Malcev variety

#Contents#
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##Definition

A __Malcev operation__ on a [[set]] $X$ is a ternary operation, a [[function]]

$$ t:X\times X\times X\to X,\,\,\,(x,y,z)\mapsto t(x,y,z) ,$$

which satisfies the identities $t(x,x,z)=z$ and $t(x,z,z)=x$.
An important motivating example is the operation $t$ of a [[heap]], for example the operation on a [[group]] defined by $t(x, y, z) = x y^{-1} z$.

An [[algebraic theory]] $T$ is a __Malcev theory__ when $T$ contains a Malcev operation. An algebraic theory is Malcev iff one of the following equivalent statements is true:

1. in the category of $T$-algebras, every [[internal relation|internal]] [[reflexive relation]] is a [[congruence]];

2. in the category of $T$-algebras, the composite (as internal relations) of any two congruences is a congruence;

3. in the category of $T$-algebras, the composition of equivalence relations is commutative.

Statement (1) is one of the motivations to introduce the notion of [[Malcev category]].

A __Malcev variety__ is the category of $T$-algebras for a Malcev theory $T$, thought of as a [[variety of algebras]].

## Proofs of equivalence

If $R \hookrightarrow X \times Y$ is a binary relation on sets, write $R(x, y)$ to say that $(x, y) \in R$. If $X$, $Y$ are $T$-algebras, then $R$ is an internal relation in $T$-$Alg$ if the conditions $R(x_1, y_1) \wedge \ldots \wedge R(x_n, y_n)$, and $\theta(x_1, \ldots, x_n) = x$, $\theta(y_1, \ldots, y_n) = y$ for any $n$-ary operation $\theta$ of $T$, jointly imply $R(x, y)$.

The set-theoretic composite of two internal relations in $T$-$Alg$ is also an internal relation, and the equality relation is always internal, so we may (and will) apply ordinary set-theoretic reasoning in our proofs below.

+-- {: .un_prop}
######Proposition 1
If $T$ is a Malcev theory, then any internal reflexive relation in $T$-$Alg$ is an internal equivalence relation.
=--

+-- {: .proof}
######Proof
If $t$ is a Malcev operation and $R$ is any internal reflexive relation on a $T$-algebra $X$, then $R$ is transitive because given $R(x, y) \wedge R(y, z)$, we infer $R(x, y) \wedge R(y, y) \wedge R(y, z)$, and this together with $t(x, y, y) = x$ and $t(y, y, z) = z$ gives $R(x, z)$ since $R$ is internal. Also $R$ is symmetric, because if $R(x, y)$, we infer $R(x, x) \wedge R(x, y) \wedge R(y, y)$, which together with $t(x, x, y) = y$ and $t(x, y, y) = x$ gives $R(y, x)$.
=--

+-- {: .un_prop}
######Proposition 2
If every internal reflexive relation is an internal equivalence relation, then the composite of any two internal equivalence relations is also an internal equivalence relation.
=--

+-- {: .proof}
######Proof
The hypothesis is that internal reflexive relations and internal equivalence relations coincide. But (internal) reflexive relations are
clearly closed under composition: $\Delta = \Delta \circ \Delta \subseteq R \circ S$.
=--

+-- {: .un_prop}
######Proposition 3
If internal equivalence relations are closed under composition, then composition of internal equivalence relations is commutative.
=--

+-- {: .proof}
######Proof
If $R$ and $S$ are equivalence relations and so is $S \circ R$, then
$$S \circ R = (S \circ R)^{op} = R^{op} \circ S^{op} = R \circ S,$$
as desired.
=--

+-- {: .un_prop}
######Proposition 4
If composition of internal equivalence relations in $T$-$Alg$ is commutative, then the theory $T$ has a Malcev operation $t$.
=--

+-- {: .proof}
######Proof
According to the yoga of (Lawvere) [[Lawvere theory|algebraic theories]], $n$-ary operations are identified with elements of $F(n)$, the free $T$-algebra on $n$ generators (more precisely, the Lawvere theory is the category opposite to the category of finitely generated free $T$-algebras). Thus we must exhibit a suitable element $t$ of $F(3)$.

Let $x, y, z$ be the generators of $F(3)$, and let $a, b$ be the generators of $F(2)$. Let $\phi$ be the unique algebra map $F(3) \to F(2)$ taking $x$ and $y$ to $a$ and $z$ to $b$, and let $\psi$ be the unique algebra map $F(3) \to F(2)$ taking $x$ to $a$ and $y$ and $z$ to $b$. An operation $t \in F(3)$ is Malcev precisely when

$$\phi(t) = b \qquad \psi(t) = a$$

Let $R$ be the equivalence relation on $F(3)$ given by the kernel pair of $\phi$, and let $S$ be the kernel pair of $\psi$. Then $R(x, y)$ and $S(y, z)$, so $(S \circ R)(x, z)$. Then, since composition of equivalence relations is assumed commutative, $(R \circ S)(x, z)$. This means there exists $t$ such that $S(x, t)$ and $R(t, z)$, or that $\psi(x) = \psi(t)$ and $\phi(t) = \phi(z)$. This completes the proof.
=--

## Examples

* The theory of [[group]]s, where $t(x, y, z) = x y^{-1} z$, is Malcev.

* The theory of [[Heyting algebra]]s, where
$$t(x, y, z) = ((z \Rightarrow y) \Rightarrow x) \wedge ((x \Rightarrow y) \Rightarrow z),$$
is Malcev.

* If $T$ is Malcev, and if $T \to T'$ is a morphism of algebraic theories, then $T'$ is Malcev. From this point of view, the theory of groups is Malcev because the theory of heaps is Malcev, and the theory of Heyting algebras is Malcev because the theory of [[cartesian closed category|cartesian closed]] [[meet-semilattice]]s is Malcev.

See also [[Malcev category]].

## The lattice of congruences $Equiv(X)$

### $Equiv(X)$ is a modular lattice

In any [[finitely complete category]], the intersection of two congruences (equivalence relations) on an object $X$ is a congruence, so that the set of equivalence relations $Equiv(X)$ is a meet-semilattice.

In a [[regular category]] such as a variety of algebras, where there is a sensible calculus of relations and relational composition, it is a simple matter to prove that if $Equiv(X)$ is closed under relational composition, then $R \circ S$ is the join $R \vee S$ in $Equiv(X)$. For, if $R, S \in Equiv(X)$, then

$$R = R \circ \Delta \subseteq R \circ S, \qquad S = \Delta \circ S \subseteq R \circ S$$

while if $R, S \subseteq T$ in $Equiv(X)$, then

$$R \circ S \subseteq T \circ T \subseteq T.$$

+-- {: .un_prop}
######Proposition
In a regular category, if $Equiv(X)$ is closed under relational composition (equivalently, if composition of equivalence relations is commutative), then $Equiv(X)$ is a [[modular lattice]].
=--

+-- {: .proof}
######Proof
The (poset-enriched) category of relations in a regular category is an [[allegory]], and hence satisfies Freyd's modular law

$$R \wedge (S \circ T) \subseteq S \circ ((S^{op} \circ R) \wedge T)$$

whenever $T: X \to Y$, $S: Y \to Z$, $R: X \to Z$ are relations. As we have just seen, the hypothesis implies that joins in $Equiv(X)$ are given by composition (so $Equiv(X)$ is a lattice), and so for $R, S, T \in Equiv(X)$ we have

$$R \wedge (S \vee T) \subseteq S \vee ((S \vee R) \wedge T).$$

Therefore, if $S \subseteq R$, we have both

$$R \wedge (S \vee T) \subseteq S \vee (R \wedge T)$$

and also $S \vee (R \wedge T) \subseteq R \wedge (S \vee T)$. Thus $S \subseteq R$ implies $R \wedge (T \vee S) = (R \wedge T) \vee S$: the modular law is satisfied in $Equiv(X)$.
=--

+-- {: .un_cor}
######Corollary
If $T$ is a Malcev theory, then the lattice of congruences $Equiv(X)$ on any $T$-algebra $X$ is a modular lattice.
=--

### $Equiv(X)$ is a Desarguesian lattice

A similar argument shows that congruence lattices for $T$-algebras $X$, for $T$ a Malcev theory, satisfy the following property (stronger than the modular property):

* [[Desarguesian axiom|Desarguesian property]]: if $R_i, S_i, T_i \in Equiv(X)$ for $i = 1, 2$, then
$$(R_1 \vee R_2) \wedge (S_1 \vee S_2)) \subseteq T_1 \vee T_2 \qquad implies \qquad (R_1 \vee S_1) \wedge (R_2 \vee S_2) \subseteq ((R_1 \vee T_1) \wedge (R_2 \vee T_2)) \vee ((S_1 \vee T_1) \wedge (S_2 \vee T_2))$$

Freyd-Scedrov's [[Categories, Allegories]] (2.157, pp. 206-207) gives the following argument: given relations $R_1, S_1, T_1: X \to Y$, $R_2, S_2, T_2: Y \to Z$ between sets, it is "easily verified" that

$$R_2 R_1 \cap S_2 S_2 \subseteq T_2 T_1 \qquad implies \qquad S_1 R_{1}^{op} \cap S_{2}^{op} R_2 \subseteq (S_1 T_{1}^{op} \wedge S_{2}^{op} T_2)(T_1 R_{1}^{op} \cap T_{2}^{op}R_2)$$

Then, under the assumption that equivalence relations internal to $T$-$Alg$ commute (so that the join of equivalence relations $R, S$ on $X$ is their relational composite $R S = R \circ S$), the Desarguesian axiom follows immediately.

## Generalized Malcev operations and connection to unorderability and recursion

An algebra $A$ for a theory is _unorderable_ if there is no non-trivial compatible [[partial order]] structure on it. It is _absolutely unorderable_ if whenever there is an embedding $A \to B$, then $B$ is unorderable.

Every algebra in a Malcev variety is absolutely unorderable. For if $t\leq u$, then

$$u= M(t,t,u) \leq M(t,u,u) = t,$$

hence $t=u$. More generally:

+-- {: .un_prop}
######Theorem
An algebra has a _generalized_ Malcev operation if and only if it is absolutely unorderable.
=--

Here, a generalized Malcev operation is a finite sequence of ternary operations $M_1\dots M_n$ such that

$$ t = M_1(t,u,u);\ \  M_1(t,t,u)=M_2(t,u,u);\ \  M_2(t,t,u) = M_3(t,u,u);\ \  \dots M_n(t,t,u)=u.$$

### Recursion and the lambda calculus

Orders are often used to explain [[recursion]], by saying that a recursive definition has a solution via [[Kleene's fixed point theorem]]. Since the [[lambda calculus]] is a famous calculus for computation and recursion, there is a question (attributed to [[Gordon Plotkin]]) of whether the [[lambda calculus]] is necessarily orderable to some extent. [[Peter Selinger]] then asked, is the lambda calculus inconsistent with Malcev operators?

