\tableofcontents

\section{Idea}

Just as the category [[Cat]] of categories can be equipped with a [[canonical model structure]], namely the [[canonical model structure on categories]], so too can [[2Cat]], the category of strict 2-categories. This extends to weak 2-categories too, although in order to actually have a [[category]] on which to put the model structure, one has to work with [[strict 2-functor|strict 2-functors]].

The characterising feature of a canonical model structure on [[2Cat]] is that the weak equivalences should be [[equivalence of 2-categories|equivalences of 2-categories]], categorifying the fact that the equivalences in the [[canonical model structure on categories]] are [[equivalence of categories|equivalences of categories]]. What the fibrations and cofibrations should be, or more specifically how to categorify [[isofibration|isofibrations]] and [[iso-cofibration|iso-cofibrations]], is one of the main points to resolve to obtain the model structure.

What is usually referred to as the canonical model structure on 2-categories is a model structure first described by [[Stephen Lack|Lack]] in [Lack2002](#Lack2002) and [Lack2004](#Lack2004). In this model structure, the fibrations are [[Lack fibration|Lack fibrations]], and every object is [[fibrant object|fibrant]], but not every object is [[cofibrant object|cofibrant]], in contrast to the [[canonical model structure on categories]].

However, it is possible to put a different canonical model structure on [[2Cat]] in which every object is both fibrant and cofibrant, using the thesis [Williamson2011](#Williamson2011). The fibrations in this case are, roughly speaking, those [[2-functor|2-functors]] in which [[semi-strict equivalence|semi-strict equivalences]] can be lifted.

\section{Lack model structure}

The following is Theorem 4 in [Lack2004](#Lack2004).

\begin{thm} There is a model structure on [[2Cat]] in which the equivalences are [[equivalence of 2-categories|equivalences of 2-categories]], and the fibrations are [[Lack fibration|Lack fibrations]]. \end{thm}

\section{Canonical model structure in which every object is both fibrant and cofibrant}

Throughout this section, let $\mathcal{E}_{semi}$ denote the [[free-standing semi-strict equivalence]].

\begin{defn} A _semi-strict equiv-fibration_ is a [[2-functor]] $p: \mathcal{A} \rightarrow \mathcal{B}$ such that, for any (strictly) commutative diagram

\begin{tikzcd}
\mathcal{X} \ar[r, "f"] \ar[d, swap, "id \times 0"] & \mathcal{A} \ar[d, "p"] \\
\mathcal{X} \times \mathcal{E}_{semi} \ar[r, swap, "g"] & \mathcal{B}
\end{tikzcd}

in [[2Cat]], there is a functor $l: \mathcal{X} \times \mathcal{E}_{semi} \rightarrow \mathcal{A}$ such that the following diagram in [[2Cat]] (strictly) commutes.

\begin{tikzcd}
\mathcal{X} \ar[r, "f"] \ar[d, swap, "id \times 0"] & \mathcal{A} \ar[d, "p"] \\
\mathcal{X} \times \mathcal{E}_{semi} \ar[r, swap, "g"] \ar[ur, "l"] & \mathcal{B}
\end{tikzcd}

\end{defn}

\begin{defn} A _semi-strict equiv-cofibration_ is a [[2-functor]] $j: \mathcal{A} \rightarrow \mathcal{B}$ such that, for any (strictly) commutative diagram

\begin{tikzcd}
\mathcal{A} \ar[r, "id \times 0"] \ar[d, swap, "j"] & \mathcal{A} \times \mathcal{E}_{semi} \ar[d, "h"] \\
\mathcal{B} \ar[r, swap, "f"] & \mathcal{X}
\end{tikzcd}

in [[2Cat]], there is a functor $k: \mathcal{B} \times \mathcal{E}_{semi} \rightarrow \mathcal{X}$ such that the following diagram in [[2Cat]] (strictly) commutes.

\begin{tikzcd}
\mathcal{A} \ar[r, "id \times 0"] \ar[d, swap, "j"] & \mathcal{A} \times \mathcal{E}_{semi} \ar[d, "j \times id"] \ar[ddr, bend left, "h"] & \\
\mathcal{B} \ar[r, swap, "id \times 0"] \ar[drr, swap, bend right, "f"] & \mathcal{B} \times \mathcal{E}_{semi} \ar[dr, "k"] & \\
& & \mathcal{X}
\end{tikzcd}

\end{defn}

\begin{defn} A [[2-functor]] $f: \mathcal{A} \rightarrow \mathcal{B}$ is a _semi-strict equivalence of 2-categories_ if there is a 2-functor $h: \mathcal{A} \times \mathcal{E}_{semi} \rightarrow \mathcal{B}$ such that the 2-functor $h \circ \left( id \times 0 \right): \mathcal{A} \rightarrow \mathcal{B}$ is equal to $f$. \end{defn}

\begin{thm} The category [[2Cat]] can be equipped with a model structure in which the weak equivalences are semi-strict equivalences of 2-categories, the fibrations are semi-strict equiv-fibrations, and the cofibrations are semi-strict equiv-cofibrations. \end{thm}

\begin{proof} By the section 'Structured interval' of the page [[walking equivalence]], $\mathcal{E}_{semi}$ can be equipped with the structure of an [[interval object]], and embellished with all the structures of [Williamson2011](#Williamson2011) which are required for Corollary XV.6 and/or Corollary XV.7 of this work, satisfying all the hypotheses of these corollaries. Applying these corollaries, we immediately obtain the theorem. \end{proof}

## Related concepts

* [[model structure on DblCat]]

\section{References}

* {#Lack2002} [[Stephen Lack]], _A Quillen model structure for 2-categories_, K-Theory 26, No. 2, 171-205 (2002). [Zentralblatt review](https://zbmath.org/?q=an%3A1017.18005) [author's webpage](http://maths.mq.edu.au/~slack/papers/qmc2cat.html)

* {#Lack2004} [[Stephen Lack]], _A Quillen model structure for bicategories_, K-Theory 33, No. 3, 185-197 (2004). [Zentralblatt review](https://zbmath.org/?q=an%3A1069.18008) [author's webpage](http://maths.mq.edu.au/~slack/papers/qmcbicat.html)

* {#Williamson2011} [[Richard Williamson]], _Cylindrical model structures_, DPhil thesis, University of Oxford, 2011. [author's webpage](http://rwilliamson-mathematics.info/cylindrical_model_structures/cylindrical_model_structures.html) [arXiv:1304.0867](https://arxiv.org/abs/1304.0867)

[[!redirects canonical model structures on 2-categories]]

[[!redirects canonical model structure on 2Cat]]
[[!redirects canonical model structures on 2Cat]]


canonical model structure on Cat

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Model category theory
+-- {: .hide}
[[!include model category theory - contents]]
=--
#### 2-Category theory
+-- {: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## The canonical model structure on $Cat$

The [[canonical model structure]] on [[Cat]] is a [[model structure]] which encapsulates part of [[category theory]] as a version of [[homotopy theory]].  It is a special case of the general notion of [[canonical model structure]] on categorical structures, and is also called the **trivial model structure** or the **categorical model structure**.  Its [[weak equivalences]] are the [[equivalences of categories]] and its [[homotopy category]] is [[Ho(Cat)]], the category obtained from the 1-category $Cat$ by identifying naturally isomorphic functors. See the
_[[joyalscatlab:Model structures on Cat|Catlab]]_ for the theory of this structure.

Assuming the [[axiom of choice]], the canonical model is the _unique_ model structure on $Cat$ such that the weak equivalences are categorical equivalences (thus justifying the word 'canonical').  It is different from the [[Thomason model structure]], where weak equivalences are functors that give a weak equivalence of simplicial sets when we take the [[nerve]].

On this page we give a concise construction of the canonical model structure, as well as two variants that make sense in the absence of the full [[axiom of choice]].

### The classical version

For purposes of this page, [[Cat]] will denote the [[1-category]] of [[small categories]] and [[functors]], and our categories are all [[strict categories]] as in ordinary set-theoretic [[foundations]].  We write $C_0$ for the set of [[objects]] of a [[small category]] $C$.

\begin{definition}
\label{ClassesOfMorphisms}
Declare a functor to be:

* a [[weak equivalence]] if it is an [[equivalence of categories]],

  (equivalently: [[fully faithful functor|fully faithful]] and [[essentially surjective functor|essentially surjective]]),

* a [[cofibration]] if it is [[injective-on-objects functor|injective on objects]], i.e. an [[isocofibration]],

* a [[fibration]] if it is an [[isofibration]].

\end{definition}

\begin{proposition}
\label{ExistenceOfTheCanonicalModelStructure}
The classes in Def. \ref{ClassesOfMorphisms} constitute a [[model category]]-[[structure]] $Cat_{can}$ on [[Cat]].
\end{proposition}
\begin{proof}
It is easy to verify that the weak equivalences satisfy the [[2-out-of-3 property]]; thus it remains to show that (acyclic cofibrations, fibrations) and (cofibrations, acyclic fibrations) are [[weak factorization systems]].

Suppose given a square
$$\array{A & \overset{f}{\to} & C\\
  ^i\downarrow && \downarrow^p\\
  B& \underset{g}{\to} & D}$$
in which $i$ is a cofibration and $p$ a fibration.

Suppose first that $p$ is acyclic.  It is easy to see that the acyclic fibrations are precisely the equivalences of categories that are surjective on objects.  Thus, since (mono, epi) is a [[weak factorization system on Set]], we can define $h_0\colon B_0\to C_0$ filling the square, and then full-faithfulness of $p$ gives a unique definition of $h$ on arrows.

Now suppose that $i$ is acyclic, so that $i_0 \colon A_0 \to B_0$ is injective.  Since $i$ is essentially surjective, we can choose, for each $b\in B_0 \setminus A_0$, an isomorphism $\phi_b\colon i(a_b) \cong b$.  We then have $g(\phi_b)\colon p(f(a_b)) = g(i(a_b)) \cong g(b)$, so since $p$ is an isofibration, we can also choose, for each $b\in B_0$, an isomorphism $\psi_b\colon f(a_b) \cong c_b$ such that $p(\psi_b) = g(\phi_b)$.  Define $h_0\colon B_0\to C_0$ to be $f_0$ on the image of $i_0$ and to take $b\in B_0 \setminus A_0$ to $c_b$.  We can define $h$ on arrows by composing with the isomorphisms $\psi$ to make it a lifting.

It remains to prove the factorization axioms.  Suppose given a functor $f\colon A\to B$.  First, define $C_0 = A_0 + B_0$, and make $C$ into a category in the unique way such that the map $C\to B$ induced by $f$ and $1_B$ is fully faithful.  Since it is surjective on objects, it is an acyclic fibration, and clearly the induced map $A\to C$ is injective on objects, i.e. a cofibration.

Next, define $D$ to be the category of triples $(a,b,\phi)$ where $\phi\colon f(a)\cong b$ is an isomorphism in $B$.  In other words, it is the [[comma object|strict iso-comma category]] $(f/_\cong 1_b)$.  The projection $D\to B$ is easily shown to be an isofibration, while the functor $A\to D$ defined by $a \mapsto (a, f(a), 1_{f(a)})$ is an injective equivalence.

This completes the proof.
\end{proof}

\begin{remark}
The two factorizations constructed above are in fact [[functorial factorizations]].

