The adjoint functor theorem Oliver Kullmann

Review Special morphisms Comma categories Universal arrows The adjoint functor theorem Natural transformations Adjoint functors Limits and colimits Transfers via functors 1 Finding initial and Oliver Kullmann terminal objects

The adjoint functor 1Computer Science, University of Wales Swansea, UK theorem http://cs.swan.ac.uk/~csoliver Reflections and coreflections

The fundamental PhD Theory Seminar (Hauptseminar) examples

Swansea, March 14, 2007 Topological examples Reminders Compact spaces The Cech-Stoneˇ compactification Concrete: Ultrafilters The adjoint functor Introduction theorem Oliver Kullmann

Review Special morphisms Comma categories The main topic is: Universal arrows Natural transformations after a review of notions and theorems relevant to Adjoint functors Limits and colimits adjunctions Transfers via functors Finding initial and we prove the “adjoint functor theorem”, a necessary terminal objects

and sufficient criterion for the existence of (left- or The adjoint functor right-) adjoints of a given functor. theorem Reflections and With this criterion the construction of adjoint functors is coreflections often considerably facilitated. The fundamental examples We will also outline main corollaries and main Topological examples applications / examples. Reminders Compact spaces The Cech-Stoneˇ compactification Concrete: Ultrafilters The adjoint functor The adjoint functor theorem theorem Oliver Kullmann

Review Theorem Consider a complete category C with small Special morphisms Comma categories morphism-sets, and a functor V : C S. Universal arrows → Natural transformations Then V has a left adjoint if and only if V respects all limits Adjoint functors Limits and colimits (is “continuous”), and for every every X Obj(S) we Transfers via functors ∈ have a “solution set”, that is, a family A of objects in Finding initial and ( i )i∈I terminal objects

C together with a family (fi )i∈I of morphisms The adjoint functor theorem fi : X V (Ai ) in S such that → Reflections and coreflections for all A Obj(C) and all f : X V (A) there is some i I ∈ → ∈ The fundamental together with some morphism g : A A in C with examples i → f = V (g) fi . Topological ◦ examples Reminders (Remark: For I = 1 we nearly get the condition, that A is Compact spaces The Cech-Stoneˇ | | compactification universal for X to V (uniqueness is missing).) Concrete: Ultrafilters The adjoint functor Overview theorem Oliver Kullmann

Review Special morphisms 1 Review Comma categories Universal arrows Natural transformations Adjoint functors 2 Finding initial and terminal objects Limits and colimits Transfers via functors

Finding initial and 3 The adjoint functor theorem terminal objects The adjoint functor theorem 4 Reflections and Reflections and coreflections coreflections

The fundamental examples 5 The fundamental examples Topological examples Reminders 6 Topological examples Compact spaces The Cech-Stoneˇ compactification Concrete: Ultrafilters The adjoint functor Cancellation and inversion theorem Oliver Kullmann Consider a category C and a morphism f in C: Review Regarding cancellation: Special morphisms Comma categories f is a monomorphism if f is left-cancellable. Universal arrows f is an epimorphism if f is right-cancellable. Natural transformations Adjoint functors Regarding invertibility: Limits and colimits Transfers via functors

f is a coretraction if f is left-invertible. Finding initial and f is a retraction if f is right-invertible. terminal objects The adjoint functor It follows, that f is an isomorphism iff f is a theorem retraction and a coretraction. Reflections and coreflections

Necessary conditions for cancellability: The fundamental Every coretraction is a monomorphism examples Topological Every retraction is an epimorphism. examples The following conditions are equivalent: Reminders Compact spaces The Cech-Stoneˇ f is an isomorphism. compactification f is a monomorphism and a retraction. Concrete: Ultrafilters f is an epimorphism and a coretraction. The adjoint functor Bimorphisms theorem Oliver Kullmann

A morphism is called a bimorphism if it is a Review Special morphisms monomorphism and an epimorphism. Comma categories Universal arrows Every isomorphism is a bimorphism. Natural transformations Adjoint functors Limits and colimits A category is called balanced if every bimorphism is an Transfers via functors isomorphism. Finding initial and terminal objects

In a balanced concrete category, the isomorphisms The adjoint functor are exactly the bijections. theorem Reflections and A concrete category, where every bijection is an coreflections isomorphism, does not need to be balanced, as The fundamental examples

MON shows (the canonical injection Topological j :(Z, , 1) (Q, , 1) is a bimorphism). examples → Reminders (The category of groups, on the other hand, is balanced.) Compact spaces The Cech-Stoneˇ compactification If V : C S is faithful, and C is balanced, then V is discrete Concrete: Ultrafilters → (i.e., the fibres of V are discrete orders). The adjoint functor Equalisers and coequalisers theorem Oliver Kullmann Consider two morphisms f , g : X Y . → Review Special morphisms α : E X is an equaliser for f , g if f α = g α, and Comma categories → ◦ ◦ Universal arrows for every α′ : E′ X with f α′ = g β′ there is a → ◦ ◦ Natural transformations unique β : E′ E with α′ = α β. Adjoint functors → ◦ Limits and colimits α : Y C is an coequaliser for f , g if α f = α g, Transfers via functors → ′ ′ ′ ◦ ◦ Finding initial and and for every α : Y C with α f = α g there is terminal objects ′→ ′ ◦ ◦ a unique β : E E with α = β α. The adjoint functor → ◦ theorem We have Reflections and coreflections 1 Every equaliser is a monomorphism. The fundamental examples 2 Every coequaliser is an epimorphism. Topological To see this, let α be an equaliser of f , g, and consider examples Reminders ϕ,ψ with α ϕ = α ψ. Trivially f (αϕ)= g(αϕ), and the Compact spaces ◦ ◦ The Cech-Stoneˇ (unique(!)) β with αβ = αϕ is β = ϕ. Now αψ = αϕ, and compactification Concrete: Ultrafilters thus ψ = β = ϕ. √ The adjoint functor Upon an object theorem Let V : C S be a functor, and X Obj(S). The Oliver Kullmann → ∈ comma-category (X ↓ V ) has Review Special morphisms objects (C,α) where C Obj(C) and α : X V (C); Comma categories ∈ → Universal arrows morphisms from (C,α) to (D,β) are f : C D with Natural transformations → Adjoint functors commuting Limits and colimits V (f ) Transfers via functors V (C) / V (D) bDD z< Finding initial and DDα β zz terminal objects DD zz DD zz The adjoint functor z theorem X Reflections and identities and composition as in C; check the coreflections commutativity of The fundamental examples

