Monads and their Algebras
5.4. Canonical presentations via free algebras
`
CL
UL
~~
a C
L
__
FL
>>
fromDto the category of algebrasCLfor the induced idempotent monadLis an equivalence of categories.
Proof. An L-algebra is an objectC ∈ Ctogether with a map c: LC → C that is a retraction of the unit componentηC. In fact,ηC andcare inverse isomorphisms. By naturality of η, ηC ·c = Lc·ηLC, butηLC = LηC, as both maps are left inverse to the isomorphismµC, and soLc·ηLC=Lc·LηC =1LC. So ifC∈Cadmits the structure of an L-algebra, then its unit component must be invertible. In fact, this necessary condition also suffices: the mapη−1C : LC→Cautomatically satisfies the associativity condition required to define an L-algebra structure on the objectC. In conclusion, there is no additional structure provided by a lift of an objectC∈Cto anL-algebra; rather being anL-algebra is aconditionon the objectC, namely whether the unit componentηCis invertible.
Exercise 4.5.vii reveals that an objectCis in the essential image ofD,→Cif and only ifηCis an isomorphism. This proves thatK: D→CLis essentially surjective. Naturality of the unitη implies that this functor is fully faithful, so we conclude thatK defines an
equivalence of categories.
In §5.5, we prove the monadicity theorem, which allows one to easily detect when the comparison functor defines an equivalence or isomorphism. In particular, the reader need not remain concerned for very long about the remaining details in the sketched proof of the monadicity of abelian groups.
Exercises.
Exercise 5.3.i. Reading between the lines in the proof of Proposition 5.3.3, prove that the category of algebras for an idempotent monad onCdefines a reflective subcategory ofC and moreover is equivalent to the Kleisli category for that monad.
5.4. Canonical presentations via free algebras
The reason for our particular interest in monadic functors, such asU:Ab →Set, is that the forgetful functorUT:CT →Cassociated to the category of algebras for a monadT onChas certain special properties, which are enumerated in Propositions 5.4.3 and 5.4.9, Lemma 5.6.1, Theorem 5.6.5, and Proposition 5.6.11. These properties are transferred along the equivalence to any monadic functor. The first of these, discussed in this section and used in the next to prove the monadicity theorem, demonstrate that any algebra for a monad admits a canonical presentation as a quotient of free algebras.
On account of the isomorphism of categoriesAbSetZ[−], the adjunction
(5.4.1) Set
Z[−] //
⊥ Ab
U
oo
is monadic, and so we build intuition by first introducing these canonical presentations in the special case of abelian groups. To avoid inelegant pedantry, we use the same notation for the monad Z[−] on Set and for the free abelian group functor Z[−] : Set → Ab.
5.4. CANONICAL PRESENTATIONS VIA FREE ALGEBRAS 169
The context—sets and functions or groups and homomorphisms—will always indicate the appropriate target category.
Any abelian groupAhas a presentation that can be defined in terms of the free and forgetful functors (5.4.1). IfG is any set of elements of an abelian groupA, there is a canonical homomorphism Z[G] → A that sends a finite Z-linear combination of these elements to the element of Athat is their sum. The set Gis a set ofgeneratorsfor A precisely when this map is surjective. Relationsinvolving these generators, meaning the terms that are to be set equal to zero, are elements of the groupZ[G], so again there is a canonical “evaluation” homomorphismZ[R] → Z[G]from the free group on a setR of relations to the free group on the generators. The setsG⊂Aof generators andR⊂Z[G]
of relations give apresentationofAif the quotient mapZ[G]Ais a coequalizer of the evaluation map and the zero homomorphism
(5.4.2) Z[R]
evaluation0 ////Z[G] ////A, in which case one often writesA=G
R
.
Ad hoc presentations as described by (5.4.2) can be a useful way to define abelian groups, but they are unlikely to be functorial: a homomorphismϕ: A → A0 is unlikely to carry the chosen presentation forAto sets of generators and relations forA0. There is, however, a canonical—by which we meanfunctorial—presentation of any abelian group.
