LUCA SPADA | UNIVERSAL ALGEBRA AND CATEGORY THEORY
LUCA SPADA UNIVERSAL ALGEBRA AND CATEGORY THEORY
cod. 0522200054
UNIVERSAL ALGEBRA AND CATEGORY THEORY
| 0522200054 | |
| DEPARTMENT OF MATHEMATICS | |
| EQF7 | |
| MATHEMATICS | |
| 2024/2025 |
| YEAR OF COURSE 2 | |
| YEAR OF DIDACTIC SYSTEM 2018 | |
| AUTUMN SEMESTER |
| SSD | CFU | HOURS | ACTIVITY | |
|---|---|---|---|---|
| MAT/01 | 6 | 48 | LESSONS |
| Exam | Date | Session | |
|---|---|---|---|
| ALGEBRA UNIVERSALE E TEORIA DELLE CATEGO | 07/01/2025 - 09:30 | SESSIONE ORDINARIA | |
| ALGEBRA UNIVERSALE E TEORIA DELLE CATEGO | 07/01/2025 - 09:30 | SESSIONE DI RECUPERO | |
| ALGEBRA UNIVERSALE E TEORIA DELLE CATEGO | 27/01/2025 - 09:30 | SESSIONE ORDINARIA | |
| ALGEBRA UNIVERSALE E TEORIA DELLE CATEGO | 27/01/2025 - 09:30 | SESSIONE DI RECUPERO | |
| ALGEBRA UNIVERSALE E TEORIA DELLE CATEGO | 17/02/2025 - 09:30 | SESSIONE ORDINARIA | |
| ALGEBRA UNIVERSALE E TEORIA DELLE CATEGO | 17/02/2025 - 09:30 | SESSIONE DI RECUPERO |
| Objectives | |
|---|---|
| CRITICAL KNOWLEDGE OF THE FUNDAMENTAL LAWS OF ALGEBRA THAT ARE COMMON TO ALL EQUATIONALLY DEFINED CLASSES OF STRUCTURES AND OF THE MAIN TOOLS FOR THEIR GENERAL STUDY. ** EXPECTED LEARNING OUTCOMES: * KNOWLEDGE AND UNDERSTANDING - KNOWLEDGE OF THE CONCEPT OF LATTICE AND ITS MAIN CLASSIFICATIONS. - KNOWLEDGE OF THE GENERAL CONCEPT OF HOMOMORPHISM, SUBALGEBRA, QUOTIENT. - KNOWLEDGE OF THE CONGRUENCE LATTICE OF AN ALGEBRA AND ITS CONNECTIONS WITH THE ALGEBRA ITSELF. - KNOWLEDGE OF THE MAIN CATEGORICAL CONSTRUCTIONS: PRODUCTS, COMPRODUCTS, DIRECT LIMITS AND INVERSE LIMITS. - KNOWLEDGE OF THE MAIN CATEGORICAL CONCEPR: CATEGORY, FUNCTOR, ADJUNCTION, NATURAL TRANSFORMATION. - KNOWLEDGE OF THE CONCEPT OF FREE ALGEBRA AND ALGEBRA OF POLYNOMIALS. * ABILITY TO APPLY KNOWLEDGE AND UNDERSTANDING - ABILITY TO RIGOROUSLY REPRODUCE THE PROOFS OF THE MAIN RESULTS OF THE COURSE. - ABILITY TO APPLY THE TECHNIQUES AND TOOLS OF THE COURSE IN SIMILAR CASES. - ABILITY TO CLASSIFY EQUATIONALLY DEFINED CLASSES OF ALGEBRAS ON THE BASIS OF THEIR MAIN PROPERTIES. - ABILITY TO EXTRAPOLATE THE CRUCIAL ASPECTS OF AN ARBITRARY CLASS OF EQUATIONAL STRUCTURES. ** AUTONOMY OF JUDGMENT THE STUDENT MUST BE ABLE TO CONNECT THE COURSE TOPICS WITH THOSE OF ANALYSIS, ALGEBRA, GEOMETRY AND THEORETICAL COMPUTER THEMES. ** COMMUNICATION SKILLS THE STUDENT WILL BE ABLE TO HOLD CONVERSATIONS WITH PRECISION AND RIGOR ON ISSUES RELATED TO UNIVERSAL ALGEBRA AND CATEGORY THEORY. |
| Prerequisites | |
|---|---|
| BASIC KNOWLEDGE OF ALGEBRA AND SET THEORY |
| Contents | |
|---|---|
