Schreiber Seminar on simplicial methods

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

• Greg Friedman, An elementary illustrated introduction to simplicial sets (arXiv:0809.4221)

Basics definitions and examples

• simplex category, simplicial identities

• simplicial object, simplicial set, sSet

examples:

• nerve

• singular simplicial complex

• internal hom (function complexes)

Geometric realization

• coend

• nerve and realization

• geometric realization

examples:

• $\Delta[1] \times \Delta[1]$

• geometric realization of categories

Fibrations

• lifting property

• boundary, horn

• Kan fibration, Kan complex

• anodyne morphism

Simplicial homotopy

• simplicial homotopy

• simplicial homotopy group

• weak homotopy equivalence

Model categories

• category with weak equivalences

• homotopy category

• factorization system

• model category

Examples

• canonical model structure

• model structure on chain complexes

Serre-Model structure on topological spaces

• model structure on topological spaces

Kan-model structure on simplicial sets

• model structure on simplicial sets

Derived functors and the homotopy theorem

• derived functor

• Quillen adjunction, Quillen equivalence

• Brown functor

• homotopy hypothesis-theorem

Homotopy type theory

• type theory, intensional type theory

• homotopy type theory,

• categorical semantics

Univalence

• associated infinity-bundle

• univalence axiom

• function extensionality