V (idC ) V (f ) V (g) Topological V (C) / V (C) , V (C) / V (D) / V (E) aCC {= bFF O x< examples CCα α {{ FFα γ xx Reminders C { F β x Compact spaces CC {{ F x C { FF xx The Cech-Stoneˇ { x compactification X X Concrete: Ultrafilters So (X V ) is a concrete category over C via the ↓ projection P :(X V ) C, P((C,α)) := C. ↓ → The adjoint functor The other side (into an object) theorem The comma-category (V ↓ X) has Oliver Kullmann Review objects (C,α) where C Obj(C) and α : V (C) X; Special morphisms ∈ → Comma categories morphisms from (C,α) to (D,β) are f : C D with Universal arrows → Natural transformations commuting Adjoint functors V (f ) Limits and colimits V (C) / V (D) Transfers via functors DD z DD zz Finding initial and α DD zz terminal objects DD zz β " |z The adjoint functor X theorem

Reflections and identities and composition as in C; check the coreflections

commutativity of The fundamental examples V (idC ) V (f ) V (g) V (C) / V (C) , V (C) / V (D) / V (E) Topological CC { FF x examples CC { F x C {{ F β x Reminders α C { α α FF xxγ CC {{ FF xx Compact spaces ! }{ "  |x The Cech-Stoneˇ X X compactification Concrete: Ultrafilters So (V X) is a concrete category over C via the ↓ projection P :(X V ) C, P((C,α)) := C. ↓ → The adjoint functor Universality via commas theorem Let V : C S be a functor, and X Obj(S). Oliver Kullmann → ∈ Review A universal arrow from X to V is an initial object of Special morphisms Comma categories (X V ). Universal arrows ↓ Natural transformations A universal arrow from V to X is a terminal object of Adjoint functors Limits and colimits (V X). Transfers via functors ↓ In [Borceux] the terminology “(co)reflection of X along V ” is used; universal Finding initial and terminal objects

constructions were (under the above name) made well-known in the 1950s, The adjoint functor and since then they carried this name. theorem Reflections and The definitions carried out: coreflections The fundamental A universal arrow from X to V is a pair (C,α) with examples C Obj(C) and α : X V (C) such that for every Topological ∈ → examples (D,β) there is a unique f : C D with β = V (f ) α. Reminders → ◦ Compact spaces The Cech-Stoneˇ A universal arrow from V to X is a pair (C,β) with compactification C Obj(C) and β : V (C) X such that for every Concrete: Ultrafilters ∈ → (D,α) there is a unique f : C D with α = β V (f ). → ◦ The adjoint functor Functorial morphisms theorem Given functors F, G : C D,a natural transformation Oliver Kullmann → η : F G is a family η =(ηX )X∈Obj(C) with Review → Special morphisms ηX : F(X) G(X) such that for all objects A, B in C and Comma categories → Universal arrows morphisms f : A B the following diagram commutes: Natural transformations → Adjoint functors Limits and colimits G(f ) G(A) / G(B) Transfers via functors O O Finding initial and terminal objects ηA ηB The adjoint functor F(f ) theorem F(A) / F(B) Reflections and coreflections

Natural transformations η : F G are morphisms The fundamental → examples between functors; the “vertical composition” of Topological natural transformations yields the category examples Reminders FUN(C, D) of functors from C to D. Compact spaces The Cech-Stoneˇ The isomorphisms η : F ֒ G in FUN(C, D) are compactification →→ Concrete: Ultrafilters exactly the natural transformations η which are .(pointwise isomorphisms” η : F(X) ֒ G(X“ X →→ The adjoint functor Adjunctions via natural isomorphisms theorem Consider functors V : C S and F : S C between Oliver Kullmann → → categories with small morphism-sets. We get two functors Review Special morphisms t Comma categories Mor(F( 1), 2): S C SET Universal arrows Natural transformations − − t × → Mor( 1, V ( 2)) : S C SET. Adjoint functors − − × → Limits and colimits Transfers via functors

(F, V ) is called an adjoint functor pair if there is a Finding initial and natural isomorphism terminal objects The adjoint functor theorem .((η : Mor(F( 1), 2) ֒ Mor( 1, V ( 2 − − →→ − − Reflections and coreflections

The triple (F, V ,η) is called an adjunction, and η is The fundamental called the adjunction-isomorphism. examples Topological F is called a left adjoint of V , and V is called a right examples Reminders adjoint of F. Compact spaces The Cech-Stoneˇ V is called right adjoint (at all) if there exists some compactification Concrete: Ultrafilters left adjoint F of V , while F is called left adjoint (at all) if there exists some right adjoint V of F. The adjoint functor Units theorem Oliver Kullmann Consider an adjunction (F, V ,η) with V : C S. For → Review X Obj(S) and Y Obj(C) we get the (natural) bijection Special morphisms Comma categories ∈ ∈ Universal arrows Natural transformations η(X,Y ) : Mor(F(X), Y ) ֒ Mor(X, V (Y )). Adjoint functors →→ Limits and colimits Transfers via functors

So for Y := F(X) we get Finding initial and terminal objects

η(X,F(X)) : Mor(F(X), F(X)) ֒ Mor(X, V (F(X))). The adjoint functor →→ theorem Reflections and Thus coreflections

ϕX := η(X,F(X))(idF(X)): X V (F(X)). The fundamental → examples

Topological It follows, that ϕ : idS V F is a natural examples → ◦ Reminders transformation, called the unit of (F, V ,η). Compact spaces The Cech-Stoneˇ For every X Obj(S) the pair (F(X),ϕX ) is a compactification ∈ Concrete: Ultrafilters universal arrow from X to V . The adjoint functor Counits theorem For X Obj(S) and Y Obj(C) from the bijection Oliver Kullmann ∈ ∈ Review η(X,Y ) : Mor(F(X), Y ) ֒ Mor(X, V (Y )). Special morphisms →→ Comma categories Universal arrows For X := V (Y ) we get Natural transformations Adjoint functors η(V (Y ),Y ) : Mor(F(V (Y )), Y ) ֒ Mor(V (Y ), V (Y )). Limits and colimits →→ Transfers via functors Thus Finding initial and −1 terminal objects ψY := η(V (Y ),Y )(idV (Y )): F(V (Y )) Y . → The adjoint functor theorem It follows, that ψ : F V idC is a natural Reflections and ◦ → coreflections transformation, called the counit of (F, V ,η). The fundamental examples For every Y Obj(C) the pair (V (Y ),ψY ) is a ∈ Topological universal arrow from F to Y . examples Reminders For an adjunction (F, V ,ν) we get the adjunction Compact spaces t t −1 t t −1 The Cech-Stoneˇ (V , F ,η ), and the unit of (V , F ,η ) is the counit of compactification Concrete: Ultrafilters (F, V ,ν) (and vice versa). (Using F t : St Ct and V t : Ct St.) → → The adjoint functor Obtaining the adjunction-isomorphism from theorem the (co)units Oliver Kullmann Review We have defined the adjoined pair (F, V ) via the Special morphisms Comma categories existence of the adjunction-isomorphism Universal arrows Natural transformations Adjoint functors ηX,Y : MorC(F(X), Y ) ֒ MorS(X, V (Y )) Limits and colimits →→ Transfers via functors which is natural in X Obj(S) and Y Obj(C). So we Finding initial and ∈ ∈ terminal objects have mappings The adjoint functor theorem

f : F(X) Y ηX,Y (f ): X V (Y ) Reflections and → → → coreflections −1 g : X V (Y ) ηX,Y (g): F(X) Y . The fundamental → → → examples Now these mappings are realised by the units Topological examples ϕX : X V (F(X)) and the counits ψY : F(V (Y )) Y : Reminders → → Compact spaces The Cech-Stoneˇ ηX,Y (f )= V (f ) ϕX compactification ◦ Concrete: Ultrafilters η−1 (g)= ψ F(g). X,Y Y ◦ The adjoint functor Adjoints via universal arrows to the functor theorem Oliver Kullmann

