I.  The Dimensional Ladder

0.  Introduction

a.  The Ladder of n-Categories

A story about climbing the dimensional ladder, without detailed
definitions but with pictures of:

sets
functions
diagrams of sets and functions
categories 
functors
natural transformations
diagrams of categories, functors and natural transformations
2-categories
2-functors
2-natural transformations
modifications
diagrams of 2-cats, 2-functors, 2-nat trans and modifications
3-categories

The categorical imperative: understanding n-categories requires
that we use (n+1)-categories.  The expository difficulties this
causes.  For example: a fairly general concept of monoid lives in
a monoidal category, but a monoidal category is itself a kind of
weakened monoid in Cat.  (This is an example of the microcosm 
principle.)  

b.  The Periodic Table

Picture of the periodic table.

A category with one object is a monoid: picture.
The natural numbers as the free monoid on one generator: picture of
arrows -

x --->--- x

x --->--- x --->--- x

x --->--- x --->--- x --->--- x

and Poincare dual picture of rocks on a line.

A 2-category with one object is a monoidal category: pictures.
The free monoidal category on one generator - objects as above,
and only rather boring morphisms (e.g. associators).  The laws
of a monoid hold up to isomorphism.  Coherence law for associator.

A 2-category with one object and one morphism is a commutative monoid:
picture.   The natural numbers as the free commutative monoid on one
generator: dots in a square.  The commutative law arises because we
can switch rocks past each other.

A 3-category with one object and one morphism is a braided monoidal
category.  Here we see the process of switching some rocks past each
other - the braiding.  The coherence law satisfied by the braiding
and associator.  

Warning about subtleties (already in commutative monoid case!)


1.  Categories

a. Categories of Mathematical Objects 

Definition of category

There are many categories of
mathematical gadgets, but we'll consider three: Set, Vect and Top,
the latter two because they don't arise very quickly from category
theory itself.  We will downplay this aspect of category theory 
because it's traditionally been emphasized to the exclusion of all
other aspects.   

Definition of inverse morphism, isomorphism.

Universal properties and how they specify an object up to
a specified isomorphism.  In each case show what they amount
to for Set, Vect and Top.

Terminal object, initial object

Product, coproduct 

Concept of opposite category - unifying the above pairs

Equalizer, coequalizer

Pullback, pushout

Limit, colimit

Theorem: a category with finite products and equalizers has
finite limits.  

Theorem: a category with pullbacks and a terminal object has
finite limits.  

Definition of decategorification: the class |C| of isomorphism
classes of objects in C.  |FinSet| = |FinVect| = N, |Set| = Card

In Set, initial object and coproduct decategorify to 0 and +

In Set, terminal object and product decategorify to 1 and x

We shall study this "categorified arithmetic" in much more detail
in Part III.  Subtraction and division are much subtler.

Very often mathematical structures arise via decategorification!

b. Categories as Mathematical Objects  

kinds of categories: groupoids, 
                     monoids 
                     groups,
                     discrete categories,
                     preorders,  
                     partially ordered sets (or posets)

presenting a category by listing objects, morphisms and
          equations.  Just like presenting a group but with an
          extra "layer" - the objects.

examples: the free category on an object, a morphism,
          an isomorphism, an endomorphism (N) and an automorphism (Z).

example: the free category on a composable chain of n morphisms
          is called [n], and it looks like an (n-1)-simplex.

exercise: the free category on an object is all of the above
          the free category on a morphism is a preorder
          the free category on an iso is a groupoid
          the free category on an endo is a monoid
          the free category on an auto is a monoid, a groupoid and a group
(make a 5x5 grid to fill out)

c. Categories from Spaces 

The fundamental groupoid of a topological space

The fundamental group of a pointed space

categories from chain complexes: a 2-term chain complex is
a category in AbGp.  Homology from homotopy: take the fundamental 
groupoid, linearize it and impose relations saying that composition
of morphisms equals addition to get a 2-category in AbGp.  Do this 
very sketchily!

