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Hσn ) = h( s11 , . . , sk11 , . . , s1n , . . , sknn ) and h( a ) = a for a ∈ A I claim that this is the same thing as a group structure on A, with multiplication a · b = h( a, b ). The unit element is given by h( ); the inverse of a ∈ A is h( a−1 ) since h( a, h( a−1 ) ) = h( h( a ), h( a−1 ) ) = h( a, a−1 ) = h( ), the unit element Try to see for yourself how the associativity of the monad and its algebras transforms into associativity of the group law. Exercise 109 Finish the proof of the theorem: for the group monad T , there is an equivalence of categories between T -Alg and Grp.

Since there is a natural bijection: ∼ Set(X, P (Y )) → Set(Y, P (X)) = Setop (P¯ (X), Y ) we have an adjunction P¯ P. Exercise 95 A general converse to the last example. Suppose that F : Setop → Set is a functor, such that for the corresponding functor F¯ : Set → Setop we have that F¯ F . Then there is a set A such that F is naturally isomorphic to Set(−, A). 48 F Exercise 96 Suppose that C ← D is a functor and that for each object C of C there is an object GC of D and an arrow εC : F GC → C with the property that f for every object D of D and any map F D → C, there is a unique f˜ : D → GC such that f /C F Dq < qq yy qq y y qq yyεC F f˜ qq# yy F GC commutes.

10 (Soundness theorem) Suppose T = Cn(S) and all sequents of S are true under the interpretation in the category C. Then all sequents of T are true under that interpretation. 11 Suppose t is substitutable for x in ϕ. There is an obvious map [t] : [[ F V (ϕ) \ {x} ∪ F V (t) ]] = [[ F V (ϕ[t/x]) ]] → [[ F V (ϕ) ]] induced by [[ t ]]; the components of [t] are projections except for the factor of [[ ϕ ]] corresponding to x, where it is [[ t ]] [[ F V (ϕ[t/x]) ]] → [[ F V (t) ]] → [[ {x} ]] There is a pullback diagram: [[ ϕ[t/x] ]] / [[ F V (ϕ[t/x]) ]]  [[ ϕ ]]  [t] / [[ F V (ϕ) ]] Exercise 72 Prove this lemma [not trivial.

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