on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
For $\mathcal{T}$ a topos, a Cisinski model structure on $\mathcal{T}$ is a model category structure on $\mathcal{T}$ such that
the cofibrations are precisely the monomorphisms;
it is a cofibrantly generated model category.
Since every topos is a locally presentable category, a Cisinski model structure is in particular a combinatorial model category structure.
Since every topos is an adhesive category, monomorphisms are automatically preserved by pushout.
Say a class $W \in Mor(\mathcal{T})$ is a localizer on $\mathcal{T}$ if it is a class of weak equivalences in a Cisinski model structure on $\mathcal{T}$.
Every small set of morphisms $\Sigma \in Mor(\mathcal{T})$ is contained in a smallest localizer, def. 2, $W(\Sigma)$.
One says that $W(\Sigma)$ is the localiser generated by $\Sigma$.
So in particular a Cisinski model structure always exists.
We discuss how on a presheaf topos equipped with a suitable notion of cylinder objects Cisinski model structures can be characterized fairly explicitly. After some preliminaries, the main statement is theorem 1 below.
(This follows sections 1.2 and 1.3 of Cisinski 06).
Let $A$ be a small category. Write $PSh(A)$ or $[A^{op}, Set]$ for the category of presheaves over $A$. We introduce here, culminating in def. 11 below, the ingredients of a homotopical structure on $PSh(A)$, which is a choice of functorial cylinder object together with a compatible notion of anodyne extensions. Further below in def. 12 this defines a model category structure on $PSh(A)$.
For $X$ in $PSh(A)$, a cylinder on $X$ in the following means a cylinder object, denoted $I \otimes X$, factoring the codiagonal
such that the first morphism is a monomorphism.
A homomorphism of such cyclinders is a pair of morphisms $X \to Y$ and $I \otimes X \to I \otimes Y$ in $PSh(A)$, making the evident squares commute. This defines a category $Cyl(A)$ of cylinder objects on presheaves on $A$, equipped with a forgetful functor $Cyl(A) \to PSh(A)$ that sends a cylinder $I \otimes X$ to its underlying object $X$.
A functorial cylinder object over $A$ is a section of this functor.
This is (Cisinski 06, def. 1.3.1).
In the following, let $J : PSh(A) \to Cyl(A)$ be a choice of functorial cylinder object. Equivalently, this is a choice of endofunctor
equipped with natural transformations
where $(\partial I) \otimes X := X \coprod X$, such that the composite is the functorial codiagonal, and where the first transformation is a monomorphism.
For $f,g : X \to Y$ two morphisms in $PSh(A)$, we say an elementary $J$-homotopy $f \Rightarrow g$ from $f$ to $g$ is left homotopy from $f$ to $g$ with respect to the chosen cylinder object $J$, hence a morphism $\eta : I \otimes X \to Y$ fitting into a diagram
We say $J$-homotopy for the equivalence relation generated by this.
$J$-homotopy is compatible with composition in $PSh(A)$.
It is sufficient to show that elementary $J$-homotopies are compatible with composition.
So for
an elementary $J$-homotopy $f \Rightarrow g$, and for
one $f' \Rightarrow g'$, we obtain an elementary homotopy $f' \circ f \Rightarrow g' \circ f$ by forming
and then an elementary $J$-homotopy $g' \circ f \Rightarrow g '\circ g$ by forming
Together this generates a $J$-homotopy $f' \circ f \Rightarrow g' \circ g$.
Hence the following is well defined.
Write $Ho_J(A)$ for the category whose objects are those of $PSh(A)$, and whose morphisms are $J$-homotopy equivalence classes of morphisms in $PSh(A)$ – the $J$-homotopy category. Write
for the projection functor.
A morphism $f : X \to Y$ in $PSh(A)$ is called a $J$-homotopy equivalence if it is sent by $Q$ to an isomorphism.
An object $X \in PSh(A)$ is called $J$-contractible if $X \to *$ is a $J$-homotopy equivalence.
A morphism $f : X \to Y$ in $PSh(A)$ is called an acyclic fibration if it has the right lifting property against all monomorphisms.