The nerve of a category (done quickly)

example: the nerve of [n] is a simplex

The classifying space of a group (done quickly)

d.  Functors 

Definition 

Examples: a functor from the free category on an object, a 
          morphism, an isomorphism, an endo, an auto 

Example: a functor from a group G to Set is a set acted on by G, or 
G-set.  

Example: a functor from a monoid M to C is an object of C acted on
by M.

Examples: the forgetful functor from Vect to Set and its adjoint,
with sketchy definition of adjoint using the fact that there's a
"natural" one-to-one correspondence between hom(Fx,y) and hom(x,Uy)

the forgetful functor from Top to Set and its adjoints 

left adjoints tend to preserve colimits (and conversely);
right adjoints tend to preserve limits (and conversely) - 
reference to later section.  

e.  The Category of Categories

Cat: a slightly mindboggling concept, laying the ground for lots
of fun self-reference and potential level slips.  A little digression 
on size issues.

The full subcategories Mon, Preord, Poset, Grp, Gpd

exercise: what's a functor from a group to Grp?   or from a monoid
to Mon?   More fun with of level slips!

Example: the fundamental groupoid functor from Top to Gpd

Example: the nerve functor from Gpd to Top.  Not quite adjoint
to the previous one, but "morally" so.  More generally, the nerve
functor from Cat to Top.  

Example: the underlying graph of a category as a functor U: Cat -> Grph
         the free category on a graph as a functor F: Grph -> Cat

Example: the terminal and initial categories

Example: the product and coproduct of categories 

Example: the hom-functor hom: C^{op} x C -> Set

f.  Natural Transformations

Definition with Pictures

Examples: a homotopy between maps between spaces gives a natural 
transformation between functors between their fundamental groupoids!

Natural transformations between mathematical constructions:

Natural isomorphism between identity functor on Vect and double dual.

Natural isomorphism between various "ordered pair" functors 
from Set^2 -> Set... i.e., various ways of making the product into
a functor!

Fundamental group versus first homology group of a pointed space.

Natural transformations as functors F: I x C -> C where I
is the free category on a morphism.

g.  Equivalence of Categories

Definition

Examples from duality:

 Vect is equivalent to Vect^{op}

 some sort of finite posets are equivalent to distributive
 lattices^{op}

 Compact Hausdorff spaces is equivalent to commutative C*-algebras^{op}

 Locally compact abelian groups is equivalent to itself^{op}

Skeletal categories

Theorem: a category is equivalent to any of its skeleta.

h.  Functor Categories

We've seen that given categories C and D, there's a set hom(C,D)
consisting of functors from C to D.  But in fact we can do better:
there's a category hom(C,D) whose objects are functors from C to
D and whose morphisms are natural transformations between these!

A functor F: C -> D gives a "representation" of C in D, which is like
a picture of C in D (draw a picture!) and we can understand things 
about C from this picture.  A natural transformation gives a "change 
of pictures".  

Example: category of actions of a group G is hom(G, Set).  Objects
are sets acted on by G, or G-sets.  Eventually we'll show how to 
completely recover a group from all its actions (?).

Example: category of representations of a group is hom(G, Vect).
Doplicher-Roberts theorem says one can completely recover a 
finite group from this category (with its extra structure).

Example: a functor from a group to Top is a continuous action

Example: more generally, a functor from a monoid to C is an action
of the monoid on some object of C.  

example: category of representations of various quivers 
(free categories on graphs)

i.  Cat as the Primordial 2-Category

Composition of functors, vertical and horizontal composition
of natural transformations, and the rules they satisfy, done using
diagrams.

Or: composition of functors, vertical composition of natural
transformations, and left/right whiskering, and the rules they
satisfy.

Show the two setups are equivalent.


2. 2-Categories

a. Strict 2-Categories

Nuts-and-bolts definition of strict 2-category based on 
example of Cat, in both styles.

Examples: 

The 2-category of groupoids, the 2-category of
categories with finite products and product-preserving functors,
the 2-category of categories with finite limits and 
finite-limit-preserving functors, and other "doctrines".

The strict fundamental 2-groupoid of a space: "taking the
pictures seriously".  A brief treatment, pointing out nuisances 
due to strictness.

