Mordell conjecture

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## Idea

>Il s’agit là d’une question qu’un arithméticien ne peut guère manquer de se poser; on n’aperçoit d’ailleurs aucun motif sérieux de parier pour ou contre. Peut-être dira-t-on que l’existence d’une infinité de solutions rationnelles pour une équation $f(x,y)=0$, en l’absence d’une raison algébrique qui la justifie, est infiniment peu probable. Mais ce n’est pas un argument.... (André Weil, [Commentaire 1979](#Weil79))

The _Mordell conjecture_ or _Falting's theorem_ is a statement about the [[finite set|finiteness]] of [[rational points]] on an [[algebraic curve]] over a [[number field]] of [[genus]] $g \gt 1$.

Its statement motivated the introduction of [[anabelian geometry]] ([Grothendieck](#Grothendieck)).

The Mordell conjecture completed a classification of the behavior of rational points on curves over $\mathbb{Q}$. For genus 0 one has no points, or something isomorphic to $\mathbb{P}^1$ and hence infinitely many. For genus 1, the [[Mordell-Weil theorem]] tells us it is either empty or a finitely generated [[abelian group]] (sometimes finite, sometimes infinite). To finish off this type of classification the Mordell conjecture shows that only the finite case can occur for higher genus.

This result also implies many non-trivial results. For example, fix a finite set of primes $S$, a dimension $n$, and a [[polarization degree]] $d$. There are only finitely many [[abelian varieties]] of dimension $n$ and polarization degree $d$ with [[bad reduction]] inside $S$. This result is so interesting that a whole industry popped up asking what other varieties this type of behavior occurs for. It seems to work for [[K3]]s.

## Sketch of Faltings' proof

We follow the seminar notes [BhattSnowden](#BhattSnowden).

The proof proceeds along the following steps:

1. Using _Parshin's trick_ and passing to the [[Jacobian variety]] to reduce the Mordell conjecture to the _Shafarevich conjecture_ that there are only finitely many [[abelian varieties]] of dimension $g$ and polarization degree $d$ with good reduction outside a fixed set of places $S$.

2. Using the semisimplicity of the [[Tate module]] and the _isogeny theorem_
$$Hom_{K}(A,B)\otimes\mathbb{Z}_{\ell}\cong Hom_{G_{K}}(T_{\ell}(A),T_{\ell}(B))$$
for abelian varieties $A$ and $B$ over $K$, to prove the Shafarevich conjecture. The idea is that there are only finitely many possibilities for the Tate module, which shows finiteness up to isogeny; a height argument then shows finiteness up to isomorphism.

3. Showing that the semisimplicity of the Tate module and the isogeny theorem will follow if there are only finitely many abelian varieties (up to isomorphism) isogenous to a fixed abelian variety $A$ (follows a strategy originally used by Tate to prove the analogous result over finite fields).

4. Showing that there are only finitely many abelian varieties (up to isomorphism) isogenous to a fixed abelian variety $A$ by relating the Faltings height to the moduli-theoretic height (for which there are only finitely many points of bounded height) using [[Arakelov theory]] then studying how the Faltings height changes under isogeny. This makes use of the Hodge-Tate decomposition from [[p-adic Hodge theory]].

## Other proofs

Other proofs of the Mordell conjecture using different methods have been given by [[Paul Vojta]] ([Vojta91](#Vojta91)) and [[Enrico Bombieri]] ([Bombieri90](#Bombieri90)). Recently another proof involving period mappings together with o-minimality methods was given by [[Brian Lawrence]] and [[Akshay Venkatesh]] ([LawrenceVenkatesh18](#LawrenceVenkatesh18)).

## Effective Mordell conjecture

Faltings' proof of the Mordell conjecture determines the finiteness of the rational points but it does not find the set of rational points. The latter problem is also known as the _effective Mordell conjecture_. An approach towards this has been proposed by [[Minhyong Kim]] and is related to his proposed [[arithmetic gauge theory]] (see also [Balakrishnan19]({#Balakrishnan19}) for a survey).

## Relation to other statements

The Mordell conjecture is implied by the [[abc conjecture]]. (See there.)

The Mordell conjecture implies [[Tate's isogeny theorem]].

See also [[Vojta's conjecture]].

## References

The Mordell conjecture originates in

* [[Louis Mordell]],  _On the rational solutions of the indeterminate equation of the third and fourth degrees_, Proc. Cambridge Philos. Soc. 21: 179–192 (1922)

It was proven in

* {#Faltings83} [[Gerd Faltings]], *Endlichkeitssätze für abelsche Varietäten über Zahlkörpern*, Inventiones Mathematicae **73** 3 (1983) 349-366  [[doi:10.1007/BF01388432](https://doi.org/10.1007/BF01388432)]

Seminar notes on Faltings' proof:

* {#BhattSnowden} [[Bhargav Bhatt]], [[Andrew Snowden]] (org.): _Faltings' proof of the Mordell conjecture_ (seminar notes, 2016) [[pdf](https://web.math.princeton.edu/~takumim/Mordell.pdf), [[BhattSnowden-FaltingsProofofMordell.pdf:file]]]

Other proofs after Faltings:

* {#Vojta91} [[Paul Vojta]], 1991, _Siegel's theorem in the compact case_, Ann. of Math. 133 (3): 509–548.

* {#Bombieri90} [[Enrico Bombieri]], _The Mordell conjecture revisited_, _Annali della Scuola Normale Superiore di Pisa - Classe di Scienze_,  Sér. 4, 17 no. 4 (1990), p. 615-640 [[numdam:ASNSP_1990_4_17_4_615_0](http://www.numdam.org/item?id=ASNSP_1990_4_17_4_615_0)]

* {#LawrenceVenkatesh18} [[Brian Lawrence]], [[Akshay Venkatesh]]: _Diophantine problems and $p$-adic period mappings_,  Invent. math. **221** (2020) 893–999  [[arxiv:1807.02721](https://arxiv.org/abs/1807.02721), [doi:10.1007/s00222-020-00966-7](https://doi.org/10.1007/s00222-020-00966-7)]

Reviews:

* [[Spencer Bloch]], _The Proof of the Mordell Conjecture_, Math. Int. **6** 2 (1984) 41-47 [[pdf](https://link.springer.com/content/pdf/10.1007/BF03024155.pdf), [doi:10.1007/BF03024155](https://doi.org/10.1007/BF03024155)]

* [[Barry Mazur]], _Abelian varieties and the Mordell-Lang conjecture_ ([pdf](http://library.msri.org/books/Book39/files/mazur.pdf))

A survey of two approaches to the effective Mordell conjecture:

* {#Balakrishnan19} [[Jennifer Balakrishnan]],  A. J. Best, F. Bianchi, [[Brian Lawrence]], J. S. Müller, N. Triantafillou, J. Vonk: *Two recent $p$-adic approaches to the (effective) Mordell conjecture*, in: *Arithmetic L-Functions and Differential Geometric Methods* Progress in Mathematics **338** Birkhäuser (2021) [[arxiv:1910.12755](https://arxiv.org/abs/1910.12755), [doi:10.1007/978-3-030-65203-6_2]( https://doi.org/10.1007/978-3-030-65203-6_2)]

Encyclopedia entries:

* [[Encyclopedia of Mathematics]], _[Mordell conjecture](http://www.encyclopediaofmath.org/index.php/Mordell_conjecture)_

* Wikipedia, _[Faltings' theorem](http://en.wikipedia.org/wiki/Mordell_conjecture#CITEREFMordell1922)_

The relation to [[anabelian geometry]] originates in

* [[Alexandre Grothendieck]], letter to [[Faltings]] (1983) ([pdf](http://www.math.jussieu.fr/~leila/grothendieckcircle/GtoF.pdf))
 {#Grothendieck}

The Weil quote stems from

* {#Weil79}[[André Weil]], _Œuvres Scientifiques - Collected Papers Volume III_, Springer Heidelberg 1979. (p.454)

[[!redirects Mordell's conjecture]]

[[!redirects Mordell Conjecture]]


Mordell-Weil theorem

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## Statement

The _Mordell-Weil theorem_ states that for an [[elliptic curve]] $E$ over a [[number field]] $K$, the group of $K$-rational points (also known as the [[Mordell-Weil group]]) $E(K)$ form a finitely generated abelian group.

## References

* Wikipedia, *[Mordell-Weil theorem](https://en.wikipedia.org/wiki/Mordell%E2%80%93Weil_theorem)*


Morgan Opie

* [webpage](http://people.math.harvard.edu/~opie/)

## Selected writings

On [[localization of a space|localization]] (in [[algebraic topology]]) formulated in [[modal homotopy type theory|modal]] [[homotopy type theory]]:

* [[J. Daniel Christensen]], [[Morgan Opie]], [[Egbert Rijke]], [[Luis Scoccola]], _Localization in Homotopy Type Theory_, Higher Structures, 4(1) (2020), 1-32 ([arXiv:1807.04155](https://arxiv.org/abs/1807.04155))

category: people


Morihiko Saito

* [webpage](http://www.kurims.kyoto-u.ac.jp/en/list/SAITO,%20Morihiko.html)

## related $n$Lab entries

* [[D-module]]

category: people


Morita context

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## Idea

A __Morita context__ or, in some authors (e.g. Bass), the __pre-equivalence data__, is a generalization of [[Morita equivalence]] between [[categories of modules]]. In the case of right [[modules]], for two [[associative algebras|associative k-algebras]] (or, in the case of $k = \mathbb{Z}$, [[ring]]s) $A$ and $B$, it consists of [[bimodule]]s ${}_A P_B$, ${}_B Q_A$, and bimodule [[homomorphisms]]
$f: P\otimes Q\to A$, $g: Q\otimes P\to B$ satisfying mixed [[associativity]] conditions.

## Properties

Theorem. (Bass II.3.4) If $f$ is surjective, then:

(i) $f$ is an isomorphism

(ii) $P$ and $Q$ are generators in the categories of $A$-modules

(iii) $P$ and $Q$ are finitely generated and projective

(iv) $g$ induces isomorphisms of bimodules $P\cong Hom_B(Q,B)$ and
$Q\cong Hom_B(P,B)$

(v) homomorphisms of $A$-algebras $End_B(P)\leftarrow A\rightarrow End_B(Q)$ are isomorphisms

(Bass II.4.1) A Morita context can be constructed from an $A$-algebra $B$ and a right $B$-module $P$. Then set $A = End_B(P)$ and $Q=Hom_B(P,B)$. Then $f = f_P$ and $g = g_P$ are defined by $(b q) p = b (q p)$ and $(q a) p = q(a p)$.

(Bass II.4.4) (i) $f_P$ s surjective iff $P$ is finitely generated projective $B$-module. Then $f_P$ s iso.

(ii) $g_P$ is surjective iff $P$ s a generator of $mod_B$, then $g_P$ is iso

(iii) The Morita context $(A,B,P,Q,f,g)$ is a Morita equivalence iff $P$ is both projective and a generator. Then $\otimes_P : mod_A\to mod_B$ and its right adjoint $Hom_B(P,-)$ form the equivalence.

