Cisinski model structure

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

##Idea
In [[Pursuing Stacks]], [[Grothendieck]] introduced the idea of a [[test category]]. These are by definition small categories on which the presheaves of sets are models for homotopy types of CW-complexes, thus generalising the situation for the category of simplices, for which the category of presheaves is that of [[simplicial sets]]. The resulting theory was completed and generalised by Cisinski: in order to prove, following Grothendieck's prediction, that presheaves on a test category form a [[model category]], he constructed all possible cofibrantly generated model structures on a given topos for which the cofibrations are the monomorphisms.

## General

+-- {: .num_defn}
###### Definition

For $\mathcal{T}$ a [[topos]], a **Cisinski model structure** on $\mathcal{T}$ is a [[model category]] structure on $\mathcal{T}$ such that

1. the [[cofibrations]] are precisely the [[monomorphisms]];

1. it is a [[cofibrantly generated model category]].

=--

+-- {: .num_remark}
###### Remark

Since every [[topos]] is a [[locally presentable category]], a Cisinski model structure is in particular a [[combinatorial model category]] structure.

=--

+-- {: .num_remark}
###### Remark

Since every [[topos]] is an [[adhesive category]], monomorphisms are automatically preserved by [[pushout]].

=--

+-- {: .num_defn #Localizer}
###### Definition

Say a class $W \subset Mor(\mathcal{T})$ is an accessible **[[localizer]]** on $\mathcal{T}$ if it is a class of [[weak equivalences]] in a Cisinski model structure on $\mathcal{T}$.

=--

+-- {: .num_prop}
###### Proposition

Every [[small set]] of [[morphisms]] $\Sigma \subset Mor(\mathcal{T})$ is contained in a smallest localizer, def. \ref{Localizer}, $W(\Sigma)$.

=--

One says that $W(\Sigma)$ is the
localiser _generated_ by $\Sigma$.

So in particular a Cisinski model structure always exists.

## On presheaf toposes
 {#OnPresheafToposes}

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](#PreliminaryNotions)_, the main statement is theorem \ref{ModelStructureFromHomotopicalStructure} below.

(This follows sections 1.2 and 1.3 of [Cisinski 06](#Cisinski06)).

### Preliminary notions
 {#PreliminaryNotions}

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. \ref{HomotopicalStructure} 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. \ref{ModelStructureMorphismsFromHomotopicalStructure} this defines a [[model category]] structure on $PSh(A)$.

+-- {: .num_defn #FunctorialCylinder}
###### Definition

For $X$ in $PSh(A)$, a **cylinder** on $X$ in the following means a [[cylinder object]], denoted $I \otimes X$, factoring the [[codiagonal]]

$$
  X \coprod X \stackrel{(\partial_X^0, \partial_X^1)}{\to} I \otimes X \stackrel{\sigma_X}{\to} X
$$

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](#Cisinski06)).

In the following, let $J : PSh(A) \to Cyl(A)$ be a choice of functorial cylinder object. Equivalently, this is a choice of endofunctor

$$
  I \otimes (-) : PSh(A) \to PSh(A)
$$

equipped with [[natural transformations]]

$$
  (\partial I)\otimes (-)
  \stackrel{(\partial^0_{(-)}, \partial^1_{(-)})}{\to}
  I \otimes (-)
  \to
  Id
$$

where $(\partial I) \otimes X := X \coprod X$, such that the composite is the functorial codiagonal, and where the first transformation is a monomorphism.

+-- {: .num_defn #ElementaryJHomotopy}
###### Definition

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 a [[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

$$
  \array{
    X &&
    \\
    \downarrow^{\mathrlap{\partial^0_X}} & \searrow^f
    \\
    I \otimes X &\stackrel{\eta}{\to}& Y
    \\
    \uparrow^{\mathrlap{\partial^1_X}} & \nearrow_g
    \\
    X
  }
  \,.
$$

We say **$J$-homotopy** for the [[equivalence relation]] _generated_ by this.

=--

([Cisinski 06, def. 1.3.3](#Cisinski06)).

+-- {: .num_prop}
###### Proposition

$J$-homotopy is compatible with composition in $PSh(A)$.

=--

+-- {: .proof}
###### Proof

It is sufficient to show that elementary $J$-homotopies are compatible with
composition.

So for

$$
  \array{
    X &&
    \\
    \downarrow & \searrow^f
    \\
    I \otimes X &\stackrel{\eta}{\to}& Y
    \\
    \uparrow & \nearrow_g
    \\
    X
  }
$$

an elementary $J$-homotopy $f \Rightarrow g$, and for

$$
  \array{
    Y &&
    \\
    \downarrow & \searrow^{f'}
    \\
    I \otimes Y &\stackrel{\eta}{\to}& Z
    \\
    \uparrow & \nearrow_{g'}
    \\
    Y
  }
$$

one $f' \Rightarrow g'$, we obtain an elementary homotopy $f' \circ f \Rightarrow g' \circ f$ by forming

$$
  \array{
    X & \stackrel{f}{\to} & Y &&
    \\
    \downarrow && \downarrow & \searrow^{f'}
    \\
    I \otimes X &\stackrel{I \otimes f}{\to}& I \otimes Y &\stackrel{\eta}{\to}& Z
    \\
    \uparrow && \uparrow & \nearrow_{g'}
    \\
    X &\stackrel{f}{\to}& Y
  }
$$

and then an elementary $J$-homotopy $g' \circ f \Rightarrow g '\circ g$ by forming

$$
  \array{
    X &&
    \\
    \downarrow & \searrow^f
    \\
    I \otimes X &\stackrel{\eta}{\to}& Y &\stackrel{g'}{\to}& Z
    \\
    \uparrow & \nearrow_g
    \\
    X
  }
  \,.
$$

Together this generates a $J$-homotopy $f' \circ f \Rightarrow g' \circ g$.

=--

Hence the following is well defined.

+-- {: .num_defn #JHomotopyCategory}
###### Definition

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

$$
  Q : PSh(A) \to Ho_J(A)
$$

for the projection functor.

=--

+-- {: .num_defn #JHomotopyEquivalence}
###### Definition

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.

=--

([Cisinski 06, def. 1.3.4](#Cisinski06)).

+-- {: .num_defn #AcyclicFibration}
###### Definition

A morphism $f : X \to Y$ in $PSh(A)$ is called an **acyclic fibration** if it has the [[right lifting property]] against all monomorphisms.

=--

([Cisinski 06, def. 1.2.18](#Cisinski06)).

+-- {: .num_prop #SomePropertiesOfAcyclicFibrations}
###### Proposition

Every acyclic fibration in $PSh(A)$ is a $J$-homotopy equivalence.

More is true: every trivial fibration $p : X \to Y$

1. has a [[section]] $s : Y \to X$;

1. which is also a $J$-homotopy left inverse;

1. by an elementary $J$-homotopy $h : id \Rightarrow s \circ p$ which satisfies
   $p \circ h = p \circ \sigma_X$.

=--

([Cisinski 06, lemma 1.3.5](#Cisinski06)).

+-- {: .proof}
###### Proof

The existence of the section $s : Y \to X$ follows by right lifting against the monomorphism $\emptyset \to Y$ (out of the [[initial object]])

$$
  \array{
     \emptyset &\to& X
     \\
     \downarrow &\nearrow_s& \downarrow^{\mathrlap{p}}
     \\
     Y &\stackrel{=}{\to}& Y
  }
  \,.
$$

The $J$-homotopy $h$ is obtained by lifting in the diagram

$$
  \array{
     X \coprod X &\stackrel{(id_X, s \circ p)}{\to}& X
     \\
     {}^{\mathllap{(\partial^0_X, \partial^1_X)}}\downarrow
     &&
     \downarrow^{\mathrlap{p}}
     \\
     I \otimes X
     &\stackrel{p \circ \sigma_X}{\to}& Y
  }
  \,.
$$

=--

+-- {: .num_defn #ElementaryHomotopicalDatum}
###### Definition

An **elementary homotopical datum** on $PSh(A)$ is a functorial cylinder $J$, def. \ref{FunctorialCylinder}, such that

1. the functor $I \otimes (-)$ commutes with small [[colimits]] and preserves monomorphisms;

1. for all [[monomorphisms]] $f : K \to L$ the [[diagrams]]

   $$
     \array{
       K &\stackrel{(id, f)}{\to}& L
       \\
       {}^{\mathllap{(e,id)}}\downarrow && \downarrow^{\mathrlap{(e,id)}}
       \\
       I \otimes K &\stackrel{I \otimes f}{\to}& I \otimes L
     }
   $$

   for $e \in \{0,1\}$ is a [[pullback]] square.

=--

([Cisinski 06, def. 1.3.6](#Cisinski06)).

+-- {: .num_remark #PullbackOfMonosAndUnions}
###### Remark

For

$$
  \array{
    R &\to & S
    \\
    \downarrow && \downarrow
    \\
    T &\to& U
  }
$$

a [[pullback]] square in $PSh(A)$ of two monomorphisms $S \hookrightarrow U$ and $T \hookrightarrow U$, the universal morphism out of the [[pushout]]

$$
  T \coprod_R S \to U
$$

is also a monomorphism, usually written as the morphism out of the _[[union]]_

$$
  T \cup S \hookrightarrow U
  \,.
$$

=--

+-- {: .proof}
###### Proof

All this follows, for instance, from the corresponding statements in [[Set]], over each object of $A$.

=--

+-- {: .num_defn}
###### Definition

Let

$$
  \partial^\epsilon : \{\epsilon\}\otimes(-) \hookrightarrow I \otimes(-)
  \,,
$$

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

$$
  \partial^\epsilon \otimes id_X : \{\epsilon\} \otimes X \to I \otimes X
  \,.
$$

=--

+-- {: .num_remark}
###### Remark

The second condition on an _elementary homotopical datum_, def. \ref{ElementaryHomotopicalDatum} implies that the canonical morphism

$$
  J \otimes K \cup (\partial J) \otimes L \to J \otimes L
$$

is a [[monomorphism]].

=--

([Cisinski 06, remark 1.3.7](#Cisinski06)).

+-- {: .proof}
###### Proof

The condition implies that

$$
  \array{
    \{\epsilon\} \otimes K &\stackrel{id_\{\epsilon\} \otimes j}{\to}&
    \{\epsilon\} \otimes L
    \\
    {}^{\mathllap{\partial^\epsilon \otimes id_K}}\downarrow && \downarrow^{\mathrlap{\partial^\epsilon \otimes id_L}}
    \\
    I \otimes K &\stackrel{(id_I) \otimes j}{\to}& I \otimes L
  }
$$

is a pullback, for $\epsilon = 0,1$, hence that

$$
  \array{
    \partial I \otimes K &\stackrel{id_{\partial I} \otimes j}{\to}&
    \partial I \otimes L
    \\
    {}^{\mathllap{i \otimes id_K}}\downarrow && \downarrow^{\mathrlap{i \otimes id_L}}
    \\
    I \otimes K &\stackrel{(id_I) \otimes j}{\to}& I \otimes L
  }
$$

is a pullback. So the statement follows with remark \ref{PullbackOfMonosAndUnions}.

=--

+-- {: .num_example #SegmentObject}
###### Example

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

$$
  \array{
    \emptyset &\to& *
    \\
    \downarrow && \downarrow^{\mathrlap{\partial^0}}
    \\
    * &\stackrel{\partial^1}{\to}& I
  }
$$

is a [[pullback]] square (here $\emptyset$ is the [[initial object]] in $PSh(A)$). This induces a functorial cylinder by the assignment

$$
  X \mapsto I \times X
  \,,
$$

where on the right we have the cartesian product.

This defines an _elementary homotopical datum_ in the sense of def. \ref{ElementaryHomotopicalDatum}.

=--

([Cisinski 06, example 1.3.8](#Cisinski06)).

+-- {: .proof}
###### Proof

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. \ref{ElementaryHomotopicalDatum} is satisfied.

Similarly, the second axiom of def. \ref{ElementaryHomotopicalDatum} holds because [[limits]] commute over each other.