2-categories from chain complexes: a 2-term chain complex is
a category in AbGp, while a 3-term chain complex is a 2-category
in AbGp.   Homology from homotopy: take the fundamental 2-groupoid,
linearize it and impose relations saying that composition (of
all forms) equals addition to get a 2-category in AbGp.  Do this 
very sketchily!

The 2-category generated by a 2-computad.

2-categories of planar string diagrams - brief treatment, pointing out
nuisances due to strictness.  Describe the 2-category generated by a 
2-computad

Definition: a strict monoidal category is a 2-categories with one object. 
A commutative monoid is a strict monoidal category with one object.
Also give nuts-and-bolts definitions!

Strict monoidal categories of planar diagrams, pointing out nuisances
due to strictness.

b. Weak 2-Categories (Bicategories)

Point out problems of strictness.
For example, how the fundamental 2-groupoid doesn't want to be strict: 
introduce the associator, left/right uniters, and their coherence laws.

Definition of weak 2-category.

Example: the weak fundamental 2-groupoid of a space.

Weak 2-categories of planar diagrams, pointing out nice features
due to weakening.  

Example of Mod, the weak 2-category of rings, bimodules and 
bimodule morphisms.  We could also use monoids, biactions and
biaction morphisms.

Example of Span, the weak 2-category of sets, spans and maps of 
spans.

Definition of weak monoidal category as weak 2-category with one
object.  Also, nuts-and-bolts definition.

Examples of Vect, RMod (R commutative) or RBiMod with its tensor product.

Examples coming from categories with finite products / coproducts.

The "all diagrams commute" version of MacLane's theorem - 
sketch of proof using associahedron.

c.  Monads and Adjunctions

Monads in a bicategory.  

Example: monads in Vect are algebras.

Adjunctions in a bicategory.  Monads from adjunctions.

Examples: 

Adjunctions in Vect are dual vector spaces; they give matrix algebras.

Other examples of monoids in a monoidal category:

  Top gives topological monoids

  Vect gives algebras

  AbGp with its tensor product gives rings

  Cat gives strict monoidal categories!

Monads in a bicategory:

  Span gives categories

Adjoint functors in Cat give monads.

Algebras of monads

Comonads and their coalgebras

d.  Limits and Colimits

How limits and colimits are examples of adjunctions and
vice versa

Kan extensions; how these are example of adjunctions and
vice versa.

As much as possible do all this "formally" in a 2-category?

e.  BiCat as the Primordial 3-Category

Weak versus strict functors between bicategories.

Weak versus strict natural transformations.

f.  2Cat versus BiCat - coherence theorems

MacLane's coherence theorem for 2-categories

One version: "all diagrams commute".  The associahedron.

Another version: Every weak 2-category is equivalent to
a strict one.  Proof via categorified Yoneda???

Another version: the strict 2-category

[strict 2-categories, strict 2-functors, pseudonatural transformations] 

is equivalent to the strict 2-category

[weak 2-categories, weak 2-functors, weak natural transformations]?

Perhaps be a bit sketchy and save details of proofs for later?

g.  Enriched Categories

The definition of enriched category; examples.

2-Categories as enriched categories

The category VCat of V-categories and V-functors

The 2-category of V-categories, V-functors and V-natural transformations 

If V has finite products so does VCat

If V is Cartesian closed so is VCat, and V is a V-category

If V has finite limits so does VCat?

If V is a distributive category so is VCat??

If V is a braided monoidal category we can define a monoidal V-category

If V is a symmetric monoidal category we can define a symmetric monoidal
V-category

Hints of higher patterns...

Exercise: under what conditions does a category enriched over commutative
monoids get enriched over abelian groups?   This is discussed in Freyd's
work on abelian categories and Ab-categoriges, and/or MacLane's Cats for
the Working Mathematician.

h.  Internal Categories

The definition of internal category; examples

double categories as internal categories

2-categories versus double categories


3.  3-Categories

a.  Strict 3-Categories

General definition of strict n-categories... and unravelled in case
n = 3.

3-categories from chain complexes

A strict 2-category with one object is a strict monoidal 2-category
(also a monoid in $(2Cat, \times)$).