## Related entries

* [[Morita equivalence]], [[Eilenberg-Watts theorem]]

## Literature

* [[Hyman Bass]], _Algebraic K-theory_, chapter 2
* [[Tomasz Brzeziński]], Adrian Vazquez Marquez, Joost Vercruysse, _The Eilenberg-Moore category and a Beck-type theorem for a Morita context_, Appl. Categ. Structures __19__ (2011), no. 5, 821–858 [MR2836546](https://www.ams.org/mathscinet-getitem?mr=2836546)
[doi](https://doi.org/10.1007/s10485-009-9217-0)
* Bruno J. Müller, _The quotient category of a Morita context_, J. Algebra __28__ (1974), 389–407 [MR0447336](http://www.ams.org/mathscinet-getitem?mr=0447336) doi
* A. I. Kashu, _On equivalence of some subcategories of modules in Morita contexts_, Discrete Math. 2003, no. 3, 46–53, [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=adm&paperid=383&what=fullt&option_lang=rus)
* Y. Doi, _Generalized smash products and Morita contexts for arbitrary Hopf algebras_, in: J. Bergen, S. Montgomery (Eds.), Advances in Hopf algebras, in: Lecture Notes in Pure and Appl. Math. __158__, Dekker 1994 [doi](https://doi.org/10.1016/j.jalgebra.2004.02.002)

* [[Stefaan Caenepeel]], [[Joost Vercruysse]], Shuanhong Wang, _Morita theory for corings and cleft entwining structures_, J. Algebra __276__1 (2004) 210-235 [doi](https://doi.org/10.1016/j.jalgebra.2004.02.002)

There are generalizations in more general bicategories:

* [[Laiachi El Kaoutit]], _Wide Morita contexts in bicategories_,  Arab. J. Sci. Eng. __33__ (2008) 153--173

* Bertalan Pecsi, _On Morita context in bicategories_, [pdf](http://www.renyi.hu/~aladar/MrtCtx.pdf)

category: algebra

[[!redirects Morita contexts]]


Morita equivalence

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## Idea

In the original sense, a _Morita equivalence_ between two [[rings]] is two [[bimodules]] between them that behave as [[inverses]] to each other under [[tensor product of modules]], up to isomorphism of bimodules. This is just the [[equivalence in a 2-category|2-categorical concept of equivalence]] for the two rings regarded as [[objects]] in the [[2-category]] of rings, with [[bimodules]] as [[1-morphisms]] and bimodule homomorphisms ([[intertwiners]]) as [[2-morphisms]] (see [here](bimodule#AsMorphismsInA2Category)).
By the [[Eilenberg-Watts theorem]] any such pair of bimodules equivalently induces a suitably "linear" [[equivalence of categories]] between the [[categories of modules]] over the given rings.

This is the original concept equivalence of rings due to [[Kiiti Morita]] (see [below](#ClassicalMoritaTheorem)).

Nowadays, the term is applied in different but closely related senses in a wide range of mathematical fields, and one speaks of _Morita equivalent_ [[categories]], [[algebraic theories]], [[geometric theories]] and so on.

Typically, such Morita situations involve three ingredients: a 'syntactic' ground level to which the respective concept of _Morita equivalence_ applies, a 'hypersyntactic' level obtained via [[Cauchy completion]], and a second process of completion to a 'semantic' level where the equivalence relation for the syntactic ground level is defined by plain equivalence of categories.  For example Morita equivalence for _small categories_ is defined as equivalence of their presheaf categories, with Cauchy completion as intermediate hypersyntactic level.

So the broad intuition is that Morita equivalence is a _coarse grained semantic equivalence that obtains between syntactic gadgets_ - basically two [[theories]] that have up to equivalence the same category of [[models]]. The role of the intermediate hypersyntactic level in this analogy is that of an 'ideal syntax' (syntax classifier) that already reflects the relations at the semantic level. The categorical equivalence (via bimodules)  from the semantic level then shows up at the intermediate level as a ('Cauchy convergent'$\sim$ 'fgp-module') bidirectional _translation_ from one syntax into another.

An important weakening of Morita equivalence is [[Morita context]] (in older literature sometimes called pre-equivalence).

## Classical Morita theorem
 {#ClassicalMoritaTheorem}

Given (associative unital) [[associative unital algebra|algebras]] $R$ and $S$ over a [[commutative ring]] $k$, the following properties are equivalent:

1. The categories of left $S$-[[modules]] and left $R$-modules are [[equivalence of categories|equivalent]];

2. The categories of right $S$-modules and right $R$-modules
are equivalent;

3. There are [[bimodules]] ${}_R M_S$ and ${}_S N_R$ such that the functors $- \otimes_R M$ and $- \otimes_S N$ form an [[adjoint equivalence]] between the category of right $S$-modules and the category of right $R$-modules;

4. The algebra $R$ is isomorphic to the endomorphism algebra of a finitely generated projective [[generator]] in the category of left (or right) $S$-modules;

5. The algebra $S$ is isomorphic to the endomorphism algebra of a finitely generated projective [[generator]] in the category of left (or right) $R$-modules.

As usual, these results give results for rings when specialized to the case $k = \mathbb{Z}$, with an algebra over $\mathbb{Z}$ being a ring.

## Definitions

### In algebra

Two algebras over a commutative ring $k$ are **Morita equivalent** if the equivalent statements in the Morita theorem above are true. A **Morita equivalence** is an [[equivalence in a 2-category|equivalence]] in the [[bicategory]] $\mathrm{Alg}_k$ with

* [[associative unital algebras]] over $k$ as objects,

* [[bimodules]] as [[1-morphisms]], and

* bimodule homomorphisms ("[[intertwiners]]" as [[2-morphisms]]).

A [theorem in ring theory](center+of+an+additive+category#examples) says that the [[center]] of an algebra is isomorphic to the center of its category of modules and that Morita equivalent algebras have isomorphic centers. In particular, two _commutative_ algebras are Morita equivalent precisely when they are isomorphic!

This shows that the property of _having center $Z$ up to isomorphism_ is stable within Morita equivalence classes. Properties of this kind are sufficiently important to deserve a special name:

A property $P$ of rings is called a **Morita invariant** iff whenever $P$ holds for a ring $R$, and $R$ and $S$ are Morita equivalent then $P$ also holds for $S$. Other Morita invariant properties are the property of being [[simple ring|simple]], [[semisimple ring|semisimple]], left or right [[Noetherian ring|Noetherian]], and left or right [[Artinian ring|Artinian]] (cf. Anderson and Fuller (1992)).

The appearance of finitely generated projective modules in the [classical Morita theorem](#ClassicalMoritaTheorem) may seem unexpected, but it emerges from the following

\begin{lemma}
Suppose $R,S$ are algebras over a commutative ring $k$ and $M$ is an $R,S$-bimodule. Then $M$ is a left adjoint in the bicategory $Alg_k$ of $k$-algebras, bimodules and bimodule morphisms if and only if $M$ is finitely generated projective as an $S$-module.
\end{lemma}

There is a slick proof of this using [[enriched category theory]], working with categories enriched over $V = k Mod$.  It makes use of these facts: a $k$-algebra $R$ is a one-object $V$-enriched category, a bimodule of $k$-algebras is a $V$-enriched profunctor, and the [[Cauchy completion]] of $R$, viewed as a one-object $V$-enriched category, is the category of finitely generated projective $R$-modules.

### In homotopy theory

In any [[homotopy theory]] framework a **Morita equivalence** between objects $C$ and $D$ is  a span
$$
  C \lt \stackrel{\simeq}{\leftarrow}
  \hat C
  \stackrel{\simeq}{\to} \gt
  D
$$
where both legs are acyclic fibrations.

In particular, if the ambient [[homotopical category]] is a [[category of fibrant objects]], then the _factorization lemma_ (see there) ensures that _every_ weak equivalence can be factored as a span of acyclic fibrations as above.

Important fibrant objects are in particular [[infinity-groupoid]]s (for instance [[Kan complex]]es are fibrant in the standard [[model structure on simplicial sets]] and [[omega-groupoid]]s are fibrant with respect to the Brown-[[Golasinski]] [[folk model structure]]). And indeed, Morita equivalences play an important role in the theory of groupoids with extra structure:

### In Lie groupoid theory

A **[[Morita morphism]] equivalence** of [[Lie groupoids]] is an [[anafunctor]] that is invertible, equivalently an invertible [[Hilsum-Skandalis morphism]]/[[bibundle]].

Lie groupoids up to Morita equivalence are equivalent to [[differentiable stack]]s. This relation between Lie groupoids and their stacks of torsors is analogous to the relation between algebras and their categories of modules, which is probably the reason for the choice of terminology.

### In tensor category theory

A [[fusion category]] $\mathcal{C}$ is Morita equivalent to another fusion category $\mathcal{D}$ if there exist a $\mathcal{C}$-module category $\mathcal{M}$ such that $\mathcal{D}$ is equivalent to the [[reverse category]] of $\mathcal{C}$-module functors from $\mathcal{M}$ to itself, namely
$$\mathcal{D}\simeq (\mathcal{C}_{\mathcal{M}}^{\vee})^{rev}:=\mathsf{Fun}_{\mathcal{C}}(\mathcal{M},\mathcal{M})^{rev}.$$
By the 'reverse category' of a monoidal category $\mathcal{C}$ we mean the monoidal category with reversed [[tensor product]] $x\otimes^{rev}y:=y \otimes x$.

## Related Concepts

* [[derived Morita equivalence]]
* [[Morita context]]
* [[Eilenberg-Watts theorem]]
* [[generator]]
* [[bimodule]]
* [[projective module]]
* [[Picard group]]
* [[bicategory]]
* [[Cauchy completion]], [[Karoubi envelope]]

## References and Literature

The concept is named after [[Kiiti Morita]].

A beautiful classical exposition is in

* [[Hyman Bass]], chapter II of _Algebraic K-theory_, Benjamin 1968.

The concept should be covered in any decent textbook on algebra and ring theory, e.g.:

* F. W. Anderson and K. R. Fuller, _Rings and Categories of Modules_, Graduate Texts in Mathematics Vol. 13, Springer, New York, 1992.

