CGP DERIVED SEMINAR

GABRIEL C. DRUMMOND-COLE

1. March 14: Damien Lejay

Thank you for being here, for supporting it by coming to the ﬁrst lecture. I’ve

named this the derived seminar and set up a webpage for the seminar, with pdfs

that would be the skeleton of the seminar and I will put information like who is

going to talk next and what it’s going to be about. I’ll upload the pdfs every week.

If you want to recall anything then it’s just for you.

I’ve been thinking about the roadmap and how to articulate the things on the

wishlist. Today is the ﬁrst day and I’ll talk for two hours about category theory.

There will be another session where we do category theory again. Then one session

of introduction to the problematics of diﬀerential graded categories and why we are

interested in these tools. One session of introduction, one session just on diﬀerential

graded categories, then two sessions on model categories. Then we’ll study diﬀer-

ential graded categories again with those tools. This will take us through April.

Then we’ll have a session on triangulated categories. Already we will have seen

these things. then we will have ∞-categories, stable ∞-categories, and comparison

theorems. I won’t plan more than that but by that time we will want to change

the wishlist and can add and change things.

[Discussion of timing]

In the ﬁrst hour I will give deﬁnitions, examples, and vocabulary, and in the

second hour we’ll do computations. Many many very important things will be said

in the next week.

Category theory is a language, a theory that helps you to write down mathe-

matics. It’s like set theory. You don’t do it for its own sake but rather do it to

help you do mathematics. To deﬁne categories and work with them we will use the

language of sets, and I’ll start with some fun about set theory.

When you start with set theory, you talk about sets and it’s great and then you

try to take the set of all sets and it’s not so great. So when you take the “set” of all

sets, this is a “class”. Classes are much bigger and that means you can’t do quite

the same operations on them.

What happens if I want to talk about the collection of all classes. You could

could call this a “superclass”, it’s not inside the theory of sets. What can I can do

with this? I don’t know. What if I take a collection of collections? I can’t really

do anything here anyway. So we haven’t really solved the problem. There’s some

boundary, but I want to be able to consider big objects. There’s a nice solution to

this, called Grothendieck universes.

Deﬁnition 1.1. A universe is a set U with some properties:

● if x is in U then the power set P (x) is in U;

● if x is in y and y is in U then x is in U (transitivity);

2 GABRIEL C. DRUMMOND-COLE

● if {x

}

i∈I

have I ∈ U and x

∈ U then

⋃

∈ U;

● the natural numbers are in the universe (or equivalently, U contains at least

one inﬁnite set).

These properties tell you that all the operations of set theory work in U. You

don’t need to go elsewhere because all you do uses these sets.

The empty set is one example. The second one is natural numbers, where each

natural number is the union of the numbers before. You never go outside of this

universe. But these don’t match the ﬁnal axiom. Already when we said the natural

numbers we use the axiom of set theory that there is an inﬁnite set. In fact, if you

just take the ZF axioms and the axiom of choice, you cannot prove that there is a

universe bigger than N, you don’t know. So I should add an axiom that for every

set X there exists a universe U with X ∈ U. This is a harmless axiom. You can put

it next to ZF and it will still be mathematics, but it will be helpful.

Something about universes, they are sets, so they have a cardinal, the cardinal

of a universe, because of the stability under these operations, you get that they are

strongly inaccessible. If κ < card(U) then 2

< card(U). The existence of universes

is equivalent to the existence of strongly inaccessible cardinals. This is completely

harmless and transparent in the rest of your life in mathematics.

Now there’s a bonus. Since N is a set, there is a universe U containing N. But

then there is a universe V containing U. Then the set of all ‘sets’ lives in U. If I

want the set of all things like this, I move to W. Every time I do something illegal

in set theory I just jump up a level of universes.

With this said, I will stop set fun and start categorical fun. I’ll give the deﬁnition

of a category and then some examples. At some point we’ll take a break before

computations.

Deﬁnition 1.2. A category C is a set of objects Ob C (I have a universe and a

“set” is inside my universe). For x and y in Ob C I have a new object Hom(x, y),

the arrows or homomorphisms from x to y, and I’ll picture these like x

Ð→ y. I have

a special arrow for x ∈ Ob C, I have a special arrow Id

∈ Hom(x, x), which goes

Ð→ x, a special arrow. When I have arrows that I think of as functions, if I have

Ð→ y

Ð→ z I get an arrow g ○f ∈ Hom(x, z). I want

● Id ○f = f and f ○Id = f.

