# Generation time for biexact functors and Koszul objects in triangulated
categories

Janina C. Letz  Janina C. Letz, Faculty of Mathematics, Bielefeld University,
PO Box 100 131, 33501 Bielefeld, Germany
UCLA Department of Mathematics, PO Box 951555, Los Angeles, CA 90095, United
States<EMAIL_ADDRESS>and  Marc Stephan  Marc Stephan, Paderborn
University, Warburger Strasse 100, 33098 Paderborn, Germany
Faculty of Mathematics, Bielefeld University, PO Box 100 131, 33501 Bielefeld,
Germany<EMAIL_ADDRESS>
###### Abstract.

This paper concerns the generation time that measures the number of cones
necessary to obtain an object in a triangulated category from another object.
This invariant is called level. We establish level inequalities for enhanced
triangulated categories: One inequality concerns biexact functors of
topological triangulated categories, another Koszul objects. In particular,
this extends inequalities for the derived tensor product from commutative
algebra to enhanced tensor triangulated categories. We include many examples.

###### Key words and phrases:

Biexact functor, triangulated category, Verdier structure, Koszul object,
cofibration category

###### 2020 Mathematics Subject Classification:

18G80 (primary), 55U35

##  1\. Introduction

Triangulated categories appear in many areas, such as algebraic topology,
representation theory or commutative algebra. One approach to understand a
triangulated category is to study how objects generate each other. An object
$X$ generates an object $Y$, if $Y$ can be obtained from $X$ by taking cones,
suspensions and retracts.

Following [ABIM10] we call the generation time _level_. Loosely speaking the
$X$-level measures the number of cones required to build objects from $X$. It
generalizes some well-known invariants: When ${\mathcal{T}}={\rm
D}(\operatorname{Mod}(R))$ the derived category of modules over a ring $R$,
then the $R$-level of a module coincides with its projective dimension $+1$;
see for example [Chr98]. For a local ring $R$ with residue field $k$, the
$k$-level of a module coincides with its Loewy length. The $k$-level of a
perfect complex $F$ has been used in [ABIM10] to establish a rank inequality
for the homology of $F$.

Further, level is closely connected to the Rouquier dimension of a
triangulated category introduced in [Rou08]. For triangulated categories with
finite Rouquier dimension, there are some Brown representability theorems; see
[BvdB03, Rou08, Let23]. For many examples of triangulated categories it is
known whether the Rouquier dimension is finite or infinite, but in the former
case the exact value is rarely known. We expect that studying level will help
computing the Rouquier dimension.

Besides the properties following from the definition [ABIM10, Lemma 2.4], not
much is known about the behavior of level. Many estimates for level are rather
rough. For example the transitivity of finite building yields a product
inequality for level. In this paper we give a refinement for biexact functors.
The following inequality is optimal in that equality can be achieved.

######  Theorem A.

Let $\mathsf{F}\colon{\mathcal{S}}\times{\mathcal{T}}\to{\mathcal{U}}$ be a
biexact functor of topological triangulated categories. Then

$\operatorname{level}_{{\mathcal{U}}}^{\mathsf{F}(X,X^{\prime})}(\mathsf{F}(Y,Y^{\prime}))\leq\operatorname{level}_{{\mathcal{S}}}^{X}(Y)+\operatorname{level}_{{\mathcal{T}}}^{X^{\prime}}(Y^{\prime})-1$

for $X,Y\in{\mathcal{S}}$ and $X^{\prime},Y^{\prime}\in{\mathcal{T}}$.

For this result see Theorem 3.6 and Theorem 6.4. Theorem A extends and
generalizes [AIN18, Lemma 2.4] where the inequality is shown for
${\mathcal{S}}={\mathcal{T}}={\mathcal{U}}={\rm D}(\operatorname{Mod}(R))$ the
derived category of a commutative ring $R$, $\mathsf{F}$ the derived tensor
product and $X=X^{\prime}=R$. In Section 7 we provide many examples of biexact
functors and triangulated categories for which the above inequality holds.
These include the tensor product on the derived category of a commutative ring
or a group ring, on the stable module category of a cocommutative Hopf
algebra, and on various categories of spectra.

In the proof of Theorem A we construct the triangles building
$\mathsf{F}(Y,Y^{\prime})$ by taking homotopy pushouts. In a general
triangulated category one does not have enough control over the homotopy
pushout and its compatibilities, since it is neither a honest pushout nor
captures the full homotopical information. For this reason we work with
biexact functors that admit a strong Verdier structure; see [KN02, Theorem
3.5] and [AL23, Definition 3.31]. This extra structure ensures that the
homotopy pushout in $3\times 3$ diagrams induced by applying the biexact
functor to exact triangles in each component has the desired compatibilities.

In practice, many biexact functors of triangulated categories admit a strong
Verdier structure. We show that whenever a triangulated category has an
enhancement and the biexact functor respects the enhancement, the biexact
functor admits a strong Verdier structure. For monoidal products this was
shown by [May01, GPS14a, GŠ18]. We consider general bifunctors. Explicitly, we
prove that for any biexact functor of stable cofibration categories the
induced bifunctor on the homotopy categories is biexact and admits a strong
Verdier structure; see Theorem 6.4. Stable cofibration categories provide a
convenient setting, since their homotopy categories are precisely the
topological triangulated categories [Sch13], and the homotopy theory of
cofibration categories is equivalent to the homotopy theory of finitely
cocomplete $\infty$-categories [Szu17].

We review cofibration categories in Section 5 and combine arguments of May and
Schwede to establish the strong Verdier structure for biexact functors induced
by biexact functors on stable cofibration categories in Section 6. A further
application is a new proof that the homotopy category of any symmetric
monoidal stable model category is tensor triangulated. Moreover, we show that
in addition to the derived monoidal product, the derived internal hom functor
admits a strong Verdier structure as well; see Theorem 6.21.

An essential class of biexact functors on triangulated categories is the class
of actions of a tensor triangulated category
$({\mathcal{S}},\otimes,\mathbbm{1})$ on a triangulated category
${\mathcal{T}}$. If the action is induced by a biexact functor of stable
cofibration categories, we say that the action is _topological_. Any graded
endomorphism of the unit $\mathbbm{1}$ in ${\mathcal{S}}$ induces a natural
transformation on ${\mathcal{T}}$, which is compatible with the suspension. In
fact, this yields a ring homomorphism from the graded endomorphism ring of
$\mathbbm{1}$ in ${\mathcal{S}}$ to the center of ${\mathcal{T}}$.

For a sequence of elements in the center, one can define a Koszul object. This
generalizes the Koszul complex over a ring. For Koszul objects we show the
following level inequality. Again, the bound can be achieved.

######  Theorem B.

Let $\alpha_{1},\ldots,\alpha_{c}$ be elements in the center of a topological
triangulated category ${\mathcal{T}}$ that are induced by a topological action
on ${\mathcal{T}}$. Then

$\operatorname{level}_{{\mathcal{T}}}^{X}(X/\\!\\!/(\alpha_{1},\ldots,\alpha_{c}))\leq
c+1$

for any $X\in{\mathcal{T}}$.

This result is contained in Theorem 4.10 combined with Theorem 6.4. Koszul
objects have been used to obtain bounds of Rouquier dimension in [BIKO10], and
they are connected to support [BIK08, Section 5].

######  Acknowledgements.

We thank Jan Šťovíček for helpful discussions.

The authors were partly supported by the Deutsche Forschungsgemeinschaft (SFB-
TRR 358/1 2023 - 491392403). Letz was also partly supported by the Alexander
von Humboldt Foundation in the framework of a Feodor Lynen research fellowship
endowed by the German Federal Ministry of Education and Research. Part of this
work was done at the Hausdorff Research Institute for Mathematics, Bonn, when
the authors were at the “Spectral Methods in Algebra, Geometry, and Topology”
trimester, funded by the Deutsche Forschungsgemeinschaft under Germany’s
Excellence Strategy–EXC-2047/1–390685813.

##  2\. Homotopy cartesian squares

We fix a triangulated category ${\mathcal{T}}$ with suspension functor
$\Sigma$. We consider ${\mathcal{T}}^{\rm{op}}$ as a triangulated category in
which
$X\xrightarrow{f^{\rm{op}}}Y\xrightarrow{g^{\rm{op}}}Z\xrightarrow{h^{\rm{op}}}\Sigma^{-1}X$
is an exact triangle in ${\mathcal{T}}^{\rm{op}}$ if and only if
$\Sigma^{-1}X\xrightarrow{-h}Z\xrightarrow{-g}Y\xrightarrow{-f}X$ is an exact
triangle in ${\mathcal{T}}$. While this convention is not significant in this
section, it ensures that the internal hom functor of a closed tensor
triangulated category is biexact in Section 6.15.

######  2.1.

Following [Nee01, Definition 1.4.1], we call a commutative square

(2.1.1)
${T}$${V}$${U}$${X}$$\scriptstyle{g}$$\scriptstyle{f}$$\scriptstyle{f^{\prime}}$$\scriptstyle{g^{\prime}}$

_homotopy cartesian_ , if there is an exact triangle

$T\xrightarrow{\left(\begin{smallmatrix}f\\\
-g\end{smallmatrix}\right)}U\oplus
V\xrightarrow{\left(\begin{smallmatrix}g^{\prime}&f^{\prime}\end{smallmatrix}\right)}X\xrightarrow{\partial}\Sigma
T\,.$

We say $\partial$ is a _connecting morphism_ of the homotopy cartesian square;
other sources use the term _differential_.

The homotopy cartesian property is symmetric in that the square Eq. 2.1.1 is
homotopy cartesian if and only if its reflection

${T}$${U}$${V}$${X}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{g^{\prime}}$$\scriptstyle{f^{\prime}}$

is homotopy cartesian. However, if $\partial$ a connecting morphism of the
homotopy cartesian square Eq. 2.1.1, then $-\partial$ is a connecting morphism
of the reflected square.

Further, the homotopy cartesian property is self-dual; that is, a square Eq.
2.1.1 is homotopy cartesian in ${\mathcal{T}}$ if and only if its dual is
homotopy cartesian in ${\mathcal{T}}^{\rm{op}}$.

Homotopy cartesian squares are strongly connected to morphisms of triangles in
which one morphism is the identity morphism.

######  2.2.

Given a morphism of exact triangles

${T}$${V}$${Z}$${\Sigma T}$${U}$${X}$${Z}$${\Sigma U}$$\scriptstyle{=}$

where the first square is homotopy cartesian, a connecting morphism is given
by the composite

$\partial=(X\to Z\to\Sigma T)\,.$

If either the morphism $X\to Z$ or $Z\to\Sigma T$ is not given, it can be
constructed such that the diagram is a morphism of triangles; see [Nee01,
Lemma 1.4.4]. Similarly, if either the morphism $V\to X$ or $T\to U$ is not
given, it can be constructed such that the diagram commutes and the first
square is homotopy cartesian; see [Nee01, Lemma 1.4.3].

While homotopy cartesian squares are in general neither pullback nor pushout
squares, they still satisfy a weakened pasting property.

######  2.3.

Consider two commutative squares and their composition:

  1. (1)

If Eqs. 2.3.1a and 2.3.1b are homotopy cartesian, then Eq. 2.3.1c is homotopy
cartesian; see [CF22, Proposition 6.11].

  2. (2)

If Eqs. 2.3.1a and 2.3.1c are homotopy cartesian, then there exists a morphism
$X\xrightarrow{\tilde{h}^{\prime}}Z$, such that the square

${V}$${Y}$${X}$${Z}$$\scriptstyle{f^{\prime}}$$\scriptstyle{h}$$\scriptstyle{f^{\prime\prime}}$$\scriptstyle{\tilde{h}^{\prime}}$

is homotopy cartesian, and
$h^{\prime}g^{\prime}=\tilde{h}^{\prime}g^{\prime}$; see [SZ19, Lemma 9].

  3. (3)

If Eqs. 2.3.1b and 2.3.1c are homotopy cartesian, then there exists a morphism
$T\xrightarrow{\tilde{g}}V$ such that the square

${T}$${V}$${U}$${X}$$\scriptstyle{\tilde{g}}$$\scriptstyle{f}$$\scriptstyle{f^{\prime}}$$\scriptstyle{g^{\prime}}$

is homotopy cartesian, and $hg=h\tilde{g}$; this is (2) in
${\mathcal{T}}^{\rm{op}}$.

If the squares Eqs. 2.3.1a, 2.3.1b and 2.3.1c are homotopy cartesian, then
there exist connecting morphisms $\partial_{1}$, $\partial_{2}$ and
$\partial_{3}$, respectively, such that

$\partial_{3}h^{\prime}=\partial_{1}\quad\text{and}\quad(\Sigma
f)\partial_{3}=\partial_{2}\,.$

###  2.4. Homotopy pushouts

If Eq. 2.1.1 is homotopy cartesian, we say $X$ is the _homotopy pushout_ of
the span $U\leftarrow T\to V$, and we write $X=U+_{T}V$. A homotopy pushout is
equipped with morphisms $U\to U+_{T}V$ and $V\to U+_{T}V$. Just as the cone of
a morphism, the homotopy pushout is unique up to non-canonical isomorphism; in
fact we have $U+_{T}0=\operatorname{cone}(T\to U)$.

We emphasize that a homotopy pushout is typically not a pushout. In fact given
morphisms $U\to Z$ and $V\to Z$ that coincide after pre-composition with $T\to
U$ and $T\to V$, respectively, with each other, there exists a _non-unique_
morphism $U+_{T}V\to Z$. Later we investigate situations in which there exists
a morphism $U+_{T}V\to Z$ with a cone compatible with a given $3\times 3$
diagram containing the square with $T$, $U$, $V$ and $Z$. We say such
situations “admit a (strong) Verdier structure”; see 3.2.

######  2.5.

We take the homotopy pushout of spans $U\leftarrow S\to V$ and $V\leftarrow
T\to W$, and obtain the commutative diagram

${T}$${W}$${S}$${V}$${V+_{T}W\makebox[0.0pt][l]{\,.}}$${U}$${U+_{S}V}$

In this diagram we complete the square in the lower right corner to a homotopy
cartesian square. Then all rectangles, in particular the horizontal and
vertical one, are homotopy cartesian by 2.3, and we obtain

$(U+_{S}V)+_{T}W=(U+_{S}V)+_{V}(V+_{T}W)=U+_{S}(V+_{T}W)\,.$

That is, the construction of the homotopy pushout is associative.

######  Lemma 2.6.

For any diagram $X\leftarrow U\leftarrow S\rightarrow V\rightarrow Y$ there
exists a morphism $U+_{S}V\to X+_{S}Y$ such that its cone is
$\operatorname{cone}(U\to X)\oplus\operatorname{cone}(V\to Y)$.

###### Proof.

We consider the commutative diagram

${S}$${U\oplus V}$${U+_{S}V}$${\Sigma S}$${S}$${X\oplus Y}$${X+_{S}Y}$${\Sigma
S}$$\scriptstyle{=}$$\scriptstyle{=}$

in which the rows are exact triangles. By 2.2, the dashed arrow exists such
that the diagram is a morphism of exact triangles and the second square is
homotopy cartesian. Then the dashed arrow is the desired morphism and, by 2.2,
it has the desired cone. ∎

Alternatively, we can change the base of the span.

######  Lemma 2.7.

Given $S\to T$ and a span $X\leftarrow T\rightarrow Y$, then there exists a
morphism $X+_{S}Y\to X+_{T}Y$ such that its cone is
$\Sigma\operatorname{cone}(S\to T)$.

###### Proof.

We consider the commutative diagram

${S}$${X\oplus Y}$${X+_{S}Y}$${\Sigma S}$${T}$${X\oplus Y}$${X+_{T}Y}$${\Sigma
T}$$\scriptstyle{=}$

in which the rows are exact triangles. By 2.2, the dashed arrow exists such
that the diagram is a morphism of exact triangles and the third square is
homotopy cartesian. Then the dashed arrow is the desired morphism and, by 2.2,
it has the desired cone. ∎

######  Remark 2.8.

Given a commutative diagram

${T}$${V}$${U}$${X}$

there exists a morphism $U+_{T}V\to X$ such that the compositions $U\to
U+_{T}V\to X$ and $V\to U+_{T}V\to X$ recover the given morphisms. Using 2.2
we obtain homotopy cartesian squares

${U+_{T}V}$${\operatorname{cone}(T\to V)}$${X}$${\operatorname{cone}(U\to X)}$
and ${U+_{T}V}$${\operatorname{cone}(T\to U)}$${X}$${\operatorname{cone}(V\to
X)\makebox[0.0pt][l]{\,.}}$

In particular, this yields that the dashed vertical arrows have the same cone
by 2.2.

In general, morphisms $\operatorname{cone}(T\to V)\to\operatorname{cone}(U\to
X)$ and $\operatorname{cone}(T\to U)\to\operatorname{cone}(V\to X)$ that are
compatible with the commutative square need not have the same cone. In the
next section we discuss a setting in which there is a natural choice for these
morphisms, and there exists a morphism $U+_{T}V\to X$ that is compatible with
this natural choice.

##  3\. Biexact functors between triangulated categories

In this section we show the level inequality for a biexact functor
$\mathsf{F}$. If we apply the biexact functor $\mathsf{F}$ to a triangle in
each component, we obtain a $3\times 3$ diagram in which each row and each
column is an exact triangle. In general, this is not enough to be able to
construct a compatible morphism as discussed in Remark 2.8. This will be
resolved by the notion of a strong Verdier structure.

