Below is a list of The Catsters hits in approximate order of upload
together with their description of the contents. First though, some
suggestions about viewing order. You need to know what categories and
functors are before you start.
BASICS OF CATEGORY THEORY
Natural transformations 1-3A and String diagrams 1-2 (these overlap)
Representables and Yoneda 1-3
Adjunctions 1,2 and 4
2-categories 1-2
ALL ABOUT MONADS
Monads 1-4
Adjunctions 3, 5-7
String diagrams 3-5
Distributive laws 1-4
LIMITS AND COLIMITS
Listed together below: terminal and initial objects (TI), products and
coproducts (PC), pullbacks and pushouts (PP), Coequalizers (CE), General
limits and colimits (GL).
After pullbacks and pushouts, check out Slice and comma categories 1-2.
TWO APPLICATIONS
Adjunctions from morphisms 1-5
Metrics spaces and enriched categories 1-2
MONOID AND GROUP OBJECTS
Monoid objects 1
Eckmann-Hilton 1
Monoid objects 2
Eckmann-Hilton 2
Group objects and Hopf algebras 1-6
GENERALIZED CATEGORIES
Spans 1-2
Multicategories 1-2
Double categories
Now for the list...
Monads
1. An introduction to monads including the definition and a look at
the monoid monad.
2. Continuation of the monoid monad example and introduction of the
category monad.
3. The definition of algebras for monads. The example of monoids as
algebras for the monoid monad.
3A. An appendix to Monads 3: more on monoids as algebras for the monoid monad.
4. Morphisms between algebras and the category of algebras. A first
look at the question of monadicity.
Adjunctions
1. The notion of an adjunction. Definition via unit and counit natural
transformations and the triangle identities.
2. Definition of adjunction via natural isomorphism between
hom-sets. Getting the unit and counit from this.
3. Adjunctions give rise to monads.
4. The two notions of adjunction coincide.
5. Every monad comes from an adjunction via its category of algebras
(the Eilenberg-Moore category).
6. Definition of the Kleisli category
7. The adjunction coming from the Kleisli category. The category of
adjunctions for a monad: the Eilenberg-Moore and Kleisli categories as
terminal and initial objects. More on monadicity.
String diagrams
1. A first look at the string diagram notation for representing
categories, functors and natural transformations.
2. The interchange law and whiskering. The last dull bits before
getting onto adjunctions. (Apologies for the drastic editing at the
end.)
3. The definition of adjunctions in string diagram language - the
snake/zig-zag relation.
4. Monads in the string diagram notation. The unit and associativity
identities as topological moves.
5. Adjunctions give rise to monads.
Natural transformations
1. The definition of natural transformations. An analogy with homotopy.
2. More on natural transformations: vertical and horizontal composition.
3. The interchange law for horizontal and vertical composition,
"proof" using whiskering. This video is a bit quiet and a bit fuzzy
and we'll probably replace it with an improved version soon.
3A. Addendum to natural transformations 3: a bit more about whiskering,
and where the interchange law really comes from.
Distributive laws
1. Definition of distributive law of one monad over another, and the
example of multiplication distributing over addition.
2. The key result about how a distributive law of S over T gives
relationships between S-algebras, T-algebras and TS-algebras.
3. We introduce the idea of monads *in* a general 2-category C (where
putting C = Cat gives the usual notion of monad *on* a category), and
define the 2-category Mnd(C) of monads, monad functors and m...
4. Distributive laws as monads in the 2-category Mnd(C).
Spans
1. Definition of the bicategory of spans in a category.
2. We show that monads in the bicategory Span(Set) are small categories.
Multicategories
1. Definition of (non-symmetric) multicategories, and the bicategory
of generalised spans in which mutlticategories are monads.
2. We show that monads in T-span (with T the free monoid monad) are
multicategories, and for general T we get T-multicategories.
2-categories
1. The definition of (strict) 2-category.
2. The middle four interchange law in a 2-category comes from
functoriality of the composition functor.
Double categories
Definition of double categories as internal categories in Cat, and
brief unravelling of this definition.
Eckmann-Hilton
1. We present and prove the Eckmann-Hilton argument: given a set with
two binary, unital operations that distribute over one another, in
fact the two operations must be the same and commutative. Proved using
the Eckmann-Hilton "clock".
