Theory and Applications of Categories, Vol. 32, No. 2, 2017, pp. 31–52.

SIMPLICIAL NERVE OF AN A_{∞}-CATEGORY

GIOVANNI FAONTE

Abstract. We introduce a functor called the simplicial nerve of an A_{∞}-category defined on the category of A_{∞}-categories with values in simplicial sets. We show that the nerve of an A_{∞}-category is an (∞, 1)-category in the sense of J. Lurie [Lur1]. This construction generalizes the nerve construction for differential graded categories given in [Lur2]. We prove that if a differential graded category is pretriangulated in the sense of A.I. Bondal and M. Kapranov [Bo-Ka] then its nerve is a stable (∞, 1)-category in the sense of J. Lurie [Lur2].

1. Introduction

A_{∞}-algebras were introduced by J.D. Stasheff [Sta] in order to encode the notion of a binary operation associative up to a coherent system of homotopies. An A_{∞}-algebra is a ℤ-graded vector space A over some base field 𝕂 together with degree 2 − k morphisms

m_{k} : A^{⊗n} [figure] A, k ≥ 1

satisfying the equation

∑_{n=r+t+s}(−1)^{sr+t}m_{r+t+1}(Id^{⊗^{r}} ⊗ m_{s} ⊗ Id^{⊗^{t}}) = 0  (1)

for n ≥ 1. This equation for n = 1 tells that m_{1} is a differential on A and for n = 2 provides the compatibility of the binary operation m_{2} with the differential m_{1} in terms of the Leibniz rule. For n = 3 the equation is

m_{2}(m_{2} ⊗ Id) − m_{2}(Id ⊗ m_{2}) = m_{1}(m_{3}) + ∑_{2=r+t} m_{3}(Id^{⊗^{r}} ⊗ m_{1} ⊗ Id^{⊗^{t}})

which expresses the fact that m_{2} is an associative operation up to the data provided by m_{3}. Values of n > 3 in equation (1) encode all the coherences needed to be satisfied by such associativity constraints. Similarly an A_{∞}-morphism is a morphism of differential graded spaces

f_{1} : A [figure] B

The author would like to thank his doctoral adviser M. Kapranov and T. Dyckerhoff for useful discussions and precious insights given through the realization of this work.

Received by the editors 2015-09-25 and, in final form, 2017-01-06. Transmitted by James Stasheff. Published on 2017-01-15. 2010 Mathematics Subject Classification: 18G30. Key words and phrases: A_{∞}-categories, nerve, higher categories, pretriangulated dg-categories. \begin{matrix}c \\ ◯ \end{matrix} Giovanni Faonte, 2017. Permission to copy for private use granted.

which preserves the binary operations only up to the data provided by

f_{2} : A ⊗ A [figure] B

whose coherences are controlled by degree 1 − k morphisms f_{k} : A^{⊗k} [figure] B for k > 3. The notion of a dg-algebra (morphisms of dg-algebras) is recovered by considering A_{∞}-algebras with vanishing m_{k} for k > 2 ( A_{∞}-morphisms with vanishing f_{k} for k > 1). K. Fukaya and M. Kontsevich-Y. Soibelman later considered A_{∞}-categories and A_{∞}-functors as a natural generalization of these notions which have been essential tools in the formulation of homological mirror symmetry [Kon-Soi],[Fuk]. Namely the Fukaya category F(X) is an A_{∞}-category associated to a symplectic manifold X and it corresponds to the A-side of such symmetry. Similarly to the case of algebras, differential graded categories (dgfunctors) can be identified with A_{∞}-categories (A_{∞}-functors) for which the composition of morphisms is strictly associative (strictly preserving the composition of morphisms).

In the first part of this paper we define a functor (Definition 2.8) called the simplicial nerve of an A_{∞}-category

N_{A_{∞}} : A_{∞}Cat [figure] SSet

defined on the category A_{∞}Cat of A_{∞}-categories with A_{∞}-functors and values in simplicial sets. We prove (Proposition 2.15) that the simplicial nerve of an A_{∞}-category is an (∞, 1)-category in the sense of J. Lurie [Lur1]. For an A_{∞}-category A its nerve is the simplicial set of A_{∞}-functors from a certain cosimplicial A_{∞}-category A[Δ^{−}], generated by the standard simplices, into A. The restriction of this functor to dg-categories provides a functorial description of the differential graded nerve N_{dg} introduced in [Lur2] by J. Lurie and earlier defined in [Hin-Sch] by V.A. Hinich and V.V. Schechtman. The existence of a model category structure without limits on A_{∞}Cat was shown in [Le-Ha] and the study of its relationships with the nerve construction presented in this paper will be subject of future work.

SIMPLICIAL NERVE OF AN A_{∞}-CATEGORY  33

In the second part we establish a connection between pretriangulated differential graded categories in the sense of A.I. Bondal and M. Kapranov [Bo-Ka] and stable (∞, 1)-categories in the sense of J. Lurie [Lur2]. Pretriangulated dg-categories provide a natural setting to address the lack of functoriality of the cone construction for triangulated categories following from the axioms of J.-L. Verdier [Ver]. To a dg-category with a zero object D it is possible to associate the dg-category of twisted complexes of D, denoted by P reT r(D), whose construction has to be understood as a triangulated hull of D. P reT r(D) has a shift functor and a functorial notion of cones inducing a triangulated structure on its homotopy category H^{0}(P reT r(D)). In particular when D is pretriangulated the dg-embedding D [figure] P reT r(D) is a quasi-equivalence of dg-categories and hence D inherits shift and cones from P reT r(D) making H^{0}(D) into a triangulated category [Bo-Ka]. On the (∞, 1)-categorical side J. Lurie in [Lur2] introduced the notion of stable (∞, 1)-category as an axiomatization of the properties of stable homotopy theory. The relevant feature for our purposes is that in a stable (∞, 1)-category the notion of an exact triangle is replaced by the one of homotopy fiber of a morphism. Moreover stable (∞, 1)- categories have canonically defined loop and suspension functors playing the role of the shift functor and its inverse for triangulated categories. This data induces a structure of triangulated category on the homotopy category of a stable (∞, 1)-category [Lur2]. We give an explicit proof (Theorem 3.18) that if D is a pretriangulated dg-category the nerve N_{dg}(D) is a stable (∞, 1)-category and H^{0}(D) is identified, as a triangulated category, with the homotopy category of N_{dg}(D). The proof is based on a direct computation performed on an (∞, 1)-category equivalent to N_{dg}(D), that we call the big dg-nerve N_{dg}^{big}(D), defined as the nerve of a certain simplicial category D_{Δ} whose simplicial set of morphisms is obtained by applying the Dold-Kan correspondence [McL] to a truncation of the cochain complex of morphisms in D.

