associative unital algebra

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

#Contents#
* table of contents
{:toc}

## Idea

Similar to the way [[modules]] generalize [[abelian groups]] by adding the operation of taking non-integer multiples, an $R$-algebra can be thought of as a generalization of a ring $S$, where the operation of taking integer multiples (seen as iterated addition) has been extended to taking arbitrary multiples with coefficients in $R$. In the trivial case, a $\mathbb{Z}$-algebra is simply a ring.

## Definition

### Over ordinary rings
 {#OverOrdinaryRings}

For $R$ a [[commutative ring]], an **associative unital $R$-algebra** is equivalently:

* a [[monoid in a monoidal category|monoid]] [[internalization|internal to]] the [[category]] [[Mod|$R Mod$]] of $R$-[[modules]] equipped with the [[tensor product of modules]] $\otimes$;

* a [[pointed object|pointed]] single-[[object]] [[enriched category|category enriched over]] $(R Mod, \otimes)$;

* a pointed $R$-[[algebroid]] with a single object;

* an  $R$-[[module]] $A$ equipped with $R$-[[linear maps]] $p \colon A \otimes A \longrightarrow A$ and $i \colon R \to A$ satisfying [[associativity]] and [[unitality]];

* a [[ring]] $A$ [[under category|under]] $R$ such that the corresponding map $R \to A$ lands in the [[center]] of $A$.

If there is no danger for confusion, one often says simply 'associative algebra', or even only '[[algebra]]'.

More generally:

* a (merely) **associative algebra** need not have a [[unit]] $i \colon R \to A$; that is, it is a [[semigroup]] instead of a [[monoid]];

* an [[ring over a ring|$R$-ring]] is a [[monoid object]] [[internalization|in]] the category [[Bimod|$R BiMod$]] of $R$-[[bimodules]] equipped with, crucially, the [[tensor product of bimodules]].

Less generally, a **[[commutative algebra]]** (where [[associativity]] and [[unitality]] are usually assumed) is a [[commutative monoid in a symmetric monoidal category|commutative monoid objecy]] [[internalization|in]] [[Mod|$R Mod$]].

For a given ring the algebras form a category, [[Alg]], and the commutative algebras a subcategory, [[CommAlg]].

### Over semi-rings

Note that everywhere rings can be replaced by [[semi-rings]] in the previous paragraph. For example a commutative associative unital $\mathbb{Q}^{+}$-algebra is nothing more than a commutative semi-ring $R$ with a [[semi-ring homomorphism]] $\mathbb{Q}^{+} \rightarrow R$.

### Over monoids in a monoidal category
 {#OverMonoidsInAMonoidalCategory}

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

Given a [[monoidal category]] $(\mathcal{C}, \otimes, 1)$, then a **[[monoid in a monoidal category|monoid internal to]]** $(\mathcal{C}, \otimes, 1)$ is

1. an [[object]] $A \in \mathcal{C}$;

1. a morphism $e \;\colon\; 1 \longrightarrow A$ (called the _[[unit]]_)

1. a morphism $\mu \;\colon\; A \otimes A \longrightarrow A$ (called the _product_);

such that

1. ([[associativity]]) the following [[commuting diagram|diagram commutes]]

   $$
     \array{
       (A\otimes A) \otimes A
         &\underoverset{\simeq}{a_{A,A,A}}{\longrightarrow}&
       A \otimes (A \otimes A)
         &\overset{id \otimes \mu}{\longrightarrow}&
       A \otimes A
       \\
       {}^{\mathllap{\mu \otimes id}}\downarrow
         && &&
       \downarrow^{\mathrlap{\mu}}
       \\
       A \otimes A
         &\longrightarrow&
         &\overset{\mu}{\longrightarrow}&
       A
     }
     \,,
   $$

   where $a$ is the associator isomorphism of $\mathcal{C}$;

1. ([[unitality]]) the following [[commuting diagram|diagram commutes]]:

   $$
     \array{
       1 \otimes A
         &\overset{e \otimes id}{\longrightarrow}&
       A \otimes A
         &\overset{id \otimes e}{\longleftarrow}&
       A \otimes 1
       \\
       & {}_{\mathllap{\ell}}\searrow
       & \downarrow^{\mathrlap{\mu}} &
       & \swarrow_{\mathrlap{r}}
       \\
       && A
     }
     \,,
   $$

   where $\ell$ and $r$ are the left and right unitor isomorphisms of $\mathcal{C}$.

Moreover, if $(\mathcal{C}, \otimes , 1)$ has the structure of a [[symmetric monoidal category]]  $(\mathcal{C}, \otimes, 1, B)$ with symmetric [[braiding]] $\tau$, then a monoid $(A,\mu, e)$ as above is called a **[[commutative monoid in a symmetric monoidal category|commutative monoid in]]** $(\mathcal{C}, \otimes, 1, B)$ if in addition

* (commutativity) the following [[commuting diagram|diagram commutes]]

  $$
    \array{
      A \otimes A
        && \underoverset{\simeq}{\tau_{A,A}}{\longrightarrow} &&
      A \otimes A
      \\
      & {}_{\mathllap{\mu}}\searrow && \swarrow_{\mathrlap{\mu}}
      \\
      && A
    }
    \,.
  $$

A [[homomorphism]] of monoids $(A_1, \mu_1, e_1)\longrightarrow (A_2, \mu_2, f_2)$ is a morphism

$$
  f \;\colon\; A_1 \longrightarrow A_2
$$

in $\mathcal{C}$, such that the following two [[commuting diagram|diagrams commute]]

$$
  \array{
    A_1 \otimes A_1
      &\overset{f \otimes f}{\longrightarrow}&
    A_2 \otimes A_2
    \\
    {}^{\mathllap{\mu_1}}\downarrow && \downarrow^{\mathrlap{\mu_2}}
    \\
    A_1 &\underset{f}{\longrightarrow}& A_2
  }
$$

and

$$
  \array{
    1_{\mathcal{c}} &\overset{e_1}{\longrightarrow}& A_1
    \\
    & {}_{\mathllap{e_2}}\searrow & \downarrow^{\mathrlap{f}}
    \\
    && A_2
  }
  \,.
$$

Write $Mon(\mathcal{C}, \otimes,1)$ for the [[category of monoids]] in $\mathcal{C}$, and $CMon(\mathcal{C}, \otimes, 1)$ for its subcategory of commutative monoids.

=--

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

Given a [[monoidal category]] $(\mathcal{C}, \otimes, 1)$, and given $(A,\mu,e)$ a [[monoid in a monoidal category|monoid in]] $(\mathcal{C}, \otimes, 1)$ (def. \ref{MonoidsInMonoidalCategory}), then a **left [[module object]]** in $(\mathcal{C}, \otimes, 1)$ over $(A,\mu,e)$ is

1. an [[object]] $N \in \mathcal{C}$;

1. a [[morphism]] $\rho \;\colon\; A \otimes N \longrightarrow N$ (called the _[[action]]_);

such that

1. ([[unitality]]) the following [[commuting diagram|diagram commutes]]:

   $$
     \array{
       1 \otimes N
         &\overset{e \otimes id}{\longrightarrow}&
       A \otimes N
       \\
       & {}_{\mathllap{\ell}}\searrow
       & \downarrow^{\mathrlap{\rho}}
       \\
       && N
     }
     \,,
   $$

   where $\ell$ is the left unitor isomorphism of $\mathcal{C}$.

1. (action property) the following [[commuting diagram|diagram commutes]]

   $$
     \array{
       (A\otimes A) \otimes N
         &\underoverset{\simeq}{a_{A,A,N}}{\longrightarrow}&
       A \otimes (A \otimes N)
         &\overset{A \otimes \rho}{\longrightarrow}&
       A \otimes N
       \\
       {}^{\mathllap{\mu \otimes N}}\downarrow
         && &&
       \downarrow^{\mathrlap{\rho}}
       \\
       A \otimes N
         &\longrightarrow&
         &\overset{\rho}{\longrightarrow}&
       N
     }
     \,,
   $$

A [[homomorphism]] of left $A$-module objects

$$
  (N_1, \rho_1) \longrightarrow (N_2, \rho_2)
$$

is a morphism

$$
  f\;\colon\; N_1 \longrightarrow N_2
$$

in $\mathcal{C}$, such that the following [[commuting diagram|diagram commutes]]:

$$
  \array{
    A\otimes N_1 &\overset{A \otimes f}{\longrightarrow}& A\otimes N_2
    \\
    {}^{\mathllap{\rho_1}}\downarrow
      &&
    \downarrow^{\mathrlap{\rho_2}}
    \\
    N_1 &\underset{f}{\longrightarrow}& N_2
  }
  \,.
$$

For the resulting **[[category of modules]]** of left $A$-modules in $\mathcal{C}$ with $A$-module homomorphisms between them, we write

$$
  A Mod(\mathcal{C})
  \,.
$$

This is naturally a (pointed) [[topologically enriched category]] itself.

=--

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

Given a (pointed) [[topologically enriched category|topological]] [[symmetric monoidal category]] $(\mathcal{C}, \otimes, 1)$, given $(A,\mu,e)$ a [[commutative monoid in a symmetric monoidal category|commutative monoid in]] $(\mathcal{C}, \otimes, 1)$ (def. \ref{MonoidsInMonoidalCategory}), and given $(N_1, \rho_1)$ and $(N_2, \rho_2)$ two left $A$-[[module objects]] (def.\ref{MonoidsInMonoidalCategory}), then the **[[tensor product of modules]]** $N_1 \otimes_A N_2$ is, if it exists, the [[coequalizer]]

$$
  N_1 \otimes A \otimes N_2
  \underoverset
    {\underset{\rho_{1}\circ (\tau_{N_1,A} \otimes N_2)}{\longrightarrow}}
    {\overset{N_1 \otimes \rho_2}{\longrightarrow}}
    {\phantom{AAAA}}
  N_1 \otimes N_1
    \overset{coequ}{\longrightarrow}
  N_1 \otimes_A N_2
$$

=--

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

Given a [[symmetric monoidal category]] $(\mathcal{C}, \otimes, 1)$ (def. \ref{SymmetricMonoidalCategory}), and given $(A,\mu,e)$ a [[commutative monoid in a symmetric monoidal category|commutative monoid in]] $(\mathcal{C}, \otimes, 1)$ (def. \ref{MonoidsInMonoidalCategory}). If all [[coequalizers]] exist in $\mathcal{C}$, then the [[tensor product of modules]] $\otimes_A$ from def. \ref{TensorProductOfModulesOverCommutativeMonoidObject} makes the [[category of modules]] $A Mod(\mathcal{C})$ into a [[symmetric monoidal category]], $(A Mod, \otimes_A, A)$ with [[tensor unit]] the object $A$ itself.

=--

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

Given a [[monoidal category|monoidal]] [[category of modules]] $(A Mod , \otimes_A , A)$ as in prop. \ref{MonoidalCategoryOfModules}, then a [[monoid in a monoidal category|monoid]] $(E, \mu, e)$ in  $(A Mod , \otimes_A , A)$ (def. \ref{MonoidsInMonoidalCategory}) is called an **$A$-[[associative algebra|algebra]]**.

