internalization

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

# Contents
* table of contents
{: toc}

## Idea

Any [[mathematical structure]] whose traditional [[Bourbaki]]-style definition is formulated within [[set theory]] can be formulated *internally* ([Grothendieck 1960](#Grothendieck60), [61](#Grothendieck61)) to any [[category]] that admits all those types of operations (typically: [[universal constructions]]) on its [[objects]] that the traditional definition applies to [[sets]], hence to the [[objects]] of the [[base topos|base]] category of [[Sets]].

This *internalization* serves to combine mathematical structures in a compatible way. For example a [[group object]] internal to the [[category]] [[Top]] of [[topological spaces]] is a [[topological group]] (while internal to [[Sets]] it is just a [[discrete group]]).

There is a [[diagram|diagrammatic]] and a [[syntax|syntactic]] way of systematizing the process of internalization: The former is essentially the topic of [[categorical algebra]], in terms of [[Lawvere theories]] ([Lawvere 1963](#Lawvere63)) or, more generally, of [[sketches]] ([Bastiani & Ehresmann 1972](#BastianiEhresmann72)), the latter of [[categorical semantics]] of [[type theories]].

An archetypical example of internalization is the notion of *[[internal groups]]* (e.g. [Eckmann & Hilton 1961](#EckmannHilton61)), which easily serves to illustrate the general idea:

Where the [[Bourbaki]]-style definition of a [[group]] is, of course:

* A *group* is a [[set]] $G$ equipped with [[functions]] $e \,\colon\,\ast \to G$ and $m \colon G \times G \to G$ and $(-)^{-1} \colon G \to G$  satisfying three conditions: [[unitality]], [[associativity]], [[inverses]].

one observes that here:

1. [[sets]] and [[functions]] are the [[objects]] and [[morphisms]] in the [[category]] of [[Sets]];

1. in invoking the [[singleton set]] $\ast$ and the [[Cartesian product]] set $G \times G$ one is making use of the fact that [[Sets]] admit the [[universal construction]] of [[finite products]];

and that this is _only_ [[mathematical structure|structure]] on [[Sets]] that the definition of [[groups]] is making use of.
Therefore, we may abstract away from [[Sets]], consider _any_ [[category]] $\mathcal{C}$ with [[finite products]] and declare that:

* An _[[internal group]]_ in $\mathcal{C}$ is an [[object]] $G$ of $\mathcal{C}$ equipped with [[morphisms]] $e \colon \ast \to G$ (i.e. out of the [[terminal object]]) and $m \colon G \times G \to G$ (i.e. out of the [[Cartesian product]]-object) and $(-)^{-1} \colon G \to G$ such that the following [[commuting diagram|diagrams commute]] in $\mathcal{C}$:

  * [[unitality]]

    $$
      \array{
        G \times \ast
        &
        \overset{
          id \times e
        }{\longrightarrow}
        &
        G \times G
        \\
        {}^{\mathllap{
           \simeq
        }}
        \big\downarrow
        &&
        \big\downarrow
        {}^{{}_{\mathrlap{ m }}}
        \\
        G
        &=&
        G
      }
      \;\;\;\;\;\;\;\;
      \array{
        \ast \times G
        &
        \overset{
          e \times id
        }{\longrightarrow}
        &
        G \times G
        \\
        {}^{\mathllap{
           \simeq
        }}
        \big\downarrow
        &&
        \big\downarrow
        {}^{{}_{\mathrlap{ m }}}
        \\
        G
        &=&
        G
      }
    $$

  * [[associativity]]

    $$
      \array{
        G \times G \times G
          &\overset{m \times id}{\longrightarrow}&
        G \times G
        \\
        {}^{{}_{\mathllap{ id \times m }}}
        \big\downarrow
        &&
        \big\downarrow
        {}^{{}_{m}}
        \\
        G \times G
        &\underset{ m }{\longrightarrow}&
        G
      }
    $$

  * [[inverses]]

    $$
      \array{
        G
        &
        \overset{
          diag
        }{
          \longrightarrow
        }
        &
        G \times G
        &
          \overset{ id \times (-)^{-1} }{\longrightarrow}
        &
        G \times G
        \\
        &
        \searrow
        &
        &&
        \big\downarrow {}^{_{\mathrlap{m}}}
        \\
        &&
        \ast
        &
        \underset{ e }{\longrightarrow}
        &
        G
      }
      \;\;\;\;\;\;\;\;\;
      \array{
        G
        &
        \overset{
          diag
        }{
          \longrightarrow
        }
        &
        G \times G
        &
          \overset{ (-)^{-1} \times id }{\longrightarrow}
        &
        G \times G
        \\
        &
        \searrow
        &
        &&
        \big\downarrow {}^{_{\mathrlap{m}}}
        \\
        &&
        \ast
        &
        \underset{ e }{\longrightarrow}
        &
        G
     }
    $$

If here $\mathcal{C} = $ [[Sets]] then this recovers the [[Bourbaki]]-definition of plain [[groups]], while if $\mathcal{C} = $ [[TopologicalSpaces]] this yields the notion of [[topological groups]]; for $\mathcal{C} = $ [[SmoothManifolds]] it yields [[Lie groups]], for $\mathcal{C} = $ [[Varieties]] it yields [[algebraic groups]], etc.

Similarly one defines [[action objects|internal actions]] of internal groups, [[formally principal actions]], etc.

In fact, realizing that [[groups]] are equivalently [[pointed object|pointed]] [[n-connected object of an (infinity,1)-topos|connected]] [[groupoids]] one may readily generalize the example of [[internal groups]] to a notion of [[internal groupoids]] and then of [[internal categories]]. This way [[category theory]] and with it much of [[mathematics]]  may be considered internally to suitable ambient [[categories]] $\mathcal{C}$ (see also at _[[internal logic]]_ and _[[internal language]]_).

Moreover, if the ambient category is in fact a [[higher category theory|higher category]], then internalization goes along with ([[vertical categorification|vertical]]) _[[categorification]]_. For example an [[internal group]] in the [[2-category]] of [[Groupoids]] is a [[2-group]] (a [[strict 2-group]] if one considers just the three [[diagrams]] shown above, or a general [[2-group]] of one adds higher [[coherence]]-diagrams.)

How much mathematics/[[internal logic]] actually exists in a given ambient [[category]] depends on how much [[extra properties]] this has. For example if $\mathcal{C}$ is a [[topos]] then *all* [[constructive mathematics]] has an internal incarnation in $\mathcal{C}$. For more on this see also at _[[relation between type theory and category theory]]_.

Here is a list matching some extra structure/property of the ambient category $\mathcal{C}$ to the kind of mathematics that exists internal to it:

* If the ambient category is equipped with the structure of a [[site]], then geometrical notions, such as defining [[morphisms]] locally on the domain with patching conditions (or more generally [[descent]] theory), exist inside it.

* If it is a [[finitely complete category]], the existence of [[finite products]] and [[terminal objects]] means that [[variety of algebras|varieties of algebras]] can be defined.

* If the category is a [[topos]] with a [[natural numbers object]], then one can do a certain sort of "ordinary" mathematics inside it, specifically [[predicative mathematics|impredicative]] [[constructive mathematics]] without the fancier tools of [[model theory]] or (ironically) [[category theory]].

* Further extra conditions on the category, such as being a [[Boolean topos]] or being a [[superextensive site]], bring the internal mathematics closer to that of $Set$.

* If the category is a [[(infinity,1)-topos]], then one could do most of ordinary constructive mathematics inside of it, including category theory, set theory, model theory, and higher category theory. Adding the [[principle of excluded middle]] or the [[axiom of choice|axiom]] of [[choice object|choice set]]/[[0-groupoids]] results in internal classical mathematics.

The [[extra structure]] required on the ambient category $\mathcal{C}$ is sometimes referred to as a _[[doctrine]]_ for internalization.   For example, there is a doctrine of [[monoidal categories]], a doctrine of [[finitely complete category|categories with finite limits]], a doctrine of [[cartesian closed categories]], and so on.

{#Externalization} Like [[vertical categorification|categorification]] or [[horizontal categorification|oidification]], there is currently no completely general formal definition of the process of internalization in a [[doctrine]], although there are one or two fairly general theorems.  However, its reverse is precise: given a doctrine $\mathcal{D}$ to which [[Sets]] (or some canonical Set-like category) belongs and a definition of foo internalized in the doctrine $\mathcal{D}$, if this definition of foo in $Sets$ reduces to the usual definition of foo, then the definition is acceptable; foos are a _deinternalization_, or [[externalization]], of internal foos.

## Examples

* [[monoid|Monoids]] can be internalized in the doctrine of [[monoidal category|monoidal categories]] ([[monoid objects]]).  For example:

  * A [[ring]] is a [[monoid object]] in [[Ab]] with its usual tensor product.

  * An [[algebra]] is a [[monoid object]] internal to [[Vect]], with its usual tensor product.

  * A strict monoidal category is a monoid internal to [[Cat]], with its cartesian product.

* Since any category with finite [[product|products]] is a [[cartesian monoidal category]], monoids can also be internalized in the doctrine of categories with finite products.

* [[group|Groups]] can also be internalized in the doctrine of categories with finite products, but not in the doctrine of monoidal categories, since diagonal maps are necessary to define what it means to have inverses.

* Monoids can also be internalized in the doctrine of [[multicategory|multicategories]].

* Commutative monoids can be internalized in the doctrine of [[symmetric monoidal category|symmetric monoidal categories]].

* Categories and groupoids can be internalized in the doctrine of [[finitely complete category|lex categories]], producing the notion of [[internal category]].

  * The classical fact that monoids can be identified with one-object categories has the following internal analogue: monoids in a lex category $C$ (qua category with finite products) can be identified with categories internal to $C$ whose object-of-objects is [[terminal object|terminal]].  However, monoids in a non-cartesian monoidal category (such as $Ab$ or $Vect$) cannot, in general, be identified with any sort of internal category.

* [[ring|Rings]] can be internalized in the doctrine of categories with finite products.

* More generally, the algebras for any [[Lawvere theory]] can be internalized in the doctrine of categories with finite products, and the algebras for any (symmetric) [[operad]] (in $Set$) can be internalized in the doctrine of (symmetric) monoidal categories.

* Pretty much any structure at all in mathematics can be internalized in a [[topos]].  Note, though, that since the [[internal logic]] of a topos is constructive, differences in axiomatization that make no difference classically can result in actual differences in behavior in a topos.  See [[constructive mathematics]] for some examples.  On the other hand, if the topos satisfies the [[axiom of choice]] (and in particular is [[Boolean topos|Boolean]]), then this complication won\'t happen.

* Categories themselves can be internalised, as algebras of an [[essentially algebraic theory]] (giving [[strict categories]]), in any [[finitely complete category]]; see [[internal category]].

## General Results

Often, the structure on the ambient category $C$ allowing a certain type of structure to be internalized in it is itself a [[vertical categorification|categorified]] version of that same structure (for example, monoids in monoidal categories).  This is an example of the [microcosm principle](http://golem.ph.utexas.edu/category/2008/12/the_microcosm_principle.html).

As one formal result along these lines, [[Tom Leinster]] has shown that for any [[cartesian monad]] $T$, $T$-algebras can be naturally internalized/enriched in $T$-multicategories, and in particular in $T$-structured categories.  For example, when $T$ is the monad whose algebras are monoids, this says that monoids can be internalized in multicategories and monoidal categories.

On the other hand, this is not always true.  For example, the categorification of a [[rig]] is a [[rig category]], but it is difficult to see how to define a rig internal to a rig category, and the usual definition of rig in $Set$ does not use the rig-category structure of $Set$ but only that it has finite products.

A very different sort of general result has to do with the [[internal logic]] of certain categories.  If a category has enough structure to interpret a certain fragment of first-order logic, then any first-order theory definable in that fragment can be internalized in that category. For example, the theory of categories can be formulated in [[cartesian logic]] and thus is interpretable in any cartesian (= lex) category. The statement that almost anything can be internalized in a topos is the high-end case of this, since the internal logic of a topos is full intuitionistic type theory.