* P. M. Cohn, _Further Algebra and Applications_, Springer Heidelberg 2003. (sec. 4.4-4.5 pp.148ff)

* [[Ross Street]], _Quantum Groups - A Path to Current Algebra_ , Cambridge UP 2007. ([ps-draft](http://www-texdev.ics.mq.edu.au/Quantum/Quantum.ps))

For an early extension to domains other than ring theory see

* H. Lindner, _Morita equivalences of enriched categories_ , Cah. Top. Géom. Diff. Cat **15** no.4 (1974) pp.377-397. ([pdf](http://archive.numdam.org/article/CTGDC_1974__15_4_377_0.pdf))

The generalizations to graded rings, Hopf algebras and corings are studied in references

* A. Marcus, _Equivalences induced by graded bimodules_, Comm. Algebra 26 (1998) 713–731; _Homology of fully graded algebras, Morita and derived equivalences_, J. Pure Appl. Alg. __133__:1–2 (1998) 209-218 doi
* S. Caenepeel, J.Vercruysse, Shuanhong Wang, _Morita theory for corings and cleft entwining structures_, J. Algebra __276__1 (2004) 210-235 [doi](https://doi.org/10.1016/j.jalgebra.2004.02.002)
* [[Stefaan Caenepeel]], Septimiu Crivei, Andrei Marcus, [[Mitsuhiro Takeuchi]], _Morita equivalences induced by bimodules over Hopf–Galois extensions_, J. Algebra __314__ (2007) 267–302 [pdf](https://core.ac.uk/download/pdf/81180359.pdf)
* [[Gabriella Böhm]], [[Joost Vercruysse]], _Morita theory for comodules over corings_,  3207-3247 (2009)  Commun. Alg. __37__:9 (2009) [doi](https://doi.org/10.1080/00927870902747993 )

The case of algebraic theories is covered in

* [[Francis Borceux|F. Borceux]], _Handbook of Categorical Algebra 2_ , CUP 1994. (sec. 3.12)

* [[Jiri Adamek|J. Adámek]], M. Sobral, L. Sousa, _Morita equivalence of many-sorted algebraic theories_ , JA **297** (2006) pp.361-371. ([preprint](https://estudogeral.sib.uc.pt/bitstream/10316/4616/1/filee8df8c9585d34e1a8d56fdaf0460d008.pdf))

For the use in O. Caramello's 'toposes as bridges'- approach that brings out the logical side of the concept:

* [[Olivia Caramello|O. Caramello]], _Topos-theoretic background_ , ms. 2014. ([pdf](http://www.oliviacaramello.com/Unification/ToposTheoreticPreliminariesOliviaCaramello.pdf))

Other references include

* [[Ralf Meyer]], _Morita equivalence in algebra and geometry_ . ([[MeyerMoritaEquivalence-2.pdf:file]])

* [[I. Dell'Ambrogio]], [[Goncalo Tabuada]], _A Quillen model structure for classical Morita theory and a tensor categorification of the Brauer group_ , arXiv:1211.2309 (2012). ([pdf](http://arxiv.org/pdf/1211.2309v1))

* [[Hans Porst]], _Generalized Morita Theories: The power of categorical algebra_, ([pdf](http://www.math.uni-bremen.de/~porst/dvis/SAMSnoticesCorr.pdf))

* [[Francis Borceux]] and [[Enrico Vitale]], _On the notion of bimodel for functorial semantics_, Appl. Categ. Structures, __2__:283–295 (1994) [pdf](http://perso.uclouvain.be/enrico.vitale/BimodAPCS.pdf)

See also

* Wikipedia, _[Morita equivalence](http://en.wikipedia.org/wiki/Morita_equivalence)_

[[!redirects Morita equivalences]]
[[!redirects Morita equivalent]]


Morita morphism

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The most useful notion of [[homomorphism]] between [[Lie groupoid]]s is not that of a [[functor]] [[internalization|internal to]] the category [[SmoothMfd]] of [[smooth manifold]]s (even though [[Lie groupoids]] can be regarded as [[internal groupoids]] in [[SmoothMfd]]), but is that of a morphism of [[smooth stacks]] ([[differentiable stacks]]) on the [[site]] [[SmoothMfd]]. To amplify this distinction in the literature a wealth of names and terms for these non-naïve homomorphisms is used, such as _Morita morphism_ . In particular a Morita morphism which is a [[homotopy equivalence]] is a _[[Morita equivalence]]_.

Ways to present the correct notion of homomorphism include

* [[anafunctors]]

* [[groupoid principal bundle|groupoid principal]]  $\,$ [[bibundles]], [[Hilsum-Skandalis morphisms]]

[[!redirects Morita morphisms]]


Moritz Groth

* [website](https://personen.uni-mainz.de/public/person/8032)

## related entries

* [[derivator]], [[pointed derivator]], [[stable derivator]]

category: people


Moritz Hauptmann

**Moritz Hauptmann** (1792-1868) was a German composer and music theorist. In the succession of Johann Sebastian Bach he was _Thomaskantor_ in Leipzig.

In his writings he tried to develop tonal music from first principles based on the progresive mediation of units and their opposites within themselves. Traditionally (Hugo Riemann, [[Karl Rosenkranz]], Peter Rummenhöller) this process of self-mediation of musical units has been identified with the [[Aufhebung]] from Hegelian dialectics whence Hauptmann's approach has broadly been characterized as "Hegelian" but this view was challenged by Wilhelm Seidel and Lothar Schmidt who stress the Goethean background of Hauptmann's aesthetics instead.

## Writings

* _Die Natur der Harmonik und Metrik_, Breitkopf & Härtel Leipzig 1853. ([wikisource](https://de.wikisource.org/wiki/Moritz_Hauptmann))

* _The nature of harmony and metre_, Da Capo Press New York 1991[1893].

## Links

* [Wikipedia entry (German)](http://de.wikipedia.org/wiki/Moritz_Hauptmann)

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

## Related entries

* [[music theory]]

## References

* William Caplin, _Moritz Hauptmann and the Theory of Suspensions_, Journal of Music Theory **28** No. 2 (1984) pp. 251-269.

* Maryam A. Moshaver, _Structure as Process: Rereading Hauptmann's Use of Dialectical Form_, Music Theory Spectrum **31** No. 2 (2009) pp. 262-283.

* Wilhelm Seidel, _Moritz Hauptmanns organische Lehre: Tradition, Inhalt und Geltung ihrer Prämisse_, International Review of the Aesthetics and Sociology of Music **2** No. 2 (1971), pp. 243-266.

category:people


Moritz Kerz

* [webpage](http://www.mathematik.uni-regensburg.de/kerz/)

## related $n$Lab entries

* [[algebraic K-theory]]

* [[Gersten conjecture]]

category: people


morphism

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## Idea

The notion of _morphism_ in [[category theory]] is an abstraction of the notion of [[homomorphism]].

In a general [[category]], a morphism is an [[arrow]] between two [[objects]]. The word *[[map]]* is also sometimes used as a synonym.

## Definition

Given two objects in a (locally small) category, say $x$ and $y$, there is a set $hom(x,y)$, called a [[hom-set]], whose elements are morphisms from $x$ to $y$.  Given a morphism $f$ in this hom-set, we write $f:x \to y$ to indicate that it goes from $x$ to $y$.

More generally, a morphism is what goes between objects in any [[n-category]].

## Examples

The most familiar example is the category [[Set]], where the objects are sets and the morphisms are functions.  Here if $x$ and $y$ are sets, a morphism $f: x \to y$ is a function from $x$ to $y$.

## Related concepts

* [[object]]

* **morphism**, [[multimorphism]]

  * [[inverse morphism]]

  * [[adjoint morphism]]

  * [[heteromorphism]]

  * [[endomorphism]], [[automorphism]], [[monomorphism]], [[epimorphism]]

* [[1-morphism]]

  [[horizontal morphism]], [[vertical morphism]]

* [[2-morphism]]

* [[3-morphism]]

* [[k-morphism]]

[[!redirects morphisms]]
[[!redirects Morphism]]
[[!redirects Morphisms]]


morphism of (∞,1)-operads

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* table of contents
{:toc}

## Definition

In the sense of [Lurie, 2.1.2](#Lurie)

For $\mathcal{A}^\otimes, \mathcal{B}^\otimes$ two [[(∞,1)-operads]] regarded as their [[(∞,1)-categories of operators]] regarded as [[quasi-categories]], a [[homomorphism]] between them is a map in the [[slice category]] of [[sSet]] over $N(FinSet_*)$ such that it respects inert morphisms.

([Lurie, def. 2.1.2.7](#Lurie))

## Related concepts

* **morphism of $(\infty,1)$-operads**

  * [[fibration of (∞,1)-operads]]

  * [[coCartesian fibration of (∞,1)-operads]]

## References

Section 2.1.2 of

* [[Jacob Lurie]], _[[Higher Algebra]]_
 {#Lurie}

[[!redirects morphisms of (∞,1)-operads]]
[[!redirects morphism of (infinity,1)-operads]]
[[!redirects morphisms of (infinity,1)-operads]]


morphism of finite presentation

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#Contents#
* table of contents
{:toc}

## Definition

A [[homomorphism]] of [[schemes]] $f \colon X\to Y$ is

1. __finitely presented at__ $x\in X$ if there is an affine [[open neighborhood]] $U$ containing $x$ and an affine open set $V\subseteq Y$ with $f(U)\subseteq V$ such that $\mathcal{O}_X(U)$ is [[finitely presented algebra|finitely presented]] as an $\mathcal{O}_Y(V)$-[[associative algebra|algebra]].

1. __locally finitely presented__ if it is finitely presented at each $x\in X$.

1. __finitely presented__ if it is locally finitely presented, [[quasicompact morphism|quasicompact]] and [[quasiseparated morphism|quasiseparated]].

1. __essentially finitely presented__ if it is a localization of a finitely presented morphism.

## Example

A standard open $Spec(R[\{s\}^{-1}]) \longrightarrow Spec(R)$ ([[Zariski topology]]) is of finite presentation. More generally, an [[étale morphism of schemes]] is of finite presentation (though essentially by definition so).

## Related concepts

* [[morphism of finite type]]

## References

* [[The Stacks Project]], Section 01TO: _[Morphisms of finite presentation](https://stacks.math.columbia.edu/tag/01TO)_

* [wikipedia: Finite, quasi-finite, finite type, and finite presentation morphisms](http://en.wikipedia.org/wiki/Glossary_of_scheme_theory#Finite.2C_quasi-finite.2C_finite_type.2C_and_finite_presentation_morphisms)

* [[David Rydh]], _Why are unramified maps not required to be locally of finite presentation?_, [MO/206333/2503](http://mathoverflow.net/a/206333/2503).

[[!redirects finitely presented morphism]]
[[!redirects locally of finite presentation]]


morphism of finite type

> This article is about [[homomorphisms]] between [[schemes]] which are locally of finite type. For the notion of locally finite type in [[dependent type theory]], see [[locally finite type]].

---

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## Idea

A [[homomorphism]] between [[schemes]] is said to be _(locally) of finite type_ if it behaves like a [[finite number|finite]] [[covering space]].

## For schemes

A morphism $f : X \to Y$ of [[schemes]] is **locally of finite type** if

* for every [[open cover]] $\{U_i \to Y\}$ by [[affine scheme]]s, $U_i \simeq Spec B_i$;

* and every cover $\{U_{i j_i} \to X\}$ by affine schemes $U_{i j_i} = Spec A_{i j_i}$, fitting into a [[commuting diagram]] (this always exists, see [[coverage]])

  $$
    \array{
      U_{i j_i} &\to& U_i
      \\
      \downarrow && \downarrow
      \\
      X &\stackrel{f}{\to}& Y
    }
  $$

  for all $i,j$,

we have that the morphism of algebras $B_i \to A_{i j}$ formally dual to $U_{i j} \to U_i$ exhibits $A_{i j}$ as a [[finitely generated algebra]] over $B_i$.

If for fixed $i$ the $j_i$ range only over a [[finite set]], then the morphism is said to be of **finite type**.

## Related concepts

* [[morphism of finite presentation]]

## References

Introductory disucssoon over the [[complex numbers]] (with an eye towards [[GAGA]]) is in

* {#Neeman07} [[Amnon Neeman]], section 3.10 _Algebraic and analytic geometry_, London Math. Soc. Lec. Note Series __345__, 2007 ([publisher](http://www.cambridge.org/us/academic/subjects/mathematics/geometry-and-topology/algebraic-and-analytic-geometry))

[[!redirects locally of finite type]]

[[!redirects morphisms of finite type]]

[[!redirects morphism locally of finite type]]
[[!redirects morphisms locally of finite type]]

[[!redirects finite type morphism]]
[[!redirects finite type morphisms]]

[[!redirects scheme of finite type]]
[[!redirects schemes of finite type]]


morphism of multicategories

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#Contents#
* table of contents
{:toc}

## Idea

The canonical kind of [[morphisms]] between [[multicategories]] generalize the concept of *[[functors]]* between [[categories]] and that of *[[monoidal functors]]* between [[monoidal categories]].