=--

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

Given a [[monoidal category|monoidal]] [[category of modules]] $(A Mod , \otimes_A , A)$ in a [[monoidal category]] $(\mathcal{C},\otimes, 1)$ as in prop. \ref{MonoidalCategoryOfModules}, and an $A$-algebra $(E,\mu,e)$ (def. \ref{AAlgebra}), then there is an [[equivalence of categories]]

$$
  A Alg_{comm}(\mathcal{C})
    \coloneqq
  CMon(A Mod)
   \simeq
  CMon(\mathcal{C})^{A/}
$$

between the [[category of commutative monoids]] in $A Mod$ and the [[coslice category]] of commutative monoids in $\mathcal{C}$ under $A$, hence between commutative $A$-algebras in $\mathcal{C}$ and commutative monoids $E$ in $\mathcal{C}$ that are equipped with a homomorphism of monoids $A \longrightarrow E$.

=--

(e.g. [EKMM 97, VII lemma 1.3](#EKMM97))

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

In one direction, consider a $A$-algebra $E$ with unit $e_E \;\colon\; A \longrightarrow E$ and product $\mu_{E/A} \colon E \otimes_A E \longrightarrow E$. There is the underlying product $\mu_E$

$$
  \array{
    E \otimes A \otimes E
    &
    \underoverset
      {\underset{}{\longrightarrow}}
      {\overset{}{\longrightarrow}}
      {\phantom{AAA}}
    &
    E \otimes E
     &\overset{coeq}{\longrightarrow}&
    E \otimes_A E
    \\
    && & {}_{\mathllap{\mu_E}}\searrow & \downarrow^{\mathrlap{\mu_{E/A}}}
    \\
    && && E
  }
  \,.
$$

By considering a diagram of such coequalizer diagrams with middle vertical morphism $e_E\circ e_A$, one find that this is a unit for $\mu_E$ and that $(E, \mu_E, e_E \circ e_A)$ is a commutative monoid in $(\mathcal{C}, \otimes, 1)$.

Then consider the two conditions on the unit $e_E \colon A \longrightarrow E$. First of all this is an $A$-module homomorphism, which means that

$$
  (\star)
  \;\;\;\;\;
  \;\;\;\;\;
  \array{
    A \otimes A &\overset{id \otimes e_E}{\longrightarrow}& A \otimes E
    \\
    {}^{\mathllap{\mu_A}}\downarrow && \downarrow^{\mathrlap{\rho}}
    \\
    A &\underset{e_E}{\longrightarrow}& E
  }
$$

[[commuting diagram|commutes]]. Moreover it satisfies the unit property

$$
  \array{
    A \otimes_A E
      &\overset{e_A \otimes id}{\longrightarrow}&
    E \otimes_A E
    \\
    & {}_{\mathllap{\simeq}}\searrow & \downarrow^{\mathrlap{\mu_{E/A}}}
    \\
    && E
  }
  \,.
$$

By forgetting the tensor product over $A$, the latter gives

$$
  \array{
    A \otimes E
      &\overset{e \otimes id}{\longrightarrow}&
    E \otimes E
    \\
    \downarrow && \downarrow^{\mathrlap{}}
    \\
    A \otimes_A E
      &\overset{e_E \otimes id}{\longrightarrow}&
    E \otimes_A E
    \\
    {}^{\mathllap{\simeq}}\downarrow
      &&
    \downarrow^{\mathrlap{\mu_{E/A}}}
    \\
    E &=& E
  }
  \;\;\;\;\;\;\;\;
   \simeq
  \;\;\;\;\;\;\;\;
  \array{
    A \otimes E
      &\overset{e_E \otimes id}{\longrightarrow}&
    E \otimes E
    \\
    {}^{\mathllap{\rho}}\downarrow && \downarrow^{\mathrlap{\mu_{E}}}
    \\
    E &\underset{id}{\longrightarrow}& E
  }
  \,,
$$

where the top vertical morphisms on the left the canonical coequalizers, which identifies the vertical composites on the right as shown. Hence this may be [[pasting|pasted]] to the square $(\star)$ above, to yield a [[commuting square]]

$$
  \array{
    A \otimes A
     &\overset{id\otimes e_E}{\longrightarrow}&
    A \otimes E
      &\overset{e_E \otimes id}{\longrightarrow}&
    E \otimes E
    \\
    {}^{\mathllap{\mu_A}}\downarrow
      &&
    {}^{\mathllap{\rho}}\downarrow
      &&
    \downarrow^{\mathrlap{\mu_{E}}}
    \\
    A &\underset{e_E}{\longrightarrow}& E &\underset{id}{\longrightarrow}& E
  }
  \;\;\;\;\;\;\;\;\;\;
   =
  \;\;\;\;\;\;\;\;\;\;
  \array{
    A \otimes A
     &\overset{e_E \otimes e_E}{\longrightarrow}&
    E \otimes E
    \\
    {}^{\mathllap{\mu_A}}\downarrow
      &&
    \downarrow^{\mathrlap{\mu_E}}
    \\
    A &\underset{e_E}{\longrightarrow}& E
  }
  \,.
$$

This shows that the unit $e_A$ is a homomorphism of monoids $(A,\mu_A, e_A) \longrightarrow (E, \mu_E, e_E\circ e_A)$.

Now for the converse direction, assume that $(A,\mu_A, e_A)$ and $(E, \mu_E, e'_E)$ are two commutative monoids in $(\mathcal{C}, \otimes, 1)$ with $e_E \;\colon\; A  \to E$ a monoid homomorphism. Then $E$ inherits a left $A$-[[module]] structure by

$$
  \rho
    \;\colon\;
  A \otimes E
    \overset{e_A \otimes id}{\longrightarrow}
  E \otimes E
    \overset{\mu_E}{\longrightarrow}
  E
  \,.
$$

By commutativity and associativity it follows that $\mu_E$ coequalizes the two induced morphisms $E \otimes A \otimes E \underoverset{\longrightarrow}{\longrightarrow}{\phantom{AA}} E \otimes E$. Hence the [[universal property]] of the [[coequalizer]] gives a factorization through some $\mu_{E/A}\colon E \otimes_A E \longrightarrow E$. This shows that $(E, \mu_{E/A}, e_E)$ is a commutative $A$-algebra.

Finally one checks that these two constructions are inverses to each other, up to isomorphism.

=--

## Variants

* A [[cosimplicial algebra]] is a [[cosimplicial object]] in the category of algebras.

* A [[dg-algebra]] is a [[monoid]] not in [[Vect]] but in the category of (co)[[chain complex]]es.

* A [[smooth algebra]] is an associative $\mathbb{R}$-algebra that has not only the usual binary product induced from the product $\mathbb{R}\times \mathbb{R} \to \mathbb{R}$, but has a $n$-ary product operation for every [[smooth function]] $\mathbb{R}^n \to \mathbb{R}$.

  This may be understood as a special case of an [[algebra over a Lawvere theory]], here the [[Lawvere theory]] [[CartSp]].

## Examples

* [[function algebra]]

## Properties

### Tannaka duality

[[!include structure on algebras and their module categories - table]]

## Related concepts

* [[noncommutative algebra]]

* [[nonassociative algebra]]

* [[nonunital algebra]]

* [[finitely generated algebra]], [[finitely presented algebra]]

* [[power-associative algebra]]

* [[augmented algebra]]

* [[unitisation of C*-algebras]]

* [[differential algebra]]

* [[differential graded algebra]], [[A-infinity algebra]]

## References

See most references on *[[algebra]]*.

See also:

* Wikipedia, *[Associative algebra](https://en.wikipedia.org/wiki/Associative_algebra)*

Discussion in the generality of [[brave new algebra]]:

* {#EKMM97} [[Anthony Elmendorf]], [[Igor Kriz]], [[Michael Mandell]], [[Peter May]], _[[Rings, modules and algebras in stable homotopy theory]]_, AMS Mathematical Surveys and Monographs Volume 47 (1997) ([pdf](http://www.math.uchicago.edu/~may/BOOKS/EKMM.pdf))

[[!redirects associative unital algebra]]
[[!redirects associative unital algebras]]
[[!redirects unital associative algebra]]
[[!redirects unital associative algebras]]
[[!redirects associative algebra]]
[[!redirects associative algebras]]
[[!redirects unital algebra]]
[[!redirects unital algebras]]


associative Yang-Baxter equation

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

#Contents#
* table of contents
{:toc}

## Idea

The __associative Yang-Baxter equation__ (AYBE) is an [[associativity|associative]] analogue of the [[classical Yang-Baxter equation]].

## Definition

If $A$ is a $k$-algebra then, in Leningrad notation, the AYBE for a matrix $r\in A\otimes A$ is the condition

$$
r_{1 3} r_{1 2} - r_{1 2} r_{2 3} + r_{2 3} r_{1 3} = 0
$$

If $r = \sum a_i\otimes b_i$, then the operator $P:A\to A$ given by

$$
P(x) = \sum a_i x b_i
$$

is a Rota-Baxter operator of weight $0$, i.e. $(A,P)$ is a [[Rota-Baxter algebra]] of weight $\lambda = 0$.

The skew-symmetric solutions ($r_{1 2} = - r_{2 1}$) of AYBE give rise to

* an algebra with the trace quadratic Poisson bracket

* double Poisson structures on a free associative algebra

* an anti-Frobenius associative subalgebra of a matrix algebra

## Related concepts

[[!include Yang-Baxter equations -- contents]]

## References

* V.N. Zhelyabin, _Jordan bialgebras of symmetric elements and Lie bialgebras_, Siberian Mathematical Journal __39__ (1998, 261-276, [open access pdf in Russian](http://www.mathnet.ru/links/da921ffaf25317abe8ca0e15e9c91eda/smj275.pdf))

* [[Marcelo Aguiar]], _Infinitesimal Hopf algebras_,  Contemporary Mathematics, 267 (2000) 1-29; _Pre-Poisson algebras_, Lett. Math. Phys. 54 (2000) 263-277, [doi](http://dx.doi.org/10.1023/A:1010818119040); _On the associative analog of Lie bialgebras_,  Journal of Algebra __244__ (2001, 492-532, [open access pdf](http://www.sciencedirect.com/science/article/pii/S0021869301988775/pdf?md5=322e83568195266edf53989698f3ae78&pid=1-s2.0-S0021869301988775-main.pdf)

* A. Polishchuk, _Classical  Yang-Baxter equation  and the $A_\infty$-constraint_, Adv. Math. __168__ (2002, 56-95) [open access pdf](http://ac.els-cdn.com/S000187080192047X/1-s2.0-S000187080192047X-main.pdf?_tid=39dd1e78-93f8-11e7-9f5c-00000aab0f01&acdnat=1504808179_dfd4e9b508ad159abe651d8258654b58)

* Chengming Bai, Li Guo, Xiang Ni, _$\mathcal{O}$-operators on associative algebras and associative Yang-Baxter equations_, Pacific J. Math. __256__ (2012) 257-289, [arxiv/0910.3261](http://arxiv.org/abs/0910.3261)

* Travis Schedler, _Poisson algebras and Yang-Baxter equations_, in: Advances in quantum computation (Tyler, TX, 2007) (Providence, RI) (K. Mahdavi and D. Koslover, eds.), Contemp. Math. __482__ (2009) 91--106
arXiv:[math.QA/0612493](https://arxiv.org/abs/math.QA/0612493)

* V. Sokolov, _Classification of constant solutions for associative Yang-Baxter on $gl(3)$_, [arxiv/1212.6421](http://arxiv.org/abs/1212.6421)