2. We use the Eckmann-Hilton argument to show that a bicategory with
only one object and only one 1-cell is a commutative monoid. TYPO
ALERT: at 4.13 mins I write "only one 2-cell" but I mean "only one
1-cell".
Monoid objects
1. We define monoid objects in monoidal categories. We start with the
motivating example of ordinary monoids, re-expressing it using only
the objects and morphisms in the category Set.
2. A monoid object in the category of monoids is a commutative
monoid. We use the Eckmann-Hilton argument.
Group objects and Hopf algebras
1. Definition of a group using only the objects and morphisms of the
category of sets and functions, with a view to defining group objects
in other categories.
2. More on the definition of group objects, in the setting of monoidal
categories where the monoidal product is given by categorical product.
3. Examples of group objects in Cartesian categories other than Set,
and the category of vector spaces as a non-Cartesian monoidal
category.
4. Group objects in a non-Cartesian monoidal category.
5. Beginning of definition of Hopf algebra by string diagrams.
6. End of definition of Hopf algebras by string diagrams.
Metrics spaces and enriched categories
1. The definition of an enriched category in preparation for the
definition of generalized metric spaces.
2. The definition of a generalized metric space as an enriched
category. The definition of a metric map as an enriched functor.
Limits and colimits
TI 1. A first look at universal properties: definition of terminal
object and some examples, sketch of proof that terminal objects are
unique up to unique isomorphism.
TI 2. Full proof that terminal objects are unique up to unique
isomorphism, and some examples of categories without terminal objects.
TI 3. Definition of initial object and some examples.
PC 1. Definition of product, example of cartesian product of sets, BxA
and AxB as two products, introducing uniqueness up to unique
isomorphism (no details).
PC 2. Uniqueness up to unique isomorphism, various examples and
non-example (tensor product of vector spaces).
PC 3. Definition of coproduct, various examples .
PC 4. The morphisms (f,g) and fxg, the diagonal, and Ax1.
PP 1. Definition of pullback, example: pullbacks in Set.
PP 2. Definition of pushouts, pushouts in Set, and the
pullback/pushout example of intersection/union of sets.
CE 1. Definition of coequaliser, examples in Set: coequalisers can be
constructed as equivalence relations, and equivalence relations can be
expressed as coequalisers.
CE 2. Quotient groups as coequalisers in the category of groups.
LC 1. Idea of a limit of an arbitrary diagram in a category: cones and
universal cones
LC 2. Explanation of simple limits as universal cones: terminal
objects, products, pullbacks and equalisers.
LC 3. Formal definition of limits: first formalising the notion of a
diagram of a given shape, and expressing a cone over it as a natural
transformation
LC 4. Formal definition of limits as certain natural isomorphisms.
LC 5. The notion of a category having all limits of a certain shape,
via a right adjoint.
LC 6. Colimits as universal cones, and the natural isomorphism formula
Adjunctions from morphisms
1. Motivation for the construction of adjoint functors for bundles
over sets.
2. The category of bundles on a set as a slice category and as a
functor category into sets.
3. The definition of the pull-back and its right adjoint for bundles
over sets.
4. A proof that the push-forward is right adjoint to pull-back.
5. Description of the left adjoint to the pull-back.
Slice and comma categories
1. Definition of slice categories C/X and X/C, products in C/X as
pullbacks in C.
2. Definition of comma categories D over and under F for a functor F,
and F over and under G for functors F and G with the same target
category.
Representables and Yoneda
1. Definition of representable functors and the Yoneda embedding
(though without calling it the Yoneda embedding yet).
2. Further explanation of the Yoneda embedding (including calling it
that, but not yet proving it's an embedding), checking naturality for
H_f.
3. Statement of Yoneda lemma and explanation of "why" it is true.
Miscellaneous topology
An open-closed cobordism 1: A look at a pipe-cleaner open-closed cobordism.
An open closed cobordism 2: Another open-closed cobordism in
pipecleaners - the "zig-zag-ator".
An open closed cobordism 3: The cut-off pair of pants.
Klein bottle: A Klein bottle made from pipe cleaners