It is common to speak here of "multifunctors" (e.g. [Elmendorf & Mandell 2004, p. 2](#ElmendorfMandell04), review in [Tate 2018, p. 2](#Tate18)) but see at *[[multifunctor]]* for other meanings of this term.

## Examples

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

The construction of [[K-theory spectra of permutative categories]] constitutes a morphism of multicategories

$$
  \mathbb{K}
  \;\colon\;
  PermCat
    \longrightarrow
  Spectra
$$

between [[multicategories]] of [[permutative categories]] (under [[Deligne tensor product]]) and [[spectra]] (under [[smash product of spectra]]).

Hence for a multilinear functor of permutative categories

$$
  F
    \;\colon\;
  \mathcal{A}_1 \times \cdots \times \mathcal{A}_n
  \longrightarrow
  \mathcal{B}
$$

there is a compatibly induced morphism of K-[[spectra]] out of the [[smash product of spectra|smash product]]

$$
  \mathbb{K}(\mathcal{A}_1)
    \wedge
    \cdots
    \wedge
  \mathbb{K}(\mathcal{A}_n)
  \longrightarrow
  \mathbb{K}(\mathcal{B})
  \,.
$$

This implies that the construction further extends to a [[2-functor]] from the [[2-category]] $PermCat Cat$ of [[enriched categories]] over the [[multicategory]] of [[permutative categories]] to that of [[enriched categories]] of [[spectra]]:

$$
  \mathbb{K}_\bullet
  \;\colon\;
  PermCat Cat \longrightarrow Spectra Cat
$$

which applies $\mathbb{K}$ to each [[hom-object]].

=--

([May 80, theorem 1.6, Theorem 2.1](K-theory+of+a+permutative+category#May80), [Elmendorf-Mandell 04, theorem 1.1](#ElmendorfMandell04), [Guillou 10, Theorem 1.1](K-theory+of+a+permutative+category#Guillou10)

## References

Review:

* {#Tate18} Ross Tate, p. 2 of: *Multicategories* (2018) [[pdf](http://www.cs.cornell.edu/courses/cs6117/2018sp/Lectures/Multicategories.pdf), [[Tate-Multicategories.pdf:file]]]

See also:

* {#ElmendorfMandell04} [[Anthony Elmendorf]], [[Michael Mandell]], _Rings, modules and algebras in infinite loop space theory_, Adv. in Math. **205** 1 (2006) 163-228 [[arXiv:math/0403403](https://arxiv.org/abs/math/0403403), [doi:10.1016/j.aim.2005.07.007](https://doi.org/10.1016/j.aim.2005.07.007)]

[[!redirects morphisms of multicategories]]


morphism of schemes

## Definition

If $(X,\mathcal{O}_X)$ and $(Y,\mathcal{O}_Y)$ a __morphism of schemes__ $(X,\mathcal{O}_X)\to(Y,\mathcal{O}_Y)$ is a pair $(f,f^\sharp):(X,\mathcal{O}_X)\to(Y,\mathcal{O}_Y)$ where $f:X\to Y$ is the continuous map of topological spaces and a morphism of sheaves of local rings $f^\sharp:\mathcal{O}_Y\to f_*\mathcal{O}_X$.

In other words, schemes form a full subcategory of the category of [[locally ringed space]]s. The morphism of sheaves of local rings $f^\sharp:\mathcal{O}_Y\to f_*\mathcal{O}_X$ is also called the [[comorphism]].

## Properties of morphisms

Since Grothendieck school, the relative point of view is emphasized in algebraic geometry. In particular, the study of classes of schemes is generalized to the study of classes of morphisms of schemes. Thus a property of some schemes can often be generalized to a property of some morphisms and this is the proper generality. For example, the affine $k$-schemes are the same as affine morphisms into the spectrum of $k$; thus affine morphisms (otherwise known as relative affine schemes) generalize affine schemes.

Much of the foundations of modern algebraic geometry is thus devoted to the study of classes of morphisms of schemes, as seen for example from major monographs like EGA, or the recent Stacks Project.

* [[affine morphism]]
* [[flat morphism]], [[faithfully flat morphism]]
* [[closed immersion of schemes]], [[open immersion of schemes]]
* [[projective morphism]]
* [[proper morphism of schemes]]
* [[quasicompact morphism]]
* [[etale morphism]]
* [[morphism of finite presentation]]
* [[morphism of finite type]]
* [[separated morphism]], [[quasiseparated morphism]]

category: geometry
[[!redirects morphisms of schemes]]


morphism of sites

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# Morphisms of sites

* table of contents
{: toc}

## Idea

A *morphism of sites* is, unsurprisingly, the appropriate sort of [[morphism]] between [[sites]].  It is defined exactly so as to induce a [[geometric morphism]] between [[toposes of sheaves]] (or, more generally, [[exact completions]]).

## Definition

Let $C$ and $D$ be [[sites]].

+-- {: .num_defn}
###### Definition
A [[functor]] $f:C\to D$ is a **morphism of sites** if

1. $f$ is [[flat functor#SiteValuedFunctors|covering-flat]], and

1. $f$ preserves covering families, i.e. for every [[covering]] $\{p_i : U_i \to U\}$ of an object $U \in C$, the family $\{f(p_i) : f(U_i) \to f(U)\}$ is a covering of $f(U) \in D$.
=--

+-- {: .num_remark}
###### Remark
If $C$ has [[finite limit]]s and all covering families in $D$ are [[strong epimorphism|strong epic]], then covering-flatness of $f$ is equivalent to $f$ preserving finite limits, i.e. being a [[left exact functor]], or equivalently to being a [[representably flat functor]].  Thus, frequently in the literature one finds a definition of a morphism of sites as being representably flat and preserving covering families.

For general $C$ and $D$, however, being representably flat *implies* being covering-flat, but not conversely.  Thus, the above definition of morphism of sites is more general than the common one.  There are few practical examples where the distinction matters, but our definition has better formal properties (see below).
=--

## Examples

+-- {: .num_example}
###### Example
If $A$ and $B$ are [[frame]]s regarded as sites via their canonical coverages, then a morphism of sites $A \to B$ is equivalently a frame [[homomorphism]], a function preserving finite [[meet]]s and arbitrary [[join]]s.
=--

+-- {: .num_example}
###### Example
**(slice sites)**

For $C$ a site and $U \in C$, the [[comma category]] $(C / U)$ inherits a topology from $C$, such that the [[forgetful functor]] $(C/U) \to C$ constitutes a morphism of sites.  This is also called the [[big site]] of $U$.  There are natural operations for [[restriction and extension of sheaves]] from a sub-site $U$ to $X$ and back.

For instance, if $X$ is a topological space and $U \in Op(X)$ is an open subset, then $U$ regarded as a topological space in its own right has corresponding to it the site $Op(U) = Op(X) \downarrow U$.
=--

+-- {: .num_example}
###### Example
For $C$ and $D$ [[regular category|regular categories]] equipped with their [[regular coverage]]s, a morphism of sites is the same as a [[regular functor]], i.e. a functor preserving finite limits and covers.

More generally, if $C$ and $D$ are [[∞-ary regular categories]] with their $\kappa$-canonical topologies, then a morphism of sites is the same as a $\kappa$-ary regular functor (preserving finite limits and $\kappa$-ary effective-epic families).
=--

+-- {: .num_example}
###### Example
For $C$ any site with finite limits, there is canonically a morphism of sites to its [[tangent category]]. See there for details.
=--

## Properties

### Relation to geometric morphisms
 {#RelationToGeometricMorphisms}

We discuss how morphisms of sites induce [[geometric morphism]]s of the corresponding [[sheaf topos]]es, and conversely.  The reader might want to first have a look at the discussion of _[geometric morphisms -- between presheaf toposes](geometric+morphism#BetweenPresheafToposes)_.

Let $f : (\mathcal{C},J) \to (\mathcal{D},K)$ be a morphism of sites, with $\mathcal{C}$ and $\mathcal{D}$ small.  Then precomposition with $f$ defines a [[functor]] between categories of presheaves $(-)\circ f : PSh(\mathcal{D}) \to PSh(\mathcal{C})$.

+-- {: .num_prop #MorphismsOfSitesInduceGeometricMorphisms}
###### Proposition
There is a [[geometric morphism]] between the [[categories of sheaves]]
$$
  (f^* \dashv f_*) : Sh(\mathcal{D},K) \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}
    Sh(\mathcal{C},J)
$$
where $f_*$ is the restriction of $(-)\circ f$ to sheaves.
=--

For the classical definition of morphisms of sites, using representably-flat functors, this appears for instance as ([Johnstone, lemma C2.3.3, cor. C2.3.4](#Johnstone)).  We give the proof in this special case; for the general case see ([Shulman](#Shulman)).

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

By the assumption that $f$ preserves covers we have that the restriction of $(-)\circ f$ to $Sh_K(\mathcal{D})$ indeed factors through $Sh(\mathcal{C}) \hookrightarrow PSh(\mathcal{C})$.

Because for $\{U_i \to U\}$ a [[cover]] in $\mathcal{C}$ and $F$ a [[sheaf]] on $\mathcal{D}$, we have that (assuming here for simplicity that $\mathcal{C}$ has finite limits)

$$
  \begin{aligned}
    PSh_{\mathcal{C}}( \underset{\to}{\lim} ( \coprod_{i, j} U_i \times_{U} U_j)
    \stackrel{\to}{\to}
      \coprod_{i} U_i )
    \;,\;
    F(f(-))
    )
    & \simeq
    \lim_{\leftarrow}
     \left(
      \prod_{i,j} PSh_{\mathcal{C}}( U_i \times_U U_j, F(f(-)))
       \stackrel{\leftarrow}{\leftarrow}
       \prod_i PSh_C(U_i , F(f(-)))
     \right)
    \\
    & \simeq
    \lim_{\leftarrow}
     \left(
      \prod_{i,j} F(f(U_i \times_U U_j)))
       \stackrel{\leftarrow}{\leftarrow}
       \prod_i F(f(U_i))
     \right)
    \\
    & \simeq
    \lim_{\leftarrow}
     \left(
      \prod_{i,j} F(f(U_i) \times_{f(U)} f(U_j))))
       \stackrel{\leftarrow}{\leftarrow}
       \prod_i F(f(U_i))
     \right)
    \\
    & \simeq
    PSh_{\mathcal{D}}( \underset{\to}{\lim} ( \coprod_{i, j} f(U_i) \times_{f(U)} f(U_j))
    \stackrel{\to}{\to}
      \coprod_{i} f(U_i) )
    \;,\;
    F
   )
  \end{aligned}
  \,,
$$

where we used the [[Yoneda lemma]], the fact that the [[hom functor]] $PSh(-,-)$ sends [[colimit]]s in the first argument to [[limit]]s, and the assumption that $f$ preserves the [[pullback]]s involved.

