Adams-Quillen theorem

#Contents#
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## Idea

The Adams-Quillen theorem describes the structure of a [[Hopf algebroid over a commutative base]] on the dual [[BP]]-[[Steenrod algebra]] $BP_\bullet(BP)$.

This is the analog for [[BP]] of what the [[Landweber-Novikov theorem]] is for [[MU]].

## Related concepts

* [[Landweber exact functor theorem]]

## References

* [[Doug Ravenel]], chapter 4, section 1 of _[[Complex cobordism and stable homotopy groups of spheres]]_, [pdf](http://www.math.rochester.edu/people/faculty/doug/mybooks/ravenelA1.pdf)


Adams–Novikov spectral sequence

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#Contents#
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## Idea

The _Adams-Novikov spectral sequence_ is the $E$-[[Adams spectral sequence]], for $E = $ [[MU]].

More in detail, the _Adams-Novikov spectral sequence_ is a class of [[spectral sequences]], which refine the _[[classical Adams spectral sequence]]_ by replacing [[ordinary cohomology]] with [[coefficients]] in [[Fp]] by [[complex cobordism cohomology theory]] [[MU]]. This makes the Adams-Novikov spectral sequence converge to the full [[homotopy groups]] of [[connective spectra]], hence in particular to the full [[stable homotopy groups of spheres]] ([Novikov 67, Theorems 3.1, 3.3](#Novikov67), reviewed as [Ravenel 78, Thm. 3.1](#Ravenel78)).

More generally for $E$ any suitable [[E-infinity ring]] there is an Adams-Novikov-type spectral sequence involving $E$-[[generalized cohomology]]/[[generalized homology]]. This fully general notion is often again just referred to as the _$E$-[[Adams spectral sequence]]_. Accordingly, see there for more. For detailed introduction, see at _[[Introduction to the Adams Spectral Sequence]]_.

## Properties

### Adams Chart
 {#AdamsChart}

[Isaksen 14](#Isaksen14):

\begin{imagefromfile}
  "file_name": "IsaksenANSSE2p.jpg"
\end{imagefromfile}

\begin{imagefromfile}
  "file_name": "IsaksenANSSEinfinityp.jpg"
\end{imagefromfile}

### Relation to Brown-Peterson spectrum

The $p$-component of the $E^2$-term of the Adams-Novikov spectral sequence for the [[sphere spectrum]], hence the one converging to the
[[stable homotopy groups of spheres]] $\pi_\ast(\mathbb{S})$ is

$$
  Ext_{BP_\ast(BP)}(BP_\ast, BP_\ast)
  \,,
$$

where $BP$ denotes the [[Brown-Peterson spectrum]] at prime $p$.

recalled e.g. as [Ravenel, theorem 1.4.2](#Ravenel)

## Related concepts

* [[stable homotopy groups of spheres]]

* [[chromatic spectral sequence]]

* [[EHP spectral sequence]]

## References

The generalization of the [[Adams spectral sequence]] from $E = $ [[HA]] to $E = $ [[MU]] is due to

* {#Novikov67} [[Sergei Novikov]], _The methods of algebraic topology from the viewpoint of cobordism theories_, Izv. Akad. Nauk. SSSR. Ser. Mat. 31 (1967), 855–951 (Russian) ([[Novikov67.pdf:file]])

While that article was still being translated to English, [[Frank Adams]] learned of and then lectured about this work in 1967 in Chicago. This lecture together with two later lectures in 1970 and in 1971 constitute the book

* {#Adams74} [[Frank Adams]],  _[[Stable homotopy and generalized homology]]_, Chicago Lectures in mathematics, 1974

Further early development of this Adams-Novikov spectral sequence:

* {#Zahler72} [[Raphael Zahler]], _The Adams-Novikov Spectral Sequence for the Spheres_, Annals of Mathematics Second Series, Vol. 96, No. 3 (Nov., 1972), pp. 480-504 ([doi:10.2307/1970821](https://doi.org/10.2307/1970821))

Review:

* {#Ravenel78} [[Douglas Ravenel]], _A Novice's guide to the Adams-Novikov spectral sequence_, in: [[Michael Barratt]], [[Mark Mahowald]] (eds.) _Geometric Applications of Homotopy Theory II_. Lecture Notes in Mathematics, vol 658, Springer 1978 ([doi:10.1007/BFb0068728](https://doi.org/10.1007/BFb0068728), [pdf](https://people.math.rochester.edu/faculty/doug/mypapers/Novice.pdf))

* {#Ravenel} [[Douglas Ravenel]], _[[Complex cobordism and stable homotopy groups of spheres]]_, page 11 of chapter I _An introduction to the homotopy groups of spheres_ ([pdf](http://www.math.rochester.edu/people/faculty/doug/mybooks/ravenel1.pdf))

* [[Doug Ravenel]], _[Complex cobordism and stable homotopy groups of spheres](http://www.math.rochester.edu/people/faculty/doug/mu.html)_, chapter IV _$B P$-Theory and the Adams-Novikov Spectral Sequence_ ([pdf](http://www.math.rochester.edu/people/faculty/doug/mybooks/ravenel4.pdf))

* {#Kochman96} [[Stanley Kochman]], sections 4.7 and 5.6 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* [[Jacob Lurie]], _Localizations and the Adams-Novikov Spectral Sequence_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture30.pdf))

The Adams chart of the ANSS for large order has been presented in

* {#Isaksen14} [[Daniel Isaksen]], _Classical and motivic Adams-Novikov charts_ ([arXiv:1408.0248](http://arxiv.org/abs/1408.0248))

[[!redirects Adams-Novikov spectral sequence]]

[[!redirects Adams-Novikov spectral sequences]]


additive (infinity,1)-category

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#### $(\infty,1)$-Category theory
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## Idea

The analog in [[(∞,1)-category]] theory of the notion of [[additive category]] in [[category theory]].

## Related concepts

* [[semiadditive (∞,1)-category]]

* [[stable (∞,1)-category]]

## References

* {#GepnerGrothNikolaus13} [[David Gepner]], [[Moritz Groth]], [[Thomas Nikolaus]], _Universality of multiplicative infinite loop space machines_, [arXiv:1305.4550](http://arxiv.org/abs/1305.4550).
[[!redirects additive (infinity,1)-categories]]

[[!redirects additive (∞,1)-category]]
[[!redirects additive (∞,1)-categories]]


additive analytic geometry

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* table of contents
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## Idea

Additive analytic geometry over the [[complex number]] is an analog of classical (called here multiplicative) [[complex analytic geometry]] obtained by replacing the convolution algebra $\ell^1(\mathbb{N},\mathbb{C})$ of converging power series of radius $1$ (discrete [[Mellin transforms]]) by the algebra $L^1(\mathbb{R}_+,\mathbb{C})$ of integrable functions (continuous Mellin transforms) or other variants.
The analog of the complex unit disc $D_m(0,1)=D(0,1)$ is the hyperbolic/additive unit disc
$$D_a(-\infty,0)=\{s,Re(s)\leq 0\}\cup \{-\infty\},$$
(if one choses the class of integrable function, we actually have $-\infty=\infty$, since the Fourier transform of an $L^1$ function vanishes at infinity; this may not be the case if we work with another class of functions/distributions).
The main interest of this new geometry is that it has a built in flow (action of the additive monoid $\mathbb{R}_+$ on itself) and its functions are locally given by continuous Mellin transforms.

One may work, if needed with other kinds of functions/distributions on $\mathbb{R}_+$, for example if one wants to study distributional trace formulas.
In any case, one needs to use [[Dirac distribution|Dirac measures/distributions]] to relate directly the above two types of analytic geometries, by the relation on coordinates given by $z=e^s$.

Higher dimensional spaces may also be considered, but the lack of a unit element in $L^1$ makes the extension of the semi and sub-analytic topologies to this geometry a bit tricky. Unitarization may be necessary.

A full complex geometry with various Grothendieck topologies seems to emerge from the consideration of the simple analogy/link with classical complex analysis given by the complex exponential map, but it is not so easy to find the right class of functions/distributions to be used here. One may simply try out with $L^1$ functions, and see where one arrives, but a regularity condition on an additive disk may also be imposed by asking that $(\log x)^n f(x)x^\delta$ is $L^1$ for every $n$, in order to obtain smoothness of functions on the boundary; using schwartz functions or overconvergent ones may also be a solution to the problem of having a nice notion of differential calculus on rational domains, that one must also solve in classical (multiplicative) analytic geometry.

Additive analytic geometry also has a non-commutative and automorphic extension
(formulated in spectral terms), that may be useful to study and formulate some problems of automorphic representation theory in a geometric context. For example, the results of Godement and Jacquet may look like a kind of pasting of two "Mellin automorphic spectra" along the usual automorphic spectrum, in this context.

## A simple example

One may paste two additive unit discs along their boundary $\mathbb{R}$,
that is the spectrum of $L^1(\mathbb{R})^+$ by $s\mapsto -s$ on $\mathbb{C}$,
to get the additive version of the projective space
$\mathbb{P}^1_a(\mathbb{C})=\{\infty\}\cup \mathbb{C}$. One may also paste two different discs along an annulus.
Remark that there is also a way to use other pastings, such as
$s\mapsto 1-s$.

## Remarks

To be able to work with $L$-functions, seen as additive functions, it seems one needs to pass to the setting of distributions of a certain kind, that may be analytic on a half space and distributional on the boundary. The study of the spectrum of distribution convolution algebras probably requires further work.

If such a function may be extended to a type of additive projective space using a functional equation, then the complement of its set $S$ of zeroes gives an open subspace $U$ of an additive projective space. In good situations, this gives an additive subspace whose sheaf of functions is equipped with the given flow, and its embedding into projective space induces an identification between the associated additive de Rham cohomology spaces in degree $0$ and $2$. The Poincaré group of this space is generated by looping around each of the given points in the set $S$, that may be infinite.

It seems that the L-function has to be automorphic so that one may study it using additive geometry, because the functional equation of theta functions is at the heart of the above construction. Indeed, even if higher dimensional complex varieties allow one to define various types of generalized multiplicative spaces (using Mellin transforms of $\delta$ distributions, for example), there is yet no clear additive versions of global arithmetic schemes, because there is no convolution operation on $L^1(\mathbb{R}_+,\mathbb{Z})$.

A possible approach to this problem of finding a higher dimensional version of the above construction is to first try to give a spectral interpretation for theta functions, so that one may study more general $L$ functions. This corresponds to decomposing the $L$-function of an arithmetic scheme with respect to the motivic weight (cohomology relative to the spectrum of $\mathbb{Z}$), and not its absolute motivic weight (which would allow to interpret, for example, the factor $s$ in the $\xi$ function).

Proving that the given spectral $\theta$ functions fulfils a functional equation is not an easy task and can't be solved directly without showing that they are decomposed in $L$-functions of cuspidal automorphic type for some $\GL_m$, and using Godement and Jacquet's generalization of Tate's thesis.

## Additive Hodge theory

One may develop an analog of [[Hodge theory]] in this context (working with real analytic additive functions in the $s$ and $\bar{s}$ coordinates, and the corresponding analog of the Laplacian; the formal theory with [[Sobolev spaces]] seems to carry through to this context as one may see on wikipedia). This will not give finite dimensionality of [[de Rham cohomology]], because of the above remark on the Poincaré group. However, such a Hodge theory may still have interesting applications related to its interaction with Poincaré duality in additive de Rham cohomology, since, in good cases, all the given construction are compatible with the given flow.


additive and abelian categories

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#### Category theory
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#### Homological algebra
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#### Additive and abelian categories
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* table of contents
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## Definitions

There is a sequence of extra [[stuff, structure, property|structure and property]] on a [[category]] that makes this category behave like a general context for [[homological algebra]]. In order of increasing structure and property this is:

1. **[[Ab-enriched category]]**:
  a [[category]] that is [[Ab]]-[[enriched category|enriched]];

2. **[[pre-additive category]]**:
  an $Ab$-enriched category that has a [[terminal object]] or [[initial object]] and therefore a [[zero object]]; Notice however that many authors (e.g. Weibel, Popescu) by preadditive (or pre-additive) simply mean $Ab$-enriched.