+-- {: .un_prop}
######Theorem
(Plotkin, Simpson, Selinger)
There can be no generalized Malcev operator with $n=1$ or $n=2$ in the lambda calculus.
=--

+-- {: .un_prop}
######Question
For $n\gt 2$, it is an open question whether generalized Malcev operators are consistent with the lambda calculus.
=--

## References

See the monograph [[Borceux-Bourn]].

For the connection to unorderability and recursion, see the following article and references therein:

* [[Peter Selinger]], [_Order-Incompleteness and Finite Lambda Reduction Models_](https://www.mscs.dal.ca/~selinger/papers/cmaa.pdf), Theoretical Computer Science 309 (2003) 43–63.

For the characterisation of [[categories]] which are [[localizations]] of [[Maltsev varieties]]:

* [[Marino Gran]] and [[Enrico Vitale]], _Localizations of Maltsev varieties_, Theory and Applications of Categories **5** 12 (1999) 281-291 [[TAC](http://www.tac.mta.ca/tac/volumes/1999/n12/5-12abs.html)]

## Spelling

The original is 'Мальцев'; Malcev spelled his name as 'Malcev' in his non-Russian papers.
This has also been transliterated 'Mal'cev', 'Mal'tsev' and 'Maltsev'.

## Related concepts

* [[invertible quasigroup]]

* [[pseudo-torsor]], [[heap]]

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Malcev, protomodular, homological and semi-abelian categories

This entry is about the book:

* [[Francis Borceux]], [[Dominique Bourn]],

  **Mal'cev, protomodular, homological and semi-abelian categories**

  Mathematics and Its Applications __566__,

  Kluwer 2004

  [doi:10.1007/978-1-4020-1962-3](https://link.springer.com/book/10.1007/978-1-4020-1962-3)

on [[nonabelian homological algebra]], such as its [[diagram chasing lemmas]], in [[Mal'cev categories]], [[protomodular categories]], [[semi-abelian categories]] and [[homological categories]].

From the introduction:

> The most striking successes of category theory, as far as clarification of mathematical situations is concerned, are probably the theory of [[abelian categories]] and the theory of [[toposes]]. This is not too amazing since both theories are closely related to the development of sheaf theory, a context in which it is desirable to get rid of the usual notion of element.

> But up to recently, category theory did not provide any comparable insight in General Algebra, a domain in which element-based mathematics remains the slogan. In particular, category theory could not provide a structural tool able to grasp, even in the most representative category of classical algebra - namely, the category [[Grp|$\mathrm{Gp}$]] of groups - the deep essence of the notion of normal [[subobject]]: namely, an equivalence class for a congruence and not just the kernel of a morphism.

> And category theory could not grasp either the conceptual foundations of the [[diagram chasing lemmas - contents|homological lemmas]]: the Nine Lemma, the [[snake lemma|Snake Lemma]], which remain valid and strongly meaningful in the category $\mathrm{Gp}$ of groups, even if this category does not belong to the abelian setting in which these lemmas are generally proved in a significant categorical way.

> Of course, there have been since a long time attempts to provide an axiomatic context in which to get the isomorphism theorems, the decomposition theorems or the previous homological lemmas for the varieties of Universal Algebra: Baer (1947, [6]), Goldie (1952, [48]), Atiyah (1956, [5]), Higgins (1956, [54]), Kurosh (1959, [73]), Hilton-Ledermann (1960, [55]), Eckmann-Hilton (1962, [40]), Tsalenko (1967, [92]), but also Hofmannn (1960, [56]), Fröhlich (1961, [46]), Huq (1968, [57]), Gerstenhaber (1970, [47]), Burgin (1970, [34]), Orzech (1972, [83]).

> These first attempts, despite their interest, consist generally in a long list of axioms whose independence is certainly not clear. But more importantly, these axioms look desperately heavy and complicated in comparison with the elegance of the characterization of abelian models. We refer the reader to the introduction of the paper by Janelidze-Márki-Tholen (2002, [60]) for a reliable historical approach to this topic.
...
> Establishing an organic and synthetic connection between all these attempts the ambition of this book. To achieve this, an additional ingredient was necessary, of purely categorical nature: the [[fibration of points]]. This fibration allows representing every category as a fibration whose fibres are pointed categories, i.e. categories with a zero object (see Bourn, 1996, [17]). This book will give evidence that the fibration of points emphasizes the importance of split epimorphisms in the context of algebraic theories, but also that this fibration of points has a very strong classification power: see on page 466 the table summarizing these classification properties.
...

## Contents

(with links to the related $n$lab entries)

* Preface

1. Metatheorems

2. [[Mal'cev categories]]

3. [[protomodular category|Protomodular categories]]

4. [[homological category|Homological categories]]

5. [[semi-abelian category|Semi-abelian categories]]

6. [[protomodular category|Strongly protomodular categories]]

7. [[essentially affine category|Essentially affine categories]]

* Appendix

  1. [[algebraic theory|Algebraic theories]]

  2. [[internal relation|Internal relations]]

  3. [[internal groupoid|Internal groupoids]]

  4. Variations on [[epimorphisms]]

  5. [[regular category|Regular]] and [[exact categories]]

  6. [[monad|Monads]]

  7. [[Grothendieck fibration|Fibrations]]

* Classification table of fibrations of points

* Bibliography

* Index of symbols

* Index of definitions

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78. F.E.J. Linton, Applied functorial semantics, Springer LNM __80__ (1969) 53-74

79. S. Mac Lane, Homology, Springer Verlag (1963)

80. S. Mac Lane, Categories for the working mathematician, 2nd edition,
Springer Verlag (1998)

81. S. Mac Lane, I. Moerdijk, [[Sheaves in Geometry and Logic]], Universitext,
Springer Verlag (1992)

82. A. I. Mal'cev, On the general theory of algebraic systems, Mat. Sbornik N. S. 35 (1954) 3-20

83. G. Orzech, Obstruction theory in algebraic categories, I and H, J. Pure Appl. Alg. 2 (1972) 287-340

84. M. C. Pedicchio, Maltsev categories and Maltsev operations, J. Pure Appl. Alg. 98 (1995) 67-71

85. M. C. Pedicchio, A categorical approach to commutator theory, J. Algebra
17'7' (1995) 647-657

86. M. C. Pedicchio, Arithmetical categories and commutator theory, Appl.
Categorical Structures 4 (1996) 297-305

87. R.S. Pierce, Modules over commutativc regular rings, Mere. Am. Math. Soc. 70 (1967)

88. A. F. Pixley, Distributivity and permutability of congruences in equational classes of algebras, Proc. Amer. Math. Soc. 14 (1963) 105-109

89. J. D. H. Smith, Mal'cev varieties, Springer Lect. Notes in Math. 554 (1976)

90. J. D. H. Smith, Centrality, Abstract of the Amer. Math. Soc. I (1980) 774-821

91. J. D. H. Smith, On the characterization of Maltsev and Jdnsson-Tarski
algebras, Preprint Iowa State Univ. (2001)

92. M. S. Tsalenko, _Correspondences over a quasi-exact category_ (Russian),
Mat. Sbornik __73__ (1967) 564-584

93. A. Ursini, _On subtractive varieties, I_, Algebra Universalis __31__ (1994) 204-222

94. A. Ursini, _On subtractive varieties, V: congruence modularity and commutators_, Algebra Universalis 43 (2000) 51-78

95. V.V. Uspenskii, _The Mal'tsev operation on countably compact spaces_, Comment. mat. Univ. Carolinæ 30 (1989) 395-402

category: reference

[[!redirects Mal'cev, protomodular, homological and semi-abelian categories]]

[[!redirects Borceux-Bourn]]
[[!redirects Borceux--Bourn]]


Malcolm G. Boshier

* [institute page](https://cquic.unm.edu/people/malcolm-boshier/index.html)

## Selected writings

On [[atomtronics]]:

* [[Luigi Amico]], [[Malcolm G. Boshier]], *Atomtronics* [[arXiv:1511.07215](https://arxiv.org/abs/1511.07215)]

* [[Luigi Amico]], [[Malcolm G. Boshier]], [[Juan Polo Gomez]] et al.: *Roadmap on Atomtronics: State of the art and perspective*, AVS Quantum Sci. **3** 039201 (2021) [[arXiv:2008.04439](https://arxiv.org/abs/2008.04439), [doi:10.1116/5.0026178](https://doi.org/10.1116/5.0026178)]

* [[Luigi Amico]], Dana Anderson, [[Malcolm Boshier]], Jean-Philippe Brantut, Leong-Chuan Kwek, Anna Minguzzi, Wolf von Klitzing: *Colloquium: Atomtronic circuits: From many-body physics to quantum technologies*, Rev. Mod. Phys. **94** (2022) 041001 [[doi:10.1103/RevModPhys.94.041001](https://doi.org/10.1103/RevModPhys.94.041001)]

category: people

[[!redirects Malcolm Boshier]]


Malcolm Perry

* [webpage](http://www.damtp.cam.ac.uk/people/m.j.perry/)

## Selected writings

Introducing [[Kaluza-Klein monopoles]]:

* {#GrossPerry83} [[David Gross]], [[Malcolm Perry]], _Magnetic Monopoles in Kaluza-Klein Theories_,  Nucl. Phys. B226 (1983) 29 (doi:10.1016/0550-3213(83)90462-5)

On the [[Perry-Schwarz action]] for [[self-dual higher gauge fields]] in 6dand the [[M5-brane]] [[Green-Schwarz sigma-model]] upon [[KK-compactification]] to 5-dimensions:

* {#PerrySchwarz96} [[Malcolm Perry]], [[John Schwarz]], _Interacting Chiral Gauge Fields in Six Dimensions and Born-Infeld Theory_, Nucl. Phys. B489 (1997) 47-64 ([arXiv:hep-th/9611065](http://arxiv.org/abs/hep-th/9611065))

On [[black brane|black]] [[D8-brane]]-solutions in [[massive type IIA supergravity]]/[[massive type IIA string theory]]:

* {#ChamblinPerry97} A. Chamblin, [[Malcolm Perry]], _Dynamic D8-branes in IIA string theory_ ([arXiv:hep-th/9712112](https://arxiv.org/abs/hep-th/9712112))

On [[Yukawa couplings]] in the [[G2-MSSM]]:

* Eric Gonzalez, [[Gordon Kane]], Khoa Dang Nguyen, [[Malcolm Perry]], _Quark and lepton mass matrices from localization in M-theory on $G_2$ orbifold_ ([arXiv:2002.11820](https://arxiv.org/abs/2002.11820))

On [[boundary conditions]] for interacting [[M2-branes]] in regarding through the [[ABJM model]]:

* {#BPST09} [[David Berman]], [[Malcolm J. Perry]], [[Ergin Sezgin]], [[Daniel C. Thompson]], *Boundary Conditions for Interacting Membranes*, JHEP 1004:025, 2010 [[arXiv:0912.3504](https://arxiv.org/abs/0912.3504), doi:10.1007/JHEP04(2010)025]

## Related entries

* [[M5-brane]]

* [[self-dual higher gauge field]]

category: people

[[!redirects Malcolm J. Perry]]


Malte Dehling

* [personal page](https://www.uni-math.gwdg.de/mdehling/)

* [GoogleScholar page](https://scholar.google.co.uk/citations?user=tBkoie8AAAAJ&hl=de)

## Selected writings

On [[weak Lie 3-algebras]]:

* [[Malte Dehling]], *On weak Lie 3-algebras* [[arXiv:1710.11104](https://arxiv.org/abs/1710.11104)]

category: people


Malte Lackmann

* [webpage](https://www.hcm.uni-bonn.de/people/profile/malte-mario-lackmann/)

## Selected writings

On the [[octonionic projective plane]]:

* {#Lackmann19} [[Malte Lackmann]], _The octonionic projective plane_, in MATRIX Book Series **4**, Springer (2021) [[doi:10.1007/978-3-030-62497-2_6](https://doi.org/10.1007/978-3-030-62497-2_6), [arXiv:1909.07047](https://arxiv.org/abs/1909.07047)]

category: people


Mamuka Jibladze

__Mamuka Jibladze__ is a Georgian mathematician with expertise in [[K-theory]], especially algebraic, [[homotopy theory]], [[category theory]], including its applications to computer science, type theory and logics, universal algebra, [[nonabelian cohomology]]. In homotopy theory he collaborated extensively with H.-J. Baues and with T. Pirashvili.

* [homepage](http://www.rmi.ge/~jib)

* Louvain thesis: _Homotopy types for "gros" toposes_, [pdf](http://www.rmi.ge/~jib/pubs/thesis.pdf)

## Selected writings

On [[cohomology]] of [[monoids in monoidal categories]] (specifically in [[distributive monoidal categories]]):

* [[Hans-Joachim Baues]], [[Mamuka Jibladze]], [[Andy Tonks]], *Cohomology of monoids in monoidal categories*, in: *Operads: Proceedings of Renaissance Conferences*, Contemporary Mathematics **202**, AMS (1997) 137-166 [[doi:10.1090/conm/202](https://doi.org/10.1090/conm/202), preprint:[pdf](https://archive.mpim-bonn.mpg.de/id/eprint/1484/1/preprint_1995_121.pdf)]

category: people


Man-Duen Choi

* [institute page](https://cqiqc.physics.utoronto.ca/people/cqiqc-members/man-duen-choi/)

* [GoogleScholar page](https://scholar.google.com/citations?user=Dxm1ZbkAAAAJ&hl=en)

## Selected writings

Early discussion of [[completely positive maps]] ([[quantum channels]]):

* {#Choi75} [[Man-Duen Choi]], *Completely positive linear maps on complex matrices*, Linear Algebra and its Applications **10** 3 (1975) 285-290 [doi:10.1016/0024-3795(75)90075-0]

category: people


Mandelbrot set

#Contents#
* table of contents
{:toc}

## Idea

The _Mandelbrot set_ is the [[subset]] of the [[complex plane]] on those points $c \in \mathbb{C}$ on which the iteration of the operation "square and add $c$" does not diverge.

This is a famous example of a [[fractal]].

## Definition

For $c \in \mathbb{C}$ a [[complex number]], consider the [[function]] on the [[complex plane]] that squares its argument and adds $c$ to the result:

$$
  \array{
     \mathbb{C} &\overset{\phantom{AA}f_c\phantom{AA}}{\longrightarrow}& \mathbb{C}
     \\
     z &\overset{\phantom{AA}\phantom{AA}}{\mapsto}& z^2 + c
  }
  \,.
$$

For $n \in \mathbb{N}$ write

$$
  f_c^n \coloneqq \underset{n\,\text{factors}}{\underbrace{ f \circ \cdots \circ f \circ f }}
$$

for the $n$-fold [[composition]] of $f_c$ with itself ($f_c^0 \coloneqq id_{\mathbb{C}}$).

Starting with value $0 \in \mathbb{C}$, this defines a [[sequence]] of points in the complex plane $(f_c^n(0))_{i \in \mathbb{C}}$ for each [[complex number]] $c \in \mathbb{C}$.

The _Mandelbrot set_ $Mdlbrt \subset \mathcal{C}$ is the [[subset]] of the [[complex plane]] on those values of $c$ for which the corresponding [[sequence]] $(f_c^n)_{n \in \mathbb{B}}$ is [[bounded set|bounded]]

$$
  Mdlbrt
   \coloneqq
  \left\{
     c \in \mathbb{C}
     \;\colon\;
     (f_c^n(0))_{n \in \mathbb{N}}\, \text{is bounded}
  \right\}
  \;\subset\;
  \mathbb{C}
  \,.
$$

Globally, at low resolution, the Mandelbrot set looks like this:

<img src="https://ncatlab.org/nlab/files/MandelbrotSet.png" width="400" >

## Properties

### Topological properties

+-- {: .num_defn #MandelbrotSpace}
###### Definition
**(Mandelbrot space)**

Regard the Mandelbrot set as a [[topological space]]

$$
  (Mdlbrt, \tau_{sub})
$$

via the [[subspace topology]] $\tau_{sub}$ inherited from the [[Euclidean space|Euclidean]] [[metric topology]] of $\mathbb{C} \simeq \mathbb{R}^2$.

=--

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

The Mandelbrot space $(Mdlbrot, \tau_{sub})$ (def. \ref{MandelbrotSpace}) is a [[compact topological space]].

=--

We **prove** this [below](#ProofCompactMandelbrot), after the following lemma:

+-- {: .num_lemma #EscapeRadius}
###### Lemma
**(escape radius)**

For ${\vert c\vert} \gt 2$ then the sequence $(f_c^n(0))_{n \in \mathbb{N}}$ is not bounded, hence the sequence of [[absolute values]] $( {\vert f_c^n(0)\vert} )_{n \in \mathbb{N}}$ diverges for ${\vert c \vert} \gt 2$.

In fact in this case the absolute values increase monotonically:

If ${\vert c\vert} \gt 2$ then  for all $n \gt 0$ we have

$$
  {\vert f_c^{n+1}(0)\vert } \gt {\vert f_c^n(0)\vert}
  \,.
$$

=--

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

So assume ${\vert c \vert} \gt 2$.

We prove the last statement by [[induction]].

Observe that it is true for $n = 1$, where we have

$$
  \begin{aligned}
    {\vert f_c^2(0) \vert}
    & =
    {\vert c^2 + c \vert }
    \\
    & \geq
    {\vert c^2\vert } - {\vert c \vert }
    \\
    & =
    {\vert c\vert}\underset{\gt 1}{\underbrace{({\vert c\vert}-1)}}
    \\
    & \gt {\vert c\vert }
    \\
    & =
    {\vert f_c(0) \vert }
  \end{aligned}
  \,.
$$

Now assume that there is $n \in \mathbb{N}$ such that ${\vert f_c^n(0) \vert} \gt {\vert c\vert}$. Then it follows that

$$
  \begin{aligned}
  \frac{
    {\vert f_c^{n+1}(0)\vert}
  }{
    \vert  f_c^n(0) \vert
  }
  & =
  \frac{
     {\vert (f_c^n(0))^2  + c \vert}
  }{
    f_c^n(0)
  }
  \\
  & \geq
  \frac{
     {\vert f_c^n(0)\vert}^2 - {\vert c \vert}
  }{
    {\vert f_c^n(0)\vert}
  }
  \\
  & =
  {\vert f_c^n(0) \vert}
  -
  \frac{
    {\vert c \vert}
  }{
    {\vert f_c^n(0) \vert}
  }
  \\
  &
  \gt
  \vert c \vert - 1
  \\
  & \gt
  1
  \end{aligned}
  \,.
$$

Here the first inequality is due to the [[triangle inequality]], the second is due to the induction assumption, and the last one is due to the initial assumption that ${\vert c \vert} \gt 2$.

=--

+-- {: .proof #ProofCompactMandelbrot}
###### Proof
that the Mandelbrot space is compact (prop. \ref{MndlbrtIsCompact})

By lemma \ref{EscapeRadius} the Mandelbrot set is a [[bounded subset]] of 2d [[Euclidean space]]. Hence by the [[Heine-Borel theorem]] is is now sufficient to show that it is a [[closed subset]], this will imply that it is compact.

The subset is closed if for every point $c \in \mathbb{C} \backslash Mdlbrt$ not contained in the Mandelbrot set there is an [[open neighbourhood]] of $c$ which still does not intersect the Mandelbrot set.

Now that $c \notin Mdlbrt \subset \mathbb{R}^2$ means by definition that for every positive real number $r$ there is an $n \in \mathbb{N}$ such that $\vert f_c^n(0)\vert \gt r$.

Pick such an $n$ for $r = 2$. Let then

$$
  \epsilon \coloneqq {\vert f_c^n(0) \vert} - 2
  \,.
$$

and consider the subset

$$
  U_{c,n}
    \coloneqq
  \left\{
    z \in \mathbb{C}
    \;\vert\;
    \left(
      \left({\vert f_c^n(0) \vert} - \epsilon \right)
       \lt
      \left(  {\vert z\vert} \lt {\vert f_c^n(0) \vert} + \epsilon \right)
    \right)
  \right\}
  \,.
$$

This is clearly an [[open neighbourhood]] of $f_c^n(0)$. Hence by continuity of the function $f_c^n \colon \mathbb{C} \to \mathbb{C}$, the [[pre-image]]

$$
  (f_c^{n-1})^{-1}(U_{c,n})
$$

is an open neighbourhood of $c \in \mathbb{C}$, and by lemma \ref{EscapeRadius} this does not intersect the Mandelbrot set.