While this model structure is [[cofibrantly generated model category|cofibrantly generated]] (Prop. \ref{CofibrantGeneration}), the above factorizations are not those constructed from the [[small object argument]] (though they are closely related to the [[algebraic weak factorization systems]] produced by the [[algebraic small object argument]].
\end{remark}

### Without choice

In the absence of the [[axiom of choice]], one must distinguish between *strong equivalences of categories*, which come with an inverse up to isomorphism, and *weak equivalences of categories*, which are merely fully faithful and essentially surjective on objects.  Since weak equivalences of categories still "preserve all categorical information," we might hope to find a model structure on $Cat$ whose weak equivalences are the *weak* equivalences of categories.  The notion of [[anafunctor]] also suggests such an approach, since an anafunctor (the "right" replacement for a functor in the absence of choice) is a particular sort of generalized morphism: a [[span]] $A\leftarrow F \to B$ of functors in which $F\to A$ is a surjective equivalence.

If there is to be such a model structure, however, then since generalized morphisms between fibrant-and-cofibrant objects are all represented by ordinary ones, there must exist "cofibrant categories" and "fibrant categories" such that every anafunctor between fibrant-and-cofibrant categories is equivalent to an honest functor, and every category can be replaced by a fibrant and cofibrant one.  It seems unlikely that this would be true without *any* choice-like axioms, but notably weaker axioms than full AC do suffice.

#### The projective model structure

For the existence of the model structure in this case, we assume [[COSHEP]], aka the "presentation axiom," namely that the category [[Set]] has enough [[projective objects]].  Define a functor $f\colon A\to B$ to be:

* a [[weak equivalence]] if it is a weak equivalence of categories, i.e. [[fully faithful functor|fully faithful]] and [[essentially surjective functor|essentially surjective on objects]].

* a [[cofibration]] if it is injective on objects, and $B_0\setminus f(A_0)$ is a projective object in $Set$.

* a [[fibration]] if it is an [[isofibration]].

As before, the acyclic fibrations are precisely the weak equivalences that are literally surjective on objects.  Now recall that assuming COSHEP, (monics with projective complement, epics) is a [[weak factorization system on Set]].  This supplies the lifting of cofibrations against acyclic fibrations.  Likewise, the factorization of $f\colon A\to B$ into a cofibration followed by an acyclic fibration is given by first factoring $f_0$ as $A_0 \to A_0 + B_0' \to B_0$, where $B_0'\to B_0$ is a projective cover.

The other factorization works exactly as before, while for lifting acyclic cofibrations against fibrations, we notice that in the original proof, we only needed to apply choice for sets indexed by $B_0\setminus i(A_0)$, which we have assumed to be projective when $i\colon A\to B$ is a cofibration.

Weak equivalences of categories are easily seen to satisfy the 2-out-of-3 property, so we have a model category.  Note that all categories are fibrant in this model structure, while the cofibrant categories are those whose set of objects is projective.

The existence of this model structure implies, in particular, that under COSHEP the category $Ana(C,D)$ is [[essentially small category|essentially small]], being in fact equivalent to the category $Fun(C',D)$ of ordinary functors where $C'$ is a cofibrant replacement for $C$.

#### The injective model structure

Is there a dual model structure in which all categories are cofibrant?  This seemingly has to do with [[stack]] completion: the fibrant objects would be *stacks* for the [[regular coverage]] of $Set$.  (Without AC, not all small categories are stacks.)  Is Makkai's [[axiom of small cardinality selection]] (which he uses, instead of COSHEP, to prove that $Ana(C,D)$ is essentially small) sufficient for the existence of an "injective" model structure on Cat?

## Properties

### Basic properties
 {#BasicProperties}

\begin{remark}\label{CategoryOfBifibrantObjects}
**(category of bifibrant objects)**
\linebreak
  It is obvious that all objects in the canonical model structure (Prop. \ref{ExistenceOfTheCanonicalModelStructure}) are [[bifibrant object|bifibrant]], i.e.:

* both [[cofibrant object|cofibrant]]

  since a functor $\varnothing \to \mathcal{C}$ from the [[empty category]] is evidently an [[injective function]] $\varnothing \to Obj(\mathcal{C})$ on objects,

* and [[fibrant objects|fibrant]],

  since a functor $\mathcal{C} \to \ast$ to the [[terminal category]] is evidently an [[isofibration]] since the only morphism in the terminal category to lift is an [[identity morphism]].

Hence, in particular, $Cat_{can}$ is a [[category of fibrant objects]] and a [[cofibration category]].
\end{remark}

\begin{remark}\label{TheAcyclicFibrations}
**(acyclic (co)fibrations)**
\linebreak
In the canonical model structure (of Prop. \ref{ExistenceOfTheCanonicalModelStructure}):

* the [[acyclic cofibrations]] are precisely the [[injective-on-objects functor|injective-on-objects]] [[equivalences of categories]]

* the [[acyclic fibrations]] are precisely the [[surjective-on-objects functor|surjective on objects]] [[equivalences of categories]].

  Namely, being [[equivalences of categories]] makes them be [[essentially surjective functor|essentially surjective]] and [[fully faithful functor|fully faithful]] ([this Prop.](equivalence+of+categories#ViaSplitEssentiallySurjectiveAndFullyFaithful)), but being also [[isofibrations]] then clearly requires them to actually be [[surjective on objects functor|surjective on objects]], which by [[fully faithful functor|fully faithfulness]] is then already sufficient for isofibrancy.

\end{remark}

\begin{proposition}\label{Properness}
**(properness)**
\linebreak
 The canonical model structure (Prop. \ref{ExistenceOfTheCanonicalModelStructure}) is a [[proper model category]].
\end{proposition}
\begin{proof}
  In view of Rem. \ref{CategoryOfBifibrantObjects} this follows by [this Prop.](proper+model+category#AllObjectsFibrantImpliesRightProper).
\end{proof}

\begin{proposition}\label{CofibrantGeneration}
**(cofibrant generation)**
\linebreak
  The canonical model structure $Cat_{can}$ (Prop. \ref{ExistenceOfTheCanonicalModelStructure}) is [[cofibrantly generated model category|cofibrantly generated]].
The sets $I$ of generating cofibrations and $J$ of generating acyclic cofibrations may be taken to consist of the following evident functors:
$$
  I
  \;\coloneqq\;
  \left\{
    \begin{array}{ccc}
      \varnothing &\longrightarrow& \{ 0 \}
      \,,
      \\
      \{0\} \sqcup \{1\} &\longrightarrow& \{ 0 \to 1 \}
      \,,
      \\
      \big\{
        0 \rightrightarrows 1
      \big\}
      &\longrightarrow& \{ 0 \to 1\}
    \end{array}
  \right\}
  \,,
$$
$$
 J
 \;\coloneqq\;
 \left\{
   \array{
     \{ 0 \}
      &\longrightarrow&
   \{ 0 \xleftrightarrow{\sim} 1\}
   }
 \right\}
  \,.
$$
\end{proposition}
([Rezk 1996, Prop. 4.1](#Rezk96))
\begin{proof}
  The functors in $I$ are evidently cofibrations. Hence by [this Prop](cofibrantly+generated+model+category#RetractsOfCellComplexesExchaustLLPOfRLP) we need to show that:

1. [[right lifting property|$RLP(I)$]] is the class of [[acyclic fibrations]].

   One sees immediately that:

   1. right lifting against $\varnothing \to \{0\}$ characterizes [[surjective on objects functors]],

   1. right lifting against $\{0\} \sqcup \{1\} \longrightarrow \{ 0 \to 1 \}$ characterizes [[full functors]],

   1. right lifting against $\big\{0 \rightrightarrows 1 \big\}\longrightarrow \{ 0 \to 1\}$ characterizes [[faithful functors]].

   Together this characterizes the acyclic fibrations, by Rem. \ref{TheAcyclicFibrations}.

1. [[right lifting property|$RLP(J)$]] is the class of [[fibrations]].

   But right lifting against $\{0\} \longrightarrow \{0 \xleftrightarrow{\sim} 1\}$ is immediately the condition of path lifting which defines [[isofibrations]].

\end{proof}

\begin{proposition}\label{CartesianMonoidalModelStructure}
**(cartesian monoidal model structure)**
\linebreak
  The canonical model structure $Cat_{can}$ (from Prop. \ref{ExistenceOfTheCanonicalModelStructure}) is a [[cartesian monoidal model category]].
\end{proposition}
([Rezk 1996, Thm. 5.1](#Rezk96), [Joyal 2010, Thm. 1.3](#Joyal10))
\begin{proof}
Since forming [[product categories]] with a fixed category $\mathcal{C}$ evidently preserves [[equivalence of categories|equivalences of categories]] and hence the canonical weak equivalences, the [unit axiom](monoidal+model+category#UnitAxiom) follows immediately. It remains to check the [pushout-product axiom](monoidal+model+category#PushoutProductAxiom).

So consider a diagram in $Cat$ of this form:
\begin{tikzcd}
  &
  \mathcal{C} \times \mathcal{D}
  \ar[
    dl,
    "{ F \times \mathrm{Id} }"{swap}
  ]
  \ar[
    dr,
    "{ \mathrm{Id} \times G }"
  ]
  \ar[
    dd,
    phantom,
    "{ \scalebox{.7}{ (po) } }"
  ]
  \\
  \mathcal{C}' \times \mathcal{D}
  \ar[
    ddr,
    "{ \mathrm{Id} \times G }"{swap}
  ]
  \ar[
    dr,
    "{ q_l }"{swap}
  ]
  &&
  \mathcal{C} \times \mathcal{D}'
  \ar[
    ddl,
    "{ F \times \mathrm{Id} }"
  ]
  \ar[
    dl,
    "{ q_r }"
  ]
  \\
  &
  \mathcal{C}' \!\times\! \mathcal{D}
  \underset{
    \mathclap{
      \mathcal{C} \times \mathcal{D}
    }
  }{\coprod}
  \mathcal{C} \!\times\! \mathcal{D}'
  \ar[
    d,
    dashed,
    "{ F \widehat \times G }"{description}
  ]
  \\
  &
  \mathcal{C}' \times \mathcal{D}'
\end{tikzcd}

First we need to show that the [[pushout-product]] $F \widehat{\times} G$ is a cofibration if both $F$ and $G$ are. By the definition of cofibrations, this means to show that on objects $(F \widehat{\times} G)_0$ is an injective map if both $F_0$ and $G_0$ are injective maps.

Now, observe (see [here](Cat#OrdinaryLimitsAndColimits)) that the [[underlying]]-object functor
$$
  (-)_0
    \,\colon\,
  Cat \to Set
$$
[[preserved limit|preserves]] both [[limits]] and [[colimits]] and hence in particular preserves [[pushout products]]. Therefore the first statement we are after is that pushout-products of injections of sets are themselves injections, and that is the case, by [this example](pushout-product#InjectionsOfSets).

Second, if $F$ is in addition a weak equivalence, we need to show that then also $F \widehat{\times} G$ is a weak equivalence.

