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University ofCambridge
Mathematics Tripos
Part III
Category Theory
Michaelmas, 2018
Lectures byP. T. Johnstone
Notes byQiangru Kuang
Contents
Contents
1 Definitions and examples 2
2 The Yoneda lemma 10
3 Adjunctions 16
4 Limits 23
5 Monad 35
6 Cartesian closed categories 47
7 Toposes 557.1 Sheaves and local operators* . . . . . . . . . . . . . . . . . . . . 62
Index 67
1
1 Definitions and examples
1 Definitions and examples
Definition (category). A category C consists of
1. a collection obC of objects ๐ด, ๐ต, ๐ถ, โฆ,
2. a collection morC of morphisms ๐, ๐, โ, โฆ,
3. two operations dom and cod assigning to each ๐ โ morC a pair ofobjects, its domain and codomain. We write ๐ด
๐โโ ๐ต to mean ๐ is a
morphism and dom ๐ = ๐ด, cod ๐ = ๐ต,
4. an operation assigning to each ๐ด โ obC a morhpism ๐ด1๐ดโโ ๐ด,
5. a partial binary operation (๐, ๐) โฆ ๐๐ on morphisms, such that ๐๐ isdefined if and only if dom ๐ = cod ๐ and let dom ๐๐ = dom ๐, cod ๐๐ =cod ๐ if ๐๐ is defined
satisfying
1. ๐1๐ด = ๐ = 1๐ต๐ for any ๐ด๐โโ ๐ต,
2. (๐๐)โ = ๐(๐โ) whenever ๐๐ and ๐โ are defined.
Remark.
1. This definition is independent of any model of set theory. If weโre given aparticuar model of set theory, we call C small if obC and morC are sets.
2. Some texts say ๐๐ means ๐ followed by ๐ (we are not).
3. Note that a morphism ๐ is an identity if and only if ๐๐ = ๐ and โ๐ = โwhenever the compositions are defined so we could formulate the defini-tions entirely in terms of morphisms.
Example.
1. The category Set has all sets as objects and all functions between sets asmorphisms (strictly, morphisms ๐ด โ ๐ต are pairs (๐, ๐ต) where ๐ is a settheoretic function).
2. The category Gp where objects are groups, morphisms are group homo-morphisms. Similarly Ring is the category of rings and Mod๐ is thecategory of ๐ -modules.
3. The category Top has all topological spaces as objects and continuousfunctions as morphisms. Similarly Unif the category of uniform spaceswith uniformly continuous functions and Mf the category of manifoldswith smooth maps.
4. The cateogry Htpy has same objects as Top but morphisms are homotopyclasses of continuous functions.More generally, given C we call an equivalence relation โ on morC acongruence if ๐ โ ๐ implies dom ๐ = dom ๐, cod ๐ = cod ๐ and ๐โ โ ๐โ
2
1 Definitions and examples
and vice versa. Then we have a set C/ โ with the same objects as C butcongruence classes as morphisms.
5. Given C, the opposite category Cop has the same objects and morphismsas C, but domain and codomain interchanged and ๐๐ in Cop is ๐๐ in C.This leads to duality principle: if ๐ is a valid statement about categoriesso is ๐ โ attained by reversing all the arrows.
6. A small category with one object is a monoid, i.e. a semigroup with 1.In particular, a group is a small category with one object, in which everymorphism is an isomorphism (i.e. for all ๐ there exists ๐ such that ๐๐ and๐๐ are identities).
7. A groupoid is a category in which every morphism is an isomorphism. Forexample, for a topological space ๐, the fundamental groupoid ๐(๐) hasall points of ๐ as objects and morphisms ๐ฅ โ ๐ฆ as homotopy classes rel0, 1 of paths ๐พ โถ [0, 1] โ ๐ with ๐พ(1) = ๐ฆ.
8. A discrete category is one whose only morphisms are identities. A preorderis a category C in which for any pair (๐ด, ๐ต) there exists at most one mor-phism ๐ด โ ๐ต. A small preorder is a set equipped with a binary relationwhich is reflexive and transitive. In particular a partially ordered set is apartially ordered set is a small preorder in which the only isomorphismsare identities.
9. The category Rel has the same objects as Set but morphisms ๐ด โ ๐ต arearbitrary relations. Given ๐ โ ๐ด ร ๐ต, ๐ โ ๐ต ร ๐ถ, we define
๐ โ ๐ = {(๐, ๐) โ ๐ด ร ๐ถ โถ โ๐ โ ๐ต s.t. (๐, ๐) โ ๐ , (๐, ๐) โ ๐}.
The identity 1๐ด โถ ๐ด โ ๐ด is {(๐, ๐) โถ ๐ โ ๐ด}. Similarly, the category Partof sets and partial functions (i.e. relations such that for all (๐, ๐), (๐, ๐โฒ) โ๐ , ๐ = ๐โฒ) can be defined.
10. Let ๐พ be a field. The category Mat๐พ has natural numbers as objects andmorphisms ๐ โ ๐ are (๐ ร ๐) matrices with entries from ๐พ. Compositionis matrix multiplication.
Definition (functor). Let C, D be categories. A functor ๐น โถ C โ Dconsists of
1. a mapping ๐ด โฆ ๐น(๐ด) from obC to obD,
2. a mapping ๐ โฆ ๐น(๐) from morC to morD such that
dom(๐น(๐)) = ๐น(dom ๐), cod(๐น๐) = ๐น cod(๐)1๐น(๐ด) = ๐น(1๐ด), (๐น (๐))(๐น(๐)) = ๐น(๐๐)
wherever ๐๐ is defined.
Example. We write Cat for the category whose objects are all small categoriesand whose morphisms are functors between them.
3
1 Definitions and examples
Example.1. We have forgetful functors ๐ โถ Gp โ Set, Ring โ Set, Top โ Set โฆ
and Ring โ AbGp (forget multiplication), Ring โ Mon (forget addi-tion).
2. Given ๐ด, the free group ๐น(๐ด) has the property given any group ๐บ, any
๐ด๐โโ ๐๐บ, there is a unique homomorphism ๐น๐ด
๐โโ ๐บ extending ๐.
๐น is a functor Set โ Gp: given any ๐ด๐โโ ๐ต, we define ๐น(๐) to be
the unique homomorphism extending ๐ด๐โโ ๐ต โช ๐๐น๐ต. Functoriality
follows from uniqueness: given ๐ต๐โโ ๐ถ, ๐น(๐๐) and (๐น๐)(๐น๐) are both
homomorphisms extending
๐ด๐โโ ๐ต
๐โโ ๐ถ โช ๐๐น๐ถ.
3. Given a set ๐ด, we write ๐๐ด for the set of all subsets of ๐ด. We can make๐ into a functor Set โ Set: given ๐ด
๐โโ ๐ต, we define
๐๐(๐ดโฒ) = {๐(๐) โถ ๐ โ ๐ดโฒ}
for ๐ดโฒ โ ๐ด.But we also have a functor ๐ โ โถ Set โ Setop defined by
๐ โ๐(๐ตโฒ) = {๐ โ ๐ด โถ ๐(๐) โ ๐ตโฒ}
for ๐ตโฒ โ ๐ต.
4. By a contravariant functor C โ D we mean a functor C โ Dop (orCop โ D. (A covariant functor is one that doesnโt reverse arrows)Let ๐พ be a field. We have a functor โ โ โถ Mod๐พ โ Modop
๐พ defined by
๐ โ = {linear maps ๐ โ ๐พ}
and if ๐๐โโ ๐, ๐โ(๐) = ๐๐.
5. We have a functor โ op โถ Cat โ Cat which is the identity on morphisms.(note that this is covariant)
6. A functor between monoids is a monoid homomorphism.
7. A functor between posets is an order-preserving map.
8. Let ๐บ be a group. A functor ๐น โถ ๐บ โ Set consists of a ast ๐ด = ๐นโtogether with an action of ๐บ on ๐ด, i.e. permutation representation of ๐บ.Similarly a functor ๐บ โ Mod๐พ is a ๐พ-linear representation of ๐บ.
9. The construction of the fundamental gropu ๐1(๐, ๐ฅ) of a space ๐ withbasepoint ๐ฅ is a functor
Topโ โ Gpwhere Topโ is the category of spaces with a chosen basepoint.Similarly, the fundamental groupoid is a functor
Top โ Gpd
where Gpd is the category of groupoids and functors between them.
4
1 Definitions and examples
Definition (natural transformation). Let C, D be categories and ๐น, ๐บ โถC โ D be two functors. A natural transformation ๐ผ โถ ๐น โ ๐บ consists of anassignment ๐ด โฆ ๐ผ๐ด from obC to morD such that dom๐ผ๐ด = ๐น๐ด, cod๐ผ๐ด =๐บ๐ด for all ๐ด, and for all ๐ด
๐โโ ๐ต in C, the square
๐น๐ด ๐น๐ต
๐บ๐ด ๐บ๐ต
๐น๐
๐ผ๐ด ๐ผ๐ต
๐บ๐
commutes, i.e. ๐ผ๐ต(๐น๐) = (๐บ๐)๐ผ๐ด.
Example.
1. Given categories C, D, we write [C, D] for the category whose objects arefunctors C โ D and whose morphisms are natural transformations.
2. Let ๐พ be a field and ๐ a vector space over ๐พ. There is a linear map๐ผ๐ โถ ๐ โ ๐ โโ given by
๐ผ๐(๐ฃ)(๐) = ๐(๐ฃ)
for ๐ โ ๐ โ. This is the ๐-component of a natural transformation
1Mod๐พโ โ โโ โถ Mod๐พ โ Mod๐พ.
3. For any set ๐ด, we have a mapping ๐๐ด โถ ๐ด โ ๐๐ด sending ๐ to {๐}. If๐ โถ ๐ด โ ๐ต then ๐๐({๐}) = {๐(๐)}. So ๐ is a natural transformation1Set โ ๐.
4. Let ๐น โถ Set โ Gp be the free group functor and ๐ โถ Gp โ Set theforgetful functor. The inclusions ๐ด โ ๐๐น๐ด is a natural transformation1Set โ ๐๐น.
5. Let ๐บ, ๐ป be groups and ๐, ๐ โถ ๐บ โ ๐ป be two homomorphisms. Then anatural tranformation ๐ผ โถ ๐ โ ๐ corresponds to an element โ = ๐ผโ suchthat โ๐(๐ฅ) = ๐(๐ฅ)โ for all ๐ฅ โ ๐บ, or equivalently ๐(๐ฅ) = โโ1๐(๐ฅ)โ, i.e. ๐and ๐ are conjugate group homomorphisms.
6. Let ๐ด and ๐ต be two ๐บ-sets, regarded as functors ๐บ โ Set. A naturaltransformation ๐ด โ ๐ต is a function ๐ satisfying ๐(๐.๐) = ๐.๐(๐) for all๐ โ ๐ด, i.e. a ๐บ-equivariant map.
When we say โnatural isomorphismโ, it is ambiguous and can formally meantwo different things: one could mean there is a natural transformation goingthe other way which when composed produces identity, or each component isan isomorphism. It turns out they coincide:
Lemma 1.1. Let ๐น, ๐บ โถ C โ D be two functors and ๐ผ โถ ๐น โ ๐บ a naturaltransformation. Then ๐ผ is an isomorphism in [C, D] if and only if each ๐ผ๐ดis an isomorphism in D.
5
1 Definitions and examples
Proof. Only if is trivial. For if, suppose each ๐ผ๐ด has an inverse ๐ฝ๐ด. We need toprove the ๐ฝโs satisfy the naturality condition: give ๐ โถ ๐ด โ ๐ต in C, we need toshow that
๐บ๐ด ๐บ๐ต
๐น๐ด ๐น๐ต
๐บ๐
๐ฝ๐ด ๐ฝ๐ต
๐น๐
commutes. But
(๐น๐)๐ฝ๐ด = ๐ฝ๐ต๐ผ๐ต(๐น๐)๐ฝ๐ด = ๐ฝ๐ต(๐บ๐น)๐ผ๐ด๐ฝ๐ด = ๐ฝ๐ต(๐บ๐)
by naturality of ๐ผ.
In study of algebraic theories (for example), we are interested in isomor-phisms of objects and investigate the properties of objects โup to isomorphismโ.However, in category theory a weaker notion of isomorphism is often more use-ful:
Definition (equivalence). Let C and D be categories. By an equivalencebetween C and D we mean a pair of functors ๐น โถ C โ D, ๐บ โถ D โ Ctogether with natural isomorphisms ๐ผ โถ 1C โ ๐บ๐น, ๐ฝ โถ ๐น๐บ โ 1D. We writeC โ D if C and D are equivalent.
We say a property ๐ of categories is a category propery if whenever Chas ๐ and C โ D then D has ๐.
For example, being a groupoid or a preorder are categorical properies, butbeing a group or a partial order are not.
Example.
1. The category Part is equivalent to the category Setโ of pointed sets (andbasepoint preserving functions). We define
๐น โถ Setโ โ Part(๐ด, ๐) โฆ ๐ด \ {๐}
and if ๐ โถ (๐ด, ๐) โ (๐ต, ๐) then ๐น๐(๐ฅ) = ๐(๐ฅ) if ๐(๐ฅ) โ ๐ and undefinedotherwise, and
๐บ โถ Part โ Setโ
๐ด โฆ ๐ด+ = (๐ด โช {๐ด}, ๐ด)
and if ๐ โถ ๐ด โ ๐ต is a partial function , we define
๐ฅ โฆ {๐(๐ฅ) if ๐ฅ โ ๐ด and ๐(๐ฅ) defined๐ต otherwise
Then ๐น๐บ is the identity on Part but ๐บ๐น is not. However there is anisomorphism
(๐ด, ๐) โ ((๐ด \ {๐})+, ๐ด \ {๐})sending ๐ to ๐ด \ {๐} and everything else to itself. This is natural.Note that there can be no isomorphism Setโ โ Part since Part has a1-element isomorphism class {โ } and Setโ doesnโt.
6
1 Definitions and examples
2. The category fdMod๐พ of finite-dimensional vector spaces over ๐พ is equiv-alent to fdModop
๐พ : the functors in both directions are โ โ and both isomor-phisms are the natural transformations given by double dual.
3. fdMod๐พ is also equivalent to Mat๐พ: we write ๐น โถ Mat๐พ โ fdMod๐พ by๐น(๐) = ๐พ๐, and ๐น(๐ด) is the map represented by ๐ด with respect to thestandard basis. To define functor ๐บ the other way, choose a basis for eachfinite-dimensional vector space and define
๐บ(๐ ) = dim๐
๐บ(๐๐โโ ๐) = matrix representing ๐ w.r.t. chosen bases
๐บ๐น is the identity, provided we choose the standard bases for the space ๐พ๐.๐น๐บ โ 1 but the chosen bases give isomorphisms ๐น๐บ(๐ ) = ๐พdim ๐ โ ๐for each ๐, which form a natural isomorphism.
Example 3 illustrates a general principle: when constructing a pair of func-tors between equivalent categories, ususally one is โcanonicalโ and the otherrequires some choice, and a clever choice results in a particularly simple formfor one way of composition. The next theorem abstracts away the โchoiceโ andtells us when a functor is part of an equivalence purely by its properties.
The criterion is stated in term of โbijectivityโ of functors, informally. It isgenerally a bad idea to look at sur/injectivity of functors on objects. Insteadthe correct way is to look at their behaviour on morphisms.
Definition (faithful, full, essentially surjective). Let C๐นโโ D be a functor.
1. ๐น is faithful if given ๐, ๐ โฒ โ morC with dom ๐ = dom ๐ โฒ, cod ๐ = cod ๐ โฒ
and ๐น๐ = ๐น๐ โฒ then ๐ = ๐ โฒ.
2. ๐น is full if given ๐น๐ด๐โโ ๐น๐ต in D then there exists ๐ด
๐โโ ๐ต in C with
๐น๐ = ๐.
3. ๐น is essentially surjective if for every ๐ต โ obD there exists ๐ด โ obCand an isomorphism ๐น๐ด โ ๐ต in D.
Definition. A subcategory Cโฒ โ C is full if the inclusion Cโฒ โ C is a fullfunctor.
Example. Gp is a full subcategory of Mon but Mon is not a full subcategoryof the category SGp of semigroups.
Lemma 1.2. Assuming the axiom of choice. A functor ๐น โถ C โ D ispart of an equivalence C โ D if and only if itโs full, faithful and essentiallysurjective.
Proof. Suppose given ๐บ, ๐ผ, ๐ฝ as in the definition of equivalence of categories.Then for each ๐ต โ obD, ๐ฝ๐ต is an isomorphism ๐น๐บ๐ต โ ๐ต so ๐น is essentiallysurjective.
7
1 Definitions and examples
Given ๐ด๐โโ ๐ต in C, we can recover ๐ from ๐น๐ as the via conjugation by ๐ผ:
๐บ๐น๐ด ๐บ๐น๐ต
๐ด ๐ต
๐บ๐น๐
๐ผ๐ด
๐
๐ผ๐ต
Hence if ๐ด๐โฒ
โโ ๐ต satisfies ๐น๐ = ๐น๐ โฒ then ๐ = ๐ โฒ.Similarly there is a natural preimage given a morphism in D. Given ๐น๐ด
๐โโ
๐น๐ต, define ๐ to be the composite
๐บ๐น๐ด ๐บ๐น๐ต
๐ด ๐ต
๐บ๐
๐ผ๐ด ๐ผ๐ต
Then ๐บ๐น๐ = ๐ผ๐ต๐๐ผโ1๐ด = ๐บ๐. As ๐บ is faithful for the same reasons as ๐น, ๐น๐ = ๐.
Conversely, for each ๐ต โ obD, choose ๐บ๐ต โ obC and an isomorphism๐ฝ๐ต โถ ๐น๐บ๐ต โ ๐ต in D. Given ๐ต
๐โโ ๐ตโฒ, define ๐บ๐ โถ ๐บ๐ต โ ๐บ๐ตโฒ to be the unique
morphism whose image under ๐น is the composition
๐ต ๐ตโฒ
๐น๐บ๐ต ๐น๐บ๐ด
๐
๐ฝ๐ต ๐ฝ๐ตโฒ
Faithfulness implies functoriality: given ๐ตโฒ๐โฒ
โโ ๐ตโณ, (๐บ๐โฒ)(๐บ๐) and ๐บ(๐โฒ๐) havethe same image under ๐น so they are equal.
By construction, ๐ฝ is a natural transformation ๐น๐บ โ 1D.Given ๐ด โ obC, define ๐ผ๐ด โถ ๐ด โ ๐บ๐น๐ด to be the unique morphism whose
image under ๐น is
๐น๐ด๐ฝโ1
๐น๐ดโโโ ๐น๐บ๐น๐ด.
๐ผ๐ด is an isomorphism since ๐ฝ๐น๐ด also has a unique preimage under ๐น. Finally๐ผ is a natural transformation, since any naturality square for ๐ผ is mapped by๐น to a commutative square (corresponding to naturality square for ๐ฝ) and ๐น isfaithful.
Note that axiom of choice is only used in the if part. The lemma is usefulas it saves us from making explicit choices when showing an equivalence byexhibiting inverses. However note that the choice is always required.
Definition (skeleton). By a skeleton of a category C we mean a full sub-category C0 containing one object from each isomorphism class. We say Cis skeletal if itโs a skeleton of itself.
Example. Mat๐พ is skeletal and the image of ๐น โถ Mat๐พ โ fdMod๐พ is askeleton of fdMod๐พ (essentially because ๐น is full and faithful).Remark. Almost any assertion about skeletons is equivalent to the axiom ofchoice. See example sheet 1 Q2. This is one reason why we should not restrictout attention to skeletal categories.
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1 Definitions and examples
Definition (monomorphism, epimorphism). Let ๐ด๐โโ ๐ต be a morphism in
C.
1. We say ๐ is a monomorphism or monic if given any pair ๐, โ โถ ๐ถ โ ๐ด,๐๐ = ๐โ implies ๐ = โ.
2. We say ๐ is a epimorphism or epic if it is a monomorphism in Cop, i.e.given any pair ๐, โ โถ ๐ต โ ๐ถ, ๐๐ = โ๐ implies ๐ = โ.
We denote monomorphisms by ๐ โถ ๐ด โฃ ๐ต and epimorphisms by ๐ โถ๐ด โ ๐ต.
Any isomorphism is monic and epic. More generally if ๐ has a left inversethen itโs monic. We call such monomorphisms split.
Definition (balanced). We say C is a balanced category if any morphismwhich is both monic and epic is an isomorphism.
Example.
1. In Set, monomorphism is precisely an injection (one direction is easy andfor the other direction take ๐ถ = 1 = {โ}) and epimorphism is precisely asurjection (use morphisms ๐ต โ 2 = {0, 1}). Thus Set is balanced.
