- Terminal and Initial objects
- Products and Coproducts
- Pullbacks and Pushouts
- Natural transformations
- Representable functors and the Yoneda Lemma
- Adjunctions (part 1)
- Adjunctions and monads
- General Limits and Colimits
- Slice and comma categories
- Monoid objects
- Adjunctions from morphisms
In an attempt to solidify and extend my knowledge of category theory, I have been working my way through the excellent series of category theory lectures posted on Youtube by Eugenia Cheng and Simon Willerton, aka the Catsters.
Edsko de Vries used to have a listing of the videos, but it is no longer available. After wresting a copy from a Google cache, I began working my way through the videos, but soon discovered that Edsko’s list was organized by subject, not topologically sorted. So I started making my own list, and have put it up here in the hopes that it may be useful to others. Suggestions, corrections, improvements, etc. are of course welcome!
As far as possible I have tried to arrange the order so that each video only depends on concepts from earlier ones. Along with each video you can also find my cryptic notes; I make no guarantee that they will be useful to anyone (even me!), but hopefully they will at least give you an idea of what is in each video. (For some of the earlier videos I didn’t take notes, so I have just copied the description from YouTube.)
I have a goal to watch two videos per week (at which rate it will take me about nine months to watch all of them); I will keep this list updated with new video links and notes as I go.
Terminal and Initial objects
Terminal and initial objects 1
- Definition and examples of terminal objects
- Sketch of proof that terminal objects are unique up to unique isomorphism
Terminal and initial objects 2
- Proof that terminal objects are unique
- Examples of categories without terminal objects
Terminal and initial objects 3
- Definition and examples of initial objects
Products and Coproducts
Products and coproducts 1
- Definition of products
- Example: cartesian product of sets
- and as two (isomorphic) products
- Uniqueness up to unique isomorphism
Products and coproducts 2
- More on uniqueness up to unique isomorphism
- Examples and non-examples of products
Products and coproducts 3
- Definition and example of coproduct
Products and coproducts 4
- Definition of the morphisms and
- The diagonal
- Products with the terminal object
Pullbacks and Pushouts
Pullbacks and pushouts 1
- Definition of pullback
- Example: pullbacks in
Pullbacks and pushouts 2
- Definition of pushouts
- Example: pushouts in
- Pullback/pushout example of intersection/union of sets
Natural transformations 1
- Definition of natural transformations.
- Naturality squares.
- Intuition about natural transformations based on homotopy.
- Alternative definition of natural transformation analogous to usual homotopy definition: a natural transformation is a functor where is the “categorical interval”, i.e. the two-object category with a single nontrivial morphism.
Natural transformations 2
- Vertical composition of natural transformations.
- Functor categories.
- Horizontal composition of natural transformations
- Note there are two ways to define horizontal composition, and they are equal by naturality.
Natural transformations 3
- Whiskering (though they don’t call it that yet).
- Horizontal composition as vertical composition of two whiskerings (in two different ways, which are equal by naturality).
- Interchange law: proof by commutativity of whiskering.
Natural transformations 3A
- Define terminology “whiskering”.
- Note vertical composition of “right-whiskered fish” works because of functoriality (of functor we whiskered by).
- Vertical composition of “left-whiskered fish” comes by definition of vectical composition for natural transformations.
- So in the end, interchange depends on three things: definition of vertical composition; functoriality; naturality.
Representable functors and the Yoneda Lemma
Representables and Yoneda 1
- Definition of representable functors (co- or contravariant): those which are naturally isomorphic to or for some .
- Contravariant: ; .
- Covariant: ; .
- Is there a functor ? Yes, the Yoneda embedding :
- , where is postcomposition with . is a natural transformation; its component at has type . Postcomposing an arrow in with yields an arrow in .
Representables and Yoneda 2
- Proof that Yoneda embedding sends morphisms to natural transformations . Comes down to the fact that composition in the category is associative.
Representables and Yoneda 3
- Look at natural transformations from to some other (contravariant) functor . Big idea: such natural transformations are entirely determined by where sends .