## Examples

* [[monoid object]]

* [[group object]]

* [[action object]]

* [[pseudo-torsor|pseudo-torsor object]] ($\sim$ [[principal bundle]]-object)

* [[ring object]]

* [[monoid object in an (∞,1)-category]]

* [[Lie algebra object]]

* [[internal locale]]

* [[internal category]]

  * [[internal diagram]]

  * [[internal site]],

  * [[internal groupoid]]

  * [[internal pseudocategory]]

  * [[groupoid object in an (∞,1)-category]],

  * [[category object in an (∞,1)-category]]

* [[internal (co-)limit]]

## Related concepts

* [[internal logic]]

## References
 {#References}

The general notion of internalization is due to

* {#Grothendieck60} [[Alexander Grothendieck]], p. 370 (3 of 23) in: [[FGA]] *Technique de descente et théorèmes d'existence en géométrie algébriques. II: Le théorème d'existence en théorie formelle des modules*, Séminaire Bourbaki : années 1958/59 - 1959/60, exposés 169-204, Séminaire Bourbaki, no. 5 (1960), Exposé no. 195 ([numdam:SB_1958-1960__5__369_0](http://www.numdam.org/item/SB_1958-1960__5__369_0), [pdf](http://www.numdam.org/item/SB_1958-1960__5__369_0.pdf), English translation: [pdf](https://labs.thosgood.com/builds/fga-3-II.pdf), [web version](https://thosgood.com/fga/book/FGA-3-II)).

with specialization to [[internal groups]], [[internal actions]], [[internal categories]] and [[internal groupoids]] made explicit in:

* {#Grothendieck61} [[Alexander Grothendieck]], Sections 3, 4 of: [[FGA]] _Techniques de construction et théorèmes d'existence en géométrie algébrique III: préschémas quotients_, Séminaire Bourbaki: années 1960/61, exposés 205-222, Séminaire Bourbaki, no. 6 (1961), Exposé no. 212,   ([numdam:SB_1960-1961__6__99_0](http://www.numdam.org/item/?id=SB_1960-1961__6__99_0), [pdf](http://www.numdam.org/item/SB_1960-1961__6__99_0.pdf), English translation: [pdf](https://labs.thosgood.com/builds/fga-3-III.pdf), [web version](https://thosgood.com/fga/book/FGA-3-III)).

and to the case of internal [[principal bundles]] (see at *[[pseudo-torsor]]*) in:

* {#Grothendieck60} [[Alexander Grothendieck]], p. 312 (15 of 30) in: *Technique de descente et théorèmes d’existence en géométrie algébrique. I. Généralités. Descente par morphismes fidèlement plats*, Séminaire N. Bourbaki exp. no190 (1960) 299-327 [[numdam:SB_1958-1960__5__299_0](http://www.numdam.org/item/?id=SB_1958-1960__5__299_0)]

Discussion with focus on [[H-spaces]], [[internal monoids]] and [[internal groups]] and proving the [[Eckmann-Hilton argument]] originates with:

* {#EckmannHilton61} [[Beno Eckmann]], [[Peter Hilton]], *Structure maps in group theory*, Fundamenta Mathematicae 50 (1961), 207-221 ([doi:10.4064/fm-50-2-207-221](https://doi.org/10.4064/fm-50-2-207-221))

* {#EckmannHilton62} [[Beno Eckmann]], [[Peter Hilton]], _Group-like structures in general categories I.  Multiplications and comultiplications_, Math. Ann. 145, 227–255 (1962) ([doi:10.1007/BF01451367](https://doi.org/10.1007/BF01451367))

* [[Beno Eckmann]], [[Peter J. Hilton]], _Group-like structures in general categories II.  Equalizers, limits, lengths_.  Mathematische Annalen 151:2 (1963), 150–186.  [doi:10.1007/bf01344176](https://doi.org/10.1007/bf01344176).

* [[Beno Eckmann]], [[Peter Hilton]], _Group-like structures in general categories III.  Primitive categories_,  Math. Ann. **150** 165–187 (1963) ([doi:10.1007/BF01470843](https://doi.org/10.1007/BF01470843))

Highlighting the role of the [[Yoneda lemma]] in internalization:

* [[Saunders MacLane]], Section III.6 in: _[[Categories for the Working Mathematician]]_, Springer (1971)

Moreover with discussion of [[action objects]]:

* {#MacLaneMoerdijk92} [[Saunders Mac Lane]], [[Ieke Moerdijk]], Section V.6 of: _[[Sheaves in Geometry and Logic]]_, Springer 1992 ([doi:10.1007/978-1-4612-0927-0](https://link.springer.com/book/10.1007/978-1-4612-0927-0))

* [[Francis Borceux]], [[George Janelidze]], [[Gregory Maxwell Kelly]], around p. 8 of: _Internal object actions_, Commentationes Mathematicae Universitatis Carolinae (2005) Volume: 46, Issue: 2, page 235-255 ([dml:249553](https://eudml.org/doc/249553))

Discussion of [[internal categories]] is often attributed to originate around (though the simple idea can hardly be recognized here):

* [[Charles Ehresmann]], _Catégories structurées_, Annales scientifiques de l'École Normale Supérieure, Série 3, Tome 80 (1963) no. 4, pp. 349-426 ([numdam:ASENS_1963_3_80_4_349_0](http://www.numdam.org/item/ASENS_1963_3_80_4_349_0))

Review of internal category theory ([[internal categories]], [[internal functors]], [[internal limits]], [[internal sites]]), typically without attribution to more original sources:

* {#Johnstone77} [[Peter Johnstone]], Chapter 2 of: _Topos theory_, London Math. Soc. Monographs __10__, Acad. Press 1977, xxiii+367 pp. (Available as Dover Reprint, Mineola 2014)

* [MacLane-Moerdijk 92, Sec. V.7](#MacLaneMoerdijk92)

* [[Francis Borceux]], Chapter 8 in Vol 1 _Basic Category Theory_ of: _[[Handbook of Categorical Algebra]]_, Cambridge University Press (1994)  ([doi:10.1017/CBO9780511525858](https://doi.org/10.1017/CBO9780511525858))

* [[Peter Johnstone]], Chapter B2 in: Vol. 1 of: _[[Sketches of an Elephant -- A Topos Theory Compendium]]_, Oxford University Press (2002)([ISBN:9780198534259](https://global.oup.com/academic/product/sketches-of-an-elephant-9780198534259))

Review of internal [[group objects]], [[action objects]], [[internal categories]] (though without attribution to the above original articles):

* [[John Michael Boardman]], *Algebraic objects in categories*, Chapter 7 of: _Stable Operations in Generalized Cohomology_ [[pdf](https://hopf.math.purdue.edu/Boardman/stabop.pdf)] in: [[Ioan Mackenzie James]] (ed.) _[[Handbook of Algebraic Topology]]_ Oxford (1995) [[doi:10.1016/B978-0-444-81779-2.X5000-7](https://doi.org/10.1016/B978-0-444-81779-2.X5000-7)]

* [[Magnus Forrester-Barker]], *Group Objects and Internal Categories* [[arXiv:math/0212065](https://arxiv.org/abs/math/0212065)]

More general internal algebraic structures, known as [[algebra over a Lawvere theory|algebras]] over [[Lawvere theories]] originate with:

* {#Lawvere63} [[William Lawvere]], *[[Functorial Semantics of Algebraic Theories]]*, Ph.D. thesis Columbia University (1963).  Published with an author's comment and a supplement in: Reprints in Theory and Applications of Categories **5** (2004) pp 1--121. ([tac:tr5](http://www.tac.mta.ca/tac/reprints/articles/5/tr5abs.html))

Yet more general internal structures via [[sketches]]:

* {#BastianiEhresmann72} [[Andrée Bastiani]], [[Charles Ehresmann]], _Categories of sketched structures_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Tome 13 (1972) no. 2, pp. 104-214 ([numdam:CTGDC_1972__13_2_104_0](http://www.numdam.org/item/?id=CTGDC_1972__13_2_104_0))

* [[Michael Barr]], [[Charles Wells]], Section 4 of: _[[Toposes, Triples, and Theories]]_, Originally published by: Springer-Verlag, New York, 1985, republished in: Reprints in [Theory and Applications of Categories, No. 12 (2005) pp. 1-287](http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html)

For more see at:

* [[algebra over a Lawvere theory|algebras over]]$\,$ [[algebraic theories]]

* [[algebra over a monad|algebras over]]$\,$ [[monads]]

* [[algebra over an operad|algebras over]]$\,$ [[operads]]

[[!redirects Internalization]]
[[!redirects internal]]
[[!redirects internal to]]
[[!redirects internally]]
[[!redirects internally to]]
[[!redirects internalisation]]
[[!redirects internalization]]


Internalization > history

< [[Internalization]]
[[!redirects Internalization -- history]]


internally projective object

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

#Contents#
* table of contents
{:toc}

## Idea

An [[object]] of a [[category]] (usually a [[topos]] or [[pretopos]]) is _internally projective_ if it satisfies the [[internalization]] of the condition on a [[projective object]].

## Definition

An object $E$ of a [[topos]] $\mathcal{T}$ is called **internally projective** if the [[internal hom]]/[[exponential object]] [[functor]]

$$
  (-)^E \colon \mathcal{T} \to \mathcal{T}
$$

preserves [[epimorphisms]].

## Equivalent characterizations

Some of these facts, and more, can be found stated in Exercises 15 and 16 in Chapter IV of [[Sheaves in Geometry and Logic]].

+-- {: .num_theorem #DepProd}
###### Theorem
$E$ is internally projective if and only if the [[dependent product]] functor (the [[right adjoint]] to [[pullback]])
$$ \Pi_E \colon \mathcal{T}/E \to \mathcal{T}$$
preserves epimorphisms.
=--
+-- {: .proof}
###### Proof
The functor $(-)^E$ is the composite $\Pi_E \circ E^*$, and pullback $E^*\colon \mathcal{T} \to \mathcal{T}/E$ preserves epis, so if $\Pi_E$ preserves epis then so does $(-)^E$.

Conversely, for any $f:A\to B$ in $\mathcal{T}/E$, we have a pair of pullback squares:
$$\array{
  \Pi_E A & \to & A^E \\
  \downarrow & & \downarrow\\
  \Pi_E B & \to & B^E \\
  \downarrow & & \downarrow\\
  1 & \xrightarrow{id_E} & E^E
}$$
The lower square and outer rectangle are easily seen to be pullbacks, hence so is the upper square.  Since epimorphisms are closed under pullback, the other implication follows.
=--

+-- {: .num_lemma #Stable}
###### Lemma
If $E$ is internally projective in $\mathcal{T}$, then $I^* E$ is internally projective in $\mathcal{T}/I$.
=--
+-- {: .proof}
###### Proof
This proof was given by [[Alex Simpson]] [here](http://mathoverflow.net/a/140262/49).  Let $q:B\to A$ be an epimorphism from $v:B\to I$ to $u:A\to I$.  Then the exponential $u^{I^*E}$ in $\mathcal{T}/I$ is the equalizer of
$$ I\times A^E \xrightarrow{pr_2} A^E \xrightarrow{u} I^E$$
and
$$ I\times A^E \xrightarrow{pr_1} I \xrightarrow{const} I^E $$
equipped with the obvious projection to $I$.  Now we have a pullback square
$$\array{
  v^{I^*E} & \to & I\times B^E \\
  \downarrow & & \downarrow \\
  u^{I^*E} & \to & I\times A^E
}$$
in which the right-hand map is epi since $E$ is internally projective; hence so is the left-hand map.
=--

+-- {: .num_theorem #StackSem}
###### Theorem
$E$ is internally projective if and only if the statement "$E$ is projective" is true in the [[stack semantics]] of $\mathcal{T}$.
=--
+-- {: .proof}
###### Proof
By definition, truth of "$E$ is projective" in the stack semantics means that for any $I$ and any epimorphism $A\to I\times E$, there exists an epimorphism $p:J\to I$ and a [[section]] of $(p\times id)^*A \to J\times E$.  (We have used the characterization that an object $X$ is projective just when every epimorphism with codomain $X$ is split.)

If $E$ is internally projective, then given an epimorphism $A\to I\times E$ as above, let $J = \Pi_{pr_1}(A)$, where $pr_1:I\times E \to I$ is the projection.  Since $I\times E$ is internally projective in $\mathcal{T}/I$ by Lemma \ref{Stable}, $p:J\to I$ is an epimorphism.  And $(p\times id)^*A \to J\times E$ is split by the [[counit]] of the adjunction $(pr_1)^* \dashv \Pi_{pr_1}$.

Conversely, suppose the above condition holds, and let $e:B\to A$ be an epimorphism in $\mathcal{T}/E$.  Let $I = \Pi_E(A)$, and let $C$ be the pullback of $e$ along the counit $\Pi_E(A) \times E\to A$.  Then $\Pi_E(B)$ is equivalently $\Pi_{pr_1}(C)$, where $pr_1:I \times E \to I$ is the projection.

Moreover, $C \to I\times E$ is epi, so by assumption there is an epi $p:J\to I$ such that $(p\times id)^*C \to J\times E$ is split.  Since pullback along epis reflects epis, it suffices to show that $p^* \Pi_{pr_1}(C)$ is split.  However, we have a pullback square
$$ \array{
  J\times E & \to & I\times E \\
  ^{pr_1}\downarrow & & \downarrow^{pr_1} \\
  J & \to & I
}$$
so by the [[Beck-Chevalley condition]], $p^* \Pi_{pr_1}(C)$ is equivalently $\Pi_{pr_1}(p\times id)^* C$.  But $(p\times id)^*C$ is split, and all functors preserve split epis.
=--

Note that Lemma D4.5.3(iii) of [[Sketches of an Elephant]] is the special case of the above stack-semantics version of internal projectivity when $I=1$.  This is insufficient for the implication (iii)$\Rightarrow$(ii) of that lemma to hold, since if so, then every projective object would be internally projective, which as we show below is not the case.

