By Asperti A.

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**Example text**

The connecting point is the construction of categories of sets where a suitable notion of limit gives an abstract notion of computability. 1), in the way explained below (in short, the computable elements are the limits of recursively enumerable indexed 55 3. ) This suggests several CCC's, such as limit spaces (L-spaces) and filter spaces (FIL) with their relevant subcategories. 4, toghether with other ideas for higher type computations. 1 Definition A limit space (L-space) (X,↓ ) is a set X and a relation (convergence) between countable sequences {xi}i∈ω ⊆ X and elements x∈X (notation: {xi}↓x) such that 1.

Exercise One clearly has to check that, for f∈YXc, ∀x∈Xc f(x)∈Yc. By the Cartesian closedness of ED, CD also is a CCC. e. sets. Typical objects in CD are RE, the recursively enumerable sets, and PR (= Pc), the partial recursive functions (see the exercise above). 23 2. Constructions Thus, in a rather indirect way, that is by topological and order properties, we obtained a CCC of countable (and numbered) sets. The proof that CD is a full sub-CCC of EN requires an important generalization, in higher types, of the classical Myhill-Shepherdson theorem for enumeration operators.

The previous property suggests an interesting generalization of a common set-theoretic construction. If f is a function from a set A to a set B, and C is a subset of B, then the inverse image of C under f, denoted f-1(C) is that subset of A defined by f-1(C) = {x/ x∈A, f(x)∈C }. It is easy to show that the diagram is a pullback square in Set. 30 2. Constructions In general, given a monic g: c →b and a morphism f: a→b, the inverse image of g under f is the subobject of a (if it exists) obtained by pulling back g along f.