Also $(-)\circ f$ preserves all [[limit]]s, because for presheaves these are computed objectwise. And since the inclusion $Sh_K(\mathcal{D}) \to PSh(\mathcal{D})$ is [[right adjoint]] (to [[sheafification]]) we have that

$$
  f_* : Sh_K(\mathcal{D}) \hookrightarrow PSh(\mathcal{D}) \stackrel{(-)\circ f}{\to} Sh_J(\mathcal{C})
$$

preserves all limits. Therefore by the [[adjoint functor theorem]] it has a [[left adjoint]]. Explicitly, this is the composite of the left adjoint to $(-)\circ f$ and to sheaf inclusion. The first is left [[Kan extension]] $Lan_f$ along $f$ and the second is [[sheafification]] $L_J$ on $(\mathcal{C},J)$, so the left adjoint is the composite

$$
  f^*  :Sh_J(\mathcal{C}) \hookrightarrow PSh(\mathcal{C}) \stackrel{Lan_f}{\to}S
   PSh(\mathcal{D}) \stackrel{L_J}{\to} Sh_J(\mathcal{D})
  \,.
$$

Here the first morphism preserves all limits, the last one all finite limits. Hence the composite preserves all finite limits if the left [[Kan extension]] $Lan_f$ does. This is the case if $f$ is a [[flat functor]].

(Because the left [[Kan extension]] is given by the [[colimit]]
$Lan_f X : d \mapsto {\underset{\to}{\lim}}((f^{op}/d) \to {\mathcal{C}}^{op} \stackrel{X}{\to} Set)$ over the [[comma category]] $f^{op}/d$ which is a [[filtered category]] if $f$ is flat, and [[filtered colimit]]s are precisely those that commute with [[finite limit]]s. For more details on this argument see the discussion at [Geometric morphisms between presheaf toposes](geometric+morphism#BetweenPresheafToposes).)

=--

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

This is a "contravariant" construction in that a morphism of sites going one way gives a geometric morphisms of toposes going the opposite way. Another condition, the _[[covering lifting property]]_ gives a covariant assignment.

=--

Conversely, any geometric morphism which restricts and corestricts to a functor between sites of definition is induced by a morphism between those sites:

+-- {: .num_prop #GeometricMorphismsComeFromMorphismsOfSites}
###### Proposition
Let $(\mathcal{C}, J)$ be a small site and let $(\mathcal{D}, K)$ be a [[small-generated site]].  Then a [[geometric morphism]]
$$
  f : Sh(\mathcal{D}, K) \to Sh(\mathcal{C}, J)
$$
is induced by a morphism of sites $(\mathcal{D}, K) \leftarrow (\mathcal{C}, J) : F$ precisely if the [[inverse image functor]] $f^*$ respects the [[Yoneda embedding]]s, i.e. there is a functor $F$ making the following diagram commute:
$$
  \array{
    \mathcal{D } &\stackrel{F}{\leftarrow}& \mathcal{C}
    \\
    {}^{\mathllap{j_{\mathcal{D}}}}\downarrow && \downarrow^{\mathrlap{j_{\mathcal{C}}}}
    \\
    Sh(\mathcal{D}, K) &\stackrel{f^*}{\leftarrow}&
    Sh(\mathcal{C}, J)
  }
  \,.
$$
=--

In the special case when $\mathcal{C}$ and $\mathcal{D}$ have finite limits and $\mathcal{D}$ is subcanonical, so that morphisms of sites can be defined using representably flat functors, this appears as ([Johnstone, lemma C2.3.8](#Johnstone)).  We give the proof in this case; for the general case see ([Shulman, Prop. 11.14](#Shulman)).  Note that the general case would not be true for the classical definition of "morphism of sites".

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

It suffices to show that given $f$, the factorization $F$ is, if it exists, necessarily a morphism of sites: because since $f^*$ is [[left adjoint]] and thus preserves all [[colimit]]s and every object in $Sh(C)$ is a colimit of [[representable functor|representables]], $f^*$ is fixed by the factorization. By uniqueness of [[adjoint functor]]s this means then that together with its [[right adjoint]] it is the geometric morphism induced from the morphism of sites, by prop. \ref{MorphismsOfSitesInduceGeometricMorphisms}.

So we show that $F$ is necessarily a morphisms of sites:

1. since the [[Yoneda embedding]] and [[sheafification]] as well as [[inverse image]]s preserve finite limits, so does $f^* j_{\mathcal{C}}$ and hence $F$ preserves finite limits, hence is a [[flat functor]];

1. $f^* h_{\mathcal{C}}$ preserves [[covering]]s (maps them to [[epimorphism]]s in $Sh(D, K)$) and since $K$ is assumed to be subcanonical it follows from [this prop.](site#CharacterizationOfSubcanonicalSites) that $j_{\mathcal{D}}$ also reflects covers. Therefore $F$ preserves covers.

=--

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

Let $(\mathcal{C},J)$ be a [[small category|small]] site and let $\mathcal{E}$ be any [[sheaf topos]]. Then we have an [[equivalence of categories]]

$$
  Topos(\mathcal{E}, Sh(\mathcal{C}, J))
  \simeq
  Site((\mathcal{C}, J), \mathcal{E})
$$

between the [[geometric morphism]]s from $\mathcal{E}$ to $Sh(\mathcal{C}, J)$ and the morphisms of [[site]]s from $(\mathcal{C}, J)$ to the [[big site]] $(\mathcal{E}, C)$ for $C$ the [[canonical coverage]] on $\mathcal{E}$.

=--

This appears as ([Johnstone, cor. C2.3.9](#Johnstone)).

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

Since for the [[canonical coverage]] the [[Yoneda embedding]] is the [[identity]], this follows directly from prop. \ref{GeometricMorphismsComeFromMorphismsOfSites}.

=--

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

Corollary \ref{LemmaForClassifyingToposes} leads to the notion of
[[classifying toposes]]. See there for more details.

=--

### Between $\kappa$-ary sites

+-- {: .num_prop}
###### Proposition
If $C$ and $D$ are [[∞-ary sites]], then a functor $f:C\to D$ is a morphism of sites if and only if it preserves finite local $\kappa$-prelimits.
=--
+-- {: .proof}
###### Proof
See ([Shulman, Prop. 4.8](#Shulman))
=--

## References

* Kashiwara-Schapira, _[[Categories and Sheaves]]_, section 17.2 (in terms of [[local isomorphisms]]).

* [[Saunders MacLane]] [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_, section VII. 10 of (in terms of covering [[sieves]]).

* [[Peter Johnstone]], _[[Elephant|Sketches of an Elephant]]_, section C2.3.

* [[Michael Shulman]], "Exact completions and small sheaves".  *Theory and Applications of Categories*, Vol. 27, 2012, No. 7, pp 97-173.  [Free online](http://www.tac.mta.ca/tac/volumes/27/7/27-07abs.html)
{#Shulman}

[[!redirects morphism of sites]]
[[!redirects morphisms of sites]]


Morris Hirsch

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

## Selected writings

On [[differential topology]]:

* [[Morris Hirsch]], _Differential topology_, GTM **33**, Springer (1976) [[doi:10.1007/978-1-4684-9449-5](https://link.springer.com/book/10.1007/978-1-4684-9449-5), [gBooks](http://books.google.com/books/about/?id=iSvnvOodWl8C)]

## Related entries

* [[differential topology]]

category: people

[[!redirects Morris W. Hirsch]]


Morris Weifeld

[Math genealogy page](https://www.mathgenealogy.org/id.php?id=4309)


Morris-Pareigis formal scheme

The [[formal schemes]] are typically studied over noetherian local rings, or in Grothendieck school in somewhat larger generality of [[pseudocompact ring]]s. However some very useful rings do not have any pseudocompact topology. Morris and Pareigis invented an extension of the theory of formal schemes over an arbitrary ground ring which involves some neat categorical ideas.

* Robert A. Morris, [[Bodo Pareigis]], _Formal groups and Hopf algebras over discrete rings_, Trans. Amer. Math. Soc.  197  (1974), 113--129, [doi](http://dx.doi.org/10.2307/1996930).

[[!redirects Morris-Pareigis formal scheme]]
[[!redirects Morris-Pareigis formal group]]
[[!redirects Morris–Pareigis formal scheme]]
[[!redirects Morris–Pareigis formal group]]
[[!redirects Morris--Pareigis formal scheme]]
[[!redirects Morris--Pareigis formal group]]


Morse function

#Contents#
* table of contents
{:toc}

## Definition

Consider a [[real numbers|real]]-valued [[smooth function]]

\[
  \label{TheFunction}
  f \colon M \longrightarrow \mathbb{R}
\]

on a [[smooth manifold]] $M$.

\begin{definition}
  \label{CriticalPoint}
A point $p\in M$ is a **critical point** of $f$ (eq:TheFunction) if for any smooth [[curve]] $\gamma : (-\epsilon, \epsilon)\to M$ with $\gamma(0)=p$, the [[tangent vector]]

$$
  \frac{d(f\circ\gamma)}{dt} |_{t=0} = 0
  \,.
$$

The critical point is **regular** if for one (or equivalently any) [[chart]] $\phi : U^{\open}\to \mathbb{R}^n$, where $p\in U$ and $\phi(p) = 0\in \mathbb{R}^n$, the **Hessian matrix**

$$\left(\frac{\partial^2 (f\circ \phi^{-1})}{\partial x^i\partial x^j}(0)\right)_{i,j=1,\ldots, n} $$

is a nondegenerate (i.e. maximal rank) matrix.
\end{definition}

\begin{definition}
The function $f$ is called a **Morse function** if every [[critical point]] of $f$ is [[regular value|regular]] (Def. \ref{CriticalPoint}).
\end{definition}

A choice of a Morse function on a compact manifold is often used to study topology of the manifold. This is called the [[Morse theory]].

## Properties

One of the basic tools of [[Morse theory]] is the _[[Morse lemma]]_.

## Related concepts

* [[perfect Morse function]]

## References

See also:

* Wikipedia, _[Morse theory](http://en.wikipedia.org/wiki/Morse_theory)_

[[!redirects Morse functions]]


Morse homology

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## Idea

(...)

## Related concepts

* [[Morse theory]]

* [[Floer homology]]

## References

Surveys:

* _A rapid course in Morse homology_ ([pdf](http://www.msri.org/attachments/sgw/477/lecture%202.pdf))

* Juanita Pinzón Caicedo, _Morse homology_ , thesis ([pdf](http://matematicas.uniandes.edu.co/~buribe/Estudiantes/MorseHomology.pdf))

* Yanfeng Chen, _A Brief History of Morse Homology_ ([pdf](http://math.berkeley.edu/~alanw/240papers03/chen.pdf))


Morse lemma

Let $M$ be a smooth manifold and $f: M\to \mathbb{R}$ a real valued function.

$p\in M$ is a **critical point** of $f$ if for any curve $\gamma : (-\epsilon, \epsilon)\to M$ with $\gamma(0)=p$, the vector

$$
\frac{d(f\circ\gamma)}{dt} |_{t=0} = 0.
$$

The critical point is **regular** if for one (or equivalently any) chart $\phi : U^{\open}\to \mathbb{R}^n$, where $p\in U$ and $\phi(p) = 0\in \mathbb{R}^n$, the Hessian matrix

$$\left(\frac{\partial^2 (f\circ \phi^{-1})}{\partial x^i\partial x^j}|_{0}\right)_{ij} $$

is a nondegenerate (i.e. maximal rank) matrix.

