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Presheaves Over a Join Restriction Category

  • Daniel Lin
Article
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Abstract

Just as the presheaf category is the free cocompletion of any small category, there is an analogous notion of free cocompletion for any small restriction category. In this paper, we extend the work on restriction presheaves to presheaves over join restriction categories, and show that the join restriction category of join restriction presheaves is equivalent to some partial map category of sheaves. We then use this to show that the Yoneda embedding exhibits the category of join restriction presheaves as the free cocompletion of any small join restriction category.

Keywords

Join restriction categories Join restriction presheaves Sheaves Cocompletion 

Mathematics Subject Classification

18B99 

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References

  1. 1.
    Cockett, J.R.B., Guo, X.: Stable meet semilattice fibrations and free restriction categories. Theory Appl. Categ. 16, 307–341 (2006)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Cockett, J.R.B., Lack, S.: Restriction categories I: categories of partial maps. Theor. Comput. Sci. 270, 223–259 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cockett, J.R.B., Lack, S.: Restriction categories II: partial map classification. Theor. Comput. Sci. 294, 61–102 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cockett, J.R.B., Manes, E.: Boolean and classical restriction categories. Math. Struct. Comput. Sci. 19, 357–416 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    DeWolf, D.: Restriction category perspectives of partial computation and geometry. Ph.D. Thesis, Dalhousie University (2017)Google Scholar
  6. 6.
    Di Paola, R., Heller, A.: Dominical categories: recursion theory without elements. J. Symb. Log. 52, 594–635 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Garner, R., Lin, D.: Cocompletion of restriction categories (under review)Google Scholar
  8. 8.
    Grandis, M.: Cohesive categories and manifolds. Ann. Mat. Pura Appl. 157, 199–244 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Guo, X.: Products, joins, meets and ranges in restriction categories. Ph.D. Thesis, University of Calgary (2012)Google Scholar
  10. 10.
    Johnstone, P.: Sketches of an Elephant: A Topos Theory Compendium, vol. 1. Oxford University Press, New York (2002)zbMATHGoogle Scholar
  11. 11.
    Kelly, G.M.: Basic Concepts of Enriched Category Theory. Cambridge University Press, Cambridge (1972) (Republished in Theory and Applications of Category, No. 10, 2005)Google Scholar
  12. 12.
    Mac Lane, S., Moerdijk, I.: Sheaves in Geometry and Logic. Springer, New York (1994)CrossRefzbMATHGoogle Scholar
  13. 13.
    Robinson, E., Rosolini, G.: Categories of partial maps. Inf. Comput. 79, 95–130 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Rosolini, G.: Continuity and effectiveness in topoi. Ph.D. Thesis, University of Oxford (1986)Google Scholar
  15. 15.
    Street, R.: Cauchy characterization of enriched categories. Repr. Theory Appl. Categ. 4, 1–16 (2004)zbMATHGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMacquarie UniversityNorth RydeAustralia

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