Review Special morphisms We have seen that given an adjunction (F, V ,η), where Comma categories Universal arrows V : C S, for every X S (the “sets”) we obtain a Natural transformations → ∈ Adjoint functors universal arrow (F(X),ϕX ) from X to V . Limits and colimits Transfers via functors

Consider the situation where we just have given the Finding initial and terminal objects functor V : C S, but not F, and for every X we have a → The adjoint functor universal arrow (F(X),ϕX ) to V , where now we have just theorem given F : Obj(S) Obj(C) and ϕ : Obj(S) Mor(S). Reflections and → → coreflections This data already uniquely determines a left adjoint F of The fundamental examples V (that is, V has some left adjoint F ′, and V has exactly Topological one left adjoint F which extends the given object map F examples Reminders and such that the unit of the adjunction is ϕ). Compact spaces The Cech-Stoneˇ compactification Concrete: Ultrafilters The adjoint functor Adjoints via universal arrows from the functor theorem Oliver Kullmann

Review Special morphisms Comma categories Dually, we have seen that given an adjunction (F, V ,η), Universal arrows Natural transformations where F : S C, for every X C we obtain a universal Adjoint functors → ∈ Limits and colimits arrow (V (X),ψX ) from F to X. Transfers via functors Assume we have the functor F : S C, but not V , and Finding initial and → terminal objects for every X we have a universal arrow (V (X),ψX ) from F The adjoint functor (where V : Obj(C) Obj(S) and ψ : Obj(C) Mor(C)). theorem → → Reflections and This data already uniquely determines a right adjoint V of coreflections ′ The fundamental F (that is, F has some right adjoint V , and F has exactly examples one right adjoint V which extends the given object map V Topological examples and such that the counit of the adjunction is ψ). Reminders Compact spaces The Cech-Stoneˇ compactification Concrete: Ultrafilters The adjoint functor Basic properties of adjunctions theorem Oliver Kullmann

Review Special morphisms Comma categories Universal arrows Natural transformations Adjoint functors 1 Limits and colimits Uniqueness of adjoints: Transfers via functors

Two left adjoints of a functor are isomorphic. Finding initial and Two right adjoints of a functor are isomorphic. terminal objects The adjoint functor 2 Preservation of limits and colimits: theorem Left adjoints preserve colimits. Reflections and coreflections Right adjoints preserve limits. The fundamental examples

Topological examples Reminders Compact spaces The Cech-Stoneˇ compactification Concrete: Ultrafilters The adjoint functor Some mnemonics theorem Oliver Kullmann

The uniqueness of adjoint functors is an extension of the Review fact, that universal arrows are unique up to isomorphism. Special morphisms Comma categories Universal arrows To remember which kind of adjoint (left? right??) respects Natural transformations Adjoint functors limits or colimits, consider the category MON of monoids Limits and colimits Transfers via functors and the adjoint pair (F, V ), where V is the forgetful Finding initial and functor and F the “free” functor: terminal objects The adjoint functor 1 The set 1 is terminal in SET, which maps via F to theorem { } the monoid N0, which is not terminal in MON. Reflections and coreflections 2 The monoid 0 is initial in MON, but the set 0 is { } { } The fundamental not initial in SET. examples Topological Thus F does not respect limits (terminal objects are examples Reminders special products, which are special limits), while V does Compact spaces The Cech-Stoneˇ not respect colimits (initial objects are special coproducts, compactification Concrete: Ultrafilters which are special colimits). The adjoint functor Properties of the unit and the counit theorem Oliver Kullmann

Review Special morphisms Comma categories Consider an adjunction (F, V ,η): Universal arrows Natural transformations Regarding F: Adjoint functors Limits and colimits F is faithful iff the unit if pointwise monomorph. Transfers via functors

F is full iff the unit is a pointwise retraction. Finding initial and F is fully faithful iff the unit is an isomorphism. terminal objects The adjoint functor Regarding V : theorem V is faithful iff the counit if pointwise epimorph. Reflections and V is full iff the counit is a pointwise coretraction. coreflections The fundamental V is fully faithful iff the counit is an isomorphism. examples So “typically” units are (pointwise) monomorph and Topological examples counits are (pointwise) epimorph. Reminders Compact spaces The Cech-Stoneˇ compactification Concrete: Ultrafilters The adjoint functor Normal examples theorem Oliver Kullmann

Review Special morphisms Comma categories Universal arrows Natural transformations Adjoint functors Recall for example the units and counits of the Limits and colimits Transfers via functors MON SET situation: → Finding initial and the units are the canonical injections X V (F(X)); terminal objects → The adjoint functor the counits are the homomorphisms F(V (M)) M, theorem → which assign to every word over a monoid M the Reflections and coreflections

composition of its “letters”.) The fundamental examples

Topological examples Reminders Compact spaces The Cech-Stoneˇ compactification Concrete: Ultrafilters The adjoint functor (Co)Limits and universal arrows theorem Oliver Kullmann Consider a category J (the “diagram”) and a functor Review Special morphisms D : J C, that is, D FUN(J, C). Comma categories Universal arrows → ∈ Natural transformations Adjoint functors Recall that we have a canonical functor Limits and colimits Transfers via functors ∆: C FUN(J, C) Finding initial and → terminal objects The adjoint functor which assigns to every object X the constant functor with theorem value X. ( is an embedding except of the cases where J Reflections and ∆ coreflections

is empty while C has at least two objects). The fundamental examples A limit of D is a universal object from ∆ to D. Topological A colimit of D is a universal object from D to ∆. examples Reminders Compact spaces (As usual with natural transformations, we take the liberty The Cech-Stoneˇ compactification here to strip off the attached domain- and Concrete: Ultrafilters codomain-categories in the triple-representation.) The adjoint functor Cones and cocones theorem Oliver Kullmann Regarding limits of D : J C: → Review The objects of (∆ D), which are pairs (P,η), where Special morphisms ↓ Comma categories η is a natural transformation from ∆(P) to D, are Universal arrows Natural transformations called cones with base D (the top of the cone is on Adjoint functors Limits and colimits the left, the base on the right; the direction is dictated Transfers via functors by the direction of the arrows). Finding initial and terminal objects