=--

+-- {: .num_example #LawvereCylinder}
###### Example
**([[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 $U$)

$$
  \Omega : U \mapsto Sub(U)
$$

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

$$
  \partial^0, \partial^1 := \top, \bot: * \to \Omega
$$

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 \ref{SegmentObject} (the "[[Bill Lawvere|Lawvere]]-segment").

=--

([Cisinski 06, example 1.3.9](#Cisinski06)).

+-- {: .proof}
###### Proof

That the two points are separated, in that

$$
  \array{
    \emptyset &\to& *
    \\
    \downarrow && \downarrow^{\mathrlap{\top}}
    \\
    * &\stackrel{\bot}{\to}& \Omega
  }
$$

is a pullback, is the defining property of the [[subobject classifier]].

=--

+-- {: .num_defn #AnodyneExtensions}
###### Definition

For $J$ an elementary homotopical datum on $PSh(A)$, a **[[class]] of [[anodyne extensions]]** is a class $AnExt \subset Mor(PSh(A))$ such that

1. there exists a [[small set]] of monomorphisms $S$ with

   $$
     AnExt = LLP(RLP(S))
   $$

1. for $K \hookrightarrow L$ a monomorphism, the [[pushout-product axiom|pushout product]] morphisms

   $$
     (I \otimes K) \cup (\{e\} \otimes L) \to I \otimes L
   $$

   (by [[Joyal-Tierney calculus]] to be thought of as "$(K \to L) \bar \otimes (\{e\} \to I)$")

   are in $AnExt$, for $e \in \{0,1\}$;

1. if $K \to L$ is in $AnExt$, then so is

   $$
     (I \otimes K) \cup ((\partial I) \otimes L)
    \to
     I \otimes L
     \,.
   $$

   (hence "$(K \to L) \bar \otimes (\partial I \to I)$").

=--

([Cisinski 06, def. 1.3.10](#Cisinski06)).

+-- {: .num_prop #PropertiesOfAnodyneExtensions}
###### Proposition

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](#Cisinski06)).

+-- {: .proof}
###### Proof

By prop. \ref{PresheavesAreCompact},
$X \in PSh(A)$ is $|Mor(A/X)|$-[[compact object|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. \ref{AnodyneExtensions} to be $\emptyset \to K$ and using that by def. \ref{ElementaryHomotopicalDatum} 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

$$
  \array{
    K &\stackrel{j}{\to}& L
    \\
    \downarrow^{\mathrlap{\partial^0_k}} && \downarrow
    \\
    I \otimes K &\stackrel{j'}{\to}& (I \otimes K) \cup \{0\}\otimes L
    \\
    & {}_{I \otimes j} \searrow & \downarrow^{\mathrlap{k}}
    \\
    && I \otimes L
  }
  \,,
$$

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. \ref{AnodyneExtensions}. Therefore also their composite $I \otimes j$ is anodyne.

=--

+-- {: .num_defn #HomotopicalStructure}
###### Definition

A **homotopical structure** on $PSh(A)$ is a choice of elementary homotopical datum $J$, def. \ref{ElementaryHomotopicalDatum} and a corresponding choice of a class of anodyne extensions $AnExt$, def. \ref{AnodyneExtensions}.

=--

([Cisinski 06, def. 1.3.14](#Cisinski06)).

\begin{defn} \label{DefinitionHomotopicalDatum} A _homotopical datum_, or _donnée homotopique_, on $PSh(A)$ is a choice of elementary homotopical datum (Definition \ref{ElementaryHomotopicalDatum}) together with a set of [[monomorphisms]] in $PSh(A)$. \end{defn}

([Cisinski 06, def. 1.3.14](#Cisinski06)).

\begin{rmk} A homotopical datum $(J, S)$ _generates_ a homotopical structure for which $AnExt$ is the smallest class of anodyne extensions relative to $J$ which contains $S$. \end{rmk}

### The model structure

+-- {: .num_defn #ModelStructureMorphismsFromHomotopicalStructure}
###### Definition

Let $A$ be a [[small category]] and $(J, AnExt)$ a homotopical structure, def. \ref{HomotopicalStructure}, 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 object|terminal]] morphism $X \to *$ has the [[right lifting property]] against the anodyne extensions (the morphisms in $AnExt$, def. \ref{AnodyneExtensions});

* 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. \ref{JHomotopyCategory},

  $$
    Ho_J(f,X) : Ho_J(B,X) \to Ho_J(A, X)
  $$

  is an [[isomorphism]] (a [[bijection]] of sets).

=--

([Cisinski 06, def. 1.3.21](#Cisinski06)).

+-- {: .num_theorem #ModelStructureFromHomotopicalStructure}
###### Theorem

With the classes of morphisms as in def. \ref{ModelStructureMorphismsFromHomotopicalStructure}, $PSh(A)$ is a [[cofibrantly generated model category]].

=--

([Cisinski 06, theorem 1.3.22](#Cisinski06)).

+-- {: .proof}
###### Proof

The retract properties are clear, as is the 2-out-of-3 property for weak equivalences, see lemma \ref{JHomotopyEquivsAreWeakEquivs} below.

The lifting properties hold by prop. \ref{AcyclicFibrationsAreAcyclicFibrations} below, the proof of which is in the section _[Lifting](#Lifting)_ below.

The factorization properties hold by cor. \ref{FirstFactorizationEstablishes} and cor. \ref{FactorizationAcyclicCofibFib} below, which are in the section _[Factorization](#Factorization)_ below.

The existence of a set of generating cofibrations is prop. \ref{CellularStructuresExist} below, that of generating acyclic cofibrations is prop. \ref{GeneratingAcyclicCofibrationsExist} below.

=--

+-- {: .num_prop #HomotopyCatIsFullSubcatgOfJHo}
###### Proposition

The [[homotopy category]] of $(PSh(A), W)$ is (up to [[equivalence of categories|equivalence]]) the [[full subcategory]] of $Ho_J$, def. \ref{JHomotopyCategory}, on the fibrant objects.

=--

([Cisinski 06, 1.3.23](#Cisinski06)).

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.

+-- {: .num_prop #AcyclicFibrationsAreAcyclicFibrations}
###### Proposition

The morphisms called _acyclic fibrations_ in def. \ref{AcyclicFibration} are indeed precisely the acyclic fibrations with respect to the model structure of theorem \ref{ModelStructureFromHomotopicalStructure}.

=--

([Cisinski 06, theorem 1.3.27](#Cisinski06)).

+-- {: .num_prop #EveryAnodyneExtensionIsWeakEquivalence}
###### Proposition

Every anodyne extension, def. \ref{AnodyneExtensions}, is a weak equivalence in the model structure of theorem \ref{ModelStructureFromHomotopicalStructure}.

=--

+-- {: .num_remark }
###### Remark

What is not true in general is the converse of prop. \ref{EveryAnodyneExtensionIsWeakEquivalence}, that every acyclic cofibration is an anodyne extension. (A counterexample derives from chapter X, remark 2.4 in [[Paul Goerss|Goerss]], [[Rick Jardine|Jardine]] _[[Simplicial homotopy theory]]_.)

=--

([Cisinski 06, remark 1.3.46](#Cisinski06)).

+-- {: .num_prop #ConditionsForCompleteness}
###### Proposition

For the model structure from theorem \ref{ModelStructureFromHomotopicalStructure}, the
following are equivalent:

1. every acyclic cofibration is an anodyne extension;

1. every morphism with right lifting against anodyne extensions is a fibration;

1. every weak equivalence with right lifting against anodyne extensions is an acyclic fibration;

1. every morphism with right lifting against anodyne extensions factors as an anodyne extension followed by a fibration.

=--

([Cisinski 06, prop. 1.3.47](#Cisinski06)).

+-- {: .num_defn }
###### Definition

A homotopical structure, def. \ref{HomotopicalStructure}, on a presheaf category
is called **complete** if the model structure from theorem \ref{ModelStructureFromHomotopicalStructure} satisfies the equivalent conditions of prop. \ref{ConditionsForCompleteness}.

=--

([Cisinski 06, def. 1.3.48](#Cisinski06)).

We discuss the proof of prop. \ref{ConditionsForCompleteness} below in _[Completeness](#Completeness)_.

### Proof

We collect lemmas to prove theorem \ref{ModelStructureFromHomotopicalStructure} and related statements. A little bit of work is required for demonstrating the lifting axioms, which we do below in _[Lifting](#Lifting)_. A little bit more work is required for demonstrating the factorization axioms, which we do below in _[Factorization](#Factorization)_. Finally, the proof of the equivalence of the conditions of completeness is in _[Completeness](#Completeness)_.

+-- {: .num_lemma #JHomotopyEquivsAreWeakEquivs}
###### Lemma

Every $J$-homotopy equivalence, def. \ref{JHomotopyEquivalence}, is a weak equivalence.

The weak equivalences satisfy the [[two-out-of-three]]-property and are stable under [[retracts]].

=--

([Cisinski 06, remark 1.3.24](#Cisinski06)).

+-- {: .proof}
###### Proof

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

$$
  Ho_J(C,X) \stackrel{g^*}{\to} Ho_J(B,X) \stackrel{f^*}{\to} Ho_J(A,X)
$$

two of three are isomorphisms, then so is the third.

=--

#### Lifting
 {#Lifting}

We discuss the lifting properties in the model structure of def. \ref{ModelStructureMorphismsFromHomotopicalStructure}.

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. \ref{AcyclicFibrationsAreAcyclicFibrations}. This we do now, after a lemma.

+-- {: .num_defn #DualDeformationRetract}
###### Definition

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 $f \circ k = f \circ \sigma_X$.

=--

+-- {: .num_lemma #SectionsOfAcyclicFibrationsAreDefRetracts}
###### Lemma

Every acyclic fibration is a dual [[strong deformation retract]], def. \ref{DualDeformationRetract}.

Every [[section]] of an acyclic fibration is a [[strong deformation retract]].

=--

([Cisinski 06, prop. 1.3.26](#Cisinski06)).

+-- {: .proof}
###### Proof

The first statement is a direct consequence of prop \ref{SomePropertiesOfAcyclicFibrations}.

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

$$
  \array{
    (I \otimes Y) \cup ((\partial I) \otimes X)
    &\stackrel{(s\circ \sigma_Y,(id_X, s \circ p))}{\to}& X
    \\
    \downarrow &\nearrow_{h}& \downarrow^{\mathrlap{p}}
    \\
    I \otimes X
    &\stackrel{ p \circ \sigma_X}{\to}& Y
  }
  \,,
$$

where the lift $h$ exists by assumption on $p$ ($s$ is necessarily a [[monomorphism]], being a [[section]]).

The resulting component triangle

$$
  \array{
    (\partial I) \otimes X
    &\stackrel{(id_X, s \circ p)}{\to}& X
    \\
    \downarrow &\nearrow_{h}&
    \\
    I \otimes X
  }
$$

exhibits $s$ as a [[deformation retract]], and the other resulting component triangle

$$
  \array{
    I \otimes Y
    &\stackrel{s\circ \sigma_Y}{\to}& X
    \\
    \downarrow^{\mathrlap{I \otimes s}} &\nearrow_{h}
    \\
    I \otimes X
  }
$$

says that $h\circ (I \otimes s) = s \circ \sigma_Y$, hence by [[natural transformation|naturality]] of cylinders $\cdots = \sigma_X \circ (I \otimes s)$, hence that the deformation retract is indeed strong.