But with $F$ certainly also $F \times \mathrm{Id}$ is an injective-on-objects equivalence of categories, hence an [[acyclic cofibration]] in the canonical model structure. By the model category properties, it follows that also the [[pushout]] $q_r$ (in the above diagram) is an acyclic cofibration, hence in particular a weak equivalence; and by [[2-out-of-3]] applied to the bottom right triangle in the above diagram it follows that with $F \times Id$ and $q_r$ also $F \widehat{\times} G$ is a weak equivalence.
\end{proof}

### Uniqueness of the model structure

Remarkably:[^fine]

\begin{proposition}\label{Uniqueness}
The model structure $Cat_{can}$ (from Prop. \ref{ExistenceOfTheCanonicalModelStructure}) is the unique model structure on $Cat$ whose [[weak equivalences]] are the usual [[equivalences of categories]].
\end{proposition}

This result justifies the term "canonical".

The following **proof** is adapted (with minor changes) from [Schommer-Pries 2012](#Schommer-Pries12). See also this MathOverflow [thread](http://mathoverflow.net/questions/18744/is-model-structure-on-catset-unique), particularly the [answer](http://mathoverflow.net/a/46471/2926) given by [[Steve Lack]] (with a pertinent comment by [[Denis-Charles Cisinski]]).

[^fine]: Along similar lines, one can [prove](https://www.matem.unam.mx/~omar/notes/modelcatsets.html) (assuming [[axiom of choice|AC]]) that there are nine -- count 'em, nine -- Quillen model structures on [[Set|$Set$]].

Let $\mathbf{M}$ denote any model structure on $Cat$ whose weak equivalences are categorical equivalences. We will prove that $\mathbf{M}$-fibrations are exactly canonical fibrations and that $\mathbf{M}$-cofibrations are exactly canonical cofibrations.

+-- {: .num_lemma}
###### Lemma
The terminal object $1$ is $\mathbf{M}$-cofibrant, i.e., the inclusion $0 \to 1$ in $Cat$ is an $\mathbf{M}$-cofibration.
=--

+-- {: .proof}
###### Proof
Let $C$ be any noninitial category; by a standard result of model category theory, there is an $\mathbf{M}$-cofibrant replacement $\tilde{C} \to C$, a weak equivalence such that $0 \to \tilde{C}$ is a cofibration. This $\tilde{C}$ is noninitial and therefore has $1$ as a retract; thus $1$ is $\mathbf{M}$-cofibrant since cofibrant objects are closed under retracts.
=--

+-- {: .num_prop #canonical}
###### Proposition
Each acyclic $\mathbf{M}$-fibration is a canonical acyclic fibration.
=--

+-- {: .proof}
Each acyclic $\mathbf{M}$-fibration $f: E \to X$ has the right lifting property with respect to $\mathbf{M}$-cofibrations. The right lifting property with respect to the $\mathbf{M}$-cofibration $0 \to 1$ is exactly the condition of being surjective on objects. Thus acyclic $\mathbf{M}$-fibrations are necessarily categorical equivalences that are surjective on objects, i.e., are necessarily canonical acyclic fibrations.
=--

Before giving the next result, we recall that the lifting relation on morphisms gives a [[Galois connection]]. Specifically, suppose $c: A \to B$ and $f: c \to d$ are functors, and define $c \perp f$ if for every morphism from $c$ to $f$ in the arrow category $Cat^\mathbf{2}$, i.e., for every commutative diagram

$$\array{
A & \to & C \\
_\mathllap{c} \downarrow & & \downarrow_\mathrlap{f} \\
B & \to & D
}$$

of functors, there is a lifting $B \to C$ filling in to make two commutative triangles. As any relation does, this lifting relation $\perp$ gives a Galois connection on subclasses of $Mor(Cat)$. General facts about Galois connections may then be applied.

+-- {: .num_cor #sub}
###### Corollary
Every canonical cofibration is an $\mathbf{M}$-cofibration. Every $\mathbf{M}$-fibration is a canonical fibration.
=--

+-- {: .proof}
###### Proof
By the Galois connection induced by the lifting relation, Proposition \ref{canonical} implies that canonical cofibrations form a subset of $\mathbf{M}$-cofibrations, and therefore that canonical acyclic cofibrations are a subset of acyclic $\mathbf{M}$-cofibrations. Again by the Galois connection, this in turn implies that $\mathbf{M}$-fibrations form a subset of canonical fibrations.
=--

At this point, we would like to show conversely that every $\mathbf{M}$-cofibration is a canonical cofibration (i.e., is injective on objects); another appeal to Galois connections would then allow us to deduce that every canonical fibration is an $\mathbf{M}$-fibration, and we would be done. Let us suppose otherwise, that there exists an $\mathbf{M}$-cofibration that is not injective on objects, and derive a contradiction.

For a set $S$, let $K(S)$ be the category whose objects are the elements of $S$, with exactly one morphism $x \to y$ for any $x, y \in S$. This gives the _codiscrete_ (or *chaotic*) functor
$K: Set \to Cat$, which is right adjoint to the forgetful functor $U: Cat \to Set$ that takes a category to its underlying set of objects. For each inhabited set $S$, we have that $K(S)$ is equivalent to $1$, and conversely any category equivalent to $1$ is isomorphic to some $K(S)$.

+-- {: .num_prop #key}
###### Proposition
If there is any $\mathbf{M}$-cofibration $f: A \to B$ that is not injective on objects, then the map $K(2) \to 1$ ($2 = \{0, 1\}$) is an (acyclic) $\mathbf{M}$-cofibration.
=--

+-- {: .proof}
###### Proof
First we observe that for any category $E$, the unit map $\eta_E: E \to K U(E)$ for the adjunction $U \dashv K$ is an $\mathbf{M}$-cofibration. For, the map $\eta_E$ is an isomorphism on objects and therefore a canonical cofibration; it is an $\mathbf{M}$-cofibration by Corollary \ref{sub}.

By hypothesis, $f$ maps two objects $a, a'$ of $A$ to the same object $b$ of $B$, so there is a commutative diagram

$$\array{
2 & \stackrel{(a, a')}{\hookrightarrow} & U A \\
\downarrow & & \downarrow_\mathrlap{f} \\
1 & \underset{b}{\to} & U B.
}$$

Let $r: U A \to 2$ be a retraction of the injection $2 \to U A$. By the adjunction $U \dashv K$, the map $r$ corresponds to a map $s: A \to K(2)$. We form a pushout square

$$\array{
A & \stackrel{s}{\to} & K(2) & & \\
_\mathllap{f} \downarrow & & \downarrow_\mathrlap{g} & & \\
B & \underset{h}{\to} & E & \underset{\eta_E}{\to} & K U(E)
}$$

where $g$ is an $\mathbf{M}$-cofibration (being the pushout of a cofibration $f$). Thus we have a composite cofibration $t \coloneqq \eta_E \circ g: K(2) \to K U(E)$. It may be verified that $K(2) \to 1$ is a retract of $t$, i.e., there is a commutative square

$$\array{
K(2) & \stackrel{id}{\to} & K(2) \\
_\mathllap{t} \downarrow & & \downarrow \\
K U(E) & \underset{j}{\leftarrow} & 1
}$$

where $j = \eta_E \circ h \circ b$; this diagram commutes on objects by construction, and it commutes on morphisms because all diagrams commute in $K U(E)$. Thus $K(2) \to 1$, being a retract of an $\mathbf{M}$-cofibration, is also an $\mathbf{M}$-cofibration.
=--

The conclusion of Proposition \ref{key} now leads to a contradiction:

+-- {: .num_prop}
###### Proposition
If $K(2) \to 1$ is an acyclic $\mathbf{M}$-cofibration, then for any category $C$, every automorphism of $C$ is an identity (which is absurd!).
=--

+-- {: .proof}
###### Proof
The object $C$ of $Cat$ has an $\mathbf{M}$-fibrant replacement $\hat{C}$ equivalent to $C$. For any isomorphism $\phi$ of $\hat{C}$, let $e: K(2) \to \hat{C}$ be the unique functor taking $0 \to 1$ in $K(2)$ to $\phi$. Then we have a commutative diagram

$$\array{
K(2) & \stackrel{e}{\to} & \hat{C} \\
_\mathllap{acyc.\; cof.} \downarrow & & \downarrow _\mathrlap{fib.} \\
1 & \to & 1
}$$

and the existence of a lift $1 \to \hat{C}$ filling in this diagram means that $\phi$ is an identity. In particular, every automorphism of $\hat{C}$ is an identity; since $C \simeq \hat{C}$, the same is true of $C$.
=--

### Relation with bisimplicial sets

Recall that Rezk's classifying diagram for a (small) category $C$ is the [[bisimplicial set]] $N (C)$ defined by $N (C)_{n, m} = Fun ([n] \times \mathbf{I}[m], C)$, where $[n]$ is the standard $n$-[[simplex]] considered as a category and $\mathbf{I}$ is the [[localization|groupoid completion]] functor (i.e. $D \mapsto D [D^{-1}]$). There is then an [[adjunction]]
$$\tau_1 \dashv N : Cat \to ssSet$$
and it can be shown that the canonical model structure on Cat is the model structure obtained by [[transferred model structure|transferring]] the [[model structure on simplicial presheaves|projective model structure]] on $ssSet$.

## Related concepts

* [[canonical model structure]]

* **canonical model structure on $Cat$**

* [[canonical model structure on groupoids|canonical model structure on Grpd]]

* [[canonical model structure on Operad]]

* [[weak equivalence of internal categories]]

## References

Original discussion in the generality of [[internal category|categories internal to]] [[topoi]] (hence of [[2-sheaves]]):

* {#JoyalTierney91} [[André Joyal]], [[Myles Tierney]], Thm. 4 (p. 232) of: *Strong stacks and classifying spaces*, in: *Category Theory* Lecture Notes in Mathematics **1488**, Springer (1991) 213-236 [[doi:10.1007/BFb0084222](https://doi.org/10.1007/BFb0084222)]

More on the ordinary case (categories internal to [[Sets]]):

* {#Rezk96} [[Charles Rezk]], *A model category for categories* (1996) [[[Rezk_ModelCategoryForCategories.pdf:file]]]

* {#Joyal10} [[André Joyal]], *[[joyalscatlab:Model structures on Cat]]* (2010)

and proof of the uniqueness of the model structure:

* {#Schommer-Pries12} [[Chris Schommer-Pries]], *[The canonical model structure on Cat](https://sbseminar.wordpress.com/2012/11/16/the-canonical-model-structure-on-cat/)*, Secret Blogging Seminar (2012)

Discussion for [[internal categories]] more generally in [[finitely complete categories]]:

* [[Tomas Everaert]], [[Rudger W. Kieboom]], [[Tim Van der Linden]], _Model structures for homotopy of internal categories_, Theory and Applications of Categories **15** 3 (CT2004) 66-94 [[tac:15-03](http://www.tac.mta.ca/tac/volumes/15/3/15-03abs.html)]

[[!redirects canonical model structure on Cat]]
[[!redirects canonical model structure on categories]]
[[!redirects natural model structure on Cat]]
[[!redirects natural model structure on categories]]
[[!redirects folk model structure on Cat]]
[[!redirects folk model structure on categories]]


canonical model structure on groupoids

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#Contents#
* table of contents
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## Idea

The [[canonical model structure]] on the 1-[[category]] [[Grpd]] of [[groupoids]] (with [[functors]] between them) is a presentation of the [[(2,1)-category]] of [[groupoids]], [[functors]] and [[natural isomorphisms]].