Limits of D are limit cones with base D. The adjoint functor theorem Regarding colimits of D : J C: → Reflections and The objects of (D ∆), which are pairs (S,η), where coreflections ↓ The fundamental η is a natural transformation from D to ∆(S), are examples

called cocones with base D (the base of the cocone Topological examples is on the left, the top on the right). Reminders Compact spaces Colimits of D are colimit cocones with base D. The Cech-Stoneˇ compactification Cones are also called projective cones, and cocones Concrete: Ultrafilters inductive cones. The adjoint functor Special limits theorem To specify J, we may just specify a general directed Oliver Kullmann

graph, and then the free category associated with this Review Special morphisms dgg is meant. Comma categories Universal arrows 1 If J is discrete, then limits are called products and Natural transformations Adjoint functors colimits are called coproducts. Limits and colimits Transfers via functors

2 If J is a doublet (the directed general graph Finding initial and consisting of the vertices 0, 1 and a bunch of arrows terminal objects The adjoint functor from 0 to 1), then limits are called equalisers, and theorem colimits are called coequalisers. Reflections and coreflections 3 • If J is , then limits are called pullbacks. The fundamental }} examples • ~} `AA Topological A examples • Reminders Compact spaces 4 • If J is , then colimits are called pushouts. The Cech-Stoneˇ }}> compactification } Concrete: Ultrafilters • A AA • The adjoint functor Complete and cocomplete categories theorem Oliver Kullmann

A category C is complete if it has all small limits, Review Special morphisms while it is cocomplete if it has all small colimits. Comma categories Universal arrows C is complete iff it has all small products and all Natural transformations Adjoint functors binary equalisers, while C is cocomplete iff it has all Limits and colimits Transfers via functors small coproducts and all binary coequalisers. Finding initial and For a given diagram category J, the existence of all terminal objects The adjoint functor limits resp. colimits in C with source J is equivalent to theorem ∆: C FUN(J, C) having a right adjoint resp. a left Reflections and → coreflections adjoint. The fundamental examples A nice property of small categories, generalising that a Topological quasi-order has all infimums iff it has all supremums: examples Reminders Compact spaces The Cech-Stoneˇ A small category is complete iff it is cocomplete. compactification Concrete: Ultrafilters (In complete small categories all limits and colimits exist.) The adjoint functor Transportation of structures theorem Oliver Kullmann

Review Special morphisms Comma categories Universal arrows Consider a functor F : C D. Natural transformations → Adjoint functors We assume a notion of a certain “structure” for C as Limits and colimits S Transfers via functors well as for D, involving the objects and morphisms as well Finding initial and as other (set-theoretical) objects; the structures in C are terminal objects denoted by (C), those in D by (D). The adjoint functor S S theorem A structure S (C) can be transported by F, obtaining Reflections and ∈S coreflections F(S), that is, applying F to all involved objects and The fundamental morphisms, and leaving the rest intact. If F(S) (D), examples ∈S Topological then we say that S is transportable via F. examples Reminders Compact spaces The Cech-Stoneˇ compactification Concrete: Ultrafilters The adjoint functor Examples theorem Oliver Kullmann Examples for structures transportable via all functors: Review 1 Single objects X F(X) and single morphisms Special morphisms → Comma categories f F(f ). Universal arrows → Natural transformations 2 Diagrams D : J C are transported to Adjoint functors → Limits and colimits F D : J D. Transfers via functors ◦ → Finding initial and 3 Natural transformations η : D D for terminal objects 1 → 2 D1, D2, : J C are transported to The adjoint functor → theorem F η : F D1 F D2 (a special case of the ◦ ◦ → ◦ Reflections and “horizontal” composition of natural transformations). coreflections The fundamental 4 An object of the comma category (D ∆C), where examples ↓ ∆C : C FUN(J, C) (in other words, a “cocone” for Topological → examples the diagram D), is transported to an “object” of Reminders ′ Compact spaces ′ (F D ∆D ), where D is the image of F. So here The Cech-Stoneˇ ◦ ↓ compactification the transport creates some “damage”, but this can be Concrete: Ultrafilters easily repaired by extending D′ to all of D. The adjoint functor Preservation and detection of properties theorem Oliver Kullmann

Review For a functor F : C D, a “structure notion” , and a Special morphisms → S Comma categories property P of such structures: Universal arrows Natural transformations Adjoint functors 1 F preserves (or respects) P for if for each Limits and colimits S Transfers via functors S (C) transportable via F we have, that if S has ∈S Finding initial and property P, then also F(S) has property P. terminal objects 2 F detects (or reflects) P for if for each S (C) The adjoint functor S ∈S theorem transportable via F we have, that if F(S) has Reflections and property P then also S has property P. coreflections The fundamental These two properties are relatively simple, because they examples focus on a structure S in C. After some examples we then Topological examples consider the case that we want to “transport back” some Reminders Compact spaces structure in D. The Cech-Stoneˇ compactification Concrete: Ultrafilters The adjoint functor Basis examples theorem Oliver Kullmann

Each functor F respects retractions, coretractions Review Special morphisms and isomorphisms. Comma categories Universal arrows Each faithful F detects monomorphisms and Natural transformations Adjoint functors epimorphisms. Limits and colimits Transfers via functors

F is called (co)continuous if F respects (co)limits. Finding initial and terminal objects F is (co)continuous iff F respects (co)products and The adjoint functor (co)equalisers. theorem Reflections and If F coreflections

respects pullbacks, then F respects monomorphisms The fundamental respects pushouts, then F respects epimorphisms. examples Topological (Note that a morphism f is mono (epi) iff id is a pullback examples Reminders (pushout) of (f , f ).) Compact spaces The Cech-Stoneˇ Thus left adjoints respect epimorphisms, and right compactification Concrete: Ultrafilters adjoints respect monomorphisms. The adjoint functor A note on representable functors theorem A functor F : C SET is called representable if there is Oliver Kullmann → some object A Obj(C) such that F is isomorphic to the Review ∈ Special morphisms (covariant) hom-functor Comma categories Universal arrows Natural transformations Mor A X Obj Mor A X Adjoint functors C( , ): C SET, (C) ( , ). Limits and colimits − → ∈ → Transfers via functors A contravariant functor F : Ct SET is called Finding initial and → terminal objects representable if there is some object A Obj(C) such ∈ The adjoint functor that F is isomorphic to the contravariant hom-functor theorem Reflections and t coreflections MorC( , A): C SET, X Obj(C) Mor(X, A). − → ∈ → The fundamental examples F is representable iff there is a universal arrow from Topological examples 0 to F. Reminders { } Compact spaces The Cech-Stoneˇ Every representable functor is continuous. compactification Concrete: Ultrafilters Representable contravariant functors on the other hand transform colimits into limits. The adjoint functor Lifting and creation of limits theorem Oliver Kullmann For a functor F : C D and a diagram D : J C: → → Review 1 Special morphisms F lifts limits for D if for a limit P of F D there is a Comma categories ◦ Universal arrows limit P0 of D which is transported via F to P. Natural transformations Adjoint functors 2 F uniquely lifts limits for D if for a limit P of F D Limits and colimits ◦ Transfers via functors there is a unique limit P0 of D transported via F to P. Finding initial and 3 F creates limits for D if if for a limit P of F D there terminal objects ◦ The adjoint functor is a unique cone P0 with base D transported via F to theorem

