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

1 Cofree objects

Consider a functor F : C D . or: under → Defn. Given an object D in D , a cofree object over D w.r.t. F

is an object C in C with an arrow ! D : FC D in D →(the counit arrow) such that for every B in C with an arrow f : FB D → there exists a unique arrow f ! : B C s.t. ! Ff ! = f . → D ◦ F C / D B FB C CC f ! f ! Ff! CC ∃ CC   C! C / FC !D D

2 Examples - For a monotonic function f : C D between posets, → the cofree element over d D is ∈ the greatest element c C such that f ( c ) d . ∈ ≤ Make counit arrows explicit! Exercise: What is the cofree object over (A, B) wrt. the diagonal functor ∆ : C C C ? → × - Fix a set A . For a functor A : Sets Sets , ×− → a cofree set over a set B is the set of functions B A .

- A cofree set over a monoid ( M, ! , 1) wrt. the free monoid functor ( )∗ : Sets Mon − → is M .

3 Facts about cofree objects

Fact: For a functor F : C D , cofree objects over D D → ∈ | | are final objects in the comma category F K ↓ D where K : 1 D is the functor constant at D . D → Corollary: Cofree objects, if they exist, are unique up to isomorphism.

Fact: If C is cofree over D wrt. F : C D → then for each B C there is a bijection ∈ | | ( )! : C(B,C) = D(FB,D) − ∼

4 Cofree objects are functorial

Consider a functor F : C D . → If every D D has a cofree object GD C wrt.F ∈ | | ∈ | | then the mapping D GD !→ ! f : D D! (f !D) → !→ ◦ defines a functor G : D C . → Further, ! : FG Id is a natural transformation. → D F C / D

!D GD F GD / D

Gf=(f ! )! F Gf f ◦ D    GD" F GD! ! / D! D!

5 Right adjoints

Defn. A functor G : D C is right adjoint to F : C D → → with counit ! : FG Id D if for every D D , → ∈ | | GD with ! D is cofree over D wrt. F . Examples: - “the” product functor : C C C is right adjoint to the diagonal× × functor→ ∆ : C C C . → × - For any set A , a right adjoint to A : Sets Sets ×− → is denoted ( ) A and defined by: − BA - the set of functions from A to B for f : B C , f A = f : BA CA → ◦− → Defn: A category is cartesian closed if it has final objects, products and if each functor A has a right adjoint. ×− 6