* [[A. Odesskii]], [[V. Rubtsov]], V. Sokolov, _Double Poisson brackets on free associative algebras_, in: Noncommutative Birational Geometry, Representations and Combinatorics, Contemp. Math. __592__, Amer. Math. Soc. (2013) 225--239 [doi](https://arxiv.org/abs/1208.2935) [arxiv/1208.2935](https://arxiv.org/abs/1208.2935)
*  [[A. Odesskii]], [[V. Rubtsov]], V. Sokolov, _Parameter-dependent associative Yang–Baxter equations and Poisson brackets_, Int. J. Geom. Meth. Mod. Phys. __11__:09, 1460036 (2014) Proc. XXII IFWGP, Univ. of Évora, Portugal, 2013 [doi](https://doi.org/10.1142/S0219887814600366)

category: algebra, physics

[[!redirects AYBE]]


associativity

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

# Contents
* table of contents
{:toc}

## Definition

In a [[category]] **associativity** is the condition that the two ways to use binary [[composition]] of [[morphisms]] to compose a sequence of three morphisms are _equal_

$$
  h \circ (g \circ f) = (h \circ g) \circ f
  \phantom{AAAA}
$$

$$
  \array{
    c_2 &\overset{ \phantom{AA}g\phantom{AA} }{\longrightarrow}& c_3
    \\
    {}^{\mathllap{f}}\Big\uparrow
      & \searrow^{\mathrlap{h \circ g}} &
    \Big\downarrow{}^{\mathrlap{h}}
    \\
    c_1 &\underset{ (h \circ g) \circ f }{\longrightarrow}& c_4
  }
  \;\;\;\;=\;\;\;\;
  \array{
    c_2 &\overset{ \phantom{AA}g\phantom{AA} }{\longrightarrow}& c_3
    \\
    {}^{\mathllap{f}}\Big\uparrow
      & {}^{\mathllap{ g \circ f }}\nearrow &
    \Big\downarrow{}^{\mathrlap{h}}
    \\
    c_1 &\underset{ h \circ (g \circ f) }{\longrightarrow}& c_4
  }
$$

If the category has a single [[object]] it is the [[delooping]] $\mathcal{B} A$ a [[monoid]] $A$, and then this condition is the _associativity_ condition on the binary operation of [[monoids]] such as [[groups]], [[rings]], [[algebras]], etc.

More generally, in [[higher category theory]], _associativity_ of composition of morphisms in an [[n-category]] means that the different ways to use binary composition for composing collections of [[k-morphisms]] form a [[contractible]] [[infinity-groupoid]]. This is a [[coherence law]].

For instance the associativity law in an [[A-infinity algebra]] is the special case of associativity in a 1-object [[A-infinity-category]].

## Examples

* See [[associahedron]].

* In a [[monoidal category]] associativity is the statement that the [[associator]] satisfies its [[coherence law]], which is true by a [[coherence theorem]].

## Related concepts

* [[unitality]]

* [[co-associativity]]

* [[commutative monoidal category]]

## References

The [[coherence law]] of associativity is stated in

* [[Hermann Grassmann]], §3 of _[[Ausdehnungslehre]]_, 1844

[[!redirects associativity]]
[[!redirects associativity law]]
[[!redirects associativity laws]]
[[!redirects associative]]
[[!redirects associative law]]
[[!redirects associative laws]]


associator

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--

* **associator**, [[unitor]], [[Jacobiator]]

* [[pentagonator]], [[hexagonator]]

***

#Contents#
* table of contents
{:toc}

## Idea

In [[algebra]], given any [[non-associative algebra]] $A$, then the trilinear map

$$
  [-,-,-] \;-\; A \otimes A \otimes A \longrightarrow A
$$

given on any elements $a,b,c \in A$ by

$$
  [a,b,c] \coloneqq (a b) c - a (b c)
$$

is called the _[[associator]]_ (in analogy with the [[commutator]] $[a,b] \coloneqq a b - b a$ ).

An algebra for which the associator vanishes is hence an _[[associative algebra]]_. If the associator is possibly non-vanishing but completely anti-symmetric (in that for any [[permutation]] $\sigma$ of three elements then $[a_{\sigma_1}, a_{\sigma_2}, a_{\sigma_3}] = (-1)^{\vert \sigma\vert} [a_1, a_2, a_3]$ for $\vert \sigma \vert$ the [[signature of a permutation|signature of the permutation]]) then $A$ is called an _[[alternative algebra]]_.

In [[category theory]] and [[higher category theory]] (for [[monoidal categories]], [[bicategories]] and their higher versions) one considers relaxing the [[equation]] that exhibits the vanishing of the associator

$$
  (a b) c = a (b c)
$$

to a [[natural equivalence]]

$$
  \alpha_{a,b,c} \;\colon\; (a b) c \overset{\simeq}{\longrightarrow}  a (b c)
  \,.
$$

By slight mismatch with the terminology in [[algebra]], it is then this equivalence which is called _the associator_.

### In Bicategories

In a [[bicategory]] the [[composition]] of [[1-morphism]]s does not satisfy [[associativity]] as an equation, but there are natural _associator_ 2-morphisms

$$
  h \circ (g \circ f) \stackrel{\simeq}{\Rightarrow} (h \circ g) \circ f
$$

that satisfy a [[coherence law]] among themselves.

If one thinks of the bicategory as obtained from a [[geometric definition of higher categories|geometrically defined 2-category]] $C$, then the composition operation of 1-morphisms is a _choice_ of  2-[[horn]]-fillers and the associator is a _choice_ of filler of the spheres $\partial \Delta[3] \to C$ formed by these.

### In monoidal categories

By the [[periodic table of higher categories]] a [[monoidal category]] is a pointed [[bicategory]] with a single object, its [[object]]s are the 1-morphisms of the bicategory. Accordingly, here the  associator is a [[natural isomorphism]]

$$ a_{x,y,z} : (x \otimes y) \otimes z \to x \otimes (y \otimes z) $$

relating the triple [[tensor products]] of these objects.

## Examples

* In a [[tensor category]] of [[representations]] the associator is called the _[[Wigner 6-j symbol]]_.

## Related concepts

* [[commutator]]

* [[coherence]]

[[!redirects associators]]


Astra Kolomatskaia

* [website](https://www.math.stonybrook.edu/cards/kolomatskaiaastra.html)

* [github](https://github.com/FrozenWinters)

* [Youtube](https://www.youtube.com/astradiol)

## Selected writings

* [[Matthew Kennedy]], [[Astra Kolomatskaia]], [[Daniel Spivak]] *An infinite quantum Ramsey theorem*, Journal of Operator Theory, Volume 84, Issue 1, Summer 2020  pp. 49-65. ([doi:10.7900/jot.2018dec11.2259](http://dx.doi.org/10.7900/jot.2018dec11.2259), [arXiv:1711.09526](https://arxiv.org/abs/1711.09526))

* [[Astra Kolomatskaia]], [[Michael Shulman]], *Displayed Type Theory and Semi-Simplicial Types* ([arXiv:2311.18781](https://arxiv.org/abs/2311.18781))

## Talks

On [[semi-simplicial types]]:

* [[Astra Kolomatskaia]], *Semi-Simplicial Types*, [[Homotopy Type Theory Electronic Seminar Talks]], 15 December 2022 ([slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Kolomaskaia-2022-12-15-HoTTEST.pdf), [video](https://www.youtube.com/watch?v=fQv2FpeFxew))

category:people


astronomy

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related entries

* [[galaxy]], [[Miky way]]

* [[cosmology]], [[observable universe]]

## References

See also

* Wikipedia, *[Astronomy](https://en.wikipedia.org/wiki/Astronomy)*


Astérisque

>Astérisque journal was created in 1973 for the SMF's first centenary. Astérisque is a top-level international journal. It publishes various monographs, renowned seminars or reports of the main international colloquiums ( available in French and English). The articles are chosen for their ability to show a research branch under a new perspective. Each volume deals with only one subject. The whole annual collection covers all the different fields of mathematics. Seven or eight volumes are published in a year. One of them is completely dedicated to Bourbaki Seminar notes.

>The editorial staff committee insists that published works should meet the highest standards as to both contents and presentation. Every manuscript is closely examined by a reporter: results, either summarised or not proved, will not be accepted. The articles are required to either contain some unusual results or to give a new viewpoint on a specific subject.

[web](http://smf4.emath.fr/en/Publications/Asterisque/Presentation/)


asymmetric relation

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

A (binary) [[relation]] $\sim$ on a set $A$ is __asymmetric__ if no two elements are related in both orders:
$$\forall (x, y: A),\; x \sim y \;\Rightarrow\; y \nsim x$$

In the language of the $2$-poset-with-duals [[Rel]] of sets and relations, a relation $R: A \to A$ is __asymmetric__ if it is disjoint from its dual:
$$R \cap R^{op} \subseteq \empty$$
Of course, this containment is in fact an equality.

An asymmetric relation is necessarily [[irreflexive relation|irreflexive]].

[[!redirects asymmetric]]
[[!redirects asymmetric relation]]
[[!redirects asymmetric relations]]


asymptotic boundary

## Related concepts

* [[boundary]], [[manifold with boundary]]

* [[asymptotic isometry]]

* [[anti de Sitter spacetime]]

* [[black brane]], [[near horizon geometry]]

* [[AdS-CFT duality]]

* [[singleton representation]]

[[!redirects asymptotic boundaries]]


asymptotic C-star-homomorphism

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Operator algebra
+-- {: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[noncommutative topology]] the standard notion of [[homomorphism]] of [[C*-algebras]] is too restrictive for some applications, related to the fact that some noncommutative $C^\ast$-algebras correspond to "locally badly behaved" noncommutative topological spaces. The notion of _asymptotic $C^\ast$-homomorphism_ is more flexible than that of plain $C^\ast$-homomorphisms and designed to correct this problem. Homotopy classes of asymptotic $C^\ast$-homomorphisms are the [[hom-sets]] in a [[category]] called _[[E-theory]]_. See there for more details.

## Definition

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

For $A,B$ two [[C*-algebras]], an **asymptotic** homomorphism between them is a $[0,\infty)$-parameterized collection of [[continuous functions]] $\{\phi_t \colon A \to B\}_{t \in [0, \infty)}$, such that

* for each $a \in A$, the [[function]] $t \mapsto \phi_t(a)$ is a [[continuous function]];

* in the [[limit of a sequence|limit]] $t \to \infty$, $\phi_t$ becomes a [[star-algebra]] [[homomorphism]].

=--

As for ordinary $C^\ast$-algebra homomorphisms one puts:

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

For $f_t, g_t \colon A \to B $ to asymptotic $C^\ast$-homomorphisms, def. \ref{AsymptoticHomomorphism}, a (right) [[homotopy]] between them is an asyptotic homomorphism $\eta_t \colon A \to C([0,1],B)$ which restricts to $f$ at 0 and to $g$ at $1$, hence such that it fits into a [[commuting diagram]] of the form

$$
  \array{
     && B
     \\
     & {}^{\mathllap{f_t}}\nearrow & \uparrow^{\mathrlap{ev_0}}
     \\
    A &\stackrel{\eta_t}{\to}& C([0,1], B)
    \\
    & {}_{\mathllap{g_t}}\searrow & \downarrow^{\mathrlap{ev_1}}
    \\
    && B
  }
  \,.
$$

Homotopy of asymptotic $C^\ast$-homomorphisms is clearly an [[equivalence relation]]. Write $[A,B]$ for the [[set]] of homotopy-[[equivalence classes]] of asymptotic homomorphisms $A \to B$.

