continuum hypothesis

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### Context
#### Foundations
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# The Continuum Hypothesis
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## Idea

> Cantor's continuum problem is simply the question: How many points are there on a straight line in Euclidean space? In other terms, the question is: How many different sets of integers do there exist?  K. Gödel ([1947](#Goedel47), p.515)

The **continuum hypothesis** is a famous problem of [[set theory]] concerning the cardinality of the [[Dedekind real numbers|Dedekind]] [[real numbers]] (the "[[continuum]]"). The hypothesis in its classical form goes back to [[Georg Cantor|G. Cantor]] and was on top of [[Hilbert's problems|Hilbert's list]] of open problems in mathematics in 1900.

In concise form the continuum hypothesis ($CH$) reads: $\quad 2^{\aleph_0}=\aleph _1\quad$; which roughly says that every [[subset]] of the Dedekind real numbers is either [[countable set|countable]] or has the same [[cardinality]] as the set of all Dedekind real numbers.

The *generalized continuum hypothesis* ($GCH$) states more generally: $\quad 2^{\aleph_k}=\aleph _{k+1}\quad$. (But see also Remark \ref{GCHinZF} below.)

The **independence of the continuum hypothesis** from the [[ZFC]] axioms of set theory has been established in landmark papers by [[Kurt Gödel|K. Gödel]] and [[Paul J. Cohen|P. J. Cohen]], the former proving the consistency of $ZFC+CH$ relative to $ZFC$ in 1938, and the latter proving the consistency of $ZFC+\neg CH$ relative to $ZFC$ in 1963. For fully [[formal proof]] see [Han 18](#Han18).

The broader implications of the independence results for set theory in general and $ZFC$ in particular are somewhat controversial. They are widely taken as a pointer towards the deficiency of $ZFC$ and the need for _new axioms_ of set theory. This position has been voiced famously in [Gödel (1947)](#Goedel47) from a [[platonism|platonist perspective]].

[[William Lawvere|W. Lawvere]] in [2003](#Lawvere03) interpreted Cantor's original point of view as saying that $CH$ holds for 'sufficiently structureless' sets and, accordingly, viewed Gödel's 1938 result as a proof of $CH$, whereas in [[Patrick Dehornoy|P. Dehornoy]]'s 2003 reinterpretation based on work of _Woodin_, $CH$ is actually conjectured to be false. [[Solomon Feferman|S. Feferman]] has argued more recently that $CH$ is essentially _mathematically indefinite_ and has made notions of 'indefiniteness' explicit that indeed enable to back this point of view with technical results (cf. [Feferman (2011)](#Feferman11)).  Similarly, [[Joel David Hamkins]] has argued (cf. [Hamkins 2012](#Hamkins12)) that under the [[set-theoretic multiverse|multiverse]] view of set theory, $CH$ is settled by our extensive knowledge of models of set theory satisfying both $CH$ and $\neg CH$ which all seem fully set-theoretic, so that no principle implying either one can possibly be considered 'obviously true'.

The attempt to give categorical accounts of the [[forcing]] methods introduced by Cohen provided a strong impetus in the development of (elementary) [[topos theory]] in the work of [[Peter Freyd|Freyd]], [[Miles Tierney|Tierney]], Lawvere and later Scedrov. The following exposition follows this categorical  approach.

## Statement

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

Let $E$ be an [[elementary topos]] with [[subobject classifier]] $\Omega$ and [[natural numbers object]] $N$.  The (external) **continuum hypothesis** in $E$ asserts that if there is a sequence of [[monomorphisms]]

$$N \hookrightarrow A\hookrightarrow \Omega^N$$

then either there exists a monomorphism $A \hookrightarrow N$ or a monomorphism $\Omega^N \hookrightarrow A$.
=--

When the topos is Boolean, so the [[Cantor-Schroeder-Bernstein theorem]] holds, this implies the existence of an isomorphism $A \simeq N$ or $\Omega^N \simeq A$. Note, however, that neither of the original monomorphisms is necessarily an isomorphism (one of them could be the successor map on $N$, for example).

In the classical case (that is, in the topos [[Set]] with the [[axiom of choice]]), this equivalently asserts that there is no strict inequality of [[cardinal numbers]]

$${|\mathbb{N}|} \lt \alpha\lt {|\Omega^\mathbb{N}|}$$

which it is more common to write as

$$ \aleph_0 \lt \alpha \lt 2^{\aleph_0} $$

In the formal proof outlined in ([Han–van Doorn 20](#HanvanDoorn20)), the statement is fomulated in [[Lean]] as:

$$\forall x, Ord(x)\Rightarrow x \leq \omega \vee P(\Omega) \leq x$$

where $x\leq y$ means $x$ is a [[subquotient]] of $y$, and $Ord(x)$ is a predicate that returns True if and only if $x$ is an [[ordinal]]. This says that no ordinal can be between $\omega$ and $P(\omega)$ in size.
In the presence of classical logic and the well-ordering principle, this then implies that no set can have cardinality between $\omega$ and $P(\omega)$.

In [[neutral constructive mathematics]], it is not provable that one has bijections $\mathbb{R} \cong L^\mathbb{N}$ between the [[real numbers]] and any set of sequences of elements of a [[trivial object|non-trivial]] [[distributive lattice|distributive sublattice]] of the [[lattice of truth values]] $L \subseteq \Omega$. In particular, it is not provable that there is a bijection $\mathbb{R} \cong 2^\mathbb{N}$ between the real numbers and [[Cantor space]] or a bijection $\mathbb{R} \cong \Omega^\mathbb{N}$ between the real numbers and the [[power set]] of the [[natural numbers]] (cf. [Bauer 2022](#Bauer22)). However, we do have $2^\mathbb{N} \subseteq \mathbb{R}$ and $2^\mathbb{N} \subseteq L^\mathbb{N}$. In addition, there are different notions of the real numbers used in [[constructive mathematics]], such as the [[Cauchy real numbers]], the [[HoTT book real numbers]], the [[Dedekind real numbers]], etc, which cannot in general be proven to coincide. Thus, there are actually different notions of the continuum hypothesis, depending upon whether one uses some notion of the [[real number|real]] [[continuum]] or some non-trivial distributive sublattice $L$ of the lattice of truth values:

* Given a notion of [[real number]] $\mathbb{R}$, the **(Cauchy/Dedekind/etc...) real continuum hypothesis** states that if there is a series of monomorphisms such that

$$N \hookrightarrow A\hookrightarrow \mathbb{R}$$

then either there exists a monomorphism $A \hookrightarrow N$ or a monomorphism $\mathbb{R} \hookrightarrow A$.

* Given a non-trivial distributive sublattice $L \subseteq \Omega$ of the [[lattice of truth values]], the **$L^\mathbb{N}$-continuum hypothesis** states that if there is a series of monomorphisms such that

$$N \hookrightarrow A\hookrightarrow L^\mathbb{N}$$

then either there exists a monomorphism $A \hookrightarrow N$ or a monomorphism $L^\mathbb{N} \hookrightarrow A$.

Note that for any notion of real number $\mathbb{R}$ it is impossible for $\mathbb{R}$ to be in bijection with the [[natural numbers]], since it is impossible for $\mathbb{R}$ to be both [[subcountable]] and [[countably indexed]]. Similarly, it is impossible for $2^\mathbb{N}$ to be in bijection with $\mathbb{N}$, since it is impossible for $2^\mathbb{N}$ to be both [[subcountable]] and [[countably indexed]].

## Unprovability

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###### Theorem

There exists a [[boolean topos]] in which the [[axiom of choice]] holds and the continuum hypothesis fails.

=--

One topos for which the theorem holds is called the *Cohen topos*; it is the topos of sheaves with respect to the [[dense topology]] (also called the $\neg\neg$-topology) on the Cohen [[poset]].  Thus, in this topos, there exist monomorphisms $\mathbb{N} \hookrightarrow A\hookrightarrow 2^{\mathbb{N}}$ but no monomorphism $A \hookrightarrow N$ or $\Omega^N \hookrightarrow A$.

The Cohen topos will be constructed from the topos [[Set]] of sets.  For this, recall that the subobject classifier of $Set$ is $2\coloneqq \{0,1\}$. The technique of constructing such a topos is called [[forcing]].

+-- {: .num_defn}
###### Definition
**(Cohen poset)**

Let $\mathbb{N}$ be the set of natural numbers; i.e. the [[natural numbers object]] in $Set$. Let $A$ be a set with strictly larger cardinality ${|A|}\gt {|2^\mathbb{N}|}$; e.g. $A\coloneqq 2^{2^{\mathbb{N}}}$ will do because of the [[diagonal argument]].  Then the *Cohen poset* $P$ is defined to be the set of morphisms

$$p:F_p\to 2$$

where $F_p\subseteq A\times \mathbb{N}$ is any [[finite set|finite]] subset.
The order relation on $P$ is defined by

$$q\le p\; iff\; F_q\supseteq F_p\;and\;q|_{F_p}=p$$

where the right-hand condition means that $q$ restricted to $F_p$ must coincide with $p$.
=--

We think of each element of $P$ as an approximation to the function $F:A\times\mathbb{N} \to 2$ that is the [[exponential|transpose]] of the putative monomorphism

$$f:A\to 2^\mathbb{N}$$

with "smaller" elements considered as better approximations. The very rough intuition is that $p\to q\to \dots$ (if $p\ge q \ge \dots$) forms a [[codirected diagram]] of monomorphisms with domains of increasing size whose colimit is $f$, and that by [[free cocompletion]] (i.e. forming (pre)sheaves) we obtain a topos in which this colimit exists.

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###### Lemma
The [[dense topology|dense]] [[Grothendieck topology]] on $P$ is [[subcanonical topology|subcanonical]].  In other words: For any $p\in P$ we have $y(p)=hom(-,p)\in\Sh(P,\neg\neg)$
=--

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###### Lemma
Let $k_{A\times\mathbb{N}}:\begin{cases}P\to Set \\ p \mapsto A\times\mathbb{N}\end{cases}$ denote the functor constant on $A\times\mathbb{N}$. Let

$$C:\begin{cases}
P\to Set
\\
p\mapsto \{(b,n)|p(b,n)=0\}\subseteq A\times \mathbb{N}
\end{cases}$$

Then we have ${\neg}{\neg} C=C$ in $Sub(k_{A\times\mathbb{N}})$; i.e. $C$ is a closed subobject with respect to the dense topology ${\neg}{\neg}$ in the [[algebra of subobjects]] of $k_{A\times\mathbb{N}}$.
=--

Let $\Omega$ denote the [[subobject classifier]] of $Psh(P)$. Let $\Omega_{{\neg}{\neg}}$ denote the subobject classifier of $Sh(P,\neg\neg)$. Recall that $\Omega_{{\neg}{\neg}}$ is given by the equalizer $\Omega_{{\neg}{\neg}}=eq(id_\Omega,{\neg}{\neg})$.

By the preceding lemma, the [[characteristic morphism]] $\chi_a$ of the subobject $a \colon C\hookrightarrow k_{A\times\mathbb{N}}=k_A\times\k_\mathbb{N}$ factors through some $f \colon k_{A\times\mathbb{N}}\to \Omega_{\neg\neg}$.

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###### Lemma
The adjoint $g:k_A\to \Omega_{{\neg}{\neg}}^{k_{\mathbb{N}}}$ of $f$ is a monomorphism.
=--

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###### Corollary
The associated-sheaf functor sends $g$ to a monomorphism in the Cohen topos.
=--

## Unrefutability

If $V$ is a model of [[ZF]], then the continuum hypothesis and the [[axiom of choice]] both hold in Gödel's [[constructible universe]] $L$ built from $V$. Actually, the GCH holds in $L$ as well.

+-- {: .num_remark #GCHinZF}
###### Remark
Regarding the statement of the *generalized* continuum hypothesis in *ZF* (not ZFC), one should distinguish various possibilities. One might leave the statement $2^{\aleph_n} = \aleph_{n+1}$ unchanged, so that the GCH becomes a statement just about ordinals or well-ordered sets. But then one could argue such a generalized continuum hypothesis is not as general or strong as it might be, since not all sets can be well-ordered using ZF alone. The more general statement would say that if there are monomorphisms $X \to Y$ and $Y \to P(X)$ for infinite sets $X$ and $Y$, then $Y$ is bijective with one of $X, P(X)$.

For example, Sierpiński proved that over ZF, the generalized continuum hypothesis implies AC. (See [[Hartogs number]].) For this result, he certainly used the stronger formulation.