=--

+-- {: .proof}
###### Proof
**of prop. \ref{AcyclicFibrationsAreAcyclicFibrations}**

First to see that the acyclic fibrations of def. \ref{AcyclicFibration} are indeed fibrations and weak equivalences:

By lemma \ref{SectionsOfAcyclicFibrationsAreDefRetracts} every acyclic fibration is in particular a $J$-homotopy equivalence, hence by lemma \ref{JHomotopyEquivsAreWeakEquivs} a weak equivalence. Moreover, by def. \ref{AcyclicFibration} 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. \ref{CellularStructuresExist}, 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 \ref{JHomotopyEquivsAreWeakEquivs} so is $j$.
Therefore, since $p$ is a fibration, we have a lift $\sigma$ in

$$
  \array{
    &\stackrel{id}{\to}&
    \\
    \downarrow^{\mathrlap{j}}
    & \nearrow_\sigma&
    \downarrow^{p}
    \\
    &\stackrel{q}{\to}&
  }
  \,.
$$

This equivalently exhibits $p$ as a [[retract]] of $q$

$$
  \array{
     &\stackrel{j}{\to} & & \stackrel{\sigma}{\to} &
   \\
   \downarrow^{\mathrlap{p}} && \downarrow^{\mathrlap{q}}
    && \downarrow^{\mathrlap{p}}
   \\
   & \stackrel{id}{\to} & & \stackrel{id}{\to} &
  }
  \,.
$$

So by lemma \ref{PropertiesOfAnodyneExtensions} with $q$ also $p$ is an acyclic fibration.

=--

([Cisinski 06, remark 1.3.28](#Cisinski06)).

#### Factorization
 {#Factorization}

We discuss the two factorization axioms for the [[model category]] structure from def. \ref{ModelStructureMorphismsFromHomotopicalStructure} to be established. First for factorizations into [cofibrations followed by acyclic fibrations](#CofibrationFollowedByAcyclicFibration), then for factorizations into [acyclic cofibrations followed by fibrations](AcyclicCofibrationFollowedByFibration).

##### Cofibration followed by acyclic fibration
 {#CofibrationFollowedByAcyclicFibration}

Every [[partial map classifier]] is an injective object, and so we have a functoral factorization of $X \to Y$ into $X \to X_\bot \times Y \to Y$. However, we can say more by appealing to general machinery.

For showing that every morphism factors as a monomorphism followed by an acyclic fibration, it is by prop. \ref{AcyclicFibrationsAreAcyclicFibrations} 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](#Cisinski06)).

We start with some entirely general statements about [[compact objects]].

+-- {: .num_prop #LimitsOverASmallDiagramAreCompact}
###### Proposition

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

$$
  \lim_\leftarrow : Func(A,Set) \to Set
$$

commutes with $\alpha$-[[filtered colimits]] / $\alpha$-[[directed colimits]].

=--

([Cisinski 06, prop. 1.2.9](#Cisinski06)).

+-- {: .num_prop #CompactnessOfColimitsOverCompacts}
###### Proposition

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 $A$.

=--

([Cisinski 06, prop. 1.2.10](#Cisinski06)).

+-- {: .proof}
###### Proof

Let $G : I \to C$ be a $\lambda$-[[filtered category|filtered diagram]].

Then by prop. \ref{LimitsOverASmallDiagramAreCompact} we have [[natural isomorphisms]]

$$
  \begin{aligned}
    \lim_{\underset{I}{\to}}
    C(\lim_{\underset{A}{\to}} F, G)
    & \simeq
  \lim_{\underset{I}{\to}}
  \lim_{\underset{A}{\leftarrow}}
  C(F, G)
   \\
  & \simeq
  \lim_{\underset{A}{\leftarrow}}
  \lim_{\underset{I}{\to}}
  C(F, G)
  \end{aligned}
  \,,
$$

because the $\lambda$-filtered diagram is at least ${\vert Mor(A)\vert}$-filtered and hence, by prop. \ref{LimitsOverASmallDiagramAreCompact}, 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

$$
  \begin{aligned}
    \cdots &
    \simeq
   \lim_{\underset{A}{\leftarrow}}
    C(F, \lim_{\underset{I}{\to}} G)
    \\
    & \simeq
    C(\lim_{\underset{A}{\to}} F, \lim_{\underset{I}{\to}} G)
  \end{aligned}
  \,.
$$

=--

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]]).

+-- {: .num_prop #PresheavesAreCompact}
###### Proposition

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]].

=--

([Cisinski 06, cor. 1.2.11](#Cisinski06)).

+-- {: .proof}
###### Proof

By the [[co-Yoneda lemma]] $X$ is the [[colimit]] over its elements

$$
  X \simeq \lim_\to (A/X \to PSh(A))
 \,.
$$

Since the image of the functor $A/X \to PSh(A)$ is in representables, which are maximally compact, the stament follows with prop. \ref{CompactnessOfColimitsOverCompacts}.

=--

+-- {: .num_defn #CellularModel}
###### Definition

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

$$
  Monos = LLP(RLP(I))
  \,.
$$

=--

([Cisinski 06, def. 1.2.26](#Cisinski06)).

+-- {: .num_remark }
###### Remark

By prop. \ref{PresheavesAreCompact} 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.

+-- {: .num_lemma #LemmaForGenerationOfMonos}
###### Lemma

Let $C \subset Mor(PSh(A))$ be a class of morphisms, and $D \subset PSh(A)$ a [[small set]] of objects, such that

1. $C$ is closed under [[pushouts]], [[retracts]] and [[transfinite composition]];

1. if $f,g \in Mor(PSh(A))$ are two composable morphisms with $f$ and $g \circ f$ in $C$, then also $g$ is in $C$;

1. every $X \in PSh(A)$ is the [[union]] of those of its [[sub-objects]] isomorphic to an object in $D$;

1. 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

$$
  C = LLP(RLP(I))
  \,.
$$

=--

([Cisinski 06, lemma. 1.2.24](#Cisinski06)).

+-- {: .proof}
###### Proof

A bit of work...

=--

+-- {: .num_prop #CellularStructuresExist}
###### Proposition

There exists a cellular structure, def. \ref{CellularModel}, on $PSh(A)$.

The set $I$ can be chosen to consist of morphisms into [[quotient objects]] of [[representable functors|representables]].

=--

([Cisinski 06, prop. 1.2.27](#Cisinski06)), also sketched at [[cellular model]].

+-- {: .proof}
###### Proof

Take in lemma \ref{LemmaForGenerationOfMonos} $C$ to be the class of monomorphisms
and $D$ to be the class of quotients of representables.

=--

+-- {: .num_cor #FirstFactorizationEstablishes}
###### Corollary

There exists a functorial factorization of morphisms in $PSh(A)$ into a monomorphism followed by an acyclic fibration.

=--

([Cisinski 06, cor. 1.2.28](#Cisinski06)).

+-- {: .proof}
###### Proof

By prop. \ref{PresheavesAreCompact} every object is $\alpha$-[[compact object|small]],
for some $\alpha$. Therefore by prop. \ref{CellularStructuresExist} we can
apply the [[small object argument]].

=--

##### Acyclic cofibration followed by fibration
 {#AcyclicCofibrationFollowedByFibration}

We show now for def. \ref{ModelStructureMorphismsFromHomotopicalStructure} 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 [[cofibrantly generated model category|generating acyclic cofibrations]]. This is the statement of prop. \ref{GeneratingAcyclicCofibrationsExist} below. Establishing this takes a few technical lemmas.

+-- {: .num_lemma #FactorAnodyneFollowedByRLPAnodyne}
###### Lemma

Every morphism admits a factorization into an anodyne extension, followed by a morphism having the right lifting property against anodyne extensions.

=--

([Cisinski 06, remark 1.3.29](#Cisinski06)).

+-- {: .proof}
###### Proof

By the [[small object argument]], in view of prop. \ref{PropertiesOfAnodyneExtensions}.

=--

+-- {: .num_lemma #ElementaryJHomotopyIntoFibIsEquivRel}
###### Lemma

If $T \in PSh(A)$ is fibrant, then for any $K \in PSh(A)$ elementary $J$-homotopy, def. \ref{ElementaryJHomotopy}, is already an equivalence relation on $Hom_{PSh(A)}(K,T)$ and coincides with $J$-homotopy.

=--

([Cisinski 06, lemma 1.3.30](#Cisinski06)).

+-- {: .num_prop #EveryAnodyneExtensionIsWeakEquivalence}
###### Proposition

Every anodyne extension is a weak equivalence.

=--

([Cisinski 06, lemma 1.3.31](#Cisinski06)).

+-- {: .proof}
###### Proof

For $j : K \to L$ an anodyne extension and $T$ a fibrant object, we need to show that

$$
  Ho_J(j,T) :  Ho_J(L,T) \to Ho_J(K,T)
$$

is a bijection.

It is surjective by the defining lifting property, which provides $\sigma$ in

$$
  \array{
    K &\stackrel{}{\to}& T
    \\
    \downarrow^{\mathrlap{j}}
    & \nearrow_{\mathrlap{\sigma}}
    \\
    L
  }
  \,.
$$

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 \ref{ElementaryJHomotopyIntoFibIsEquivRel} 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

$$
  \array{
    (I \otimes K) \cup ((\partial I) \otimes L)
    &\stackrel{(h,(l_0,l_1))}{\to}&
    T
    \\
    \downarrow & \nearrow_{\mathrlap{\eta}}
    \\
    I \otimes L
  }
  \,,
$$

where the left morphism is anodyne, by the second clause of def. \ref{AnodyneExtensions}, 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.

=--

+-- {: .num_lemma #WeakEquivalencesBetweenFibrantObjects}
###### Lemma

A morphism between fibrant objects is a weak equivalence precisely if it is a $J$-homotopy equivalence, def. \ref{JHomotopyEquivalence}.

=--

([Cisinski 06, lemma 1.3.32](#Cisinski06)).

+-- {: .proof}
###### Proof

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. \ref{JHomotopyEquivalence} 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.

=--

+-- {: .num_lemma #NaiveFibAndDualStrongDefRetractIsAcyclFib}
###### Lemma

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]].

=--

([Cisinski 06, lemma 1.3.33](#Cisinski06)).

+-- {: .proof}
###### Proof

That the former implies the latter was the statement of lemma \ref{SectionsOfAcyclicFibrationsAreDefRetracts}. 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

$$
  \array{
    K &\stackrel{a}{\to}& X
    \\
    \downarrow^{\mathrlap{i}} && \downarrow^{\mathrlap{p}}
    \\
    L &\stackrel{b}{\to}& Y
  }
$$

a commuting diagram with $i$ a monomorphism, there is a lift. To this end, observe that the given structures induce a morphism

$$
  (I \otimes K) \cup (\{1\} \otimes L) \to X
$$

such that

$$
  \array{
    (I \otimes K) \cup (\{1\} \otimes L)
    &&\stackrel{(k \circ (I \otimes a), s \circ b)}{\to}&&
    X
    \\
    \downarrow^{\mathrlap{j}} && && \downarrow^{\mathrlap{p}}
    \\
    I \otimes L &\stackrel{\sigma_L}{\to}& L
    &\stackrel{b}{\to}& Y
  }
  \,.
$$

By the second clause of def. \ref{AnodyneExtensions} 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.

=--

+-- {: .num_lemma #NaiveFibIntoFibIsWeakEquivIffItIsAcyclicFib}
###### Lemma

A morphism into a fibrant object with [[right lifting property]] against anodyne extensions is a weak equivalence precisely if it is an acyclic fibration.

=--

([Cisinski 06, lemma 1.3.34](#Cisinski06)).

+-- {: .proof}
###### Proof

We already know from prop. \ref{AcyclicFibrationsAreAcyclicFibrations} 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 \ref{NaiveFibAndDualStrongDefRetractIsAcyclFib} it is sufficient to show that it is a dual strong deformation retract.

By lemma \ref{WeakEquivalencesBetweenFibrantObjects} $p$ is also a $J$-homotopy equivalence. By lemma \ref{ElementaryJHomotopyIntoFibIsEquivRel} this is exhibited by an elementary $J$-homotopy $k : I \otimes Y \to Y$, which in particular gives a commuting diagram

$$
  \array{
    Y &\stackrel{t}{\to}& X
    \\
    \downarrow^{\mathrlap{\partial^1_Y}} && \downarrow^{\mathrlap{p}}
    \\
    I \otimes Y &\stackrel{k}{\to}& Y
  }
  \,,
$$

from which we obtain a lift $k' : I \otimes Y \to X$. Set then

$$
  s := k' \circ \partial^0_Y
  \,.
$$

One finds then $p \circ s = id_Y$.
As $p$ is a $J$-homotopy equivalence, it follows that $s$ is its homotopic inverse, in particular, there is a $J$-homotopy $h : id_X \Rightarrow s \circ p$.