=--

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

For $A,B \in $ [[C*Alg]], the set $[A, C_0((0,1), B)]$ is naturally an [[abelian group]] under the composition operation which sends the homotopy classes presented by $f,g \colon A \times (0,1) \to B$ to the homotopy class of

$$
  f + g \;\colon\; A \times (0,1) \stackrel{\cdot 2}{\to} A \times (0,2) \stackrel{x \mapsto \left\{  \array{f(x) & x \lt 1 \\ g(x-1) & x \gt 1 } \right.  }{\to} B
  \,.
$$

=--

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

The $t$-wise composition of two asymptotic $C^\ast$-homomorphisms is not in general itself an asymptotic $C^\ast$-homomorphims. However, every asympotic homomorphism is homotopic to one which is an [[equicontinuous function]], and $t$-wise composition of equicontinuous asymptotic $C^\ast$-homomorphisms is again an asymptotic homomorphism.

=--

## Examples

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

Two asymptotic $C^\ast$-homomorphisms which differe just by a reparameterization of $[0,\infty)$ while having the same [[limit of a sequence|limit]] can be related by a homotopy, def. \ref{AsymptoticHomotopy}.

=--

+-- {: .num_example }
###### Example
**(self-adjointification homotopy)**

For $f_t \colon A \to B $ an asymptotic $C^\ast$-homomorphism, there is a homotopy to the asymptotic morphism

$$
  \tilde f_t(a) \coloneqq \tfrac{1}{2}\left(f(a) + f(a^\ast)^\ast\right)
  \,.
$$

=--

## References

The notion was introduced in

* [[Alain Connes]], [[Nigel Higson]], _Déformations, morphismes asymptotiques et $K$-théorie bivariante_, C. R. Acad. Sci. Paris Sér. I Math. __311__ (1990), no. 2, 101–106, [MR91m:46114](http://www.ams.org/mathscinet-getitem?mr=1065438), [pdf](ftp://ftp.bnf.fr/578/N5781521_PDF_107_112DM.pdf)
 {#ConnesHigson90}

A review is for instance around p. 23 of

* _Introduction to KK-theory and E-theory_, Lecture  notes (Lisbon 2009) ([pdf slides](http://oaa.ist.utl.pt/files/cursos/courseD_Lecture4_KK_and_E1.pdf))
 {#Introduction}

[[!redirects asymptotic C-star-homomorphisms]]

[[!redirects asymptotic C*-homomorphism]]
[[!redirects asymptotic C*-homomorphisms]]


asymptotic dimension

Related $n$Lab entries: [[dimension]], [[coarse topology]]

* [[Mikhail Gromov]], _Hyperbolic groups_, Essays in group theory, Math. Sci. Res. Inst. Publ. __8__, Springer, New York, 1987, pp. 75–263. MR MR919829 (89e:20070) 61; _Asymptotic invariants of infinite groups_, in: Geometric group theory, vol. 2 (Sussex, 1991), London Math. Soc. Lecture Note Ser. __182__, Cambridge Univ. Press 1993, pp. 1–295. MR1253544 (95m:20041)

* G. Bell, A. Dranishnikov, _Asymptotic dimension_, [math.GR/0703766](http://arxiv.org/abs/math/0703766)

> The asymptotic dimension theory was founded by Gromov in the early 90s. In this paper we give a survey of its recent history where we emphasize two of its features: an analogy with the dimension theory of compact metric spaces and applications to the theory of discrete groups.

* B. Grave, _Asymptotic dimension of coarse spaces_, New York J. Math. 12 (2006) 249–256

category: topology


asymptotic expansion

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

#Contents#
* table of contents
{:toc}

## Idea

### General

An _asymptotic expansion_ of a [[function]] is a [[formal power series]] that may not [[convergence|converge]], but whose terms decrease fast enough such that the truncation of the series at any finite order still provides a controlled approximation to a given [[function]].

A key class of examples of asymptotic expansions are the [[Taylor series]] of [[smooth functions]] (example \ref{TaylorSeriesOfSmoothFunctionIsAsymptoticSeries} below) around any point. Beware that by [[Borel's theorem]] this means that _every_ [[formal power series]] is the asymptotic expansion of _some_ smooth function and of more than one smooth function (remark \ref{TheoremBorel} below).

In [[resurgence theory]] one tries to re-identify from an asymptotic expansion the corresponding [[analytic function|non-analytic]] contributions.

### In perturbative quantum field theory

The concept of asymptotic expansions plays a key role in the interpretation of [[perturbative quantum field theory]] (pQFT): This computes [[quantum observables]] as [[formal power series]] (in the [[coupling constant]] and in [[Planck's constant]]) whose [[radius of convergence]] necessarily vanishes in cases of interest ([Dyson 52](#Dyson52)).

Nevertheless, for examples such as [[quantum electrodynamics]] and [[quantum chromodynamics]] as in the [[standard model of particle physics]], the truncation of these series to the first handful of [[loop orders]]  happens to agree with [[experiment]] (such as at the [[LHC]] collider) to high precision (for [[QED]]) or at least good precision (for [[QCD]]). Therefore one interprets the [[scattering matrix]] in [[perturbative quantum field theory]] as an asymptotic expansion of what should be the true [[non-perturbative field theory|non-perturbative]] result.

With [[resurgence theory]] one may try to deduce from the [[Feynman perturbation series]] regarded as an asymptotic expansion the hidden [[non-perturbative effects]].

## Definition

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

Given a [[function]] $f \colon \mathbb{R} \to \mathbb{R}$, a [[formal power series]] $\sum_{n = 0}^\infty a_n x^n$ is an
**asymptotic expansion** of $f$ at $x = 0$ if for each $n \in \mathbb{N}$ the [[limit of a sequence|limit]] of the difference between $f$ and the [[sum]] of the first $n$ terms of the series divided by $x^n$ is zero as $x$ tends to 0:

$$
  \underset{x \to 0}{\lim}
  \left(
    \frac{1}{x^n}
    \left(
      f(x)  - \sum_{k = 0}^n a_k x^k
    \right)
  \right)
  \;=\;
  0
  \,.
$$

=--

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

This definition makes no statement about the behaviour as $n \to \infty$. In particular an asymptotic expansion may have vanishing [[radius of convergence]] (and nevertheless provide useful approximate information).

=--

## Examples
 {#Examples}

+-- {: .num_example #TaylorSeriesOfSmoothFunctionIsAsymptoticSeries}
###### Example
**([[Taylor series]] of [[smooth function]] is asymptotic series)**

The [[Taylor series]] of a [[smooth function]] $f \colon \mathbb{R} \to \mathbb{R}$ at any point is always an asymptotic expansion of $f$ around that point, regardless of whether its [[radius of convergence]] vanishes or not.

=--

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

This follows from the [[Hadamard lemma]], which says that for each $n \in \mathbb{N}$ and each expansion point $x_0 \in \mathbb{R}$ (which we may without restrict of generality assume to be $x_0 = 0$) there exists a smooth function $h_n \colon \mathbb{R} \to \mathbb{R}$ such that

$$
  f(x) = f(0) + x f^{(0)}(0) + \frac{1}{2} x^2 f^{(2)}(0)
  + \cdots +
  \frac{1}{n!} x^n f^{n}(0)
  +
 \frac{1}{(n+1)!} x^{n+1} h_n(x)
  \,,
$$

where $f^{(k)} \colon \mathbb{R} \to \mathbb{R}$ denotes the $k$th [[derivative]] of $f$.

Therefore with

$$
  (a_k)_{k \in \mathbb{N}}
  \coloneqq
  \left(
    \frac{1}{k!} f^{(k)}(0)
  \right)_{k \in \mathbb{N}}
$$

the [[coefficients]] of the [[Taylor series]] of $f$ at $x_0 = 0$, we have

$$
  \begin{aligned}
    \underset{x \to 0}{\lim}
    \left(
      \frac{1}{x^n}
      \left(
        f(x)  - \sum_{k = 0}^n  a_k x^k
      \right)
    \right)
    & =
    \underset{x \to 0}{\lim}
    \left(
      \frac{1}{x^n}
      \left(
        \frac{1}{(n+1)!} x^{n+1} h_n(x)
      \right)
    \right)
    \\
    & =
    \underset{x \to 0}{\lim}
    \left(
      x \frac{1}{(n+1)!} h_n(x)
    \right)
    \\
    & =
    0 \cdot \frac{1}{(n+1)!} h_n(0)
    \\
    & = 0
  \end{aligned}
  \,.
$$

Here in taking the [[limit of a sequence|limit]] we used from [[Hadamard's lemma]] that $h_n(x)$ and hence also $x h_n(x)$ is a [[smooth function]], hence in particular a [[continuous function]], on all of $\mathbb{R}$, hence that its limit as $x \to 0$ is just the value of the function at $x = 0$.

=--

+-- {: .num_remark #TheoremBorel}
###### Remark
**([[Borel's theorem]])**

Beware that by [[Borel's theorem]], _every_ [[formal power series]] is the [[Taylor series]] of _some_ [[smooth function]], and of more than one smooth function; hence by example \ref{TaylorSeriesOfSmoothFunctionIsAsymptoticSeries} every formal power series is the asymptotic expansion of _some_ smooth function, and of more than one smooth function.

=--

## Properties

### Optimal truncation and superasymptotics
 {#OptimalTruncation}

A rule-of-thumb for where to truncate an asymptotic series so that the resulting finite [[sum]] is as close as possible to the "actual" value is to truncate at the term that gives the _smallest_ contribution. This rule-of-thumb is called _optimal truncation_, or _superasymptotics_ ([Berry-Howls 90](#BerryHowls90), see [Berry 91](#Berry91)).

For some classes of asymptotic series there are [[proofs]] that "optimal truncation" indeed works, see the references [below](#ReferencesOptimalTruncation).

## History

From [Suslov 05](#Suslov05):

> Classical books on diagrammatic techniques $[$in [[perturbative quantum field theory]]$]$ describe the construction of [[Feynman diagram|diagram]] series as if they were well defined. However, almost all important [[Feynman perturbation series|perturbation series]] are hopelessly [[divergent series|divergent]] since they have zero [[radii of convergence]]. The first argument to this effect was [given by Dyson](#Dyson52) with regard to [[quantum electrodynamics]].

> $[$...$]$

> Even though Dyson's argument is unquestionable, it was hushed up or decried for many years: the scientific community was not ready to face the problem of the hopeless divergency of perturbation series.

> $[$...$]$

> The modern status of divergent series suggests that techniques for manipulating them should be included in a minimum syllabus for graduate students in theoretical physics. However, the theory of divergent series is almost unknown to physicists, because the corresponding parts of standard university courses in calculus date back to the mid-nineteenth century, when divergent series were virtually banished from mathematics.