In weakly predicative mathematics, power sets do not exist, so the stronger version of the continuum hypothesis for Dedekind real numbers would have to be rephrased in terms of the Sierpinski space $\mathbb{S}$ and function sets: if there are monomorphisms $X \to Y$ and $Y \to \mathbb{S}^X$ for infinite sets $X$ and $Y$, then $Y$ is bijective with one of $X, \mathbb{S}^X$, or equivalently the version involving natural numbers and function sets: if there are monomorphisms $X \to Y$ and $Y \to \mathbb{N}^X$ for infinite sets $X$ and $Y$, then $Y$ is bijective with one of $X, \mathbb{N}^X$.
=--

## Generalization: Easton's theorem
 {#EastonTheorem}

Just how flexible can the power [[operation]] $\kappa \mapsto 2^\kappa$ be? There are of course some constraints. Obvious ones are that $\kappa \lt 2^\kappa$ and $2^\kappa \leq 2^\lambda$ whenever $\kappa \leq \lambda$. A more refined one is a consequence of [[König's theorem]], namely that

* $\kappa \lt cof(2^\kappa)$

where the right side is the [[cofinality]] of $2^\kappa$.

A remarkable illustration of the power of the forcing method is [[Easton's theorem]], which says that as far as [[regular cardinals]] go, these are really the *only* constraints.

+-- {: .num_theorem}
###### Theorem
**([Easton 70](#Easton70))**

Suppose $\mathcal{M}$ is a [[model]] of [[ZFC]] in which the generalized continuum hypothesis (GCH) holds. Let $F$ be a [[partial function]] from the class of infinite [[regular cardinals]] to the class of [[cardinals]] such that

* $F$ is strictly increasing and preserves the order $\leq$;

* $\kappa$ is less than the cofinality of $F(\kappa)$ for all $\kappa \in dom(F)$.

Then there is a generic extension $\mathcal{M}[G]$ of $\mathcal{M}$ with the same cardinals and cofinalities, such that $\mathcal{M}[G] \models 2^\kappa = F(\kappa)$ for all $\kappa \in dom(F)$.
=--

On the other hand, the behavior of the power operation on [[regular cardinal|singular cardinals]] is not so unconstrained. For example, in a model of ZFC, the smallest cardinal for which GCH fails can never be singular. The so-called "[[pcf theory]]" (or "possible cofinalities theory"), due to [[Saharon Shelah]], gives some information on possible bounds for the power operation on singular cardinals (among other things).

## Links

* Stanford Encyclopedia of Philosophy, _[The Continuum Hypothesis](http://plato.stanford.edu/entries/continuum-hypothesis/)_

* MO _Solutions to the Continuum Hypothesis_ . ([link](http://mathoverflow.net/questions/23829/solutions-to-the-continuum-hypothesis))

## References

* [[John Bell|J. L. Bell]], _Set Theory - Boolean-Valued Models and Independence Proofs_ , Oxford Logic Guides **47** 3rd ed. Oxford UP 2005.

* [[Alonzo Church|A. Church]], _Paul J. Cohen and the Continuum Problem_, pp.15-20 in Proceedings ICM Moscow 1966. ([pdf](http://www.mathunion.org/ICM/ICM1966.1/Main/icm1966.1.0015.0020.ocr.pdf))

* [[Paul J. Cohen|P. J. Cohen]], _The independence of the continuum hypothesis I_, Proc. Nat. Acad. Sci. **50** (1963) pp.1143-1148. ([pdf](http://www.ncbi.nlm.nih.gov/pmc/articles/PMC221287/pdf/pnas00240-0135.pdf))

* [[Paul J. Cohen|P. J. Cohen]], _The independence of the continuum hypothesis II_, Proc. Nat. Acad. Sci. **51** (1963) pp.105-110. ([pdf](http://www.ncbi.nlm.nih.gov/pmc/articles/PMC300611/pdf/pnas00175-0117.pdf))

* [[Paul J. Cohen|P. J. Cohen]], _Set Theory and the Continuum Hypothesis_ , Benjamin New York 1966. (Dover reprint 2008)

* [[Patrick Dehornoy|P. Dehornoy]], _Progrès récents sur l'hypothèse du continu (d'après Woodin)_ , Séminaire Bourbaki exposé **915** (2003). ([English version](https://www.lmno.cnrs.fr/archives/dehornoy/Surveys/DgtUS.pdf))

* {#Feferman11}[[Solomon Feferman]], _The Continuum Hypothesis is neither a definite mathematical problem nor a definite logical problem_ , Harvard lectures 2011. ([pdf](http://math.stanford.edu/~feferman/papers/CH_is_Indefinite.pdf))

* {#Hamkins12} [[Joel David Hamkins]], *The set-theoretic multiverse*, Review of Symbolic Logic 5:416-449 (2012), [arxiv](https://arxiv.org/abs/1108.4223)

* M.C. Fitting, _Intuitionistic Logic, Model Theory and Forcing_, North-Holland Amsterdam 1969.

* {#Goedel47}[[K. Gödel]], _What is Cantor's continuum problem?_ , Am. Math. Monthly **54** no. 9 (1947) pp.515-25. ([pdf](http://www.personal.psu.edu/ecb5/Courses/M475W/Readings/Week06-Sep30-IntoTheTwentiethCentury/01-WhatisCantorsContinuumProblembyKurtGodel.pdf))

* {#Lawvere03} [[William Lawvere|F. W. Lawvere]], _Foundations and Applications: Axiomatization and Education_, Bulletin of Symbolic Logic **9** no.2 (2003) pp.213-224. ([ps-preprint](https://www.math.ucla.edu/~asl/bsl/0902/0902-006.ps))

* [[Saunders Mac Lane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_ , Springer Heidelberg 1994. (sections VI.2, VI.3)

* [[W. Hugh Woodin]], _The Continuum Hypothesis, Part I_ , Notices AMS **48** no.6 (2001) pp.567-576. ([pdf](http://www.ams.org/notices/200106/fea-woodin.pdf))

* [[W. Hugh Woodin]], _The Continuum Hypothesis, Part II_ , Notices AMS **48** no.7 (2001) pp.681-690. ([pdf](http://www.ams.org/notices/200107/fea-woodin.pdf))

* {#Easton70} W. Easton, _Powers of regular cardinals_, Ann. Math. Logic, 1 (2): 139–178, (1970) [doi:10.1016/0003-4843(70)90012-4](https://dx.doi.org/10.1016%2F0003-4843%2870%2990012-4)

A [[formal proof]] of the independence of the continuum hypothesis from [[ZFC]] is in

* {#Han18} [[Jesse Han]], _Flypitch project -- Formal proof of the independence of CH_ ([github:flypitch](https://github.com/flypitch), [pdf](https://github.com/flypitch/flypitch-notes/blob/master/forcing-notes.pdf))

published as

* {#HanvanDoorn20} [[Jesse Han|Jesse Michael Han]] and [[Floris van Doorn]],  _A Formal Proof of the Independence of the Continuum Hypothesis_, Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs (2020) pp 353–366, doi:[https://doi.org/10.1145/3372885.3373826](https://doi.org/10.1145/3372885.3373826), arXiv:[2102.02901](https://arxiv.org/abs/2102.02901).

For the failure of the continuum to be equivalent to the powerset of the natural numbers in [[constructive mathematics]], see:

* {#Bauer22} [[Andrej Bauer]], *The countable reals* ([video](https://www.youtube.com/watch?v=4CBFUojXoq4)).

[[!redirects CH]]
[[!redirects Continuum hypothesis]]
[[!redirects GCH]]
[[!redirects generalized continuum hypothesis]]

[[!redirects Easton's theorem]]
[[!redirects Easton theorem]]


continuum mechanics

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### Context
#### Physics
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#Contents#
* table of contents
{:toc}

## Idea

_Continuum mechanics_ refers to the [[physics]]/[[mechanics]] of continuous bodies such as notably [[fluids]] but also [[solids]] ([[solid state physics]]). As opposed to the mechanics of point [[particles]] (to which it is of course related by various limiting processes).

## Related concepts

* [[classical mechanics]], [[quantum mechanics]]

  * [[kinematics]], [[dynamics]]

  * [[rigid body dynamics]]

* [[Hamiltonian mechanics]], [[Lagrangian mechanics]]

* [[cohesion]], [[elasticity]], [[solidity]]

## References

### General

* Wikipedia, _[Continuum mechanics](http://en.wikipedia.org/wiki/Continuum_mechanics)_

* Gérard A Maugin, _Continuum Mechanics Through the Twentieth Century: A Concise Historical Perspective_, 2013

### Formalization

The [[axiom|axiomatization]] of [[continuum mechanics]] [[internal logic|internal]] to [[toposes]] has led [[William Lawvere]] to the introduction of [[synthetic differential geometry]]

* [[William Lawvere]], _Toposes of laws of motion_ , transcript of a talk in Montreal, Sept. 1997 ([pdf](http://www.acsu.buffalo.edu/~wlawvere/ToposMotion.pdf))

and to the notion of [[cohesion]]

* [[William Lawvere]], _[[Some Thoughts on the Future of Category Theory]]_ in A. Carboni, M. Pedicchio, G. Rosolini, _Category Theory_  , [[Como|Proceedings of the International Conference held in Como]], Lecture Notes in Mathematics 1488, Springer (1991)

* [[William Lawvere]], _Axiomatic cohesion_ Theory and Applications of Categories, Vol. 19, No. 3, 2007, pp. 41–49. ([pdf](http://www.tac.mta.ca/tac/volumes/19/3/19-03.pdf))

For more on this see also the collection

* [[William Lawvere]], [[Stephen Schanuel]] (eds.), _[[Categories in Continuum Physics]]_, Lectures given at a Workshop held at SUNY, Buffalo 1982, Lecture Notes in Mathematics 1174, 986

and

* [[William Lawvere]], _Variable sets, topoi and etendu_ Notices Amer. Math. Soc.
22 (1975) A675.

* W. Noll, _The foundations of mechanics and thermodynamics_ Springer, 1974.

### History

Early precursor of continuum mechanics has been argued to be recognizable in [[Descartes]]'s mechanism ([Dusek 99, p.23](#Dusek99)).

* {#Dusek99} V. Dusek, _The Holistic Inspirations of Physics_ NewBrunswick: Rutgers University Press (1999) ([review](http://www.jstor.org/stable/188634?seq=1))

[[!redirects continuum physics]]

[[!redirects classical continuum mechanics]]


contour integration > history

< [[contour integration]]


contractible > history

< [[contractible]]
[[!redirects contractible -- history]]


contractible chain complex

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###Context###
#### Homological algebra
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#Contents#
* table of contents
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## Definition

A [[chain complex]] $V$ is called _contractible_ if the unique [[terminal object|terminal]] [[morphism]] $V \to 0$ to the [[zero object|zero complex]] is a [[homotopy equivalence]], hence if the [[identity morphism]] on $V$ is [[null homotopy|null homotopic]]. One also says that $V$ is _null-homotopic_.

It is called _weakly contractible_ if $V \to 0$ is a [[quasi-isomorphism]].

## Examples

* Every [[mapping cone]] on an [[identity morphism|identity]] [[chain map]] is contractible.

* In [[quantization of Yang-Mills theory]] [[gauge fixing]] is typically implementd in [[BV-BRST formalism]] after introducing a contractible chain complex of [[field bundles]] for [[field (physics)|fields]] called the _[[Nakanishi-Lautrup field]]_ and the _[[antighost field]]_.

## Related concepts

* [[contractible space]]

* [[contractible type]]

[[!include homotopy n-types - table]]

[[!redirects contractible chain complexes]]

[[!redirects null homotopic chain complex]]
[[!redirects null homotopic chain complexes]]


contractible space

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#### Topology
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#### Higher category theory
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[[!include higher category theory - contents]]
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# Contents
* table of contents
{: toc}

## Definition

### For topological spaces

A [[topological space]] $X$ is **contractible** if the canonical map $X \to \ast$ is a [[homotopy equivalence]]. It is **weakly contractible** if this map is a [[weak homotopy equivalence]], hence if all [[homotopy groups]] of $X$ are trivial.

Where the [[Whitehead theorem]] does not apply, we may find examples of weakly contractible but not contractible spaces, such as the [double comb space](http://topospaces.subwiki.org/wiki/Double_comb_space) in [[Top]].