1.1. Conventions and notations. From now on we fix a field 𝕂 of characteristic 0. The category V ect_{ℤ}(𝕂) is the category whose objects are ℤ-graded vector spaces over 𝕂

V = ⨁_{n∈ℤ} V^{n}

and morphisms are degree preserving 𝕂-linear maps. The tensor product of graded vector spaces is defined as

(V ⊗ W)^{n} = ⨁_{p+q=n} V^{p} ⊗ W^{q}

and the graded space of morphisms as

Hom_{V ect_{ℤ}(𝕂)}^{n}(V, W) = ∏_{p∈ℤ} Hom_{V ect(𝕂)}(V^{p}, W^{p+n})

We say that a morphism is of degree n if it belongs to such graded component. The tensor product of two morphisms is defined according to the convention

(f ⊗ g)(x ⊗ y) = (−1)^{deg(x)deg(g)}f(x) ⊗ g(y)  (2)

A cochain complex is an object V ∈ Ob(V ect_{ℤ}(𝕂)) together with a morphism d ∈ Hom_{V ect_{ℤ}(𝕂)}^{1}(V, V) such that d^{2} = 0. We call d the differential of the cochain complex. A morphism of cochain complexes f : V [figure] W is a morphism f ∈ Hom_{V ect_{ℤ}(𝕂)}^{0}(V, W) such that d ∘ f = f ∘ d. The category Ch^{•}(𝕂) is the category whose objects are cochain complexes and morphisms are morphisms of cochain complex. The cohomology of a cochain complex (V, d) is the ℤ-graded vector space defined as

H^{•}(V) = \frac{Ker(d)}{Im(d)}

A morphism of cochain complexes f : V [figure] W is called a quasi-isomorphism if the morphism induced in cohomology H^{•}(f) : H^{•}(V) [figure] H^{•}(W) is an isomorphism. The tensor product of ℤ-graded vector spaces extends to a functor on Ch^{•}(𝕂) defining a symmetric monoidal structure (Ch^{•}(𝕂), ⨂, 𝕂) where, for cochain complexes V and W, we have

d_{V ⊗W} = d_{V} ⊗ Id_{W} + Id_{V} ⊗ d_{W}

Similarly the category Ch_{•}(𝕂) of chain complexes has objects ℤ-graded vector spaces together with a morphism d ∈ Hom_{V ect_{ℤ}(𝕂)}^{−1}(V, V) such that d^{2} = 0 and analogue notions of homology and quasi-isomorphism. We denote by op : Ch^{•}(𝕂) [figure] Ch_{•}(𝕂) the functor associating to a cochain complex the chain complex V_{p}^{op} = V^{−p} with the same differential and by τ_{≥0} : Ch_{•}(𝕂) [figure] Ch_{•}^{≥0}(𝕂) the truncation functor

τ_{≥0}(V)_{p} = \begin{cases}0 & if p < 0 \\ Ker(d_{|_{V_{0}}}) & if p = 0 \\ V_{p} & if p > 0. \end{cases}

For a category C the category of simplicial objects in C, denoted by S(C), is the category of functors F un(Δ^{op}, C) where Δ is the standard simplex category. Similarly, the category of cosimplicial objects in C is the category of functors F un(Δ, C). We refer to [Lur1, Chap. 1-2-3] for the theory of (∞, 1)-categories and related constructions. The smallness assumptions necessary for the consistency of the results presented in this work are implicitly assumed.

2. The simplicial nerve of an A_{∞}-category

2.1. A_{∞}-categories and A_{∞}-functors. We recall now the notion of A_{∞}-category and of A_{∞}-functor. We refer to [Le-Ha] for an extensive survey of the subject. For the purposes of this work we will refer to an A_{∞}-category meaning a strictly unital A_{∞}- category and to an A_{∞}-functor meaning a strictly unital A_{∞}-functor.

2.2. Definition. [A_{∞}-category] Let 𝕂 be a field, an A_{∞}-category A is the data of:

• A set of objects Ob(A)

• For every pair of objects x, y ∈ Ob(A) a graded space of morphisms Hom_{A}^{•}(x, y)

• For k ≥ 1 and a sequence of objects x_{0}, x_{1}, …, x_{k}, a morphism of degree 2 − k

m_{k} : Hom_{A}^{•}(x_{k−1}, x_{k}) ⊗ ⋯ ⊗ Hom_{A}^{•}(x_{0}, x_{1}) [figure] Hom_{A}^{•}(x_{0}, x_{k})

such that, for every n ≥ 1

∑_{n=r+t+s}(−1)^{sr+t}m_{r+t+1}(Id^{⊗^{r}} ⊗ m_{s} ⊗ Id^{⊗^{t}}) = 0  (3)

• For every object x ∈ Ob(A) a degree 0 element 1_{x} ∈ Hom_{A}^{•}(x, x), called the identity

at x, such that

m_{1}(1_{x}) = 0
m_{2}(1_{x} ⊗ a) = a = m_{2}(a ⊗ 1_{x})
m_{n}(a_{1} ⊗ ⋯ ⊗ a_{j−1} ⊗ 1_{x} ⊗ a_{j+1} ⊗ … a_{n}) = 0

for n > 2, 1 ≤ j ≤ n.

2.3. Definition. [A_{∞}-functor] Let A and B be two A_{∞}-categories, an A_{∞}-functor

f : A [figure] B

is the data of:

• A map of sets f_{0} : Ob(A) [figure] Ob(B)

• For k ≥ 1 and a sequence of objects x_{0}, x_{1}, …, x_{k}, a morphism of degree 1 − k

f_{k} : Hom_{A}^{•}(x_{k−1}, x_{k}) ⊗ ⋯ ⊗ Hom_{A}^{•}(x_{0}, x_{1}) [figure] Hom_{B}^{•}(f_{0}(x_{0}), f_{0}(x_{k}))

such that, for n ≥ 1

∑_{n=r+t+s}(−1)^{sr+t}f_{r+t+1}(Id^{⊗^{r}} ⊗ m_{s} ⊗ Id^{⊗^{t}}) = ∑_{\begin{matrix}1≤r≤n \\ i_{1}+···+i_{r}=n \end{matrix}}(−1)^{ϵ_{r}}m′_{r}(f_{i_{1}} ⊗ ⋯ ⊗ f_{i_{r}}) (4)

where

ϵ_{r} = ϵ_{r}(i_{1}, …, i_{r}) = ∑_{2≤k≤r} ((1 − i_{k}) ∑_{1≤l≤k−1} i_{l})  (5)

and satisfying the strict unitality conditions:

f_{1}(1_{x}) = 1_{f_{0}(x)}
f_{n}(a_{1} ⊗ ⋯ ⊗ a_{j−1} ⊗ 1_{x} ⊗ a_{j+1} ⊗ … a_{n}) = 0

for n > 1, 1 ≤ j ≤ n.

2.4. Remark. A_{∞}-categories and A_{∞}-functors can be defined in a more canonical way using the Bar construction [Le-Ha]. The sign convention adopted in Definition 2.2 and 2.3 follows from the convention (2) defining the tensor product of graded morphisms after considering such construction.