Every acyclic fibration in $PSh(A)$ is a $J$-homotopy equivalence.
More is true: every trivial fibration $p : X \to Y$
has a section $s : Y \to X$;
which is also a $J$-homotopy left inverse;
by an elementary $J$-homotopy $h : id \Rightarrow s \circ p$ which satisfies $p \circ h = p \circ \sigma_X$.
The existence of the section $s : Y \to X$ follows by right lifting against the monomorphism $\emptyset \to Y$ (out of the initial object)
The $J$-homotopy $h$ is obtained by lifting in the diagram
An elementary homotopical datum on $PSh(A)$ is a functorial cylinder $J$, def. 3, such that
the functor $I \otimes (-)$ commutes with small colimits;
for all monomorphisms $f : K \to L$ the diagrams
for $e \in \{0,1\}$ is a pullback square.
For
a pullback square in $PSh(A)$ of two monomorphisms $S \hookrightarrow U$ and $T \hookrightarrow U$, the universal morphism out of the pushout
is also a monomorphism, usually written as the morphism out of the union
All this follows, for instance, from the corresponding statements in Set, over each object of $A$.
Let
for $\epsilon = 0,1$, be the subfunctor which is the image of $\partial^\epsilon : Id_{PSh(A)} \to I \otimes (-)$.
This way for any $X \in PSh(A)$ the boundary inclusions $\partial^\epsilon_X : X \to I \otimes X$ are identified with
The second condition on an elementary homotopical datum, def. 8 implies that the canonical morphism
is a monomorphism.
The condition implies that
is a pullback, for $\epsilon = 0,1$, hence that
is a pullback. So the statement follows with remark 3.
Let $I \in PSh(A)$ be any object equipped with two points (global sections) $\partial^0, \partial^1 : * \to I$ which are disjoint in that
is a pullback square (here $\emptyset$ is the initial object in $PSh(A)$). This induces a functorial cylinder by the assignment
where on the right we have the cartesian product.
This defines an elementary homotopical datum in the sense of def. 8.
The disjointness of the two points ensures that $* \coprod * \stackrel{(\partial^0, \partial^1)}{\to} I$ is a monomorphism.
The interval commutes with colimits as these and the product are computed objectwise, and products in Set commute with colimits. More abstractly: by the Giraud theorem valid in the presheaf topos $PSh(A)$ we have “universal colimits”: they are preserved by pullback, and in particular by cartesian product. Therefore the first clause of def. 8 is satisfied.
Similarly, the second axiom of def. 8 holds because limits commute over each other.
(Lawvere cylinder)
Let $I := \Omega$ be the subobject classifier in the presheaf topos $PSh(A)$. This is the presheaf which to an object $U$ of $A$ assigns the set of subobjects of the representable functor given by $U$ (the sieves on $U$)
and which to a morphism $f : U_1 \to U_2$ assigns the pullback functor $f^* : Sub(U_2) \to Sub(U_1)$.
Let then
be the morphisms that classify top and bottom, respectively, the terminal and the initial subobject of the terminal object.
This is a segment object in the sense of example 1 (the “Lawvere-segment”).
That the two points are separated, in that
is a pullback, is the defining property of the subobject classifier.
For $J$ an elementary homotopical datum on $PSh(A)$, a class of anodyne extensions is a class $AnExt \subset Mor(PSh(A))$ such that
there exists a small set $S$ with
for $K \hookrightarrow L$ a monomorphism, the pushout product morphisms
(by Joyal-Tierney calculus to be thought of as “$(K \to L) \bar \otimes (\{e\} \to I)$”)
are in $AnAxt$, for $e \in \{0,1\}$;
if $K \to L$ is in $AnExt$, then so is
(hence “$(K \to L) \bar \otimes (\partial I \to I)$”).
A class of anodyne extensions
is generated under pushouts, transfinite composition and retracts from the morphisms in $\Lambda$;
is a subclass of the monomorphisms;
contains all morphisms of the form $e : K \to (I \otimes K)$;
is closed under $I\otimes(-)$;
(Cisinski 06, remark. 1.3.11).