   * [[pseudo-abelian category]]: a pre-additive category such that every [[idempotent]] has a [[kernel]] and [[cokernel]]

3. **[[additive category]]**: a pre-additive category that admits binary [[product]]s or binary [[coproduct]]s and therefore binary [[biproduct]]s (equivalently, an $Ab$-enriched category with all finite products or coproducts);

4. **[[pre-abelian category]]**:
  an additive category that admits [[kernel]]s and  [[cokernel]]s (equivalently, an $Ab$-enriched category with all finite [[limit]]s and [[colimit]]s);

5. **[[abelian category]]**:
  a pre-abelian category such that every [[monomorphism]] is a kernel and every [[epimorphism]] is a cokernel (and many other equivalent definitions).

Pre-abelian and abelian categories are sometimes called (AB1) and (AB2) categories, after the sequence of additional axioms on top of additive categories introduced by Grothendieck in [[Tohoku]].  AB1 and AB2 are self-dual axioms (AB1 is existence of kernels and cokernels, and AB2 requires that, for any $f$, the canonical morphism $\mathrm{Coim}\,f\to \mathrm{Im}\,f$ is an isomorphism). These continue in non-selfdual manner:

* AB3: an abelian category with all coproducts (hence with all [[colimit]]s);

* AB4: an (AB3) category in which coproducts of monomorphisms are monic;

* AB5: an (AB3) category in which [[filtered colimits]] of [[exact sequence]]s exist and are exact;

* AB6: an (AB3) category such that
  * for every object $A$ in $C$ and any family $B^j$ with $j \in J$ of directed families $B^j = B^j_i$ with $i in I_j$ the intersections of [[subobject]]s  over $j$ commute with direct sums over $j$.
  * Notice that this implies that inf for any family of subobjects exists.

The concepts (AB3--AB6) also have dual forms (AB3\*--AB6\*).

There are further refinements along these lines. In particular

* [[Grothendieck category]]: an (AB5) category with a [[generator]].

## Further refinements

Various further axiom structures are considered for additive (sometimes abelian) categories.

* [[suspended category]]

* [[triangulated category]]

* [[Quillen exact category]]

* [[Grothendieck category]]:  an AB5-category with a [[generator]]

* Gabriel's [[property sup]]

## Generalizations

* Abelian groups are to general [[group]]s as [[abelian category|abelian categories]] are to [[semi-abelian category|semi-abelian categories]].

## Examples

Various generic classes of examples of additive and abelian categories are of relevance:

## Related concepts

* [[delta-functor]]
* [[category of chain complexes]]
* [[derived category]]
* [[stable (infinity,1)-category]]

[[!redirects additive and abelian categories]]
[[!redirects additive or abelian categories]]
[[!redirects additive or abelian category]]

[[!redirects AB5-category]]
[[!redirects AB5-categories]]


additive and abelian categories - contents

**[[additive and abelian categories]]**

## Context and background

* [[enriched category theory]]

* [[homological algebra]]

* [[stable homotopy theory]]

## Categories

* [[Ab-enriched category]]

* [[pre-additive category]]

  * [[pseudo-abelian category]]

* [[additive category]],

* (AB1) [[pre-abelian category]]

* (AB2) [[abelian category]]

* (AB5) [[Grothendieck category]]

## Functors

* [[additive functor]]

* left/right [[exact functor]]

## Derived categories

* [[derived category]]

* [[derived functor]]

* [[homotopy category]]

* [[triangulated category]]

[Edit this sidebar](/nlab/edit/additive+and+abelian+categories+-+contents)


additive category

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## Definition

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

An **additive category** is a [[category]] which is

1. an [[Ab-enriched category]];

   (sometimes called a [[pre-additive category]]--this means that each [[hom-set]] carries the structure of an [[abelian group]] and composition is [[bilinear map|bilinear]])

1. which admits [[finite limit|finite]] [[coproducts]]

   (and hence, by prop. \ref{ProductsAreBiproducts} below, finite [[products]] which coincide with the coproducts, hence finite [[biproducts]]).

The natural [[morphisms]] *between* additive categories are [[additive functors]].

=--

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

A [[pre-abelian category]] is an additive category which also has [[kernels]] and [[cokernels]].  Equivalently, it is an Ab-enriched category with all [[finite limits]] and finite [[colimits]].  An especially important sort of additive category is an [[abelian category]], which is a pre-abelian one satisfying the extra exactness property that all [[monomorphisms]] are [[kernels]] and all [[epimorphisms]] are [[cokernels]].  See at _[[additive and abelian categories]]_ for more.

=--

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

The [[Ab]]-enrichment of an additive category does not have to be given a priori.  Every [[semiadditive category]] (a category with finite [[biproducts]]) is automatically  [[enriched category|enriched]] over [[commutative monoids]] (as described at _[[biproduct]]_), so an additive category may be defined as a category with finite biproducts whose [[hom-object|hom-monoids]] happen to be [[groups]].  (The requirement that the hom-monoids be groups can even be stated in elementary terms without discussing enrichment at all, but to do so is not very enlightening.)  Note that the entire $Ab$-enriched structure follows automatically for [[abelian category|abelian categories]].

=--

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

Some authors use **additive category** to simply mean an Ab-enriched category, with no further assumptions. It can also be used to mean a $CMon$-enriched (commutative monoid enriched) category, with or without assumptions of products.

=--

## Properties

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

In an [[Ab-enriched category]] (or even just a $CMon$-enriched category), a [[finite product|finite]] [[product]] is also a [[coproduct]], and dually (hence a [[biproduct]]).

This statement includes the zero-ary case: any [[terminal object]] is also an [[initial object]], hence a [[zero object]] (and dually), hence every additive category has a [[zero object]].

More precisely, for $\{X_i\}_{i \in I}$ a [[finite set]] of objects in an Ab-enriched category, the unique morphism

$$
  \underset{i \in I}{\coprod} X_i \longrightarrow \underset{j \in I}{\prod} X_j
$$

whose components are identities for $i = j$ and are [[zero morphism|zero]] otherwise is an [[isomorphism]].

=--

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

Consider first the nullary (i.e., zero-ary) case. Given a [[terminal object]] $\ast$, the unique morphism $id_\ast: \ast \to \ast$ is the zero morphism $0$ in its hom-object. For any object $A$, the zero morphism $0_A: \ast \to A$ must equal any morphism $f: \ast \to A$ on account of $f = f id_\ast = f 0 = 0_A$ where the last equation is by $CMon$-enrichment. Hence $\ast$ is initial. (N.B.: this argument applies more generally to categories [[enriched category|enriched]] in [[pointed sets]], and is self-[[formal duality|dual]].)

Consider now the case of binary (co-)products. Using [[zero morphisms]], in addition to its canonical [[projection]] maps $p_i \colon X_1 \times X_2 \to X_i$, any binary [[product]] also admits "injection" maps $X_i \to X_1 \times X_2$, and dually for the [[coproduct]]:

$$
  \array{
    X_1 && && X_2
    \\
    & \searrow^{\mathrlap{(id,0)}} && {}^{\mathllap{(0,id)}}\swarrow
    \\
    {}^{\mathllap{id_{X_1}}}\downarrow && X_1 \times X_2 && \downarrow^{\mathrlap{id_{X_2}}}
    \\
    & \swarrow_{\mathrlap{p_{X_1}}} && {}_{\mathllap{p_{X_2}}}\searrow
    \\
    X_1 && && X_2
  }
  \;\;\;\;\;\;\;\;\;\;\;\;\,,\;\;\;\;\;\;\;\;\;\;\;\;
  \array{
    X_1 && && X_2
    \\
    & \searrow^{\mathrlap{i_{X_1}}} && {}^{\mathllap{i_{X_2}}}\swarrow
    \\
    {}^{\mathllap{id_{X_1}}}\downarrow
     && X_1 \sqcup X_2 && \downarrow^{\mathrlap{id_{X_2}}}
    \\
    & \swarrow_{\mathrlap{(id,0)}} && {}_{\mathllap{(0,id)}}\searrow
    \\
    X_1 && && X_2
  }
  \,.
$$

Observe some basic compatibility of the $Ab$-enrichment with the product:

First, for $(\alpha_1,\beta_1), (\alpha_2, \beta_2)\colon R \to X_1 \times X_2$ then

$$
  (\star)
  \;\;\;\;\;\;
  (\alpha_1,\beta_1) + (\alpha_2, \beta_2)
    =
  (\alpha_1+ \alpha_2 , \; \beta_1 + \beta_2)
$$

(using that the projections $p_1$ and $p_2$ are linear and by the universal property of the product).

Second, $(id,0) \circ p_1$ and $(0,id) \circ p_2$ are two projections on $X_1\times X_2$ whose sum is the identity:

$$
  (\star\star)
  \;\;\;\;\;\;
  (id, 0) \circ p_1
  +
  (0, id) \circ p_2
    =
  id_{X_1 \times X_2}
  \,.
$$

(We may check this, via the [[Yoneda lemma]] on [[generalized elements]]: for $(\alpha, \beta) \colon R \to X_1\times X_2$ any morphism, then $(id,0)\circ p_1 \circ (\alpha,\beta) = (\alpha,0)$ and $(0,id)\circ p_2\circ (\alpha,\beta) = (0,\beta)$, so the statement follows with equation $(\star)$.)

Now observe that for $f_i \;\colon\; X_i \to Q$ any two morphisms, the sum

$$
  \phi
    \;\coloneqq\;
  f_1 \circ p_1 + f_2 \circ p_2
    \;\colon\;
  X_1 \times X_2
    \longrightarrow
  Q
$$

gives a morphism of [[cocones]]

$$
  \array{
    X_1 && && X_2
    \\
    & \searrow^{\mathrlap{(id,0)}} && {}^{\mathllap{(0,id)}}\swarrow
    \\
    {}^{\mathllap{id_{X_1}}}\downarrow && X_1 \times X_2 && \downarrow^{\mathrlap{id_{X_2}}}
    \\
    &  &&
    \\
    X_1 && \downarrow^{\mathrlap{\phi}} && X_2
    \\
    & {}_{\mathllap{f_1}}\searrow && \swarrow_{\mathrlap{f_2}}
    \\
    && Q
  }
  \,.
$$

Moreover, this is unique: suppose $\phi'$ is another morphism filling this diagram, then, by using equation $(\star \star)$, we get

$$
  \begin{aligned}
    \phi
    & =
    \phi \circ id_{X_1 \times X_2}
    \\
    &=
    \phi \circ ( (id_{X_1},0) \circ p_1 + (0,id_{X_2})\circ p_2 )
    \\
    & =
    \phi \circ (id_{X_1}, 0) \circ p_1 + \phi \circ (0, id_{X_2}) \circ p_2
    \\
    & =
    f_1 \circ p_1 + f_2 \circ p_2
    \\
    & =
    \phi' \circ (id_{X_1}, 0) \circ p_1 + \phi' \circ (0, id_{X_2}) \circ p_2
    \\
    & =
    \phi' \circ ( (id_{X_1},0) \circ p_1 + (0,id_{X_2})\circ p_2 )
    \\
    &=
    \phi' \circ id_{X_1 \times X_2}
    \\
    &=
    \phi'
  \end{aligned}
  \,.
$$

This means that $X_1\times X_2$ satisfies the [[universal property]] of a [[coproduct]].

