Spring 2012
Seminar on simplicial methods
The notion of simplicial set is a powerful tool for studying – by combinatorial means – topological spaces up to weak homotopy equivalence. It has fundamental applications throughout mathematics, whenever homotopy theory plays a role. One speaks of simplicial homotopy theory.
The seminar starts with looking at the basics of simplicial sets and their geometric realization to topological spaces. From this we motivate fundamental notions like Kan fibration of simplicial sets, simplicial homotopy and simplicial homotopy groups. These are the ingredients for the model structure on simplicial sets which allows to grasp their relation to topological spaces via the central theorem that establishes a Quillen equivalence between the homotopy theory of simplicial sets and that of topological spaces.
Standard references include
Pierre Gabriel, Michel Zisman, Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35. Springer, New York (1967)
Edward Curtis, Simplicial Homotopy Theory, Advances in Math., 6, (1971), 107 – 209.
Paul Goerss, Rick Jardine, Simplicial homotopy theory, Progress in Mathematics, Birkhäuser (1996)
A pedagogical introduction to the basic notions is in
examples:
internal hom (function complexes)
examples:
$\Delta[1] \times \Delta[1]$
Examples
Last revised on May 29, 2012 at 22:04:00. See the history of this page for a list of all contributions to it.