=--

## Related concepts

* [[Julia set]]

[[!redirects Mandelbrot sets]]

[[!redirects Mandelbrot space]]
[[!redirects Mandelbrot spaces]]


Manfred Hartl

Manfred Hartl is at the Université de Valenciennes et du Hainaut Cambrésis. He
is research is on polynomial functors and a form of functor calculus.

* [Research themes (in French)](http://www.univ-valenciennes.fr/lamav/Hartl/recherche.html)

category: people


Manfred Herbst

* [webpage](http://www.math.uni-augsburg.de/geo/mitarbeiter/herbst/)

category: people


Manfred Lehn

[[!redirects Mafred Lehn]]

* [webpage](http://www.mathematik.uni-mainz.de/Members/lehn)

category: people


Manfred Scheunert

* [webpage](http://www.th.physik.uni-bonn.de/People/scheuner/)

## related $n$Lab entries

* [[super Lie algebra]]

category: people


Manfredo do Carmo

* [webpage](http://en.wikipedia.org/wiki/Manfredo_do_Carmo)

## related $n$Lab entries

* [[good open cover]]

category: people


Manfredo P. do Carmo

* [Wikipedia entry](https://en.wikipedia.org/wiki/Manfredo_do_Carmo)

## Selected writings

On [[differential forms]]:

* [[Manfredo P. do Carmo]], *Differential Forms and Applications*, Springer (1994) [[doi:10.1007/978-3-642-57951-6](https://doi.org/10.1007/978-3-642-57951-6)]

category: people

[[!redirects Manfredo P. Carmo]]
[[!redirects Manfredo Carmo]]


manifold

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Manifolds and cobordisms
+-- {: .hide}
[[!include manifolds and cobordisms - contents]]
=--
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A _manifold_ is a [[topological space]] which looks locally like a [[Cartesian space]], commonly a finite-dimensional Cartesian space $\mathbb{R}^n$, in which case one speaks of a _manifold of [[dimension]] $n$_ or _$n$-fold_, but possibly an infinite-dimensional [[topological vector space]], in which case one has an [[infinite-dimensional manifold]].

What "locally looks like" means depends on what sort of structure we are considering a Cartesian space to embody. At one extreme, we can think of $\mathbb{R}^n$ as merely a [[topological space]]. Or, $\mathbb{R}^n$ may be considered as carrying more rigid types of structure, such as $C^k$-[[differential structure]], [[smooth structure]], [[PL structure|piecewise-linear (PL) structure]], real [[analytic function|analytic structure]], affine structure, hyperbolic structure, foliated structure, etc., etc. Accordingly we have notions of _[[topological manifold]]_, _[[differentiable manifold]]_, _[[smooth manifold]]_, _[[analytic manifold]]_ etc. By default these are modeled on [[finite number|finite]] [[dimension|dimensional]] spaces, but most notions have generalizations to a corresponding notion of [[infinite dimensional manifold]].

> graphics grabbed from [[The Geometry of Physics - An Introduction|Frankel]]

In any case, the type of geometry embodied in a particular flavor of manifold is controlled by a particular [[groupoid]] or, more generally, [[category]] of transformations which preserves whatever geometric features one is interested in; cf. Felix Klein's _[[Erlangen program|Erlanger Programm]]_.

## Definitions

Here we will focus on the general notion of a manifold. More concrete examples can be found in individual pages such as [[topological manifold]] and [[smooth manifold]].

We will present two possible definitions. The first, via [[pseudogroups]], has a simpler definition, but has two (rather serious) drawbacks:

1. There is in general no notion of morphisms between manifolds. At best, we can only talk about isomorphisms of manifolds.
1. We can only define smooth $n$-manifolds for each $n$ separately, but not smooth manifolds in general (and the same applies to complex manifolds etc.)

The second definition via cartologies was proposed by [[Todd Trimble]] to solve the above two problems.

### Definition via pseudogroups

The setting is a [[topological space]] $X$ together with a [[pseudogroup]] $G$ on $X$. For the sake of concreteness, the reader may as well focus on the case $X = \mathbb{R}^n$ and $G$ is the [[groupoid]] of [[diffeomorphisms]] between open subsets of $X$.

#### Definitions

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

A **[[chart]]** on a [[topological space]] $M$ is an [[open subset]] $U$ of $M$ together with an open embedding
$$
  \phi: U \to X\,.
$$

Two charts $\phi: U \to X$ and $\psi: V \to X$ are  $G$-**compatible** if
$$\psi \circ \phi^{-1}: \phi(U \cap V) \to \psi(U \cap V)$$
belongs to $G$.
=--

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

A $G$-**atlas** on a [[topological space]] $M$ is a family of  $G$-compatible charts $(\phi_\alpha: U_\alpha \to X)_\alpha$, def. \ref{Chart}, such that $(U_\alpha)_\alpha$ covers $M$.  The (restricted) maps $\phi_{\alpha \beta} = \phi_\beta \circ \phi_{\alpha}^{-1}$ are called **transition functions** between the charts of the atlas.

=--

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

A $G$-**manifold** is a [[topological space]] equipped with a $G$-atlas (definition \ref{Atlas}).

=--

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

This means that we can think of a $G$-manifold as a space which is _locally modeled_ on $X$ according to the geometry $G$.

=--

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

It is almost invariably the case in classical manifold theory that one requires some technical [[nice topological space|niceness]] properties on the [[topological space]] underlying a manifold.

Usually, in the definition of manifold it is understood that the underlying topological space

1. is a [[Hausdorff topological space]] (if not one usually speaks explicitly of a _[[non-Hausdorff manifold]]_)

1. is a [[paracompact topological space]];

Often it is also assumed that the topology has a countable [[basis of a topology|basis]] as well.

In many of the typical cases, this will mean that $M$ is [[metric space|metrizable]]. In many studies, for example in
[[cobordism theory]], one goes even further and assumes the manifolds are [[compact space|compact]].

=--

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

An atlas is _not_ considered an essential part of the structure of a manifold: two different atlases may yield the same manifold structure. This is encoded by the following definition \ref{IsomorphismOfManifolds} of [[isomorphisms]] between manifolds.

=--
#### Examples

If the term "manifold" appears without further qualification, what is usually meant is a **smooth $n$-manifold** of some [[natural number]] **dimension** $n$: a $G$-manifold where $G$ is the pseudogroup of invertible $C^{\infty}$ maps between open sets of $\mathbb{R}^n$. Replacing $\mathbb{R}^n$ here by a half-space $\{x \in \mathbb{R}^n: x_1 \geq 0\}$, one obtains the notion of smooth **manifold with boundary**. Or, replacing $\mathbb{R}^n$ here by the $n$-cube $I^n$, one obtains the notion of (smooth) $n$-**manifold with (cubical) corners**.  Morphisms of manifolds are here called **smooth maps**, and isomorphisms are called **diffeomorphisms**.  (In manifold theory, one usually reserves the term **smooth function** for smooth maps to $\mathbb{R}$.)

A **topological $n$-manifold** is a manifold with respect to the pseudogroup of homeomorphisms between open sets of $\mathbb{R}^n$. Any continuous function between topological manifolds is a morphisms, and any homeomorphism is an isomorphism. A **[[piecewise-linear manifold|piecewise-linear (PL) $n$-manifold]]** is where the pseudogroup consists of piecewise-linear homeomorphisms between such open sets; morphisms are called **piecewise-linear (PL) maps**.

One can go on to define, in a straightforward way, real [[analytic manifolds]], complex analytic manifolds, elliptic manifolds, hyperbolic manifolds, and so on, using the general notion of pseudogroup.

Any space $X$ can always be turned into a manifold modelled on itself, using any pseudogroup $G$. Simply take the inclusions of open sets as charts.

#### Isomorphisms of manifolds
+-- {: .num_defn #IsomorphismOfManifolds}
###### Definition

An **[[isomorphism]]** of $G$-manifolds $f: M \to N$ (defined by chosen atlas structures, def. \ref{Atlas}) is a [[homeomorphism]] $f$ such that

$$
  \phi(U \cap f^{-1}(V)) \overset{\phi^{-1}}{\to} U \cap f^{-1}(V) \overset{f}{\to} f(U) \cap V \overset{\psi}{\to} \psi(f(U) \cap V)
$$

is in $G$ whenever $(U, \phi)$ is a [[coordinate chart]], def. \ref{Chart} of $x \in M$, and $(V, \psi)$ is a coordinate chart of $f(x) \in N$.

If $M_1$ and $M_2$ are two $G$-manifold structures on the same topological space $M$, then $M_1$ and $M_2$ are considered **equal** as $G$-manifolds if $id: M \to M$ is an isomorphism from $M_1$ to $M_2$ (and hence also from $M_2$ to $M_1$).

=--

Alternatively, atlases are ordered by inclusion, and two atlases define the same manifold structure on $M$ if they have a common upper bound. Equivalently, two atlases define the same manifold structure if each chart of one is compatible with each chart of the other. Or, one could extend any atlas to the (unique) maximal atlas containing it, which consists of all charts compatible with each of the charts in the original atlas, and simply identify a manifold structure with a maximal atlas.

+-- {: .query}

  _Rafael_: Can one define a manifold object in a category C as a G-manifold with G related to C? What would the relation between G and C be to obtain G-manifolds in C as manifold objects?

_Toby_:  Yes, I think that this would make perfect sense; I think that we\'d want $G$ to be an [[internal groupoid]] in $C$.  Note that defining things like 'smooth manifold' in $C$ might still be difficult, but we\'ve reduced it to internalising [[Cart Sp]] in $C$.  (There\'s also the matter that the above definition takes a notion of [[space]] for granted, so you\'d have to internalise that into $C$ too, but I\'m not sure how important that really is, when I think about how the topology on a smooth manifold can be recovered from the smooth structure.)

_Rafael_: Can someone that knows more than me about this add the result of this question to this article so nobody have to ask again.

_Toby_:  I\'d rather not, since it\'s all 'I think' and 'might be difficult'; it\'s better as a query box, moved to the bottom if necessary.  But if Todd agrees with me, then maybe he\'ll add it.
=--

### Definition via cartologies

+-- {: .query}
Note: the following is tentative "original research". It is prompted by the desire to extend the pseudogroup approach for defining general notions of manifold, so as to cover also an appropriate general notion of "map". Comments, improvements, and corrections are encouraged -- _Todd_.

I\'ve read through it once, and it makes sense.  I\'ll read through it again more carefully later.  ---Toby
=--

We begin by defining the [[2-poset]] (i.e., locally [[preorder]]ed [[bicategory]]) of **regions**, denoted $Reg$. The objects are topological spaces (or locales if you prefer); the morphisms are [[partial functions]] with open domain, that is spans
$$X \overset{i}{\leftarrow} U \overset{f}{\to} Y$$
where $f$ is continuous and $i$ is an open embedding. The spans are locally (that is, for fixed $X$ and $Y$) ordered by inclusion.