By a [[formal dual|dual]] argument, the binary coproduct $X_1 \sqcup X_2$ is seen to also satisfy the universal property of the binary product. By [[induction]], this implies the statement for all finite (co-)products.  (If a particular finite (co-)product exists but binary ones do not, one can adapt the above argument directly to that case.)

=--

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

Such products which are also coproducts as in prop. \ref{ProductsAreBiproducts} are sometimes called _[[biproducts]]_ or _[[direct sums]]_; they are [[absolute limit|absolute limits]] for [[Ab]]-[[enriched category|enrichment]].

=--

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

The coincidence of products with biproducts in prop. \ref{ProductsAreBiproducts} does _not_ extend to infinite products and coproducts.) In fact, an [[Ab-enriched category]] is [[Cauchy complete category|Cauchy complete]] just when it is additive and moreover its [[idempotents]] split.

=--

Conversely:

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

A **[[semiadditive category]]** is a [[category]] that has all [[finite products]] which, moreover, are [[biproducts]] in that they coincide with finite [[coproducts]] as in def. \ref{ProductsAreBiproducts}.

=--

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

In a [[semiadditive category]], def. \ref{SemiadditiveCategory}, the [[hom-sets]] acquire the structure of [[commutative monoids]] by defining the sum of two morphisms $f,g \;\colon\; X \longrightarrow Y$ to be

$$
  f + g
    \;\coloneqq\;
  X \overset{\Delta_X}{\to} X \times X
  \simeq
  X \oplus X
  \overset{f \oplus g}{\longrightarrow}
  Y \oplus Y
  \simeq
  Y \sqcup Y
  \overset{\nabla_Y}{\to}
  Y
  \,.
$$

With respect to this operation, [[composition]] is [[bilinear map|bilinear]].

=--

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

The [[associativity]] and commutativity of $+$ follows directly from the corresponding properties of $\oplus$. Bilinearity of composition follows from [[natural transformation|naturality]] of the [[diagonal]] $\Delta_X$ and [[codiagonal]] $\nabla_X$:

$$
  \array{
    W
      &\overset{\Delta_W}{\longrightarrow}&
    W \times W
      &\overset{\simeq}{\longrightarrow}&
    W \oplus W
    \\
    \downarrow^{\mathrlap{e}}
      &&
    \downarrow^{\mathrlap{e \times e}}
      &&
    \downarrow^{\mathrlap{e \oplus e}}
    \\
    X
      &\overset{\Delta_X}{\to}&
    X \times X
      &\simeq&
    X \oplus X
      &\overset{f \oplus g}{\longrightarrow}&
    Y \oplus Y
      &\simeq&
    Y \sqcup Y
      &\overset{\nabla_X}{\to}&
    Y
    \\
      &&
      &&
      &&
    \downarrow^{\mathrlap{h \oplus h}}
      &&
    \downarrow^{\mathrlap{h \sqcup h}}
      &&
    \downarrow^{\mathrlap{h}}
    \\
      &&
      &&
      &&
    Z \oplus Z
      &\simeq&
    Z \sqcup Z
      &\overset{\nabla_Z}{\to}&
    Z
  }
$$

=--

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

Given an additive category according to def. \ref{AdditiveCategory}, then the enrichment in [[commutative monoids]] which is induced on it via prop. \ref{ProductsAreBiproducts} and prop. \ref{SemiAdditivityInducesAbelianMonoidEnrichment} from its underlying [[semiadditive category]] structure coincides with the original enrichment.

=--

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

By the proof of prop. \ref{ProductsAreBiproducts}, the [[codiagonal]] on any object in an additive category is the sum of the two projections:

$$
  \nabla_X \;\colon\; X \oplus X \overset{p_1 + p_2}{\longrightarrow} X
  \,.
$$

Therefore (checking on [[generalized elements]], as in the proof of prop. \ref{ProductsAreBiproducts}) for all morphisms $f,g \colon X \to Y$ we have [[commuting squares]] of the form

$$
  \array{
    X &\overset{f+g}{\longrightarrow}& Y
    \\
    {}^{\mathllap{\Delta_X}}\downarrow && \uparrow^{\mathrlap{\nabla_Y =}}_{\mathrlap{p_1 + p_2}}
    \\
    X \oplus X
      &\underset{f \oplus g}{\longrightarrow}&
   Y\oplus Y
  }
  \,.
$$

=--

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

Prop. \ref{SemiaddtiveStructureUnderlyingAdditiveInducesOriginalEnrichment} says that being an [[additive category]] is an extra [[property]] on a category, not extra [[structure]]. We may ask whether a given category is additive or not, without specifying with respect to which abelian group structure on the hom-sets.

=--

## Related concepts

* [[semiadditive category]]

* [[abelian category]]

* [[additive (∞,1)-category]]

* [[triangulated category]]

## References

Textbook accounts:

* [[Masaki Kashiwara]], [[Pierre Schapira]], Section 8 of: *[[Categories and Sheaves]]*, Grundlehren der Mathematischen Wissenschaften **332**, Springer (2006) [[doi:10.1007/3-540-27950-4](https://link.springer.com/book/10.1007/3-540-27950-4), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/kashiwara2.pdf)]

Discussion in [[homological algebra]]:

* [[Charles Weibel]], *[[An Introduction to Homological Algebra]]*, Cambridge University Press (1994) [[doi:10.1017/CBO9781139644136](https://doi.org/10.1017/CBO9781139644136), [pdf](https://web.math.rochester.edu/people/faculty/doug//otherpapers/weibel-hom.pdf)]

See also:

* {#IntroductionToLinearCategories} [[William Lawvere]], *Introduction to Linear Categories and Applications*, course lecture notes (1992) [[pdf](https://github.com/mattearnshaw/lawvere/blob/192dac273e8bf352f307f87b9ec4fe8ef7dc85b9/pdfs/1992-introduction-to-linear-categories-and-applications.pdf), [[Lawvere-LinearCategories.pdf:file]]]

Discussion of [[model category]] structures on additive categories is around def. 4.3 of

* {#Beligiannis} Apostolos Beligiannis, _Homotopy theory of modules and Gorenstein rings_, Math. Scand. 89 (2001) ([pdf](http://users.uoi.gr/abeligia/mathscand.pdf))

Formalization of additive categories as [[univalent categories]] in [[univalent foundations of mathematics]] ([[homotopy type theory]]):

* [[unimath]], *Additive categories* [[UniMath.CategoryTheory.Additive](https://unimath.github.io/doc/UniMath/4dd5c17/UniMath.CategoryTheory.Additive.html)]

A characterisation of preadditive categories in terms of commutative monoids in cartesian multicategories is given in:

* [[Claudio Pisani]], _Sequential multicategories_, Theory and Applications of Categories 29.19 (2014), [arXiv:1402.0253](https://arxiv.org/abs/1402.0253)

[[!redirects additive categories]]


additive disjunction

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

#Contents#
* table of contents
{:toc}

## Idea

In the context of [[linear logic]] or [[linear type theory]], the *additive disjunction* "$\oplus$" is a form of [[disjunction]] whose [[categorical semantics]] is typically that of the [[direct sum]] (e.g. [Mihályi & Novitzká (2013), p. 9](#MihályiNovitzká13)).

## Related concepts

* [[direct sum]], [[quantum parallelism]]

[[!include logic symbols -- table]]

## References

* {#MihályiNovitzká13} Daniel Mihályi, Valerie Novitzká, *What about Linear Logic in Computer Science?*, Acta Polytechnica Hungarica **10** 4 (2013) 147-160 [[pdf](http://acta.uni-obuda.hu/Mihalyi_Novitzka_42.pdf), [[MihalyiNovitzka-LinearLogic.pdf:file]]]

[[!redirects additive disjunctions]]

[[!redirects linear disjunction]]
[[!redirects linear disjunctions]]


additive envelope

The additive envelope of an [[Ab-enriched category]] $A$ is defined by taking the objects as formal direct sums of objects in $A$, and morphisms as matrices of coefficients, giving an [[additive category]].  This is a universal construction.

For instance, a [[semisimple category]] is the additive envelope of its full subcategory consisting of simple objects.  Also, any additive category is equivalent to its own additive envelope.

By further taking the [[Karoubi envelope]] (i.e. formally adding images of idempotent elements), one constructs a [[Karoubian category]] called the pseudo-abelian envelope of $A$.  In general, the pseudo-abelian envelope of $A$ is not abelian.

The combination of additive envelope and Karoubi envelope gives the [[Cauchy completion]] in the case of [[Ab-enriched categories]].


additive functor

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Enriched category theory
+--{: .hide}
[[!include enriched category theory - contents]]
=--
#### Additive and abelian categories
+--{: .hide}
[[!include additive and abelian categories - contents]]
=--
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

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

A [[functor]] $F: \mathcal{A} \to \mathcal{B}$ between [[additive category|additive categories]] is itself called __additive__ if it preserves finite [[biproducts]].

That is,

1. $F$ maps a [[zero object]] to a zero object, $F(0) \simeq 0 \in \mathcal{B}$;

1. given any two [[objects]] $x, y \in \mathcal{A}$, there is an [[isomorphism]] $F(x \oplus y) \cong F(x) \oplus F(y)$, and this respects the inclusion and projection maps of the [[direct sum]]:

$$
\array { x &          &            &          & y \\
           & {}_{\mathllap{i_x}}\searrow &            & \swarrow_{\mathrlap{i_y}} \\
           &          & x \oplus y \\
           & {}^{\mathllap{p_x}}\swarrow &            & \searrow^{\mathrlap{p_y}} \\
         x &          &            &          & y }
\quad\quad\stackrel{F}{\mapsto}\quad\quad
\array { F(x) &          &                                      &          & F(y) \\
              & {}_{\mathllap{i_{F(x)}}}\searrow &   & \swarrow_{\mathrlap{i_{F(y)}}} \\
              &          & F(x \oplus y) \cong F(x) \oplus F(y) \\
              & {}^{\mathllap{p_{F(X)}}}\swarrow &                                      & \searrow^{\mathrlap{p_{F(y)}}} \\
         F(x) &          &                                      &          & F(y) }
$$

=--

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

In practice, functors between additive categories are generally assumed to be additive.

=--

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

Each of the following conditions is sufficient for guaranteeing that a functor $\mathcal{A} \to \mathcal{B}$ preserves biproducts (where $\mathcal{A}$ and $\mathcal{B}$ are categories with a zero object):

* The functor preserves finite products (for instance, because it's a right adjoint) and any product in $\mathcal{B}$ is a biproduct.
* The functor preserves finite coproducts (for instance, because it's a left adjoint) and any coproduct in $\mathcal{B}$ is a biproduct.
* The functor preserves finite products and coproducts.

=--

## Examples

\begin{example}
\label{HomFunctor}
The [[hom-functor]] $Hom(-,-) \colon \mathcal{A}^{op}\times \mathcal{A} \to Ab$ is additive in both arguments separately (using the nature of [[biproducts]] and that [[hom-functors preserve limits]] in each variable separately).
\end{example}

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

For $\mathcal{A} = R$[[Mod]] and $N \in \mathcal{A}$, the  functor that forms [[tensor product of modules]] $(-)\otimes N \colon \mathcal{A} \to \mathcal{A}$.

=--

In fact these examples are generic, see prop. \ref{RightExactAdditiveIsTensor} below.

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

Every [[solid abelian group]] is by definition an [[additive functor]].
=--

## Properties

### Relation to $Ab$-enriched functors

An [[additive category]] canonically carries the structure of an [[Ab-enriched category]] where the $Ab$-enrichment structure is induced from the biproducts as described at _[[biproduct]]_.

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

With respect to the canonical [[Ab-enriched category]]-structure on additive categories $\mathcal{A}$, $\mathcal{B}$, additive functors $F : \mathcal{A} \to \mathcal{B}$ are equivalently [[Ab]]-[[enriched functors]].