######  3.1.

Recall that an exact functor $\mathsf{F}\colon{\mathcal{S}}\to{\mathcal{T}}$
of triangulated categories is equipped with a natural isomorphism
$\tau\colon\mathsf{F}\Sigma\to\Sigma\mathsf{F}$ such that for any exact
triangle $X\xrightarrow{f}Y\xrightarrow{g}Z\xrightarrow{h}\Sigma X$ in
${\mathcal{S}}$ the triangle

$\mathsf{F}(X)\xrightarrow{\mathsf{F}(f)}\mathsf{F}(Y)\xrightarrow{\mathsf{F}(g)}\mathsf{F}(Z)\xrightarrow{\tau\mathsf{F}(h)}\Sigma\mathsf{F}(X)$

is exact in ${\mathcal{T}}$.

We call a bifunctor
$\mathsf{F}\colon{\mathcal{S}}\times{\mathcal{T}}\to{\mathcal{U}}$ of
triangulated categories _biexact_ , if it is equipped with natural
isomorphisms

$\vartheta\colon\mathsf{F}(\Sigma-,-)\to\Sigma\mathsf{F}(-,-)\quad\text{and}\quad\zeta\colon\mathsf{F}(-,\Sigma-)\to\Sigma\mathsf{F}(-,-)$

such that

  1. (1)

for any $X\in{\mathcal{S}}$ the functor $\mathsf{F}(X,-)$ with $\zeta(X,-)$ is
exact,

  2. (2)

for any $X^{\prime}\in{\mathcal{T}}$ the functor $\mathsf{F}(-,X^{\prime})$
with $\vartheta(-,X^{\prime})$ is exact, and

  3. (3)

the following square anti-commutes

(3.1.1) ${\mathsf{F}(\Sigma X,\Sigma X^{\prime})}$${\Sigma\mathsf{F}(X,\Sigma
X^{\prime})}$${\Sigma\mathsf{F}(\Sigma
X,X^{\prime})}$${\Sigma^{2}\mathsf{F}(X,X^{\prime})\makebox[0.0pt][l]{\,.}}$$\scriptstyle{\vartheta}$$\scriptstyle{\zeta}$${(-1)}$$\scriptstyle{\Sigma\zeta}$$\scriptstyle{\Sigma\vartheta}$

In [KS06, Definition 10.3.6] such a functor is called a triangulated
bifunctor.

######  3.2.

Following [KN02, Theorem 3.5] and [AL23, Definition 3.31], we say a biexact
functor $\mathsf{F}\colon{\mathcal{S}}\times{\mathcal{T}}\to{\mathcal{U}}$
_admits a Verdier structure_ , if for all exact triangles

(3.2.1) $X\xrightarrow{f}Y\xrightarrow{g}Z\xrightarrow{h}\Sigma
X\quad\text{and}\quad
X^{\prime}\xrightarrow{f^{\prime}}Y^{\prime}\xrightarrow{g^{\prime}}Z^{\prime}\xrightarrow{h^{\prime}}\Sigma
X^{\prime}$

in ${\mathcal{S}}$ and ${\mathcal{T}}$, respectively, there exists an object
$W\in{\mathcal{U}}$ and and exact triangles

(3.2.2)
${\mathsf{F}(X,Y^{\prime})}$${W}$${\mathsf{F}(Z,X^{\prime})}$${\Sigma\mathsf{F}(X,Y^{\prime})}$${W}$${\mathsf{F}(Y,Y^{\prime})}$${\mathsf{F}(Z,Z^{\prime})}$${\Sigma
W}$${\mathsf{F}(Y,X^{\prime})}$${W}$${\mathsf{F}(X,Z^{\prime})}$${\Sigma\mathsf{F}(Y,X^{\prime})\makebox[0.0pt][l]{\,,}}$$\scriptstyle{j}$$\scriptstyle{q}$$\scriptstyle{\vartheta\mathsf{F}(h,f^{\prime})}$$\scriptstyle{i}$$\scriptstyle{\mathsf{F}(g,g^{\prime})}$$\scriptstyle{p}$$\scriptstyle{j^{\prime}}$$\scriptstyle{q^{\prime}}$$\scriptstyle{\zeta{\mathsf{F}(f,h^{\prime})}}$

such that the following diagram commutes

(3.2.3)
${\mathsf{F}(X,X^{\prime})}$${\mathsf{F}(Y,X^{\prime})}$${\mathsf{F}(Z,X^{\prime})}$${\Sigma\mathsf{F}(X,X^{\prime})}$${W}$${W}$${\mathsf{F}(X,Y^{\prime})}$${\mathsf{F}(Y,Y^{\prime})}$${\mathsf{F}(Z,Y^{\prime})}$${\Sigma\mathsf{F}(X,Y^{\prime})}$${W}$${\Sigma
W}$${\mathsf{F}(X,Z^{\prime})}$${\mathsf{F}(Y,Z^{\prime})}$${\mathsf{F}(Z,Z^{\prime})}$${\Sigma\mathsf{F}(X,Z^{\prime})}$${\Sigma
W}$${\Sigma
W}$${(-1)}$${\Sigma\mathsf{F}(X,X^{\prime})}$${\Sigma\mathsf{F}(Y,X^{\prime})}$${\Sigma\mathsf{F}(Z,X^{\prime})}$${\Sigma^{2}\mathsf{F}(X,X^{\prime})\makebox[0.0pt][l]{\,.}}$${\rm(I)}$$\scriptstyle{\rm(iv)}$$\scriptstyle{\rm(iii)}$${\rm(II)}$$\scriptstyle{\rm(ii)}$$\scriptstyle{\rm(i)}$${\rm(IV)}$${\rm(III)}$${\rm(V)}$$\scriptstyle{\rm(vi)}$$\scriptstyle{\rm(v)}$${\rm(VI)}$

In the diagram all rows and columns are the exact triangles obtained by
applying $\mathsf{F}$ to Eq. 3.2.2. The anti-commutativity of the bottom right
square is due to Eq. 3.1.1. The morphisms involving $W$ are those that appear
in the triangles Eq. 3.2.2, or the suspension of those; this means for example
that the morphism $\Sigma W\to\Sigma\mathsf{F}(X,Z^{\prime})$ is the morphism
$\Sigma q^{\prime}$.

Abridged the commutativity of Eq. 3.2.3 means the squares (I–VI) and the
triangles (i–vi) commute.

Further, we say $\mathsf{F}$ admits a _strong_ Verdier structure, if the
squares (I–VI) are homotopy cartesian; for (VI) we mean it is homotopy
cartesian after we replace any one of the morphisms by its negative.

If $\mathsf{F}$ admits a strong Verdier structure, then the connecting
morphisms are given by the appropriate compositions of morphisms in Eq. 3.2.2;
cf. 2.2.

######  Remark 3.3.

If a biexact functor $\mathsf{F}$ admits a (strong) Verdier structure, then
the diagram Eq. 3.2.3 can be extended in any direction by rotation. In
particular, a biexact functor $\mathsf{F}$ admits a (strong) Verdier structure
if and only if $\mathsf{F}(\Sigma-,-)$, or equivalently
$\mathsf{F}(-,\Sigma-)$, does.

A bifunctor $\mathsf{F}\colon{\mathcal{S}}\times{\mathcal{T}}\to{\mathcal{U}}$
of triangulated categories is biexact if and only if its opposite
$\mathsf{F}^{\rm{op}}\colon{\mathcal{S}}^{\rm{op}}\times{\mathcal{T}}^{\rm{op}}\to{\mathcal{U}}^{\rm{op}}$
is biexact. Moreover, a biexact functor $\mathsf{F}$ admits a (strong) Verdier
structure if and only if $\mathsf{F}^{\rm{op}}$ does.

The symmetric monoidal product of a tensor triangulated category is a special
case of a biexact functor. In this situation 3.2 is (TC3) of [May01]. May
provided an informal argument for establishing this condition if the tensor
triangulated category comes from a symmetric monoidal, stable model category
and checked it for certain families of such model categories. In [GPS14a] the
same condition was shown for tensor triangulated categories induced by a
stable monoidal derivator. In Section 6 we show that any biexact functor
between stable cofibration categories induces a biexact functor admitting a
strong Verdier structure on homotopy categories. In particular, this includes
many bifunctors on algebraic triangulated categories; for examples see Section
7.4.

######  Remark 3.4.

For any biexact functor
$\mathsf{F}\colon{\mathcal{S}}\times{\mathcal{T}}\to{\mathcal{U}}$ and exact
triangle $X\to Y\to Z\to\Sigma X$ in ${\mathcal{S}}$, any morphism
$X^{\prime}\to Y^{\prime}$ in ${\mathcal{T}}$ induces a morphism of exact
triangles in ${\mathcal{U}}$. This morphism of exact triangles is middling
good in the sense of [Nee91, Definition 2.4]. Moreover, if $\mathsf{F}$ admits
a Verdier structure, then the morphism of exact triangles is Verdier good in
the sense of [CF22, Definition 3.1].

Before we establish the level inequality for a biexact functor, we recall the
definition of level.

######  3.5.

For an object $X$ of ${\mathcal{T}}$, we denote by
$\operatorname{thick}^{0}_{{\mathcal{T}}}(X)$ the full subcategory of zero
objects. We denote the smallest full subcategory that contains $X$ and is
closed under (de)suspension, finite coproducts, and retracts by
$\operatorname{thick}^{1}_{{\mathcal{T}}}(X)$. For $n\geq 2$ we inductively
set

$\operatorname{thick}^{n}_{{\mathcal{T}}}(X)\colonequals\left\\{Y\in{\mathcal{T}}\,\middle|\,\begin{gathered}\text{there
is an exact triangle}\\\ Y^{\prime}\to Y\oplus\tilde{Y}\to
Y^{\prime\prime}\to\Sigma Y^{\prime}\\\ \text{with
}Y^{\prime}\in\operatorname{thick}^{1}_{{\mathcal{T}}}(X)\text{ and
}Y^{\prime\prime}\in\operatorname{thick}^{n-1}_{{\mathcal{T}}}(X)\end{gathered}\right\\}\,.$

The subcategories give an exhaustive filtration of the smallest thick
subcategory containing $X$. The construction of these subcategories is robust:
In the definition we may assume that
$Y^{\prime}\in\operatorname{thick}^{i}_{{\mathcal{T}}}(X)$ and
$Y^{\prime\prime}\in\operatorname{thick}^{j}_{{\mathcal{T}}}(X)$ for any
$i+j=n$. Moreover, if $Y\in\operatorname{thick}^{n}_{{\mathcal{T}}}(X)$, then
it is enough to take a retract in the last step.

The subcategories $\operatorname{thick}^{n}_{{\mathcal{T}}}(X)$ were first
introduced in [BvdB03, 2.2] where they were denoted by $\langle X\rangle_{n}$.
With the notation $\operatorname{thick}^{n}_{{\mathcal{T}}}(X)$ we follow
[ABIM10, 2.2].

Let $X,Y\in{\mathcal{T}}$. The _$X$ -level of $Y$_ is

$\operatorname{level}_{{\mathcal{T}}}^{X}(Y)\colonequals\inf\left\\{n\geq
0\,\middle|\,Y\in\operatorname{thick}^{n}_{{\mathcal{T}}}(X)\right\\}\,;$

see [ABIM10, 2.3].

######  Theorem 3.6.

Let $\mathsf{F}\colon{\mathcal{S}}\times{\mathcal{T}}\to{\mathcal{U}}$ be a
biexact functor that admits a strong Verdier structure. Then

$\operatorname{level}_{{\mathcal{U}}}^{\mathsf{F}(X,X^{\prime})}(\mathsf{F}(Y,Y^{\prime}))\leq\operatorname{level}_{{\mathcal{S}}}^{X}(Y)+\operatorname{level}_{{\mathcal{T}}}^{X^{\prime}}(Y^{\prime})-1$

for any $X,Y$ in ${\mathcal{S}}$ and $X^{\prime},Y^{\prime}$ in
${\mathcal{T}}$.

###### Proof.

For $i,j\geq 1$, let

(3.6.1) ${Y_{i-1}}$${Y_{i}}$${X_{i}}$${\Sigma
Y_{i-1}}$${Y^{\prime}_{j-1}}$${Y^{\prime}_{j}}$${X^{\prime}_{j}}$${\Sigma
Y^{\prime}_{j-1}}$ with
${X_{i}\in\operatorname{thick}^{1}_{{\mathcal{S}}}(X)}$${X^{\prime}_{i}\in\operatorname{thick}^{1}_{{\mathcal{T}}}(X^{\prime})}$

be exact triangles in ${\mathcal{S}}$ and ${\mathcal{T}}$, respectively, with
$Y_{0}=0$ and $Y^{\prime}_{0}=0$. We set
$Z_{i,j}\colonequals\mathsf{F}(Y_{i},Y^{\prime}_{j})$. Since $\mathsf{F}$
admits a strong Verdier structure, there exists a homotopy pushout of the span
$Z_{i-1,j}\leftarrow Z_{i-1,j-1}\rightarrow Z_{i,j-1}$ that fits into an exact
triangle

$Z_{i-1,j}+_{Z_{i-1,j-1}}Z_{i,j-1}\to
Z_{i,j}\to\mathsf{F}(X_{i},X^{\prime}_{j})\to\Sigma(Z_{i-1,j}+_{Z_{i-1,j-1}}Z_{i,j-1})$

and makes the following diagram commute

(3.6.2)
${Z_{i-1,j-1}}$${Z_{i,j-1}}$${Z_{i-1,j}}$${Z_{i-1,j}+_{Z_{i-1,j-1}}Z_{i,j-1}}$${Z_{i,j}\makebox[0.0pt][l]{\,.}}$

We let

$W_{k}\colonequals
Z_{1,k}+_{Z_{1,k-1}}Z_{2,k-1}+_{Z_{2,k-2}}\dots+_{Z_{k-1,1}}Z_{k,1}\quad\text{for
}k\geq 1$

be the iterated homotopy pushout over the diagonal $i+j=k+1$. By the
associativity of the homotopy pushout we can write

$\displaystyle W_{k}$
$\displaystyle=Z_{1,k}+_{Z_{1,k}}(Z_{1,k}+_{Z_{1,k-1}}Z_{2,k-1})+_{Z_{2,k-1}}\ldots$
$\displaystyle\quad\quad+_{Z_{k-1,2}}(Z_{k-1,2}+_{Z_{k-1,1}}Z_{k,1})+_{Z_{k,1}}Z_{k,1}\,.$

Expressed in this way $W_{k}$ is an iterated homotopy pushout over the same
bases as $W_{k+1}$. Since Eq. 3.6.2 commutes, we can inductively apply Lemma
2.6 to obtain a morphism $W_{k}\to W_{k+1}$ with

$\operatorname{cone}(W_{k}\to
W_{k+1})=\bigoplus_{i+j=k+2}\mathsf{F}\left(X_{i},X^{\prime}_{j}\right)\in\operatorname{thick}^{1}_{{\mathcal{U}}}(\mathsf{F}(X,X^{\prime}))\,.$

As
$W_{1}=\mathsf{F}(X_{1},X^{\prime}_{1})\in\operatorname{thick}^{1}_{{\mathcal{U}}}(\mathsf{F}(X,X^{\prime}))$,
it follows that

$\operatorname{level}_{{\mathcal{U}}}^{\mathsf{F}(X,X^{\prime})}(W_{k})\leq
k\,.$

Now we use this inequality to establish the claim. We may assume that
$m\colonequals\operatorname{level}_{{\mathcal{S}}}^{X}(Y)$ and
$n\colonequals\operatorname{level}_{{\mathcal{T}}}^{X^{\prime}}(Y^{\prime})$
are finite. Then there exist sequences of triangles Eq. 3.6.1, such that $Y$
is a retract of $Y_{m}$, and $Y^{\prime}$ is a retract of $Y^{\prime}_{n}$,
and the sequences stabilize afterwards: $Y_{i}=Y_{m}$ and $X_{i+1}=0$ for
$i\geq m$ and $Y^{\prime}_{j}=Y^{\prime}_{n}$ and $X^{\prime}_{j+1}=0$ for
$j\geq n$. In particular, the morphisms $Z_{i-1,j}\to Z_{i,j}$ are identities
for $i>m$ and the morphisms $Z_{i,j-1}\to Z_{i,j}$ are identities for $j>n$.
Thus the iterated homotopy pushout $W_{m+n-1}$ simplifies to
$W_{m+n-1}=\mathsf{F}(Y_{m},Y^{\prime}_{n})$. It follows that
$\operatorname{level}_{{\mathcal{U}}}^{\mathsf{F}(X,X^{\prime})}(\mathsf{F}(Y,Y^{\prime}))\leq
m+n-1$. ∎

We will treat cofibration categories in Section 5 to provide a large class of
examples of biexact functors admitting a strong Verdier structure in Section 6
and hence where Theorem 3.6 holds. Concrete examples are given in Sections 6
and 7. First we establish another level inequality in a closely related
setting.

##  4\. Koszul objects

Koszul objects generalize Koszul complexes to triangulated categories. They
are defined for a sequence of elements in a ring that acts on the triangulated
category.