If f : A [figure] B and g : B [figure] C are A_{∞}-functors, their composition g ∘ f : A [figure] C has graded components given by

(g ∘ f)_{k} = ∑_{r=1}^{k} ∑_{i_{1}+···+i_{r}=k}(−1)^{ϵ_{r}(i_{1},...,i_{r})}g_{r}(f_{i_{1}} ⊗ ⋯ ⊗ f_{i_{r}})  (6)

This composition is strictly associative with unit given by Id_{A} and defines a category A_{∞}Cat of A_{∞}-categories over 𝕂.

A differential graded category (dg-category) over a field 𝕂 is a category enriched over the symmetric monoidal category (Ch^{•}(𝕂), ⨂, 𝕂) of cochain complexes and a dg-functor is a functor of enriched categories (see [Ke] for the notion of enriched category). In particular dg-categories are identified with A_{∞}-categories having m_{k} = 0, for k > 2, and dg-functors with A_{∞}-functors having f_{k} = 0, for k > 1. This means that there exists a faithful functor

i : dgCat [figure] A_{∞}Cat  (7)

where dgCat is the category of differential graded categories over 𝕂.

For a dg-category D its underlying 𝕂-linear category D_{un} has the same objects of D and morphisms

Hom_{D_{un}}(x, y) = Ker(d_{0} : Hom_{D}^{0}(x, y) [figure] Hom_{D}^{1}(x, y))

We refer to a morphism in a dg-category, without specifying the degree, meaning a morphism in D_{un} and we say that a dg-category D is a dg-enhancement of a 𝕂-linear category V if there exists an isomorphism V ≃ D_{un}. The homotopy category H^{0}(D) of D is the 𝕂-linear category with the same objects of D and vector space of morphisms given by:

Hom_{H^{0}(D)}(x, y) = H^{0}(Hom_{D}^{•}(x, y))

A dg-functor f : D [figure] E is a quasi-equivalence of dg-categories if the induced functor on the homotopy categories H^{0}(f) : H^{0}(D) [figure] H^{0}(E) is an equivalence and the induced morphisms of cochain complexes f : Hom_{D}^{•}(x, y) [figure] Hom_{E}^{•}(f(x), f(y)) are quasi-isomorphisms of complexes, for every x, y ∈ Ob(D).

2.5. Construction of the nerve.

2.6. Definition. The A_{∞}-category (in fact dg-category) A_{∞}[Δ^{n}] generated by the standard n-simplex is the A_{∞}-category whose objects are the integers {0, 1, …, n}, morphisms spaces given, for 0 ≤ i, j ≤ n, by

Hom_{A_{∞}[Δ^{n}]}^{•}(i, j) = \{ \begin{matrix}𝕂 · (i, j) & i ≤ j \\ ∅ & i > j \end{matrix}

with deg((i, j ))=0 and A_{∞}-structure determined by the maps

\{ \begin{matrix}m_{1} = 0 \\ m_{2}((j, k), (i, j)) = (i, k), & for i ≤ j ≤ k \\ m_{n} = 0, & for n > 2 \end{matrix}

with identities 1_{i} = (i, i) ∈ Hom_{A_{∞}[Δ^{n}]}^{•}(i, i), for i = 0, …, n.

2.7. Proposition. The construction [n] [figure] A_{∞}[Δ^{n}] defines a cosimplicial A_{∞}-category

A_{∞}[Δ^{−}] : Δ [figure] A_{∞}Cat

Proof. Consider the standard cofaces and codegeneracies morphisms in Δ

δ_{j}^{n} : [n − 1] [figure] [n],  0 ≤ j ≤ n
σ_{j}^{n} : [n] [figure] [n − 1],  0 ≤ j ≤ n − 1

with

δ_{j}^{n}(k) = \{ \begin{matrix}k, & 0 ≤ k ≤ j − 1 \\ k + 1, & j ≤ k ≤ n − 1 \end{matrix}

σ_{j}^{n}(k) = \{ \begin{matrix}k, & 0 ≤ k ≤ j \\ k − 1, & j + 1 ≤ k ≤ n \end{matrix}

The induced cofaces A_{∞}-functors

(δ_{j}^{n})_{⋆} : A_{∞}[Δ^{n−1}] [figure] A_{∞}[Δ^{n}]

are defined by

\{ \begin{matrix}(δ_{j}^{n})_{⋆,0}(k) = δ_{j}^{n}(k) \\ (δ_{j}^{n})_{⋆,1}(i, k) = (δ_{j}^{n}(i), δ_{j}^{n}(k)) \\ (δ_{j}^{n})_{⋆,p} = 0, & p > 2 \end{matrix}

Similarly the codegeneracies A_{∞}-functors

(σ_{j}^{n})_{⋆} : A_{∞}[Δ^{n}] [figure] A_{∞}[Δ^{n−1}]

are

\{ \begin{matrix}(σ_{j}^{n})_{⋆,0}(k) = σ_{j}^{n}(k) \\ (σ_{j}^{n})_{⋆,1}(i, k) = (σ_{j}^{n}(i), σ_{j}^{n}(k)) \\ (σ_{j}^{n})_{⋆,p} = 0, & p > 2 \end{matrix}

The fact that this assignment determines a cosimplicial A_{∞}-category follows from the standard cosimplicial structure of Δ^{−}.

2.8. Definition. [Simplicial Nerve of an A_{∞}-category] For an A_{∞}-category A its simplicial nerve N_{A_{∞}}(A) is the simplicial set whose n-simplices are given by

N_{A_{∞}}(A)_{n} = Hom_{A_{∞}Cat}(A_{∞}[Δ^{n}], A)

and simplicial structure induced by applying the functor Hom_{A_{∞}Cat}(−, A) to the cosimplicial A_{∞}-category A_{∞}[Δ^{−}].

2.9. Proposition. An n-simplex of the simplicial nerve of an A_{∞}-category A is determined by n + 1 objects

x_{i} ∈ Ob(A), i = 0, …, n

and by a collection of elements

f_{i_{0}...i_{k}} ∈ Hom_{A}^{1−k}(x_{i_{0}}, x_{i_{k}})

38  GIOVANNI FAONTE

for 1 ≤ k ≤ n and 0 ≤ i_{0} < i_{1} < ⋯ < i_{k} ≤ n, satisfying the conditions

f_{i_{0},i_{0}} = Id_{x_{i_{0}}}

f_{i_{0},...,i_{p},i_{p},...,i_{l}} = 0, for 2 ≤ l ≤ n

m_{1}(f_{i_{0}...i_{k}}) = ∑_{0<j<n}(−1)^{j−1}f_{i_{0}...î_{j}...i_{k}} + ∑_{0<j<n}(−1)^{1+k(j−1)}m_{2}(f_{i_{j}...i_{k}}, f_{i_{0}...i_{j}})+