=--

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

An $Ab$-enriched functor preserves all finite biproducts that exist, since finite biproducts in [[Ab-enriched category|Ab-enriched categories]] are [[Cauchy complete category|Cauchy colimits]].

=--

### Characterization of right exact additive functors
 {#CharacterizationByExactness}

Let $R, R'$ be [[rings]].

The following is the _[[Eilenberg-Watts theorem]]_. See there for more.

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

If an additive functor $F : R$[[Mod]] $\to R'$[[Mod]] is a [[right exact functor]], then there exists an $R'$-$R$-[[bimodule]] $B$ and a [[natural isomorphism]]

$$
  F \simeq B \otimes_R (-)
$$

with the functor that forms the [[tensor product]] with $B$.

=--

This is ([Watts, theorem 1](#Watts)),

## Related concepts

* [[additive monad]]

* [[enriched functor]]

* In the context of [[derived functors in homological algebra]] one considers functors that are additive and in addition left/right [[exact functors]], as discussed above in _[Characterization by exactness](#CharacterizationByExactness)_.

* [[satellite]]

## References

* {#Watts} Charles Watts, _Intrinsic characterizations of some additive functors_, Proceedings of the American Mathematical Society, **11** 1 (1960) 5-8 (1959) [[jstor:2032707](http://www.jstor.org/stable/2032707)]

[[!redirects additive functors]]


additive K-theory

Term __additive K-theory__ is a synonym for [[cyclic homology]], used in early articles mainly of Russian and a bit by French school. Sometimes, additive K-theory denotes more specifially a different packing of cyclic homology, namely with appropriately shifted degrees (see the reference by Loday-Quillen at [[Loday-Quillen-Tsygan theorem]] and the article by Kuribayashi below).

__Additive K-theory__ is also a title of a historical article

* [[Boris Tsygan]], [[Boris Feigin]],  _Additive K-theory_, in K-theory, arithmetic and geometry, LNM 1289 (1987), edited by Yu. I. Manin, pp. 67--209, seminar 1984-1986 in Moscow), [MR89a:18017](http://www.ams.org/mathscinet-getitem?mr=923136)

The additive K-theory is here studied in relation to the algebraic K-theory and Hochschild homology. Like there is a K-theory spectrum, one also constructs an additive K-theory/cyclic homology spectrum.

Contents of Tsygan-Feigin

* Introduction.
* Ch. 1. Additive K-functors.
* Ch. 2. Derived functors and relative additive K-functors.
* Ch. 3. Generalized free products.
* Ch. A. Lie algebra homology.
* Ch. 5. Operations in additive K-theory.
* Ch. 6. Additive K-functors of the commutative noetherian algebras.
* Ch. 7. Characteristic classes.
* Appendix. Cyclic objects.

Related articles include

* Б. Л. Фейгин, Б. Л. Цыган, "Аддитивная K-теория и кристальные когомологии", Функц. анализ и его прил., 19:2 (1985), 52–-62, [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=faa&paperid=1358&what=fullt&option_lang=rus), [MR88e:18008](http://www.ams.org/mathscinet-getitem?mr=800920); Engl. transl. in B. L. Feĭgin, B. L. Tsygan, _Additive $K$-theory and crystalline cohomology_, Functional Analysis and Its Applications, 1985, 19:2, 124–132.

* Katsuhiko Kuribayashi, Toshihiro Yamaguchi, _On additive K-theory with the Loday-Quillen $\ast$-product_, Math. Scand. __87__, No 1 (2000) [article](http://www.mscand.dk/article/view/14296) [doi](http://dx.doi.org/10.7146/math.scand.a-14296)

[[!redirects Additive K-theory]]


additive monad

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Enriched category theory
+--{: .hide}
[[!include enriched category theory contents]]
=--
#### Additive and abelian categories
+--{: .hide}
[[!include additive and abelian categories - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[monad]] $\mathbf{G}=(G,\mu,\eta)$ on an [[additive category]] $A$ is **additive** if its underlying [[endofunctor]] $G:A\to A$ is an [[additive functor]]. One defines an **additive [[comonad]]** in the same vein.

Note that every additive category is [[Ab-enriched category|Ab-enriched]], and an additive monad is then the same as an [[Ab]]-[[enriched monad]].

## Related concepts

* [[strong monad]]

[[!redirects additive monads]]

[[!redirects additive comonad]]
[[!redirects additive comonads]]


ADE -- table

**[[ADE classification]]** and **[[McKay correspondence]]**

| [[Dynkin diagram]]/
 [[Dynkin quiver]]  | [[dihedron]],
 [[Platonic solid]] | [[classification of finite rotation groups|finite subgroups of SO(3)]] | [[classification of finite rotation groups|finite subgroups of SU(2)]] | [[simple Lie group]] |
|---------------------|---|-----|---------------|----------------------|
|  $A_{n \geq 1}$  |                   | [[cyclic group]]
 $\mathbb{Z}_{n+1}$ | [[cyclic group]]
 $\mathbb{Z}_{n+1}$ | [[special unitary group]]
 $SU(n+1)$ |
|  [[A1]]  |   | [[cyclic group of order 2]]
 $\mathbb{Z}_2$  | [[cyclic group of order 2]]
 $\mathbb{Z}_2$  | [[SU(2)]]  |
|  [[A2]]  |   | [[cyclic group of order 3]]
 $\mathbb{Z}_3$  | [[cyclic group of order 3]]
 $\mathbb{Z}_3$  | [[SU(3)]]  |
|  [[A3]]
 =
 [[D3]]  |   | [[cyclic group of order 4]]
 $\mathbb{Z}_4$  | [[cyclic group of order 4]]
 $2 D_2 \simeq \mathbb{Z}_4$  | [[SU(4)]]
 $\simeq$
 [[Spin(6)]] |
|  [[D4]]  | [[dihedron]] on
 [[bigon]] | [[Klein four-group]]
 $D_4 \simeq \mathbb{Z}_2 \times \mathbb{Z}_2$ |  [[quaternion group]]
 $2 D_4 \simeq$ [[Q8]] | [[SO(8)]], [[Spin(8)]] |
|  [[D5]]  | [[dihedron]] on
 [[triangle]] | [[dihedral group of order 6]]
 $D_6$ |  [[binary dihedral group of order 12]]
 $2 D_6$  | [[SO(10)]], [[Spin(10)]] |
|  [[D6]]  | [[dihedron]] on
 [[square]] | [[dihedral group of order 8]]
 $D_8$ |  [[binary dihedral group of order 16]]
 $2 D_{8}$  | [[SO(12)]], [[Spin(12)]] |
|  $D_{n \geq 4}$  | [[dihedron]],
 [[hosohedron]] | [[dihedral group]]
 $D_{2(n-2)}$ | [[binary dihedral group]]
 $2 D_{2(n-2)}$ | [[special orthogonal group]], [[spin group]]
 $SO(2n)$, $Spin(2n)$ |
|  $E_6$  | [[tetrahedron]] | [[tetrahedral group]]
 $T$ | [[binary tetrahedral group]]
 $2T$ | [[E6]] |
|  $E_7$  | [[cube]],
 [[octahedron]] | [[octahedral group]]
 $O$ | [[binary octahedral group]]
 $2O$ | [[E7]] |
|  $E_8$  | [[dodecahedron]],
 [[icosahedron]] | [[icosahedral group]]
 $I$ | [[binary icosahedral group]]
 $2I$ | [[E8]] |


ADE classification

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

#Contents#
* table of contents
{:toc}

## Idea

A long list of mathematical structures happens to have a classification that is in [[bijection]] with the [[simply laced Dynkin diagrams]] of types A, D and E (but excluding type B and C), for instance

* [[Platonic solids]]

* [[finite group|finite]] [[subgroups]] of the [[special orthogonal group]] $SO(3)$ and of the [[special unitary group]] $SU(2)$ (see at [[classification of finite rotation groups]]) ([Milnor 57](#Milnor57), see e.g. [Keenan 03, theorem 4](#Keenan03))

* [[simply laced Dynkin diagrams]] (their [[simple Lie groups]])

\begin{imagefromfile}
  "file_name": "ADEDynkin.jpg",
  "width": 490
\end{imagefromfile}

* connected [[quivers]] with a [[finite number]] of [[indecomposable object|indecomposable]] [[quiver representations]] over an [[algebraically closed field]]

  ([[Gabriel's theorem]])

* 7d [[spherical space forms]] with [[spin structure]] carrying $N \geq 4$ [[Killing vectors]] (see at [spherical space form -- 7d with spin structure](spherical+space+form#7DSphericalSpaceFormsWithSpinStructure))

  equivalently: [[near horizon geometries]] of smooth (i.e. non-[[orbifold]]) $\geq \tfrac{1}{2}$ [[BPS state|BPS]] [[black brane|black]] [[M2-brane]]-solutions of the [[equations of motion]] of [[11-dimensional supergravity]]

  This is due to [MFFGME 09](#MFFGME09).

[[!include 7d spherical space forms -- table]]

* [[du Val singularities]] (see [[ADE singularity]])

* [[singularities]] of [[elliptic fibrations]] (see there are _[classification of singular fibers](elliptic+fibration#ClassificationOfSingularFibers)_)

* certain 4d [[ALE spaces]] ([Kronheimer 89](#Kronheimer89))

* certain [[2d CFTs]]

* certain [[6d (2,0)-supersymmetric QFT|6d CFTs]]

* intersection diagrams of vanishing 2-cycles in [[K3]]s (e.g. [BBS 07, p.423](#BBS07))

and many more.

[[!include ADE -- table]]

The obvious question for what might be the conceptual origin of this joint classification is attributed to ([Arnold 76](#Arnold76)).

Starting with ([Douglas-Moore 96](#DouglasMoore96)) is the observation that many of these structures are naturally aspects of the description of [[string theory]] [[KK-compactification|KK-compactified]] on _[[orbifolds]] with [[ADE singularities]]_ of the form $\mathbb{C}^n \sslash \Gamma$ for $\Gamma$ a [[finite group|finite]] [[subgroup]] of $SL_2(\mathbb{C})$.

## Via $N=2 $ super Yang-Mills theory
 {#ViaSuperYangMillsTheory}

Various seemingly unrelated structures in mathematics fall into an "ADE classification". Notably [[finite group|finite]] [[subgroups]] of [[special unitary group|SU(2)]] and [[compact Lie group|compact]] [[simple Lie groups]] do. The way this works usually is that one tries to classify these structures somehow, and ends up finding that the classification is governed by the combinatorics of [[Dynkin diagrams]] (see also _[[McKay correspondence]]_).

While that does explain a bit, it seems the statement that both the [[icosahedral group]] and the Lie group [[E8]] are related to the same [[Dynkin diagram]] somehow is still more a question than an answer. Why is that so?

The first key insight is due to [Kronheimer 89](ADE+singularity#Kronheimer89a). He showed that the (resolutions of) the [[orbifold]] quotients $\mathbb{C}^2/\Gamma$ for finite subgroups $\Gamma$ of $SU(2)$ are precisely the generic form of the [[gauge group|gauge]] [[orbits]] of the [[direct product group]] of $U(n_i)$s acting in the evident way on the [[direct sum]] of $Hom(\mathbb{C}^{n_i}, \mathbb{C}^{n_j})$-s, where $i$ and $j$ range over the vertices of the [[Dynkin diagram]], and $(i,j)$ over its edges.

This becomes more illuminating when interpreted in terms of [[gauge theory]]: in a [[quiver gauge theory]] the [[gauge group]] is a [[direct product group]] of  $U(n_i)$ factors associated with vertices of a [[quiver]], and the [[particles]] which are [[charged particle|charged]] under this gauge group arrange, as a [[linear representation]], into a [[direct sum]] of $Hom(\mathbb{C}^{n_i}, \mathbb{C}^{n_j})$-s, for each edge of the quiver.