######  4.1.

The _(graded) center_ of a triangulated category ${\mathcal{T}}$ is

$\operatorname{Z}^{*}({\mathcal{T}})\colonequals\bigoplus_{d\in\mathbb{Z}}\left\\{\alpha\colon\operatorname{id}_{\mathcal{T}}\to\Sigma^{d}\,\middle|\,\alpha\Sigma=(-1)^{d}\Sigma\alpha\right\\}\,.$

This is a graded-commutative graded ring; that means

$(\Sigma^{|\alpha|}\beta)\circ\alpha=(-1)^{|\alpha||\beta|}(\Sigma^{|\beta|}\alpha)\circ\beta\,;$

see [BF08]. An action of a graded-commutative graded ring $R$ on
${\mathcal{T}}$ is equivalent to a graded ring homomorphism
$R\to\operatorname{Z}^{*}({\mathcal{T}})$.

######  4.2.

For $X\in{\mathcal{T}}$ and a sequence
$\bm{\alpha}=\alpha_{1},\ldots,\alpha_{c}$ in
$\operatorname{Z}^{*}({\mathcal{T}})$, the _Koszul object of $\bm{\alpha}$ on
$X$_ is

$X/\\!\\!/\bm{\alpha}\colonequals\begin{cases}X&c=0\\\
\operatorname{cone}(X\xrightarrow{\alpha_{1}(X)}\Sigma^{|\alpha_{1}|}X)&c=1\\\
(X/\\!\\!/(\alpha_{1},\ldots,\alpha_{c-1}))/\\!\\!/\alpha_{c}&c>1\,.\end{cases}$

The Koszul object is unique up to non-unique isomorphism.

From the construction of the Koszul object we immediately get the inequality
$\operatorname{level}_{{\mathcal{T}}}^{X}(X/\\!\\!/(\alpha_{1},\ldots,\alpha_{c}))\leq
2^{c}$. We can improve this bound when the elements $\alpha_{i}$ arise from an
action of a monoidal triangulated category on ${\mathcal{T}}$.

###  4.3. Action by monoidal categories

We recall the definition and fix notation for a monoidal structure on a
triangulated category; see [HPS97, Definition A.2.1]. For details on the
coherence axioms see [ML98, XI.1] and also [Kel64].

######  4.4.

A _monoidal triangulated category_ $({\mathcal{S}},\otimes,\mathbbm{1})$
consists of a triangulated category ${\mathcal{S}}$ with a monoidal product
$(\otimes,\mathbbm{1})$ where
$\otimes\colon{\mathcal{S}}\times{\mathcal{S}}\to{\mathcal{S}}$ is a biexact
functor and $\mathbbm{1}\in{\mathcal{S}}$ the _unit_ object. This means there
are isomorphisms

$\alpha\colon X\otimes(Y\otimes Z)\to(X\otimes Y)\otimes
Z\,,\quad\lambda\colon\mathbbm{1}\otimes X\to X\quad\text{and}\quad\rho\colon
X\otimes\mathbbm{1}\to X\,,$

that are natural transformations of exact functors in each variable satisfying
the coherence axioms of a monoidal category.

A monoidal triangulated category $({\mathcal{S}},\otimes,\mathbbm{1})$ is
_symmetric_ if it is equipped with an isomorphism

$\sigma\colon X\otimes Y\to Y\otimes X$

that is a natural transformation of exact functors in each variable satisfying
the coherence axioms of a symmetric monoidal category. Symmetric monoidal
triangulated categories are also called _tensor triangulated categories_. Some
sources additionally assume that the monoidal structure is closed; we discuss
closed monoidal structures on triangulated categories in 6.20.

######  4.5.

Let $({\mathcal{S}},\otimes,\mathbbm{1})$ be a monoidal triangulated category.
A _(left) action_ of ${\mathcal{S}}$ on a triangulated category
${\mathcal{T}}$ consists of a biexact bifunctor
$\mathsf{F}\colon{\mathcal{S}}\times{\mathcal{T}}\to{\mathcal{T}}$ together
with isomorphisms

$\alpha\colon\mathsf{F}(X,\mathsf{F}(Y,Z))\to\mathsf{F}(X\otimes
Y,Z)\quad\text{and}\quad\lambda\colon\mathsf{F}(\mathbbm{1},Z)\to Z\,,$

that are natural transformations of exact functors in each variable satisfying
coherence axioms analogous to the ones of a monoidal category; see for example
[Hov99, Definition 4.1.6] and [BKSS20, Section 1] for details. In particular,
any monoidal triangulated category $({\mathcal{S}},\otimes,\mathbbm{1})$ acts
on itself via the tensor product
$\otimes\colon{\mathcal{S}}\times{\mathcal{S}}\to{\mathcal{S}}$.

######  4.6.

Let $({\mathcal{S}},\otimes,\mathbbm{1})$ be a monoidal triangulated category
and $\mathsf{F}$ an action of ${\mathcal{S}}$ on a triangulated category
${\mathcal{T}}$. We denote by

$\operatorname{End}^{*}_{{\mathcal{S}}}(X)\colonequals\bigoplus_{n\in\mathbb{Z}}{\mathcal{S}}(X,\Sigma^{n}X)$

the graded endomorphism ring. Then the monoidal structure induces a
homomorphism of graded rings

$\operatorname{End}^{*}_{{\mathcal{S}}}(\mathbbm{1})\to\operatorname{Z}^{*}({\mathcal{T}})\,,\quad
f\mapsto\alpha_{f}\colonequals(X\cong\mathsf{F}(\mathbbm{1},X)\xrightarrow{\mathsf{F}(f,X)}\mathsf{F}(\Sigma^{|f|}\mathbbm{1},X)\cong\Sigma^{|f|}X)\,;$

see for example [BKSS20, Proposition 2.1]. The isomorphisms involve $\lambda$
from the ${\mathcal{S}}$-action and $\vartheta$ from 3.1. We say
$\alpha\in\operatorname{Z}^{*}({\mathcal{T}})$ is _induced by $\mathsf{F}$_,
if $\alpha=\alpha_{f}$ for some
$f\in\operatorname{End}^{*}_{{\mathcal{S}}}(\mathbbm{1})$. We can choose

(4.6.1) $X/\\!\\!/\alpha_{f}=\mathsf{F}(\operatorname{cone}(f),X)$

as the Koszul object of $\alpha_{f}$ on $X$; in particular the Koszul object
is functorial in $X$ for $\alpha_{f}$. Moreover, for
$f,g\in\operatorname{End}^{*}_{{\mathcal{S}}}(\mathbbm{1})$ we have an
isomorphism

$\displaystyle X/\\!\\!/(\alpha_{f},\alpha_{g})$
$\displaystyle\cong\mathsf{F}(\operatorname{cone}(g),\mathsf{F}(\operatorname{cone}(f),X))\cong\mathsf{F}(\operatorname{cone}(g)\otimes\operatorname{cone}(f),X)$
$\displaystyle\cong\mathsf{F}(\operatorname{cone}(f)\otimes\operatorname{cone}(g),X)\cong
X/\\!\\!/(\alpha_{g},\alpha_{f})\,.$

If the monoidal product is symmetric, then this isomorphism is induced by the
natural isomorphism $\sigma$. Otherwise, the isomorphism
$\operatorname{cone}(g)\otimes\operatorname{cone}(f)\cong\operatorname{cone}(f)\otimes\operatorname{cone}(g)$
is not canonical, though its existence follows from the $3\times 3$ diagram
obtained by applying $\otimes$ to the triangles involving $f$ and $g$.

In general, the objects $X/\\!\\!/(\alpha,\beta)$ and
$X/\\!\\!/(\beta,\alpha)$ need not be isomorphic, as illustrated by the
following example.

######  Example 4.7.

Let $k$ be a field and $A=k[x]/(x^{2})$. By [KY11, Proposition 5.4], the
center of ${\rm D}_{b}(\operatorname{mod}(A))$ is the trivial extension ring
$k[\zeta]\ltimes\prod_{r\geqslant 0}k$ where $\zeta$ is of degree $2$ if the
characteristic of $k$ is not $2$ and of degree $1$ if $k$ is of characteristic
$2$, and $k=k[\zeta]/(\zeta)$ as a $k[\zeta]$-module. The ring $k[\zeta]$ is
the Hochschild cohomology of $A$ over $k[x]$ and the elements of $k[\zeta]$
are induced by a bifunctor; we discuss the action of Hochschild cohomology in
Example 7.10. We focus on the elements in the center coming from
$\prod_{r\geqslant 0}k$. The category ${\rm D}_{b}(\operatorname{mod}(A))$ is
a Krull-Schmidt category where the indecomposable objects are the complexes
$A_{m}^{n}$ that have $A$ in degrees $m\leq d\leq n$ connected by
differentials $x\operatorname{id}_{A}$ and are zero elsewhere. We take
$\eta_{0},\eta_{1}\in\prod_{r\geqslant 0}k$ determined by

$\eta_{r}(A_{m}^{n})=\begin{cases}x_{m}^{n}&n-m=r\\\ 0&n-m\neq
0\,,\end{cases}$

where $x_{m}^{n}$ is the morphism with
$(x_{m}^{n})_{m}=x\operatorname{id}_{A}$ and zero otherwise. This morphism is
not homotopic to zero or $\operatorname{id}_{A}$. For these objects we obtain

$A_{n}^{n}/\\!\\!/\eta_{0}=A_{n}^{n+1}\quad\text{and}\quad
A_{n}^{n}/\\!\\!/\eta_{1}=A_{n}^{n}\oplus\Sigma A^{n}_{n}\,.$

In particular we obtain

$A_{n}^{n}/\\!\\!/(\eta_{0},\eta_{1})=A_{n}^{n+2}\oplus A_{n+1}^{n+1}\not\cong
A_{n}^{n+1}\oplus\Sigma A_{n}^{n+1}=A^{n}_{n}/\\!\\!/(\eta_{0},\eta_{1})\,.$

Moreover, we have

$A_{n}^{n}/\\!\\!/(\eta_{0}+\eta_{1})=A_{n}^{n+1}\quad\text{and}\quad(\eta_{0}+\eta_{1})(A_{n}^{n}/\\!\\!/(\eta_{0}+\eta_{1}))=x_{n}^{n+1}\neq
0\,.$

Thus elements in the center of ${\mathcal{T}}$ are not necessarily trivial on
their Koszul objects.

###  4.8. Level inequalities involving Koszul objects

For Koszul objects, that are induced by an action, we obtain:

######  Lemma 4.9.

Let ${\mathcal{T}}$ be a triangulated category and
$\bm{\alpha}=\alpha_{1},\ldots,\alpha_{c}$ a sequence of elements in
$\operatorname{Z}^{*}({\mathcal{T}})$ each induced by an action of a monoidal
triangulated category
$\mathsf{F}_{i}\colon{\mathcal{S}}_{i}\times{\mathcal{T}}\to{\mathcal{T}}$.
Then

$\operatorname{level}_{{\mathcal{T}}}^{X/\\!\\!/\bm{\alpha}}(Y/\\!\\!/\bm{\alpha})\leq\operatorname{level}_{{\mathcal{T}}}^{X}(Y)$

for any objects $X$ and $Y$ in ${\mathcal{T}}$.

###### Proof.

Let $\alpha=\alpha_{f}$ be an element induced by an action $\mathsf{F}$. Then
by Eq. 4.6.1 we have

$\operatorname{level}_{{\mathcal{T}}}^{X/\\!\\!/\alpha}(Y/\\!\\!/\alpha)=\operatorname{level}_{{\mathcal{T}}}^{\mathsf{F}(\operatorname{cone}(f),X)}(\mathsf{F}(\operatorname{cone}(f),Y))\leq\operatorname{level}_{{\mathcal{T}}}^{X}(Y)\,,$

since $\mathsf{F}(\operatorname{cone}(f),-)$ is an exact functor. Hence the
desired inequality holds by induction on $c$. ∎

######  Theorem 4.10.

Let ${\mathcal{T}}$ be a triangulated category and
$\bm{\alpha}=\alpha_{1},\ldots,\alpha_{c}$ a sequence of elements in
$\operatorname{Z}^{*}({\mathcal{T}})$ each induced by an action of a monoidal
triangulated category
$\mathsf{F}_{i}\colon{\mathcal{S}}_{i}\times{\mathcal{T}}\to{\mathcal{T}}$. If
each $\mathsf{F}_{i}$ admits a strong Verdier structure, then

$\operatorname{level}_{{\mathcal{T}}}^{X}(Y/\\!\\!/(\alpha_{1},\ldots,\alpha_{c}))\leq\operatorname{level}_{{\mathcal{T}}}^{X}(Y)+c$

for any $X,Y\in{\mathcal{T}}$.

###### Proof.

By induction, it suffices to establish the inequality

$\operatorname{level}_{{\mathcal{T}}}^{X}(Y/\\!\\!/\alpha)\leq\operatorname{level}_{{\mathcal{T}}}^{X}(Y)+1$

for just one element $\alpha=\alpha_{f}$ induced by an action of a monoidal
triangulated category
$\mathsf{F}\colon{\mathcal{S}}\times{\mathcal{T}}\to{\mathcal{T}}$ admitting a
strong Verdier structure.

Since $X\cong\mathsf{F}(\mathbbm{1},X)$ and
$Y/\\!\\!/\alpha=\mathsf{F}(\operatorname{cone}(f),Y)$, the desired inequality
follows from Theorem 3.6 using that
$\operatorname{level}_{{\mathcal{S}}}^{\mathbbm{1}}(\operatorname{cone}(f))\leq
2$. ∎

######  Example 4.11.

Let $R$ be a commutative ring. Then there is a natural embedding
$R\to\operatorname{Z}^{*}({\rm D}(\operatorname{Mod}(R)))$. For
$x_{1},\ldots,x_{c}\in R$ we denote by
$\operatorname{Kos}^{R}(x_{1},\ldots,x_{c})$ the Koszul complex on
$x_{1},\ldots,x_{c}$. Then

$Y/\\!\\!/(x_{1},\ldots,x_{c})=Y\otimes^{\mathbf{L}}_{R}\operatorname{Kos}^{R}(x_{1},\ldots,x_{c})$

for any $Y\in{\rm D}(\operatorname{Mod}(R))$. The inequality in Theorem 4.10
yields

$\operatorname{level}^{X}(Y\otimes^{\mathbf{L}}_{R}\operatorname{Kos}^{R}(x_{1},\ldots,x_{c}))\leq\operatorname{level}^{X}(Y)+c\,.$

When $Y$ is an $R$-module and $x_{1},\ldots,x_{c}$ is a $Y$-regular sequence,
then

$Y\otimes^{\mathbf{L}}_{R}\operatorname{Kos}^{R}(x_{1},\ldots,x_{c})\simeq
Y/(x_{1},\ldots,x_{c})\,.$

Thus we recover the inequality

$\operatorname{projdim}_{R}(Y/(x_{1},\ldots,x_{c}))\leq\operatorname{projdim}_{R}(Y)+c$

for $X=R$; see for example [BH98, Exercise 1.3.6].

######  Remark 4.12.

The upper bound in Theorem 4.10 can be achieved. In fact, [BIKO10, Theorem
3.3] provides conditions such that

$\operatorname{level}_{{\mathcal{T}}}^{X}(X/\\!\\!/(\alpha_{1},\ldots,\alpha_{c}))\geq
c+1$

for some sequence $\alpha_{1},\ldots,\alpha_{c}$.

##  5\. Stable cofibration categories and their homotopy categories

We are interested in triangulated categories that arise as homotopy categories
of stable cofibration categories. Such triangulated categories are called
topological and they encompass all algebraic triangulated categories. Similar
to Quillen model categories, cofibration categories are a model for homotopy
theory. They are dual to Brown’s notion of categories of fibrant objects in
[Bro73, I.1.] and correspond to precofibration categories in which all objects
are cofibrant in the terminology of [RB09]. Szumiło proved in [Szu17] that the
homotopy theory of cofibration categories is equivalent to the homotopy theory
of finitely cocomplete $\infty$-categories. In this section we discuss
definitions and important properties of cofibration categories and stable
cofibration categories; for a general reference see [Sch13]. Schwede proved
that the homotopy category of a stable cofibration category is triangulated.
We will supplement his proof in Proposition 5.8 to show that the triangulation
is strong in the sense of May [May01]. Lemma 5.7 extends [May01, Lemma 5.7] to
relate pushouts in a stable cofibration category to homotopy pushouts in the
associated triangulated category.

######  5.1.

Any _cofibration category_ ${\mathcal{C}}$ comes with two classes of
morphisms; a class of _cofibrations_ and a class of _weak equivalences_. These
are subject to the following axioms:

  1. (1)

Every isomorphism is a weak equivalence and weak equivalences satisfy the
$2$-out-of-$3$ property: For composable morphisms $f$ and $g$, if two out of
$f$, $g$, $gf$ are weak equivalences, then so is the third.

  2. (2)

Every isomorphism is a cofibration and cofibrations are closed under
composition.

  3. (3)

Any diagram $Y\leftarrow X\rightarrow Z$ in which $Y\leftarrow X$ is a
cofibration has a pushout $Y\cup_{X}Z$ and the morphism $Z\to Y\cup_{X}Z$ is a
cofibration. If additionally, $Y\leftarrow X$ is a weak equivalence then so is
$Z\to Y\cup_{X}Z$.

  4. (4)

The category ${\mathcal{C}}$ has an initial object and every morphism from an
initial object is a cofibration.