● For three composable arrows I want (f ○g)○h = f ○(g ○h).

Next I have to to give an example. Instead I will give the example, the category

of sets. I’ll call it U −Set, and for this I need the set of objects. The objects of my

category will be U. If I have two sets x and y in U, I need to know Hom

U −Set

(x, y),

this will be the set of functions from x to y. I have to give my identity element,

which will be the identity function, and I have to give composition, and that’s

composition of functions.

We don’t want to speak about universes all the time so I will talk about Set

instead of U − Set. This is the most basic example, you’ll have more complicated

categories, and a lot of the time they will be built like this because they will have

underlying sets.

I’ll give some basic examples and classical notation for them. We’re going to see.

There’s a category called Ab which is the category of Abelian groups. The objects

of Ab are the Abelian groups, but I cannot take “all” Abelian groups, that’s too

DERIVED SEMINAR 3

big, so I want the underlying set to be in U. If I have two Abelian groups A and

B, then Hom

(A, B) is the linear (additive) functions from A to B. I should be

sure that composition of additive functions is an additive function and so on. No

problem, so this is a category.

There’s a category Rings of rings, whose objects are rings such that the under-

lying set is in U. Then Hom

Rings

(R, S) is the set of ring homomorphisms from R

to S.

So these are natural. Most concepts in mathematics can be described in this

language. Let me give a category Ban of Banach spaces, so the objects are Banach

spaces whose underlying sets are in U. Then Hom

Ban

(V, W ) are the continuous

linear functions from V to W .

I could also take a category Ban

≤1

, which has the same objects, Banach spaces,

but diﬀerent arrows. Here Hom

Ban

≤1

(V, W ) is the set of continuous linear maps

f ∶ V → W such that ∣∣f∣∣ ≤ 1. If you compose two contracting morphisms you still

get a contracting morphism, and the identity is a contracting morphism.

Let me give two other well-known categories. Vect

is the category of real vector

spaces, here the objects are vector spaces with underlying set in U. I’ll end the list of

known categories by Top, again the objects are topological spaces with underlying

set in U. The topology will then be inside U. There is no problem here. The

functions will be continuous functions.

Immediately from the deﬁnition is a principle that allows you to build twice as

many categories, the duality principle. In categories, there is a diﬀerence between

left and right, and what I can do is build a category by swapping the arrows.

Let C be a category. Let C

be the category with the same objects and the

arrows reversed, for x, y ∈ Ob C I want Hom

(x, y) = Hom

(y, x). So let’s give an

example. So say you have a category with only the following arrows

0 1

and you take the opposite category you get

0 1

If you take Rings

you get a useful and well-known category, the category of aﬃne

schemes. Taking the opposite category is useful because of things that depend on

the side of the arrow. So if you have property P on C

, that means you have

“co”-P on C. If you change the direction of the arrow, things are swapped.

Since Descartes, we have tried to embed geometry in algebra, and here we see

the principle that “geometry” is “algebra”

. I know how to do this computation

on my algebra and I hope it will give me what I want on my geometry.

Now I’ll give a deﬁnition speciﬁc to category theory. If I have two arrows

x y

whose compositions are equal to the identity, g ○f = Id and f ○g = Id, then we call

f an isomorphism and say that x ≅ y, and I want to give a table of diﬀerences of

how you think in set and in category theory.

Sets Categories

set object

∈ x → y

= ≅

4 GABRIEL C. DRUMMOND-COLE

You never want to talk about elements in category theory, you want to describe

things in terms of arrows. It’s forbidden. Equality is awkward, I’ll say isomorphic

or equivalent. There are plenty of examples but probably it’s good to take a break

right now. I’ll see you in ﬁve minutes.

Deﬁnition 1.3. D is a subcategory of C if Ob(D) ⊂ Ob(C) and Hom

(x, y) ⊂

Hom

(x, y) and identities and compositions are compatible with the inclusion.

Examples include Ban

≤1

⊂ Ban, or ﬁnite sets inside sets. I could take the objects

of U-sets with surjective morphisms. The identity is surjective and compositions of

surjective morphisms are surjective. I could take torsion Abelian groups.

Deﬁnition 1.4. D ⊂ C is full if for x, y ∈ Ob(D), we have Hom

(x, y) = Hom

(x, y).

I will say D ⊂ C is wide if Ob(D) = Ob(C).