By prop. 11, $X \in PSh(A)$ is $|Mor(A/X)|$-compact. Therefore the set $\Lambda$ admits a small object argument, which shows the first statement (see there).
Since monomorphisms are closed under these operations, the second statement follows.
The third statement follows by choosing the morphism in the second item of def. 10 to be $\emptyset \to K$ and using that by def. 8 the interval commutes with colimits, so that $I \otimes \emptyset \simeq \emptyset$.
Finally, to see that with $j : K \to L$ also $I \otimes j : I \otimes L \to I \otimes L$ is anodyne, consider the naturality diagram of the endpoint inclusion
factored through the top pushout square, as indicated. Here $j'$ is anodyne, being a pushout of an anodyne morphism, and $k$ is anodyne by the second clause in def. 10. Therefore also their composite $I \otimes j$ is anodyne.
A homotopical structure on $PSh(A)$ is a choice of elementary homotopical datum $J$, def. 8 and a corresponding choice of a class of anodyne extensions $AnExt$, def. 10.
Let $A$ be a small category and $(J, AnExt)$ a homotopical structure, def. 11, on $PSh(A)$. Define the following classes of objects and morphisms in $PSh(A)$:
the cofibrations are the monomorphisms;
the fibrant objects are those $X \in PSh(A)$ for which the terminal morphism $X \to *$ has the right lifting property against the anodyne extensions (the morphisms in $AnExt$, def. 10);
the weak equivalences are those morphisms $f : A \to B$, such that for all fibrant objects $X$ the induced morphism in the $J$-homotopy category, def. 5,
is an isomorphism (a bijection of sets).
With the classes of morphisms as in def. 12, $PSh(A)$ is a cofibrantly generated model category.
(Cisinski 06, theorem 1.3.22).
The retract properties are clear, as is the 2-out-of-3 property for weak equivalences, see lemma 1 below.
The lifting properties hold by prop. 6 below, the proof of which is in the section Lifting below.
The factorization properties hold by cor. 1 and cor. 4 below, which are in the section Factorization below.
The existence of a set of generating cofibrations is prop. 12 below, that of generating acyclic cofibrations is prop. 15 below.
The homotopy category of $(PSh(A), W)$ is (up to equivalence) the full subcategory of $Ho_J$, def. 5, on the fibrant objects.
Before coming to the proof of these lemmas, the following two statements say that the terminology introduced so far is indeed consistent with the meaning of this theorem.
The morphisms called acyclic fibrations in def. 7 are indeed precisely the acyclic fibrations with respect to the model structure of theorem 1.
(Cisinski 06, theorem 1.3.27).
Every anodyne extension, def. 10, is a weak equivalence in the model structure of theorem 1.
What is not true in general is the converse of prop. 13, that every acyclic cofibration is an anodyne extension. (A counterexample derives from chapter XI, remark 2.4 in Goerss, Jardine Simplicial homotopy theory.)
For the model structure from theorem 1, the following are equivalent:
every acyclic cofibration is an anodyne extension;
every morphism with right lifting against anodyne extensions is a fibration;
every weak equivalence with right lifting against anony extensions is an acyclic fibration;
every morphism with right lifting against anodyne extensions factors as an anodyne extension followed by a fibration.
A homotopical structure, def. 11, on a presheaf category is called complete if the model structure from theorem 1 satisfies the equivalent conditions of prop. 8.
We discuss the proof of prop. 8 below in Completeness.
We collect lemmas to prove theorem 1 and related statements. A little bit of work is required for demonstrating the lifting axioms, which we do below in Lifting. A little bit more work is required for demonstrating the factorization axioms, which we do below in Factorization. Finally, the proof of the equivalence of the conditions of completeness is in Completeness.
Every $J$-homotopy equivalence, def. 6, is a weak equivalence.
The weak equivalences satisfy the two-out-of-three-property and are stable under retracts.
The first statement holds by definition of $Ho_J$.
The second statement also follows directly from the definition. If for $A \stackrel{f}{\to} B \stackrel{g}{\to} C$ and fibrant $X$ in the composite
two of three are isomorphisms, then so is the third.