P, and this unique cone is a limit cone for D. Reflections and coreflections If no diagram is specified, then the lifting- or The fundamental creation-property holds (at all) if it holds for all D : J C examples → with small . Topological J examples F creates limits iff Reminders Compact spaces F detects limits and F uniquely lifts limits. The Cech-Stoneˇ compactification Concrete: Ultrafilters For colimits we have the corresponding dual notions. The adjoint functor Basis usage of lifts theorem Consider a functor F : C D: Oliver Kullmann → If F lifts limits, and D is complete, then is complete. Review Special morphisms C Comma categories If F lifts colimits, and D is cocomplete, then C is Universal arrows Natural transformations cocomplete. Adjoint functors Limits and colimits It seems that there are no general criterions for lifting Transfers via functors (co)limits, but that lifts are given for special (general) Finding initial and terminal objects

situations. For us the main example is, using categories The adjoint functor C, S, objects X Obj(S) and a functor V : C S: theorem ∈ → Reflections and 1 V continuous P :(X V ) C creates limits. coreflections ⇒ ↓ → 2 V cocontinuous P :(V X) C creates colimits. The fundamental ⇒ ↓ → examples Thus: Topological examples If V is continuous and C complete, then all comma Reminders Compact spaces categories (X V ) are complete. The Cech-Stoneˇ ↓ compactification If V is cocontinuous and C cocomplete, then all Concrete: Ultrafilters comma categories (V X) are cocomplete. ↓ The adjoint functor Regarding full embeddings theorem Of course, isomorphisms between categories respect and Oliver Kullmann create limits. Close to isomorphisms are full embeddings. Review Special morphisms Comma categories So consider a full embedding F : C D (that is, F is an Universal arrows → Natural transformations isomorphism of C to a full subcategory F(C) of D; note Adjoint functors Limits and colimits however that the codomain of F is D). Transfers via functors F detects limits and colimits. Finding initial and terminal objects

F respects (co)limits iff every (co)limit in F(C) is also The adjoint functor a (co)limit in D. theorem Reflections and If F lifts (co)limits, then the lifting is unique, and F coreflections The fundamental actually creates (co)limits. examples Now assume that F respects (co)limits and that C is Topological examples (co)complete: Reminders Compact spaces The Cech-Stoneˇ F lifts (co)limits iff F(C) is isomorphism-closed. compactification Concrete: Ultrafilters (To show the direction from left to right, consider (co)products of one-object-families.) The adjoint functor Weakening initiality and terminality theorem Oliver Kullmann

Consider a category C. Call an object A Obj(C) Review ∈ Special morphisms a source if for all Y Obj(C) there is f : A Y ; Comma categories ∈ → Universal arrows a sink if for all X Obj(C) there is f : X A. Natural transformations ∈ → Adjoint functors More generally, call A Obj(C) (where A is small(!)) Limits and colimits ⊆ Transfers via functors a set-source if for all Y Obj(C) there is A A and Finding initial and ∈ ∈ terminal objects f : A Y ; → The adjoint functor a set-sink if for all X Obj(C) there is A A and theorem ∈ ∈ Reflections and f : X A. coreflections → The fundamental Obviously: examples 1 A category having a source does not need to have Topological examples an initial object. Reminders Compact spaces 2 The Cech-Stoneˇ A category having a sink does not need to have a compactification terminal object. Concrete: Ultrafilters The adjoint functor Solution set condition for initiality/terminality theorem Oliver Kullmann

Lemma A complete category with some set-source has Review Special morphisms an initial object. Comma categories Universal arrows The dual version is: A cocomplete category with Natural transformations Adjoint functors some set-sink has a terminal object. Limits and colimits Transfers via functors

This lemma is the germ of what could be considered Finding initial and the underlying theme of this talk (and of the adjoint terminal objects The adjoint functor functor theorem): “Connecting the two sides”, i.e., theorem switching from the “unnamed side” of category Reflections and coreflections theory to the “co-side”, or, more specifically, The fundamental switching from the limits to the colimits. examples Topological Let us start the proof; let C be a complete category and A examples Reminders some set-source. The first strike is easy: S := Q A is a Compact spaces The Cech-Stoneˇ source of C. In the remainder of the proof only S will be compactification Concrete: Ultrafilters used. The adjoint functor Killing endomorphisms theorem Oliver Kullmann

In general S need not to be an initial object. Review Special morphisms Comma categories In order to be initial, S must not have non-trivial Universal arrows Natural transformations endomorphisms. So consider an equaliser Adjoint functors Limits and colimits Transfers via functors

σ : E S Finding initial and → terminal objects for the set of endomorphisms of S. We want to show that The adjoint functor theorem E now is an initial object. So consider some object X and Reflections and two morphisms f , g : E X. We need to show that f = g coreflections → The fundamental holds. examples Let α : E′ E be an equaliser of f , g. We need to show Topological → examples that α is an isomorphism. Since every equaliser is a Reminders Compact spaces monomorphism, this is equivalent to showing that is a The Cech-Stoneˇ α compactification retraction. Concrete: Ultrafilters The adjoint functor Round the circle theorem Oliver Kullmann

′ Review Since S is a source, we get some morphism π : S E , Special morphisms → Comma categories yielding the diagram Universal arrows ′ Natural transformations E @ Adjoint functors ` @ Limits and colimits @ π α @@ Transfers via functors @@  @ Finding initial and E / S terminal objects ~ AA σ ~~ AA The adjoint functor ~~ AA theorem ~~ f g AA ~ Reflections and X X coreflections

The fundamental examples For the two endomorphisms σ α π and idS of S: ◦ ◦ Topological (σ α π) σ = idS σ, σ (α π σ)= σ idE . examples ◦ ◦ ◦ ◦ ⇒ ◦ ◦ ◦ ◦ Reminders Compact spaces The Cech-Stoneˇ σ is a monomorphism, so α π σ = idE . QED compactification ◦ ◦ Concrete: Ultrafilters The adjoint functor The adjoint functor theorem again theorem Oliver Kullmann