So the Banach and surjection examples are wide and the ﬁnite set and torsion

examples are full.

Deﬁnition 1.5. A category is a groupoid if every morphism is an isomorphism.

Take G a group, then we can build a category BG with one object and then

Hom

(∗, ∗) = G. It’s a point with G-many arrows, with composition given by

composition in the group. Because it’s a group you always can invert arrows.

If you have an action of G on X, then you can build a category a bit like G, where

the objects of the category are the elements of X and Hom(x, y) = {g ∈ G ∶ y = g.x}.

An arrow means there is an action of an element of G that goes from x to y. I can

always go back since G is a group.

If C is a category, we have the interior groupoid of C, the objects are the objects

of C, and the morphisms are the isomorphisms of C.

Now I want to spend the last forty-ﬁve minutes making computations of one of

the major things in category theory, called limits and colimits. This is a key big

thing in category theory.

So far I don’t have a lot of things in the structure. I have objects and arrows.

Maybe I have something like this:

∗ ∗

∗

In a category you have a notion of approximation of a diagram by a single object

of your category, and you have a problem of what it means to approximate. Close

means arrows, and arrows have a side. I can approximate on the left or on the

right.

If I have an approximation on the left, I call this the limit and denote it lim

←Ð

and on the right I call it the colimit and write lim

Ð→

D. Let me simplify my diagram.

I want an arrow from my limit A to every element of my category and I want the

DERIVED SEMINAR 5

compositions to be commutative:

∗

A ∗

∗

I will say I have a limit if I have a universal thing like this. If A is my limit, it’s

closer to my diagram than any other one. If I have another approximation B then

I should get an arrow B → A which is unique.

For colimits it’s the same picture. An approximation on the right is something

like this Z:

∗

∗ Z

∗

The “best” approximation is one Z so that whenever you have an approximation

on the right, W , you get a map Z → W , unique.

This is called the colimit because it’s on the right. If I swap directions, in the

opposite category limits become colimits and vice versa.

Let X be an object of C. The best approximation on both the left and right is

X, lim

←Ð

D = X and lim

Ð→

D = X. What if D is a single arrow, X

Ð→ Y . To make an

approximation on the left, Z, I need a map ϕ ∶ Z → X and a map ψ ∶ Z → Y . Then

ψ = f ○ϕ by compatibility. I want an approximation by only one object, and now I

want just a map Z

Ð→ X, and so my limit is X. So lim

←Ð

D = X.

What about my approximation on the right? This will be an object Z, with

maps ψ ∶ X → Z and ϕ ∶ Y → Z. Compatibility says that ψ = ϕ ○f so ψ is useless.

Then I want an approximation by one object and Y looks like a good candidate.

So the colimit is Y .

When you have an arrow X → Y then the approximation on the left is X and

on the right is Y .

If I have two arrows X

Ð→ Y

Ð→ Z, what are the best approximations on the left

and on the right? On the left it’s X with the identity map because the maps from

the limit to Y and Z are useless. On the right it’s Z with the identity.

What about the empty diagram? Can you have a best approximation on the

left and on the right? On the left it’s called a ﬁnal object ∗, every object in the

category has a morphism to this object, a unique one, to ∗. An initial object is a

ﬁnal object in the opposite category. An initial object is one, notated ∅, an object

so that Hom(∅, c) is always of cardinality one.

In the category of sets, the empty set is initial, there is a unique map from the

empty set to any set, and a set with only one element is ﬁnal. In the category of

Abelian groups, the Abelian group 0 is initial, and it’s also ﬁnal because you always

have a map from any Abelian group to 0.

6 GABRIEL C. DRUMMOND-COLE

I want to give you more types of diagrams now. What about two objects and

no arrows. What is the limit or colimit if you have just two objects A and B. You

have some approximation Z, maybe, with arrows f ∶ Z → A and g ∶ Z → B. If this

exists we call it A × B. In sets this is the product.

So for the colimit, you need two maps, one from A → W and one from B → W .

In sets it’s called A ⊔B, what about in Ab, there you have A ⊕B. In the category

Ab, the coproduct of A and B is A ⊕ B also. Then you can get the product and

the coproduct are the same.

