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The Adjoint Functor Theorem Oliver Kullmann

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The Adjoint Functor Theorem Oliver Kullmann The adjoint functor theorem Oliver Kullmann Review Special morphisms Comma categories The adjoint functor theorem Universal arrows Natural transformations Adjoint functors Limits and colimits Transfers via functors Oliver Kullmann Finding initial and Computer Science Department terminal objects The adjoint functor University of Wales Swansea theorem Reflections and coreflections The fundamental PhD Theory Seminar (Hauptseminar) examples Topological Swansea, March 14, 2007 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 : C → S Comma categories morphism-sets, and a functor V . 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 for all A ∈ Obj(C) and all f : X → V (A) there is some i ∈ I coreflections The fundamental together with some morphism g : Ai → A in C with examples 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 Reminders The following conditions are equivalent: 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 :( , ·, 1) → ( , ·, 1) is a bimorphism). examples Z Q 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 0 0 0 0 Universal arrows for every α : E → X with f ◦ α = g ◦ β there is a Natural transformations 0 0 Adjoint functors unique β : E → E with α = α ◦ β. Limits and colimits α : Y → C is an coequaliser for f , g if α ◦ f = α ◦ g, Transfers via functors 0 0 0 Finding initial and and for every α : Y → C with α ◦ f = α ◦ g there is terminal objects a unique β : E → E0 with α0 = 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) bD < Finding initial and DD α β zz terminal objects DD zz DD zz The adjoint functor D zz theorem X Reflections and identities and composition as in C; check the coreflections The fundamental commutativity of examples V (id ) C V (f ) V (g) Topological V (C) / V (C) , V (C) / V (D) / V (E) aC = bF O < examples CC α α {{ FF α γ xx Reminders C { F β x CC {{ FF xx Compact spaces CC { F x The Cech-Stoneˇ { F x compactification X { 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 ( , α) ∈ (C) α : ( ) → Review objects C where C Obj and 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 C F CC {{ FF xx examples C { F β x Reminders α C {α α F xγ C { F x Compact spaces CC {{ FF xx ! }{ " |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 this name) made well-known in the 1950s, and The adjoint functor 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 G(f ) Limits and colimits 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 between functors; the “vertical composition” of examples Topological natural transformations yields the category examples Reminders FUN(C, D) of functors from C to D.
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