CGP DERIVED SEMINAR
GABRIEL C. DRUMMOND-COLE
1. March 28: Seongjin Choi: Motivation
[Seminar dinner leaves at 6:10.]
Thank you for coming today. I will talk about loclization of categories with
respect to S. The situation is as follows. Let C be a category and S a subset of the
morphisms. Then what we want is a “localization category functor” C → S
−1
C such
that for any category D and functor F ∶ C → D such that F (s) is an isomorphism
for s ∈ S, there is a unique functor S
−1
C → D so that the diagram commutes.
The claim is that there exists such a functor unique up to isomorphism. Unique-
ness is not hard so I’ll focus on existence. Before constructing the localization
category, I’ll define notation.
Definition 1.1. A quiver is a quadruple (E, V, s, t) where E is a set of edges, V
is a set of vertices, and s and t are source and target maps from E to V .
A morphism between Q = (E, V, s, t) and Q
′
= (E
′
, V
′
, s
′
, t
′
) is a pair f = (f
e
, f
v
)
where f
e
is a map E → E
′
and f
v
a map V → V
′
such that the diagrams commute
E
f
e
//
s
E
′
s
′
E
f
e
//
t
E
′
t
′
V
f
v
//
V
′
V
f
v
//
V
′
We can think of this, roughly, as a category without identities morphisms or
a composition structure. I’ll denote Cat as the category of all categories, using
Grothendieck universes, and there is a forgetful functor from Cat to Quiv.
There is a functor in the other direction called the free or path functor.
Definition 1.2. Let Q = (E, V, s, t) be a quiver. Then I define the PaQ, the path
category of Q to have objects V and morphisms between a and b finite paths of
directed edges in Q, that is, (a
n
= b, f
n
, . . . , a
1
, f
1
, a
0
= a) with s(f
i
) = a
i−1
and
t(f
i
) = a
i
. The composition is concatenation; the identity is the path (a), so
(a
n
, f
n
, . . . , f
1
, a
0
)○(b
m
, g
m
, . . . , b
1
, g
1
, b
0
) = (a
n
, f
n
, . . . , f
1
, a
0
= b
m
, g
m
, . . . , b
1
, g
1
, b
0
)
when a
0
= b
m
.
So given the quiver Q I define Pa Q as described, so using this path category I
will describe the localization category.
Define a quiver Q = (E, V, s, t) as follows. We are given C and the set S. Using
C, the vertices are the objects of C. The edges are Hom C ∐ S (call the inclusions i
1
and i
2
) and so s ○ i
1
= s
C
and t ○ i
1
= t
C
whereas s ○ i
2
= t
C
and t ○ i
2
= s
C
. We want
to define S
−1
C as Pa Q/ ∼, where I’ll explain the equivalence relation ∼, the objects
are the objects of Pa Q and the morphisms are equivalence classes where we have
1
2 GABRIEL C. DRUMMOND-COLE
● i
1
(v) ○ i
1
(u) ∼ i
1
(v ○ u) when v ○ u is defined,
● i
1
(id
C
a) = Id
Pa Q
(a)
● i
2
σ ○ i
1
σ = Id
Pa Q
(s
C
(σ))
● i
1
σ ○ i
2
σ = Id
Pa Q
(t
C
(σ))
So for example, let C be the category ●
f
Ð→ ● with two objects and one arrow f and
S = {f }. Then first I construct a quiver
a b
i
1
f
i
2
f
Next I construct Pa Q modulo the equivalence relations. Then I get only one
morphism from a to b and likewise from b to a.
Let me introduce some examples. We want to construct a localization category
adding some invertible morphisms.
● For S a set of isomorphisms, then we have S
−1
C = C and ` = id
C
.
● For S = Hom C, then S
−1
C is the groupoid completion of C.
● For A an Abelian category (where the morphisms between two objects are
an Abelian group and composition is bilinear, you have finite direct sums
and products, and every morphism has a kernel and cokernel, and the
natural map coker ker f → ker coker is an isomorphism), such as modules
over R, but not Set. Given an Abelian category, define C(A) as the category
of complexes in A, that is, sets {X
n
, d
n
} for n ∈ Z with d
n
∶ X
n
→ X
n+1
and
X
n
∈ Ob(A) with d
n+1
○ d
n
= 0, with morphisms between two complexes a
map from X
n
to Y
n
for each n commuting with d
n
and d
n+1
. Then you
can define a functor H
n
∶ C(A) → A by {X
n
, d
n
} → ker d
n
/Id
n+1
. Define S
as the set of morphisms in C(A) which become isomorphisms under these
functors for all n. These are called quasi-isomorphisms. Then I’ll call the
localization category at S the derived category of A.
