Kleisli Monoids
Proefschrift ingediend met het oog op het behalen van de graad Master in de Wiskunde
Stijn T´ oth
Promotor: Prof. Dr. Eva Colebunders
Academiejaar 2013-2014
For a Set-monad T we consider (T, 2)-algebras as a generalization of Barr’s presentation of topological spaces in terms of ultrafilter convergence. These lax algebras require the existence of lax extensions of T. Through these lax algebras we describe different topological structures via a convergence struc- ture.
By generalizing the description of topological spaces by neighborhood systems, we obtain the notion of Kleisli monoids. In general the category of Kleisli monoids is a neighborhood-like presentation of categories of lax algebras. A notable advantage of this approach is that it does not require explicitly the lax extension of the associated monad. In this thesis, we present, specifically, ordered sets, topological spaces and closure spaces as Kleisli monoids and lax algebras.
i
Voor een Set-monadT beschouwen we (T, 2)-algebra’s als een veralgemeening van Barr’s voorstelling van topologische ruimten in termen van ultrafilter convergentie. Deze lax algebra’s vereisen het bestaan van lax extensies van T. Via deze lax algebra’s beschrijven we verschillende topologische structuren door een convergentiestructuur.
Door de beschrijving van topologische ruimten via omgevingensystemen te veralgemenen, verkrijgen we de notie van Kleisli mono¨ıden. In het alge- meen is de categorie van Kleisli mono¨ıden een omgevingen-achtige presen- tatie van categorie¨en van lax algebra’s. Een noemenswaardig voordeel van deze benadering is dat het bestaan van een lax extensie van de ge¨associeerde monad niet expliciet gebruikt wordt. In deze thesis beschrijven we specifiek geordende verzamelingen, topologische ruimten en closure ruimten als Kleisli mono¨ıden en lax algebra’s.
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Abstract i
Samenvatting ii
Contents iii
Acknowledgments v
Introduction vi
Inleiding x
1 Basic concepts 1
1.1 The compositional structure of relations . . . 1
1.2 Orders . . . 3
1.3 Modules . . . 3
1.4 Adjunctions for ordered sets . . . 5
1.5 Closure operation and interior operation . . . 6
1.6 Completeness and lattices . . . 7
1.7 Filters, ultrafilters, cliques and ultracliques . . . 8
2 Monads 15 2.1 Monads . . . 15
2.2 Kleisli triples . . . 26
3 Lax extensions 30 3.1 Lax functors and their transformations . . . 30
3.2 Lax extensions of functors . . . 31
3.3 Lax extensions of monads . . . 36
3.4 The Barr extension . . . 37
3.5 The Beck-Chevalley condition . . . 41
3.6 The Barr extension of a monad . . . 43
iii
4 Algebra structures and unitary relations 46
4.1 Eilenberg-Moore algebra . . . 46
4.2 Lax algebra . . . 48
4.3 The Kleisli category . . . 54
4.4 Kleisli convolution . . . 56
4.5 Unitary (T, 2)-relations . . . 59
4.6 Associativity of unitary (T, 2)-relations . . . 60
4.7 Fundamental example of a lax algebra . . . 64
4.8 -algebras . . . 69
4.9 Induced order on lax algebras . . . 71
5 Kleisli Monoids 73 5.1 Topological spaces via neighborhood filters . . . 74
5.2 Power-enriched monads . . . 77
5.3 T-monoids . . . 82
5.4 The Kleisli extension . . . 87
5.5 Topological spaces via filter convergence . . . 92
6 Initial extensions 98 6.1 Algebraic functors . . . 98
6.2 Initial extensions . . . 99
6.3 Sup-dense and interpolating monad morphisms . . . 103
6.4 (S, 2)-categories as Kleisli monoids . . . 106
6.5 Compact Hausdorff closure spaces . . . 112
Bibliography 117
Index 119
In the first place, I would like to thank my promotor professor Eva Colebunders for introducing me to the research of monoidal topology and for the profes- sional guidance she has given me the past year. I also want to thank all the professors I have encountered in the past five years for teaching and inspiring me in mathematics.
Next, I want to thank my parents, brothers and partner, for standing by my side and motivating me, although they don’t always understand the topics I am talking about.
For the support and moments of relaxation they provide, I want to thank all my friends and classmates.
Finally, I wish you take pleasure in reading my thesis.
Stijn T´oth May 2014
v
Monoidal Topology describes an active research that provides a powerful uni- fying framework and theory for ordered, metric and topological structures.
For this framework we use category theory as language. By this theory, we will describe, inter alia, ordered1 spaces and topological spaces by a conver- gence structure.
Monoids are a simple structure, i.e. a set with an associative multipli- cation and a unit. By viewing these conditions in diagrams we can generalize this notion. Instead of a set, we use an endofunctor T : Set → Set and the multiplication and unit become natural transformations m : T T → T and e : 1Set → T. We have a monad, when the diagrams for the natural transfor- mations, similar to the diagrams of the multiplication and the unit of a monoid, commute. In Chapter 2 we give different examples of monads.
With a monad T = (T, m, e) Eilenberg and Moore (1965) constructed a category SetT with objects sets X equipped with a map a : T X → X making the two diagrams
T T X T a //
mX
T X
a
X eX //
1X !!
T X
a
T X a //X X
commutative and with appropriate morphisms. This category will be called the Eilenberg-Moore category and its objects are the Eilenberg-Moore alge- bras.
The starting point for monoidal topology is the proof by Manes [10] in 1969 that compact Hausdorff spaces are precisely the Eilenberg-Moore alge- bras of the ultrafilter monad = (β, m, e). In this description, the map a : βX → X of a compact Hausdorff space X assigns to every ultrafilter its unique point of convergence in X, requiring the two basic axioms of an Eilenberg-Moore algebra a(βa(U )) = a(mX(U )) and a(eX(x)) = x for all
1By an order on a space X, we mean a reflexive and transitive relation. An order does
not need to satisfy the antisymmetric axiom.
vi
U ∈ ββX and x ∈ X. We will rewrite these axioms by using the principal ultrafilter on x
eX(x) = ˙x = {A ⊆ X | x ∈ A}
and the Kowalsky sum of U
mX(U ) = ΣU = {A ⊆ X | {U ∈ βX | A ∈ U} ∈ U }.
These ultrafilters, together with the notations V → z instead of a(V) = z and U → V instead of βa(U ) = V, give a more intuitive form of the basic axioms, i.e.
U → V & V → z =⇒ ΣU → z and
˙x → x for all U ∈ ββX, V ∈ βX and x, z ∈ X.
For metric spaces and convergence of sequences, we can describe the first axiom in the following way. Take a sequence (xn)n∈Nthat converges to x ∈ X, then for a collections of sequences, where the first sequence converges to x1, the second one to x2, and so on, we take the diagonal sequence, i.e.
the first element of the first sequence, the second element of the second sequence, . . . Then the diagonal sequence converge to x. The Kowalsky sum is a generalization of the diagonal sequence.
In 1970 Barr [1] observed that in an arbitrary topological space it is no longer guaranteed that every ultrafilter has a point of convergence (compact- ness) and that there is at most one such point (Hausdorffness) and therefore we will need a relation instead of a map. Such a relation a describing conver- gence has to satisfy a(βa(U )) ≤ a(mX(U )) (transitivity) and x ≤ a(eX(x)) (reflexivity) for allU ∈ ββX and x ∈ X. Thus a topology on X is described by two simple axioms on a convergence relation between ultrafilters and points. In the discussion above we glossed over an important point: having the Set-functor β, one knows what βa is when a is a map, but not necessar- ily when a is just a relation. To define βa, when a is a relation, there is a straightforward solution.
