CGP DERIVED SEMINAR
GABRIEL C. DRUMMOND-COLE
1. July 25: Tae-Su Kim: Triangulated dg categories
Let T be a dg category over k. I’ll remind you that T
op
− mod is the category
of dg functors T
op
→ C(k). We have h ∶ T → T
op
− mod , the Yoneda embedding,
which takes x to y ↦ T (y, x) and morphism via compositions.
Now some facts about this Yoneda embedding are:
● h(x) is cofibrant and fibrant for all x,
● That is, h is a functor from T to Int(T
op
− mod ), the fibrant and cofibrant
objects here; we’ll denote Int(T
op
− mod ) by
̂
T .
● h is quasi-fully faithful, so
● [h] ∶ [T ] → [
̂
T ] is fully faithful.
Definition 1.1. F in T
op
− mod is quasi-representable if F is in the essential
image of [h], that is, there exists x in T so that F ≅ h
x
in [
̂
T ].
For F in T
op
− mod , define a functor χ
F
∶ T
op
− mod → k − mod takes G to
Hom
[T
op
− mod ]
(F, G) and morphism by compositions.
Definition 1.2. F in T
op
− mod is compact if χ
F
(
⊕
i
G
i
) ≅
⊕
i
χ
F
(G
i
)
Here the direct sum of functors T
op
→ C(k) is evaluated as the direct sum
objectwise in T
op
.
We claim that quasi-representability implies compactness. How to show this?
Assume that we have χ
h
x
(G) ≅ H
0
(G
x
) and G =
⊕
i
G
i
and F is quasi-representable
with x as its representing object, then χ
F
(G) is χ
h
x
(G) which is H
0
(G
x
) which is
H
0
((
⊕
G
i
)
x
) which is
⊕
H
0
((G
i
)
x
) which is eventually isomorphisc to
⊕
χ
F
(G
i
).
So we should show this isomorphism χ
h
(G) ≅ H
0
(G
x
)? I’ll construct an iso-
morphism between Hom
[
̂
T ]
(h
x
, G) and H
0
(G
x
). So first we’ll construct Φ from
Hom
̂
T
(h
x
, G) → G
x
. So such a morhpism is α a natural transformation, which is
{α
y
∶ T (y, x) → G
y
}. We send this α to α
x
(id
x
). This is a chain map because
dα = {dα
y
} ↦ (dα)
x
(id
x
) = d(α
x
(id
x
)) ± α
x
(did
x
); the second term vanishes be-
cause id
x
is always closed. So this is d(α
x
(id
x
)). So this Φ is a chain map, also,
degree zero.
This then induces a map on cohomology, and taking the degree zero part, we
get a map from H
0
(Hom
̂
T
(h
x
, G)) → H
0
(G
x
), and by definition the left hand side
here is Hom
[
̂
T ]
(h
x
, G).
1
2 GABRIEL C. DRUMMOND-COLE
How do we define an inverse Ψ? Take a ∈ G
x
to Ψ(a)
y
∶ (h
x
)
y
→ G
y
, where this
sends φ in T (y, x) = (h
x
)
y
to G(φ)(a). We can show that this is a chain map.
d(Ψ(a)
y
)(φ) = d(Ψ(a)
y
(φ)) ± Ψ(a)
y
(dφ)
= d(G(φ)(a)) ± G(dφ)(a)
= d(G(φ)(a)) ± dG(φ)(a)
= ±G(φ)(da) = ±Ψ(da)
y
(φ)
so we have that this is a chain map up to a sign that I’ll leave as an exercise.
Then we can define the inverse Ψ as [a] ↦ [{Ψ(a)
y
}]. Now for x in T , we can
consider the composition H
0
(G) → Hom
[
̂
T ]
(h
x
, G) → H
0
(G
x
), which sends [a] first
to the class of {φ ↦ G(φ)(a)} which goes to G(id
x
)(a) which is [id(a)] = [a]. We
checked that Φ ○ Ψ is the identity. Similarly we can show the other direction is the
identity but I’ll omit that.
Definition 1.3. A dg category T is triangulated if every compact object in
̂
T is
quasi-representable.
We showed that quasi-representables are compact but we consider here the op-
posite direction. If every compact object is quasi-representable, then we call this
triangulated. We call the full category of triangulated dg categories dg − cat
tr
. It’s
easy to see that we have the inclusion ι ∶ Ho(dg − cat
tr
) ↪ Ho(dg − cat).