+ ∑_{\begin{matrix}1≤r≤n \\ 0<j_{1}<···<j_{r−1}<n \end{matrix}}(−1)^{1+ϵ_{r}}m_{r}(f_{i_{j_{r−1}}...i_{k}}, …, f_{i_{0}...i_{j_{1}}})

where ϵ_{r} as in equation (5). Proof. An n-simplex f ∈ N_{A_{∞}}(A)_{n} is by definition an A_{∞}-functor

f : A_{∞}[Δ^{n}] [figure] A

This determines n + 1 objects of

x_{i} = f_{0}(i) ∈ Ob(A)

and for 1 ≤ k ≤ n a morphism of degree 1 − k

f_{k} : Hom_{A_{∞}[Δ^{n}]}^{•}(x_{i_{k−1}}, x_{i_{k}}) ⊗ ⋯ ⊗ Hom_{A_{∞}[Δ^{n}]}^{•}(x_{i_{0}}, x_{i_{1}}) [figure] Hom_{A}^{•}(f_{0}(x_{i_{0}}), f_{0}(x_{i_{k}})) which corresponds to the choice, for every string 0 ≤ i_{0} < i_{1} < ⋯ < i_{k} ≤ n, of an element

f_{i_{0}...i_{k}} = f_{k}((i_{k−1}, i_{k}) ⊗ ⋯ ⊗ (i_{0}, i_{1})) ∈ Hom_{A}^{1−k}(x_{i_{0}}, x_{i_{k}})

The conditions satisfied by f_{i_{0}...i_{k}} follow from Definition 2.3. 2.10. Corollary. For a dg-category D its differential graded nerve [Lur2, §1.3] equals the simplicial nerve of i(D)

N_{dg}^{sm}(D) = N_{A_{∞}}(i(D))

where i is defined in Remark 7. Proof. The proof follows from Proposition 2.9.

2.11. Proposition. The simplicial structure of N_{A_{∞}}(A) can be described as follows: for f ∈ N_{A_{∞}}(A)_{n} the components of j-th face map d_{j}^{n}(f) are given by

d_{j}^{n}(f)_{i_{0}...i_{k}} = \{ \begin{matrix}f_{i_{0}...i_{p−1}(i_{p}+1)...(i_{k}+1)}, & j ≤ i_{p}, 0 ≤ p ≤ k \\ f_{i_{0}...i_{k}}, & j > i_{k} \end{matrix}

for 1 ≤ k ≤ n − 1 and a string 0 ≤ i_{0} < i_{1} < ⋯ < i_{k} ≤ n − 1. The components of the j-th degeneracy map s_{j}^{n}(f) are

s_{j}^{n}(f)_{i_{0}i_{1}} = \{ \begin{matrix}f_{(i_{0}−1)(i_{1}−1)}, & j ≤ i_{0} − 1 \\ f_{i_{0}(i_{1}−1)}, & i_{0} < j < i_{1} − 1 \\ Id_{x_{i_{0}}}, & i_{0} = j, i_{1} = j + 1 \\ f_{i_{0}i_{1}}, & j ≥ i_{1} \end{matrix}  (8)

for 0 ≤ i_{0} < i_{1} ≤ n + 1 and

s_{j}^{n}(f)_{i_{0}...i_{k}} = \{ \begin{matrix}f_{(i_{0}−1)...(i_{k}−1)}, & j ≤ i_{0} − 1 \\ f_{i_{0}...i_{p}(i_{p+1}−1)...(i_{k}−1)}, & i_{p} < j < i_{p+1} − 1, 0 < p < k \\ 0, & i_{p} = j, i_{p+1} = j + 1 \\ f_{i_{0}...i_{k}}, & j ≥ i_{k} \end{matrix}  (9)

for 2 ≤ k ≤ n + 1, 0 ≤ i_{0} < i_{1} < ⋯ < i_{k} ≤ n + 1.

Proof. The j-th face map

d_{j}^{n} : N_{A_{∞}}(A)_{n} [figure] N_{A_{∞}}(A)_{n−1}

evaluated on an n-simplex f ∈ N_{A_{∞}}(A) is by definition the A_{∞}-functor

d_{j}^{n}(f) = f ∘ (δ_{j}^{n})_{⋆}

The composition law for A_{∞}-functors (see Equation (6)) gives

(f ∘ (δ_{j}^{n})_{⋆})_{k} = f_{k}((δ_{j}^{n})_{⋆,1} ⊗ ⋯ ⊗ (δ_{j}^{n})_{⋆,1})

and hence, for 1 ≤ k ≤ n − 1 and a string 0 ≤ i_{0} < i_{1} < ⋯ < i_{k} ≤ n − 1

d_{j}^{n}(f)_{i_{0}...i_{k}} = \{ \begin{matrix}f_{i_{0}...i_{p−1}(i_{p}+1)...(i_{k}+1)}, & j ≤ i_{p}, 0 ≤ p ≤ k \\ f_{i_{0}...i_{k}}, & j > i_{k} \end{matrix}

A similar computation shows that the j-th degeneracy map is determined by the formulas (8) and (9).

2.12. Proposition. The simplicial nerve construction defines a functor

N_{A_{∞}} : A_{∞}Cat [figure] SSet

Proof. Any A_{∞}-functor g : A [figure] B induces a map of simplicial sets (g)_{⋆} : N_{A_{∞}}(A) [figure] N_{A_{∞}}(B) by the assignment

(g)_{⋆}(f) = f ∘ g

where f ∈ Hom_{A_{∞}Cat_{𝕂}}(A_{∞}[Δ^{n}], A) is an n-simplex in N_{A_{∞}}(A). More explicitly, if 1 ≤ k ≤ n and 0 ≤ i_{0} < i_{1} < ⋯ < i_{k} ≤ n, we have from Equation (6)

((g)_{⋆}(f))_{i_{0}...i_{k}} = ∑_{r=1}^{k} ∑_{j_{1}+···+j_{r}=k}(−1)^{ϵ_{r}(j_{1},...,j_{r})}g_{r}(f_{i_{jr+···+j_{2}}...i_{k}}, …, f_{i_{0}...i_{jr}})

The functoriality of N_{A_{∞}} hence follows from its definition.

2.13. Definition. [Lur1] An (∞, 1)-category (or weak Kan complex) is a simplicial set X such that, for any 0 < p < n and any map of simplicial sets f : Λ_{p}^{n} [figure] X, there exists an extension to the full n-simplex g : Δ^{n} [figure] X

Λ_{p}^{n} X Δ^{n} if g

where Λ_{p}^{n} is the p-th inner horn.

2.14. Remark. A simplicial category is a category enriched over the symmetric monoidal category (SSet, ×, pt) of simplicial sets where the monoidal structure is the point-wise cartesian product of simplices. Simplicial categories can be related to simplicial sets through a pair of adjoint functors

N_{SCat}: SCat ⇄ SSet :C[−]  (10)

where SCat is the category of simplicial categories with the obvious notion of morphisms and the functor N_{SCat} is called the nerve of a simplicial category. This adjunction lifts to a Quillen adjunction of model categories between the Kan model structure on SCat and the Joyal model structure on SSet whose fibrant objects are (∞, 1)-categories. We refer to [Lur1, §2.2] for a more detailed discussion about this construction and for the definition of the homotopy category of an (∞, 1)-category.