Next we lift the trivial $J$-homotopy $p \Rightarrow p$
to transform $h$ into a dual stronf deformation:
$$
\array{
  \partial I \otimes I \otimes X \cup I \otimes \{1\} \otimes X &
  \stackrel{([h, s \circ p \circ h], s \circ p \circ \sigma_X)}{\to} & X \\
  \downarrow & H \nearrow & \downarrow^{\mathrlap{p}} \\
  I \otimes I \otimes X &
  \stackrel{p \circ h \circ \sigma_{I \otimes X}}{\to} &
  Y
}
$$
Now $H \circ (I \otimes \partial^0_{X})$ is a $J$-homotopy showing that $p$ is a dual strong deformation retract.

=--

+-- {: .num_cor }
###### Corollary

A cofibration into a fibrant object is a weak equivalence precisely if it is an anodyne extension.

=--

([Cisinski 06, cor 1.3.35](#Cisinski06)).

+-- {: .proof}
###### Proof

By lemma \ref{EveryAnodyneExtensionIsWeakEquivalence} 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. \ref{FactorAnodyneFollowedByRLPAnodyne} 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 \ref{NaiveFibIntoFibIsWeakEquivIffItIsAcyclicFib} $q$ is an acyclic fibration.

Therefore we have a lift $s$ in

$$
  \array{
     &\stackrel{id}{\to}&
    \\
    \downarrow^{\mathrlap{q}}
    &\nearrow_s&
    \downarrow^{\mathrlap{i}}
    \\
    &\stackrel{j}{\to}&
  }
$$

and this exhibits $i$ as a [[retract]] of $j$. Hence with $j$ also $i$ is an anodyne extension.

=--

+-- {: .num_prop }
###### Proposition

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.

=--

([Cisinski 06, prop.  1.3.36](#Cisinski06)).

+-- {: .proof}
###### Proof

(...)

=--

+-- {: .num_cor }
###### Corollary

The acyclic cofibrations are stable under [[transfinite composition]] and [[pushouts]]

=--

([Cisinski 06, cor.  1.3.37](#Cisinski06)).

+-- {: .num_lemma }
###### Lemma

Every [[strong deformation retract]] is an anodyne extension.

=--

([Cisinski 06, cor.  1.3.38](#Cisinski06)).

+-- {: .num_lemma }
###### Lemma

Every anodyne extension between fibrant objects is a [[strong deformation retract]].

=--

([Cisinski 06, lemma  1.3.39](#Cisinski06)).

+-- {: .num_prop }
###### Proposition

Pour tout cardinal assez grand $\alpha$, si on pose $\beta = 2^\alpha$, pour toute cofibration triviale $i : C \to D$, et pour tout [[subobject|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.

=--

([Cisinski 06, prop.  1.3.40](#Cisinski06)).

+-- {: .proof}
###### Proof

Three pages of work...

=--

+-- {: .num_prop #GeneratingAcyclicCofibrationsExist}
###### Proposition

There exists a set of generating acyclic cofibrations.

=--

([Cisinski 06, prop.  1.3.42](#Cisinski06)).

+-- {: .proof}
###### Proof

Use lemma \ref{LemmaForGenerationOfMonos} with $C$ the class of acyclic cofibrations and $D = Acc_\alpha(A)$ the set of $\alpha$-accessible presheaves for a sufficiently large cardinal $\alpha$.

=--

+-- {: .num_cor #FactorizationAcyclicCofibFib}
###### Corollary

There is a functorial factorization of every morphism into an acyclic cofibration followed by a fibration.

=--

([Cisinski 06, cor.  1.3.43](#Cisinski06)).

+-- {: .proof}
###### Proof

By prop. \ref{GeneratingAcyclicCofibrationsExist} and prop. \ref{PresheavesAreCompact} we may apply the [[small object argument]].

=--

#### Completeness
 {#Completeness}

We list lemmas to show prop. \ref{ConditionsForCompleteness}.

(...)

### Localizers
 {#ALocalizers}

Continuing to let $A$ be a [[small category]], write $PSh(A)$ for its [[category of presheaves]].

+-- {: .num_defn #ALocalizer}
###### Definition

An **$A$-localizer** is a [[class]] of morphisms $W \subset Mor(PSh(A))$ satisfying the following axioms

1. [[2-out-of-3]];

1. every acyclic fibration, def. \ref{AcyclicFibration}, is in $W$;

1. 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 $W(\emptyset)$.

=--

([Cisinski 06, def. 1.4.1](#Cisinski06)))

+-- {: .num_prop}
###### Proposition

For $(J,AnExt)$ a homotopical structure, def. \ref{HomotopicalStructure} on $PSh(A)$, the class $W$ of weak equivalences of the induced model category structure of theorem \ref{ModelStructureFromHomotopicalStructure} is an $A$-localizer.

If $\Lambda$ is a small set generating the anodyne extensions, $AnExt = LLP(RLP(\Lambda))$, then $W = W(\Lambda)$.

=--

([Cisinski 06, prop. 1.4.2](#Cisinski06)))

\begin{thm} \label{TheoremCharacterisationOfAccessibleLocalisersForPresheafCategories} Let $W \subset Mor(PSh(A))$. The following are equivalent.

1. $W$ is an accessible $A$-localizer, def. \ref{ALocalizer};

1. 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 \ref{ModelStructureFromHomotopicalStructure} from the homotopical structure, def. \ref{HomotopicalStructure}, given by the [[Lawvere cylinder]], def. \ref{LawvereCylinder}, and $AnExt := S$.

1. There is some homotopical structure, def. \ref{HomotopicalStructure}, on $PSh(A)$, such that $W$ is the class of weak equivalences of the model structure corresponding to it by theorem \ref{ModelStructureFromHomotopicalStructure}.

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 \ref{LawvereCylinder} and the empty set.

\end{thm}

([Cisinski 06, theorem 1.4.3](#Cisinski06))

+-- {: .num_remark}
###### Remark

This may be compared to [[Jeff Smith's theorem]], which constructs a model structure on a [[locally presentable category]].

=--

([Cisinski 06, scholie 1.4.6](#Cisinski06)))

### Simplicial completion
 {#SimplicialCompletion}

Give a localizer $W$ on $PSh(A)$, there is a localizer $W_{horiz}$ on $PSh(A \times \Delta)$

$$
  W_{horiz} \coloneqq
  \{
    X \times q^* (\Delta_1) \to X
  \}
  \,.
$$

(...)

See ([Ara, p. 9](#Ara)).

## Examples
 {#Examples}

* The archetypical and motivating example is the [[classical model structure on simplicial sets]], which is a Cisinski model structure on the [[presheaf topos]] over the [[simplex category]] ([Cisinski 06, section 2](#Cisinski06)).

* 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 [[Bousfield localization of model categories|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](#SimplicialCompletion) of this model structure. (see [Ara](#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 model structure|transferred]] from one that is.

* For any small category $A$, the homotopical datum (Definition \ref{DefinitionHomotopicalDatum}) given by the $(J, \emptyset)$, where $J$ is the [[Lawvere cylinder ]](Definition \ref{LawvereCylinder}), generates a Cisinki model structure on the category of presheaves on $A$. By  ([Cisinski 06, rem. 1.3.15](#Cisinski06)), a set of generating anodyne extensions is $A\times \Omega \cup_A B \to B \times \Omega$ where $A \to B$ are generators for the class of monomorphisms. This is sometimes known as the [[minimal Cisinski model structure]]. See also Theorem \ref{TheoremCharacterisationOfAccessibleLocalisersForPresheafCategories}.

## Properties

### Properness

+-- {: .num_theorem #RightProper}
###### Theorem
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 model category|right proper]] if and only if pullback along fibrations between fibrant objects in this model structure takes morphisms in $\Sigma$ to weak equivalences.
=--

This is [Cisinski 02, Théorème 4.8](#Cisinski02). This is closely related to (but not a consequence of) a theorem of Bousfield that applies to any model category whatsoever; see [[proper model category#FibrationsBetweenFibrantObjectsSuffice|this proposition]].

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.

## Related concepts

* [[test topos]]

## References
 {#References}

The original articles are

* {#Cisinski02} [[Denis-Charles Cisinski]] , _Théories homotopiques dans les topos_, JPAA, Volume 174 (2002), p.43-82

* {#Cisinski06} [[Denis-Charles Cisinski]], *Les préfaisceaux comme types d'homotopie*, Astérisque **308** Soc. Math. France (2006), 392 pages [[numdam:AST_2006__308__R1_0](http://www.numdam.org/item/?id=AST_2006__308__R1_0) [pdf](http://www.math.univ-toulouse.fr/~dcisinsk/ast.pdf)]

A more recent, English presentation of much of this material appears in section 2.4 of

* {#Cisinski20} [[Denis-Charles Cisinski]], _Higher category theory and homotopical algebra_ ([pdf](http://www.mathematik.uni-regensburg.de/cisinski/CatLR.pdf))

Further developments are in

* [[Rick Jardine]], _Categorical homotopy theory_ (2003) ([K-theory](http://www.math.uiuc.edu/K-theory/0669/))

* [[Marc Olschok]], _On constructions of left determined model structures_, PhD thesis (2009) ([pdf](http://is.muni.cz/th/183259/prif_d/diss.pdf))

* [[Simon Henry]], _Minimal model structures_, [arXiv:2011.13408](https://arxiv.org/abs/2011.13408).

Some work on generalizing from presheaf toposes to all toposes is in

* [[Denis-Charles Cisinski]], _Faisceaux localement asphériques_ (preliminary version), 2003, [pdf](http://www.mathematik.uni-regensburg.de/cisinski/mtest2.pdf)

See also

* [[André Joyal]], _[[joyalscatlab:Cisinski's theory]]_

* [[Dimitri Ara]], _Higher quasi-categories vs higher Rezk spaces_ ([arXiv:1206.4354](http://arxiv.org/abs/1206.4354))
 {#Ara}

[[!redirects Cisinski model structures]]

[[!redirects Cisinski model category]]
[[!redirects Cisinski model categories]]


CKM matrix

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In the [[standard model of particle physics]], the _CKM matrix_, arising from [[diagonal matrix|diagranomalizing]] the [[Yukawa couplings]], measures the [[interaction]] of the [[quarks]] via [[W-bosons]] and hence via the [[weak nuclear force]].