The **Morse lemma** states that for any regular critical point $p$ of $f$ there is a chart $\phi: U\to \mathbb{R}^n$ around $p$ such that the function in these coordinates is quadratic:

$$(f\circ\phi^{-1})(x^1,\ldots,x^n) = f(p) +\sum_{i=1}^k x_i^2 - \sum_{j=k+1}^n x_j^2$$

and the number $k$ is determined by the Hessian matrix. While the Morse lemma is proved by Morse, the modern proof is by Moser's deformation method.  The Morse lemma can be generalized to smooth functions on a Hilbert manifold, in which case there is a linear operator $A$ such that in suitable local coordinates, the quadratic functional $f\circ\phi^{-1}$ can be written as $f(p)+$<$A x,x$>.

* V. Guillemin, S. Sternberg, _Geometric asymptotics_, [Appendix 1](http://www.ams.org/online_bks/surv14/surv14-appI.pdf)


Morse potential

Morse potential is one of the standard 1-dimensional potentials which are used mainly in quantum mechanics (especially in atomic and molecular physics). If $r$ is the coordinate (say, the distance between atoms) in $\mathbb{R}_+$, and $l, D$ constants ($l$ being the scale of the system) then one of the standard forms of Morse potential is

$$
V(r) = D (e^{- 2 x/l} - 2 e^{-x/l})
$$

up to addition of physically irrelevant constant summand this is the same as $D (1 - exp(-x/l))^2$.

* wikipedia [Morse potential](https://en.wikipedia.org/wiki/Morse_potential)
* P. M. Morse, _Diatomic molecules according to the wave mechanics. II. Vibrational levels_, Phys. Rev. __34__, 57-64 (1929)
* B Molnár, M G Benedict, J Bertrand, _Coherent states and the role of the affine group in the quantum mechanics of the Morse potential_, J. Phys. __A34__, n.14, 3139-3151 (2001) [doi](https://doi.org/10.1088/0305-4470/34/14/318)
* Jens Peder Dahl, Michael Springborg, _The Morse oscillator in position space, momentum space, and phase space_, J. Chem. Physics __88__, 4535 (1988) [doi](http://dx.doi.org/10.1063/1.453761)
* C. Quesne, _Revisiting (quasi-)exactly solvable rational extensions of the Morse potential_, Internat. J. Modern Phys. A 27 (2012) no. 13, 1250073, 18 pp. [MR2926659](http://www.ams.org/mathscinet-getitem?mr=2926659) [doi](https://doi.org/10.1142/S0217751X1250073X)

category: physics

[[!redirects Morse Hamiltonian]]


Morse theory

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## Idea

**Morse theory** is the method of studying the [[topology]] of a [[smooth manifold]] $M$ by the study of [[Morse functions]] $M\to\mathbb{R}$ and their associated [[gradient flows]].

Classical Morse theory centered around simple statements like Morse inequalities, concerning just the [[Betti number]]s. It is useful not only for studying manifolds, but also for studying infinite CW-type spaces *homotopically filtered in manifolds*, as by Milnor and Bott (especially the *stable homotopy of the classical groups*) for spaces of paths in a smooth manifold.

Novikov--Morse theory is a variant using [[multivalued functions]]. There is also a [[discrete Morse theory]] for combinatorial cell complexes.

There are some infinite-dimensional generalizations like Floer instanton homology for 3-dimensional manifolds and also the Hamiltonian variant of [[Floer homology]] (and cohomology) for (finite dimensional) [[symplectic manifolds]]. An especially well-studied case is that of the [[cotangent bundle]] of a differentiable manifold with its standard symplectic structure; this is sometimes called Floer--Oh homology. Floer homology has been partly motivated by Arnold's conjecture on periodic trajectories in [[classical mechanics]]. The symplectic variant of Floer cohomology is related to [[quantum cohomology]].

The founders of Morse theory were [[Marston Morse]], [[Raoul Bott]] and [[Albert Schwarz]].

## Definitions

On a smooth manifold $M$, a [[smooth map|smooth]] function $\varphi: M \to \mathbb{R} $ is said to be *Morse* (or *a Morse function*) if

* the zero set of $ d \varphi $ consists of isolated points, and
* the [[Hessian]] of $\varphi$ at these points is nondegenerate.

The Morse functions on $M$ are dense in most reasonable topologies you could put on $C^{\infty}(M)$.
A further condition which is useful, in case $M$ is not compact, is

* if the (closed!) preimage of $( -\infty , \lambda ] $ under $\varphi$ is compact for all $\lambda$, then $\varphi$ is said to be *coercive*, whether or not it is Morse.

Together with a (smooth) Riemann structure $g =\langle \cdot,\cdot\rangle$, any real function $\varphi$ on $M$ defines a [[flow]] on $M$ by the equation
$$ - \langle \dot x, Y_x \rangle = Y_x \varphi = d\varphi(Y_x).$$
The Morse functions are notable in that the flows they define have isolated fixed-points with trivially linearizable dynamics, and <nowiki>...[fixme: less vague?]...</nowiki> no other stable cyles.

When $\varphi$ is Morse and coercive, the unstable manifolds of the fixpoints can be arranged into a [[CW complex]] $C_{unstable} (\varphi,g)$, canonically homeomorphic to $M$.  When $M$ is compact, $\varphi$ and $-\varphi$ are automatically both coercive, and $-\varphi$ induces a [[Poincare duality|dual]] CW complexe $ C_{stable} (\varphi,g) $.  Concretely, ....<nowiki>[details]</nowiki>.

### Sketch of a trivial application

Let $ X \to Y $ be a surjective submersion of compact smooth manifolds, and assume $Y$ is connected.  By suitable implicit function theorems, the preimage of any parametrized non-stationary curve $\gamma :(0,1)\to Y$ is a submanifold of $X$, and furthermore the parameter is a Morse function on this submanifold, having *no critical points*. (It is *not* coercive).  By a little more analysis, the Morse gradient flow is therefore a smooth family of homotopy equivalences.  A trivial adjustment of the Riemann structure further allows that the Morse flow sends fibers to fibers diffeomorphically, so that in fact the fibers over neighboring points of $Y$ are diffeomorphic.  But since $Y$ is connected, this implies that *all* the fibers are diffeomorphic, so that $X\to Y$ is a smooth fiber bundle over $Y$.

### Slightly less-trivial example

The restriction to the [[unit sphere]] in $\mathbb{R}^{n+1}$ of a generic quadratic form is Morse with $2(n+1)$ critical points --- two of each [[bilinear form|index]]; and furthermore this restriction clearly descends to $\mathbb{RP}^n$ as a Morse function with $n+1$ critical points, one of each index.  It can be shown that this is indeed the minimal collection of critical points supported by real projective space.

## Related concepts

* [[level set]]

* [[Reeb graph]]

## References

### General

* [[Raoul Bott]], _[Morse theory indomitable](http://archive.numdam.org/article/PMIHES_1988__68__99_0.pdf)_, _Publications Mathématiques de L'IHÉS_, 1988, Volume 68, Number 1, Pages 99-114.

* [[Raoul Bott]], [Lectures on Morse Theory, Old and New](http://www.ams.org/journals/bull/1982-07-02/S0273-0979-1982-15038-8/),  Bull. Amer. Math. Soc. 7 (1982), 331-358.

* [[Raoul Bott]], The stable homotopy of the classical groups.
 Ann. of Math. (2) 70 1959 313–337.

* [[Daniel Freed]], [Commentary](http://www.ams.org/journals/bull/2011-48-04/S0273-0979-2011-01349-0/) on "Lectures on Morse Theory, Old and New", Bull. Amer. Math. Soc., 48(4), October 2011, 517–523

* [[Marco Gualtieri]], _[Course page](http://www.math.toronto.edu/mgualt/Morse%20Theory/Morse_2009.html)_, lecture notes and links.

* [[Martin A. Guest|Martin Guest]], Morse theory in the 1990s, [arXiv:math/0104155](http://arxiv.org/abs/math/0104155).

* [[Loring Tu]], _Morse theory_ [node](http://www.math.harvard.edu/history/bott/bottbio/node9.html) on online bio of R. Bott

* [[M. M. Postnikov]], Введение в теорию Морса — М.: Наука, 1971

### Morse complex and homology

* [[John Milnor]], _Lectures on the h-cobordism theorem_, Notes by L. Siebenmann & J. Sondow, Princeton Univ.
Press, 1965.
* Matthias Schwarz, _Morse homology_, Progress in Mathematics __111__, 1993

There is also a variant due to Barannikov, and in a more abstract form due to Viterbo:

* S. Barannikov, _The framed Morse complex and its invariants_, Advances in Soviet Math. 21 (1994), 93-115.

* François Laudenbach, _On an article by S. A. Barannikov_, [arxiv/1509.03490](http://arxiv.org/abs/1509.03490)

* Dorian Le Peutrec, Francis Nier, Claude Viterbo, [_Precise Arrhenius Law for p-forms: The Witten Laplacian and Morse–Barannikov Complex_](https://link.springer.com/article/10.1007/s00023-012-0193-9), Annales Henri Poincaré 14 (2013), 567–610.

### Relation to supersymmetric quantum mechanics

The relation to [[supersymmetric quantum mechanics]] is due to

* [[Edward Witten]]. _Supersymmetry and Morse theory_. J. Diff Geom. 17(4): 661-692 (1982). ([doi](https://doi.org/10.4310/jdg/1214437492))

Reviews include

* [[Gábor Pete]], section 2 of _Morse theory_, lecture notes 1999-2001   ([pdf](http://www.math.bme.hu/~gabor/morse.pdf))


Morse-Kelley set theory

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## Idea

The **Morse-Kelley set theory** or **Morse-Kelley class theory** ($MK$) is an axiomatic approach to [[class theory]] and [[set theory]] which has both [[classes]] and [[sets]]. Whereas [[NBG]] (which also has both classes and sets) is [[conservative extension|conservative]] over [[ZFC]], Morse--Kelley is not a conservative extension of $NBG$. The principal difference from $NBG$ is that $MK$ allows arbitrary formulas $\phi$ appearing in the class [[axiom of specification|comprehension axiom schema]] (in particular, formulae with quantifiers ranging over classes themselves).

The approach is explained in the appendix to [[John Kelley]]'s 1955 book _General Topology_.