By changing the ultrafilter monad = (β, m, e) by any other monad T = (T, m, e), these conditions on a relation a : T X →7 X give rise to the notion of lax algebras. Together with the appropriate morphisms, they form a category of lax algebras. Again, we glossed over the issue of defining T a when a is a relation. While there is for the ultrafilter monad a straight- forward solution, in general we are confronted with the issue of extending a monad T = (T, m, e) on Set to Rel. For our purposes it suffices that this
extension is lax, not a genuine monad on Rel. The definition and examples of lax extensions of monads can be found in Chapter 3.
In section 4.8 we demonstrate the connection between the Eilenberg- Moore algebras and the lax algebras for the ultrafilter monad. We will prove the result of Manes [10], but in a different way. We use the result of Barr [1], altough, historically seen, Barr based himself on the paper of Manes.
With the concept of lax algebras developed, one tries to describe uni- formly all structures that seem to matter in topology, inter alia, ordered sets, closure spaces and topological spaces. Ordered sets are simple to describe as lax algebras. Considering the case where T is the identity monad on Set and the extension of the identity monad on Set is the identity monad on Rel, we see that the lax algebra requirements can be denoted by a · a ≤ a and 1X ≤ a, with ≤ to be read as set-theoretical inclusion if the relation a is pre- sented as a ⊆ X × X. These requirements are precisely the transitivity and the reflexivity of an order on X. The appropriate morphisms for lax algebras are in this case precisely the order-preserving maps. In Chapter 4 and the following chapters we give more examples of lax algebras.
When we look at topological spaces and their description in terms of ultrafilter convergence, one might ask if it is possible to trade the ultrafilter convergence for filter convergence. To answer this question, it is useful to axiomatize topologies on a set X in terms of neighborhood filters of points.
Therefore we consider maps ν : X → F X into the set F X of filters on X, to be thought of as assigning to each point its neighborhood filter. Ordering such maps pointwise by reverse inclusion and considering them as morphisms of the Kleisli category SetFof the filter monadF = (F, m, e), i.e using the Kleisli composition ◦, one obtains another monoidal characterization of topologies on a set X by the axioms:
ν ◦ ν ≤ ν and eX ≤ ν.
In this way topological spaces are represented as Kleisli monoids (X, ν), or as F-monoids [13].
In the previous description the filter monad F may be replaced by any monad T on Set when the sets T X carry a complete-lattice order, suitably compatible with the monad operations. Such a monad T may be charac- terized via a monad morphism τ : P → T, with P the powerset monad, and is called power-enriched. The equivalence between neighborhood systems and filter convergence, given by
ν(x) ⊆ F ⇔ F → x
can be generalized at the level of a power-enriched monad T.
With an appropiate lax extension ofT to Rel, this correspondence yields a presentation of Kleisli monoids as lax algebras. By applying this presentation to the filter monad, we can see that the description of topological spaces remains valid if we trade ultrafilter convergence for filter convergence. The results regarding Kleisli monoids are explained in Chapter 5.
Since we can describe a topological space with two axioms in terms of ultrafilter convergence and with two axioms in terms of filter conver- gence, we have an isomorphism between the category of lax algebras of the ultrafilter monad and the category of lax algebras of the filter monad. In Chapter 6 we will try to generalize this interaction for an arbitrary monad morphism α :S → T. We discover which conditions are sufficient on the monad morphism in order to obtain an isomorphism between the category of lax algebras of S and T, respectively.
Closure spaces are spaces with a closure operation, that is a monotone map c : P X → P X such that c is reflexive (A ⊆ c(A)) and idempotent (c(c(A)) = c(A)). When such a closure operation c is finitely additive and c(∅) = ∅, it defines a topological space. Closure spaces, on their own, are an interesting structure to investigate. In 1940 Birkhoff’s motivation [2] for considering closure spaces came from the observation that the collection of closed sets of a closure space forms a complete lattice. Closure spaces have now applications in data analysis, knowledge representation, quantum logic and in the representation theory of physical systems.
Having at hand lax algebraic characterizations of topological spaces (with either the ultrafilter or filter monad), we wonder whether a similar description is possible for closure spaces. To find such a description, we will introduce the concept of a clique on X, as a collection C of subsets of X such that two elements of C have a non-empty intersection if ∅ /∈ C and that C is an up-set. We call a clique C proper if ∅ /∈ C and we call a clique an ultraclique if it is proper and maximal with respect to the inclusion. By defining these concepts we can write closure spaces in the language of monoidal topology, as lax algebras and Kleisli monoids.
As a final result we investigate the Eilenberg-Moore algebras for the ultra- clique monad. We define compactness and Hausdorffness for closure spaces and prove that the category of Eilenberg-Moore algebras of the ultraclique monad is isomorphic to the category of compact Hausdorff closure spaces, inspired by the result of Manes [10].
Monoidale Topologie beschrijft een actief onderzoek dat een kader en een theorie biedt, die geordende, metrische en topologische stucturen op eenzelfde manier beschrijft. Als taal voor dit kader gebruiken we categorie theorie. Aan de hand van deze theorie zullen we, o.a., geordende2 ruimten en topologische ruimten door een convergentiestructuur beschrijven.
Mono¨ıden zijn eenvoudige structuren, i.e. een verzameling met een asso- ciative vermenigvuldiging en een eenheid. Door deze voorwaarden in dia- grammen te bekijken, kunnen we dit begrip veralgemenen. In plaats van een verzameling gebruiken we een endofunctor T : Set → Set en de vermenig- vuldiging en eenheid worden natuurlijke transformaties m : T T → T en e : 1Set → T. We krijgen een monad als de diagrammen voor de natuurlijke transformaties, gelijkaardig aan de diagrammen voor de vermenigvuldiging en de eenheid van een mono¨ıde, commuteren. In Hoofdstuk 2 geven we verschillende voorbeelden van monads.
Met een monad T = (T, m, e) construeerden Eilenberg en Moore (1965) een categorie SetT met als objecten verzamelingen X uitgerust met een functie a : T X → X die de twee diagrammen
T T X T a //
mX
T X
a
X eX //
1X !!
T X
a
T X a //X X
commutatief maakt en met gepaste morfismes. Deze categorie noemen we de Eilenberg-Moore categorie en zijn objecten zijn de Eilenberg-Moore algebra’s.
Het beginpunt voor monoidale topologie is het bewijs door Manes [10] in 1969 dat compacte Hausdorff ruimte exact de Eilenberg-Moore algebra’s voor de ultrafilter monad = (β, m, e) zijn. In deze beschrijving wijst de functie a : βX → X van een compacte Hausdorff ruimte X aan elke ultrafilter zijn uniek convergentiepunt toe, die voldoet aan de twee basis axioma’s van een
2Met een orde op een ruimte X bedoelen we een reflexieve en transitieve relatie. Een
orde moet niet voldoen aan het axioma van de anti-symmetrie.
x
Eilenberg-Moore algebra a(βa(U )) = a(mX(U )) en a(eX(x)) = x voor alle U ∈ ββX en x ∈ X. We zullen deze axioma’s herschrijven door gebruik te maken van de principale ultrafilter op x
eX(x) = ˙x = {A ⊆ X | x ∈ A}
en de Kowalsky som van U
mX(U ) = ΣU = {A ⊆ X | {U ∈ βX | A ∈ U} ∈ U }.
Deze ultrafilters, samen met de notatie V → z in plaats van a(V) = z en U → V in plaats van βa(U ) = V, geven een intu¨ıtievere vorm van de basis axioma’s, i.e.