Pick one such particle, and follow it around as the gauge group transforms it. The space swept out is its gauge [[orbit]], and [Kronheimer 89](#Kronheimer89) says that if the quiver is a Dynkin diagram, then this gauge orbit looks like $\mathbb{C}^2/\Gamma$.

On the other extreme, gauge theories are of interest whose gauge group is not a big direct product, but is a [[simple Lie group]], such as [[special unitary group|SU(N)]] or [[E8]]. The mechanism that relates the two classes of examples is [[spontaneous symmetry breaking]] ("[[Higgs field|Higgsing]]"): the ground state energy of the field theory may happen to be achieved by putting the fields at any one point in a higher dimensional space of field configurations, acted on by the gauge group, and fixing any one such point "spontaneously" singles out the corresponding [[stabilizer subgroup]].

Now here is the final ingredient: it is [[N=2 D=4 super Yang-Mills theory]] ("[[Seiberg-Witten theory]]") which have a potential that is such that its [[vacua]] break a simple gauge group such as $SU(N)$ down to a Dynkin diagram [[quiver gauge theory]]. One place where this is reviewed, physics style, is in [Albertsson 03, section 2.3.4](N=2+D=4+super+Yang-Mills+theory#Albertsson03).

More precisely, these theories have two different kinds of vacua, those on the "[[Coulomb branch]]" and those on the "[[Higgs branch]]" depending on whether the scalars of the "[[vector multiplets]]" (the gauge field sector) or of the "[[hypermultiplet]]" (the matter field sector) vanish. The statement above is for the Higgs branch, but the Coulomb branch is supposed to behave "dually".

So that then finally is the relation, in the ADE classification, between the simple Lie groups and the finite subgroups of SU(2): start with an N=2 super Yang Mills theory with gauge group a simple Lie group. Let it spontaneously find its vacuum and consider the orbit space of the remaining spontaneously broken symmetry group. That is (a resolution of) the orbifold quotient of $\mathbb{C}^2$ by a discrete subgroup of $SU(2)$.

## Related concepts

* [[24]]
* [[McKay correspondence]]

## References

### General Surveys

* {#Arnold76} [[Vladimir Arnold]], _Problems in present day mathematics_, (1976) in Felix E. Browder, _Mathematical developments arising from Hilbert problems_, Proceedings of symposia in pure mathematics 28, American Mathematical Society, p. 46, _Problem VIII. The A-D-E classifications_ (V. Arnold).

A survey is in

* Michael Hazewinkel, W Hesseling, Dirk Siersma, and Ferdinand Veldkamp, _The ubiquity of Coxeter Dynkin diagrams (an introduction to the ADE problem)_, Nieuw Archief voor Wiskunde 25 (1977), 257-307. ([pdf](http://oai.cwi.nl/oai/asset/10039/10039A.pdf))

which in turn is summarized in

* {#Siegel14} Kyler Siegel, _The Ubiquity of the ADE classification in Nature_ , 2014 ([pdf](http://math.stanford.edu/~ksiegel/ADEClassifications.pdf))

See also

* Wikipedia, _[ADE classification](http://en.wikipedia.org/wiki/ADE_classification)_

Discussion of the free finite group actions on spheres goes back to

* {#Milnor57} [[John Milnor]], _Groups which act on $S^n$ without fixed points_, American Journal of Mathematics Vol. 79, No. 3 (Jul., 1957), pp. 623-630 ([JSTOR](http://www.jstor.org/stable/2372566))

Review inclues

* {#Keenan03} [[Adam Keenan]], _Which finite groups act freely on spheres?_, 2003 ([pdf](http://www.math.utah.edu/~keenan/actions.pdf))

Discussion of [[ALE spaces]] via ADE include

* {#Kronheimer89} [[Peter Kronheimer]], _The construction of ALE spaces as hyper-Kähler quotients_, J. Differential Geom. Volume 29, Number 3 (1989), 665-683. ([Euclid](https://projecteuclid.org/euclid.jdg/1214443066))

Related stuff includes...

on [[immersions]] of [[3-spheres]] into $\mathbb{R}^4$:

* Shumi Kinjo, _Immersions of 3-sphere into 4-space associated with Dynkin diagrams of types A and D_ ([arXiv:1309.6526](http://arxiv.org/abs/1309.6526))

### In string theory

The original articles explaining the appearance of ADE classification from within [[string theory]] include

* {#DouglasMoore96} [[Michael Douglas]], [[Gregory Moore]], _D-branes, Quivers, and ALE Instantons_ ([arXiv:hep-th/9603167](http://arxiv.org/abs/hep-th/9603167))

* [[Clifford Johnson]], [[Robert Myers]], _Aspects of Type IIB Theory on ALE Spaces_, Phys.Rev. D55 (1997) 6382-6393 ([arXiv:hep-th/9610140](http://arxiv.org/abs/hep-th/9610140))

* [[Michael Douglas]], [[Brian Greene]], [[David Morrison]], _Orbifold Resolution by D-Branes_, Nucl.Phys. B506:84-106,1997 ([arXiv:hep-th/9704151](http://arxiv.org/abs/hep-th/9704151))

* [[Brian Greene]], [[Calin Lazaroiu]], Mark Raugas, _D-branes on Nonabelian Threefold Quotient Singularities_, Nucl.Phys. B553 (1999) 711-749 ([arXiv:hep-th/9811201](http://arxiv.org/abs/hep-th/9811201))

* Andrea Cappelli, Jean-Bernard Zuber, _A-D-E Classification of Conformal Field Theories_ ([arXiv:0911.3242](http://arxiv.org/abs/0911.3242))

Surveys include

* MO discussion, _[ADE classification from string theory](http://mathoverflow.net/a/34680/381)_

Discussion of an ADE-classification of BPS [[Freund-Rubin compactifications]] is in

* {#MFFGME09} [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], [[Sunil Gadhia]], [[Elena Méndez-Escobar]], _Half-BPS quotients in M-theory: ADE with a twist_, JHEP 0910:038,2009 ([arXiv:0909.0163](http://arxiv.org/abs/0909.0163), [pdf slides](http://www.maths.ed.ac.uk/~jmf/CV/Seminars/YRM2010.pdf))

Specifically the ADE classfication involved in the [[6d (2,0)-supersymmetric QFT]] on the [[M5-brane]] is discussed in

* {#Witten95} [[Edward Witten]], _Some Comments On String Dynamics_ ([arXiv:hep-th/9507121](http://arxiv.org/abs/hep-th/9507121))

* {#HeckmannMorrisonVafa13} [[Jonathan Heckman]], [[David Morrison]], [[Cumrun Vafa]], _On the Classification of 6D SCFTs and Generalized ADE Orbifolds_ ([arXiv:1312.5746](http://arxiv.org/abs/1312.5746))

Discussion in the context of [[M-theory on G2-manifolds]] includes

* {#Acharya02} [[Bobby Acharya]], section 3.1.1 of _M Theory, $G_2$-manifolds and Four Dimensional Physics_,  Classical and Quantum Gravity Volume 19 Number 22, 2002  ([pdf](http://users.ictp.it/~pub_off/lectures/lns013/Acharya/Acharya_Final.pdf))

* {#BBS07} [[Katrin Becker]], [[Melanie Becker]], [[John Schwarz]], p. 423 of _String Theory and M-Theory: A Modern Introduction_, 2007

### In solid state physics
 {#ReferencesInSolidStatePhysics}

On ADE-classifications in/of [[topological phases of matter]]/[[topological order]]:

* Mayukh Nilay Khan, Jeffrey C. Y. Teo, Taylor L. Hughes, *Anyonic Symmetries and Topological Defects in Abelian Topological Phases: an application to the ADE Classification*, Phys. Rev. B **90** 235149 (2014) $[$[arXiv:1403.6478](https://arxiv.org/abs/1403.6478)$]$

[[!redirects ADE classifications]]

[[!redirects ADE-classification]]
[[!redirects ADE-classifications]]

[[!redirects A-D-E classification]]
[[!redirects A-D-E classifications]]


ADE singularity

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#### Geometry
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#Contents#
* table of contents
{:toc}

## Idea

An _ADE singularity_ is an [[orbifold]] [[fixed point]] locally of the form $\mathbb{C}^2\sslash\Gamma$ with $\Gamma \hookrightarrow SU(2)$ a [[finite subgroup of SU(2)]] given by the [[ADE classification]] (and $SU(2)$ is understood with its defining linear [[action]] on the [[complex numbers|complex]] [[vector space]] $\mathbb{C}^2$).

## Properties

### Resolution by spheres touching along a Dynkin diagram
 {#ResolutionBySpheresTouchingAlongADynkinDiagram}

The [[blow-up]] of an ADE-singularity is given by a [[union]] of [[Riemann spheres]] that touch each other such as to form the shape of the [[Dynkin diagram]] whose A-D-E label corresponds to that of the given [[finite subgroup of SU(2)]].

This statement is originally due to ([duVal 1934 I, p. 1-3 (453-455)](#duVal1934I)). A description in terms of [[hyper-Kähler geometry]] is due to [Kronheimer 89a](#Kronheimer89a).

Quick survey of this fact is in [Reid 87](#Reid87), a textbook account is [Slodowy 80](#Slodowy80).

{#InStringTheory} In [[string theory]] this fact is interpreted in terms of [[gauge enhancement]] of the [[M-theory]]-lift of coincident [[black brane|black]] [[D6-branes]] to an  [[MK6]] at an ADE-singularity ([Sen 97](enhanced+gauge+symmetry#Sen97)):

\begin{center}
\begin{imagefromfile}
  "file_name": "ADESingularity.jpg",
  "width": 760
\end{imagefromfile}
\end{center}

> graphics grabbed from [HSS18](#HSS18)

See at _[[M-theory lift of gauge enhancement on D6-branes]]_ for more.

$\,$

[[!include ADE -- table]]

### From coincident KK-monopoles

[[!include KK-monopole geometries -- table]]

### Bridgeland stability conditions

For $G_{ADE} \subset SU(2)$ a [[finite subgroup of SU(2)]],
let $\tilde X$ be the [[resolution of singularities|resolution]] of the corresponding ADE-singularity as [above](#ResolutionBySpheresTouchingAlongADynkinDiagram).

Then the [[connected component]] of the space of [[Bridgeland stability conditions]] on the bounded [[derived category]] of [[coherent sheaves]] over $\tilde X$ can be described explicitly ([Bridgeland 05](#Bridgeland05)).

Specifically for type-A singularities the space of stability conditions is in fact [[connected topological space|connected]] and [[simply-connected topological space]] ([Ishii-Ueda-Uehara 10](#IshiiUedaUehara10)).

Brief review is in [Bridgeland 09, section 6.3](#Bridgeland09).

## Related concepts

* [[singularity]]

* [[conical singularity]]

* [[quiver gauge theory]]

* [[M-theory on G2-manifolds]]

* [[D7-brane]]

## References

### General

Original articles include

* {#duVal1934I} [[Patrick du Val]], _On isolated singularities of surfaces which do not affect the conditions of adjunction. I_, Proceedings of the Cambridge Philosophical Society, 30 (4): 453–459  (1934a) ([doi:10.1017/S030500410001269X](https://doi.org/10.1017/S030500410001269X))

* {#duVal1934II} [[Patrick du Val]], _On isolated singularities of surfaces which do not affect the conditions of adjunction. II_, Proceedings of the Cambridge Philosophical Society, 30 (4): 460–465 (1934) ([doi:10.1017/S0305004100012706](https://doi.org/10.1017/S0305004100012706))

* {#duVal1934III} [[Patrick du Val]], _On isolated singularities of surfaces which do not affect the conditions of adjunction. III_, Proceedings of the Cambridge Philosophical Society, 30 (4): 483–491 (1934) ([doi:10.1017/S030500410001272X](https://doi.org/10.1017/S030500410001272X))

Textbook accounts include

* {#Durfee79} Alan H. Durfee, _Fifteen characterizations of rational double points and simple critical points_, L'Enseignement Mathématique Volume: 25 (1979) ([doi:10.5169/seals-50375](http://dx.doi.org/10.5169/seals-50375), [pdf](http://www.maths.ed.ac.uk/~v1ranick/papers/durfee15.pdf))

* {#Slodowy80} [[Peter Slodowy]], _Simple singularities and simple algebraic groups_, in Lecture Notes in Mathematics 815, Springer, Berlin, 1980.