We discuss the lifting properties in the model structure of def. 12.
Since fibrations are defined to be the morphisms satisfying the right lifting property against acyclic cofibration, we only need to show that the fibrations which are also weak equivalences have the right lifting property against the monomorphisms. For this it is sufficient to show prop. 6. This we do now, after a lemma.
A deformation retract $f : X \to Y$ in $PSh(A)$ is a retract with retraction $g : Y \to X$, which is also a section of $f$ up to a $J$-homotopy $h : id_Y \Rightarrow f \circ g$. It is strong if $h \circ (I \otimes f) = \sigma_Y \circ (I \otimes f)$.
A dual deformation restract $f : X \to Y$ has a section by a morphism $g : Y \to X$ and is also a retract up to a $J$-homotopy $k : id_X \Rightarrow g \circ f$. Is is strong if $f \circ k = f \circ \sigma_X$.
Every acyclic fibration is a dual strong deformation retract, def. 14.
Every section of an acyclic fibration is a strong deformation retract.
The first statement is a direct consequence of prop 3.
For the second statement, let $p : X \to Y$ be an acyclic fibration, and let $s : Y \to X$ be a section. This induces a commuting square
where the lift $h$ exists by assumption on $p$ ($s$ is necessarily a monomorphism, being a section).
The resulting component triangle
exhibits $s$ as a deformation retract, and the other resulting component triangle
says that $h\circ (I \otimes s) = s \circ \sigma_Y$, hence by naturality of cylinders $\cdots = \sigma_X \circ (I \otimes s)$, hence that the deformation retract is indeed strong.
of prop. 6
First to see that the acyclic fibrations of def. 7 are indeed fibrations and weak equivalences:
By lemma 2 every acyclic fibration is in particular a $J$-homotopy equivalence, hence by lemma 1 a weak equivalence. Moreover, by def. 7 the acyclic fibrations right-lift against monomorphisms, hence in particular against the acyclic cofibrations, hence are fibrations.
Conversely, let $p : X \to Y$ be a fibration which is also a weak equivalence. We need to show that it has the right lifting property against all monomorphisms.
By prop. 12, proven below, we may apply the small object argument to factor $p = q \circ j$ as a monomorphism $j$ followed by an acyclic fibration $q$. By the previous argument, $q$ is a weak equivalence, and so by lemma 1 so is $j$. Therefore, since $p$ is a fibration, we have a lift $\sigma$ in
This equivalently exhibits $p$ as a retract of $q$
So by lemma 4 with $q$ also $p$ is an acyclic fibration.
We discuss the two factorization axioms for the model category structure from def. 12 to be established. First for factorizations into cofibrations followed by acyclic fibrations, then for factorizations into acyclic cofibrations followed by fibrations.
For showing that every morphism factors as a monomorphism followed by an acyclic fibration, it is by prop. 6 sufficient to show that the monomorphisms are generated by a small set that admits the small object argument. This we do now.
This section follows (Cisinski 06, section 1.2).
We start with some entirely general statements about compact objects.
For $A$ a small category, let $\alpha = {\vert Mor(A)\vert}$ be the smallest regular cardinal $\geq$ the cardinality of the set of morphisms of $A$. Then the limit functor
commutes with $\alpha$-filtered colimits / $\alpha$-directed colimits.
Let $C$ be a category with all small colimits, let $\alpha$ be a cardinal, let $A$ be a small category and finally let $F : A \to C$ be a functor with values in $\alpha$-compact objects in $C$. Then the colimit $\lim_\to F$ is $\lambda$-compact object, for $\lambda$ the maximum of $\alpha$ and the cardinality ${\vert Mor A\vert}$ of the set of morphisms of $A$.
Let $G : I \to C$ be a $\lambda$-filtered diagram.
Then by prop. 9 we have natural isomorphisms
because the $\lambda$-filtered diagram is at least ${\vert Mor(A)\vert}$-filtered and hence, by prop. 9, its colimit commutes with the limit over $A$.