These local posets are not cocomplete, but they admit certain obvious [[joins]]: given a family of regional maps
$$(U_\alpha, f_\alpha): X \to Y$$
the join $\bigvee_\alpha (U_\alpha, f_\alpha)$ exists iff we have local compatibility:
$$f_{\alpha}|_{U_\alpha \cap U_\beta} = f_{\beta}|_{U_\alpha \cap U_\beta}$$
for all $\alpha, \beta$. Notice that composition on either side with a $1$-cell preserves any local joins which exist.

Every coreflexive morphism $r \leq 1_X$ in $Reg$ [[retract|splits]]: there is a map in $Reg$,
$$Ext(r) \overset{id}{\leftarrow} Ext(r) \overset{i}{\to} X,$$
whose opposite $i^op: X \to Ext(r)$ also belongs to $Reg$ (that is, $i$ is an open embedding), and the equations
$$i^{op} \circ i = 1_{Ext(r)} \qquad i \circ i^{op} = r$$
hold. The object $Ext(r)$ may be called the _extension_ of $r$. This splitting is a kind of [[comprehension principle]] familiar from the theory of [[allegory|allegories]], among other things.

A **cartology** is a (locally [[full subcategory|full]]) subbicategory $i: C \hookrightarrow Reg$ such that

* (Closure under open subspaces) If $X \in Ob(C)$ and $r \leq 1_X$ in $Reg$, then $i: Ext(r) \to X$ and its opposite $i^{op}$ are morphisms of $C$.
* ("Sheaf condition") The inclusion $i: C \to Reg$ reflects and preserves local joins.

Intended examples include the case where the objects of $C$ are Euclidean spaces $\mathbb{R}^n$, and morphisms are spans
$$(U, f): \mathbb{R}^m \to \mathbb{R}^n$$
where $f$ is smooth.

Given a cartology $C$, a morphism $r = (U, f): X \to Y$ in $C$ is **pseudo-invertible** if there exists $s = (V, g): Y \to X$ such that $s \circ r = 1_U$ and $r \circ s = 1_V$.

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

In a cartology, the pseudo-invertible morphisms from an object $X$ to itself form a pseudogroup (as defined earlier).
=--

The notion of a $C$-manifold modeled on an object $X$ of $C$ is defined just as before, using the pseudogroup on $X$ implied by the previous lemma. In particular, we have $C$-**charts** of an atlas structure on $M$, which are morphisms in $Reg$
$$X \overset{i}{\leftarrow} U \overset{\phi}{\to} M$$
satisfying the expected properties. We can thus speak of $C$-**manifolds** (or $(C, X)$-manifolds if we want to make explicit the modeling space $X$).

Now, given a cartology $C$, we define the category of $C$-manifolds. Let $M$ be a $(C, X)$-manifold and $N$ a $(C, Y)$-manifold. Then, a $C$-**morphism** from $M$ to $N$ is a continuous map $f: M \to N$ such that the $Reg$-composite
$$\array{
&& U &&&& M &&&& V && \\
& i \swarrow && \searrow \phi && 1 \swarrow && \searrow  f && \psi \swarrow && \searrow j & \\
X &&&& M &&&& N &&&& Y
}$$
belongs to $C$, for every pair of charts $(U, \phi): X \to M$ and $(V, \psi): Y \to N$.

These definitions need to be carefully checked against known examples (e.g., the categories $Top$, $PL$, and $Smooth$, among others).

## Generalizations

* [[generalized smooth space]]

* [[differentiable stack]]

* [[derived smooth manifold]]

* [[supermanifold]]

  * [[Euclidean supermanifold]] (notice that the definition of that is very much along the lines of the Klein-program-style definition above).

* [[semialgebraic manifold]]

## Related concepts

* [[dimension of a manifold]]

* [[low-dimensional topology]]

* [[projective manifold]]

* [[manifold with boundary]]

* [[closed manifold]]

* [[product manifold]]

## References

> For more see the references at *[[differentiable manifold]]* and *[[smooth manifold]]*.

The vision of the modern notion of manifolds is attributed to:

* [[Bernhard Riemann]], *[[Über die Hypothesen, welche der Geometrie zu Grunde liegen]]*, Göttingen (1845) [[doi:10.1007/978-3-642-35121-1](https://doi.org/10.1007/978-3-642-35121-1)]

  Engl. transl: [[William Clifford]]: *[[On the hypotheses which underlie geometry]]*, Nature **VIII** (1873) 183-184 [[doi:10.1007/978-3-319-26042-6](https://doi.org/10.1007/978-3-319-26042-6)]

See also:

* John Loftin, _The real definition of a smooth manifold_ ([pdf](http://andromeda.rutgers.edu/~loftin/difffal03/manifold.pdf))

[[!redirects manifold]]
[[!redirects manifolds]]
[[!redirects manfold]]


Manifold Atlas

The _Manifold Atlas Project_ is a wiki on [[differential topology]] and [[algebraic topology]]:

* [website](http://www.map.mpim-bonn.mpg.de/Main_Page)

and a fully refereed online, open access journal

* [Bulletin of the Manifold Atlas](http://www.boma.mpim-bonn.mpg.de/)

housing refereed pages from the wiki. Pages from the wiki are subsequently independent of a published version, so can continue to develop.


manifold calculus

#Contents#
* table of contents
{:toc}

## Idea

>Let $M$ be a smooth manifold without boundary and denote by $\mathcal{O}(M)$ the poset of open subsets of $M$, as defined in [Wei99](#Wei99), ordered by inclusion. Manifold calculus is a way to study (say, the homotopy type of) contravariant functors $F$ from $\mathcal{O}(M)$ to spaces which take isotopy equivalences to (weak) homotopy equivalences. In essence, it associates to such a functor a tower - called the Taylor tower - of polynomial approximations which in good cases converges to the original functor, very much like the approximation of a function by its Taylor series. ([BriWei](#BriWei))

In [BriWei](#BriWei) the authors develop an enriched version.

##Related concepts

* [[Goodwillie calculus]]
* [[orthogonal calculus]]
* [[Weiss topology]]

##References

* {#Wei99} [[Michael Weiss]], _Embeddings from the point of view of immersion theory I_, Geom. Topol. 3 (1999), 67–101
*  [[Thomas Goodwillie]], [[Michael Weiss]], _Embeddings from the point of view of immersion theory, Part II_, Geometry and Topology 3 (1999), 103-118.

* {#BriWei} [[Pedro Boavida de Brito]], [[Michael Weiss]], _Manifold calculus and homotopy sheaves_, Homology, Homotopy and Applications, vol. 15(2), 2013, pp.361–383 ([arXiv:1202.1305](http://arxiv.org/abs/1202.1305))

* [[Hiro Lee Tanaka]], _Manifold calculus is dual to factorization homology_, [pdf](http://math.mit.edu/conferences/talbot/2012/notes/14_Tanaka_FactorizationHomology%28hiro%29.pdf)
* [[Brian Munson]], _Introduction to the manifold calculus of Goodwillie-Weiss_ ([arXiv:1005.1698](http://arxiv.org/abs/1005.1698))
* [[Thomas Willwacher]], _Configuration spaces of points and real Goodwillie-Weiss calculus_, [talk](https://www.newton.ac.uk/seminar/20181204100011001) at Isaac Newton Institute, 2018.


manifold diagram

# Contents
* automatic toc
{: toc}

## Idea

*Manifold $n$-diagrams* generalize [[string diagram|string diagrams]] to higher dimensions---they recover string diagrams in dimension $n = 2$. For $n = 3$, they specialize to the notion of [[surface diagram|surface diagrams]].

The idea of manifold diagrams has its roots in the relation between [[stratified space|stratified]] [[manifold|manifolds]] and [[higher category theory|higher categories]]. One way to think about this relation is by noting that stratified manifolds arise after dualizing (in the sense of [[Poincaré duality]]) [[pasting diagram|pasting diagrams of directed cells]] in higher categories.

A well-known instance of this relation materializes in the [[generalized tangle hypothesis]]: indeed, [[tangle|tangles]] can (in a canonical way) be regarded as examples of manifold diagrams.

Figure 1: m-Tangles in dimension n can be regarded as manifold diagrams (picture reproduces part of Fig. 30 in BaezDolan95); note that k = n-m is the codimension of the tangle).

From the perspective of the tangle hypothesis, manifold diagrams can be used to express the structure of coherently dualizable objects. More precisely, this translates between "manifold singularities" and relations satisfied by the higher morphisms associated to a coherently dualizable objects. For instance, the manifold diagram in Figure 2 expresses the "swallowtail identity" satisfied by [[biadjunction|coherent biadjunctions]] (this is a _manifold 4-diagram_: think about the two surfaces as embedded in 3-dimensional space, and in the 4th spatial dimension we deform the top surface into the bottom one). The terminology derives from Thom's classification of classical singularities (cf. [this picture](https://mathworld.wolfram.com/SwallowtailCatastrophe.html)).

Figure 2: The swallowtail identity as a manifold 4-diagram (reproducing the title image of MSRI program 323: higher categories and categorification).

Many other interesting categorical and [[higher algebra|higher-algebraic identities]] can be expressed in manifold diagrams in a similar way. As an example, the [Yang–Baxter equation](https://en.wikipedia.org/wiki/Yang%E2%80%93Baxter_equation) (or, relatedly, the Reidemeister moves) can be expressed in manifold 4-diagrams as well, see Figure 3.

Figure 3: The Yang-Baxter identity as a manifold 4-diagram, illustrated as movie of 1-tangles in dimension 3 read from left to right.

In higher dimensions the types of identites become more and more complicated: Figure 4 shows a _manifold 5-diagram_ representing the [Zamolodchikov tetrahedron equation](https://blogs.ams.org/visualinsight/2016/03/15/zamolodchikov-tetrahedron-equation/)). Much diagrammatic algebra of surfaces along these lines as been developed by [Carter](#Carter) and collaborators.

Figure 4: The Zamolodchikov tetrahedron equation as a manifold 5-diagram, illustrated as a movie from left to right (but the movie only has 2 frames).

Manifold diagrams are meant to provide a definitional framework for higher algebraic laws such as those above in _all_ dimensions. Moreover, while the above examples only illustrate such laws for tangles (i.e. for embedded ordinary manifolds), in full generality manifold diagrams are diagrams of [[stratified space|stratified]] [[manifold|manifolds]].