## See also

* [[class theory]]

* [[set theory]]

## References

A definitive source (by one of the authors of the theory) is

* [[Anthony P. Morse]], _A theory of sets_, Pure and Applied Mathematics XVIII, Academic Press (1965), xxxi+130 pp.   Second Edition, Pure and Applied Mathematics 108, Academic Press (1986), xxxii+179 pp. ISBN: 0-12-507952-4

For discussion about the [[category of classes]] in Morse-Kelly class theory, see

* [[Henrik Forssell]], *Categorical Models of Intuitionistic Theories of Sets and Classes*. Master's thesis, Carnegie Mellon University, 2004. ([PDF](https://www.phil.cmu.edu/projects/ast/Papers/forssell_thesis.pdf))

[[!redirects Morse-Kelley set theory]]
[[!redirects Morse–Kelley set theory]]
[[!redirects Morse--Kelley set theory]]
[[!redirects Morse-Kelley class theory]]
[[!redirects Morse–Kelley class theory]]
[[!redirects Morse--Kelley class theory]]
[[!redirects MK]]


Morten Brun

* [Homepage](http://folk.uib.no/mbr085/)

Related webpages

* [[Tambara functor]]

category: people


Morton Brown

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

## Selected writings

* [[Morton Brown]], _Locally flat embeddings of topological manifolds_ in  _Topology of 3–manifolds  and related topics_ (Proceedings of the University of Georgia Institute, 1961),pp. 83 – 91. Prentice – Hall, Englewood Cliffs, N.J., 1962

* [[Morton Brown]], _Locally flat imbeddings of topological manifolds_, Annals of Mathematics, Vol. 75 (1962), p. 331-341 ([jstor:1970177](https://www.jstor.org/stable/1970177))

  (on the [[collar neighbourhood theorem]])

## Related $n$Lab entries

* [[collar neighbourhood theorem]]

category: people


Moses Schönfinkel

[[!redirects Moses Schonfinkel]]

* [Wikipedia entry](https://de.wikipedia.org/wiki/Moses_Sch%C3%B6nfinkel)

## related $n$Lab entries

* [[combinatory logic]]

* [[currying]]

category: people


Moshe Rozali

* [webpage](http://www.phas.ubc.ca/~rozali/)

category: people


Moshé Flato

* [MacTutor History of Mathematics Archive entry](https://mathshistory.st-andrews.ac.uk/Biographies/Flato/)

* [Wikipedia entry](http://de.wikipedia.org/wiki/Moshé_Flato)

## Selected writings

Introducing [[formal deformation quantization]]:

* {#BayenFlatoFronsdalLichnerowiczSternheimer78} [[François Bayen]], [[Moshé Flato]], [[Christian Fronsdal]], , [[André Lichnerowicz]], [[Daniel Sternheimer]], _Deformation theory and quantization. I. Deformations of symplectic structures._, Annals of Physics **111** 1 (1978) 61-110 [doi:10.1016/0003-4916(78)90224-5]

* {#BayenFlatoFronsdalLichnerowiczSternheimer78b} [[François Bayen]], [[Moshé Flato]], [[Christian Fronsdal]], [[André Lichnerowicz]], [[Daniel Sternheimer]], _Deformation theory and quantization. II. Physical applications_, Annals of Physics **111** 1 (1978) 111-151 [doi:10.1016/0003-4916(78)90225-7]

On [[superconformal multiplets]] and the [[singleton representation]]:

* {#FlatoFronsdal81} [[Moshé Flato]], [[Christian Fronsdal]], _Quantum field theory of singletons. The rac_, J. Math. Phys. **22** (1981) 1100 [[doi:10.1063/1.524993](https://doi.org/10.1063/1.524993)]

## Related entries

* [[formal deformation quantization]]

* [[singleton representation]]

category: people

[[!redirects Moshe Flato]]


Moss Sweedler

* [webpage](http://www.math.cornell.edu/People/Faculty/sweedler.html)

Moss E. Sweedler was an early researcher in [[Hopf algebras]], authoring

* [[M. E. Sweedler]], __Hopf algebras__, Mathematics Lecture Note Series, Benjamin 1969.

and is a professor emeritus at Cornell. His work on noncommutative extensions dealing with [[Sweedler coring]]s and related work of his colleague Chase from Cornell on noncommutative [[Galois theory]], predating [[Hopf-Galois theory]] from the 1970s.

## Related entries

* [[Sweedler coring]]

* [[Sweedler notation]]

* [[measuring]]

category: people
[[!redirects M. E. Sweedler]]
[[!redirects Moss E. Sweedler]]


Mostow rigidity theorem

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## Statement

If a [[3-manifold]] $\Sigma$ (or more generally a [[smooth manifold]] of [[dimension]] $\geq 3$) admits the [[mathematical structure|structure]] of a [[hyperbolic manifold]] of [[finite number|finite]] [[volume]], then this structure is unique up to [[isometry]], and in fact it is uniquely determined already by the [[fundamental group]] $\pi_1(\Sigma)$.

([Mostow 68](#Mostow68)).

## References

The original article:

* {#Mostow68} [[George Mostow]], _Quasi-conformal mappings in $n$-space and the rigidity of hyperbolic space forms_,  Publications Mathématiques de l'IHÉS, Volume 34  (1968), p. 53-104 ([numdam:PMIHES_1968__34__53_0](http://www.numdam.org/item/PMIHES_1968__34__53_0))

Review:

* Marc Bourdon, _Mostow type rigidity theorems_,
In: _Handbook of Group Actions (Vol. IV)_, Advanced Lectures in Mathematics 41, Ch. 4, pp. 139–188, International Press 2018 ([pdf](http://math.univ-lille1.fr/~bourdon/papiers/Mostow.pdf), [book toc pdf](https://www.intlpress.com/site/pub/files/preview/bookpubs/00000587.pdf))

See also

* Wikipedia, _[Mostow rigidity theorem](http://en.wikipedia.org/wiki/Mostow_rigidity_theorem)_

[[!redirects Mostow's rigidity theorem]]


Mostowski set theory

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## Idea

**Mostowski set theory** is a [[well-founded relation|well-founded]] [[material set theory]] which is [[material-structural adjunction|equivalent in strength]] to the [[structural set theories]] of [[ETCS]] and bounded [[SEAR]] with [[axiom of choice|choice]].

## Definition

### In classical logic

We assume that we are working in full [[classical logic]]. Mostowski set theory has the following [[axioms]] and [[axiom schemata]]:

1. [[axiom of extensionality|Extensionality]]: for any set $x$ and $y$, $x = y$ if and only if for all $z$, $z \in x$ if and only if $z \in y$.

2. [[empty set|Empty set]]: there is a set $\varnothing = \{\}$.

3. [[axiom of pairing|Pairing]]: for all sets $x$ and $y$, there is a set $\{x, y\}$

4. [[axiom of union|Union]]: for all sets $x$, there is a set $\{z \vert \exists y \in x.z \in y\}$

5. [[axiom schema of bounded separation|Schema of $\Delta_0$-separation]]: for any $\Delta_0$-formula $\phi(x)$ and for any set $a$, there is a set $\{x \in a \vert \phi(x)\}$

6. Schema of limited $\Delta_0$-replacement: for any $\Delta_0$-formula $\phi(x, y)$, for all sets $a$ and $b$, if for any set $x \in a$ there is a unique set $y$ such that $\phi(x, y)$, and for all $z \in y$, $z \subseteq b$, then there is a set $\{y \vert \exists x \in a.\phi(x, y)\}$

7. [[power set|Power sets]]: for any set $x$, there is a set $\mathcal{P}(x) = \{y \vert y \subseteq x\}$

8. [[axiom of infinity|Infinity]]: there is a set $\omega$ such that for all sets $x$, $x \in \omega$ if and only if $x = \emptyset$ or there exists a set $y \in \omega$ such that $y \cup \{y\} = x$.

9. [[axiom of choice|Choice]]: for any set $a$, if for all $x \in a$ there is a set $y \in x$, then there is a function $f$ from $a$ to $\bigcup a$ such that $f(x) \in x$ for all $x \in a$.

10. [[axiom of regularity|Regularity]]: if there is a set $y \in x$, then there is a set $y \in x$ such that $x \cap y = \emptyset$

11. [[transitive closure|Transitive closure]]: every set is a subset of a smallest transitive set

12. [[Mostowski's principle]]: every well-founded extensional relation is isomorphic to a transitive set equipped with the relation $\in$.

This implies that Mostowski set theory is equivalent in strength to [[ETCS]], a [[well-pointed topos]] with [[natural numbers object]] and the [[axiom of choice]].

#### Variations of Mostowski set theory in classical logic

By removing axiom 9 from Mostowski set theory, one gets **Mostowski set theory without choice**, which is equivalent in strength to a [[well-pointed]] [[Boolean topos]] with [[natural numbers object]] and the [[axiom of well-founded materialization]].

By removing axiom 7 from Mostowski set theory, one gets **[[predicative mathematics|predicative]] Mostowski set theory**, which is equivalent in strength to a [[well-pointed]] [[Boolean pretopos]] with [[natural numbers object]] and the [[axiom of choice]].

By removing axiom 8 from Mostowski set theory, one gets **[[finite mathematics|weakly finitist]] Mostowski set theory**, which is equivalent in strength to a [[well-pointed]] [[Boolean topos]] with the [[axiom of choice]].

If one replaces axiom 8 with the **[[axiom of finiteness]]** (for any formula $\phi$, if $\phi(\emptyset)$ and for all sets $x$, $\phi(x)$ implies that $\phi(x \cup \{x\})$, then for all sets $x$, $\phi(x)$), one gets **strongly finitist Mostowski set theory**, which is equivalent in strength to the category [[FinSet]]. Axioms 7 and 9 are redundant in this formulation, since both are implied by the axiom of finiteness.

Thus, the most general variation of Mostowski set theory in classical logic is **weakly finitist predicative Mostowski set theory without choice**, which consists of axioms 1-6 and 10-12, and is equivalent in strength to a general [[well-pointed]] [[Boolean pretopos]] with the [[axiom of well-founded materialization]].

### In intuitionisitc logic

Now, we assume that we are working in [[intuitionistic logic]]. Then **$\Delta_0$-classical Mostowski set theory** has the following [[axioms]] and [[axiom schemata]]:

1. [[axiom of extensionality|Extensionality]]: for any set $x$ and $y$, $x = y$ if and only if for all $z$, $z \in x$ if and only if $z \in y$.

2. [[empty set|Empty set]]: there is a set $\varnothing = \{\}$.

3. [[axiom of pairing|Pairing]]: for all sets $x$ and $y$, there is a set $\{x, y\}$

4. [[axiom of union|Union]]: for all sets $x$, there is a set $\{z \vert \exists y \in x.z \in y\}$

5. [[axiom schema of bounded separation|Schema of $\Delta_0$-separation]]: for any $\Delta_0$-formula $\phi(x)$ and for any set $a$, there is a set $\{x \in a \vert \phi(x)\}$

6. Schema of limited $\Delta_0$-replacement: for any $\Delta_0$-formula $\phi(x, y)$, for all sets $a$ and $b$, if for any set $x \in a$ there is a unique set $y$ such that $\phi(x, y)$, and for all $z \in y$, $z \subseteq b$, then there is a set $\{y \vert \exists x \in a.\phi(x, y)\}$

7. [[power set|Power sets]]: for any set $x$, there is a set $\mathcal{P}(x) = \{y \vert y \subseteq x\}$

8. [[axiom of infinity|Infinity]]: there is a set $\omega$ such that for all sets $x$, $x \in \omega$ if and only if $x = \emptyset$ or there exists a set $y \in \omega$ such that $y \cup \{y\} = x$.

9. [[axiom of choice|Choice]]: for any set $a$, if for all $x \in a$ there is a set $y \in x$, then there is a function $f$ from $a$ to $\bigcup a$ such that $f(x) \in x$ for all $x \in a$.

10. [[axiom of foundation|Regularity]]: if there is a set $y \in x$, then there is a set $y \in x$ such that $x \cap y = \emptyset$

11. [[transitive closure|Transitive closure]]: every set is a subset of a smallest transitive set

12. [[Mostowski's principle]]: every well-founded extensional relation is isomorphic to a transitive set equipped with the relation $\in$.

This implies that Mostowski set theory is equivalent in strength to [[ETCS]] in intuitionistic logic, a [[constructively well-pointed topos]] with [[natural numbers object]] and the [[axiom of choice]].