2.15. Proposition. For an A_{∞}-category A its simplicial nerve N_{A_{∞}}(A) is an (∞, 1)- category.

Proof. Consider a morphism of simplicial sets f : Λ_{p}^{n} [figure] N_{A_{∞}}(A), where n > 0 and 0 < p < n are fixed. Such morphism can be identified with an n-simplex of N_{A_{∞}}(A) whose components f_{0...n} and f_{0...p̂...n} are not given. The morphism g : Δ^{n} [figure] N_{A_{∞}}(A) defined by

g_{0...n} = 0

g_{0...p̂...n} = ∑_{0<j<n,j≠p}(−1)^{j−1+p}f_{0...ĵ...n} + ∑_{0<j<n}(−1)^{1+n(j−1)+p}f_{j...n} ∘ f_{0...j}+

+ ∑_{\begin{matrix}1≤r≤n \\ 0<j_{1}<···<j_{r−1}<n \end{matrix}}(−1)^{1+ϵ_{r}(j_{1},...,j_{r−1})+p}m_{r}(f_{i_{j_{r−1}}...i_{k}}, …, f_{i_{0}...i_{j_{1}}})

g_{|Λ_{p}^{n}} = f

provides an extension of f.

3. Comparison between pretriangulated dg-categories and stable (∞, 1)- categories

3.1. Pretriangulated dg-categories.

3.2. Definition. A zero object of a dg-category D is an object 0 ∈ Ob(D) such that for every X ∈ Ob(D)

Hom_{D}^{•}(X, 0) = 0^{•} = Hom_{D}^{•}(0, X)

where 0^{•} is the cochain complex having 0 in each degree and zero differential.

3.3. Definition. [dg-category of twisted complexes, [Bo-Ka]] Let D be a dg-category with a zero object 0. A twisted complex of D is the data consisting of a pair K = (K_{i}, q_{ij})_{i,j∈ℤ} where:

• K_{i} ∈ Ob(D) are equal to 0 for all but a finite number of indices i ∈ ℤ

• q_{ij} ∈ Hom_{D}^{i−j+1}(K_{i}, K_{j}) are morphisms satisfying the Maurer-Cartan equation

d(q_{ij}) + ∑_{k∈ℤ} q_{kj}q_{ik} = 0

for every i, j ∈ ℤ.

The dg-category P reT r(D) of twisted complexes of D is the differential category whose objects are twisted complexes of D and cochain complex of morphisms

Hom_{P reT r(D)}^{k}(K, K′) = ⨁_{l+j−i=k}Hom_{D}^{l}(K_{i}, K′_{j})

with differential d evaluated on f ∈ Hom_{D}^{l}(K_{i}, K′_{j}) given by the expression

d(f) = d(f) + ∑_{m}(q′_{jm}f + (−1)^{l(i−m+1)}fq_{mi})

42  GIOVANNI FAONTE

3.4. Remark. For a dg-category D the category of dg-functors dgF un(D, Ch^{•}(𝕂)) has a dg-enhancement (as described in [Bo-Ka]) in such a way that the diagram of dg-functors

D dgF un(D, Ch^{•}(V ect_{k})) P reT r(D) ϵα h

is commutative. Here h is the dg-Yoneda functor defined on objects by

h(X)(Y) = Hom_{D}^{•}(X, Y)

ϵ is the dg-functor sending an object X ∈ Ob(D) to the twisted complex concentrated in degree 0 and α is the dg-functor that associates to a twisted complex K = (K_{i}, q_{ij})_{i,j∈ℤ} the dg-functor

α(K)(Y) = ⨁_{i∈ℤ} Hom_{D}^{•}(Y, K_{i})[−i]

with twisted differential d+q. Moreover both dg-categories P reT r(D) and dgF un(D, Ch^{•}(𝕂)) have shift functors and functorial cones which are preserved under the dg-functor α. For a twisted complex K its shift by 1 is given explicitly by the formula

K[1]_{i} = K_{i+1}
q[1]_{ij} = q_{i+1,j+1}

and for a morphism of twisted complexes f : K [figure] K′ its cone is the twisted complex

Cone(f) = (K_{i+1} ⨁ K′_{i}, q′′_{ij})

where q′′_{ij} is the matrix

q′′_{ij} = |\begin{matrix}q_{i+1,j+1} & f_{i+1,j} \\ 0 & q′_{ij} \end{matrix}|

This definition of shift functor and cone construction determines a triangulated structure on the homotopy category H^{0}(P reT r(D)) [Bo-Ka] where exact triangles are given by sequences in H^{0}(P reT r(D)) of the form

K K′ Cone(f) K[1] f

3.5. Definition. [Pretriangulated dg-category, [Bo-Ka]] A dg-category D is called pretriangulated if for every twisted complex K ∈ P reT r(D), the dg-functor α(K) is isomorphic to h(X) for some object X ∈ D.

3.6. Remark. If D is a pretriangulated dg-category the dg-functor ϵ : D [figure] P reT r(D) is a quasi-equivalence of dg-categories [Bo-Ka]. In this situation it is possible to transfer the shift functor and the cone construction of P reT r(D) to D. Namely the shift by +1 of X ∈ Ob(D) is defined as

X[1] = T (ϵ(X)[1])  (11)

where T is the inverse equivalence of ϵ and T (ϵ(X)[1]) is an object of D representing the dg-functor α(T (ϵ(X)[1])). Similarly the cone of a morphism f : X [figure] Y is

Cone(f) = T (Cone(ϵ(f)))

Moreover there exist canonical quasi-isomorphisms of complexes

Hom_{D}^{•}(X[1], Y) ≃ Hom_{D}^{•}(X, Y [−1]) ≃ Hom_{D}^{•}(X, Y)[−1]

and for a morphism f : X [figure] Y one has

Hom_{D}^{k}(Cone(f), Z) = Hom_{D}^{k}(Y, Z) ⊕ Hom_{D}^{k−1}(X, Z)  (12)

with differential

d^{k} = |\begin{matrix}d_{(Y,Z)}^{k} & 0 \\ (−1)^{k}(− ∘ f) & d_{(X,Z)}^{k−1} \end{matrix}|

where d_{(X,Y)}^{k} = d_{Hom_{D}^{k}(X,Y)}, and

Hom_{D}^{k}(Z, Cone(f)) = Hom_{D}^{k}(Z, Y) ⊕ Hom_{D}^{k+1}(Z, X)  (13)

with differential

d^{k} = |\begin{matrix}d_{(Z,Y)}^{k} & (− ∘ f) \\ 0 & d_{(Z,X)}^{k+1} \end{matrix}|

3.7. Proposition. [Bo-Ka] The homotopy category H^{0}(D) of a pretriangulated dg-category D is triangulated in the sense of [Ver] with shift functor

[1] : H^{0}(D) [figure] H^{0}(D)

defined on objects by Equation (11) and class of exact triangles of the form

K K′ Cone(f) K[1] f

where f is a morphism of twisted complexes. Moreover the functor

H^{0}(T) : H^{0}(P reT r(D)) [figure] H^{0}(D)

is an equivalence of triangulated categories.