U → V & V → z =⇒ ΣU → z en
˙x → x voor alle U ∈ ββX, V ∈ βX en x, z ∈ X.
Voor metrische ruimten en de convergentie van rijen kunnen we het eerste axioma op volgende manier beschrijven. Neem een rij (xn)n∈Ndie naar x ∈ X convergeert. Dan nemen we voor een collectie rijen, waarvan de eerste rij naar x1 convergeert, de tweede rij naar x2, etc. , de diagonaalrij, i.e. het eerste element van de eerste rij, het tweede element van de tweede rij, . . . Dan convergeert de diagonaalrij naar x. De Kowalsky som is de veralgemening van de diagonaalrij.
In 1970 observeerde Barr [1] dat in een willekeurige topologische ruimte het niet langer gegarandeerd is dat elke ultrafilter een convergentiepunt heeft (compactheid) en dat er hoogstens ´e´en zo’n punt is (Hausdorff) en we daarom een relatie nodig zullen hebben in plaats van een functie. Zo’n relatie a dat convergentie beschrijft moet voldoen aan a(βa(U )) ≤ a(mX(U )) (transi- tiviteit) en x ≤ a(eX(x)) (reflexiviteit) voor alle U ∈ ββX en x ∈ X.
Dus een topologie op X is beschreven door twee eenvoudige axioma’s op een convergentierelatie tussen ultrafilters en punten. In de bovenstaande discussie verdoezelden we een belangrijk punt: wanneer we de Set-functor β kennen, weten we wat βa is, als a een functie is, maar niet noodzakelijk als a slechts een relatie is. Om βa te bepalen, als a een relatie is, bestaat er een rechtstreekste oplossing.
Als we de ultrafilter monad = (β, m, e) vervangen door een andere monadT = (T, m, e), dan geven deze voorwaarden op een relatie a : T X →7 X aanleiding tot het begrip van lax algebra’s. Samen met geschikte morfismes, vormen ze de categorie van lax algebra’s. Opnieuw moeten we nog T a
defini¨eren als a een relatie is. Terwijl er voor de ultrafilter monad een recht- streekse oplossing bestaat, worden we in het algemeen geconfronteerd met de kwestie van een monadT = (T, m, e) op Set uit te breiden naar Rel. Voor onze doeleinden volstaat het dat deze extensie lax is en niet een volwaardige monad op Rel. De definitie en voorbeelden van lax extensies van monads zijn te vinden in Hoofdstuk 3.
In Sectie 4.8 tonen we de connectie tussen Eilenberg-Moore algebra’s en lax algebra’s voor de ultrafilter monad aan. We zullen het resultaat van Manes [10] bewijzen op een andere manier dan Manes het zelf heeft bewezen.
We gebruiken het resultaat van Barr [1], alhoewel, historisch gezien, Barr zich gebaseerd heeft op de paper van Manes.
Met het begrip “lax algebra” ontwikkeld, probeert men alle structuren die er toe doen in de topologie, zoals, geordende verzamelingen, closure ruimten en topologische ruimten, uniform te beschrijven. Geordende verzamelingen zijn eenvoudig te beschrijven als lax algebra’s. Beschouwen we het geval waar T de identiteit monad op Set is en de extensie van de identiteit monad op Set de identiteit monad op Rel is, dan kunnen de voorwaarden om een lax algebra te zijn beschreven worden door a · a ≤ a en 1X ≤ a, waarbij ≤ moet gelezen worden als de verzameling-theoretische inclusie als de relatie a voorgesteld wordt als a ⊆ X × X. Deze voorwaarden geven precies de transitiviteit en de reflexiviteit van een orde op X. De gepaste morfismes voor lax algebra’s zijn in dit geval precies de orde-bewarende functies. In Hoofdstuk 4 en de daarop volgende hoofdsukken geven we meer voorbeelden van lax algebra’s.
Wanneer we naar topologische ruimten en hun beschrijving in termen van ultrafilter convergentie kijken, kunnen we ons afvragen of het mogelijk is om de ultrafilter convergentie te veranderen door filter convergentie. Om deze vraag te beantwoorden, is het nuttig om topologie¨en op een verzameling X te axiomatiseren in termen van omgevingenfilters van punten. Daarom beschouwen we functies ν : X → F X naar de verzameling F X van alle filters op X, die we kunnen zien als aan elk punt zijn omgevingenfilter toe te kennen. Ordenen we zo’n functies puntgewijs door omgekeerde inclusie en beschouwen we ze als morfismes van de Kleisli categorie SetF van de filter monad F = (F, m, e), i.e. gebruik maken van de Kleisli samenstelling ◦, dan verkrijgen we een andere monoidale karakterisatie van topologie¨en op een verzameling X door de axioma’s:
ν ◦ ν ≤ ν and eX ≤ ν.
Op deze manier worden topologische ruimten voorgesteld als Kleisli mono¨ıden (X, ν), of als F-mono¨ıden [13].
In de voorgaande beschrijving mag de filter monad F vervangen worden door elke andere monad T op Set als de verzamelingen T X een volledige
tralie orde dragen dat compatibel is met de monad operaties. Zo’n monad T kan gekarakteriseerd worden via een monad morfisme τ : P → T, met P de powerset monad en noemen we power-enriched. De equivalentie tussen omgevingensystemen en filter convergentie, gegeven door
ν(x) ⊆ F ⇔ F → x
kan veralgemeend worden voor een power-enriched monad T.
Met een geschikte lax extensie van T naar Rel, houdt deze correspon- dentie een presentatie van Kleisli mono¨ıden als lax algebra’s in. Door deze presentatie toe te passen op de filter monad, zien we dat de beschrijving van topologische ruimten geldig blijft als we ultrafilter convergentie ruilen voor filter convergentie. De resultaten met betrekking tot Kleisli mono¨ıden zijn uitgelegd in Hoofdstuk 5.
Omdat we topologische ruimten kunnen beschrijven met twee axioma’s in termen van ultrafilter convergentie en met twee axioma’s in termen van filter convergentie, hebben we een isomorfisme tussen de categorie van lax alge- bra’s van de ultrafilter monad en de categorie van lax algebra’s van de filter monad. In Hoofdstuk 6 proberen we deze interactie te veralgemenen voor een willekeurig monad morfisme α :S → T. We ontdekken welke voorwaarden op de monad morfisme voldoende zijn om een isomorfisme te verkrijgen tussen de categorie van lax algebra van respectievelijk S en T.
Closure ruimten zijn ruimten met een closure operatie, i.e. een mono- tone functie c : P X → P X zodat c reflexief is (A ⊆ c(A)) en idempotent is (c(c(A)) ⊆ c(A)). Als zo’n closure operatie c eindig additief is en c(∅) = ∅, dan definieert c een topologische ruimte. Closure operaties, op zichzelf, zijn een interessante structuur om te onderzoeken. In 1940 kwam Birkhoff’s motivatie [2] om closure ruimten te beschouwen van de observatie dat de verzameling van gesloten verzamelingen van een closure ruimte een volledige tralie vormt. Closure ruimten hebben toepassingen in data analyse, knowl- edge representation, quantum logica en in de representatietheorie van fysische systemen.
Omdat we de lax algebra¨ısche karakterisaties van topologische ruimten (met ofwel de ultrafilter ofwel de filter monad) kunnen beschrijven, vragen we ons af of er een gelijkaardige beschrijving mogelijk is voor closure ruimten.