## Projectivity versus internal projectivity

If the [[terminal object]] in $\mathcal{T}$ is [[projective object|projective]], then every internally projective object is projective. In the converse direction,

+-- {: .num_prop #enough}
###### Proposition
If $\mathcal{T}$ has [[projective object|enough projectives]] and projectives are closed under binary [[products]], then every projective object is internally projective. (In particular, if all objects of $\mathcal{T}$ are projective then all objects are internally projective.)
=--

+-- {: .proof}
###### Proof
Let $P$ be a projective object. To show that $e^P \colon E^P \to B^P$ is epic whenever $e \colon E \to B$ is epic, choose an epi $\phi \colon P' \to B^P$ where $P'$ is projective (using the assumption of enough projectives). Since $P \times P'$ is projective, there exists a lift through $e$ of the horizontal composite as shown:

$$\array{
 & & & & E \\
 & & & & \downarrow ^\mathrlap{e} \\
P' \times P & \underset{\phi \times 1}{\to} & B^P \times P & \underset{eval}{\to} & B;
}$$

this, by [[currying]], provides a lift of $\phi \colon P' \to B^P$ through $e^P$. Since $\phi$ is epic, this immediately implies $e^P$ is epic, as desired.
=--

+-- {: .num_remark}
###### Remark
Proposition \ref{enough} may fail without the assumption that projective objects are closed under binary products. An example is given [here](/nlab/show/presentation+axiom#counter).
=--

+-- {: .num_remark}
###### Remark
The _internal axiom of choice_ (that is, the [[axiom of choice]] interpreted in the [[internal logic]] of the topos) is equivalent to the statement that every object is internally projective.  This is strictly weaker than the "external" axiom of choice that every [[epimorphism]] in the topos is [[split epimorphism|split]].
=--

+-- {: .num_cor #presheaf}
###### Corollary
In a presheaf topos $Set^{C^{op}}$, if $C$ has binary products, then every projective presheaf is internally projective.
=--

+-- {: .proof}
###### Proof
Representables, and arbitrary coproducts of representables, are projective, and every presheaf is covered by some coproduct of representables. This implies that projective presheaves are precisely retracts of coproducts of representables. Under the assumption that $C$ has binary products, coproducts of representables, and also their retracts, are also closed under binary products. Thus projective presheaves are closed under binary products. Now apply Proposition \ref{enough}.
=--

## Discussion

* Forum discussions [one](http://nforum.ncatlab.org/discussion/3008/internally-projective-objects/), [two](http://nforum.ncatlab.org/discussion/4342/internally-projective-objects/).

## Related concepts

* [[projective object]], [[projective presentation]], [[projective cover]], [[projective resolution]]

  * [[projective module]]

  * [[internally projective object]]

* [[injective object]], [[injective presentation]], [[injective envelope]], [[injective resolution]]

  * [[injective module]]

* [[free object]], [[free resolution]]

  * [[free module]]

* flat object, [[flat resolution]]

  * [[flat module]]

[[!redirects internally projective objects]]


interpolation


interpretation

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

#Contents#
* table of contents
{:toc}

## Idea

In [[formal logic]] and [[model theory]], _interpretation_ refers to equipping the [[syntax]] of some [[theory]] with a [[semantics]]. Naïvely this means finding a model (in the category of sets) for the theory. This is subsumed by treating interpretations as functors out of [[syntactic categories]].

## Definition

In the language of [[categorical logic]], interpretations are representations of theories inside some category $\mathbf{C}$. Depending on the kind of theory ([[cartesian logic|cartesian]], [[regular logic|regular]], [[coherent logic|coherent]], [[first-order logic|first-order]], [[geometric logic|geometric]]), an interpretation of $T$ in $\mathbf{C}$ is just a ([[cartesian logic|cartesian]], [[regular logic|regular]], [[coherent logic|coherent]], [[first-order logic|first-order]], [[geometric logic|geometric]]) functor to $\mathbf{C}$.

When $\mathbf{C}$ is [[Set]], $T$ is first-order, and the functor (say $M$) is _logical_ (in the sense of Makkai-Reyes, equivalently coherent if we take the [[Morleyization]] of $T$), we get _models_ in the sense usually studied in [[model theory]].

## Interpretations of theories in each other
Let $T_1$ and $T_2$ be ([[cartesian logic|cartesian]], [[regular logic|regular]], [[coherent logic|coherent]], [[first-order logic|first-order]], [[geometric logic|geometric]]) theories. A ([[cartesian logic|cartesian]], [[regular logic|regular]], [[coherent logic|coherent]], [[first-order logic|first-order]], [[geometric logic|geometric]]) _interpretation_ $T_1 \to T_2$ is just a functor between the [[syntactic categories]] $\mathbf{Def}(T_1) \to \mathbf{Def}(T_2)$.

Elsewhere, interpretations have been defined as assignments of symbols in the language $\mathcal{L}_1$ of $T_1$ to definable sets of $T_2$ satisfying various coherence conditions (usually at least product-preserving) which amount to functoriality.

Note that via the duality between taking [[syntactic category|syntactic categories]] and [[internal logic|internal logics]], a model of $T$ in $\mathbf{Set}$ is just an interpretation of $T$ in the theory $\mathsf{Lang}(\mathbf{Set})$.

### Bi-interpretations of theories
We say that $T_1$ and $T_2$ are _bi-interpretable_ if there are functors (of appropriate logical strength) $\mathbf{Def}(T_1) \leftrightarrows \mathbf{Def}(T_2)$ forming an [[equivalence of categories]]. One direction of [[conceptual completeness]] is that bi-interpretable theories have equivalent categories of models. This follows from the fact that ([[cartesian logic|cartesian]], [[regular logic|regular]], [[coherent logic|coherent]], [[first-order logic|first-order]], [[geometric logic|geometric]]) functors are closed under composition, and that equivalent categories induce equivalences of functor categories.

Many notions from [[geometric stability theory]] and [[classification theory]] are invariant under bi-interpretability, e.g. [[stability in model theory|stability]], [[quantifier elimination]], [[elimination of imaginaries]], etc.

Since bi-interpretations induce equivalences of categories of models, monsters of bi-interpretable first-order theories $T_1, T_2$ will have isomorphic automorphism groups, with the isomorphism induced by restrictions along reducts. This indicates the bi-interpretation extends to the [[imaginaries]] of $T_1$ and $T_2$ also, so that $T_1^{\operatorname{eq}} \simeq T_2^{\operatorname{eq}}$, in fact uniquely---which is the universal property of the [[pretopos completion]].

## Interpretations of models in each other
This notion has appeared in the model-theoretic literature, and is what some model theorists mean when they say "interpretation." Specialize to first-order logic and models in [[Set]], and fix models $M_1 \models T_1, M_2 \models T_2$. An _interpretation_ of $M_1$ in $M_2$ is a surjection $U \overset{f}{\twoheadrightarrow} M_1$ for $U$ some subset of $M_2^k$, some $k \in \mathbb{N}$, such that the pullback of $f^*X$ of any definable (with parameters) set $X$ of $M_1$ along $f$ is again definable in $Y$. This is enough to induce a logical functor $\mathbf{Def}(T_1) \to \mathbf{Def}(T_2)$ (surjectivity implies witnessed existentials continue to be witnessed, and the functor being induced by pullback implies logicalness), in fact a logical functor $\mathbf{Def}(T_1(M_1)) \to \mathbf{Def}(T_2(M_2))$ of $T_1$ and $T_2$ enriched with the elementary diagrams of $M_1$ and $M_2$.

Facts:

* Every interpretation between theories can be realized as being induced by a concrete interpretation between sufficiently saturated models of those theories.

* Given an $f : U \twoheadrightarrow M_1$, any other $g : U \twoheadrightarrow M_1$ which is also an interpretation $M_1 \to M_2$ is of the form $\sigma \circ f$ for some $\sigma \in \operatorname{Aut}(M_1)$.

### Examples

* The construction of the [[Grothendieck group of a commutative monoid]] implements a canonical interpretation of the ring $\mathbb{Z}$ in the semiring $\mathbb{N}$ (in fact inside $\mathbb{N}$ viewed as the standard model of [[Peano arithmetic]]).

* The complex field $\mathbb{C}$ is canonically interpreted in the real field $\mathbb{R}$ by identifying $\mathbb{C}$ with $\mathbb{R}^2$ and noting that the multiplication of complex numbers is definable from multiplication on the reals.

## Classifying toposes
An interpretation $T \to T'$ is precisely the inverse image part of a [[geometric morphism]] of [[classifying topos|classifying toposes]] $\mathcal{E}(T') \to \mathcal{E}(T)$. In particular, [[structure in model theory|models]] of $T$ are [[points of a topos | points]] of [[Set]] in $\mathcal{E}(T)$.

## Related entries

* [[formula]]

* [[model theory]], [[structure (model theory)]]

* [[geometric morphism]]

## References

* wikipedia [interpretation (model theory)](http://en.wikipedia.org/wiki/Interpretation_%28model_theory%29)

* Wilfrid Hodges, _A shorter model theory_, Cambridge Univ. Press 1997

* Olivia Caramello, _Topos-theoretic preliminaries_.


interpretation of quantum mechanics

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

> *when it comes to atoms, language can be used only as in poetry.* [[[Niels Bohr|Bohr]], [1920](https://en.wikiquote.org/wiki/Niels_Bohr)]

# Contents
* table of contents
{: toc}

## Idea

In the words of the [Stanford Encyclopedia of Philosophy](#SEP):

> [[quantum mechanics|Quantum mechanics]] is, at least at first glance and at least in part, a mathematical machine for predicting the behaviors of microscopic [[particles]] — or, at least, of the [[measurement|measuring]] instruments we use to explore those behaviors — and in that capacity, it is spectacularly successful: in terms of power and precision, head and shoulders above any [[theory (physics)|theory]] we have ever had. [[mathematics|Mathematically]], the theory is well understood; we know what its parts are, how they are put together, and why, in the mechanical sense (i.e., in a sense that can be answered by describing the internal grinding of gear against gear), the whole thing performs the way it does, how the information that gets fed in at one end is converted into what comes out the other. The question of what kind of a world it describes, however, is controversial; there is very little agreement, among physicists and among philosophers, about what the world is like according to [[quantum mechanics]]. Minimally interpreted, the [[theory (physics)|theory]] describes a set of facts about the way the microscopic world impinges on the macroscopic one, how it affects our [[measurement|measuring instruments]], described in everyday language or the language of [[classical mechanics]]. Disagreement centers on the question of what a microscopic world, which affects our apparatuses in the prescribed manner, is, or even could be, like intrinsically; or how those apparatuses could themselves be built out of microscopic parts of the sort the theory describes.

> That is what an interpretation of the theory would provide: a proper account of what the world is like according to quantum mechanics, intrinsically and from the bottom up.

The need for and proper nature of such an interpretation may be subtle and is the subject of much debate. Generally it is maybe worthwhile to keep in mind that all [[physical theory]], well-confirmed as it may be, may usefully be subjected to deeper inspection. This point is well made by [[Richard Feynman]] in _The character of physical law_ (1967):

> For those people who insist that the only thing that is important is that the theory agrees with experiment, I would like to imagine a discussion between a Mayan astronomer and his student. The Mayans were able to calculate with great precision predictions, for example, for eclipses and for the position of the moon in the sky, the position of Venus, etc. It was all done by arithmetic. They counted a certain number, and subtracted some numbers, and so on. There was no discussion of what the moon was. There was no discussion even of the idea that it went around. They just calculated the time when there would be an eclipse, or when the moon would rise at the full, and so on. Suppose that a young man went to the astronomer and said 'I have an idea. Maybe those things are going around, and there are balls of something like rocks out there, and we could calculate how they move in a completely different way from just calculating what time they appear in the sky', 'Yes', says the astronomer, 'and how accurately can you predict eclipses?' He says, 'I haven't developed the thing very far yet', Then says the astronomer, 'Well, we can calculate eclipses more accurately than you can with your model, so you must not pay any attention to your idea because obviously the mathematical scheme is better'. There is a very strong tendency, when someone comes up with an idea and says, 'Let's suppose that the world is this way', for people to say to him, 'What would you get for the answer to such and such a problem?' And he says 'I haven't developed it far enough'. And they say, 'Well, we have already developed it much further, and we can get the answers very accurately'. So it is a problem whether or not to worry about philosophies behind ideas.

## Examples

* SEP: [Copenhagen interpretation of qm](http://plato.stanford.edu/entries/qm-copenhagen)

* SEP: [qm:collapse theories](http://plato.stanford.edu/entries/qm-collapse),

  [[Bub-Clifton theorem]]

* [[hidden variable theory]]

  * [[Bohmian mechanics]]

* SEP: [many-world interpretation of qm](http://plato.stanford.edu/entries/qm-manyworlds),

* SEP: [modal-interpretations of qm](http://plato.stanford.edu/entries/qm-modal)

* SEP: [Everett's relative-state formulation of qm](http://plato.stanford.edu/entries/qm-everett)

* [[Bayesian interpretation of quantum mechanics]]

## Bohr's standpoint
 {#BohrStandpoint}

In ([Bohr 49](#Bohr49)) it is argued that

> _however far the phenomena transcend the scope of classical physical explanation, the account of all evidence must be expressed in classical terms_ . The argument is simply that by the word 'experiment' we refer to a situation where we can tell others what we have done and what we have learned and that, therefore, the account of the experimental arrangement and the results of the observations must be expressed in unambiguous language with suitable application of the terminology of classical physics.