3.8. Stable (∞, 1)-categories.

3.9. Definition. An (∞, 1)-category X is pointed if there exists an object 0 ∈ X_{0} called the zero object such that for every A ∈ X_{0}

Map_{X}(A, 0) ≃ ∗ ≃ Map_{X}(0, A)

where Map_{X}(−, −) is the Kan complex of morphisms in X (see [Lur1, §2.2]).

3.10. Definition. A triangle in a pointed (∞, 1)-category X is a diagram Δ^{1} ×Δ^{1} [figure] X of the form

\begin{matrix}C \\ B \\ 0 \\ A \\ g \\ f \end{matrix}

We say that a triangle is a fiber sequence (the fiber of g) if it is homotopy cartesian and is a cofiber sequence (the cofiber of f) if it is homotopy cocartesian.

3.11. Definition. [Stable (∞, 1)-category, [Lur2]] An (∞, 1)-category is stable if

• It is pointed

• Every morphism admits fiber and cofiber

• A triangle is a fiber sequence if and only if it is a cofiber sequence

3.12. Remark. A stable (∞, 1)-category X has canonical constructions of the suspension and loop functors

Σ, Ω : X [figure] X

which are equivalences of (∞, 1)-categories [Lur2, Chap. 1]. Explicitly these functors are given on objects by

Σ(A) = 0 ⨿_{A}^{h} 0
Ω(A) = 0 ×_{A}^{h} 0

Moreover its homotopy category h(X) is an additive category and it comes equipped with a notion of distinguished triangles which are diagrams of the form

\begin{matrix}A \\ B \\ C \\ A[1] = ΣA \\ f \\ g \\ h \end{matrix}

induced by a diagram Δ^{1} × Δ^{2} [figure] X

D 0C B 0′ A h̃ g̃ f̃

where • 0 and 0’ are both zero objects

• Both squares are pushout diagrams in X

• The morphisms f̃ an g̃ represent f and g respectively

• The map h is the composition with the homotopy class of h̃ with an equivalence D ≃ A[1].

3.13. Proposition. [Lur2] The homotopy category h(X) of a stable (∞, 1)-category X is a triangulated category in the sense of [Ver] with shift functor induced by the suspension functor

Σ : h(X) [figure] h(X)

and class of distinguished triangles as described in Remark 3.12.

3.14. The Dold-Kan correspondence and computation of homotopy limits. Recall that the category S(V ect_{𝕂}) of simplicial vector spaces over 𝕂 has a model structure called the Quillen model structure [Qui]. Weak-equivalences and fibrations of simplicial vector spaces are morphisms inducing weak-homotopy equivalences and Kan fibrations on the underlying simplicial sets. The category Ch_{•}^{≥0}(𝕂) of non-negatively graded chain complexes has a model structure called the projective model structure [Qui]. Weakequivalences are quasi-isomorphisms of chain complexes and fibrations are chain maps which are epimorphisms in each positive degree. Cofibrant objects in this model structure are retract of complexes of projective 𝕂-modules. The Dold-Kan correspondence

DK: Ch_{•}^{≥0}(𝕂) ⇄ S(V ect_{𝕂}) :N  (14)

establishes an equivalence of categories between the category Ch_{•}^{≥0}(𝕂) of positively graded chain complexes over 𝕂 and the category S(V ect_{𝕂}) [McL]. According to S. Schwede and B. Shipley [Sch-Shi] these functors are both left and right adjoint of a Quillen adjunction between the Quillen model structure on S(V ect_{𝕂}) and the projective model structure on Ch_{•}^{≥0}(V ect_{𝕂}). Moreover homology groups and homotopy groups are identified under this correspondence.

Recall that for a model category C and a Reedy indexing category I, the Reedy model structure on the category of functors F un(I, C) (or I-shaped diagrams in C) is the model structure for which a morphism f : X [figure] Y is a weak-equivalence if it is an object-wise weak-equivalence in C, a cofibration if the relative latching morphisms at every object of I are cofibrations in C and a fibration if the relative matching morphisms at every object of I are fibrations in C. Quillen equivalences of model categories induce Quillen equivalences in the respective categories of functors with the Reedy model structures [Hir]. We say that C has I-shaped limits if the functor

(−)_{const.} : C [figure] F un(I, C)

taking an object x ∈ C to the constant functor with value x, has a right adjoint

lim : F un(I, C) [figure] C

If C has I-shaped limits the total right derived functor of lim exists [Hir] and defines the homotopy limit functor

holim : h(F un(I, C)) [figure] h(C)

given explicitly on an object X ∈ h(F un(I, C)) by

holim(X) = ℝ(lim)(X) = lim(P (X))

where P (X) is the fibrant replacement for X in the Reedy model structure on F un(I, C). These observations lead to the following lemma.

3.15. Lemma. Let I be a Reedy indexing category and consider the functor

DK_{∗} : F un(I, Ch_{•}^{≥0}(𝕂)) [figure] F un(I, S(V ect_{𝕂}))

defined by applying object-wise the functor DK of (14) to an I-shaped diagram in Ch_{•}^{≥0}(𝕂). For X ∈ F un(I, Ch_{•}^{≥0}(𝕂)) we have

holim(DK_{∗}(X)) ≃ DK(lim(P (X)))

where P (X) is the fibrant replacement of X in the Reedy model category on F un(I, Ch_{•}^{≥0}(𝕂)).

Proof. By definition we have

holim(DK_{∗}(X)) ≃ lim(P (DK_{∗}(X)))

The functor DK is the right and left adjoint of a Quillen equivalence [Sch-Shi] hence

P (DK_{∗}(X)) ≃ DK_{∗}(P (X))

Moreover DK preserves limits which implies that

lim(P (DK_{∗}(X))) ≃ lim(DK_{∗}(P (X))) ≃ DK(lim(P (X)))

3.16. Example. We give an explicit construction of the fibrant replacement in the Reedy model structure in order to compute homotopy limits when the indexing category I is the category

0 1 2

Given f : X [figure] Y a morphism in F un(I, C) it is easy to check that relative matching morphisms are

M_{0}(f) : X_{0} [figure] Y_{0}

and for i = 1, 2

M_{i}(f) : X_{i} [figure] Y_{i} ×_{Y_{0}} X_{0}

In particular a morphism f is a fibrations if the morphisms M_{i}(f) are fibrations in C, for i = 0, 1, 2, and an object X ∈ F un(I, C) is a fibrant object if X_{0} is a fibrant object in C and the morphisms X_{1} [figure] X_{0}, X_{2} [figure] X_{0} are fibrations. Hence a fibrant replacement of a diagram

\begin{matrix}X_{0} \\ X_{1} \\ X_{2} \\ f_{10} \\ f_{20} \end{matrix}

where X_{0} fibrant is given by

\begin{matrix}X_{0} \\ P (f_{10}) \\ P (f_{20}) \end{matrix}

where

\begin{matrix}X_{i} \\ X_{0} \\ P (f_{i0}) \\ ≃ \\ f_{i1} \end{matrix}

is a trivial cofibration-fibration factorization of the morphisms f_{i0}, i = 1, 2. In particular we get the following expression for the homotopy fibre product of X

