model structure on topological spaces

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#### Model category theory
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[[!include model category theory - contents]]
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#### Topology
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[[!include topology - contents]]
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#### Homotopy theory
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[[!include homotopy - contents]]
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#Contents#
* table of contents
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## Idea

Philosophically, [[model category|model structures]] allow one to [[localization|localize]] a [[category]] at a particular collection of [[category with weak equivalences|weak equivalences]], which one would like to formally invert.

For [[topological space]]s, there are two natural candidates for the collection $W$ of weak equivalences:

1. the [[weak homotopy equivalence]]s

1. and the [[homotopy equivalence]]s.

Both of these have accompanying model structures.  Interestingly, these two model structures can also be combined to form what's known as the _mixed_ model structure.

All of these model structures exist not only on the category of all topological spaces, but also on most [[convenient categories of topological spaces]].  Using a nice category instead is sometimes important, such as if we want the model structure to be [[monoidal model category|monoidal]].

## Definition

### Classical Quillen Model Structure
  {#QuillenModelStructure}

The first, and most prevalent, [[model category]] structure on [[Top]], called the _[[classical model structure on topological spaces]]_ ([Quillen67, II.3](#Quillen67)) has

* weak equivalences are the [[weak homotopy equivalences]];

* fibrations are the [[Serre fibrations]], maps which have the [[right lifting property]] with respect to all inclusions of the form $i_0 : D^n \hookrightarrow D^n \times I$ that include the $n$-disk as $D^n \times \{0\}$.

* cofibrations are the "[[retracts]] of [[relative cell complexes]]".

This is a [[cofibrantly generated model category]]. The cofibrations $C$ are generated by the set of [[boundary]] inclusions $S^{n-1} \hookrightarrow D^n$ for all $n \in \mathbb{N}$ in the sense that they are the smallest saturated class containing these morphisms. As a consequence, of Quillen's [[small object argument]], all cofibrations have the form described above, where a [[relative cell complex]] is a [[transfinite composition|transfinite composite]] of [[pushouts]] of [[coproducts]] of these generating maps.

This model structure is sometimes called the **[[Quillen model structure on topological spaces]]** or **$q$-model structure** on [[Top]].

### Hurewicz (or Strøm) Model Structure
 {#StromModelStructure}

A second model structure -- the *[[Strøm model structure]]* -- has

* weak equivalences are the [[homotopy equivalences]];

* fibrations are the **[[Hurewicz fibration]]s**, which are defined to be maps that have the right lifting property with respect to _all_ inclusions $i_0 : A \hookrightarrow A \times I$ for any topological space $A$.

* cofibrations are determined by these classes and are called the **closed [[Hurewicz cofibration]]s**.

This model structure is sometimes called the **Hurewicz model structure**, since it uses Hurewicz fibrations and cofibrations, or also the $h$-model structure, where $h$ can stand for either "Hurewicz" or "homotopy equivalence."  However, it is also sometimes called the [[Strøm model structure]], since it was first proven to exist by [[Arne Strøm]].

### Mixed Model Structure

From the definitions, [[Hurewicz fibration]]s are necessarily [[Serre fibration]]s. It is well-known that [[homotopy equivalence]]s are [[weak homotopy equivalence]]s. If we write $(C_1, F_1, W_1)$ for the classes of the first model structure and $(C_2, F_2, W_2)$ for the classes of the second, we have $W_2 \subset W_1$ and $F_2 \subset F_1$.

In general given two model structures with these inclusions, we get a third **[[mixed model structure]]** $(C_m, F_2, W_1)$ where the cofibrations $C_m$ are determined by the other two classes.

On topological spaces, this model structure has

* weak equivalences the [[weak homotopy equivalence]]s

* fibrations the [[Hurewicz fibration]]s;

*  cofibrant spaces (the [[m-cofibrant space]]s) are precisely those spaces that are homotopy equivalent to [[CW complex]]es.

## Properties

### Restriction to convenient categories of topological spaces
 {#OnNiceTopologicalSpaces}

For the discussion of the [[homotopy theory]] given by the model structure on topological spaces, it is necessary or at least useful to pass to [[convenient categories of topological spaces]].

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###### Definition

Write

* $k Top \hookrightarrow Top$ for the [[full subcategory]] of [[k-spaces]];

* $CGTop \hookrightarrow Top$ for the full subcategory of [[compactly generated spaces]].

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###### Proposition
**([[model structure on compactly generated topological spaces]])**

There is a [[model category]] structure $kTop_{Qu}$ on $k Top$ in which a morphism is a cofibration, fibration or weak equivalence, respectively, precisely if it is so under the inclusion $k Top \hookrightarrow Top$ in the [[classical model structure on topological spaces]]. This inclusion is the [[left adjoint]] in a [[Quillen equivalence]]

$$
  k Top
  \underoverset
    {\underset{k}{\longleftarrow}}
    {\overset{}{\hookrightarrow}}
    {\;\;\;\; \simeq_{\mathrlap{Qu}} \;\;\;\;}
  Top
$$

=--

This appears for instance as ([Hovey, theorem 2.4.23](#Hovey))

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###### Proposition

There is a [[model category]] structure $CGTop_{Quillen}$ on $CGTop$ in which a morphism is a cofibration, fibration or weak equivalence, respectively, precisely if it is so under the inclusion $CGTop \hookrightarrow Top$. And this inclusion is the [[right adjoint]] in a [[Quillen equivalence]]

$$
  CGTop_{Qu}
  \underoverset
    {\underset{}{\hookrightarrow}}
    {\overset{ h }{\longleftarrow}}
    {\;\;\;\; \simeq_{\mathrlap{Qu}} \;\;\;\; }
  k Top_{Qu}
  \mathrlap{\,.}
$$

=--

(e.g. [Hovey 1999, Thm. 2.4.25](#Hovey99))

To appreciate the following, notice that [[classical model structure on topological spaces|$Top_{Qu}$]] is _not_ a [[monoidal model category]], because [[Top]] itself is not ([[cartesian closed category|cartesian]]) [[closed monoidal category|closed]].

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###### Proposition

Both $k Top_{Qu}$ and $CGTop_{Qu}$ are [[symmetric monoidal category|symmetric]] [[monoidal model categories]].

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This appears as ([Hovey, prop. 4.2.11](#Hovey)).

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###### Proposition

In fact $CGTop_{Quillen}$ is a [[cartesian closed model category]].

=--

(This is briefly mentioned in [Berger-Moerdijk 2003, Sec. 3.3.2](#BergerMoerdijk03).)

### Relation between $Top_{Quillen}$ and $sSet_{Quillen}$

The [Quillen model structure](#QuillenModelStructure) $Top_{Qullen}$ is [[Quillen equivalence|Quillen equivalent]] to the standard (Quillen) [[model structure on simplicial sets]] via the total [[fundamental infinity-groupoid|singular complex]] and [[geometric realization]] functors.

$$
  (\vert-\vert \dashv Sing)
  \colon
  Top_{Quillen}
    \stackrel{\overset{|-|}{\leftarrow}}{\underset{Sing}{\to}}
  sSet_{Quillen}
  \,.
$$

Since the standard [[model structure on simplicial sets]] is a [[presentable (infinity,1)-category|presentation]] of the [[(∞,1)-category]] [[∞Grpd]] of [[∞-groupoid]]s realized as [[Kan complex]]es, this identifies topological spaces with [[∞-groupoid]]s in an [[(∞,1)-category|(∞,1)-categorical]] sense. Notably it says that every $\infty$-groupoid is, up to equivalence, the [[fundamental ∞-groupoid]] of some topological space.

This statement is called the [[homotopy hypothesis]] (which here is a theorem). See there for more details.

### Relation between $Top_{Quillen}$ and $Top_{Strom}$

The identity functor constitutes a [[Quillen adjunction]]

$$
  (Id \dashv Id)
   :
   Top_{Strom}
    \stackrel{\leftarrow}{\to}
   Top_{Quillen}
$$

between the [Quillen model structure](#QuillenModelStructure) and the [Strom model structure](#StromModelStructure) on $Top$. Here $Top_{Strom} \to Top_{Quillen}$ is the [[Quillen adjunction|right Quillen functor]].

## Related concepts

* [[homotopical structure on C*-algebras]]

* [[model structure on topological spaces]]

  * [[classical model structure on topological spaces]]

    * [[classical model structure on pointed topological spaces]]

  * [[Strøm model structure]]

  * [[model structure on Delta-generated topological spaces]]

## References

The original "Quillen" or "q-" model structure is due to

* {#Quillen67} [[Daniel Quillen]], Ch. II, Sec. 3 of *[[Homotopical Algebra]]*, Lecture Notes in Mathematics **43**, Springer 1967([doi:10.1007/BFb0097438](https://doi.org/10.1007/BFb0097438))

An expository, concise and comprehensive writeup is in

* [[Philip Hirschhorn]], _The Quillen model category of topological spaces_, 2015 ([arXiv:1508.01942](http://arxiv.org/abs/1508.01942))

Standard textbooks references include

* {#Hovey}  [[Mark Hovey]], *[[Model Categories]]*, Mathematical Surveys and Monographs, Volume 63, AMS (1999) ([ISBN:978-0-8218-4361-1](https://bookstore.ams.org/surv-63-s), [doi:10.1090/surv/063](https://doi.org/http://dx.doi.org/10.1090/surv/063), [pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/hovey-model-cats.pdf), [Google books](http://books.google.co.uk/books?id=Kfs4uuiTXN0C&printsec=frontcover))

* {#Hirschhorn02} [[Philip Hirschhorn]], _Model Categories and Their Localizations_, AMS Math. Survey and Monographs Vol 99 (2002) ([ISBN:978-0-8218-4917-0](https://bookstore.ams.org/surv-99-s/), [pdf toc](http://www.gbv.de/dms/goettingen/360115845.pdf), [pdf](http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf))

* [[Kate Ponto]], [[Peter May]], section 17 of _More concise algebraic topology_ ([pdf](http://www.maths.ed.ac.uk/~aar/papers/mayponto.pdf))

For the "Hurewicz," "h-" or "[[Strøm model structure]]":

* [[Arne Strøm]], _The homotopy category is a homotopy category_, Archiv der Mathematik 23 (1972) ([pdf](https://www.uio.no/studier/emner/matnat/math/MAT9580/v17/documents/strom-the-homotopy-category-is-a-homotopy-category-1972.pdf), [[Strom_HomotopyCategory.pdf:file]])

For the "m-" or "[[mixed model structure]]":

* Michael Cole, _Mixing model structures_, Topology Appl. 153 no. 7 (2006) [doi](http://dx.doi.org/10.1016/j.topol.2005.02.004).

The generalization to the [[model structure on topological operads]] is due to

* {#BergerMoerdijk03} [[Clemens Berger]], [[Ieke Moerdijk]], _Axiomatic homotopy theory for operads_, Comment. Math. Helv. Vol. 78 (2003), no. 4 ([arXiv:math/0206094](http://arxiv.org/abs/math/0206094))

[[!redirects model category of topological spaces]]

[[!redirects model structures on topological spaces]]
[[!redirects model structures on Top]]

[[!redirects q-model structure]]
[[!redirects q-model structures]]

[[!redirects m-model structure]]
[[!redirects m-model structures]]


model structures on chain complexes of sheaves of abelian groups

[[!redirects model structure on chain complexes of sheaves of abelian groups]]
\tableofcontents

## Introduction

The purpose of this article is to explain how the general abstract techniques for computing [[sheaf cohomology]] described in articles like [[model structures on simplicial presheaves]] connect to the more traditional techniques for computing [[abelian sheaf cohomology]] such as injective resolutions, acyclic resolutions, Čech cohomology, Cartan and Leray theorems, etc.

## Model structures

Fix some [[site]] S such as Zariski, Nisnevich, etc., or one of the differential-geometric or complex-geometric sites.
We are interested in the category of [[chain complexes]] of [[sheaves]] or [[presheaves]] on S,
and how to compute [[abelian sheaf cohomology]] on S using [[model structures]] on these categories.

The following three [[model categories]] are [[Quillen equivalent]]:

1. The local [[projective model structure]] on [[chain complexes]] of [[presheaves]] of abelian groups.

2. The local [[injective model structure]] on chain complexes of presheaves of abelian groups;

3. The local [[injective model structure]] on chain complexes of sheaves of abelian groups;

For typical sites, it is not possible to construct a [[projective model structure]] or a local projective model structure on chain complexes of sheaves of abelian groups, which is often expressed by saying that the category of sheaves of abelian groups does not have enough [[projective objects]].

Here “local” means the same thing in all three cases: the class of [[weak equivalences]]
of these [[model structures]] is precisely the class of [[natural transformations]] of presheaves
that induce isomorphisms on sheaves of [[homology]] groups.
The latter can be computed as [[associated sheaves]] of presheaves of homology groups, which are computed objectwise on the site.

Furthermore, for a site with enough points (as is the case with the Zariski and Nisnevich site, as well as pretty much any site used in differential or complex geometry), the class of weak equivalences can also be described as the class of [[natural transformations]] of presheaves that at every point of the site induces a quasi-isomorphism of [[stalks]] at this point.

Next, if the site is [[hypercomplete]] (as is the case for any site with enough points), the following [[model categories]] are [[Quillen equivalent]] to each other and to the model categories 1-3),
in fact we have equalities 1=4, 2=5:

4\.  The [[projective model structure]] on chain complexes of presheaves of abelian groups left Bousfield localized at [[Čech nerves]] of covering families.

5\.  The [[injective model structure]] on chain complexes of presheaves of abelian groups left Bousfield localized at Čech nerves of covering families;

The [[model structures]] 1) and 2) can in general also be expressed as a [[left Bousfield localization]] of the projective/injective model structure on presheaves,
this time with respect to the (larger) class of [[hypercovers]],
or, equivalently, maps inducing isomorphisms on homology sheaves.
If the site is [[hypercomplete]], this process yields the same weak equivalence as in 4) and 5).

The model structure 3) is typically not a localization of a global model structure,
however, it can be easily constructed directly using, e.g., the [[Smith recognition theorem]].

There is an analogous statement in the setting of sheaves of abelian groups:
the category of sheaves of abelian groups is the [[category of fractions]], or, equivalently, [[reflective localization]],
of the category of presheaves of abelian groups with respect to the class of maps given by local isomorphisms of presheaves of abelian groups.
Left Bousfield localizations are precisely the homotopy coherent analogues of [[reflective localizations]].

To emphasize again: if the site has enough points, all three notions of a local weak equivalence:
isomorphisms on homology sheaves, stalkwise quasi-isomorphisms, and Čech-local weak equivalences coincide.
They are quite different from ordinary (objectwise) weak equivalences of presheaves, i.e., objectwise [[quasi-isomorphisms]].
(For sites that do not have enough points, such as the [[etale site]], it is important to distinguish [[Čech descent]] from [[hyperdescent]], which are quite different in general.)

The advantage of 4) and 5) is that [[fibrant objects]] are now easier to describe:
they are precisely [[fibrant objects]] in the original model structure (before a [[left Bousfield localization]]) that are local.
Here “local” amounts precisely to the homotopy coherent Čech descent property:
for any covering family in the site, the induced augmented cosimplicial diagram
exhibits its apex as the [[homotopy limit]] of the remaining cosimplicial diagram.
This is simply the homotopy coherent version of the usual gluing axiom for [[sheaves]] of abelian groups.

Any presheaf $F$ is weakly equivalent (in the localized model structure) to a local presheaf $L(F)$
via a (local) weak equivalence $F\to L(F)$, which is simply a homotopy coherent analogue of the [[associated sheaf]] map.
It is important not confuse the map $F\to L(F)$ with an entirely different map $F\to K(F)$
that is given by taking the [[associated sheaf]] of abelian groups of $F_n$ separately in every degree $n$.
The functor $K$ in fact gives a [[Quillen equivalence]] 2→3,
and in general has nothing to do with the functor $L$, which is related to the [[fibrant replacement]] functor in the model structures 1-5).

Indeed, the [[fibrant replacement]] functor in 1-5) can be computed by applying first $L$, then any [[fibrant replacement functor]] in the original (global) model structure.
For [[projective model structures]], the global fibrant replacement can be simply computed as the objectwise fibrant replacement,
which in the case of presheaves of chain complexes is a trivial operation, i.e., every presheaf of chain complexes is projectively fibrant.
For [[injective model structures]], the global fibrant replacement can be computed as the injective resolution.

The relevance of the [[fibrant replacement]] functor is explained by its relevance for computing the derived mapping chain complex of presheaves of chain complexes.
Recall that given [[presheaves]] $F$ and $G$ of chain complexes of abelian groups,
we can define the mapping chain complex $Map(F,G)$ using the usual formula with [[kernels]] (or [[equalizers]]).

What we really are interested in is the derived mapping chain complex $RMap(F,G)$,
which can be computed as $Map(Q F, R G)$, where $Q$ is a cofibrant replacement functor and $R$ is a fibrant replacement functor, in any of the [[model structures]] 1-5).
In the more traditional language, (the cohomology groups of) RMap are also known as the [[hypercohomology]] (groups) of $F$ with coefficients in $G$.

The derived mapping chain complex depends only on the class of [[weak equivalences]], not on the class of fibrations or cofibrations,
so in what follows we write $RMap_{local}(F,G)$ and $RMap_{global}(F,G)$ to distinguish the cases when we use local weak equivalence versus objectwise weak equivalences.

The fundamental relation between two chain complexes is
$$RMap_{local}(F,G) = RMap_{global}(F,L(G)),$$
where $G\to L(G)$ is the localization map described above.
In the [[injective model structure]] (local or not) all objects are cofibrant, so in this case we can take $Q=id$.
In the global [[projective model structure]], all objects are fibrant, so in this case we can take $R=id$.

For the local [[injective model structures]] 2, 3, 5 all objects are cofibrant, so $Q$ can be taken to be the identity functor,
whereas $R$ is (for example) the [[injective resolution]] functor.
In these cases, we obtain the fairly traditional recipe for computing [[abelian sheaf cohomology]]: replace the sheaf $F$ of coefficients by its fibrant replacement,
e.g., the injective resolution of $F$, then compute the ordinary mapping chain complex.

For the [[projective model structures]] 1, 4 fibrancy boils down to locality, i.e., the homotopy coherent descent condition, so $R=L$ is the homotopy coherent analogue of the [[associated sheaf]] functor,
whereas cofibrant objects are the same in the local and global model structure
and are degreewise [[retracts]] of [[coproducts]] of representables

Now the [[abelian sheaf cohomology]] of an object $X$ in the site with coefficients in a sheaf of abelian groups $F$ can be computed as follows.
First, convert $X$ to its representable presheaf of sets, then convert the resulting presheaf of sets into a presheaf of abelian groups
by applying the free abelian group functor objectwise.
We will abuse the language and continue to refer to such presheaves of abelian groups as representable presheaf.

In fact, the same technique continues to work for more general objects than objects of the site, e.g., for [[algebraic spaces]],
which can also be converted into presheaves of abelian groups by applying the free abelian group functor objectwise.
Now $X$ and $F$ are presheaves of abelian groups, which we can convert into presheaves of [[chain complexes]] by placing them in chain degree 0.
The $n$th [[sheaf cohomology]] group of $X$ with coefficients in $F$ is now simply the nth cohomology group of the [[chain complex]] $RMap(X,F)$.

## Traditional recipes for abelian sheaf cohomology

We can now easily recover all traditional recipes for computing [[abelian sheaf cohomology]]:

1.  In the model structure 1), we can replace F by its [[injective resolution]], which yields a local [[fibrant replacement]].
Since any object (including $X$) is cofibrant, we conclude by evaluating the mapping chain complex and taking its $n$th cohomology group.
In particular, if $X$ comes from a representable presheaf, then we can simply evaluate the [[injective resolution]] on $X$ and take its cohomology.

2.  We can also replace $F$ by its acyclic resolution $A$ via a map $F\to A$, which is a local weak equivalence.
Any acyclic resolution is a local object, but not necessarily a [[fibrant object]].
Thus, $RMap_{local}(X,F)$ can be computed as $RMap_{global}(X,A)$.
In typical applications, $X$ is representable, which means that $RMap(X,A)$ can be computed by evaluating $A$ on $X$,
which can be seen (for example) by switching to the [[projective model structure]], where $X$ is cofibrant and any object (including $A$) is fibrant.
This recovers the [[de Rham theorem]].

3.  In the model structure 3) we can fibrantly replace $F$ by taking its associated homotopy coherent sheaf $L(F)$.
Then we cofibrantly replace $X$ by taking the [[Čech nerve]] $\check C(U)$ of its covering family $U$.
Now $RMap(X,F)$ can be computed as $RMap(\check C(U),L(F))$, i.e., the Čech cohomology of $L(F)$.

4.  To recover Cartan's and Leray's theorems, first replace $X$ by $\check C(U)$
and compute
$$RMap(X,F) = RMap(\check C(U),F) = RMap(hocolim_n \check C(U)_n, F) = holim_n RMap(\check C(U)_n, F).$$
In the mapping chain complex on the right, the object $\check C(U)_n$ is a coproduct of representable presheaves
given by iterated [[fiber products]] of maps in the covering family $U$.
In particular, $RMap(\check C(U)_n, F)$ can be computed as the product of individual derived mapping chain complexes
from individual [[fiber products]] to $F$.
The conditions of Cartan's and Leray's theorems guarantee that these derived mapping chain complexes
can be simply computed by evaluating $F$ on the corresponding representable objects.
(Here we must emphasize that it is not necessary to replace $F$ by $L(F)$, since instead we computed the derived mapping chain complexes directly.)
Therefore, $RMap(\check C(U),F)$ can be computed as the [[total complex]] of the Čech bicomplex of $F$ with respect to $U$.

## Related concepts

* [[simplicial presheaf]], [[model structures on simplicial presheaves]]

* [[chain complex]], [[model structures on chain complexes]]

* [[abelian sheaf cohomology]]


model structures on dg-categories > history

[[!redirects model structures on dg-categories]]


model theory

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### Context
#### Model theory
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[[!include model theory - contents]]
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# Contents
* table of contents
{:toc}

## Idea

On the one hand, there is [[internal language|syntax]].  On the other hand, there is [[structure|semantics]].  Model theory is (roughly) about the relations between the two: model theory studies classes of [[models]] of [[theories]], hence classes of "[[structures in model theory|mathematical structures]]".

A model theory for a particular [[logic]] typically works within a given [[universe]], and specifies a notion of "[[mathematical structure]]" in this context, namely a _[[structure in model theory|structure for a language]]_, and a _[[definition]] of [[truth]]_. Logic is typically specified by [[signature (in logic)|language]] (function symbols, relations symbols and constants), and formulas (formed from symbols, [[variables]], Boolean operations and [[quantifiers]]). Language together with a choice of a set of formulas without [[free variables]] (viewed as [[axioms]]) is called a [[theory]]. A structure is an interpretation of a language $L$ via a given set $M$ together with interpretation of the symbols of the language. A __[[model]]__ of a [[theory]] $\mathcal{T}$ in the language $L$ is an [[structure in model theory|L-structure]] which satisfies each formula in $\mathcal{T}$.

The two main problems of model theory are

* classification of a given theory $\mathcal{T}$ in a language $L$
* study the family of all [[definable sets]] of a [[structure (in model theory)|structure]] $M$ of a [[language (in model theory)|language]] $L$

In all memorable examples, the collection of structures for a language will form an interesting [[category]] (see [Beke-Rosicky 11](#BekeRosicky11) for a characterization of these), and the subcollection of those structures verifying a given collection $Th$ of [[propositions]] in the language are an interesting [[subcategory]] again.

Model theory as currently conceived has strong analogies with the classical theory of [[algebraic varieties]] and hence with the part of [[algebraic geometry]] that deals with these. One such analogy is that [[definable subsets]] are likened to [[zero loci]] of [[equations]], and another consists of various notions of [[dimension]] which can be likened to Krull dimension. One can also say that classical algebraic geometry often provides a testing ground for more general developments in model theory. (For the most part, model theory does not deal however with more global concepts of modern algebraic geometry such as [[sheaves]] or [[schemes]].)

## Scope
 {#Scope}

In full generality, _model theory_ would study all kinds of [[models]] over all kinds of [[theories]], hence pretty much everything that is considered (in) [[mathematics]]. However, in order to find effective classification results for models, one traditionally restricts attention to very special kinds of theories only.

Traditionally the default subject of model theory (e.g. ([Hodges 93](#Hodges93))) are [[first-order theories]] (only) and their models in [[Set]] (only), these are the traditional [[structures in model theory]]. For emphasis the study of this specific case is also called _first-order model theory_ or _classical model theory_ ([SEP](#SEPEntry)). Among the fundamental classfication theorems provable in this context are the [[compactness theorem]] and the [[Löwenheim-Skolem theorem]]. (See also _[[geometric stability theory]]_.)

More modern developments in model theory consider also wider classes of [[theories]] than just [[first-order theories]] (e.g [Tent-Ziegler 12](#TentZiegler12)). In particular there is [categorical model theory](#CategoricalModelTheory) ([Makkai-Paré 89](#MakkaiPare89), [Beke-Rosicky 11](#BekeRosicky11)).

## Examples

Classical model theory is concerned with models of [[theories]] in [[first-order logic]], this is what we discuss first, below in

* [First-order logic](#FirstOrderLogic).

More generally one can consider models over theories in [[higher-order logic]], this we discuss below in

* [Higher-order logic](#HigherOrderLogic).

### Classical Model theory in First-order logic
 {#FirstOrderLogic}

See also/first [[theory]]/[[first-order theory]].

+-- {: .standout}
**Caveat Lector.**
This may duplicate/contradict other nLab accounts of [[FOL]], e.g. [[theory]]; it is present here for illustrative purposes only.  We attempt to obviate the trouble of quantifier scope by using _addressing_ rather than _naming_ of variables; specifically, the variable $x_n$ occurs _bound_ in a formula if it is nested within more than $n$ quantifiers, and otherwise _free_.
=--

We describe an [[operad]] $L$ in [[Set]] over two [[type]]s, $W$ and $P$ for "words" and "propositions", respectively.  The full operad is freely generated by various disjoint suboperads

* a free suboperad $O$ of operations, or functions, of types $W^n\to W$;
* an $\mathbb{N}$-indexed suboperad $X$ of variables of types $1\to W$;
* a (free) suboperad $R$ of predicates, or relations, of types $W^n\to P$;
* the equality operad, of type $W^2\to P$
* the boolean operad, $B$: it's got $\wedge, \vee,\neg,\dots$ of types $P^{k}\to P$ for $k=1,2$, and perhaps $k=0$ if you want to include $\bot$; these compose the way you think they should.
* the quantifier $\forall:$, usually written $\forall x_j$, where $x_j\in X_0$; \forall is of type $P\to P$.

Notable subsets of $L$ include $O[X]$, generated by $O\cup X$, the suboperad of parametrized words, and $R[X]$, the elements of types $? \to P$ in the suboperad generated by $O,R,X$.

Abusively, an  _$L$-structure_, or _interpretation_ of the functions and relations is an [[algebra over an operad|algebra]] $(W,P,O,R)\mapsto (M,2,O_M,R_M)$ for the suboperad $Q$ of $L$ generated by $O\cup R$ with the type $P$ interpreted by the initial [[boolean algebra]] $2$.

The _Tarski Definition of Truth_ is a [[functor|natural]] extension of the $Q$-algebra $M$ to an $L$-algebra, $(M,\dots)$ such that
* for $\Phi$ in $R[X]$ and for all functions $m:X\to M$, the sensible thing $\Phi[m]$ is true in $M$.
* $\Phi \wedge \Theta$ is true if for all functions $m:X\to M$, both $\Phi[m]$ and $\Theta[m]$ are true, and $\neg \Phi$ is true if and only if $\Phi$ is _not_.
* $(\forall:\Phi)[m]$ is true in $M$ precisely when $\Phi[m_a]$ is true for all $a\in M$, where
$$m_a(x_i) = \left\{ \array{ a & i=0 \\ m(x_{i-1}) & else }\right. $$

(Again, this really should be written more clearly, but it's a start.)

+-- {: .query}
[[JCMcKeown]]: is there some nicer way to say the quantifier nonsense? I'm thinking along the lines
> there are two actions of $\mathbb{N}$ on $L$: one shifts all the variables by 1, the other adds a $\forall$ quantifier; and the Tarski extension is the one that makes these commute somehow

_ibid_: maybe it's more right to say that $P_M$ should be the boolean algebra $\mathcal{P}(|M|^\mathbb{N})$? This again has that natural action of $\mathbb{N}$ on it...
=--

#### Definition and remarks
{: #definition_theory_and_model_fol}

An $L$-structure $M$, as an $L$-algebra with extra properties, defines a **complete first-order theory** $Th(M)$, that subset of $L$ which $M$ interprets as $\top\in 2$, or _true_.  Conversely, given a collection $T$ of elements of $L$ of type $?\to P$, we say that $M\models T$, or in words $M$ **is a model of the theory** $T$ whenever $T\subset Th(M)$.  There is an obvious Galois connection between theories $T$ and the collections of $L$-structures that are models.  Much of deeper model theory studies the fine structure of this connection.

### Operads, Monads and Lawvere theories
 {#OperadsMonadsAndLawvereTheories}

Structures from [[universal algebra]], [[higher algebra]] and [[categorical logic]] that conveniently allow tospeak about [[theories]] and their [[models]] are [[operads]], [[monads]] and [[Lawvere theories]]. Each of these may be understood as characterizing a [[theory]]. Its [[models]] then are the corresponging algebras, see at _[[algebra over an operad]]_, _[[algebra over a monad]]_ and _[[algebra over a Lawvere theory]]_, respectively.

### Higher-order logic
 {#HigherOrderLogic}

Remaining within [[Set]], we can also generalize beyond first-order logic to various [[higher-order logic|higher-order logics]].

((insert your favourite variant here))

### Infinitary logic and Categorical model theory
 {#CategoricalModelTheory}

For a reasonable fragment of [[infinitary logic]], then the [[category]] of all [[models]] over [[theories]] in this [[language]] is an [[accessible category]] and moreover all accessible categories arise this way. ([Makkai-Pare 89](#MakkaiPare89), [Beke-Rosicky 11](#BekeRosicky11)).

## Important theorems

The following are closely interrelated, and depend on having a suitable universe $V$.  We can view them as theorems of $ZFC$ or as (relatively mild) conditions on $V$.

*(... clarify ...)*

###### Gödel's completeness theorem

Given a first-order theory $T$ in some language $L$, $T$ is consistent iff there is a model of $T$ in $V$ --- that is, iff $M\models T$ for some $M\in V$.

See at _[[completeness theorem]]_.

###### Gödel's compactness theorem

Under the same hypotheses, $T$ is consistent iff every finite subset of $T$ is consistent; expressed semantically, a theory $T$ has a model iff every finite subset of $T$ has a model.

See at _[[compactness theorem]]_.

###### Łoś ultraproduct theorem

*(...think of a good way to state this...)*

See at [[Łoś ultraproduct theorem]].

##### Corollaries worth thinking about

It follows that first-order theories are quite permissive; or in other words that they're inefficient at pinning down particular structures.

For example, consider the complete first-order theory $Th(V_\omega,\in)$, and any total order $(X,\lt)$.  If one *expands* the language (coresponding to an injective morphism of operads) to include constant symbols $c_x$ for $x\in X$, then for any  subset $s$ of $X$ of finite size $n + 1$, one has
$$(V_\omega,\in,0,1,\cdots,n)\models Th(V_\omega,\in)\cup \{c_x\in c_y \mid x\lt y;x,y\in s\}$$
so that the *finite extensions* of $Th(V_\omega,\in)$ by suborders of $X$ are all consistent; by compactness, the fully extended theory $Th(V_\omega,\in)\cup \{c_x\in c_y \mid x\lt y;x,y\in X\}$ is also consistent; thus by completeness there is a structure $(M,\epsilon,\cdots,c_x,\cdots)$ such that
*  $(M,\epsilon)\models Th(V_\omega,\in)$
*  $c_x\epsilon c_y$ for all $x\lt y$ in $X$

By a similar argument, (if ZFC is consistent) there are models $M'$ of classical set theory satisfying the (higher-order) property that the natural numbers object $\omega_{M'}$ of $M'$ includes your favourite total order $(X,\lt)$ *as a suborder* --- of course, $M'$ isn't allowed to *know* this --- notably, there is **no object** $\xi$ in $M$ such that $\{y\mid y\epsilon \xi\} = \{c_x \mid x\in X\}$.

## Related entries

* [[model theory - contents]]

* [[definable set]], [[definable groupoid]]

* [[type (in model theory)]]

* [[stability in model theory]], [[geometric stability theory]]

* [[categoricity]]

* [[abstract model theory]]

* [[generalized quantifier]]

* [[nonstandard analysis]], [[set theory]], [[foundations and logic]], [[algebraic set theory]], [[forcing]], [[o-minimal structure]]

* [[Lascar group]]

* [[Birkhoff's HSP theorem]]

* [[model theory and physics]]

* entries on researchers in model theory: [[Alfred Tarski]], [[Boris Zilber]], [[Kenneth Kunen]], [[Saharon Shelah]]

## References
 {#References}

Early original texts"

* {#Kochen61} [[Simon Kochen]], p. 223 in: *Ultraproducts in the Theory of Models*, Ann. Math.  **74** 2 (1961) 221-261 [[doi:10.2307/1970235](https://doi.org/10.2307/1970235)]

  > (also introducing the notation for "$\vDash$-[[judgements]]", cf. [there](judgment#History))

Discussion of classical first-order model theory includes

* {#SEPEntry} Stanford Encyclopedia of Philosophy, _[First-order Model Theory](http://plato.stanford.edu/entries/modeltheory-fo/)_

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

* {#Hodges93} [[Wilfrid Hodges]], _Model Theory_, Cambridge University Press 1993; _A shorter model theory_, Cambridge UP 1997

Other basic texts on model theory include

* {#TentZiegler12} Katrin Tent, [[Martin Ziegler]], _A course in model theory_, Lecture Notes in Logic, Cambridge University Press, April 2012

* Chen Chung Chang, H. Jerome Keisler, _Model Theory. Studies in Logic and the Foundations of Mathematics_. 1973, 1990, Elsevier.

* David Marker, _Model Theory: An Introduction_ Graduate Texts in Mathematics __217__ (2002)

* C U Jensen, H Lenzing, _Model theoretic algebra: with particular emphasis on fields, rings, modules_ (1989)

* [Shelah archive](http://shelah.logic.at), [Shelah's books](http://shelah.logic.at/books.html)

* Gerald E Sacks, _Saturated model theory_, Benjamin 1972

Specific discussion of generalization of classical first-order model theory to more general kinds of [[logic]] include...

...to [[modal logic]]:

* Valentin Goranko, Martin Otto, _Model theory of modal logic_, [pdf](http://www.mathematik.tu-darmstadt.de/~otto/papers/mlhb.pdf)

...to [[infinitary logic]]:

* H. Keisler. _Model theory for infinitary logic_, North-Holland, Amsterdam, 1971.

Discussion of aspects of [[category theory]] and [[categorical logic]] ([[accessible categories]]) in model theory includes

* {#MakkaiPare89} [[Michael Makkai]], [[Robert Paré]], _Accessible categories: the foundations of categorical model theory_, Contemporary mathematics vol 104, AMS 1989

* {#BekeRosicky11} [[Tibor Beke]], [[Jiří Rosický]], _Abstract elementary classes and accessible categories_, 2011   ([pdf](http://www.math.muni.cz/~rosicky/papers/elem7.pdf))

* B.Hart et. al (eds.), _Models, Logics, and Higher-dimensional Categories,  A Tribute to the Work of Mihali Makkai_, AMS (2011)

Discussion of [[geometric stability theory]] includes

* Anand Pillay, _Geometric stability theory_

* [[Boris Zilber]], _Elements of geometric stability theory_, lecture notes, [pdf](http://people.maths.ox.ac.uk/zilber/est.pdf); _On model theory, non-commutative geometry and physics_, (survey draft) [pdf](http://people.maths.ox.ac.uk/zilber/bul-survey.pdf); _Zariski geometries_, book, [draft pdf](http://people.maths.ox.ac.uk/zilber/s.pdf);  On model theory, noncommutative geometry and physics, conference talk, [video](http://zomobo.net/play.php?id=wnhbpIQBpiE)

* John T. Baldwin, _Fundamentals of stability theory_

Discussion of [[motivic integration]] includes

* [[David Kazhdan]], _Lecture notes in motivic integration_, with intro to logic and model theory, [pdf](http://www.ma.huji.ac.il/~kazhdan/Notes/motivic/b.pdf)

* R. Cluckers, J. Nicaise, J. Sebag (Editors), _Motivic Integration and its Interactions with Model Theory and Non-Archimedean Geometry_, 2 vols.
London Mathematical Society Lecture Note Series __383__, __384__


model theory - contents

**[[model theory]]**

## Basic concepts and techniques

* [[language]], [[signature]]

* [[theory]], [[first-order theory]]

* [[model]]

* [[interpretation]]

* [[structure in model theory]]

* [[elementary embedding]]

* [[type in model theory]]

* [[compactness theorem]]

* [[diagram of a first-order structure]]

* [[definable set]]

* [[matroid|(combinatorial) pregeometry]]

* [[ultraproduct]], [[ultraroot]], [[ultrapower]]

* [[indiscernible sequence]]

* [[Morleyization]]

* [[Morley sequence]]

* [[Ramsey theorem]]

* [[Erdos-Rado theorem]]

* [[Ehrenfeucht-Fraïssé games]] (back-and-forth games)

## Dimension, ranks, forking

* [[forking and dividing]]

* [[Morley rank]]

* [[Shelah 2-rank]]

* [[Lascar U-rank]]

* [[Vapnik–Chervonenkis dimension]]

## Universal constructions

* [[Fraïssé limits]]

* [[monster model]]

* [[Hrushovski construction]]

* [[pretopos | product of theories]]

## [[stuff, structure, property|Extra stuff, structure, properties]]

* [[definable group]]

* [[definable groupoid]]

* [[definable category]]

* [[elimination of imaginaries]]

* [[hyperimaginary element | elimination of hyperimaginaries]]

* [[elimination of quantifiers]]

* [[omega-categorical structure]]

* [[existentially closed model]]

* [[model completeness]]

* [[stability in model theory | stable theories]]

* [[geometric stability theory]]

* [[generic predicate]]

* [[ultracategory]]

## Examples

* [[ACF]]

* [[DLO]]

* [[countable random graph]]

* [[ACVF]]

* [[RCF]]

* [[nonstandard analysis]]

* [[Peano arithmetic]]

* [[Robinson arithmetic]]

* [[Nelson arithmetic]]

## Theorems

* [[Ax-Kochen-Ershov theorem]]

* [[Keisler-Shelah isomorphism theorem]]

* [[conceptual completeness | conceptual completeness]]

* [[Makkai duality]]

* [[Los ultraproduct theorem]]

* [[Morley categoricity theorem]]

* [[omitting types theorem]]

* [[Deligne completeness theorem]]


model theory and physics

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#### Model theory
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#### Physics
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[[!include physicscontents]]
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#Contents#
* table of contents
{:toc}

## Idea

Application of [[model theory]] to foundations of [[physics]]...

## Related concepts

* [[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]]

* [[motivation for cohesion]]

## References

* [[Boris Zilber]], _On model theory, non-commutative geometry and physics_, (survey draft) ([pdf](http://people.maths.ox.ac.uk/zilber/bul-survey.pdf))

* [[John Alexander Cruz Morales]], _Interplay between model theory, noncommutative geometry and physics after Boris Zilber_ [Workshop "Philosophy of Mechanics: Mathematical Foundations"](http://phil.physico-math.gie.im/indexPMMF) February 12-14, 2014 - Université Paris Diderot ([pdf](http://phil.physico-math.gie.im/static/slides/Cruz-Zilber.pdf))

* {#MoralesZilber14} [[John Alexander Cruz Morales]], [[Boris Zilber]], _The geometric semantics of algebraic quantum mechanics_ ([arXiv:1410.7277](http://arxiv.org/abs/1410.7277))


model theory of fields

### Idea

Model theory of fields is a branch of [[model theory]] dedicated to study of (axiomatic, mainly first order) theories (in the sense of logic) which are dedicated to [[field]]s and, in more recently, to [[differential field]]s.

### Basic theories of fields

First of all there is the __theory of fields__, which is a first order theory with equality in the language of rings $\{+, -, .,0,1\}$ and whose axioms are axioms of commutative integral domains together with an axiom
$$
 (\forall x)(x\neq 0\implies (\exists y)(x\cdot y = 1))
$$

Then there are more specific theories, e.g. the __theory of algebraically closed fields__ ACF and the __theory of [[real closed field]]s__ RCF.

ACF has quantifier elimination and is model complete.

### Literature

* David Marker, _Model theory, an introduction_, Springer Grad. Texts in Math. __217__ (2002)
* Model theory of fields, edited by Marker, Messmer, Pillay. Lecture Notes in Logic 5. ASL-Peters, 2006
* James Ax, _The elementary theory of finite fields_, Annals of Math. Series 2. 88 (2): 239–271 (1968) [doi](https://doi.org/10.2307/1970573)

category: model theory, algebra


model topos

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#### Homotopy theory
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* table of contents
{:toc}

## Idea

A _model topos_ is a [[model category]] that [[presentable (∞,1)-category|presents]] an [[(∞,1)-topos]].

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

A [[model category]] $\mathcal{C}$ is a **model topos** if there is a [[simplicial site]] $K$ and a [[Quillen equivalence]] $\mathcal{C} \simeq sPSh(K)_{loc}$ to the local [[model structure on sSet-presheaves]] over $K$.

=--

This appears as [Rezk, 6.1](#Rezk).

## Related concepts

* [[model site]], [[model structure on simplicial presheaves]]

* [[Ho(CombModCat)]]

[[!include flavors of higher toposes -- list]]

[[!include locally presentable categories - table]]

## References

The terminology "model topos" is due to:

* {#Rezk} [[Charles Rezk]], _Toposes and homotopy toposes_, 2010 ([pdf](http://www.math.uiuc.edu/~rezk/homotopy-topos-sketch.pdf), [[Rezk_HomotopyToposes.pdf:file]])

Essentially this idea appears earlier in:

* {#Simpson99} [[Carlos Simpson]],  *A Giraud-type characterization of the simplicial categories associated to closed model categories as $\infty$-pretopoi* ([arXiv:math/9903167](http://arxiv.org/abs/math/9903167))

* [[Bertrand Toën]], [[Gabriele Vezzosi]], _Homotopical Algebraic Geometry I: Topos theory_, Advances in Mathematics **193** 2 (2005)  257-372  [[arXiv:math.AG/0207028](http://arxiv.org/abs/math.AG/0207028), [doi:10.1016/j.aim.2004.05.004](https://doi.org/10.1016/j.aim.2004.05.004)]

In the context of [[categorical semantics]] for [[univalence|univalent]] [[homotopy type theory]], the combination of terminology "model topos" with *[[type-theoretic model category]]* to *type-theoretic model topos*:

* {#Shulman19} [[Michael Shulman]], §1.3 in: _All $(\infty,1)$-toposes have strict univalent universes_ ([arXiv:1904.07004](https://arxiv.org/abs/1904.07004))

reviewed in

* {#Riehl22} [[Emily Riehl]], §6.1 of: *On the $\infty$-topos semantics of homotopy type theory*, lecture at *[Logic and higher structures](https://conferences.cirm-math.fr/2689.html)* CIRM (Feb. 2022) [[pdf](https://emilyriehl.github.io/files/semantics.pdf), [[Riehl-InfinityToposSemantics.pdf:file]]]

[[!redirects model toposes]]
[[!redirects model topoi]]

[[!redirects type-theoretic model topos]]
[[!redirects type-theoretic model topoi]]
[[!redirects type-theoretic model toposes]]

[[!redirects type theoretic model topos]]
[[!redirects type theoretic model topoi]]
[[!redirects type theoretic model toposes]]


model-theoretic Galois theory

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* table of contents
{:toc}

## Idea
Classical [[Galois theory]] is about the classification of intermediate field extensions in a (field-theoretic) [[algebraic closure]] of a ground field according to the structure of a [[profinite group|profinite automorphism group]].

This can be adapted to the setting of a [[monster model|universal domain]] of any [[first-order theory]] which [[elimination of imaginaries|eliminates imaginaries]]: we can analogously classify [[definable closure|definably closed]] subsets of a [[definable closure|model-theoretic algebraic closure]] of some parameter set according to the structure of a profinite automorphism group.

In the language of [[Grothendieck's Galois theory]], we can sketch the next two sections as: the category of finite $A$-definable sets in a [[monster model]] $\mathbb{M}$ equipped with the forgetful functor to [[Set]] is a [[Galois category]].

## The case of finite extensions
It will be instructive to look at the case of finite extensions first, as the case of infinite extensions will be a version of the same argument but souped-up with formalities about profinite groups.

Recall the fundamental theorem of Galois theory for finite extensions of fields:

**Theorem.** For $L/K$ a finite, [[separable field extension|separable]], [[normal field extension]], there is an order-reversing bijective correspondence

$$\operatorname{Sub}_{\mathbf{Grp}} \left( \operatorname{Aut}(L/K)\right) \leftrightarrows \operatorname{Int} \left(L/K\right)$$

between the subgroups of the group of field automorphisms of $L$ fixing $K$ pointwise, and the intermediate field extensions between $K$ and $L$. The correspondence is given by sending a subgroup to its field of fixed points, and an intermediate extension to its stabilizer subgroup.

This can be translated to general model-theoretic language as follows: we work in a monster model $\mathbb{M} \models T$ for a complete first-order theory which [[elimination of imaginaries|eliminates imaginaries]]. Let $A$ be a small parameter set in $M$. We have the following

**Dictionary**, between our general $T$ eliminating imaginaries and the special case $T = [[ACF]]$ the theory of an algebraically closed field.

- $A \supseteq K$ corresponds to "$A$ is an extension of $K$".
- If $A$ is an extension of $K$, the  [[definable closure]] of $A$ corresponds to the [[perfect field|perfect hull]] of the field $K(A)$ generated by adjoining $A$.
- If $A$ is an extension of $K$, the  [[algebraic closure]] of $A$ corresponds to the separable closure of the field $K(A)$ generated by adjoining $A$.
- If $A$ is a definably closed extension of $K$ with $A = \operatorname{dcl}(\gamma \cup K)$ for $\gamma$ finite, this corresponds to $K(A)$ being finitely-generated.
- The [[orbit]] of an element $\ell \in \mathbb{M}$ under $\Aut(\mathbb{M}/K)$ corresponds to conjugates of $\ell$ in $K(\ell)$ under $\operatorname{Aut}(K(\ell)/K)$.
- The size of this orbit corresponds to the degree of the field extensions $K(\ell)/K$.
- If this orbit is finite, $\ell$ is said to be _algebraic_ over $K$.
- The condition "$\forall \ell \in L \supseteq K$, $\operatorname{Orb}_{\operatorname{Aut}(\mathbb{M}/L)}(\ell) \subseteq L$" corresponds to "$L$ is a normal extension of $K$".

When $L$ is a normal extension of $K$, $L$ splits into $\operatorname{Aut}(\mathbb{M}/K)$-orbits, and so the latter group acts via restriction on $L/K$. We call the image of the induced group homomorphism $\operatorname{Aut}(\mathbb{M}/K) \to \Sym(L/K)$ $\operatorname{Aut}(L/K)$.

With this in place, we can show:

**Theorem.** Let $K$ be a definably closed parameter set. Let $A$ be a normal extension of $K$ generated by the finite algebraic tuple $\gamma$. Then there is an order-reversing bijective correspondence between the subgroups of $\operatorname{Aut}(A/K)$ and the definably closed intermediate extensions of $A/K$. The correspondence is given by maps $\mathsf{Fix}$ sending a subgroup to its fixed points and $\mathsf{Stab}$ sending an intermediate definably closed extension to its stabilizer subgroup.

_Proof._ By saturation, $\mathsf{Fix}$ is well-defined, and $\mathsf{Stab}$ is clearly well-defined.

$\mathsf{Fix}$ is left-[[inverse]] to $\mathsf{Stab}$: by saturation in the [[monster model|monster]], any fixed points of $\mathsf{Stab}(B)$ for $K \subseteq B \subseteq A$ must be in the definable closure of $B$, so whenever $B$ is definably closed, $\mathsf{Fix} \left( \mathsf{Stab} (B) \right) \subseteq B$, with the reverse inclusion immediate.

$\mathsf{Stab}$ is left-inverse to $\mathsf{Fix}$: for $H$ a subgroup of $\operatorname{Aut}(A/K)$, note that for any $c$ a [[elimination of imaginaries|code]] for the $H$-orbit of $\gamma$ and for any $\sigma \in \operatorname{Aut}(\mathbb{M}/K)$, $\sigma$ fixes $c$ if and only if the restriction $\sigma \restriction A$ permutes $H.\gamma$. Since $\gamma$ generates $A$, any automorphism in $\operatorname{Aut}(A/K)$ is determined by where it sends $\gamma$, so $\sigma \restriction A \operatorname{.} \gamma \in H.\gamma \iff \sigma \restriction A \in H$.

In particular, since $H.\gamma$ is finite, $c$ is actually $H.\gamma$-definable, hence $A$-definable. Since $A$ is definably closed, $c \in A$, and so $c \in \mathsf{Fix}(H)$. If $g \in \mathsf{Stab}(\mathsf{Fix}(H))$, $g$ in particular fixes $c$, hence $g \in H$, and it's clear that $H \subseteq \mathsf{Stab}(\mathsf{Fix}(H))$. $\square$

## The case for infinite extensions
We'll get the case for infinite extensions by just classifying all subextensions of $\operatorname{acl}(A)/\operatorname{dcl}(A)$ at once.

### The model-theoretic absolute Galois group
Let $\mathbb{M} \models T$ be a [[monster model]]. Let $A$ be a small parameter set. $\operatorname{acl}(A)$ is a normal extension of $A$, because every finite $A$-definable set splits into $\operatorname{Aut}(\mathbb{M}/A)$-orbits. The _absolute Galois group_ $\operatorname{Gal}(A)$ of $A$ is $\operatorname{Aut}(\operatorname{acl}(A)/\operatorname{dcl}(A)$.

(For example, in $\mathsf{ACF}$, this recovers the usual absolute Galois group.)

Now, $\operatorname{acl}(A)$ is the colimit of the diagram of finite $A$-definable sets. From [[commutativity of limits and colimits]], we know that whenever $F : \mathbf{C} \to G \text{-} \mathbf{Set}$ is a cofiltered diagram of $G$-sets, then taking orbits of $\underset{\longleftarrow}{\lim}F$ is the same as taking a limit of the orbits. Dually, if we take automorphism groups, we get:

$$ \operatorname{Aut}\left(\underset{\longrightarrow}{\lim} F\right) \simeq \underset{\longleftarrow}{\lim} \left(F(c) \right).$$

So $\operatorname{Gal}(A)$ is profinite.

### The fundamental theorem of model-theoretic Galois theory

**Theorem.** Let $T$ be a first-order theory which [[elimination of imaginaries|eliminates imaginaries]], and let $\mathbb{M} \models T$ be a [[monster model|monster]]. Let $A \subseteq \mathbb{M}$ be a small parameter set. Then there is a bijective order-reversing correspondence

$$ \mathsf{Fix} : \operatorname{Sub}_{\text{Krull-closed}}\left(\operatorname{Gal}(A)\right) \leftrightarrows \operatorname{Sub}_{\text{dcl-closed}}\left(\operatorname{acl}(A)/\operatorname{dcl}(A)\right) : \mathsf{Stab}$$

given by taking a subgroup closed in the [[profinite group|profinite topology]] of $\operatorname{Gal}(A)$ to its fixed points and by taking a definably-closed intermediate extension of $\operatorname{acl}(A)/\operatorname{dcl}(A)$ to its stabilizer.

_Proof._ That $\mathsf{Fix}$ is left-inverse to $\mathsf{Stab}$ again follows from being in a monster.

On the other hand, let $H$ be a closed subgroup. $H$ is an intersection of basic open subgroups $H_i$ which are preimages of $\operatorname{Aut}(B_i/A)$, for $B_i$ finite and $A$-definable. Fixing an ordering on the $B_i$ and treating them as tuples, obtain codes $c_i$ for the orbit of each $B_i$ under $\operatorname{Aut}(B_i/A)$. Since each $B_i$ is finite $A-definable$, $c_i$ is $A$-algebraic. The stabilizer of each $c_i$ is precisely $H_i$, so $H = \bigcap_{i \in I} \mathsf{Stab}(c_i)$. Since each $c_i$ is fixed by $H$, whenever $g \in \mathsf{Stab} \left(\mathsf{Fix}(H) \right)$, then in fact $g \in H$. $\square$

## Model theory and the Tannakian formalism
In the motivating examples (see below) it turns out that Galois (i.e. relative automorphism) groups are themselves definable (i.e. arise as interpretations in the model of internal groups in $\mathbf{Def}(T)$.) In this case what you're taking the Galois group of must formally resemble an [[internal diagram]] on this [[group object|internal group]]; in model theory these are studied as what model theorists call _internal covers_. A structure theorem of Hrushovski makes this correspondence explicit: internal covers are torsors of definable groupoids and vice-versa; see [here](http://arxiv.org/abs/math/0603413).

All internal groups in $\mathbf{Def}(\mathsf{ACF})$ are in fact algebraic groups, so this is reminiscent of reconstruction results arising from [[Tannaka duality]]. As it turns out, one can make this analogy explicit and recover a slightly-weakened version of the Tannakian formalism for algebraic groups using the theory of internal covers; see the paper by Moshe Kamensky [here](https://arxiv.org/abs/0908.0604).

## Examples
If $T = \mathsf{ACF}$ the [[theory of algebraically closed fields]], this recovers the classical fundamental theorem of Galois theory.

If $T = \mathsf{DCF}$ the [[theory of differentially closed fields]], this recovers [[differential Galois theory]]. (In fact, Kolchin's work was what inspired Poizat to introduce imaginaries and work out classical Galois theory in a model-theoretic setting.)

Any theory $T$ can be conservatively interpreted inside a theory $T^{\operatorname{eq}}$ which eliminates imaginaries and hence "admits a Galois theory." This is the [[coherent logic|coherent]] special case of a result [[Olivia Caramello]] spells out in very general terms in her article ([Caramello 13](#Caramello13)) on topological (toposic) Galois theory.

## Discussion

## Related concepts
* [[Galois theory]]
* [[Grothendieck's Galois theory]]
* [[elimination of imaginaries]]
* [[definable closure|algebraic type]]
* [[Tannaka duality]]

## References
* Bruno Poizat, _Une Theorie de Galois Imaginaire_.

* Bruno Poizat, _A Course in Model Theory_.

* Alice Medvedev and Ramin Takloo-Bighash, [_An invitation to model-theoretic Galois theory_](http://arxiv.org/abs/0909.4340)

* {#Caramello13} [[Olivia Caramello]], _Topological Galois theories_, ([arXiv: 1301.0300](http://arxiv.org/abs/1301.0300))

[[!redirects model theoretic Galois theory]]


modelizer

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[[!include homotopy - contents]]
=--
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A **modelizer** is a presentation of the [[(infinity,1)-category|(∞,1)-category]] of [[infinity-groupoid|∞-groupoids]], or at least, the [[homotopy category]] thereof.

## Definition

A **modelizer** is a [[category]] $M$ and a subcategory $W$ satisfying these conditions:

* $(M, W)$ is a saturated [[homotopical category]], meaning $W$ is precisely the class of morphisms in $M$ that become invertible in the [[localization]] $M [W^{-1}]$.
* $M [W^{-1}]$ is [[equivalence of categories|equivalent]] to the category of weak [[homotopy types]], i.e. [[Ho(Top)]] (with respect to [[weak homotopy equivalences]]).

More precisely, it is a category $M$ equipped with a functor $\pi : M \to Ho(Top)$ such that, for $W$ the class of morphisms inverted by $\pi$, the induced functor $M [W^{-1}] \to Ho(Top)$ is an equivalence of categories.

A **morphism of modelizers** $(M, W) \to (M', W')$ is a functor $F : M \to M'$ such that:

* $F$ sends morphisms in $W$ to morphisms in $W'$.
* The functor $M [W^{-1}] \to M' [W'^{-1}]$ so induced is an equivalence of categories.
* The composite $M \overset{F}{\to} M' \overset{\pi}{\to} Ho(Top)$ is isomorphic to $M \overset{\pi}{\to} Ho(Top)$.

An **elementary modelizer** is a modelizer whose underlying category is the [[category of presheaves]] on a [[test category]], with the weak equivalences the ones described at the linked page.

## Examples

The main examples turn out to be [[model categories]]:

* Tautologically, [[Top]] (with weak homotopy equivalences) is a modelizer. This extends to a [[model structure on topological spaces]].
* Quillen showed that there is a [[model structure on simplicial sets]] that is [[Quillen equivalence|Quillen-equivalent]] to the previous one; in particular, $sSet$ (with weak homotopy equivaleces) is a modelizer.
* Thomason showed that [[Cat]] (with nerve equivalences) is a modelizer by exhibiting a [[Thomason model structure|model structure]] Quillen-equivalent to the on $sSet$.

## Cisinski's theorem

+-- {: .num_theorem}
###### Theorem (Cisinski)

If $A$ is a test category, then there exists a [[model structure on presheaves over a test category|model structure]] on $[A^{op}, Set]$ that is Quillen-equivalent to the standard model structure on $sSet$.

=--

## Related concepts

* [[category with weak equivalences]]

* [[localizer]]

## References

* [[Alexander Grothendieck|Grothendieck, A.]] (1983) [[Pursuing stacks]], §28.
* [[Denis-Charles Cisinski|Cisinski, D.-C.]] (2002, 2006) _Les préfaisceaux comme modèles des types d'homotopie_.

[[!redirects modelizers]]


models for concurrency

[[!redirects Models for concurrency]]
# Contents
* automatic table of contents goes here
{:toc}

(The ideas section here is partially adapted from papers of [[Eric Goubault]].  In particular the paper in HHA, see reference list. He was one of the first to propose some form of [[directed homotopy theory|directed homotopy]] as a model for aspects of concurrency.) This entry tries to list and summarise various models for concurrency and to examine some of them for (potential) interaction with other entries in the Lab, especially ones related to directed topology, and to causal sets ([[causets]]). (This comparison will initially not be deep, as we will first put down a lot of definitions in the linked pages, with a view to possible interactions later on.)

##Idea##

Concurrency theory in Theoretical Computer Science deals with systems in which several computational activities ( *processes* ) can be performed at the same time, in an asynchronous way.  Concurrent systems were introduced in order to have increased computational power and speed or so that some concurrent transactions can be handled more efficiently (and sometimes handled at all). Some of the models discussed here have been, perhaps, superceded as the technology has moved on, and the _flavour of the month_ changes, but they still provide useful mathematical models with potential for use elsewhere, so are certainly worth examining.

One of the main problems about concurrency is to have processes cooperating successfully for a common goal.  Cooperation seems to imply some form of synchronisation and information passing.   This can be done through *message passing* models, for instance. In that class of models, processes have their own local memory, which cannot be accessed by other processes.  The way to communicate values to other processes is by explicitly sending values to these other processes. One of the possible ways of doing this is via a *Rendezvous model* in which the action of sending results in the sender being blocked until the receiver actually receives the message and then, symmetrically the action of receiving blocks the receiver until the message is actually received. This method gives a partial synchronisation of the sender and receiver for the purpose of the passing of the message.

+--{: .query}
The source I am using to help me understand this does not quite make sense here and other sources that I have looked at, have not helped that much, so can someone de-blurr the meaning for me here.  [[Tim Porter|Tim]]
=--

There are variants to this,  which include _non-blocking send but blocking receive_ , _non-blocking send and receive_ (*asynchronous message passing*), _broadcast to groups of processes_ , instead of ''point to point'' communication, etc.

+--{: .query}
[[Adam]]: "Non-blocking send with blocking receive" is the rule required by Kahn process networks; this ensures that the resulting processes form a [domain](http://en.wikipedia.org/wiki/Domain_theory).  From a semantics standpoint I'm not sure that broadcast to $n$ recipients is really any different from just sending $n$ messages, one to each recipient.
=--

####Shared memory models
These make up another class of concurrent architectures.  Here processes have a local memory where they can perform local computations, but also have a common memory space which is accessible to all.  Communication between processes is essentially asynchronous and is realised by reading and writing values in this common space. Here the problem occurs that several processes may want to write values to the same part of the memory at the same time.  To avoid the problems that this implies it is necessary to 'protect' the access to shared variables by some mechanism. A classical one is the ''semaphore'' as introduced by Dijkstra in 1968. One of the usual pictures of a 'toy' situation used here is the [[progress graph|Swiss flag progress graph]], where two users attempt to change records in a database.  There is a shared resource and hence a problem of conflict, even of deadlock.

The challenge is to find adequate mathematical models so as better to understand the behaviour of such concurrent systems.

(More needs to be put here and the above is not that great.  Not sure how to rewrite it and what to add, though!)

+-- {: .query}
[[Adam]]: "Shared memory" is just a friendly way of presenting message-passing systems to a programmer -- at the end of the day, everything is message passing underneath if you dig down far enough.  If you want to __really__ dig, consider that digital circuits are point-to-point communication mechanisms, and physicists are suspicious of any "[action at a distance](http://en.wikipedia.org/wiki/Nonlocality)" -- which is what you would need in order to implement shared memory without some sort of message passing mechanism somewhere underneath.  Ours is a message-passing world; shared state is just an illusion used to make centralized-and-serial programmers feel more comfortable in a distributed-and-parallel world.  One could argue that the "mathematical models" called for are nothing more than simple translations into message-passing operations, which are the primary object of study.  (Note: I use message-passing here in the broadest sense, encompassing Petri Nets, CSP, trace structures, Kahn process networks, etc -- anything without shared state)

[[Tim Porter]]:  @Adam. Why not develop this a bit on some off shoot pages?  Also why not join n-forum and then it will be easier to discuss the points you make without discussions filling up the n-Lab pages.

I hope to get a bit more CS and TCS into the n-lab as there seem to be possibilities for interesting interactions between some topics there and the 'n-POV' as well as connections with quantum computing.

What do you think? As I suggested please do get onto the n-forum if you have the time.
=--

##Models

####Traditional Models####

* [[transition systems]], and [[transition systems|labelled transition systems]];

* [[trace monoid|trace monoids]] and [[Mazurkiewicz trace]] theory;

* [[asynchronous automata]];

* [[Petri nets]];

*  [[event structures]];

* [actor models](http://en.wikipedia.org/wiki/Actor_model);

####Geometrical models####

* [[progress graph]];

* [[higher dimensional automaton]];

* [[higher dimensional transition system]];

* [[cubical set]].

## Related entries

* [[transition system]]

* [[bisimulation]]

## References

* [[Mogens Nielsen]], [[Glynn Winskel]]: *Models for concurrency*, in: *Handbook of Logic in Computer Science* vol 4, Oxford Univ. Press (1994) 1-148 [[pdf](https://www.cl.cam.ac.uk/~gw104/winskel-nielsen-models-for-concurrency.pdf), [[WinskelNielsen-Concurrency.pdf:file]], [doi:10.5555/218623.218630](https://dl.acm.org/doi/10.5555/218623.218630), [ISBN:9780198537809](https://global.oup.com/academic/product/handbook-of-logic-in-computer-science-9780198537809?lang=en&cc=at)]

 * [[Lisbeth Fajstrup]], [[Eric Goubault]], [[Emmanuel Haucourt]], [[Samuel Mimram]], [[Martin Raussen]], [[Directed Algebraic Topology and Concurrency]],Springer International Publishing 2016 (xi+167 pages)

 *  [[Eric Goubault]], [[Some geometric perspectives in concurrency theory]],

*   [[Eric Goubault]] and [[Samuel Mimram]], [Formal Relationships Between Geometrical and Classical Models for Concurrency](http://fr.arxiv.org/abs/1004.2818), Arxiv 1004.2818.

category: computer science


+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Synthetic differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Topos Theory
+--{: .hide}
[[!include topos theory - contents]]
=--
=--
=--

This entry collects pointers related to:

* [[Ieke Moerdijk]], [[Gonzalo E. Reyes]]:

  \linebreak

  **Models for Smooth Infinitesimal Analysis**

  \linebreak

  Springer (1991)

  [doi:10.1007/978-1-4757-4143-8](https://link.springer.com/book/10.1007/978-1-4757-4143-8)

about [[smooth toposes]] [[categorical semantics|interpreting]]  [[synthetic differential geometry]] that have a [[full and faithful functor|full and faithful]] embedding of the [[category]] [[Diff]] of smooth [[manifolds]].

#Contents#
* table of contents
{:toc}

## Summary

The book discusses the construction and the properties of [[smooth topos]]es $(\mathcal{T},R)$ that model the axioms of [[synthetic differential geometry]] and are _well-adapted_ to [[differential geometry]] in that there is a [[full and faithful functor]] $Diff \to \mathcal{T}$ embedding the [[category]] [[Diff]] of smooth [[manifold]]s into the more general category $\mathcal{T}$.

All models are obtained as [[category of sheaves|categories of sheaves]] on [[site]]s whose underlying category is a [[subcategory]] of that of [[smooth loci]].

* [[de Rham theorem|synthetic de Rham theorem]]

## Models {#Models}

The following tabulates various models for [[smooth topos]]es and lists their properties.

### the topos $\mathcal{G}$

### the topos $\mathcal{F}$

### the topos $\mathcal{Z}$

The [[smooth topos]] $\mathcal{Z}$ is that of [[sheaf|sheaves]] on the category $\mathbb{L}$ of [[smooth loci]] with respect to the [[Grothendieck topology]] given by _finite_ open covers of smooth loci.

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

$\mathcal{Z} := Sh_{fin-open}(\mathbb{L})$ is the [[category of sheaves]] on the entire [[site]] $\mathbb{L}$ of [[smooth locus|smooth loci]] where the [[cover]]ing [[sieve]]s of any [[smooth locus]] $\ell A$ are those generated by covering families

$$
  \{f_i : \ell A_i  \to \ell A\}
$$

given by a _finite_ collection of elements $(a_i \in A)_{i=1}^n$ such that the ideal generated by these elements contains the unit, $1 \in (a_1, \cdots, a_n)$, and for each $i$ a commutative diagram

$$
  \array{
    \ell A_i &&\stackrel{\simeq}{\to}&&  \ell (A\{a_i^{-1}\})
    \\
    & {}_{\mathllap{f_i}}\searrow && \swarrow
    \\
    &&
    \ell A
  }
  \,,
$$

where the right diagonal morphism is the canonical inclusion of a smooth locus corresponding to a [[generalized smooth algebra|smooth ring]] with one element inverted.

=--

This is in chapter VI, 1. Inversion of elements is described around proposition 1.6 in chapter I.

For instance for $\ell A = R = \ell C^\infty(\mathbb{R})$ the real line, a covering family is given by maps from two further copies of the real line $f_{1,2} : R \to R$ determined by any two smooth functions $a_1, a_1 \in C^\infty(\mathbb{R})$ with support  $(-\infty,1)$ and $(-1,\infty)$. By proposition 1.6 in chapter I we have $C^\infty(\mathbb{R})\{a_1^{-1}\} = C^\infty((-\infty,1))$ and $C^\infty(\mathbb{R})\{a_2^{-1}\} = C^\infty((-1,\infty))$. As both these open intervals are diffeomorphic to the real line, and as [[Diff]] embeds fully, we have isomorphisms $R \stackrel{\simeq}{\to} \ell(C^\infty(\mathbb{R}/{a_i^{-1}})$. Hence the cover defined by $(a_i, a_2)$ is the ordinary open cover of the real line by the two open subsets $(-\infty,1)$ and $(-1,\infty)$.

Similarly using I.1.6, one finds the general result:

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

Let $\ell C^\infty(U)/I$ be a [[smooth locus]] with $U \subset \mathbb{R}^n$ open and $I$ an ideal of $C^\infty(U)$. Then, up to isomorphisms, its covering families are precisely those families

$$
  \{\ell C^\infty(U_i)/(I|_{U_i}) \to \ell C^\infty(U)/I\}_{i=1}^n
$$

such that the $U_i \subset U$ are an ordinary open cover of $U$.

Equivalently, such families where the $(U_i)$ need not cover all of $U$, but where there is $V \subset U$ open such that the $(U_i)$ together with $V$ do cover and $1 \in I|_V$.

=--

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

This is lemma 1.2 in chapter VI.

=--

We now list central properties of this topos.

+-- {: .um_prop}
###### Proposition
**(properties)**

For the [[topos]] $\mathcal{Z}$ the following is true.

* the [[Grothendieck topology]] is [[subcanonical coverage|subcanonical]]

  (chapter VI, lemma 1.3)

* the [[category]] [[Diff]] of smooth [[manifold]]s embeds [[full and faithful functor|full and faithfully]], $Diff \hookrightarrow \mathcal{Z}$

  (chapter VI, corollary 1.4)

* the general [[Kock-Lawvere axiom]] holds

  (chapter VI, 1.9)

* the [[integration axiom]] holds

  (chapter VI, 1.10)

* it models [[nonstandard analysis]] in that

  * since the topology is subcanonical, in particular the
    [[smooth locus]]
    $\mathbb{I} := \ell(C^\infty(\mathbb{R}-{0})/(f|germ_0(f)
      = 0))
    $
    of the ring of restrictions of [[germ]]s of functions
    at 0 to $\mathbb{R}-{0}$
    is an object: the object of **invertible infinitesimals**.

    However, even though this object exists, in the
    [[intuitionistic logic|intuitionistic]]
    [[internal logic]] of the topos one cannot prove that there
    are any infinitesimal _elements_ : all one can prove is
    that it is false that there are no elements:
    $\not \not \exists x : x \in \mathbb{I}$.

    (chapter VI, section 1.8)

    This changes when one refines to the topos $\mathcal{B}$,
    discussed below (section VI.5).

  * due the conditions that covers are finite,
    the [[smooth locus]] $N := \ell C^\infty(\mathbb{N})$
    -- which is such that functions to it are arbitrary
    locally constant $\mathbb{N}$-valued functions --
    does not coincide with the [[natural numbers object]]
    of the topos, which is the [[sheafification]] of the
    [[presheaf]] constant on $\mathbb{N} \in Set$:

    since covering families are by _finite_ covers
    it follows that maps into the sheafification of
    the presheaf constant on $\mathbb{N}$ are _bounded_
    smooth $\mathbb{N}$-valued functions,
    instead of all such functions.

    (chapter VI, 1.6)

    The object $N = \ell C^\infty(\mathbb{N})$ is called the
    object of **[[smooth natural numbers]]** . It may be thought
    of as containing "infinite natural numbers".

=--

### the topos $\mathcal{B}$

The [[smooth topos]] $\mathcal{B}$ may be motivated as a slight
refinement of the topos $\mathcal{Z}$ designed such that in the [[internal logic]] of $\mathcal{B}$ it does become true that for $\mathbb{I}$ the object of invertible infinitesimals, we have $\exists x : x \in \mathbb{I}$, internally.

+-- {: .un_defn}
###### Definition
**(chapter VI, 5.1)**

$\mathcal{B} := Sh_{fin-open/proj}(\mathbb{L})$ is the [[category of sheaves]] on the [[site]] $\mathbb{L}$ of [[smooth locus|smooth loci]] with [[cover]]ing [[sieve]]s given by

* finite open covers as above for $\mathbb{Z}$

* and in addition letting projections $\ell A \times \ell B \to \ell A$ out of [[product]]s in $\mathbb{L}$ (for $B \neq 0 $ if $A \neq 0$) be singleton covers.

=--

(This is not a [[Grothendieck topology]], as it is not closed under composition, but still a [[coverage]].)

+-- {: .um_prop}
###### Proposition
**(properties)**

The topos inherits most of the properties of $\mathcal{Z}$, notably:

* the [[Grothendieck topology]] is [[subcanonical coverage|subcanonical]]

  (chapter VI, 5.2)

* the [[category]] [[Diff]] of smooth [[manifold]]s embeds [[full and faithful functor|full and faithfully]], $Diff \hookrightarrow \mathcal{B}$

  (chapter VI, 5.2)

* the general [[Kock-Lawvere axiom]] holds

  (chapter VI, below 5.4)

* the [[integration axiom]] holds

  (chapter VI, below 5.4)

* invertible infinitesimally small and infinitesimally big numbers are realized (with some vague similarity to [[nonstandard analysis]], but without its basic features like transfer principle in full generality)

  (chapter VI, below 5.4).

A main difference is that in $\mathcal{B}$ every
[[smooth locus]], i.e.
every [[representable functor|representable]], is an [[inhabited object]].
In particular therefore there exist, in the [[internal logic]], elements of the object of invertible infinitesimals:

$$
  \exists x \in \mathbb{I}
  \,.
$$

(chpater VI, prop 5.4).

=--

category: reference


models in presheaf toposes

[[!redirects structures in presheaf toposes]]

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

#Contents#
* table of contents
{:toc}

## Statement

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

Let $\mathbb{T}$ be a [[geometric theory]] over a [[signature (in logic)|signature]] $\Sigma$ and let $\mathcal{C}$ be a [[small category]]. Then there is an [[equivalence of categories]]

$$
  \mathbb{T}Mod([\mathcal{C}, Set])
  \simeq
  [\mathcal{C}, \mathbb{T}Mod(Set)]
$$

between the [[category of models]] in the [[presheaf topos]] over $\mathcal{C}^{op}$ and the [[category of presheaves]] with values in $\mathbb{T}$-models.

=--

For instance ([Johnstone, cor. D1.2.14](#Johnstone)).

Note that this continues to work for theories which involve infinitary limits as well. (The key observation is just that limits in $[\mathcal{C}, Set]$ are taken pointwise.)

## Examples

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

A _[[group object]]_ in a [[presheaf topos]] is equivalently a presheaf of [[groups]]. This is a fact used a lot for instance in [[homological algebra]] (for [[abelian groups]]). See at _[[group object]]_ for more.

=--

## Related concepts

* [[categorical semantics]]

## References

Around cor. D1.2.14 in

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


This page provides linkes related to the textbook

* [[Chris Isham]]

  _Modern Differential Geometry for Physicists_

  ([pdf](https://cdn.preterhuman.net/texts/science_and_technology/physics/Mathematical_Physics/Modern%20Differential%20Geometry%20for%20Physicists%202nd%20ed.,%20-%20C.%20Isham.pdf))

on [[differential geometry]] and a bit of [[sheaf and topos theory]] in ([[mathematical physics|mathematical]]) [[physics]].

Related textbooks include

* [[Theodore Frankel]], _[[The Geometry of Physics - An Introduction]]_

Related $n$Lab entries include

* _[[geometry of physics]]_

category: reference


+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
=--
=--

This pages collects links related to

* {#ElmendorfKrizMay} [[Anthony Elmendorf]], [[Igor Kriz]], [[Peter May]],

  _Modern foundations for stable homotopy theory_,

  in [[Ioan Mackenzie James]] (ed.),

  _[[Handbook of Algebraic Topology]]_,

  1995

  Amsterdam: North-Holland, pp. 213–253,

  ([pdf](http://hopf.math.purdue.edu/Elmendorf-Kriz-May/modern_foundations.pdf))

on the basics of [[stable homotopy theory]] in terms of the [[model category]] of [[S-modules]] with its [[symmetric smash product of spectra]].

#Contents#
* table of contents
{:toc}

## 1. Spectra and the stable homotopy category

* [[generalized (Eilenberg-Steenrod) cohomology]]

* [[pointed object]], [[loop space]], [[suspension]]

* [[homotopy equivalence]]

* [[Brown representability theorem]]

* [[Omega spectrum]], [[coordinate-free spectrum]]

* [[suspension spectrum]]

* [[homotopy classes]]

* [[homotopy group]], [[stable homotopy groups of spheres]]

* [[weak homotopy equivalence]]

* [[stable homotopy category]]

* [[CW spectrum]]

* [[Whitehead theorem]]

## 2. Smash products and twisted half-smash products

* [[smash product]]

* [[smash product of spectra]]

* [[symmetric smash product of spectra]]

## 3. The category of $S$-modules and its derived category

* [[S-module]], [[model structure on S-modules]]

## 4. The smash product of $S$-modules

* [[symmetric smash product of spectra]]

## 5. $A_\infty$ and $E_\infty$ ring spectra and their categories of modules

* [[A-∞ ring]], [[E-∞ ring]]

* [[∞-module]]

## 6. The smash product of $R$-modules and function $R$-modules

## 7. $Tor$ and $Ext$ in topology and algebra

* [[Tor]], [[Ext]]

## 8.Universal coefficients and Künneth spectral sequence

* [[universal coefficient theorem]]

* [[Künneth spectral sequence]]

## 9. Algebraic constructions in the derived category of $R$-modules

* [[brave new algebra]]

## 10. Algebra structures on localizations and on quotients by ideals

## 11. The specialization of $MU$-modules and algebras

* [[MU]]

* [[Adams spectral sequence]]

* [[Adams-Novikov spectral sequence]]

category: reference


Moderne Algebra

A textbook by [[Bartel Leendert van der Waerden]].

* Moderne Algebra. Teil I, Die Grundlehren der mathematischen Wissenschaften, vol. 33, Berlin, New York: Springer-Verlag, 1930, ISBN 978-3-540-56799-8

* Moderne Algebra. Teil II, Die Grundlehren der mathematischen Wissenschaften, vol. 34, Springer-Verlag, 1931, ISBN 978-3-540-56801-8

Based on lectures by [[Emil Artin]] and [[Emmy Noether]],
it was the first textbook using the modern approach to [[algebra]].

category: reference


modification

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

# Modifications
* table of contents
{: toc}

## Idea

A [[transfor]] between [[pseudonatural transformation|pseudo]]/[[lax natural transformations]] is sometimes called a _modification_. Hence modifications are the [[3-morphisms]] in a [[3-category]] [[2Cat]] of [[2-categories]] and [[2-functors]] between them.

Just as a [[natural transformation]] between [[functor]]s is a
collection of $1$-cells indexed by $0$-cells, a _modification_ between
transformations is an indexed collection of 2-cells.

## Definitions

To be most general, start with [[lax natural transformation|lax transformations]]
$\alpha, \beta : F \dot{\to} G : C \to D$. Given those, a modification
$m : \alpha \ddot{\to} \beta$ assigns to each $0$-cell $A\in
C$ a $2$-cell $m_A : \alpha_A \Rightarrow \beta_A$ in $D$
that commutes suitably with the $2$-cell components of
$\alpha$ and $\beta$ (see Leinster for details).

If $\alpha,\beta$ are strict transformations, then the
complicated-looking modification condition becomes simply
a naturality square with globs $m_A : \alpha_A \Rightarrow
\beta_A$ where the $1$-cells $\alpha_A$ would normally be.

## Generalisation

When you get tired of thinking individually about $n$-categories, functors, transformations, modifications, and so on, check out [[(n,k)-transformation]].

(Some discussion from here has also been moved to there.)

## Related concepts

* [[2-functor 2-category]]

## References

* {#JohnsonYau20} [[Niles Johnson]], [[Donald Yau]], Section 4.4 of: *2-Dimensional Categories*, Oxford University Press 2021 [[arXiv:2002.06055](http://arxiv.org/abs/2002.06055), [doi:10.1093/oso/9780198871378.001.0001](https://oxford.universitypressscholarship.com/view/10.1093/oso/9780198871378.001.0001/oso-9780198871378)]

* [[Tom Leinster]], _Basic bicategories_ [[arXiv:math/9810017](http://arxiv.org/abs/math/9810017)]

[[!redirects modification]]
[[!redirects modifications]]


modified Bernoulli number

[[!redirects modified Bernoulli numbers]]

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

#Contents#
* table of contents
{:toc}

## Idea

A variant of the [[Bernoulli numbers]] introduced in [Zagier 98](#Zagier98).

## Properties

### Periodicity

The modified Bernoulli numbers $B_n$ are 12-periodic for $n$ odd [Zagier, (A.4) (A.18)](#ZagierProperties).

## References

### General

Introduced in

* {#Zagier98} [[Don Zagier]], _A modified Bernoulli number_, Nieuw Archief voor Wiskunde, 16:63–72, 1998 ([[ZagierModifiedBernoulli.pdf:file]])

* [[Don Zagier]], _Hecke operators and periods of modular forms_, Israel Math. Conf. Proc., 3 (1990), 321–336 ([hdl:21.11116/0000-0004-3936-0](http://hdl.handle.net/21.11116/0000-0004-3936-0))

Further discussion in

* {#ZagierProperties} [[Don Zagier]], _Curious and Exotic Identities for Bernoulli Numbers_ ([[ZagierPropertiesBernoulli.pdf:file]])

* Atul Dixit, Victor H. Moll, Christophe Vignat, _The Zagier modification of Bernoulli numbers and a polynomial extension. Part I_ ([arXiv:1209.4110](https://arxiv.org/abs/1209.4110))

### In Chern-Simons theory

The "[[Wheels theorem]]", saying that the  [[perturbative quantization of 3d Chern-Simons theory|perturbative Chern-Simons]] [[Wilson loop observable]] of the [[unknot]] is, as a [[universal Vassiliev invariant]], a [[series]] of wheel-shaped [[Jacobi diagrams]] with [[coefficients]] the [[modified Bernoulli numbers]], is due to

* [[Dror Bar-Natan]], Thang T Q Le, [[Dylan Thurston]], _Two applications of elementary knot theory to Lie algebras and Vassiliev invariants_, Geom. Topol. Volume 7, Number 1 (2003), 1-31 ([euclid.gt/1513883092](https://projecteuclid.org/euclid.gt/1513883092))

following

* [[Dror Bar-Natan]], [[Stavros Garoufalidis]], [[Lev Rozansky]], [[Dylan Thurston]], _Wheels, wheeling, and the Kontsevich integral of  the unknot_ ([q-alg/9703025](http://arxiv.org/abs/q-alg/9703025))


modified Reidemeister move 1 - SVG

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category: svg


modular arithmetic

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

\tableofcontents

## The modulus relation

The basic relation in modular arithmetic is the modulus relation. In [[dependent type theory]] with a [[natural numbers type]] and [[function types]], one could define the modulus relation on the natural numbers as a ternary inductive type family $m \equiv n \mod k$ indexed by natural numbers $m:\mathbb{N}$, $n:\mathbb{N}$, and $k:\mathbb{N}$, inductively defined by the following constructors:

* for each $k:\mathbb{N}$, a term
$$\mathrm{indzero}(k):0 \equiv 0 \mod k$$
* for each $k:\mathbb{N}$, $m:\mathbb{N}$ and $n:\mathbb{N}$, a function
$$\mathrm{indsucc}(m, n, k):(m \equiv n \mod k) \to (s(m) \equiv s(n) \mod k)$$
* for each $k:\mathbb{N}$, $m:\mathbb{N}$ and $n:\mathbb{N}$, a term
$$\mathrm{indmodleft}(m, n, k):m \cdot k \equiv 0 \mod k$$
* for each $k:\mathbb{N}$, $m:\mathbb{N}$ and $n:\mathbb{N}$, a term
$$\mathrm{indmodright}(m, n, k):0 \equiv n \cdot k \mod k$$

where $m \cdot n$ is multiplication of two natural numbers $m:\mathbb{N}$ and $n:\mathbb{N}$.

For each nonzero natural number $k$, the type $(\mathbb{N}, m \equiv n \mod k)$ is [[equivalence of types|equivalent]] to the [[finite type]] $\mathrm{Fin}(k)$ equipped with its canonical [[identity type]]. When $k$ is zero the type $(\mathbb{N}, m \equiv n \mod k)$ is equivalent to the type of natural numbers with its canonical identity type.

This implies that the modulus relation is valued in [[h-propositions]] and is a family of [[decidable]] [[equivalence relations]] indexed by $k$.

## Related concepts

* [[finite field]]
* [[cyclic group]]

## References

* Wikipedia, _[Modular arithmetic](https://en.wikipedia.org/wiki/Modular_arithmetic)_


modular curve

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
#### Elliptic cohomology
+-- {: .hide}
[[!include elliptic cohomology -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _modular curve_ is a  [[moduli space]] of [[elliptic curves]] over the [[complex numbers]] equipped with [[level structure on an elliptic curve|level-n structure]], for some $n \in \mathbb{N}$. Concretely this is equivalent to the [[quotient]]

$$
  \mathcal{M}_{ell}(\mathbb{C})[n] \coloneqq \mathfrak{h}/\Gamma(n)
$$

of the [[upper half plane]] $\mathfrak{h}$ [[action|acted]] on by the $n^{th}$ [[principal congruence subgroup]] $\Gamma(n)\hookrightarrow SL_2(\mathbb{Z})$ of the [[special linear group]] acting by [[Möbius transformations]].

This has a [[compactification]]

$$
  \mathcal{M}_{ell}(\mathbb{C})[n] \hookrightarrow \mathcal{M}_{\overline{ell}}(\mathbb{C})[n]
$$

and often that is referred to by default as the modular curve.

The quotients by the other two [[congruence subgroups]] are

* $\mathcal{M}_{\overline{ell}}(\mathbb{C})[n]_0 \simeq \mathfrak{h}/\Gamma_0(n)$ -- the moduli space of complex [[elliptic curves]] equipped with a [[cyclic group|cyclic]] [[subgroup]] $\mathbb{Z}/n\mathbb{Z}$ or [[order of a group|order]] $n$;

* $\mathcal{M}_{\overline{ell}}(\mathbb{C})[n]_1 \simeq \mathfrak{h}/\Gamma_1(n)$ -- the moduli space of complex [[elliptic curves]] equipped with an element (a point) in an $n$-[[torsion subgroup]].

By construction, these modular curves provide [[covers]] ([[atlases]]) for the [[moduli stack of elliptic curves]] $\mathcal{M}_{ell}(\mathbb{C})$ over the complex numbers.

## Related concepts

* [[modularity theorem]]

* The analog of a modular curve with [[elliptic curves]] generalized to more general [[abelian varieties]] are _[[Shimura varieties]]_.

## References

* Wikipedia, _[Modular curve](http://en.wikipedia.org/wiki/Modular_curve)_