## Definition

### Outline

Several definitional variants exist in dimensions $\leq 4$ (see also at [[surface diagram|surface diagrams]]). In general dimensions, manifold diagrams (introduced in [Dorn-Douglas 2022](#DornDouglas22)) can be defined as conical, compactly triangulable stratifications of [[directed topological space|directed euclidean space]].

\begin{defn} A **manifold $n$-diagram** is a conical, compactly triangulable stratification of standard $n$-dimensional directed space.
\end{defn}

Here, "standard directed space" can be taken to mean [[directed topological space#framed_spaces|standard $\mathbb{R}^n$ with standard $n$-framing]]. The term "conical" means a directed version of usual [[stratified space#conical_strat|conicality]]; namely, in the context of directed spaces one requires tubular neighborhoods to interact nicely with the directions of the underlying directed space. "Compact triangulability" means, roughly, that up to isomorphism a description by finite data can be given.

### Details

We provide details for the preceding definition.

\begin{defn} The _standard $n$-framing_ of $\mathbb{R}^n$ is the chain of oriented $\mathbb{R}$-fiber bundles $\pi_i : \mathbb{R}^i \to \mathbb{R}^{i-1}$ ($1 \leq i \leq n$) with $\pi_i$ defined to be the map that forgets the last coordinate of $\mathbb{R}^i$ (and fibers carry the standard orientation of $\mathbb{R}$ after identifying $\mathbb{R}^i = \mathbb{R}^{i-1} \times \mathbb{R}$).
\end{defn}

When considering $\mathbb{R}^n$ we tacitly always think of it as 'standard framed $\mathbb{R}^n$' and, thus, we stop mentioning the standard framing as an explicit structure all-together. Indeed, more important than defining the standard $n$-framing is to define the maps that preserve it.

{#framed_map_euclidean_case}
\begin{defn} A _framed map_ $F : \mathbb{R}^n \to \mathbb{R}^n$ is a map for which there exist (necessarily unique) maps $F_j : \mathbb{R}^j \to \mathbb{R}^j$ ($0 \leq j \leq n$) with $F_n = F$ such that $\pi_i \circ F_i = F_{i-1} \circ \pi_i$ with $F_i$ preserving orientations of fibers of $\pi_i$ (i.e. mapping fibers strictly monotously).
\end{defn}

A framed _stratified_ map $(\mathbb{R}^n,f) \to (\mathbb{R},g)$ is a [[stratified space#the_category_of_stratifications|stratified map]] whose underlying map $\mathbb{R}^n \to \mathbb{R}^n$ is framed. Moreover, when working with products $(\mathbb{R}^k,f) \times (\mathbb{R}^{n-k}, g)$ we will identitify $\mathbb{R}^n \cong \mathbb{R}^k \times \mathbb{R}^{n-k}$ in the standard way; and, when working with [[stratified space#conical_strat|cone stratifications]] $(\mathrm{Cone}(S^{n-1}), \mathrm{cone}(l))$, we will standard embed $S^{n-1} \into \mathbb{R}^n$ and identify $\mathrm{Cone}(S^{n-1}) \cong \mathbb{R}^n$ by mapping $(x \in S^{n-1},\lambda \in [0,1))$ to $\frac{\lambda}{1 - \lambda}x \in \mathbb{R}^n$.

\begin{defn} A stratification $(\mathbb{R}^n,f)$ is _framed conical_ if each point $x \in \mathbb{R}^n$ it has a framed stratified neighborhood (framed) homeomorphic to $\mathbb{R}^k \times (\mathrm{Cone}(S^{n-k-1}), \mathrm{cone}(l))$ with $x \in \mathbb{R}^k \times \{0\}$, where $0 \leq k \leq n$ and $(S^{n-1}, l)$ is some stratification.
\end{defn}

\begin{defn} A _compactly-described triangulation_ $K$ of $\mathbb{R}^n$ is a finite stratification of $\mathbb{R}^n$ by open disks whose closures are the images of [linear embeddings](https://math.stackexchange.com/questions/1326711/what-is-a-linear-embedding-from-a-simplex-deltan-to-mathbbrn) $\Delta^k \times \mathbb{R}^l_{\geq 0} \into \mathbb{R}^n$ ($k + l \leq n$). This now translates to the framed stratified case as follows: a stratification $(\mathbb{R}^n,f)$ is _framed compactly triangulable_ if it admits a framed stratified subdivision $(\mathbb{R}^n,K) \to (\mathbb{R}^n,f)$ of $f$ by a compactly-described triangulation $K$.
\end{defn}

We can now put these concepts together to obtain the following form definition of manifold diagrams.

\begin{defn}(Formalization of earlier sketch). A **manifold $n$-diagram** is a framed conical, framed compactly triangulable stratification of $\mathbb{R}^n$.
\end{defn}

The natural 'isomorphism' relation of manifold diagrams is framed stratified homeomorphism.

## Properties

The definition has several important consequences, which we list in this section. These consequences have been formally worked out in [Dorn and Douglas 22](#DornDouglas22). (Note the primary definition in the latter text differs from that given here, but a comparison is given.)

### Combinatorialization theorem

While the compact triangulability condition requires the existence of _some_ combinatorial representation, there is in fact a _canonical_ combinatorial representation for isomorphism classes of manifold diagrams. This leads to a theory of combinatorial objects called [[n-truss|trusses]]. The full 'combinatorialization theorem' is spelled out [[n-truss#manifold_diagram_classification|here]].

### Framed smooth structure of strata

Strata in manifold diagrams are, indeed, manifolds. Moreover, they have canonical smooth structures. This follows since manifold strata carry canonical PL structure, and are canonically framed, and since framed PL manifolds have unique compatible framed smooth structure.

### Well-definedness of links

Links of tubular neighborhoods in manifold diagrams are in fact well-defined. That is, there is, up to isomorphism, a unique choice of link. This stands in contrast to classical topological conical stratifications, where non-homeomorphic choices of links exist.

### Dual cellular theory of manifold diagrams

Manifold diagrams have canonical geometric duals (in the sense of Poincaré duality). This relates manifold diagrams to 'diagrams of directed cells' in the sense of classical [[pasting diagram|pasting diagrams]] (but with  cell shapes that generalize many common classes of shapes such as 'globular', 'simplicial', or 'opetopic' shapes). Details are spelled out in [Dorn and Douglas 22, Sec. 2.4](#DornDouglas22).

## Related concepts

* [[n-truss]]

* [[n-mesh]]

* [[manifold-diagrammatic n-category]]

* [[homotopy.io]]

* [[directed topological space]]

## References

* {#BaezDolan95} [[John Baez]] and [[James Dolan]], _Higher-dimensional Algebra and Topological Quantum Field Theory_, 1995 ([arXiv](http://arxiv.org/abs/q-alg/9503002))

* {#Carter} [[Scott Carter]], _An excursion in diagrammatic algebra_, 2012, World Scientific ([doi:10.1142/8315](https://doi.org/10.1142/8315))

* {#DornDouglas22} [[Christoph Dorn]] and [[Christopher Douglas]], _Manifold diagrams and tame tangles_, 2022 ([arXiv](https://arxiv.org/abs/2208.13758), [latest](https://cxdorn.github.io/manifold-diagram-paper/))

* {#Dorn23} [[Christoph Dorn]], _Nine short stories about geometric higher categories_, 2023 ([pdf](https://cxdorn.github.io/assets/pdfs/nine-stories.pdf))

Brief introductions:

* *From zero to manifold diagrams*, TallCat Seminar Talk, ([video](https://www.youtube.com/watch?v=67RAzY-13r4))

* *[An Invitation to Geometric Higher Categories](https://golem.ph.utexas.edu/category/2023/03/an_invitation_to_geometric_hig.html)*, $n$-Category Café post

* [[Christoph Dorn]], *Manifold Diagrams -- A Brief Report*, [talk at](https://ncatlab.org/nlab/show/Center+for+Quantum+and+Topological+Systems#DornNov2023) [[CQTS]] (15 Nov 2023) [[[Dorn-ManifoldDiagramsReport.pdf:file]], [YT](https://youtu.be/7KK89MuxmX8)]

[[!redirects manifold diagrams]]


manifold structure of mapping spaces

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Mapping space
+--{: .hide}
[[!include mapping space - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea ##

The category of [[smooth manifolds]] is not [[cartesian closed]], even when [[infinite-dimensional manifolds]] are allowed.  However, that does not mean that _no_ [[mapping spaces]] between certain smooth manifolds can be given the structure of a smooth manifold.  This is true when the source is [[compact topological space|compact]].  Thus, in particular, this applies to [[loop spaces]].

The method of proving this depends mostly on the structure of the target and only minimally on that of the source.  It is not hard to generalise it to [[manifolds with boundary]] (to get, for example, [[path spaces]]), or even manifolds with corners.  This raises the obvious question as to how general this result can be made.  The purpose of this page is to determine the answer.  Our conjecture is the following:

+-- {: .num_theorem #mfdconj}
###### Conjecture
Let $N$ be a [[Frölicher space]] whose [[curvaceous topology]] is [[sequentially compact]].  Let $M$ be a smooth manifold that admits a [[local addition]].  Then the Frölicher space of smooth maps from $N$ to $M$ is a smooth manifold.
=--

## Construction of smooth manifold structure on mapping space
 {#ConstructionOfSmoothManifoldStructureOnMappingSpace}

> The following needs attention. For a more recent version see ([Stacey](#Stacey)).

### Background and Remarks

The question discussed here can be viewed as the counterpoint to the oft-heard maxim (attributed to Grothendieck):

> It is better to work in a nice category with nasty objects than in a nasty category with nice objects.

Smooth manifolds are an example of "nice objects in a nasty category"; for example, one can rarely take subobjects or quotients.  The standard procedure at this point is to embed the nasty category in some larger, nicer category and work there.  In the case of smooth manifolds, this has led to all of the categories that are listed at [[generalized smooth space]].

One can now go on to study this enlarged category, and investigate how much of what is known about the original category extends to the larger one.  In this line, the original category is viewed mainly as a source of ideas.  An alternative approach, and that taken here, is to view the original category as being a subcategory of "special objects" inside the larger one.

One can make an analogy with the real and complex numbers.  Many aspects of the study of real numbers become much easier and clearer when extended to the complex numbers.  At this point, one has a choice: one can simply study the complex numbers or one can use the complex numbers as a tool to study the real ones.

Thus, to adapt a saying of Hadamard ([here](Jacques+Hadamard#TheShortestPath)), we could introduce our own maxim:

> The shortest distance between two truths about nice objects often lies in a nice category.