Om zo’n beschrijving te vinden, zullen we het begrip “clique” op X defini¨eren, als een collectie C van deelverzamelingen van X zodat twee elementen van C een niet-lege doorsnede hebben als ∅ /∈ C en dat C een up-set is. We noemen een clique C “proper” als ∅ /∈ C en we noemen een clique een ultraclique als ze proper is en maximaal ten opzichte van de inclusie. Door deze begrippen te defini¨eren, kunnen we closure ruimten schrijven in de taal van monoidale topologie, namelijk als lax algebra’s en als Kleisli mono¨ıden.
Als laatste onderzoeken we de Eilenberg-Moore algebra’s voor de ultra- clique monad. We defini¨eren compactheid en Hausdorff voor closure ruimten en bewijzen dat de categorie van Eilenberg-Moore algebra’s voor de ultra- clique monad isomorf is met de categorie van compact Hausdorff closure ruimten, ge¨ınspireerd door het resultaat van Manes [10].
Basic concepts
In the first chapter of this thesis we give a short introduction to concepts mostly known to a master student. It provides the notations, terminology and theory used in the rest of the thesis. This chapter is based on the book
“Monoidal Topology” [4], specifically on chapter II, written by Gavin J. Seal and Walter Tholen. The results on cliques and ultracliques are elaborated by the author.
1.1 The compositional structure of relations
A relation r from a set X to a set Y distinguishes those elements x ∈ X and y ∈ Y that are r-related; we write x r y if x is r-related to y. Hence, depending on whether we display r as a subset, a two-valued function or a multi-valued function via
r ⊆ X × Y, r : X × Y → {false, true} = {0, 1} = 2, r : X → P Y respectively, where P Y denotes the powerset of Y , x r y may be equivalently written as
(x, y) ∈ r, r(x, y) = true = 1, y ∈ r(x).
Writing r : X →7 Y when r is a relation from X to Y , we can “multiply” r with s : Y →7 Z via ordinary relational composition:
x (s · r) z ⇐⇒ ∃y ∈ Y : (x r y and y s z).
Writing r ≤ r0 (with r0 : X →7 Y ) when, equivalently,
r ⊆ r0, ∀x ∈ X ∀y ∈ Y : (r(x, y) |= r0(x, y)), ∀x ∈ X : (r(x) ⊆ r0(x)),
1
we see that the multiplication respects ≤, since
r ≤ r0, s ≤ s0 =⇒ s · r ≤ s0· r0. Moreover, relational composition is associative, so that
(t · s) · r = t · (s · r),
when t : Z →7 W, and for the identity relation 1X (with x 1X x0 =⇒ x = x0) one has r · 1X = r = 1Y · r. For r : X →7 Y one has the opposite (or dual) relation r◦ : Y →7 X with
y r◦ x ⇐⇒ x r y, for all x ∈ X and y ∈ Y, which satisfies
(s · r)◦ = r◦· s◦, (1X)◦ = 1X, (r◦)◦ = r, r ≤ r0 =⇒ r◦ ≤ (r0)◦.
Note that when r is the graph of a map f : X → Y (so that x r y ⇐⇒
f (x) = y), then r◦(y) = f−1(y) is simply the fibre of f over y ∈ Y.
For a map f : X → Y we have that
f · f◦ ≤ 1Y and 1X ≤ f◦· f.
The category Rel, with objects sets and morphisms relations, can be seen as an extension of Set, the category with objects sets and morphisms maps, via the functor
(−)◦ : Set → Rel : X 7→ X, f 7→ f◦, for every set X and map f : X → Y , with
f◦ : X →7 Y ; f◦(x, y) = 1 if f (x) = y 0 otherwise
Proof. To prove that (−)◦ is a functor, we check the two conditions for a functor. First we remark that (1X)◦(x, y) = 1 if and only if 1X(x) = y, thus x = y. So we know that
(1X)◦ = 1X.
Next we look at the two relations (g · f )◦ and g◦ · f◦ for f : X → Y and g : Y → Z,
(g · f )◦(x, z) = 1 if g(f (x)) = z 0 otherwise and
(g◦· f◦)(x, z) = 1 if ∃y ∈ Y : f (x) = y and g(y) = z 0 otherwise
and thus
(g · f )◦ = g◦· f◦.
1.2 Orders
An order on a set X is a relation a : X →7 X that carries a monoid structure with respect to the compositional structure of relations; that is, a satisfies
a · a ≤ a, 1X ≤ a.
Hence, a is simply a transitive and reflexive relation on X:
(x ≤ y and y ≤ z =⇒ x ≤ z), x ≤ x
for all x, y, z ∈ X, when we write x ≤ y for x a y. The order is separated if a ∩ a◦ = 1X, so that
x ≤ y and y ≤ x =⇒ x = y.
A map f : X → Y of ordered sets is monotone (or order-preserving) if
f · a ≤ b · f,
where a, b denote the orders on X, Y respectively and f is identified with its graph. Hence, if we write ≤ for both a and b,
x ≤ y =⇒ f (x) ≤ f (y) for all x, y ∈ X.
The category with objects ordered sets and monotone maps as morphisms is called Ord.
1.3 Modules
A relation r : X →7 Y between ordered sets is a module if (≤Y) · r · (≤X) ≤ r, that is, if
x0 ≤ x and x r y and y ≤ y0 =⇒ x0r y0,
for all x, x0 ∈ X, y, y0 ∈ Y. Hence, the relation r is a module if and only if the map r : Xop × Y → {true, false} is monotone (where Xop × Y is ordered componentwise). Graphically, we indicate modularity of a relation r : X →7 Y by
r : X →◦ Y.
Every monotone map f : X → Y gives rise to the modules f∗ =≤Y ·f : X →◦ Y and f∗ = f◦· ≤Y: Y →◦ X
that is, x f∗ y ⇐⇒ f (x) ≤ y and y f∗x ⇐⇒ y ≤ f (x) for all x ∈ X, y ∈ Y.
The following rules may be easily verified when g : Y → Z is monotone.
1. 1∗X = (1X)∗ = (≤X),
2. (g · f )∗ = g∗· f∗ and (g · f )∗ = g∗· f∗, 3. 1∗X ≤ f∗· f∗ and f∗· f∗ ≤ 1∗Y.
Proof.
1. By definition we have 1∗X =≤X ·1X. Thus, for x, y ∈ X, we get x (≤X ·1X) y ⇐⇒ ∃z ∈ X : x 1X z and z ≤ y
⇐⇒ x ≤ y.
The other statement is proved analogously.
2. For x ∈ X and z ∈ Z we know that x (g · f )∗ z means that g(f (x)) ≤ z.
On the other hand, we have
x (g∗ · f∗) z ⇐⇒ x (≤Z ·g· ≤Y ·f ) z
⇐⇒ ∃w ∈ Z, ∃v, y ∈ Y : x f v & v ≤ y & y g w & w ≤ z
⇐⇒ ∃w ∈ Z, ∃v, y ∈ Y : f (x) = v & v ≤ y & g(y) = w & w ≤ z
⇐⇒ f (x) ≤ y and g(y) ≤ z
⇐⇒ g(f (x)) ≤ z,
because g is monotone. The other statement is proved analogously.
3. If for x ∈ X and y ∈ Y we have that x (f∗· f∗) y, then we know that there exists v, w, z ∈ Y such that x f v, v ≤ w, w ≤ z, z f◦ y, in other words f (x) ≤ f (y). Thus the statement holds if f is monotone, which is the fact by definition. The other statement is proved analogously.
Modularity is closed under relational composition. Indeed, for modules r : X →◦ Y and s : Y →◦ Z, one has
(≤Z) · (s · r) · (≤X) ≤ (≤Z) · s · (≤Y) · (≤Y) · r · (≤X) ≤ s · r, so that s · r : X →◦ Z is again a module.
Mod is the category with ordered sets as objects and modules as maps.
