') ° f) ) as K' is a functor
= K'(εΤ(c')) ° K'(FT(f)) ) by def. of εΤ and FT = εK'(c') ° F(f) as K'εΤ=εK' and F=(K'°FT) = εF(c') ° F(f) as K'(c) = K(c) = F(c)
= K(f). by def. of K
This completes the proof. ♦
For consistency with the terminology adopted for the comparison functor, we shall denote by KT the unique arrow from the initial object (FT, UT, η , εT) in the category of all resolutions for a
monad (T, η , µ).
Consider now the comparison functor from the initial to the terminal object. Of course, it must be KT = KT; let us check this explicitly.
For any object c in CT,
KT(c) = (UT(c), UT(εΤ(c)) ) by def. of KT
= (T(c), µc)
= FT(c) by def. of FT
= KT(c). by def. of KT
And for any morphism f∈CT[c,c'],
KT(f) = UT(f) by def. of KT = µc' ° T(f) by def. of UT = εΤ(T(c'), µ(c')) ° T(f) by def. of εΤ = εΤFT(c') ° FT(f) by def. of FT = KT(f) . by def. of KT Exercises
1. Prove that the comparison functor KT = KT : CT→CT is full and faithful.
2. Prove that the Kleisli Category is isomorphic to the full subcategory of CT consisting of all free algebras.