## Related concepts

* [[series]], [[divergent series]]

* [[Taylor series]], [[Puiseux series]]

* [[asymptotic series]], [[transseries]]

* [[resummation]], [[regular summation method]], [[Borel summability]]

* [[perturbation series]], [[Stokes phenomenon]]

* [[Feynman perturbation series]]

## References

### General

An original article is

* G. Watson, _A theory of asymptotic series_, Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character
Vol. 211, (1912), pp. 279-313 ([JSTOR](http://www.jstor.org/stable/91005))

Basic introductions include

* Joel Feldman, _Taylor series and asymptotic expansions_ lecture notes [pdf](http://www.math.ubc.ca/~feldman/m321/asymptotic.pdf)

* R. Shankar Subramanian, _An Introduction to Asymptotic Expansions_ ([pdf](https://web2.clarkson.edu/projects/subramanian/ch561/notes/Asymptotic%20Expansions.pdf))
* Richard Chapling, _Asymptotic methods_, 2016 ([pdf](https://rc476.user.srcf.net/asymptoticmethods/am_notes.pdf))

* [[Gerald Dunne]], _Introduction to Resurgence, Trans-series and
Non-perturbative Physics_, 2018 ([pdf](https://www.icts.res.in/sites/default/files/NUMSTRINGS2018-2018-01-27-Gerald-Dunne.pdf))

### Non-convergence of the Feynman perturbation series

The argument that the [[S-matrix]] formal power series in all [[perturbative quantum field theories]] of interest is necessarily divergent (and hence at best an asymptotic series) is due to

* {#Dyson52} [[Freeman Dyson]], _Divergence of perturbation theory in quantum electrodynamics_, Phys. Rev. 85, 631, 1952 ([spire](http://inspirehep.net/record/29799?ln=en))

made more precise in

* {#Lipatov77} [[Lev Lipatov]], _Divergence of the Perturbation Theory Series and the Quasiclassical Theory_, Sov.Phys.JETP 45 (1977) 216–223 ([pdf](http://jetp.ac.ru/cgi-bin/dn/e_045_02_0216.pdf))

recalled for instance in

* {#Suslov05} [[Igor Suslov]], section 1 of _Divergent perturbation series_, Zh.Eksp.Teor.Fiz. 127 (2005) 1350; J.Exp.Theor.Phys. 100 (2005) 1188 ([arXiv:hep-ph/0510142](https://arxiv.org/abs/hep-ph/0510142))

* Justin Bond, last section of _Perturbative QFT is Asymptotic; is Divergent; is Problematic in Principle_ ([pdf](https://mcgreevy.physics.ucsd.edu/s13/final-papers/2013S-215C-Bond-Justin.pdf))

* {#Strocchi13} [[Franco Strocchi]], §2.2 of: _An Introduction to Non-Perturbative Foundations of Quantum Field Theory_, Oxford University Press (2013) [[doi:10.1093/acprof:oso/9780199671571.001.0001](https://doi.org/10.1093/acprof:oso/9780199671571.001.0001)]

* {#HollandsWald14} [[Stefan Hollands]], [[Robert Wald]], section 4.1 of _Quantum fields in curved spacetime_, Physics Reports Volume 574, 16 April 2015, Pages 1-35 ([arXiv:1401.2026](https://arxiv.org/abs/1401.2026))

* Marco Serone, from 2:46 on in _A look at $\phi^4_2$ using perturbation theory_ ([recording](https://www.youtube.com/watch?v=J4nxvY1rOhI))

In the example of [[phi^4 theory]] this non-convergence of the perturbation series is discussed in

* {#Helling} Robert Helling, p. 4 of _Solving classical field equations_ ([pdf](http://homepages.physik.uni-muenchen.de/~helling/classical_fields.pdf))

### Optimal truncation and superasymptotics
 {#ReferencesOptimalTruncation}

Discussion of "optimal truncation" of asymptotic series and of "superasymptotics" includes the following:

* {#BerryHowls90} [[Michael Berry]], C. J. Howls, _Hyperasymptotics_,  Proceedings: Mathematical and Physical Sciences Vol. 430, No. 1880 (Sep. 8, 1990), pp. 653-668  ([jstor:79960](https://www.jstor.org/stable/79960))

* {#Berry91} [[Michael Berry]], _Asymptotics, superasymptotics, hyperasymptotics_, in  H. Segur, S. Tanveer, and H. Levine, (eds.)  _Asymptotics Beyond All Orders_,  Plenum, Amsterdam, 1991, pp. 1-14 ([doi:10.1007/978-1-4757-0435-8_1](https://doi.org/10.1007/978-1-4757-0435-8_1))

* O. Costin, M. D. Kruskal, _On optimal truncation of divergent series solutions of nonlinear differential systems; Berry smoothing_, Proc. R. Soc. Lond. A 455, 1931-1956 (1999) ([arXiv:math/0608410](https://arxiv.org/abs/math/0608410))

[[!redirects asymptotic expansions]]

[[!redirects asymptotic series]]


asymptotic freedom

\tableofcontents

## Idea

(...)

## Related concepts

* [[Yang-Mills theory]], [[QCD]], [[lattice QCD]]

* [[confinement]], [[mass gap]]

* [[quark-gluon plasma]]

* [[Landau pole]]

## References

* Wikipedia, _[Asymptotic freedom](http://en.wikipedia.org/wiki/Asymptotic_freedom)_
* Chris Elliott, Brian Williams, Philsang Yoo, _Asymptotic freedom in the BV formalism_, J. Geom. Phys. __123__ (2018)  246-283 [arxiv/1702.05973](https://arxiv.org/abs/1702.05973) [doi](https://doi.org/10.1016/j.geomphys.2017.08.009)


asymptotic isometry

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

#Contents#
* table of contents
{:toc}

## Idea

Considering a class of [[diffeomorphisms]] which are not [[isometries]] but change metrics a tiny bit, with some [[asymptotic boundary]] conditions.

This is useful to obtain [[conformal symmetry]] algebras in some [[theory (physics)|physical theories]] of [[gravity]], esp. in 3D.

## Related concepts

* [[asymptotic boundary]]

## References

* J. Brown, [[Marc Henneaux]], _Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity_, Comm. Math. Phys. __104__ (1986) 207 (key paper!!) [euclid](http://projecteuclid.org/euclid.cmp/1104114999)

* Coussaert, [[Marc Henneaux]], P. van Driel, Class. Quant. Grav. __12__ (1995) 2961

...Brown-Henneaux boundary condition

* [[Marc Henneaux]], [[Claudio Teitelboim]], _Asymptotically anti-de Sitter spaces_,  Comm. Math. Phys. __98__, Number 3 (1985), 391-424, [euclid](http://projecteuclid.org/euclid.cmp/1103942446)

New developments

* [[Marc Henneaux]], Soo-Jong Rey, _Nonlinear W(infinity) algebra as asymptotic symmetry of 3-dimensional higher spin Anti-de Sitter gravity_, [arxiv/1008.4579](http://arxiv.org/abs/1008.4579)

[[!redirects asymptotic symmetry]]
[[!redirects asymptotic symmetries]]
[[!redirects asymptotic isometries]]


asymptotic notation

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

\tableofcontents

## Definition

### Big O

Let $\mathbb{R}$ denote the [[set]] of [[real numbers]]. Then there is a function

$$O:(\mathbb{R} \to \mathbb{R}) \to \mathcal{P}(\mathbb{R} \to \mathbb{R})$$

from the set of [[endofunctions]] on the real numbers $\mathbb{R} \to \mathbb{R}$ to the set of [[subsets]] of [[endofunctions]] on the real numbers $\mathcal{P}(\mathbb{R} \to \mathbb{R})$, such that given a real-valued endofunction $g:\mathbb{R} \to \mathbb{R}$, an real-valued endofunction $f:\mathbb{R} \to \mathbb{R}$ is said to be in $O(g)$ if and only if there merely exists positive real numbers $c:\mathbb{R}^+$ and $n_0:\mathbb{R}^+$ such that for all positive real numbers $n:\mathbb{R}^+$, if $n \geq n_0$, then $\vert f(n) \vert \leq c \vert g(n) \vert$:

$$f \in O(g) \coloneqq \exists c:\mathbb{R}^+.\exists n_0:\mathbb{R}^+.\exists n:\mathbb{R}^+.(n \geq n_0) \implies (\vert f(n) \vert \leq c \vert g(n) \vert)$$

If one doesn't have [[power sets]] in the foundations, one would have to define $O(g)$ as a [[family]] of structural [[subsets]]: for each $g:\mathbb{R} \to \mathbb{R}$, a set $O(g)$ and an [[injection]] $i_O(g):O(g) \hookrightarrow (\mathbb{R} \to \mathbb{R})$. Then $f \in O(g)$ if $f$ is in the [[image]] of $i_O(g)$.

### Big Omega

Let $\mathbb{R}$ denote the [[set]] of [[real numbers]]. Then there is a function

$$\Omega:(\mathbb{R} \to \mathbb{R}) \to \mathcal{P}(\mathbb{R} \to \mathbb{R})$$

from the set of [[endofunctions]] on the real numbers $\mathbb{R} \to \mathbb{R}$ to the set of [[subsets]] of [[endofunctions]] on the real numbers $\mathcal{P}(\mathbb{R} \to \mathbb{R})$, such that given a real-valued endofunction $g:\mathbb{R} \to \mathbb{R}$, an real-valued endofunction $f:\mathbb{R} \to \mathbb{R}$ is said to be in $\Omega(g)$ if and only if there merely exists positive real numbers $c:\mathbb{R}^+$ and $n_0:\mathbb{R}^+$ such that for all positive real numbers $n:\mathbb{R}^+$, if $n \geq n_0$, then $\vert f(n) \vert \geq c \vert g(n) \vert$:

$$f \in \Omega(g) \coloneqq \exists c:\mathbb{R}^+.\exists n_0:\mathbb{R}^+.\exists n:\mathbb{R}^+.(n \geq n_0) \implies (\vert f(n) \vert \geq c \vert g(n) \vert)$$

If one doesn't have [[power sets]] in the foundations, one would have to define $\Omega(g)$ as a [[family]] of structural [[subsets]]: for each $g:\mathbb{R} \to \mathbb{R}$, a set $\Omega(g)$ and an [[injection]] $i_\Omega(g):\Omega(g) \hookrightarrow (\mathbb{R} \to \mathbb{R})$. Then $f \in \Omega(g)$ if $f$ is in the [[image]] of $i_\Omega(g)$.

### Big theta

Let $\mathbb{R}$ denote the [[set]] of [[real numbers]]. Then there is a function

$$\Theta:(\mathbb{R} \to \mathbb{R}) \to \mathcal{P}(\mathbb{R} \to \mathbb{R})$$

defined for all real-valued endofunctions $g:\mathbb{R} \to \mathbb{R}$ as the intersection of the subsets $O(g)$ and $\Omega(g)$

$$\Theta(g) \coloneqq O(g) \cap \Omega(g)$$

By the properties of [[power sets]], given a real-valued endofunction $f:\mathbb{R} \to \mathbb{R}$, $f$ is said to be in $\Theta(g)$ if it is in both $O(g)$ and $\Omega(g)$.

$$f \in \Theta(g) \coloneqq (f \in O(g)) \wedge (f \in \Omega(g))$$

If one doesn't have [[power sets]] in the foundations, one would have to define $\Theta(g)$ as a [[family]] of structural [[subsets]]: for each $g:\mathbb{R} \to \mathbb{R}$, a set $\Theta(g)$ and an [[injection]] $i_\Theta(g):\Theta(g) \hookrightarrow (\mathbb{R} \to \mathbb{R})$. Then $f \in \Theta(g)$ if $f$ is in the [[image]] of $i_\Theta(g)$.

## Examples

Let $p$ be a [[real polynomial function]] with [[degree]] $n$. Then $p \in \Theta(\lambda x:\mathbb{R}.x^n)$.