Now since each $F(a)$ is assumed to be $\alpha$-compact and hence is also $\lambda$-compact, we conclude with the natural isomorphisms
For $X \in PSh(A)$, write $A/X$ for the category of elements of $X$. Write ${\vert Mor(A/X)\vert}$ for the cardinality of the set of morphisms of the category of elements (throughout assuming $A$ to be a small category).
For $X \in PSh(A)$ any object, the hom functor $Hom(X, -) : PSh(A) \to Set$ preserves ${\vert Mor(A/X)\vert}$-filtered colimits.
In other words: $X$ is a ${\vert Mor(A/X)\vert}$-compact object.
By the co-Yoneda lemma $X$ is the colimit over its elements
Since the image of the functor $A/X \to PSh(A)$ is in representables, which are maximally compact, the stament follows with prop. 10.
A cellular model on $PSh(A)$ is a choice of a small set $I \subset Mor(PSh(A))$ of monomorphisms, such that the class of all monomorphisms is generated from it
By prop. 11 we may apply the small object argument in $PSh(A)$ and so it follows that $LLP(RLP(I))$ is the smallest class containing $I$ that is closed under pushout, retract and transfinite composition.
The following lemma will be used to show that cellular structures always exist.
Let $C \subset Mor(PSh(A))$ be a class of morphisms, and $D \subset PSh(A)$ a small set of objects, such that
$C$ is closed under pushouts, retracts and transfinite composition;
if $f,g \in Mor(PSh(A))$ are two composable morphisms with $f$ and $g \circ f$ in $C$, then also $g$ is in $C$;
every $X \in PSh(A)$ is the union of those of its sub-objects isomorphic to an object in $D$;
for every $X \to Y$ in $C$ and every sub-object $Z$ of $Y$ in $D$, there is a sub-object $T \hookrightarrow Y$ from $D$, which contains $Z$ and such that $T \cap X \to T$ is in $C$.
Then the small set $I \subset Mor(PSh(A))$ of morphisms in $C$ with codomain in $D$, generates $C$ as
A bit of work…
There exists a cellular structure, def. 15, on $PSh(A)$.
The set $I$ can be chosen to consist of morphisms into quotient objects of representables.
Take in lemma 3 $C$ to be the class of monomorphisms and $D$ to be the class of quotients of representables.
There exists a functorial factorization of morphisms in $PSh(A)$ into a monomorphism followed by an acyclic fibration.
By prop. 11 every object is $\alpha$-small, for some $\alpha$. Therefore by prop. 12 we can apply the small object argument.
We show now for def. 12 that every morphism factors as an acyclic cofibration followed by a fibration. Since the fibrations are defined by right lifting against acylcic cofibrations, for this it is sufficient to establish a set of generating acyclic cofibrations. This is the statement of prop. 15 below. Establishing this takes a few technical lemmas.
Every morphism admits a factorization into an anodyne extension, followed by a morphism having the right lifting property against anodyne extensions.
By the small object argument, in view of prop. 4.
If $T \in PSh(A)$ is fibrant, then for any $K \in PSh(A)$ elementary $J$-homotopy, def. 4, is already an equivalence relation on $Hom_{PSh(A)}(K,T)$ and coincides with $J$-homotopy.
Every anodyne extension is a weak equivalence.
For $j : K \to L$ an anodyne extension and $T$ a fibrant object, we need to show that
is a bijection.
It is surjective by the defining lifting property, which provides $\sigma$ in
To see injectivity, let $l_0, l_1 : L \to T$ be two morphisms such that $l_0 \circ j$ and $l_1 \circ j$ coincide in $Ho_J$. By lemma 5 this is the case precisely if there is an elementary $J$-homotopy $h : I \otimes K \to T$ relating them. This induces the horizontal morphism in the diagram
where the left morphism is anodyne, by the second clause of def. 10, so that the lift denoted $\eta$ exists. This lift exhibits a $J$-homotopy $l_0 \Rightarrow l_1$, hence shows that $l_0$ was already equal to $l_1$ in $Ho_J$, hence that $j^*$ is injective.
A morphism between fibrant objects is a weak equivalence precisely if it is a $J$-homotopy equivalence, def. 6.