## Definition

From [PDG 18](#PDG18):

## Observation and anomaly

See at

* [[Cabibbo anomaly]]

* [[V_cb puzzle]]

## Related concepts

* [[standard model of particle physics]]

* [[flavour (particle physics)]]

* [[weak nuclear force]]

* [[V_cb puzzle]]

* [[Cabibbo anomaly]]

## References

* {#PDG18} [[Particle Data Group]] Review 2018, _12. CKM Quark-Mixing Matrix_ ([pdf](http://pdg.lbl.gov/2019/reviews/rpp2018-rev-ckm-matrix.pdf))

* Sébastien Descotes-Genon, [[Patrick Koppenburg]], _The CKM Parameters_, Annual Review of Nuclear and Particle Science Vol 67:97-127, 12 October 2017 ([arXiv:1702.08834](https://arxiv.org/abs/1702.08834))

See also

* Wikipedia, _[Cabibbo-Kobayashi-Maskawa matrix](https://en.wikipedia.org/wiki/Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix)_

[[!redirects Cabibbo-Kobayashi-Maskawa matrix]]


Claire Tomesch

I am a graduate student at the [University of Chicago](http://www.math.uchicago.edu).  I am interested in ([[higher category theory|higher]]) [[category theory]] and its applications.

category: people


Claire Voisin

* [webpage](http://www.math.polytechnique.fr/~voisin/)

* [Wikipedia](http://en.wikipedia.org/wiki/Claire_Voisin)

## Selected writings

On [[Hodge theory]]:

* {#Voisin02} [[Claire Voisin]] (translated bei [[Leila Schneps]]), _[[Hodge theory and Complex algebraic geometry]] I,II_, Cambridge Stud. in Adv. Math. __76, 77__, 2002/3 ([doi:10.1017/CBO9780511615344](https://doi.org/10.1017/CBO9780511615344))

* {#Voisin08} [[Claire Voisin]], *Hodge theory and the topology of compact Kähler and complex projective manifolds*, Lecture notes (2008) [[pdf](http://www.math.columbia.edu/~thaddeus/seattle/voisin.pdf), [[Voisin-HodgeTheory.pdf:file]]]

## Related $n$Lab entries

* [[Hodge theory]]

  * [[Hodge structure]]

  * [[Hodge conjecture]], [[integral Hodge conjecture]]

* [[local system]]

category: people

[[!redirects C. Voisin]]


clan

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

* table of contents
{: toc}

## Idea

A *clan* in the sense of [Joyal 2017, Def. 1.1.1](#Joyal17) (previously: *category with [[display maps]]* [[Taylor 1987, §4.3.2](#Taylor87)], or *display category* [[Taylor 1999, §8.3](#Taylor99)], a non-strict notion of [[contextual categories]]) is a minimum notion of [[fibration category]] providing [[categorical semantics for dependent types]] modelling (just) [[context extension]] and [[dependent pair type]]-[[type formation|formation]] (but not necessarily, say, [[function types]] or [[identity types]]):

## Definition

A clan is a [[category]] with a [[terminal object]] equipped with a sub-[[class]] $Fib \subset Mor$ of its [[morphisms]], to be called the *[[fibrations]]* and serving the role of [[display maps]], such that:

1. every [[isomorphism]] is in $Fib$,

1. for every [[object]] $X$ the [[terminal object|terminal map]] $X \to \ast$ is in $Fib$ (hence all objects are [[fibrant object|fibrant]], cf. *[[category of fibrant objects]]*),

1. for $f \in Fib$ and $h$ any [[cospan|coincident]] morphism, the [[base change]] ([[pullback]])  $h^\ast f$ exits and is in $Fib$ (this interprets [[context extension]])

1. for composable $f,g \in Fib$ also their [[composition]] $g \circ f \in Fib$ (this interprets [[dependent pair type]]-[[type formation]])

## Related concepts

* [[fibration category]]

* [[category of fibrant objects]]

* [[categorical semantics of dependent types]]

  * [[tribe]]

  * [[type-theoretic model category]]

## References

The definition of *categories with [[display maps]]* and their role as [[categorical semantics for dependent types]] is due to

* {#Taylor87} [[Paul Taylor]], §4.3.2 in: *Recursive Domains, Indexed Category Theory and Polymorphism*, Cambridge (1983-7) [[[Taylor-IndexedCategoryTheory.pdf:file]]]

being a non-strict version of [[contextual categories]]:

* {#Taylor99} [[Paul Taylor]], §8.3 in: _[[Practical Foundations of Mathematics]]_, Cambridge University Press (1999) [[webpage](http://www.paultaylor.eu/~pt/prafm/index.html)]

The terminology "clan" for this notion is due to:

* {#Joyal17} [[André Joyal]], _Notes on Clans and Tribes_ [[arxiv:1710.10238](https://arxiv.org/abs/1710.10238)]

but the general notion of [[categorical semantics of dependent types]] is classical, going back at least to [Seely 1984](relation+between+type+theory+and+category+theory#Seely84b) and  [Hofmann 1997](categorical+semantics+of+dependent+type+theory#Hofmann97).

The type-theory-literature traditionally refers to such categories generically as *[[display map]]/[[fibration categories]] with such-and-such types*, eg.

[North 2019, Rem. 2.4](#North19):

> The definitions in this section are relatively standard in the literature. A display map category which models [[dependent pair type|$\Sigma$-types]] coincides with Joyal’s notion of *clan*, and a display map category which models [[dependent pair type|$\sum$-types]] and [[dependent function type|$\prod$-types]] coincides with his notion of *$\pi$-clan*.

* {#North19} [[Paige Randall North]], *Identity types and weak factorization systems in Cauchy complete categories*, Mathematical Structures in Computer Science **29** 9 (2019) 1411-1427 [[doi:10.1017/S0960129519000033](https://doi.org/10.1017/S0960129519000033), [arXiv:1901.03567](https://arxiv.org/abs/1901.03567)]

[[!redirects clans]]


Clara Löh

* [personal page](https://loeh.app.uni-regensburg.de/)

* [wikipedia entry](https://de.wikipedia.org/wiki/Clara_L%C3%B6h)

## Selected writings

On [[geometric group theory]]:

* [[Clara Löh]], *Geometric Group Theory*, Springer 2017 ([doi:10.1007/978-3-319-72254-2](https://link.springer.com/book/10.1007/978-3-319-72254-2), [pdf](http://www.mathematik.uni-regensburg.de/loeh/teaching/ggt_ws1011/lecture_notes_old.pdf))

## Related $n$Lab entries

* [[Cayley graph]]

category: people

[[!redirects Clara Loh]]


Clarence I. Lewis

* [Wikipedia entry](https://en.wikipedia.org/wiki/C._I._Lewis)

* [plato.stanford.edu/entries/lewis-ci](https://plato.stanford.edu/entries/lewis-ci)

## Selected writings

On [[formal logic]] and giving the first modern formalizations of [[modal logic]]:

* [[Clarence I. Lewis]], [[Cooper H. Langford]], *Symbolic Logic* (1932), Dover (2000) [App. II [pdf](https://www.hist-analytic.com/Lewisstrict.pdf)]

Translation of [[Gottfried Leibniz]]'s original statement of the  principles of  [[identity of indiscernibles]] and [[indiscernibility of identicals]] (and then Fichte's "[[first law of thought]]"):

* {#Lewis18} [[Clarence I. Lewis]], Appendix (p. 373) of: *A Survey of Symbolic Logic*, University of California (1918) [[[LewisAppendix-LeibnizIndiscernibles.pdf:file]]]

category: people

[[!redirects Clarence Lewis]]


Clarence Wilkerson

* [Webpage at TAMU](https://www.math.tamu.edu/~cwilkers/)

* [Webpage at Purdue](http://www.math.purdue.edu/~wilker/)

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=16936)

## Selected writings

On [[Dickson invariants]]:

* {#Wilkerson83} [[Clarence Wilkerson]], _A primer on Dickson invariants_, in [[Haynes Miller]], [[Stewart Priddy]] (eds.) _Proceedings of the Northwestern Homotopy Theory Conference_, 1983  ([doi:10.1090/conm/019](http://dx.doi.org/10.1090/conm/019))

On [[p-compact groups]]:

* {#DwyerWilkerson94} [[William Dwyer]], [[Clarence Wilkerson]], _Homotopy Fixed-point Methods for Lie Groups and Finite Loop Spaces_,  Annals of Mathematics Second Series, Vol. 139, No. 2 (Mar., 1994), pp. 395-442 ([jstor:2946585](https://www.jstor.org/stable/2946585))

The [[Dwyer-Wilkerson H-space]]:

*  {#DwyerWilkerson93} [[William Dwyer]], [[Clarence Wilkerson]], _A new finite loop space at the prime two_, J. Amer. Math. Soc. 6 (1993), 37-64  ([doi:10.1090/S0894-0347-1993-1161306-9](https://doi.org/10.1090/S0894-0347-1993-1161306-9))

category: people


Clark Barwick

* [website](https://www.maths.ed.ac.uk/~cbarwick/)

## Selected writings

On [[(∞,n)-categories]]:

* {#BarwickThesis} [[Clark Barwick]], _$(\infty,n)$-$Cat$ as a closed model category_ PhD (2005)

On [[Bousfield localization of model categories|Bousfield localization of]] [[combinatorial model categories]]:

* {#Barwick10} [[Clark Barwick]], *On left and right model categories and left and right Bousfield localizations*, Homology, Homotopy and Applications **12** 2 (2010) 245–320
[[doi:10.4310/hha.2010.v12.n2.a9](https://doi.org/10.4310/hha.2010.v12.n2.a9), [euclid:1296223884](https://projecteuclid.org/journals/homology-homotopy-and-applications/volume-12/issue-2/On-left-and-right-model-categories-and-left-and-right/hha/1296223884.full), subsuming: [arXiv:0708.2067](https://arxiv.org/abs/0708.2067), [arXiv:0708.2832](https://arxiv.org/abs/0708.2832), [arXiv:0708.3435](http://arxiv.org/abs/0708.3435)]

On [[relative categories]]:

* {#BarwickKan12} [[Clark Barwick]], [[Daniel Kan]], _Relative categories: Another model for the homotopy theory of homotopy theories_. Indagationes Mathematicae **23** (2012) 42-68 [[arXiv:1011.1691](https://arxiv.org/abs/1011.1691), [doi:10.1016/j.indag.2011.10.002](https://doi.org/10.1016/j.indag.2011.10.002)]

On models for [[(∞,n)-categories]]:

* [[Clark Barwick]], [[Chris Schommer-Pries]], _On the Unicity of the Homotopy Theory of Higher Categories_ ([arXiv:1112.0040](http://arxiv.org/abs/1112.0040), [slides](http://prezi.com/w0ykkhh5mxak/the-uniqueness-of-the-homotopy-theory-of-higher-categories/))

On [[simplicial localization]] as a [[functor]] on the category of [[relative categories]]:

* {#BarwickKan12} [[Clark Barwick]], [[Daniel Kan]], *A characterization of simplicial localization functors and a discussion of DK equivalences*, Indagationes Mathematicae Volume 23, Issues 1–2, March 2012, Pages 69-79 ([doi:10.1016/j.indag.2011.10.001](https://doi.org/10.1016/j.indag.2011.10.001))

On [[exact (∞,1)-categories]]:

* [[Clark Barwick]], _On the Q construction for exact quasicategories_ ([arXiv:1301.4725](http://arxiv.org/abs/1301.4725))

and the [[theorem of the heart]]:

* [[Clark Barwick]], *On exact infinity-categories and the Theorem of the Heart*, Compositio Mathematica **151** (2015) 2160-2186 [[arXiv:1212.5232](https://arxiv.org/abs/1212.5232), [doi:10.1112/S0010437X15007447](https://doi.org/10.1112/S0010437X15007447)]

On [[spectral Mackey functors]] and [[equivariant algebraic K-theory]]:

* {#Barwick14} [[Clark Barwick]], _Spectral Mackey functors and equivariant algebraic K-theory (I)_, Adv. Math., 304:646–727,  2017 ([doi:10.1016/j.aim.2016.08.043](https://doi.org/10.1016/j.aim.2016.08.043), [arXiv:1404.0108](http://arxiv.org/abs/1404.0108))

* {#Barwick15} [[Clark Barwick]], [[Saul Glasman]], Jay Shah, _Spectral Mackey functors and equivariant algebraic K-theory (II)_ ([arXiv:1505.03098](http://arxiv.org/abs/1505.03098))

On [[cyclotomic spectra]]:

* {#BarwickGlasman16} [[Clark Barwick]], [[Saul Glasman]], _Cyclonic spectra, cyclotomic spectra, and a conjecture of Kaledin_ ([arXiv:1602.02163](http://arxiv.org/abs/1602.02163))

* [[Clark Barwick]], [[Saul Glasman]], _Noncommutative syntomic realization_ ([pdf](http://dl.dropbox.com/u/1741495/papers/tornado2.pdf))

See also:

* *SHIFTED: Stratified Homotopy Invariants, Field Theories, Exodromy, and Duality* ([github](https://shifted-project.github.io/))

## Related $n$Lab entries

* [[Bousfield localization]]

* [[n-category]]

* [[algebraic K-theory]]

* [[cyclotomic spectrum]]

* [[Parametrized Higher Category Theory and Higher Algebra]]

category: people


class

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Modalities, Closure and Reflection
+-- {: .hide}
[[!include modalities - contents]]
=--
=--
=--

\tableofcontents

## Idea

In [[set theory]], a class is a [[proposition]] or [[truth value]] in the [[context]] of a [[set]] [[free variable]]. In [[dependent type theory]] with a [[type universe]] $U$, a class is an [[h-proposition]] in the context of a [[free variable]] $x:U$.

