The aim of this article is to provide a mathematics student an outline for studying various textbooks, leading up to an understanding of higher categories, and its applications to algebraic topology. It is divided into four modules, and each module is a 6-week period of intense study. The approach we've taken is to introduce homological algebra in the first module, elementary topological notions in the second, model categories in the third unifying notions from the first two modules, finally leading up to higher categories in the last.
- Basics of mathematical reasoning. Tao's Analaysis I; chapter 12: appendix on the basics of mathematical logic.
- Set theory. Johnstone's Notes on Set Theory and Logic, in its entirety.
- Groups. Artin's Algebra; chapters on group theory taught from a linear algebra perspective.
- Rings and Modules. Dummit & Foote; chapters in Part II.
- Abelian categories. Mac Lane's Categories for the Working Mathematician, in brief.
- Basic notions of point-set topology. Munkres' Topology, chapters 2-4.
- Basic notions in classical algebraic topology. Hatcher's Algebraic Topology, chapters 0 and 1.
Module I: Homological Algebra
- Mac Lane's Homology.
- Weibel's Homological Algebra with my annotations.
- Hatcher's Algebraic Topology; Chapter 2: singular and cellular homology.
1. Algebra of
This is the definition of a left
Ideals are the prime examples of submodules. A submodule
A module can be quotiented with a submodule to obtain a quotient module. Quotienting with submodules works exactly like quotienting with ideals:
It is trivial to verify that
Broadly speaking, there are three operations on modules:
Direct sum and direct product,
and , coincide when the sum or product is finite. Direct sums of modules yield free modules, and these are used to build free resolutions.
, yields a third module. and are a covariant right-exact functors in the general case, and if is a flat module, tensoring with it preserves exactness.
, yields a third module. and are left-exact functors, for fixed module . is a covariant functor and is a contravariant functor.
There exists an additional connection between the tensor product and the Hom functor:
2. Calculus of chain complexes
A category with modules as objects and module morphisms as morphisms turns into an abelian category, in which every morphism admits a kernel and cokernel, every monic is a kernel, and every epi is a cokernel. A chain complex is defined as a sequence of morphisms satisfying the condition
The chain is said to be exact at
To make them easier to work with, every l.e.s. can be broken up into short exact sequences:
Resolutions are exact sequences that turn modules into complexes.
The free resolution is a prime example of a projective resolution, since every free module is projective.
Any two projective resolutions of a module are chain homotopic:
Dually, we can form an injective coresolution of a module using injectives:
Any two injective coresolutions of a module are chain homotopic:
4. Derived functors
Given either right-exact or left-exact functor
5. Homology of abelian groups
First, note that
Now, to compute
6. Singular and cellular homology
Module II: Simplicial homotopy
- Bruno Vallette's notes with my annnotations.
- Friedmann's survery article on Simplicial Sets.
1. Elementary homotopy theory: cylinders and cones
2. Fiber and cofiber sequences
3. Fibrations and cofibrations
4. CW complexes
5. Simplicial sets
See (b) before proceeding.
Module III: Model categories
- Dwyer-Spalinski's Model Categories with my annnotations.
- Najib Idrissi's series of 12 video lectures.
- Hirschorn's Model Categories; chapters on cofibrantly-generated model categories and simplicial model structure.
1. Motivation for model categories
Watch the first lecture in (b).
2. Definition of a model category with brief examples
A functorial factorization is an ordered pair
3. The homotopy category: constructions of
, , and
4. Showing that
is a model category
(MC1)-(MC4) are easily checked, but the "small object argument" is required to check (MC5). See (a).
5. Cofibrantly generated model categories and Quillen adjunctions
6. Simplicial sets and the Quillen model structure