This is one flavor of the various [[canonical model structures]] on classes of [[categories]] and [[higher categories]].

## Definition

\begin{definition}
\label{TheDefinition}
Let [[Grpd]] be the 1-[[category]] of [[small category|small]] [[groupoids]] with [[functors]] between them. Say that a morphism in $Grpd$ --- a [[functor]] $f \colon C \longrightarrow D$ --- is:

* a _[[weak equivalence]]_ iff it is an [[equivalence of categories]] hence a [[weak homotopy equivalence]] of groupoids,

* a _[[fibration]]_ iff it is an [[isofibration]],

* a _[[cofibration]]_ iff it is an [[isocofibration]], hence  [[injective-on-objects functor|injective on objects]].

\end{definition}

\begin{proposition}
\label{ExistenceAndBasicProperties}

Equipped with the classes from Def. \ref{TheDefinition} [[Grpd]] is a [[model category]] $Grpd_{can}$, which is

* [[combinatorial model category|combinatorial]]

  (in particular [[cofibrantly generated model category|cofibrantly generated]]),

* [[proper model category|proper]],

* [[simplicial model category|simplicial]]

  (induced under the [[simplicial nerve]] by the canonical enrichment over itself),

* [[cartesian monoidal model category|cartesian monoidal]].

\end{proposition}
The claim that the classes in Def. \ref{TheDefinition} indeed give a model structure is due to [Anderson 1978](#Anderson78), repeated by [Bousfield 1989](#Bousfield89), both without proof. Proofs appear in [Joyal & Tierney 1991, Thm. 2](#JoyalTierney91) (there in the generality of [[internal groupoid|groupoids internal to]] [[topoi]], hence: of [[stacks]]) and [Strickland 2000](#Strickland00). Properness follows immediately (by [this Prop.](proper+model+category#AllObjectsFibrantImpliesRightProper)) because all objects are evidently [[bifibrant objects|bifibrant]]. In the context of the [[model structure for (2,1)-sheaves]] the structure  is considered, [Hollander 2001, theorem 2.1](#Hollander01) who states also the [[cofibrantly generated model category|cofibrant generation]] and [[simplicial model category|simplicial enrichment]], without proof.
Since [[Grpd]] is a [[locally presentable category]] (for instance by [this Prop](sketch#LimitSketchableCategoriesAreLocallyPresentable), observing that groupoids are the models of a [[limit sketch]], namely given by diagrams as shown at *[[internal category]]*) this implies that $Grpd_{can}$ is [[combinatorial model category|combinatorial]].
Finally the [[cartesian monoidal model category|cartesian model structure]] follows just as for the [[canonical model structure on Cat]], see [there](canonical+model+structure+on+Cat#CartesianMonoidalModelStructure).

## Properties

### Relation to canonical model structure on $Cat$

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

The model structure $Grpd_{nat}$ is the restriction of the
[[canonical model structure on Cat]] from [[categories]] to groupoids.

=--

See at *[[canonical model structure]]* for more.

### Relation to classical model structure on $sSet$

Consider the pair of [[adjoint functors]]
\[
  \label{NerveAdjunction}
  (\tau \dashv N)
  \,\colon\,
  Grpd
    \underoverset
      { \underset{N}{\longrightarrow} }
      { \overset{\tau}{\longleftarrow} }
      { \bot }
  sSet
\]
where $N$ is the [[simplicial nerve]] with values in the category [[sSet]] of [[simplicial sets]].

One readily checks that:

\begin{proposition}
With the canonical model structure on $Grpd$ (from Prop. \ref{ExistenceAndBasicProperties}) and the [[classical model structure on simplicial sets]], (eq:)NerveAdjunction is a [[Quillen adjunction]]
$$
  (\tau \dashv N)
  \,\colon\,
  Grpd_{can}
    \underoverset
      { \underset{N}{\longrightarrow} }
      { \overset{\tau}{\longleftarrow} }
      { \bot_{\mathrlap{Qu}} }
  sSet_{Qu}
  \,.
$$
\end{proposition}
(cf. [Hollander 2001, Cor. 2.3](#Hollander01))

In fact:
\begin{proposition}
$Grpd_{can}$ is the [[transferred model structure]] obtained from [[classical model structure on simplicial sets|$sSet_{Qu}$]] under (eq:NerveAdjunction).
\end{proposition}
(cf. [Hollander 2001, Lem. 2.4](#Hollander01))

## Related concepts

* [[canonical model structure]]

* [[canonical model structure on Cat]]

* **canonical model structure on $Grpd$**

  * [[model structure for n-groupoids]]

  * [[model structure for ∞-groupoids]]

* [[canonical model structure on Operad]]

## References

Some aspects (like the pullback stability of fibrations of groupoids in its prop. 2.8) appeared in

* {#Brown70} [[Ronnie Brown]], *Fibrations of groupoids*, Journal of Algebra **15** 1 (1970) 103-132 \[doi:10.1016/0021-8693(70)90089-X, [pdf](https://groupoids.org.uk/pdffiles/fibrationsgpds.pdf)\]

The existence of the model structure is stated (without proof) in:

* {#Anderson78} [[Donald W. Anderson]], p. 783 (12 of 37) in: *Fibrations and geometric realization*,  Bull. Amer. Math. Soc. **84** 5 (1978) 765-788 [[euclid:1183541139](http://projecteuclid.org/euclid.bams/1183541139)]

and (by referencing Anderson, still without proof) in:

* {#Bousfield89} [[Aldridge Bousfield]], _Homotopy Spectral Sequences and Obstructions_ , Israel Journal of Math., **66** 1-3 (1989) 54-105 [[doi:10.1007/BF02765886](https://doi.org/10.1007/BF02765886)

Proofs are spelled out in:

* {#JoyalTierney91} [[André Joyal]], [[Myles Tierney]], Thm. 2 (p. 221) *Strong stacks and classifying spaces*, in: *Category Theory* Lecture Notes in Mathematics **1488**, Springer (1991) 213-236 [[doi:10.1007/BFb0084222](https://doi.org/10.1007/BFb0084222)]

  > (in the generality of [[internal groupoids]] in a [[topos]], hence of *[[stacks]]*)

* {#Strickland00} [[Neil Strickland]], §6.1 of: _$K(n)$-local duality for finite groups and groupoids_ , Topology **39** 4 (2000) [[arXiv:math/0011109](https://arxiv.org/abs/math/0011109), doi:10.1016/S0040-9383(99)00031-2]

The model structure on functors with values in $Grpd_{nat}$ (a [[model structure for (2,1)-sheaves]]):

* {#Hollander01} [[Sharon Hollander]], §2.1 in: *A homotopy theory for stacks*, Israel Journal of Mathematics **163** 1 (2008) 93-124 [[arXiv:math.AT/0110247](http://arxiv.org/abs/math.AT/0110247), [doi:10.1007/s11856-008-0006-5](https://doi.org/10.1007/s11856-008-0006-5)]

A model structure on $Cat$ but localized such as to make the fibrant objects be groupoids:

* {#Sharma23} [[Amit Sharma]], *Picard groupoids and $\Gamma$-categories*, [[Cahiers]] **LXIV** 3 (2023) [[arXiv:2002.05811](https://arxiv.org/abs/2002.05811), [pdf](http://cahierstgdc.com/wp-content/uploads/2023/07/SHARMA_LXIV-3.pdf)]

[[!redirects natural model structure on groupoids]]

[[!redirects canonical model structure on Grpd]]

[[!redirects model structure on groupoids]]
[[!redirects model structures on groupoids]]


canonical model structure on Operad

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## Idea

A [[model category]] structure on the [[category]] [[Operad]] of [[Set]]-enriched coloured [[symmetric operads]] which generalizes the [[canonical model structure]] on [[Cat]].

## Definition

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

Call a [[morphism]] of [[operads]] $f : P \to Q$  a **[[weak equivalence]]**  if

1. its underlying [[functor]] of [[categories]] is an [[essentially surjective functor]];

1. for every collection $(c_1, \cdots, c_n; c)$ of colours it induces an [[isomorphism]]

   $$
     P(c_1, \cdots, c_n; c) \to Q(f(c_1), \cdots, f(c_n); f(c))
   $$

   (the operadic analog of being [[full and faithful functor|full and faithful]]).

Call a morphism $f : P \to Q$ a **fibration** if for every [[isomorphism]] in $Q$ and a lift of its source object to $P$ there is an isomorphism in $P$ covering it under $f$.

Call a morphism a **cofibration** if it is an injection on objects (on colours)

=--

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

This defines a [[cofibrantly generated model category]] structure on [[Operad]].

=--

This is due to ([Weiss 07](#Weiss)).

## Related concepts

* [[canonical model structure on Cat]]

* [[model structure on operads]]

* [[model structure on dendroidal sets]]

## References

* [[Ittay Weiss]], _Dendroidal sets_, PhD thesis ([web](http://igitur-archive.library.uu.nl/dissertations/2007-0918-200833/UUindex.html))
 {#Weiss}

[[!redirects canonical model structure on operads]]


canonical momentum

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#### Geometric quantization
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#Contents#
* table of contents
{:toc}

## Idea

The archetypical example of a [[mechanical system]] is a [[particle]] propagating on a [[manifold]] $\Sigma$. The [[phase space]] of this particular system happens to be canonically identified with the [[cotangent bundle]] $X \coloneqq T^* \Sigma$ of $\Sigma$. Here the [[covector]] $(x,p)$ in $X$ over a point $x \in \Sigma$ is physically interpretd as describing a [[state]] of the system where the particle is at position $x \in \Sigma$ and has [[momentum]] (essentially: speed) as given by $p$.

Therefore locally for [[coordinate patch]] $\phi : \mathbb{R}^{2n} \simeq \mathbb{R}^n \times \mathbb{R}^n \to T^* \Sigma$ the $2n$-canonical coordinates of the [[Cartesian space]] $\mathbb{R}^n$ are naturally thought of as decomposed into $n$ "canonical coordinates" on the first $n$ factors and a set of "canonical momenta", being the canonical coordinates on the second $\mathbb{R}^n$-factor.

Notice that "canonical" here refers (at best) to the canonical coordinates of the [[Cartesian space]] $\mathbb{R}^n$ _once $\phi$ has been chosen_. The choice of $\phi$ however is arbitrary. Hence, despite the (standard) term, there is nothing much canonical about these "canonical coordinates" and "canonical momenta".

In general, the [[phase space]] of a [[physical system]] is a [[symplectic manifold]] which need not be a [[cotangent bundle]] as for the [[particle]] [[sigma-model]].

But _locally_ over a [[coordinate patch]] every [[symplectic manifold]] looks like $\mathbb{R}^{2n} \simeq \mathbb{R}^n \times \mathbb{R}^n$ such that under this identification the [[symplectic form]] reads $\sum_{i = 1}^n d q_i \wedge d p^i$, for $\{q_i\}$ the canonical [[coordinates]] on one $\mathbb{R}^n$ and $\{p^i\}$ for the other.

Therefore generally, in the context of [[mechanics]], with such a local  identification one calls _$p^i$_ the **canonical momentum** of the coordinate (or sometimes "canonical coordinate") $q_i$.