- Yoneda lemma: (natural in and ). I.e. the set of objects in is isomorphic to the hom-set of natural transformations between and .
Adjunctions (part 1)
- Given categories and and functors and , we have the following situations:
- Isomorphism: ,
- Equivalence: ,
- Adjunction: , So we can think of an adjunction as a “weaker sort of equivalence”.
- and are subject to triangle identities: is the identity, and similarly for .
- These laws can be expressed as commuting diagrams of 2-cells: draw and as 2-cells and paste them in two different ways.
- Alternate definition of adjunction : an isomorphism natural in and .
- What “natural in and ” means here.
- Hint: sending identity morphisms across the iso gives us and from the first definition. Proof deferred to Adjunctions 4.
- Note: Adjunctions 4, not 3, follows on to 2.
- Given: an isomorphism which is natural in and .
- Notation: write application of the isomorphism as an overbar.
- Construct the two squares implied by naturality. Follow them each around in both directions (since they involve a natural isomorphism) to get four equations in total governing how the iso interacts.
- Define and by applying the isomorphism to appropriate identity morphisms. Naturality and the triangle identities follow from the above four equations.
- Monads give us a way to talk about algebraic theories (monoids, categories, groups, etc.).
- Definition of a monad:
- with unit and associativity laws.
- Note what is meant by a commutative diagram of natural transformations
- Example: monad for monoids (aka the list monad)
- , maps a set to set of words in (i.e. lists)
- is singleton
- is concatenation
- Note, unit and associativity for monad is different than unit and associativity of monoids, which has already been encoded in the definition of .
- Proof that the list monad (“monad for monoids”) is in fact a monad
- Example: monad for small categories
- , category of graphs
- makes the free category on a graph (morphisms = paths in the underlying graph)
- With only one object, this reduces to the monad for monoids.
- Proof of monads laws is basically the same as for the list monad.
- Algebras for monads. Monads are supposed to be like algebraic theories; algebras are models.
- An algebra for a monad is an object (the “underlying object”) equipped with an “action” , satisfying the “obvious” axioms ( must interact “sensibly” with and ).
- Example: , = list monad (“monad for monoids”)
- An algebra is a set equipped with
- First axiom says must simply project element out of length-one list.
- Other axiom is essentially associativity.
- That is, algebras for the list monad are monoids.
- Example for monad of categories (from last time) works the same way.
More on monoids as monad algebras of the list monad.
- Given a monad algebra , construct the monoid:
- whose underlying set is
- The monad algebra law for (a triangle) just says that can’t do anything interesting on one-element lists: it has to just return the single element.
- Identity and associativity laws for the monoid come from the other monad algebra law, saying how interacts with (a square), and from how the list functor is defined. We start with a way of mapping lists down to values, which bakes in the idea that it doesn’t matter how we associate the list.
Monad algebras form a category (called ).
Given two monad algebras and , a morphism between them consists of a morphism of underlying objects, , such that the obvious square commutes.
Example. List monad again. . A morphism of monoids is a function such that . See how this equation arises from the commuting square for monad morphisms, by starting with a 2-element list in upper left and following it around.
Given a particular mathematical theory, can it be expressed as the category of algebras for some monad? I.e. given a category , is it equivalent to for some ? (Answer: no, not in general, e.g. category of topological spaces can’t.)
But this is still an interesting question, more or less the question of “monadicity”. Category said to be monadic over category if can be expressed as category of algebras of monads over .
Adjunctions and monads
- Note: depends on monads.
- Examples of adjunctions:
- between the category of sets and the category of monoids:
- similarly between category of graphs and category of (small) categories.
In general, free functors are left adjoint to forgetful functors. (How to remember the direction: “left” has four letters, just like “free”.)
- Every adjunction gives rise to a monad . Check monad laws:
- Monad triangle laws are just adjunction triangle laws with extra or everywhere.
- Monad associativity law is naturalty for , or something like that.
“Every monad comes from an adjunction via its category of algebras.”
Last time we showed every adjunction gives rise to a monad. What about the converse?
