MODEL STRUCTURE OF DCAT
Abstract. This is a reading note of the paper by Tabuada’s article
“Une structure de cat´egorie de mod`eles de Quillen sur la cat`egorie des
dg-cat´egories”, C. R. Acad. Sci. Paris, Ser. I 340 (2005), 15 - 19 for my
own purpose of learning the language of dg-categories and the model
structure on the category of dg-categories. It first translates French into
English and secondly adds full details of the proof for the beginning
learner (like myself) on dg-categories – prepared by Yong-Geun Oh –
– Reading note on Tabuada’s paper –
(K) Let K be the dg-category with two objects 1, 2 and whose morphisms
are generated by
f ∈ Hom
0
K
(1, 2), g ∈ Hom
0
K
(2, 1), r
1
∈ Hom
−1
K
(1, 1),
r
2
∈ Hom
−1
K
(2, 2), r
12
∈ Hom
−2
K
(1, 2)
with relations
df = 0 = dg, dr
1
= gf − 1
1
, dr
2
= fg − 1
2
, dr
12
= fr
1
− r
2
f;
(A) Let A the dg-category with a single object 3 with Hom
A
(3, 3) = k;
(F ) Let F : A → K be the dg-functor which maps 3 to 1;
(B) Let B be the dg-category with two objects 4 and 5 such that
Ham
B
(4, 4) = k1
4
, Hom
B
(5, 5) = k1
5
, Hom
B
(4, 5) = 0 = Hom
B
(5, 4);
(S
n−1
, D
n
) For each n ∈ Z, we denote by S
n−1
the complex k[n − 1] and by D
n
the cone of the identity morphism of S
n−1
. In other words,
D
n
= Cone(1 : S
n−1
→ S
n−1
) = k[n] ⊕ k[n − 1];
(P(n)) We denote by P(n) the dg-category with two objects 6 and 7, with
Hom
P(n)
(6, 6) = k1
6
, Hom
P(n)
(7, 7) = k1
7
,
Hom
P(n)
(7, 6) = 0, Hom
P(n)
(6, 7)
∼
=
D
n
;
(R(n)) Let R(n) be the functor from B to P(n) sending 4 on 6 and 5 on 7;
(C(n)) Next we consider the dg-category C(n) with two objects 8, 9 with
Hom
C(n)
(8, 8) = k1
8
, Hom
C(n)
(9, 9) = k1
9
,
Hom
C(n)
(9, 8) = 0, Hom
C(n)
(8, 9)
∼
=
S
n−1
;
(S(n)) Let S(n) : C(n) → P(n) be the dg-functor which maps 8 to 6, 9 to
7 and the morphism space S
n−1
to the morphism space D
n
through
the identity of k[n − 1];
1
2 MODEL STRUCTURE OF DCAT
(Q) Finally let Q be the dg-functor mapping the empty dg-category ∅ —
which is the initial object in DCAT — to A.
Theorem 0.1 (Main Theorem). If we consider for C the category DCAT
for W the subcategory of quasi-equivalences, for J the dg functors F and
R(n), n ∈ Z, and for I the dg functors Q and S(n), n ∈ Z, then the condi-
tions of the recognition theorem [H, 2.1.19] are fulfilled. Thus the category
DCAT admits a Quillen model structure whose weak equivalences are the
quasi-equivalences.
Out of this theorem, one can easily show that for the above model struc-
ture, all objects are fibrant and that a dg-functor F : C → D is a fibration
if and only if the followings hold:
(1) for all objects c
1
and c
2
in C, a morphism from Hom
C
(c
1
, c
2
) to
Hom
D
(F (c
1
), F (c
2
)) is surjective in each degree,
(2) for all object c
1
of C and isomorphism v : F (c
1
) → d in H
0
(D), there
exists a morphism u : c
1
→ c
2
inducing an isomorphism in H
0
and
satisfying F (u) = v.
Recall the statement of [H, 2.1.19]. To make the statements thereof, we
need some preparations.
Definition 0.2. Let C be a category, and let I is a set of maps in C.
(1) The subcategory of I-injectives is the subcategory of maps that have
the right lifting property with respect to every element of I.
(2) The subcategory of I-cofibrations is the subcategory of maps that
have the left lifting property with respect to every I-injective. An
object is I-cofibrant if the map to it from the initial object of C is
an I-cofibration.
Example 0.3. (1) If I is the set of inclusions ∂∆(n) → ∆(n) is sSet,
then the I-injectives are the trivial fibrations, and the I-cofibrations
are the inclusions of simplicial sets.
