An outline for studying higher category theory


Paris

Overture

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.

Prerequisites:

  1. Basics of mathematical reasoning. Tao's Analaysis I; chapter 12: appendix on the basics of mathematical logic.
  2. Set theory. Johnstone's Notes on Set Theory and Logic, in its entirety.
  3. Groups. Artin's Algebra; chapters on group theory taught from a linear algebra perspective.
  4. Rings and Modules. Dummit & Foote; chapters in Part II.
  5. Abelian categories. Mac Lane's Categories for the Working Mathematician, in brief.
  6. Basic notions of point-set topology. Munkres' Topology, chapters 2-4.
  7. Basic notions in classical algebraic topology. Hatcher's Algebraic Topology, chapters 0 and 1.

Module I: Homological Algebra

  1. Mac Lane's Homology.
  2. Weibel's Homological Algebra with my annotations.
  3. Hatcher's Algebraic Topology; Chapter 2: singular and cellular homology.

1. Algebra of -modules

-modules are abelian groups associated with the ring . Let be one such abelian group. Then, the following axioms are satisfied for , :

This is the definition of a left -module; to obtain the definition of a right -module, simply swap and in the equations. An -bimodule is simultaneously a left -module and right- module.

Ideals are the prime examples of submodules. A submodule of an -module a subset of , which is closed under multiplication with an element of the ring:

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 is a module:

Broadly speaking, there are three operations on modules:

  1. 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.
  2. Tensor product, , 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.
  3. Hom functor, , 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: and form an adjoint pair.

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 , often abbreviated as :

Equivalently, . With this condition, it is possible to define the homology group .

The chain is said to be exact at if . A long exact sequence is exact at every :

To make them easier to work with, every l.e.s. can be broken up into short exact sequences:

3. Resolutions

Resolutions are exact sequences that turn modules into complexes.

A module is defined to be projective if:

A module is defined to be injective if:

A free -module is one that can be written as a direct sum of isomorphic copies of . A free resolution of module is the following exact sequence:

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 , we can define left or right derived functors as (co)homology groups of lifting a projective or injective (co)chain.

is the prime example of a left derived functor, obtained by setting , and is the prime example of a right derived functor, obtained by setting .

5. Homology of abelian groups

First, note that can be viewed as a bimodule.

To compute and for some simple cases, consider the standard projective resolution of :

Now, to compute , with an abelian group, tensor with to obtain the following projective chain:

Hence, , , with vanishing for all higher . Flipping arrows, we obtain the injective resolution, and applying functor , we obtain and , similarly vanishing for all higher .

An extension of by is defined to be:

We write to mean "homology of group G, with coefficients in B", and it is defined to be . Cohomology of groups can be defined similarly, in terms of .

6. Singular and cellular homology

See (c).

Module II: Simplicial homotopy

  1. Bruno Vallette's notes with my annnotations.
  2. Friedmann's survery article on Simplicial Sets.

1. Elementary homotopy theory: cylinders and cones

See (a).

2. Fiber and cofiber sequences

See (a).

3. Fibrations and cofibrations

See (a).

4. CW complexes

See (a).

5. Simplicial sets

See (b) before proceeding.

Module III: Model categories

  1. Dwyer-Spalinski's Model Categories with my annnotations.
  2. Najib Idrissi's series of 12 video lectures.
  3. 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

See (a).

is a retract of iff .

A functorial factorization is an ordered pair of functors such that for all . In particular, the domain of is the domain of , the codomain of is the domain of , and the codomain of is the codomain of .

has left-lifting property with respect to if there is a lift such that and .

Category is a model category if it is equipped with three classes of morphisms that satisfy the following axioms:

[(Co)complete] is complete and cocomplete.

[2 out of 3] If are three morphisms such that any two are in , then so is the third.

[Retracts]

[Lifting]

[Functorial factorization]

3. The homotopy category: constructions of , , and

See (a).

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

See (c).

6. Simplicial sets and the Quillen model structure

See (c).

7. Anodyne extensions and simplicial homotopy groups

8. Some rational homotopy theory, with applications: CGDAs, and Sullivan algebras

Module IV: Higher categories