Review Special morphisms Theorem Consider a complete category C with small Comma categories Universal arrows morphism-sets, and a functor V : C S. Natural transformations → Adjoint functors Limits and colimits Then V has a left adjoint if and only if V is “continuous, Transfers via functors and for every every X Obj(S) we have a “solution set”, Finding initial and ∈ terminal objects that is, a set-source in (X V ); explicitely stated, we ↓ The adjoint functor require a family (Ai )i∈I of objects in C together with a theorem family (fi )i∈I of morphisms fi : X V (Ai ) in S such that Reflections and → coreflections for all A Obj(C) and all f : X V (A) there is some i I The fundamental ∈ → ∈ examples together with some morphism g : Ai A in C with Topological → examples f = V (g) fi . Reminders ◦ Compact spaces The Cech-Stoneˇ compactification Concrete: Ultrafilters The adjoint functor Finally the proof theorem Oliver Kullmann

Review Consider X Obj(C). Special morphisms ∈ Comma categories 1 We have to show the existence of a universal arrow Universal arrows Natural transformations from X to V . Adjoint functors Limits and colimits 2 In other words, we have to show the existence of an Transfers via functors Finding initial and initial object in (X V ). terminal objects ↓ 3 Since V is continuous and is complete, also The adjoint functor C theorem

(X V ) is complete. Reflections and ↓ coreflections 4 Now the solution set condition just means that The fundamental (X V ) has a set-source. examples ↓ 5 Topological Thus by the previous lemma we get the initial object examples Reminders in (X V ). Compact spaces ↓ The Cech-Stoneˇ QED compactification Concrete: Ultrafilters The adjoint functor Generators and cogenerators theorem Oliver Kullmann

Consider a category C. A set A of objects in C is a Review Special morphisms generator set, if for every pair f , g : X Y of Comma categories → Universal arrows morphisms in C we have that f = g if and only if there Natural transformations Adjoint functors is some A A and a morphism α : A X with Limits and colimits ∈ → Transfers via functors f α = g α. Finding initial and ◦ ◦ terminal objects cogenerator set, if for every pair f , g : X Y of → The adjoint functor morphisms in C we have that f = g if and only if there theorem is some A A and a morphism α : Y Ai with Reflections and ∈ → coreflections α f = α g. ◦ ◦ The fundamental If A = 1, then we just speak of a generator resp. a examples | | Topological cogenerator. For example, in the category SET of sets: examples Reminders 1 0 is a generator. Compact spaces The Cech-Stoneˇ { } compactification 2 0, 1 is a cogenerator. Concrete: Ultrafilters { } The adjoint functor Eliminating the solution set condition theorem Oliver Kullmann

Review Special morphisms Comma categories Universal arrows Lemma Consider a complete category C with small Natural transformations Adjoint functors morphism-sets, and a functor V : C S. Furthermore Limits and colimits → Transfers via functors assume that Finding initial and terminal objects 1 C has a cogenerator set. The adjoint functor 2 C is concrete. theorem Reflections and 3 Every monomorphism in C is injective. coreflections 4 Every bijective morphism in C is an isomorphism. The fundamental examples

Then V has a left adjoint if and only if V is continuous. Topological examples Reminders Compact spaces The Cech-Stoneˇ compactification Concrete: Ultrafilters The adjoint functor Left adjoints for embeddings theorem Oliver Kullmann

Review Special morphisms We consider now the problem of finding a left adjoint for Comma categories Universal arrows some embedding into a category C. Natural transformations Adjoint functors Limits and colimits W.l.o.g. we can take this embedding as the canonical Transfers via functors embedding of a sub-category U of C. Finding initial and terminal objects A left adjoint F : C U of this embedding now is called a The adjoint functor → theorem reflector for U, and is characterised by canonical Reflections and isomorphisms for coreflections The fundamental examples MorU(F(X), Y ) = MorC(X, Y ) ∼ Topological examples Reminders for X Obj(C) and Y Obj(U). Compact spaces ∈ ∈ The Cech-Stoneˇ compactification Concrete: Ultrafilters The adjoint functor Reflective and coreflective subcategories theorem Oliver Kullmann

Review Special morphisms Comma categories Universal arrows Natural transformations A reflector for a subcategory U of some category C is Adjoint functors Limits and colimits a left-adjoint of the canonical embedding, and if such Transfers via functors a reflector exists, then U is called a reflective Finding initial and terminal objects subcategory of C. The adjoint functor Dually, a coreflector for a subcategory U of some theorem Reflections and category C is a right-adjoint of the canonical coreflections embedding, and if such a coreflector exists, then U is The fundamental examples called a coreflective subcategory of C. Topological examples Reminders Compact spaces The Cech-Stoneˇ compactification Concrete: Ultrafilters The adjoint functor Left adjoints for full embeddings theorem Oliver Kullmann Since every embedding is faithful, the counits of the Review adjunction are always epimorphisms. Special morphisms Comma categories Now in the important situation, that is a full Universal arrows U Natural transformations subcategory, the counits are isomorphisms, and then Adjoint functors Limits and colimits w.l.o.g. the counits can be assumed to be the identities Transfers via functors Finding initial and and F is identical on U. terminal objects

The adjoint functor So in this situation the inverse of the theorem

adjunction-isomorphism takes a very simple form: Reflections and coreflections g : X Y ֒ F(g): F(X) Y . The fundamental → →→ → examples Topological The other direction is given by examples Reminders Compact spaces The Cech-Stoneˇ f : F(X) Y ֒ f ϕX : X Y , compactification → →→ ◦ → Concrete: Ultrafilters using the unit ϕ : X F(X) of the adjunction. X → The adjoint functor Meaning of full (co)reflective subcategories theorem Oliver Kullmann Assume that U is a full subcategory of C. Review 1 U is reflective iff there exists a functor F : C U, Special morphisms → Comma categories identical on U, such that for every general object Universal arrows Natural transformations X Obj(C) and for every special object S Obj(U) Adjoint functors ∈ ∈ Limits and colimits the morphisms from X to S are, via F, in 1-1 Transfers via functors Finding initial and correspondence to the morphisms from F(X) to S — terminal objects

F(X) is the most general specialisation of X loosing The adjoint functor nothing w.r.t. morphisms on X into special spaces. theorem Reflections and 2 U is coreflective iff there exists a functor G : C U, coreflections → identical on U, such that for every general object The fundamental examples Y Obj(C) and for every special object S Obj(U) ∈ ∈ Topological the morphisms from S to Y are, via G, in 1-1 examples Reminders correspondence to the morphisms from S to G(Y ) — Compact spaces The Cech-Stoneˇ so G Y is the most general specialisation loosing compactification ( ) Concrete: Ultrafilters nothing w.r.t. morphisms into Y from special spaces. The adjoint functor Inheriting (co)limits theorem Oliver Kullmann

