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Database queries and constraints via lifting problems

Published online by Cambridge University Press:  11 October 2013

DAVID I. SPIVAK
DAVID I. SPIVAK*
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, United States of America Email: dspivak@mit.edu
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Abstract

Previous work has demonstrated that categories are useful and expressive models for databases. In the current paper we build on that model, showing that certain queries and constraints correspond to lifting problems, as found in modern approaches to algebraic topology. In our formulation, each SPARQL graph pattern query corresponds to a category-theoretic lifting problem, whereby the set of solutions to the query is precisely the set of lifts. We interpret constraints within the same formalism, and then investigate some basic properties of queries and constraints. In particular, to any database π, we can associate a certain derived database Qry(π) of queries on π. As an application, we explain how giving users access to certain parts of Qry(π), rather than direct access to π, improves the ability to manage the impact of schema evolution.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 24 , Issue 6 , December 2014 , e240602
Copyright
Copyright © Cambridge University Press 2013 

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References

Awodey, S. and Warren, M. A. (2009) Homotopy theoretic models of identity types. Mathematical Proceedings of the Cambridge Philosophical Society 146 (1)45–55.CrossRefGoogle Scholar
Bancilhon, F and Spyratos, N. (1981) Update semantics of relational views. ACM TODS 6 557–575.CrossRefGoogle Scholar
Barr, M. and Wells, C. (2005) Toposes, triples, and theories (corrected reprint of the 1985 original published by Springer-Verlag), Reprints in Theory and Applications of Categories 12 1–287.Google Scholar
Borceux, F. (1994) Handbook of categorical algebra 1–3, Encyclopedia of Mathematics and its Applications 50–52, Cambridge University Press.Google Scholar
Carlsson, G., Zomorodian, A., Collins, A. and Guibas, L. (2004) Persistence barcodes for shapes. In: Scopigno, R. and Zorin, D. (eds.) Eurographics Symposium on Geometry Processing 127–138.Google Scholar
Deus, H. F.et al. (2010) Provenance of microarray experiments for a better understanding of experiment results. Proceedings of The Second International Workshop on the role of Semantic Web in Provenance Management, Shanghai, China.Google Scholar
Deutsch, A., Nash, A. and Remmel, J. (2008) The Chase Revisited. Proceedings of Symposium on Principles of Database Systems (PODS), ACM.Google Scholar
Diskin, Z. and Kadish, B. (1994) Algebraic graph-oriented=category-theory-based – manifesto of categorizing data base theory. Technical report, Frame Inform Systems.Google Scholar
Dugger, D. (2008) A primer on homotopy colimits. ePrint available at http://math.uoregon.edu/~ddugger/hocolim.pdf.Google Scholar
Ehresmann, C. (1968) Esquisses et types des structures algèbriques. Buletinul Institutului Politehic din Iasi (N.S.) 14 (18) (1–2) 1–14.Google Scholar
Gambino, N. and Kock, J. (2013) Polynomial functors and polynomial monads. Mathematical Proceedings of the Cambridge Philosophical Society 154 153–192.Google Scholar
Garner, R. (2009) Understanding the small object argument. Applied Categorical Structures 17 (3)247–285.Google Scholar
Ghrist, R. (2008) Barcodes: the persistent topology of data. Bulletin of the American Mathematical Society (N.S.) 45 (1)61–75.Google Scholar
Hartshorne, R. (1977) Algebraic Geometry, Graduate Texts in Mathematics 52, Springer-Verlag.Google Scholar
Hirschhorn, P. (2003) Model categories and their localizations, Mathematical surveys and monographs, American Mathematical Society 99.Google Scholar
Johnson, M. (2001) On Category Theory as a (meta) Ontology for Information Systems Research. Proceedings of the international conference on Formal Ontology in Information Systems.Google Scholar
Johnson, M., Rosebrugh, R. and Wood, R. J. (2002) Entity-relationship-attribute designs and sketches. Theory and Applications of Categories 10 94–112.Google Scholar
Johnstone, P. (2002) Sketches of an elephant 1-2, Oxford logic guides 43–44, The Clarendon Press.Google Scholar
Joyal, A. (2002) Quasi-categories and Kan complexes. Journal of Pure and Applied Algebra 175 (1–3)207–222.Google Scholar
Joyal, A. (2010) Catlab. (Available online at http://ncatlab.org/joyalscatlab/show/Factorisation+systems.)Google Scholar
Kato, A. (1983) An abstract relational model and natural join functors. Bulletin of Informatics and Cybernetics 20 95–106.Google Scholar
Kelly, G. M. (1974) On clubs and doctrines. In: Kelly, G. M. (ed.) Category Seminar. Springer-Verlag Lecture Notes in Mathematics 420 181–256.Google Scholar
Lurie, J. (2009) Higher topos theory, Annals of Mathematical Studies 170, Princeton University Press.CrossRefGoogle Scholar
Mac Lane, S. (1988) Categories for the working mathematician (second edition), Graduate texts in mathematics 5, Springer Verlag.Google Scholar
Mac Lane, S. and Moerdijk, I. (1994) Sheaves in Geometry and Logic: a first introduction to topos theory, Universitext, Springer-Verlag.Google Scholar
Makkai, M. (1997) Generalized sketches as a framework for completeness theorems I. Journal of Pure and Applied Algebra 115 (1)49–79.Google Scholar
May, J. P. (1999) A concise course in Algebraic Topology, Chicago Lectures in Mathematics, University of Chicago Press.Google Scholar
Morava, J. (2012) Theories of anything. (Available at http://arxiv.org/abs/1202.0684v1.)Google Scholar
Prud'hommeaux, E. and Seaborne, A. (eds.) (2008) SPARQL Query Language for RDF: W3C Recommendation 2008/01/15. (Available at http://www.w3.org/TR/2008/REC-rdf-sparql-query-20080115/.)Google Scholar
Quillen, D. G. (1967) Homotopical Algebra. Springer-Verlag Lecture Notes in Mathematics 43.Google Scholar
Spivak, D. I. (2009) Simplicial databases. (Available at http://arxiv.org/abs/0904.2012.)Google Scholar
Spivak, D. I. (2012) Functorial data migration. Information and Computation 217 31–51.CrossRefGoogle Scholar
Spivak, D. I. and Kent, R. E. (2012) Ologs: A Categorical Framework for Knowledge Representation. PLoS ONE 7 (1).CrossRefGoogle ScholarPubMed
Tuijn, C. and Gyssens, M. (1992) Views and decompositions from a categorical perspective. In: Biskup, J. and Hull, R. (eds.) Database Theory – ICDT '92: Proceedings 4th International Conference. Springer-Verlag Lecture Notes in Computer Science 646 99–112.Google Scholar
Voevodsky, V. (2006) A very short note on the homotopy λ-calculus. Unpublished note.Google Scholar