(2) If J is the set of inclusions Λ[n, k] → ∆[n] in sSet, then the J-
injectives are the Kan fibrations, and J-cofibrations are the trivial
cofibrations.
We recall the notion of transfinite composition of maps.
Definition 0.4. Suppose C is a category with all small colimits, and λ is
an ordinal. A λ-sequence in C is a colimit-preserving functor X : λ → C,
written as X
0
→ X
1
→ · · · → X
β
→ · · · . Since X preserves colimits, for all
limit ordinals γ < λ, the induced map
colimi
β<γ
X
β
→ X
γ
is an isomorphism. We refer to the map X
0
→ colimi
β<γ
X
β
as the compo-
sition of the λ-sequence.
If D is a collection of morphims of C and every map X
β
→ X
β+1
for
β + 1 < λ is in D, we refer to the composition X
0
→ colimt
β<λ
X
β
as a
transfinite composition of maps of D.
MODEL STRUCTURE OF DCAT 3
Actually the composition is not unique, but only unique up to isomor-
phism under X, sicne the colimit is not unique.
Definition 0.5. Suppose that C is a category that is closed under small
colimits and I is a set of maps in C. Then
(1) the subcategory of relative I-cell complexes is the subcategory of
maps that can be constructed as a transfinite composition of pushouts
of elements of I,
(2) an object is an I-cell complex if the map to it from the initial object
of C is a relative I-cell complex, and
(3) a map is an inclusion of I-cell complexes if it is a relative I-cell
complex whose domain is an I-cell complex.
Theorem 0.6 (Theorem 2.1.19 [H]). Let C be a category with all small
colimits and limits. Suppose W is a subcategory of C, and I and J are sets
of maps of C. Then there is a cofibrantly generated model structure on C
with I as the set of generating cofibrations, J as the set of generating trivial
cofibrations, and W as the subcategory of weak equivalences if and only if
the following conditions are satisfied.
(1) The subcategory W has two out of three property and is closed under
retracts.
(2) The domains of I are small relative to I-cell.
(3) The domains of J are small relative to J -cell.
(4) J − cell ⊂ W ∩ I − cof.
(5) I − inj ⊂ W ∩ J − inj.
(6) Either W ∩ I − cof ⊂ J − cof or W ∩ J − inj ∩ I − inj.
In the rest of the note, we give complete details of the proof of Theorem
0.1.
Two out of three property is apparent. To prove closedness under retracts,
consider a morphism F : X → Y in W and the commuting diagram
A
g
i
//
X
f
j
//
A
g
B
i
//
Y
j
//
B
satisfying j ◦ i = 1
A
.
Take the Ho-functor and get the diagram
A
[g]
[i]
//
X
∼
=
[j]
//
A
[g]
B
//
Y
//
B
By considering the left square, we find [g] is injective and from the right
square the surjectivity of [g] follows, and hence [g] is an isomorphism. This
proves g ∈ W.
4 MODEL STRUCTURE OF DCAT
For Statement (2), we note that the domains of I are the union
Obj C(n)
a
{∅}.
We in fact prove the following
Lemma 0.7. The domains of C(n) are small relative to I-cell. Similar for
Statement (3).
Proof. Note that the domains of I are the categories C(n) and the empty
category ∅. The empty category ∅ is clearly 1-small, and so we will only
consider the categories C(n) for n ∈ Z.
It will be enough to prove that C(n) is ω-small, where ω is the first infinite
ordinal. This is because by definition the collection D of I-cells is that of
transfinite composition of pushouts of {C(n) | n ∈ Z} ∪ ∅: the latter is
filtered, |D|-small for its cardinality |D| and |D| ≥ ω, which follows from the
fact that I-cell is closed under the transfinite compositions. (See [H, Lemma
2.1.12].) Therefore it is enough to prove that C(n) is ω-small.
Consider C(n) for given n ∈ Z. Let X : I → DGCAT be a (small) filtered
diagram of dg-categories. By construction of filtered colimits that we give
in the appendix, it follows that given a dg-functor f : C(n) → colim
i∈I
X
i
,
since I is filtered we can find an i
0
such that both 8 and 9 lend in X
i
0
. Then
by using the fact that S
n−1
is ω-small in the category of chain complexes,
we may find an object i
0
→ j such that f factors through X
j
. The fact that
this factorisation is essentially unique is a consequence of the definition of
the filtered colimit of dg-categories.
Next we prove the following.