* {#Lamotke86} [[Klaus Lamotke]], chapter IV of _Regular Solids and Isolated Singularities_, Vieweg, Braunschweig, Wiesbaden 1986.

* {#Reid87} [[Miles Reid]], _Young persons guide to canonical singularities, in [[Spencer Bloch]] (ed.), _[Algebraic geometry -- Bowdoin 1985, Part 1](https://www.ams.org/books/pspum/046.1/)_, Proc. Sympos. Pure Math. 46 Part 1, Amer. Math. Soc., Providence, RI, 1987, pp. 345-414 ([pdf](http://www.maths.ed.ac.uk/cheltsov/quotient/pdf/reid3.pdf))

  (The last formula on page 409 has a typo: there should be no $r$ in the [[denominator]].)

Discussion of [[resolution of singularities|resolution]] of ADE-singularities in terms of [[hyper-Kähler geometry]]:

* {#Kronheimer89a} [[Peter Kronheimer]], _The construction of ALE spaces as hyper-Kähler quotients_, J. Differential Geom. Volume 29, Number 3 (1989), 665-683. ([euclid:1214443066](https://projecteuclid.org/euclid.jdg/1214443066))

* {#Kronheimer89b} [[Peter Kronheimer]], _A Torelli-type theorem for gravitational instantons_, J. Differential Geom. Volume 29, Number 3 (1989), 685-697 ([euclid:1214443067](https://projecteuclid.org/euclid.jdg/1214443067))

and in terms of preprojective algebras:

* {#CBH98} William Crawley-Boevey, Martin P. Holland, _Noncommutative deformations of Kleinian singularities_, Duke Math. J. Volume 92, Number 3 (1998), 605-635 ([euclid:1077231679](https://projecteuclid.org/euclid.dmj/1077231679))

Reviews and lecture notes:

* _[Classification of singularities](http://www.mathematik.uni-kl.de/~zca/Reports_on_ca/29/paper_html/node10.html)_

* {#Burban} Igor Burban, _Du Val Singularities_ ([pdf](http://www.mi.uni-koeln.de/~burban/singul.pdf))

* [[Miles Reid]], _The Du Val Singularities $A_n$, $D_n$, $E_6$, $E_7$, $E_8$_ ([pdf](http://homepages.warwick.ac.uk/~masda/surf/more/DuVal.pdf))

* Anda Degeratu, _Crepant Resolutions of Calabi-Yau Orbifolds_, 2004 ([pdf](https://home.mathematik.uni-freiburg.de/degeratu/crepant.pdf))

* {#Siegel14} Kyler Siegel, section 6 of _The Ubiquity of the ADE classification in Nature_ , 2014 ([pdf](http://math.stanford.edu/~ksiegel/ADEClassifications.pdf))

* MathOverflow, _[Resolving ADE singularities by blowing up](http://mathoverflow.net/q/186368/381)_

On the [[Chen-Ruan cohomology|Chen-Ruan]] [[orbifold cohomology]] of ADE-singularities:

* [[Fabio Perroni]], *Orbifold Cohomology of ADE-singularities* ([arXiv:0510528](https://arxiv.org/abs/math/0510528))

* [[Fabio Perroni]], *Chen-Ruan cohomology of ADE-singularities*, International Journal of Mathematics, Vol. 18, No. 9 (2007) 1009-1059 ([arXiv:math/0605207](https://arxiv.org/abs/math/0605207), [doi:10.1142/S0129167X07004436](https://doi.org/10.1142/S0129167X07004436))

Families of examples of [[G2 manifolds|G2 orbifolds]] with ADE singularities are constructed in

* {#Reidegeld15} [[Frank Reidegeld]], _$G_2$-orbifolds from K3 surfaces with ADE-singularities_ ([arXiv:1512.05114](http://arxiv.org/abs/1512.05114))

* [[Frank Reidegeld]], _$G_2$-orbifolds with ADE-singularities_ ([pdf](https://core.ac.uk/download/pdf/159317626.pdf))

[[Riemannian geometry]] of manifolds with ADE singularities is discussed in

* Boris Botvinnik, [[Jonathan Rosenberg]], _Positive scalar curvature on manifolds with fibered singularities_ ([arXiv:1808.06007](https://arxiv.org/abs/1808.06007))

See also

* Wikipedia, _[du Val singularity](https://en.wikipedia.org/wiki/Du_Val_singularity)_

### Via Bridgeland stability

Discussion of [[Bridgeland stability conditions]] for ([[resolution of singularities|resolutions of]]) ADE singularities includes:

* {#Bridgeland05} [[Tom Bridgeland]], _Stability conditions and Kleinian singularities_, International Mathematics Research Notices 2009.21 (2009): 4142-4157 ([arXiv:0508257](https://arxiv.org/abs/math/0508257))

* {#IshiiUedaUehara10} Akira Ishii, Kazushi Ueda, Hokuto Uehara, _Stability conditions on $A_n$-singularities_, Journal of Differential Geometry 84 (2010) 87-126 ([arXiv:math/0609551](https://arxiv.org/abs/math/0609551))

and specifically over [[Dynkin quivers]]

* {#Qiu15} [[Yu Qiu]], Def. 2.1 _Stability conditions and quantum dilogarithm identities for Dynkin quivers_, Adv. Math., 269 (2015), pp 220-264 ([arXiv:1111.1010](https://arxiv.org/abs/1111.1010))

* [[Tom Bridgeland]], [[Yu Qiu]], Tom Sutherland, _Stability conditions and the $A_2$ quiver_ ([arXiv:1406.2566](https://arxiv.org/abs/1406.2566))

### In string theory
  {#ReferencesInStringTheory}

Discussion of [[ADE-singularities]] in [[string theory]] on [[orbifolds]]:

#### M-theory on ADE-orbifolds reducing to D6-branes in type II

[[M-theory lift of gauge enhancement on D6-branes]]:

* {#Sen97} [[Ashoke Sen]], _A Note on Enhanced Gauge Symmetries in M- and String Theory_, JHEP 9709:001,1997 ([arXiv:hep-th/9707123](http://arxiv.org/abs/hep-th/9707123))

* {#IbanezUranga12} [[Luis Ibáñez]], [[Angel Uranga]], Section 6.3.3 of: _[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]_, Cambridge University Press 2012

#### Heterotic M-theory on ADE-orbifolds

[[heterotic M-theory on ADE-orbifolds]]:

* {#Sen97} [[Ashoke Sen]], _A Note on Enhanced Gauge Symmetries in M- and String Theory_, JHEP 9709:001,1997 ([arXiv:hep-th/9707123](http://arxiv.org/abs/hep-th/9707123))

* {#FLO99} Michael Faux, [[Dieter Lüst]], [[Burt Ovrut]], _Intersecting Orbifold Planes and Local Anomaly Cancellation in M-Theory_, Nucl. Phys. B554: 437-483, 1999 ([arXiv:hep-th/9903028](https://arxiv.org/abs/hep-th/9903028))

* {#FLO00a} Michael Faux, [[Dieter Lüst]], [[Burt Ovrut]], _Local Anomaly Cancellation, M-Theory Orbifolds and Phase-Transitions_, Nucl. Phys. B589: 269-291, 2000 ([arXiv:hep-th/0005251](https://arxiv.org/abs/hep-th/0005251))

* {#FLO00b} Michael Faux, [[Dieter Lüst]], [[Burt Ovrut]], _An M-Theory Perspective on Heterotic K3 Orbifold Compactifications_, Int. J. Mod. Phys. A18:3273-3314, 2003 ([arXiv:hep-th/0010087](https://arxiv.org/abs/hep-th/0010087))

* {#FLO00c} Michael Faux, [[Dieter Lüst]], [[Burt Ovrut]], _Twisted Sectors and Chern-Simons Terms in M-Theory Orbifolds_, Int. J. Mod. Phys. A18: 2995-3014, 2003 ([arXiv:hep-th/0011031](https://arxiv.org/abs/hep-th/0011031))

* {#KSTY99} [[Vadim Kaplunovsky]], J. Sonnenschein, [[Stefan Theisen]], S. Yankielowicz, _On the Duality between Perturbative Heterotic Orbifolds and M-Theory on $T^4/Z_N$_, Nuclear Physics B Volume 590, Issues 1–2, 4 December 2000, Pages 123-160 Nuclear Physics B ([arXiv:hep-th/9912144](https://arxiv.org/abs/hep-th/9912144), doi:10.1016/S0550-3213(00)00460-0)

* {#GKSTY02} E. Gorbatov, [[Vadim 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))

* {#HSS18} [[John Huerta]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Equivariant homotopy and super M-branes|Real ADE-equivariant (co)homotopy and Super M-branes]]_, CMP (2019) ([arXiv:1805.05987](https://arxiv.org/abs/1805.05987), [doi:10.1007/s00220-019-03442-3](http://link.springer.com/article/10.1007/s00220-019-03442-3))

#### Heterotic string theory on ADE-orbifolds

[[heterotic string theory]] on ADE-orbifolds:

* [[Paul Aspinwall]], [[David Morrison]], _Point-like Instantons on K3 Orbifolds_,  Nucl. Phys. B503 (1997) 533-564 ([arXiv:hep-th/9705104](https://arxiv.org/abs/hep-th/9705104))

* {#Witten99} [[Edward Witten]], _Heterotic String Conformal Field Theory And A-D-E Singularities_, JHEP 0002:025, 2000 ([arXiv:hep-th/9909229](https://arxiv.org/abs/hep-th/9909229))

#### Type II-string theory on ADE-orbifolds
 {#ReferencesTypeIIStringsOnADE}

The observation that the [[worldsheet]] [[2d CFT]] correspoding to a string probing (a [[D-brane]] on) an $A_{\kappa-1}$-type singularity $\mathbb{H}/_{C_{\kappa + 2}}$ is the chiral [[WZW model]] for  the [[affine Lie algebra]] [[su(2)]] at level $\kappa - 2$ (plus some trivial summands):

* [[Hirosi Ooguri]] [[Cumrun Vafa]], p. 10-12 of: *Two-Dimensional Black Hole and Singularities of CY Manifolds*, Nucl. Phys. B **463** (1996) 55-72  ([arXiv:hep-th/9511164](https://arxiv.org/abs/hep-th/9511164), [doi:10.1016/0550-3213%2896%2900008-9](https://doi.org/10.1016/0550-3213%2896%2900008-9))

* [[Wolfgang Lerche]], [[Carsten Andrew Lütken]], [[Christoph Schweigert]], p. 4 of: *D-Branes on ALE Spaces and the ADE Classification of Conformal Field Theories*, Nucl.Phys. B **622** (2002) 269-278  ([doi:10.1016/S0550-3213%2801%2900613-7](https://doi.org/10.1016/S0550-3213%2801%2900613-7), [arXiv:hep-th/0006247](https://arxiv.org/abs/hep-th/0006247))

On how this $\widehat{\mathfrak{su}(2)}^{\kappa-2}$-CFT encodes the [[BPS states]] of [[special unitary group|$SU(\kappa)$]]-[[SYM]] on [[D3-branes]] transverse to the singularity:

* [[Wolfgang Lerche]], *On a Boundary CFT Description of Nonperturbative $\mathcal{N} = 2$ Yang-Mills Theory* ([arXiv:hep-th/0006100](https://arxiv.org/abs/hep-th/0006100))

