幾何学特論II
The course gives an introduction to homotopy theory centered around
the space of a category. We develop the techniques to analyze this
space, including basic category theory, simplicial sets, and model
category theory.
Here is a more detailed syllabus:
Text: Lecture notes will be handed out for each class.
Time and place: Friday 10:30-12:00 in Science Building 1, room 309.
Lecture notes:
- Lecture 1: The space of a
category.
- Lecture 2: Limits and
colimits. Filtered colimits and finite limits of sets commute.
- Lecture 3: Adjoint functors.
- Lecture 4: Characterizing the geometric
realization by maps from it.
- Lecture 5: The skeleton filtration of a
simplicial set.
- Lecture 6: The geometric realization of
a simplicial set is a CW-complex. The category of k-spaces.
- Lecture 7: Geometric realization
preserves finite products. Pointed k-spaces.
- Lecture 8: Higher homotopy groups.
- Lecture 9: The mapping fiber and the
long exact sequence of homotopy groups.
- Lecture 10: Weak equivalences, Serre
fibrations, Serre cofibrations.
- Lecture 11: Quillen model
categories.
- Lecture 12: The homotopy
category; Quillen functors and their derived functors.
- Lecture 13: The Reedy model structure.
Quillen's Theorem A and Theorem B.
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