A proposal for a formalization of this point of view is the concept of the _[[Bohr topos]]_ associated with the [[kinematics]] of a [[quantum mechanical system]]. For more arguments for this see also at _[[order-theoretic structure in quantum mechanics]]_.

As ([Peres 97](#Peres97)) highlights:

> Bohr was very careful and never claimed that there were in nature two different types of physical systems. All he said was that we had to use two different (classical or quantum) languages in order to describe different parts of the world.

and continues:

> The peculiar property of the quantum measuring process is that we have to use both descriptions for the same object: namely, the measuring apparatus obeys quantum dynamics while it interacts with the quantum system under study, and at a later stage the same apparatus is considered as a classical object, when it becomes the permanent depositary of information. This dichotomy is the root of the quantum measurement dilemma: there can be no unambiguous classical-quantum dictionary.

## Theorems

* [[order-theoretic structure in quantum mechanics]]

  * [[Kochen-Specker theorem]]

  * [[Alfsen-Shultz theorem]]

  * [[Harding-Döring-Hamhalter theorem]]

* [[Wigner theorem]]

* [[Bell's theorem]]

* [[Bub-Clifton theorem]]

## Related entries

* [[wave function collapse]], [[quantum measurement]]

  [[propositions as projections]]

* [[EPR paradox]]

* [[Gell-Mann principle]]

* [[quantum logic]]

* [[probability amplitude]]

* [[quantum probability]]

* [[quantum measurement]], [[decoherence]], [[consistent histories approach to quantum mechanics]]

* [[philosophy of physics]]

* [[epistemology]]

* [[coordination]]

* [[computable quantum physics]]

## References
 {#References}

### Historical arguments

Discussion by the founders of quantum mechanics includes the following.

In

* {#EinsteinPodoldskyRosen35} [[Albert Einstein]], [[Boris Podolsky]], [[Nathan Rosen]], _Can the Quantum-Mechanical Description of Physical Reality be Considered Complete?_ Physical Review **47** 10 (1935) 777-780  [[doi:10.1103/PhysRev.47.777](https://doi.org/10.1103/PhysRev.47.777)]

it was argued (see at *[[EPR paradox]]*) that quantum mechanics cannot be a complete description of fundamental physics. The contrary was argued in:

* [[Nils Bohr]], _Can Quantum-Mechanical Description of Physical Reality be Considered Complete?_ Physical Review 48, 696–702 (1935)  ([publisher](http://prola.aps.org/abstract/PR/v48/i8/p696_1))

The famous assertion (pp. 209) by Bohr that all experiments in quantum mechanics must be possible to describe in "classical terms" (cf. *[[Bohr topos]]*):

* {#Bohr49} [[Nils Bohr]], *Discussion with Einstein on Epistemological Problems in Atomic Physics*, in: P. A. Schilpp (ed.), *Albert Einstein, Philosopher-Scientist*, The Library of Living Philosophers **VII** Evanston (1949) 201-241, and *Niels Bohr Collected Works* **7** (1996) 339-381 [doi:10.1016/S1876-0503(08)70379-7]

{#PrimasStatementOfCopenhagen} The first systematic statement of what should count as the *Copenhagen interpretation* (broadly attributed to [[Werner Heisenberg]] and [[Nils Bohr]]) may be due to [Primas (1983) p. 99](#Primas83) (reproduced in [Omnès (1994) p. 85](#Omnès94)):

The [[many-worlds interpretation of quantum mechanics]]:

* {#Everett57a} [[Hugh Everett III]], *The Theory of the Universal Wavefunction* PhD thesis, Princeton (1957) [[pdf](https://www.pbs.org/wgbh/nova/manyworlds/pdf/dissertation.pdf)] reprinted in: [De Witt & Graham 1973](#DeWittGraham73)

  > (for discussion of the key argument here see at *[[deferred measurement principle]]* the section *[Deferred measurement and Interpretations](deferred+measurement+principle#DeferredMeasurementAndInterpretations)*)

* {#Everett57b} [[Hugh Everett III]], *"Relative State" Formulation of Quantum Mechanics*, Rev. Mod. Phys. **29** (1957) 454-462 [[doi:10.1103/RevModPhys.29.454](https://doi.org/10.1103/RevModPhys.29.454)] (reprinted in [De Witt & Graham 1973](#DeWittGraham73))

  > "This total lack of effect of one branch on another also implies that no observer will ever be aware of any 'splitting' process. Arguments that the world picture presented by this theory is contradicted by experience, because we are unaware of any branching process, are like the criticism of the Copernican theory that the mobility of the earth as a real physical fact is incompatible with the common sense interpretation of nature because we feel no such motion. In both cases the argument fails when it is shown that the theory itself predicts that our experience will be what it in fact is."

* {#DeWittGraham73} [[Bryce S. DeWitt]], [[Neill Graham]] (eds.), *The Many-Worlds Interpretation of Quantum Mechanics*, Princeton Series in Physics, Princeton University Press (1973, 2016) [[ISBN:9780691645926](https://press.princeton.edu/books/hardcover/9780691645926/the-many-worlds-interpretation-of-quantum-mechanics), [jstor:j.ctt13x0wwk](https://www.jstor.org/stable/j.ctt13x0wwk)]

The catch-phrase "shut up and calculate" for the agnostic interpretation is due to

* [[N. David Mermin]], *What's Wrong with this Pillow?*, Physics Today **42** 4 (1989) 9–11 [[doi:10.1063/1.2810963](https://doi.org/10.1063/1.2810963)]

even though often mis-attributed to [[Richard Feynman]]:

* [[N. David Mermin]], *Could Feynman Have Said This?*, Physics Today **57** 5 (2004) 10–11 [[doi:10.1063/1.1768652](https://doi.org/10.1063/1.1768652)]

### Review and textbooks

* [[Jagdish Mehra]], *The quantum principle: Its interpretation and epistemology*, Dialectica **27** 2 (1973) 75-157 [[jstor:42968519](https://www.jstor.org/stable/42968519)]

* {#Primas83} [[Hans Primas]], *Pioneer Quantum Mechanics and its Interpretation*, Ch. 3 in: *Chemistry, Quantum Mechanics and Reductionism*, Springer (1983) [[doi:10.1007/978-3-642-69365-6](https://doi.org/10.1007/978-3-642-69365-6)]

* {#Scheibe73} [[Erhard Scheibe]], *The logical analysis of quantum mechanics*, Pergamon Press (1973)

* [[Eugene P. Wigner]], *Epistemology of Quantum Mechanics*, Part I in: [[Jagdish Mehra]] (ed.), *Philosophical Reflections and Syntheses* Vol B.VI in: *The Collected Works of Eugene Wigner*, Springer (1973) [[doi:10.1007/978-3-642-78374-6_5](https://doi.org/10.1007/978-3-642-78374-6_5)]

* {#Omnès94} [[Roland Omnès]], *[[The Interpretation of Quantum Mechanics]]*, Princeton University Press (1994) [[ISBN:9780691036694](http://press.princeton.edu/titles/5525.html)]

* [[Max Tegmark]], *The Interpretation of Quantum Mechanics: Many Worlds or Many Words?*, Fortsch. Phys. **46** (1998) 855-862 [[arXiv:quant-ph/9709032](https://arxiv.org/abs/quant-ph/9709032)]

* {#Peres97} [[Asher Peres]], *Interpreting the Quantum World*, Stud. History Philos. Modern Physics **29** (1998) 611 [[arXiv:quant-ph/9711003](http://arxiv.org/abs/quant-ph/9711003)]

* [[Jeffrey Bub]], *Interpreting the quantum world*, Cambridge University Press (1999) [[ISBN:9780521653862](https://www.cambridge.org/ae/academic/subjects/physics/quantum-physics-quantum-information-and-quantum-computation/interpreting-quantum-world?format=PB&isbn=9780521653862)]

  > (focus on the [[Bub-Clifton theorem]])

* [[Daniel Greenberger]], [[Wolfgang L. Reiter]], [[Anton Zeilinger]], *Epistemological and Experimental Perspectives on Quantum Physics*, Vienna Circle Institute Yearbook (VCIY) **7** (1999) [[doi:10.1007/978-94-017-1454-9](https://doi.org/10.1007/978-94-017-1454-9)]

* Florian J. Boge, *Quantum Mechanics Between Ontology and Epistemology*,  European Studies in Philosophy of Science (ESPS) **10**, Springer (2008) [[doi:10.1007/978-3-319-95765-4](https://doi.org/10.1007/978-3-319-95765-4)]

* {#Wallace07} [[David Wallace]], *Philosophy of Quantum Mechanics*, Part 2 in: Dean Rickles (ed.), *The Ashgate Companion to Contemporary Philosophy of Physics* (2008) 16-98 [[ISBN:9780754655183](https://www.routledge.com/The-Ashgate-Companion-to-Contemporary-Philosophy-of-Physics/Rickles/p/book/9780754655183), [arXiv:0712.0149](http://arxiv.org/abs/0712.0149)]

* [[Maximilian Schlosshauer]], *Decoherence, the measurement problem, and interpretations of quantum mechanics*,  	Rev. Mod. Phys. **76** (2004) 1267-1305 [[arXiv;quant-ph/0312059](https://arxiv.org/abs/quant-ph/0312059), [doi:10.1103/RevModPhys.76.1267](https://doi.org/10.1103/RevModPhys.76.1267)]

  > (in view of dynamical [[quantum decoherence]])

* [[Edward MacKinnon]], *Interpreting Physics -- Language and the Classical/Quantum Divide*, Springer (2012) [[doi:10.1007/978-94-007-2369-6](https://doi.org/10.1007/978-94-007-2369-6)]

* [[Robert Spekkens]], Lectures on _Foundations of quantum mechanics_ (2012) [[web](https://pirsa.org/C12002)]

  also: _The Quantum Puzzle_ [[web](http://www.rwspekkens.com/index.php?Show=page&PI=2)]

An approach to [[wave function collapse]] via macroscopic [[decoherence]]:

* [[Wojciech Zurek]], *Decoherence, einselection, and the quantum origins of the classical*, Rev. Mod. Phys. **75** (2003) 715-775 [[quant-ph/0105127](http://arxiv.org/abs/quant-ph/0105127), [doi:10.1103/RevModPhys.75.715](http://dx.doi.org/10.1103/RevModPhys.75.715)]

See also:

* [[Arnold Neumaier]], *Coherent Quantum Physics: A Reinterpretation of the Tradition*, 1st edition, De Gruyter (2019) [ISBN:9783110667295, [doi:10.1515/9783110667387](https://doi.org/10.1515/9783110667387)]

One quote above is taken from the first paragraphs of

* {#SEP} Stanford Encyclopedia of Philosophy, _[Quantum mechanics](http://plato.stanford.edu/entries/qm/)_

[[!include possible-worlds and many-worlds -- references]]

[[!redirects interpretation of quantum mechanics]]
[[!redirects interpretations of quantum mechanics]]
[[!redirects interpretation of quantum physics]]
[[!redirects interpretations of quantum physics]]

[[!redirects many-worlds interpretation]]
[[!redirects many-worlds interpretation of quantum mechanics]]

[[!redirects Copenhagen interpretation]]
[[!redirects Copenhagen interpretation of quantum mechanics]]


intersecting D-brane model

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[string phenomenology]] for [[type IIA string theory]], quasi-realistic [[model (physics)|models]] (i.e. close to the [[standard model of particle physics]], or an [[MSSM]]) may be obtained by [[KK-compactifications]] from the 10-[[dimensions]]  down to 4d with configurations of [[D6-branes]] which fill all of 4d spacetime and [[brane intersection|intersect]] in a certain way in the 6-dimensional [[fiber]] space.

> graphics grabbed from [Uranga 12a, p. 12](#Uranga12a)

The [[Chan-Paton gauge field]] on the [[D6-branes]] yields the [[gauge fields]] in 4d, and the precise [[brane intersection|intersection]] pattern determines the effective [[fundamental particle]] content in 4d.

See also at [[string phenomenology]] the section _[Models in type II with intersecting branes](string+phenomenology#ModelsInTypeIIWithIntersectingBranes)_.

## Bottom-up and Top-down approaches

One distinguishes [[bottom-up and top-down model building]] strategies:

>  snippet grabbed from [Aldazabal-Ibáñez-Quevedo-Uranga 00](#AldazabalIbanezQuevedoUranga00)

## Properties

### Chiral fermions
  {#ChiralFermions}

One of the most striking special properties of the [[standard model of particle physics]] is that its content of [[fermionic]] [[fundamental particles]] is "[[chiral fermion|chiral]]", in that the left and right [[Weyl spinor]]-components of the would-be [[Dirac spinor]]-representation of the [[quarks]], [[electrons]] and [[neutrinos]] couple _differently_ to the [[gauge fields]] (for review see e.g. [Ibanez-Uranga 12, section 1.1](#IbanezUranga12)).

