2022/2023
Research Seminar "Introduction to Homotopical Algebra and Simplicial Topology"
Type:
Optional course (faculty)
Delivered by:
Laboratory of Algebraic Geometry and Its Applications
When:
1, 2 module
Open to:
students of one campus
Language:
English
ECTS credits:
6
Contact hours:
64
Course Syllabus
Abstract
n the 1960s, Dan Quillen developed a formalism convenient for doing homotopy theory in amuch broader context. Quillen introduced model categories — a kind of abstract categories with distinguishedclasses of morphisms mimicking the category of topological spaces in its homotopical aspect. However, despitethe topological appearance of model categories, they are applicable in various situations, including homolog-ical algebra and rational homotopy theory. A few years earlier, Dan Kan constructed another very importanttheory. That is the homotopy theory of simplicial sets. Simplicial sets are combinatorial analogs of topologicalspaces. However, unlike other combinatorial models, such as simplicial complexes, they are flexible enough todo various constructions such as taking mapping spaces and various colimits. Still, simplicial sets are combi-natorial enough to define algebraic invariants directly without any need for external analytic instruments. Wewill discuss both topics and later apply them to a wonder of modern homotopy theory —∞-categories. Thecourse will be quite unusual in format. We will meet on two separate days. One of which will be «homotopicalalgebra day» when we discuss abstract theory. Another will be «simplicial topology day» when we discuss themain source of applications and inspiration of homotopical algebra — simplicial topology.PREREQUISITES:Courses «Introduction to Homological Algebra and Category Theory» (taught by A. Goro-dentsev or C. Brav), «Algebraic topology» (taught by M.Finkelberg or M.Kasaryan)