  5. (5)

Any morphism $X\to Y$ in ${\mathcal{C}}$ can be factored as a cofibration
followed by a weak equivalence.

We write $X\rightarrowtail Y$ for a cofibration and $X\xrightarrow{\sim}Y$ for
a weak equivalence. Maps that are both cofibrations and weak equivalences are
called _acyclic cofibrations_. We denote the localization of ${\mathcal{C}}$
at the weak equivalences by
$\gamma\colon{\mathcal{C}}\to\operatorname{Ho}({\mathcal{C}})$; see [GZ67,
I.1.]. Given a zig-zag

$Y_{1}\xleftarrow{s_{1}}X_{1}\xrightarrow{f_{1}}Y_{2}\xleftarrow{s_{2}}X_{2}\rightarrow\ldots\leftarrow
X_{n}\xrightarrow{f_{n}}Y_{n}$

in ${\mathcal{C}}$, where each $s_{i}$ is a weak equivalence, we write

$\gamma(Y_{1}\xleftarrow{\sim}X_{1}\rightarrow
Y_{2}\xleftarrow{\sim}X_{2}\rightarrow\ldots\xleftarrow{\sim}X_{n}\rightarrow
Y_{n})$

for the composite
$\gamma(f_{n})\ldots\gamma(s_{2})^{-1}\gamma(f_{1})\gamma(s_{1})^{-1}$ in
$\operatorname{Ho}({\mathcal{C}})$.

######  Remark 5.2.

The homotopy category $\operatorname{Ho}({\mathcal{C}})$ of a cofibration
category may not be locally small; see [Sch13, Remark A.2]. For any model
category ${\mathcal{M}}$, the full subcategory of cofibrant objects together
with the cofibrations and weak equivalences of ${\mathcal{M}}$ between
cofibrant objects is a cofibration category. This is the main family of
examples of interest for us. In this case, the homotopy category is locally
small, since the homotopy category of cofibrant objects in ${\mathcal{M}}$ is
equivalent to the homotopy category of ${\mathcal{M}}$ and the latter is
locally small.

Let ${\mathcal{C}}$ be a pointed cofibration category. Pointed means that
every initial object is also a terminal object. We write $\ast$ for the
terminal object. When $X\to Y$ is a cofibration we write $Y/X$ for the pushout
$Y\cup_{X}\ast$.

######  5.3.

An object $C$ is _weakly contractible_ if $C\to\ast$ is a weak equivalence. A
cofibration $X\to C$ with $C$ weakly contractible is called a _cone (of $X$)_.

A morphism between objects extends to a morphism between cones in the
following way: For any morphism $f\colon X\to Y$ and cones $X\to C_{X}$ and
$Y\to C_{Y}$ in ${\mathcal{C}}$, there exists a morphism $\bar{f}\colon
C_{X}\to\bar{C}$ and an acyclic cofibration $s\colon C_{Y}\to\bar{C}$ such
that

${X}$${Y}$${C_{X}}$${\bar{C}}$${C_{Y}}$$\scriptstyle{f}$$\scriptstyle{\bar{f}}$$\scriptstyle{s}$$\scriptstyle{\sim}$

commutes in ${\mathcal{C}}$ and $C_{X}\cup_{X}C_{Y}\to\bar{C}$ is a
cofibration; see [Sch13, Lemma A.3]. The pair $(\bar{f},s)$ is called a _cone
extension_ of $f\colon X\to Y$. Instead of $(\bar{f},s)$ we often say
$\bar{C}$ is the cone extension. Note, that if $f$ is a cofibration, then so
is $\bar{f}$. The composite

$\gamma(C_{X}/X\xrightarrow{\bar{f}/f}\bar{C}/Y\xleftarrow{s/\operatorname{id}_{Y}}C_{Y}/Y)$

in the homotopy category is independent of the chosen cone extension.

######  5.4.

In a pointed cofibration category ${\mathcal{C}}$ we fix a cone $X\to CX$ for
every object $X$. The _suspension_ of $X$ is $\Sigma X\colonequals CX/X$. By
[Sch13, Proposition A.4] this defines a functor
$\Sigma\colon{\mathcal{C}}\to\operatorname{Ho}({\mathcal{C}})$ which is given
on morphisms as

$\Sigma(X\xrightarrow{f}Y)\colonequals\gamma(CX/X\to\bar{C}/Y\leftarrow CY/Y)$

where $\bar{C}$ is a cone extension of $f$. Moreover, the suspension functor
$\Sigma$ takes weak equivalences to isomorphisms and thus induces a functor
$\Sigma\colon\operatorname{Ho}({\mathcal{C}})\to\operatorname{Ho}({\mathcal{C}})$.

######  5.5.

Let $f\colon X\rightarrowtail Y$ be a cofibration in ${\mathcal{C}}$. The
_connecting morphism_ of $f$ is

$\delta(f)\colonequals\gamma(Y/X\xleftarrow{\sim}CX\cup_{X}Y\to\Sigma X)\,.$

We call

$X\xrightarrow{\gamma(f)}Y\to Y/X\xrightarrow{\delta(f)}\Sigma X$

an _elementary exact triangle_.

We will use repeatedly that the connecting morphism is natural in the sense of
[Sch13, Proposition A.11]: For any commutative square

${X}$${Y}$${X^{\prime}}$${Y^{\prime}}$$\scriptstyle{f}$$\scriptstyle{f^{\prime}}$

in $\mathcal{C}$, where $f$ and $f^{\prime}$ are cofibrations, the diagram

${Y/X}$${\Sigma X}$${Y^{\prime}/X^{\prime}}$${\Sigma
X^{\prime}}$$\scriptstyle{\delta(f)}$$\scriptstyle{\delta(f^{\prime})}$

commutes in $\operatorname{Ho}({\mathcal{C}})$.

###  5.6. Stable cofibration categories

If the induced endofunctor $\Sigma$ on $\operatorname{Ho}({\mathcal{C}})$ is
an equivalence, then the pointed cofibration category ${\mathcal{C}}$ is said
to be _stable_. Schwede proved in [Sch13, Theorem A.12] that the homotopy
category of any stable cofibration category is triangulated where the exact
triangles are those isomorphic to an elementary exact triangle.

May [May01, Lemma 5.7] proved for some families of stable model categories
that a pushout square of cofibrant objects, where two parallel morphisms are
cofibrations, induces a homotopy cartesian square in the homotopy category.
Groth, Ponto, and Shulman established the result for stable derivators in
[GPS14b, Theorem 6.1], in particular extending May’s result to all stable
model categories. Combining May’s arguments with Schwede’s results we provide
a proof of the analogous result for stable cofibration categories so that
there is no need to change frameworks.

######  Lemma 5.7.

Let

${X}$${Y}$${Z}$${P}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{f^{\prime}}$$\scriptstyle{g^{\prime}}$

be a pushout square in a stable cofibration category ${\mathcal{C}}$ with
$f\colon X\to Y$ a cofibration. Then its image in
$\operatorname{Ho}({\mathcal{C}})$ is a homotopy cartesian square.

###### Proof.

By rotation, it is enough to construct an exact triangle

$Y\oplus
Z\xrightarrow{\left(\begin{smallmatrix}g^{\prime}&f^{\prime}\end{smallmatrix}\right)}P\xrightarrow{\partial}\Sigma
X\xrightarrow{\Sigma\left(\begin{smallmatrix}-f\\\
g\end{smallmatrix}\right)}\Sigma(Y\oplus Z)\,.$

Let $\operatorname{cyl}(X)$ be a cylinder object for $X$; that is we factor
the fold map $X\sqcup X\to X$ into a cofibration followed by a weak
equivalence

${X\sqcup
X}$${\operatorname{cyl}(X)}$${X\,;}$$\scriptstyle{(i_{0},i_{1})}$$\scriptstyle{q}$$\scriptstyle{\sim}$

see for example [RB09, Section 1.5]. Let

$M(f,g)\colonequals\operatorname{cyl}(X)\cup_{X\sqcup X}(Y\sqcup
Z)=Y\cup_{X}\operatorname{cyl}(X)\cup_{X}Z$

be the double mapping cylinder of $(f,g)$. The gluing lemma applied to

${\operatorname{cyl}(X)\cup_{X}Z}$${\operatorname{cyl}(X)}$${X}$${Y}$${Z}$${X}$${Y}$$\scriptstyle{\sim}$$\scriptstyle{i_{0}}$$\scriptstyle{f}$$\scriptstyle{=}$$\scriptstyle{=}$$\scriptstyle{g}$$\scriptstyle{f}$

provides a weak equivalence $M(f,g)\xrightarrow{\sim}P$; see [RB09, Lemma
1.4.1(1)(b)]. Moreover, $M(f,g)/(Y\sqcup Z)\cong\operatorname{cyl}(X)/(X\sqcup
X)$ and we have weak equivalences

$\operatorname{cyl}(X)/(X\sqcup
X)=\ast\cup_{X}\operatorname{cyl}(X)\cup_{X}\ast\xleftarrow{\sim}\ast\cup_{X}\operatorname{cyl}(X)\cup_{X}CX\xrightarrow{\sim}\ast\cup_{X}CX=\Sigma
X.$

It follows that the elementary exact triangle arising from the cofibration
$Y\sqcup Z\rightarrowtail M(f,g)$ yields an exact triangle

$Y\oplus
Z\xrightarrow{\left(\begin{smallmatrix}g^{\prime}&f^{\prime}\end{smallmatrix}\right)}P\rightarrow\Sigma
X\xrightarrow{\delta}\Sigma(Y\oplus Z)\,.$

It remains to check that $\delta$ is the suspension of $(-f,g)^{T}\colon X\to
Y\oplus Z$. We show that its projection to $\Sigma Y$ is $-\Sigma f$. Choosing
a cone extension $CX\to C\leftarrow CZ$ of $g\colon X\to Z$, this follows from
the naturality of the connecting morphism 5.5 applied to

${Y\sqcup Z}$${Y\cup_{X}\operatorname{cyl}(X)\cup_{X}Z}$${Y\sqcup
C}$${Y\cup_{X}\operatorname{cyl}(X)\cup_{X}C}$${Y}$${Y\cup_{X}\operatorname{cyl}(X)\cup_{X}CX}$${Y}$${Y\cup_{X}X\cup_{X}CX\makebox[0.0pt][l]{\,.}}$$\scriptstyle{\operatorname{id}_{Y}\cup
q\cup_{X}\operatorname{id}_{CX}}$

The connecting morphism of the cofibration $Y\to Y\cup_{X}CX$ is indeed
$-\Sigma f$ by the proof of the rotation axiom in [Sch13, Theorem A.12].

Similarly, the projection of $\delta\colon\Sigma X\to\Sigma(Y\oplus Z)$ to
$\Sigma Z$ is $\Sigma g$. The sign of $-\Sigma g$ cancels with the sign
arising from

$\gamma(\Sigma
X=\ast\cup_{X}CX\xleftarrow{\sim}CX\cup_{X}CX\xrightarrow{\sim}CX\cup_{X}\ast=\Sigma
X)=-\operatorname{id}_{\Sigma X}\,;$

see [Sch13, Proposition A.8(iii)]. ∎

The following supplements the proof of the octahedral axiom in
$\operatorname{Ho}({\mathcal{C}})$; see [Sch13, Theorem A.12]. In particular,
we show that the triangulation of $\operatorname{Ho}({\mathcal{C}})$ is
_strong_ in the sense of [May01, Definition 3.8].

######  Proposition 5.8.

Let ${X}$${Y}$${Z}$$\scriptstyle{f}$$\scriptstyle{g}$ be a composition of
cofibrations in a stable cofibration category ${\mathcal{C}}$. Then

${Y}$${Y/X}$${Z}$${Z/X}$$\scriptstyle{\gamma(g)}$ and ${Z/X}$${\Sigma
X}$${Z/Y}$${\Sigma Y}$$\scriptstyle{\delta(gf)}$$\scriptstyle{\Sigma
f}$$\scriptstyle{\delta(g)}$

are homotopy pushout squares in $\operatorname{Ho}({\mathcal{C}})$.

###### Proof.

The first square considered in $\mathcal{C}$ fits into a commutative diagram

${X}$${Y}$${Z}$${\ast}$${Y/X}$${Z/X}$

and it follows from the pasting lemma for pushouts that it is a pushout
square. Thus its image is a homotopy pushout square in
$\operatorname{Ho}({\mathcal{C}})$ by Lemma 5.7.

We show that the second square is homotopy cartesian. We pick a cone extension
$\bar{C}$ of $f$. This fits into a diagram

${Z/X}$${CX\cup_{X}Z}$${CX/X}$${Z/Y}$${\bar{C}\cup_{Y}Z}$${\bar{C}/Y}$${Z/Y}$${CY\cup_{Y}Z}$${CY/Y}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{=}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

consisting of four commutative squares in ${\mathcal{C}}$. Since weak
equivalences become isomorphisms in $\operatorname{Ho}({\mathcal{C}})$, it is
enough to show that the upper right-hand square is homotopy cartesian in
$\operatorname{Ho}({\mathcal{C}})$. We will show that it is a pushout square
in ${\mathcal{C}}$ and that $CX\cup_{X}Z\to\bar{C}\cup_{Y}Z$ is a cofibration
in order to apply Lemma 5.7.

The pasting lemma for pushouts yields diagrams of pushout squares

${X}$${Z}$${\ast}$${CX}$${CX\cup_{X}Z}$${\Sigma X}$ and
${Y}$${Z}$${\ast}$${CX\cup_{X}Z}$${\Sigma
X}$${\bar{C}}$${\bar{C}\cup_{Y}Z}$${\bar{C}/Y\makebox[0.0pt][l]{\,.}}$

By 5.3 the morphism $CX\cup_{X}CY\to\bar{C}$ is a cofibration, and hence so is
$CX\cup_{X}Y\to\bar{C}$. We apply [RB09, Lemma 1.4.1(1)(a)] to

${CX}$${X}$${Z}$${\bar{C}}$${Y}$${Z}$$\scriptstyle{=}$

and obtain that $CX\cup_{X}Z\to\bar{C}\cup_{Y}Z$ is a cofibration. Thus Lemma
5.7 applied to the square with corners $CX\cup_{X}Z$, $\bar{C}\cup_{Y}Z$,
$\Sigma X$, $\bar{C}/Y$ provides the desired homotopy cartesian square. ∎

######  Corollary 5.9.

The homotopy category of a stable cofibration category is a strongly
triangulated category in the sense of [May01, Definition 3.8].

###### Proof.

By the same argument as in the proof of the octahedral axiom in [Sch13,
Theorem A.12] it is enough to consider a composition
$X\xrightarrow{f}Y\xrightarrow{g}Z$ in the homotopy category where
$f=\gamma(f^{\prime})$ and $g=\gamma(g^{\prime})$ for cofibrations
$f^{\prime}$ and $g^{\prime}$. Then the claim holds by Proposition 5.8. ∎

##  6\. Biexact functors between stable cofibration categories

In this section we show that a biexact functor between stable cofibration
categories induces a biexact functor of triangulated categories and that the
induced functor admits a strong Verdier structure; see Theorem 6.4. Analogous
results have been proved for monoidal products; for some families of monoidal
stable model categories see May [May01, Section 6] and for strong, stable
monoidal derivators see [GPS14a, Theorem 6.2]. In particular, we show that
May’s arguments extend to any Quillen bifunctor between stable model
categories as the bifunctors can be restricted to biexact functors between
categories of cofibrant objects.

######  6.1.

A functor $\mathsf{F}\colon{\mathcal{C}}\to{\mathcal{D}}$ between cofibration
categories is _exact_ if it preserves initial objects, cofibrations, weak
equivalences, and pushouts of diagrams, where one leg is a cofibration. Since
$\mathsf{F}$ preserves weak equivalences, it induces a functor
$\operatorname{Ho}(\mathsf{F})\colon\operatorname{Ho}({\mathcal{C}})\to\operatorname{Ho}({\mathcal{D}})$
on the homotopy categories. If ${\mathcal{C}}$ and ${\mathcal{D}}$ are stable,
then $\operatorname{Ho}(\mathsf{F})$ is an exact functor of triangulated
categories by [Sch13, Proposition A.14].

The natural isomorphism
$\tau\colon\operatorname{Ho}(\mathsf{F})\Sigma\to\Sigma\operatorname{Ho}(\mathsf{F})$
is given as follows: We fix an object $X$ and let $C$ be a cone extension of
the identity on $\mathsf{F}(X)$ with respect to the cones $C\mathsf{F}(X)$ and
$\mathsf{F}(CX)$. Then

$\tau(X)=\gamma(\mathsf{F}(CX)/\mathsf{F}(X)\to
C/\mathsf{F}(X)\xleftarrow{\sim}C\mathsf{F}(X)/\mathsf{F}(X))\,.$

######  Lemma 6.2.

Let $\mathsf{F},\mathsf{G}\colon{\mathcal{C}}\to{\mathcal{D}}$ be exact
functors between stable cofibration categories and
$\eta\colon\mathsf{F}\to\mathsf{G}$ a natural transformation. Then
$\operatorname{Ho}(\eta)\colon\operatorname{Ho}(\mathsf{F})\to\operatorname{Ho}(\mathsf{G})$
is a natural transformation of exact functors; that is
$(\Sigma\operatorname{Ho}(\eta))\tau^{\mathsf{F}}=\tau^{\mathsf{G}}(\operatorname{Ho}(\eta)\Sigma)$.