### For $\infty$-groupoids

Since the [[Whitehead theorem]] applies in [[∞Grpd]] (and generally in any [[hypercomplete (∞,1)-topos]]), being weakly equivalent to the point is the same as there being a contraction. So an [[∞-groupoid]] is **weakly contractible** if and only if it is **contractible**.

$$
  (C \;\text{is weakly contractible}) \Leftrightarrow
  (C \stackrel{\simeq}{\to} *)
  \,.
$$

In this context one tends to drop the "weakly" qualifier.

Sometimes one allows also the empty object $\emptyset$ to be contractible. To distinguish this, we say

* an $\infty$-groupoid is **(-1)-[[truncated]]** (is a [[(-1)-groupoid]]) if it is either empty or equivalent to the point;

* an $\infty$-groupoid is **(-2)-[[truncated]]** (is a [[(-2)-groupoid]]) if it is equivalent to the point.

### For cohesive $\infty$-groupoids

[[cohesive infinity-groupoid|Cohesive $\infty$-groupoids]] could be contractible in two different ways: topologically contractible in the first sense, or homotopically contractible in the second sense. A cohesive $\infty$-groupoid $S$ is **homotopically contractible** if its underlying $\infty$-groupoid $\Gamma(S)$ is contractible. A cohesive $\infty$-groupoid is **topologically contractible** if its [[fundamental infinity-groupoid]] $\Pi(S)$ is contractible. These two notions of contractibility are not equivalent to each other: in [[Euclidean-topological infinity-groupoids]] the [[unit interval]] is topologically contractible, but homotopically the unit interval is only [[0-truncated]].

## Related concepts

* [[locally contractible space]]

* [[contractible chain complex]]

* [[contractible type]]

[[!include homotopy n-types - table]]

[[!redirects contractible]]

[[!redirects contractible space]]
[[!redirects contractible spaces]]

[[!redirects weakly contractible space]]
[[!redirects weakly contractible spaces]]

[[!redirects contractible topological space]]
[[!redirects contractible topological spaces]]

[[!redirects weakly contractible topological space]]
[[!redirects weakly contractible topological spaces]]

[[!redirects contractible homotopy type]]
[[!redirects contractible homotopy types]]

[[!redirects weakly contractible homotopy type]]
[[!redirects weakly contractible homotopy types]]


contractible type

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#### Type theory
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# Contractible types
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## Idea

In [[homotopy type theory]], the notion of **contractible [[type]]** is an internalization of the notion of [[contractible space]] / [[(-2)-truncated]] object.

Contractible types are also called of **[[h-level]] $0$**. They represent the notion _[[true]]_ in homotopy type theory.

## Definition

We work in [[intensional type theory|intensional]] [[type theory]] with [[dependent sums]], [[dependent products]], and [[identity types]],

\begin{definition}
For $X$ a [[type]], let

$$
  isContr(X) \coloneqq \sum_{x\colon X} \prod_{y\colon X} (y=x)
$$

be the [[dependent sum]] in one [[variable]] $x : X$ over the [[dependent product]] on the other [[variable]] $y \colon X$ of the $x,y$-[[dependent type|dependent]] [[identity type]] $(x = y)$.

We say that $X$ is a **contractible type** if $isContr(X)$ is an [[inhabited type]].
\end{definition}

\begin{remark}
In [[propositions as types]] language, this can be pronounced as "there exists a point $x\colon X$ such that every other point $y\colon X$ is equal to $x$."

Under the [[homotopy theory|homotopy-theoretic]] interpretation, it should be thought of as the type of *contractions* of $X$ --- since the dependent product describes *continuous* functions, the paths from $y$ to $x$ depend continuously on $y$ and thus exhibit a contraction of $X$ to $x$.
\end{remark}

\begin{definition}
A provably [[equivalence in homotopy type theory|equivalent]] definition is
the [[product type]] of $X$ with the [[isProp]]-type of $X$:

$$
  isContr(X) \coloneqq X \;\times\; isProp(X)
  \,.
$$
\end{definition}

(Here of course we have to use a definition of [[isProp]] which doesn't refer to $isContr$).

\begin{remark}
This now says that $X$ is contractible iff $X$ is [[inhabited type|inhabited]] and an [[h-proposition]].
\end{remark}

There is a third definition of a contractible type, provably equivalent to the others.

\begin{definition}\label{SingletonInduction}
Let $(A, a:A)$ be a [[pointed type]]. $A$ satisfies **singleton induction** if for every [[type family]] $B$ over $A$ the dependent function
$$\mathrm{evpt}(a):\left(\prod_{x:A} B(x)\right) \to B(a)$$
has a [[section]]. A **contractible type** is a pointed type which satisfies singleton induction.
\end{definition}

### Rules for isContr

If the [[dependent type theory]] only has rules for [[identity types]] and [[dependent identity types]], one could define the isContr modality by the following rules:

Formation rules for isContr types:
$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{isContr}(A) \; \mathrm{type}}$$

Introduction rules for isContr types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a:A \quad \Gamma, x:A \vdash \tau(x):a =_A x}{\Gamma \vdash w(a, \lambda x.\tau(x)):\mathrm{isContr}(A)}$$

Elimination rules for isContr types:
$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, p:\mathrm{isContr}(A) \vdash \mathrm{pt}(p):A} \quad \frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, p:\mathrm{isContr}(A), x:A \vdash c(p, x):\mathrm{pt}(p) =_A x}$$

Computation rules for isContr types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a:A \quad \Gamma, x:A \vdash \tau(x):a =_A x}{\Gamma \vdash \beta_{\mathrm{isContr}(A)}^\mathrm{pt}(a):\mathrm{pt}(w(a, \lambda x.\tau(x))) =_A a}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash a:A \quad \Gamma, x:A \vdash \tau(x):a =_A x}{\Gamma, x:A \vdash \beta_{\mathrm{isContr}(A)}^w(x):c(w(a, \lambda x.\tau(x)))(x) =_{(-) =_A x}^{\beta_{\mathrm{isContr}(A)}^\mathrm{pt}(a)} \tau(x)}$$

Uniqueness rules for isContr types:
$$\frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma, p:\mathrm{isContr}(A) \vdash \eta_{\mathrm{isContr}(A)}(p):w(\mathrm{pt}(p), \lambda x.c(p, x)) =_{\mathrm{isContr}(A)} p}$$

## Properties

\begin{proposition}
For any type $A$, the type $isContr(A)$ is an [[h-proposition]].  In particular, we can show $isContr(A) \to isContr(isContr(A))$: if a type is contractible, then its space of contractions is also contractible.
\end{proposition}

\begin{proposition}
A type is contractible if and only if it is [[equivalence in homotopy type theory|equivalent]] to the [[unit type]].
\end{proposition}

### Relation to dependent sum types and equivalences

A family of elements $x:A \vdash f(x):B$ is an [[equivalence of types]] if its family of [[fiber types]] $y:B \vdash \sum_{x:A} f(x) =_B y$ is a family of [[contractible types]].

The [[dependent sum type]] of a family of contractible types $x:A \vdash B(x)$ with witnesses $x:A \vdash p(x):\mathrm{isContr}(B(x))$ is equivalent to the index type $A$ itself given by the first pair projection:

$$z:\sum_{x:A} B(x) \vdash \pi_1(z):A$$

$$x:A \vdash p(x):\mathrm{isContr}\left(\sum_{z:\sum_{x:A} B(x)} \pi_1(z) =_A x\right)$$

This is due to the fact that for any family of types $x:A \vdash B(x)$, there is an equivalence

$$x:A \vdash q(x):\left(\sum_{z:\sum_{x:A} B(x)} \pi_1(z) =_A x\right) \simeq B(x)$$

which means that if $B(x)$ is contractible, then
$$\sum_{z:\sum_{x:A} B(x)} \pi_1(z) =_A x$$
is contractible.

## Categorical semantics
 {#CategoricalSemantics}

We discuss the [[categorical semantics]] of contractible types.

Let $\mathcal{C}$ be a [[locally cartesian closed category]] with [[categorical semantics of homotopy type theory|sufficient structure]] to intepret all the above type theory.  This means that $\mathcal{C}$ has a [[weak factorization system]] with [[stable path objects]], and that [[acyclic cofibrations]] are preserved by pullback along fibrations between fibrant objects.  (We ignore questions of [[coherence]], which are not important for this discussion.)

In this [[categorical semantics]], the interpretation of a type $\vdash A : Type$ is a [[fibrant object]] $[\vdash A : Type]$, which for short we will just write $A$. The interpretation of the [[identity type]] $x,y : A \vdash (x = y) : Type$ is as the [[path space object]] $A^I \to A \times A$.
The interpretation of $isContr(A)$ is the object obtained by taking the [[dependent product]] of the [[path space object]] along one [[projection]] $p_2 : A\times A\to A$ and then forgetting the remaining morphism to $A$.

$$
  [isContr(A)]
  \;\;
  =
  \;\;
  \array{
    \prod_{p_2} A^I
    \\
    \downarrow
    \\
    A
    \\
    \downarrow
    \\
    *
  }
  \,.
$$

The interpretation $[\hat a : isContr(A)]$ of a [[term]] of $isContr(A)$ is precisely a morphism $\hat a : * \to \prod_{p_2} A^I$.

\begin{proposition}
Let $\mathcal{C}$ be a [[type-theoretic model category]]. Write $[isContr(A)]$ for the [[categorical semantics|interpretation]] of $isContr(A)$ in $\mathcal{C}$. Then:

Global points $* \to [isContr(A)]$ in $\mathcal{C}$ are in bijection with [[contraction]] [[right homotopies]] of the object $A$, hence to [[diagrams]] in $\mathcal{C}$ of the form

$$
  \array{
    A &\stackrel{\eta}{\to}& A^I
    \\
    & {}_{\mathllap{(id, const_a)}}\searrow & \downarrow
    \\
    && A \times A
  }
  \,,
$$

where $const_a$ is a morphism of the form $A \to * \stackrel{a}{\to} A$ and where $A^I$ is the [[path space object]] of $A$ in $\mathcal{C}$.
\end{proposition}

\begin{proof}
Given a global point $\hat a : * \to \prod_{p_2} A^I$, write $a : * \to A$ for the corresponding composite

$$
  \array{
     * &\stackrel{\hat a}{\to} & \prod_{p_2} A^I
     \\
     &{}_{\mathllap{a}}\searrow & \downarrow
     \\
     && A
  }
  \,.
$$

in $\mathcal{C}$. This is an element in the [[hom set]] $\mathcal{C}_{/A}(a, \prod_{p_2} A^I)$ of the [[slice category]] over $A$. By the ([[base change]] $\dashv$ [[dependent product]])-[[adjunction]] this is equivalently an element in $\mathcal{C}_{/A \times A}( p_2^* a, A^I )$.

Notice that the pullback $p_2^* a$ is the left morphism in

$$
  \array{
    A &\to& *
    \\
    {}^{\mathllap{(id,const_a)}}\downarrow && \downarrow^{\mathrlap{a}}
    \\
    A \times A &\stackrel{p_2}{\to}& A
  }
  \,.
$$

Therefore a morphism $p_2^* a \to  A^I$ in $\mathcal{C}_{/A \times A}$ is
equivalently in $\mathcal{C}$ a diagram of the form

$$
  \array{
    A &&\stackrel{\eta}{\to}&& A^I
    \\
    & {}_{\mathllap{(id,const_a)}}\searrow && \swarrow
    \\
    && A \times A
  }
  \,.
$$

This is by definition a [[contraction]] right [[homotopy]] of $A$.
\end{proof}

\begin{remark}
Thus if $isContr(A)$, then
$A\to 1$ is a (right) [[homotopy equivalence]], and hence (since $A$ is [[fibrant]]) an [[acyclic fibration]].

Conversely, if $\mathcal{C}$ is a [[model category]], $A$ and $1$ are cofibrant, and $A\to 1$ is an acyclic fibration, then $A\to 1$ is a right homotopy equivalence, and hence $isContr(A)$ has a global element.  Thus, in most cases, the existence of a global element of $isContr(A)$ (which is unique up to homotopy, since $isContr(A)$ is an [[h-proposition]]) is equivalent to $A\to 1$ being an acyclic fibration.