Globally the notion of canonical momenta may not exist at all. The notion that does exist globally is that of a [[polarization]] of a symplectic manifold. See there for more details.

## Properties

### On a symplectic vector space

Discussion of how there is a flat connection on the bundle of [[spaces of quantum states]] over the space of choices of [[polarizations]] of a [[symplectic vector space]] and how this reproduces the traditional relation between canonical coordinates and canonical momenta by [[Fourier transform]] is in ([Kirwin-Wu 04](#KirwinWu04)).

## Related concepts

* [[momentum]]

* [[canonical coordinates]], [[canonical transformation]]

* [[phase space]]

* [[Hamiltonian mechanics]]

* [[geometric quantization]]

## References

* [[Marc Henneaux]], [[Claudio Teitelboim]], §1.1.1 in: _[[Quantization of Gauge Systems]]_, Princeton University Press (1992) [[ISBN:9780691037691](https://press.princeton.edu/books/paperback/9780691037691/quantization-of-gauge-systems), [jstor:j.ctv10crg0r](https://www.jstor.org/stable/j.ctv10crg0r)]

* {#KirwinWu04} William Kirwin, Siye Wu, _Geometric Quantization, Parallel Transport and the Fourier Transform_,  	Comm. Math. Phys. 266 (2006), no. 3, pages 577 -- 594 ([arXiv:math/0409555](http://arxiv.org/abs/math/0409555))

[[!redirects canonical momenta]]


canonical presentation

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# Contents
* the following line creates the automatic table of contents
{: toc}

## Definition

Given an [[adjunction]] $\langle F,G,\eta,\varepsilon\rangle\colon
X\to A$, the __canonical presentation__ of an object
$a\in\operatorname{obj}(A)$ is the [[fork]]
$$
F G F Ga\underoverset{F G\varepsilon_a}{\varepsilon_{F G a}}{\rightrightarrows}F G a
\overset{\varepsilon_a}{\rightarrow}a
$$
(this is indeed a fork, by the [[natural transformation|naturality]] of $\varepsilon$).

## Properties

* In general, this fork need not be a [[coequalizer]], but if $G$ is [[monadic functor|monadic]],
then we do get a coequalizer.  To see this, note that the above pair
is $G$-split:  When applying
$G$ to the fork, we get the [[split coequalizer]]
$$
T^2 x\underoverset{T h}{\mu_x}{\rightrightarrows}T x\overset{h}{\rightarrow}{x}
$$
for the [[monad]] $\mathbb{T} := \langle T = G F,\eta,\mu=G\varepsilon F
\rangle$ corresponding to the given adjunction and for the
$\mathbb{T}$-[[algebra of a monad|algebra]] $\langle x,h\rangle = \langle G a , G\varepsilon_a
\rangle$.  Hence, by the [[monadicity theorem]], $G$ (in particular) [[reflected colimit|reflects]]
coequalizers for our pair.

* The two [[parallel arrows]] $F G\varepsilon_a$ and $\varepsilon_{F G a}$
appearing in the canonical presentation have a common
[[section]], namely, $F \eta_{G a}$ (by the [[triangle identities]]).  Hence,
whenever $G$ is monadic, the resulting coequalizer is a [[reflexive coequalizer]].

## Example

If $A = \mathbf{Grp}$, $X = \mathbf{Set}$ and $G$ is the forgetful
functor, then for a group $a$, $F G a$ is just the [[free group]] on the
elements of $a$, and $\varepsilon$ is the projection, taking a
''formal product'' of elements of $a$ to the actual product in $a$
(since by a triangle identity we have $G\varepsilon_a(\langle
t\rangle)=t$ where $t\in Ga$ and $\langle t\rangle=\eta_{G a}(t)=$ the
reduced word with one letter $t$).

Since the coequalizer of
$\cdot\underoverset{f}{g}{\rightrightarrows}\cdot$ in $\mathbf{Grp}$ is
the familiar [[quotient group|quotient]] by the normal subgroup generated by elements like
$f(t) g(t)^{-1}$, the canonical presentation (a coequalizer in this
case) is indeed a [[group presentation|presentation]] of $a$ in terms of generators and
relations.

## References

* _[[Categories Work]]_, p. 153.

* [[John Baez]], ''Universal algebra and diagrammatic reasoning,'' slides [available online](http://math.ucr.edu/home/baez/universal)

[[!redirects canonical presentation]]
[[!redirects canonical presentations]]


canonical resolution

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##Idea

Let $K$ be the strict [[2-category]] [[Cat]] and $C$ a [[category]] in $K$. Then we can identify a [[monad]] on $C$ with an [[endofunctor]] $T \colon C\to C$ (a [[monoid]] in the [[endofunctor category]] $K(C,C)$). We recall that for any [[object]] $c$ of $C$, we obtain a certain sort of [[resolution]] of $c$. Note that the canonical resolution is not a resolution in the sense of the above cited nLab page because it is not necessarily acyclic or [[contractible]] in the relevant sense. In other words, it does not always have trivial [[cohomology]], especially since our category may not even be equipped with such a notion.

##Definitions

The _canonical resolution_ of $c$ in $C$ with respect to a [[monad]] $T \colon C\to C$ is a [[augmented simplicial set|coaugmented]] [[simplicial object|cosimplicial object]] $CanRes_T:\Delta_+\to C$ such that $CanRes_T([-1])=c$ and $CanRes_T([n])=T^{n+1}c$ for each $n\geq 0$. It has [[coface]] [[morphisms]] $T^{n}c\to T^{n+1}c$ given by the unit map of $T$, and codegeneracies given by the monoid structure $T T c\to T c$.

###Monadic Cohomology
{#monadiccohomology}

Given a suitable functor $E:C\to A$, for $A$ an [[additive category]], we can define the [[monadic cohomology]] of an object $c$ to be the [[cohomology]] of the [[cochain complex]] associated to the composition of [[functors]] $E\circ CanRes_T:\Delta_+\to A.$ This has been introduced by Godement who called monad and the resolution from it the standard construction (thus, standard resolution is a term which is still used). By suitably dualizing, we can define _comonadic homology_.

###Resolution of a $T$-algebra

Suppose that $c$ is an object of $C$ and also an [[module over a monad|algebra]] for $T$. Thus there is a [[morphism]] $T c\to c$ satisfying certain properties. In particular, this map provides an extra codegeneracy which defines an [[augmented simplicial set|augmented]] [[simplicial object]] living inside of the canonical resolution of $c$. This simplicial object is frequently referred to as the [[bar construction]] of $c$.

Note that we can consider this internal [[bar construction]] to also be the canonical resolution associated to the induced [[comonad]] on the [[Eilenberg-Moore category]] of algebras over $T$.

This [[bar construction]] can also be obtained by precomposing $CanRes_T$ with the functor that includes $\Delta^{op}_+\hookrightarrow \Delta$ as the subcategory whose morphisms preserve the minimal and maximal elements of finite totally ordered sets.

## Examples

### The bar resolution for abelian groups

There is a monad on [[Set]], the category of sets, comprising the [[free abelian group]] functor $F \colon Set\to Ab$ and the [[forgetful functor]] $U \colon Ab\to Set$. The [[Eilenberg-Moore category|category of algebras]] of this monad is precisely the category of [[abelian groups]] (in other words, the [[free abelian group]] functor $F$ is [[monadic]]).

Thus we may produce the cosimplicial canonical resolution of any set $X$. If $X$ supports (at least one) abelian group structure, then we can add a codegeneracy to the canonical resolution which defines the usual [[bar construction]] on $X$ with that particular abelian group structure.

## Related concepts

* [[homological resolution]]

* [[monadic cohomology]]

* [[simplicial resolution]]

* [[bar and cobar construction]]

## References

The original articles:

* {#Godement58} [[Roger Godement]], Appendix of: *Topologie algébrique et theorie des faisceaux*, Actualités Sci. Ind. **1252**, Hermann, Paris (1958) [[webpage](https://www.editions-hermann.fr/livre/topologie-algebrique-et-theorie-des-faisceaux-roger-godement), [[Godement-TopologieAlgebrique.pdf:file]]]

* {#Huber61} [[Peter J. Huber]], *Homotopy theory in general categories*, Mathematische Annalen **144** (1961) 361–385 [[doi:10.1007/BF01396534](https://doi.org/10.1007/BF01396534)]

Further discussion:

* [[Michael Barr]], [[Jon Beck]], _Homology and Standard Constructions_, in _[[Seminar on Triples and Categorical Homology Theory]]_, Lecture Notes in Maths. **80**, Springer (1969), Reprints in Theory and Applications of Categories __18__ (2008) 186-248 [[TAC:18](http://www.tac.mta.ca/tac/reprints/articles/18/tr18abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/18/tr18.pdf)]

*  {#Duskin75} [[John Duskin]], *Simplicial methods and the interpretation of "triple" cohomology"*  Memoirs  of the AMS **163**, Amer. Math. Soc. (1975) [[ISBN:978-1-4704-0645-5](https://bookstore.ams.org/memo-3-163)]

See also:

* [[Michael Barr]], *Cartan-Eilenberg cohomology and triples*, J. Pure Applied Algebra **112** 3 (1996) 219–238 \[doi:10.1016/0022-4049(95)00138-7, [pdf](https://www.math.mcgill.ca/barr/papers/coho.pdf), [[Barr-CECohomology.pdf:file]]\]

* [[Michael Barr]], *Algebraic cohomology: the early days*, in *Galois Theory, Hopf Algebras, and Semiabelian Categories*, Fields Institute Communications **43** (2004) 1–26 \[doi:10.1090/fic/043, [pdf](https://www.math.mcgill.ca/barr/papers/algcohom.pdf), [[Barr-AlgebraicCohomology.pdf:file]]\]

Textbook account:

* [[Charles Weibel]], _[[An Introduction to Homological Algebra]]_

[[!redirects canonical resolutions]]

[[!redirects standard resolution]]
[[!redirects standard resolutions]]
[[!redirects standard construction]]

[[!redirects cotriple homology]]
[[!redirects triple cohomology]]


canonical sheaf

the [[sheaf]] of [[sections]] of a [[canonical bundle]]

[[!redirects canonical sheaves]]


canonical topology

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## Definition

The **canonical topology** on a [[category]] $C$ is the [[Grothendieck topology]] on $C$ which is the largest [[subcanonical site|subcanonical]] topology.  More explicitly, a [[sieve]] $R$ is a [[covering]] for the canonical topology iff every [[representable functor]] is a [[sheaf]] for every [[pullback]] of $R$.  Such sieves are called *universally effective-epimorphic*.

## Examples

### On a Grothendieck topos

If $C$ is a [[Grothendieck topos]], then the canonical [[covering]] [[sieves]] are those that are jointly [[epimorphism|epimorphic]].  Moreover, in this case the canonical topology is generated by *small* jointly epimorphic families, since $C$ has a small [[generator|generating set]].

The canonical topology of a Grothendieck topos is also special in that every [[sheaf]] is [[representable functor|representable]]; that is, $C \simeq Sh_{canonical}(C)$.