An interpretation of this phenomenon, under the expected [[K-theory classification of D-brane charge]], as due to the (somewhat neglected) sector of [[twisted equivariant K-theory]] where the twist is by an [[inner local system]] which may appear inside an A-type singularity:

* [[Hisham Sati]], [[Urs Schreiber]], (around p. 23 of) *[[schreiber:Anyonic defect branes in TED-K-theory]]* ([arXiv:2203.11838](https://arxiv.org/abs/2203.11838))

#### Type $I'$-string theory on ADE-orbifolds

[[type I' string theory]] on ADE-orbifolds

* {#BergmanRodriguezGomez12} [[Oren Bergman]], Diego Rodriguez-Gomez, Section 3 of: _5d quivers and their $AdS_6$ duals_, JHEP07 (2012) 171 ([arxiv:1206.3503](https://arxiv.org/abs/1206.3503))

#### Type $I$-string theory on ADE-orbifolds

[[type I string theory]] on ADE-orbifolds

* [[Kenneth Intriligator]], _New String Theories in Six Dimensions via Branes at Orbifold Singularities_, Adv. Theor. Math. Phys.1:271-282, 1998 ([arXiv:hep-th/9708117](https://arxiv.org/abs/hep-th/9708117))

#### M-theory on $G_2$-orbifolds with ADE-singularities

[[M-theory on G2-manifolds]] $\,$ [with ADE-singularities](M-theory+on+G2-manifolds#EnhancedGaugeGroups):

* {#Acharya98} [[Bobby Acharya]], _M theory, Joyce Orbifolds and Super Yang-Mills_, Adv. Theor. Math. Phys. 3 (1999) 227-248 ([arXiv:hep-th/9812205](http://arxiv.org/abs/hep-th/9812205))

* {#Acharya00} [[Bobby Acharya]], _On Realising $N=1$ Super Yang-Mills in M theory_ ([arXiv:hep-th/0011089](http://arxiv.org/abs/hep-th/0011089))

* {#AcharyaSpence00} [[Bobby Acharya]], B. Spence, _Flux, Supersymmetry and M theory on 7-manifolds_ ([arXiv:hep-th/0007213](http://arxiv.org/abs/hep-th/0007213))

* {#Acharya02} [[Bobby Acharya]], _M Theory, $G_2$-manifolds and Four Dimensional Physics_,  Classical and Quantum Gravity Volume 19 Number 22, 2002  ([pdf](http://users.ictp.it/~pub_off/lectures/lns013/Acharya/Acharya_Final.pdf))

* {#AtijayMaldacenaVafa00} [[Michael Atiyah]], [[Juan Maldacena]], [[Cumrun Vafa]], _An M-theory Flop as a Large N Duality_, J. Math. Phys.42:3209-3220, 2001 ([arXiv:hep-th/0011256](https://arxiv.org/abs/hep-th/0011256))

* {#BeasleyWitten02} [[Chris Beasley]], [[Edward Witten]], *A Note on Fluxes and Superpotentials in M-theory Compactifications on Manifolds of $G_2$ Holonomy*, JHEP 0207:046 (2002) ([arXiv:hep-th/0203061](http://arxiv.org/abs/hep-th/0203061))

* {#AtiyahWitten01} [[Michael Atiyah]], [[Edward Witten]] *M-Theory dynamics on a manifold of $G_2$-holonomy*, Adv. Theor. Math. Phys. 6 (2001) ([arXiv:hep-th/0107177](http://arxiv.org/abs/hep-th/0107177))

* {#Witten01} [[Edward Witten]], _Anomaly Cancellation On Manifolds Of $G_2$ Holonomy_ ([arXiv:hep-th/0108165](http://arxiv.org/abs/hep-th/0108165))

* {#AcharyaWitten01} [[Bobby Acharya]], [[Edward Witten]], _Chiral Fermions from Manifolds of $G_2$ Holonomy_ ([arXiv:hep-th/0109152](http://arxiv.org/abs/hep-th/0109152))

#### F-theory with ADE-singularities

[[F-theory]] with ADE-singularities

* [[Paul Aspinwall]], [[David Morrison]], _Point-like Instantons on K3 Orbifolds_,  Nucl. Phys. B503 (1997) 533-564 ([arXiv:hep-th/9705104](https://arxiv.org/abs/hep-th/9705104))

See also at _[[F-branes -- table]]_

[[!redirects ADE singularities]]

[[!redirects ADE-singularity]]
[[!redirects ADE-singularities]]

[[!redirects ADE orbifold]]
[[!redirects ADE orbifolds]]

[[!redirects ADE-orbifold]]
[[!redirects ADE-orbifolds]]

[[!redirects du Val singularity]]
[[!redirects du Val singularities]]


Adeel Khan

* [website](https://www.preschema.com)

Master's thesis:

* [[Adeel Khan]], _On derived categories and noncommutative motives of varieties_, [arXiv](http://arxiv.org/abs/1401.7222).

## Related entries

* [[Chow motive]]
* [[noncommutative motive]]
* [[derived algebraic geometry]]
* [[algebraic stacks]]

[[!redirects Adeel Khan Yusufzai]]
[[!redirects Adeel Ahmad Khan]]


Adel Bilal

* [webpage](http://www.lpt.ens.fr/?page=membre&id=25&lang=fr)

## Selected writings

On the [[discrete light cone quantization]] formulation of [[M-theory]] in view of the [[BFSS matrix model]]:

* [[Adel Bilal]], *DLCQ of M-Theory as the Light-Like Limit*, Phys. Lett. B **435** (1998) 312-318 [[arXiv:hep-th/9805070](https://arxiv.org/abs/hep-th/9805070), doi:10.1016/S0370-2693(98)00811-9]

Review of the [[BFSS matrix model]]:

* [[Adel Bilal]], *M(atrix) Theory : a Pedagogical Introduction*, Fortsch. Phys. **47** (1999) 5-28 [[arXiv:hep-th/9710136](https://arxiv.org/abs/hep-th/9710136), doi:10.1002/(SICI)1521-3978(199901)47:1/3<5::AID-PROP5>3.0.CO;2-B]

On the [[Green-Schwarz mechanism]] in [[heterotic M-theory]]:

* [[Adel Bilal]], Jean-Pierre Derendinger, Roger Sauser, _M-Theory on $S^1/\mathbb{Z}_2$ : New Facts from a Careful Analysis_, Nucl. Phys. B576 (2000) 347-374 ([arXiv:hep-th/9912150](https://arxiv.org/abs/hep-th/9912150))

On the [[non-abelian DBI action]]:

* [[Eric Bergshoeff]], [[Adel Bilal]], [[Mees de Roo]], A. Sevrin, _Supersymmetric non-abelian Born-Infeld revisited_, JHEP 0107, 029 (2001) [[arXiv:hep-th/0105274](https://arxiv.org/abs/hep-th/0105274)]

category: people


adelic integral

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#Contents#
* table of contents
{:toc}

## Idea

The concept of _adelic integration_ is a concept of [[integration]] in [[arithmetic geometry]] over [[rings of adeles]].

This was introduced for [[arithmetic curves]] in ([Tate 50](#Tate50), [Iwasawa 52](#Iwasawa52), [Iwasawa 92](#Iwasawa92)) and used to establish [[analytic continuation]], [[functional equations]] and [[Euler product]]-form of [[L-functions]] and [[zeta functions]], see [below](#ZetaFunctions).  Here one speaks of _[[Iwasawa-Tate theory]]_. In ([Fesenko 08, chapter 3](#Fesenko08)) adelic integration is generalized to [[higher arithmetic geometry]]. A textbook account is in  ([Goldfeld-Hundley 11, section 2,2](#GoldfeldHundley11))).

## Examples

### Zeta functions and L-functions
 {#ZetaFunctions}

* The [completed Riemann zeta function](Riemann+zeta+function#TheCompletedZetaFunction) (see there) is naturally an adelic integral (reviewed e.g. in [Goldfeld-Hundley 11, (2.2.6)](#GoldfeldHundley11)).

For L-functions: [Wikipedia Rankin–Selberg method -- Modern adelic theory](http://en.wikipedia.org/wiki/Rankin–Selberg_method#Modern_adelic_theory)

### $p$-Adic string scattering amplitudes

In the context of [[p-adic string theory]] (see there) one considers adelic integral version of the [[Veneziano amplitude]] and of some further [[string scattering amplitudes]] for the [[open string]].

### Adelic path integral

Analogies with the [[Feynman path integral]] are pointed out in ([Fesenko 06, section 18](#Fesenko06))...

## Related entries

* [[Iwasawa-Tate theory]]

* [[Mellin transform]]

## References

The original articles are

* {#Tate50} [[John Tate]], _[[Fourier analysis in number fields, and Hecke's zeta-functions]]_, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., pp. 305–34 1950

* {#Iwasawa52} {#Iwasawa92} [[Kenkichi Iwasawa]], _A note on functions_, Proc. ICM 1950, [link](http://www.mathunion.org/ICM/ICM1950.1) [MR0044534](http://www.ams.org/mathscinet-getitem?mr=0044534); _On the rings of valuation vectors_, Ann. Math. (II) 57:2 (Mar., 1953), pp. 331-356 [jstor](http://www.jstor.org/stable/1969863); Letter to [[Jean Dieudonne]] [link](http://books.google.com/books?ei=jyALTq-_L4nkiAL6zrHXAQ)

The generalization to [[higher arithmetic geometry]] and hence to [[arithmetic zeta functions]] is due to

* {#Fesenko06} [[Ivan Fesenko]], _Measure, integration and elements of harmonic analysis on generalized loop spaces_, Proc. of
St Petersburg Math. Soc. 12(2005), 179–199; English transl. in AMS Transl. Series 2, 219 (2006), 149–164 ([pdf](http://www.maths.nott.ac.uk/personal/ibf/aoh.pdf))

* {#Fesenko08} [[Ivan Fesenko]], chaper 3 of _Adelic approch to the zeta function of arithmetic schemes in dimension two_, Moscow Math. J. 8 (2008), 273–317 ([pdf](https://www.maths.nottingham.ac.uk/personal/ibf/ada.pdf))

* {#FesenkoAnalysisI} [[Ivan Fesenko]], _Analysis on arithmetic schemes I_ ([pdf](https://www.maths.nottingham.ac.uk/personal/ibf/a1.pdf))

* {#Fesenko10} [[Ivan Fesenko]], _Analysis on arithmetic schemes II_, Journal of K-theory 5: 437–557, 2010 ([pdf](https://www.maths.nottingham.ac.uk/personal/ibf/a2.pdf))

A textbook account is in

* {#GoldfeldHundley11} [[Dorian Goldfeld]], [[Joseph Hundley]], chapter 2 of _Automorphic representations and L-functions for the general linear group_, Cambridge Studies in Advanced Mathematics 129, 2011 ([pdf](https://www.maths.nottingham.ac.uk/personal/ibf/text/gl2.pdf))

[[!redirects adelic integrals]]

[[!redirects adelic integration]]

[[!redirects idelic integral]]
[[!redirects idelic integrals]]

[[!redirects idelic integration]]


adequate equivalence relation

An [[algebraic cycle]] on a [[scheme]] $X$ of finite type over a [[field]] $k$ is a finite linear combination $\sum_{i=1}^r n_i Z_i$ of integral closed subschemes $Z_i\subset X$ with integral coefficients $n_i$.  The algebraic cycles form a group $\mathcal{Z}= \mathcal{Z}_*$ of algebraic cycles on $X$ which is graded by the dimensions of the cycles. Sometimes (for equidimensional $X$) one looks at the grading by codimension $\mathcal{Z}_*$.