More generally, we may apply this locally.  Suppose that $A\to B$ is a fibration, which we can regard as a dependent type
$$x\colon B \vdash A(x)\colon Type.$$
Then we have a dependent type
$$x\colon B \vdash isContr(A(x))\colon Type$$
represented by a fibration $isContr(A)\to B$.  By applying the above argument in the [[slice category]] $\mathcal{C}/B$, we see that (if $\mathcal{C}$ is a model category, and $A$ and $B$ are cofibrant) $isContr(A)\to B$ has a [[section]] exactly when $A\to B$ is an acyclic fibration.

We can also construct the type
$$\prod_{x\colon B} isContr(A(x))$$
in global context, which has a global element precisely when $isContr(A)\to B$ has a section.  Thus, a global element of this type is also equivalent to $A\to B$ being an acyclic fibration.
\end{remark}

## Examples

* The [[unit type]] is a contractible type.

* The [[interval type]] is a contractible type.

* Every [[cone type]] is a contractible type, of which the unit type (cone type of the empty type) and the interval type (cone type of the unit type) are examples of cone types.

## Related concepts

[[!include homotopy n-types - table]]

* [[isEquiv]]

* **isContr**

* [[isProp]]

* [[uniqueness quantifier]]

* [[contractible space]], [[contractible chain complex]]

* [[empty type]], [[falsehood]]

* [[center of contraction]]

## References

The notion of contractible types (and with it the modern notion of [[equivalence in homotopy type theory]]) originates around:

* {#Voevodsky10} [[Vladimir Voevodsky]], p. 8, 10 of: *Univalent Foundations Project* (2010) [[pdf](http://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/univalent_foundations_project.pdf), [[Voevodsky-UFP2010.pdf:file]]]

Textbook account:

* {#UFP13} [[Univalent Foundations Project]], §3.11 in: *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]* (2013) [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)]

[[Coq]]-code for contractible types:

* [HoTT repository](https://github.com/HoTT/HoTT/blob/master/Coq/Contractible.v)

[[!redirects singleton induction]]

[[!redirects contractible type]]
[[!redirects contractible types]]
[[!redirects h-level 0]]
[[!redirects h-level 0 type]]
[[!redirects (-2)-truncated type]]
[[!redirects (-2)-truncated types]]
[[!redirects type of contractions]]
[[!redirects h-contractible type]]
[[!redirects isContr]]

[[!redirects singleton contractibility]]


contraction

The term *contraction* can mean:

* in [[metric space]] theory: see *[[short map]]*

* in [[fixed point]] theory and [[topological vector spaces]]: dilation with coefficient $\lambda\lt 1$

* in [[differential geometry]]/[[linear algebra]]: *[[tensor contraction]]* refers to the [[evaluation]] of [[tensors]] on each other (such as of [[differential forms]] on [[vector fields]]).

* in [[formal logic]] and [[type theory]]: the *[[contraction rule]]* is one of the [[structural rules]],

* [[homotopy theory]]: see at [[contractible homotopy type]]

  for the corresponding notion in

  [[homotopy type theory]] see: *[[center of contraction]]*

category: disambiguation

[[!redirects contraction]]
[[!redirects contractions]]


contraction rule

# The contraction rule
* table of contents
{: toc}

## Idea

In [[logic]], the _contraction rule_ states that any valid [[deduction]] that uses a [[premise]] more than once remains valid using that premise only once.  Along with the [[weakening rule]] and the [[exchange rule]], it is one of the most commonly adopted [[structural rules]].

The contraction rule is not used in all [[logical frameworks]]. For instance in [[linear logic]] it is discarded.

## Statements

Exactly how this looks depends on the [[logic]] used.

...

## Semantics

This rule corresponds to having [[monoidal category with diagonals|diagonals for the monoidal structure]] corresponding to the logical binary operator at hand.

## Related concepts

* [[inference rule]]

* [[structural rule]]

* [[weakening rule]]

* [[exchange rule]]

## References

The notion of contraction as a [[structural inference rule]] originates (under the German name *Zusammenziehung*) with:

* {#Gentzen35} [[Gerhard Gentzen]], §1.2.1 in: _Untersuchungen über das logische Schließen I_ _Mathematische Zeitschrift_ **39** 1 (1935) [[doi:10.1007/BF01201353](http://dx.doi.org/10.1007/BF01201353)]

* {#Gentzen69} [[Gerhard Gentzen]], §1.21 in: *Investigations into Logical Deduction*, in M. E. Szabo (ed.), *The Collected Papers of Gerhard Gentzen*, Studies in Logic and the Foundations of Mathematics **55**, Springer (1969) 68-131 [[ISBN:978-0-444-53419-4](https://www.sciencedirect.com/bookseries/studies-in-logic-and-the-foundations-of-mathematics/vol/55), [pdf](https://logic-teaching.github.io/prop/texts/Gentzen%201969%20-%20Investigations%20into%20Logical%20Deduction.pdf)]

On the [[categorical semantics]]:

* [[Bart Jacobs]], *Semantics of weakening and contraction*, Annals of Pure and Applied Logic **69** 1 (1994) 73-106 [doi:10.1016/0168-0072(94)90020-5]

Discussion in [[dependent type theory]]:

* {#Jacobs98} [[Bart Jacobs]], p. 122, 585 in: *Categorical Logic and Type Theory*, Studies in Logic and the Foundations of Mathematics **141**, Elsevier (1998)  [[ISBN:978-0-444-50170-7](https://www.sciencedirect.com/bookseries/studies-in-logic-and-the-foundations-of-mathematics/vol/141), [pdf](https://people.mpi-sws.org/~dreyer/courses/catlogic/jacobs.pdf)]

  > (emphasis on the [[categorical model of dependent types]])

* {#UFP13} [[Univalent Foundations Project]], §A of *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]* (2013) [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)]

  > (in the context of [[homotopy type theory]])

* {#Rijke18} [[Egbert Rijke]], *Dependent type theory* [[pdf](https://www.andrew.cmu.edu/user/erijke/hott/dtt.pdf)], Lecture 1 in: *[[Introduction to Homotopy Type Theory]]*, lecture notes, CMU (2018) [[pdf](http://www.andrew.cmu.edu/user/erijke/hott/hott_intro.pdf), [[Rijke-IntroductionHoTT-2018.pdf:file]], [webpage](https://www.andrew.cmu.edu/user/erijke/hott/)]

  > (in the context of [[homotopy type theory]])

[[!redirects contraction rule]]
[[!redirects contraction rules]]


contradiction

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

#Contents#
* table of contents
{:toc}

## Idea

Most of the time, a [[proposition]] $\phi$ is a _contradiction_ if assuming it there is a [[proof]] of [[false]], hence if the [[hypothetical judgement]]

$$
 \phi \vdash \bot
$$

is valid.

In [[paraconsistent logic]], however, one sometimes reserves the term "contradiction" for the situation when both a proposition and its negation hold.  This is often a contradiction in the above sense, but not always.

A system of [[formal logic]] that proves a contradiction is called [[inconsistency|inconsistent]].

## Examples

* [[Russell's paradox]]

* [[Burali-Forti's paradox]]

* [[Cantor's paradox]]

## Related concepts

* [[ex falso quodlibet]], [[empty type]]

* [[paradox]]

* [[inconsistency]]

[[!redirects contradictions]]


contramodule

Given a $k$-[[associative algebra|algebra]] $A$ and an $A$-[[coring]] $C$ one can introduce  right $C$-[[comodules]] $(M,\rho)$ as right $A$-[[modules]] $M$ equipped with a  [[coaction]] which is a right $A$-module map $\rho: M\to M\otimes_A C$ satisfying the standard axioms (comodules for [[coalgebras]] were introduced by Cartier in the 1950s). There is a rather more recent dualization of the concept. A __$C$-contramodule__ is a right $A$-module $M$ equipped with a right $A$-module map $\alpha: Hom_A(C,M)\to M$ such that the diagrams

$$\array{
Hom_A(C,Hom_A(C,M)) &&\stackrel{Hom_A(C,\alpha)}\to && Hom_A(C,M)\\
\downarrow \cong &&&&\downarrow \alpha\\
Hom_A(C\otimes_A C,M)&\stackrel{Hom_A(\Delta,M)}\to&Hom_A(C,M)&\stackrel{\alpha}\to& M\\}$$

$$\array{
Hom_A(A,M) &&\stackrel{Hom_A(\epsilon,M)}\to && Hom_A(C,M)\\
&\searrow \cong &&\swarrow \alpha &\\
&&M&& \\}$$

commute.

Historically contramodules seem to have first been mentioned a few times in the Eilenberg--Moore treatments of [[homological algebra]] in the 1960s and by category theorists (Vazquez, Garcia, Barr). Positselski has used them in his approach to [[semiinfinite cohomology]].

* Leonid Positselski, _Homological algebra of semimodules and semicontramodules. Semi-infinite homological algebra of associative algebraic structures_, [arxiv/0708.3398](http://arxiv.org/abs/0708.3398);
_Contramodules_, [arxiv/1503.00991](http://arxiv.org/abs/1503.00991); _Contraherent cosheaves_, [arxiv/1209.2995](http://arxiv.org/abs/1209.2995)

* [[T. Brzeziński]], _Contramodules_, [pdf](http://www.irb.hr/korisnici/zskoda/BrzezinskiSplitSlides.pdf), slides at International Conference on "Categories in Geometry", Split, September 24-28, 2007.

* [[T. Brzeziński]], _Hopf-cyclic homology with contramodule coefficients_, To appear in Quantum Groups and Noncommutative Spaces, M Marcolli and D Parashar (eds) Vieweg Verlag (Max-Planck Series), Preprint 2008, [arxiv:0806.0389](http://arxiv.org/abs/0806.0389)

[[!redirects contramodule]]
[[!redirects contramodules]]


contrapositive

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#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
=--
=--

In [[propositional logic]], the __contrapositive rule__ states that it is valid to derive $\neg{Q} \to \neg{P}$ from $P \to Q$ (where $\neg$ is [[negation]] and $\to$ is [[implication]]).  In symbols:
$$ \frac {P \to Q} {\neg{Q} \to \neg{P}} \;CP $$
The combination of this rule, followed by modus ponens (the [[elimination rule]] for implication) was traditionally called _modus tollens_:
$$ \frac {\displaystyle\frac{P \to Q} {\neg{Q} \to \neg{P}} \;CP \;\;\; \neg{Q}} {\neg{P}} \; \to\mathcal{E} $$

The contrapositive rule is valid in [[intuitionistic logic]], not just in [[classical logic]]; however, the reverse rule is valid only in classical logic.

Another intuitionistically valid rule, this one reversible, is
$$ \frac {P \to \neg{Q}} {Q \to \neg{P}} $$
as both statements are equivalent to the negation of $P \wedge Q$ (where $\wedge$ is [[conjunction]]).  However, the superficially similar
$$ \frac {\neg{P} \to Q} {\neg{Q} \to P} $$
is again valid only classically.

[[!redirects contrapositive]]
[[!redirects contrapositives]]

[[!redirects contrapositive rule]]
[[!redirects contrapositive rules]]

[[!redirects contraposition]]
[[!redirects contrapositions]]


contravariant 2-functor

A [[2-functor]] can be [[contravariant functor|contravariant]] in three ways, depending on whether it reverses the [[1-morphisms]], the [[2-morphisms]], or both; this corresponds to the three kinds of [[opposite 2-category]].  All kinds of contravariant 2-functor appear in a [[3-category with contravariance]].


contravariant functor

# Contravariant functors
* table of contents
{: toc}

## Idea

A contravariant functor is like a [[functor]] but it reverses the directions of the [[morphisms]].  (Between [[groupoids]], contravariant functors are essentially the same as functors.)

## Between categories

A **contravariant functor** $F$ from a [[category]] $C$ to a category $D$ is simply a [[functor]] from the [[opposite category]] $C^op$ to $D$.

To emphasize that one means a functor $C \to D$ as stated and _not_ as a functor $C^{op} \to D$ one sometimes says **covariant functor** when referring to non-contravariant functors, for emphasis.

Equivalently, a contravariant functor from $C$ to $D$ may be thought of as a functor from $C$ to $D^op$, but the version above generalises better to functors of many variables.

Also notice that while the objects of the [[functor category]] $[C^{op}, D]$ are in canonical bijection with those in the functor category $[C, D^{op}]$ (both are contravariant functors from $C$ to $D$), the morphisms in the two functor categories are in general different, as

$$
  [C^{op}, D] \simeq [C, D^{op}]^{op}
  \,.
$$

This matters when discussing a [[natural transformation]] from one contravariant functor to another.