2. In Gp, monomorphism is precisely an injection (use homomorphism fromthe free group with one generator, i.e. Z โ ๐ด) and epimorphism is pre-cisely a surjection (use free product with amalgamation). Thus Gp isbalanced.
3. In Rng, monomorphism is precisely an injection (similarly to free group)but the inclusion Z โ Q is an epimorphism, since if ๐, ๐ โถ Q โ ๐ agreeon all integers they agree everywhere. So Rng is not balanced.
4. In Top, monomorphism is precisely an injection and epimorphism is pre-cisely a precisely surjection (same argument as Set) but Top is not bal-anced since a continuous bijection need not to have a continuous inverse.
9
2 The Yoneda lemma
2 The Yoneda lemmaIt may seem weird to devote an entire chapter to a lemma. However, althoughthe Yoneda lemma is indeed a lemma, with a simple statement and straight-forward proof, it is much more than a normal lemma and underlies the entirecategory theory.
Here is a little story about Yoneda lemma. It is named after Nobuo Yoneda,who is better known as a computer scientist than a mathematician. The resultis likely not due to him and he wasnโt the first person to write it down. In fact,he never actually wrote down the lemma, as opposed to some books claiming thelemma was first to be found in one of Yonedaโs papers. The story, accordingto Saunders Mac Lane, is that after a conference he met Yoneda on a trainplatform. While exchanging conversations Yoneda told him the result, whichMac Lane was not aware of at that moment but immediately recognised itsimportance. Mac Lane later attributed the lemma to Yoneda and because ofhis standing in category theory, the name retains. Perhaps this is the first andonly result in mathematics to be enunciated on a train platform!
Definition (locally small category). We say a category C is locally smallif, for any two objects ๐ด, ๐ต, the morphisms ๐ด โ ๐ต in C form a set C(๐ด, ๐ต).
If we fix ๐ด and let ๐ต vary, the assignment ๐ต โฆ C(๐ด, ๐ต) becomes a functorC(๐ด, โ) โถ C โ Set: given ๐ต
๐โโ ๐ถ, C(๐ด, ๐) is the mapping ๐ โฆ ๐๐. Similarly,
๐ด โฆ C(๐ด, ๐ต) defines a functor C(โ, ๐ต) โถ Cop โ Set.
Lemma 2.1 (Yoneda lemma). Let C be a locally small category, ๐ด โ obCand ๐น โถ C โ Set is a functor. Then natural transformations C(๐ด, โ) โ ๐นare in bijection with elements of ๐น๐ด.
Moreover, this bijection is natural in both ๐ด and ๐น.
Proof. We prove the first part now. The second part follows very easily once wehave gained some intuitions so weโll come back to it later. Given ๐ผ โถ C(๐ด, โ) โ๐น, we define
ฮฆ(๐ผ) = ๐ผ๐ด(1๐ด) โ ๐น๐ด.
Conversely, given ๐ฅ โ ๐น๐ด, we define ฮจ(๐ฅ) โถ C(๐ด, โ) โ ๐น by
ฮจ(๐ฅ)๐ต(๐ด๐โโ ๐ต) = ๐น๐(๐ฅ) โ ๐น๐ต
which is natural: given ๐ โถ ๐ต โ ๐ถ, we have
ฮจ(๐ฅ)๐ถC(๐ด, ๐)(๐) = ฮจ(๐ฅ)๐ถ(๐๐) = ๐น(๐๐)(๐ฅ)(๐น๐)ฮจ(๐ฅ)๐ต(๐) = (๐น๐)(๐น๐)(๐ฅ) = ๐น(๐๐)(๐ฅ)
10
2 The Yoneda lemma
C(๐ด, ๐ต) C(๐ด, ๐ถ)
๐ ๐๐
๐น๐(๐ฅ) ๐น(๐๐)(๐ฅ)
๐น๐ต ๐น๐ถ
C(๐ด,๐)
ฮจ(๐ฅ)๐ต ฮจ(๐ฅ)๐ถ
๐น๐
by functoriality of ๐น. Now left to show they are inverses to each other.
ฮฆฮจ(๐ฅ) = ฮจ(๐ฅ)๐ด(1๐ด) = ๐น(1๐ด)(๐ฅ) = ๐ฅฮจฮฆ(๐ผ)๐ต(๐) = ฮจ(๐ผ๐ด(1๐ด))๐ต(๐) = ๐น๐(๐ผ๐ด(1๐ด)) = ๐ผ๐ตC(๐ด, ๐)(1๐ด) = ๐ผ๐ต(๐)
so ฮจฮฆ(๐ผ) = ๐ผ.
Corollary 2.2 (Yoneda embedding). The assignment ๐ด โฆ C(๐ด, โ) definesa full faithful functor Cop โ [C, Set].
Proof. Put ๐น = C(๐ต, โ) in the above proof, we get a bijection between C(๐ต, ๐ด)and morphisms C(๐ด, โ) โ C(๐ต, โ) in [C, Set]. We need to verify that this isfunctorial. But it sends ๐ โถ ๐ต โ ๐ด to the natural transformation ๐ โฆ ๐๐. Sofunctoriality follows from associativity.
We call this functor (or the functor C โ [Cop, Set] sending ๐ด to C(โ, ๐ด))the Yoneda embedding of C and typically denote it by ๐. At first glance, [C, Set]seems like a much more complicated entity and is much more unwieldy. However,it stands out as being more concrete and thus easier to deal with. It is in analogywith group representation: instead of an abstract group, we consider its actionon a set, which is more explicit and concrete. Weโll come back to this point ina minute.
Now return to the second part of the lemma. Suppose for the momentthat C is small, so that [C, Set] is locally small. Then we have two functorsC ร [C, Set] โ Set: one sends (๐ด, ๐น) to ๐น๐ด, and the other is the composite
C ร [C, Set]๐ ร1โโโ [C, Set]op ร [C, Set]
[C,Set](โ,โ)โโโโโโโโโ Set
where the last map maps a pair of functors to the set of natural transformationsbetween them, and the naturality in the statement of Yoneda lemma says thatthese are naturally isomorphic. We can translate this into an elementary state-ment, making sense even when C isnโt small: given ๐ด
๐โโ ๐ต and ๐น
๐ผโโ ๐บ, there
are two ways of producing an element of ๐บ๐ต from the natural transformation.For example, given ๐ฝ โถ C(๐ด, โ) โ ๐น, the two ways give the same result, namely
๐ผ๐ต(๐น๐)๐ฝ๐ด(1๐ด) = (๐บ๐)๐ผ๐ด๐ฝ๐ด(1๐ด)
11
2 The Yoneda lemma
which is equal to ๐ผ๐ต๐ฝ๐ต(๐).
[C, Set](C(๐ด, โ), ๐น) [C, Set](C(๐ต, โ), ๐น)
๐น๐ด ๐น๐ต
๐บ๐ต ๐บ๐ต
[C, Set](C(๐ด, โ), ๐บ) [C, Set](C(๐ต, โ), ๐บ)
ฮฆ๐ด,๐น
ฮฆ๐ต,๐น๐น๐
๐ผ๐ด ๐ผ๐ต
๐บ๐
ฮฆ๐ด,๐บ
ฮฆ๐ต,๐บ
Definition (representable functor, representation). We say a functor ๐น โถC โ Set is representable if itโs isomorphic to C(๐ด, โ) for some ๐ด.
By a representation of ๐น, we mean a pair (๐ด, ๐ฅ) where ๐ฅ โ ๐น๐ด is suchthat ฮจ(๐ฅ) is an isomorphism. We also call ๐ฅ a universal element of ๐น.
Corollary 2.3. If (๐ด, ๐ฅ) and (๐ต, ๐ฆ) are both representations of ๐น, then thereis a unique isomorphism ๐ โถ ๐ด โ ๐ต such that (๐น๐)(๐ฅ) = ๐ฆ.
Proof. Consider the composite
C(๐ต, โ)ฮจ(๐ฆ)โโโโ ๐น
ฮจ(๐ฅ)โ1
โโโโโ C(๐ด, โ),
By Yoneda embedding, this is of the form ๐ (๐) for a unique isomorphism ๐ โถ๐ด โ ๐ต and the diagram
C(๐ต, โ) C(๐ด, โ)
๐น
๐ (๐)
ฮจ(๐ฆ) ฮจ(๐ฅ)
commutes if and only if (๐น๐)(๐ฅ) = ๐ฆ.
Example.
1. The forgetful functor Gp โ Set is representable by (Z, 1). Similarly theforgetful functor Rng โ Set is representable by (Z[๐ฅ], ๐ฅ). The forgetfulfunctor Top โ Set is representable by ({โ}, โ).
2. The functor ๐ โ โถ Setop โ Set is representable by ({0, 1}, {1}). This isthe bijection between subsets and characteristic functions.
3. Let ๐บ be a group. The unique (up to isomorphism) representable functor๐บ(โ, โ) โถ ๐บ โ Set is the Cayley representation of ๐บ, i.e. the set ๐๐บ with๐บ acting by left multiplication.
12
2 The Yoneda lemma
4. Let ๐ด and ๐ต be two objects of a locally small category C. Then we havea functor Cop โ Set sending ๐ถ to C(๐ถ, ๐ด) ร C(๐ถ, ๐ต) (note that it is apurely categorical product and require only cartesian product of the mor-phism sets). A representation of this, if it exists, is called a (categorical)product of ๐ด and ๐ต, and denoted
๐ด ร ๐ต
๐ด ๐ต
๐1 ๐2
This pair has the property that, for any pair (๐ถ๐โโ ๐ด, ๐ถ
๐โโ ๐ต), there is a
unique ๐ถโโโ ๐ด ร ๐ต with ๐1โ = ๐ and ๐2โ = ๐.
๐ด ร ๐ต
๐ด ๐ถ ๐ต
๐1 ๐2
๐๐
โ
Products exist in many categories of interest: in Set, Gp, Rng, Top, โฆthey are โjustโ cartesian products. In posets they are binary meets.Dually, we have the notion of coproduct
๐ด + ๐ต
๐ด ๐ต
๐1 ๐2
These also exist in many categories of interest.
5. Let ๐, ๐ โถ ๐ด โ ๐ต be morphisms in a locally small category C. We have afunctor ๐น โถ Cop โ Set defined by
๐น(๐ถ) = {โ โ C(๐ถ, ๐ด) โถ ๐โ = ๐โ},
which is a subfunctor of C(โ, ๐ด). A representation of ๐น, if it exists, iscalled an equaliser of (๐, ๐). It consists of an object ๐ธ and a morphism๐ธ
๐โโ ๐ด such that ๐๐ = ๐๐ and every โ with ๐โ = ๐โ factors uniquely
through ๐. In Set, we take ๐ธ = {๐ฅ โ ๐ด โถ ๐(๐ฅ) = ๐(๐ฅ)} and ๐ to beinclusion. Similar constructions work in Gp, Rng, Top, โฆ
๐ธ ๐ด ๐ต
๐ถ
๐ ๐
๐โ
Dually we have the notion of coequaliser .Remark. If ๐ occurs as an equaliser then it is a monomorphism, since any โfactors through it in at most one way. We say a monomorphism is regular if itoccurs as an equaliser.
Split monomorphisms are regular (see example sheet 1 Q6 (i)). Note that aregular mono that is also epic implies isomorphism: if the equaliser ๐ of (๐, ๐)is epic then ๐ = ๐ so ๐ โ 1cod ๐.
13
2 The Yoneda lemma
Definition (separating/detecting family, seperator, detector). Let C be acategory, ๐ข a class of objects of C.
1. We say ๐ข is a separating family for C if, given ๐, ๐ โถ ๐ด โ ๐ต such that๐โ = ๐โ for all ๐บ
โโโ ๐ด with ๐บ โ ๐ข then ๐ = ๐. (i.e. the functor
C(๐บ, โ) where ๐บ โ ๐ข are collectively faithful)
2. We say ๐ข is a detecting family for C if, given ๐ด๐โโ ๐ต such that every
๐บโโโ ๐ต with ๐บ โ ๐ข factors uniquely through ๐, then ๐ is an isomor-
phism.
If ๐ข = {๐บ} then we call ๐บ a separator or detector.
Lemma 2.4.
1. If C is a balanced category then any separating family is detecting.
2. If C has equalisers (i.e. every pair has an equaliser) then any detectingfamily is separating.
Proof.
1. Suppose ๐ข is separating and ๐ด๐โโ ๐ต satisfies condition in definition 2. If
๐, โ โถ ๐ต โ ๐ถ satisfy ๐๐ = โ๐, then ๐๐ฅ = โ๐ฅ for every ๐บ๐ฅโโ ๐ต, so ๐ = โ,
i.e. ๐ is epic.
Similarly if ๐, โ โถ ๐ท โ ๐ด satisfy ๐๐ = ๐โ then ๐๐ฆ = โ๐ฆ for any ๐บ๐ฆโโ ๐ท,
since both are factorisations of ๐๐๐ฆ through ๐. So ๐ = โ, i.e. ๐ is monic.
2. Suppose ๐ข is detecting and ๐, ๐ โถ ๐ด โ ๐ต satisfy definition 1. Then theequaliser ๐ธ
๐โโ ๐ด of (๐, ๐) is isomorphism so ๐ = ๐.
Example.
1. In [C, Set] the family {C(๐ด, โ) โถ ๐ด โ obC} is both separating and de-tecting. This is just a restatement of Yoneda lemma.
2. In Set, 1 = {โ} is both a separator and a detector since it represents theidentity functor Set โ Set. Similarly Z is both in Gp since it representsthe forgetful functor Gp โ Set.Dually, 2 = {0, 1} is a coseparator and a codetector in Set since it repre-sents ๐ โ โถ Setop โ Set.
3. In Top, 1 = {โ} is a separator since it represents the forgetful functorTop โ Set, but not a detector. In fact Top has no detecting set ofobjects: for any infinite cardinality ๐ , let ๐ be a discrete space of cardi-nality ๐ , and ๐ the same set with โco-๐ โ topology, i.e. ๐น โ ๐ closed if andonly if ๐น = ๐ or ๐น has cardinality smaller ๐ . The identity map ๐ โ ๐is continuous but not a homomorphism. So if {๐บ๐ โถ ๐ โ ๐ผ} is any set ofspaces, taking ๐ larger than cardinality of ๐บ๐ for all ๐ yields an exmapleto show that the set is not detecting.
14
2 The Yoneda lemma
4. Let C be the category of pointed conntected CW-complexes and homotopyclasses of (basepoint-preserving) continuous maps. J. H. C. Whiteheadproved that if ๐
๐โโ ๐ in this category induces isomorphisms ๐๐(๐) โ
๐๐(๐ ) for all ๐ then it is an isomorphism in C. This says that {๐๐ โถ ๐ โฅ 1}is a detecting set for C. But P. J. Freyd showed there is no faithful functorC โ Set, so no separating set: if {๐บ๐ โถ ๐ โ ๐ผ} were separating then
๐ โฆ โ๐โ๐ผ
C(๐บ๐, ๐)
would be faithful.
Note that any functor of the form C(๐ด, โ) preserves monos, but they donโtpreserve epis. We give a name to those special functors.
Definition (projective, injective). We say an object ๐ is projective if given
๐
๐ด ๐ต
๐๐
๐
there exists ๐๐โโ ๐ด with ๐๐ = ๐. If C is locally small, this says C(๐ , โ)
preserves epimorphisms.Dually an injective object of C is a projective object of Cop.Given a class โฐ of epimorphisms, we say ๐ is โฐ-projective if it satisfies
the condition for all ๐ โ โฐ.
Lemma 2.5. Representable functors are (pointwise) projectives in [C, Set].
Proof. Suppose givenC(๐ด, โ)
๐น ๐บ
๐ฝ๐พ
๐ผ
where ๐ผ is pointwise surjective. By Yoneda, ๐ฝ corresponds to some ๐ฆ โ ๐บ๐ด andwe can find ๐ฅ โ ๐น๐ด with ๐ผ๐ด(๐ฅ) = ๐ฆ. Now if ๐พ โถ C(๐ด, โ) โ ๐น corresponds to ๐ฅ,then naturality of the Yoneda bijection yields ๐ผ๐พ = ๐ฝ.
15
3 Adjunctions
3 Adjunctions
Definition (adjunction). Let C and D be two categories and ๐น โถ C โD, ๐บ โถ D โ C be two functors. By an adjunction between ๐น and ๐บ wemean a bijection between morphisms ๐ โถ ๐น๐ด โ ๐ต in D and ๐ โถ ๐ด โ ๐บ๐ต inC, which is natural in ๐ด and ๐ต, i.e. given ๐ดโฒ
๐โโ ๐ด and ๐ต
โโโ ๐ตโฒ, โ ๐(๐น๐) =
(๐บโ)๐๐ โถ ๐น๐ดโฒ โ ๐ตโฒ
๐น๐ด ๐ต
๐น๐ดโฒ ๐ตโฒ
๐
โ๐น๐
๐ด ๐บ๐ต
๐ดโฒ ๐บ๐ตโฒ
๐
๐บโ๐
We say ๐น is left adjoint to ๐บ and write ๐น โฃ ๐บ.
Example.
1. The functor ๐น โถ Set โ Gp is left adjoint to the forgetful functor ๐ โถGp โ Set since any function ๐ โถ ๐ด โ ๐๐ต extends uniquely to a grouphomomorphism ๐ โถ ๐น๐ด โ ๐ต and any homomorphism induces a set func-tion. Naturality in ๐ต is easy and naturality in ๐ด follows from the definitionof ๐น as a functor. Similar for Rng, Mod๐พ, โฆ
2. The forgetful functor ๐ โถ Top โ Set has a left adjoint ๐ท which equipsany set with the discrete topology and a right adjoint ๐ผ which equips anyset with the indiscrete topology so ๐ท โฃ ๐ โฃ ๐ผ.
3. The functor ob โถ Cat โ Set (recall that Cat is the category of smallcategories) has a left adjoint ๐ท sending ๐ด to the discrete category withob(๐ท๐ด) = ๐ด and only identity morphisms, and a right adjoint ๐ผ sending๐ด to the category with ob(๐ผ๐ด) = ๐ด and one morphism ๐ฅ โ ๐ฆ for each(๐ฅ, ๐ฆ) โ ๐ด ร ๐ด. In this case ๐ท in turn has a left adjoint ๐0 sending a smallcategory C to its set of connected components, i.e. the quotient of obCby the smallest equivalence relation identifying dom ๐ with cod ๐ for all๐ โ morC. So ๐0 โฃ ๐ท โฃ ob โฃ ๐ผ.
4. Let ๐ be the monoid {1, ๐} with ๐2 = ๐. An object of [๐, Set] is apair (๐ด, ๐) where ๐ โถ ๐ด โ ๐ด satisfying ๐2 = ๐. We have a functor ๐บ โถ[๐, Set] โ Set sending (๐ด, ๐) to
{๐ฅ โ ๐ด โถ ๐(๐ฅ) = ๐ฅ} = {๐(๐ฅ) โถ ๐ฅ โ ๐ด}
and a functor ๐น โถ Set โ [๐, Set] sending ๐ด to (๐ด, 1๐ด). Claim that
๐น โฃ ๐บ โฃ ๐น.
Given ๐ โถ (๐ด, 1๐ด) โ (๐ต, ๐), it must take values in ๐บ(๐ต, ๐) and any ๐ โถ(๐ต, ๐) โ (๐ด, 1๐ด) is determined by its values on the image of ๐. In someway this is due to the two ways in which the fixed point of ๐ can be written.
5. Let 1 be the discrete category with one object โ. For any C, there is aunique functor C โ 1. A left adjoint for this picks out an initial object of
16
3 Adjunctions
C, i.e. an object ๐ผ such that there exists a unique ๐ผ โ ๐ด for each ๐ด โ ob๐ถ.Dually a right adjoint for C โ 1 corresponds to a terminal object of C.
6. Let ๐ด๐โโ ๐ต be a morphism in Set. We can regard ๐๐ด and ๐๐ต as posets,
and we have functors ๐๐ โถ ๐๐ด โ ๐๐ต and ๐ โ๐ โถ ๐๐ต โ ๐๐ด. Claim๐๐ โฃ ๐ โ๐: we have ๐๐(๐ดโฒ) โ ๐ตโฒ if and only if ๐(๐ฅ) โ ๐ตโฒ for all ๐ฅ โ ๐ดโฒ,if and only if ๐ดโฒ โ ๐ โ๐(๐ตโฒ).