Is is clear that every $J$-homotopy is a weak equivalence. Conversely, let $f : X \to Y$ be a weak equivalence between fibrant objects. Write $Ho_J^{fib} \hookrightarrow Ho_J$ for the full subcategory of $Ho_J$, def. 6 on the fibrant objects. The localization $Q(f)$ is by definition in $Ho_J^{fib}$ and for all objects $T \in Ho_J^{fib}$ the morphism $Ho_J^{fib}(Q(f), T)$ is an isomorphism. By the Yoneda lemma, therefore, $Q(f)$ itself is an isomorphism in $Ho_J^{fib}$, hence also in $Ho_J$, hence is a weak equivalence.
If a morphism has the right lifting property against the anodyne extensions, then it is an acyclic fibration precisely if it is a dual strong deformation retract.
That the former implies the latter was the statement of lemma 2. Conversely, let $p$ be a dual strong deformation retract, meaning that there is $s : Y \to X$ with $p \circ s = id$, as well as a morphism $k : I \otimes X \to X$ exhibiting a $J$-homotopy $id \Rightarrow s \circ p$. This being strong means that $p \circ k = p \circ \sigma_X$.
We need to show that this implies for
a commuting diagram with $i$ a monomorphism, there is a lift. To this end, observe that the given structures induce a morphism
such that
By the second clause of def. 10 the morphism on the left is an anodyne extension, and so this diagram admits a lift $h : I \otimes L \to X$. One see that $l := h \circ \partial^0_L$ is a lift of the original square above.
A morphism into a fibrant object with right lifting property against anodyne extensions is a weak equivalence precisely if it is an acyclic fibration.
We already know from prop. 6 that acyclic fibrations are weak equivalences.
So let $Y$ be fibrant and let $p : X \to Y$ be a weak equivalence that has rlp against anodyne extensions. We need to show that $p$ is an acyclic fibration. By lemma 7 it is sufficient to show that it is a dual strong deformation retract.
By lemma 6 $p$ is also a $J$-homotopy equivalence. By lemma 5 this is exhibited by an elementary $J$-homotopy $k : I \otimes Y \to Y$, which in particular gives a commuting diagram
from which we obtain a lift $k' : I \otimes Y \to X$. Set then
One finds then $p \circ s = id_Y$.
Next …
A cofibration into a fibrant object is a weak equivalence precisely if it is an anodyne extension.
By lemma 13 we already know that every anodyne extension is a weak equivalence. So we need to show that a cofibration $i : A \to T$ into a fibrant object $T$ which is a weak equivalence is also an anodyne extension. By prop. 4 we may factor this as $i = q \circ j$, with $j$ an anodyne extension and $Q$ having RLP against anodyne extensions. Since $j$ is a weak equivalence, by 2-out-of-3 so is $q$. By lemma 8 $q$ is an acyclic fibration.
Therefore we have a lift $s$ in
and this exhibits $i$ as a retract of $j$. Hence with $j$ also $i$ is an anodyne extension.
A cofibration is a weak equivalence precisely if it has the left lifting property against morphisms into a fibrant object that have the right lifting property against anodyne extensions.
(…)
The acyclic cofibrations are stable under transfinite composition and pushouts
Every strong deformation retract is an anodyne extension.
Every anodyne extension between fibrant objects is a strong deformation retract.
Pour tout cardinal assez grand $\alpha$, si on pose $\beta = 2^\alpha$, pour toute cofibration triviale $i : C \to D$, et pour tout sous-objet $\beta$-accessible $J$ de $D$, il existe un sous-objet $\beta$-accessible $K$ de $D$, qui contient $J$, tel que l’inclusion canonique $C \cap K \to K$ soit une cofibration triviale.
Three pages of work…
There exists a set of generating acyclic cofibrations.
Use lemma 3 with $C$ the class of acyclic cofibrations and $D = Acc_\alpha(A)$ the set of $\alpha$-accessible presheaves for a sufficiently large cardinal $\alpha$.
There is a functorial factorization of every morphism into an acyclic cofibration followed by a fibration.
By prop. 15 and prop. 11 we may apply the small object argument.
We list lemmas to show prop. 8.
