## Containers: Constructing strictly positive types.(English)Zbl 1077.68015

Summary: We introduce the notion of a Martin-Löf category – a locally cartesian closed category with disjoint coproducts and initial algebras of container functors (the categorical analogue of W-types) – and then establish that nested strictly positive inductive and coinductive types, which we call strictly positive types, exist in any Martin-Löf category.
Central to our development are the notions of containers and container functors. These provide a new conceptual analysis of data structures and polymorphic functions by exploiting dependent type theory as a convenient way to define constructions in Martin-Löf categories. We also show that morphisms between containers can be full and faithfully interpreted as polymorphic functions (i.e. natural transformations) and that, in the presence of W-types, all strictly positive types (including nested inductive and coinductive types) give rise to containers.

### MSC:

 68N18 Functional programming and lambda calculus

Epigram
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### References:

 [1] M. Abbott, Categories of containers, Ph.D. Thesis, University of Leicester, 2003. · Zbl 1029.68096 [2] M. Abbott, T. Altenkirch, N. Ghani, Categories of containers, in: Proc. Foundations of Software Science and Computation Structures, Lecture Notes in Computer Science, Vol. 2620, Springer, Berlin, 2003, pp. 23-38. · Zbl 1029.68096 [3] M. Abbott, T. Altenkirch, N. Ghani, C. McBride, Derivatives of containers, in: Sixth International Conference on Typed Lambda Calculi and Applications, Lecture Notes in Computer Science, Vol. 2701, Springer, Berlin, 2003, pp. 16-30. · Zbl 1039.68078 [4] M. Abbott, T. Altenkirch, N. Ghani, Representing nested inductive types using W-types, in: International Colloquium on Automata, Languages and Programming, ICALP, 2004, pp. 59-71. · Zbl 1099.03058 [5] M. Abbott, T. Altenkirch, N. Ghani, C. McBride, Constructing polymorphic programs with quotient types, in: Seventh International Conference on Mathematics of Program Construction (MPC 2004), Lecture Notes in Computer Science, Vol. 3125, February 2004, pp. 2-15. · Zbl 1106.68335 [6] M. Abbott, T. Altenkirch, N. Ghani, C. McBride, $$\partial$$ for data, February 2004, submitted for publication. [7] A. Abel, T. Altenkirch, A predicative strong normalisation proof for a $$\lambda$$-calculus with interleaving inductive types, in: Types for Proof and Programs, TYPES ’99, Lecture Notes in Computer Science, Vol. 1956, Springer, Berlin, 2000, pp. 1-18. · Zbl 0988.03029 [8] Aczel, P., On relating type theories and set theories, Lecture notes in comput. sci., 1657, 1-18, (1999) · Zbl 0944.03056 [9] T. Altenkirch, Constructions, Inductive types and strong normalization, Ph.D. Thesis, University of Edinburgh, November 1993. [10] T. Altenkirch, Extensional equality in intensional type theory, in: 14th Symposium on Logic in Computer Science, 1999, pp. 412-420. [11] T. Altenkirch, B. Reus, Monadic presentations of lambda terms using generalized inductive types, in: J. Flum, M. Rodríguez-Artalejo (Eds.), CSL’99, Lecture Notes in Computer Science, Vol. 1683, Springer, Berlin, 1999, pp. 453-468. · Zbl 0944.03011 [12] Bénabou, J., Fibrations petites et localement petites, C.R. acad. sci. Paris, 281, A831-A834, (1975) · Zbl 0349.18006 [13] Bénabou, J., Fibred categories and the foundations of naive category theory, J. symbol. logic, 50, 1, 10-37, (1985) · Zbl 0564.18001 [14] Bird, R.; Paterson, R., Generalised folds for nested datatypes, Formal aspects comput., 11, 3, 200-222, (1999) · Zbl 0937.68027 [15] F. Borceux, Handbook of Categorical Algebra 2, Encyclopedia of Mathematics, Vol. 51, Cambridge University Press, Cambridge, 1994. · Zbl 0843.18001 [16] Crole, R.L., Categories for types, (1993), Cambridge University Press Cambridge · Zbl 0837.68077 [17] Dybjer, P., Representing inductively defined sets by wellorderings in martin-Löf’s type theory, Theoret. comput. sci., 176, 329-335, (1997) · Zbl 0898.68047 [18] N. Gambino, M. Hyland, Wellfounded trees and dependent polynomial functors, in: S. Berardi, M. Coppo, F. Damiani (Eds.), Types for Proofs and Programs (TYPES 2003), Lecture Notes in Computer Science, Springer, Berlin, 2004, pp. 210-225. · Zbl 1100.03055 [19] Hasegawa, R., Two applications of analytic functors, Theoret. comput. sci., 272, 1-2, 112-175, (2002) · Zbl 0984.68030 [20] M. Hofmann, On the interpretation of type theory in locally cartesian closed categories, in: Computer Science Logic, CSL94, 1994, pp. 427-441 · Zbl 1044.03544 [21] Hofmann, M., Extensional constructs in intensional type theory, (1997), Springer Berlin · Zbl 1411.03001 [22] M. Hofmann, Syntax and semantics of dependent types, in: A.M. Pitts, P. Dybjer (Eds.), Semantics and Logics of Computation, Vol. 14, Cambridge University Press, Cambridge, 1997, pp. 79-130. · Zbl 0919.68083 [23] Hoogendijk, P.; de Moor, O., Container types categorically, J. function. program., 10, 2, 191-225, (2000) · Zbl 0959.68023 [24] Jacobs, B., Categorical logic and type theory, () · Zbl 0911.03001 [25] Johnstone, P.T., Topos theory, (1977), Academic Press New York · Zbl 0368.18001 [26] A. Joyal, Foncteurs analytiques et espèces de structures, in: Combinatoire Énumérative, Lecture Notes in Mathematics, Vol. 1234, Springer, Berlin, 1986, pp. 126-159. [27] Martin-Löf, P., An intuitionistic theory of types: predicative part, (), 73-118 [28] Martin-Löf, P., Intuitionistic type theory, (1984), Bibliopolis Napoli [29] McBride, C., Epigram: practical programming with dependent types, (2004), Lecture Notes of the Advanced Functional Programming Summerschool in Tartu Estonia · Zbl 1158.68356 [30] McBride, C.; McKinna, J., The view from the left, J. function. program., 14, 1, 16-111, (2004) · Zbl 1069.68539 [31] Moerdijk, I.; Palmgren, E., Wellfounded trees in categories, Ann. pure appl. logic, 104, 189-218, (2000) · Zbl 1010.03056 [32] B. Nordström, K. Petersson, J.M. Smith, Programming in Martin-Löf’s Type Theory, International Series of Monographs on Computer Science, Vol. 7, Oxford University Press, Oxford, 1990. [33] R. Paré, D. Schumacher, Abstract families and the adjoint functor theorems, in: P.T. Johnstone, R. Paré (Eds.), Indexed Categories and Their Applications, Lecture Notes in Mathematics, Vol. 661, Springer, Berlin, 1978, pp. 1-125. [34] A. Poigné, Basic category theory, in: S. Abramsky, D.M. Gabbay, T.S.E. Maibaum (Eds.), Handbook of Logic in Computer Science, Handbook of Logic in Computer Science, Vol. 1, Oxford University Press, Oxford, 1992. pp. 413-640. [35] Seely, R.A.G., Locally Cartesian closed categories and type theory, Math. proc. camb. phil. soc., 95, 33-48, (1984) · Zbl 0539.03048 [36] Streicher, T., Semantics of type theory, (1991), Progress in Theoretical Computer Science Birkhäuser [37] D. Turner, Elementary strong functional programming, in: R. Plasmeijer, P. Hartel (Eds.), First International Symposium on Functional Programming Languages in Education, Lecture Notes in Computer Science, Vol. 1022, Springer, Berlin, 1996, pp. 1-13. [38] B. van den Berg, F. de Marchi, Non-well-founded trees in categories, 2004, arXiv math.CT/0409158. · Zbl 1166.03042
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