Answer: yes. In fact, given a monad, there is an entire category of adjunctions which give rise to it, which always has initial and terminal objects: these are the constructions found by Kleisli and by Eilenberg-Moore, respectively. Intuitively, any other adjunction giving rise to the monad can be described by the morphisms between it and the Kleisli and Eilenberg-Moore constructions.
Let be a monad.
Terminal solution (Eilenberg-Moore): consider category of -algebras, also written . We construct an adjunction . (Intuition: “freely” constructs a -algebra; “forgets” the algebra structure.)
is easy to construct: .
What about ? Sends to the “free” -algebra on , with underlying set . Then evaluation map is . That is, . Need to check that this definition of really gives a monad algebra as a result. In this case the monad algebra laws are just the monad laws for !
Now define a unit and counit. is just the for the monad. is an algebra morphism from the free algebra on (i.e. ) to : in fact, itself is such a morphism, by the second algebra law.
Prove triangle laws for and : exercise for the watcher/reader.
This time, initial solution to “does a monad give rise to any adjunctions”: Kleisli.
- The Kleisli category for a monad on category , written or
- Objects: objects of .
- Morphisms: .
- Composition: given and , produce .
- Identity: .
- Category axioms come from monad axioms. Associativity comes from associativity and naturality of ; unit laws come from unit laws for .
Intuition: this is the category of free algebras: is equivalent, under the adjunction, to , morphism between free algebras.
Note, for the Eilenberg-Moore category (last time) it was complicated to define the objects and simple to define the morphisms. For Kleisli, it’s the other way around. “Conservation of complicatedness.”
The adjunction that comes from the Kleisli category, giving rise to the original monad .
Again, let be a monad. We will construct , where is the Kleisli category defined in Adjunctions 6, with .
- sends objects to “free algebras”
- Identity on objects.
- On morphisms, sends to (equivalently ).
- sends a “free algebra” to its “underlying object”
- Sends to .
- Sends to .
- Unit and counit
- we can take as the of the monad.
- we can take to be id.
- Adjunction laws come down to monad laws (left to viewer).
Given a monad on , we have a category of adjunctions giving rise to (morphisms are functors making everything commute). is the initial object and is terminal.
Question of monadicity: given an adjunction , is ? If so, say “ is monadic over ”, i.e. everything in can be expressed as monad algebras of . Or can say the adjunction is a “monadic adjunction”. Can also say that the right adjoint (forgetful functor ) “is monadic”. Monadic adjunctions are particularly nice/canonical.
General Limits and Colimits
General limits and colimits 1
Defining limits in general, informally.
- The thing we take a limit of is called a diagram (a collection of objects and morphisms). A limit of a diagram is a universal cone.
- A cone over a diagram is an object (vertex) together with morphisms (projection maps) to all objects in the diagram, such that all triangles commute.
- Universal cone is the “best” one, through which all other cones factor, i.e. there is a unique morphism from the vertex of one to the other such that all the relevant triangles commute.
General limits and colimits 2
Examples of limits.
- Terminal objects: limit over the empty diagram.
- Products: limit over discrete diagram on two objects.
- Pullback: limit over a “cospan”, i.e. a diagram like . Note that we usually ignore the edge of the cone to , since it is uniquely determined by the edges to and .
- Equalizer: limit over a parallel pair of arrows.
General limits and colimits 3
- Note: requires natural transformations.
- Formal definitions of:
- Diagram (functor from an index category)
- Cone (natural transformation from constant functor to diagram).
General limits and colimits 4
- Requires Yoneda.
Formal definition of a limit: given a diagram , a limit for is an object together with a family of isomorphisms natural in . I.e. a natural correspondence between morphisms (the “factorization” from one cone to another) and morphisms (i.e. natural transformations) from to in the functor category (i.e. cones over with vertex ). That is, every cone with vertex has a unique factorization morphism, and vice versa!. The “vice versa” part is the surprising bit. If we have a limit then every morphism is the factorization for some cone to the universal cone.
