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Download e-book for iPad: Topos Theory by I. Moerdijk and J. van Oosten

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By I. Moerdijk and J. van Oosten

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Show that ({∗}, R) is a terminal object 1 in Eff . b) Prove that on Eff , Γ can be described like this: Γ(X, R) = X + / ∼, where X + = {x ∈ X | E(x) = ∅}, and x ∼ x iff R(x, x ) = ∅. Define also the effect of Γ on arrows. c) Prove that Γ is left adjoint to ∇ : Set → Eff . Here we see a contrast with the category of sheaves on a site. The ‘constant sheaves functor’ ∆ : Set → Sh(C, Cov) is not right adjoint, but left adjoint to Γ. Later we shall see, that ∇ does not have a right adjoint. On the other hand, the embedding S → Eff does have a left adjoint, the ‘separated reflection’: it sends (X, R) to (Γ(X, R), E) where E([x]) = x∈X R(x, x).

Tn , Y1 , . . , Yn and xX 1 , . . , xm are as in the previous item, then [[ t1 , . . , tn ]] is the unique map from [[ X1 × · · · × Xm ]] to [[ Y1 × · · · × Yn ]] such that its composition with the i-th projection is [[ t i ]], for each i. 54 Exercise 55 Write out t where t = f (f (x, y), g(x, u, c)) where x, y, u are variables, c a constant and f and g function symbols of appropriate arities. Next, we extend the interpretation to formulas. Formulas are built up from: atomic formulas (which are of the form R(t 1 , .

The definition of [[ ∀y Y ψ ]] is quite similar and uses ∀π . The following lemma holds. 8 Suppose ϕ(xX , x1 , . . , xn ) is a formula and t(y) is a term of sort X with a variable of sort Y . Suppose that t is substitutable for x in ϕ. Then we have a pullback diagram [[ ϕ[t/x] ]] / [[ ϕ ]]   / [[ X1 × · · · × Xn ]] [[ Y × X1 × · · · × Xn ]] [[ t(y,x) ]] The proof is a straightforward induction on ϕ, where for the quantifier steps one uses the Beck-Chevalley condition. Summing up, we have defined an interpretation [[ ϕ ]] as subobject of Xn 1 [[ X1 × · · · × Xn ]], for any sequence of variables xX 1 , .

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Topos Theory by I. Moerdijk and J. van Oosten


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