Having mentioned the plethora of extensions of the category of smooth manifolds, we should comment on our choice of [[Frölicher spaces]].
The inclusion of the category of smooth manifolds into each of the extensions factors through the category of Frölicher spaces.  Therefore, if we work in, say, the category of [[diffeological spaces]] then we can split the question "Is the diffeological space $X$ a smooth manifold?" into "Is $X$ a Frölicher space?" and "Is the resulting Frölicher space a manifold?".  Moreover, as we are interested in $C^\infty(N,M)$ with $M$ a smooth manifold (and thus a  Frölicher space), then if we are working with one of the "maps in" approaches, we can replace the $N$ in $C^\infty(N,M)$ by its "Frölicherification" without changing the set.  Thus the key piece of the puzzle is to study $C^\infty(N,M)$ for $N$ a Frölicher space and the rest will follow by applying "general nonsense".

Another remark worth saying is that the conjecture stated is not the most general statement that could be considered.  It is simple to extend this conjecture to a relative version whereby $M$ is equipped with a family of submanifolds and $N$ with a family of subsets and the maps are constrained to take the subsets to the corresponding submanifolds.

Finally, let us note that the main results about the linear model spaces are recorded on the page [[linear mapping spaces]].

### Charts
 {#Charts}

Let $M$ be a [[smooth manifold]] (possibly [[infinite dimensional smooth manifold|infinite dimensional]]) modelled on the [[convenient vector space]] $V$.  Let $N$ be a [[sequentially compact]] [[Frölicher space]].  Let $\{P_i : P_i \subseteq M\}$ be a family of submanifolds of $M$.  Let $\{Q_i : Q_i \subseteq N\}$ be a [[family of subsets]] of $N$ with the same indexing set.

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

We write $C^\infty(N,M;Q_i,P_i)$ for the set of [[smooth functions]] $N \to M$ which map each $Q_i$ into the corresponding $P_i$.

=--

As a smooth manifold, $M$ naturally has the structure of a Frölicher space so this mapping space is well-defined.

We assume that the _pair_ $(M,\{P_i\})$ admits a [[local addition]].  By that, we mean that $M$ admits a local addition, say $\eta$, with the property that it restricts to a local addition on each $P_i$.  We shall also assume, for simplicity, that the domain of $\eta$ is $T M$.

Let $g \colon N \to M$ be a smooth map with $g(Q_i) \subseteq P_i$.  Let $E_g$ be the space of sections of $g^* T M$ with the property that the sections over each $Q_i$ are constrained to lie in the corresponding $g^* T P_i$.  In more detail, we define $g^* T M$ in the usual manner:

$$
g^* T M \coloneqq \{(x,v) \in N \times T M : g(x) = \pi(v)\}
$$

and then take the space of smooth maps $f \colon N \to g^* T M$ with the property that the composition $N \to g^* T M \to N$ is the identity.  Within that space, we further restrict to those $f$ such that the image of the map $Q_i \to g^* T M \to T M$ lies in $T P_i$.

Although $N$ could be quite complicated, because $T M \to M$ is a vector bundle, $E_g$ is a vector space.  Furthermore, by trivialising $g^* T M$ using a finite number of trivialisations (possible as $N$ is sequentially compact), we can embed $E_p$ as a closed subspace of $C^\infty(N,V^n)$ for some $n$.  This embedding shows that $E_p$ is a [[convenient vector space]], in the sense of [Kriegl and Michor](#km).

+-- {: .query}
[[Andrew Stacey]] This, I think, is the crucial part: that $E_p$ is a convenient vector space.  I need to expand on this and check that all is as I think it is.
=--

We define a map for $\Phi \colon E_g \to C^\infty(N,M;\{Q_i\},\{P_i\})$ as follows.  Let $f \in E_p$.  Then $f$ is a section of $g^* T M$ and so is a map $N \to g^* T M$.  By the definition of $g^* T M$, we can think of $f$ as a map $N \to N \times T M$ which projects to the identity on the first factor.  By applying the projection to the second factor, we obtain a map $\hat{f} \colon N \to T M$.  Composing with $\eta$ produces a map $\eta \circ \hat{f} \colon N \to M$.  As $f \in E_g$, the restriction of $\hat{f}$ to $Q_i$ lands in $T P_i$, whence $\eta \circ \hat{f}$ takes $Q_i$ into $P_i$.  The map $f \mapsto \eta \circ \hat{f}$ is what we call $\Phi$.

Let us identify its image.  Let $V \subseteq M \times M$ be the image of the local addition.  Define $U_g \subseteq C^\infty(N,M;\{Q_i\},\{P_i\})$ to be the set of those functions $h$ such that $(g,h) \colon N \to M \times M$ takes values in $V$.  We claim that the image of $\Phi$ is $U_g$ and that $\Phi$ is a bijection $E_g \to U_g$.

Let us start with the image.  Let $h \in U_g$.  Then $(g,h) \colon N \to M \times M$ takes values in $V$, so we can compose with $(\pi \times \eta)^{-1}$ to get a map $\check{h} \colon N \to T M$.  Together with the identity on $N$, we get a map $N \to N \times T M$.  By construction, $\pi \check{h} = g$ and so this map ends up in $g^* T M$ (which has the subspace structure).  Again by construction, the projection of this map to $N$ is the identity and so it is a section of $g^* T M$.  That it takes $Q$ to $T P$ follows from the fact that $\eta$ restricts to a local addition on $P$, whence as $h(Q) \subseteq P$, $\check{h}(Q) \subseteq T P$.  Hence $\Phi$ is onto.  Moreover, this construction yields the inverse of $\Phi$ and so it is a bijection.

Thus we have charts for $C^\infty(N,M;Q,P)$.

### Transition functions

The next step is the transition functions.  To prove this in full generality, we assume not just two different functions at which to base our charts, but also two different local additions to define them.  This will show that our resulting manifold structure is independent of this choice.  We could go further than we do, and allow our local additions to be in the most general form given at [[local addition]], but this would crowd the notation with little benefit.

Thus we start with $g_1, g_2 \in C^\infty(N,M;Q,P)$ and two local additions $\eta_1, \eta_2 \colon T M \to M$.  Let us write $V_1$ and $V_2$ for the images of $(\pi \times \eta_1)$ and $(\pi \times \eta_2)$.  Let $E_1 = E_{g_1}$ and $E_2 = E_{g_2}$.

We define $W_{1 2} \in g_1^* T M$ as follows.  We describe a point in $g_1^* T M$ by specifying its point in $N \times T M$.

$$
W_{1 2} \coloneqq \{(x,v) \in g_1^* T M : (g_2(x),\eta_1(v)) \in V_2\}
$$

and $W_{2 1} \subseteq g_2^* T M$ similarly.

+-- {: .num_lemma #diff}
###### Lemma
$W_{1 2} \cong W_{2 1}$
=--

+-- {: .proof}
###### Proof
The map $g_2 \times \eta_1 \colon N \times T M \to M \times M$ is smooth and so the preimage of $V_2$ under this map is open in $N \times T M$.  Thus $W_{1 2}$ is open in $g^* T M$ and the map $(x, v) \mapsto (g_2(x), \eta_1(v))$ is a smooth map $W_{1 2} \to V_2$.  Since $(\pi \times \eta_2) \colon T M \to V_2$ is a diffeomorphism, we can define a smooth map $\theta_1 \colon W_{1 2} \to T M$ by

$$
\theta_1(x,v) = (\pi \times \eta_2)^{-1}(g_2(x), \eta_1(v))
$$

Now

$$
\pi \theta_1(x,v) = \pi(\pi \times \eta_2)^{-1}(g_2(x),\eta_1(v)) = g_2(x)
$$

so $\theta_1(x,v) \in T_{g_2(x)} M$ and thus $(x,\theta_1(x,v)) \in g_2^* T M$ for all $(x,v) \in W_{1 2}$.  Then

$$
\eta_2 \theta_1(x,v) = \eta_2(\pi \times \eta_2)^{-1}(g_2(x),\eta_1(v)) = \eta_1(v)
$$

so $(g_1(x),\eta_2\theta_1(x,v)) = (g_1(x),\eta_1(v))$ which is in $V_1$.  Hence $(x,\theta_1(x,v)) \in W_{2 1}$.  Thus we have a map

$$
\phi_{2 1} \colon W_{1 2} \to W_{2 1}, \qquad \phi_{1 2}(x,v) = (x, \theta_1(x,v))
$$

Similarly, we have a map $\phi_{1 2}$ in the other direction.  Both of these maps are smooth since they are smooth into $N \times T M$.

Let us consider $\phi_{1 2}\phi_{2 1}(x,v)$.  Expanding this out yields:

$$
\begin{aligned}
\phi_{1 2}\phi_{2 1}(x,v) &= \phi_{1 2}(x, \theta_1(x,v)) \\
&=(x, \theta_2(x, \theta_1(x,v))) \\
&=(x, (\pi \times \eta_1)^{-1}(g_1(x), \eta_2\theta_1(x,v))) \\
&=(x, (\pi \times \eta_1)^{-1}(g_1(x), \eta_1(v))) \\
&=(x, (\pi \times \eta_1)^{-1}(\pi(v),\eta_1(v))) \\
&=(x,v)
\end{aligned}
$$

where we have used the fact that $(x,v) \in g_1^*T M$ so $\pi(v) = g_1(x)$.  Thus $\phi_{2 1}$ and $\phi_{1 2}$ are inverses, whence they are diffeomorphisms.
=--

+-- {: .num_lemma #trans}
###### Lemma
The transition function is $f \mapsto \phi_{2 1} \circ f$.
=--

+-- {: .proof}
###### Proof
Let us start with the domain and codomain of the transition function.  The domain is $\{f \in E_1 : \Phi_1(f) \in U_2\}$.  The set $U_2$ consists of those functions $h \colon N \to M$ such that $(g_2, h)$ takes values in $V_2$.  Thus $\Phi_1(f) \in U_2$ if and only if $(g_2, \Phi_1(f)) \in V_2$.  Since $\Phi_1(f) = \eta_1 \circ \hat{f}$, we see that for $x \in N$, $v \coloneqq \hat{f}(x) \in T M$ must be such that $(g_2(x), \eta_1(v)) \in V_2$.  This is precisely the condition that $(x,v)$ be in $W_{1 2}$.  Thus the domain of the transition function is the set of sections $f \in E_1$ such that $f(x) \in W_{1 2}$ for each $x \in N$.

The transition function, $\Psi_{2 1}$, is given by $\Psi_{2 1} = \Phi_2^{-1} \Phi_1$.  It is therefore completely characterised by the fact that $\Phi_2 \Psi_{2 1} = \Phi_1$.