## Between higher categories

Since [[n-categories]] (and also [[(infinity,n)-categories]]) have $2^n$ different kinds of [[opposite category]] depending on which of the $k$-morphisms are reversed for $1\le k\le n$ (see for instance [[opposite 2-category]]), they also have $2^n$ different kinds of "contravariant functor".

## Abstractly

Categories, covariant functors, and [[natural transformations]] form a [[2-category]] [[Cat]].  To include the contravariant functors as well, we can equip $Cat$ with a [[duality involution]], or we can generalize to a [[2-category with contravariance]], or some more general structure that also includes [[extranatural transformations]] or [[dinatural transformations]].  There could also be higher-categorical versions, such as a [[3-category with contravariance]].

## Related concepts

* [[presheaf]]

[[!include notions of pullback -- contents]]

## References

Textbook accounts:

* [[Saunders MacLane]], §II.2 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

* [[Francis Borceux]], Section 1.4 in: *[[Handbook of Categorical Algebra]]* Vol. 1: *Basic Category Theory* [[doi:10.1017/CBO9780511525858](https://doi.org/10.1017/CBO9780511525858)]

[[!redirects contravariant functor]]
[[!redirects contravariant functors]]


contravariant type family

+-- {: .rightHandSide}
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### Context
#### Directed homotopy type theory
+-- {: .hide}
[[!include directed homotopy type theory - contents]]
=--
=--
=--

\tableofcontents

## Definition

In [[simplicial homotopy type theory]], given a [[type]] $A$ and a [[type family]]
$$x:A \vdash B(x) \; \mathrm{type}$$
let us define the type family
$$x:A, y:A, f:\mathrm{hom}_A(x, y), u:B(x), v:B(y) \vdash \mathrm{hom}_{B(f)}(u, v) \; \mathrm{type}$$
as
$$\mathrm{hom}_{B(f)}(u, v) \coloneqq \left\langle\prod_{t:\mathbb{2}} C(f(t)) \vert_{[u,v]}^{\partial \Delta^1} \right\rangle$$
where $\mathbb{2}$ is the directed interval primitive, $\Delta^1$ is the $1$-simplex probe shape, and $\partial \Delta^1$ is its boundary.

Then a **contravariant type family** over $A$ is a [[type family]] $x:A \vdash B(x) \; \mathrm{type}$ such that given elements $x:A$, $y:A$, $f:\mathrm{hom}_A(x, y)$, and $v:B(y)$, the type
$$\sum_{u:B(x)} \mathrm{hom}_{B(f)}(u, v)$$
is a [[contractible type]].

## Related concepts

* [[hom type]]

* [[covariant type family]]

## References

* {#RiehlShulman17} [[Emily Riehl]], [[Michael Shulman]], *A type theory for synthetic $\infty$-categories* $[$[arXiv:1705.07442](https://arxiv.org/abs/1705.07442)$]$


Contributors > history


contributors to algebraic geometry

Disclaimer: This page is created mainly to have more navigability within the $n$lab, and is not meant to be a balanced list of historically important contributors (though some attention/effort to live partly to this aspect is made). Naturally, we list more people whose work is closer to foundational, categorical and cohomological study of algebraic varieties and generalizations and pay less attention to the classical themes. We will also list some algebraic geometers of less importance, but who wrote major historical texts quoted and playing role in the lab.

* [[Niels Henrik Abel]], [[Abhyankar|Shreeram Shankar Abhyankar]], [[Yves André]], [[Suren Arakelov]] ([[Arakelov geometry]]), [[Donu Arapura]], [[Michael Artin]]

* Arnaud Beauville, [[Kai Behrend]], [[Alexander Beilinson]], [[Joseph Bernstein]], [[Pierre Berthelot]], [[Roman Bezrukavnikov]], [[Bhargav Bhatt]], [[Spencer Bloch]], [[Fedor Bogomolov]], [[Enrico Bombieri]], [[Armand Borel]], [[Alexei Bondal]], [[Paul Bressler]]

* [[Pierre Cartier]], [[Guido Castelnuovo]], [[Denis-Charles Cisinski]], Herbert Clemens

* [[Pierre Deligne]], [[Christopher Deninger]], [[Duiliu-Emanuel Diaconescu]], Ron Donagi, [[Vladimir Drinfeld]], [[Nikolai Durov]]

* [[Federigo Enriques]], [[Hélène Esnault]]

* [[Gerd Faltings]], [[Edward Frenkel]], [[William Fulton]]

* [[Ofer Gabber]], [[Dennis Gaitsgory]], [[Victor Ginzburg]], [[Alexander Goncharov]], [[Phillip Griffiths]], [[Alexandre Grothendieck]]

* [[Günter Harder]], [[Joseph Harris]], [[Robin Hartshorne]], [[David Hilbert]], [[Heisuke Hironaka]], [[André Hirschowitz]], [[Friedrich Hirzebruch]], [[Nigel Hitchin]], [[William Hodge]], [[Daniel Huybrechts]]

* [[Luc Illusie]]

* Uwe Jannsen, [[Jacobi|Charles Gustave Jacob Jacobi]], Aise Johan de Jong, [[Dominic Joyce]]

* [[Dmitry Kaledin]], [[Mikhail Kapranov]], [[Anton Kapustin]], [[Masaki Kashiwara]], [[Kazuya Kato]], [[Ludmil Katzarkov]], [[Yujiro Kawamata]], [[Kiran Kedlaya]], [[Bernhard Keller]], [[Alexander Kuznetsov]], [[Felix Klein]],[[Steven Kleiman]], [[David Knudson]], [[Kunihiko Kodaira]], Janos Kollár, [[Maxim Kontsevich]]

* [[Laurent Lafforgue]], [[Serge Lang]], [[Robert Langlands]], Yves Laszlo, [[Gérard Laumon]], [[Solomon Lefschetz]], [[Marc Levine]], [[Joseph Lipman]],[[Valery Lunts]], [[Jacob Lurie]]

* [[Robert Macpherson]], [[Yuri Manin]], [[Barry Mazur]], [[James Milne]], [[Takuro Mochizuki]], [[Shinichi Mochizuki]], [[Boris Moishezon]], [[Joseph Mordell]], [[John Morgan]], [[Shigefumi Mori]], [[Fabien Morel]], [[Shigeru Mukai]], [[David Mumford]]

* [[Hiraku Nakajima]], [[Mudumbai Seshachalu Narasimhan]], [[Amnon Neeman]], [[Madhav Nori]]

* [[Arthur Ogus]], [[Andrei Okounkov]], [[Martin Olsson]], [[Dmitri Orlov]]

* [[Rahul Pandharipande]], [[Tony Pantev]], [[Alexander Polishchuk]]

* [[Daniel Quillen]]

* [[Michael Rapoport]], [[Bernhard Riemann]], [[Alexander Rosenberg]], [[Markus Rost]]

* [[Mikio Sato]], [[Corrado Segre]], [[Jean-Pierre Serre]], [[Francesco Severi]] [wikipedia](http://it.wikipedia.org/wiki/Francesco_Severi), [[Peter Scholze]], [[Igor Shafarevich]], [[Gorō Shimura]], [[Vyacheslav Shokurov]] (Vyacheslav Vladimirovich), [[Carlos Simpson]], [[Christophe Soulé]], [[Andrei Suslin]]

* [[Yutaka Taniyama]], [[Richard Taylor]], [[Bertrand Toen]], [[Robert Thomason]], Yuri Tschinkel, [[Boris Tsygan]]

* [[Michel van den Bergh]], [[Gabriele Vezzosi]], [[Angelo Vistoli]], Claire Voisin, [[Vladimir Voevodsky]]

* [[André Weil]], [[Andrew Wiles]], [[Mariusz Wodzicki]], [[Edward Witten]]

* [[Shing-Tung Yau]]

* [[Oscar Zariski]], [[Eric Zaslow]]


control theory

* Michel Fliess, Jean Lévine, Philippe Martin, Pierre Rouchon, _Flatness and defect of non-linear systems: introductory theory and examples_, Int. J. of control 61:6,
1327-1361 [doi](http://dx.doi.org/10.1080/00207179508921959) RGate: [pdf](https://www.researchgate.net/profile/Philippe_Martin/publication/292446786_Flatness_and_defect_of_nonlinear_systems_Introductory_theory_and_applications/links/56b1cd1908ae5ec4ed49eea5.pdf)
* E. Brion, N. Akulin et al., _Non-holonomic control I_, Proc. SPIE 5833, Quantum Informatics 2004, 62 (July 28, 2005); [doi](http://dx.doi.org/10.1117/12.620307)
* Jan Willem Polderman, Jan C. Willems, _Introduction to the mathematical theory of systems and control_

Quantum control

* Clément Sayrin, Igor Dotsenko et al. _Real-time quantum feedback prepares and stabilizes photon number states_, Nature 477, Numéro 7362, pages 73-77 [arXiv:1107.4027](http://arxiv.org/pdf/1107.4027)


controlled quantum gate

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Computation
+-- {: .hide}
[[!include constructivism - contents]]
=--
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

##  Idea

In [[quantum information theory]], a *controlled* [[quantum gate]] is a quantum gate whose operation on a given [[space of states]] is conditioned on the data in another space of "control states".

### The conceptual problem

The analogous situation for classical [[logic gates]] is without subtlety: Here the parameterization of a [[logic]] gate on $(n_{in}, n_{out})$-[[bits]] by $n_{ctrl}$ control bits is a choice of function

$$
  Bool^{n_{ctrl}}
  \xrightarrow{\;\;\; F_{\bullet} \;\;\;}
  Map
  \big(
    Bool^{n_{in}}
    ,\,
    Bool^{n_{out}}
  \big)
$$

from the intended set of controls to the [[set of functions]] between the intended input/output; and by the [[internal hom]]-[[adjoint functor|adjunction]] in the [[cartesian monoidal category]] of [[Sets]]  this is in [bijective correspondence ](adjoint+functor#InTermsOfHomIsomorphism) with a single function out of the [[Cartesian product]] set of the control bits with the input bits:
$$
  \array{
    Bool^{ n_{ctrl} + n_{in} }
    &\simeq&
    Bool^{ n_{ctrl} }
    \times
    Bool^{ n_{in} }
    &\xrightarrow{\;\;\;}&
    Bool^{ n_{out} }
    \\
    &&
    ( b_{ctrl}, b_in )
    &\mapsto&
    F_{b_{ctrl}}\big( b_{in} \big)
  }
  \,.
$$

This function would/could be called the "controlled logic gate". Notice that the control bits in the above function are "discarded after use", in that they cannot in general be recovered from the output of the above function.

The subtleties with generalizing this situation to [[quantum logic gates]] are:

1. The control of quantum gates may be either by quantum data or by classical data, which is not quite the same.

   \linebreak

   (In a *quantum-controlled* quantum logic gate, the control itself may be in a [[quantum superposition]], while a *classically-controlled* quantum gate is effectively an [[indexed set]] of quantum gates, indexed by classical data.)

1. A quantum-controlled quantum gate should, like any pure [[quantum logic gate]], be [[unitary]], in particular [[invertible]], hence it "must not discard" its control qbits.

   \linebreak

   This *rules out* the otherwise evident non-cartesian analog of the above classical situation (with the product operation replaced by the non-cartesian [[tensor product]] of [[finite dimensional vector space|finite-dimensional]] [[Hilbert spaces]], typically, and hence with the [[internal hom]] now being the corresponding linear space of linear maps)):

   $$
     \frac{
       QBit^{n_{ctrl}}
       \xrightarrow{\;\;\; F_\bullet \;\;\;}
       Map
       \big(
         QBit^{ n }
         ,\,
         QBit^{ n }
       \big)
     }{
       QBit^{n_{ctrl}}
       \otimes
       QBit^{n}
       \xrightarrow{\phantom{----}}
       QBit^{ n }
     }
   $$

   While the linear map at the bottom always exists, it cannot be invertible (unless the control is trivial, with $n_{ctrl} = 0$, or the state space is trivial, with $n = 0$) and hence does not qualify as a [[quantum logic gate]].

### The traditional notion
 {#IdeaTraditionalNotion}

A general and precise definition of controlled quantum gates is hard to find in traditional literature (cf. [Gurevich & Blass (2021) §1](deferred+measurement+principle#GurevichBlass21)).