X_{2} ×^{h}_{X_{0}} X_{1} := holim(X) ≃ P (f_{20}) ×_{X_{0}} P (f_{10})

When C = Ch_{•}^{≥0}(𝕂) with the projective model structure a trivial cofibration-fibration factorization of a chain map f_{•} : A_{•} [figure] B_{•} is given by

\begin{matrix}A_{•} \\ B_{•} \\ P (f_{•}) \\ i \\ p \\ f_{•} \end{matrix}

where P (f_{•}) is the chain complex in degree n > 0

P (f_{•})_{n} = A_{n} ⨁ B_{n+1} ⨁ B_{n}

and in degree 0

P (f_{•})_{0} = A_{0} ⨁ B_{1} ⨁ D_{0}

with D_{0} ⊆ B_{0} defined by the equation b_{0} = d(b_{1}) + f_{0}(a_{0}), with b_{1} ∈ B_{1} and a_{0} ∈ A_{0}. The differential is

d_{n} = |\begin{matrix}d_{A_{n}} & 0 & 0 \\ −f_{n} & −d_{B_{n+1}} & Id_{B_{n}} \\ 0 & 0 & d_{B_{n}} \end{matrix} |

and the morphisms i and p are

i_{n} = |Id_{A_{n}} 0 f_{n}|

p_{n} = |\begin{matrix}0 \\ 0 \\ Id_{B_{n}} \end{matrix} |

One can easily check that H_{∗}(P (f_{•})) ≃ H_{∗}(A_{•}) and that p is a fibration, being degree-wise surjective. The fact that i is a cofibration follows from the fact that every 𝕂-vector space is free and hence projective.

3.17. Pretriangulated dg-categories are stable (∞, 1)-categories under the nerve construction.

3.18. Theorem. For a pretriangulated dg-category D the dg-nerve N_{dg}(D) is a stable (∞, 1)-category. Moreover H^{0}(D) is identified with h(N_{dg}(D)) as triangulated categories.

Proof. Recall that the big dg-nerve N_{dg}^{big}(D) is the (∞, 1)-category defined as the nerve of the simplicial category D_{Δ} (see Remark 2.14) having the same objects of D and simplicial set of morphisms given by

Map_{D_{Δ}}(X, Y) := DK(τ_{≥0}(Hom_{D}(X, Y)^{op}))

where DK is the functor of the Dold-Kan correspondence (14), τ_{≥0} and (−)^{op} are the functors defined in Section 1.1. The big dg-nerve and the dg-nerve are equivalent (∞, 1)- categories [Lur2, §1.3] hence it is enough to show that the big dg-nerve is a stable (∞, 1)- category. Let 0 be the zero object of D then

Map_{D_{Δ}}(X, 0) ≃ ∗ ≃ Map_{D_{Δ}}(0, X)

for every object X of D hence N_{dg}^{big}(D) is pointed. For a 1-simplex f : X [figure] Y in N_{dg}^{big}(D) we show that it admits fiber and cofiber. Consider the case of the cofiber first. Let j : Y [figure] Cone(f) be the degree 0 morphism corresponding to (Id_{Y} , 0) according to the equality (see Equation (13))

Hom_{D}^{0}(Y, Cone(f)) = Hom_{D}^{0}(Y, Y) ⊕ Hom_{D}^{1}(Y, X)

This is a closed morphism because

d(Id_{Y} , 0) = (d(Id_{Y}) + f ∘ 0, d(0)) = (0, 0)

hence j identifies a 1-simplex of N_{dg}^{big}(D). The composition j ∘ f is null-homotopic in the sense that for

h = (0, Id_{Y}) ∈ Hom_{D}^{−1}(Y, Cone(f)) = Hom_{D}^{−1}(Y, Y) ⊕ Hom_{D}^{0}(Y, X)

we have

d(h) = (d(0) + f ∘ Id_{Y} , d(0)) = (f, 0) = j ∘ f

These data determines a triangle in N_{dg}^{big}(D)

\begin{matrix}X \\ Y \\ 0 \\ Cone(f) \\ j \\ 0 \\ f \\ 0 \end{matrix}  (15)

In order to show that Cone(f) is the cofiber of f we need to construct a weak-equivalence

Map_{D_{Δ}}(Cone(f), Z) [figure] Map_{D_{Δ}}(Y, Z) ×^{h}_{Map_{D_{Δ}}(X,Z)} ∗  (16)

for every object Z ∈ Ob(D). According to Lemma 3.15 this is equivalent to exhibit a quasi-isomorphism of chain complexes

τ_{≥0}(Hom_{D}(Cone(f), Z)^{op}) [figure] τ_{≥0}(Hom_{D}(Y, Z)^{op}) ×^{h}_{τ_{≥0}(Hom_{D}(X,Z)^{op})} 0  (17)

Following Example 3.16 the right hand side of Equation (17) can be identified with the fibre product

P (− ∘ f) ×_{τ_{≥0}(Hom_{D}^{•}(X,Z)^{op})} P (0)

where the chain complex P (− ∘ f) is

\begin{cases}P (− ∘ f)_{0} = Ker(d_{Hom_{D}^{0}(Y,Z)}) ⊕ Hom_{D}^{−1}(X, Z) ⊕ D^{0}, & if k = 0 \\ P (− ∘ f)_{k} = Hom_{D}^{−k}(Y, Z) ⊕ Hom_{D}^{−k−1}(X, Z) ⊕ Hom_{D}^{−k}(X, Z), & if k > 0. \end{cases}

with D^{0} ⊆ Hom_{D}^{0}(X, Z) defined by the equation g^{0} = h^{0}∘f+d(g^{−1}), for h^{0} ∈ Ker(d_{Hom_{D}^{0}(Y,Z)}) and g^{−1} ∈ Hom_{D}^{−1}(X, Z). The differential for k > 0 is

d_{k} = |\begin{matrix}d_{(Y,Z)}^{−k} & 0 & 0 \\ −(− ∘ f) & −d_{(X,Z)}^{−k−1} & Id_{(X,Z)} \\ 0 & 0 & d_{(Y,Z)}^{−k} \end{matrix}|

and for k = 0

d_{0} = |\begin{matrix}d_{(Y,Z)}^{0} & 0 & 0 \\ −(− ∘ f) & −d_{(X,Z)}^{−1} & Id_{(X,Z)} \\ 0 & 0 & d_{(Y,Z)}^{0} \end{matrix}|

where d_{(X,Y)}^{k} = d_{Hom_{D}^{k}(X,Y)}. The chain complex P (0) is

\begin{cases}P (0)_{0} = Hom_{D}^{−1}(X, Z) ⊕ Im(d_{Hom_{D}^{−1}(X,Z)}), & if k = 0 \\ P (0)_{k} = Hom_{D}^{−k−1}(X, Z) ⊕ Hom_{D}^{−k}(X, Z), & if k > 0. \end{cases}

with differential for k > 0

d_{k} = |\begin{matrix}−d_{(X,Z)}^{−k−1} & Id_{(X,Z)} \\ 0 & d_{(Y,Z)}^{−k} \end{matrix}|

and for k = 0

d_{0} = |\begin{matrix}−d_{(X,Z)}^{−1} & Id_{(X,Z)} \\ 0 & d_{(Y,Z)}^{0} \end{matrix}|