Review Consider a (co)complete category C and a subcategory Special morphisms Comma categories U. We want to construct a (co)reflection for U, that is a left Universal arrows Natural transformations resp. right adjoint for J : U ֒ C. To apply the adjoint Adjoint functors → Limits and colimits functor theorem, we need U to be (co)complete, and J to Transfers via functors respect (co)limits. Finding initial and terminal objects The easiest way here is to inherit all (co)limits from C (if The adjoint functor theorem

possible): Reflections and coreflections

U inherits (co)limits from C if every limit in C The fundamental for objects and morphisms in U can be realised in U. examples Topological examples If U inherits (co)limits, then U is (co)complete and J is Reminders Compact spaces (co)continuous. The Cech-Stoneˇ compactification Concrete: Ultrafilters The adjoint functor Constructing reflections theorem Oliver Kullmann

Review Special morphisms Comma categories Universal arrows Natural transformations Lemma Consider a concrete and complete category C. Adjoint functors Limits and colimits Let U be a subcategory such that Transfers via functors Finding initial and 1 U inherits limits from C. terminal objects

2 The adjoint functor Every monomorphism in U is injective. theorem 3 Every bijective morphism in U is an isomorphism. Reflections and coreflections 4 U has a cogenerator set. The fundamental examples Then U is a reflective subcategory of C. Topological examples Reminders Compact spaces The Cech-Stoneˇ compactification Concrete: Ultrafilters The adjoint functor Direct examples for reflections I theorem Oliver Kullmann

Review Special morphisms Comma categories Universal arrows Natural transformations Adjoint functors Limits and colimits Transfers via functors The category of all preordered (or thin) categories is Finding initial and a reflective full i-c subcategory of CAT. terminal objects The adjoint functor The category PORD of all partially ordered sets is a theorem reflective full i-c subcategory of QORD. Reflections and coreflections

The fundamental examples

Topological examples Reminders Compact spaces The Cech-Stoneˇ compactification Concrete: Ultrafilters The adjoint functor Direct examples for reflections II theorem Oliver Kullmann

Review Special morphisms Comma categories The category of all T0-spaces is a reflective full i-c Universal arrows Natural transformations subcategory of TOP. Adjoint functors Limits and colimits MON can be canonically identified with the (non-full) Transfers via functors subcategory of SGR given by all semigroups with Finding initial and terminal objects neutral element (then uniquely determined) together The adjoint functor with neutral-element preserving homomorphisms. theorem Then is a reflective sub-category of . Reflections and MON SGR coreflections

(Note that here the reflector is not identical on the The fundamental monoids, and the counit is only pointwise epimorph; examples Topological the reflector is faithful (actually an embedding), and examples Reminders accordingly all units are pointwise monomorph.) Compact spaces The Cech-Stoneˇ compactification Concrete: Ultrafilters The adjoint functor Basic (direct) examples for coreflections theorem Oliver Kullmann

Review Special morphisms Comma categories Universal arrows The category SET of all sets has a canonical i-c full Natural transformations Adjoint functors embedding into PORD (using discrete orders). Limits and colimits Then SET is a coreflective subcategory of PORD. Transfers via functors Finding initial and The category of all discrete categories is a terminal objects The adjoint functor coreflective (i-c full) subcategory of CAT. theorem QORD is canonically isomorphic to STOP, the i-c Reflections and coreflections

full subcategory of TOP given by the condition, that The fundamental arbitrary intersections of open sets are open. STOP examples Topological is a coreflective subcategory of TOP. examples Reminders Compact spaces The Cech-Stoneˇ compactification Concrete: Ultrafilters The adjoint functor Reminder theorem Oliver Kullmann A topological space is a pair (X,τ), where X is a set and τ P(X) is closed under finite intersections and Review ⊆ Special morphisms arbitrary unions. Comma categories Universal arrows Natural transformations The category TOP of topological spaces has a Adjoint functors Limits and colimits natural i-c full embedding (via dualisation) into the Transfers via functors

category of all closure systems (pairs (X, M), where Finding initial and M P(X) is closed under arbitrary intersections), terminal objects ⊆ − The adjoint functor which in turn is an i-c full subcategory of S(P ). theorem

The category HTOP of T2-topological spaces (two Reflections and coreflections distinct points can be separated by disjoint open The fundamental sets) is an i-c full subcategory of TOP. examples Topological A topological space is quasi-compact if every open examples Reminders cover has a finite sub-cover. A topological space X is Compact spaces The Cech-Stoneˇ compact if X is T2 and quasi-compact. compactification Concrete: Ultrafilters The category COMP of compact topological spaces is an i-c full subcategory of TOP. The adjoint functor The forgetful functor for all spaces theorem Oliver Kullmann Consider V : TOP SET. → Review Special morphisms D : SET TOP, S (S, P(X)) is the canonical left Comma categories → → Universal arrows adjoint of V . Natural transformations Adjoint functors (D is also an i-c full embedding, and a right-inverse of V .) Limits and colimits I : SET TOP, S (S, , S ) is the canonical Transfers via functors → → {∅ } Finding initial and right adjoint of V . terminal objects (I is also an i-c full embedding, and a right-inverse of V .) The adjoint functor theorem

Thus V respects all limits and colimits. Reflections and coreflections

However, V for example does neither detect products The fundamental nor coproducts. examples Topological On the other hand, by considering initial and final examples Reminders structures, we see that V uniquely lifts limits and Compact spaces The Cech-Stoneˇ colimits. compactification Concrete: Ultrafilters So TOP is complete and cocomplete. The adjoint functor The subcategories of (in)discrete spaces theorem Oliver Kullmann

We have seen the adjoint pairs (D, V ) and (V , I), Review Special morphisms Regarding the images Comma categories Universal arrows D(SET), the i-c full subcategory of discrete spaces Natural transformations Adjoint functors (every subset is open) Limits and colimits Transfers via functors

I(SET), the i-c full subcategory of indiscrete spaces Finding initial and (no subset other than the empty set and the full set is terminal objects The adjoint functor open) theorem we can consider the functors Reflections and coreflections

VD : TOP D(SET) The fundamental → examples VI : TOP I(SET). → Topological examples We obtain that Reminders Compact spaces D(SET) is a bi-coreflective subcategory of TOP The Cech-Stoneˇ compactification I(SET) is a bi-reflective subcategory of TOP. Concrete: Ultrafilters The adjoint functor Remarks regarding monos and epis theorem Since TOP is a concrete category, every monomorphism Oliver Kullmann is injective, every epimorphism is surjective (and this is Review Special morphisms inherited for every sub-category). By the existence of the Comma categories Universal arrows left- and right-adjoint we also get the inverse. However, Natural transformations Adjoint functors for that we do not need such strong means: Limits and colimits Transfers via functors

