Skip to main content
SpringerLink
Log in

Approximate Injectivity

  • Published:
Applied Categorical Structures Aims and scope Submit manuscript

Abstract

In a locally \(\lambda \)-presentable category, with \(\lambda \) a regular cardinal, classes of objects that are injective with respect to a family of morphisms whose domains and codomains are \(\lambda \)-presentable, are known to be characterized by their closure under products, \(\lambda \)-directed colimits and \(\lambda \)-pure subobjects. Replacing the strict commutativity of diagrams by “commutativity up to \(\mathcal {\varepsilon }\)”, this paper provides an “approximate version” of this characterization for categories enriched over metric spaces. It entails a detailed discussion of the needed \(\mathcal {\varepsilon }\)-generalizations of the notion of \(\lambda \)-purity. The categorical theory is being applied to the locally \(\aleph _1\)-presentable category of Banach spaces and their linear operators of norm at most 1, culminating in a largely categorical proof for the existence of the so-called Gurarii Banach space.

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adámek, J., Herrlich, H., Rosický, J., Tholen, W.: On a generalized small object argument for the injective subcategory problem. Cahiers Topologie Géom. Différentielle Catégoriques 43, 83–106 (2002)

    MathSciNet  MATH  Google Scholar 

  2. Adámek, J., Rosický, J.: Locally Presentable and Accessible Categories. Cambridge University Press, Cambridge (1994)

    Book  MATH  Google Scholar 

  3. Avilés, A., Sánchez, F.C., Castillo, J.M.F., Gonzáles, M., Moreno, Y.: Banach spaces of universal disposition. J. Fun. Anal. 261, 2347–2361 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Beke, T.: Sheafifiable homotopy model categories. Math. Proc. Cambr. Phil. Soc. 129, 445–475 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Borceux, F.: Handbook of Categorical Algebra 2. Categories and Structures. Cambridge University Press, Cambridge (1994)

    Book  MATH  Google Scholar 

  6. Eagle, C.J., Farah, I., Hart, B., Kadets, B., Kalashnyk, V., Lupini, M.: Fraissé limits of \(C^\ast \)-algebras. J. Symb. Logic 81, 755–773 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Garbulińska, J., Kubiś, W.: Remarks on Gurarii spaces. Extracta Math. 26, 235–269 (2011/2012)

  8. Hušek, M.: Urysohn universal space, its development and Hausdorff’s approach. Top. Appl. 155, 1493–1501 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kelly, G.M.: Basic Concepts of Enriched Category Theory. Cambridge University Press, Cambridge (1982)

    MATH  Google Scholar 

  10. Kelly, G.M.: Structures defined by finite limits in the enriched context, I. Cahiers Topologie Géom. Différentielle Catégoriques 23, 3–42 (1982)

    MathSciNet  MATH  Google Scholar 

  11. Kubiś, W.: Metric enriched categories and approximate Fraissé limits. arXiv:1210.6506

  12. Kubiś, W.: Katětov functors. Appl. Categ. Struct. 25, 561–602 (2017)

    MATH  Google Scholar 

  13. Kubiś, W.: Game-theoretic characterization of the Gurarii space. Arch. Math. (2017). https://doi.org/10.1007/s00013-017-1088-2

    MATH  Google Scholar 

  14. Kurz, A., Rosický, J.: Weak factorizations, fractions and homotopies. Appl. Cat. Struct. 13, 141–160 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lieberman, M., Rosický, J.: Metric abstract elementary classes as accessible categories. J. Symb. Logic 82, 1022–1318 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lupini, M.: Fraissé limits in functional analysis. arXiv:1510.05188

  17. Rosický, J., Adámek, J., Borceux, F.: More on injectivity classes in locally presentable categories. Th. Appl. Categ. 10, 148–161 (2002)

    MATH  Google Scholar 

  18. Rosický, J.: Accessible categories, saturation and categoricity. J. Symb. Logic 62, 891–901 (1997)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Faculty of Sciences, Masaryk University, Kotlářská 2, 611 37, Brno, Czech Republic

    J. Rosický

  2. Department of Mathematics and Statistics, Faculty of Science, York University, 4700 Keele Street, Toronto, Ontario, M3J 1P3, Canada

    W. Tholen

Authors
  1. J. Rosický
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. W. Tholen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. Rosický.

Additional information

Communicated by M. M. Clementino.

J. Rosický: Supported by the Grant Agency of the Czech Republic under the Grant P201/12/G028. W. Tholen: Supported by the Natural Sciences and Engineering Research Council of Canada under the Discovery Grants program.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rosický, J., Tholen, W. Approximate Injectivity. Appl Categor Struct 26, 699–716 (2018). https://doi.org/10.1007/s10485-017-9510-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10485-017-9510-2

Keywords

Mathematics Subject Classification

Search

Navigation