1.4 Adjunctions for ordered sets
For ordered sets X, Y, the set Ord(X, Y ) = {f | f : X → Y monotone} is itself ordered pointwise by
f ≤ f0 ⇐⇒ ∀x ∈ X : f (x) ≤ f0(x).
This order is preserved by composition on either side; whenever h : W → X and k : Y → Z are monotone then
f ≤ f0 =⇒ k · f · h ≤ k · f0· h.
A monotone map g : Y → X is called
1. right adjoint if there is a monotone map f : X → Y with 1X ≤ g · f, f · g ≤ 1Y;
2. an isomorphism if there is a monotone map f : X → Y with 1X = g · f, f · g = 1Y.
If g is right adjoint, the corresponding f is called left adjoint to g and one writes
f a g.
The terminology becomes more plausible when we consider the following fact:
Proposition 1.4.1. A map g : Y → X is right adjoint if and only if there is a map f : X → Y such that f (x) ≤ y ⇐⇒ x ≤ g(y) for all x ∈ X and y ∈ Y (where f is not assumed to be monotone a priori).
Proof. The necessity of the condition is obvious since x ≤ g(y) implies f (x) ≤ f · g(y) ≤ y, and dually for “ =⇒ ”. For its sufficiency, observe that f (x) ≤ f (x) implies x ≤ g · f (x) and dually f · g(y) ≤ y. The monotonicity of f (and also of g) follows, since x ≤ x0 ≤ g · f (x0) yields f (x) ≤ f (x0). Corollary 1.4.2. A right adjoint map g (with left adjoint f ) is fully faithful, i.e.
g(x) ≤ g(y) =⇒ x ≤ y, if and only if f · g ≤ 1Y and 1Y ≤ f · g.
Proof. We have f · g ≤ 1Y because g and f are adjoint. Since g and f are adjoint, it also follows that 1X ≤ g · f , and we get
g(y) ≤ g · f · g(y).
Since g is fully faithful, we know y ≤ f · g(y) and thus 1Y ≤ f · g.
Conversely assume that g(x) ≤ g(y), then we have x ≤ f · g(x) ≤ f · g(y) ≤ y.
1.5 Closure operation and interior operation
A closure operation is a monotone map c : X → X such that c · c ≤ c, 1X ≤ c.
We call the first condition idempotency and the second one reflexivity.
A closure space is a set X which comes with a closure operation on the powerset P X, ordered by inclusion. A map f : X → Y is continuous if
f (cX(A)) ⊆ cY(f (A))
for all A ⊆ X. A interior operation is a monotone map d : X → X such that
d ≤ d · d, d ≤ 1X.
A interior space is a set X which comes with a interior operation on the powerset P X, ordered by inclusion. A map f : X → Y is continuous if
f−1(dY(B)) ⊆ dX(f−1(B)) for all B ⊆ Y.
Via the order isomorphism (−){ : (P X)op → P X (which maps A ∈ P X to its complement A{ := X \ A in X), any closure operation c on P X corresponds to an interior operation d on P X, and vice-versa:
c(A){= d(A{), for all A ⊆ X.
If we call Cls the category of all closure spaces and continuous functions and Int the category of interior spaces and continuous functions, we have
Cls ∼= Int.
1.6 Completeness and lattices
Let X be an ordered set. For an element x ∈ X we define the up-set of x in X as follows
↑X x = {y ∈ X | y ≥ x}.
The up-closure of A ⊆ X is
↑X A = [
x∈A
↑X x.
We say that A is up-closed (or an up-set) if
↑X A = A.
With UpX we denote all the up-sets of X. We can define the dual notions analogously. The down-set of x is
↓X x = {y ∈ X | y ≤ x}.
The rest of the notions are down-closure, down-closed, down-set, DnX.
We say that an ordered set X is complete if and only if the map
↓: X → DnX is right adjoint; equivalently, if there is a map W
X = W : DnX → X which for every A ∈ DnX satisfies
∀x ∈ X :_
A ≤ x ⇐⇒ A ⊆↓ x.
We may call W A the supremum (or least upper bound) of A.
The dual notion of the adjunctions between the down-set and the suprema, can also be defined. But note that, for ↑X to be monotone, UpX is ordered by reverse inclusion and we get the notion of an infimum (or greatest lower bound):
∀x ∈ X : x ≤^
A ⇐⇒ A ⊆↑ x.
A monotone map f : X → Y of ordered sets preserves the supremum W A of A ⊆ X if f (W A) is a supremum of {f (x) | x ∈ A} in Y . Moreover, f is a sup-map if it preserves every existing supremum in X. The dual notions are: preserves an infimum, inf-map.
A lattice is a separated ordered set X with finite infima and finite suprema (for evere finite subset A of X, there exists a infimum and supre- mum). So there exists a top (>) and a bottom (⊥) element. A homomor- phism of lattices is a map that preserves the infima, the suprema and the top and bottom elements.
We call Sup the category with objects the complete lattices and with morphisms the sup-maps. We call Inf the category with objects the complete lattices and with morphisms the inf-maps.
A semilattice is a commutative monoid (see Definition 2.1.1) where every element is idempotent, so that x · x = x for all x ∈ X. One denotes such monoids by (X, ·, e). A homomorphism of semilattices is a map that preserves the operations ·, e. The category of semilattices and semilattice homomorphisms is denoted by
SLat.
1.7 Filters, ultrafilters, cliques and ultracliques
Definition 1.7.1. We call F a filter on the set X, if it is a collection of subsets of X satisfying
(1) A, B ∈ F ⇒ A ∩ B ∈ F , (2) X ∈ F ,
(3) A ∈ F , A ⊆ B ⇒ B ∈ F .
We call F a proper filter if F is a filter and satisfies (4) ∅ /∈ F .
For every map f : X → Y and every filter F on X, one defines the image filter f [F ] on Y by
f [F ] :=↑P X {f (A) | A ∈ F } = {B ⊆ Y | f−1(B) ∈ F }.
Thus f [F ] is the filter generated by the filterbase {f (A) | A ∈ F }.
To show that this definition is properly, we prove the last equality of the definition. That is why we take a subset B ⊆ Y such that there exist a subset A ∈ F and f (A) ⊆ B. Then we see immediately f−1(B) ∈ F , because a filter is an up-set. Conversely, for a subset B ⊆ Y such that f−1(B) ∈ F , we have f · f−1(B) ⊆ B. Thus B ∈ f [F ].
For every A ⊆ X, one has the following principal filter on X:
A =↑˙ P X A = {B ⊆ X | A ⊆ B}.
Definition 1.7.2. An ultrafilter U on a set X is a maximal element within the set of proper filters on X, ordered by inclusion. That is, U is a proper filter on X such that if F is a proper filter on X with U ⊆ F , then U = F .
A more convenient characterization for ultrafilters is the following one:
Lemma 1.7.3. For a proper filter U on X, the following statements are equivalent:
(i) U is an ultrafilter on X;
(ii) for all A, B ⊆ X, if A ∪ B ∈ U then A ∈ U or B ∈ U ;
(iii) for every subset A ⊆ X, one has A ∈ U or A{ ∈ U (where A{= X \ A denotes the complement of A in X).
Proof. This is proven in the course “Topology” [5]. For a map f : X → Y and an ultrafilter U on X, the image f [U ] is also an ultrafilter on Y . For every x ∈ X, the principal filter ˙x =↑P X {x} is an ultrafilter on X.
We now give some important propositions on ultrafilters, which we will use later on.
Proposition 1.7.4. Every proper filter F on X is contained in an ultrafilter U on X.