In [[set-level type theory|set-level]] [[dependent type theory]] with a separate [[set]] or [[type]] [[judgment]] but no [[type universes]], there are no [[set]]/[[type]] free variables. However, one could nevertheless interpret a class in [[dependent type theory]] as a generalized [[modal operator]] on [[sets]]/[[types]] which takes sets to [[h-propositions]] by the [[propositions as some types]] interpretation. If we use the [[propositions as types]] interpretation of [[dependent type theory]], then a class in any [[dependent type theory]] is just a generalized [[modal operator]] on [[sets]]/[[types]]. The generalized modal operators here are not required to be either [[idempotent]] or [[monadic]].

One could internalize the notion of class inside of the foundations, and this internal notion of class is used to address size issues in foundations, and in particular, are used in [[category theory]] for defining various [[locally small categories]] and [[large categories]].

## Definitions

There are multiple ways to define a class in different [[foundations of mathematics]]. Let us work in [[natural deduction]].

* Suppose we are given an [[unsorted set theory]] as [[logic over type theory]] whose objects are [[sets]]. A **class** $C$ is a [[proposition]] or [[truth value]] in the [[context]] of a [[free variable]] $A$.
$$\Gamma, A \vert \Phi \vdash C \; \mathrm{prop}$$

* Suppose we are given a [[simply sorted set theory]] or [[dependently sorted set theory]] as [[logic over type theory]] with a [[sort]] $\mathrm{Set}$ of [[sets]]. A **class** $C$ is a [[proposition]] or [[truth value]] in the [[context]] of a [[free variable]] for a set $A:\mathrm{Set}$.
$$\Gamma, A:\mathrm{Set} \vert \Phi \vdash C \; \mathrm{prop}$$

* Suppose we are given an [[unsorted set theory]] as [[logic over type theory]] whose objects are "probable sets", with a [[predicate]] $\mathrm{isSet}(A)$ saying whether a "probable set" $A$ is a set. A **class** $C$ is a [[proposition]] or [[truth value]] in the [[context]] of a [[free variable]] $A$ and the true proposition $\mathrm{isSet}(A) \; \mathrm{true}$.
$$\Gamma, A \vert \Phi, \mathrm{isSet}(A) \; \mathrm{true} \vdash C \; \mathrm{prop}$$

* Suppose we are given a [[simply sorted set theory]] or [[dependently sorted set theory]] as [[logic over type theory]] with a [[sort]] $\mathrm{ProbSet}$ of "probable sets" with a [[predicate]] $\mathrm{isSet}(A)$ on $\mathrm{ProbSet}$ saying whether a term $A:\mathrm{ProbSet}$ is a set. A **class** $C$ is a [[proposition]] or [[truth value]] in the [[context]] of a [[free variable]] for a probable set $A:\mathrm{ProbSet}$ and the true proposition $\mathrm{isSet}(A) \; \mathrm{true}$.
$$\Gamma, A:\mathrm{ProbSet} \vert \Phi, \mathrm{isSet}(A) \; \mathrm{true} \vdash C \; \mathrm{prop}$$

* Suppose we are given a [[dependent type theory]] with a [[type universes]] $\mathcal{U}$ and a separate [[type]] [[judgment]], as well as a [[type of all propositions]] $\Omega$. Then a **class** $C$ is a [[proposition]] in the [[context]] of a type $A:\mathcal{U}$
$$\Gamma, A:\mathcal{U} \vdash C:\Omega$$

* Suppose we are given a [[dependent type theory]] with a [[type universes]] $\mathcal{U}$ and a separate [[type]] [[judgment]], and are using the [[propositions as some types]] interpretation of dependent type theory. Then a **class** $C$ is a [[type]] in the [[context]] of a type $A:\mathcal{U}$
$$\Gamma, A:\mathcal{U} \vdash C \; \mathrm{type}$$
with the rule
$$\frac{\Gamma, A:\mathcal{U} \vdash C \; \mathrm{type} \quad \Gamma \vdash B:\mathcal{U}}{\Gamma \vdash p^C(B):\mathrm{isProp}(C[B/A])}$$

* Suppose we are given a [[dependent type theory]] with a [[type universes]] $\mathcal{U}$ and a separate [[type]] [[judgment]], and are using the [[propositions as types]] interpretation of dependent type theory. Then a **class** $C$ is a [[type]] in the [[context]] of a type $A:\mathcal{U}$
$$\Gamma, A:\mathcal{U} \vdash C \; \mathrm{type}$$

* Suppose we are given a [[dependent type theory]] with a [[sequence]] of [[Russell universes]] $\mathcal{U}_i$ but no separate [[type]] [[judgment]], and are using the [[propositions as some types]] interpretation of dependent type theory. Then a **class** $C$ is an [[h-proposition]] in the successor universe $\mathcal{U}_{s_\mathcal{N}(i)}$ in the [[context]] of a type $A:\mathcal{U}_i$
$$\Xi \vert \Gamma, A:\mathcal{U}_i \vdash C:\mathrm{Prop}_{\mathcal{U}_{s_\mathcal{N}(i)}}$$

* Suppose we are given a [[dependent type theory]] with a [[sequence]] of [[Russell universes]] $\mathcal{U}_i$ but no separate [[type]] [[judgment]], and are using the [[propositions as types]] interpretation of dependent type theory. Then a **class** $C$ is a [[type]] in the successor universe $\mathcal{U}_{s_\mathcal{N}(i)}$ in the [[context]] of a type $A:\mathcal{U}_i$
$$\Xi \vert \Gamma, A:\mathcal{U}_i \vdash C:\mathcal{U}_{s_\mathcal{N}(i)}$$

* Suppose we are given a [[dependent type theory]] with a separate [[type]] [[judgment]] but no universes. Then a **class** $C$ is a generalized [[modal operator]] on the type theory, which isn't required to be either [[idempotent]] or [[monadic]], such that given a [[type]] $A$, the [[type]] $C(A)$ is an [[h-proposition]].
$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash C(A) \; \mathrm{type}} \qquad \frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash p_A^C:\mathrm{isProp}(C(A))}$$

* Suppose we are given a [[dependent type theory]] with a separate [[type]] [[judgment]] but no [[type universes]], and are using the [[propositions as types]] interpretation of dependent type theory. Then a **class** $C$ is a generalized [[modal operator]] on the type theory which isn't required to be either [[idempotent]] or [[monadic]].
$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash C(A) \; \mathrm{type}}$$

## Examples

* In [[set theory]], there is a class $\exists a.a \in A$ for whether the set $A$ is an [[inhabited set]]. In [[dependent type theory]], the class is the [[propositional truncation]] [[modal operator]] $[A]$.

* In [[set theory]], there is a class $\forall a.a \in A \wedge \forall b.b \in A \wedge a = b$ for whether the set $A$ is a [[subsingleton]]. In [[dependent type theory]], the class is the [[isProp]] [[modal operator]] $\Pi (a:A).\Pi (b:A).(a =_A b)$.

* In [[set theory]], there is a class $\exists a.a \in A \wedge \forall b.b \in A \wedge a = b$ for whether the set $A$ is a [[singleton]]. In [[dependent type theory]], the class is the [[isContr]] [[modal operator]] $\Sigma (a:A).\Pi (b:A).(a =_A b)$.

## Internal classes

It is possible to [[internalization|internalize]] the notion of class inside of the [[foundations]] itself. Classes are either primitive, such as in [[class theory]], or a derived concept from a notion of [[universe]] in the foundations, such as in [[set theory]] or [[type theory]]. These internal classes are useful to address size issues in foundations, and in particular, are used in [[category theory]] for defining various [[locally small categories]] and [[large categories]].

### Classes as primitive

There are foundational theories called [[class theories]] where classes are primitives, rather than propositions in the context of a free variable $x:\mathrm{Set}$. This is similar to [[dependently sorted set theory|dependently sorted]] [[allegorical set theories]], where [[relations]] are primitives, rather than propositions in the context of free variables $a:A$ and $b:B$.

* Material class theory: Classes are primitives of the theory, and sets are defined as specific kinds of classes.  Examples include [[Morse-Kelley class theory]] and [[von Neumann–Bernays–Gödel class theory]].

* Structural class theory: A **class** is an object of a [[category of classes]], and sets are defined as specific kinds of classes. Examples include [[category with class structure]] and the branch of [[algebraic set theory]]

### Classes as derived from universes

#### In set theory

There are many notions of [[universe]] in [[set theory]], including [[Grothendieck universes]], [[well-pointed pretopos|well-pointed]] [[Heyting pretopoi]], and [[well-pointed category|well-pointed]] [[division allegories]].

Let $U$ be a [[universe]]. Then a **class** relative to $U$ is a [[subset]] $C \subseteq U$ with a given injection $i:C \hookrightarrow U$. If one has choice, any subset comes with a given injection via the [[axiom of choice]]. Thus, by this definition, it is a injective [[family of sets|family of $U$-small sets]].

Equivalently, a class $C$ relative to $U$ is an [[endofunction]] $C:U \to U$ such that for all $U$-small sets $A \in U$, $C(A)$ is a [[subsingleton]] subset.

#### In dependent type theory

In [[dependent type theory]], let $(U, T)$ be a [[univalent universe|univalent]] [[Tarski universe]], let $(\mathrm{Set}_U, \mathrm{Set}_T)$ be the univalent [[h-groupoid]] of $U$-small [[h-sets]], let $V_U$ be the [[material set theory|material]] [[cumulative hierarchy]] [[higher inductive type]] relative to $U$. Then a **class** is an [[endofunction]] $C:\mathrm{Set}_U \to \mathrm{Set}_U$ such that for all $U$-small sets $A:\mathrm{Set}_U$, $(C(A))$ is a $U$-small [[h-proposition]], and a **material class** is a function $C:V_U \to \mathrm{Set}_U$ such that for all material sets $A:V_U$, $(C(A))$ is a $U$-small [[h-proposition]].

### Proper classes

A **proper class** is a class which is not a [[set]]. What a set is differs from foundations to foundations.

What not being a set means depends upon the foundation; in [[material set theory]], one would use the property of not being equal to any sets, while in [[structural set theory]], one would use the property of not being in [[bijection]] with any sets.

Given a notion of [[universe]], a **[[proper class]] relative to $U$** is a class relative to $U$ which is not a set. If classes are defined as subsets of $U$ with an injection into $U$, then a proper class is a class which is not a [[singleton subset]].

In the context of the [[global axiom of choice]], a proper class is a class which can be put in [[bijection]] with the class of all [[ordinal number|ordinals]], $Ord$.

## Category of classes

The [[category of classes]] is closed under all large [[colimits]]
and [[small limits]].
See the linked article for more information and precise definitions.

Just as an [[elementary topos]] is an axiomatization of basic properties of the category [[Set]], a [[category with class structure]] is an axiomatization of basic properties of the category $Class$.  See also [[algebraic set theory]].

## See also

* [[class object]]

* [[class theory]]

* [[category of classes]]

* [[type of classes]]

* [[family of sets]]

* [[large set]], [[small set]]

## References

For the definition of a material class relative to a universe $U$ in [[homotopy type theory]], see section 10.5.3 of:

* [[Univalent Foundations Project]], *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]*, Institute for Advanced Study (2013) [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)]

A paper detailing one approach to the technical side of how classes appear in [[category theory]] (namely using [[Grothendieck universes]]) is

* {#Levy18} [[Paul Blain Levy]], _Formulating Categorical Concepts using Classes_ [[arXiv:1801.08528](https://arxiv.org/abs/1801.08528)]

[[!redirects class]]
[[!redirects classes]]
[[!redirects internal class]]
[[!redirects internal classes]]
[[!redirects class relative to a universe]]
[[!redirects classes relative to a universe]]

[[!redirects proper class]]
[[!redirects proper classes]]

[[!redirects structural class]]
[[!redirects structural classes]]

[[!redirects material class]]
[[!redirects material classes]]


class equation

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--

# Contents #
* table of contents
{: toc}

## Idea

The term *class equation* (or *class formula*, or *orbit decomposition* formula) refers to a basic type of counting argument that comes about by decomposing a [[finite set|finite]] [[G-set]] as a [[union]] of its [[orbits]]. This has a number of fundamental applications in [[group theory]].

