By van Oosten J.

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Show that the /D, F functor Cone(M ) → Cone(GM ) induced by G has a left adjoint. From the theorem on preservation of (co)limits by adjoint functors one can often conclude that certain functors cannot have a right or a left adjoint. 2. In chapter 3 we’ve seen that f is epi iff is a pushout; since left adjoints preserve identities and pushouts, they preserve epis; therefore the forgetful functor Mon → Set does not have a right adjoint; b) The functor (−) × X : Set → Set does not preserve the terminal object unless X is itself terminal in Set; therefore, it does not have a left adjoint for non-terminal X.

Exercise 116 Define unit and counit; check FT GT . Exercise 117 Let T be a monad on D. Call an object of T -Alg free if it is in the image of F T : D → T − Alg. Show that the Kleisli category DT is equivalent to the full subcategory of T -Alg on the free T -algebras. Now for every adjunction C o F G / D with GF = T , there is a unique compar- ison functor L : DT → C such that GL = GT and LFT = F . L sends the object X to F (X) and f : X → Y (so f : X → T (Y ) = GF (Y ) in D) to its transpose f˜ : F (X) → F (Y ).

8 A labelled sequent is an expression of the form ψ σ ϕ or σ ϕ where ψ and ϕ are the formulas of the sequent (but ψ may be absent), and σ is a finite set of variables which includes all the variables which occur free in a formula of the sequent. Let [[ σ ]] = [[ S1 ]] × · · · × [[ Sn ]] if σ = {xS1 1 , . . , xSnn }; there are projections πψ πϕ [[ σ ]] → [[ F V (ϕ) ]] and (in case ψ is there) [[ σ ]] → [[ F V (ψ) ]]; we say that the sequent ψ σ ϕ is true for the interpretation if (πψ )∗ ([[ ψ ]]) ≤ (πϕ )∗ ([[ ϕ ]]) as subobjects of [[ σ ]], and σ ϕ is true if (πϕ )∗ ([[ ϕ ]]) is the maximal subobject of [[ σ ]].