The observation that such [[Weyl spinor|chiral fermions]] do indeed appear when [[D6-branes]]  [[brane intersection|intersect]] at an angle on an $\mathbb{R}^{3,1}$ is due to ([Berkooz-Douglas-Leigh 96](#BerkoozDouglasLeigh96)), see also ([AFIRU 00, section 4](#AFIRU00), [BCLS 05, section 2.3](#BCLS05)). For review with an eye towards [[RR-field tadpole cancellation]] on [[toroidal orbifold|toroidal]] [[orientifolds]] see also [Forste-Honecker-Schreyer 01](#ForsteHoneckerSchreyer01), [Honecker 01, Sec. 2](#Honecker01), [Honecker 02](#Honecker02).

> graphics grabbed from [Ibáñez-Uranga 12](#IbanezUranga12)

Vague review is in ([Ibáñez-Uranga 12, section 10.2.1](#IbanezUranga12), [Uranga 12a, section 2.3](#Uranga12a)).

The lift of this situation to [[M-theory on G2-manifolds]] is discussed in ([Berglund-Brandhuber 02](#BerglundBrandhuber02), [Bourjaily-Espahbodi 08](#BourjailyEspahbodi08)).

### Generations of fermions
 {#GenerationsOfFermions}

While (presently) intersecting D-brane models don't explain why there are _precisely_ 3 [[generations of fundamental particles]] in the [[standard model of particle physics]], they do have the property that generically any such model does feature several [[generations of fundamental particles]].

The reason is that in these models there is one copy of a set of fundamental particles at each intersection point of two 3-manifolds (the internal part of the D6-branes) in a compact 6-dimensional space, and generically these intersection numbers are greater than one and hence induce a finite number of [[generations of fundamental particles|generations]] ([BCLS 05, section 2.3](#BCLS05), [Ibanez-Uranga 12, p. 307](#IbanezUranga12), [Uranga 12a, p. 12](#Uranga12a)):

> graphics grabbed from [Uranga 12a, p. 13](#Uranga12a)

### Yukawa couplings

In [[intersecting D-brane models]] [[Yukawa couplings]] are encoded by [[worldsheet instantons]] of open strings stretching between the [[brane intersection|intersecting]] [[D-branes]] (see [Marchesano 03, Section 7.5](#Marchesano03)). Mathematically this is encoded by  [[derived hom-spaces]] in a [[Fukaya category]] (see [Marchesano 03, Section 7.5](#Marchesano03)).

\begin{center}

\end{center}

> table grabbed from [Marchesano 03](#Marchesano03)

Realistic [[Yukawa couplings]] and [[fermion]] [[masses]] in an [[MSSM]] [[Pati-Salam GUT model]] with 3 [[generations of fermions]] realized on [[intersecting D-brane model|intersecting]] [[D6-branes]] [[KK-compactification|KK-compactified]] on a [[toroidal orbifold]] $T^6\sslash (\mathbb{Z}_2 \times \mathbb{Z}_2)$ are claimed in
in [Mayes 19](#Mayes19), [Gemill-Howington-Mayes 19](#GemillHowingtonMayes19), based on [Chen-Li-Mayes-Nanopoulos 07a](#ChenLiMayesNanopoulos07a), [Chen-Li-Mayes-Nanopoulos 07b](#ChenLiMayesNanopoulos07b).

\linebreak

### Higgs mechanism
 {#HiggsMechanism}

The [[Higgs mechanism]] naturally arises in [[intersecting D-brane models]]: The [[Higgs field]] appears here as the [[scalar field]] that witnesses in [[perturbative quantum field theory|perturbation theory]] the process of _brane recombination_ at the intersection locus of the D-branes ([Cremades-Ibanez-Marchesano 02, section 7](#CremadesIbanezMarchesano02)):

Further developments in [Ibanez-Marchesano-Rabadan 01](#IbanezMarchesanoRabadan01), [Hebecker-Knochel-Weigand 13](#HebeckerKnochelWeigand13) and specifically via [[string field theory]] in [HINSS 18](#HINSS18).

For review see [Ibanez-Uranga 12, fig 10.2](#IbanezUranga12):

\linebreak

### RR-Tadpole cancellation and Orientifolding
 {#Orientifolding}

Consistent intersecting D-brane models have to be in [[type I string theory]], or generally in [[type II string theory]] with [[orientifold]] backgrounds, to achieve [[RR-field tadpole cancellation]].

This is a key consistency condition in intersecting D-brane model building (e.g. [BCLS 05, section 2.4](#BCLS05), [Ibanez-Uranga 12, section 4.4](#IbanezUranga12))

[[!include RR-field tadpole cancellation on toroidal orientifolds -- table]]

### Intersections of D6s with D8/O8s
 {#IntersectionOfD6WithO8}

A [[black brane|black]] [[D6-brane]] may end on a [[black brane|black]] [[NS5-brane]], and in fact each [[black brane|brane]] [[NS5-brane]] at a non-trivial [[ADE singularity]] has to be the junction of two [[black brane|black]] [[D6-branes]]. For details see at _[D6-branes ending on NS5-branes](NS5-brane#D6BranesEndingOnNS5Branes)_.

> from [GKSTY 02](#GKSTY02)

If in addition the [[black brane|black]] [[NS5-brane]] sits at an [[O8-plane]], hence at the [[orientifold]] [[fixed point]]-locus (see [above](#Orientifolding)), then in the ordinary $\mathbb{Z}/2$-[[quotient]] it appears as a "[[half-brane]]" with only one copy of [[D6-branes]] ending on it:

> from [GKSTY 02](#GKSTY02)

(In [Hanany-Zaffaroni 99](#HananyZaffaroni99) this is interpreted in terms of the [['t Hooft-Polyakov monopole]].)

The lift to [[M-theory]] of this situation is an [[M5-brane]] intersecting an [[M9-brane]]:

> from [GKSTY 02](#GKSTY02)

Alternatively, the [[O8-plane]] may intersect the [[black brane|black]] [[D6-branes]] away from the [[black brane|black]] [[NS5-brane]]:

> from [HKLY 15](#HKLY15)

In general, some of the NS5 sit away from the [[O8-plane]], while some sit on top of it:

> from [Hanany-Zaffaroni 98](#HananyZaffaroni98)

### Relation to M-theory on $G_2$-manifolds

Lift to [[M-theory on G2-manifolds]] (e.g. [[G2-MSSM]]): see references [below](#ReferencesLiftToMTheory)

$\,$

### Cosmology and Holography
 {#Cosmology}

Since the [[near horizon geometry]] of [[BPS state|BPS]] [[black branes]] is conformal to the [[Cartesian product]] of [[anti de Sitter spaces]] with the unit $n$-sphere around the brane, the [[cosmology]] of intersecting D-brane models realizes the [[observable universe]] on the [[asymptotic boundary]] of an _approximately_ [[anti de Sitter spacetime]] (see for instance [Kaloper 04](#Kaloper04), [Flachi-Minamitsuji 09](#FlachiMinamitsuji09)). The basic structure is hence that of _[[Randall-Sundrum models]]_, but details differ, such as notably in _warped throat_ geometries, see [Uranga 05, section 18](#Uranga05).

These warped throat models go back to [Klebanov-Strassler 00](#KlebanovStrassler00) which discusses aspects of [[confinement]] in [[Yang-Mills theory]] on conincident ordinary and _[[fractional D-brane|fractional]]_ [[D3-branes]] at the [[singularity]] of a warped [[conifold]]. See also [Klebanov-Witten 98](#KlebanovWitten98).

> snippet grabbed from [Uranga 05, section 18](#Uranga05)

> here: "RS"=[[Randall-Sundrum model]]; "KS"=[Klebanov-Strassler 00](#KlebanovStrassler00)

In particular this means that [[AdS-CFT duality]] applies in _some approximation_ to intersecting D-brane models (e.g. [Soda 10](#Soda10), [GHMO 16](#GHMO16)), thus allowing to compute, to some approximation, [[non-perturbative effects]] in the [[Yang-Mills theory]] on the intersecting branes in terms of [[gravity]] on the ambient warped throat $\sim$ [[anti de Sitter spacetime|AdS]] ([Klebanov-Strassler 00, section 6](#KlebanovStrassler00))

Such approximate versions of [[AdS-CFT]] for gauge theories realized on intersecting D-branes are used for instance to estimate [[non-perturbative effects]] in [[QCD]], such as the [[shear viscosity]] of the [[quark-gluon plasma]] (see the references [there](quark-gluon+plasma#ReferencesViaAdSCFT)). See at _[[AdS-QCD correspondence]]_ for more on this.

$\,$

### Computer scan of Gepner-model compactifications
 {#ComputerScanOfGepnerModelCompactifications}

Discussion of [[string phenomenology]] of [[intersecting D-brane models]] [[KK-compactification|KK-compactified]] with non-geometric [[fibers]] such that the would-be string [[sigma-models]] with these [[target spaces]] are in fact [[Gepner models]] (in the sense of _[Spectral Standard Model and String Compactifications](https://www.physicsforums.com/insights/spectral-standard-model-string-compactifications/)_) is in ([Dijkstra-Huiszoon-Schellekens 04a](#DijkstraHuiszoonSchellekens04a), [Dijkstra-Huiszoon-Schellekens 04b](#DijkstraHuiszoonSchellekens04a)):

>  A plot of [[standard model of particle physics|standard model]]-like [[coupling constants]] in a computer scan of [[Gepner model]]-[[KK-compactification]] of [[intersecting D-brane models]] according to [Dijkstra-Huiszoon-Schellekens 04b](#DijkstraHuiszoonSchellekens04b).

>  The blue dot indicates the couplings in $SU(5)$-[[GUT]] theory. The faint lines are NOT drawn by hand, but reflect increased density of Gepner models as seen by the computer scan.

## Examples

### General

[[!include brane bound states -- table]]

### Intersecting D6-brane models

(...)

* {#IshiharaKataokaSato99} S. Ishihara, H. Kataoka, Hikaru Sato, _$D=4$, $N=1$, Type IIA Orientifolds_, Phys. Rev. D60 (1999) 126005 ([arXiv:hep-th/9908017](https://arxiv.org/abs/hep-th/9908017))

(...)

### Intersecting D4-brane models
 {#IntersectingD4BraneModels}

Only [[D4-branes]]:

* {#BKL01} D. Bailin, G. V. Kraniotis, A. Love, _Standard-like models from intersecting D4-branes_, Phys. Lett. B530 (2002) 202-209 ([arXiv:hep-th/0108131](https://arxiv.org/abs/hep-th/0108131))

* {#KataokaShimojo01} H. Kataoka, M. Shimojo, _$SU(3) \times SU(2) \times U(1)$ Chiral Models from Intersecting D4-/D5-branes_, Progress of Theoretical Physics, Volume 107, Issue 6, June 2002, Pages 1291–1296 ([arXiv:hep-th/0112247](https://arxiv.org/abs/hep-th/0112247), [doi:10.1143/PTP.107.1291](https://doi.org/10.1143/PTP.107.1291))

* {#Bailin02} D. Bailin, _Standard-like models from D-branes_, J Phys (2003) 60: 199 ([arXiv:hep-th/0210227](https://arxiv.org/abs/hep-th/0210227))

* D. Bailin, G. V. Kraniotis, A. Love, _New Standard-like Models from Intersecting D4-Branes_, Phys. Lett. B547 (2002) 43-50 ([arXiv:hep-th/0208103](https://arxiv.org/abs/hep-th/0208103))

[[D4-D8 brane bound states]]:

give the [[Witten-Sakai-Sugimoto model]] for [[holographic QCD]]

> graphics grabbed from [Erlich 09, section 1.1](WSS-model#Erlich09)

> graphics grabbed from [Rebhan 14](WSS-model#Rebhan14)

(...)