## Related concepts

* [[algorithm]]
* [[analytic number theory]]

## References

* Wikipedia, [Asymptotic notation](https://en.wikipedia.org/wiki/Asymptotic_notation)

[[!redirects big O]]
[[!redirects big omega]]
[[!redirects big Omega]]
[[!redirects big theta]]
[[!redirects big Theta]]

[[!redirects big O notation]]
[[!redirects big omega notation]]
[[!redirects big Omega notation]]
[[!redirects big theta notation]]
[[!redirects big Theta notation]]

[[!redirects asymptotic notation]]
[[!redirects Bachmann-Landau notation]]


asymptotic representation theory

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

#Contents#
* table of contents
{:toc}

## Idea

*Asymptotic representation theory* is [[representation theory]] in the [[limit of a sequence|limit]] of large [[dimension]] of [[linear representation]] and/or of large [[groups]] being represented.

Much of asymptotic representation theory is concerned specifically with the [[symmetric group]] and studies asymptotics of shapes of [[Young diagrams]] and of numbers of [[Young tableaux]] under [[measures]] such as the [[Plancherel measure]] and/or the [[Schur-Weyl measure]].

## References

* [[Anatoly Vershik]], *Two lectures on the asymptotic representation theory and statistics of Young diagrams*,  In: Vershik A.M., Yakubovich Y. (eds) *Asymptotic Combinatorics with Applications to Mathematical Physics* Lecture Notes in Mathematics, vol 1815. Springer 2003 ([doi:10.1007/3-540-44890-X_7](https://doi.org/10.1007/3-540-44890-X_7))

* [[G. Olshanski]], *Asymptotic representation theory*, Lecture notes 2009-2010 ([webpage](https://lpetrov.cc/art), [pdf 1](https://storage.lpetrov.cc/Olshanski_ART_course_1.pdf), [pdf 2](https://storage.lpetrov.cc/Olshanski_ART_course_2.pdf))

* Piotr Śniady, *Combinatorics of asymptotic representation theory*, European Congress of Mathematics 2012 ([arXiv:1203.6509](https://arxiv.org/abs/1203.6509))


asymptotic safety

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In the context of [[quantum field theory]] (QFT) the term _asymptotic safety_ ([Weinberg 79](#Weinstein79)) refers to the situation where a QFT may not be [[renormalization|renormalizable]] (is not defined on all scales by the choice of a [[finite number]] of [[coupling constants]]), but has the property that its [[renormalization group flow]] has a non-trivial [[non-perturbative effect|non-perturbative]] [[fixed point]] such that the subspace of the space of [[coupling constants]] whose RG-flow converges to this fixed point is [[finite number|finite]] [[dimension|dimensional]].

> One defines the "UV-critical hypersurface" as the set of all those points in the infinite-dimensional theory space which are "pulled" into the fixed point by the inverse RG flow: trajectories lying in this surface approach the fixed point for increasing momentum scales. General arguments and known examples suggest that the UV-critical hypersurface has a finite dimensionality. This dimensionality equals the number of (infrared-) relevant couplings, i.e. couplings which get attracted to the fixed point in the UV. The important point is that once the value of these (few) couplings are known at some scale all other (irrelevant) couplings are fixed by requiring an asymptotically safe theory, that is a trajectory which lies entirely in the UV-critical hypersurface. By this means we achieve that, first, the couplings are determined by a finite number of measurements rendering the theory predictive, and, second, the UV behavior is unproblematic without any unphysical divergences. ([Nink-Reuter 12, p. 2](#NinkReuter12))

A key example of a non-renormalizable QFT is [[Einstein gravity]], and there is speculation that it might be asymptotically safe, and that this might be the solution to the construction of [[quantum gravity]] ([Reuter 96](#Reuter96)).

## Issues
{#Issues}

Issues that the program of asymptotic safety of gravity is facing include the following:

### Theoretical issues

1. All existing computations that see hints for a UV-fixed point do so by first applying a drastic truncation to the space of couplings, and then checking only whether there is a UV-fixed point for the RG-flow in the remaining small subspace. It seems unclear to which extent these approximate considerations may be extrapolated.

1. Most existing computations consider only pure Einstein gravity without matter coupling. It seems unclear to which extent these results may be extrapolated to the situation where matter is taken account of (but see [Biemans-Platania-Saueressig 17](#BiemansPlataniaSaueressig17)).

1. {#DonoghueSummarizes} [Donoghue 19](#Donoghue19) claims technical problems with the handling of [[renormalization group flow]] in Newton's constant $G_N$ as currently practiced in the asymptotic safety program:

    1. The "[[running coupling constant|running]]" $G_N(E)$ comes from power-divergent corrections that vanish in [[dimensional regularization]].

      1. Computing different [[Minkowski spacetime]] processes gives different power-divergent corrections: there is no generic common $G_N(E)$.

      1. There is no generic [[energy]] $E$: some processes depend on $s \sim + E^2$, others on $t \sim -E^2$ and (unlike in the usual log running) this makes a difference.

1. If one assumes (as is widely, but no generally believed) that [[Bekenstein-Hawking entropy]] seen in classical gravity is to correspond to a microscopic [[entropy]] of its quantum degrees of freedom, then the scaling of this entropy with area as opposed to volume contradicts the assumption that [[quantum gravity]] is a [[local field theory]] at small scales and higher energies ([Shomer 07, section IV](#Shomer07)) and hence then it contradicts the asymptotic safety of gravity.

   Similarly, if one trusts the [[AdS/CFT correspondence]] then gravity is fundamentally not a local field theory, only its boundary [[CFT]] is, in contradiction with asymptotic safety of gravity ([Shomer 07, section IV](#Shomer07)). But see [FR22](#FR22).

### Experimental issues

The near criticality of the [[Higgs field]] [[vacuum]] (see there at _[Higgs mass and vacuum (in-)stability](Higgs+field#MassAndVacuumInstability)_) implies that the [[coefficient]] $\lambda$ of the quartic part of the Higgs potential is close to zero after [[renormalization group flow]] ("RGE") to around the [[Planck scale]] of about $10^{19}$ [[GeV]] (e.g. [BDGGSSS 13, p. 17-18](#BDGGSSS13)):

In fact also the [[beta function]] $\beta_\lambda$ of the quartic coupling $\lambda$ (i.e. its logarithmic [[derivative]] with respect to [[scale]]) is close to zero around the [[Planck scale]] of about $10^{19}$ [[GeV]] ([BDGGSSS 13, p. 18](#BDGGSSS13)):

Earlier it has been suggested that this reflects the principle of asymptotic safety ([Shaposhnikov-Wetterich 09](#ShaposhnikovWetterich09)). But this would mean that not only $\lambda$ and its [[beta-function|RGE-derivative]] $\beta_\lambda$ vanish around the [[Planck scale]], but that in fact all higher derivatives do, too (see e.g [Niedermaier 06, equation (1.5)](#Niedermaier06)) hence that $\beta_\lambda$ asymptotes to zero. But this does not seem to be the case; in ([BDGGSSS 13, p. 17-18](#BDGGSSS13)) it says:

>  As shown in fig. 2 (upper right), the corresponding Higgs quartic [[beta-function]] vanishes at a [[scale]] of about $10^{17}$-$10^{18}$ [[GeV]]. In order to quantify the degree of cancellation in the β-function, we plot in fig. 2 (lower right) $\beta_\lambda$ in units of its pure [[top quark]] contribution.  The vanishing of $\beta_\lambda$ looks  more  like  an  accidental  cancellation  between  various  large  contributions, rather than an asymptotic approach to zero.

## References

The idea of asymptotic safety as such and as a cure for [[quantum gravity]] is due to

* {#Weinstein79} [[Steven Weinberg]], _Ultraviolet divergences in quantum theories of gravitation_, in "General Relativity: An Einstein centenary survey", ed. S. W. Hawking and W. Israel. Cambridge University Press. pp. 790–831 (1979) ([spire](https://inspirehep.net/record/159043/))

It gained new popularity with this result:

* {#Reuter96} [[Martin Reuter]], _Nonperturbative Evolution Equation for Quantum Gravity_, Phys.Rev. D57 (1998) 971-985 ([arXiv:hep-th/9605030](https://arxiv.org/abs/hep-th/9605030))

An attempt to conceptually explain why [[gravity]] might have a UV-fixed point is in this article:

* {#NinkReuter12} Andreas Nink, [[Martin Reuter]], _On quantum gravity, Asymptotic Safety, and paramagnetic dominance_, Int. J. Mod. Phys. D22 (2013) 1330008 ([arXiv:1212.4325](https://arxiv.org/abs/1212.4325))

Observation of a special role of spacetime dimension 2:

* {#LauscherReuter06} O. Lauscher, [[Martin Reuter]], _Asymptotic Safety in Quantum Einstein Gravity: nonperturbative renormalizability and fractal spacetime structure_,  In: Fauser B., Tolksdorf J., Zeidler E. (eds.) )Quantum Gravity_ Birkhäuser Basel 2006 ([arXiv:hep-th/0511260](https://arxiv.org/abs/hep-th/0511260), [doi:10.1007/978-3-7643-7978-0_15](https://doi.org/10.1007/978-3-7643-7978-0_15))

and speculation of this being related to the [[Connes-Lott model]] (see [here](2-spectral+triple#Connes06OnRelationToStringVacua)):

* {#Connes06} [[Alain Connes]],  p. 8 of _Noncommutative Geometry and the standard model with neutrino mixing_, JHEP0611:081,2006 ([arXiv:hep-th/0608226](http://arxiv.org/abs/hep-th/0608226))

Review:

* {#Niedermaier06} [[Max Niedermaier]], _The Asymptotic Safety Scenario in Quantum Gravity -- An Introduction_, Class.Quant.Grav.24:R171-230,2007 ([arXiv:gr-qc/0610018](https://arxiv.org/abs/gr-qc/0610018))

* [[Max Niedermaier]], [[Martin Reuter]], _The Asymptotic Safety Scenario in Quantum Gravity_, Living Reviews in Relativity December 2006, 9:5 [doi:10.12942/lrr-2006-5](http://link.springer.com/article/10.12942/lrr-2006-5)

* Riccardo Martini, Gian Paolo Vacca, Omar Zanusso, *Perturbative approaches to non-perturbative quantum gravity*, in: *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2210.13910](https://arxiv.org/abs/2210.13910)]

* Astrid Eichhorn, Marc Schiffer, *Asymptotic safety of gravity with matter*, in: [[Handbook of Quantum Gravity]], Springer (2023) [[arXiv:2212.07456](https://arxiv.org/abs/2212.07456)]

* Alexander Bednyakov, Alfiia Mukhaeva, *Perturbative Asymptotic Safety and Its Phenomenological Applications* [[arXiv:2309.08258](https://arxiv.org/abs/2309.08258)]

* Jan M. Pawlowski, Manuel Reichert, *Quantum Gravity from dynamical metric fluctuations* in *[[Handbook of Quantum Gravity]]* [[arXiv:2309.10785](https://arxiv.org/abs/2309.10785)]

Critical review:

* {#Shomer07} Assaf Shomer, _A pedagogical explanation for the non-renormalizability of gravity_ ([arXiv:0709.3555](https://arxiv.org/abs/0709.3555))

* {#Donoghue19} [[John Donoghue]], _A Critique of the Asymptotic Safety Program_,  Frontiers in Physics 8, 56 (2020) ([arXiv:1911.02967](https://arxiv.org/abs/1911.02967), [doi:10.3389/fphy.2020.00056](https://www.frontiersin.org/articles/10.3389/fphy.2020.00056/full))

See also:

* [[Roberto Percacci]], _Asymptotic Safety FAQs_ ([web](http://www.percacci.it/roberto/physics/as/faq.html))