What traditional texbooks state (e.g. [Nielsen & Chuang (2000) §4.3](#NielsenChuang00)) is that for
$$
  G \;\colon\; \mathscr{K} \longrightarrow \mathscr{K}
$$
a given [[quantum logic gate]], its version controlled by a single [[qbit]] is its [[direct sum]] with the [[identity map]], which may be thought of as the block-[[diagonal matrix|diagonal]]  [[matrix]]-notation with diagonal blocks being $G$ and the [[identity function]]:

$$
  \underset{
    QBit
  }{
    \underbrace{
      (\mathbb{C} \oplus \mathbb{C})
    }
  }
  \otimes
  \mathscr{K}
  \;\simeq\;
  \mathscr{K} \oplus \mathscr{K}
  \xrightarrow{
    \;
    id \oplus G
    \,=\,
    \left[
    \begin{array}{cc}
      id & 0
      \\
      0 & G
    \end{array}
    \right]
    \;
  }
  \mathscr{K} \oplus \mathscr{K}
  \;\simeq\;
  \underset{
    QBit
  }{
    \underbrace{
      (\mathbb{C} \oplus \mathbb{C})
    }
  }
  \otimes
  \mathscr{K}
$$

If we understand, as usual, that

$$
  QBit
  \;\;\simeq\;\;
  \mathbb{C}\cdot \vert 0 \rangle
  \;\oplus\;
  \mathbb{C}\cdot \vert 1 \rangle
$$

then this gives a (invertible!) quantum gate on the tensor product space $QBit \otimes \mathscr{K}$, with the property that

1. $\vert 0 \rangle \,\otimes\, \vert \psi \rangle \;\;\mapsto\;\; \vert 0 \rangle \,\otimes\, \phantom{G}\vert \psi \rangle$

1. $\vert 1 \rangle \,\otimes\, \vert \psi \rangle \;\;\mapsto\;\; \vert 0 \rangle \,\otimes\, G\vert \psi \rangle$

In words: "*If the control Qbit is definitely set, then we operate with $G$, if it is definitely not set then we operate trivially, and in general we operate with the respective superposition of these two actions.*"

A basic example of this construction is the [[controlled quantum NOT gate]].

## Definition
 {#Definition}

The following is a more general definition along the lines of *[[quantum circuits via dependent linear types]]*.

Given a ([[finite set|finite]]) [[set]] $W$ of "control parameter values" (typically $W =$ [[Bool]] = $\{0, 1\}$), a $W$-[[indexed set]] of [[quantum logic gates]] is a [[morphism]]

$$
  \begin{array}{rccc}
    &
    \mathscr{H}_\bullet
    &
    \xrightarrow{\;\;\; G_\bullet \;\;\;}
    &
    \mathscr{H}_\bullet
    \\
    w \colon W \;\; \vdash \;
    &
    \mathscr{H}_w
    &
    \xrightarrow{\;\;\; G_w \;\;\;}
    &
    \mathscr{H}_w
  \end{array}
$$

in a [[category]] $LinType_B$ of $B$-[[dependent linear types]] (typically: [[complex vector bundles]] over $W$), which is [[fiber]]-wise a [[linear map]] $G_w$ with given desired properties ([[invertible]], [[unitary]], ...).

As such, we may call this the **classically controlled quantum gate**, whose classical control parameter is $w \in W$ and whose quantum gate action for a given such parameter is $G_w$.

\begin{imagefromfile}
    "file_name": "ControlledQuantumGateTyping-230808.jpg",
    "width": "750",
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

Here in the bottom left we assume that the state spaces  themselves are independent of $w \colon W$, hence $\mathscr{H}_\bullet \,=\, \mathscr{H}$, which is typically the case in application.

## Examples

### Controlled NOT gate
 {#ExampleCNOT}

The following shows the [[CNOT gate]] in its two incarnations as a classically or a quantumly controlled quantum gate:

\begin{imagefromfile}
    "file_name": "CNOTgate-typing-230808.jpg",
    "width": "800",
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

### Quantum teleportation

Also the correction-operation in the [[quantum teleportation protocol]] ("Bob's operation") is given by classically-controlled quantum gates.

## Properties

* The *[[deferred measurement principle]]* interchanges classically-controlled with quantumly-controlled quantum gates.

## References
 {#References}

The symbolic notation for controlled quantum gates originates with:

* [[Richard Feynman]], *Quantum mechanical computers*, Foundations of Physics **16** (1986) 507–531 [[doi:10.1007/BF01886518](https://doi.org/10.1007/BF01886518)]

* {#BBCDMSSSW95} [[Adriano Barenco]], [[Charles H. Bennett]], [[Richard Cleve]], [[David P. DiVincenzo]], [[Norman Margolus]], [[Peter Shor]], [[Tycho Sleator]], [[John A. Smolin]], [[Harald Weinfurter]], *Elementary gates for quantum computation*, Phys. Rev. A**52** (1995) 3457 [[arXiv:quant-ph/9503016](https://arxiv.org/abs/quant-ph/9503016), [doi:10.1103/PhysRevA.52.3457](https://doi.org/10.1103/PhysRevA.52.3457)]

Traditional textbook accounts of controlled quantum gates:

* {#NielsenChuang00} [[Michael A. Nielsen]], [[Isaac L. Chuang]], §4.3 *Quantum computation and quantum information*, Cambridge University Press (2000) [[doi:10.1017/CBO9780511976667](https://doi.org/10.1017/CBO9780511976667), [pdf](http://csis.pace.edu/~ctappert/cs837-19spring/QC-textbook.pdf), [[NielsenChuangQuantumComputation.pdf:file]]]

See also at:

* [quantum computation -- References -- Classically controlled quantum computing](quantum+computation#ReferencesClassicallyControlledQuantumComputing)

Discussion in terms of [[quantum information theory via dagger-compact categories]] (cf. at *[[quantum reader monad]]*):

* {#HeunenVicary19} [[Chris Heunen]], [[Jamie Vicary]], around Lem.  5.61 in: *Categories for Quantum Theory*, Oxford University Press 2019 $[$[ISBN:9780198739616](https://global.oup.com/academic/product/categories-for-quantum-theory-9780198739616)$]$

The above discussion and graphics follows:

* {#CQTS22} [[CQTS]], *[Quantum Data Types via Linear HoTT](https://ncatlab.org/schreiber/show/QDataInLHoTT#QTML2022)* (Nov 2022)

[[!redirects controlled quantum gates]]

[[!redirects controlled quantum logic gate]]
[[!redirects controlled quantum logic gates]]

[[!redirects classically controlled quantum gate]]
[[!redirects classically controlled quantum gates]]

[[!redirects classically controlled quantum logic gate]]
[[!redirects classically controlled quantum logic gates]]

[[!redirects classically controlled quantum computing]]
[[!redirects parameterized quantum circuits]]


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

# Contents
* table of contents
{: toc}

## Idea

The term [[convenient category of topological spaces]] is used (e.g. [Steenrod 1967](#Steenrod67)) for a [[category]] of [[topological spaces]] [[nice topological space|nice]] enough to address many of the needs of [[topology|working topologists]], notably including the condition of being a [[cartesian closed category]]. As such, they are examples of [[nice category of spaces|nice categories of spaces]].

A primary example is the category of [[compactly generated spaces]].

## Definition

While the authors of this article don't know whether there exists in the literature a widely accepted definition of "convenient category of topological spaces", we propose the following definition as reasonable and convenient (see also the discussion [below](#NvC) on the distinction between "nice" and "convenient"):

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

A **convenient category of topological spaces** is a [[full subcategory|full]] [[replete subcategory]] $C$ of the category of all topological spaces [[Top]] such that the following conditions 1-3 hold:

1. Every [[CW complex]] is an object of $C$;

2. $C$ is [[cartesian closed]];

3. $C$ is [[complete category|complete]] and [[cocomplete category|cocomplete]].
=--

Frequently it is also felt desirable to add closure under certain types of [[subspaces]]. For instance, in the well-known examples one has

* $C$ is closed under [[closed subspaces]] in [[Top]], i.e., if $X$ belongs to $C$ and $A \subseteq X$ is a closed subspace (in $Top$), then $A$ also belongs to $C$.

At times one might hope that $C$ is closed under [[open subspaces]] as well, but this does not hold for all objects in some of the well-known examples of convenient categories.

It may be well to note that [[colimits]] and [[limits]] in $C$ need not agree with the corresponding colimits and limits in $Top$, except under certain conditions.  Some convenient categories are [[reflective subcategory|reflective]] or [[coreflective subcategory|coreflective]] in $Top$, in which case they are closed under limits or colimits respectively.  Moreover, because $C$ is a full subcategory of $Top$ which contains all CW complexes, the usual sorts of colimits used to present CW complexes are the same whether interpreted in $Top$ or in $C$.  Also, if $C$ is closed under closed subspaces, then an [[equalizer]] of a pair of maps between [[Hausdorff spaces]] in $C$ (being a closed subspace) is the same whether computed in $Top$ or in $C$.

On the other hand, [[products]] of $C$-objects in $Top$ need not land in $C$, so in that situation the product in $Top$ and the product in $C$ do not agree.  This is in particular the case for compactly generated spaces.  In fact, the "compactly generated product" is sometimes preferable to the $Top$-product for more explicit reasons: for instance, if $X$ and $Y$ are CW complexes, then $X \times Y$ need not be a CW complex in the usual product topology, but it is in the compactly generated topology.

## Examples
 {#Examples}

### Categories of colimits of generating spaces
 {#CategoriesOfColimitsOfGeneratingSpaces}

The original example of a convenient category of topological spaces is that of

* [[compactly generated topological space]]

and another example is that of

* [[Delta-generated spaces]] ([[Euclidean-generated spaces]]).

Both of these are special cases of the general class of convenient *$GenSp$-generated topological spaces*:

These are those topological spaces $X$ which agree with the [[colimit]] ([in TopSp](Top#UniversalConstructions)) of all continuous images inside them of *generating spaces* in a specified [[full subcategory]] $GenSp \,\subset\, $ [[TopSp]] ([Vogt 1971, Sec. 1](#Vogt71)):

\[
  \label{Gfication}
  G(X)
  \,\coloneqq\,
  \Big(
  \underset{
    \underset{
      { T \in GenSp }
      \atop
      {T \to X}
    }{\longrightarrow}
  }{\lim}
  \,
  T
  \Big)
  \xrightarrow{\;\;\; \simeq \;\;\;}
  X
\]

The [[full subcategories]] of such $GenSp$-generated spaces are [[coreflective subcategory|coreflective]] ([Vogt 1971, Cor. 1.4](#Vogt71))

$$
  TopSp
    \underoverset
      {\underset{G}{\longrightarrow}}
      {\hookleftarrow}
      {\;\;\;\;\bot\;\;\;\;}
  G TopSp
  \,.
$$

If one demands of the category $GenSp \,\hookrightarrow TopSp\,$ of generating spaces, for any pair $G_1, G_2 \,\in\, GenSp$ that ([Vogt 1971, Axiom 2](#Vogt71)):

1. their [[product topological space]] is again in $GenSp$;

1. the [[compact-open topology]] on the set of maps $G_1 \to G_2$ has a [[continuous function|continuous]] [[evaluation map]]

then $G TopSp$ is a [[Cartesian closed category]] ([Vogt 1971, Thm. 3.6](#Vogt71)), with [[internal hom]] given by the image under $G$-fication (eq:Gfication) of the [[compact open topology]].

If one in addition demands that $GenSp$ contains all [[topological simplices]] (or all [[Cartesian spaces]] $\mathbb{R}^n$), then $G(X) \to X$ is a [[weak homotopy equivalence]] ([Vogt 1971, Prop. 1.2 (g)](#Vogt71)).

As special cases of this general construction, the above examples of

* [[compactly generated topological spaces]]  is obtained by

  taking $GenSp$ to be given by all [[compact Hausdorff spaces]]

* [[Delta-generated topological spaces]] is obtained by

  taking $GenSp$ to be given by all [[Cartesian spaces]] (equivalently by all [[topological simplices]]).

See also [Gaucher 2007, Sec. 2](#Gaucher07).

Further developments along these lines are in[Escardo, Lawson &Simpson 2004](#EscardoLawsonSimpson04):

The topological spaces $X$ for which there is an [[exponential law for spaces|exponentiable space]] in [[Top]] (i.e such that $X \times (-)  \;\colon\; Top \to Top$ has a [[right adjoint]])
may be described more concretely as _core-compact_ spaces (spaces whose topology is a [[continuous lattice]]). Suppose given a collection $\mathcal{C}$ of core-compact spaces, with the property that the product of any two spaces in $\mathcal{C}$ is a [[colimit]] in [[Top]] of spaces in $\mathcal{C}$. Such a collection $\mathcal{C}$ is called **productive**. Spaces which are $Top$-colimits of spaces in $\mathcal{C}$ are called $\mathcal{C}$-**generated**.