###### Proof.

We pick cone extensions $C_{\mathsf{F}}$ and $C_{\mathsf{G}}$ of the identity
on $\mathsf{F}(X)$ and $\mathsf{G}(X)$, respectively, as above. Then
$\mathsf{F}(CX)\to C_{\mathsf{F}}$ and $\mathsf{G}(CX)\to C_{\mathsf{G}}$ are
cones, and we pick a cone extension $C$ of $\eta(CX)$ with respect to these
cones. Then we obtain a commutative diagram

${\mathsf{F}(CX)/\mathsf{F}(X)}$${C_{\mathsf{F}}/\mathsf{F}(X)}$${C\mathsf{F}(X)/\mathsf{F}(X)}$${C/\mathsf{G}(X)}$${\mathsf{G}(CX)/\mathsf{G}(X)}$${C_{\mathsf{G}}/\mathsf{G}(X)}$${C\mathsf{G}(X)/\mathsf{G}(X)\makebox[0.0pt][l]{\,.}}$$\scriptstyle{\sim}$$\scriptstyle{\eta}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

It remains to observe, that in $\operatorname{Ho}({\mathcal{D}})$ the
morphisms in the top and bottom row are $\tau^{\mathsf{F}}$ and
$\tau^{\mathsf{G}}$, respectively, the morphism in the left column is
$\operatorname{Ho}(\eta)(\Sigma X)$ and the morphism in the right column is
$\Sigma\operatorname{Ho}(\eta)(X)$. ∎

######  Definition 6.3.

Let ${\mathcal{C}}$, ${\mathcal{D}}$, ${\mathcal{E}}$ be cofibration
categories. A functor
$\mathsf{F}\colon{\mathcal{C}}\times{\mathcal{D}}\to{\mathcal{E}}$ is
_biexact_ , if it is exact in each variable and for any morphisms $f\colon
X\to Y$ in ${\mathcal{C}}$ and $f^{\prime}\colon X^{\prime}\to Y^{\prime}$ in
${\mathcal{D}}$, the induced morphism

(6.3.1)
$\mathsf{F}(X,Y^{\prime})\cup_{\mathsf{F}(X,X^{\prime})}\mathsf{F}(Y,X^{\prime})\to\mathsf{F}(Y,Y^{\prime})$

is a cofibration provided that $f$ and $f^{\prime}$ are cofibrations.

A biexact functor
$\mathsf{F}\colon{\mathcal{C}}\times{\mathcal{D}}\to{\mathcal{E}}$ preserves
weak equivalences and thus induces a functor
$\operatorname{Ho}(\mathsf{F})\colon\operatorname{Ho}({\mathcal{C}})\times\operatorname{Ho}({\mathcal{D}})\to\operatorname{Ho}({\mathcal{E}})$
which coincides with $\mathsf{F}$ on objects.

######  Theorem 6.4.

Let $\mathsf{F}\colon{\mathcal{C}}\times{\mathcal{D}}\to{\mathcal{E}}$ be a
biexact functor between stable cofibration categories. Then the induced
functor $\operatorname{Ho}(\mathsf{F})$ is a biexact functor of triangulated
categories and $\operatorname{Ho}(\mathsf{F})$ admits a strong Verdier
structure.

###### Proof.

By 6.1 and 6.2, we have natural isomorphisms

$\vartheta\colon\operatorname{Ho}(\mathsf{F})(\Sigma-,-)\to\Sigma\operatorname{Ho}(\mathsf{F})(-,-)\quad\text{and}\quad\zeta\colon\operatorname{Ho}(\mathsf{F})(-,\Sigma-)\to\Sigma\operatorname{Ho}(\mathsf{F})(-,-)\,.$

We will show that $\operatorname{Ho}(\mathsf{F})$ admits a strong Verdier
structure. This will imply the compatibility condition Eq. 3.1.1.

It suffices to establish the strong Verdier structure for elementary exact
triangles

$X\xrightarrow{\gamma(f)}Y\xrightarrow{\gamma(g)}Z\xrightarrow{\delta(f)}\Sigma
X\quad\text{and}\quad
X^{\prime}\xrightarrow{\gamma(f^{\prime})}Y^{\prime}\xrightarrow{\gamma(g^{\prime})}Z^{\prime}\xrightarrow{\delta(f^{\prime})}\Sigma
X^{\prime}$

in $\operatorname{Ho}({\mathcal{C}})$ and $\operatorname{Ho}({\mathcal{D}})$,
respectively.

We set
$W\colonequals\mathsf{F}(X,Y^{\prime})\cup_{\mathsf{F}(X,X^{\prime})}\mathsf{F}(Y,X^{\prime})$.
By Eq. 6.3.1 the morphism $W\to\mathsf{F}(Y,Y^{\prime})$ is a cofibration.
Using the pasting lemma repeatedly, we obtain the following diagram in which
every square is a pushout square:

(6.4.1)
${\mathsf{F}(X,X^{\prime})}$${\mathsf{F}(Y,X^{\prime})}$${\ast}$${\mathsf{F}(X,Y^{\prime})}$${W}$${\mathsf{F}(X,Z^{\prime})}$${\ast}$${\mathsf{F}(Y,Y^{\prime})}$${\mathsf{F}(Y,Z^{\prime})}$${\mathsf{F}(Z,Z^{\prime})\makebox[0.0pt][l]{\,.}}$

This yields the elementary exact triangles

${W}$${\mathsf{F}(Y,Y^{\prime})}$${\mathsf{F}(Z,Z^{\prime})}$${\Sigma
W}$${\mathsf{F}(Y,X^{\prime})}$${W}$${\mathsf{F}(X,Z^{\prime})}$${\Sigma\mathsf{F}(Y,X^{\prime})\makebox[0.0pt][l]{\,,}}$$\scriptstyle{i}$$\scriptstyle{p=\delta(i)}$$\scriptstyle{j^{\prime}}$$\scriptstyle{q^{\prime}}$

and similarly, we obtain an elementary exact triangle

${\mathsf{F}(X,Y^{\prime})}$${W}$${\mathsf{F}(Z,X^{\prime})}$${\Sigma\mathsf{F}(X,Y^{\prime})\makebox[0.0pt][l]{\,.}}$$\scriptstyle{j}$$\scriptstyle{q}$

The dotted arrow in the first triangle is $\gamma(\mathsf{F}(g,g^{\prime}))$.
In the latter two triangles the dotted arrows are
$(\Sigma\gamma(\mathsf{F}(f,X^{\prime})))\delta(\mathsf{F}(X,f^{\prime}))$ and
$(\Sigma\gamma(\mathsf{F}(X,f^{\prime})))\delta(\mathsf{F}(f,X^{\prime}))$,
respectively, by applying the naturality of the connecting morphism 5.5 to the
defining pushout square of $W$. Since
$\delta(\mathsf{F}(X,f^{\prime}))=\zeta\operatorname{Ho}(\mathsf{F})(X,\delta(f^{\prime}))$,
and similarly for $\delta(\mathsf{F}(f,X^{\prime}))$, these are the required
triangles Eq. 3.2.2.

We need to check, that the morphisms satisfy the compatibility conditions in
Eq. 3.2.3, and that the required squares are homotopy cartesian. We use the
labeling from Eq. 3.2.3.

By construction (i–iv) commute. The triangles (v) and (vi) commute by the
naturality of the connecting morphism in 5.5. Further by Eqs. 6.4.1 and 5.7
the squares (I–III) commute and are homotopy pushout squares.

Next we show that the desuspension of (VI) is anti-commutative and homotopy
cartesian when replacing one arrow by its negative. Pasting pushouts in the
cube with top face the defining pushout diagram of $W$ and bottom face the
pushout of
$\mathsf{F}(X,Z^{\prime})\leftarrow\ast\rightarrow\mathsf{F}(Z,X^{\prime})$,
shows that
$W/\mathsf{F}(X,X^{\prime})\cong\mathsf{F}(X,Z^{\prime})\sqcup\mathsf{F}(Z,X^{\prime})$.
By construction the composition
$W\to\mathsf{F}(X,Z^{\prime})\sqcup\mathsf{F}(Z,X^{\prime})\to\mathsf{F}(X,Z^{\prime})\sqcup\ast\cong\mathsf{F}(X,Z^{\prime})$
coincides with the morphism $W\to\mathsf{F}(X,Z^{\prime})$ above; and
similarly for the second summand. Thus we have an elementary exact triangle

$\mathsf{F}(X,X^{\prime})\to
W\to\mathsf{F}(X,Z^{\prime})\oplus\mathsf{F}(Z,X^{\prime})\xrightarrow{\delta}\Sigma\mathsf{F}(X,X^{\prime})$

in $\operatorname{Ho}({\mathcal{E}})$. It follows from 5.5 that the connecting
morphism $\delta$ restricts to $\delta(\mathsf{F}(X,f^{\prime}))$ and
$\delta(\mathsf{F}(f,X))$ on the summands $\mathsf{F}(X,Z^{\prime})$ and
$\mathsf{F}(Z,X^{\prime})$, respectively. Rotating this triangle shows that
the square with corners $W$, $\mathsf{F}(X,Z^{\prime})$,
$\mathsf{F}(Z,X^{\prime})$, $\Sigma\mathsf{F}(X,X^{\prime})$ anti-commutes,
and it is homotopy cartesian if we replace one arrow by its negative.

Applying Proposition 5.8 to the compositions of cofibrations
$\mathsf{F}(X,Y^{\prime})\to W\to\mathsf{F}(Y,Y^{\prime})$ and
$\mathsf{F}(Y,X^{\prime})\to W\to\mathsf{F}(Y,Y^{\prime})$ shows that (IV) and
(V) are commutative and homotopy cartesian, respectively.

It remains to observe, that Eq. 3.1.1 is a consequence of Eq. 3.2.2 by using
the exact triangles

$X\to 0\to\Sigma X\xrightarrow{-\operatorname{id}_{\Sigma X}}\Sigma
X\quad\text{and}\quad X^{\prime}\to 0\to\Sigma
X^{\prime}\xrightarrow{-\operatorname{id}_{\Sigma X^{\prime}}}\Sigma
X^{\prime}\,.$

Hence $\operatorname{Ho}(\mathsf{F})$ is a biexact functor that admits a
strong Verdier structure. ∎

######  Remark 6.5.

Let $X\to Y\to Z\to\Sigma X$ and $X^{\prime}\to Y^{\prime}\to
Z^{\prime}\to\Sigma X^{\prime}$ be exact triangles in
$\operatorname{Ho}({\mathcal{C}})$, and $W$ the object constructed from these
triangles as in the proof of Theorem 6.4. When we replace the triangle $X\to
Y\to Z\to\Sigma X$ by $Y\to Z\to\Sigma X\to\Sigma Y$ (with appropriate signs)
in the proof of Theorem 6.4, we obtain an object $V$. Then there exist
homotopy pushout squares

${V}$${\Sigma\mathsf{F}(Y,X^{\prime})}$${\mathsf{F}(Z,Y^{\prime})}$${\Sigma
W}$ and
${W}$${\mathsf{F}(Y,Y^{\prime})}$${\mathsf{F}(X,Z^{\prime})\oplus\mathsf{F}(Z,X^{\prime})}$${V\makebox[0.0pt][l]{\,.}}$

in $\operatorname{Ho}({\mathcal{C}})$. By [KN02, Theorem 4.1], this is also a
direct consequence of the strong Verdier structure and the octahedral axiom.

The second homotopy pushout square also appears in [May01] as (TC4). Using
Lemma 6.10 below, it is straightforward to check that instead of replacing
$X\to Y\to Z\to\Sigma X$ by $Y\to Z\to\Sigma X\to\Sigma Y$, one can obtain
$\Sigma^{-1}V$ from $\Sigma^{-1}Z\to X\to Y\to Z$ and
$\Sigma^{-1}Z^{\prime}\to X^{\prime}\to Y^{\prime}\to Z^{\prime}$.

###  6.6. Naturality

In the proof of Theorem 6.4 we constructed the object $W$ as a pushout in the
stable cofibration category. We will show that the strong Verdier structure on
the bifunctor on the homotopy categories inherits some of the naturality from
the bifunctor on the stable cofibration categories. These compatibility
results may be of interest for future applications, but we do not use them
here.

######  6.7.

When we work in the homotopy category $\operatorname{Ho}({\mathcal{C}})$ of a
cofibration category ${\mathcal{C}}$, it would be helpful to lift commutative
diagrams in $\operatorname{Ho}({\mathcal{C}})$ to commutative diagrams in
${\mathcal{C}}$ to use the structure of ${\mathcal{C}}$. This works for
commutative squares: As in the proof of (T4) in [Sch13, Theorem A.12], any
morphism in $\operatorname{Ho}({\mathcal{C}})$ factors as the localization
functor $\gamma\colon{\mathcal{C}}\to\operatorname{Ho}({\mathcal{C}})$ applied
to a cofibration followed by an isomorphism. Thus any commutative square in
$\operatorname{Ho}({\mathcal{C}})$ is isomorphic to a commutative square such
that two parallel arrows are $\gamma$ of cofibrations. Schwede then shows in
the proof of (T3) in [Sch13, Theorem A.12], that any such commutative square
in $\operatorname{Ho}({\mathcal{C}})$ is isomorphic to $\gamma$ applied to a
commutative square in ${\mathcal{C}}$ in which two parallel arrows are
cofibrations.

######  Lemma 6.8.

Let $\mathsf{F}\colon{\mathcal{C}}\times{\mathcal{D}}\to{\mathcal{E}}$ be a
biexact functor of stable cofibration categories and let

${X_{1}}$${Y_{1}}$${X_{2}}$${Y_{2}}$ and
${X^{\prime}_{1}}$${Y^{\prime}_{1}}$${X^{\prime}_{2}}$${Y^{\prime}_{2}}$

be commutative squares in $\operatorname{Ho}({\mathcal{C}})$ and
$\operatorname{Ho}({\mathcal{D}})$, respectively. Then there exist homotopy
pushouts $W_{i}$ of
$\mathsf{F}(X_{i},Y^{\prime}_{i})\leftarrow\mathsf{F}(X_{i},X^{\prime}_{i})\rightarrow\mathsf{F}(Y_{i},X^{\prime}_{i})$
in $\operatorname{Ho}({\mathcal{E}})$ satisfying 3.2 and a morphism $W_{1}\to
W_{2}$ in $\operatorname{Ho}({\mathcal{E}})$ that is compatible with the exact
triangles Eq. 3.2.2.

Moreover, the cone of the morphism $W_{1}\to W_{2}$ is isomorphic to

$\operatorname{cone}(\mathsf{F}(X_{1},Y^{\prime}_{1})\to\mathsf{F}(X_{2},Y^{\prime}_{2}))+_{\operatorname{cone}(\mathsf{F}(X_{1},X^{\prime}_{1})\to\mathsf{F}(X_{2},X^{\prime}_{2}))}\operatorname{cone}(\mathsf{F}(Y_{1},X^{\prime}_{1})\to\mathsf{F}(Y_{2},X^{\prime}_{2}))\,.$

###### Proof.

By 6.7, we can assume that the squares lift to commutative squares in
${\mathcal{C}}$ and ${\mathcal{D}}$, respectively, with the horizontal
morphisms being cofibrations. Then we obtain the morphism $W_{1}\to W_{2}$
from the morphism of the spans. The pushout property yields morphisms of the
diagrams Eq. 6.4.1 and thus a morphism of triangles in
$\operatorname{Ho}({\mathcal{E}})$.

For the second claim, let $C_{1}$ be a cone extension of
$\mathsf{F}(X_{1},X^{\prime}_{1})\to\mathsf{F}(X_{1},Y^{\prime}_{1})$ and
$C_{2}$ a cone extension of
$\mathsf{F}(X_{1},X^{\prime}_{1})\to\mathsf{F}(Y_{1},X^{\prime}_{1})$. We
consider the commutative diagram

${C_{1}}$${C\mathsf{F}(X_{1},X^{\prime}_{1})}$${C_{2}}$${\mathsf{F}(X_{1},Y^{\prime}_{1})}$${\mathsf{F}(X_{1},X^{\prime}_{1})}$${\mathsf{F}(Y_{1},X^{\prime}_{1})}$${\mathsf{F}(X_{2},Y^{\prime}_{2})}$${\mathsf{F}(X_{2},X^{\prime}_{2})}$${\mathsf{F}(Y_{2},X^{\prime}_{2})\makebox[0.0pt][l]{\,.}}$

Whether we first take pushouts horizontally and then vertically, or the other
way around, we obtain the same object. By [RB09, Lemma 1.4.1(1)(a)] these
pushouts exist. In $\operatorname{Ho}({\mathcal{E}})$ the former yields
$\operatorname{cone}(W_{1}\to W_{2})$, and the latter the desired homotopy
pushout using Lemma 5.7. ∎

One should compare the previous Lemma 6.8 to Lemmas 2.6 and 2.7, as the latter
treat special cases, though in a more general setting.

Next we show that the construction of the strong Verdier structure in Theorem
6.4 is invariant under rotation. Namely, if we rotate the triangle in the
first component of the biexact functor, then the three exact triangles Eq.
3.2.2 from the corresponding strong Verdier structure are up to rotation
canonically isomorphic to the three exact triangles obtained from the strong
Verdier structure after rotating the triangle in the second component.

We will use Schwede’s proof of the rotation axiom from [Sch13, Theorem A.12].