Rather than choose a proper subset of generators, we take all of the elements of Ato be generators; the canonical evaluation homomorphismα:Z[A] Ais certainly surjective.
Similarly, rather than choose any particular set of relations inZ[A], we take all of the elements ofZ[A]to be “relations.” Here we do not intend to send every formal sum of elements ofA to zero; the result would be the trivial group. Instead, we generalize the meaning of presentation, making use of the fact that coequalizers are more flexible than cokernels. Proposition 5.4.3 will demonstrate that the diagram
Z[Z[A]]
µA
//Z[α] //Z[A] α ////A
is always a coequalizer. That is, any abelian group is the quotient of the free abelian group on its underlying set modulo the relation that identifies a formal sum with the element to which it is evaluated.
In general:
Proposition 5.4.3. Let(T, η, µ)be a monad onCand let(A,T A−→α A)be aT-algebra.
Then
(5.4.4) (T2A, µT A)
µA //
Tα //(T A, µA) α ////(A, α)
is a coequalizer diagram inCT.
Proposition 5.4.3 can be proven directly by a diagram chase, but we prefer to deduce it as a special case of a more general result, the statement of which requires some new definitions.
Definition 5.4.5. Asplit coequalizerdiagram consists of maps x
f //
g //y h //
t
]] z
s
gg
so thath f =hg,hs=1z,gt=1y, and f t=sh.
The conditionh f =hgsays that this triple of maps defines afork, i.e., his a cone under the parallel pair f,g: x⇒y.
Lemma 5.4.6. The underlying fork of a split coequalizer diagram is a coequalizer. More-over, it is anabsolute colimit: any functor preserves this coequalizer.
Proof. Given a mapk:y→wso thatk f =kg, we must show thatkfactors through h; uniqueness of a hypothetical factorization follows becausehis a split epimorphism. The factorization is given by the mapks:z→w, as demonstrated by the following easy diagram chase:
ksh=k f t=kgt=k.
Now clearly split coequalizers, which are just diagrams of a particular shape, are preserved by any functor, so the equationally-witnessed universal property of the underlying
fork is also preserved by any functor.
Example 5.4.7. For any algebra(A, α:T A→A)for a monad(T, η, µ)onC, the diagram T2A Tα //
µA //T A α //
ηT A
`` A
ηA
jj
defines a split coequalizer diagram inC.
Note that while the fork of Example 5.4.7 lifts alongUT:CT→Cto a diagram (5.4.4) of algebras, the splittingsηA andηT Ado not. This situation is captured by the following general definition.
Definition 5.4.8. Given a functorU:D→C:
• AU-split coequalizeris a parallel pair f,g: x⇒yinDtogether with an extension of the pairU f,Ug:U x⇒Uyto a split coequalizer diagram
U x
U f //
Ug //Uy h //
t
__ z
s
jj
inC.
• Ucreates coequalizers ofU-split pairsif anyU-split coequalizer admits a coequalizer inD whose image under U is isomorphic to the fork underlying the givenU-split coequalizer diagram inC, and if any such fork inDis a coequalizer.
• Ustrictly creates coequalizers ofU-split pairsif anyU-split coequalizer admits a unique lift to a coequalizer inDfor the given parallel pair.
Proposition 5.4.9. For any monad(T, η, µ)acting on a categoryC, the monadic forgetful functorUT:CT →Cstrictly creates coequalizers ofUT-split pairs.
Proof. Suppose given a parallel pair f,g: (A, α) ⇒ (B, β)inCT that admits aUT -splitting:
A
f //
g //B h //
t
]] C.
s
hh
We must show thatC lifts to an algebra (C, γ) and h lifts to an algebra map that is a coequalizer of f andginCT, and that moreover these lifts are unique with this property.
To define the algebra structure mapγ, note that the functorT: C→Cpreserves the split
5.4. CANONICAL PRESENTATIONS VIA FREE ALGEBRAS 171
coequalizer diagram; in particular, by Lemma 5.4.6,T his the coequalizer ofT f andT g. The algebra structure mapsαandβdefine a diagram
T A
in which the square defined byf andT fand the square defined bygandT gboth commute.