## References

### General

The [[bottom-up model building|bottom-up approach]] to intersecting D-brane model building was initiated in

* {#AldazabalIbanezQuevedoUranga00} G. Aldazabal, [[Luis Ibáñez]], F. Quevedo, [[Angel Uranga]], _D-Branes at Singularities : A Bottom-Up Approach to the String Embedding of the Standard Model_, JHEP 0008:002 2000 ([arXiv:hep-th/0005067](http://xxx.lanl.gov/abs/hep-th/0005067))

The observation that [[Weyl spinor|chiral fermions]] appear when [[D6-branes]]  intersect at an angle is due to

* {#BerkoozDouglasLeigh96} [[Micha Berkooz]], [[Michael Douglas]], [[Robert Leigh]], _Branes Intersecting at Angles_, Nucl. Phys. B480:265-278, 1996 ([arXiv:hep-th/9606139](https://arxiv.org/abs/hep-th/9606139))

development for [[toroidal orbifold|toroidal]] [[orientifolds]] is due to

* {#ForsteHoneckerSchreyer01} Stefan Forste, [[Gabriele Honecker]], Ralph Schreyer, _Orientifolds with branes at angles_, JHEP 0106:004, 2001 ([arXiv:hep-th/0105208](https://arxiv.org/abs/hep-th/0105208))

* {#Honecker01} [[Gabriele Honecker]], _Non-supersymmetric Orientifolds with D-branes at Angles_, Fortsch.Phys. 50 (2002) 896-902 ([arXiv:hep-th/0112174](https://arxiv.org/abs/hep-th/0112174))

* {#Honecker02} [[Gabriele Honecker]], _Non-supersymmetric orientifolds and chiral fermions from intersecting D6- and D8-branes_, thesis 2002 ([[HoneckerIntersectingDBraneModels02.pdf:file]])

Review includes

* G. Aldazabal, [[Luis Ibáñez]], F. Quevedo, _On Realistic Brane Worlds from Type I Strings_ ([arXiv:hep-ph/0005033](https://arxiv.org/abs/hep-ph/0005033))

* {#Marchesano03} [[Fernando Marchesano]], _Intersecting D-brane Models_ ([arXiv:hep-th/0307252](https://arxiv.org/abs/hep-th/0307252))

* {#BCLS05} [[Ralph Blumenhagen]], [[Mirjam Cvetic]], Paul Langacker, [[Gary Shiu]], _Towards Realistic Intersecting D-Brane Models_, Ann. Rev. Nucl. Part. Sci. 55:71-139, 2005 ([arXiv:hep-th/0502005](http://arxiv.org/abs/hep-th/0502005))

A textbook account is

* {#IbanezUranga12} [[Luis Ibáñez]], [[Angel Uranga]], _[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]_, Cambridge 2012

Some chapters of which appear separately:

for [[D6-branes]]:

* [Aldazabal-Ibanez-Quevedo-Uranga 00, section 10.2](#AldazabalIbanezQuevedoUranga00)

* {#Uranga12a} [[Angel Uranga]], _Model building in IIA: Intersecting brane worlds_, 2012 ([pdf](http://scgp.stonybrook.edu/wp-content/uploads/2012/05/Uranga-Lecture-Notes-1.pdf), [[Uranga12.pdf:file]])

for [[D7-branes]]:

* {#Uranga12b} [[Angel Uranga]], _Model building in type IIB string theory_, 2012 ([pdf](http://scgp.stonybrook.edu/wp-content/uploads/2012/05/Uranga-Lecture-Notes-2.pdf))

* Anshuman Maharana, [[Eran Palti]], _Models of Particle Physics from Type IIB String Theory and F-theory: A Review_ ([arXiv:1212.0555](http://arxiv.org/abs/1212.0555))

Discussion of the [[Higgs mechanism]] in intersecting D-brane models:

* {#IbanezMarchesanoRabadan01} [[Luis Ibáñez]], [[Fernando Marchesano]], R. Rabadan, section 7 of _Getting just the Standard Model at Intersecting Branes_, JHEP 0111:002, 2001 ([arXiv:hep-th/0105155](https://arxiv.org/abs/hep-th/0105155))

* {#CremadesIbanezMarchesano02} D. Cremades, [[Luis Ibáñez]], [[Fernando Marchesano]], _Intersecting brane models of particle physics and the Higgs mechanism_, JHEP, 0207, 022 2002 ([arXiv:hep-th/0203160](https://arxiv.org/abs/hep-th/0203160))

* {#HebeckerKnochelWeigand13} Arthur Hebecker, Alexander K. Knochel, [[Timo Weigand]], _The Higgs mass from a String-Theoretic Perspective_,  Nucl.Phys. B874 (2013) 1-35 ([arXiv:1304.2767](https://arxiv.org/abs/1304.2767))

* {#HINSS18} Manami Noumi Hashi, Hiroshi Isono, Toshifumi Noumi, Gary Shiu, Pablo Soler, _Higgs Mechanism in Nonlocal Field Theories_, JHEP08 (2018) 064 ([arXiv:1805.02676](https://arxiv.org/abs/1805.02676))

See also

* {#AFIRU00} G. Aldazabal, S. Franco, [[Luis Ibáñez]], R. Rabadan, A. M. Uranga, _$D=4$ Chiral String Compactifications from Intersecting Branes_, J.Math.Phys.42:3103-3126, 2001 ([arXiv:hep-th/0011073](https://arxiv.org/abs/hep-th/0011073))

* [[Ralph Blumenhagen]], [[Volker Braun]], Boris Kors, [[Dieter Lüst]], _The Standard Model on the Quintic_, Summary of Talks at SUSY02, 1st Intl. Conference on String Phenomenology in Oxford, Strings 2002 and 35th Ahrenshoop Symposium. ([arXiv:hep-th/0210083](http://arxiv.org/abs/hep-th/0210083))

* {#Lust04} [[Dieter Lüst]], _Intersecting Brane Worlds -- A Path to the Standard Model?_, Class. Quant. Grav.21 : S1399-1424, 2004 ([arXiv:hep-th/0401156](http://arxiv.org/abs/hep-th/0401156))

* Ching-Ming Chen, Tianjun Li, Dimitri V. Nanopoulos, _Standard-Like Model Building on Type II Orientifolds_, Nucl.Phys.B732:224-242,2006 ([arXiv:hep-th/0509059](http://arxiv.org/abs/hep-th/0509059))

* [[Angel Uranga]], _The standard model from D-branes in string theory_, talk at Padova, January 2008 ([pdf](http://active.pd.infn.it/g4/seminars/2008/files/uranga.pdf))

* {#DolanKrippendorfQuevedo11} Matthew J. Dolan, Sven Krippendorf, Fernando Quevedo, _Towards a Systematic Construction of Realistic D-brane Models on a del Pezzo Singularity_, JHEP 1110 (2011) 024 ([arXiv:1106.6039](http://arxiv.org/abs/1106.6039))

* Yuta Hamada, Tatsuo Kobayashi, Shohei Uemura, _Standard Model-like D-brane models and gauge couplings_, Nuclear Physics B Volume 897, August 2015, Pages 563-582 ([arXiv:1409.2740](https://arxiv.org/abs/1409.2740))

* Jill Ecker, [[Gabriele Honecker]], Wieland Staessens, _D6-Brane Model Building on $\mathbb{Z}_2 \times \mathbb{Z}_6$: MSSM-like and Left-Right Symmetric Models_, Nuclear Physics B Volume 901, December 2015, Pages 139-215,  ([arXiv:1509.00048](https://arxiv.org/abs/1509.00048))

Intersection with [[O8-planes]]/[[D8-branes]] ([[M-theory on S1/G_HW times H/G_ADE]]) is discussed in

* {#HananyZaffaroni98} [[Amihay Hanany]], [[Alberto Zaffaroni]], _Branes and Six Dimensional Supersymmetric Theories_, Nucl.Phys. B529 (1998) 180-206 ([arXiv:hep-th/9712145](https://arxiv.org/abs/hep-th/9712145))

* {#HananyZaffaroni99} [[Amihay Hanany]], [[Alberto Zaffaroni]], _Monopoles in String Theory_, JHEP 9912 (1999) 014 ([arXiv:hep-th/9911113](https://arxiv.org/abs/hep-th/9911113))

* {#GKSTY02} E. Gorbatov, V.S. Kaplunovsky, J. Sonnenschein, [[Stefan Theisen]], S. Yankielowicz, _On Heterotic Orbifolds, M Theory and Type I' Brane Engineering_, JHEP 0205:015, 2002 ([arXiv:hep-th/0108135](https://arxiv.org/abs/hep-th/0108135))

* {#HKLY15} Hirotaka Hayashi, Sung-Soo Kim, Kimyeong Lee, Futoshi Yagi, _6d SCFTs, 5d Dualities and Tao Web Diagrams_ ([arXiv:1509.03300](https://arxiv.org/abs/1509.03300))

### The landscape of models

On the [[landscape of string theory vacua|landscape]] of intersecting D-brane models:

Computer scan of [[Gepner model]]-[[KK-compactifications]] of intersecting D-brane models:

* {#DijkstraHuiszoonSchellekens04a} T.P.T. Dijkstra, L. R. Huiszoon, [[Bert Schellekens]], _Chiral Supersymmetric Standard Model Spectra from Orientifolds of Gepner Models_, Phys.Lett. B609 (2005) 408-417 ([arXiv:hep-th/0403196](https://arxiv.org/abs/hep-th/0403196))

* {#DijkstraHuiszoonSchellekens04b} T.P.T. Dijkstra, L. R. Huiszoon, [[Bert Schellekens]], _Supersymmetric Standard Model Spectra from RCFT orientifolds_, Nucl.Phys.B710:3-57,2005 ([arXiv:hep-th/0411129](https://arxiv.org/abs/hep-th/0411129))

Computer scan of [[toroidal orbifold]]-[[KK compactifications]] is discussed in

* [[Ralph Blumenhagen]], [[Florian Gmeiner]], [[Gabriele Honecker]], [[Dieter Lüst]], [[Timo Weigand]], _The Statistics of Supersymmetric D-brane Models_, Nucl.Phys.B713:83-135, 2005 ([arXiv:hep-th/0411173](https://arxiv.org/abs/hep-th/0411173))

* {#GBHLW05} [[Florian Gmeiner]], [[Ralph Blumenhagen]], [[Gabriele Honecker]], [[Dieter Lüst]], [[Timo Weigand]], _One in a Billion: MSSM-like D-Brane Statistics_, JHEP 0601:004, 2006 ([arXiv:hep-th/0510170](https://arxiv.org/abs/hep-th/0510170))

### Detailed realistic models
 {#ReferencesDetailedRealisticModels}

Realistic [[Yukawa couplings]] and [[fermion]] [[masses]] in an [[MSSM]] [[Pati-Salam GUT model]] with 3 [[generations of fermions]] realized on [[intersecting D-brane model|intersecting]] [[D6-branes]] [[KK-compactification|KK-compactified]] on a [[toroidal orbifold]] $T^6\sslash (\mathbb{Z}_2 \times \mathbb{Z}_2)$ are claimed in

* {#ChenLiMayesNanopoulos07a} Ching-Ming Chen, Tianjun Li, [[Van Eric Mayes]], [[Dimitri Nanopoulos]], _A Realistic World from Intersecting D6-Branes_, Phys.Lett.B665:267-270, 2008 ([arXiv:hep-th/0703280](https://arxiv.org/abs/hep-th/0703280), [doi:10.1016/j.physletb.2008.06.024](https://doi.org/10.1016/j.physletb.2008.06.024))

* {#ChenLiMayesNanopoulos07b} Ching-Ming Chen, Tianjun Li, [[Van Eric Mayes]], [[Dimitri Nanopoulos]], _Realistic Yukawa Textures and SUSY Spectra from Intersecting Branes_, Phys.Rev.D77:125023, 2008 ([arXiv:0711.0396](https://arxiv.org/abs/0711.0396))

* {#Mayes19} [[Van Eric Mayes]], _All Fermion Masses and Mixings in an Intersecting D-brane World_ ([arXiv:1902.00983](https://arxiv.org/abs/1902.00983))

* {#GemillHowingtonMayes19} Jordan Gemmill, Evan Howington, [[Van Eric Mayes]], _One String to Rule Them All: Neutrino Masses and Mixing Angles_ ([arXiv:1907.07106](https://arxiv.org/abs/1907.07106))

* Tianjun Li, Adeel Mansha, Rui Sun, _Revisiting the Supersymmetric Pati-Salam Models from Intersecting D6-branes_ ([arXiv:1910.04530](https://arxiv.org/abs/1910.04530))

* Tianjun Li, Adeel Mansha, Rui Sun, _Generalized Supersymmetric Pati-Salam Models from Intersecting D6-branes_ ([arXiv:1912.11633](https://arxiv.org/abs/1912.11633))

### Lift to M-theory
 {#ReferencesLiftToMTheory}

Lift of intersecting D-brane models to [[M-theory on G2-manifolds]] with [[ADE-singularities]] is discussed in the following articles

* {#BerglundBrandhuber02} Per Berglund, Andreas Brandhuber, _Matter from $G_2$-manifolds_, Nucl.Phys. B641 (2002) 351-375 ([arXiv:hep-th/0205184](https://arxiv.org/abs/hep-th/0205184))

* K. Behrndt, [[Gianguido Dall'Agata]], [[Dieter Lüst]], S. Mahapatra, _Intersecting 6-branes from new 7-manifolds with G_2 holonomy_, JHEP 0208:027,2002 ([arXiv:hep-th/0207117](https://arxiv.org/abs/hep-th/0207117))

* [BCLS 05, section 2.9](#BCLS05)

* {#BourjailyEspahbodi08} Jacob L. Bourjaily, Sam Espahbodi, _Geometrically Engineerable Chiral Matter in M-Theory_ ([arXiv:0804.1132](https://arxiv.org/abs/0804.1132))

### Cosmology
 {#ReferencesCosmology}

Discussion for [[cosmology]] of [[intersecting D-brane models]] (ambient $\sim$ [[anti de Sitter spacetimes]] with the $\sim$ conformal intersecting branes at the [[asymptotic boundary]]) includes the following (see also at _[[Randall-Sundrum model]]_):

* {#KlebanovStrassler00} [[Igor Klebanov]], [[Matthew Strassler]], _Supergravity and a Confining Gauge Theory: Duality Cascades and $\chi^{SB}$-Resolution of Naked Singularities_, JHEP 0008:052, 2000 ([arXiv:hep-th/0007191](https://arxiv.org/abs/hep-th/0007191))

* {#KlebanovWitten98} [[Igor Klebanov]], [[Edward Witten]], _Superconformal Field Theory on Threebranes at a Calabi-Yau Singularity_, Nucl.Phys.B536:199-218, 1998 ([arXiv:hep-th/9807080](https://arxiv.org/abs/hep-th/9807080))

* Luis Anchordoqui, Jose Edelstein, Carlos Nunez, Santiago Perez Bergliaffa, Martin Schvellinger, Marta Trobo, Fabio Zyserman, _Brane Worlds, String Cosmology, and AdS/CFT_, Phys. Rev. D64:084027, 2001 ([arXiv:hep-th/0106127](https://arxiv.org/abs/hep-th/0106127))

* {#Kaloper04} Nemanja Kaloper, _Origami World_, JHEP 0405 (2004) 061 ([arXiv:hep-th/0403208](https://arxiv.org/abs/hep-th/0403208))

* {#Uranga05} [[Angel Uranga]], section 18 of _TASI lectures on String Compactification, Model Building, and Fluxes_, 2005 ([pdf](http://cds.cern.ch/record/933469/files/cer-002601054.pdf), [cern:933469](http:cds.cern.ch/record/933469))

* {#FlachiMinamitsuji09} Antonino Flachi, Masato Minamitsuji, _Field localization on a brane intersection in anti-de Sitter spacetime_, Phys.Rev.D79:104021, 2009 ([arXiv:0903.0133](https://arxiv.org/abs/0903.0133))

* {#Soda10} Jiro Soda, _AdS/CFT on the brane_, Lect.Notes Phys.828:235-270, 2011 ([arXiv:1001.1011](https://arxiv.org/abs/1001.1011))

* {#GHMO16} Gianluca Grignani, Troels Harmark, Andrea Marini, Marta Orselli, _The Born-Infeld/Gravity Correspondence_, Phys. Rev. D 94, 066009 (2016) ([arXiv:1602.01640](https://arxiv.org/abs/1602.01640))

Survey is in

* {#Padilla02} Antonio Padilla, _Braneworld Cosmology and Holography_ ([arxiv:hep-th/0210217](https://arxiv.org/abs/hep-th/0210217))

* {#Teraguchi07} Shunsuke Teraguchi, around slide 21 of: _String theory and its relation to particle physics_, 2007 ([pdf](http://phys.cts.ntu.edu.tw/ppp7/talks/PPP7_Shunsuke_Teraguchi.pdf))

Discussion of 4d [[de Sitter spacetime]] [[cosmology]] models on brane worlds in $\sim$ [[anti de Sitter spacetime|AdS]] [[bulk]] spacetimes:

* {#BDDGS18} Souvik Banerjee, [[Ulf Danielsson]], Giuseppe Dibitetto, Suvendu Giri, Marjorie Schillo, _Emergent de Sitter cosmology from decaying AdS_ ([arXiv:1807.01570](https://arxiv.org/abs/1807.01570))

* Souvik Banerjee, [[Ulf Danielsson]], Giuseppe Dibitetto, Suvendu Giri, Marjorie Schillo, _de Sitter Cosmology on an expanding bubble_ ([arXiv:1907.04268](https://arxiv.org/abs/1907.04268))

### Experimental tests

Discussion of observational/[[experiment|experimental]] test of brane world models:

by analysis of the [[event horizon]] of the [[black hole]] in the center of the [[galaxy]] [[Messier 87]]:

* Indrani Banerjee, Sumanta Chakraborty, Soumitra SenGupta, _Silhouette of M87${}^*$: A new window to peek into the world of hidden dimensions_ ([arXiv:1909.09385](https://arxiv.org/abs/1909.09385))

[[!redirects intersecting D-brane models]]

[[!redirects intersecting brane model]]
[[!redirects intersecting brane models]]

[[!redirects intersecting D-brane model building]]

[[!redirects branes at angles]]

[[!redirects brane world model]]
[[!redirects brane world models]]


intersection

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

# Contents
* table of contents
{: toc}

## Definition

An **intersection** is a [[meet]] of [[subset]]s or (more generally) [[subobject]]s. Dually, a **cointersection** is a [[union]]/[[join]] of cosubobjects.

This includes the traditional [[set theory|set-theoretic]] intersection of subsets of some ambient set, as well as intersections of completely arbitrary sets (which are subsets of the [[universe]]) in [[material set theory]].

In a [[finitely complete category]], the intersection of two [[monomorphism]]s $A\hookrightarrow X$ and $B\hookrightarrow X$ can be computed by a [[pullback]] of the [[cospan]] $A\to X \leftarrow B$. Dually, the cointersection of two [[epimorphisms]] is their [[pushout]].

The __nullary intersection__ of the subsets of $X$ is $X$ itself.  A __binary intersection__ is the intersection of two sets, and a __finitary intersection__ is the intersection of [[finite set|finitely]] many sets.  Finitary intersections may be built out of binary and nullary intersections.

## Properties

* [[interactions of images and pre-images with unions and intersections]]

  * [[the image of an intersection is contained in the intersection of the images]]

  * [[pre-images preserve intersections]],

## Related concepts

* [[product]]

* [[union]]

* [[intersection product]], [[intersection theory]]

* [[brane intersection]]

[[!redirects intersection]]
[[!redirects intersections]]

[[!redirects cointersection]]
[[!redirects cointersections]]

[[!redirects nullary intersection]]
[[!redirects nullary intersections]]

[[!redirects binary intersection]]
[[!redirects binary intersections]]

[[!redirects finitary intersection]]
[[!redirects finitary intersections]]


intersection cohomology

#Contents#
* table of contents
{:toc}

## Idea

While for [[smooth manifolds]] there is a [[Poincaré duality]] relating [[singular homology]] and [[singular cohomology]], this fails for singular spaces. One can extend the idea of duality by taking into account (co)dimensions of singular strata. This idea can be concretized in a definition of **intersection cohomology** which has an additional parameter called **perversity**. In two special cases of perversity, it boils down to either homology or cohomology; thus some call it intersection homology. Main applications are in study of topology of singular spaces and in [[geometric representation theory]]. In particular properties of intersection of cycles of various codimension, including transversality are neatly accounted for.

## References

* [[Markus Banagl]], *Topological invariants of stratified spaces*, Springer Monographs in Mathematics, Springer (2000) [[doi:10.1007/3-540-38587-8](https://doi.org/10.1007/3-540-38587-8)]

  > (on [[stratified spaces]])

* [[Markus Banagl]], *Rational generalized intersection homology theories*, Homology, Homotopy and Applications, **12** 1 (2010) 157-185  [[pdf](http://www.mathi.uni-heidelberg.de/~banagl/pdfdocs/spectrumqih.pdf), [euclid](https://projecteuclid.org/journals/homology-homotopy-and-applications/volume-12/issue-1/Rational-generalized-intersection-homology-theories/hha/1296223826.full)]

* [[Robert MacPherson]], _Intersection homology and perverse sheaves_,  1990 notes [pdf](http://faculty.tcu.edu/gfriedman/notes/ih.pdf)

* R. MacPherson, _Global questions in the topology of singular spaces_, Proc. ICM Warsaw 1983, [pdf](http://www.mathunion.org/ICM/ICM1983.1/Main/icm1983.1.0213.0236.ocr.pdf)

* M. Goresky, G. MacPherson, _Morse theory and intersection homology theory_, Astérisque 101-102 parts II-III (1983), 135-192.

* [[Greg Friedman]], *Singular Intersection Homology*,  Cambridge University Press 2020 ([doi:10.1017/9781316584446](https://doi.org/10.1017/9781316584446), [pdf](http://faculty.tcu.edu/gfriedman/IHbook.pdf))

* Wikipedia, *[Intersection homology](https://en.wikipedia.org/wiki/Intersection_homology)*

* {#KirwanWoolf06} F. Kirwan, [[Jonathan Woolf]], _An introduction to intersection homology_,  Chapman & Hall/CRC, 2006. [MR2006k:32061](http://www.ams.org/mathscinet-getitem?mr=2207421)

* A. Borel et al. _Intersection cohomology_, Notes on the seminar held at the University of Bern, 1983. Progress in Math. __50__, Birkhäuser 1984 [MR88d:32024](http://www.ams.org/mathscinet-getitem?mr=788171)

* Springer Online Enc. of Math., [intersection homology](http://eom.springer.de/I/i052000.htm)

* MO question: [what happened to intersection homology](http://mathoverflow.net/questions/6874/what-if-anything-happened-to-intersection-homology)

* Steven L. Kleiman, _The development of intersection homology theory_, Pure Appl. Math. Q. __3__ (2007), no. 1, Special Issue: In honor of Robert D. MacPherson. Part 3, 225–282, [MR2008e:55006](http://www.ams.org/mathscinet-getitem?mr=2330160)

* T. A. Springer, _Quelques applications de la cohomologie d'intersection_,  Séminaire Bourbaki __24__ (1981-1982), Exp. No. 589, 25 p.
[MR689533](http://www.ams.org/mathscinet-getitem?mr=689533),  [Zbl0526.22014](http://www.zentralblatt-math.org/zmath/en/search/?q=an:0526.22014)

* [[George Lusztig]], _Intersection cohomology methods in representation theory_, Proc. ICM 1990, [pdf](http://mathunion.org/ICM/ICM1990.1/Main/icm1990.1.0155.0174.ocr.pdf)

In the generality of [[matroids]]:

* {#Huh22} [[June Huh]], _Combinatorics and Hodge theory_, Proc. Int. Cong. Math. **1** (2022)
 [[doi:10.4171/ICM2022/205](), [pdf](https://www.mathunion.org/fileadmin/IMU/Prizes/Fields/2022/jh.pdf), [[Huh-CombinatoricsAndHodgeTheory.pdf:file]]]

[[!redirects intersection cohomology]]
[[!redirects intersection homology]]


intersection number

## References

* Wikipedia, _[Intersection number](https://en.wikipedia.org/wiki/Intersection_number)_

[[!redirects intersection numbers]]


intersection pairing

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

On a [[space]] of suitable even [[dimension]] the _[[cup product]]_ on suitable mid-dimensional [[cohomology]] is often called the _intersection product_ -- this, or its evaluation on the [[fundamental class]] of the whole space.

Under [[Poincaré duality]] these cohomology classes may correspond to [[cycles]] and then under suitable conditions or in a suitable sense, the cup product dually counts (or otherwise detects) literally the [[intersection]] points of the two subspaces, whence the name. It is the topic of _[[intersection theory]]_ to make this statement precise, classical results to this extent include [[Bézout's theorem]] and its refinement to the [[Serre intersection formula]].

If here [[cohomology]] is replaced by [[differential cohomology]] then [[quadratic refinements]] of the intersection product provide the [[Lagrangians]] for [[higher dimensional Chern-Simons theory]] and govern the structure of [[self-dual higher gauge theory]]. See there for more.

In a little more detail:  For $X$ a [[space]] of [[dimension]] $2k$ and $H^k(X)$ a [[cohomology group]] on a space $X$ equipped with [[orientation in generalized cohomology|H-orientation]] in degree $k$ with coefficients in some $A$, the **intersection pairing** on [[cohomology]] is the map

$$
  H^k(X) \times H^k(X) \to A
$$

given by [[fiber integration]]

$$
  (\lambda, \omega) \mapsto \int_X (\lambda \cup \omega)
  \,,
$$

of the [[cup product]]

$$
  \cup : H^k(X) \times H^k(X) \to H^{2k}(X)
  \,.
$$

## Examples

### In integral cohomology

The [[signature of a quadratic form]] of the [[quadratic form]] induced by the intersection pairing in [[integral cohomology]] is the _[[signature genus]]_.

In dimension 4, see also:

* Wikipedia, _[Intersection form (4-manifold)](http://en.wikipedia.org/wiki/Intersection_form_%284-manifold%29)_.

### In $\mathbb{Z}_2$-cohomology

Over a [[Riemann surface]] $X$, the intersection pairing on $H^1(X, \mathbb{Z}_2)$ has a [[quadratic refinement]] by the function that sends a [[Theta characteristic]] to the mod 2-dimension of its space of sections. See _[Theta characteristic -- Over Riemann surfaces](Theta+characteristic#OverRiemannSurfaces)_.

### In ordinary differential cohomology: higher abelian Chern-Simons theory
 {#SecondaryIntersectionPairing}

For the case that the cohomology in question is [[ordinary differential cohomology]],

* a [[cocycle]] in degree $k$ is a [[circle n-bundle with connection|circle (k-1)-bundle with connection]],

* the cup product is the [[Beilinson-Deligne cup product]];

* and the required notion of orientation is now _[[orientation in differential cohomology]]_: a [[differential Thom class]].

The differentially refined intersection pairing is non-trivial and interesting also on manifolds of dimension less than $2k$, where the integral intersection pairing vanishes: it provides a _[[secondary characteristic class]]_, a **secondary intersection pairing**.

Notably, the [[diagonal]] of the intersection pairing in in dimension $2k-1$ is the [[action functional]] of quadratic abelian [[higher dimensional Chern-Simons theory]].

Its [[quadratic refinement]] is discussed in
([Hopkins-Singer](#HopkinsSinger)), induced from/modeled on [[characteristic elements of a bilinear form|characteristic elements]] given by [[integral Wu structures]].

### On framed manifolds

On a [[framed manifold]] the intersection pairing has a canonical [[quadratic refinement]], leading to the [[Kervaire invariant]].

## Related concepts

* [[functional cup product]]

* [[homotopy Whitehead integral formula]]

[[!include quadratic invariants - table]]

[[!include square roots of line bundles - table]]

## References

Introductions and surveys include

* [[Akhil Mathew]], _Intersection theory on surfaces_, 2013 ([web](http://amathew.wordpress.com/2013/01/27/intersection-theory-on-surfaces/))

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

Discussion of the intersection pairing in [[ordinary differential cohomology]] and especially its [[quadratic refinement]] is in

* {#HopkinsSinger} [[Mike Hopkins]], [[Isadore Singer]], _[[Quadratic Functions in Geometry, Topology, and M-Theory]]_

[[!redirects intersection pairings]]

[[!redirects secondary intersection pairing]]
[[!redirects secondary intersection pairings]]

[[!redirects intersection product]]
[[!redirects intersection products]]

[[!redirects intersection number]]
[[!redirects intersection numbers]]


intersection product > history

< [[intersection product]]
[[!redirects intersection product >history]]


intersection theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

Intersection theory studies literally the [[intersection]] of pairs of sub-spaces inside an ambient [[space]]. Historically [[Bézout's theorem]] described the signed ([[integer]]-valued) [[number]] of points at which two [[algebraic curves]] meet transversally inside an ambient [[algebraic surface]]. Dually, under [[Poincaré duality]], this integer is the evaluation of the [[cup product]] of the duals of the two [[fundamental classes]] of the curves on that of the ambient spaces, and hence the [[cup product]] on mid-degree cohomology classes is also called the _[[intersection product]]_ and is one central aspect of intersection theory.

However, if the sub-spaces do not intersect sufficiently transversally, then their plain set-theoretic number of intersection points will not agree with the cohomological [[intersection product]] thus defined. Historically [[Serre's intersection formula]] was a first step to remedy this by replacing the plain [[tensor product]] of [[structure sheaves]] (which is what [[Isbell duality|dually]] describes the [[intersection]] of the [[varieties]]) by a [[Tor]], hence by its [[derived functor]].

In the modern version of the theory (as indicated e.g. in the introduction of ([Lurie-Spaces](#LurieSpaces))) this is interpreted as saying that the intersection is to be taken in _[[derived algebraic geometry]]_ (and the fundamental classes are to be taken to be [[virtual fundamental classes]]).

With intersection theory interpreted in [[(infinity,1)-topos theory]] this way then the nature of Bézout's original statement should hold in full generality essentially without further assumptions and this would be the central statement of intersection theory:

_the [[cup product]] of two [[virtual fundamental classes]] of mid-dimensional subspaces  equals the fundamental class of their derived intersection_.

## Related concepts

* [[intersection product]]

* [[Gromov-Witten invariant]],

* [[Riemann-Roch theorem]],

* [[enumerative geometry]],

* [[Bezout theorem]],

* [[virtual fundamental class]], [[Mumford class]], [[homological mirror symmetry]], [[Schubert calculus]]

## References

Introductions and surveys include

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

* [[William Fulton]], _Introduction to intersection theory in algebraic geometry_, CBMS 54, AMS, 1996, second edition

* [[Akhil Mathew]], _Intersection theory on surfaces_, 2013 ([web](http://amathew.wordpress.com/2013/01/27/intersection-theory-on-surfaces/))

* [[Günther Trautmann]], *Introduction to intersection theory* (2007) $[$[[IntersTh.pdf:file]]$]$

The perspective in [[derived algebraic geometry]] was clearly articulated in the introduction of

* {#LurieSpaces} [[Jacob Lurie]], _[[Structured Spaces]]_

See also

* [[David Mumford]], _Towards an enumerative geometry of the moduli space of curves_, Arithmetic and geometry, Vol. II, Birkhäuser Boston, Boston, MA, 1983, pp. 271–328, [MR85j:14046](http://www.ams.org/mathscinet-getitem?mr=85j:14046)

* [[Maxim Kontsevich]], _Intersection theory on the moduli spaces of curves and the matrix. Airy function_, Comm.Math.Phys.,vol.147(1992),1-23, [pdf](http://193.51.104.7/~maxim/TEXTS/intersection_theory_6.pdf); _Intersection theory on the moduli spaces of curves_, Functional.Anal.Appl.,25:2(1991) [pdf](http://193.51.104.7/~maxim/TEXTS/Intersection%20theory%20and%20Airy%20function.pdf)


intertwiner

## Idea

The term _intertwiner_ is a synonym for _[[homomorphism]] of [[representations]]/[[actions]]/[[modules]]_.

Notably for [[linear representations]] $\rho_1$, $\rho_2$ of some [[group]] or [[associative algebra]] $A$ on [[vector spaces]] $V_1, V_2$, a [[linear map]] $u : V_1 \to V_2$ is said to _intertwine_ $\rho_1$ with $\rho_2$ if for all $a \in A$ we have that

$$
  u \circ \rho_1(a) = \rho_2(a)\circ u
  \,.
$$

[[!redirects intertwiners]]

[[!redirects bimodule intertwiner]]
[[!redirects bimodule intertwiners]]


interval

> This entry is about the notion in [[order theory]]. For the related concept in [[topology]] see at _[[topological interval]]_, and for  concept in [[homotopy theory]] see at _[[interval object]]_.

***

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## In posets

### Idea

In the general context of [[posets]], an _interval_ is an [[under category]], [[over category]], or under-over category. They are closed under betweenness: if two points belong to an interval and a third point is between them, then that third point also belongs to the interval.

### Definitions

Given a [[poset]] $P$ and an element $x$ of $P$, the __upwards unbounded interval__ $[x,\infty[$ (also $[x,\infty)$, $[x,\infty[_P$, etc) is the [[subset]]
$$ {[x, \infty[} = \{ y : P \;|\; x \leq y \} ;$$
the __downwards unbounded interval__ $]{-\infty}, x]$ (also $(-\infty,x]$, $]{-\infty},x]_P$, etc) is the subset
$$ ]{-\infty}, x] = \{ y : P \;|\; y \leq x \} ;$$
and given an element $y$ of $P$, the __bounded interval__ $[x,y]$ (also $[x,y]_P$) is the subset
$$ [x,y] = \{ z : P \;|\; x \leq z \leq y \} .$$

Thinking of $P$ as a [[category]] and subsets of $P$ as [[full subcategory|subcategories]], $[x,\infty[$ is the [[coslice category]] $(x/P)$, $]{-\infty},x]$ is the [[slice category]] $(P/x)$, and $[x,y]$ is the bislice category $(y/P/x)$.

An interval with distinct [[top]] and [[bottom]] element in a [[total order]] is also called a **linear interval**. (Sometimes this is called a **strict linear interval** and just "linear interval" then refers to the situation where top and bottom may coincide.)

Besides the __closed intervals__ above, we also have the __open intervals__

*  $ {]x, \infty[} = {[x,\infty[} \setminus \{x\} = \{ y : P \;|\; x \lt y \} ,$
*  $ {]{-\infty}, x[} = {]{-\infty}, x]} \setminus \{x\} = \{ y : P \;|\; y \lt x \} ,$
*  $ {]x, y[} = [x, y] \setminus \{x, y\} = \{ z : P \;|\; x \lt z \lt y \} ,$

as well as the __half-open intervals__

*  $ {[x,y[} = [x,y] \setminus \{y\} = \{ z : P \;|\; x \leq z \lt y \} ,$
*  $ ]x,y] = [x,y] \setminus \{x\} = \{ z : P \;|\; x \lt z \leq y \} .$

These are important in analysis, and more generally whenever the [[quasiorder]] $\lt$ is at least as important as the [[partial order]] $\leq$.

The entire poset $P$ is also considered an __unbounded interval__ in itself.

### Examples

#### Intervals in the real line

Intervals of [[real numbers]] are important in [[analysis]] and [[topology]]. They may be succinctly characterized as the [[connected space|connected]] [[subspaces]] of the real line. The bounded closed intervals in the real line are the original [[compact spaces]].

The interval in the reals has a [[universal property|universal]] characterization: it [is the terminal coalgebra](http://ncatlab.org/nlab/show/terminal+coalgebra+of+an+endofunctor#UnitInterval) of the [[endofunctor]] on the category of all intervales that glues an interval end-to-end to itself.

The __unit interval__ $[0,1]$ is primary in [[homotopy theory]]; specifically in [[topological homotopy theory]] a __[[left homotopy]]__ from a [[continuous function]] $f$ to a continuous function $g$ is a continuous function  $h \colon A \times I \to B$ out of the [[product topological space]] with the [[topological interval]] $I = [0,1]$ such that $h(x,0) = f(x)$ and $h(x,1) = g(x)$. More generally this is the concept of left homotopy for an [[interval object]] in a suitable ([[model category|model]]) [[category]].

The usual [[integral]] in ordinary calculus is done over an interval in the real line, a compact interval for a 'proper' integral, or any interval for an 'improper' integral.  The theory of [[Lebesgue measure]] removes this restriction and allows integrals over any [[measurable subset]] of the real line.  Still, the Lebesgue measure on intervals (even compact intervals) generates all of the rest.

To integrate a $1$-[[differential form|form]] on the real line requires orienting an interval; the standard orientation is from $x$ to $y$ in $[x,y]$.  If $x \gt y$, then $[x,y]$ (which by the definition above would be [[empty set|empty]]) may also be interpreted as $[y,x]$ with the reverse orientation.  This also matches the traditional notation for the integral.

### Properties

#### Classifying topos

The [[classifying topos]] for linear intervals is the category [[sSet]] of [[simplicial sets]]. See the section _[For intervals](http://ncatlab.org/nlab/show/classifying+topos#ForIntervals)_ at _[[classifying topos]]_.

#### Relation to simplices
 {#RelationToSimplices}

Let $\mathbb{I}$ be the category of _finite_ linear intervals.

There is an [[equivalence of categories]]

$$
  \widehat{(-)} : \Delta^{op} \stackrel{\simeq}{\to} \mathbb{I}
$$

from the [[opposite category]] of the [[simplex category]] to $\mathbb{I}$.

Here

$$
  \widehat{[n]} \coloneqq Hom_{\Delta}([n],[1]) \simeq [n+1]
$$

and the inverse is

$$
  [n] \mapsto Hom_{\mathbb{I}}([n],[1])
  \,.
$$

See also at _[Simplex category -- Duality with intervals](simplex+category#DualityWithIntervals)_.

#### Intervals as generators of the incidence algebra

Recall that the [[incidence algebra]] $I(P)$ of a poset $P$ (relative to some commutative ring $R$) is an [[associative unital algebra]] containing all functions $f : P \times P \to R$ such that $x \nleq y$ implies $f(x,y) = 0$. For any pair of elements related by the order $x \leq y$, we can define an element $\epsilon_{x,y}$ of the incidence algebra by:
$$
\epsilon_{x,y}(u,v) = \begin{cases}1 & u = x, w = y \\ 0 & \text{otherwise}\end{cases}
$$
and the collection of such functions $\epsilon_{x,y}$ form a [[basis]] of $I(P)$ as an $R$-[[module]].
So, the [[dimension]] of the incidence algebra $I(P)$ is equal to the total number of (non-empty) intervals in $P$.

Information about the number of intervals in a finite poset is also encoded in its [[zeta polynomial]].

## In homotopy theory

In [[homotopy theory]], "cellular" models for the intervals play a central role. See [[interval object]].

## In geometry

The [[geometry]] (for instance [[differential geometry]]) of intervals, for instance in the [[real line]], are often relevant.

See for instance [Geometric spaces and their homotopy types](http://ncatlab.org/nlab/show/cohesive+homotopy+type+theory#GeometricSpacesAndTheirHomotopyTypes) at [[cohesive homotopy type theory]].

## Related concepts

* [[interval arithmetic]]

* [[interval type]], [[interval object]]

* [[abstract circle]]

* [[twisted arrow category]]

* [[t-norm]]

[[!redirects interval]]
[[!redirects intervals]]

[[!redirects unit interval]]

[[!redirects under-over category]]

[[!redirects linear interval]]
[[!redirects linear intervals]]

[[!redirects open interval]]
[[!redirects open intervals]]

[[!redirects closed interval]]
[[!redirects closed intervals]]

[[!redirects half-open interval]]
[[!redirects half-open intervals]]