Proof. Suppose F is a proper filter and define
X := {G | G a proper filter on X, F ⊆ G}.
Then (X , ⊆) is an partially ordered set with the property that every totally ordered subset has an upperbound in X . Thus Zorn’s Lemma gives us the
existence of the ultrafilter.
In fact, this statement can be used to formulate a formally finer assertion:
Corollary 1.7.5. For a filter G and a proper filter F on X such that F ( G, there is an ultrafilter U on X with F ⊆ U but G * U.
Proof. Indeed, for some B ∈ G with B /∈ F , one considers the filter F0 =↑ {B{∩ A | A ∈ F },
which is proper since B{∩ A = ∅ would imply A ⊆ B ∈ F . So there exists an ultrafilter U containing F0, and therefore also F ; as B{ ∈ U , we must have
B /∈ U .
As an important consequence, we obtain:
Corollary 1.7.6. Every filter F on X is the intersection of all ultrafilters on X containing F .
Proof. We must show that every filter F ∈ F X may be obtained as F =\
{U ∈ βX | F ⊆ U }.
The inclusion “⊆” is clear. Moreover, when F is proper, so is the filter G obtained on the right-hand side; therefore, if F ( G, there exists an ultrafilter U with F ⊆ U but G * U, contradicting the definition of G.
If ∅ ∈ F , then there do not exist ultrafilters finer than F , so {U ∈ βX | F ⊆ U } = ∅.
In the complete lattice of all filters the infinum of the empty set is the biggest element. For filters, it is the improper filter. Thus the corollary holds.
We can also define the dual notion of a filter. We call it an ideal.
Definition 1.7.7. A collection of subsets J of X is an ideal if (1) A, B ∈ J ⇒ A ∪ B ∈ J ,
(2) ∅ ∈ J ,
(3) A ∈ J , B ⊆ A ⇒ B ∈ J . We call the ideal proper if
(4) X /∈ J .
Corollary 1.7.8. For a proper filter F and an ideal J on X such that F ∩ J = ∅, there is an ultrafilter U with F ⊆ U and U ∩ J = ∅.
Proof. Since F is an up-set, the fact that F is disjoint from J translates as A * J for all A ∈ F and J ∈ J , or equivalently as A ∩ J{ 6= ∅ for all A ∈ F , J ∈ J . Thus,
G := {A ∩ J{ | A ∈ F , J ∈ J }
is a proper filter containing F , and Proposition 1.7.4 yields the existence of an ultrafilter U with G ⊆ U , and consequently F ⊆ U . If there was J ∈ J ∩U , one would conclude J{ ∈ U , a contradiction./ We now define new structures, called cliques and ultracliques, which have as a structure a lot in common with filters and ultrafilters. The notions of
cliques and ultracliques are introduced and used independently by a range of authors and under different names. One of the main sources we used is
“Neighborhood spaces” by D.C. Kent and Won Keun Min [8]. They call a clique a p-stack and an ultraclique an ultra-p-stack. Thampuran [15] in turn calls it extended filters with the pair-wise intersection property. De Groot, Jensen and Verbeek [6] named cliques and ultracliques respectively linked systems and maximal linked systems, because the elements of cliques and ultracliques are pair-wise linked. We use the name Gavin J. Seal [4] has chosen.
Definition 1.7.9. A clique C on a set X is a subset of P X such that for all A, B ∈ P X :
(1) if ∅ /∈ C and A, B ∈ C, then A ∩ B 6= ∅, (2) A ∈ C, A ⊆ B ⇒ B ∈ C.
A clique is proper if ∅ /∈ C. The set of all cliques on X is denoted by CX.
An ultraclique D on X is a maximal element in the set of proper cliques on X, ordered by inclusion. That is, D is a proper clique on X and for all proper cliques C on X with D ⊆ C, we have D = C.
For ultracliques we have the following characterization, which can be found in “Neighborhood spaces” [8].
Lemma 1.7.10. For a proper clique D ∈ CX the following statements are equivalent:
(1) D is an ultraclique;
(2) if A ∩ D 6= ∅, for all D ∈ D, then A ∈ D;
(3) B /∈ D ⇒ X \ B ∈ D.
Proof. (1) ⇒ (2). If A ∩ D 6= ∅, for all D ∈ D and A /∈ D, then D ∪ {B ⊆ X | A ⊆ B}
would be a clique strictly larger then D.
(2) ⇒ (3). If B /∈ D, then by (2), there is D ∈ D such that B ∩ D = ∅.
Thus D ⊆ X \ B, which implies X \ B ∈ D.
(3) ⇒ (1). Suppose D ⊆ C, with C ∈ CX and C ∈ C. If C /∈ D, then by (3) X \ C ∈ D and hence X \ C ∈ C, a contradiction.
Similar to filters we define for a clique C ∈ CX and a map f : X → Y the image clique
f [C] :=↑P X {f (C) | C ∈ C} = {B ⊆ Y | f−1(B) ∈ C},
where the last equality is proven entirely analogously as for the filter case.
We now prove that f [C] is a clique, or an ultraclique, if C is one.
Lemma 1.7.11. For a map f : X → Y and C ∈ CX, f [C] is a clique. If C is an ultraclique, so is f [C].
Proof. First, we remark that f (∅) = ∅ and therefore the image clique of a proper clique will be proper. Take A, B ∈ f [C] for a proper clique C, then there exists C, D ∈ C such that f (C) ⊆ A and f (D) ⊆ B. By the definition of a clique is C ∩ D 6= ∅, and so is f (C ∩ D) 6= ∅. Therefore A ∩ B 6= ∅.
Suppose that C is an ultraclique. If B ∩ f (C) 6= ∅ for all C ∈ C, then f−1(B) ∩ C 6= ∅ for all C ∈ C and by Lemma 1.7.10, f−1(B) ∈ C, which implies B ∈ f [C] and thus f [D] is an ultraclique. The existence of “sufficiently many” ultraclique requires the axiom of choice (see 1.7.12), but these structures appear to be less diffcult to achieve then ultrafilters. For example, if the set X has three distinct elements x, y, z ∈ X, then a non-principal ultraclique is given by
↑P X {{x, y}, {x, z}, {y, z}}.
Assuming the lemma of Zorn (or the axiom of choice), we can prove there exist many ultracliques.
Lemma 1.7.12. Every proper clique is contained in an ultraclique.
Proof. Suppose C is a proper clique and define
X := {D | D a proper clique on X, C ⊆ D}.
Then (X , ⊆) is a partially ordered set with the property that every totally ordered subset has an upperbound in X. Zorn’s lemma gives us the existence
of the ultraclique.
Every clique C can be written as the intersection of all ultracliques con- taining C, as with filters as the intersection of all ultrafilters finer than it.
Lemma 1.7.13. Every clique C on X can be written as C =\
{D ∈ CX | C ⊆ D, D an ultraclique}.
Proof. If C is proper, then C is a subset of the intersection of all ultracliques that contain C. Conversely, we suppose that A /∈ C, then we have for all C ∈ C that C * A. This means that for all C ∈ C there exists a point in C ∩ (X \ A). Because of Lemma 1.7.12 there exists an ultraclique D such that C ⊆ D and X \ A ∈ D. Thus A /∈ D.
If C is improper, then there are no ultracliques finer than C. In the com- plete lattice of all cliques the infinum of the empty set is equal to the biggest
element, the improper clique.
Notes on Chapter 1
In the definition of a closure operation 1.5 one of the defining properties is the idempotency of the map c. In most books the definition of idempotency is c·c = c, but, like in our case, when we have reflexivity, both interpretations of idempotency are equal, since
c · c ≤ c ≤ c · 1X ≤ c · c.