(…)
Let still $A$ be a small category and write $PSh(A)$ for its category of presheaves.
An $A$-localizer is a class of morphisms $W \subset Mor(PSh(A))$ satisfying the following axioms
every acyclic fibration, def. 7, is in $W$;
The class of monomorphisms that is in $W$ is stable under pushout and transfinite composition.
The elements of $W$ we call $W$-equivalences. For $S \subset Mor(PSh(A))$ a class of morphisms, the smallest $A$-localizer containing $S$ is called the $A$-localized generated by $S$. If an $A$-localizer is generated from a small set, we call it accessible. The minimal $A$-localizer is $W(\emptyset)$.
For $(J,AnExt)$ a homotopical structure, def. 11 on $PSh(A)$, the class $W$ of weak equivalences of the induced model category structure of theorem 1 is an $A$-localizer.
If $\Lambda$ is a small set generating the anodyne extensions, $AnExt = LLP(RLP(\Lambda))$, then $W = W(\Lambda)$.
Let $W \subset Mor(PSh(A))$. The following are equivalent.
$W$ is an $A$-localizer, def. 16;
There is a set $S \subset Mor(PSh(A))$ of monomorphisms, such that $W$ is the class of weak equivalences of the model structure induced by theorem 1 from the homotopical structure, def. 11, given by the Lawvere cylinder, def. 2, and $AnExt := S$.
There is some homotopical structure, def. 11, on $PSh(A)$, such that $W$ is the class of weak equivalences of the model structure corresponding to it by theorem 1.
There exists a cofibrantly generated model category on $PSh(A)$ such that $W$ is its class of weak equivalences, and such that the cofibrations are the monomorphisms.
In particular, $PSh(A)$ admits a model structure whose cofibrations are the monomorphisms and whose weak equivalences are the minimal localizer, $W(\emptyset)$. This is called the minimal model structure on $PSh(A)$. It is generated from the homotopical datum given by the Lawvere cylinder, example 2 and the empty set.
This may be compared to Jeff Smith's theorem, which constructs a model structure on a locally presentable category.
Give a localizer $W$ on $PSh(A)$, there is a localizer $W_{horiz}$ on $PSh(A \times \Delta)$
(…)
See (Ara, p. 9).
The archetypical and motivating example is the standard model structure on simplicial sets, which is a Cisinski model structure on the presheaf topos on the simplex category (Cisinski 06, section 2).
Accordingly, the injective model structure on simplicial presheaves over a site $C$ is a Cisinski model structure, namely on the presheaf topos $PSh(C \times \Delta)$. Moreover, every left Bousfield localization of such a model structure is still a Cisinski model structure, since left Bousfield localization preserves the class of cofibrations.
Notice that, as discussed there, every presentable (infinity,1)-category has a presentation by such a localization, hence by a Cisinski model structure.
Also the model structure for quasi-categories is a Cisinski model structure on sSet, induced by the localizer given by the spine inclusions.
Moreover, the model structure for complete Segal spaces is the simplicial completion of this model structure. (see Ara).
As a cellular set-variant of this, the model structure on cellular sets is a Cisinski model structure on the category of presheaves over the Theta category restricted to $n$-cells.
The model structure on dendroidal sets is not exactly a Cisinki model structure, but is transferred from one that is.
If $W(\Sigma)$ is the localizer generated by a set $\Sigma$ in a Grothendieck topos, then (the Cisinski model structure whose weak equivalences are) $W(\Sigma)$ is right proper if and only if pullback along fibrations between fibrant objects in this model structure takes morphisms in $\Sigma$ to weak equivalences.
We can use this to prove that a locally presentable (∞,1)-category is locally cartesian closed (∞,1)-category if and only if it has a presentation by a right proper Cisinski model structure. See locally cartesian closed (∞,1)-category for details.
The original articles are
Further developments are in
Rick Jardine, Categorical homotopy theory (2003) (K-theory)
Marc Olschok, On constructions of left determined model structures, PhD thesis (2009) (pdf)
See also
Dimitri Ara, Higher quasi-categories vs higher Rezk spaces (arXiv:1206.4354)