LUCA SPADA | TEORIA DELLE CATEGORIE
LUCA SPADA TEORIA DELLE CATEGORIE
cod. 8803000025
TEORIA DELLE CATEGORIE
| 8803000025 | |
| DIPARTIMENTO DI FISICA "E.R. CAIANIELLO" | |
| P.H.D. COURSE | |
| MATHEMATICS, PHYSICS AND APPLICATIONS | |
| 2020/2021 |
| YEAR OF COURSE 1 | |
| YEAR OF DIDACTIC SYSTEM 2020 | |
| ANNUALE |
| SSD | CFU | HOURS | ACTIVITY | |
|---|---|---|---|---|
| MAT/01 | 4 | 20 | LESSONS |
| Objectives | |
|---|---|
| CRITICAL KNOWLEDGE OF THE LANGUAGE AND OF THE MAIN TOOLS OF CATEGORY THEORY. ####### EXPECTED LEARNING RESULTS: *** KNOWLEDGE AND UNDERSTANDING - KNOWLEDGE OF THE MAIN UNIVERSAL CONSTRUCTIONS (PRODUCTS, CO-PRODUCTS, PUSHOUTS, PULLBACKS, ETC.) AND THEIR ROLE IN MATHEMATICS. - KNOWLEDGE OF THE CONCEPT OF FUNCTOR BETWEEN CATEGORIES, ADDITIONS AND EQUIVALENCES, AND OF THEIR ROLE IN MATHEMATICS. *** ABILITY TO APPLY KNOWLEDGE AND UNDERSTANDING - ABILITY TO RECOGNIZE CATEGORICAL CONSTRUCTIONS IN MATHEMATICS. - ABILITY TO APPLY THE TECHNIQUES AND TOOLS OF THE COURSE IN SIMILAR CASES NOT DISCUSSED IN CLASS. - ABILITY TO REPRODUCE THE PROOFS OF THE MAIN RESULTS OF THE COURSE. *** AUTONOMY OF JUDGMENT - THE STUDENT SHOULD BE ABLE TO CONNECT THE TOPICS OF THE COURSE WITH THOSE OF ANALYSIS, ALGEBRA, GEOMETRY AND WITH THEORETICAL COMPUTER SCIENCES. *** COMMUNICATION SKILLS - THE STUDENT WILL BE ABLE TO SUPPORT CONVERSATIONS WITH PRECISION AND RIGOR ON TOPICS RELATED TO THE THEORY OF CATEGORIES |
| Prerequisites | |
|---|---|
| BASIC KNOWLEDGE OF ALGEBRA AND SET THEORY |
| Contents | |
|---|---|
| - CATEGORIES, UNIVERSAL PROPERTIES, FUNCTORS (4 HOURS). - NATURAL TRANSFORMATIONS, ADJOINT FUNCTORS AND CATEGORICAL EQUIVALENCES (4 HOURS). - CONCRETE DUALITIES (4 HOURS). - YONEDA LEMMA (4 HOURS). - SHEAVES AND TOPOS (4 HOURS). |
| Teaching Methods | |
|---|---|
| THE 20-HOUR COURSE TAKE PLACE IN THE II SEMESTER. THE COURSE INCLUDES THEORETICAL LESSONS WITH GROUP DISCUSSIONS (4CFU) IN THE CLASSROOM. DURING THE DISCUSSIONS, PHD STUDENTS (POSSIBLY IN GROUPS) SOLVE THEORETICAL PROBLEMS WHICH WILL THEN BE USED TO ACHIEVE MORE INVOLVED RESULTS. THIS LAST PHASE PROMOTES THE ABILITY TO IMAGINE POSSIBLE STRATEGIES TO FORMALIZE INTUITIONS AND TO BUILD COMPLEX CONCEPTS STARTING FROM BASIC ONES. |
| Verification of learning | |
|---|---|
| THE EXAMINATION TEST IS AIMED TO ASSESS THE KNOWLEDGE AND UNDERSTANDING OF THE CONCEPTS PRESENTED IN THE LESSON AS A WHOLE, AS WELL AS THE RIGOUR AND AUTONOMY IN THE USE OF SUCH TOOLS. THE PHD STUDENT CAN CHOOSE TO TAKE THE FINAL EXAM IN ONE OF THE FOLLOWING WAYS: - A SHORT ORAL INTERVIEW (ABOUT 30 MINUTES) IN WHICH THE KNOWLEDGE ACQUIRED ABOUT THE BASIC CONCEPTS AND THE MOST ADVANCED CONCEPTS OF CATEGORY THEORY WILL BE ASSESSED. - THE PRESENTATION OF A TOPIC, AGREED WITH THE TEACHER AND NOT COVERED IN THE COURSE, IN THE FORM OF A SHORT SEMINAR OPEN ALSO TO OTHER DOCTORAL STUDENTS LASTING ABOUT 45 MINUTES. - SOLVING A SET OF EXERCISES AT HOME. THE FINAL EVALUATION IS EXPRESSED AS A JUDGMENT (SUFFICIENT, GOOD, ETC.). THE MAXIMUM GRADE CAN BE ATTRIBUTED TO DOCTORAL STUDENTS WHO SHOW TO BE ABLE TO USE THE MOST ADVANCED KNOWLEDGE AND SKILLS ACQUIRED DURING THE COURSE, EVEN BY FINDING CONNECTIONS WITH CONTEXTS DIFFERENT FROM THOSE PROPOSED IN THE LESSON. |
| Texts | |
|---|---|
| HAROLD SIMMONS. AN INTRODUCTION TO CATEGORY THEORY. CAMBRIDGE UNIVERSITY PRESS, 2011. ROBERT GOLDBLATT TOPOI: THE CATEGORIAL ANALYSIS OF LOGIC, DOVER PUBLICATIONS 2006. SAUNDERS MAC LANE. CATEGORIES FOR THE WORKING MATHEMATICIAN (SECOND EDITION). SPRINGER. 1988. |
| More Information | |
|---|---|
| LECTURER'S WEBSITE : HTTP://LOGICA.DIPMAT.UNISA.IT/LUCASPADA/ |
BETA VERSION Data source ESSE3 [Ultima Sincronizzazione: 2022-05-23]