######  Remark 6.9.

Let $f\colon X\to Y$ be a cofibration in a stable cofibration category
${\mathcal{C}}$. Then

${X}$${CX\cup_{X}Y}$${\Sigma X}$${\Sigma Y}$${X}$${Y/X}$${\Sigma X}$${\Sigma
Y\makebox[0.0pt][l]{\,.}}$$\scriptstyle{=}$$\scriptstyle{\delta}$$\scriptstyle{=}$$\scriptstyle{=}$$\scriptstyle{-\Sigma(f)}$

is an isomorphism of exact triangles. If $C_{X}$ is another cone for $X$ and
$\overline{C}$ a cone extension for $CX$ and $C_{X}$, then the latter exact
triangle is isomorphic to

${X}$${Y/X}$${C_{X}/X}$${\Sigma Y,}$$\scriptstyle{\delta}$

via $\gamma(CX/X\rightarrow\overline{C}/X\leftarrow C_{X}/X)$ and the identity
morphisms on the other objects.

######  Lemma 6.10.

Let $\mathsf{F}\colon{\mathcal{C}}\times{\mathcal{D}}\to{\mathcal{E}}$ be a
biexact functor of stable cofibration categories and $f\colon X\to Y$ and
$f^{\prime}\colon X^{\prime}\to Y^{\prime}$ cofibrations in ${\mathcal{C}}$
and ${\mathcal{D}}$, respectively. Let $W_{1}$ be the pushout of
$\mathsf{F}(X,CX^{\prime}\cup_{X^{\prime}}Y^{\prime})\leftarrow\mathsf{F}(X,Y^{\prime})\to\mathsf{F}(Y,Y^{\prime})$
and $W_{2}$ the pushout of
$\mathsf{F}(CX\cup_{X}Y,X^{\prime})\leftarrow\mathsf{F}(Y,X^{\prime})\to\mathsf{F}(Y,Y^{\prime})$.
Then there is a canonical isomorphism $e\colon W_{1}\to W_{2}$ in
$\operatorname{Ho}({\mathcal{E}})$ that yields canonical isomorphisms of the
exact triangles Eq. 3.2.2; that is for $Z=Y/X$ and
$Z^{\prime}=Y^{\prime}/X^{\prime}$ the following diagrams commute

(6.10.1) ${\mathsf{F}(Y,Y^{\prime})}$${W_{1}}$${\mathsf{F}(X,\Sigma
X^{\prime})}$${\Sigma\mathsf{F}(Y,Y^{\prime})}$${\mathsf{F}(Y,Y^{\prime})}$${W_{2}}$${\mathsf{F}(\Sigma
X,X^{\prime})}$${\Sigma\mathsf{F}(Y,Y^{\prime})\makebox[0.0pt][l]{\,,}}$$\scriptstyle{j^{\prime}_{1}}$$\scriptstyle{=}$$\scriptstyle{q^{\prime}_{1}}$$\scriptstyle{e}$$\scriptstyle{\cong}$$\scriptstyle{\zeta\mathsf{F}(f,\Sigma
f^{\prime})}$$\scriptstyle{\cong}$$\scriptstyle{=}$$\scriptstyle{j_{2}}$$\scriptstyle{q_{2}}$$\scriptstyle{\vartheta\mathsf{F}(\Sigma
f,f^{\prime})}$ (6.10.2)
${W_{1}}$${\mathsf{F}(Y,Z^{\prime})}$${\mathsf{F}(Z,\Sigma
X^{\prime})}$${\Sigma
W_{1}}$${W_{2}}$${\mathsf{F}(Y,Z^{\prime})}$${\Sigma\mathsf{F}(Z,X^{\prime})}$${\Sigma
W_{2}}$$\scriptstyle{i_{1}}$$\scriptstyle{e}$$\scriptstyle{\cong}$$\scriptstyle{\mathsf{F}(g,h^{\prime})}$$\scriptstyle{=}$$\scriptstyle{p_{1}}$$\scriptstyle{\zeta}$$\scriptstyle{\cong}$$\scriptstyle{\Sigma
e}$$\scriptstyle{\cong}$$\scriptstyle{q^{\prime}_{2}}$$\scriptstyle{\zeta\mathsf{F}(g,h^{\prime})}$$\scriptstyle{-\Sigma
j^{\prime}_{2}}$

and analogously, there is a third isomorphism of exact triangles.

###### Proof.

Using the pasting lemma we can write $W_{1}$ and $W_{2}$ as pushouts over
$\mathsf{F}(X,X^{\prime})$. Then we obtain the zig-zag of weak equivalences

${W_{1}}$${\mathsf{F}(X,CX^{\prime})\cup_{\mathsf{F}(X,X^{\prime})}\mathsf{F}(Y,Y^{\prime})}$${\mathsf{F}(CX,CX^{\prime})\cup_{\mathsf{F}(X,X^{\prime})}\mathsf{F}(Y,Y^{\prime})}$${W_{2}}$${\mathsf{F}(CX,X^{\prime})\cup_{\mathsf{F}(X,X^{\prime})}\mathsf{F}(Y,Y^{\prime})\,}$${=}$$\scriptstyle{\sim}$${=}$$\scriptstyle{\sim}$

and we let $e\colon W_{1}\to W_{2}$ be the corresponding isomorphism in the
homotopy category. By a diagram chase one obtains commutativity of Eq. 6.10.1.

For Eq. 6.10.2 we consider the commutative diagram

${W_{1}}$${\mathsf{F}(X,CX^{\prime})\cup_{\mathsf{F}(X,X^{\prime})}\mathsf{F}(Y,Y^{\prime})}$${\mathsf{F}(Y,CX^{\prime})\cup_{\mathsf{F}(Y,X^{\prime})}\mathsf{F}(Y,Y^{\prime})}$${W}$${\mathsf{F}(CX,CX^{\prime})\cup_{\mathsf{F}(X,X^{\prime})}\mathsf{F}(Y,Y^{\prime})}$${\mathsf{F}(CX\cup_{X}Y,CX^{\prime})\cup_{\mathsf{F}(Y,X^{\prime})}\mathsf{F}(Y,Y^{\prime})}$${W_{2}}$${\mathsf{F}(CX,X^{\prime})\cup_{\mathsf{F}(X,X^{\prime})}\mathsf{F}(Y,Y^{\prime})}$${\mathsf{F}(CX\cup_{X}Y,CX^{\prime})\cup_{\mathsf{F}(Y,X^{\prime})}\mathsf{F}(Y,Y^{\prime})\makebox[0.0pt][l]{\,.}}$${=}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$${\colonequals}$${=}$$\scriptstyle{\sim}$$\scriptstyle{=}$

Note that the horizontal arrow in the middle is a cofibration using that
$\mathsf{F}$ is biexact and [RB09, Lemma 1.4.1(1)(a)]. The morphism $W_{2}\to
W$ is a pushout of the cofibration
$\mathsf{F}(CX,X^{\prime})\to\mathsf{F}(CX,CX^{\prime})$ and thus a
cofibration as well. Hence their composite, the bottom horizontal arrow, is a
cofibration. We obtain a commutative diagram of connecting morphisms

${\mathsf{F}(Y/X,CX^{\prime}/X^{\prime})}$${\Sigma
W_{1}}$${\mathsf{F}(Y/X,CX^{\prime}/X^{\prime})}$${\Sigma
W}$${\mathsf{F}(CX\cup_{X}Y,CX^{\prime}/X^{\prime})}$${\Sigma
W_{2}}$$\scriptstyle{\delta}$$\scriptstyle{=}$$\scriptstyle{\delta}$$\scriptstyle{\delta}$

in $\operatorname{Ho}({\mathcal{E}})$ and isomorphisms of three exact
triangles. Since $\mathsf{F}(CX\cup_{X}Y,CX^{\prime})$ is a cone for
$\mathsf{F}(CX\cup_{X}Y,X^{\prime})$ we can use Remark 6.9 to obtain an
isomorphism from the bottom exact triangle to the exact triangle

${W_{2}}$${\mathsf{F}(Y,Y^{\prime}/X^{\prime})}$${\Sigma\mathsf{F}(CX\cup_{X}Y,X^{\prime})}$${\Sigma
W_{2}\,,}$

in which the connecting morphism is
$-\Sigma(\mathsf{F}(CX\cup_{X}Y,X^{\prime})\to W_{2})$ and the isomorphism on
the third object is $\zeta$. Finally, using the weak equivalences
$CX\cup_{X}Y\to Z$ and $CX^{\prime}\cup_{X}^{\prime}Y^{\prime}\to Z^{\prime}$
we obtain the commutative diagram Eq. 6.10.2. ∎

###  6.11. Quillen bifunctors

Recall that a bifunctor
$\mathsf{F}\colon{\mathcal{L}}\times{\mathcal{M}}\to{\mathcal{N}}$ between
model categories is a _Quillen bifunctor_ if it has a right adjoint in each
variable and it satisfies the pushout-product axiom: For any cofibrations
$f\colon X\to Y$ in ${\mathcal{L}}$ and $f^{\prime}\colon X^{\prime}\to
Y^{\prime}$ in ${\mathcal{M}}$, the induced morphism

$\mathsf{F}(X,Y^{\prime})\cup_{\mathsf{F}(X,X^{\prime})}\mathsf{F}(Y,X^{\prime})\to\mathsf{F}(Y,Y^{\prime})$

is a cofibration which is an acyclic cofibration if $f$ or $f^{\prime}$ is an
acyclic cofibration. It follows that $\mathsf{F}$ has a total left derived
functor
$\mathrm{L}\mathsf{F}\colon\operatorname{Ho}({\mathcal{L}})\times\operatorname{Ho}({\mathcal{M}})\to\operatorname{Ho}({\mathcal{N}})$;
see [Hov99, Proposition 4.3.1]. For cofibrant objects $X$ of ${\mathcal{L}}$
and $Y$ of ${\mathcal{M}}$ it can be computed by
$\mathrm{L}\mathsf{F}(X,Y)=\mathsf{F}(X,Y)$.

######  Remark 6.12.

The condition that
$\mathsf{F}\colon{\mathcal{L}}\times{\mathcal{M}}\to{\mathcal{N}}$ has a right
adjoint in each variable implies, that the right adjoints uniquely extend to
bifunctors
$\operatorname{hom}^{\ell}\colon{\mathcal{L}}^{\rm{op}}\times{\mathcal{N}}\to{\mathcal{M}}$
and
$\operatorname{hom}^{r}\colon{\mathcal{M}}^{\rm{op}}\times{\mathcal{N}}\to{\mathcal{L}}$
such that the isomorphisms

${\mathcal{L}}(X,\operatorname{hom}^{r}(Y,Z))\cong{\mathcal{N}}(\mathsf{F}(X,Y),Z)\cong{\mathcal{M}}(Y,\operatorname{hom}^{\ell}(X,Z))\,.$

are natural in $X$, $Y$ and $Z$; see [ML98, IV.7,Theorem 3]. This is called an
_adjunction of two variables_ ; see [Hov99, Definition 4.1.12].

If $\mathsf{F}$ is a Quillen bifunctor, then so are the cyclic shifts

$\displaystyle{\mathcal{N}}^{\rm{op}}\times{\mathcal{L}}$
$\displaystyle\to{\mathcal{M}}^{\rm{op}}\,,\quad$ $\displaystyle(Z,X)$
$\displaystyle\mapsto\operatorname{hom}^{\ell}(X,Z)\,,\quad\text{and}$
$\displaystyle{\mathcal{M}}\times{\mathcal{N}}^{\rm{op}}$
$\displaystyle\to{\mathcal{L}}^{\rm{op}}\,,$ $\displaystyle(Y,Z)$
$\displaystyle\mapsto\operatorname{hom}^{r}(Y,Z)\,;$

see [Hov99, Lemma 4.2.2]. Moreover, the adjunction of two variables induces an
adjunction of two variables
$(\mathrm{L}\mathsf{F},\operatorname{Rhom}^{\ell},\operatorname{Rhom}^{r})$ on
homotopy categories; see [Hov99, Proposition 4.3.1].

######  Lemma 6.13.

Let $\mathsf{F}\colon{\mathcal{L}}\times{\mathcal{M}}\to{\mathcal{N}}$ be a
Quillen bifunctor between model categories. Then $\mathsf{F}$ restricts to a
biexact functor of the corresponding categories of cofibrant objects
$\mathsf{F}\colon{\mathcal{L}}_{c}\times{\mathcal{M}}_{c}\to{\mathcal{N}}_{c}$.

###### Proof.

For any cofibrant object $X$ in ${\mathcal{L}}$, the functor
$\mathsf{F}(X,-)\colon{\mathcal{M}}\to{\mathcal{N}}$ is a left Quillen functor
and thus restricts to an exact functor of cofibration categories
$\mathsf{F}(X,-)\colon{\mathcal{M}}_{c}\to{\mathcal{N}}_{c}$. Similarly,
$\mathsf{F}(-,Y)\colon{\mathcal{L}}_{c}\to{\mathcal{N}}_{c}$ is an exact
functor of cofibration categories for any cofibrant object $Y$ of
${\mathcal{M}}$. It now follows from the pushout-product axiom that
$\mathsf{F}\colon{\mathcal{L}}_{c}\times{\mathcal{M}}_{c}\to{\mathcal{N}}_{c}$
is a biexact functor. ∎

Recall that any pointed model category ${\mathcal{M}}$ comes with a suspension
functor
$\Sigma\colon\operatorname{Ho}({\mathcal{M}})\to\operatorname{Ho}({\mathcal{M}})$
on the homotopy category. Moreover, ${\mathcal{M}}$ is called _stable_ if
$\Sigma$ is an equivalence. This is equivalent to the condition that the
category of cofibrant objects ${\mathcal{M}}_{c}$ is stable, as the
suspensions commute with the equivalence
$\operatorname{Ho}({\mathcal{M}}_{c})\to\operatorname{Ho}({\mathcal{M}})$. If
${\mathcal{M}}$ is stable, then we use this equivalence to equip
$\operatorname{Ho}({\mathcal{M}})$ with the triangulated structure coming from
$\operatorname{Ho}({\mathcal{M}}_{c})$. This agrees with the triangulated
structure on $\operatorname{Ho}({\mathcal{M}})$ from [Hov99, §7]; see the
proof of [Hov99, Proposition 6.3.4] to verify that the connecting morphisms
coincide.

The triangulated structure on $\operatorname{Ho}({\mathcal{M}})$ induces a
triangulated structure on its opposite category
$\operatorname{Ho}({\mathcal{M}})^{\rm{op}}$. Alternatively, we can equip
$\operatorname{Ho}({\mathcal{M}})^{\rm{op}}\cong\operatorname{Ho}({\mathcal{M}}^{\rm{op}})$
with the triangulated structure coming from
$\operatorname{Ho}(({\mathcal{M}}^{\rm{op}})_{c})$. The cofibrant objects in
${\mathcal{M}}^{\rm{op}}$ are the fibrant objects in ${\mathcal{M}}$. These
two triangulated structures on $\operatorname{Ho}({\mathcal{M}})^{\rm{op}}$
agree by [Hov99, Theorem 7.1.11] with our convention for the triangulated
structure on the opposite of a triangulated category.

######  Corollary 6.14.

Let $\mathsf{F}\colon{\mathcal{L}}\times{\mathcal{M}}\to{\mathcal{N}}$ be a
Quillen bifunctor between stable model categories. Then the total left derived
functor
$\mathrm{L}\mathsf{F}\colon\operatorname{Ho}({\mathcal{L}})\times\operatorname{Ho}({\mathcal{M}})\to\operatorname{Ho}({\mathcal{N}})$
is biexact and admits a strong Verdier structure. Moreover, the functors
$\operatorname{Rhom}^{\ell}$ and $\operatorname{Rhom}^{r}$ are biexact and
admit strong Verdier structures.

###### Proof.

This follows from Theorem 6.4 and Lemma 6.13. ∎

###  6.15. Monoidal cofibration categories

Monoidal products on triangulated categories are essential examples of biexact
functors. In Section 7.1 we see that many tensor triangulated categories arise
as homotopy categories of stable, monoidal model categories. The unit of the
monoidal product on the model category is often not a cofibrant object, so
that the cofibrant replacement of the unit takes the role of the unit in the
corresponding cofibration category. However, the cofibrant replacement of the
unit need not satisfy the same properties as the unit. This motivates the
following definition.

######  6.16.

A _(weakly unital) monoidal cofibration category_
$({\mathcal{C}},\otimes,\mathbbm{1})$ consists of a cofibration category
${\mathcal{C}}$, an object $\mathbbm{1}$ and a biexact functor
$-\otimes-\colon{\mathcal{C}}\times{\mathcal{C}}\to{\mathcal{C}}$ with a
natural isomorphism

$\alpha\colon X\otimes(Y\otimes Z)\to(X\otimes Y)\otimes Z$

and natural weak equivalences

$\lambda\colon\mathbbm{1}\otimes X\to X\quad\text{and}\quad\rho\colon
X\otimes\mathbbm{1}\to X$

satisfying the following coherence axioms: The two canonical ways to compose
the natural transformations along

(6.16.1) $\begin{gathered}((W\otimes X)\otimes Y)\otimes Z\to
W\otimes(X\otimes(Y\otimes Z))\,,\quad(\mathbbm{1}\otimes Y)\otimes Z\to
Y\otimes Z\,,\\\ (X\otimes\mathbbm{1})\otimes Z\to X\otimes
Z,\quad\text{and}\quad(X\otimes Y)\otimes\mathbbm{1}\to X\otimes
Y\end{gathered}$

are equal.

