توبوس

توبوس topos (تجمع بالإنكليزية على "topoi" أو "toposes") أحد انماط التصنيفات التي تسلك سلوك تصنيف من الحزم sheaves لمجموعات على فضاء طوبولوجي. تفصيلات نظرية التوبوس ستناقش في خلفية ونشأة نظرية التوبوس، في الرياضيات.

يعود تاريخ التوبوس إلى إدخال فكرة الحزم في الرياضيات في الأربعينات من القرن العشرين لدراسة فضاء رياضي ما عن طريق دراسة الحزم على هذا الفضاء. تم توسيع هذه الفكرة لاحقا من قبل ألكسندر غروتينديك بإدخال مصطلح التوبوس.

مراجع

• جون سي بايز: Topos theory in a nutshell, http://math.ucr.edu/home/baez/topos.html. A gentle introduction.
• Stephen Vickers: Toposes pour les nuls and Toposes pour les vraiment nuls. Available at Vickers’ website. Elementary and even more elementary introductions to toposes as generalized spaces.

The following textbooks provide easy paced first introductions (including basics of category theory). They should be suitable for students of various—even non-mathematical—disciplines:

• F. William Lawvere and Stephen H. Schanuel: Conceptual Mathematics: A First Introduction to Categories, Cambridge University Press, Cambridge, 1997. An "introduction to categories for computer scientists, logicians, physicists, linguists, etc." (cited from cover text).
• F. William Lawvere and Robert Rosebrugh: Sets for Mathematics, Cambridge University Press, Cambridge, 2003. Discusses the foundations of mathematics from a categorical perspective. A book "for students who are beginning the study of advanced mathematical subjects".

الأعمال الأصلية لغروتينديك

• ألكسندر غروتينديك and Verdier: Théorie des topos et cohomologie étale des schémas (known as SGA4)". New York/Berlin: Springer, ??. (Lecture notes in mathematics, 269–270)

Interesting research books that are provide introductions to topos theory (or to a specific aspect of it), but which do not primarily cater to students. The given order roughly (!) reflects the difficulty of the level of exposition:

• Colin McLarty: Elementary Categories, Elementary Toposes, Clarendon Press, Oxford, 1992. Includes a nice introduction of the basic notions of category theory, topos theory, and topos logic. Assumes very few prerequisites.
• Robert Goldblatt: Topoi, the Categorial Analysis of Logic. North-Holland, New York, 1984. (Studies in logic and the foundations of mathematics, 98.). A good start.

This book has been reprinted by Dover Publications, Inc (2006). The book can also be accessed freely on Robert Goldblatt's homepage: Topoi, the Categorial Analysis of Logic.

• John L. Bell: The development of categorical logic. http://publish.uwo.ca/~jbell/catlogprime.pdf (PDF)
• سوندرز ماكلين and Ieke Moerdijk: Sheaves in Geometry and Logic: a First Introduction to Topos Theory, Springer, New York, 1992. More complete, and more difficult to read.
• Michael Barr and Charles Wells: Toposes, Triples and Theories, Springer, 1985. Corrected online version at http://www.cwru.edu/artsci/math/wells/pub/ttt.html. More concise than Sheaves in Geometry and Logic, but not an easy reading for the beginner.

كتب للمتخصصين

• Francis Borceux: Handbook of Categorical Algebra 3: Categories of Sheaves, Volume 52 of the Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1994. The third part of "Borceux' remarkable magnum opus", as Johnstone has labelled it. Still suitable as an introduction, though beginners may find it hard to recognize the most relevant results among the huge amount of material given.
• Peter T. Johnstone: Topos Theory, L. M. S. Monographs no. 10, Academic Press, 1977. For a long time the standard compendium on topos theory. However, it has also been described as "far too hard to read, and not for the faint-hearted", as quoted by Johnstone himself.
• Peter T. Johnstone: Sketches of an Elephant: A Topos Theory Compendium, Oxford Science Publications, Oxford, 2002. Johnstone’s overwhelming compendium. As of early 2006, two of the scheduled three volumes were available.

كتب تستهدف تطبيقات خاصة لنظرية التوبوس

• Maria Cristina Pedicchio and Walter Tholen (editors): Categorical Foundations: Special Topics in Order, Topology, Algebra, and Sheaf Theory. Volume 97 of the Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2004. Includes many interesting special applications.

موسوعات أخرى

• Topos على بلانيت ماث