Proposition 0.8. We have J − cell ⊂ W .
Proof. Consider the co-Cartesian diagrams
B
T
//
R(n)
J
inc
P(n)
//
U
A
N
//
F
L
inc
K
//
M
in DCAT. A J-cell is a transfinite composition of pushouts with R(n)
and/or F . We want to prove inc ∈ W in either case, i.e., that it is a
quasi-equivalence.
We start with the first diagram. The category U is the disjoint union
U = J
a
P(n) : R(n)(4) ∼ T (4), R(n)(5) ∼ T (5)
with the addition of new morphisms j : T (4) → T (5) of degree n − 1 and
: T (4) → T (5) of degree n satisfying d = j. We mention that in the level
of objects there is a natural one-one correspondence between that of J and
MODEL STRUCTURE OF DCAT 5
that of U. But in the level of morphisms, we have the decomposition
Hom
U
(X, Y ) =
M
m≥0
Hom
(m)
U
(X, Y )
with
Hom
(m)
U
(X, Y ) = (T (5), Y ) ⊗ D
n
(T (5), T (4)) ⊗ D
n
⊗ · · · ⊗ D
n
⊗ (X, T (4)),
where (, ) denotes Ham
J
(, ). Since the complex D
n
is contractible, the
inclusion
Hom
J
(X, Y ) → Hom
U
(X, Y )
is a quasi-isomorphism. Since the inclusion functor is the identity in objects,
it is a quasi-equivalence.
Next we consider the co-Cartesian diagram
A
N
//
F
L
inc
K
//
M
in DCAT. Again we want to show that inc is a quasi-equivalence. In the
level of objects, one more object corresponding to 2 in K is added to L.
We consider the category L
0
obtained from L by adding a morphism
s : N(3) → H where H is a new object and ds = 0. Then we consider
Mod L
0
the category of right dg-modules on L
0
and the Yoneda embedding
c
( . ) : L
0
→ Mod L
0
.
We denote by
b
X := Hom
L
0
(, X).
Let L
1
be the full subcategory of Mod L
0
whose objects are the cone
C := Cone(bs :
[
N(3) →
b
H) and the dg-representable functors.
Define the category L
2
to be the one obtained by adding to L
1
a morphism
h : C → C of degree 1 in End
L
2
(C) such that dh = 1
C
.
Lemma 0.9. The category M is naturally identified with the full dg-subcategory
of L
2
whose objects are the images of L
0
in L
2
.
Proof. We first note that the set of objects of
c
L
0
is formed by
b
X = Hom
L
2
(, X), X ∈ L
0
.
We note objectwise
b
H is added to
b
L in
c
L
0
. We need to examine their
morphism spaces. We only need to consider morphisms on
[
N(3), H and the
cone C = Cone(bs :
[
N(3) →
b
H).
Let us first analyze the condition dh = 1
C
. Using the fact C =
[
N(3)[1]⊕
b
H
which is the cone of bs :
[
N(3) →
b
H, we write d
C
and h : C → C as the
matrices
d
C
=
d
[
N(3)[1]
0
bs d
[
N(3)
!
, h =
k
m n
6 MODEL STRUCTURE OF DCAT
where k : N (3)[1] → H, : H → N(3)[1], m : H → N(3)[1] and n : H → H.
More precisely, we have
d
n
C
=
−d
n+1
[
N(3)
0
bs
n+1
d
n
b
H
!
, h
n
=
−k
n+1
−
n
m
n+1
n
n
where k
n+1
: N(3)
n+1
→ H
n
,
n
: H
n
→ N(3)
n
, m
n+1
: N(3)
n+1
→ H
n−1
and n
n
: H
n
→ H
n−1
. We note that k and n has degree −1, has degree 0
and m has degree −2, which is responsible for the assignment of the signs
in the matrix recalling the sign rule
f
n
C[k]
= (−1)
k|f |
f
k+n
C
in the definition of the shifted map on the shifted complex C[k] for general
graded linear map f : C → C of degree |f|.
Recall that given a chain complex (C, d
C
) and a map of chain complexes
f : C → C, the differential d of f in Hom
C
is defined as df − (−1)
|f|
fd.