The name $\Delta_0$-classical Mostowski set theory comes from the fact that the [[law of excluded middle]] is only valid for $\Delta_0$-formulas in the theory, being a consequence of the [[axiom of choice]], rather than for all formulas in the theory, as in the case for Mostowski set theory in classical logic.

#### Variations of Mostowski set theory in intuitionistic logic

By removing axiom 9 from $\Delta_0$-classical Mostowski set theory, one gets **[[intuitionistic logic|intuitionistic]] Mostowski set theory**, which is equivalent in strength to a [[constructively well-pointed topos]] with [[natural numbers object]] and the [[axiom of well-founded materialization]].

By replacing axiom 7 by the [[axiom of fullness]] (for all sets $a$ and $b$, the set of [[entire relations]] from $a$ to $b$ exists) in intuitionistic Mostowski set theory, one gets **constructive Mostowski set theory**.

By removing axiom 7 from $\Delta_0$-classical Mostowski set theory, one gets **[[predicative mathematics|predicative]] $\Delta_0$-classical Mostowski set theory**, which is equivalent in strength to a [[constructively well-pointed]] [[Heyting pretopos]] with [[natural numbers object]] and the [[axiom of choice]].

By removing axiom 7 from intuitionistic Mostowski set theory, one gets **[[predicative mathematics|strongly predicative]] constructive Mostowski set theory**, which is equivalent in strength to a [[constructively well-pointed]] [[Heyting pretopos]] with a [[natural numbers object]] and the [[axiom of well-founded materialization]].

By replacing axiom 7 by the axiom of exponentiation (for all sets $a$ and $b$, the set $b^a$ of functions from $a$ to $b$ exists) in intuitionistic Mostowski set theory, one gets **(neutral) [[predicative mathematics|weakly predicative]] constructive Mostowski set theory**, which is equivalent in strength to a [[constructively well-pointed]] [[ΠW-pretopos]] with the [[axiom of well-founded materialization]].

By removing axiom 8 from any $X$ Mostowski set theory, with $X$ being one of ($\Delta_0$-classical, intuitionistic, predicative $\Delta_0$-classical, weakly predicative constructive, strongly predicative constructive), one gets **[[finite mathematics|weakly finitist]] $X$ Mostowski set theory**, which is equivalent in strength to a [[constructively well-pointed]] $C$, where $C$ is one of ([[Boolean topos]] with the [[axiom of choice]], [[elementary topos]] with the [[axiom of well-founded materialization]], [[Boolean pretopos]] with the [[axiom of choice]], [[Π-pretopos]] with the [[axiom of well-founded materialization]], [[Heyting pretopos]] with the [[axiom of well-founded materialization]]).

If one replaces axiom 8 in any of the above with the **[[axiom of finiteness]]** (for any formula $\phi$, if $\phi(\emptyset)$ and for all sets $x$, $\phi(x)$ implies that $\phi(x \cup \{x\})$, then for all sets $x$, $\phi(x)$), one gets **strongly finitist Mostowski set theory**, which is equivalent in strength to the category [[FinSet]]. Axioms 7 and 9 are redundant in this formulation, since both are implied by the axiom of finiteness.

Thus, the most general variation of Mostowski set theory in intuitionistic logic is **weakly finitist, strongly predicative, constructive Mostowski set theory**, which consists of axioms 1-6 and 10-12, and is equivalent in strength to a general [[constructively well-pointed]] [[Heyting pretopos]] with the [[axiom of well-founded materialization]].

## Material-structural adjunctions

Given any material set theory $V$ which satisfies axioms 1-6. Then one could construct the category $\mathbb{Set}(V)$ of [[sets]] and [[functions]] in $V$, and $\mathbb{Set}(V)$ is a [[constructively well-pointed]] [[Heyting pretopos]].

Given any constructively well-pointed Heyting pretopos $\mathcal{E}$, we can construct the type $\mathbb{V}(\mathcal{E})$ as the [[type]] of well-founded extensional accessible pointed graph objects in $\mathcal{E}$. $\mathbb{V}(\mathcal{E})$ is a model of material set theory satisfying axioms 1-6 and 10-12.

## See also

* [[material set theory]]
* [[ETCS]]

## References

* [[Michael Shulman]] (2018). Comparing material and structural set theories. [arXiv:1808.05204](https://arxiv.org/abs/1808.05204).

[[!redirects intuitionistic Mostowski set theory]]
[[!redirects constructive Mostowski set theory]]


Mostowski's collapsing lemma

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## Idea

**Mostowski's collapsing lemma** states that any [[extensional relation|extensional]] [[well-founded relation]] is isomorphic to a (necessarily unique) [[transitive set]].

Mostowski's lemma can be proven in [[ZF]] set theory using the [[axiom of replacement]] and [[axiom of separation]].  In set theories that are not powerful enough to prove the lemma, it can be adopted as a separate axiom; in this case it is sometimes called **Mostowski's principle**.

Mostowski's principle is sort of a "dual" of the [[axiom of foundation]] that the membership relation is well-founded (and, by the [[axiom of extensionality]], extensional).  A related principle for [[structural set theory]] is the [[axiom of well-founded materialization]].

## See also

* [[Mostowski set theory]]

## References

* [[Michael Shulman]] (2018). Comparing material and structural set theories. [arXiv:1808.05204](https://arxiv.org/abs/1808.05204).

category: foundational axiom

[[!redirects Mostowski's collapsing lemma]]
[[!redirects Mostowski collapsing lemma]]
[[!redirects Mostowski's lemma]]
[[!redirects Mostowski lemma]]
[[!redirects Mostowski's principle]]
[[!redirects Mostowski principle]]
[[!redirects Mostowski's axiom]]
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motivation for cohesive toposes

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#### Discrete and concrete objects
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This page is meant to provide non-technical motivation for the notion of _[[cohesion]]_ as in formalized in the notions

* _[[cohesive topos]]_,

* _[[cohesive (∞,1)-topos]]_, _[[cohesive homotopy type theory]]_ .

#Contents#
* table of contents
{:toc}

## Motivation

A notion of _[[cohesion]]_ on a collection $\mathbf{H}$ of [[spaces]] is supposed to be a means to specify how [[points]] in any space $X \in \mathbf{H}$ "hang together" or "cohere", analogous to how water molecules in a droplet of water are held together by [cohesion (in the sense of chemistry)](http://en.wikipedia.org/wiki/Cohesion_%28chemistry%29). (The inclined reader may, or may not, recognize this also in _[WdL, Form und Inhalt](Science+of+Logic#FormUndInhalt)_, see the discussion there.)

A basic example arises for [[topological spaces]] or [[manifolds]]: here the "droplet of water" is an _[[open ball]]_ of points. Indeed, one of the central examples of cohesive spaces is that of [[smooth spaces]] and these are spaces characterized by the fact that they can be _probed_ by smooth open balls (in the sense described at _[[motivation for sheaves, cohomology and higher stacks]]_), such that these smooth open balls are the _basic_ "cohesive droplets" out of which any smooth space is built (this roughly in the sense of [[topological base|basis of a topology]], but a bit more generally than that).

That intuition should be evident enough. The question is which formal axioms capture this droplet-intuition accurately and efficiently. The crucial insight of [[Bill Lawvere]] (see the references at _[[cohesive topos]]_) was that a rather minimalistic set of axioms already does the job:

1. there has to be an assignment $\Pi : \mathbf{H} \to Set$ that sends every cohesive space $X$ to its _set of cohesively [[connected components]]_. For instance a single open ball as above, a basic droplet, is sent to the set $\{*\}$ with a single element.

1. We should be entitled to regard every [[set]] $S \in Set$ trivially as a cohesive space in two ways:

    * either we regard every element in $S$ as an atomic cohesive droplet itself, cohesively disconnected to any other point. Call the resulting cohesive space $Disc(X)$. For [[smooth space|smooth cohesion]], $Disc(X)$ is simply the [[discrete space|discrete manifold]] being the [[disjoint union]] of one point per element in $S$.

    * or, at the opposite extreme, we regard all the elements in $S$ as being cohesively connected, hence regard all of $S$ itself as one single big cohesive droplet. We write $coDisc(X)$ for $S$ regarded as a cohesive space this way, because in the context of [[Euclidean-topological infinity-groupoid|topological cohesion]] this is the [[codiscrete space]] on the given set.

1. The collection of discrete and codiscrete cohesive spaces should sit nicely inside the collection of all cohesive spaces, essentially in just the obvious way that one intuitively expects. For instance cohesive maps between two discretely cohesive spaces should be simply maps between the underlying sets, and so on.

   In particular the notion of "discrete" exhibited by the collection of discrete objects has indeed to be compatible with the notion of "cohesive" as seen by that map $\Pi$ above. This is just the evident consistency condition: for instance if $Disc(S)$ is a set regarded as a discrete cohesive space, then $\Pi(Disc(S))$, which is supposed to be its set of cohesively connected points, should be just $S$ itself, since no point in $Disc(S)$ is supposed to be cohesively connected to any other.

   In particular $\Pi(Disc(*))$ should be the point again.

That's essentially it, already. It sounds very simple (hopefully) and indeed it is very simple. Once one knows the notion of an [[adjoint functor]] it is pretty straightforward to formalize the above text in terms of that notion, to arrive at the concept of _[[cohesive topos]]_.

The interesting observation is that simple as this idea is, it has very powerful consequences ... once it is formalized not just with adjoint functors but with [[adjoint (∞,1)-functors]]. This is conceptually a very simple step. Moreover, in the [[foundations]] provided by [[homotopy type theory]] this is actually the simpler and more natural step. It leads to _[[cohesive (∞,1)-toposes]]_ and _[[cohesive homotopy type theory]]_.

## Special aspects

### Local contractibility / local $\infty$-connectedness
 {#LocalContractibility}

While cohesive spaces subsume several familiar notions of [[geometry]], there are some constraints.

In particular, a cohesive space is always _locally contractible_ or rather _locally $\infty$-connected_ in some sense. The local contractions are those of the "basic cohesive droplets", in the spirit of the above discussion. (In the [[(∞,1)-topos theory|(∞,1)-topos theoretic]] formalization this is reflected in the fact that every [[cohesive (∞,1)-topos]] is in particular a _[[locally ∞-connected (∞,1)-topos]]_.)

For instance [[locally contractible topological ∞-groupoids]] are cohesive, as are [[Euclidean-topological ∞-groupoids]].

But a geometry modeled on a [[small category|small]] [[full subcategory]] of [[Top]] that contains [[locally contractible topological space|locally non-contractible topological spaces]] will in general not be cohesive. In particular for instance general [[topological stacks]] do not live in a cohesive $(\infty,1)$-topos. But for instance [[differentiable stacks]] do, and generally [[smooth ∞-groupoids]], because every [[manifold]] and hence in particular every [[smooth manifold]] is locally contractible.

## Related entries

* [[geometry of physics]]

* [[fiber bundles in physics]]

* [[higher category theory and physics]]

* [[string theory FAQ]]

* [[twisted smooth cohomology in string theory]]

* [[motives in physics]]

* [[Hilbert's sixth problem]]

* [[model theory and physics]]

* [[L-infinity algebras in physics]]

* [[motivation for sheaves, cohomology and higher stacks]]

* [[applications of (higher) category theory]]

* [[motivation for higher differential geometry]]

category: motivation

[[!redirects motivation for cohesion]]


motivation for directed homotopy > history