## Statement

Let $G$ be a [[group]] and let $A$ be a [[G-set]] (given by a [[homomorphism]] $G \to \hom_{Set}(A, A)$ of [[monoids]], with which is associated an [[action]] $\alpha: G \times A \to A$). Recall that $A$ is [[connected object|connected]] in the [[category of G-sets]] if $A$ is [[inhabited set|inhabited]] and the action is [[transitive action|transitive]]; in this case, choosing an [[element]] $a \in A$, there is a [[surjection]] of $G$-sets $G \to A$ sending $1 \mapsto a$, and this induces an [[isomorphism]] $G/Stab(a) \cong A$ where $Stab(a)$ is the [[stabilizer]] of $a$ and $G/Stab(a)$ is the $G$-set consisting of left [[cosets]] of $Stab(a)$.

More generally, each $G$-set $A$ admits a canonical decomposition as a [[coproduct]] of its connected components; the components are usually called the [[orbits]] of the action. Choosing a representative element $a_x$ in each orbit $x$, this means we have an isomorphism of $G$-sets

$$A \cong \sum_{orbits x} G/Stab(a_x).$$

By taking $G$ and $A$ to be [[finite set|finite]] and counting elements, we get an equation of the form

$${|A|} = \sum_{orbits x} \frac{{|G|}}{{|Stab(a_x)|}}.$$

We call an instance of this equation a *class equation*. By judicious choice of groups $G$ and $G$-sets $A$, often in combination with [[number theory|number-theoretic]] arguments, one can derive many useful consequences; some sample applications are given below.

Notice that reading the class equation equivalently as

$$\sum_{orbits x} \frac{{1}}{{|Stab(a_x)|}} = \frac{|A|}{|G|} $$

it expresses the [[groupoid cardinality]] of the [[action groupoid]] of $G$ acting on $A$.

## Applications

### Centers of $p$-groups

Let $p$ be a [[prime number|prime]]; recall that a $p$-[[p-primary group|group]] is a [[finite group]] whose [[order of a group|order]] is a power of $p$. A basic structural result is the following.

+-- {: .num_prop #pgroup}
###### Proposition
A non-trivial $p$-group $G$ has a nontrivial [[center]] $Z(G)$.
=--

+-- {: .proof}
###### Proof
Let $G$ act on itself by the [[conjugation action]] $G \times G \to G$, $(g, h) \mapsto g h g^{-1}$. In this case an orbit $Orb(h)$ is usually called the [[conjugacy class]] of $h$, and $Orb(h)$ is trivial (consists of exactly one element $h$) iff $h$ belongs to $Z(G)$. In any case ${|Orb(h)|} = \frac{{|G|}}{{|Stab(h)|}}$ divides ${|G|} = p^n$, and therefore $p$ divides ${|Orb(h)|}$ if $h$ is noncentral. In this case the class equation takes the form

$${|G|} = {|Z(G)|} \; + \sum_{nontrivial\; orbits x} \frac{{|G|}}{{|Stab(a_x)|}}$$

and now since $p$ divides ${|G|}$ as well as each term in the sum over nontrivial orbits, it must also divide ${|Z(G)|}$. In particular, $Z(G)$ has more than one element.
=--

It follows by [[induction]] that $p$-groups are [[solvable group|solvable]], since the center is a [[normal subgroup]] and the quotient $G/Z(G)$ is also a $p$-group.  Since a group obtained from an abelian group by repeated central extensions is nilpotent, $p$-groups are in fact [[nilpotent group|nilpotent]].

### Number of fixed points

An elementary observation that is frequently useful is that the number of [[fixed points]] of an [[involution]] on a [[finite set]] $S$ has the same parity as $S$. This is a statement about $\mathbb{Z}/(2)$-sets; we generalize this to a statement about $G$-sets for general $p$-groups $G$. (Again, a fixed point of a $G$-set is an element whose orbit is a singleton.)

+-- {: .num_prop #pparity}
###### Proposition
If $G$ is a $p$-group acting on a set $A$, then

* ${|A|} \equiv {|Fix(A)|} \; mod p$.

As special cases, if there is just one fixed point, then ${|A|} \equiv 1 \; mod p$, and if $p$ divides ${|A|}$, then $p$ divides ${|Fix(A)|}$.
=--

+-- {: .proof}
###### Proof
The class equation takes the form

$${|A|} = {|Fix(A)|} \; + \sum_{nontrivial\; orbits x} \frac{{|G|}}{{|Stab(a_x)|}}$$

where $p$ divides each summand over nontrivial orbits on the right, since $G$ is a $p$-group. Now reduce mod $p$.
=--

### Wedderburn's theorem

In the theory of finite [[projective planes]], an important result is that a projective plane is Pappian if it is Desarguesian. The purely algebraic version of this is Wedderburn's theorem:

+-- {: .num_theorem}
###### Theorem
A finite [[division ring]] $D$ is commutative.
=--

+-- {: .proof}
###### Proof
**(Witt)**
The center of $D$ is a [[field]]; if $D$ is of characteristic $p \gt 0$, then the center $F$ has $q = p^f$ elements for some $f$. We may regard $D$ as an $F$-vector space of dimension $n$, whence the number of elements of its multiplicative group $D^\times$ is $q^n-1$.

The conjugation action of $D^\times$ on itself yields a decomposition

$$D^\times \cong F^\times \; + \sum_{nontrivial\; orbits x} \frac{{|D^\times|}}{{|Stab(a_x)|}}$$

where again the elements of the center $F^\times = Z(D^\times)$ correspond to the trivial orbits. The stabilizer of any $a_x$, together with $0$, forms a division ring (strictly) intermediate between $F$ and $D$; usually this is called the [[centralizer]] $C(a_x)$ of $a_x$. Putting $d_x = dim_F(C(a_x))$, the division ring $C(a_x)$ has $q^{d_x}$ elements, and notice $d_x$ divides $n$ because $n/d_x$ is just the dimension of $D$ seen as a vector space (module) over $C(a_x)$. Thus ${|Stab(a_x)|} = q^{d_x} - 1$, and we have a class equation

$$q^n - 1 = q - 1 + \sum_x \frac{q^n - 1}{q^{d_x} - 1}.$$

Now let $\zeta \in \mathbb{C}$ be any primitive $n^{th}$ [[root of unity]]. Since $z - \zeta$ divides each polynomial $z^n-1$ and $\frac{z^n - 1}{z^{d_x} - 1}$, so does $\prod_{prim.\; \zeta} (z-\zeta)$. It follows that the algebraic integer $\prod_{prim.\; \zeta} (q - \zeta)$ divides each of the integers $q^n-1$ and $\frac{q^n - 1}{q^{d_x} - 1}$, and hence divides $q-1$ according to the class equation. But also ${|q - \zeta|} \geq {|q-1|}$ for any root of unity $\zeta$. Thus ${|q - \zeta|} = {|q-1|}$ and $n = 1$, i.e., $F = Z(D)$ is all of $D$ as was to be shown.
=--

### Sylow theorems

Let $G$ be a finite group of order $n$, and suppose that $p$ is a prime that divides $n$; say $n = p^f u$ where $p$ does not divide $u$. Recall that a [[Sylow p-subgroup]] is a $p$-subgroup of maximal order $p^f$.

A fundamental fact of group theory is that Sylow $p$-subgroups exist and they are all conjugate to one another; also the number of Sylow $p$-subgroups is $\equiv 1 \; mod p$.

Existence of Sylow $p$-subgroups can be proven by exploiting the same type of argument as in the proof of Proposition \ref{pgroup}:

+-- {: .num_theorem #ppower}
###### Theorem
If $G$ has order $n$ and $p^k$ is a prime power dividing $n$, then there is a subgroup of $G$ of order $p^k$.
=--

+-- {: .proof}
###### Proof
First we show that Sylow subgroups exist. We start by observing that if a group $H$ has a $p$-Sylow subgroup $P$, then so does any subgroup $G$. To prove this, first note that if we let $G$ act on $H/P$ by left translation, then the stabilizer of any element $h P$ is $G \cap h P h^{-1}$, a $p$-group since $h P h^{-1}$ is. Then note that since $H/P$ has cardinality prime to $p$, so must one of its [[connected object|connected components]] $G/Stab(a_x)$ in its $G$-set decomposition

$$H/P \cong \sum_{orbits\; x} G/Stab(a_x),$$

and this makes $Stab(a_x)$ a $p$-Sylow subgroup of $G$.

Then, if $G$ is of order $n$, apply this observation to the embedding

$$G \stackrel{Cayley}{\hookrightarrow} Perm({|G|}) \cong S_n \hookrightarrow GL_n(\mathbb{Z}/(p)) = H$$

where we embed the [[symmetric group]] $S_n$ via [[permutation matrices]] into the group $H$ of $n \times n$ invertible matrices over $\mathbb{Z}/(p)$. The group $H$ has order $(p^n - 1)(p^n - p)\ldots (p^n - p^{n-1})$, with maximal $p$-factor $p^{n(n-1)/2}$. It thus has a $p$-Sylow subgroup given by unitriangular matrices, i.e., upper-triangular matrices with all $1$\'s on the diagonal. Therefore $p$-Sylow subgroups $P$ exist for any finite group $G$.

Finally, note that by Proposition \ref{pgroup}, $P$ is solvable and therefore has a [[composition series]]

$$\{1\} = P_0 \subset P_1 \subset \ldots \subset P$$

where each $P_k$ has order $p^k$.
=--

+-- {: .num_theorem #SylowConj}
###### Theorem
If $H$ is a $p$-subgroup of $G$ and $P$ is a Sylow $p$-subgroup, then $g^{-1} H g \subseteq P$ for some $g \in G$. In particular, all Sylow $p$-subgroups are conjugate to one another.
=--

+-- {: .proof}
###### Proof
$G$ acts on the set of cosets $G/P$ as usual by left translation, and we may restrict the action to the $p$-subgroup $H$. By maximality of $P$, we see ${|G/P|}$ is prime to $p$, and so by Proposition \ref{pparity}, ${|Fix_H(G/P)|}$ is also prime to $p$. In particular, $Fix_H(G/P)$ has at least one element, say $g P$. We infer that $h g P = g P$ for all $h \in H$, or that $g^{-1} h g P = P$ for all $h \in H$, and this implies that $g^{-1} H g \subseteq P$.
=--

+-- {: .num_theorem #numberSylow}
###### Theorem
The number of Sylow $p$-subgroups of $G$ is $\equiv 1 \; mod p$.
=--

+-- {: .proof}
###### Proof
Let $Y$ be the set of Sylow $p$-subgroups; $G$ acts on $Y$ by conjugation. As all Sylow $p$-subgroups are conjugate, there is just one orbit of the action, and the stabilizer of an element $P \in Y$ is just the [[normalizer]] $N_G(P)$ (by definition of normalizer). Thus $Y \cong G/N_G(P)$ as $G$-sets.

Restrict the action to the subgroup $P$. Of course the element $P \in Y$ is a fixed point of this restricted action, and if $Q$ is any other fixed point, it means $x Q x^{-1} = Q$ for all $x \in P$, whence $P \subseteq N_G(Q)$. Now: $P, Q$ are both Sylow $p$-subgroups of $N_G(Q)$ and are therefore conjugate to each other (as seen within the group $N_G(Q)$). But $Q$ is already fixed by the conjugation action in its stabilizer $N_G(Q)$, so we conclude $P = Q$. We conclude $Fix_P(Y)$ has exactly one element. From ${|Y|} \equiv {|Fix_P(Y)|} \; mod p$ (Proposition \ref{pparity}), the theorem follows.
=--

+-- {: .num_cor #numbertwoSylow}
###### Corollary
If $G$ is a group of order $n = p^k m$ where $p$ is prime to $m$, then the number $n_p$ of $p$-Sylow subgroups divides $m$.
=--

+-- {: .proof}
###### Proof
Because $G$ acts transitively on $p$-Sylow subgroups (Theorem \ref{SylowConj}) , the number $n_p$ divides ${|G|} = p^k m$. From Theorem \ref{numberSylow}, we have $a n_p + b p^k = 1$ for some integers $a, b$. Since $n_p$ divides both terms of the left side of $a n_p m + b p^k m = m$, it divides $m$.
=--

The Sylow theorems are routinely used throughout group theory. As a sample application: if $p, q$ are distinct primes, with $p^2 \nequiv 1\; \mod q$ and $q \nequiv 1\; \mod p$, then any group of order $p^2 q$ is abelian. (For example, a group of order $2023 = 7 \cdot 17^2$ must be commutative.)