Let $\mathrm{SmProj}_k$ be the category of smooth projective [[algebraic variety|varieties]] over $k$.  A rule giving an [[equivalence relation]] $\mathcal{Z}^*(X)$ for every $X$ in $\mathrm{SmProj}_k$, and which is compatible with grading is an **adequate equivalence relation** such that

1. For every pair of cycles $a,b\in \mathcal{Z}^*(X)$, there exists $a'\sim a$ such that $a'$ is transversal to $b$.

2. Consider a product $X\times Y$ in $\mathrm{SmProj}_k$, denote by $p_X,p_Y$ its projections. Consider cycles $a\in \mathcal{Z}^*(X)$ and $b\in \mathcal{Z}^*(X\times Y)$ such that $b$ and $p_X^*(a)$ intersect properly. Then $a\sim 0$ implies $p_Y_*(p_X^*(a)\cdot b) \sim 0$ where $p_X^*(a)\cdot b$ denotes the intersection product.

The [[intersection product]], which is associative but only partially defined on $\mathcal{Z}^*$, then becomes globally defined on $\mathcal{Z}^*/{\sim}$.

Typical choices are rational, algebraic and numerical adequate equivalence relations.  The rational is the finest and the numerical is the coarsest nonzero adequate equivalence relation.


adequate subcategory > history

See [[dense functor]].


adhesive category

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###Context###
#### Category theory
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[[!include category theory - contents]]
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# Adhesive categories
* table of contents
{: toc}

## Idea

An *adhesive category* is a [[category]] in which [[pushouts]] of [[monomorphisms]] exist and "behave more or less as they do in the category of [[Set|sets]]", or equivalently in any [[topos]].

## Definition

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

The following conditions on a [[category]] $C$ are equivalent.  When they are satisfied, we say that $C$ is **adhesive**.

1. $C$ has [[pullbacks]] and [[pushouts]] of [[monomorphisms]], and pushout squares of monomorphisms are also pullback squares and are stable under pullback.

1. $C$ has pullbacks, and pushouts of monomorphisms, and the latter are also [[2-colimit|(bicategorical)]] pushouts in the [[bicategory]] of [[spans]] in $C$.

1. (If $C$ is small) $C$ has pullbacks and pushouts of monomorphisms, and admits a [[fully faithful functor|full embedding]] into a [[Grothendieck topos]] preserving pullbacks and pushouts of monomorphisms.

1. $C$ has pullbacks and pushouts of monomorphisms, and in any cubical diagram:

   ![cube](https://i.imgur.com/doD9CSD.png)

   if $X\to Y$ is a monomorphism, the bottom square is a pushout, and the left and back faces are pullbacks, then the top face is a pushout if and only if the front and right face are pullbacks.  In other words, pushouts of monomorphisms are [[van Kampen colimits]].

=--

## Properties

Notice that generally [[monomorphisms]] are preserved by [[pullback]] (see [there](monomorphism#MonomorphismsArePreservedByPullback)) but in a general category they may not need to be preserved by [[pushout]]. In an adhesive category, however, they are:

+-- {: .num_prop}
###### Proposition
In an adhesive category, suppose given a [[pushout]] square
$$
  \array{
    C
    & \overset{m}{\longrightarrow}
    &
    A
    \\
    \mathllap{{}^f}\big\downarrow
    &&
    \big\downarrow\mathrlap{{}^g}
    \\
    B
    &
    \underset{n}{\longrightarrow}
    &
    D
  }
$$
such that $m$ is a [[monomorphism]].  Then:

1. $n$ is also a monomorphism.

2. The square is also a [[pullback]] square.

3. The square is also a [[distributivity pullback]] around $(g,m)$; hence in particular $n = \forall_g m$ is the [[universal quantification]].
=--

For a proof of the above proposition, see ([Lack, prop. 2.1](#Lack)) and ([Lack-Sobocinski, Lemmas 2.3 and 2.8](#LS1)).  The latter Lemma 2.8 states only that $n = \forall_g m$ (a weaker universal property since it refers only to other *monomorphisms* into $D$), but the proof applies more generally.

+-- {: .num_prop}
###### Proposition
An adhesive category with a [[strict initial object]] is automatically an [[extensive category]].
=--

We define a [[pushout complement]] of $m:C\to A$ and $g:A\to D$ to be a pair of arrows $f:C\to B$ and $n:B\to D$ such that $n f = g m$ and this [[commutative square]] is a [[pushout]].  The following proposition is crucial in [[double pushout graph rewriting]].

+-- {: .num_prop}
###### Proposition
In an adhesive category, if $m:C\to A$ is mono and $g:A\to D$ is any morphism, then if a pushout complement exists, it is unique up to unique isomorphism.
=--
+-- {: .proof}
###### Proof
We give only a sketch; details are in [(LS, Lemma 4.5)](#LS).  If $(f,n)$ and $(f',n')$ are two pushout complements, consider the two pushout squares as morphisms in the [[arrow category]] with target $g$, and take their pullback.  The resulting commutative cube can be viewed as a morphism in the category of commutative squares from the pullback square of $m$ against itself (which is again $m$, since $m$ is mono) to the pullback square of $n$ against $n'$.  Denote the vertex of the latter pullback square by $U$.  Applying the van Kampen property in two directions, we find that the maps $U\to B$ and $U\to B'$ are both pushouts of $1_C$, hence isomorphisms.  This gives an isomorphism between the pushout complements; it is unique since $n$ and $n'$ are mono (being pushouts of the mono $m$).
=--

+-- {: .num_prop}
###### Proposition
In an adhesive category, every monomorphism is [[regular monomorphism|regular]].
In particular, every adhesive category is [[balanced category|balanced]].
=--

+-- {: .proof}
###### Proof
Let $m: A \to B$ be a monomorphism.
By adhesiveness, the cokernel pair (pushout)
$$
  \array{
    A
    & \overset{m}{\longrightarrow}
    &
    B
    \\
    \mathllap{{}^m}\big\downarrow
    &&
    \big\downarrow\mathrlap{{}^u}
    \\
    B
    &
    \underset{v}{\longrightarrow}
    &
    P
    \rlap{.}
  }
$$
is a pullback.
This exhibits $m$ as an equalizer of $u$ and $v$.
=--

## Examples
 {#Examples}

\begin{example}
Any [[topos]] is adhesive ([Lack-Sobocisnki](#LSToposes)).  For [[Grothendieck toposes]] this is easy, because $Set$ is adhesive and adhesivity is a condition on colimits and finite limits, hence preserved by functor categories and left-exact localizations.  For [[elementary toposes]] it is a theorem of [Lack and Sobocinski](#LSToposes).

The fact that [[monomorphisms]] are preserved by [[pushouts]] in toposes plays a central role for [[Cisinski model structures]] such as notably the [[classical model structure on simplicial sets]], where the [[class]] of monomorphisms is identified with the class of [[cofibrations]] and as such required to be closed under pushout (in particular).
\end{example}

\begin{example}
Some [[counter-examples]]:

* The [[1-category]] [[Cat]] of [[strict categories]] and [[functors]] is _not_ adhesive.

* Neither are the categories of [[posets]], [[topological spaces]], and [[groupoids]].

\end{example}
([Lack and Sobocinski,  Counterexample 7](#LS)).

## Related concepts

Adhesiveness is an [[exactness property]], similar to being a [[regular category]], an [[exact category]], or an [[extensive category]].  In particular, it can be phrased in the language of "lex colimits".

## References

* {#LS1} [[Steve Lack]] and [[Pawel Sobocinski]], *Adhesive and quasiadhesive categories*, RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 39 (2005) no. 3, pp. 511-545, [Numdam](http://www.numdam.org/item/ITA_2005__39_3_511_0/),
[author PDF](https://www.ioc.ee/~pawel/papers/adhesivejournal.pdf)

* {#LS} [[Steve Lack]] and [[Pawel Sobocinski]], *Adhesive categories*,
In: Walukiewicz I. (eds) Foundations of Software Science and Computation Structures. FoSSaCS 2004. Lecture Notes in Computer Science, vol 2987. Springer, Berlin, Heidelberg. doi:[10.1007/978-3-540-24727-2_20](https://doi.org/10.1007/978-3-540-24727-2_20),
[author PDF](https://www.ioc.ee/~pawel/papers/adhesive.pdf)

* {#LSToposes} [[Steve Lack]] and [[Pawel Sobocinski]], *Toposes are adhesive*, In: Corradini A., Ehrig H., Montanari U., Ribeiro L., Rozenberg G. (eds) Graph Transformations. ICGT 2006. Lecture Notes in Computer Science, vol 4178. Springer, Berlin, Heidelberg. doi:[10.1007/11841883_14](https://doi.org/10.1007/11841883_14),
[author PDF](https://www.ioc.ee/~pawel/papers/toposesAdhesive.pdf)

* {#Lack} [[Steve Lack]], *An embedding theorem for adhesive categories*, Theory and Applications of Categories, Vol. 25, 2011, No. 7, pp 180-188. [journal page](http://www.tac.mta.ca/tac/volumes/25/7/25-07abs.html), arXiv:[1103.0600](https://arxiv.org/abs/1103.0600)

* {#GarnerLack12} [[Richard Garner]], [[Steve Lack]]: *On the axioms for adhesive and quasiadhesive categories*, Theory and Applications of Categories, **27**  3 (2012) 27-46 [[arXiv:1108.2934](http://arxiv.org/abs/1108.2934), [tac:27-03](http://www.tac.mta.ca/tac/volumes/27/3/27-03abs.html)]

[[!redirects adhesive categories]]


ADHM construction

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#### Differential cohomology
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[[!include differential cohomology - contents]]
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#### Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
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#Contents#
* table of contents
{:toc}

## Idea

A method for constructing [[Yang-Mills instanton]] solutions. ([ADHM](#ADHM))

## Related concepts

* [[Nahm transform]]

* [[Skyrmion]]

## References

The original articles:

* {#ADHM} [[Michael Atiyah]], [[Nigel Hitchin]], [[Vladimir Drinfeld]], [[Yuri Manin]], *Construction of instantons*, Physics Letters A **65**  3 (1978) 185-187  [doi:10.1016/0375-9601(78)90141-X]

* Norman H. Christ, Erick J. Weinberg, and Nancy K. Stanton,  *General self-dual Yang-Mills solutions*, Phys. Rev. D **18** 6 (1978) [[doi:10.1103/PhysRevD.18.2013](https://doi.org/10.1103/PhysRevD.18.2013)]

  > (emphasis on well-separated instanton solutions)

Surveys:

* [[Bert Lindenhovius]], *Instantons and the ADHM construction*, Amsterdam (2011) [[[Lindenhovius-Instantons.pdf:file]]]

* {#DoreyHollowoodKhozeMattis02} Nick Dorey, Timothy J. Hollowood, Valentin V. Khoze, Michael P. Mattis, *The Calculus of Many Instantons*, Phys. Rept. **371** (2002) 231-459 [[arXiv:hep-th/0206063](https://arxiv.org/abs/hep-th/0206063), doi:10.1016/S0370-1573(02)00301-0]

In 8 dimensions:

* Atsushi Nakamula, Shin Sasaki, Koki Takesue, *ADHM construction of (anti-)self-dual instantons in eight dimensions*, Nuclear Physics B **910** (2016) 199-224 [[arXiv:1604.01893](https://arxiv.org/abs/1604.01893), [doi:10.1016/j.nuclphysb.2016.06.026](https://doi.org/10.1016/j.nuclphysb.2016.06.026)]

For the relation to [[Skyrmions]] see there.

[[!redirects ADHM instanton]]
[[!redirects ADHM instantons]]


adiabatic

For the moment there is:

* [[adiabatic switching]]

* [[quantum adiabatic theorem]]

## References

See also

* Wikipedia, *[List of adiabatic concepts](https://en.wikipedia.org/wiki/List_of_adiabatic_concepts)*

category: disambiguation