7. Galois connection: suppose givens sets ๐ด and ๐ต and a relation ๐ โ ๐ดร๐ต.We define mappings โ โ, โ ๐ between ๐๐ด and ๐๐ต by
๐๐ = {๐ฆ โ ๐ต โถ โ๐ฅ โ ๐, (๐ฅ, ๐ฆ) โ ๐ }, ๐ โ ๐ด๐ โ = {๐ฅ โ ๐ด โถ โ๐ฆ โ ๐ , (๐ฅ, ๐ฆ) โ ๐ }, ๐ โ ๐ต
These mappings are order-reversing, i.e. contravariant functors, and ๐ โ๐๐ if and only if ๐ ร ๐ โ ๐ , so by symmetry if and only if ๐ โ ๐ โ. Wesay โ ๐ and โ โ are adjoint on the right.
8. The functor ๐ โ โถ Setop โ Set is self-adjoint on the right since a func-tion ๐ด โ ๐๐ต corresponds bijectively to subsets of ๐ด ร ๐ต, and hence bysymmetry to functions ๐ต โ ๐๐ด.
Theorem 3.1. Let ๐บ โถ D โ C be a functor. Then specifying a left adjointfunctor for ๐บ is equivalent to specifying an initial object of (๐ด โ ๐บ) foreach ๐ด โ obC where (๐ด โ ๐บ) has objects pairs (๐ต, ๐) with ๐ด
๐โโ ๐บ๐ต and
morphisms (๐ต, ๐) โ (๐ตโฒ, ๐ โฒ) are morphisms ๐ต๐โโ ๐ตโฒ such that the following
diagram commutes๐ด ๐บ๐ต
๐บ๐ตโฒ
๐
๐โฒ๐บ๐
Proof. Suppose given ๐น โฃ ๐บ. Consider the morphism ๐๐ด โถ ๐ด โ ๐บ๐น๐ด corre-sponding to ๐น๐ด
1โโ ๐น๐ด. Then (๐น๐ด, ๐๐ด) is an object of (๐ด โ ๐บ). Moreover,
given ๐ โถ ๐น๐ด โ ๐ต and ๐ โถ ๐ด โ ๐บ๐ต, the diagram
๐ด ๐บ๐น๐ด
๐บ๐ต
๐๐ด
๐๐บ๐
commutes if and only if๐น๐ด ๐น๐ด
๐ต
1๐ด
๐๐
commutes by naturality condition in adjunction, i.e. ๐ = ๐. So (๐น๐ด, ๐๐ด) isinitial to (๐ด โ ๐บ).
17
3 Adjunctions
Conversely, suppose given an initial object (๐น๐ด, ๐๐ด) for each (๐ด โ ๐บ). Given๐ โถ ๐ด โ ๐ดโฒ, we define ๐น๐ โถ ๐น๐ด โ ๐น๐ดโฒ to be the unique morphism making
๐ด ๐บ๐น๐ด
๐ดโฒ ๐บ๐น๐ดโฒ
๐๐ด
๐ ๐บ๐น๐๐๐ดโฒ
commutes. Functoriality follows from uniqueness: given ๐ โฒ โถ ๐ดโฒ โ ๐ดโณ, both๐น(๐ โฒ๐) and (๐น๐ โฒ)(๐น๐) are both morphisms (๐น๐ด, ๐๐ด) โ (๐น๐ดโณ, ๐๐ดโณ๐ โฒ๐) in (๐ด โ๐บ). To show ๐น โฃ ๐บ: given ๐ด
๐โโ ๐บ๐ต, we define ๐ โถ ๐น๐ด โ ๐ต to be the unique
morphism (๐น๐ด, ๐๐ด) โ (๐ต, ๐) in (๐ด โ ๐บ). This is a bijection with inverse
(๐น๐ด๐โโ ๐ต) โฆ (๐ด
๐๐ดโโ ๐บ๐น๐ด
๐บ๐โโโ ๐บ๐ต).
The latter mapping is natural in ๐ต since ๐บ is a functor, and in ๐ด since byconstruction ๐ is a natural transformation 1C โ ๐บ๐น.
Corollary 3.2. If ๐น and ๐น โฒ are both left adjoint to ๐บ โถ D โ C then theyare naturally isomorphic.
Proof. It basically follows from the fact that initial object in any category, ifexists, is unique up to isomorphism. For any ๐ด, (๐น๐ด, ๐๐ด) and (๐น โฒ๐ด, ๐โฒ
๐ด) areboth initial in (๐ด โ ๐บ) so there is a unique isomorphism
๐ผ๐ด โถ (๐น๐ด, ๐๐ด) โ (๐น โฒ๐ด, ๐โฒ๐ด).
In any naturality square for ๐ผ, the two ways round are both morphisms in(๐ด โ ๐บ) whose domain is initial, so theyโre equal.
Lemma 3.3. GivenC D E
๐น
๐บ
๐ป
๐พ
with ๐น โฃ ๐บ and ๐ป โฃ ๐พ, we have
๐ป๐น โฃ ๐บ๐พ.
Proof. We have bijections between morphisms ๐ด โ ๐บ๐พ๐ถ, morphisms ๐น๐ด โ๐พ๐ถ and morphisms ๐ป๐น๐ด โ ๐ถ, which are both natural in ๐ด and ๐ถ.
Corollary 3.4. Given a commutative square
C D
E F
of categories and functors, if the functors all have left adjoints, then thediagram of left adjoints commutes up to natural isomorphsms.
18
3 Adjunctions
Proof. By Lemma 3.3, both ways round the diagram of left adjoints are leftadjoint to the composite C โ F, so by Theorem 3.1 they are isomorphic.
Actually, we didnโt use the full strength of Lemma 3.3: if we require merelyinstead that the original commutative diagram is only up to natural isomor-phism, then weโll get the same conclusion. In practice, however, the weakerversion stated above will usually suffice.
Definition (unit, counit). Given an adjunction ๐น โฃ ๐บ, the natural trans-formation ๐ โถ 1C โ ๐บ๐น emerging in the proof of Theorem 3.1 is called theunit of the adjunction.
Dually we have a natural transformation ๐ โถ ๐น๐บ โ 1D such that ๐๐ต โถ๐น๐บ๐ต โ ๐ต corresponds to ๐บ๐ต
1๐บ๐ตโโโ ๐บ๐ต, is called the counit.
Theorem 3.5. Given ๐น โถ C โ D, ๐บ โถ D โ C, specifying an adjunction๐น โฃ ๐บ is equivalent to specifying two natural transformations
๐ โถ 1C โ ๐บ๐น๐ โถ ๐น๐บ โ 1D
satisfying the commutative diagrams
๐น ๐น๐บ๐น
๐น
๐น๐
1๐น
๐๐น
๐บ ๐บ๐น๐บ
๐บ
๐๐บ
1๐บ๐บ๐
which are called the triangular identities.
Proof. First suppose given ๐น โฃ ๐บ. Define ๐ and ๐ as in Theorem 3.1 and itsdual. Now consider the composite
๐น๐ด๐น๐๐ดโโโ ๐น๐บ๐น๐ด
๐๐น๐ดโโโ ๐น๐ด.
Under the adjunction this corresponds to
๐ด๐๐ดโโ ๐บ๐น๐ด
1๐บ๐น๐ดโโโโ ๐บ๐น๐ด
but this also corresponds to 1๐น๐ด so ๐๐น๐ด๐น๐๐ด = 1๐น๐ด. The other identity is dual.Conversely, suppose ๐ and ๐ satisfying the trianglular identities. Given ๐ด
๐โโ
๐บ๐ต, let ฮฆ(๐) be the composite
๐น๐ด๐น๐โโ ๐น๐บ๐ต
๐๐ตโโ ๐ต
and given ๐น๐ด๐โโ ๐ต, let ฮจ(๐) be
๐ด๐๐ดโโ ๐บ๐น๐ด
๐บ๐โโโ ๐บ๐ต.
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3 Adjunctions
Then both ฮฆ and ฮจ are both natural. Need to show that ฮฆฮจ and ฮจฮฆ areidentity mappings. But
ฮจฮฆ(๐ด๐โโ ๐บ๐ต)
=๐ด๐๐ดโโ ๐บ๐น๐ด
๐บ๐น๐โโโโ ๐บ๐น๐บ๐ต
๐บ๐๐ตโโโ ๐บ๐ต
=๐ด๐โโ ๐บ๐ต
๐๐บ๐ตโโโ ๐บ๐น๐บ๐ต
๐บ๐๐ตโโโ ๐บ๐ต
=๐ด๐โโ ๐บ๐ต
where the second equality is naturality of ๐ and the third equality is triangularequation. Dually ฮฆฮจ(๐) = ๐.
Sometimes this is taken to be the definition of adjunction.Obviously two inverse functors form an adjunction. We have seen before a
weaker notion of inverse, namely a pair of functors forming an equivalence ofcategories. The question is, do they always from an adjunction? The answer isyes, but sometimes we canโt see it since weโve chosen the wrong isomorphism.
Lemma 3.6. Given functors ๐น โถ C โ D, ๐บ โถ D โ C and natural isomor-phisms
๐ผ โถ 1C โ ๐บ๐น๐ฝ โถ ๐น๐บ โ 1D
there are isomorphism
๐ผโฒ โถ 1C โ ๐บ๐น๐ฝโฒ โถ ๐น๐บ โ 1D
which satisfy the triangular identities so ๐น โฃ ๐บ and ๐บ โฃ ๐น.
This is often summarised as โevery equivalence is an adjoint equivalenceโ.
Proof. We fix ๐ผโฒ = ๐ผ and modify ๐ฝ. We have to change the domain andcodomain of ๐ฝ by conjugation.
Let ๐ฝโฒ be the composite
๐น๐บ(๐น๐บ๐ฝ)โ1
โโโโโโ ๐น๐บ๐น๐บ(๐น๐ผ๐บ)โ1
โโโโโโ ๐น๐บ๐ฝโโ 1D.
Note that ๐น๐บ๐ฝ = ๐ฝ๐น๐บ since
๐น๐บ๐น๐บ ๐น๐บ
๐น๐บ 1D
๐น๐บ๐ฝ
๐ฝ๐น๐บ ๐ฝ๐ฝ
commmutes by naturality of ๐ฝ and ๐ฝ is monic.
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3 Adjunctions
Now (๐ฝโฒ๐น)(๐น๐ผโฒ) is the composite
๐น๐น๐ผโโโ ๐น๐บ๐น
(๐ฝ๐น๐บ๐น)โ1
โโโโโโโ ๐น๐บ๐น๐บ๐น(๐น๐ผ๐บ๐น)โ1
โโโโโโโ ๐น๐บ๐น๐ฝ๐นโโ ๐น
=๐น(๐ฝ๐น)โ1
โโโโโ ๐น๐บ๐น๐น๐บ๐น๐ผโโโโโ ๐น๐บ๐น๐บ๐น
(๐น๐ผ๐บ๐น)โ1
โโโโโโโ ๐น๐บ๐น๐ฝ๐นโโ ๐น
=๐น(๐ฝ๐น)โ1
โโโโโ ๐น๐บ๐น๐ฝ๐นโโ ๐น
=1๐น
since ๐บ๐น๐ผ = ๐ผ๐บ๐น. Similarly (๐บ๐ฝโฒ)(๐ผโฒ๐บ) is
๐บ๐ผ๐บโโโ ๐บ๐น๐บ
(๐บ๐น๐บ๐ฝ)โ1
โโโโโโโ ๐บ๐น๐บ๐น๐บ(๐บ๐น๐ผ๐บ)โ1
โโโโโโโ ๐บ๐น๐บ๐บ๐ฝโโโ ๐บ
=๐บ(๐บ๐ฝ)โ1
โโโโโ ๐บ๐น๐บ๐ผ๐บ๐น๐บโโโโ ๐บ๐น๐บ๐น๐บ
(๐บ๐น๐ผ๐บ)โ1
โโโโโโโ ๐บ๐น๐บ๐บ๐ฝโโโ ๐บ
=๐บ(๐บ๐ฝ)โ1
โโโโโ ๐บ๐น๐บ๐บ๐ฝโโโ ๐บ
=1๐บ
Lemma 3.7. Suppose ๐บ โถ D โ C has a left adjoint ๐น with counit ๐ โถ ๐น๐บ โ1D, then
1. ๐บ is faithful if and only if ๐ is pointwise epic,
2. ๐บ is full and faithful is and only if ๐ is an isomorphism.
Proof.
1. Given ๐ต๐โโ ๐ตโฒ, ๐บ๐ corresponds under the adjunction to the composite
๐น๐บ๐ต๐๐ตโโ ๐ต
๐โโ ๐ตโฒ.
Hence the mapping ๐ โฆ ๐บ๐ is injective on morphisms with domain ๐ต (andspecified codomain) if and only if ๐ โฆ ๐๐๐ต is injective, if and only if ๐๐ต isepic.
2. Similarly, ๐บ is full and faithful if and only if ๐ โฆ ๐๐๐ต is bijective. If๐ผ โถ ๐ต โ ๐น๐บ๐ต is such that ๐ผ๐๐ต = 1๐น๐บ๐ต, i.e. ๐ผ is left inverse of ๐๐ต, then
๐๐ต๐ผ๐๐ต = ๐๐ต,
whence ๐๐ต๐ผ = 1๐ต. So ๐๐ต is an isomophism. Thus ๐ is an isomorphism.
Definition (reflection, reflective subcategory). By a reflection we mean anadjunction in which the right adjoint is full and faithful (equivalently thecounit is an isomorphism).
We say a full subcategory Cโฒ โ C is reflective if the inclusion Cโฒ โ Chas a left adjoint.
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3 Adjunctions
Example.
1. The category AbGp of abelian groups is reflective in Gp: the left adjointsends a group ๐บ to its abelianization ๐บ/๐บโฒ, where ๐บโฒ is the subgroupgenerated by all commutators [๐ฅ, ๐ฆ] = ๐ฅ๐ฆ๐ฅโ1๐ฆโ1 for ๐ฅ, ๐ฆ โ ๐บ. (The unitof the adjunction is the quotient map ๐บ โ ๐บ/๐บโฒ)This is where the name โreflecting subcategoryโ comes from: mirror is 2Dso reflection in a mirror cannot create a copy of a 3D object. However,it keeps every 2D detail fully and faithfully. Similarly abelianization doesnot tell you everything about ๐บ but as much as an abelian group can. c.f.universal property.
2. Given an abelian group, let ๐ด๐ denote the torsion subgroup, i.e. the sub-group of elements of finite orders. The assigment ๐ด โฆ ๐ด/๐ด๐ gives a leftadjoint to the inclusion tfAbGp โ AbGp, where tfAbGp is the fullsubcategory of torsion-free abelian groups.On the other hand ๐ด โฆ ๐ด๐ is the right adjoint to the inclusion tAbGp โAbGp from torsion abelian groups to abelain groups, so this subcate-gory is coreflective. There are many many examples in algebra involving(co)reflective subcategories, and the constructions all give rise to impor-tant universal properries.
3. Let KHaus โ Top be the full subcategory of compact Hausdorff spaces.The inclusion KHaus โ Top has a left adjoint ๐ฝ, the Stone-ฤech com-pactification. It is a reflective subcategory. Weโll revisit this example laterin the course.
4. Let ๐ be a topological space. We say ๐ด โ ๐ is sequentially closed if๐ฅ๐ โ ๐ฅโ and ๐ฅ๐ โ ๐ด for all ๐ implies ๐ฅโ โ ๐ด. Note that closedimplies sequentially closed but not vice versa. We say ๐ is sequentialif all sequentially closed sets are closed, e.g. a metric space. Given anon-sequential space ๐, let ๐๐ be the same set with topology given bythe sequentially open sets (complements of sequentially closed sets) in ๐.Certainly the identity ๐๐ โ ๐ is continuous, and defines the counit ofan adjunction between the inclusion Seq โ Top and its right adjoint๐ โฆ ๐๐ .
5. If ๐ is a topological space, the poset ๐ถ๐ of closed subsets of ๐ is reflectivein ๐๐ with reflector given by closure and the poset ๐๐ of open subsetsis coreflective with coreflector given by interior.
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4 Limits
4 Limits
Definition (diagram, cone, limit).
1. Let J be a category (almost always small, often finite). By a diagramof shape J in C we mean a functor ๐ท โถ J โ C. The objects ๐ท(๐) for๐ โ obJ are called vertices of the diagram and the morphisms ๐ท(๐ผ)๐ผ โ morJ are called edges of ๐ท.
2. Given ๐ท โถ J โ C, a cone over ๐ท consists of an object ๐ด of C, called
the apex of the cone, together with morphisms ๐ด๐๐โโ ๐ท(๐) for each
๐ โ obJ, called the legs of the cone, such that
๐ด
๐ท(๐) ๐ท(๐โฒ)
๐๐ ๐๐โฒ
๐ท(๐ผ)
commutes for all ๐๐ผโโ ๐โฒ in morJ.
Given cones (๐ด, (๐๐)๐โob J) and (๐ต, (๐๐)๐โob J), a morphism of cones
between them is a morphism ๐ด๐โโ ๐ต such that
๐ด ๐ต
๐ท(๐)๐๐
๐
๐๐
commutes for all ๐.We write Cone(๐ท) for the category of all cones over ๐ท.
3. A limit for ๐ท is a terminal object of Cone(๐ท), if this exists.Dually we have the notion of cone under a diagram (sometime calledcocone) and of colimit (i.e. initial cone under ๐ท).
For example, if J is the category
โ โ
โ โ
with 4 objects and 5 non-identity morphisms, a diagram of shape J is a com-mutative square
๐ด ๐ต
๐ถ ๐ท
๐
๐ โ
๐
On the other hand, to express a not-necessarily-commutative square, we use
23
4 Limits
the shapeโ โ
โ โ
An alternative way to understand limits is that, if C is locally small and Jis small, we have a functor Cop โ Set sending ๐ด to the set of cones with apex๐ด. A limit for ๐ท is a representation of this functor.
A thrid way to visualise cones: if ฮ๐ด denotes the constant diagram of shapeJ with all vertices ๐ด and all edges 1๐ด, then a cone over ๐ท with apex ๐ด is thesame thing as a natural transformation ฮ๐ด โ ๐ท. ฮ is a functor C โ [J, C]and Cone(๐ท) is the arrow category (ฮ โ ๐ท). So to say that every diagram ofshape J in C has a limit is equivalent to saying that ฮ has a right adjoint. (Wesay C has limits of shape J)
Dually C has colimits of shape J if and only if ฮ โถ C โ [J, C] has a leftadjoint.
Example.
1. Suppose ๐ฝ = โ . There is a unique diagram of shape J in C; a cone overit is just an object, and a morphism of cones is a morphism of C. So alimit for the empty diagram is a terminal object of C. We defined limitsin terms of terminal object but now the terminal object is also a speciallimit. Dually a colimit for it is an initial object.
2. Let J be the categoryโ โ
A diagram of shape J is a pair of objects ๐ด, ๐ต; a cone over it is a span
๐ถ
๐ด ๐ต
and a limit for it is a product
๐ด ร ๐ต
๐ด ๐ต
๐1 ๐2
as defined in example 4 on page 12. Dually a colimit for it is a coproduct.More generally, if J is a small discrete category, a diagram of shape J isan indexed family (๐ด๐ โถ ๐ โ J), and a limit for it is a product (โ๐โJ ๐ด๐
๐๐โโ
๐ด๐ โถ ๐ โ J). Dually, (๐ด๐
๐๐โโ โ๐โJ ๐ด๐ โถ ๐ โ J), sometimes also written as
โ๐โJ ๐ด๐.
3. Let J be the categoryโ โ
24
4 Limits
A diagram of shape J is a parallel pair
๐ด ๐ต๐
๐
a cone over it is๐ถ
๐ด ๐ต
โ ๐
satisfying ๐โ = ๐ = ๐โ, or equivalently a morphism โ โถ ๐ถ โ ๐ด satisfying๐โ = ๐โ. A (co)limit for the diagram is a (co)equaliser as defined inexample 5 on page 12.
4. Let J be the categoryโ
โ โ A diagram of shape J is a cospan
๐ด
๐ต ๐ถ
๐๐
a cone over it is๐ท ๐ด
๐ต ๐ถ
๐
๐ ๐
satisfying ๐๐ = ๐ = ๐๐, or equivalently a span (๐, ๐) completing the dia-gram to a commutative square. A limit for the diagram is called a pullbackof (๐, ๐). In Set, the apex of the pullback is the โfibre productโ
๐ด ร๐ถ ๐ต = {(๐ฅ, ๐ฆ) โถ ๐ด ร ๐ต โถ ๐(๐ฅ) = ๐(๐ฆ)}.
Dually, colimits of shape Jop are pushouts. Given
๐ด ๐ต
๐ถ
๐
๐
we โpush ๐ along ๐โ to get the RHS of the colimt square.
5. Let J be the poset of natural numbers. A diagram of shape J is a directedsystem
๐ด0๐0โโ ๐ด1
๐1โโ ๐ด2
๐2โโ ๐ด3
๐3โโ โฆ
A colimit for this is called a direct limit: it consists of ๐ดโ equipped withmorphisms ๐ด๐
๐๐โโ ๐ดโ satisfying ๐๐ = ๐๐+1๐๐ for all ๐ and universal
among such. Dually we have inverse system and inverse limit.