+-- {: .proof}
###### Proof
We have $n_p|q$ by Corollary \ref{numbertwoSylow}, but $n_p \neq q$ (using Theorem \ref{numberSylow} and $q \nequiv 1\; \mod p$), so $n_p = 1$. Arguing similarly we have $n_q|p^2$ but $n_q \neq p$ and $n_q \neq p^2$, so $n_q = 1$. The $p$-Sylow subgroup $P$ of order $p^2$ and the $q$-Sylow subgroup $Q$ of order $q$ are both abelian. $P \cap Q = \{1\}$ since $p, q$ are relatively prime, and $P, Q$ are normal subgroups of $G$ since $n_p, n_q$ are both $1$. It follows that $P Q$ is a subgroup of order $p^2 q$, hence $P Q = G$. Thus to prove $G$ abelian, it suffices to show that if $x \in P$ and $y \in Q$, then $x$ and $y$ commute, i.e., $x y x^{-1} y^{-1} = 1$. But by normality of $Q$, the element $(x y x^{-1}) y^{-1}$ belongs to $Q$; similarly, the element $x(y x^{-1} y^{-1})$ belongs to $P$, and so $x y x^{-1} y^{-1} \in P \cap Q = \{1\}$. The result follows.
=--

## Related concepts

* [[class function]]

* [[Klein geometry]]

[[!redirects class equations]]

[[!redirects class formula]]
[[!redirects class formulas]]
[[!redirects conjugation class formula]]
[[!redirects conjugation class formulas]]
[[!redirects orbit decomposition formula]]

[[!redirects Sylow theorem]]
[[!redirects Sylow theorems]]

[[!redirects Sylow's theorem]]
[[!redirects Sylow's theorems]]


class field theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Arithmetic geometry
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

__Class field theory__ studies finite-dimensional abelian [[field extensions]] of [[number fields]] and of [[function fields]], hence of [[global fields]] by relating them to the [[idele class group]].

Class field theory clarifies the origin of various [[reciprocity]] laws in [[number theory]]. The basic (one dimensional) class field theory stems from the ideas of Kronecker and Weber, and results of Hilbert soon after them. Main results of the theory belong to the first half of the 20th century (Hilbert, Artin, Tate, Hasse...) and are quite different for the [[local field]] from the [[global field]] case. One of the basic objects the class group, is related to the [[Picard group]] in algebraic geometry.
Generalizations for higher dimensional fields came later under now active higher class field theory, which is usually formulated in terms of algebraic K-theory and is closely related to deep questions of algebraic geometry (Paršin, Tate, Kato, Saito etc.).

Given an algebraic [[number field]] $k$ one defines a (congruence divisor) class field group $A/H$ in $k$; according to Weber

> An algebraic extension $K/k$ is called a __class field__ to $A/H$, if exactly those prime divisors in $k$ of first degree which belong to the principal class $H$ split completely in $K$

Some of the basic results of the class field theory are the Artin reciprocity theorem, existence theorem, uniqueness theorem, ordering theorem, Weber isomorphy theorem and the decomposition law of class field theory.

## The Hilbert Class Field

For a number field $K$, the _Hilbert class field_ $H$ is defined to be its maximal unramified abelian extension. In this special case, class field theory tells us that the Galois group $Gal(H/K)$ is isomorphic to the [[ideal class group]] of $K$.

In [[arithmetic topology]], this is seen as an analogue of the [[Hurewicz theorem]] (see [Morishita12](#Morishita12)).

## Related concepts

* [[Langlands correspondence]]

* [[geometric class field theory]]

* [[Artin reciprocity law]]

## References

* {#FroehlichCassels67} [[Albrecht Fröhlich]], [[J. W. S. Cassels]] (eds.), _Algebraic number theory_, Acad. Press 1967, with many reprints; Fröhlich, Cassels, Birch, Atiyah, Wall, Gruenberg, Serre, Tate, Heilbronn, Rouqette, Kneser, Hasse, Swinerton-Dyer, Hoechsmann, systematic lecture notes from the instructional conference at Univ. of Sussex, Brighton, Sep. 1-17, 1965 (ISBN:9780950273426, [pdf](https://www.math.arizona.edu/~cais/scans/Cassels-Frohlich-Algebraic_Number_Theory.pdf), [errata pdf](https://www.ma.imperial.ac.uk/~buzzard/errata.pdf) by [[Kevin Buzzard]])

* [[Albrecht Fröhlich]], Martin J. Taylor, _Algebraic number theory_, Cambridge Studies in Advanced Mathematics 27, 1993

* wikipedia [class field theory](http://en.wikipedia.org/wiki/Class_field_theory)

* A. N. Paršin A. N. Local class field theory, Trudy Mat. Inst. Steklov __165__
1984;  _Galois cohomology and Brauer group of local fields_, Trudy Mat.
Inst. Steklov __183__, 1984.

The following survey of Connes-Marcolli work has a quick introduction to algebraic number theory including basic notions of CFT

* P. Almeida, _Noncommutative geometry and arithmetics_, Russian Journal of Mathematical Physics 16, No. 3, 2009, pp. 350–362, [doi](http://dx.doi.org/10.1134/S1061920809030030), see also nLab:arithmetic and noncommutative geometry

The following article sketches the geometric intuition behind the reciprocity laws as the relation between two approaches to the maximal abelian quotient of the fundamental group, mimicking the ideas of Galois theory

* Alexander Schmidt, _Higher dimensional class field theory from a topological point of view_, [page](http://www.mathi.uni-heidelberg.de/~schmidt/papers/schmidt21-en.html)

The following 2 articles make parallel between some notions of [[QFT]] and of number theory and in particular about the analogy between the [[Weil reciprocity law]] for [[function field]]s and the Takahashi-Ward identities of field theory:

* [[Leon Takhtajan]], _Quantum field theories on algebraic curves and A. Weil reciprocity law_, [arxiv/0812.0169](http://arxiv.org/abs/0812.0169); _Quantum field theories on an algebraic curve_, [pdf](http://www.math.sunysb.edu/~leontak/Takhtajan-00.pdf), 2000

Some history is in

* {#Cogdell12} James W. Cogdell, _L-functions and non-abelian class field theory, from
Artin to Langlands_, 2012 ([pdf](https://people.math.osu.edu/cogdell.1/Artin-www.pdf))

The analogy between class field theory for the Hilbert class field and the Hurewicz theorem can be found in

* {#Morishita12} [[Masanori Morishita]], _Knots and Primes: An Introduction to Arithmetic Topology_, 2012, Springer, ([web](https://books.google.co.uk/books?id=DOnkGOTnI78C&pg=PA156#v=onepage&q&f=false))


class function

> This entry is about the term of "class function" as it is used in algebraic contexts. (Such class functions are usually set-maps.) For the notion of the same name in [[set theory]] see at _[[class function (set theory)]]_.

#Contents#
* table of contents
{:toc}

## Definition

A [[function]] on a [[group]] is a **class function** if it is [[invariant]] under [[conjugation action]], i.e. it is equivalent to a function on the set of [[conjugacy classes]].

## Examples

Every [[group character]] is in particular a class function.

## Related concept

* [[class formula]]

* [[Schur orthogonality relation]]

* [[representation ring]]

* [[rational equivariant K-theory]], [[equivariant Chern character]]

## References

See also

* Wikipedia, _[Class function](https://en.wikipedia.org/wiki/Class_function)_

[[!redirects class functions]]


class function (set theory)

> This entry is about the term of "class function" as it is used in set-theoretical contexts. For the notion of the same name in [[algebra]] see at _[[class function]]_.

#Contents#
* table of contents
{:toc}

## Idea

Recall that in [[set theory]] a [[function]] $f$ from a [[set]] $S_1$ to a set $S_2$ may be encoded in terms of a [[relation]] on the [[Cartesian product]] $S_1 \times S_2$ of the two sets, namely the subset $ R \subset S^1 \times S^2$ with $R = \{  (x,y) \vert y = f(x)\}$ (the [[graph]] of the function).

This concept has an evident generalization to the case where $S_1$ and $S_2$ are allowed to be proper classes. In this case one speaks of _class functions_.

## Definition

A class function is a [[class]] $R$  which is a [[relation]] with the property that if $(x,y)\in R$ and $(x,y')\in R$, then $y=y'$.

A class function need not be a [[set]].

Class functions are an important concept when formalizing [[category theory]] on [[set theory|set-theoretic]] [[foundations]] without using [[universes]], especially when

* constructing functors step by step, e.g. by first constructing an assignment between two classes of objects, and then compatibly augmenting this assignment to make it a functor,

* most especially when constructing _functors on [[functor categories]]_, whose classes of objects tend to be proper classes.

## Example in Category Theory

An example is Mac Lane's discussion  [p. 23](#MacLane1998) of the category of all [[classes]] in his chapter on [[Foundations]]. There, the class of [[arrow]]s is defined by writing "its arrows [are] all functions $f\colon C\rightarrow C'$ between classes".

Class functions are the [[morphism]]s in a [[category with class structure]].

## Example in Set Theory

Arguably the prototypical example of an essential use of the concept of class functions in [[set theory]] is [[Easton's theorem]] [Theorem 15.18](#Jech2002).
on the behaviour of the class function $x\mapsto 2^x$ on the class of all cardinals. (In Easton's theorem, the domain of the class function in question is the proper class of all [[regular cardinals]].)

Often, authors resort to synonyms like [[operation]] or [[assignment]] when writing about proper classes, with [p. 23](#MacLane1998) being a counterexample.

## References

* {#MacLane1998} [[Saunders Mac Lane]]. _[[Categories for the Working Mathematician]]_ Second Edition. Springer (1998)

* {#Jech2002} T. Jech. Set theory. Third Edition. Springer (2002)


class number

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Arithmetic geometry
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

In [[number theory]], the _class number_ of a [[number field]] is the [[order of a group|order]] of its [[ideal class group]].

## Properties

### Relation to the pole of the zeta function

The [[Dedekind zeta function]] $\zeta_K$ of $K$ has a [[simple pole]] at $s = 1$. The _[[class number formula]]_ says that its [[residue]] there is proportional the the product of the [[regulator of a number field|regulator]] with the class number of $K$

$$
  \underset{s\to 1}{\lim} (s-1) \zeta_K(s)
  \propto
  ClassNumber_K \cdot Regulator_K
  \,.
$$

## References

* Wikipedia, _[Ideal class group](http://en.wikipedia.org/wiki/Ideal_class_group)_

[[!redirects class numbers]]


class number formula

#Contents#
* table of contents
{:toc}

## Idea

Given a [[number field]] $K$, the [[Dedekind zeta function]] $\zeta_K$ of $K$ has a [[simple pole]] at $s = 1$. The _class number formula_ says that its [[residue]] there is proportional the product of the [[regulator of a number field|regulator]] with the [[class number]] of $K$

$$
  \underset{s\to 1}{\lim} (s-1) \zeta_K(s)
  \propto
  ClassNumber_K \cdot Regulator_K
  \,.
$$

Variants of this for [[arithmetic varieties]] over $\mathbb{Q}$ are the _[[Birch and Swinnerton-Dyer conjecture]]_ and the _[[Beilinson conjecture]]_.

## References

* Wikipedia, _[Class number formula](http://en.wikipedia.org/wiki/Class_number_formula)_

[[!redirects class number formulas]]

[[!redirects analytic class number formulas]]
[[!redirects analytic class number formula]]