In many books and courses the notion of a filter is somewhat different from the one we use here. We have chosen to make a difference between proper and improper filters. We need the improper filters, to construct the set of all filters on the empty set. This is a pathological case we need to take into account in the theory about Kleisli monoids (see 5).
Monads
This chapter defines one of our most important structures troughout this thesis, namely the concept of a monad. To define this structure, we base our- selfs on a well-known algebra structure, the monoid. To emphasize the simi- larities between monoids and monads more clearly, we first give the definition of a monoid. For the monads we use a broader language, namely category theory, but these structures are constructed in the same way. Every monad is actually a monoid in a specific category. This chapter is based on Chapter II [4]. The proofs in this chapter are elaborated by the author.
2.1 Monads
As promised we now repeat the definition of a monoid and its morphisms, followed by the definition of a monad and its morphisms.
Definition 2.1.1. A monoid M is a set M that comes with a binary and a nullary operation
m : M × M → M, e : {?} → M
that are associative and make e = e(?) a neutral element of M ; equivalently, the diagrams
M × (M × M ) ∼= //
1M×m
(M × M ) × Mm×1M//M × M
m
M × M m //M
15
and
{?} × M e×1M//
pr2
∼= &&
M × M
m
M × {?}
1M×e
oo
pr1
∼=
xxM commute.
A homomorphism f : M → N of monoids preserves both operations:
f · mM = mN · (f × f ) and f · eM = eN.
Definition 2.1.2. A monad T = (T, m, e) on the category Set is given by a functor T : Set → Set and two natural transformations, the multiplication and unit of the monad
m : T T → T, e : 1Set → T,
satisfying the multiplication law and the left and right unit laws:
m · mT = m · T m, m · eT = 1T = m · T e;
equivalently, these equalities mean that the diagrams T T T T m //
mT
T T
m
T T m //T and
T eT //
1T !!
T T
m T e T
oo
1T
}}T commute.
A morphism of monads α :S → T (where S = (S, n, d)) is a natural transformation α : S → T that preserves the monad structure:
α · n = m · (α ∗ α), α · d = e
(with ∗ the horizontal composition or the Godement product).
When we look at the category of all endofunctors on Set. Then T is a monoid in this category [9].
We now give some examples of monads, using structures we know or have defined in Chapter 1.
Examples 2.1.3. 1. The identity functor
1Set : Set → Set : X 7→ X, f 7→ f (with f : X → Y ) together with the map
1X : X → X : x 7→ x
forms a monad in a trivial way, namely I = (1Set, 1, 1) with 1 = (1X)X∈|Set|. We call it the identity monad.
2. The powerset functor
P : Set → Set : X 7→ P X, f 7→ P f,
with P X the powerset of X and P f : P X → P Y : A 7→ {f (a) | a ∈ A}
for every f : X → Y, together with the union map mX : P P X → P X : A 7→[
A and singleton map
eX : X → P X : x 7→ {x}
forms the powerset monad P = (P, m, e).
Proof. First we notice that the functions mX and eX are well-defined, because the union of a collection of subsets of X is a subset of X, also the singleton of an element of a set is a set.
Now we will prove that e and m are natural transformations. We take a function f : X → Y and we need to prove that
X eX //
f
P X
P f
Y e
Y //P Y and
P P X mX //
P P f
P X
P f
P P Y m
Y //P Y are commutative.
So take an element x ∈ X and we can see that (P f · eX)(x) = P f ({x})
= {f (x)}
= eY(f (x))
= (eY · f )(x).
For the second diagram we take a collection A of subsets of X (A ∈ P P X) and we compute
(P f · mX)(A) = P f ([ A)
=[
{P f (A) | A ∈ A}
= (mY · P P f )(A).
Now we can verify that this functor and these maps give a monad.
First we prove the multiplication law. Therefore we take a set X and look at the following diagram
P P P X
mP X
P mX//P P X
mX
P P X m
X
//P X
.
We have for A ∈ P P P X
(mX · P mX)(A ) = mX({mX(A) | A ∈ A })
= mX({[
A | A ∈A })
=[ {[
A | A ∈A }
= {a ∈ X | ∃A ∈ A, ∃A ∈A : a ∈ A}
=[ [ A
= mX([ A )
= (mX · mP X)(A ).
Finally we prove the left and right unit law, P X eP X //
1P X $$
P P X
mX
P eX P X
oo
1P X
zz
P X
Look at a subset A of X, then we get (mX · eP X)(A) = mX({A}) =[
{A} = A = 1P X(A), and
(mX·P eX)(A) = mX({{a} | a ∈ A}) =[
{{a} | a ∈ A} = A = 1P X(A).
Thus P is a monad.
3. Take for T the filter functor
F : Set → Set : X 7→ F X, f 7→ F f,
with F X the set of all filters on X and the image of f : X → Y the function F f : F X → F Y : F 7→ {B ⊆ Y | f−1(B) ∈ F }. Take for e the principal filter map
eX : X → F X : x 7→ ˙x = {A ⊆ X | x ∈ A}
and for m the map
mX : F F X → F X :F 7→ ΣF = {A ⊆ X | AF∈F }
with AF = {F ∈ F X | A ∈ F }. We call the map m the Kowalsky sum. Then F = (F, m, e) is a monad. We call F the filter monad.
Proof. First we prove that F is a functor. In order to be well-defined (F f )(F ) should be a filter. This is the case, because
(F f )(F ) =↑P X {f (F ) | F ∈ F }.
Take B ⊆ Y such that f−1(B) ∈ F , then f (f−1(B)) ⊆ B and so B ∈↑P X {f (F ) | F ∈ F }. For the other inclusion we look at f (F ) with F ∈ F . Then F ⊆ f−1(f (F )) and thus f (F ) ∈ (F f )(F ).
The two conditions of a functor are easy to check. For a set X, we have F (1X)(F ) = {B ⊆ X | 1−1X (B) ∈ F } = {B ⊆ X | B ∈ F } = F = 1F X(F ).
And for two functions f : X → Y and g : Y → Z and a filter F on X, we have
(F g · F f )(F ) = F g({A ⊆ Y | f−1(A) ∈ F })
= {B ⊆ Z | g−1(B) ∈ {A ⊆ Y | f−1(A) ∈ F }}
= {B ⊆ Z | f−1(g−1(B)) ∈ F }
= F (g · f )(F ).
It is obvious that the function eX is well-defined. In order to see that the function mX is well-defined, we need to prove that mX(F ) = ΣF , with F ∈ F2X, is a filter.
First we remark that every F ∈ F2X contains F X. If we prove that XF = F X, then we get that XF ∈ F and thus X ∈ mX(F ). Conse- quently ΣF is not empty. The statement XF = F X is very easy to see, because XFis always a subset of F X by definition and every filter on X contains X and is therefore an element of XF.
Let A and B be elements of mX(F ), then we need to prove that A ∩ B ∈ mX(F ). We know that A and B ∈ mX(F ), so AF and BF are elements of the filter F . Therefore is AF∩ BF ∈ F . If we prove that (A ∩ B)F = AF∩ BF, then we know that A ∩ B ∈ mX(F ). Thus take a filter F ∈ (A ∩ B)F, then A ∩ B ∈ F . Because F is a filter, we have A ∈ F and B ∈ F . Thus F ∈ AF and F ∈ BF and so F ∈ AF∩ BF. Conversely, take a filter F ∈ AF∩ BF, then we know that F ∈ AF and F ∈ BF. By definition we have A ∈ F and B ∈ F . Because F is a filter, A ∩ B is in F . Thus F ∈ (A ∩ B)F.
Now let A and B be subsets of X such that A ⊆ B and A ∈ mX(F ).