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4 Limits
Theorem 4.1.
1. Suppose C has equalisers and all finite (respectively small) products.Then C has all finite (respectively small) limits.
2. Suppose C has pullbacks and a terminal object, then C has all finitelimts.
Proof.
1. Suppose given ๐ท โถ J โ C. Form the industrial strength products
๐ = โ๐โob J
๐ท(๐), ๐ = โ๐ผโmor J
๐ท(cod๐ผ).
We have morphisms ๐, ๐ โถ ๐ โ ๐ defined by
๐๐ผ๐ = ๐cod ๐ผ, ๐๐ผ๐ = ๐ท(๐ผ)๐dom ๐ผ
for all ๐ผ. Let ๐ โถ ๐ธ โ ๐ be an equaliser of (๐, ๐). The composites
๐๐ = ๐๐๐ โถ ๐ธ โ ๐ท(๐)
form a cone over ๐ท: given ๐ผ โถ ๐ โ ๐โฒ in J,
๐ท(๐ผ)๐๐ = ๐ท(๐ผ)๐๐๐ = ๐๐ผ๐๐ = ๐๐ผ๐๐ = ๐๐โฒ๐ = ๐๐โฒ .
Given any cone (๐ด, (๐๐ โถ ๐ โ obJ)) over ๐ท, there is a unique ๐ โถ ๐ด โ ๐with ๐๐๐ = ๐๐ for each ๐ and
๐๐ผ๐๐ = ๐cod ๐ผ = ๐ท(๐ผ)๐dom ๐ผ = ๐๐ผ๐๐
for all ๐ผ, and hence ๐๐ = ๐๐, so exists unique ๐ โถ ๐ด โ ๐ธ with ๐๐ = ๐. So(๐ธ, (๐๐ โถ ๐ โ obJ)) is a limit cone.
2. It is enough to construct finite products and equalisers. But if 1 is theterminal object, then a pullback for
๐ด
๐ต 1
has the universal property of a product ๐ด ร ๐ต and we can form โ๐๐=1 ๐ด๐
inductively as
๐ด1 ร (๐ด2 ร (๐ด3 ร โฏ (๐ด๐โ1 ร ๐ด๐)) โฏ).
Now to form the equalisers of ๐, ๐ โถ ๐ด โ ๐ต, consider the cospan
๐ด
๐ด ๐ด ร ๐ต
(1๐ด,๐)(1๐ด,๐)
26
4 Limits
A cone over this consists of
๐ ๐ด
๐ด
โ
๐
satisfying (1๐ด, ๐)โ = (1๐ด, ๐)๐ or equivalently 1๐ดโ = 1๐ด๐ and ๐โ = ๐๐, orequivalently a morphism โ โถ ๐ โ ๐ด satisfying ๐โ = ๐โ. So a pullback for(1๐ด, ๐) and (1๐ด, ๐) is an equaliser of (๐, ๐).
Definition (complete, cocomplete). We say a category C is complete if ithas all small limits. Dually, C is cocomplete if has all small colimts.
For example, Set is both complete and cocomplete: products are cartesianproducts and coproducts are disjoint unions. Similarly Gp, AbGp, Rng, Mod๐พare all complete and cocomplete.1 Top is also complete and cocomplete, withboth product and coproduct given by the underlying set.
Definition (preserve limit, reflect limit, create limit). Let ๐น โถ C โ D be afunctor.
1. We say ๐น preserves limits of shape J if, given ๐ท โถ J โ C and a limitcone (๐ฟ, (๐๐ โถ ๐ โ obJ)) in C, (๐น๐ฟ, (๐น๐๐ โถ ๐ โ obJ)) is a limit for ๐น๐ท.
2. We say ๐น reflects limits of shape J if, given ๐ท โถ J โ C and a cone(๐ฟ, (๐๐ โถ ๐ โ obJ)) such that (๐น๐ฟ, (๐น๐๐ โถ ๐ โ obJ)) is a limit for ๐น๐ทthen (๐ฟ, (๐๐ โถ ๐ โ obJ)) for ๐ท.
3. We say ๐น creates limts of shape J if, given ๐ท โถ J โ C and a limit(๐, (๐๐ โถ ๐ โ obJ)) for ๐น๐ท, there exists a cone (๐ฟ, (๐๐ โถ ๐ โ obJ))over ๐ท whose image under ๐น is isomorphic to the limit cone, and anysuch cone is a limit for ๐ท.
Remark.
1. If C has limits of shape J and ๐น โถ C โ D preserves them and reflectsisomorphisms then ๐น reflects limits of shape J.
2. ๐น reflects limits of shape 1 if and only if ๐น reflects isomorphism.
3. If D has limits of shape J and ๐น โถ C โ D creates them, then ๐น bothpreserves and reflects them.
4. In any of the statement of Theorem 4.1, we may replace both instancesof โC hasโ by either โC has and ๐น โถ C โ D preservesโ or โD has and๐น โถ C โ D createsโ.
Example.1Note that the products have underlying set the cartesian products of those of each com-
ponent, but coproducts tend to be different. Weโll discuss this later.
27
4 Limits
1. ๐ โถ Gp โ Set creates all small limits: given a family (๐บ๐ โถ ๐ โ ๐ผ)of groups, there is a unique group structure on โ๐โ๐ผ ๐๐บ๐ making theprojections homomorphisms, and this makes it a product in Gp. Similarlyfor equalisers.But ๐ doesnโt preserve coproducts: ๐(๐บ โ ๐ป) โ ๐๐บ โจฟ ๐๐ป.
2. ๐ โถ Top โ Set preserves all small limits and colimits but doesnโt reflectthem: if ๐ฟ is a limit for ๐ท โถ J โ Top and ๐ฟ is not discrete, there isanother cone with apex ๐ฟ๐, which is the same underlying space as ๐ฟ withdiscrete topology, mapped to the same limit in Set.
3. The inclusion functor ๐ผ โถ AbGp โ Gp reflects coproducts, but doesnโtpreserve them. The direct sum ๐ดโ๐ต (coproduct in AbGp) is not normallyisomorphic to the free product ๐ด โ ๐ต, which is not abelian unless either ๐ดor ๐ต is trivial, but if ๐ด โ {๐} then ๐ด ร ๐ต โ ๐ด โ ๐ต โ ๐ต.
The following lemma tells us how to construct limit in functor categories:
Lemma 4.2. If D has limits of shape J then so does the functor category[C, D] for any C, and the forgetful functor [C, D] โ Dob C creates them.
Proof. Suppose given a diagram of shape J in [C, D]. Think of it as a functor๐ท โถ J ร C โ D. For each ๐ด โ obC, let (๐ฟ๐ด, (๐๐,๐ด โถ ๐ โ obJ)) be a limit conefor the diagram ๐ท(โ, ๐ด) โถ J โ D.
Given ๐ด๐โโ ๐ต in C, the composition
๐ฟ๐ด๐๐,๐ดโโโ ๐ท(๐, ๐ด)
๐ท(๐,๐)โโโโโ ๐ท(๐, ๐ต)
form a cone over D(โ, ๐ต), since the square
๐ท(๐, ๐ด) ๐ท(๐, ๐ต)
๐ท(๐โฒ, ๐ด) ๐ท(๐โฒ, ๐ต)
๐ท(๐,๐)
๐ท(๐ผ,๐ด) ๐ท(๐ผ,๐ต)๐ท(๐โฒ,๐)
commutes. So there is a unique ๐ฟ๐ โถ ๐ฟ๐ด โ ๐ฟ๐ต making
๐ฟ๐ด ๐ท(๐, ๐ด)
๐ฟ๐ต ๐ท(๐, ๐ต)
๐๐,๐ด
๐ฟ๐ ๐ท(๐,๐)๐๐,๐ต
commute for all ๐. Uniqueness follows from functoriality: given ๐ โถ ๐ต โ ๐ถ,๐ฟ(๐๐) and ๐ฟ(๐)๐ฟ(๐) are factorisations of the same cone through the limit ๐ฟ๐ถ.And this is the unique functor structure on ๐ด โฆ ๐ฟ๐ด making the ๐๐,โ intonatural transformations.
The cone (๐ฟ, (๐๐,โ โถ ๐ โ obJ)) is a limit: suppose given another cone(๐, (๐๐,โ โถ ๐ โ obJ)), then for each ๐ด, (๐๐ด, (๐๐,๐ด โถ ๐ โ obJ)) is a coneover D(โ, ๐ด), so induces a unique ๐ผ๐ด โถ ๐๐ด โ ๐ฟ๐ด. Naturality of ๐ผ followsfrom uniqueness of factorisation through a limit. So (๐, (๐๐ฝ)) factors uniquelythrough (๐ฟ, (๐๐)). (This is creation is the strict sense, i.e. equality instead ofisomorphism)
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4 Limits
Remark. In any category, a morphism ๐ด๐โโ ๐ต is monic if and only if
๐ด ๐ด
๐ด ๐ต
1๐ด
1๐ด ๐๐
is a pullback. Hence any functor which preserves pullbacks preserves monomor-phisms. In particular if D has pullbacks momomorphisms in [C, D] are justpointwise monos. See example sheet 1 for a counterexample for the necces-sity of the pullback condition. Now we can delete the word โpointwiseโ in thestatement of Lemma 2.5.
Theorem 4.3 (right adjoint preserves limits). Suppose ๐บ โถ D โ C has aleft adjoint. Then ๐บ preserves all limits which exist in D.
Proof 1 with additional assumption. Suppose C and D both have limits of shapeJ. We have a commutative diagram
C D
[J, C] [J, D]
๐น
ฮ ฮ[J,๐น ]
where ฮ sends an object to its constant diagram and [J, ๐น ] is composition with๐น. All functors in it have right adjoints (in particular, [J, ๐น ] โฃ [J, ๐บ]). So byCorollary 3.4 the diagram of right adjoints
D C
[J, D] [J, C]
๐บ
limJ
[J,๐บ]
limJ
commutes up to isomorphism, i.e. ๐บ preserves limits of shape J.
Proof 2. Suppose given ๐ท โถ J โ D and a limit cone (๐ฟ, (๐๐ โถ ๐ฟ โ ๐ท(๐) โถ๐ โ obJ)). Given a cone (๐ด, (๐ผ๐ โถ ๐ด โ ๐บ๐ท(๐) โถ ๐ โ obJ)) over ๐บ๐ท, the
morphisms ๐น๐ด๏ฟฝ๏ฟฝ๐โโ ๐ท(๐) form a cone over ๐ท, so they induce a unique ๐น๐ด
๐ฝโโ ๐ฟ
such that ๐๐๐ฝ = ๐ผ๐ for all ๐. Then ๐ด
๐ฝโโ ๐บ๐ฟ is the unique morphism satisfying
(๐บ๐๐)๐ฝ = ๐ผ๐ for all ๐. So (๐บ๐ฟ, (๐บ๐๐ โถ ๐ โ obJ)) is a limit cone in C.
The last major theorem in this chapter is adjoint functor theorem. It saysthat morally the converse of the above theorem is also true, i.e. a functor pre-serving all limits ought to have a left adjoint. It may only fail so if some limitsdo not exist. The โprimevalโ adjoint functor theorem is exactly this: if D hasand ๐บ โถ D โ C preserves all limits, then ๐บ has a left adjoint. However, this istoo strong a condition as the categories having all limits can be shown to be pre-orders. Thus there are two more versions cut down on the all limits requirementand use some set theory to replace part of it.
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4 Limits
Lemma 4.4. Suppose D has and ๐บ โถ D โ C preserves limits of shape J.Then for any ๐ด โ obC the arrow category (๐ด โ ๐บ) has limits of shape J,and the forgetful functor ๐ โถ (๐ด โ ๐บ) โ D creates them.
Proof. Suppose given ๐ท โถ J โ (๐ด โ ๐บ). Write ๐ท(๐) as (๐๐ท(๐), ๐๐). Let(๐ฟ, (๐๐ โถ ๐ฟ โ ๐๐ท(๐))๐โob J) be a limit for ๐๐ท. Then (๐บ๐ฟ, (๐๐)๐โob J) is a limitfor ๐บ๐๐ท. Since the edges of ๐๐ท are morphisms in (๐ด โ ๐บ), the ๐๐ form a coneover ๐บ๐๐ท so there is a unique โ โถ ๐ด โ ๐บ๐ฟ such that (๐บ๐๐)โ = ๐๐ for all ๐, i.e.thereโs a unique โ such that the ๐๐โs are all morphisms in (๐ฟ, โ) โ (๐๐ท(๐), ๐๐)in (๐ด โ ๐บ). We need to show that ((๐ฟ, โ), (๐๐)๐โob J) is a limit cone in (๐ด โ ๐บ).If (๐ถ, (๐๐)๐โob J) is any cone over D then (๐ถ, (๐๐)๐โob J) is a cone over ๐๐ท sother is a unique โ โถ ๐ถ โ ๐ฟ with ๐๐โ = ๐๐ for all ๐. We need to show (๐บโ)๐ = โ.But
(๐บ๐๐)(๐บโ)๐ = (๐บ๐๐)๐ = ๐๐ = (๐บ๐๐)โ
for all ๐ so (๐บโ)๐ = โ by uniqueness of factorisations through limits.
Recall that we have seen a limit for the empty diagram is a terminal object.We can also consider dually the diagram of โmaximal sizeโ, namely that of acategory over itself, to realise inital object as a limit. Think for example posets,in which limit for an empty diagram is maximimum while limit for the
Lemma 4.5. A category C has an initial object if and only if 1C โถ C โ C,regarded as a diagram of shape C in C, has a limit.
Note that this is an exception to J being small.
Proof. First suppose C has an initial object ๐ผ. Then the unique morphisms(๐ผ โ ๐ด โถ ๐ด โ obC) form a cone over 1C and given any cone (๐๐ด โถ ๐ถ โ ๐ด โถ ๐ด โobC), for any ๐ด the triangle
๐ถ ๐ผ
๐ด
๐๐ผ
๐๐ด
commutes so ๐๐ผ is the unique factorisation of (๐๐ด โถ ๐ด โ obC) through (๐ผ โ ๐ด โถ๐ด โ obC).
Conversely, suppose (๐ผ, (๐๐ด โถ ๐ผ โ ๐ด)๐ดโob C) is a limit. Then for any ๐ โถ ๐ผ โ๐ด the diagram
๐ผ ๐ผ
๐ด
๐๐ผ
๐๐ด๐
commutes. ๐ผ is weakly initial as we donโt know if it is unique, i.e. if ๐๐ผ = 1๐ผ. Inparticular, putting ๐ = ๐๐ด, we see that ๐๐ผ is a factorisation of the limit conethrough itself so ๐๐ผ = 1๐ผ. Hence every ๐ โถ ๐ผ โ ๐ด satisfies ๐ = ๐๐ด.
The primeval adjoint functor theorem follows immediately from the previoustwo lemmas and 3.3. However, it only applies to functors between preorders.See example sheet 2 Q6.
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4 Limits
Theorem 4.6 (general adjoint functor theorem). Suppose D is locally smalland complete. Then ๐บ โถ D โ C has a left adjoint if and only if ๐บ preservesall small limits and satisfies the solution set condition, which says that foreach ๐ด โ obC, there exists a set of morphisms {๐๐ โถ ๐ด โ ๐บ๐ต๐ โถ ๐ โ ๐ผ} suchthat every โ โถ ๐ด โ ๐บ๐ถ factors as
๐ด๐๐โโ ๐บ๐ต๐
๐บ๐โโโ ๐บ๐ถ
for some ๐ and some ๐ โถ ๐ต๐ โ ๐ถ.
Proof. If ๐น โฃ ๐บ then ๐บ preserves limits. To obtain the solution set, note that{๐๐ด โถ ๐ด โ ๐บ๐น๐ด} is a singleton solution set by 3.3 since it is initial.
Conversely, by 4.10 (๐ด โ ๐บ) is complete and it inherits local smallness fromD. We need to show that if ๐ is complete and locally small and has a weaklyinitial set of objects {๐ต๐ โถ ๐ โ ๐ผ} then ๐ has an initial object. First form๐ = โ๐โ๐ผ ๐ต๐; then ๐ is weakly initial. Now form the limit of
๐ ๐โฎ
where edges are all the endomorphisms of ๐. Denote it ๐ โถ ๐ผ โ ๐. ๐ผ is alsoweakly initial in ๐. Suppose given ๐, ๐ โถ ๐ผ โ ๐ถ. Form the equaliser ๐ โถ ๐ธ โ ๐ผof (๐, ๐). Then there exists โ โถ ๐ โ ๐ธ since ๐ is weakly initial. ๐๐โ โถ ๐ โ ๐and 1๐ are edegs of the diagram so ๐ = ๐๐โ๐. But ๐ is monic so ๐โ๐ = 1๐ผ so ๐ issplit epic so ๐ = ๐. Thus ๐ผ is initial.
Example.
1. This example comes from Mac Lane. He asked the question that, if we arenot given the free group functor, how can we to construct the left adjointof forgetful functor? Consider the forgetful functor ๐ โถ Gp โ Set. By4.6a ๐ creates all small limits so Gp has them and ๐ preserves them. Gpis locally smal, given a set ๐ด, any ๐ โถ ๐ด โ ๐๐บ factors as
๐ด โ ๐๐บโฒ โ ๐๐บ
where ๐บโฒ is the subgroup generated by {๐(๐ฅ) โถ ๐ฅ โ ๐ด} and card๐บโฒ โคmax{โต0, card๐ด}. Let ๐ต be a set of this cardinality. Consider all subsets๐ตโฒ โ ๐ต, all group structures on ๐ตโฒ and all mappings ๐ด โ ๐ตโฒ. These giveus a solution set at ๐ด.
2. Consider the category CLat of complete lattices (posets with arbitrarymeets and joints). Again the forgetful functor ๐ โถ CLat โ Set creates allsmall limits. But A. W. Hales showed in 1964 that for any cardinal ๐ thereexists complete lattices of cardinality โฅ ๐ generated by three elements. Sothe solution set condition fails at ๐ด = {๐ฅ, ๐ฆ, ๐ง} as we cannot bound thecartinality of the solution set, and ๐ doesnโt have a left adjoint.
The general adjoint functor theorem is general in the sense that it appliesto all categories, although it imposes solution set condition, which is a ratherstrong condition on the functor. Special adjoint functor theorem aims to get ridof the condition on the functor.
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4 Limits
Definition (subobject, quotient object). By a subobject of an object ๐ด ofC we mean a monomorphism ๐ดโฒ โฃ ๐ด. The subobjects of ๐ด are preorderedby ๐ดโณ โค ๐ดโฒ if there is a factorisation
๐ดโณ ๐ดโฒ
๐ด
Dually we have quotient objects.
Definition (well-powered). We say C is well-powered if each ๐ด โ obC hasa set of subobjects {๐ด๐ โฃ ๐ด โถ ๐ โ ๐ผ} such that every subobject of ๐ด isisomorphic to some ๐ด๐.
Dually if Cop is well-powered, we say C is well-copowered. (not cowell-powered, as it implies badly powered. It also sounds like something beingpowered by Simon Cowell, which is not quite what we study in this course)
For example in Set we can take inclusions {๐ดโฒ โช ๐ด โถ ๐ดโฒ โ ๐๐ด}. This isalso where the name โwell-poweredโ comes from as in the Set case it simplymeans power set exists.
Before stating and proving the special functor theorem we point out a simpleyet powerful observation about pullback square.
Lemma 4.7. Given a pullback square
๐ ๐ด
๐ต ๐ถ
โ
๐ ๐๐
with ๐ monic. Then ๐ is monic.
Proof. Suppose ๐ฅ, ๐ฆ โถ ๐ท โ ๐ satisfy ๐๐ฅ = ๐๐ฆ. Then
๐โ๐ฅ = ๐๐๐ฅ = ๐๐๐ฆ = ๐โ๐ฆ.
By ๐ is monic so โ๐ฅ = โ๐ฆ. So ๐ฅ, ๐ฆ are factorisations of the same cone throughthe limit cone (โ, ๐).
Theorem 4.8 (special adjoint functor theorem). Suppose C and D are bothlocally small, and that D is complete and well-powered and has a coseparatingset. Then a functor ๐บ โถ D โ C has a left adjoint if and only if it preservesall small limits.