+-- {: .num_theorem}
######Theorem
**(Escardó, Lawson, Simpson)**

If $\mathcal{C}$ is a productive class, then the full subcategory of $Top$ whose objects are $\mathcal{C}$-generated is a [[coreflective subcategory]] of [[Top]] (hence [[complete category|complete]] and [[cocomplete category|cocomplete]]) that is [[cartesian closed category|cartesian closed]].
=--

The other convenience conditions listed in this article (inclusion of [[CW-complexes]], closure under [[closed subspaces]]) are in practice usually satisfied as well. For example, if closed subspaces of objects of $\mathcal{C}$ are $\mathcal{C}$-generated, then closed subspaces of $\mathcal{C}$-generated spaces are also $\mathcal{C}$-generated. If the unit interval $I$ is $\mathcal{C}$-generated, then so are all CW-complexes.

### Subcategories of toposes
 {#ExamplesSubcategoriesOfToposes}

The [above](#CategoriesOfColimitsOfGeneratingSpaces) example of the category of [[Delta-generated topological spaces]] [above](#CategoriesOfColimitsOfGeneratingSpaces) has the remarkable property that it is

1. a [[full subcategory]] of the [[quasi-topos]] of [[diffeological spaces]] (see [there](diffeological+space#RelationToTopologicalSpaces)),

1. which is in turn a [[full subcategory]] of the [[cohesive topos]] of [[smooth sets]] (see [there](geometry+of+physics+--+smooth+sets#DiffeologicalSpacesAreTheConcreteSmoothSets));

1. which in turn is a [[full sub-(infinity,1)-category|full sub-$(\infty,1)$-category]] of the [[cohesive (infinity,1)-topos|cohesive $(\infty,1)$-topos]] of [[smooth infinity-groupoids|smooth $\infty$-groupoids]]

such that the canonical [[shape modality]] (the [[shape via cohesive path ∞-groupoid|smooth path ∞-groupoid construction]]) still sees the correct underlying [[homotopy type]] of topological spaces ([[schreiber:Proper Orbifold Cohomology|SS20, Ex. 3.18]], see also at [[model structure on Delta-generated topological spaces]]):

\begin{imagefromfile}
    "file_name": "TopologicalConvenienceBySmoothGroupoids210930.jpg",
    "width": 800,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "From [SS21](https://ncatlab.org/schreiber/show/Equivariant+principal+infinity-bundles)"
\end{imagefromfile}

## Counterexamples

A naive approach to the problem of constructing "convenient categories" usually runs into problems. For example, one could try to work with the full subcategory of $Top$ consisting of [[exponentiable spaces]]; the problem is that even if $X$ and $Y$ are exponentiable, the exponential $Y^X$ may not be:

* The category of exponentiable spaces is not cartesian closed. (It is however [[cartesian monoidal category|cartesian]]: the product of two exponentiable spaces under the usual product topology is exponentiable.)

To see this, we recall that a [[Hausdorff space]] is exponentiable if and only if it is [[locally compact space|locally compact]], and that an exponential $Y^X$ (provided it exists) is Hausdorff if $Y$ is. Thus, it is enough to exhibit two locally compact Hausdorff spaces $X$, $Y$ whose exponential is not locally compact.

Take $X = \mathbb{R}$ with its usual topology and $Y = \mathbb{N}$ (the set of [[natural numbers]]) with the [[discrete topology]]. Suppose that an exponential $\mathbb{R}^\mathbb{N}$ exists in the category of locally compact Hausdorff spaces. Then it must be a countable product of copies of $\mathbb{R}$ by the following calculation:

$$LCH(X \times \mathbb{N}, \mathbb{R}) \cong LCH(\sum_{\mathbb{N}} X, \mathbb{R}) \cong \prod_{\mathbb{N}} LCH(X, \mathbb{R})$$

where the last functor in $X$, in order to be [[representable functor|representable]], would have to be represented by a product $\prod_{\mathbb{N}} \mathbb{R}$. Using the universal property of products, one may easily exhibit a scalar multiplication

$$\mathbb{R} \times \mathbb{R}^{\mathbb{N}} \to \mathbb{R}^{\mathbb{N}}$$

rendering $\mathbb{R}^{\mathbb{N}}$ a [[topological vector space]] (a TVS) over the real numbers. But it is well-known that locally compact TVS are finite-dimensional, and we have reached an absurdity.

Similarly, the topological product $\mathbb{N}^{\mathbb{N}}$ is homeomorphic to [[Baire space of irrational numbers|Baire space]], and this is not locally compact either.

## "Nice" versus "convenient" categories of spaces
{#NvC}

A related entry is [[nice category of spaces]]. Here we explain the difference between "convenient" and "nice" categories of spaces.

"A convenient category of topological spaces" is the title of [a well-known paper](#Steenrod) by [[Norman Steenrod]], who emphasized particularly [[function space]] constructions (i.e., cartesian closure) as a great convenience for working algebraic topologists, on top of (generally less problematic) completeness and cocompleteness assumptions. (See also the historical remarks that follow.)

It is well-known that $Top$ is not [[cartesian closed category|cartesian closed]].  One can characterize the [[exponential law for spaces|exponentiable spaces]], which include all [[locally compact space|locally compact]] [[Hausdorff space|Hausdorff]] spaces, but as we saw above, the naive idea of simply cutting down to some of these does not give a good cartesian closed category either, since firstly it need not be complete and cocomplete, and secondly even if $X$ and $Y$ are exponentiable, the exponential $Y^X$ need not be.

However, one can cut down to *some* full subcategory of spaces which does admit function spaces.  This typically involves the subtle and delicate interplay between compactness conditions and openness conditions.  It comes at a cost -- that limits and/or colimits in the subcategory might not be computed as they are in $Top$ -- but this is generally considered a very small price to pay in exchange for the great convenience of these assumptions.

That a convenient category of topological spaces contains all the CW complexes was not explicitly declared by Steenrod, but we feel certain that [[algebraic topology|algebraic topologists]] want these as part of their convenient category. In practice this is a mild assumption, because the usual examples certainly contain all finite topological products of the [[unit interval]] $I$, and are closed under those $Top$-colimits used to present CW complexes, as built inductively from the basic spaces $I^n \cong D^n$.

"Nice categories of spaces" should be thought of as a wider and vaguer term; it really means the category of spaces has "nice" categorical properties for some mathematical purpose at hand. Certainly any convenient category of spaces should be considered a nice category of spaces for the general purposes of algebraic topology. On the other hand, there are categories of spaces which are not "convenient" in the technical sense described above, but which may be very nice for certain purposes.

For example, the category of [[compact Hausdorff spaces]] can be considered as being very nice for certain purposes. For instance, it is [[monadic category|monadic]] over $Set$ (!) and a [[pretopos]]. It is, however, not "convenient", as it is very far from being cartesian closed. In a different direction, there is the (complete, cocomplete) cartesian closed category of [[equilogical spaces]], but this is not a full subcategory of $Top$, and the core concerns of mathematicians working with equilogical spaces are somewhat different from those of algebraic topologists.

It should also be noted that "[[space]]" itself has a wider meaning than the technical notion of "[[topological space]]", even if topological intuitions come into play. For example, in [[domain theory]], one often considers certain types of [[posets]] (for example, [[directed-complete partial order|dcpos]]) as certain types of "spaces". In this direction, we have that the category of dcpos and [[Scott-continuous map]]s between them forms a complete, cocomplete, cartesian closed category. However, these types of spaces are quite far removed from the traditional concerns of topology, hence are not "convenient" in the sense given above (despite the manifold relations between Scott continuity and the point-set topology which underlies the classical results on compactly generated spaces). In another direction, the category of [[locales]] is in some respects "nice", but these are not topological spaces either.

Along with the entry on [[nice category of spaces]] and the examples above, see [[Johnstone's topological topos]] and Spanier's [[quasitopological space]]s ([ref](#Spanier)). None of these is convenient in the precise sense above. However there are advantages in having a category which is not only cartesian closed but also locally cartesian closed. In algebraic topology, this has led to Peter Booth's work on "fibred mapping spaces" ([ref](#Booth)), ([ref](#Crabb-Jamea)).

There are possible advantages in homotopy theory of using a "topological topos", (Johnstone [ref](#Johnstone)), Harasani [ref](#Harasani)).

## Historical remarks
 {#HistoricalRemarks}

The phrase "convenient category of topological spaces" predates [Steenrod 1967](#Steenrod67) by a number of years;  [Brown 1963, Introduction](#Brown63) says:

> It may be that the category of Hausdorff $k$-spaces is adequate and convenient for all purposes of topology.

The requirements for convenience were spelled out in [Brown 1964](#Brown64).

In fact, Brown (not Steenrod, as has sometimes been assumed) was the first to prove that Hausdorff [[Kelley space|k-spaces]] (with the "kelleyfied" weakening of the [[compact-open topology|compact--open topology]] on function spaces) formed what is now called a [[cartesian closed category]] (see [Brown 1961](#Brown61)), influenced by [Cohen 1954](#Cohen54).

Note Brown's work predates the formal introduction of cartesian closed categories in [[Bill Lawvere]]'s thesis by a couple of years; in fact there is clear anticipation in Brown's thesis of the notions of [[closed monoidal category|monoidal closed]] and cartesian closed categories, which was to attract much attention throughout the sixties. Further,  Brown's results  on the topological case are more precise than Steenrod's, since the 1964 paper deals with functions continuous on compact subsets, and for these obtains a homeomorphism without kellyfication. An account of this may be found in the book [Topology and Groupoids](#Brown06).

Of course, as has often been emphasized by Lawvere, the need for and convenience of considering function spaces is a very old idea in [[geometry]] (going back to the roots of the [[calculus of variations]], for example). The constructions of exponentials of topological spaces via the compact--open topology had been known for a long time; see for example [[John Kelley]]'s _General Topology_ (1955). The relevance of what are today called [[Kelley space]]s (or k-spaces[^1]) had also been recognized; for example, Kelley's book indicates the [[complete space|completeness]] of function spaces (wrt the compact--open topology) when the base is a complete [[uniform space]] and the exponent is a k-space. However, the problem of obtaining a class of spaces _closed_ under function spaces wasn't solved prior to Brown's thesis. Brown also observed the relevance of k-spaces to studies of how products interact with [[quotient space]]s. The general appreciation of the connection between cartesian closure and preservation of quotients under products came with the appreciation of the conceptual simplicity of categorical [[adjunctions]], namely the point that the functor $- \times X$ preserves colimits if it has an exponential right adjoint $(-)^X$. The question also solved in Brown's 1964 paper was to give, in the Hausdorff case,  a left adjoint to  the functor $ (-)^Y $ when the function spaces have the compact-open topology, and so to give a monoidal closed structure on the category of Hausdorff spaces; the generalisation to the non-Hausdorff case is given in the paper of Booth and Tillotson listed below.

Appreciation of the role of convenient categories was in full force by the early seventies (for a sample, see [[Peter May]]'s _Geometry of Iterated Loop Spaces_, where the category of Hausdorff k-spaces plays a foundational role). The notion of a "convenient category" is recognized elsewhere too (and not just within the categorical and algebraic topology communities); see for example the book by [Kriegl and Michor](#KM).

[^1]: The '$k$' certainly  refers to 'kompakt' rather than Kelley's initial.