Let us consider $\Phi_2$ applied to $\psi_{2 1} \circ f$ for $f \in E_1$ such that $f$ takes values in $W_{1 2}$.  Expanding out the definition, we have:

$$
\begin{aligned}
(\psi_{2 1} \circ f)(x) &= \psi_{2 1}(f(x)) \\
&= (x, \theta_1(x, \hat{f}(x))) \\
&= (x, (\pi \times \eta_2)^{-1}(g_2(x), \eta_1\hat{f}(x)))
\end{aligned}
$$

Now the result applying $\Phi_2$ to $h$ is $\eta_2 \circ h$ where $h(x) = (x, \hat{h}(x))$.  Thus the result of applying $\Phi_2$ to $\psi_{2 1} \circ f$ is the function

$$
x \mapsto \eta_2 (\pi \times \eta_2)^{-1}(g_2(x), \eta_1\hat{f}(x)) = \eta_1\hat{f}(x)
$$

which is exactly the same function as $\Phi_1(f)$.  Hence

$$
\Psi_{2 1}(f) = \psi_{2 1} \circ f
$$

and thus $\Psi_{2 1}$ is a diffeomorphism.
=--

### Conclusion

In conclusion we have

+-- {: .num_corollary #cordiff}
###### Corollary

The mapping space $C^\infty(N,M;\{Q_i\},\{P_i\})$
of def. \ref{TheMappingSpace} equipped with charts as discussed [above](#Charts)
is a smooth manifold.

=--

## Properties

### Relation between diffeological and Fréchet manifold structure
 {#RelationBetweenDiffeologicalAndFrechetStructure}

Since smooth manifolds form a [[full subcategory]] of [[diffeological spaces]], the mapping space $C^\infty(X,Y)$ between two manifolds always exists canonically as a diffeological space:

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

For $U \in $ [[CartSp]], a smooth plot of $C^\infty(X,Y)_{diff}$ over $U$ is a [[smooth function]] $U \times X \to Y$ (hence a $U$-parameterized smooth collection of smooth functions $X \to Y$).

=--

If $X$ is a [[compact manifold]] then there is also the structure of a [[Fréchet manifold]] $C^\infty(X,Y)_{Fr}$ on the mapping space. We discuss that and how these two smooth structures coincide.

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

Define a functor

$$
  \iota \colon FrechetManifolds \to DiffeologicalSpaces
$$

in the evident way by taking for $X$ a [[Fréchet manifold]] for any $U \in $ [[CartSp]] the set of $U$-plots of $\iota(X)$ to be the set of smooth functions $U \to X$.

=--

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

The functor $\iota \colon FrechetManifolds \hookrightarrow DiffeologicalSpaces$ is a [[full and faithful functor]].

=--

This appears as ([Losik, theorem 3.1.1](#Losik)).

Since moreover diffeological spaces are fully faithful in [[smooth sets]] (them being precisely the [[concrete objects]] in smooth sets), this implies in particular that Frechet manifolds are fully faithful in [[smooth sets]]

$$
  FrechetManifolds
    \hookrightarrow
  DiffeologicalSpaces
    \hookrightarrow
  SmoothSets
  \,.
$$

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

Let $X, Y \in SMoothManifold$ with $X$ a [[compact manifold]].

Then under this embedding, the diffeological mapping space structure $C^\infty(X,Y)_{diff}$ on the mapping space coincides with the Fréchet manifold structure $C^\infty(X,Y)_{Fr}$:

$$
  \iota(C^\infty(X,Y)_{Fr})
  \simeq
  C^\infty(X,Y)_{diff}
  \,.
$$

=--

This appears as ([Waldorf, lemma A.1.7](#Waldorf)).

## References

* {#Losik} M. V. Losik, _Categorical Differential Geometry_ Cah. Topol. Géom. Différ.
Catég., 35(4):274–290, 1994.

* {#Waldorf} [[Konrad Waldorf]], _Transgression to Loop Spaces and its Inverse I_ ([arXiv:0911.3212](http://arxiv.org/abs/0911.3212))

* {#Stacey}[[Andrew Stacey]], _Yet more smooth mapping spaces and their smoothly local properties_ arXiv:[1301.5493](http://arxiv.org/abs/1301.5493)

Generalising the Lie group structure on the diffeomorphisms of a manifold, the case of non-compact [[orbifolds]] is in

* [[Alexander Schmeding]], _The diffeomorphism group of a non-compact orbifold_, PhD thesis, Paderborn 2013, Dissertationes Math. (Rozprawy Mat.) **507** (2015), DOI: [10.4064/dm507-0-1](http://dx.doi.org/10.4064/dm507-0-1), [nbn-resolving.de/urn:nbn:de:hbz:466:2-12166](http://digital.ub.uni-paderborn.de/ubpb/urn/urn:nbn:de:hbz:466:2-12166), arXiv:[1301.5551](https://arxiv.org/abs/1301.5551)

The generalization to [[mapping stacks]] of [[differentiable stacks]] is discussed in

* {#RobertsVozzo16} [[David Roberts]], [[Raymond Vozzo]], _The smooth Hom-stack of an orbifold_ ([arXiv:1610.05904](https://arxiv.org/abs/1610.05904))

[[!redirects mapping spaces that are manifolds]]

[[!redirects smooth mapping space]]
[[!redirects smooth mapping spaces]]
[[!redirects manifolds of mapping spaces]]


manifold with boundary

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[manifold]] is a [[topological space]] that is locally isomorphic to a [[Cartesian space]] $\mathbb{R}^n$.

A **manifold with [[boundary]]** is a topological space that is locally isomorphic either to an $\mathbb{R}^n$ or to a **half-space** $H^n = \{ \vec x \in \mathbb{R}^n | x^n \geq 0\}$.

A **manifold with corners** is a topological space that is locally isomorphic to an $H^n_i = \{ \vec x \in \mathbb{R}^n | x^{i+1}, \cdots, x^n \geq 0\}$ for $0 \leq i \leq n$.

For details see at _[[manifold]]_.

## Properties

### Collar neighbourhood theorem

* [[collar neighbourhood theorem]]

### Embedding into the category of diffeological spaces

+-- {: .num_prop #SmoothManifoldsWithBoundaryEmbedIntoDiffeologicalSpaces}
###### Proposition
**([[manifolds with boundaries and corners]] form [[full subcategory]] of [[diffeological spaces]])**

The evident [[functor]]

$$
  SmthMfdWBdrCrn \overset{\phantom{AAAA}}{\hookrightarrow} DiffeologicalSpaces
$$

from the [[category]] of [[smooth manifold|smooth]] [[manifolds with boundaries and corners]] to that of [[diffeological spaces]] is [[fully faithful functor|fully faithful]], hence is a [[full subcategory]]-embedding.

=--

([Iglesias-Zemmour 13, 4.16](#IglesiasZemmour13), [Gürer & Iglesias-Zemmour 19](#GurerIZ19))

## Related concepts

* [[asymptotic boundary]]

* [[manifold with corners]]

* [[closed manifold]]

* [[MOFr]], [[MUFr]], [[MSUFr]]

## References

### On manifolds with boundary

* _Manifolds with boundary_ ([pdf](http://math.ucr.edu/~res/math260s10/manwithbdy.pdf), [[ManifoldsWithBoundary.pdf:file]])

On [[cobordism theory]] of [[MUFr]]-[[manifolds with boundaries]], their [[e-invariant]] and their appearance in the first line of the [[Adams-Novikov spectral sequence]]:

* {#ConnerFloyd66} [[Pierre Conner]], [[Edwin Floyd]], Section 16 of: _The relation of cobordism to $K$-theories_, Lecture Notes in Mathematics __28__ Springer 1966 ([doi:10.1007/BFb0071091](https://link.springer.com/book/10.1007/BFb0071091), [MR216511](http://www.ams.org/mathscinet-getitem?mr=216511))

### On manifolds with corners

* [[Klaus Jänich]], Section 1.2 of: _On the classification of $O(n)$-manifolds_, Math. Ann. 176, 53–76 (1968) ([doi:10.1007/BF02052956](https://doi.org/10.1007/BF02052956))

* [[Dominic Joyce]], _On manifolds with corners_ ([arXiv:0910.3518](http://arxiv.org/abs/0910.3518))

The [[full subcategory]]-embedding of manifolds with boundaries and corners into that of [[diffeological spaces]] is discussed in:

* {#IglesiasZemmour13} [[Patrick Iglesias-Zemmour]], section 4.16 of _Diffeology_, Mathematical Surveys and Monographs, AMS (2013) ([web](http://math.huji.ac.il/~piz/Site/The%20Book.html), [publisher](http://www.ams.org/bookstore-getitem/item=SURV-185))

* {#GurerIglesiasZemmour17} [[Serap Gürer]],  [[Patrick Iglesias-Zemmour]], _Differential forms on corners_, 2017  ([pdf](http://math.huji.ac.il/~piz/documents/DBlog-Rmk-DFOC.pdf))

* {#GurerIZ19} [[Serap Gürer]],  [[Patrick Iglesias-Zemmour]], _Differential forms on manifolds with boundary and corners_, Indagationes Mathematicae, Volume 30, Issue 5, September 2019, Pages 920-929 ([doi:10.1016/j.indag.2019.07.004](https://doi.org/10.1016/j.indag.2019.07.004))

On [[cobordism theory]] of [[manifolds with corners]]:

* Gerd Laures, _On cobordism of manifolds with corners_, Trans. Amer. Math. Soc. 352 (2000) ([doi:10.1090/S0002-9947-00-02676-3](https://doi.org/10.1090/S0002-9947-00-02676-3))

  > (their [[f-invariant]] and their appearance in the second line of the [[Adams-Novikov spectral sequence]])

* {#Genauer12} Josh Genauer, _Cobordism categories of manifolds with corners_, Transactions of the American Mathematical Society Vol. 364, No. 1 (2012), pp. 519-550 ([arXiv:0810.0581](https://arxiv.org/abs/0810.0581), [jstor:41407770](https://www.jstor.org/stable/41407770), [doi:10.1090/S0002-9947-2011-05474-7](https://doi.org/10.1090/S0002-9947-2011-05474-7))

[[!redirects manifold with boundaries]]
[[!redirects manifolds with boundary]]
[[!redirects manifolds with boundaries]]

[[!redirects manifold with corners]]
[[!redirects manifolds with corners]]

[[!redirects manifold with boundaries and corners]]
[[!redirects manifolds with boundaries and corners]]

[[!redirects smooth manifold with boundaries and corners]]
[[!redirects smooth manifolds with boundaries and corners]]