Hence the right hand side of Equation (17) is computed by the cochain complex having degree k > 0 component

Hom_{D}^{−k}(Y, Z) ⊕ Hom_{D}^{−k−1}(X, Z) ⊕ Hom_{D}^{−k−1}(X, Z) ⊕ Hom_{D}^{−k}(X, Z)

with differential

d_{k} = |\begin{matrix}d_{(X,Z)}^{−k} & 0 & 0 & 0 \\ −(− ∘ f) & −d_{(X,Z)}^{−k−1} & 0 & 0 \\ 0 & 0 & −d_{(X,Z)}^{−k−1} & Id_{(X,Z)} \\ 0 & 0 & 0 & d_{(X,Z)}^{−k} \end{matrix}|

and degree 0 component the subspace of the direct sum P (− ∘ f)_{0} ⊕ P (0)_{0} corresponding to D^{0} ⊕ Im(d_{Hom_{D}^{−1}(X,Z)}) via the inclusion in Hom_{D}^{0}(X, Z). This chain complex is quasiisomorphic to the chain complex whose degree k > 0 component is

Hom_{D}^{−k}(Y, Z) ⊕ Hom_{D}^{−k−1}(X, Z)

with differential

d_{k} = |\begin{matrix}d_{(X,Z)}^{−k} & 0 \\ −(− ∘ f) & −d_{(X,Z)}^{−k−1} \end{matrix}|

and degree 0 component

Ker(d_{Hom_{D}^{0}(Y,Z)}) ⊕ E^{−1}

where E^{−1} ⊆ Hom_{D}^{−1}(X, Z) is defined by the equation d(g^{−1}) + g^{0} ∘ f = 0, for g^{0} ∈ Ker(d_{Hom_{D}^{0}(Y,Z)}). This is quasi-isomorphic to the truncation τ_{≥0}(Hom_{D}^{•}(Cone(f), Z)^{op}) and hence it provides the weak-equivalence of equation (17). For computing the fiber of f, a similar computation shows that the morphism i : Cone(f)[−1] [figure] X corresponding to (Id_{X}, 0) according to the equality (see Equation (12))

Hom_{D}^{0}(Cone(f)[−1], X) = Hom_{D}^{0}(X, X) ⊕ Hom_{D}^{1}(Y, X)

induces and homotopy cartesian triangle in N_{dg}^{big}(D)

\begin{matrix}Cone(f)[1] \\ X \\ 0 \\ Y \\ f \\ 0 \\ j \\ 0 \end{matrix}  (18)

We show now that the diagram (15) is also cartesian. Let h and g be the closed degree 0 morphisms corresponding respectively to (0, π_{X}) and (0, (−f) ⊕ i_{X}) according to the equalities

Hom_{D}^{0}(X, Cone(j)[−1]) = Hom_{D}^{−1}(X, Y) ⊕ Hom_{D}^{0}(X, Y ⊕ X)
Hom_{D}^{0}(Cone(j)[−1], X) = Hom_{D}^{1}(Y, X) ⊕ Hom_{D}^{0}(Y ⊕ X, X)

where π_{X} : Y ⊕ X [figure] X and i_{X} : X [figure] Y ⊕ X are the canonical projection and inclusion morphisms. Let α corresponding to (0, −i_{Y} , 0, 0) according to the identification of Hom_{D}^{−1}(Cone(j)[−1], Cone(j)[−1]) with

Hom_{D}^{−1}(Y, Y) ⊕ Hom_{D}^{0}(Y, Y ⊕ X) ⊕ Hom_{D}^{−2}(Y ⊕ X, Y) ⊕ Hom_{D}^{−1}(Y ⊕ X, Y ⊕ X)

We have that d(α) = g ∘ h − Id_{Cone(j)[−1]} because the differential in degree −1 is given by (see Equations (12) and (13))

d_{−1}(c^{−1}, c^{0}, c^{−2}, d^{−1}) = |\begin{matrix}d(c^{−1}) + (Id_{Y} ⊕ f) ∘ c^{0} \\ d(c^{0}) \\ d(c^{−2}) + c^{−1} ∘ (Id_{Y} ⊕ f) + (Id_{Y} ⊕ f) ∘ d^{−1} \\ d(d^{−1}) + c^{0} ∘ (Id_{Y} ⊕ f) \end{matrix}|

On the other hand h ∘ g = Id_{X} which implies that X and Cone(j)[−1] are homotopy equivalent in N_{dg}^{big}(D). A similar argument shows that the diagram (18) is also cocartesian. The fact that H^{0}(D) is equivalent to h(N_{dg}^{big}(D)) as triangulated categories follows from the arguments in this proof.

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Department of Mathematics, Yale University, 10 Hillhouse Avenue, New Haven CT 06520 USA.

Email: giovanni.faonte@yale.edu
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Transmitting editors.

Clemens Berger, Université de Nice-Sophia Antipolis: cberger@math.unice.fr Richard Blute, Université d’ Ottawa: rblute@uottawa.ca Lawrence Breen, Université de Paris 13: breen@math.univ-paris13.fr Ronald Brown, University of North Wales: ronnie.profbrown(at)btinternet.com Valeria de Paiva: Nuance Communications Inc: valeria.depaiva@gmail.com Ezra Getzler, Northwestern University: getzler(at)northwestern(dot)edu Kathryn Hess, Ecole Polytechnique Fédérale de Lausanne: kathryn.hess@epfl.ch Martin Hyland, University of Cambridge: M.Hyland@dpmms.cam.ac.uk Anders Kock, University of Aarhus: kock@imf.au.dk Stephen Lack, Macquarie University: steve.lack@mq.edu.au F. William Lawvere, State University of New York at Buffalo: wlawvere@buffalo.edu Tom Leinster, University of Edinburgh: Tom.Leinster@ed.ac.uk Ieke Moerdijk, Utrecht University: i.moerdijk@uu.nl Susan Niefield, Union College: niefiels@union.edu Robert Paré, Dalhousie University: pare@mathstat.dal.ca Jiri Rosicky, Masaryk University: rosicky@math.muni.cz Giuseppe Rosolini, Università di Genova: rosolini@disi.unige.it Alex Simpson, University of Ljubljana: Alex.Simpson@fmf.uni-lj.si James Stasheff, University of North Carolina: jds@math.upenn.edu Ross Street, Macquarie University: ross.street@mq.edu.au Walter Tholen, York University: tholen@mathstat.yorku.ca Myles Tierney, Université du Québec à Montréal : tierney.myles4@gmail.com R. J. Wood, Dalhousie University: rjwood@mathstat.dal.ca