By recalling deg h = 1 and expanding the equation,
(dh)
n
= 1
C
n
: N(3)
n+1
⊕ H
n
→ N(3)
n+1
⊕ H
n
,
we obtain
1
N
n+1
= d
n
[
N(3)
k
n+1
+ k
n+2
d
n+1
[
N(3)
+
n+1
bs
n+1
,
0 = −d
n
[
N(3)
n
+
n+1
d
n
b
H
0 = d
n−1
b
H
m
n+1
− m
n+2
d
n+1
[
N(3)
− bs
n
k
n+1
−
n+1
bs
n+1
,
id
H
n
= −d
n−1
b
H
n
n
− n
n+1
d
n
b
H
+ bs
n
n
.
We set
f
0
= bs, g
0
= , r
0
1
= −k, r
0
2
= n, r
0
12
= −m
Then we have
f
0
g
0
− 1
H
= dr
0
2
.
g
0
f
0
− 1
N(3)
= dr
0
1
and
r
0
2
f
0
− f
0
r
0
1
= dr
0
12
.
Therefore the collection of morphisms {f
0
, g
0
, r
0
1
, r
0
2
, r
0
12
} on the set {
b
N(3),
b
H}
of objects presents the category K. This completes the proof.
Therefore we have Hom
M
(X, Y )
∼
=
Hom
L
2
(
b
X,
b
Y ). On the other hand, we
have decomposition
Hom
L
2
(
b
X,
b
Y ) =
M
n≥0
Hom
(n)
L
2
(
b
X,
b
Y )
where
Hom
(n)
L
2
(
b
X,
b
Y ) = Hom
L
1
(C,
b
Y )⊗S
2
⊗Hom
L
1
(C, C)⊗S
2
⊗· · ·⊗S
2
⊗Hom
L
1
(
b
X, C).
MODEL STRUCTURE OF DCAT 7
We remark that this decomposition is not a direct sum of complexes but only
as a graded group. Now we compute the differential of g
n+1
·h·g
n
·h · · · h·g
1
∈
Hom
(n)
L
2
(
b
X,
b
Y ) using dh = 1. It becomes
d(g
n+1
) · h · g
n
· h · · · h · g
1
+ (−1)
|g
n+1
|
g
n+1
· 1 · g
n
· h · · · h · g
1
+ · · ·
This shows that the sum ⊕
m
n≥0
Hom
(n)
L
2
(
b
X,
b
Y ) is a subcomplex for all m ≥ 0
and Hom
L
2
(
b
X,
b
Y ) which gives rise to an exhaustive filtration of Hom
L
2
(
b
X,
b
Y ).
Its n-th associated graded group is Hom
(n)
L
2
(
b
X,
b
Y ).
On the other hand, the complex Ham
L
1
(
b
X, C) can be identified with the
cone of the isomorphism
s
∗
: Hom
L
0
(
b
X,
[
N(3)) → Hom
L
0
(
b
X,
b
H)
and so contractible.
Combing the above, we derive that the inclusion
Hom
L
(X, Y ) → Hom
M
(X, Y )
∼
=
Hom
L
2
(
b
X,
b
Y )
is a quasi-isomorphism. Since the inclusion functor is the identity in the
level of objects, it becomes a quasi-equivalence.
This completes the proof of the proposition.
Finally we will show the following proposition which will prove Statements
(5) and (6) simultaneously.
Proposition 0.10. We have
I − inj = J − inj ∩ W.
For the proof of this proposition, we introduce another relevant category:
Denote by Surj the category of functors G : H → I in DCAT satisfying
(1) G induces a surjection in objects,
(2) G induces a quasi-isomorphism which is also surjective in the com-
plex of morphisms.
Then we will prove the following
(0.1) I − inj = Surj = J − inj ∩ W.
Lemma 0.11. We have I − inj = Surj.
Proof. Consider the commutative diagram
C(n)
S(n)
D
//
H
G
P(n)
E
//
I
8 MODEL STRUCTURE OF DCAT
of dg-cageogories. This corresponds to the commutative diagram in the
category of complexes
S
n−1
Hom
H
(D(8), D(9))
D
n
Hom
I
(E(6), E(7))
D
i
n
G
E
where D(8), D(9) are objects of H and E(6), E(7) are those of I. In the
category of complexes, surjectivity of G is equivalent to the lifting property
of E by the following
Proposition 0.12 (Proposition 2.3.4 [H]). A map p : X → Y in Ch(R) is
a fibration i.e has the lifting property with respect to the maps 0 → D
n
iff
p
n
: X
n
→ Y
n
is surjective for all n.
Next we prove the following
Lemma 0.13. We have J − inj ∩ W = Surj.
Proof. We will first show Surj ⊂ J − inj ∩ W .
For each functor H from N to E in class Surj, it follows from definition
that H ∈ W . Therefore it remains to showSurj ⊂ J − inj.