Notice that if $(D,J)$ is a site of definition for the topos $C$, then this says that

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

$$
  Sh_J(D) \simeq Sh_{canonical}(Sh_J(D))
  \,.
$$

=--

(e.g. [Johnstone, prop C 2.2.7](#Johnstone), [Makkai-Reyes, lemma 1.3.14](#Makkaireyes))

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

For simplicity, assume $(\mathbb{C}, J)$ is a [[small site|small]] [[subcanonical topology|subcanonical]] [[site]]. The quasi-inverse of the [[Yoneda embedding]] $\mathbf{Sh}(\mathbb{C}, J) \to \mathbf{Sh}(\mathbf{Sh}(\mathbb{C}, J))$ has a very simple description: it is the [[functor]] that sends a [[sheaf]] $F : \mathbf{Sh}(\mathbb{C}, J)^\mathrm{op} \to \mathbf{Set}$ to its restriction along the [[Yoneda embedding]] $\mathbb{C} \to \mathbf{Sh}(\mathbb{C}, J)$.

Indeed, suppose $F : \mathbf{Sh}(\mathbb{C}, J)^\mathrm{op} \to \mathbf{Set}$ is a sheaf. We claim that $F$ is determined up to unique [[isomorphism]] by its restriction along the embedding $\mathbb{C} \to \mathbf{Sh}(\mathbb{C}, J)$. Indeed, let $X : \mathbb{C}^\mathrm{op} \to \mathbf{Set}$ be a $J$-sheaf. Then $X$ is the [[colimit]] of a canonical small [[diagram]] of [[representable functor|representable]] [[sheaves]] on $(\mathbb{C}, J)$ in a canonical way. Consider the colimiting [[cocone]] on $X$: it is a universal [[effective epimorphism|effective epimorphic]] family and is therefore a [[covering]] family in the canonical topology on $\mathbf{Sh}(\mathbb{C}, J)$. Thus, $F (X)$ is indeed determined up to unique isomorphism by the restriction of $F$ to $\mathbb{C}$. We must also show that the restriction is actually a sheaf on $(\mathbb{C}, J)$; but this is true because $J$-covering [[sieves]] in $\mathbb{C}$ become universal effective epimorphic families in $\mathbf{Sh}(\mathbb{C}, J)$.

Thus we obtain a functor $\mathbf{Sh}(\mathbf{Sh}(\mathbb{C}, J)) \to \mathbf{Sh}(\mathbb{C}, J)$ that is left quasi-inverse to the embedding $\mathbf{Sh}(\mathbb{C}, J) \to \mathbf{Sh}(\mathbf{Sh}(\mathbb{C}, J))$, and the argument above shows that it is also a right quasi-inverse.

=--

## Related concepts

* [[subcanonical topology]]

* [[trivial topology]]

## References
 {#References}

A textbook account in [[topos theory]] is in

* {#Johnstone} [[Peter Johnstone]], _[[Sketches of an Elephant]]_

* {#Makkaireyes} Makkai, [[Gonzalo Reyes]], _First Order Categorical Logic_

Discussion of examples:

* [[Cynthia Lester]], *Covers in the Canonical Grothendieck Topology* [[arXiv:1909.03384](https://arxiv.org/abs/1909.03384)]

* [[Cynthia Lester]], *The canonical Grothendieck topology and a homotopical analog*, PhD thesis (2019) [[uoregon:1794/24924](https://scholarsbank.uoregon.edu/xmlui/handle/1794/24924) [pdf](https://scholarsbank.uoregon.edu/xmlui/bitstream/handle/1794/24924/Lester_oregon_0171A_12522.pdf?sequence=1&isAllowed=y)]

* [[Cynthia Lester]], *The canonical Grothendieck topology and a homotopical analog* [[arXiv:1909.03188](https://arxiv.org/abs/1909.03188)]

Discussion in the refined context of [[higher topos theory]] is in

* [[Jacob Lurie]], section 6.2.4 and around prop. 5.5.2.2., remark 6.3.5.17 of _[[Higher Topos Theory]]_

* [[David Carchedi]], MO discussion _[Canonical topology for infinity topoi revisited.](http://mathoverflow.net/questions/95172/canonical-topology-for-infinity-topoi-revisited)_

[[!redirects canonical coverage]]
[[!redirects canonical Grothendieck topology]]

[[!redirects canonical site]]
[[!redirects canonical sites]]


canonical transformation

> This entry is about the notion in [[Hamiltonian mechanics]]/[[symplectic geometry]]. For a different notion of the same name in [[category theory]] see at _[[canonical morphism]]_.

> To be merged with [[generating function in classical mechanics]].

***

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[physics]] and specifically in [[Hamiltonian mechanics]] what is called a _canonical transformation_ is an [[equivalence]] or at least a [[homomorphism]] of [[phase spaces]]. The term is to be read as short for "transformation of [[canonical coordinates]]".

Since phase spaces are mathematically identified with [[symplectic manifolds]], canonical transformations are [[symplectomorphisms]]. In fact, the notion as used in physics is a bit more general, and subsumes also certain [[Lagrangian correspondences]] between symplectic manifolds $(X_{1,2}, \omega_{1,2})$. The "generating functions" for canonical transformations as used in the physics literture correspond to functions on the [[correspondence space]] $(X_1 \times X_2, p_1^\ast \omega_1 - p_2^\ast \omega_2)$. ([Weinstein 83](#Weinstein83))

## References

The standard definition, interpreted in [[symplectic geometry]] appears for instance on p. 206 of

* [[Vladimir Arnold]], _[[Mathematical methods of classical mechanics]]_

Discussion of generating functions starts on p. 266 there.

The formalization in terms of [[Lagrangian correspondences]] is discussed on p. 5 of

* [[Alan Weinstein]], _Lectures on Symplectic Manifolds_, volume 29 of CBMS Regional Conf. Series in Math. Amer. Math. Soc., 1983. third
printing.
  {#Weinstein83}

[[!redirects canonical transformations]]

[[!redirects canonical coordinate transformation]]
[[!redirects canonical coordinate transformations]]


Cantor cube

A __Cantor cube__ is a [[topological group]] of the form  of a Cartesian product $\{0,1\}^K$ where $K$ is some index set. In particular it is a direct product of $K$ copies of the 2-element group with [[Tihonov topology]].

* wikipedia: [Cantor cube](https://en.wikipedia.org/wiki/Cantor_cube)

category: topology


Cantor space

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

# Contents
* table of contents
{: toc}

## Idea

What is called _Cantor space_, after [[Georg Cantor]], is the [[topological space]] obtained from the [[closed interval]] $[0,1]$ by

1. removing the middle third, retaining only $[0,\tfrac{1}{3}]$ and $[\tfrac{2}{3}, 1]$;

1. removing from  these two pieces _their_ middle thirds, to retain the pieces $[0,\tfrac{1}{9}]$, $[\tfrac{2}{9},\tfrac{1}{3}]$, $[\tfrac{2}{3}, \tfrac{7}{9}]$, $[\tfrac{8}{9},1]$;

1. and so ever on.

$$
  \array{
    \mathrlap{[0} && &,& &&  \mathllap{1]}
    \\
    \mathrlap{[0} &,& \mathllap{\tfrac{1}{3}]}
     &\phantom{(\tfrac{1}{2}, \tfrac{2}{3})}
     &  \mathrlap{[\tfrac{2}{3}} &,& \mathllap{1]}
    \\
    \mathrlap{[0,\tfrac{1}{9}]}
    &
    \phantom{AAAAAAAA}
    &
    \mathllap{[\tfrac{2}{9}, \tfrac{1}{3} ]}
    &
    \phantom{  (\tfrac{1}{3}, \tfrac{2}{3}) }
    &
    \mathrlap{[\tfrac{2}{3}, \tfrac{7}{9}]}
    &
    \phantom{AAAAAAAA}
    &
    \mathllap{[\tfrac{8}{9}, 1]}
    \\
    &&& \vdots
  }
$$

Cantor space is a fundamental object of [[descriptive set theory]]; some indications of its use may be found at [[Polish space]]. Among its applications is a simple construction of a "[[Peano curve|space-filling curve]]" (q.v.).

## Definition
 {#Definition}

Traditionally the Cantor space was conceived of

* [as a subspace of the real line](#AsSubspaceOfTheRealLine)

but of course it may also be described

* [as an abstract space](#AsAbstractTopologicalSpace)

in itself.

### As an abstract space
 {#AsAbstractTopologicalSpace}

In brief, Cantor space may be abstractly described as the [[topological space|topological]] [[product]] of countable many copies of the [[discrete space]] $\{0, 1\}$. In more concrete detail:

Recall that a __[[binary digit]]__ is either $0$ or $1$; the [[set]] (or [[discrete space]]) of binary digits is the [[Boolean domain]] $\mathbb{B}$.

A __point__ in Cantor space is an [[infinite sequence]] of binary digits.  Accordingly, Cantor space may be denoted $\mathbb{B}^{\mathbb{N}}$, since its set of points is a [[function set]].

An __open__ in Cantor space is a collection $G$ of [[list|finite sequences]] of binary digits (that is a [[subset]] of the [[free monoid]] $\mathbb{B}^*$) such that:

*  If $u \in G$ and $v$ is an extension of $u$ (that is $u$ with possibly additional digits added to the end), then $v \in G$;

*  If $u:0 \in G$ and $u:1 \in G$ (where $u:i$ is the immediate extension of $u$ by the digit $i$), then $u \in G$.

A point $\alpha$ __belongs__ to an open $G$ if, for some $u$ in $G$, $\alpha$ is an extension of $u$.

An alternative characterization of Cantor space is as the [[terminal coalgebra for an endofunctor|terminal coalgebra]] for the endofunctor on [[Top]], $X \mapsto X + X$.

### What kind of space?

Traditionally, Cantor space is understood as a [[topological space]].  We start with the points, as defined above, then specify which sets of points are [[open subset|open]].  Although there are other ways to state which sets are open, we may define a set to be open if it is the set of points that belong to some open $G$ as defined above.

A newer approach is to understand Cantor space as a [[locale]].  Then we start with the opens and define an [[order]] relation on them to define a [[frame]].  In this case, the order relation is the obvious one, that $G \leq H$ if $G \subseteq H$ as subsets of $\mathbb{B}^*$.  Then the [[point of a locale|points]] come for free, and correspond precisely to the points as defined above.

In [[classical mathematics]], these two approaches are equivalent; a point is determined by its opens, and an open is determined by its points.  The theorem that a point is determined by its opens (so that Cantor space, as a topological space, is [[sober space|sober]]) is valid [[internalisation|internal]] to any [[pretopos]] with an [[exponential object|exponentiable]] [[natural numbers object]]; as such, it applies even in [[predicative mathematics|predicative]] and [[constructive mathematics|constructive]] mathematics.  However, the theorem that an open is determined by its points (so that Cantor space, as a locale, is [[spatial locale|topological]]) is equivalent to the [[fan theorem]]; it is true in some pretoposes and accepted by some schools of constructivism but false in other pretoposes and rejected, or even refuted, by other constructivists.

When the fan theorem is not valid, the localic approach is probably better; it allows more of the useful properties of Cantor space to hold.

### As a subspace of the real line
 {#AsSubspaceOfTheRealLine}

Cantor space is usually conceived of as a [[topological subspace]] of the [[real line]]:

Write $Disc(\{0,1\})$ for the the [[discrete topological space]] with two points. Write $\underset{n \in \mathbb{N}}{\prod} Disc(\{0,1\})$ for the [[product topological space]] of a [[countable set]] of copies of this discrete space with itself (i.e. the corresponding [[Cartesian product]] of sets $\underset{n \in \mathbb{N}}{\prod} \{0,1\}$ equipped with the [[Tychonoff topology]] induced from the [[discrete topology]] of $\{0,1\}$).

Then consider the [[function]]

$$
  \array{
    \underset{n \in \mathbb{N}}{\prod}
      &\overset{\kappa}{\longrightarrow}&
    [0,1]
    \\
    (a_i)_{i \in \mathbb{N}}
    &\overset{\phantom{AAAA}}{\mapsto}&
    \underoverset{i = 1}{\infty}{\sum} \frac{2 a_i}{3^i}
  }
$$

which sends an element in the product space, hence a [[sequence]] of binary digits, to the value of the [[power series]] as shown on the right.

One checks that this is a [[continuous function]] (from the [[product topology]] to the [[Euclidean space|Euclidean]] [[metric topology]] on the [[closed interval]]). Moreover with its [[image]] $\kappa\left( \underset{n \in \mathbb{N}}{\prod} \{0,1\}\right) \subset [0,1]$ equipped with its [[subspace topology]], then this is a [[homeomorphism]] onto its image:

$$
  \underset{n \in \mathbb{N}}{\prod} Disc(\{0,1\})
    \overset{\phantom{AA}\simeq\phantom{AA}}{\longrightarrow}
  \kappa\left(
    \underset{n \in \mathbb{N}}{\prod} Disc(\{0,1\})
  \right)
    \overset{\phantom{AAAA}}{\hookrightarrow}
  [0,1]
  \,.
$$

This image is the _Cantor space_ as a subspace of the closed interval.

From the localic perspective, a [[continuous map]] is given by a [[homomorphism]] of frames in the opposite direction.  Given an open $\sim$ in $\mathbb{R}$ (as a [[binary relation]] on [[rational numbers]], as described at [[locale of real numbers]]), this is mapped to the open $G$ in Cantor space such that $u \in G$ if and only if
$$ \sum_{i=1}^{len(u)} \frac { 2 u_i } { 3^i } \sim \sum_{i=1}^{len(u)} \frac { 2 u_i } { 3^i } + \frac 1 { 3^{-len(u)} } .$$
One then checks that this is an embedding.

+-- {: .query}
I should check this some day; for the moment, I am taking it on faith.  ---Toby
=--

In either case, the idea is:

*  A point of Cantor space corresponds to a number written in base $3$ with infinitely many digits, using only the digits $0$ and $2$ (which are the options for $2 a_i$ when $a_i \in \{0,1\}$); while

*  An open corresponds to a union of intervals, each of which is given by approximating a number in base $3$ to a finite number of digits, using only the digits $0$ and $2$.

One sometimes speaks of the __Cantor set__ to stress that one is considering Cantor space as a subspace of the real line.

As we can also consider Cantor space as a product space $(\mathbb{Z}/2)^n$ of countably many copies of $\mathbb{Z}/(2)$, which carries a group structure, we can view Cantor space $C$ as a [[topological group]]. In particular, it is a [[homogeneous space]] (its group of self-homeomorphisms acts transitively on the space).

## Properties

Cantor space, especially in its guise as a subspace of the real line, is quite famous; see [Wikipedia](http://en.wikipedia.org/wiki/Cantor_set).  Here are some headline properties:

*  Cantor space is a [[compact Hausdorff space]].  (For the topological space, this statement is again equivalent to the [[fan theorem]]; for the locale, it holds regardless.)

*  Cantor space is [[connected space|totally disconnected]].

* Thus Cantor space is a [[Stone space]].

*  Cantor space is [[metric space|metrizable]], and every compact metrizable space is a [[quotient space]] of Cantor space (see Theorem \ref{HA} below).

*  As a subspace of $\mathbb{R}$, the Cantor set is [[perfect space|perfect]] and [[uncountable set|uncountable]] but of [[Lebesgue measure|Lebesgue]] [[null set|measure zero]].

*  The Cantor set is a precisely self-similar [[fractal]] with [[Hausdorff dimension]] $\log_3 2 \approx 0.631$.

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

A [[topological space]] is [[homeomorphism|homeomorphic]] to Cantor space if and only if it is [[inhabited set|nonempty]], [[compact topological space|compact]], [[totally disconnected topological space|totally disconnected]], [[metric  space|metrizable]], and [[perfect topological space|perfect]].

=--

This result is sometimes called **[[Brouwer]]'s theorem**. It can be seen from the perspective of [[Stone duality]], where the dual result is that any two [[countable set|countable]] [[atom|atomless]] [[Boolean algebras]] are [[isomorphism|isomorphic]]; this dual result can be proven by a [[back-and-forth argument]].

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

The  [[one-point compactification]] $\widebar{X}$ of a space $X$ that is [[second-countable space|second-countable]] [[local compactum|locally compact]] [[Hausdorff topological space|Hausdorff]], [[totally disconnected space|totally disconnected]] and perfect, is homeomorphic to Cantor space (provided $X$ is not itself compact).

=--

+-- {: .proof}
###### Proof
$\widebar{X}$ is also second-countable, compact Hausdorff and therefore compact regular, and so by the [[Urysohn metrization theorem]] it is compact metrizable. The point $p$ at infinity is not isolated since we assume $X$ is not compact, so $\widebar{X}$ is perfect. If $V$ is any open neighborhood of $p$, so $V = \neg K$ for some compact $K \subset X$, then we claim there exists a clopen that contains $K$; in that case $V$ contains a clopen whence $\{p\}$ is the quasi-component of $p$ (hence also the connected component since we're in a compact Hausdorff space). But the argument [here](/nlab/show/compact+Hausdorff+rings+are+profinite#compopen) shows that for each $x \in K$ there is a clopen neighborhood of $x$ contained in $\neg \{p\}$; finitely many of these clopens cover $K$, and the claim follows by considering their union.
=--

It follows from this result that all such spaces $X$ are homeomorphic: they all have Cantor space as their one-point compactifications, and so they are all homeomorphic to the space obtained obtained by removing a single point from Cantor space. This applies for example to spaces obtained by removing a finite number $n \geq 1$ of points from Cantor space.

Cantor space is also a "universal" compact metric space in the following sense.

+-- {: .num_theorem #HA}
###### Theorem
**(Hausdorff-Alexandroff)**
Every compact metric space is a continuous image of Cantor space.
=--

This implies that every compact metric space is a [[quotient space]] of Cantor space, since a surjective map between [[compact Hausdorff spaces]] is a closed surjection, and closed surjections are quotient maps.

+-- {: .proof}
###### Proof
First, every compact metric space $X$ is [[separable space|separable]]: has a [[countable set|countable]] [[dense set]] $\{x_0, x_1, \ldots\}$. Assume, as we may, that the metric $d: X \times X \to [0, \infty)$ is valued in $[0, 1]$. Then the map $y: X \to [0, 1]^\mathbb{N}$ to the [[Hilbert cube]], defined by

$$y(x) = (d(x, x_n))_{n \in \mathbb{N}}$$

(a type of restricted "[[Yoneda embedding]]", regarding [[metric spaces]] as [[enriched categories]]), is continuous and maps $X$ onto a [[closed subspace]] of $I^\mathbb{N}$. As mentioned at [Peano curve](https://ncatlab.org/nlab/show/Peano+curve#construction), there is a continuous surjection $C \to I^\mathbb{N}$. Taking the [[pullback]] in [[Top]]

$$\array{
K & \hookrightarrow & C \\
\mathllap{surj} \downarrow & & \downarrow \mathrlap{surj} \\
X & \stackrel{y}{\hookrightarrow} & I^\mathbb{N}
}$$

we see that to produce a continuous surjection $C \to X$, it suffices to exhibit a continuous surjection $C \to K$.

In fact, every closed subspace $K \hookrightarrow C$ admits a retraction. There is a clever trick for seeing this: represent Cantor space $C$ instead as the subspace of $[0, 1]$ whose points, when written in base $6$, have just $0$\'s and $5$\'s in their representation. This subspace has the geometric property that if $x, y \in C$, then $\frac{x+y}{2} \notin C$. As a result, for $x, y, z \in C$ we have $d(x, y) = d(x, z)$ only if $y = z$ and so: given a closed subspace $K$ of $C$, there is for each $x \in C$ a *unique* element $k_x \in K$ such that $d(x, k_x) = d(x, K)$. The assignment $x \mapsto k_x$ is continuous (in fact a [[locally constant function]] on $C \setminus K$, and continuous on $K$ as is easily seen) and provides the desired retraction.
=--

## Related concepts

* [[Peano curve]]

* [[Sierpinski space]]

* [[long line]], [[Warsaw circle]],

* [[line with two origins]]

## References

* [[Friedhelm Waldhausen]], p. 3-4 in _Topologie_ ([pdf](https://www.math.uni-bielefeld.de/~fw/ein.pdf))

* Proof Wiki, _[Cantor Space as Countably Infinite Product](https://proofwiki.org/wiki/Cantor_Space_as_Countably_Infinite_Product)_

[[!redirects Cantor space]]
[[!redirects Cantor set]]


Cantor's paradox

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

# Cantor\'s Paradox
* table of contents
{: toc}

## Idea

Cantor\'s Paradox is a [[paradox]] of naïve [[set theory]] based on an observation of [[Bertrand Russell]] that is similar to, but predates, [[Russell's Paradox]].  Like Russell\'s Paradox, it is essentially about [[diagonalization|diagonalisation]] and [[Cantor's Theorem]].

## Argument

Assume a [[set]] $V$ of all [[sets]], and consider the [[power set]] $\mathcal{P}V$.  Since every element of $\mathcal{P}V$ is a set, $\mathcal{P}V$ is a [[subset]] of $V$.  In particular, we have an [[injection]] $\mathcal{P}V \hookrightarrow V$, which contradicts the conclusion of [[Cantor's Theorem]].

## History

Cantor\'s Paradox may be credited to [[Georg Cantor]], because he considered its implications before Russell did; however, he did not view it as a [[paradox]], but merely a demonstration that the collection of all sets was an 'inconsistent multiplicity'.  (That is also the usual modern view, if we interpret Cantor\'s inconsistent multiplicities as [[proper classes]].)

The argument above is a little opaque, because it relies on Cantor\'s Theorem, which must be proved as well to complete the argument.  However, if one applies the ([[constructive mathematics|constructive]]!) proof of Cantor\'s Theorem to Cantor\'s Paradox and evaluates all of the definitions involved, then one ends up with the set $\{x \;|\; x \notin x\}$ of [[Russell's Paradox]]; this is precisely what Russell himself did to develop his paradox.  That paradox is now the usual go-to paradox to demonstrate the [[inconsistency]] of naïve set theory, especially [[material set theory]].