Then we have that AF ∈ F . Let F be a filter in AF, thus A ∈ F . We know that B ∈ F , because F is a filter. Therefore F ∈ BF. We see that AF ⊆ BF if A ⊆ B. Due to the fact that F is a filter,we know that BF∈F . In other words B ∈ mX(F ).
This proves that mX(F ) is a filter and therefore mX is well-defined.
After checking that eX and mX are well-defined, we prove that e and m are natural transformations. So take a function f : X → Y and look at the following diagrams.
X
f
eX //F X
F f
F F X
F F f
mX //F X
F f
Y e
Y
//F Y F F Y mY //F Y For x ∈ X we have
(F f )(eX(x)) = (F f )( ˙x)
= {B ⊆ Y | f−1(B) ∈ ˙x}
= {B ⊆ Y | x ∈ f−1(B)}
= {B ⊆ Y | f (x) ∈ B}
=f (x)˙
= eY(f (x)).
For the commutativity of the second diagram we need the following equality:
(F f )−1(BF) = {F ∈ F X | F f (F ) ∈ BF}
= {F ∈ F X | B ∈ F f (F )}
= {F ∈ F X | f−1(B) ∈ F }
= f−1(B)F. For F ∈ F F X we get
(F f )(mX(F )) = F f({A ⊆ X | AF∈F })
= {B ⊆ Y | f−1(B) ∈ {A ⊆ X | AF ∈F }}
= {B ⊆ Y | f−1(B)F∈F }
= {B ⊆ Y | (F f )−1(BF) ∈F }
= {B ⊆ Y | BF∈ {C ⊆ F Y | (F f)−1(C ) ∈ F }}
= mY({C ⊆ F Y | (F f)−1(C ) ∈ F })
= mY(F F f (F )).
To finalise the proof, we show the multiplication law and the left and right unit laws. For this we need the following equalities:
(BF)F = m−1X (BF), e−1X (BF) = B.
F ∈ (BF)F ⇐⇒ BF∈F
⇐⇒ B ∈ mX(F )
⇐⇒ mX(F ) ∈ BF
⇐⇒ F ∈ m−1X (BF).
x ∈ (eX)−1(BF) ⇐⇒ eX(x) ∈ BF
⇐⇒ B ∈ eX(x) = ˙x
⇐⇒ x ∈ B.
For the multiplication law, we look at the following diagram, for a set X,
F F F X
mF X
F mX//F F X
mX
F F X m
X //F X.
Take Φ ∈ F F F X, then we get
(mX · mF X)(Φ) = mX({A ⊆ F X | AF ∈ Φ})
= {B ⊆ X | BF∈ {A ⊆ F X | AF∈ Φ}}
= {B ⊆ X | (BF)F∈ Φ}
= {B ⊆ X | m−1X (BF) ∈ Φ}
= {B ⊆ X | BF∈ {A ⊆ F X | m−1X (A ) ∈ Φ}}
= mX({A ⊆ F X | m−1X (A ) ∈ Φ})
= (mX · F mX)(Φ).
For the left and right unit laws, we look at the following diagram, for a set X,
F X eF X //
1F X $$
F F X
mX
F eX F X
oo
1F X
zzF X For every F ∈ F X we have
(mX · eF X)(F ) = mX( ˙F )
= {B ⊆ X | BF∈ ˙F }
= {B ⊆ X | F ∈ BF}
= {B ⊆ X | B ∈ F }
= F and
(mX · F eX)(F ) = mX({F ⊆ F X | e−1X (F ) ∈ F})
= {B ⊆ X | BF∈ {F ⊆ F X | e−1X (F ) ∈ F}}
= {B ⊆ X | e−1X (BF) ∈ F }
= {B ⊆ X | B ∈ F }
= F
This proves that F = (F, m, e) is a monad. 4. The ultrafilter functor
β : Set → Set : X 7→ βX, f 7→ βf,
with βX = {U ⊆ P X | U ultrafilter }, the set of all ultrafilters on X and βf (U ) = {B ⊆ Y | f−1(B) ∈ U }, together with the principal filter map
eX : X → βX : x 7→ ˙x = {A ⊆ X | x ∈ A}
and the map
mX : ββX → βX :U 7→ {A ⊆ X | A ∈U },
where A = {U ∈ βX | A ∈ U }, forms a monad = (β, m, e), called the ultrafilter monad . We call the map mX the Kowalsky sum Proof. The proof for the ultrafilter monad is almost the same as the proof for the filter monad. There are two things in addition we need to check. First we need to prove that mX(U ) is proper for every U ∈ ββX.
Suppose that ∅ ∈ mX(U ), then we have that ∅ ∈U , and
∅ = {U ∈ βX | ∅ ∈ U } = ∅.
Thus ∅ ∈U , but this is impossible when U is an ultrafilter. Therefore we have ∅ 6∈ mX(U ).
The second thing we need to prove is the ultrafilter condition of mX(U ).
So suppose we have A /∈ mX(U ) for A ⊆ X, then we have A ∈/ U . Because U is an ultrafilter, we know that βX \ A ∈U . The proof is complete, if βX \ A = (X \ A) . Indeed,
βX \ A = {U ∈ βX | U /∈ A }
= {U ∈ βX | A /∈ U }
= {U ∈ βX | X \ A ∈ U } ( U ultrafilter)
= (X \ A) .
5. The up-set functor
U : Set → Set : X 7→ U X, f 7→ U f,
with U X = {S ⊆ P X |↑P X S = S}, all the up-closed sets for the inclusion order on P X and U f (S) = {B ⊆ Y | f−1(B) ∈ S}, together with the principal up-set map
eX : X → U X : x 7→ ˙x = {A ⊆ X | x ∈ A},
and the map
mX : U U X → U X :S 7→ {A ⊆ X | AU ∈S },
where AU = {S ∈ U X | A ∈ S}, forms a monad U = (U, m, e), called the up-set monad .
Proof. To check that the functor U is well-defined, we have to prove that (U f )(S) is up-closed.
↑P X (U f )(S) = {B ⊆ Y | ∃A ∈ (U f )(S) : A ⊆ B}
= {B ⊆ Y | ∃A : f−1(A) ∈ S and A ⊆ B}
= {B ⊆ Y | f−1(B) ∈ S},
where the last equality holds due the fact that S is up-closed and f−1(A) ⊆ f−1(B) if A ⊆ B.
It is obvious that ˙x is up-closed, so we can conclude that eX is well- defined.
Finally we verify that the image of S under the map mX is up-closed.
Therefore we need to remark that if A ⊆ B, then also AU ⊆ BU. Take S ∈ AU, then we have A ∈ S and ↑P X S = S. Because A ⊆ B, we know that B ∈ S. Thus S ∈ BU.
Take an up-closed set S of UX (S ∈ UUX) and compute
↑P X mX(S ) = {B ⊆ X | ∃A ∈ mX(S ) : A ⊆ B}
= {B ⊆ X | ∃A : AU ∈S , A ⊆ B}
= {B ⊆ X | ∃A :↑P P X AU = AU, A ⊆ B}
= {B ⊆ X | ∃A : AU ⊆ BU and AU ∈S }
= {B ⊆ X | BU ∈S }
= mX(S ).
Because of the similarity with the filter monad, we don’t have to prove the conditions of a monad. The proof is completely analogous to the proof of the filter monad (changing AF by AU).
This proves that U = (U, m, e) is a monad. In the proofs above, we have seen that these monads are similar.
This is easily explained by the fact that they are submonads of each other. There exists a chain of monad morphisms all given objectwise by inclusion maps:
I → → F → U.