The class of arrows with the right lifting property against R(n) is made
of functors surjective at the level of the complexes of morphims. Hence it is
enough to show that H has the right lifting property against F . Consider
the diagram
A
F
P
//
N
H
K
U
//
E.
This provides the lower left corner of the diagram
P (3)
H
//
D
H
U(1)
U(f)
//
U(2)
and a contraction h of the cone C
1
= Cone(
\
U(f)) →
b
E). (Recall Lemma
0.9.) Since H is surjective on objects, there exists D ∈ N such that
H(D) = U (2) and H is a quasi-isomorphism surjective in the level of mor-
phism complexes. Threrefore we can lift U (f ) to U(f ) : P (3) → D. In
MODEL STRUCTURE OF DCAT 9
terms of the category of dg-modules, we obtain the diagram
\
P (3))
b
H
[
U(f)
//
b
D
b
H
//
C
2
b
H
[
U(1)
[
U(f)
//
[
U(2)
//
C
1
with contraction h of C
1
and C
2
is the cone of the morphism
[
U(f). According
to Lemma 0.9 we need to lift the contraction h to a contraction h
∗
of C
2
.
But since
b
H induces a quasi-isomorphism surjective in the endomorphism
algebras, we can lift h to h
∗
by the following proposition applied to the
diagram
C
2
b
H
//
C
2
b
H
C
1
h
//
C
1
Proposition 0.14 (Proposition 2.3.5 [H]). A map p : X → Y in Ch(R) is
a trivial fibration if and only if it has the lifting property against the map
i
n
: S
n−1
→ D
n
.
This proves Surj ⊂ J − inj ∩ W .
Now we show the opposite inclusion. Let L : D → S be a functor in
J − inj ∩ W . Then for any diagram
A
//
D
L
K
//
S
the functor L has the right lifting property. Since L ∈ W , it is enough to
show that L is surjective in objects. For any object E ∈ S, since L ∈ W ,
there exists C ∈ D and a morphism q ∈ Hom
S
(L(C), E) that induces an
isomorphism in H
0
(S)
C
L
L(C)
q
//
E.
Also q is in the image of f under the functor K to S since L ∈ J −inj. Since
L ∈ J − inj, the right lifting property above implies that we can lift the
morphism q : L(C) → E to some q : C → E for some E ∈ D. In particular,
L(E) = E. This proves surjectivity of L on objects.
This finishes the proof.
10 MODEL STRUCTURE OF DCAT
Appendix A. Colimit of dg cagories
Now we describe how to compute filtered colimits of dg-categories.
Let C : I → DGCAT be a filtered diagram,
The set of objects of the colimit Obj(colim
D
) is defined as the colimit of
the sets of objects of the categories C
β
for β ∈ I:
D
0
' colim
β
Ob(C
β
).
The only thing that is left to define is the dg-module of maps between
two objects of the colimit. To define Hom
C
(X, Y ) for X, Y ∈ D
0
we pro-
ceed as follows. By definition of filtered colimits in Set, there exists some
β(X), β(Y ) ∈ I such that
X ∈ Ob(C
β(X)
), Y ∈ Ob(C
β(Y )
).
Consider the category J defined as a full subcategory of J whose objects j
are such that there exists maps j
X
: β(X) → j, j
Y
: β(Y ) → j in I. Since I
is filtered J is also filtered.
For any j ∈ J, we can get a dg-module
Hom
dg
C
j
(C(j
X
)(X), C(j
Y
)(Y ))
and for any arrow j → j
0
a dg-map between those two dg-modules. We
define the dg-module of maps between X and Y to be the colimit of this
diagram in the category of dg-modules:
Hom
dg
D
(X, Y ) ' lim
−→
j∈J
Hom
dg
C
j
(C(j
X
)(X), C(j
Y
)(Y )).
The construction does not depend on the choice of β(X) of β(Y ) as any two
different choices will lead to two cofinal filered categories J.
The composition is then induced from the compositions of the categories
in the diagram and the fact that the tensor product of dg-modules commute
with filtered colimits.
We let the reader check that this new dg-categories has the correct uni-
versal property.
References
[H] M. Hovey, Model Categories, Math. Surveys and Monographs 63, AMS, Providence,
RI, 1999.
[T] G. Tabuada, “Une structure de cat´egorie de mod`eles de Quillen sur la cat`egorie des
dg-cat´egories, C. R. Acad. Sci. Paris, Ser. I 340 (2005), 15 - 19.
