Skip to main content
SpringerLink
Log in

Grothendieck Institutions

  • Published:
Applied Categorical Structures Aims and scope Submit manuscript

Abstract

We extend indexed categories, fibred categories, and Grothendieck constructions to institutions. We show that the 2-category of institutions admits Grothendieck constructions (in a general 2-categorical sense) and that any split fibred institution is equivalent to a Grothendieck institution of an indexed institution. We use Grothendieck institutions as the underlying mathematical structure for the semantics of multi-paradigm (heterogenous) algebraic specification. We recuperate the so-called ‘extra theory morphisms’ as ordinary theory morphisms in a Grothendieck institution. We investigate the basic mathematical properties of Grothendieck institutions, such as theory colimits, liberality (free constructions), exactness (model amalgamation), and inclusion systems by ‘globalisation’ from the ‘local’ level of the indexed institution to the level of the Grothendieck institution.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Bénabou, J.: Fibred categories and the foundations of naive category theory, J. Symbolic Logic 50 (1985), 10–37.

    Google Scholar 

  2. Borceux, F.: Handbook of Categorical Algebra, Cambridge University Press, 1994.

  3. Căzănescu V. E. and Ro¸su, G.: Weak Inclusion systems, Math. Struct. Comput. Sci.7(2) (1997), 195–206.

    Google Scholar 

  4. Diaconescu, R.: Extra TheoryMorphisms for Institutions: logical semantics for multi-paradigm languages, Appl. Categ. Struct. 6(4) (1998), 427–453. A preliminary version appeared as JAIST Technical Report IS-RR-97-0032F in 1997.

    Google Scholar 

  5. Diaconescu, R. and Futatsugi, K.: Logical foundations of CafeOBJ, Theo. Comp. Sci., to appear.

  6. Diaconescu, R. and Futatsugi, K.: CafeOBJ Report: The Language, Proof Techniques, and Methodologies for Object-Oriented Algebraic Specification, AMAST Series in Computing 6, World Scientific, 1998.

  7. Diaconescu, R., Goguen, J. and Stefaneas, P.: Logical support for modularisation, in G. Huet and G. Plotkin (eds.), Logical Environments, Proceedings of a Workshop held in Edinburgh, Scotland, May 1991, Cambridge, 1993, pp. 83–130.

  8. Diaconescu, R. and Stefaneas, P.: Categorical foundations of modularization for multiparadigm languages, Technical Report IS-RR-98-0014F, Japan Advanced Institute for Science and Technology, 1998.

  9. Goguen, J. and Burstall, R.: Institutions: Abstract model theory for specification and programming, J. Assoc. Comp. Math. 39(1) (1992), 95–146.

    Google Scholar 

  10. Grothendieck, A.: Catégories fibrées et descente, in Revêtements étales et groupe fondamental, Séminaire de Géométrie Algébraique du Bois-Marie 1960/61, Exposé VI, Institut des Hautes Études Scientifiques, 1963. Reprinted in Lecture Notes in Math. 224, Springer, 1971, pp. 145-194.

  11. Kreowski, H.-J. and Mossakowski, T.: Equivalence and difference between institutions: Simulating Horn clause logic with based algebras, Math. Struct. Comp. Sci. 5 (1995), 189–215.

    Google Scholar 

  12. MacLane, S.: Categories for the Working Mathematician, 2nd edn, Springer, 1998.

  13. Meseguer, J.: Membership algebra as a logical framework for equational specification, in F. Parisi-Pressice (ed.), Proc. WADT'97, 1998, pp. 18–61.

  14. Paré, R. and Schumacher, D.: Indexed Categories and their Applications, Lecture Notes in Math. 661, Springer, 1978, Chapt. Abstract Families and the Adjoint Functor Theorems, pp. 1–125.

  15. Tarlecki, A.: On the existence of free models in abstract algebraic institutions, Theo. Comp. Sci. 37 (1986), 269–304. Preliminary version, University of Edinburgh, Computer Science Department, Report CSR-165-84, 1984.

    Google Scholar 

  16. Tarlecki, A.: Moving between logical systems, in M. Haveraaen, O. Owe and O.-J. Dahl (eds.), Recent Trends in Data Type Specification, Proceedings of 11th Workshop on Specification of Abstract Data Types. Oslo, Norway, September 1995, 1996, pp. 478-502.

  17. Tarlecki, A.: Towards heterogeneous specifications, in Proc. Workshop on Frontiers of Combining Systems, 1998.

  18. Tarlecki, A., Burstall, R. and Goguen, J.: Some fundamental algebraic tools for the semantics of computation, Part 3: Indexed categories, Theo. Comp. Sci. 91 (1991), 239–264. Also, Monograph PRG-77, Programming Research Group, Oxford University, August 1989.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics “Simian, Stoilow”, Romania

    Răzvan Diaconescu

Authors
  1. Răzvan Diaconescu
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Diaconescu, R. Grothendieck Institutions. Applied Categorical Structures 10, 383–402 (2002). https://doi.org/10.1023/A:1016330812768

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1016330812768

Search

Navigation