## Related concepts

* [[compactly generated topological space]]

* [[Delta-generated topological space]], [[D-topological space]]

* [[quasi-topological space]]

## References

The terminology ("convenient category of topological spaces") became widely adopted with

* {#Steenrod67} [[Norman Steenrod]], _A convenient category of topological spaces_, Michigan Math. J. 14 (1967) 133--152 ([euclid:mmj/1028999711](http://projecteuclid.org/euclid.mmj/1028999711))

referring there to the category of *[[compactly generated topological space]]* (see the references [there](compactly+generated+topological+space#Refererences) for more), whose convenient [[cartesian closed category|cartesian closure]] had [previously](#HistoricalRemarks) been highlighted in:

* {#Brown61} [[Ronnie Brown]], _Some problems of algebraic topology: a study of function spaces, function complexes, and FD-complexes_, DPhil thesis, Oxford University, 1961 (Note: the Appendix of this thesis was withdrawn from the examination.)  [(pdf)](https://ora.ox.ac.uk/objects/uuid:3af55800-4be7-462f-b91d-9769a6dac2c4)

* {#Brown63} [[Ronnie Brown]], _Ten topologies for $X\times Y$_, Quart. J.Math. (2) 14 (1963), 303--319. ([doi:10.1093/qmath/14.1.303](https://doi.org/10.1093/qmath/14.1.303), [pdf](http://groupoids.org.uk/pdffiles/tentopologies.pdf))

* {#Brown64} [[Ronnie Brown]], _Function spaces and product topologies_,  Quart. J. Math. (2) 15 (1964), 238--250. ([doi:10.1093/qmath/15.1.238](https://doi.org/10.1093/qmath/15.1.238))

influenced, in turn, by:

* {#Cohen54} D. E. Cohen,  _Spaces with weak topology_, Quart. J. Math., (2) 5 (1954) 77–-80 ([doi:10.1093/qmath/5.1.77](https://doi.org/10.1093/qmath/5.1.77))

Discussion in the generality that subsumes [[compactly generated topological spaces]] and [[Delta-generated topological spaces]] and all cases of subcategory-generated spaces in between:

* {#Vogt71} [[Rainer M. Vogt]], *Convenient categories of topological spaces for homotopy theory*,  Arch. Math 22, 545–555 (1971) ([doi:10.1007/BF01222616](https://doi.org/10.1007/BF01222616))

* {#Wyler73} [[Oswald Wyler]], *Convenient categories for topology*, General Topology and its Applications **3** 3 (1973) 225-242 (<a href="https://doi.org/10.1016/0016-660X(72)90014-1">doi:10.1016/0016-660X(72)90014-1</a>)

* {#EscardoLawsonSimpson04} [[Martín Escardó]], [[Jimmie Lawson]], [[Alex Simpson]], Section 3 of: *Comparing Cartesian closed categories of (core) compactly generated spaces*, Topology and its Applications Volume 143, Issues 1–3, 28 August 2004, Pages 105-145 ([doi:10.1016/j.topol.2004.02.011](https://doi.org/10.1016/j.topol.2004.02.011))

* {#FajstrupRosicky08} [[Lisbeth Fajstrup]], [[Jiří Rosický]], Section 3 of: _A convenient category for directed homotopy_, Theory and Applications of Categories, Vol. 21, 2008, No. 1, pp 7-20. ([arXiv:0708.3937](https://arxiv.org/abs/0708.3937), [tac:21-01](http://www.tac.mta.ca/tac/volumes/21/1/21-01abs.html))

* {#Gaucher07} [[Philippe Gaucher]], Section 2 of: *Homotopical interpretation of globular complex by multipointed d-space*, Theory and Applications of Categories, vol. 22, number 22, 588-621, 2009 ([arXiv:0710.3553](https://arxiv.org/abs/0710.3553))

On the abstract construction principle of these examples:

* [[Michael Barr]], *Building closed categories*, [[Cahiers|Cahiers Topologie Géométrie Différentielle]] **19** (1978) 115–129 [[numdam:CTGDC_1978__19_2_115_0](http://www.numdam.org/item/?id=CTGDC_1978__19_2_115_0)]

See also:

* {#Spanier} [[Edwin Spanier]], _Quasi-topologies_, Duke Mathematical Journal 30 (1) (1963), 1--14.

  > (on [[quasi-topological spaces]])

* {#Booth} Peter I. Booth,  _The exponential law of maps. II._  Math. Z. 121 (1971), 311–319.

* {#BT} P. Booth, ; Tillotson, J., _Monoidal closed, Cartesian closed and convenient categories of topological spaces_. Pacific J. Math. 88 (1980), no. 1, 35--53. [project euclid](http://projecteuclid.org/euclid.pjm/1102779712)

* Michael Crabb, [[Ioan James]] ,  _Fibrewise homotopy theory_.   Springer Monographs in Mathematics. Springer-Verlag London, Ltd., London, 1998. viii+341 pp. ISBN: 1-85233-014-7 {#CrabbJames}

* {#Preuss02} Gerhard Preuß, _Foundations of topology: an approach to convenient topology_, Kluwer, Dordrecht/ Boston 2002 ([doi:10.1007/978-94-010-0489-3](https://www.springer.com/gp/book/9781402008917))

  survey in: _Convenient topology --  a new branch of topology_ ([web](http://at.yorku.ca/i/a/a/c/54.htm))

* {#Brown06} [[Ronnie Brown]] _Topology and Groupoids_, Booksurge (2006), available from amazon: Section 5.9: Spaces of functions and the compact-open topology.

Discussion via [[topos theory]]:

* {#Johnstone} [[Peter Johnstone]], _On a topological topos_, Proc. London Math. Soc. (3) 38 (1979) 237--271.

* Harasani, Hamed A. _Topos theoretic methods in general topology_, PhD Thesis, University of Wales, Bangor, (1988) [(link to pdf files)](http://groupoids.org.uk/harasani.html). {#Harasani}

Analogous considerations for [[diffeological spaces]] (subsuming [[D-topological spaces]])

* {#KM} [[Andreas Kriegl]]; [[Peter Michor]], _[[The Convenient Setting of Global Analysis]]_, Mathematical Surveys and Monographs, Volume 53. American Mathematical Society, Providence, RI (1997).

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convenient manifold

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{:toc}

## Idea

A [[manifold]], possibly [[infinite dimensional manifold|infinite-dimensional]], is called a _convenient manifold_ -- implicitly meaning: convenient for _[[differential geometry]]_ -- if it is modeled on a [[convenient vector space]].

One should note that this usage of the adjective 'convenient' is different to that in 'convenient category', for example of smooth spaces. In that case the _category_ is convenient, whereas here the _objects_ are convenient.

## Properties

### Embedding into the Cahiers-topos

Together with [[convenient vector spaces]], convenient manifods embed into the [[Cahier topos]] of [[synthetic differential geometry|synthetic differential]] [[smooth spaces]]. See at _[[Cahiers topos]]_ for more on this.

## Related concepts

* [[Banach manifold]]

* [[Hilbert manifold]]

* [[ILH manifold]]

* [[Frechet manifold]]

* [[convenient manifold]]

## References

A standard textbook reference is

* [[Andreas Kriegl]], [[Peter Michor]]: _The convenient setting of global analysis_, AMS (1997) ([pdf](http://www.mat.univie.ac.at/~michor/apbookh.pdf))
 {#KM}
* Peter Michor, _A convenient setting for differential geometry and global analysis_, Cahiers Topologie Géom. Différentielle 25 (1984), no. 1, 63–109, [MR86g:58014a](http://www.ams.org/mathscinet-getitem?mr=764972); _A convenient setting for differential geometry and global analysis. II_, Cahiers Topologie Géom. Différentielle 25 (1984), no. 2, 113–178.

A survey is for instance in the slides

* Richard Blute, _Convenient Vector Spaces, Convenient Manifolds and Differential Linear Logic_ (2011) ([pdf](http://aix1.uottawa.ca/~rblute/FMCS.pdf))
 {#Blute}
* [[John C. Baez]], Alexander E. Hoffnung, _Convenient categories of smooth spaces_, Trans. Amer. Math. Soc. 363 (2011), 5789-5825 [pdf](http://www.ams.org/journals/tran/2011-363-11/S0002-9947-2011-05107-X/S0002-9947-2011-05107-X.pdf)

[[!redirects convenient manifolds]]


convenient vector space

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# Contents
* table of contents
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##Idea

A _convenient_ vector space is a [[locally convex topological vector space]] satisfying a certain [[completeness]] property.  Loosely, the completeness property is that "all derivatives which ought to exist actually do".

The term _convenient vector space_ was introduced by Kriegl and Michor as part of their theory of [[global analysis]].  It is equivalent to the notion _locally complete_ which is more usual in [[functional analysis]].

## Definition

There are various equivalent definitions of a __convenient vector space__; a long list is given in Theorem 2.14 of [KM](#KM).  We shall give the definition that is closest to its _raison d'être_, namely the existence of derivatives.

+-- {: .num_defn #ConSp}
###### Definition
A [[locally convex topological vector space]] $E$ is said to be **convenient** or **$c^\infty$-complete** if whenever $c \colon \mathbb{R} \to E$ is a [[curve]] such that $l \circ c \colon \mathbb{R} \to \mathbb{R}$ is [[smooth map|smooth]] for all $l \in E^*$ then $c$ is smooth.
=--

We shall give one other characterisation which recasts the definition in language more common in functional analysis.

+-- {: .num_defn #LocCpt}
###### Definition
A [[locally convex topological vector space]], $E$, is said to be **locally complete** if for $B \subseteq E$ a [[bounded subset|bounded]], [[closed subset|closed]], [[absolutely convex subset]], its [[norm space]], $E_B$, is a [[Banach space]].
=--

The equivalence of these two forms part of Theorem 2.14 of [KM](#KM).

Other equivalent characterizations of convenient vector spaces $E$ are:

* $E$ is a [[locally convex vector space]] and every [[Mackey-Cauchy sequence]] in $E$ converges.

* $E$ is a [[locally convex vector space]] and for every [[smooth map|smooth]] [[curve]] $c : \mathbb{R} \to E$ there is a curve $\int c$ (an [[antiderivative]] of $c$) such that $c$ is the [[derivative]] of $\int c$.

See for instance ([Blute](#Blute)).

## Examples

* A [[Cartesian space]] $\mathbb{R}^n$ carries a unique structure of a convenient vector space. This also holds for [[Fréchet spaces]] in general and countable strict locally convex [[inductive limits]] thereof. (See [[LF-space]]).

* For $X$ and $Y$ two convenient vector spaces, the vector space of [[smooth functions]] $C^\infty(X,Y)$ is again a convenient vector space. This is to a large degree the motivating example. It makes the category of convenient vector spaces be [[cartesian-closed category|Cartesian closed]]. (See below).

## Properties

* The [[category]] of convenient vector spaces and [[smooth functions]] between them is a [[cartesian closed category]].

* The definition of a convenient vector space has a natural interpretation in terms of [[Frölicher spaces]]. It may also be related to [[synthetic differential geometry]]: convenient vector spaces form a [[full subcategory]] of the [[Cahiers topos]] ([Kock 86](#Kock86), [Kock-Reyes 86](#KockReyes86)) and in fact already of the topos of [[smooth sets]], hence (being [[concrete object|concrete]]) of the category of [[diffeological spaces]] ([Kock-Reyes 04, p. 5](#KockReyes04)).

## Related concepts

* [[convenient manifold]]

## References

A standard textbook reference is

* {#KM} [[Andreas Kriegl]], [[Peter Michor]]: _The convenient setting of global analysis_, AMS (1997) ([pdf](http://www.mat.univie.ac.at/~michor/apbookh.pdf))

A survey is for instance in the slides

* {#Blute} [[Richard Blute]], _Convenient Vector Spaces, Convenient Manifolds and Differential Linear Logic_ (2011) ([pdf](http://aix1.uottawa.ca/~rblute/FMCS.pdf))

Results on equivalent characterizations are for instance in

* Thomas E. Gilsdorf, _Strictly Webbed Convenient Locally Convex Spaces_, Int. Journal of Math. Analysis, Vol. 1, 2007, no. 16, 775 - 782 ([pdf](http://www.m-hikari.com/ijma/ijma-password-2007/ijma-password13-16-2007/gilsdorfIJMA13-16-2007.pdf))

The embedding of convenient vector spaces into the [[Cahiers topos]] and hence the treatment of their differential geometry by [[synthetic differential geometry]] is due to

* {#Kock86} [[Anders Kock]], _Convenient vector spaces embed into the Cahiers topos_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 27 no. 1 (1986), p. 3-17  ([numdam](http://www.numdam.org/item?id=CTGDC_1986__27_1_3_0))

and with a corrected definition of the site of definition in

* {#KockReyes86} [[Anders Kock]], [[Gonzalo Reyes]], _Corrigendum and addenda to: Convenient vector spaces embed into the Cahiers topos_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 27 no. 1 (1986), p. 3-17 ([numdam](http://www.numdam.org/item?id=CTGDC_1987__28_2_99_0))

The stronger statemet of embedding into [[diffeological spaces]] appears in

* {#KockReyes04} [[Anders Kock]], [[Gonzalo Reyes]], _Categorical distribution theory; heat equation_ ([arXiv:math/0407242](https://arxiv.org/abs/math/0407242))

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