Proof. The only if is given by right adjoint preserves limits. For the other direc-tion, for any ๐ด โ obC, (๐ด โ ๐บ) is locally complete by 4.10, locally small, andwell-powered since the subobjects of (๐ต, ๐) in (๐ด โ ๐บ) are just those subobjects๐ตโฒ โฃ ๐ต in D for which ๐ factors through ๐บ๐ตโฒ โฃ ๐บ๐ต. Also if {๐๐ โถ ๐ โ ๐ผ} is acoseparating set for D then the set
{(๐๐, ๐) โถ ๐ โ ๐ผ, ๐ โ C(๐ด, ๐บ๐๐)}
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4 Limits
is coseparating in (๐ด โ ๐บ): given ๐, โ โถ (๐ต, ๐) โ (๐ตโฒ, ๐ โฒ) in (๐ด โ ๐บ) with ๐ โ โ,there exists ๐ โถ ๐ตโฒ โ ๐๐ for some ๐ with ๐๐ โ ๐โ, and then ๐ is also a morphism(๐ตโฒ, ๐ โฒ) โ (๐๐, (๐บ๐)๐ โฒ) in (๐ด โ ๐บ).
So we need to show that if ๐ is complete, locally small and well-poweredand has a coseparating set {๐๐ โถ ๐ โ ๐ผ} then ๐ has an initial object. Form theproduct ๐ = โ๐ โ ๐๐. We have the industrial stength product and now we needthe industrial strength pullback. Consider the diagram
๐๐ ๐๐
โฎ
๐ โฒ ๐
whose edges are representative set of subobjects of ๐ and form its limit
๐ผ ๐๐
๐๐
โฆ
๐ โฒ
By the argument in the previous lemma, the legs of the cones are all monic;in particular ๐ผ โฃ ๐ is monic, and itโs a least subobject of ๐. Hence ๐ผ has noproper subobjects. So given ๐, ๐ โถ ๐ผ โ ๐ด their equaliser is an isomorphism andhence ๐ = ๐.
We get uniqueness for free but need to work harder to show existence. Nowlet ๐ด be any object of ๐. Form the product ๐ = โ๐โ๐ผ,๐โ๐(๐ด,๐๐) ๐๐. There is anobvious โ โถ ๐ด โ ๐ defined by ๐๐,๐นโ = ๐; and โ is monic, since the ๐๐โs are acoseparating set. We also have a morphism ๐ โถ ๐ โ ๐ defined by ๐๐,๐๐ = ๐๐.Now form the pullback
๐ต ๐ด
๐ ๐
โ
๐
by lemma ๐ is monic so ๐ต is a subobject of ๐. Hence there exists
๐ผ ๐ต
๐
and hence a morphism ๐ผ โ ๐ต โ ๐ด.
This result is due to Freyd, who first published it in a book as an exerciseto the readers (!).
Example. Consider the inclusion ๐ผ โถ KHaus โ Top, where KHaus is thefull subcategory of compact Hausdorff spaces. KHaus has and ๐ผ preservessmall products (by Tychonoffโs theorem) and equalisers (since equalisers of pairs
33
4 Limits
๐, ๐ โถ ๐ โ ๐ with ๐ Hausdorff are closed subspaces). Both cateogories arelocally small and KHaus is well-powered (subobjects of ๐ are isomorphic toclosed subspaces). The closed interval [0, 1] is a coseparateor in KHaus byUrysohnโs lemma. So by special adjoint functor theorem ๐ผ has a left adjoint ๐ฝ,the Stone-ฤech compactification.
Remark.
1. ฤechโs construction of ๐ฝ is as follow: given ๐, form
๐ = โ๐โถ๐โ[0,1]
[0, 1]
and define โ โถ ๐ โ ๐ by ๐๐โ = ๐. Define ๐ฝ๐ to be the closure of theimage of โฤechโs proof that this works is essentially the same as SAFT.
2. We could have used GAFT to construct ๐ฝ by a cardinality argument: weget a solution set at ๐ by considering all continuous ๐ โถ ๐ โ ๐ with๐ compact Hausdorff and ๐(๐) dense in ๐ and such ๐ have cardinalityโค 22card ๐ .
34
5 Monad
5 MonadThe idea of a monad is what is left in an adjunction when you cannot see oneof the categories. Suppose given ๐ โถ C โ D, ๐ โถ D โ C with ๐น โฃ ๐บ. How muchof this structure can we describe without mentioning D? We have
1. the functor ๐ = ๐บ๐น โถ C โ C,
2. the unit ๐ โถ 1C โ ๐ = ๐บ๐น,
3. and โshadowโ of counit as a natural transformation ๐ = ๐บ๐๐น โถ ๐ ๐ =๐บ๐น๐บ๐น โ ๐บ๐น = ๐
satisfying the commutative diagrams 1, 2
๐ ๐ ๐ ๐
๐
๐ ๐
1๐
๐
๐๐
1๐
by the triangle inequalities, and 3
๐ ๐ ๐ ๐ ๐
๐ ๐ ๐
๐ ๐
๐๐ ๐
๐
by naturality of ๐.
Definition (monad). A monad T = (๐ , ๐, ๐) on a category C consists of afunctor ๐ โถ C โ C and natural transofrmations ๐ โถ 1C โ ๐ , ๐ โถ ๐ ๐ โ ๐satisfying commutative diagrams 1 - 3.
๐ and ๐ are called the unit and multiplication of T.
The name โmonadโ was used because of the similarity of axioms of monadwith those of monoid. Before that, although being well-known to mathemati-cians, monad didnโt really have an identifier. It goes by the name โstandardconstructionโ, then โtripleโ, thereby the letter T. But both are confessionsof failure to come up with a meaningful name! Someone invented the nameโmonadโ and Mac Lane popularised it.
Example.
1. Any adjunction ๐น โฃ ๐บ induces a monad (๐บ๐น , ๐, ๐บ๐๐น) on C and a comonad(๐น๐บ, ๐, ๐น๐๐บ) on D.
2. Let ๐ be a monoid. The functor (๐ รโ) โถ Set โ Set has a monad struc-ture with unit given by ๐๐ด(๐) = (๐๐, ๐) and multiplication ๐๐ด(๐, ๐โฒ, ๐) =(๐๐โฒ, ๐). The monad identities follow from the monoid ones.
3. Let C be any category with finite products and ๐ด โ obC. The functor(๐ด ร โ) โถ C โ C has a comonad structure with counit ๐๐ต โถ ๐ด ร ๐ต โ ๐ตgiven by ๐2 and comultiplication ๐ฟ๐ต โถ ๐ด ร ๐ต โ ๐ด ร ๐ด ร ๐ต given by(๐1, ๐1, ๐2).
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5 Monad
Does every monad comes from an adjunction? The answer is yes and is givenindependently in 1965 by Eilenberg, Moore and Kleisli. We will cover both.
In example 2 above we have the cateogry [๐, Set]. Its forgetful functor toSet has a left adjoint, sending ๐ด to ๐ ร ๐ด with ๐ acting by multiplication onthe left factor. This adjunction gives rise to the monad.
Definition (Eilenberg-Moore algebra). Let T be a monad on C. A T-algebra is a pair (๐ด, ๐ผ) with ๐ด โ obC and ๐ผ โถ ๐ ๐ด โ ๐ด satisfying commu-tative diagrams 4, 5
๐ด ๐ ๐ด
๐ด
๐๐ด
1๐ด๐ผ
๐ ๐ ๐ด ๐ ๐ด
๐ ๐ด ๐ด
๐ ๐ผ
๐๐ด ๐ผ
๐ผ
A homomorphism ๐ โถ (๐ด, ๐ผ) โ (๐ต, ๐ฝ) is a morphism ๐ โถ ๐ด โ ๐ต suchthat diagram 6
๐ ๐ด ๐ ๐ต
๐ด ๐ต
๐ ๐
๐ผ ๐ฝ๐
commutes.The category of T-algebras is denoted CT.
Lemma 5.1. The forgetful functor ๐บT โถ CT โ C has a left-adjoint ๐นT andthe adjunction induces T.
Proof. We define a โfreeโ T-algebra functor, mimicking that in monoid case.Define ๐นT๐ด = (๐ ๐ด, ๐๐ด) (an algebra by 2, 3) and ๐นT(๐ด
๐โโ ๐ต) = ๐ ๐ (a homo-
morphism by naturality of ๐). Clearly ๐บT๐นT = ๐, the unit of the adjunction is๐. We define the counit ๐(๐ด,๐ผ) = ๐ผ โถ (๐ ๐ด, ๐๐ด) โ (๐ด, ๐ผ) (a homomorphism by5) and is natural by 6. The triangle identities
๐๐น๐ด(๐น๐๐ด) = 1๐น๐ด
๐บ๐(๐ด,๐ผ)๐๐ด = 1๐ด
are given by 1 and 4. Finally, the monad induced by ๐นT โฃ ๐บT has functor ๐and unit ๐, and
๐บT๐๐นT๐ด = ๐๐ด
by definition of ๐นT๐ด.
Kleisli took a โminimalistโ approach: if ๐น โถ C โ D, ๐บ โถ D โ C induces Tthen so does ๐น โถ C โ Dโฒ, ๐บ1Dโฒ โถ Dโฒ โ C where Dโฒ is the full subcategory ofD on objects ๐น๐ด. So in trying to construct D, we may assume ๐น is surjective(or indeed bijective) on objects. But then morphisms ๐น๐ด โ ๐น๐ต correspondbijectively to morphisms ๐ด โ ๐บ๐น๐ต = ๐ ๐ต in C. This leads us half way throughas we sitll have to specify how to compose morphisms (in general domains andcodomains donโt match up).
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5 Monad
Definition (Kleisli category). Given a monad T on C, the Kleisli cat-egory CT has obCT = obC and morphisms ๐ด ๐ต (we use bluearrow to signify morphism in CT) are ๐ด โ ๐ ๐ต in C. The composite๐ด ๐ต ๐ถ๐ ๐ is
๐ด ๐ ๐ต ๐ ๐ ๐ถ ๐ถ๐ ๐ ๐ ๐๐ถ
and the identity ๐ด ๐ด is ๐ด๐๐ดโโ ๐ ๐ด.
To verify associativity, suppose given ๐ด ๐ต ๐ถ ๐ท๐ ๐ โ then
๐ด ๐ ๐ต ๐ ๐ ๐ถ ๐ ๐ ๐ ๐ท ๐ ๐ ๐ท
๐ ๐ถ ๐ ๐ ๐ท ๐ ๐ท
๐ ๐๐ ๐ ๐ โ
๐๐ถ
๐ ๐๐ท
๐๐๐ท ๐๐ท
๐ โ ๐๐ท
commutes: the upper way monad is (โ๐)๐ and the lower is โ(๐๐). (used diagram3 in rightmost square)
The unit laws similarly follow from
๐ด ๐ ๐ต ๐ ๐ ๐ต
๐ ๐ต
๐ ๐ ๐๐ต
1๐๐ต๐๐ต
๐ด ๐ ๐ต
๐ ๐ด ๐ ๐ ๐ต ๐ ๐ต
๐
๐๐ด1๐๐ต1๐๐ต
๐ ๐ ๐๐ต
(used diagram 1 and 2 respecitively in the triangles)
Lemma 5.2. There exists an adjunction ๐นT โถ C โ CT, ๐บT โถ C๐ โ Cinducing the monad T.
Proof. We define ๐นT๐ด = ๐ด, ๐นT(๐ด๐โโ ๐ต) = ๐ด
๐โโ ๐ต
๐๐ตโโ ๐ ๐ต. ๐นT preserves
identities by definition. For composites, consider ๐ด๐โโ ๐ต
๐โโ ๐ถ, we get
๐ด ๐ต ๐ ๐ต
๐ถ ๐ ๐ถ ๐ ๐ ๐ถ
๐ ๐ถ
๐ ๐๐ต
๐ ๐ ๐๐๐ถ ๐ ๐๐ถ
1๐๐ถ
๐๐ถ
(used diagram 1 in the triangle)We define
๐บT๐ด = ๐ ๐ด
๐บT(๐ด๐โโ๐ต) = ๐ ๐ด
๐ ๐โโ ๐ ๐ ๐ต
๐๐ตโโโ ๐ ๐ต
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5 Monad
๐บT preserves identities by diagram 1. For composite, consider ๐ด ๐ต ๐ถ๐ ๐
, we get๐ ๐ด ๐ ๐ ๐ต ๐ ๐ ๐ ๐ถ ๐ ๐ ๐ถ
๐ ๐ต ๐ ๐ ๐ถ ๐ ๐ถ
๐ ๐ ๐ ๐ ๐
๐๐ต
๐ ๐๐ถ
๐๐๐ถ ๐๐ถ
๐ ๐ ๐๐ถ
(used diagram 3 is last square)Have
๐บT๐นT๐ด = ๐ ๐ด๐บT๐T๐ = ๐๐ต(๐ ๐๐ต)๐ ๐ = ๐ ๐น
so we take ๐ โถ 1C โ ๐ as the unit of ๐นT โฃ ๐บT. The counit ๐ ๐ด ๐ด๐๐ด is1๐ ๐ด. To verify naturality consider the square
๐ ๐ด ๐ ๐ต
๐ด ๐ต
๐นT๐บT๐
๐๐ด ๐๐ต
๐
This expands to
๐ ๐ด ๐ ๐ ๐ต ๐ ๐ต ๐ ๐ ๐ต
๐ ๐ต
๐ ๐ ๐๐ต ๐๐๐ต
1๐๐ต๐๐ต
(used diagram 2 in triangle) so ๐ is natural.Finally, ๐บT(๐ ๐ด
๐๐ดโโ๐ด) = ๐๐ด so
๐บT(๐๐ด)๐๐บT๐ด = ๐๐ด๐๐ ๐ด = 1๐ ๐ด
and (๐๐นT๐ด)(๐นT๐๐ด) is๐ด ๐ ๐ด ๐ ๐ ๐ด
๐ ๐ด
๐๐ด ๐๐๐ด
1๐๐ด๐๐ด
(used diagram 1 in triangle) which is (1๐นT๐ด). Also ๐บT(๐๐นT๐ด) = ๐๐ด so ๐นT โฃ ๐บTinduces T.
Theorem 5.3. Given a monad T on C, let Adj(T) be the category whoseobjects are the adjunctions ๐น โถ C โ D, ๐บ โถ D โ D inducing T, and whosemorphisms
(C D) (C Dโฒ)๐น
๐บ
๐น โฒ
๐บโฒ
are functors ๐ป โถ D โ Dโฒ satisfying ๐ป๐น = ๐น โฒ and ๐บโฒ๐ป = ๐บ. Thenthe Kleisli adjunction is an initial object of Adj(T) and Eilenberg-Mooreadjunction is terminal.
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5 Monad
Proof. Let ๐น๐บ be an object of Adj(T). We define the Eilenberg-Moore com-parison functor ๐พ โถ D โ CT by ๐พ๐ต = (๐บ๐ต, ๐บ๐๐ต) where ๐ is the counit of๐น โฃ ๐บ. Note this is an algebra by one of the triangular identities for ๐น โฃ ๐บ andnaturality of ๐, and ๐พ(๐ต
๐โโ ๐ตโฒ) = ๐บ๐, a homomorphism by naturality of ๐.
Clearly ๐บT๐พ = ๐บ and
๐พ๐น๐ด = (๐บ๐น๐ด, ๐บ๐๐น๐ด) = (๐ ๐ด, ๐๐ด) = ๐นT๐ด
๐พ๐น(๐ด๐โโ ๐ดโฒ) = ๐ ๐ = ๐นT๐
so ๐พ is a morphism of Adj(T).Suppose ๐พโฒ โถ D โ CT is another such, then since ๐บT๐พโฒ = ๐บ we know
๐พโฒ๐ต = (๐บ๐ต, ๐ฝ๐ต) where ๐ฝ is a natural transformation ๐บ๐น๐บ โ ๐บ. Also since๐พโฒ๐น = ๐นT, we have
๐ฝ๐น๐ด = ๐๐ด = ๐บ๐๐น๐ด.
Now given any ๐ต โ obD, consider the diagram.Also since ๐พโฒ๐น = ๐นT we have ๐ฝ๐น๐ด = ๐๐ด = ๐บ๐๐น๐ด.Now given any ๐ต โ obD, consider the diagram
๐บ๐น๐บ๐น๐บ๐ต ๐บ๐น๐บ๐ต
๐บ๐น๐บ๐ต ๐บ๐ต
๐บ๐น๐บ๐๐ต
๐บ๐๐น๐บ๐ต =๐ฝ๐น๐บ๐ต ๐บ๐๐ต ๐ฝ๐ต
๐บ๐๐ต
Both squares commute so ๐บ๐๐ต and ๐ฝ๐ต have the same composite with ๐บ๐น๐บ๐๐ต.But this is split epic with splitting ๐บ๐น๐๐บ๐ต so ๐ฝ = ๐บ๐. Hence ๐พโฒ = ๐พ.
We now define the Kleisli comparison functor ๐ฟ โถ CT โ D by ๐ฟ๐ด = ๐น๐ด,
๐ฟ(๐ด๐โโ๐ต) = ๐น๐ด
๐น๐โโ ๐น๐บ๐น๐ต
๐๐น๐ตโโโ ๐น๐ต.
๐ฟ preserves identities by one of the triangular identities for ๐น โฃ ๐บ. Given๐ด
๐โโ๐ต
๐โโ๐ถ, we have
๐น๐ด ๐น๐บ๐น๐ต ๐น๐บ๐น๐บ๐น๐ถ ๐น๐บ๐น๐ถ
๐น๐ต ๐น๐บ๐น๐ถ ๐น๐ถ
๐น๐ ๐น๐บ๐น๐
๐๐น๐ต
๐น๐บ๐๐น๐ถ
๐๐น๐บ๐น๐ถ ๐๐น๐ถ
๐น๐ ๐๐น๐ถ
Also
๐บ๐ฟ๐ด = ๐ ๐ด = ๐บT๐ด
๐บ๐ฟ(๐ด๐โโ๐ต) = (๐บ๐๐น๐ต)(๐น๐บ๐) = ๐๐ต(๐ ๐) = ๐บT๐
๐บ๐นT๐ด = ๐น๐ด
๐ฟ๐นT(๐ด๐โโ ๐ต) = (๐๐น๐ต)(๐น๐๐ต)(๐น๐) = ๐น๐
For future reference, note that ๐ฟ is full and faithful: its effect on morphisms(with blue domains and codmains) is that of transposition across ๐น โฃ ๐บ.
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5 Monad
Finally for uniqueness, suppose ๐ฟโฒ โถ CT โ D is a morphism of Adj(T). Wemust have ๐ฟโฒ๐ด = ๐น๐ด and ๐ฟโฒ maps the counit ๐ ๐ดโ๐ด to the counit ๐น๐บ๐น๐ด
๐๐น๐ดโโโ
๐น๐ด. For any ๐ด๐โโ๐ต, we have
๐ = 1๐ ๐ด(๐นT๐)
so ๐ฟโฒ(๐) = ๐๐น๐ด(๐น๐) = ๐ฟ๐.
If C has coproducts then so does CT since ๐นT preserves them. But in generalit has few other limits or colimits. In contrast, we have
Theorem 5.4.
1. The forgetful functor ๐บ โถ CT โ C creates all limits which exists in C.
2. If C has colimits of shape J then ๐บ โถ CT โ C creates them if and onlyif ๐ preserves them.
Proof.
1. Suppose given ๐ท โถ J โ CT. Write ๐ท(๐) = (๐บ๐ท(๐), ๐ฟ๐) and suppose(๐ฟ, (๐๐ โถ ๐ฟ โ ๐บ๐ท(๐) โถ ๐ โ obJ)) is a limit cone for ๐บ๐ท. Then thecomposites
๐ ๐ฟ๐ ๐๐โโโ ๐ ๐บ๐ท(๐)
๐ฟ๐โโ ๐บ๐ท(๐)
form a cone over ๐บ๐ท since the edges of ๐บ๐ท are homomorphisms, so theyinduce a unique ๐ โถ ๐ ๐ฟ โ ๐ฟ such that ๐๐๐ = ๐ฟ๐(๐๐) for all ๐. The factthat ๐ is a T-algebra structure on ๐ฟ follows from the fact that the ๐ฟ๐are algebra structure and uniqueness of factorisations through limits. So((๐ฟ, ๐), (๐๐ โถ ๐ โ obJ)) is the unique lifting of the limit cone over ๐บ๐ทto a cone over ๐ท, and itโs a limit, since given a conve over ๐ท with apex(๐ด, ๐ผ), we get a unique factorisation ๐ด
๐โโ ๐ฟ in C, and ๐น is an algebra
homomorphism by uniqueness of factorisation through ๐ฟ.For โธ direction, suppose given ๐ท โถ J โ CT as in 1, and a colimit cone
(๐บ๐ท(๐)๐๐โโ ๐ฟ โถ ๐ โ obJ) in C, then (๐ ๐บ๐ท(๐)
๐ ๐๐โโโ ๐ ๐ฟ โถ ๐ โ obC) is also
a colimit cone, so the composite
๐ ๐บ๐ท(๐)๐๐โโ ๐บ๐ท(๐)
๐๐โโ ๐ฟ
induces a unique ๐ โถ ๐ ๐ฟ โ ๐ฟ. The rest of the argument is like 1.
