CGP DERIVED SEMINAR

GABRIEL C. DRUMMOND-COLE

1. April 11: Wanmin Liu: Model categories I

In Chinese philosophy, there are four levels of understanding. The ﬁrst level is

that we don’t know that we don’t know something. Level three is that we know

that we don’t know something. Before this seminar, I didn’t know the word “model

category.” Maybe after some hard work, we don’t know we know something, this

is level two, and then we know we know something after fully studying it.

So today we want to talk about model category theory, introduced by Quillen

many years ago, in the late 1960s, and this one provides a general setting to study

the homotopy category, construct the basic machinery of homotopy theory. The

motivation for myself is this so-called fundamental result, which I’ll write here.

This is given by:

Theorem 1.1. (Tabuada 2005) Let k be a commutative unital ring. Then the

category of dg categories over k (let’s recall this: the objects are dg categories, and

morphisms are dg functors) admits the structure of a coﬁbrantly generated model

category where the weak equivalences are quasi-equivalences.

So my goal in this talk is to give the deﬁnition of model categories via weak

factorization systems. I’ll also give some very basic properties. In the next hour

we’ll have many examples.

So today I will have many deﬁnitions, but they’re not so hard.

1.1. Weak factorization systems (WFS). Fix a category (a small category) C.

Deﬁnition 1.1. Let ι ∶ A → X and π ∶ E → B be morphisms. Suppose we have a

commutative diagram



and there is a lift from X → E such that the two triangles commute. Then we say

that ι is a left lifting with respect to π and that π is a right lifting with respect to

ι, and denote this ι ⧄ π.

Then we also deﬁne the collection, let L and R be two classes of morphisms in

C, we have two classes. We can deﬁne L

⧄

is the collection of all π such that ι ⧄ π

for all ι in L. Similarly,

⧄

R is the collection of all ι such that ι ⧄ π for all π in R.

Deﬁnition 1.2. A weak factorization is a pair (L, R), such that

(1) (factorization) any morphism f can be written as a composition ι ⋅ π = π ○ ι

for some ι in L and π in R.

(2) (closure) L =

⧄

R and L

⧄

= R.

2 GABRIEL C. DRUMMOND-COLE

Maybe I need one example for you. You can take C to be the category of sets.

Then L could be injective functions and R could be all surjective functions.

A morphism f is a retract of a morphism g if we have the following diagram:

))



Lemma 1.1. L

⧄

and

⧄

R are closed under taking retracts.

Let’s just prove one of these. suppose π ∶ E → B is in L

⊠

and f is a retract of it.

Then since ι is in L and π is in its orthogonal we get a lifting



))



// //

Let me give a second deﬁnition, equivalent.

Deﬁnition 1.3. A pair of classes of morphisms (L, R) is a WFS if

(1) for any morphism f there is a factorization f = ι ⋅ π with ι ∈ L and π ∈ R.

(2) L ⧄ R

(3) L and R are closed under retracts.

Maybe it’s obvious that the other deﬁnition implies this one. But how do we see

that R ⊃ L

⧄

? So let π be in L

⧄

. Then we can factorize π = i ⋅ p. Then we have



So we can rewrite this as



and so since p ∈ R and R is closed under retracts, we conclude that π is in R.

1.2. Model categories.

Deﬁnition 1.4. A model cotegory M is a category with

(1) all small limits and colimits exist, Damien talked about these,

(2) equipped with three classes of morphisms, (W, C, F ), called weak equiva-

lences and denoted ∼, coﬁbrations (denoted ↣), and ﬁbrations (denoted ↠)

such that

(a) (two out of three) If f ∶ X → Y and g ∶ Y → Z then if two of {f, g, f ○g}

is a weak equivalence, so is the third.

(b) (W ∩ C, F ) is a WFS.

DERIVED SEMINAR 3

What is the meaning of this here? It’s better to give an equivalent deﬁnition.

Proposition 1.1. Let M be a category with

(1) M has all small limits and colimits

(2) M has three classes of maps (W, C, F ) so that

(a) W satisﬁes the two out of three property,

(b) W , C, and F are closed under taking retracts,



the dotted arrow exists if ι is in C and π is in F and at least one is in

W .

(d) There exist factorizations of every morphism f into a coﬁbration fol-

lowed by a ﬁbration ι ⋅ π where either one can be chosen to be acyclic

(in W ).

(From the very beginning, the lifting is not unique.)

1.3. Basic properties. Let (L, R) be a weak factorization system. Then L and

R are closed under composition. To prove this for R, take f and g in R. To show

that f ⋅ g is also in R, it’s just a diagram chase,



with h in L. But we have a lift



and then we have a lift



Another lemma is that L is closed under taking pushouts and R is closed under

taking pullbacks. This means that if you have a pushout diagram



4 GABRIEL C. DRUMMOND-COLE

if g is in L then h is in L. If this is a a pullback and h is in R then g is in R.

Let me prove this for R. We know that h is in R. We want to show g is R.

We’ll show that it has the left lifting property. We are given a diagram



with g in L; we want a lift B → W . So we will use the properties of pullbacks.

Because of this diagram, we could draw another diagram



and we have a lift ` from B to X as indicated, which gives us



by the property of the pushout. We need to see that k = j ⋅ p. If I compose the

diagram with j we get a diagram

k⋅f

k⋅g



then k is j ⋅ p by uniqueness of the map to the pushout

Let me give the last lemma.

Lemma 1.2. L is closed under colimits and R is closed under limits.

Proof. Let’s do it for L. Consider the best approximation from the right X

Ð→

Ð→ ⋯ → X

Fix a morphism π ∶ E → B. We want a lifting



But this means we want a map X

→ E with compatibilities. In the n = 0 level we

have X

to E. Then by induction, suppose we have it for X

. Then because f

DERIVED SEMINAR 5

in L and π in R we have a lift



n+1