For every sub-category of TOP including all constant Finding initial and maps (for example for all full sub-categories) the terminal objects The adjoint functor monomorphisms are exactly the injective maps. theorem For every sub-category of TOP including some Reflections and coreflections

indiscrete space with at least two objects the The fundamental epimorphisms are exactly the surjective maps. examples Topological We see a (typical) asymmetry: For many interesting examples Reminders sub-categories, monomorphisms are the injective maps, Compact spaces The Cech-Stoneˇ but often non-trivial indiscrete spaces are excluded compactification Concrete: Ultrafilters (because of separation concerns), and so whether every epimorphisms needs to be surjective is open there. The adjoint functor Regarding limits theorem Oliver Kullmann

Review Special morphisms To construct the limits in TOP, it suffices to know Comma categories Universal arrows products and equalisers: Natural transformations Adjoint functors The (canonical) product of a family X ∈ of spaces Limits and colimits ( i )i I Transfers via functors

is the smallest topology on the product set Finding initial and terminal objects Qi∈I V (Xi ), such that all projections are continuous The adjoint functor (in other words, the initial topology for the projection theorem

maps). Reflections and coreflections The (canonical) equaliser of a family f : X Y of i → The fundamental continuous maps is the sub-space given by examples x X (i I f (x)) constant together with the Topological { ∈ | ∈ → i } examples canonical embedding. Reminders Compact spaces The Cech-Stoneˇ compactification Concrete: Ultrafilters The adjoint functor Compact spaces and their category theorem Oliver Kullmann Regarding the category COMP and the forgetful functor V : COMP SET: Review → Special morphisms Comma categories COMP is an i-c full subcategory of TOP. Universal arrows Natural transformations COMP is balanced: Adjoint functors Limits and colimits The monomorphisms are exactly the injective maps Transfers via functors

(which are now closed maps, whence topological Finding initial and embeddings). terminal objects The epimorphisms are exactly the surjective maps. The adjoint functor theorem COMP is closed under products in TOP Reflections and coreflections (Tychonoff’s theorem). The fundamental COMP is closed under equaliser formation in TOP examples Topological (equalising sets are closed in Hausdorff spaces). examples Reminders Thus COMP inherits limits, whence V respects limits. Compact spaces The Cech-Stoneˇ Using balancedness appropriately, we also get compactification Concrete: Ultrafilters V respects and creates limits. The adjoint functor Applying the solution set condition theorem Oliver Kullmann

Review Special morphisms Comma categories Our aim is to get a left-adjoint for V : COMP SET. Universal arrows → Natural transformations Applying the solution set condition, for a set S we need to Adjoint functors Limits and colimits show Transfers via functors

Finding initial and the existence of a family (Ai )i∈I of compact spaces terminal objects

together with maps fi : S V (Ai ), The adjoint functor → theorem such that for all compact topological spaces A and Reflections and every map f : S V (A) there is some i I and a coreflections → ∈ continuous map g : Ai A such that f = V (G) fi . The fundamental → ◦ examples To construct such families (Ai )i∈I is left as an exercise Topological examples (we want to apply stronger means). Reminders Compact spaces The Cech-Stoneˇ compactification Concrete: Ultrafilters The adjoint functor COMP as a reflective subcategory of TOP theorem Oliver Kullmann We know that TOP is concrete and complete. Review 1 COMP inherits limits. Special morphisms Comma categories 2 The monomorphisms in COMP are exactly the Universal arrows Natural transformations injective continuous maps. Adjoint functors Limits and colimits 3 The isomorphisms in COMP are exactly the bijective Transfers via functors Finding initial and continuous maps. terminal objects

The adjoint functor To apply the lemma about the existence of reflections, we theorem

need a cogenerator (set). Consider the unit interval Reflections and I =[0, 1] (a compact space): coreflections The fundamental examples By Urysohn’s lemma for every compact space K and Topological x, y K , x = y, there exists a continuous examples ∈ Reminders f : K I with f (x)= 0 and f (y)= 1. Compact spaces → The Cech-Stoneˇ compactification Thus I is a generator for COMP. Thus COMP is a Concrete: Ultrafilters reflective subcategory of TOP. The adjoint functor The Cech-Stoneˇ compactification theorem Oliver Kullmann By the lemma about the existence of reflections, we now Review get: Special morphisms ′ Comma categories There exists a reflection β : TOP COMP. Universal arrows → Natural transformations Adjoint functors ′ X for a topological space X is called the Cech-Stoneˇ Limits and colimits β ( ) Transfers via functors

compactification of X: Finding initial and terminal objects For X completely regular β′(X) is a compactification The adjoint functor of X in the normal sense (while otherwise no theorem compactifications in the normal sense exists). Reflections and coreflections ′ Restricting β to HTOP we obtain an epi-reflector. The fundamental examples ′ Restricting β further to D(SET), via the identification Topological of discrete spaces and sets we obtain a left adjoint of examples Reminders the forgetful functor COMP SET. Compact spaces The Cech-Stoneˇ → compactification In the remainder, a concrete construction for β(X) for Concrete: Ultrafilters discrete spaces X is outlined. The adjoint functor The Cech-Stoneˇ compactification of discrete theorem spaces Oliver Kullmann Review Given a discrete topological space X, the topology of βX Special morphisms Comma categories is as follows: Universal arrows Natural transformations 1 For A X let Adjoint functors Limits and colimits ⊆ Transfers via functors [A] := F βX : A F Finding initial and { ∈ ∈ } terminal objects The adjoint functor and let B := [A]: A P(X) . theorem { ∈ } 2 B is stable under finite intersection, and thus the Reflections and coreflections

closure of B creates a topology on βX. The fundamental The basic facts about the topological space βX: examples Topological β(X) is compact. examples Reminders Compact spaces X is considered to be a subspace of βX via the The Cech-Stoneˇ compactification (topological) embedding x X GF(x) βX. Concrete: Ultrafilters ∈ → ∈ X is a discrete and dense subset of βX. The adjoint functor The left adjoint of V : COMP SET again theorem → Oliver Kullmann

Review Special morphisms Now the map X Obj(SET) βX COMP yields a Comma categories ∈ → ∈ Universal arrows functor (since there are no restrictions on maps on Natural transformations Adjoint functors discrete spaces) β : SET COMP: Limits and colimits → Transfers via functors β, V is an adjoint pair. Finding initial and ( ) terminal objects

The adjoint functor (And thus, due to the uniqueness of adjoints, β is theorem ′ isomorphic to β restricted to discrete spaces.) Reflections and coreflections

The fundamental examples

Topological End examples Reminders Compact spaces The Cech-Stoneˇ compactification Concrete: Ultrafilters