Unit:
Κατεύθυνση Θεωρητικά ΜαθηματικάLibrary of the School of Science
Supervisors info:
Ιωάννης Εμμανουήλ, Καθηγητής, Τμήμα Μαθηματικών, Εθνικό και Καποδιστριακό Πανεπιστήμιο Αθηνών
Μιχάλης Μαλιάκας, Καθηγητής, Τμήμα Μαθηματικών, Εθνικό και Καποδιστριακό Πανεπιστήμιο Αθηνών
Ιωάννης Ντόκας, Επίκουρος Καθηγητής, Τμήμα Μαθηματικών, Εθνικό και Καποδιστριακό Πανεπιστήμιο Αθηνών
Original Title:
Μία εισαγωγή στις Τριγωνισμένες Κατηγορίες και το Θεώρημα Αναπαραστασιμότητας του Brown
Translated title:
An introduction to Triangulated Categories and the Brown Representability Theorem
Summary:
The basic idea of this dissertation is to study a class of triangulated categories, which generalizes for all large regular cardinals, the class of aleph_{0}-compactly generated triangulated categories, in order to preserve their most interesting properties, e.g. the closure of the class under localizing subcategories generated by aleph_{0}-compact sets of objects, and under Verdier localization with localizing subcategories generated by aleph_{0}-compact sets of objects, the validity of Brown's Representability Theorem ( "for the cohomology" ), and its dual ( "for the homology" ). This new class, is the class of well generated triangulated categories founded by Neeman. In particular, we will prove that well generated triangulated categories satisfy Brown's Representability Theorem. We will generalize Thomason's Localization Theorem, which will allow us to conclude, inter alia, that the localizing subcategories generated by sets of objects, and the Verdier quotients of well generated triangulated categories with localizing subcategories generated by sets of objects are also well generated triangulated categories, and therefore satisfy Brown's Representability Theorem. Finally, although the definition of well generated triangulated categories is not self-dual, the method of proving Brown's Representability Theorem is generalized to the duals of well generated triangulated categories, but unfortunately under a strong hypothesis, the presence of enough injectives objects in some very specific categories of functors which we will study in detail.
Main subject category:
Science
Keywords:
triangulated categories, triangulated functors, Verdier localisation, Thomason localisation, Brown Representability Theorem