[discussion of what the homotopy category of triangulated dg categories means].
Inside
̂
T we have
̂
T
pe
, the full subcategory of compact objects in
̂
T . We have
h ∶ T →
̂
T actually has its essential image in
̂
T
pe
, because quasi-representability
implies compactness. So we’ll use the same notation h. If T is triangulated, then
every compact object is quasi-representable. Then there is an isomorphism between
T and
̂
T
pe
in the homotopy category of dg − cat. Conversely, if h ∶ T →
̂
T
pe
is
essentially surjective (this is the essential image), then compact objects are quasi-
representable so that T is triangulated.
Now I want to show that
̂
T
pe
is triangulated. One way to show this is using a
theorem from To¨en’s other paper, namely that the Yoneda embedding induces a
quasi-equivalence
R Hom(
̂
T
op
pe
, Int(C(k))) → R Hom(T
op
, Int(C(k)))
whose left hand side is
̂
pe
T inside of which we can find
̂
pe
T
pe
. So we get a functor
(̂)
pe
from Ho(dg − cat) to Ho(dg − cat
tr
), the trianguleted hull of T . This pe is “per-
fect.” For R a k-algebra and the category BR, then [
BR
pe
] ≅ H
perf
(R) ⊂ D(R).
Triangulated in this sense implies that there is a natural triangulated structure in
the ordinary sense on [T ].
Let me continue. So we have two functors between the categories Ho(dg − Cat)
and Ho(dg cat
tr
), ι and (̂)
pe
. We can show that under certain conditions ι is right
adjoint. To show this we need a lemma. For more detail you can check To¨en’s
paper, Lemma 7.3, which says
Lemma 1.1. Let T
′
be triangulated. Under certain conditions (T
′
op
− mod is
cofibrantly generated and for F ∈ T
′
op
− mod , F
x
is projective for all x. Then h
∗
∶
(T
′
op
− mod )
̂
T
pe
→ (T
′
op
− mod )
T
is a Quillen equivalence which then induces an
equivalence on homotopy categories, so we have a bijection of isomorphism classes
of objects in these homotopy categories.
DERIVED SEMINAR 3
We assumed that T
′
is triangulated, so that it is isomorphic to
̂
T
′
pe
in the
homotopy category. Consider this diagram.
Hom
Ho(dg cat
tr
)
(
̂
T
pe
, T
′
)
//
Hom
Ho(dg cat)
(
̂
T
pe
, Int(T
′ op
− mod ))
≅
Iso(Ho((T
′ op
− mod )
̂
T
pe
))
Iso(Ho((T
′ op
− mod )
T
))
Hom
Ho(dg cat)
(T, ιT
′
)
//
Hom(Ho(dg cat))(T, Int(T
′ op
− mod ))
The image of the top horizonal map are the morphisms which factor
̂
T
pe
→ T
′
→
Int(T
′ op
− mod ) and the same for the bottom map. It is actually clear that the
property of factorizing is independent of the bijection between the isomorphism
classes of the homotopy categories, so these two criteria of the inclusion, are the
same and the two hom sets are the same.
The last thing I have to talk about is Morita equivalence.
Definition 1.4. f ∶ T → T
′
a morphism in Ho(dg Cat) is a Morita equivalence if
ˆ
f
pe
∶
̂
T
pe
→
̂
T
′
pe
is an isomorphism.
Denote by W
Mor
the set of all Morita equivalences. We consider localization at
Morita equivalences. We want to consider
Ho(dg cat)
(̂)
pe
//
`
Ho(dg cat
tr
)
W
−1
Mor
Ho(dg cat)
99
and you can actually show that this dotted arrow is an equivalence. It follows from
some formal steps which is related to the localization process.
Proposition 1.1. Let f ∶ T → T
′
be morphisms in dg cat. Then the following are
equivalent:
(1) f is a Morita equivalence.
(2) For T
0
is triangulated we have that [T
′
, T
0
] → [T, T
0
] is a bijection.
(3) The induced functor f
∗
∶ D(T
′
) → D(T ) is an equivalence of categories
after restriction to compact objects.
(4) The induced functor Lf
!
∶ D(T ) → D(T
′
) is an equivalence of categories
after restriction to compact objects.
One of the steps here is hard but you can do it.
[To¨en took the simplest definition, being equivalent to the category of perfect
objects, but it’s better than being triangulated to be isomorphic to this. This is
like triangulated plus idempotent complete.]
