Skip to main content Accessibility help
×
×
Home

Database queries and constraints via lifting problems

  • DAVID I. SPIVAK (a1)

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.

Copyright

References

Hide All
Awodey, S. and Warren, M. A. (2009) Homotopy theoretic models of identity types. Mathematical Proceedings of the Cambridge Philosophical Society 146 (1)4555.
Bancilhon, F and Spyratos, N. (1981) Update semantics of relational views. ACM TODS 6 557575.
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 1287.
Borceux, F. (1994) Handbook of categorical algebra 1–3, Encyclopedia of Mathematics and its Applications 50–52, Cambridge University Press.
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 127138.
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.
Deutsch, A., Nash, A. and Remmel, J. (2008) The Chase Revisited. Proceedings of Symposium on Principles of Database Systems (PODS), ACM.
Diskin, Z. and Kadish, B. (1994) Algebraic graph-oriented=category-theory-based – manifesto of categorizing data base theory. Technical report, Frame Inform Systems.
Dugger, D. (2008) A primer on homotopy colimits. ePrint available at http://math.uoregon.edu/~ddugger/hocolim.pdf.
Ehresmann, C. (1968) Esquisses et types des structures algèbriques. Buletinul Institutului Politehic din Iasi (N.S.) 14 (18) (1–2) 114.
Gambino, N. and Kock, J. (2013) Polynomial functors and polynomial monads. Mathematical Proceedings of the Cambridge Philosophical Society 154 153192.
Garner, R. (2009) Understanding the small object argument. Applied Categorical Structures 17 (3)247285.
Ghrist, R. (2008) Barcodes: the persistent topology of data. Bulletin of the American Mathematical Society (N.S.) 45 (1)6175.
Hartshorne, R. (1977) Algebraic Geometry, Graduate Texts in Mathematics 52, Springer-Verlag.
Hirschhorn, P. (2003) Model categories and their localizations, Mathematical surveys and monographs, American Mathematical Society 99.
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.
Johnson, M., Rosebrugh, R. and Wood, R. J. (2002) Entity-relationship-attribute designs and sketches. Theory and Applications of Categories 10 94112.
Johnstone, P. (2002) Sketches of an elephant 1-2, Oxford logic guides 43–44, The Clarendon Press.
Joyal, A. (2002) Quasi-categories and Kan complexes. Journal of Pure and Applied Algebra 175 (1–3)207222.
Joyal, A. (2010) Catlab. (Available online at http://ncatlab.org/joyalscatlab/show/Factorisation+systems.)
Kato, A. (1983) An abstract relational model and natural join functors. Bulletin of Informatics and Cybernetics 20 95106.
Kelly, G. M. (1974) On clubs and doctrines. In: Kelly, G. M. (ed.) Category Seminar. Springer-Verlag Lecture Notes in Mathematics 420 181256.
Lurie, J. (2009) Higher topos theory, Annals of Mathematical Studies 170, Princeton University Press.
Mac Lane, S. (1988) Categories for the working mathematician (second edition), Graduate texts in mathematics 5, Springer Verlag.
Mac Lane, S. and Moerdijk, I. (1994) Sheaves in Geometry and Logic: a first introduction to topos theory, Universitext, Springer-Verlag.
Makkai, M. (1997) Generalized sketches as a framework for completeness theorems I. Journal of Pure and Applied Algebra 115 (1)4979.
May, J. P. (1999) A concise course in Algebraic Topology, Chicago Lectures in Mathematics, University of Chicago Press.
Morava, J. (2012) Theories of anything. (Available at http://arxiv.org/abs/1202.0684v1.)
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/.)
Quillen, D. G. (1967) Homotopical Algebra. Springer-Verlag Lecture Notes in Mathematics 43.
Spivak, D. I. (2009) Simplicial databases. (Available at http://arxiv.org/abs/0904.2012.)
Spivak, D. I. (2012) Functorial data migration. Information and Computation 217 3151.
Spivak, D. I. and Kent, R. E. (2012) Ologs: A Categorical Framework for Knowledge Representation. PLoS ONE 7 (1).
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 99112.
Voevodsky, V. (2006) A very short note on the homotopy λ-calculus. Unpublished note.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Structures in Computer Science
  • ISSN: 0960-1295
  • EISSN: 1469-8072
  • URL: /core/journals/mathematical-structures-in-computer-science
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed