Examples - For a monotonic function f : D C between posets, → the free element over c C is ∈ the least element d D such that c f ( d ) . ∈ ≤ - X ∗ is a free monoid over X Make unit wrt. the forgetful functor G : Mon Sets . arrows explicit! → - The path category of a graph G is free over G wrt. the forgetful functor U : Cat Graph . → Exercise: What is the free poset over a set X wrt. the forgetful functor G : Pos Sets ? → Exercise: What is the free object over (A, B) wrt. the diagonal functor ∆ : C C C ? → ×
1 Facts about free objects
Fact: For a functor G : D C , free objects over X C are initial objects in the→ comma category K ∈G| | X ↓ where K : 1 C is the functor constant at X . X → Corollary: Free objects, if they exist, are unique up to isomorphism.
Fact: If A is free over X wrt. G : D C → then for each B D there is a bijection ∈ | | ( )! : hom (X, GB) = hom (A, B) − C ∼ D
2 Free objects are functorial
Consider a functor G : D C . → If every C C has a free object FC D wrt. G ∈ | | ∈ | | then the mapping C FC !→ ! f : C C! (ηC f) → !→ ! ◦ defines a functor F : C D . → Further, η : Id GF is a natural transformation. C → G CDo
ηC C / GF C FC
f GF f Ff=(η f)! C! ◦ C! η / GF C! FC" C!
3 Left adjoints
Defn. A functor F : C D is left adjoint to G : D C → → with unit η : Id GF if for every C C , C → ∈ | | FC with η C is free over C wrt. G . Examples: - the free monoid functor is left adjoint to G : Mon Sets → - the path category functor is left adjoint to U : Cat Graph → - “the” coproduct functor +:C C C is left adjoint to the diagonal functor× → ∆ : C C C . → × Exercise: What is a left adjoint to a monotonic function between posets?
4 Facts about left adjoints
Theorem: Left adjoints to any fixed G , if they exist, are unique up to natural isomorphism.
Theorem: If F is left adjoint to G , then: - F preserves colimits, - G preserves limits.
Theorem: Let D be locally small & complete (ie. have all limits). A functor G : D C has a left adjoint if and only if: → - G preserves limits, - for every C C there exists a set (ie not a proper class) ∈ | | f : C GD i of arrows such that { i → i | ∈ I} for each D D and f : C GD , ∈ → there exist i and g : D i D such that f = Gg f . ∈ I → ◦ i
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