Definition (monadicity). Given an adjunctin ๐น โฃ ๐บ, we say the adjunction(or the functor ๐บ) is monadic if the comparison functor ๐พ โถ D โ CT is partof an equivalence of categories.
Note that since the Kleisi comparison CT โ D is full and faithful, itโs partof an equivalence if and only if it (equivalently, ๐น) is essentially surjective onobjects.
40
5 Monad
Remark. Given any adjunction ๐น โฃ ๐บ, for each object ๐ต of D we have adiagram
๐น๐บ๐น๐บ๐ต ๐น๐บ๐ต ๐ต๐น๐บ๐๐ต
๐๐น๐บ๐ต
๐๐ต
with equal composites. The โprimevalโ monadicity theorem asserts that CT ischaracterised in Adj(T) by the fact that these diagrams are all coequalisers.
Definition (reflexivity, split coequaliser).
1. We say a parallel pair ๐, ๐ โถ ๐ด โ ๐ต is reflexive if their exists ๐ต๐โโ ๐ด
such that ๐๐ = ๐๐ = 1๐ต.We say C has reflexive coequalisers if it has coequalisers of all reflexivepairs. Equivalently, colimits of shape
โ โ
2. By a split coequaliser diagram we mean a diagram
๐ด ๐ต ๐ถ๐
๐๐ก
โ
๐
satisfying โ๐ = โ๐, โ๐ = 1๐ถ, ๐๐ก = 1๐ต and ๐๐ก = ๐ โ.
These equalisers imply that โ is a coequaliser of (๐, ๐), if ๐ต๐ฅโโ ๐ท
satisfies ๐ฅ๐ = ๐ฅ๐ then
๐ฅ = ๐ฅ๐๐ก = ๐ฅ๐๐ = ๐ฅ๐ โ
so ๐ฅ factors through โ and the factorisation is unique since its splitmonic.Note that split coequalisers are preserved by all functors.
3. Given a functor ๐บ โถ D โ C, a parallel pair ๐ด ๐ต๐
๐is called
๐บ-split if there exists a split coequaliser diagram
๐บ๐ด ๐บ๐ต ๐ถ๐บ๐
๐บ๐๐ก
โ
๐
in C.
41
5 Monad
Note that ๐น๐บ๐น๐บ๐ต ๐น๐บ๐ต๐น๐บ๐๐ต
๐๐น๐บ๐ตis ๐บ-split, since
๐บ๐น๐บ๐น๐บ๐ต ๐บ๐น๐บ๐ต ๐ถ๐บ๐น๐บ๐๐ต
๐บ๐๐น๐บ๐ต
๐๐บ๐น๐บ๐ต
๐บ๐๐ต
๐๐บ๐ต
Note that the aforementioned pair
๐น๐บ๐น๐บ๐ต ๐น๐บ๐ต๐น๐บ๐๐ต
๐๐น๐บ๐ต
is reflexive with ๐ = ๐น๐๐บ๐ตโฒ .
Lemma 5.5. Suppose given an adjunction C D๐น
๐บwhere ๐น โฃ ๐บ,
inducing a monad T on C. Then ๐พ โถ D โ CT has a left adjoint provided,
for every T-algebra (๐ด, ๐ผ), the pair ๐น๐บ๐น๐ด ๐น๐ด๐น๐ผ
๐๐น๐ดhas a coequaliser in
D.
Proof. We define ๐ฟ โถ CT โ D by taking ๐น๐ด โ ๐ฟ(๐ด, ๐ผ) to be a coequaliserfor (๐น๐ผ, ๐๐น๐ด). Note that this is a functor CT โ D. Recall that ๐พ is definedby ๐พ๐ต = (๐บ๐ต, ๐บ๐๐ต). For any ๐ต, morphisms ๐ฟ๐ด โ ๐ต correspond bijectivelyto morphisms ๐น๐ด
๐โโ ๐ต satisfying ๐(๐น๐ผ) = ๐(๐๐น๐ด). These correspond to mor-
phisms ๐ด๐
โโ ๐บ๐ต satisfying
๐๐ผ = ๐บ๐ = ๐บ(๐๐ต(๐น ๐)) = (๐บ๐๐ต)(๐ ๐)
i.e. to algebra homomorphisms (๐ด, ๐ผ) โ ๐พ๐ต. And these bijections are naturalin (๐ด, ๐ผ) and in ๐ต.
Theorem 5.6 (precise monadicity theorem). ๐บ โถ D โ C is monadic if andonly if ๐บ has a left adjoint and creates coequalisers of ๐บ-split pairs.
Theorem 5.7 (refined/reflexive monadicity theorem). Suppose D has and๐บ โถ D โ C preserves reflexive coequalisers, and that ๐บ reflects isomorphismsand has a left adjoint. Then ๐บ is monadic.
Proof. Theorem 5.6 โน : sufficient to show that ๐บT โถ CT โ C creates co-equalisers of ๐บT-split pairs. But this follows from the argument of 5.4 (2), since
if ๐, ๐ โถ (๐ด, ๐ผ) โ (๐ต, ๐ฝ) is a ๐บT-split pairs, the coequalisers of ๐ด ๐ต๐
๐is
preserved by ๐ and ๐ ๐.Theorem 5.6 โธ and Theorem 5.7: Let T denote the monad induced by
๐น โฃ ๐บ. For any T-algebra (๐ด, ๐ผ), the pair ๐น๐บ๐น๐ด ๐ด๐น๐ผ
๐๐น๐ดis both reflexive
and ๐บ-split, so has a coequaliser in D and hence by Lemma 5.5, ๐พ โถ D โ CT
42
5 Monad
has a left adjoint ๐ฟ. Then unit of ๐ฟ โฃ ๐พ at an algebra (๐ด, ๐ผ), the coequaliserdefining ๐ฟ(๐ด, ๐ผ) is mapped by ๐พ to the diagram
๐นT๐ ๐ด ๐นT๐ด ๐พ๐ฟ(๐ด, ๐ผ)
(๐ด, ๐ผ)
๐นT๐ผ
๐๐ด ๐ผ ๐(๐ด,๐ผ)
and ๐(๐ด,๐ผ) is the factorisation of this through the ๐บT-split coequaliser ๐ผ. Buteither set of hypothesis implies that ๐บ preserves the coequaliser defining ๐ฟ(๐ด, ๐ผ),so ๐(๐ด,๐ผ) is an isomorphism. For the counit ๐ โถ ๐ฟ๐พ๐ต โ ๐ต, we have a coequaliser
๐น๐บ๐น๐บ๐ต ๐น๐บ๐ต ๐ฟ๐พ๐ต
๐ต
๐น๐บ๐๐ต
๐๐น๐บ๐ต ๐๐ต ๐๐ต
Again, either set of hypothesis implies that ๐๐ต is a coequaliser of (๐น๐บ๐๐ต, ๐๐น๐บ๐ต)so ๐๐ต is an isomorphism.
Example.
1. The forgetful functors Gp โ Set, Rng โ Set, Mod๐ โ Set โฆ all satisfythe hypothesis of refined monadicity theorem, for the relexive coequalisers,use example sheet 4 Q3 which shows that if
๐ด ๐ต ๐ถ๐
๐โ
is a reflexive coequaliser diagram in Set then so is
๐ด๐ ๐ต๐ ๐ถ๐.๐๐
๐๐
โ๐
2. Any reflection is monadic: this follows from example sheet 3 Q3, but canalso be proved using precise monadicity theorem. Let D be a relfecitve(full) subcategory of C, and suppose a pair ๐, ๐ โถ ๐ด โ ๐ต in D fits into asplit coequaliser diagram
๐ด ๐ต ๐ถ๐
๐๐ก
โ
๐
in C. Then ๐ก and ๐๐ก = ๐ โ belongs to D since D is full and hence ๐ is in Dsince itโs an equaliser of (1๐ต, ๐ โ) and D is closed under limits in C. Hencealso โ โ morD.
3. Consider the composite adjunction
Set AbGp tfAbGp๐น
๐
๐ฟ
๐ผ
These two factors are monadic by the above two examples respectively,but the composite isnโt, since the monad it induces on Set is isomorphicto that induced by ๐น โฃ ๐.
43
5 Monad
4. Consider the forgetful functor ๐ โถ Top โ Set. This is faithful and hasboth left and right adjoint (so preserves all coequalisers), but the monadinduced on Set is (1, 1, 1) and the category of algebras is Set.
5. One may get the impression that monadicty is something only shared byalgebraic construction but not topological constructions. Consider thecomposite adjunction
Set Top KHaus๐ท
๐
๐ฝ
๐ผ
We shall show that this satisfies the hypothesis of precise monadicity the-orem. Let
๐ ๐ ๐๐
๐๐ก
โ
๐
be a split coequaliser in Set where ๐ and ๐ have compact Hausdorfftopologies and ๐, ๐ are continuous. Note that the quotient topology on๐ โ ๐ /๐ is compact, so itโs the only possible candidate for a compactHausdorff topology making โ continuous.We use the lemma from general topology: if ๐ is compact Hausdorff, thena quotient ๐ /๐ is Hausdorff if and only if ๐ โ ๐ ร ๐ is closed. We note
๐ = {(๐ฆ, ๐ฆโฒ) โถ โ(๐ฆ) = โ(๐ฆโฒ)}= {(๐ฆ, ๐ฆโฒ) โถ ๐ โ(๐ฆ) = ๐ โ(๐ฆโฒ)}= {(๐ฆ, ๐ฆโฒ) โถ ๐๐ก(๐ฆ) = ๐๐ก(๐ฆโฒ)}
so if we define ๐ = {(๐ฅ, ๐ฅโฒ) โถ ๐(๐ฅ) = ๐(๐ฅโฒ)} โ ๐ ร ๐ then ๐ โ (๐ ร ๐)(๐),but this reverse inclusion also holds. But
๐ ๐ ร ๐ ๐๐๐1
๐๐2
is an equaliser, ๐ is Hausdorff, so ๐ is closed on ๐ ร๐ and hence compact.So ๐ = (๐ ร ๐)(๐) is compact and hence closed in ๐ ร ๐.Morally, a category that is monadic over Set can be thought as an alge-braic object, in the sense that it is defined by algebraic equations. Thisapplies to KHaus by including โinfinitary equationsโ.
Definition (monadic tower). Let C D๐น
๐บbe an adjunction and sup-
pose D has reflexive coequalisers. The monadic tower of ๐น โฃ ๐บ is the
44
5 Monad
diagramโฎ
D (CT)S
CT
C
๐พโฒ
๐พ
๐บ
๐ฟ
๐ฟ
๐น
where T is the monad induced by ๐น โฃ ๐บ, ๐พ is as in 5.7, ๐ฟ as in 5.11(comparison?) S is the moand induced by ๐ฟ โฃ ๐พ and so on.
We say ๐น โฃ ๐บ has monadic length ๐ if we reach an equlvalence after ๐steps.Monadic length 0: already equivalence. Monadic length 1: monadFor example, the adjunction of 5.14c has monadic length 1. The adjunction
of 5.14d has monadic length โ.We will need one more result for topos theory, which we state without proof.
(lifting of left adjoint in categorical algebra)
Theorem 5.8. Suppose given an adjunction C D๐ฟ
๐ and monads T,S
on C, D respectively, and a functor ๐ โถ DS โ CT such that
DS CT
D C
๐
๐บS ๐บT
๐
commutes up to isomorphism. Suppose also DS has reflexive coequalisers.Then ๐ has a left adjoint ๐ฟ.
Proof. Note that if ๐ฟ exists we must have ๐ฟ๐นT โ ๐น S๐ฟ by 3.6. So weโd expect
๐ฟ(๐ด, ๐ผ) to be a coequaliser of two morphisms ๐น S๐ฟ๐ ๐ด ๐น S๐ฟ๐ด.๐น S๐ฟ๐ผ
?To con-
struct the second morphism, note first that we assume wlog ๐บT๐ = ๐ ๐บG, bytransporting T-algebra structurs along the isomorphism ๐บT๐ (๐ต, ๐ฝ) โ ๐ ๐ต.
We obtain ๐ โถ ๐ ๐ โ ๐ ๐ by starting from
๐ ๐ ๐โโ ๐ ๐ = ๐ ๐บS๐น S = ๐บT๐ ๐น S,
conjugate by ๐น to get๐นT๐ โ ๐ ๐น S,
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5 Monad
and finally (?)
๐ ๐ = ๐บT๐นT๐ ๐โโ ๐บT๐ ๐น S = ๐ ๐บS๐น S = ๐ ๐.
Convert it into ๐ โถ ๐ฟ๐ โ ๐๐ฟ by
๐ฟ๐๐ฟ๐ ๐พโโโ ๐ฟ๐ ๐ ๐ฟ
๐ฟ๐๐ฟโโโ ๐ฟ๐ ๐๐ฟ
๐ฟ๐๐ฟโโโ ๐๐ฟ
where ๐พ and ๐ฟ are the unit and counit of ๐ฟ โฃ ๐ . Transposing across ๐น S โฃ ๐บS,we get ๐ โถ ๐น S๐ฟ๐
๐น S
โโ ๐ฟ. The pair (๐น S๐ฟ๐ผ, ๐๐ด) is relexive, with common splitting๐น S๐ฟ๐. (๐ is the question mark in the diagram)
It can be verified (albeit extremely tedious) that the coequaliser of this pairhas the unviersal property we require for ๐ฟ(๐ด, ๐ผ).
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6 Cartesian closed categories
6 Cartesian closed categories
Definition (exponentiable object, cartesian closed category). Let C be acategory with finite products. We say ๐ด โ obC is exponentiable if thefunctor (โ) ร ๐ด โถ C โ C has a right adjoint (โ)๐ด.
If every object of C is exponentiable, we say C is cartesian closed.
Intuitively, exponential object โliftsโ morphisms to an object, instead of aset. โinternalisationโ
Example.
1. Set is cartesian closed, with ๐ต๐ด = Set(๐ด, ๐ต). A function ๐ โถ ๐ถ ร ๐ด โ ๐ตcorresponds to ๐ โถ ๐ถ โ ๐ต๐ด.
2. Cat is cartesian closed with DC = [C, D]. In fact we have implicitly usedthis idea when discussing limits in functor categories.
3. In Top, if an exponential ๐ ๐ exists, its points must be the continuousmaps ๐ โ ๐. The compact-open topology on Top(๐, ๐ ) has the universalproperty of an exponential if and only if ๐ is locally compact.Note that finite products of exponential objects are exponentiable: since
(โ) ร (๐ด ร ๐ต) โ (โ ร ๐ด) ร ๐ต,
we have (โ)๐ดร๐ต โ ((โ)๐ต)๐ด. However, even if ๐ and ๐ are locally com-pact, ๐ ๐ neednโt be. So the exponentiable objects donโt form a cartesianclosed full subcategory.
4. A cartesian closed poset is called a Heyting semilattice: itโs a poset withfinite meet โง and a binary operation โน satisfying
๐ โค (๐ โน ๐) if and only if ๐ โง ๐ โค ๐.
For example, a complete poset is a Heyting semilattice if and only if itsatisfies the infinite distributive law
๐ โง โ๐โ๐ผ
{๐๐} = โ๐โ๐ผ
{๐ โง ๐๐}.
For any topological space, the lattice ๐ช(๐) of open sets satisfies this con-dition, since โง and โจ conincide with โฉ an โช. (However it does not satisfythe dual condition: arbitrary intersection needs to take interior).
Recall from example sheet 2 that, if ๐ต โ ob๐ถ, we define the slice categoryC/๐ต to have objects which are morphisms ๐ด โ ๐ต in C and morphisms arecommutative triangles
๐ด ๐ดโฒ
๐ต
47
6 Cartesian closed categories
The forgetful functor C/๐ต โ C will be denoted ฮฃ๐ต. If C has finite products,ฮฃ๐ต has a right adjoint ๐ตโ which sends ๐ด to
๐ด ร ๐ต๐2โโ ๐ต
since morphisms๐ถ ๐ด ร ๐ต
๐ต
(๐,๐)
๐ ๐2
correspond to morphisms ๐ โถ ฮฃ๐ต๐ = ๐ถ โ ๐ด.
Lemma 6.1. If C has all finite limits then an object ๐ต is exponentiable ifand only if ๐ตโ โถ C โ C/๐ต has a right adjoint ฮ ๐ต.
Proof.
1. โธ : The composite ฮฃ๐ต๐ตโ is equal to (โ) ร ๐ต so we take (โ)๐ต to beฮฃ๐ต๐ตโ.
2. โน : If ๐ต is exponentiable, for any ๐ โถ ๐ด โ ๐ต we define ฮ ๐ต9๐) to be thepullback
ฮ ๐ต(๐) ๐ด๐ต
1 ๐ต๐ต
๐น ๐ต
๐2
where ๐2 is the transpose of the projection (?). Then morphisms ๐ถ โฮ ๐ต(๐) corresponding to morphisms ๐ถ โ ๐ด๐ต making
๐ถ ๐ด๐ต
1 ๐ต๐ต
๐๐ต
๐2
commute, i.e. to morhpisms ๐ถ ร ๐ต โ ๐ด making
๐ถ ร ๐ต ๐ด
๐ต๐2
๐
commute.
Lemma 6.2. Suppose C has finite limits. If ๐ด is exponentiable in C then๐ตโ๐ด is exponentiable in C/๐ต for any ๐ต. Moreover ๐ตโ preserves exponen-tials.
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6 Cartesian closed categories
Proof. Given an object๐ถ
๐ต
๐ , form the pullback
๐ ๐ถ๐ด
๐ต ๐ต๐ด
๐๐ตโ๐ด ๐๐ด
๐1
Then for any๐ท
๐ต
๐ , morphisms ๐ โ ๐๐ตโ๐ด in C/๐ต correspond to morphisms
๐ทโโโ ๐ถ๐ด making
๐ท ๐ถ๐ด
๐ต ๐ต๐ด
โ
๐ ๐๐ด
๐1
commute, and hence to morphisms ๐ท ร ๐ดโโโ ๐ถ making
๐ท ร ๐ด ๐ถ
๐ต
โ
๐๐1 ๐
commmute. But๐ท ร ๐ด ๐ต ร ๐ด
๐ท ๐ต
๐ร1๐ด
๐1 ๐1
๐
is a pullback in C, i.e. a product in C/๐ต.For the second assertion, note that if ๐ถ
๐โโ ๐ต is of the form ๐ต ร ๐ธ
๐1โโ ๐ต
then the pullback defining ๐๐ตโ๐ด becomes
๐ต ร ๐ธ๐ด ๐ต๐ด ร ๐ธ๐ด
๐ต ๐ต๐ด
๐1ร1๐ธ๐ด
๐1 ๐1
๐1
so ๐๐ตโ๐ด โ ๐ตโ(๐ธ๐ด).
Remark. C/๐ต is isomorphic to the category of coalgebra for the monad struc-ture on (โ) ร ๐ต (5.2c). So the first part of the lemma could be proved using liftof adjoint (last theorem of chapter 5).
Definition (locally cartesian closed). We say C is locally cartesian closedif it has all finite limits and each C/๐ต is cartesian closed.
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6 Cartesian closed categories
Note that this includes the fact that C โ C/1 is cartesian closed. So theusuage is to the contrary of normal usage of โlocallyโ as it imposes a strictercondition.
Example.
1. Set is locally cartesian closed since Set/๐ต โ Set๐ต for any ๐ต.
2. For any small category C, [C, Set] is cartesian closed: by Yoneda
๐บ๐น(๐ด) โ [C, Set](C(๐ด, โ), ๐บ๐น) โ [C, Set](C(๐ด, โ) ร ๐น, ๐บ)
so we take RHS as a definition of ๐บ๐น(๐ด) and define ๐บ๐น on morphisms๐ด
๐โโ ๐ต by composition with C(๐, โ) ร 1๐น. Note that the class of funtors
๐ป for which we have
[C, Set](๐ป, ๐บ๐น) โ [C, Set](๐ป ร ๐น, ๐บ)
is closed under colimits. But every functor C โ Set is a colimit ofrepresentables.In fact [C, Set] is locally cartesian closed since all its slice categories[C, Set]/๐น are of the same form. See example sheet 4 Q6.
3. Any Heyting semilattice ๐ป is locally cartesian closed since ๐ป/๐ โ โ (๐),the poset of elements โค ๐, and ๐โ = (โ) โง ๐ is surjective.