Now I want to give you the deﬁnition of the coproduct in rings, but maybe, I

have two ring maps from R and S to W , then I want this to be R ⊔ S → W , and

the answer is R ⊗

S, this is ring theory. How do you build this map? You take the

product of the two things. You take R ⊗

S → W ⊗

W → W . In rings this is the

tensor product, while for Abelian groups it’s sum and for sets it’s disjoint union. So

these things behave in the same way. If you can prove something abstractly about

coproducts then it will apply in all these contexts at once, so tensor products of

rings is like disjoint unions of sets is like sum of Abelian groups.

Another type of small diagrams,

X Y

So what is the limit, if it exists? I get Z

Ð→ X and Z

Ð→ with two conditions ψ = f ○ϕ

and ψ = g ○ϕ. So I have just the data ϕ and the condition f ○ϕ = g ○ϕ. I’ll say that I

factorize through the equalizer Eq(f, g) if this is in sets, that’s {x ∈ X, f (x) = g(x)}

and then that includes in X. Any time you have an arrow like this and compose

you get this condition. And any time you have an approximation it lands inside

this subset.

What about for colimits? I want an approximation W with maps ϕ ∶ Y → W

and ψ ∶ X → W . The conditions are ϕ ○ f = ψ and ϕ ○ g = ψ. So ψ is useless and

we just need ϕ ○ f = ϕ ○ g. Sometimes there is a colimit, and here in set theory

the colimit is called the coequalizer, this is the equalizer in the opposite category.

Can you guess what is the coequalizer? It’s going to be a quotient, it will be Y / ∼,

the equivalence relation generated by y ∼ z if there exists x ∈ X with f(x) = y and

g(x) = z. We know how to compute this.

Let me give this in a context that is much more visual. Let A and B be Abelian

groups, and take the two maps 0 and g:

A B

and then the limit or equalizer is the kernel of g and the colimit or coequalizer is

the cokernel of g.

Now I want to go to ﬁber products. A ﬁber product is a diﬀerent kind of diagram,

you have

Y Z

The colimit of this is Z with the identity map, this is not very interesting. In the

other direction it will be the ﬁber product X ×

Y , which may or may not exist. In

DERIVED SEMINAR 7

sets it will exist, and there it will be pairs {(x, y) ∶ f(x) = g(y)}. You can check

that this is the correct limit in the category of sets.

Another tye of diagram, I swap the arrows and get

Z X

If this diagram has a colimit it’s called a pushout and is written X ⊔

Y , and may

not exist. In sets it exists, and is modeled by X ⊔ Y / ∼ where x ∼ y if there is z in

Z with x = f (z) and y = g(z).

In rings, the pushout is the tensor product

A B

C B ⊗

So far I have written down limits and colimits for ﬁnite diagrams but we have

them for inﬁnite diagrams. The simplest ones will have uninteresting compositions.

Consider X

⊂ X

⊂ ⋯, say I have an inﬁnite tower of sets like this. What

is the limit of this diagram? It’s X

. The colimit will be the union of these. It

receives every one and anything that receives every one accepts a map from the

union. What if I have ⋅ ⊂ ⋯ ⊂ X

⊂ X

. The colimit is X

and the colimit is the

intersection.

These are ﬁltered, a ﬁltered poset is one where you can always compare two

elements with the help of a third element, for any x and y there is z with arrows

x → z and y → z. Field extensions of Q are something like that, you can add

√

2 or

i or both. You can extend from either of the smaller ones to the big one.

√

2, i)

√

2) Q(i)

Filtered diagrams are good.

What is good about ﬁltered posets is that it is easy to compute their colimits

in sets. If I have a ﬁltered poset with at most one arrow x

Ð→ x

. The formula

for the colimit is that you take the disjoint union of the X

modulo the equivalence

relation that x

∈ X

and x

∈ X

are equivalent if there exists k and f

and f

that f

) = f

I have to give a basic idea for constructing any limit or colimit. Let me give a

deﬁnition.

Deﬁnition 1.6. A category C is complete if it has all limits. This means that every

time I give you a diagram you can ﬁnd the limit. I mean a diagram of a reasonable

size, relative to the universe U. I’ll say it’s cocomplete if it has all colimits.

Theorem 1.1. U −Set is complete and cocomplete.

8 GABRIEL C. DRUMMOND-COLE

There are several diﬀerent recipes. One is to compute inﬁnite products and

equalizers. This is a way to compute limits. In the other direction you need inﬁnite

coproducts and coequalizers. On the other hand if you have pushouts and ﬁltered

colimits you can deﬁne all colimits, a diﬀerent recipe.