A monoidal cofibration category is _symmetric_ , if it is equipped with a
natural isomorphism

$\sigma\colon X\otimes Y\to Y\otimes X$

satisfying the following coherence axioms: The two canonical ways to compose
the natural transformations along

$X\otimes\mathbbm{1}\to X\quad\text{and}\quad(X\otimes Y)\otimes Z\to
Z\otimes(X\otimes Y)$

are equal and $\sigma^{2}=\operatorname{id}$.

A _(left) action_ of a monoidal cofibration category
$({\mathcal{C}},\otimes,\mathbbm{1})$ on a cofibration category
${\mathcal{D}}$ consists of a biexact functor
$\mathsf{F}\colon{\mathcal{C}}\times{\mathcal{D}}\to{\mathcal{D}}$ with a
natural isomorphism

$\mathsf{F}(X,\mathsf{F}(Y,Z))\to\mathsf{F}(X\otimes Y,Z)$

and a natural weak equivalence

$\mathsf{F}(\mathbbm{1},X)\to X$

satisfying coherence axioms analogous to a monoidal cofibration category Eq.
6.16.1, except for the last one.

######  Remark 6.17.

A monoidal cofibration category $({\mathcal{C}},\otimes,\mathbbm{1})$ is in
general _not_ a monoidal category since we require $\lambda$ and $\rho$ to be
weak equivalences instead of isomorphisms. As a consequence, the axioms Eq.
6.16.1 are not superfluous, in contrast to the case of a monoidal category;
cf. [Kel64]. Moreover, when $\lambda$ and $\rho$ are isomorphisms, the
coherence axioms Eq. 6.16.1 imply further coherence. The same need not hold
when $\lambda$ and $\rho$ are weak equivalences. Explicitly, the identities
$\lambda(\mathbbm{1})=\rho(\mathbbm{1})$ or
$\sigma(\mathbbm{1},\mathbbm{1})=\operatorname{id}_{\mathbbm{1}\otimes\mathbbm{1}}$
need not be satisfied in a (symmetric) monoidal cofibration category.

######  Example 6.18.

A (symmetric) monoidal model category $({\mathcal{M}},\otimes,\mathbbm{1})$
consists of a model category ${\mathcal{M}}$ and a (symmetric) monoidal
structure $(\otimes,\mathbbm{1})$ on ${\mathcal{M}}$ such that $\otimes$ is a
Quillen bifunctor and the _unit axiom_ holds: For any cofibrant replacement
$Q\mathbbm{1}\to\mathbbm{1}$ and any cofibrant object $X$ the morphisms

$Q\mathbbm{1}\otimes X\to\mathbbm{1}\otimes X\quad\text{and}\quad X\otimes
Q\mathbbm{1}\to X\otimes\mathbbm{1}$

are weak equivalences.

If ${\mathcal{M}}$ is a (symmetric) monoidal model category, then its category
of cofibrant objects ${\mathcal{M}}_{c}$ together with the restriction of the
tensor product and a cofibrant replacement of the unit of ${\mathcal{M}}$ is a
(symmetric) monoidal cofibration category.

Let ${\mathcal{M}}$ be a monoidal model category and ${\mathcal{N}}$ a (left)
${\mathcal{M}}$-model category; see [Hov99, Definition 4.2.18]. Then the
restriction of the action ${\mathcal{M}}\times{\mathcal{N}}\to{\mathcal{N}}$
to the cofibrant objects together with a cofibrant replacement of the tensor
unit of ${\mathcal{M}}$ defines an action of the monoidal cofibration category
${\mathcal{M}}_{c}$ on the cofibration category ${\mathcal{N}}_{c}$.

######  Theorem 6.19.

Let $({\mathcal{C}},\otimes,\mathbbm{1})$ be a (symmetric) monoidal stable
cofibration category. Then
$(\operatorname{Ho}({\mathcal{C}}),\operatorname{Ho}(\otimes),\gamma(\mathbbm{1}))$
is a (symmetric) monoidal triangulated category, and the induced monoidal
product admits a strong Verdier structure.

###### Proof.

By Theorem 6.4, the functor $\operatorname{Ho}(\otimes)$ is biexact and admits
a strong Verdier structure. Using Lemma 6.2, the natural isomorphism $\alpha$,
the natural weak equivalences $\lambda$, $\rho$, and if available the symmetry
$\sigma$ induce natural isomorphisms of exact functors in each variable on the
homotopy category. These provide the (symmetric) monoidal structure for
$(\operatorname{Ho}({\mathcal{C}}),\operatorname{Ho}(\otimes),\gamma(\mathbbm{1}))$.
∎

######  6.20.

Let $({\mathcal{T}},\otimes,\mathbbm{1})$ by a tensor triangulated category.
We assume $X\otimes-$ has a right adjoint $\operatorname{hom}(X,-)$ for any
$X\in{\mathcal{T}}$. Then
$\operatorname{hom}\colon{\mathcal{T}}^{\rm{op}}\times{\mathcal{T}}\to{\mathcal{T}}$
is a bifunctor, called _internal hom functor_. There are induced natural
isomorphisms $\vartheta$ and $\zeta$ as in 3.1 satisfying Eq. 3.1.1 and
$\operatorname{hom}$ is exact in the second variable. We say
$({\mathcal{T}},\otimes,\mathbbm{1})$ is _closed_ , if $\operatorname{hom}$ is
also exact in the first variable; see [HPS97, Definition A.2.1].

The homotopy category of a symmetric monoidal stable model category is a
closed tensor triangulated category by [Hov99, Theorem 6.6.4] together with
Cisinski’s proof of [Hov99, Conjecture 5.7.5] in [Cis08, Corollaire 6.8]. We
provide a new proof that does not rely on [Hov99, Conjecture 5.7.5]. In
addition, we establish strong Verdier structures for the monoidal product and
the internal hom on $\operatorname{Ho}({\mathcal{M}})$.

######  Theorem 6.21.

Let $({\mathcal{M}},\otimes,\mathbbm{1})$ be a symmetric monoidal stable model
category. Then
$(\operatorname{Ho}({\mathcal{M}}),\operatorname{Ho}(\otimes),\gamma(\mathbbm{1}))$
is a closed symmetric monoidal triangulated category, and the induced monoidal
product and internal hom functor admit a strong Verdier structure.

###### Proof.

By Theorem 6.19,
$(\operatorname{Ho}({\mathcal{M}}),\operatorname{Ho}(\otimes),\gamma(\mathbbm{1}))$
is a tensor triangulated category and $\operatorname{Ho}(\otimes)$ admits a
strong Verdier structure. Since $\otimes$ is a Quillen bifunctor, the
corresponding internal hom functor on ${\mathcal{M}}$ induces a biexact
functor admitting a strong Verdier structure on
$\operatorname{Ho}({\mathcal{M}})$ by Corollary 6.14. In particular, the
tensor triangulated category
$(\operatorname{Ho}({\mathcal{M}}),\operatorname{Ho}(\otimes),\gamma(\mathbbm{1}))$
is closed. ∎

The existence of a strong Verdier structure for $\operatorname{Ho}(\otimes)$
is known by the analogous result for monoidal stable derivators [GPS14a,
Theorem 6.2, Lemma 6.8] as any symmetric monoidal stable model category
${\mathcal{M}}$ has an associated monoidal stable derivator. If in addition
${\mathcal{M}}$ is cofibrantly generated, then the associated monoidal
derivator is closed; see [GPS14a, Theorem 9.13]. In this case, the internal
hom functor admits a strong Verdier structure by the analogous result for
derivators [JY21, Proposition 4.1.9].

##  7\. Examples

In the following we discuss examples of bifunctors that admit a strong Verdier
structure so that Theorem 3.6 applies. Many of these functors are monoidal
products of a tensor triangulated category or actions of a tensor triangulated
category. Any action of a tensor triangulated category induces elements in the
center so that Theorem 4.10 holds for the corresponding Koszul objects.

###  7.1. Tensor triangulated categories

Recall from 4.4 and 6.20 that a closed tensor triangulated category
$({\mathcal{T}},\otimes,\mathbbm{1})$ consists of a triangulated category
${\mathcal{T}}$ and a compatible closed symmetric monoidal product $\otimes$
with unit $\mathbbm{1}$. In particular, the monoidal product $\otimes$ and the
internal hom are biexact functors in the sense of 3.1.

######  7.2.

Each of the following tensor triangulated categories is the homotopy category
of a symmetric monoidal stable model category. In particular, Theorem 3.6
holds for the monoidal product and the internal hom functor, and Theorem 4.10
for elements induced by endomorphisms of the unit.

  1. (1)

The derived category ${\rm D}(\operatorname{Mod}(R))$ of modules over a
commutative ring $R$ with monoidal product $\otimes^{\mathbf{L}}_{R}$ and unit
$R$. The endomorphism ring is $\operatorname{End}^{*}(R)=R$. A corresponding
monoidal model category is the category $\operatorname{Ch}(R)$ of unbounded
chain complexes over $R$ with the projective model structure; see [Hov99,
Proposition 4.2.13].

  2. (2)

The derived category ${\rm D}(\operatorname{Mod}(RG))$ for a commutative ring
$R$ and a finite group $G$ with monoidal product $\otimes^{\mathbf{L}}_{R}$
and unit $R$. The endomorphism ring is
$\operatorname{End}^{*}(R)=\operatorname{H}^{*}(G;R)$, the group cohomology
ring. A corresponding model category is the category of unbounded chain
complexes $\operatorname{Ch}(RG)$ with the projective model structure. The
monoidal product on $\operatorname{Ch}(RG)$ is $\otimes_{R}$ with the diagonal
$G$-action. There is a natural isomorphism

$\operatorname{Ch}(RG)(X\otimes_{R}Y,Z)\cong\operatorname{Ch}(R)(X,\operatorname{hom}_{R}(Y,Z))\,,$

where $\operatorname{hom}_{R}(Y,Z)$ is the hom-complex over $R$ equipped with
the conjugation action. The forgetful functor
$\operatorname{Ch}(RG)\to\operatorname{Ch}(R)$ is left Quillen and in
particular preserves cofibrations. By the definition of the projective model
structure, a morphism in $\operatorname{Ch}(RG)$ is a fibration or weak
equivalence if and only if it is so in $\operatorname{Ch}(R)$. Since
$\operatorname{hom}_{R}(-,-)$ on $\operatorname{Ch}(R)$ is a Quillen bifunctor
it now follows that $\operatorname{hom}_{R}(-,-)$ on $\operatorname{Ch}(RG)$
is a Quillen bifunctor and hence so is $\otimes_{R}$. The unit axiom can be
deduced from the unit axiom for $\operatorname{Ch}(R)$ as well.

  3. (3)

The stable module category $\underline{\operatorname{Mod}}(H)$ of a finite
dimensional cocommutative Hopf algebra $H$ over a field $k$ with monoidal
product $\otimes_{k}$ and unit $k$; see [Hov99, Proposition 4.2.15]. This
includes in particular the stable module category
$\underline{\operatorname{Mod}}(kG)$ of a group algebra $kG$ of finite group
$G$. For a group algebra, the endomorphism ring is
$\operatorname{End}^{*}(k)=\hat{\operatorname{H}}^{*}(G,k)$, the Tate
cohomology ring.

  4. (4)

The homotopy category of spectra in stable homotopy theory; see for example
[MMSS01].

  5. (5)

The homotopy category of equivariant spectra; see [MM02].

  6. (6)

The motivic stable category in $\mathbb{A}^{1}$-homotopy theory; see [Jar00].

######  7.3.

We obtain many examples for actions of tensor triangulated categories that
admit a strong Verdier structure by restricting the monoidal structures from
7.2 to suitable subcategories.

  1. (1)

Let $R$ be a commutative noetherian ring. Then $\operatorname{Perf}(R)$, the
full subcategory of bounded complexes of finitely generated projective modules
of ${\rm D}(\operatorname{Mod}(R))$, is a tensor triangulated category with
the restricted monoidal product. Further, it acts on ${\rm
D}_{b}(\operatorname{mod}(R))$, the bounded derived category of finitely
generated modules.

  2. (2)

Let $R$ be a commutative noetherian ring and $G$ a finite group. Then
$\operatorname{Perf}(RG)$ inherits the monoidal structure from ${\rm
D}(\operatorname{Mod}(RG))$ and acts on ${\rm D}_{b}(\operatorname{mod}(RG))$;
cf. [BKSS20, Section 8].

###  7.4. Tensor products of bimodules

Under reasonable conditions a stable model structure on a monoidal category
transfers to stable model structures on categories of bimodules. We will show
in 7.8 that tensoring over a monoid induces a biexact functor on homotopy
categories that admits a strong Verdier structure. Moreover, the derived
tensor product is part of an adjunction of two variables and the adjoints are
biexact functors that admit a strong Verdier structure as well.

######  7.5.

Let $({\mathcal{C}},\otimes,\mathbbm{1})$ be a biclosed monoidal category,
that is $\otimes$ has a right adjoint in each variable denoted
$\operatorname{hom}^{\ell}$ and $\operatorname{hom}^{r}$. We further assume
that equalizers and coequalizers exist. We briefly recall basic constructions
for bimodules; for more details see [Bar96].

For monoids $A$ and $B$ in ${\mathcal{M}}$, we write
$\operatorname{Bimod}(A,B)$ for the category of $A$-$B$-bimodules. If the
monoidal structure on ${\mathcal{C}}$ is symmetric, then
$\operatorname{Bimod}(A,B)$ is isomorphic to the category of left $A\otimes
B^{\rm{op}}$-modules. The category of left $A$-modules $\operatorname{Mod}(A)$
can be identified with $\operatorname{Bimod}(A,\mathbbm{1})$ where
$\mathbbm{1}$ is the trivial monoid.

We write $X\otimes_{B}Y$ for the tensor product of a right $B$-module $X$ with
a left $B$-module $Y$ defined by the coequalizer of $X\otimes B\otimes
Y\rightrightarrows X\otimes Y$. Since $\otimes$ preserves coequalizers in each
variable, we obtain a bifunctor

$-\otimes_{B}-\colon\operatorname{Bimod}(A,B)\otimes\operatorname{Bimod}(B,C)\to\operatorname{Bimod}(B,C)\,,$

where $C$ is a monoid in ${\mathcal{C}}$ as well.

If $X$ is an $A$-$B$-bimodule and $Z$ is an $A$-$C$-bimodule, then
$\operatorname{hom}^{\ell}_{A}(X,Z)$ is a $B$-$C$-bimodule with $B$-action

$B\otimes\operatorname{hom}^{\ell}_{A}(X,Z)\to\operatorname{hom}^{\ell}_{A}(X,Z)$

induced by the adjoint transpose of

$X\otimes B\otimes\operatorname{hom}^{\ell}_{A}(X,Z)\to X\otimes
B\otimes\operatorname{hom}^{\ell}(X,Z)\to
X\otimes\operatorname{hom}^{\ell}(X,Z)\to Z\,,$

and $C$-action

$\operatorname{hom}^{\ell}_{A}(X,Z)\otimes
C\to\operatorname{hom}^{\ell}_{A}(X,Z)$

induced by the adjoint transpose of

$X\otimes\operatorname{hom}^{\ell}_{A}(X,Z)\otimes C\to
X\otimes\operatorname{hom}^{\ell}(X,Z)\otimes C\to Z\otimes C\to Z\,.$

We obtain a bifunctor

$\operatorname{hom}^{\ell}_{A}(-,-)\colon\operatorname{Bimod}(A,B)^{\rm{op}}\times\operatorname{Bimod}(A,C)\to\operatorname{Bimod}(B,C)\,.$

Similarly, we have a bifunctor

$\operatorname{hom}^{r}_{C}(-,-)\colon\operatorname{Bimod}(B,C)^{\rm{op}}\times\operatorname{Bimod}(A,C)\to\operatorname{Bimod}(A,B)\,.$

The tensor product over a monoid is a left adjoint in each variable.
Explicitly, there are natural isomorphisms

$\displaystyle\operatorname{Bimod}(B,C)(Y,\operatorname{hom}^{\ell}_{A}(X,Z))$
$\displaystyle\cong\operatorname{Bimod}(A,C)(X\otimes_{B}Y,Z)$
$\displaystyle\cong\operatorname{Bimod}(A,B)(X,\operatorname{hom}^{r}_{C}(Y,Z))$

for $X\in\operatorname{Bimod}(A,B)$, $Y\in\operatorname{Bimod}(B,C)$ and
$Z\in\operatorname{Bimod}(A,C)$ defined as follows. For a morphism of
bimodules $Y\to\operatorname{hom}^{\ell}_{A}(X,Z)$, the adjoint transpose of

$Y\to\operatorname{hom}^{\ell}_{A}(X,Z)\to\operatorname{hom}^{\ell}(X,Z)$

induces a morphism $X\otimes_{B}Y\to Z$ of $A$-$C$-bimodules. It is
straightforward to check that this assignment defines a natural isomorphism.
Analogously, one defines the second natural isomorphism.