* {#DonoghuePercacci21} [[John Donoghue]] vs. [[Roberto Percacci]], *Debate on Asymptotically Safe Quantum Gravity*, Perimeter Institute, April 2021 ([video](https://pirsa.org/21040021), [slides](https://pdf.pirsa.org/files/21040021.pdf))

* {#BiemansPlataniaSaueressig17} Jorn Biemans, Alessia Platania, Frank Saueressig, _Renormalization group fixed points of foliated gravity-matter systems_ ([arXiv:1702.06539](https://arxiv.org/abs/1702.06539))

The suggestion that asymptotic safety explains the observed [[mass]] of the [[Higgs particle]] is attributed to

* {#ShaposhnikovWetterich09} [[Mikhail Shaposhnikov]], [[Christof Wetterich]], _Asymptotic safety of gravity and the Higgs boson mass_, Phys. Lett. B 683 (2010) 196 [[arXiv:0912.0208](https://arxiv.org/abs/0912.0208)]

but see p. 17 of

* {#BDGGSSS13} Dario Buttazzo, Giuseppe Degrassi, Pier Paolo Giardino, [[Gian Giudice]], Filippo Sala, Alberto Salvio, [[Alessandro Strumia]], section 7 of _Investigating the near-criticality of the Higgs boson_ ([arXiv:1307.3536](https://arxiv.org/abs/1307.3536))

and for more see at _[Higgs field -- Asymptotic safety?](https://ncatlab.org/nlab/show/Higgs%20field#AsymtoticSafetyOrNot)_

Application to [[scattering amplitudes]]:

* Jan H. Kwapisz, Krzysztof A. Meissner, _Asymptotic safety and quantum gravity amplitudes_ ([arXiv:2005.03559](https://arxiv.org/abs/2005.03559))

Unitarity, causality and stability in Asymptotic Safety:

* Alessia Platania, [[Christof Wetterich]], _Non-perturbative unitarity and fictitious ghosts in quantum gravity_ ([arXiv:2009.06637](https://arxiv.org/abs/2009.06637))

* Alessia Platania _Causality, unitarity and stability in quantum gravity: a non-perturbative perspective_ ([arXiv:2206.04072](https://arxiv.org/abs/2206.04072))

Review on early-universe cosmology in the context of asymptotic safety:

* Alessia Platania, *From renormalization group flows to cosmology*, Front.in Phys. 8 (2020) 188 [[arXiv:2003.13656](https://arxiv.org/abs/2003.13656)]

Review on [[black holes]] in the context of asymptotic safety:

* Alessia Platania, *Black Holes in Asymptotically Safe Gravity*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2302.04272](https://arxiv.org/abs/2302.04272)]

See also:

* Edoardo D'Angelo, *Asymptotic Safety in Lorentzian quantum gravity* [[arXiv:2310.20603](https://arxiv.org/abs/2310.20603)]

Relation to String Theory:

* Senarath de Alwis, et al. _Asymptotic safety, string theory and the weak gravity conjecture_ ([arXiv:1907.07894](https://arxiv.org/abs/1907.07894))

* Ivano Basile, Alessia Platania, _Cosmological α′-corrections from the functional renormalization group_ ([arXiv:2101.02226](https://arxiv.org/abs/2101.02226))

* Ivano Basile, Alessia Platania, _String tension between de Sitter vacua and curvature corrections_ ([arXiv:2103.06276](https://arxiv.org/abs/2103.06276))

* Ivano Basile, Alessia Platania, _Asymptotic Safety: Swampland or Wonderland?_ ([arXiv:2107.06897](https://arxiv.org/abs/2107.06897))

* Fei Gao, Masatoshi Yamada, _Determining holographic wave functions from Wilsonian renormalization group_ ([arXiv:2202.13699](https://arxiv.org/abs/2202.13699))

* Renata Ferrero, [[Martin Reuter]], *On the possibility of a novel (A)dS/CFT relationship emerging in Asymptotic Safety* [[arXiv:2205.12030](https://arxiv.org/abs/2205.12030)]


asymptotically flat spacetime

#Contents#
* table of contents
{:toc}

## Idea

A [[spacetime]] is _asymptotically flat_ if far enough away from any point it looks like flat [[Minkowski spacetime]].

On asymptotically flat spacetimes a total gravitational [[mass]]/[[energy]] is well defined, the _[[ADM mass]]_. See also at _[[positive energy theorem]]_.

Regarding a field configuration of [[Einstein gravity]] as a [[Cartan connection]] for the inclusion of the [[Lorentz group]] into the [[Poincaré group]] (see at _[[first-order formulation of gravity]]_), then asymptotic flatness means that the [[Cartan geometry]] asymptotivally approaches its model [[Klein geometry]]. From this perspective there are various generalizations of asymptotic flatness:

for instance for [[anti de Sitter gravity]] it means that spacetime asymptootes to [[anti de Sitter spacetime]]. For more on this see at _[[AdS-CFT correspondence]]_.

## Related concepts

* [[conformally flat manifold]]

* [[conformal compactification]]

* [[ADM mass]]

## References

### Asymptotically Minkowski spacetimes

* Wikipedia, _[Asymptotically flat spacetimes](http://en.wikipedia.org/wiki/Asymptotically_flat_spacetime)_

* Jörg Frauendiener, _Conformal Infinity_ ([living reviews](http://relativity.livingreviews.org/Articles/lrr-2000-4/))

### Asymptotically AdS spacetimes

* [[Marc Henneaux]], [[Claudio Teitelboim]], _Asymptotically Anti-de Sitter Spaces_, Comm. Math. Phys. 98, 391-424 (1985) ([pdf](http://srv2.fis.puc.cl/~mbanados/Cursos/TopicosRelatividadAvanzada/HenneauxTeitelboim.pdf))

[[!redirects asymptotically flat spacetimes]]
[[!redirects asymptotically anti de Sitter spacetime]]
[[!redirects asymptotically anti de Sitter spacetimes]]


asymptotics

## References

* Wikipedia, _[Asymptotic analysis](http://en.wikipedia.org/wiki/Asymptotic_analysis)_

[[!redirects asymptotic]]

[[!redirects asymptotic analysis]]


asynchronous automaton

Asynchronous automata are a generalisation of both [[transition systems]] and [[Mazurkiewicz traces]]. Their study has influences  other models for concurrency such as _transition systems with independence_ (also called _asynchronous transition systems_).  The idea is to decorate transition systems with an independence relation (much as in (Mazurkiewicz) [[trace alphabets]]) between actions that allow one to distinguish true concurrency from mutual exclusion (i.e. _non-determinism_).  Following the paper by Goubault and Mimram, we use a slight modification called _automata with concurrency relations_:
+--{: .un-definition}
##Definition##
An _automaton with concurrency relations_  $(S,i,E,Tran,I)$ consists of

* a [[transition system]] $(S,i,E,Tran)$, such that whenever $(s,a,s')$, and $(s,a,s'')$ are in $Tran$, then $s' = s''$;

and

* $I = \{I_s\mid s\in S\}$ is a family of irreflexive, symmetric binary relations, $I_s$ on $E$ such that whenever $a_1I_s a_2$ (with $a_1,a_2 \in E$), there exist transitions $(s,a_1,s_1)$, $(s,a_2,s_2)$, $(s_1,a_2,r)$, and $(s_2,a_1,r)$ in $Tran$.
=--

##References##

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

[[!redirects asynchronous automaton]]
[[!redirects asynchronous automata]]


AT category

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

# Contents
* table of contents
{: toc}

## Idea

[Freyd (1999)](#Freyd99) gave a sharp and clever description of the commonalities and differences between [[abelian categories]] and [[toposes]] (or, in the first place, between abelian categories and [[pretoposes]]), by introducing a finitely axiomatized theory of "AT categories". Thus abelian categories and pretoposes are examples of AT categories; underscoring how much these structures have in common, Freyd's theory hews so closely to both of these two basic examples that in fact every AT category splits cleanly as a [[product]] of a pretopos part and an abelian part.

Here, the properties held in common between abelian categories and pretoposes are all exactness conditions; in the AT set-up, the decisive difference between abelian categories and pretoposes is concentrated solely in the behavior of the [[initial object]] with respect to [[cartesian product]] (being in one case the cartesian monoidal unit, i.e. an element that is universally absorbed into any other, and in the other an absorbing element, i.e. an element that absorbs any other element into itself, aka a zero element or annihilating element).

There is no research we know of on AT categories. One might say that they were invented by Freyd really for the sole theoretical purpose of demonstrating the commonalities and differences between abelian categories and pretopos categories, not because AT categories are found in nature. Indeed, a "typical" AT category is something of a mythological creature with the head of an abelian lion and the body of a pretopos dragon, and serves no purpose as such, even though pure lions and pure dragons themselves serve a purpose. Some of the axioms, particular axiom "AE" below, seem expressly designed to fuse them together in ad hoc fashion.

## Definition

Freyd introduces some baseline assumptions on the categories in question:

* [[finitely complete category|Finite completeness]],

* Presence of an [[initial object]],

* Presence of [[pushout|pushouts]] for pairs of [[morphism|arrows]] one of which is [[monomorphism|monic]],

* Presence of pushouts for [[kernel pair]]s

Each of these assumptions obviously holds in any [[abelian category]] and in any [[pretopos]].

+-- {: .num_remark}
###### Remarks
If we introduce predicates to express these four baseline assumptions, such as a predicate formula $P(p, q; f, g)$ whose meaning is "$(p, q)$ is a pushout pair for pairs of arrows $(f, g)$", then the remaining AT axioms given below all have a simple logical structure: they can be expressed as [[Horn theory|Horn clauses]] in the language generated by the baseline predicates. The first 8 axioms are in fact universal Horn clauses in these predicates.

This might look like a side comment for logicians, but it might also signal that Freyd intends to invoke, if pressed for details of proofs, one of a battery of representation theorems, such as the representation theorem for regular categories whose precise formulation is in terms of Horn theories -- see [[Categories, Allegories]] for examples.

Furthermore, by choosing a *specific* initial object, a specific pullback for each pair of arrows with common target, etc., an AT category can be cast as a model for a certain [[essentially algebraic theory]]. In other words, such "algebraic AT categories" can be internalized in any finitely complete category.
=--

The crucial difference between "abelianness" and "toposness" is concentrated in the following definition:

\begin{definition}
\label{TypeATObjects}
Let $0$ denote the [[initial object]], and for any [[object]] $X$ let

* $\pi_1 \,\colon\, 0 \times X \longrightarrow 0$,

* $\pi_2 \,\colon\, 0 \times X \longrightarrow X$

denote the two product [[projections]].

We say $X$ is

1. of **type T** if $\pi_1$ is an isomorphism,

1. of **type A** if $\pi_2$ is an isomorphism.

\end{definition}

With this:

* A pretopos will turn out to be precisely an AT category in which every object is of type T.

* An abelian category will turn out to be an AT category where every object is of type A.

Here then are the AT exactness axioms. Again, each of them is satisfied in every abelian category and in every pretopos, and according to Freyd's thesis, any exactness condition satisfied in both classes of categories is a logical consequence of this set of axioms. Some of Freyd's remarks in his original posting are included in parentheses.

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

A category meeting the baseline assumptions above is an **AT category** if the following 8 axioms plus axiom "AE" are satisfied.

1. The category is an effective [[regular category]].

   ("Yes this can be stated as universal Horn conditions on pullbacks and the special pushouts mentioned above.")