## Related concepts

* [[Russell's paradox]]

* [[Burali-Forti's paradox]]

category: paradox

[[!redirects Cantor paradox]]
[[!redirects Cantor Paradox]]
[[!redirects Cantor's paradox]]
[[!redirects Cantor's Paradox]]
[[!redirects Cantor's paradox]]
[[!redirects Cantor's Paradox]]
[[!redirects Cantor\'s paradox]]
[[!redirects Cantor\'s Paradox]]


Cantor's theorem

# Cantor\'s Theorem
* table of contents
{: toc}

## Summary

[[Georg Cantor]] proved many theorems, but the one usually called _Cantor\'s theorem_ is the first nontrivial theorem of Cantor\'s new [[set theory]]: that some infinities are bigger than others; in particular, any infinite [[cardinal number]] (or [[infinite set]]) generates a larger one by taking the [[power set]].

(The theorem applies to all sets, not just infinite ones, although it\'s fairly obvious for [[finite sets]].)

Cantor\'s theorem should not be confused with the Cantor--Schroeder--Bernstein theorem (see [[cardinal number]]), which is different (but related, as it is important to justify Cantor\'s interpretation of his theorem).

## History

Prior to Cantor, people tended to think of infinity (whether they believed in it or not) as an absolute concept: all infinities are equivalent.  It had been noticed (by [[Galileo Galilei|Galileo]], for example) that it\'s possible to give an [[infinite set]] a [[self-function|self]]-[[injection]] that is not a [[surjection]]; for example, the inclusion of the even integers into the integers by doubling.  Thus here are two infinities ---the infinity $E$ of even integers and the infinity $N$ of all integers--- that are actually equivalent, even though at first glance it would appear that $E$ is smaller.

Cantor showed that such an equivalence *fails* with the [[real numbers]] $R$: no map from $N$ to $R$ can be surjective (so that $R$ is [[uncountable set|uncountable]]).  His first argument was ad hoc, but he then generalised this with the _diagonal argument_ to show that no map from any set $S$ to its [[power set]] $\mathcal{P}S$ could be surjective.  (This covered the uncountability of $R$, since Cantor found a [[bijection]] between $R$ and $\mathcal{P}N$, which we can now regard as an instance of the [[Cantor–Schröder–Bernstein Theorem]].)  As there is an obvious injective map (the [[singleton]] map) from $S$ to $\mathcal{P}S$, Cantor concluded that the cardinality of the one is strictly smaller than the cardinality of the other.

Cantor\'s argument, like his whole set theory, was controversial at the time.  Those who, like [[Leopold Kronecker|Kronecker]], believed that all mathematics should be [[finite mathematics]], considered it meaningless.  Even more moderate early [[constructive mathematics|constructivists]], such as [[Jan Brouwer|Brouwer]] and [[Hermann Weyl|Weyl]], found that it had little point.  (In particular, it says nothing directly about $R$, since the bijection between $R$ and $\mathcal{P}N$ is not constructively valid, although Cantor\'s original proof that $R$ is uncountable can be made to work.)

However, the versions of the theorem stated below are constructively valid, and Theorem \ref{lawvere} is even [[predicative mathematics|predicatively]] valid (as long as one has [[function sets]]); modern constructivists generally accept these as theorems.  (The interpretation, however, is not so clear.)  They are in fact theorems in the [[internal logic|internal language]] of any [[topos]] (and Theorem \ref{lawvere} is a theorem in any $\Pi$-[[pretopos]]).

## Statements and proofs

+-- {: .num_theorem #lawvere}
###### Theorem
(Lawvere\'s version.)

Given sets $S$ and $V$, suppose that there is a [[surjection]] from $S$ to the [[function set]] $S \to V$ (also written $V^S$).  Then every map $n\colon V \to V$ has a [[fixed point]]; that is, $n(x) = x$ for some $x\colon V$.
=--

(This generalizes to _[[Lawvere's fixed point theorem]]_.)

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

Let $f\colon S \to (S \to V)$ be any function, and consider the function $g\colon S \to V$ given by
$$ g(a) = n(f(a)(a)) .$$
That is, if one uses [[currying]] to interpret $f$ as a function from $S \times S$ to $V$, then $g$ is defined using the [[diagonal map]] $\Delta_S$ as
$$ S \stackrel{\Delta_S}\to S \times S \stackrel{f}\to V \stackrel{n}\to V .$$

Now suppose that $f$ is surjective.  Then there must be some element $a\colon S$ such that $f(a) = g$.  But then
$$ g(a) = n(f(a)(a)) = n(g(a)) ,$$
so $g(a)$ is a fixed point of $n$.
=--

The presence of the diagonal map $\Delta_S$ here explains why this proof is called the _diagonal argument_.  (This explanation is anachronistic but morally correct.) Lawvere's proof also explains (in fact generalizes) the $Y$‑ or [[fixed-point combinator]] in [[untyped lambda-calculus]], where $Y(n)$ is a fixed-point for any term $n$.

It immediately follows (even constructively) that if $V$ has a [[self-mapping]] with no fixed point (such a map is sometimes called a *derangement*), then no map from $S$ to $S \to V$ can be a surjection.  In fact, we have something slightly stronger than (but classically equivalent to) the failure of $f$ to be a surjection: there actually exists an element $g$ of $S \to V$ that is not equal to any value in the range of $f$.  (If $V$ has an [[apartness relation]], then you can get an even stronger result for a correspondingly stronger hypothesis on $n$, but that doesn\'t apply to the versions below.)

+-- {: .num_theorem #cantor}
###### Theorem
(Cantor\'s version.)

Given a set $S$, there is no surjection from $S$ to the [[power set]] $\mathcal{P}S$.
=--

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

Let $V$ be the set of [[truth values]], and let $n\colon V \to V$ be [[negation]].  Since $n$ has no fixed point, apply Theorem \ref{lawvere}.
=--

Note that although negation doesn\'t have all of its usual properties in constructive mathematics, $p = \neg{p}$ is still impossible.

The next version is classically equivalent to the previous version (at least if you check that $\mathcal{P}S$ is [[inhabited set|inhabited]]), but not constructively equivalent.  (Indeed, unlike Theorem \ref{lawvere}, it apparently has no constructive analogue when $\mathcal{P}S$ is replaced by $V^S$.)  This argument is from Proposition 2.8.8 of Taylor\'s _[[Practical Foundations]]_ (although I don\'t know if it really originated there).

+-- {: .num_theorem #taylor}
###### Theorem
(Taylor\'s version.)

Given a set $S$, there is no [[injection]] from $\mathcal{P}S$ to $S$.
=--

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

Let $i\colon \mathcal{P}S \to S$ be any function.  Define $f\colon S \to \mathcal{P}S$ as follows:
$$ f(a) = \{ b\colon S \;|\; \forall (U\colon \mathcal{P}S),\; i(U) = a \;\Rightarrow\; b \in U \} .$$
If $i$ is an injection, then $f$ is a [[retraction]] of $i$ and hence a surjection, which is impossible by Theorem \ref{cantor}.
=--

Of course, Cantor also proved Theorem \ref{taylor}, but his proof was not constructive.

We can combine Theorems \ref{cantor} and \ref{taylor} into the following even more general statement, taken from D4.1.8 of Johnstone\'s _[[Elephant]]_.

+-- {: .num_theorem #johnstone}
###### Theorem
(Johnstone\'s version.)

Given a set $S$, its power set $\mathcal{P}S$ cannot be expressed as a [[subquotient]] of $S$.
=--

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

Suppose we had a set $B$, an injection $i\colon B \hookrightarrow S$ and a surjection $f\colon B \to \mathcal{P}S$.  Then the [[preimage]] function $i^*\colon\mathcal{P}S \to \mathcal{P}B$ would be a surjection (because $i^\ast \exists_i = 1_{\mathcal{P}B}$), as would the [[image]] function $\exists_f\colon \mathcal{P}B \to \mathcal{P}\mathcal{P}S$ (because $\exists_f f^\ast = 1_{\mathcal{P}\mathcal{P}S}$).  Thus their composite would be a surjection $\mathcal{P}S \to \mathcal{P}\mathcal{P}S$, which is impossible by Theorem \ref{cantor}.
=--

## Interpretation

Note that there exists an [[injection]] from $S$ to $\mathcal{P}S$, given by the [[singleton]] map.  So in the arithmetic of [[cardinal numbers]], we have
$$ {|S|} \leq {|\mathcal{P}S|} .$$
But as there is no such surjection, there certainly is no [[bijection]], and we have
$$ {|S|} \ne {|\mathcal{P}S|} .$$
So we conclude that
$$ {|S|} \lt {|\mathcal{P}S|} .$$
That is, each set is strictly smaller in cardinality than its power set.

This interpretation relies on a good relationship between $\leq$ and $=$ on the class of cardinal numbers; in particular, the Cantor--Schroeder--Bernstein theorem that $\leq$ is [[antisymmetric relation|antisymmetric]].  In [[constructive mathematics]], this relationship is lacking, and it is quite possible for two sets to each be strictly smaller than each other in the sense above.  Thanks to Theorem \ref{taylor}, this is not possible for a set and its power set, but it does mean that the interpretation of $\lt$ as relative size is problematic ---indeed almost as problematic as the idea that there are fewer even integers than integers!

[[Paul Taylor]] has argued that the essential value of Cantor\'s Theorem is the lemma, implicit in Cantor\'s proof, that [[Bill Lawvere]] isolated as Theorem \ref{lawvere}.  As the main interpretation of Theorem \ref{cantor} is meaningful only in a specific set-theoretic context (particularly, one where the Cantor--Schroeder--Bernstein theorem holds), it may not survive a 'revolution' that overthrows the primacy of that context.  But Lawvere\'s lemma will survive, since it 'does the work' while Cantor\'s theorem 'takes the credit'.  (See [Taylor 2009](#Taylor2009) below for further discussion of 'Lemmas that do the work and Theorems that take the credit'.)

## In posets

A theorem analogous to Cantor's theorem for sets can be formulated for other [[cartesian closed categories]]. In particular, one may ask whether it is possible to have a surjection $X \to 2^X$ between posets, where the base $2$ is not the discrete poset $\{0, 1\}$ but rather the order $\{0 \leq 1\}$.

The answer is there is no such surjection $f: X \to 2^X$, but this does not follow from a simple application of Lawvere's fixed-point theorem, where one tries to rule out such $f$ by invoking existence of a map $2 \to 2$ that has no [[fixed point]] (there is no such poset map $2 \to 2$!). Nor can we appeal to some crude cardinality argument; for example, if $X$ is the ordinal $\omega$, then $2^X$ is the order $\bot \sqcup \omega^{op}$ (freely adjoin a bottom element $\bot$ to $\omega^{op}$), which has the same cardinality. So some other proof must be sought.

Proving there is no surjection $X \to 2^X$ is an amusing exercise. One line of attack (which internalizes to any topos) may be found [here](https://ncatlab.org/toddtrimble/published/Cantor's+theorem+for+posets).

## Related concepts

* [[Lawvere's fixed point theorem]]

## References

*  Taylor 2009 June 20:  On the categories mailing list, Message-ID [E1MJ5TV-00066b-BI@mailserv.mta.ca](http://www.mta.ca/~cat-dist/archive/2009/09-6).
   {#Taylor2009}

*  On constructivenews, a [discussion](https://groups.google.com/group/constructivenews/browse_thread/thread/5f65c7ac479c15dd) about Theorem \ref{taylor} in the case where $S$ and $V$ are both the set of [[natural numbers]].  (Apparently this fails in certain [[realizability topos]]es.)

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