Thus,
h·β·T f =h· f·α=h·g·α=h·β·T g,
which says thath·βdefines a cone under the pairT f,T g:T A⇒ T B. By the universal property of their coequalizerT h, there is a unique mapγ:T C→Cso that the right-hand square above commutes. Once we show that the pair(C, γ)is aT-algebra, this commutative square will demonstrate thath: (B, β)→(C, γ)is aT-algebra homomorphism.
It remains to check that the diagrams C ηC //
commute. This will follow from the corresponding conditions for theT-algebra(B, β)and the fact that the coequalizer maps are epimorphisms. Specifically, we have diagrams
B
in which all but the right-most face of each prism is known to commute. It follows that γ·ηC·h=1C·handγ·µC·T2h=γ·Tγ·T2h, and we conclude that the right-most faces commute by canceling the epimorphismshandT2h. Thus,(C, γ)is aT-algebra.
Finally, we show that h: (B, β) → (C, γ) is a coequalizer in CT. Given a cone
using the universal property of the coequalizer inC. To check that jlifts to a map of T-algebras, it again suffices to verify that j·γ = δ·T j after precomposing with the epimorphismT h. The result follows from an easy diagram chase, using the fact thathand kare algebra maps:
j·γ·T h= j·h·β=k·β=δ·T k=δ·T j·T h.
Proof of Proposition 5.4.3. Example 5.4.7 shows that the fork (5.4.4) is part of aUT-split coequalizer. In particular,α:T A → A is an absolute coequalizer of the pair Tα, µA:T2A⇒T AinC. The proof of Proposition 5.4.9 demonstrates that this coequalizer
lifts to a coequalizer inCT.
The results of Propositions 5.4.3 and 5.4.9 extend to general monadic functors with one caveat: as monadic adjunctions are only required to be equivalent to and not isomorphic to the adjunction involving the category of algebras, a monadic functorU: D→Cmight only create, rather than strictly create,U-split coequalizers.
Corollary 5.4.10. If C
F //
⊥ D
U
oo is a monadic adjunction, then (i) U:D→Ccreates coequalizers ofU-split pairs.
(ii) For anyD∈D, there is a coequalizer diagram
(5.4.11) FU FU D
FUD //
FU D //FU D D //D involving the counit:FU⇒1Dof the adjunction.
Proof. To say that U: D → Cis monadic is to say that there is an equivalence of categories
D
U
K ' //CT
UT
~~C
whereT is the monadU F.
For (i), if f,g: A ⇒ Bis aU-split pair inD, then commutativityU =UTKimplies that K f,Kg: KA ⇒ K Bis aUT-split pair inCT. Proposition 5.4.9 lifts the fork of the U-split coequalizer diagram inCto a coequalizer diagram inCT. Any inverse equivalence L: CT −→' Dto the functor K can be used to map this data to a coequalizer for the pair f,g: A ⇒ B. As KL 1CT, this coequalizer maps via U: D → C to a fork that is isomorphic to the fork of the givenU-split coequalizer.9 Note also thatU preserves and reflects all coequalizer diagrams that are U-split because UT has this property and K preserves and reflects all coequalizers (see Lemma 3.3.6). This completes the proof thatU createsU-split coequalizers.
For (ii), the fork (5.4.11) isU-split by the diagram U FU FU D
U FUD//
UFU D//U FU D UD //
ηU FU D
aa U D
ηU D
mm
inC. By (i), the functor U: D → Creflects U-split coequalizers, so the fork (5.4.11)
defines a coequalizer diagram inD.
Exercises.
Exercise 5.4.i. The coequalizer of a parallel pair of morphisms f andgin the category Abis equally the cokernel of the map f −g. Explain how the canonical presentation of an abelian group described in Proposition 5.4.3 defines a presentation of that group, in the usual sense.
9IfKL=1CT, as is the case if these functors define an isomorphism of categories, then this coequalizer inD is a lift of theU-split coequalizer inCandU:D→Cstrictly createsU-split coequalizers.