If we set then etc. In particular corresponds to some cone, which is THE universal cone. The Yoneda lemma says (?) that the entire natural isomorphism is determined by this one piece of data (where goes). Note that both and are functors . The Yoneda lemma says that a natural transformation from to is isomorphic to — i.e. a cone with vertex , the universal cone.
The universality of this cone apparently comes from naturality.
General limits and colimits 5
- Requires adjunctions.
- Notation for limits. Categories that “have all limits (of a given shape)”.
- The natural isomorphism defining a limit can be seen as an adjunction where , and is the functor that takes a diagram and produces its limit.
- Claim: this is an adjunction if has all -limits. Need to show that the iso is also natural in , and that is actually a functor.
General limits and colimits 6
Colimits using the same general formulation. “Just dualize everything”.
Cocone (“cone under the diagram”) is an object with morphisms from the objects in the diagram such that everything commutes.
Universal cocone: for any other cocone, there is a unique morphism from the universal cocone to the other cone which makes everything commute. Note it has to go that direction since the universal cocone is supposed to be a “factor” of other cocones.
In Eugenia’s opinion the word “cocone” is stupid.
More generally: natural isomorphism between cocones and morphisms. . Limits in are the same as colimits in , and vice versa.
All limits are terminal objects in a category of cones (and colimits are initial objects).
Since terminal objects are initial objects in (and vice versa), we can even say that all universal properties are initial objects (and terminal objects) somewhere.
Slice and comma categories
Slice and comma categories 1
Slice category. Given a category , fix an object . Then we define the slice category by
- Objects are pairs where .
- Morphisms from to are morphisms in which make the triangle commute.
Coslice category, or “slice under” category is the dual of , i.e. objects are pairs where , etc.
If has a terminal object , . (Dually, .)
Products in are pullbacks in having as a corner. (Dually, coproucts in are pushouts.)
Slice and comma categories 2
Comma categories are a generalization of slice categories. Fix a functor and an object . Then we can form the comma category .
- Objects: pairs . Image of some object under and an arrow from it to .
- Morphisms are morphisms in such that makes the relevant diagram commute.
Of course we can dualize, (“cocomma” sounds even stupider than “cocone”, perhaps).
Apparently comma categories give us nice ways to talk about adjunctions.
Let’s generalize even more! Fix the functor but not the object . Then we can form :
- Objects: triple .
- Morphism is a pair of morphisms and such that the relevant square commutes.
Can also dualize, .
An even further generalization! Start with two functors , . Form :
- Objects: triples .
- Morphisms: obvious generalization.
In fact, all of these constructions are universal and can be seen as limits/colimits from the right point of view. “Next time”. (?)
Coequalisers are a colimit. Show up all over the place. Give us quotients and equivalence relations. Also tell us about monadicity (given an adjunction, is it a monadic one?).
Definition: a coequaliser is a colimit of a diagram consisting of two parallel arrows.
More specifically, given , a coequaliser is an object equipped with such that , with a universal property: given any other with , factors uniquely through .
Example: in : coequaliser of is a quotient , where is the equivalence relation generated by for all .
Conversely, we can start with an equivalence relation and build it using a coequaliser. Given: an equivalence relation . Note we have . Coequaliser is equivalence classes of .
Quotient groups as coequalisers. Consider a group and a normal subgroup . In the category of groups, consider two parallel maps : the inclusion map , and the zero map which sends everything to the identity element . Claim: the coequaliser of these two maps is the quotient group , together with the quotient map .
Let’s see why. Suppose we have another group with a group homomorphism such that ; that is, for all . We must show there is a unique homomorphism which makes the diagram commute.
Notation: under the quotient map gets sent to ( iff ). For the homomorphism , send to . Note this is required to make things commute, which gives us uniqueness; we must check this is well-defined and a group homomorphism. If then . By definition, , and since is a group homomorphism, . Hence it is well-defined, and must additionally be a group homomorphism since and is a group homomorphism.
Monoid objects 1
Idea: take the definition of monoids from , and “plunk it” into any other category with enough structure.
- A monoid is:
- A set
- A binary operation on
- A unit
Now let’s reexpress this categorically in . Note we have been talking about elements of sets; we have to replace this with use of only objects and morphisms of .
