Simplicial Homotopy Theory

General data

Course ID:	1000-1M22STH
Erasmus code / ISCED:	11.1 Kod klasyfikacyjny przedmiotu składa się z trzech do pięciu cyfr, przy czym trzy pierwsze oznaczają klasyfikację dziedziny wg. Listy kodów dziedzin obowiązującej w programie Socrates/Erasmus, czwarta (dotąd na ogół 0) – ewentualne uszczegółowienie informacji o dyscyplinie, piąta – stopień zaawansowania przedmiotu ustalony na podstawie roku studiów, dla którego przedmiot jest przeznaczony. / (0541) Mathematics The ISCED (International Standard Classification of Education) code has been designed by UNESCO.
Course title:	Simplicial Homotopy Theory
Name in Polish:	Symplicjalna teoria homotopii
Organizational unit:	Faculty of Mathematics, Informatics, and Mechanics
Course groups:	(in Polish) Przedmioty obieralne na studiach drugiego stopnia na kierunku bioinformatyka Elective courses for 2nd stage studies in Mathematics
ECTS credit allocation (and other scores):	(not available) Basic information on ECTS credits allocation principles: the annual hourly workload of the student’s work required to achieve the expected learning outcomes for a given stage is 1500-1800h, corresponding to 60 ECTS; the student’s weekly hourly workload is 45 h; 1 ECTS point corresponds to 25-30 hours of student work needed to achieve the assumed learning outcomes; weekly student workload necessary to achieve the assumed learning outcomes allows to obtain 1.5 ECTS; work required to pass the course, which has been assigned 3 ECTS, constitutes 10% of the semester student load. view allocation of credits
Language:	English
Type of course:	elective monographs
Requirements:	Topology II 1000-134TP2
Prerequisites (description):	Familiarity with basics of topology within the scope of the Topology I course. Understanding of the fundamental notions of homotopy theory (homotopy equivalence, fundamental group) as discussed in Topology II. It will be helpful (but not required) to have familiarity with concepts of algebraic topology such as singular homology, CW-complexes and homotopy groups.
Short description:	The course is an introduction to the combinatorial methods of algebraic topology. The central concepts are simplicial sets, homotopies between simplicial maps and homotopy equivalences between simplicial sets. The purpose of the lecture is to develop purely combinatorial methods of constructing homotopy types and their invariants as well as to compare the homotopy theory of simplicial sets to the classical homotopy theory of topological spaces.
Full description:	Simplicial sets, skeletal filtrations. Limits, colimits and exponential objects of simplicial sets. Homotopies, homotopy equivalences, weak homotopy equivalences. Kan complexes and Kan fibrations, homotopy limits. Cofibrations and homotopy colimits. Fibrant replacements and homotopy groups of simplicial sets. Simplicial aproximation and equivalence of homotopy theories of simplicial sets and topological spaces. Weak factorization systems and model categories. The Kan–Quillen model structure.
Bibliography:	Paul Goerss, John F. Jardine Simplicial homotopy theory 1999 André Joyal, Myles Tierney An Introduction to Simplicial Homotopy Theory André Joyal, Myles Tierney Notes on simplicial homotopy theory Peter May Simplicial objects in algebraic topology 1967
Learning outcomes:	Familiarity with basic concepts of homotopy theory in the framework of simplicial sets: homotopies, (weak) homotopy equivalences, fibrations, cofibrations, fibrant replacements. Ability to recognize homotopy non-invariant constructions and to approximate them by homotopy invariant ones. Understanding of the analogy between the homotopy theories of simplicial sets and topological spaces.
Assessment methods and assessment criteria:	Participation in classes, written homework assignments and oral exam.

This course is not currently offered.

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