A strict 2-category with one morphism is a commutative strict 
monoidal category... a notion too strict!

b.  Semistrict 3-Categories

Gray tensor product of 2-categories; semistrict 3-categories are
categories enriched over $(2\Cat, \tensor)$.

The fundamental 3-groupoid of a topological space.  $\Pi_3(S^2)$
as an example.  

A semistrict 3-category with one object is a semistrict monoidal
2-category (also a monoid in $(2\Cat, \tensor)$).

A semistrict 3-category with one morphism is a strict braided
category.

Definition of strict symmetric category.  A glimpse of the 
periodic table.

c.  Tricategories

Definition.  Note how things are getting unmanageable, but 
also note some patterns.  The 3d associahedron shows up here.

A tricategory with one object is a monoidal bicategory.

A semistrict 3-category with one morphism is a strict braided
category.

Examples of strict braided and symmetric categories: 

 every category with finite products or coproducts gives a symmetric
 monoidal category

 RMod with R a commutative ring is a symmetric monoidal category 

d.  Weak Monads and Adjunctions

The concept of weak monad or "pseudomonad" in a tricategory, or
perhaps just in a semistrict 3-category.  Pictures!

Example: a monoidal category is a weak monad in the monoidal 
2-category $(\Cat, \times)$.

The concept of weak adjunction.  The swallowtail coherence law.

How weak adjunctions give weak monads.

e.  Weak Limits and Colimits

f.  Enrichment and Internalization?

3 levels of "internalization" for 2-categories?

g.  The Lax World

Lax 2-functors, lax monoidal functors: how a lax functor 
between bicategories is good enough to transport a monad
from one to the other.  How a lax functor from the terminal 
bicategory to C is precisely a monoid in C.   

(Oplax...)

Lax limits.  How a lax fixed point for an endomorphism T: x -> x
in Cat is an algebra for T.   How the lax equalizer (?) of T and 1
is the Eilenberg-Moore category.  

h.  Tricat as the Primordial 4-category

strict versus weak 3-functors... etc.

i.  Coherence Theorems 

Every tricategory is equivalent to a semistrict one, but not to a
strict one.


4.  4-Categories


5.  Case Studies

a.  2-Braids and 2-Tangles

Braids as the free braided monoidal category on one object

Tangles as the free braided monoidal category with duals on one object

(do 1-braids and 1-tangles in dimensions 1, 2, 3, 4)

The issue of framing.

b.  Quantum Groups

algebras, coalgebras, bialgebras (in a general monoidal category)

the category of representations of an algebra

the monoidal category of representations of a bialgebra

the braided monoidal category of representations of a quasitriangular 
bialgebra

the symmetric monoidal category of representations of a triangular
bialgebra

the monoidal (resp. braided monoidal, symmetric monoidal)
category with duals of representations of a Hopf algebra
(resp. quasitriangular Hopf algebra, triangular Hopf algebra)

Examples of quasitriangular Hopf algebras and their tangle invariants

c.  Real Numbers, Complex Numbers, Quaternions and Octonions

cross product algebras - the 0,1,3,7 theorem

d.  Electrical Circuits

linear algebra over the rig of costs

e.  Logic Gates

linear algebra over the rig of truth values

f.  2-Braids and 2-Tangles

(do 2-braids and 2-tangles in dimensions 2,3,4,5,6?

g.  Quivers and Dynkin Diagrams  (move to "Categories as Theories",
section on "Category Representations".)

Representations of quivers: proof that quivers containing certain
"bad" subquivers have wild representation type, by noting that the
dimension of the space of indecomposable representations exceeds the
dimension of the space of intertwiners.  Note that this problem is a
categorified version of the problem of (a certain sort of) Laplacians
on graphs.  It's possible that this whole section should wait until
part II, where we may introduce the incidence geometry and symmetry 
aspects of Dynkin diagrams, and have more to play with.  But, to really
get some synergy here, we'd have to figure out why the quivers with 
tame representation type are precisely those of finite-dimensional 
semisimple Lie algebras!  And this remains mysterious to me...

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