CGP DERIVED SEMINAR
GABRIEL C. DRUMMOND-COLE
1. April 18: Mehdi Tavakol: The small object argument
I want to discuss some conditions for getting factorizations for free.
I’ll start by recalling some definitions for small and compact objects. This is
confusing in the literature. Let me first say that I want to define small objects.
They behave like, they have a small amount of information. To say precisely, I
should first say something about a filtered diagram. If I have a partially ordered
set J which is an index set, and I assume it’s filtered, by which I mean that if you
have two objects i and j, there is something bigger than both of them, when you
have an object X in the category, I say it’s small if for any such diagram indexed
by J, say {Y
j
}
j∈J
with colimit Y , and I have an induced diagram which passes to
Hom
C
(X, Y
j
) → Hom(X, Y ), we say X is small if for any such diagram, the map
from the colimit of these hom sets to the hom set into the colimit is a bijection.
This, let me give some examples. If C is a set, you’ll see that it should be a finite
set. If X = N, then you can write it as a union of N
n
where N
n
= {1, . . . , n}. Then
let Y
n
= N
n
. Then we think about the identity map from N to N. Then we can’t
find a collection of maps from N to N
n
for any n that leads to this, I can’t factor
through a finite set. So from this I can say that small objects in sets are finite sets.
You can use the same kind of idea to say that if you want to look at groups,
then small objects are finitely presented groups. Then you can see that by the same
idea, these are going to be small objects.
For algebraic structures you can think of them as objects with some kind of
finiteness conditions. “Compact” is a little confusing because in topological spaces,
compact spaces are not compact objects.
Let me make one remark, I’m ignoring a cardinal condition. You can define
“κ-small objects” for κ a regular cardinal (here κ is ℵ
0
), and, well, let me do the
example later.
Now I’ll define presentable categories. There are two conditions on the category
C
(1) C admits all U-small colimits
(2) Hom(X, Y ) is U-small
(3) there is a set of ℵ
0
-small objects which generate C under U-small colimits.
So some examples. For C the category of sets or algebraic things like groups. If C is
the category of Banach spaces, then it’s not ℵ
0
-presentable, but it’s ℵ
1
-presentable,
which I’ll leave as an exercise.
Another example that is not ℵ
0
-presentable but ℵ
1
, there are some examples in
topological groups, complicated ones. So topological spaces are not presentable.
1
2 GABRIEL C. DRUMMOND-COLE
Let me say a little more about lifting problems. If I have a diagram
A
//
p
X
q
B
>>
//
Y
then we say that p has the left lifting property with respect to q and q has the
right lifting property with respect to p. Then for a collection of maps we can talk
about lifting properties. If S is any collection of maps, then
⧄
(S
⧄
) has a stability
condition (we can easily see that this contains S). Let me define another thing, a
weakly saturated class, which is, I have small colimits, and my class of morphisms
is weakly saturated if it’s closed under pushouts:
A
f
//
X
B
f
′
//
Y
where if this is a pushout and f is in S then f
′
is in S. It’s closed under infinite
composition, so if you have
D
i
φ
ij
C
φ
i
>>
φ
j
D
j
and we have this for D
i
for i ∈ I, and if D is the colimit of the diagram, then there
is, for any j, there’s a map D
j
→ D, and I have this map from C → D
j
→ D, and
this is in S.
It’s also closed under retracts, which means that if you have the following diagram
C
//
f
C
′
g
//
C
D
//
D
′
//
D
where the two horizontal compositions are the identity, and we know that g is in
S, then f is in S.
Now I can state the proposition, the small object argument.
If C is a presentable category and I have a collection A
0
of maps φ
i
which is
indexed by U-small I, and if you have f ∶ X → Y , then you can decompose f as
X → Z → Y with f
′
∶ X → Z where f
′
is in the smallest weakly saturated class of
morphisms generated by A
0
and f
′′
has the right lifting property with respect to
all elements in A
0
.
DERIVED SEMINAR 3
Let me explain the construction, we want to construct this, I just want to know
the statement, so to do the construction, I look at C
i
→ D
i
, and look at all collec-
tions of such maps, with Z
0
= X
C
i
//
X
D
i
//
Y
and I take the colimit for all guys in A
0
, and then I get a colimit Z
1
which maps
to Y . Then I can do something similar
C
i
//
Z
0
~~
Z
1
D
i
//
Y
and I play the same game. Then I take the colimit and get Z, and then it’s easy
to check that this is in the smallest weakly saturated class.
So what is this used for? This weakly saturated class is
⧄
(A
⧄
0
).
