A logic of orthogonality

Jiří Adámek; Michel Hébert; Lurdes Sousa

Archivum Mathematicum (2006)

  • Volume: 042, Issue: 4, page 309-334
  • ISSN: 0044-8753

Abstract

top
A logic of orthogonality characterizes all “orthogonality consequences" of a given class Σ of morphisms, i.e. those morphisms s such that every object orthogonal to Σ is also orthogonal to s . A simple four-rule deduction system is formulated which is sound in every cocomplete category. In locally presentable categories we prove that the deduction system is also complete (a) for all classes Σ of morphisms such that all members except a set are regular epimorphisms and (b) for all classes Σ , without restriction, under the set-theoretical assumption that Vopěnka’s Principle holds. For finitary morphisms, i.e. morphisms with finitely presentable domains and codomains, an appropriate finitary logic is presented, and proved to be sound and complete; here the proof follows immediately from previous joint results of Jiří Rosický and the first two authors.

How to cite

top

Adámek, Jiří, Hébert, Michel, and Sousa, Lurdes. "A logic of orthogonality." Archivum Mathematicum 042.4 (2006): 309-334. <http://eudml.org/doc/249785>.

@article{Adámek2006,
abstract = {A logic of orthogonality characterizes all “orthogonality consequences" of a given class $\Sigma $ of morphisms, i.e. those morphisms $s$ such that every object orthogonal to $\Sigma $ is also orthogonal to $s$. A simple four-rule deduction system is formulated which is sound in every cocomplete category. In locally presentable categories we prove that the deduction system is also complete (a) for all classes $\Sigma $ of morphisms such that all members except a set are regular epimorphisms and (b) for all classes $\Sigma $, without restriction, under the set-theoretical assumption that Vopěnka’s Principle holds. For finitary morphisms, i.e. morphisms with finitely presentable domains and codomains, an appropriate finitary logic is presented, and proved to be sound and complete; here the proof follows immediately from previous joint results of Jiří Rosický and the first two authors.},
author = {Adámek, Jiří, Hébert, Michel, Sousa, Lurdes},
journal = {Archivum Mathematicum},
keywords = {orthogonal subcategory problem},
language = {eng},
number = {4},
pages = {309-334},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {A logic of orthogonality},
url = {http://eudml.org/doc/249785},
volume = {042},
year = {2006},
}

TY - JOUR
AU - Adámek, Jiří
AU - Hébert, Michel
AU - Sousa, Lurdes
TI - A logic of orthogonality
JO - Archivum Mathematicum
PY - 2006
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 042
IS - 4
SP - 309
EP - 334
AB - A logic of orthogonality characterizes all “orthogonality consequences" of a given class $\Sigma $ of morphisms, i.e. those morphisms $s$ such that every object orthogonal to $\Sigma $ is also orthogonal to $s$. A simple four-rule deduction system is formulated which is sound in every cocomplete category. In locally presentable categories we prove that the deduction system is also complete (a) for all classes $\Sigma $ of morphisms such that all members except a set are regular epimorphisms and (b) for all classes $\Sigma $, without restriction, under the set-theoretical assumption that Vopěnka’s Principle holds. For finitary morphisms, i.e. morphisms with finitely presentable domains and codomains, an appropriate finitary logic is presented, and proved to be sound and complete; here the proof follows immediately from previous joint results of Jiří Rosický and the first two authors.
LA - eng
KW - orthogonal subcategory problem
UR - http://eudml.org/doc/249785
ER -

References

top
  1. Adámek J., Hébert M., Sousa L., A Logic of Injectivity, Preprints of the Department of Mathematics of the University of Coimbra 06-23 (2006). Zbl1184.18002MR2369160
  2. Adámek J., Herrlich H., Strecker G. E., Abstract and Concrete Categories, John Wiley and Sons, New York 1990. Freely available at www.math.uni-bremen.de/dmb/acc.pdf (1990) MR1051419
  3. Adámek J., Rosický J., Locally presentable and accessible categories, Cambridge University Press, 1994. (1994) Zbl0795.18007MR1294136
  4. Adámek J., Sobral M., Sousa L., A logic of implications in algebra and coalgebra, Preprint. Zbl1229.18001MR2565857
  5. Borceux F., Handbook of Categorical Algebra I, Cambridge University Press, 1994. (1994) 
  6. Casacuberta C., Frei A., On saturated classes of morphisms, Theory Appl. Categ. 7, No. 4 (2000), 43–46. Zbl0947.18002MR1751224
  7. Freyd P. J., Kelly G. M., Categories of continuous functors I, J. Pure Appl. Algebra 2 (1972), 169–191. (1972) Zbl0257.18005MR0322004
  8. Gabriel P., Zisman M., Calculus of Fractions and Homotopy Theory, Springer Verlag 1967. (1967) Zbl0186.56802MR0210125
  9. Hébert M., 𝒦 -Purity and orthogonality, Theory Appl. Categ. 12, No. 12 (2004), 355–371. MR2068519
  10. Hébert M., Adámek J., Rosický J., More on orthogonolity in locally presentable categories, Cahiers Topologie Géom. Différentielle Catég. 62 (2001), 51–80. MR1820765
  11. Mac Lane S., Categories for the Working Mathematician, Springer-Verlag, Berlin-Heidelberg-New York 1971. (1971) Zbl0232.18001
  12. Roşu G., Complete categorical equational deduction, Lecture Notes in Comput. Sci. 2142 (2001), 528–538. MR1908795

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.