- A monoid (take 2) is:
- An object
- A morphism (note we use Cartesian product structure of )
- A morphism
- A commutative diagram
- A commutative diagram
Now we take the definition and port it to any monoidal category.
- A monoid object in a monoidal category is:
- An object
- A morphism
- A morphism
- A commutative diagram
- A commutative diagram
Given a functor , an end is an object in which is “limit-like” in some sense.
Ends are not as common as coends (and perhaps not as intuitive?). Two particular places where ends do show up:
- natural transformations (especially in enriched setting; see Ends 2)
- reconstruction theorems (recover an algebra from category of its representations, i.e. Tannaka reconstruction, see Ends 3)
- A wedge consists of
- an object
- a family of -morphisms for all
- such that for all the obvious square with vertices , , , and commutes. (Dinaturality/extranaturality.)
- This is in some sense a generalization of a cone.
An end is a universal wedge, i.e. a wedge such that if then there exists a unique morphism through which the components of factor.
Note we write the object using the intergral notation, (the morphisms of the wedge are left implicit).
Simple example of an end: . In this case a wedge consists of:
- for each a function
- such that we have .
That is, for every we have , such that . i.e. the family are the components of a natural transformation .
Note this goes in the other direction too, that is, a wedge is precisely the same thing as a function . Therefore, the universal such is precisely this set of natural transformations. (Can be thought of as “set of symmetries” of a category. Also the Hochschild cohomology.)
More examples. First, straightforward generalization: given functors , form the bifunctor . Then we can see that
(Proof is just a small generalization of the proof in Ends 2, left as exercise.) Useful in an enriched context, can use this end to construct an object of natural transformations instead of a set.
- is a monoid in the category of sets.
- - is the category of sets that acts on
- is the forgetful functor.
- Result: . (In general, natural transformations over forgetful functor reconstructs algebraic objects.)
Proof (application of Yoneda):
- Let be monoid operation and the identity.
- An -set consists of a set together with an action . Morphisms are just functions which commute with the actions (“equivariant maps”).
- Note is representable:
- Consider the -set (“left-regular representation of ”)
- Define by .
- Note determines since is equivariant: . In fact .
- Thus .
- So , and by (co-?)Yoneda, this is just .
Combine some of the previous examples. Recall
- (Ends 2)
- (Ends 3)
What happens if we combine these two results? First, look at the end from last time:
- Let be a natural transformation on . That is, is a function , such that commutes with the underlying function of any equivariant map, i.e.
- As we showed last time, for some .
- Note is just a function , but has to commute with equivariant functions.
Now look at the end of the bare hom-functor in the category of -sets. i.e.
Now if , we have
What’s the difference? is now a family of equivariant maps. But note equivariant maps are determined by their underlying function. So any diagram of this form implies one of the previous form; the only thing we’ve added is that itself has to be equivariant (in the previous case could be any function). So in fact we have
i.e. we’re picking out some subset of . Question: which subset is it? That is, given such a we know for some ; which ’s can we get?
Consider the left-regular representation again. Then we know is just left-multiplication by some . But it has to commute with equivariant maps; picking the action on the particular element , this means for all
that is, , i.e. .
So we conclude .
Adjunctions from morphisms
Adjunctions from morphisms 1
General phenomenon: associate some category to an object . For example:
- In representation theory, to a group or algebra we associate a category of modules or representations.
- In algebraic topology, to a space we associate (category of “bundles”?)
- In algebraic geometry, to an algebraic variety associate the category of sheaves.
- In logic, to a set of terms associate a category of subsets (predicates) over the terms.
- In analysis, to a metric space associate a category of Lipschitz functions .
Question: if we have a morphism , how does that relate to the categories and associated to and ?
We often get some sort of “pullback” functor . (Also often get some sort of monoidal structure on and , and is often monoidal.)
We also get various “pushforwards” , right adjoint to . In some situation we also get a left adjoint to .
This is the beginning of the story of “Grothendieck’s 6 operations”. Lots of similar structure arises in all these different areas.