4. Cat is not locally cartesian closed, since not all strong epis are regular.c.f. example sheet 3 Q6.
Note that given๐ด
๐ต
๐ in C/๐ต, the iterated slice (C/๐ต)/๐ is isomorphic to
C/๐ด, and this identifies ๐โ โถ C/๐ต โ (C/๐ต)/๐ with the operation of pullingback morphisms along ๐. So by 6.3 C is locally cartesian closed if and only if ithas finite limits and ๐โ โถ C/๐ต โ C/๐ด has a right adjoint ฮ ๐ for every ๐ด
๐โโ ๐ต
in C. This can be taken as the definition of locally cartesian closed category.
Theorem 6.3. Suppose C is locally cartesian closed and has reflexive co-equalisers. Then every morphism ๐ด
๐โโ ๐ต factors as
๐ด๐
โ ๐ผ๐โฃ ๐ต
where ๐ is regular epic and ๐ is monic.
Proof. First form the pullback
๐ ๐ด
๐ด ๐ต
๐
๐ ๐๐
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6 Cartesian closed categories
and then form the coequaliser
๐ ๐ด ๐ผ.๐
๐
๐
Since ๐๐ = ๐๐, ๐ factors as
๐ด ๐ผ ๐ต.๐ ๐
Suppose given ๐ท ๐ผ๐
โwith ๐๐ = ๐โ, form the pullback
๐ธ ๐ท
๐ด ร ๐ด ๐ผ ร ๐ผ
๐
(๐,โ) (๐,โ)๐ร๐
Since ๐ ร ๐ factors as (๐ ร 1๐ผ)(1๐ด ร ๐) and both factors are pullbacks of ๐, it isan epimorphism and so is ๐. Now
๐๐ = ๐๐๐ = ๐๐๐ = ๐โ๐ = ๐๐โ = ๐โ
so there exists ๐ธ๐โโ ๐ with ๐๐ = ๐, ๐๐ = โ.
Now๐๐ = ๐๐๐ = ๐๐๐ = ๐โ,
i.e. ๐๐ = โ๐. But ๐ is epic so ๐ = โ. Hence ๐ is monic.
Note that this implies any strong epi ๐ด๐โโ ๐ต is regular since the monic part
of its image factorisation is an isomorphism. In particular, regular epimorphismsare stable under composition.
Definition. If C and D are cartesian closed categories and ๐น โถ C โ Dpreserves products then for each pair of objects (๐ด, ๐ต) of C, we get a naturalmorphism ๐ โถ ๐น(๐ต๐ด) โ ๐น๐ต๐น๐ด, namely the transpose of
๐น(๐ต๐ด) ร ๐น๐ด โ ๐น(๐ต๐ด ร ๐ด)๐น(ev)โโโโ ๐น๐ต
where ev is the counit of ((โ) ร ๐ด โฃ (โ)๐ด).We say ๐น is a cartesian closed functor if ๐ is an isomorphism for every pair
(๐ด, ๐ต). Note that the second part of 6.4 syas that if C is locally cartesianclosed then ๐โ โถ C/๐ต โ C/๐ด is a continuous cartesian functor for any๐ด
๐โโ ๐ต.
Theorem 6.4. Let C and D be cartesian closed categories and ๐น โถ C โ Da functor having a left adjoint ๐ฟ. Then ๐น is cartesian closed if and only ifthe canonical morphism ๐๐ด,๐ต
๐ฟ(๐ต ร ๐น๐ด) ๐ฟ๐ต ร ๐ฟ๐น๐ด ๐ฟ๐ต ร ๐ด(๐ฟ๐1,๐ฟ๐2) 1๐ฟ๐ดร๐๐ต
is an isomorphism for all ๐ด โ obC, ๐ต โ obD. This condition is called
51
6 Cartesian closed categories
Frobenius reciprocity.
Note that if C is locally cartesian closed then ๐โ โถ C/๐ต โ C/๐ด has a leftadjoint ฮฃ๐ given by composition with ๐, and itโs easy to verify that
ฮฃ๐(๐ ร ๐โโ) โ ฮฃ๐๐ ร โ.
Proof.
โข โน : Given an inverse for ๐ โถ ๐น(๐ถ๐ด) โ ๐น๐ด๐น๐ด, we define ๐โ1๐ด,๐ต to be the
composite
๐ฟ๐ต ร ๐ต ๐ฟ((๐ต ร ๐น๐ด)๐น๐ด) ร ๐ด ๐ฟ(๐น๐ฟ(๐ต ร ๐น๐ด)๐น๐ด) ร ๐ต
๐ฟ(๐ต ร ๐น๐ด) ๐ฟ(๐ต ร ๐น๐ด)๐ด ร ๐ด ๐ฟ๐น(๐ฟ(๐ต ร ๐น๐ด)๐ด) ร ๐ด
๐ฟ๐ร1 ๐ฟ(๐๐น๐ด)ร1
๐ฟ๐โ1ร1
ev ๐ร1
The verification is a tedious exercise.
โข โธ : Given an inverse for ๐, we define ๐โ1 to be
๐น((๐ฟ(๐น๐ถ๐น๐ด ร ๐ด))๐ด) ๐น๐ฟ(๐น๐ถ๐น๐ด) ๐น๐ถ๐น๐ด
๐น((๐ฟ(๐น๐ถ๐น๐ด ร ๐น๐ด))๐ด) ๐น((๐ฟ๐น๐ถ)๐ด) ๐น(๐ถ๐ด).
๐น(๐โ1๐ด)
๐น๐ ๐
๐น((๐ฟ(ev))๐ด) ๐น(๐๐ด)
Corollary 6.5. Suppose C and D are cartesian closed, and ๐น โถ C โ D hasa left adjoint ๐ฟ which preserves finite products. Then ๐น is cartesian closedif and only if ๐น is full and faithful.
Proof.
โข โน : ๐ฟ preserves 1 so if we substitute ๐ต in the definition of ๐ above,we get ๐ฟ๐น๐ด
๐๐ดโโ ๐ด. But ๐ is an isomorphism if and only if ๐น is full and
faithful.
โข โธ : If ๐ฟ preserves binary products and ๐ is an isomorphism. Then bothfactors in the definition of ๐ are isomorphisms.
Definition (exponential ideal). Let C be a cartesian closed category. Byan exponential ideal of C we mean a class of objects (or a full subcategory)โฐ such that ๐ต โ โฐ implies ๐ต๐ด โ โฐ for all ๐ด โ obC.
Example.
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6 Cartesian closed categories
1. We say ๐ด is subterminal if ๐ด โ 1 is monic. In any cartesian closed cate-gory C, the class SubC(1) is an exponential ideal since ๐ด is subterminal ifand only if there exists at most 1 morphisms ๐ต โ ๐ด for any ๐ต, if and onlyif at most 1 morphism ๐ถ ร ๐ต โ ๐ด for any ๐ต and ๐ถ, so by adjunction ifand only if at most 1 morphism ๐ถ โ ๐ด๐ต, if and only if ๐ด๐ต is subterminal.More generally, if C is locally cartesian closed then SubC(๐ด) is an expo-nential ideal in C/๐ด for any ๐ด. if C also satisfies the hypotheses of 6.7then SubC(๐ด) is reflexive in C/๐ด.
2. Let ๐ be a topological space. By a presheaf of ๐ we mean a functor๐น โถ ๐ช(๐)op โ Set where ๐ช(๐) is the partial order of open subsets of ๐.So ๐น has sets ๐น(๐) for each open ๐ and restriction maps ๐น(๐) โ ๐น(๐ ) โถ๐ฅ โฆ ๐ฅ|๐ whenever ๐ โ ๐.We say ๐น is a sheaf if whenever ๐ = โ๐โ๐ผ ๐๐ and weโre given ๐ฅ๐ โ ๐น(๐๐)for each ๐, such that
๐ฅ๐|๐๐โฉ๐๐= ๐ฅ๐|๐๐โฉ๐๐
for all (๐, ๐), then exists a unique ๐ฅ โ ๐น(๐) such that ๐ฅ|๐๐= ๐ฅ๐ for all ๐.
We write Sh(๐) โ [๐ช(๐)op, Set] for the full subcategory of sheaves. Weโllshow Sh(๐) is an exponential ideal: given presheaves ๐น, ๐บ, ๐บ๐น(๐) is theset of natural transofrmations ๐น ร ๐ช(๐)(โ, ๐) โ ๐บ or equivalently, theset of natural transformations ๐น|๐ โ ๐บ|๐ where ๐น|๐ โถ ๐ช(๐)op โ Set isthe presheaf obtained by restricting ๐น to open sets in ๐. Now suppose ๐บis a sheaf. Suppose ๐ = โ๐โ๐ผ ๐๐ and suppose given ๐ผ๐ โถ ๐น |๐๐
โ ๐บ|๐๐for
each ๐ such that๐ผ๐|๐๐โฉ๐๐
= ๐ผ๐|๐๐โฉ๐๐
for all ๐, ๐. Given ๐ฅ โ ๐น(๐ ) where ๐ โ ๐, write ๐๐ = ๐ โฉ ๐๐, then ๐ =โ๐โ๐ผ ๐๐. The elements ๐ฅ|๐๐
, ๐ โ ๐ผ, satisfying the compatibility conditionand hence so do the elements (๐ผ๐)๐๐
(๐ฅ|๐๐) โ ๐บ(๐๐) . So thereโs a unique
๐ฆ โ ๐บ(๐ ) such that๐ฆ|๐๐
= (๐ผ๐)๐๐(๐ฅ|๐๐
)
for all ๐ and we define this to be ๐ผ๐(๐ฅ). This defines natural transforma-tions ๐ผ โถ ๐น |๐ โ ๐บ|๐ and itโs the unique transformation whose restrictionto ๐๐ is ๐ผ๐ for each ๐.Note that since Sh(๐) is closed under finite products (and in fact alllimits) in [๐ช(๐)op, Set], it is itself cartesian closed.
Lemma 6.6. Suppose C is cartesian closed and D โ C is a (full) reflectivesubcategory, with reflector ๐ฟ โถ C โ D. Then D is an exponential ideal ifand only if ๐ฟ preserves binary products.
Proof.
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6 Cartesian closed categories
โข โน : Suppose ๐ด, ๐ต โ obC, ๐ถ โ obD. Then we have bijections
๐ด ร ๐ต โ ๐ถ๐ด โ ๐ถ๐ต
๐ฟ๐ด โ ๐ถ๐ต
๐ฟ๐ด ร ๐ต โ ๐ถ๐ต โ ๐ถ๐ฟ๐ด
๐ฟ๐ต โ ๐ถ๐ฟ๐ด
๐ฟ๐ด ร ๐ฟ๐ด โ ๐ถ
so ๐ฟ๐ด ร ๐ฟ๐ต has the universal property of ๐ฟ(๐ด ร ๐ต).
โข โธ : Suppose ๐ต โ obD, ๐ด, ๐ถ โ obC. We have bijections
๐ถ โ ๐ต๐ด
๐ถ ร ๐ด โ ๐ต๐ฟ๐ถ ร ๐ฟ๐ด โ ๐ฟ(๐ถ ร ๐ด) โ ๐ต
๐ฟ(๐ฟ๐ถ ร ๐ด) โ ๐ต๐ฟ๐ถ ร ๐ด โ ๐ต
๐ฟ๐ถ โ ๐ต๐ด
so every ๐ถ โ ๐ต๐ด factors throu ๐ถ โ ๐ฟ๐ถ, hence ๐ต๐ด โ obD.
54
7 Toposes
7 ToposesTopos has it orgin in the French school of algebraic geometry. In 1963, whenstuding cohomologies in gemoetry, Grothendieck studied toposes as categoriesof โgeneralised sheavesโ. As sheaves can be seen as representation of spaces andproperties of the space can be detected from sheaves, toposes become generalisedspaces. Thus the name topos: something more fundamental than topology.
J. Giraud gave a characterisation of such categories by (set-theoretic) cate-gorical properties. F. W. Lawvere and M. Tierney (1969 - 1970) investigated theelementary categorical properties of these categories and come up with the ele-mentary definition. In fact a Grothendieck topos is exactly a Lawvere-Tierneytopos which is (co)complete and locally small, and has a separating set of ob-jects (which is what Grothendieck should, but didnโt, come up with, by theway).
Definition (subobject classifier, topos, logical functor).
1. Let โฐ be a category with finite limits. A subobject classifier for โฐ is amonomorphism โค โถ ฮฉโฒ โฃ ฮฉ such that for every mono ๐ โถ ๐ดโฒ โฃ ๐ด inโฐ, there is a unique ๐๐ โถ ๐ด โ ฮฉ for which there is a pullback square
๐ดโฒ ฮฉโฒ
๐ด ฮฉ
๐ โค๐๐
Note that, for any ๐ด, there is a unique ๐ด โ ฮฉ which factors throughโค โถ ฮฉโฒ โฃ ฮฉ, so the domain of โค is actually a terminal object.If โฐ is well-powered, we have a functor Subโฐ โถ โฐ โ Set sending ๐ดto the set of (isomorphism classes) of subobjects of ๐ด and acting onmorphisms by pullback, and an subobject classifier is a representationof this functor.
2. A topos is a category which has finite limits, is cartesian closed andhas a subobject classifier.
3. If โฐ and โฑ are toposes, a logical functor ๐น โถ โฐ โ โฑ is one whichpreserves finite limits, exponentiables and the subobject classifier.
Example.1. Set is a topos, with ฮฉ = {0, 1} and โค = 1 โถ 1 โฆ {0, 1}. Have
๐(๐) = {1 ๐ โ ๐ด0 ๐ โ ๐ด
So also is the category Set๐ of finite set, or the category Set๐ of sets ofcardinality < ๐ , where ๐ is an infinite cardinal such that if ๐ < ๐ then2๐ < ๐ .
2. For any small category C, [Cop, Set] is a topos: weโve seen that itโs carte-sian, and ฮฉ is determined by Yoneda:
ฮฉ(๐ด) โ [Cop, Set](C(โ, ๐ด), ฮฉ) โ {subfunctors of C(โ, ๐ด)}.
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7 Toposes
So we define ฮฉ(๐ด) to be the set of sieves on ๐ด, i.e. sets ๐ of morphismswith codomain ๐ด such that if ๐ โ ๐ then ๐๐ โ ๐ for any ๐.Given ๐ โถ ๐ต โ ๐ด and a sieve ๐ on ๐ด, we define ๐โ๐ to be the set of ๐ withcodomain ๐ต such that ๐๐ โ ๐ . This makes ฮฉ into a functor Cop โ Set:โค โถ 1 โ ฮฉ is defined by
โค๐ด(โ) = {all morphisms with codomain ๐ด}.
Given a subfunctor ๐ โถ ๐น โฒ โฃ ๐น, we define ๐๐ โถ ๐น โ ฮฉ by
(๐๐)๐ด(๐ฅ) = {๐ โถ ๐ต โ ๐ด โถ ๐น๐(๐ฅ) โ ๐น โฒ(๐ต)}.
This is the unique natural transformation making
๐น โฒ 1
๐น ฮฉ
๐ โค๐๐
a pullback.
3. For any space ๐, Sh(๐) is a topos. Itโs cartesian closed. For the subobjectclassifier we take
ฮฉ(๐) = {๐ โ ๐ช(๐) โถ ๐ โ ๐}and ฮฉ(๐ โฒ โ ๐) is the map ๐ โฆ ๐ โ ๐ โฒ. ฮฉ is a sheaf since if we have๐ = โ๐โ๐ผ ๐๐ and ๐๐ โ ๐๐ such that ๐๐ โฉ ๐๐ = ๐๐ โฉ ๐๐ for each ๐, ๐ then๐ = โ๐โ๐ผ ๐๐ is the unique open subset of ๐ with ๐ โฉ ๐๐ = ๐๐ for each ๐.If ๐ โถ ๐น โฒ โฃ ๐น is a subsheaf then for any ๐ฅ โ ๐น(๐) the sieve
{๐ โ ๐ โถ ๐ฅ|๐ โ ๐น โฒ(๐ )}
has a greatest element since ๐น โฒ is a sheaf. So we define ๐๐ โถ ๐น โ ฮฉ tosend ๐ฅ to this object.
4. Let C be a group ๐บ. The topos structure on [๐บ, Set] is particularly simple:๐ต๐ด is the set of all ๐บ-equivariant maps ๐ โถ ๐ด ร ๐บ โ ๐ต but such an ๐ isdetermined by its values at elements of the form (๐, 1) since ๐(๐, ๐) =๐.๐(๐โ1.๐, 1), and this restriction can be any mapping ๐ด ร {1} โ ๐ต. Sowe can take ๐ต๐ด to be the set of functions ๐ด โ ๐ต with ๐บ acting by
(๐.๐)(๐) = ๐(๐(๐โ1.๐))
and ฮฉ = {0, 1} with trivial ๐บ-action. So the forgetful functor [๐บ, Set] โSet is logical, as is the functor which equips a set ๐ด with trivial ๐บ-action.Moreover, even if ๐บ is infinite, [๐บ, Set] is a topos and this inclusion[๐บ, Set๐] โ [๐บ, Set] is logical. Similarly if ๐ข is a large group (i.e. theunderlying space may not be a set), then [๐ข, Set] is a topos.
5. Let C be a category such that every C/๐ด is equivalent to a finite category.Then [Cop, Set๐] is a topos. Similarly if C is large but all C/๐ด are small,then [Cop, Set] is a topos. In partciular [On, Set] is a topos, but itโs notlocally small.
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Lemma 7.1. Suppose โฐ has finite limits and a subobject classifier. Thenevery monomorphism in โฐ is regular. In particular โฐ is balanced.
Proof. The universal monomorphism โค โถ 1 โฃ ฮฉ is split and hence regular. Butany pullback of a regular mono is regular: if ๐ is an equaliser of (๐, โ) then๐พโ(๐) is an equaliser of (๐๐, โ๐). The second assertion follows since a regularmono that is also epic is an isomorphism.
Given an object ๐ด in a topos โฐ, we write ๐๐ด for the exponential ฮฉ๐ด. andโ๐ดโฃ ๐๐ด ร ๐ด for the subobject corresponding to ev โถ ๐๐ด ร ๐ด โ ฮฉ. Thishas the property that, for any ๐ต and any ๐ โถ ๐ โฃ ๐ต ร ๐ด, there is a uniqueโ๐โ โถ ๐ต โ ๐๐ด such that
๐ โ๐ด
๐ต ร ๐ด ๐๐ด ร ๐ด
๐
โ๐โร1ฮ
is a pullback.
Definition (power-object). By a power-object for ๐ด in a category โฐ withfinite limits, we mean an object ๐๐ด equipped with โ๐ดโ ๐๐ด ร ๐ด satisfyingth above.
We say โฐ is a weak topos if every ๐ด โ obโฐ has a power-object.Similarly we say ๐น โถ โฐ โ โฑ is weakly logical if ๐น(โ๐ด) โฃ ๐น(๐๐ด) ร ๐น๐ด
is a power-object for ๐น๐ด for every ๐ด โ obโฐ.
A power-object for the termimal object is the same as a subobject classifier.This is an ad hoc defintion and we will soon show that โฐ is cartesian closed
and therefore we can safely drop the adjective โweakโ. Consequently this maybe taken as the definition of topos.
Lemma 7.2. ๐ is a functor โฐop โ โฐ . Moreover it is self-adjoint on theright.
Compare this with the contravariant power set functor on Set, which is aspecial case.
Proof. Given ๐ โถ ๐ด โ ๐ต, we define ๐๐ โถ ๐๐ต โ ๐๐ด to correspond to thepullback
๐ธ๐ โ๐ต
๐๐ต ร ๐ด ๐๐ต ร ๐ต1ร๐
For any โ๐โ โถ ๐ โ ๐๐ต, itโs easy to see that (๐๐)โ๐โ corresponds to (1๐ถร๐)โ(๐),hence ๐ โฆ ๐๐ is functorial. For any ๐ด and ๐ต, we have a bijection betweensubobjects of ๐ด ร ๐ต and of ๐ต ร ๐ด given by composition with (๐2, ๐1) โถ ๐ด ร ๐ต โ๐ต ร ๐ด. This yields a (natural) bijection between morphisms ๐ด โ ๐๐ต and๐ต โ ๐๐ด.
We write {}๐ด โถ ๐ด โ ๐๐ด (pronounced โsingletonโ) for the morphism corre-sponding to (1๐, 1๐) โถ ๐ด โฃ ๐ด ร ๐ด.
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Lemma 7.3. Given ๐ โถ ๐ด โ ๐ต, {}๐ต๐ corresponds to (1๐ด, ๐) โถ ๐ด โฃ ๐ด ร ๐ตand (๐๐){}๐ต corresponds to (๐, 1๐ด) โถ ๐ด โฃ ๐ต ร ๐ด.
Proof. The square๐ด ๐ต
๐ด ร ๐ต ๐ต ร ๐ต
๐
(1,๐) (1,1)๐ร1
is a pullback. Similarly for the second assertion.
Corollary 7.4.
1. {}๐ด โถ ๐ด โ ๐ด is monic.
2. ๐ is faithful.