| - LATTICE THEORY (6 HOURS) - ALGEBRAIC STRUCTURES. (8 HOURS) - SUBALGEBRAS, HOMOMORPHISMS AND CONGRUENCES. (4 HOURS) - CATEGORIES, FUNCTORS (8 HOURS) - PRODUCTS, CO-PRODUCTS, SUB-DIRECT PRODUCTS. (4 HOURS) - LIMITS AND CO-LIMITS (4 HOURS) - NATURAL TRANSFORMATIONS AND ADJUNCTIONS. (6 HOURS) - FREE ALGEBRAS. (4 HOURS) - EQUATIONAL CLASSES (8 HOURS) |
| Teaching Methods | |
|---|---|
| THE 48-HOUR COURSE TAKE PLACE IN THE II SEMESTER. THE COURSE INCLUDES THEORETICAL LESSONS WITH GROUP DISCUSSIONS (6CFU) IN THE CLASSROOM. DURING THE DISCUSSIONS, 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 EXAM AIMS TO EVALUATE THE WHOLE KNOWLEDGE AND UNDERSTANDING OF THE CONCEPTS INTRODUCED DURING THE LECTURES, AS WELL AS THE ACCURACY AND INDEPENDENCE IN USING SUCH TOOLS. THE EXAMINATION CONSISTS OF AN ORAL INTERVIEW (ABOUT 45 MINUTES) WHERE IT WILL BE EVALUATED THE KNOWLEDGE ACQUIRED ON BASIC AND MOST ADVANCED CONCEPTS IN UNIVERSAL ALGEBRA. STUDENTS MUST FIRST DEMONSTRATE THAT THEY KNOW THE CONCEPTS (DEFINITIONS) COVERED DURING THE COURSE AND THAT THEY UNDERSTOOD THEM, SHOWING THAT THEY CAN INDEPENDENTLY BUILD EXAMPLES. LATER THE QUESTIONS WILL BE AIMED AT UNDERSTANDING IF STUDENTS KNOW HOW TO USE THOSE CONCEPTS AND DEFINITIONS AND KNOW THE FUNDAMENTAL PROPERTIES SEEN DURING THE COURSE (THEOREMS). ONLY IF BOTH THE PREVIOUS PARTS ARE SUCCESSFULLY OVERCOME THE REASONS WHY THESE PROPERTIES HOLDS WILL BE DISCUSSED (DEMONSTRATIONS). THE FINAL MARK IS UP TO THIRTY. TO PASS THE EXAM (MINIMUM GRADE 18/30), THE STUDENT MUST SHOW TO HAVE UNDERSTANDOOD THE FUNDAMENTAL CONCEPTS AND RESULTS OF THE COURSE. LAUDE WILL BE ATTRIBUTED TO STUDENTS THAT PROVE THEMSELVES TO BE ABLE TO INDEPENDENTLY USE THE KNOWLEDGE AND SKILLS ACQUIRED ON THE MOST ADVANCED TOPICS OF THE COURSE, AND ARE ABLE TO FIND CONNECTIONS WITH CONTEXTS DIFFERENT THAN THOSE PRESENTED IN THE LECTURES. |
| Texts | |
|---|---|
| CLIFFORD BERGMAN. UNIVERSAL ALGEBRA: FUNDAMENTAL AND SELECTED TOPICS. CRC PRESS. 2011 S. BURRIS, H. P. SANKAPPANAVAR. A COURSE ON UNIVERSAL ALGEBRA. (FREELY AVAILABLE ONLINE) LEINSTER, TOM. BASIC CATEGORY THEORY. VOL. 143. CAMBRIDGE UNIVERSITY PRESS, 2014. (FREELY AVAILABLE ONLINE) HAROLD SIMMONS. AN INTRODUCTION TO CATEGORY THEORY. CAMBRIDGE UNIVERSITY PRESS, 2011. (FREELY AVAILABLE ONLINE) ROBERT GOLDBLATT TOPOI: THE CATEGORIAL ANALYSIS OF LOGIC, DOVER PUBLICATIONS 2006. (DFREELY AVAILABLE ONLINE) SAUNDERS MAC LANE. CATEGORIES FOR THE WORKING MATHEMATICIAN (SECOND EDITION). SPRINGER. 1988. (FREELY AVAILABLE ONLINE) LECTURE NOTES |
| More Information | |
|---|---|
| LECTURER'S WEBSITE : HTTP://LOGICA.DIPMAT.UNISA.IT/LUCASPADA/ EMAIL: LSPADA@UNISA.IT |
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