In addition to the monoids $A$, $B$, and $C$, consider a monoid $D$. There are
natural isomorphisms

$X\otimes_{B}(Y\otimes_{C}Z)\to(X\otimes_{B}Y)\otimes_{C}Z\,,\quad
A\otimes_{A}X\to X\quad\text{and}\quad X\otimes_{B}B\to X$

for any bimodules $X\in\operatorname{Bimod}(A,B)$,
$Y\in\operatorname{Bimod}(B,C)$, and $Z\in\operatorname{Bimod}(C,D)$. The
associativity isomorphism is induced by the associativity of ${\mathcal{C}}$
using that $\otimes$ preserves coequalizers in each variable. The unit
isomorphisms are induced by the structure maps of $X$ as an $A$-module and as
a $B$-module, respectively.

This equips the monoids in ${\mathcal{C}}$, bimodules, and morphisms of
bimodules with the structure of a biclosed bicategory; see [Bar96]. In
particular, $\operatorname{Bimod}(A,A)$ is a biclosed monoidal category with
monoidal product $\otimes_{A}$ and unit $A$.

######  7.6.

Let $({\mathcal{M}},\otimes,\mathbbm{1})$ be a cofibrantly generated monoidal
model category. In particular, the monoidal product
$\otimes\colon{\mathcal{M}}\times{\mathcal{M}}\to{\mathcal{M}}$ is a Quillen
bifunctor and thus part of an adjunction of two variables
$(\otimes,\operatorname{hom}^{\ell},\operatorname{hom}^{r})$.

We say the cofibrantly generated model structure on ${\mathcal{M}}$ _(right)
transfers_ to the category of bimodules $\operatorname{Bimod}(A,B)$, if
$\operatorname{Bimod}(A,B)$ is a cofibrantly generated model category with
generating (trivial) cofibrations $A\otimes i\otimes B$ for generating
(trivial) cofibrations $i$ of ${\mathcal{M}}$. By adjunction, this implies
that the weak equivalences and fibrations of $\operatorname{Bimod}(A,B)$ are
the bimodule morphisms whose underlying morphisms in ${\mathcal{M}}$ are weak
equivalences and fibrations, respectively. For sufficient conditions and
examples when ${\mathcal{M}}$ is symmetric monoidal see [SS00, Theorem 4.1 and
Section 5].

######  Lemma 7.7.

Suppose the cofibrantly generated model structure on ${\mathcal{M}}$ transfers
to $\operatorname{Bimod}(A,B)$ for any monoids $A$ and $B$. Then

$-\otimes_{B}-\colon\operatorname{Bimod}(A,B)\times\operatorname{Bimod}(B,C)\to\operatorname{Bimod}(A,C)$

is a Quillen bifunctor for any monoids $A$, $B$ and $C$ in ${\mathcal{M}}$.

###### Proof.

The bifunctor $-\otimes_{B}-$ is part of an adjunction of two variables by
7.5. The pushout-product axiom can be checked on generating cofibrations and
generating trivial cofibrations by [Hov99, Corollary 4.2.5]. In our situation
this holds, because for any morphisms $f$, $g$ in ${\mathcal{M}}$, the
pushout-product of $A\otimes f\otimes B$ and $B\otimes g\otimes C$ is the
pushout-product of $f$ and $g$ tensored by $A$ from the left and by $C$ from
the right, and the functor $A\otimes-\otimes C$ is a left Quillen functor from
${\mathcal{M}}$ to $\operatorname{Bimod}(A,C)$. ∎

######  Proposition 7.8.

Let $({\mathcal{M}},\otimes,\mathbbm{1})$ be a monoidal stable model category.
Suppose the model structure on ${\mathcal{M}}$ is cofibrantly generated and
transfers to $\operatorname{Bimod}(A,B)$ for any monoids $A$ and $B$ such that
$\operatorname{Bimod}(A,B)$ is a stable model category. Then the total derived
functors

$\displaystyle-\otimes^{\mathbf{L}}_{B}-\colon\operatorname{Ho}(\operatorname{Bimod}(A,B))\times\operatorname{Ho}(\operatorname{Bimod}(B,C))\to\operatorname{Ho}(\operatorname{Bimod}(A,C))\,,$
$\displaystyle\operatorname{Rhom}^{r}_{C}(-,-)\colon\operatorname{Ho}(\operatorname{Bimod}(B,C))^{\rm{op}}\times\operatorname{Ho}(\operatorname{Bimod}(A,C))\to\operatorname{Ho}(\operatorname{Bimod}(A,B))\,,$
$\displaystyle\operatorname{Rhom}^{\ell}_{A}(-,-)\colon\operatorname{Ho}(\operatorname{Bimod}(A,B))^{\rm{op}}\times\operatorname{Ho}(\operatorname{Bimod}(A,C))\to\operatorname{Ho}(\operatorname{Bimod}(B,C))$

are biexact and admit a strong Verdier structure.

###### Proof.

Since $-\otimes_{B}-$ is a Quillen bifunctor by 7.7, its total left derived
functor is biexact and admits a strong Verdier structure by Corollary 6.14.
The functors $\operatorname{hom}^{r}_{C}(-,-)$ and
$\operatorname{hom}^{\ell}_{A}(-,-)$ are up to switching arguments in the case
of $\operatorname{hom}^{\ell}_{A}(-,-)$ the opposite functors of the cyclic
shifts of $-\otimes_{B}-$; see Remark 6.12. Since the the cyclic shifts are
Quillen bifunctors, their total left derived functors are biexact and admit a
strong Verdier structure, hence so do the opposites of these left derived
functors, that is $\operatorname{Rhom}^{r}_{C}(-,-)$ and
$\operatorname{Rhom}^{\ell}_{A}(-,-)$. ∎

######  Example 7.9.

Let ${\mathcal{M}}=\operatorname{Ch}(R)$ be the model category of unbounded
chain complexes over a commutative ring $R$ equipped with the projective model
structure in which the weak equivalences are the quasi-isomorphisms and the
fibrations are the degreewise epimorphisms; see [Hov99, Theorem 2.3.11]. A
monoid $A$ in $\operatorname{Ch}(R)$ is a differential graded $R$-algebra and
the category of $A$-modules is the category of differential graded modules
over $A$. The model structure of $\operatorname{Ch}(R)$ transfers to
$\operatorname{Mod}(A)$; see [SS00, Lemma 2.3]. Since every object of
$\operatorname{Ch}(R)$ is fibrant, so is every object of
$\operatorname{Mod}(A)$. To check that the model structure on
$\operatorname{Mod}(A)$ is stable, we consider the loop space functor $\Omega$
instead of the suspension. The loop space functor is the suspension in
$\operatorname{Mod}(A)^{\rm{op}}$ and for a stable model category it yields
the inverse of the suspension functor in the homotopy category. We set
$D^{1}=\operatorname{cone}(\operatorname{id}_{R})$, the chain complex with $R$
in degrees one and zero and the identity as differential. Let $X$ be an
$A$-module. Then

$\operatorname{hom}^{r}_{R}(D^{1},X)\to\operatorname{hom}^{r}_{R}(R,X)\cong X$

is a fibration and $\operatorname{hom}^{r}_{R}(D^{1},X)$ is weakly
contractible. Its fiber $\Omega(X)$, that is the pullback along $0\to R$, is a
shift of $X$. Thus $\Omega$ is an equivalence on
$\operatorname{Ho}(\operatorname{Mod}(A))$. The homotopy category of
$\operatorname{Mod}(A)$ is the derived category ${\rm D}(A)$ of differential
graded modules over $A$.

Let $A$, $B$, $C$ be differential graded algebras over a commutative ring $R$.
The derived functors

$\displaystyle-\otimes^{\mathbf{L}}_{B}-\colon{\rm D}(A\otimes
B^{\rm{op}})\times{\rm D}(B\otimes C^{\rm{op}})\to{\rm D}(A\otimes
C^{\rm{op}})\,,$
$\displaystyle\operatorname{RHom}_{C^{\rm{op}}}(-,-)\colon({\rm D}(B\otimes
C^{\rm{op}}))^{\rm{op}}\times{\rm D}(A\otimes C^{\rm{op}})\to{\rm D}(A\otimes
B^{\rm{op}})\quad\text{and}$
$\displaystyle\operatorname{RHom}_{A}(-,-)\colon({\rm D}(A\otimes
B^{\rm{op}}))^{\rm{op}}\times{\rm D}(A\otimes C^{\rm{op}})\to{\rm D}(B\otimes
C^{\rm{op}})\,.$

are biexact and admit a strong Verdier structure.

######  Example 7.10.

Let $R$ be a commutative ring and $S$ an associative algebra over $R$. We
denote by ${S}^{e}_{R}\colonequals S\otimes^{\mathbf{L}}_{R}S^{\rm{op}}$ the
_derived enveloping algebra_ of $S$ with respect to $R$. By choosing a
projective differential graded algebra resolution $A\xrightarrow{\sim}S$ over
$R$, we can view $A\otimes_{R}A^{\rm{op}}\xrightarrow{\sim}{S}^{e}_{R}$ as a
differential graded algebra.

By Example 7.9, we have a monoidal triangulated category ${\rm
D}(\operatorname{Mod}({S}^{e}_{R}))$ with monoidal product
$\otimes^{\mathbf{L}}_{S}$ and unit $S$ acting on ${\rm
D}(\operatorname{Mod}(S))$ via $\otimes^{\mathbf{L}}_{S}$. The endomorphism
ring of ${\rm D}(\operatorname{Mod}({S}^{e}_{R}))$ is

$\operatorname{Ext}^{*}_{{S}^{e}_{R}}(S,S)\equalscolon\operatorname{HH}^{*}(S/R)\,,$

the Shukla cohomology of $S$ over $R$. When $R=k$ a field, then this coincides
with Hochschild cohomology. In particular, Theorem 4.10 holds for any sequence
in $\operatorname{HH}^{*}(S/R)$.

## References

  * [ABIM10] Luchezar L. Avramov, Ragnar-Olaf Buchweitz, Srikanth B. Iyengar, and Claudia Miller, _Homology of perfect complexes_ , Adv. Math. 223 (2010), no. 5, 1731–1781. MR 2592508
  * [AIN18] Luchezar L. Avramov, Srikanth B. Iyengar, and Amnon Neeman, _Big Cohen–Macaulay modules, morphisms of perfect complexes, and intersection theorems in local algebra_ , Doc. Math. 23 (2018), 1601–1619. MR 3890961
  * [AL23] Ettore Aldrovandi and Cynthia Lester, _On multi-determinant functors for triangulated categories_ , Theory Appl. Categ. 39 (2023), 769–803. MR 4640338
  * [Bar96] Michael Barr, _The Chu construction_ , Theory Appl. Categ. 2 (1996), No. 2, 17–35. MR 1393936
  * [BF08] Ragnar-Olaf Buchweitz and Hubert Flenner, _Global Hochschild (co-)homology of singular spaces_ , Adv. Math. 217 (2008), no. 1, 205–242. MR 2357326
  * [BH98] Winfried Bruns and Jürgen Herzog, _Cohen–Macaulay rings_ , revised ed., Cambridge Stud. Adv. Math., vol. 39, Cambridge Univ. Press, Cambridge, 1998\. MR 1251956
  * [BIK08] Dave Benson, Srikanth B. Iyengar, and Henning Krause, _Local cohomology and support for triangulated categories_ , Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), no. 4, 573–619. MR 2489634
  * [BIKO10] Petter Andreas Bergh, Srikanth B. Iyengar, Henning Krause, and Steffen Oppermann, _Dimensions of triangulated categories via Koszul objects_ , Math. Z. 265 (2010), no. 4, 849–864. MR 2652539
  * [BKSS20] Aslak Bakke Buan, Henning Krause, Nicole Snashall, and Øyvind Solberg, _Support varieties—an axiomatic approach_ , Math. Z. 295 (2020), no. 1-2, 395–426. MR 4100008
  * [Bro73] Kenneth S. Brown, _Abstract homotopy theory and generalized sheaf cohomology_ , Trans. Amer. Math. Soc. 186 (1973), 419–458. MR 341469
  * [BvdB03] Alexey I. Bondal and Michel van den Bergh, _Generators and representability of functors in commutative and noncommutative geometry_ , Mosc. Math. J. 3 (2003), no. 1, 1–36, 258. MR 1996800
  * [CF22] J. Daniel Christensen and Martin Frankland, _On good morphisms of exact triangles_ , J. Pure Appl. Algebra 226 (2022), no. 3, Paper No. 106846, 33. MR 4289720
  * [Chr98] J. Daniel Christensen, _Ideals in triangulated categories: phantoms, ghosts and skeleta_ , Adv. Math. 136 (1998), no. 2, 284–339. MR 1626856
  * [Cis08] Denis-Charles Cisinski, _Propriétés universelles et extensions de Kan dérivées_ , Theory Appl. Categ. 20 (2008), No. 17, 605–649. MR 2534209
  * [GPS14a] Moritz Groth, Kate Ponto, and Michael Shulman, _The additivity of traces in monoidal derivators_ , J. K-Theory 14 (2014), no. 3, 422–494. MR 3349323
  * [GPS14b] by same author, _Mayer-Vietoris sequences in stable derivators_ , Homology Homotopy Appl. 16 (2014), no. 1, 265–294. MR 3211746
  * [GŠ18] Moritz Groth and Jan Šťovíček, _Tilting theory via stable homotopy theory_ , J. Reine Angew. Math. 743 (2018), 29–90. MR 3859269
  * [GZ67] Peter Gabriel and Michel Zisman, _Calculus of fractions and homotopy theory_ , Springer-Verlag New York, Inc., New York,, 1967. MR 210125
  * [Hov99] Mark Hovey, _Model categories_ , Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, Providence, RI, 1999. MR 1650134
  * [HPS97] Mark Hovey, John H. Palmieri, and Neil P. Strickland, _Axiomatic stable homotopy theory_ , vol. 128, Mem. Amer. Math. Soc., no. 610, Amer.Math. Soc., Providence, RI., 1997. MR 1388895
  * [Jar00] J. F. Jardine, _Motivic symmetric spectra_ , Doc. Math. 5 (2000), 445–552. MR 1787949
  * [JY21] Fangzhou Jin and Enlin Yang, _Künneth formulas for motives and additivity of traces_ , Adv. Math. 376 (2021), Paper No. 107446, 83. MR 4178918
  * [Kel64] G. Maxwell Kelly, _On MacLane’s conditions for coherence of natural associativities, commutativities, etc_ , J. Algebra 1 (1964), 397–402. MR 0182649
  * [KN02] Bernhard Keller and Amnon Neeman, _The connection between May’s axioms for a triangulated tensor product and Happel’s description of the derived category of the quiver $D_{4}$_, Doc. Math. 7 (2002), 535–560. MR 2015053
  * [KS06] Masaki Kashiwara and Pierre Schapira, _Categories and sheaves_ , Grundlehren Math. Wiss., vol. 332, Springer-Verlag, Berlin, 2006. MR 2182076
  * [KY11] Henning Krause and Yu Ye, _On the centre of a triangulated category_ , Proc. Edinb. Math. Soc. (2) 54 (2011), no. 2, 443–466. MR 2794666
  * [Let23] Janina C. Letz, _Brown representability for triangulated categories with a linear action by a graded ring_ , Arch. Math. (Basel) 120 (2023), no. 2, 135–146.
  * [May01] J. Peter May, _The additivity of traces in triangulated categories_ , Adv. Math. 163 (2001), no. 1, 34–73. MR 1867203
  * [ML98] Saunders Mac Lane, _Categories for the working mathematician_ , second ed., Grad. Texts in Math., vol. 5, Springer-Verlag, New York, 1998. MR 1712872
  * [MM02] M. A. Mandell and J. P. May, _Equivariant orthogonal spectra and $S$-modules_, Mem. Amer. Math. Soc. 159 (2002), no. 755, x+108. MR 1922205
  * [MMSS01] M. A. Mandell, J. P. May, S. Schwede, and B. Shipley, _Model categories of diagram spectra_ , Proc. London Math. Soc. (3) 82 (2001), no. 2, 441–512. MR 1806878
  * [Nee91] Amnon Neeman, _Some new axioms for triangulated categories_ , J. Algebra 139 (1991), no. 1, 221–255. MR 1106349
  * [Nee01] by same author, _Triangulated categories_ , Ann. of Math. Stud., vol. 148, Princeton University Press, Princeton, NJ, 2001. MR 1812507
  * [RB09] Andrei Radulescu-Banu, _Cofibrations in homotopy theory_ , arXiv e-prints, 2009, arXiv:math/0610009v4, pp. 1–160.
  * [Rou08] Raphaël Rouquier, _Dimensions of triangulated categories_ , J. K-Theory 1 (2008), no. 2, 193–256. MR 2434186
  * [Sch13] Stefan Schwede, _The $p$-order of topological triangulated categories_, J. Topol. 6 (2013), no. 4, 868–914. MR 3145143
  * [SS00] Stefan Schwede and Brooke E. Shipley, _Algebras and modules in monoidal model categories_ , Proc. Lond. Math. Soc. (3) 80 (2000), no. 2, 491–511. MR 1734325
  * [SZ19] Manuel Saorín and Alexander Zimmermann, _Symmetry of the definition of degeneration in triangulated categories_ , Algebr. Represent. Theory 22 (2019), no. 4, 801–836. MR 3985141
  * [Szu17] Karol Szumiło, _Homotopy theory of cocomplete quasicategories_ , Algebr. Geom. Topol. 17 (2017), no. 2, 765–791. MR 3623671