Proof.
1. If {}๐ = {}๐ then (1๐ด, ๐) and (1๐ด, ๐) are isomorphic as subobjects of๐ด ร ๐ต, which forces ๐ = ๐.
2. Similarly if ๐๐ = ๐๐ then (๐๐){} = (๐๐){} so we again deduce ๐ = ๐.
Given a mono ๐ โถ ๐ด โฃ ๐ต is โฐ, we define โ๐ โถ ๐๐ด โ ๐๐ด to correspond tothe composite
โ๐ด ๐๐ด ร ๐ด ๐๐ด ร ๐ต.1ร๐
Then for any โ๐โ โถ ๐ถ โ ๐๐ด, (โ๐)โ๐โ correponds to
๐ ๐ถ ร ๐ด ๐ถ ร ๐ต๐ 1ร๐
so ๐ โฆ โ๐ is a functor Mono(โฐ) โ โฐ.
Lemma 7.5 (Beck-Chevalley condition). Suppose
๐ท ๐ด
๐ต ๐ถ
โ
๐ ๐๐
is a pullback with ๐ monic. Then the diagram
๐๐ด ๐๐ถ
๐๐ท ๐๐ต
โ๐
๐โ ๐๐
โ๐
commutes.
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Proof. Consider the diagram
๐ธ๐ โ๐ด
๐๐ด ร ๐ท ๐๐ด ร ๐ด
๐๐ด ร ๐ต ๐๐ด ร ๐ถ
1รโ
1ร๐ 1ร๐1ร๐
The lower square is a pullback so the upper square is a pull back. This isequivalent to the composite is a pullback.
Theorem 7.6 (Parรฉ). The functor ๐ โถ โฐop โ โฐ is monadic.
Proof. It has a left adjoint ๐ โถ โฐ โ โฐop by 7.5 (the โself-adjoint on the rightโlemma). Itโs faithful by 7.7 (ii) and hence reflects isomorphisms by 7.3. โฐop hascoequalisers since โฐ has equalisers. Suppose
๐ด ๐ต๐
๐๐
is a coreflexive pair in โฐ, then ๐ and ๐ are (split) monic and the equaliser๐ โถ ๐ธ โ ๐ด makes
๐ธ ๐ด
๐ด ๐ต
๐
๐ ๐
๐
a pullback square. Since any cone over
๐ด
๐ด ๐ต
๐
๐
has both legs equal so by Beck-Chevalley condition we have (๐๐)(โ๐) = (โ๐)(๐๐).But we also have (๐๐)(โ๐) = 1๐น๐ด since
๐ด ๐ด
๐ด ๐ต
1
1 ๐
๐
is a pullback, and similarly (๐๐ธ)(โ๐) = 1๐๐ธ. So
๐๐ต ๐๐ด ๐๐ธ๐๐
๐๐
โ๐
๐๐
โ๐
is a split coequaliser and in particular a coequaliser. Hence by 5.13 ๐ is monadic.
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Corollary 7.7.
1. A weak topos has finite colimits. Moreover if it has any infinite limitsthen it has the corresponding colimits. In particular if it is completethen it is cocomplete.
2. If a weakly logical functor has a left adjoint then it has a right adjoint.
Proof.
1. ๐ creates all limits which exist, by 5.e.
2. By definition if ๐น is weakly logical then
โฐop โop
โฐ โ
๐น
๐ ๐
๐น
commutes up to isomorphism. So this follows from 5.16.
Lemma 7.8. Let โฐ be a category with finite limits and suppose ๐ด โ obโฐhas a power-object ๐๐ด. Then, for any ๐ต, ๐ตโ(๐๐ด) is a power-object for ๐ตโ๐ดin โฐ/๐ต.
Proof. Given๐ถ
๐ต
๐ we have a pullback square
๐ถ ร ๐ด ๐ต ร ๐ด
๐ถ ๐ต
๐ร1
๐1 ๐1
๐
so ฮฃ๐ต(๐ ร ๐ตโ๐ด) โ ๐ถ ร ๐ด. Hence
Subโฐ/๐ต(๐ ร ๐ตโ๐ด) โ Subโฐ(๐ถ ร ๐ด),
but if โ โถ ๐ถ โ ๐๐ด corresponds to๐
๐ถ ร ๐ด
then the upper square of the diagram
๐ ๐ตร โ๐ด
๐ถ ร ๐ด ๐ต ร ๐๐ด ร ๐ด
๐ต
(๐,โ)ร1๐ด
๐๐1 ๐1
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is a pullback. So๐ต ร ๐๐ด
๐ต
๐1 equipped with ๐ตโ(โ๐ด) โฃ ๐ตโ(๐๐ด ร ๐ด) is a power
object for ๐ตโ๐ด.
Theorem 7.9. Suppose โฐ is a weak topos. Then for any ๐ต โ obโฐ, โฐ/๐ต isa weak topos and ๐ตโ โถ โฐ โ โฐ/๐ต is weakly logical.
Proof. The second assertion follows from the previous lemma. For the first,
we need to construct a power object for an arbitrary๐ด
๐ต
๐ in โฐ/๐ต. Then the
pullbackฮฃ๐ต(๐ ร ๐) ๐ด
๐ถ ๐ต
๐
๐
is a subobject of ๐ถ ร ๐ด, namely the equaliser of
๐ถ ร ๐ด ๐ต๐๐1
๐๐2
Define โง โถ ๐๐ด ร ๐๐ด โ ๐๐ด to correspond to the intersection of ๐โ13(โ๐ดโฃ
๐๐ดร๐ด) and ๐โ23(โ๐ดโฃ ๐๐ดร๐ด) and dfine ๐1๐ด โฃ ๐๐ดร๐๐ด to be the equaliser of
๐๐ด ร ๐๐ด ๐๐ด.โง
๐1Then, for any ๐ถ, ๐ถ ๐๐ด ร ๐๐ด(โ๐โ,โ๐โ) factors through
๐1๐ด if and only if ๐ โค ๐ in Subโฐ(๐ถ ร ๐ด). Now form the pullback
๐ ๐1๐ด
๐๐ด ร ๐ต ๐๐ด ร ๐๐ต ๐๐ด ร ๐๐ด
(โ,๐)1ร{} 1ร๐๐
Given any๐ถ
๐ต
๐ morphisms ๐โโโ ๐ in โฐ/๐ต correspond to morphisms ๐ถ
โโโโ ๐๐ด
such that the subobject named by โโ is contained in that named by (๐๐)({})๐.But the latter is indeed ฮฃ๐ต(๐ ร ๐) โฃ ๐ถ ร ๐ด so ๐ is a power object for ๐ inโฐ/๐ต.
Corollary 7.10. A weak topos is locally cartesian closed. In particular, itis a topos.
Proof. For any ๐ โถ ๐ด โ ๐ต in โฐ, we can define (โฐ/๐ต)/๐) with โฐ/๐ด and ๐โ โถโฐ/๐ต โ โฐ/๐ด with pullback along ๐. Hence all such functors are weakly logical.
But ๐โ has a left adjoint ฮฃ๐ so by 7.7 (2) (in lecture 7.10 (ii)) it has a rightadjoint ฮ ๐. Hence by 6.3 โฐ/๐ต is cartesian closed for any ๐ต.
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Remark. It can be shown that a weakly logical functor is cartesian closed, andhence logical.
Corollary 7.11.
1. Any epimorphism in a topos is regular.
2. Any ๐ โถ ๐ด โ ๐ต in a topos factors uniquely up to isomorphism as
๐ด ๐ผ ๐ต๐ ๐
Proof. โฐ is locally cartesian closed by the above corollary and has coequalisersby 7.10(i) so by 6.7 every ๐ factors uniquely as regular epi plus mono. If ๐ itselfis epic then the monic part of the factorisation is isomorphism by 7.3, so ๐ isregular epic.
7.1 Sheaves and local operators*Recall that Sh(๐) โ [๐ช(๐)op, Set] is a full subcategory closed under limits. Infact itโs reflective and the reflector ๐ฟ โถ [๐ช(๐)op, Set] โ Sh(๐) preserves finitelimits. This suggests considering reflective subcategories ๐ โ โฐ for which thereflector preserves finite limits (equivalently, pullbacks).
Lemma 7.12. Given such a reflective subcategory and a monomorphism๐ดโฒ โฃ ๐ด in โฐ, define ๐(๐ดโฒ) โฃ ๐ด by the pullback diagram
๐(๐ดโฒ) ๐ฟ๐ดโฒ
๐ด ๐ฟ๐ด๐๐ด
Then ๐ดโฒ โฆ ๐(๐ดโฒ) is a closure operation on Subโฐ(๐ด) and commutes withpullback along a fixed morphism of โฐ.
By โclosureโ we mean an order-preserving inflationary idempotent operator.
Proof. Since๐ด ๐ฟ๐ดโฒ
๐ด ๐ฟ๐ด
๐๐ดโฒ
๐๐ก๐๐ด
and ๐ดโฒ โค ๐ดโณ in Sub(๐ด) implies ๐ฟ๐ดโฒ โค ๐ฟ๐ดโณ in Sub(๐ฟ๐ด) and hence ๐(๐ดโฒ) โค๐(๐ดโณ).
Since ๐ฟ๐ is an isomorphism,
๐ฟ๐ดโฒ ๐ฟ๐ฟ๐ดโฒ
๐ฟ๐ด ๐ฟ๐ฟ๐ด
๐ฟ๐๐ดโฒ
๐ฟ๐๐ด
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7 Toposes
is a pullback, and since ๐ฟ preserves pullbacks we deduce ๐ฟ๐(๐ดโฒ) โ ๐ฟ๐ดโฒ inSub(๐ฟ๐ด). Hence ๐(๐(๐ดโฒ)) โ ๐(๐ดโฒ).
For stability under pullback, suppose
๐ดโฒ ๐ตโฒ
๐ด ๐ต๐
is a pullback. Then in the cube
๐(๐ดโฒ) ๐ฟ๐ดโฒ
๐(๐ตโฒ) ๐ฟ๐ตโฒ
๐ด ๐ฟ๐ด
๐ต ๐ฟ๐ต
๐๐ด
๐ ๐ฟ๐
๐๐ต
the front, back and right faces are pullbacks, whence the left face is too.
Definition (local operator). Let โฐ be a topos. By a local operator in โฐ wemean a morphism ๐ โถ ฮฉ โ ฮฉ satisfying the commutative diagrams
1 ฮฉ ฮฉ
ฮฉ
โค
โค๐
๐
๐
ฮฉ1 ฮฉ1
ฮฉ ร ฮฉ ฮฉ ร ฮฉ๐ร๐
where ฮฉ1 is the order relation on ฮฉ define in 7.12.Given a closure operator on subobjects in the above lemma, define ๐ฝ โฃ
ฮฉ to be the closure of โค โถ 1 โฃ ฮฉ and ๐ โถ ฮฉ โ ฮฉ to be the classifying mapof ๐ฝ โฃ ฮฉ. Then, for any ๐ดโฒ โฃ ๐ด with classifying map ๐๐ โถ ๐ด โ ฮฉ, thecomposite ๐๐๐ classifies ๐(๐ดโฒ) โฃ ๐ด.
Given a pullback-stable cloure operation ๐ on subobjects, we say ๐ดโฒ โฃ ๐ด isdense if ๐(๐ดโฒ) โฃ ๐ด is isomorphism, and closed if ๐ดโฒ โฃ ๐(๐ดโฒ) is isomorphism.
Lemma 7.13. Suppose given a commutative square
๐ตโฒ ๐ดโฒ
๐ต ๐ด
๐โฒ
๐ ๐
๐
with ๐ dense and ๐ closed. Then there is a unique ๐ โถ ๐ต โ ๐ดโฒ with ๐๐ = ๐
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(and ๐๐ = ๐ โฒ).Proof. We have ๐ โค ๐โ(๐) in Sub(๐ต) so
1๐ต โ ๐(๐) โค ๐โ(๐(๐)) โ ๐โ(๐)
so we define ๐ as๐ต ๐โ(๐ดโฒ) ๐ดโฒโ
Note that ๐(๐ดโฒ) may be characterised as the unique (up to isomorphism)subobject ๐ดโณ such that ๐ดโฒ โฃ ๐ดโณ is dense and ๐ดโณ โฃ ๐ด is closed.
Lemma 7.14. Suppose ๐ is induced as in Lemma 7.12 by a reflector ๐ฟ โถโฐ โ ๐ preserving finite limits. Then an object ๐ด of โฐ belongs to ๐ (up toisomorphism) if and only if, given any diagram
๐ตโฒ ๐ด
๐ต
๐โฒ
๐
with ๐ dense, there exists a unique ๐ โถ ๐ต โ ๐ด with ๐๐ = ๐ โฒ.
Proof. Note first that ๐ is dense if and only if ๐ฟ๐ is an isomorphism: โธfollows from the definition. โน follows since, by the proof of Lemma 7.12, weknow ๐ฟ(๐ตโฒ) and ๐ฟ(๐(๐ตโฒ)) are isomorphic in Sub(๐ต).
Given this, if ๐ด is in ๐ then the given diagram extends uniquely to
๐ตโฒ ๐ฟ๐ตโฒ ๐ด
๐ต ๐ฟ๐ต
๐๐ตโฒ
โ ๐๐ต
Conversely, suppose ๐ด satisfies the condition. Let ๐ ๐ด๐
๐be the kernel-
pair of ๐๐ด โถ ๐ด โ ๐ฟ๐ด and ๐ โถ ๐ด โฃ ๐ the factorisation of (1๐ด, 1๐ด) through(๐, ๐). Since ๐ฟ๐๐ด is isomorphism and ๐ฟ preservse pullbacks, ๐ฟ๐ is isomorphismso ๐ is dense. This forces ๐ = ๐ so ๐๐ด is monic. And ๐๐ด is dense so we get aunique ๐ โถ ๐ฟ๐ด โ ๐ด with ๐๐๐ด = 1๐ด. Now ๐๐ด๐๐๐ด = ๐๐ด and since ๐ฟ๐ด satisfies thecondition we have ๐๐ด๐ = 1๐ฟ๐ด.
We say ๐ด is a sheaf (for ๐, or for ๐) it it satisfies the condition in the previouslemma. Given a local operator ๐ on โฐ, we write sh๐(โฐ) for the full subcategoryof ๐-sheaves in โฐ.
Our aim is to show sh๐(โฐ) is a topos.
Lemma 7.15. sh๐(โฐ) is closed under limits in โฐ and an exponential ideal.
Proof. The first assertion follows since the definition involves only morphismswith codomain ๐ด. For the second, note that if ๐ โถ ๐ตโฒ โฃ ๐ต is dense the so isร1๐ถ โถ ๐ตโฒ ร ๐ถ โฃ ๐ต ร ๐ถ for any ๐ถ (since itโs ๐โ
1(๐)) and so if ๐ด is a sheaf thenany morphism ๐ตโฒ โ ๐ด๐ถ extends uniquely to a morphism ๐ต โ ๐ด๐ถ.
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Lemma 7.16. If ๐ด is a sheaf then a subobject ๐ โถ ๐ดโฒ โฃ ๐ด in โฐ is a sheafif and only if it is closed.
Proof.
โข โธ : Immediate from Lemma 7.13.
โข โน : Consider๐ดโฒ ๐(๐ดโฒ) ๐ด๐ ๐
๐ is dense so if ๐ด is a sheaf we get a unique ๐ โถ ๐(๐ดโฒ) โ ๐ดโฒ with ๐๐ = 1๐ดโฒ .But ๐(๐ดโฒ) is a sheaf so ๐๐๐ = ๐ and thus ๐๐ = 1๐(๐ดโฒ).
We define ฮฉ๐ โฃ ฮฉ to be the equaliser of ฮฉ ฮฉ.๐
1ฮฉ
Then for any ๐ด,
morphisms ๐ด โ ฮฉ๐ corresponding to closed subobjects of ๐ด.
Lemma 7.17. ฮฉ๐ is a ๐-sheaf.
Proof. We want to show that if ๐ โถ ๐ต โฃ ๐ด is a dense mono the pullback along๐ yields a bijection from closed subobjects of ๐ด to closed subobjects of ๐ต. If๐ โถ ๐ดโฒ โฃ ๐ด is closed then the pullback
๐ตโฒ ๐ดโฒ
๐ต ๐ด
๐โฒ
๐โฒ ๐
๐
๐โฒ is dense so ๐ดโฒ โฃ ๐ด is the closure of ๐ตโฒ โฃ ๐ต โฃ ๐ด. It remains to show thatif ๐ตโฒ โฃ ๐ต is closed, it is iso up to the pullback of its closure in ๐ด. But (writing๐ดโฒ โฃ ๐ด for the closure) we have a factorisation ๐ตโฒ โ ๐โ๐ดโฒ which is dense since๐ตโฒ โ ๐ดโฒ is dense, and closed since ๐ตโฒ โ ๐ต is closed.
Theorem 7.18. For any local operator ๐ on โฐ, sh๐(โฐ) is a topos. Moreoveritโs reflective in โฐ and the reflector preserves finite limits.
Proof. sh๐(โฐ) is cartesian closed and has a subobject classifier ฮฉ๐ by the lemma.To construct the reflect, consider the composite
๐ด ฮฉ๐ด ฮฉ๐ด๐
{} ๐๐ด
this corresponds to the closure (๐, ๐) โถ ๐ด โฃ ๐ด ร ๐ด of the diagonal subobject(1๐ด, 1๐ด) โถ ๐ด โฃ ๐ด ร ๐ด. I claim (with proof omitted) that ๐ด ๐ด
๐
๐is the
kernel-pair of ๐. Hence any morphism ๐ โถ ๐ด โ ๐ต where ๐ต is a sheaf satisfies๐๐ = ๐๐. So if we form the image
๐ด ๐ผ ฮฉ๐ด๐
๐ ๐
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7 Toposes
of ๐ any such ๐ factors uniquely through ๐.Now ฮฉ๐ด
๐ is a sheaf by the lemmas so if we form the closure ๐ฟ๐ด โฃ ฮฉ๐ด๐ of ๐,
we get a morphism ๐ด โ ๐ฟ๐ด through which any morphism from ๐ด to a sheaffactors uniquely. Hence ๐ฟ becomes a functor โฐ โ sh๐(โฐ), left adjoint to theinclusion.
By 6.13, we know ๐ฟ preserves finite products. In fact it preserves equalisersas well (can be checked but omitted here).
Finally, we state the relation between topos and the topos Grothendieck wasinterested in (i.e. sheaves):
Theorem 7.19. For a category โฐ, TFAE:
1. โฐ is a topos, complete and locally small and has a separating set ofobjects.
2. there exists a small category C and a local operators on [Cop, Set]such that
โฐ โ sh๐([Cop, Set]).
Sketch of proof.
1. 1 โน 2: since sh๐([Cop, Set]) has given properties.
2. 1 โธ 2: take C to be the full subcategory of โฐ on the separating set andconsider
โฐ [โฐop, Set] [Cop, Set].๐
66
Index
adjoint functor theoremgeneral, 31special, 32
adjoints on the right, 17adjunction, 16algebra, 36arrow category, 17
Beck-Chevalley condition, 58
category, 2balanced, 9cartesian closed, 47cocomplete, 27complete, 27discrete, 3locally cartesian closed, 49locally small, 10opposite, 3skeletal, 8small, 2well-powered, 32
closed, 63closure, 62coequaliser, 13, 25
split, 41comonad, 35cone, 23congruence, 2coproduct, 13, 24counit, 19create limit, 27
dense, 63detecting family, 14detector, 14diagram, 23direct limit, 25
Eilenberg-Moore comparisonfunctor, 39
epimorphism, 9equaliser, 13, 25equivalence, 6exponentiable object, 47exponential ideal, 52
Frobenius reciprocity, 52
functor, 3contravariant, 4essentially surjective, 7faithful, 7forgetful, 4full, 7representable, 12
Heyting semilattice, 47
initial object, 16injective, 15inverse limit, 25
Kleisli category, 37
local operator, 63logical functor, 55, 57
monad, 35monadic tower, 44monadicity, 40monadicity theorem, 42monoid, 3monomorphism, 9
regular, 13split, 9
multiplication, 35
natural transformation, 5
Parรฉ theorem, 59power-object, 57preserve limit, 27presheaf, 53product, 13, 24projective, 15pullback, 25pushout, 25
quotient object, 32
reflection, 21reflective subcategory, 21reflexivity, 41right adjoint preserves limits, 29
separating family, 14separator, 14sheaf, 53sieve, 56
67
Index
singleton, 57skeleton, 8slice category, 47Stone-ฤech compactification, 22,
34subobject, 32subobject classifier, 55subterminal, 53
terminal object, 17topos, 55, 57triangular identities, 19
unit, 19, 35universal element, 12
Yoneda embedding, 11Yoneda lemma, 10
68