A logic of orthogonality
Adámek, Jiří ; Hébert, Michel ; Sousa, Lurdes
Archivum Mathematicum, Tome 042 (2006), p. 309-334 / Harvested from Czech Digital Mathematics Library
Access to full text
Full (PDF)

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.

Publié le : 2006-01-01
Classification:  03C05,  03G30,  18C10,  18C35 
@article{108011,
     author = {Ji\v r\'\i\ Ad\'amek and Michel H\'ebert and Lurdes Sousa},
     title = {A logic of orthogonality},
     journal = {Archivum Mathematicum},
     volume = {042},
     year = {2006},
     pages = {309-334},
     zbl = {1156.18301},
     mrnumber = {2283016},
     language = {en},
     url = {http://dml.mathdoc.fr/item/108011}
}

Adámek, Jiří; Hébert, Michel; Sousa, Lurdes. A logic of orthogonality. Archivum Mathematicum, Tome 042 (2006) pp. 309-334. http://gdmltest.u-ga.fr/item/108011/

Adámek J.; Hébert M.; Sousa L. A Logic of Injectivity, Preprints of the Department of Mathematics of the University of Coimbra 06-23 (2006). | MR 2369160 | Zbl 1184.18002

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/$\sim $dmb/acc.pdf (1990) | MR 1051419

Adámek J.; Rosický J. Locally presentable and accessible categories, Cambridge University Press, 1994. (1994) | MR 1294136 | Zbl 0795.18007

Adámek J.; Sobral M.; Sousa L. A logic of implications in algebra and coalgebra, Preprint. | MR 2565857 | Zbl 1229.18001

Borceux F. Handbook of Categorical Algebra I, Cambridge University Press, 1994. (1994)

Casacuberta C.; Frei A. On saturated classes of morphisms, Theory Appl. Categ. 7, No. 4 (2000), 43–46. | MR 1751224 | Zbl 0947.18002

Freyd P. J.; Kelly G. M. Categories of continuous functors I, J. Pure Appl. Algebra 2 (1972), 169–191. (1972) | MR 0322004 | Zbl 0257.18005

Gabriel P.; Zisman M. Calculus of Fractions and Homotopy Theory, Springer Verlag 1967. (1967) | MR 0210125 | Zbl 0186.56802

Hébert M. $\mathcal{K}$-Purity and orthogonality, Theory Appl. Categ. 12, No. 12 (2004), 355–371. | MR 2068519

Hébert M.; Adámek J.; Rosický J. More on orthogonolity in locally presentable categories, Cahiers Topologie Géom. Différentielle Catég. 62 (2001), 51–80. | MR 1820765

Mac Lane S. Categories for the Working Mathematician, Springer-Verlag, Berlin-Heidelberg-New York 1971. (1971) | Zbl 0232.18001

Roşu G. Complete categorical equational deduction, Lecture Notes in Comput. Sci. 2142 (2001), 528–538. | MR 1908795