2 results
Dinaturality for free
- Edited by M. P. Fourman, University of Edinburgh, P. T. Johnstone, University of Cambridge, A. M. Pitts, University of Cambridge
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- Book:
- Applications of Categories in Computer Science
- Published online:
- 24 September 2009
- Print publication:
- 26 June 1992, pp 107-118
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Summary
The first aim of this paper is to attack a problem posed in [1] about uniform families of maps between realizable functors on PER's.
To put this into context, suppose that we are given a category C to serve as our category of types. The authors of [1] observe that the types representable in the second-order lambda; calculus and most extensions thereof can be regarded as being obtained from functors (Cop × C)n → C by diagonalisation of corresponding contra and covariant arguments. Terms in the calculus give rise to dinatural transformations. This suggests a general structure in which parametrised types are interpreted by arbitrary functors (Cop × C)n → C, and their elements by dinatural transformations. Unfortunately as the authors of the original paper point out, this interpretation can not be carried out in general since dinaturals do not necessarily compose.
However, suppose we are in the extraordinary position that all families of maps which are of the correct form to be a dinatural transformation between functors (Cop × C)n → C are in fact dinatural, a situation in which we have, so to speak, the dinaturality for free. In this situation dinaturals compose. The result is a structure for a system in which types can be parametrised by types (second-order lambda calculus without the polymorphic types). Suppose, in addition, the category in question is complete, then we can perform the necessary quantification (which is in fact a simple product), and obtain a model for the second-order lambda calculus.
Remarks on algebraically compact categories
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- By P. J. Freyd
- Edited by M. P. Fourman, University of Edinburgh, P. T. Johnstone, University of Cambridge, A. M. Pitts, University of Cambridge
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- Book:
- Applications of Categories in Computer Science
- Published online:
- 24 September 2009
- Print publication:
- 26 June 1992, pp 95-106
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Summary
In Algebraically Complete Categories (in the proceedings of the Category theory conference in Como '90) an ALGEBRAICALLY COMPLETE CATEGORY was defined as one for which every covariant endofunctor has an initial algebra. This should be understood to be in a 2-category setting, that is, in a setting in which the phrase “every covariant endofunctor” refers to an understood class of endofunctors.
Given an endofunctor T the category of T- INVARIANT objects is best defined as the category whose objects are triples <A,f, g> where f:TA →A, g:A→TA and fg and gf are both identity maps. T-Inv appears as a full subcategory of both T-Alg and T-Coalg, in each case via a forgetful functor. The Lambek lemma and its dual say that the initial object in T-Alg and the final object in T-Coalg may be viewed as objects in T-Inv wherein they easily remain initial and final. Of course there is a canonical map from the initial to the final. I will say that T is ALGEBRAICALLY BOUNDED if this canonical map is an isomorphism, equivalently if T-Inv is a punctuated category, that is one with a biterminator, an object that is both initial and final.
An algebraically bicomplete category is ALGEBRAICALLY COMPACT if each endofunctor is algebraically bounded. (As with algebraic completeness this should be understood to be in a 2-category setting.) In this context I will use the term FREE T-ALGEBRA rather than either initial algebra or final coalgebra.