Categories, types, and structures. Introduction to category by Andrea Asperti

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By Andrea Asperti

Class idea is a mathematical topic whose value in numerous components of desktop technological know-how, so much particularly the semantics of programming languages and the layout of courses utilizing summary info forms, is largely said. This publication introduces type idea at a degree applicable for desktop scientists and gives sensible examples within the context of programming language layout.

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We define the extension fÎEN[b,a°] of f by giving f'ÎR which represents f . That is, set ff'(n)(0) = f(n). Such an f'ÎR exists by the s-m-n (iteration) theorem. Thus, f(e'(n)) = e°(f'(n)) = if ff'(n)(0) converges then e( ff'(n)(0)) else ^ . Therefore, if f(e'(n)) = e(f'(n)) is defined, then f f'(n) (0) converges and, hence, f(e'(n)) = e(f f'(n) (0)) = f(e'(n)). Finally, set t a b(f) = f . fÎpEN[b,a] f'(n) = f g'(n) (0). (Exercise: check the due diagram). By the fact above, this defines the lifting functor in pEN.

The category pCPO is given by defining complete partial orders under the assumption that directed sets are not empty. Thus, the objects of pCPO do not need to have a bottom element. As for morphisms, take the partial continuous functions with open subsets as domains. Clearly, the lifting functor is defined as it is for pPo. 2. 2. Given a = (a,e)ÎObpEN, define a° = (a °,e°) by adding a new element ^ to the set a and by defining e°(n) = if fn(0) converges then e(f n (0)) else ^ . Clearly, e° : w®a° is surjective.

References The general notions can be found in the texts mentioned at the end of chapter 1, though their presentation and notation may be different. For example, in the case of CCC's, this is so also because these categories play a greater role in the categorical approach to Type Theory, or to denotational semantics, than in other applications of Category Theory. Notice that Grp and Top are not CCC's and consider that the origin of Category Theory is largerly indebted to algebraic geometry. Applications of universal concepts from Category Theory to Programming Language Theory have been developed by several authors, in particular for program specification.

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