The aim of this article is to collect various resources for an audience wishing to embark on higher categories, and its applications to algebraic topology. It is divided into three modules, with a timeline of six weeks for each module. The approach we've taken is to introduce homological algebra in the first module, elementary topological notions in the second, and model categories in the third. The modules should prepare the student sufficiently to embark on
- 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, modules, and vector spaces. Dummit & Foote; chapters in Part II.
- 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.
- Yoneda Lemma, adjunctions and limits in category theory. Mac Lane's Categories for the Working Mathematician, chapters 1 through 5.
Module I: Homological Algebra
- Mac Lane's Categories for the Working Mathematician, chapter 8: Abelian categories.
- Weibel's Homological Algebra; chapters 1 and 2.
- 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: Homotopy
- Bruno Vallette's notes with my annnotations.
- Friedmann's survery article on Simplicial Sets.
- Riehl's Leisurely Introduction to Simplicial Sets.
1. Elementary homotopy theory
2. Fiber and cofiber sequences
3. Fibrations and cofibrations
4. Fiber bundles
5. CW complexes
6. Simplicial sets
See (b) before proceeding.
Module III: Model categories
- Dwyer-Spalinski's Model Categories.
- 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
7. Anodyne extensions and simplicial homotopy groups
-categories and beyond
- Monoidal categories. Lurie's Higher Topos Theory, appendix A.1.3.
- Enrichment in category theory. Riehl's Categorical Homotopy Theory, the beginnings of chapter 3; Lurie's Higher Topos Theory, appendix A.1.4.
- Basics of Kan Extensions. Riehl's Category Theory in Context, chapter 6.
-categories. Lurie's Higher Topos Theory; a thorough study of chapter 1, and a cursory study of chapters 2 and 3.
Basics of stable
-categories. Lurie's Higher Algebra; a thorough study of chapter 1.