1. The arrow $0 \to 1$ is monic. ("Note that it follows that all maps from $0$ are monic.")

1. If $i\colon A \to C$ is [[monomorphism|mono]], then any [[pushout]] square
$$\array{
A & \stackrel{i}{\to} & C \\
\downarrow & & \downarrow \\
B & \underset{j}{\to} & D
}$$
is also a [[pullback]], and $j$ is mono.

1. The functor $0 \times -$ preserves pushouts of kernel pairs, and pushouts of pairs of arrows one of which is monic.

1. If $f\colon B \to 0 \times C$ is [[epimorphism|epi]] and
$$\array{
A & \to & B \\
\downarrow & & \downarrow f \\
0 & \to & 0 \times C
}$$
is a [[pullback]], then it's also a [[pushout]].

1. In the [[full subcategory]] of type T objects (Def. \ref{TypeATObjects}), pushouts of pairs of morphisms, one of which is monic, are universal (i.e., stable under pullback).

1. Define a functor $T$ by the pushout diagram
$$\array{
0 \times X & \stackrel{\pi_1}{\to} & 0 \\
\pi_2 \downarrow & & \downarrow \\
X & \to & T X
}$$
Then $T$ preserves pullbacks.

1. Given a morphism $f\colon X \to Y$, if $T f$ and $0 \times f$ are isomorphisms, then so is $f$.

=--

This is the basic list of exactness assumptions which permit a sharp comparison between pretoposes and abelian categories. From this list alone one can prove that every AT category embeds faithfully in a product of a pretopos and an abelian category. To show that every AT category is _equivalent_ to such a product, Freyd appends the following axiom involving an existential Horn condition:

+-- {: .num_defn #AxiomAE}
###### Definition
**axiom (AE)**

For every object $X$, there is a map $\xi\colon T X \to X$ such that
$$0 \times X \stackrel{\pi_1}{\to} X \stackrel{\xi}{\leftarrow} T X$$
is a coproduct diagram, and the canonical map $X \to T X$ is a retraction of $\xi$.

=--

To bring actual toposes into the picture, Freyd adds some more axioms, but let's first take stock and see what this gives us so far.

## Basic consequences of the AT axioms

+-- {: .num_prop}
###### Proposition
For any $X$ the unique map $0 \to X$ is monic.
=--

+-- {: .proof}
###### Proof
By axiom 2, the map $0 \to 1$ is monic. The pullback-preserving functor $- \times X$ preserves monos, so

$$0 \times X \stackrel{! \times 1_X}{\to} 1 \times X \cong X$$

(which is just the second projection $\pi_2: 0 \times X \to X$) is monic. The unique map $0 \to 0 \times X$ is monic since it has a retraction $\pi_1: 0 \times X \to0$. The result follows.
=--

+-- {: .num_prop}
###### Proposition
Binary coproducts exist.
=--

+-- {: .proof}
###### Proof
Because we can take the pushout of a pair of monos $0 \to X$, $0 \to Y$.
=--

+-- {: .num_prop}
###### Proposition
Coproducts are disjoint.
=--

+-- {: .proof}
###### Proof
By axiom 3, the two coprojections $i_X \colon X \to X + Y$, $i_Y \colon Y \to X + Y$ are monic, and their pullback is initial.
=--

## Category of type A objects is abelian

In this section we show that the full subcategory of type A objects forms an abelian category.

+-- {: .num_lemma #tozero}
###### Lemma
An object is of type A if and only if there exists a map to $0$.
=--

+-- {: .proof}
###### Proof
If $X$ is of type A, then we clearly have $X \cong 0 \times X \stackrel{\pi_1}{\to} 0$. Conversely, suppose there exists $p: X \to 0$. Note that since $0 \to 1$ is monic, there is at most one map $Y \to 0$ for any object $Y$. It follows quickly that maps $Y \to X$ are in natural bijective correspondence with maps $Y \to X \times 0$ by composition with $\pi_1: X \times 0 \to X$, so that $\pi_1$ is an isomorphism by the Yoneda lemma.
=--

+-- {: .num_prop #coreflective}
###### Proposition
The full subcategory of type A objects is coreflective.
=--

+-- {: .proof}
###### Proof
By Lemma \ref{tozero}, the subcategory of type A objects in an AT category $C$ is equivalent to the subcategory $C/0 \hookrightarrow C/1 \simeq C$, which is the category of coalgebras for the functor $A(X) = 0 \times X$. Thus the category of type A objects is comonadic.
=--

+-- {: .num_lemma}
###### Lemma
Objects of type A are closed under binary products, finite coproducts, subobjects, and quotient objects (= cokernels of kernel pairs).
=--

+-- {: .proof}
###### Proof
This follows from the fact that the category of type A objects is the slice category $C/0$.
=--

Before we prove the next lemma, recall that in a category with zero objects, a **kernel** of an arrow $f: A \to B$ is an equalizer of the pair $f, 0: A \stackrel{\to}{\to} B$. This is the same as a pullback

$$\array{
K & \to & A \\
\downarrow & & \downarrow f \\
0 & \to & B
}$$

Cokernels are defined dually, and can be formulated dually as certain pushouts. Since pushouts along monos exist, we can take the cokernel of any mono, and in particular the cokernel of any kernel.

+--{: .num_lemma}
###### Lemma
In the category of type A objects, every mono is the kernel of its cokernel, and every epi is the cokernel of its kernel.
=--

+-- {: .proof}
###### Proof
Of course, $0$ is a zero object in the category of type A objects. If $i\colon A \to B$ is a mono in the category of type A objects, then the pushout
$$\array{
A & \stackrel{i}{\to} & B \\
\downarrow & & \downarrow \\
0 & \to & coker(i)
}$$
exists and is also a pullback, by axiom 3, and hence $i$ is the kernel of its cokernel. Now suppose $f: A \to C$ is an epi in the category of type A objects. Since $0 \times C \cong C$, we have an epi $f\colon A \to 0 \times C$. Then, by axiom 5, the pullback
$$\array{
ker(f) & \to & A \\
\downarrow & & \downarrow f \\
0 & \to & 0 \times C
}$$
is also a pushout, so $f$ is the cokernel of its kernel.
=--

+-- {: .num_theorem}
###### Theorem
The category of type A objects is an abelian category.
=--

+-- {: .proof}
###### Proof
Any category with zero objects, binary products and coproducts, and in which every mono is the kernel of its cokernel and every epi is the cokernel of its kernel, is in fact an abelian category. See Freyd-Scedrov, _Categories, Allegories_, 1.598 (p. 95).
=--

## Category of type T objects is a pretopos

Now we show that the full subcategory of type T objects is a pretopos. It is clear that $0 \times 0 \cong 0$ (since $0 \to 1$ is monic), so $0$ is a type T object.

+-- {: .num_lemma #exactlyone}
###### Lemma
If $A$ is type A and $T$ is type T, then there exists exactly one map $A \to T$. Type T objects are characterized by this property.
=--

+-- {: .proof}
###### Proof
There is exactly one morphism $A \to 0$. Hence morphisms $A \to T$ are in bijection with maps $A \to 0 \times T \cong 0$, of which there is exactly one. For the second statement, suppose that $X$ has the property that there is exactly one map $A \to X$ for each type A object. Such objects $X$ are closed under products, and $0$ is such an object; therefore $0 \times X$ is such an object. On the other hand, $0 \times X$ is of type A since it projects to $0$. Hence there is at most one morphism $0 \times X \to 0 \times X$, and it follows that $0 \times X \to 0 \to 0 \times X$ is the identity, so that $0 \times X \cong 0$. Thus $X$ is of type T.
=--

+-- {: .num_cor #strict}
###### Corollary
The initial object $0$ is [[strict initial object|strict]] in the category of type T objects.
=--

+-- {: .proof}
###### Proof
Given $T$ of type T and $T \to 0$, we know $T$ is type A, and therefore by Lemma \ref{exactlyone} there is exactly one map $T \to T$. Hence $T \to 0 \to T$ is the identity, and of course so is $0 \to T \to 0$. So $T$ is initial.
=--

+-- {: .num_cor #closed}
###### Corollary
The full subcategory of objects of type T is closed under products, coproducts, subobjects, and quotient objects.
=--

+-- {: .proof}
###### Proof
Closure under products and subobjects is immediate from Lemma \ref{exactlyone}. Closure under quotients and coproducts follows from axiom 4.
=--

+-- {: .num_theorem}
###### Theorem
The full subcategory of objects of type T is a pretopos.
=--

+-- {: .proof}
###### Proof
Corollary \ref{closed} gives finite completeness, coproducts, and quotients of kernel pairs. Axiom 6 gives that in the full subcategory of $T$ objects, pushouts along monos are stable under pullback, and the initial object is stable under pullback in the category of T objects, because it is strict by Corollary \ref{strict}. It follows that coproducts are universal in the category of T objects. They are also disjoint by an earlier result, so the category of T objects is extensive. It is also effective regular by axiom 1, hence a pretopos.
=--

## Splitting into type A and type T objects

In this section we show that every AT category $C$ is the product of the abelian category $C_A$ of type A objects and the pretopos $C_T$ of type T objects.

Let us first observe that the coreflector (see Proposition \ref{coreflective})

$$0 \times {-}\colon C \to C_A$$

is left exact (as all coreflectors are), and

* Preserves the initial object (by axiom 2),

* Preserves pushouts of kernel pairs, and pushouts of pairs of arrows one of which is monic (axiom 4),

* And therefore also preserves coproducts.

A functor between AT categories which is left exact and which preserves such classes of finite colimits may be called a **morphism of AT categories**. Hence the coreflector is a morphism of AT categories.

Next let us observe that the subcategory $C_T \hookrightarrow C$ is _reflective_; the reflector is the functor $T$ defined above by means of a suitable pushout. The reflector clearly preserves any colimits that exist, and preserves pullbacks (by axiom 7), so

$$T\colon C \to C_T$$

is also a morphism of AT categories. Therefore we have a morphism of AT categories

$$F = \langle 0 \times {-}, T \rangle\colon C \to C_A \times C_T$$

and now we wish to prove, under a suitable additional axiom (also satisfied by every abelian category and every pretopos), that this is an equivalence.

+-- {: .num_lemma}
###### Lemma
The functor $F$ is faithful.
=--

+-- {: .proof}
###### Proof
The functor $F$ is left exact and therefore preserves kernels. By axiom 8, $F$ reflects isomorphisms. It follows immediately from these two facts that $F$ is faithful.
=--

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

With this result, Freyd's "first task" is complete: any AT category may be faithfully represented in a product of a pretopos and an abelian category.

=--

At this point we bring in axiom **AE**, def. \ref{AxiomAE}, for its debut appearance. It may be helpful to consider that $T X$ is by definition the [[cokernel]] appearing in the exact sequence

$$0 \times X \stackrel{\pi_2}{\to} X \to coker(\pi_2)$$

and that axiom AE asserts that this exact sequence splits (with splitting $\xi: coker(\pi_2) \to X$), in such a way that $X$ is the coproduct of the end terms.

+-- {: .num_prop}
###### Proposition
The functor $F$ is full.
=--

+-- {: .proof}
Suppose given maps $f: 0 \times X \to 0 \times Y$, $g: T X \to T Y$, and contemplate the diagram
$$\array{
0 \times X & \stackrel{\pi_2}{\to} & X & \to & T X \\
f \Big\downarrow & & & & \Big\downarrow g \\
0 \times Y & \underset{\pi_2}{\to} & Y & \stackrel{\overset{\xi}{\leftarrow}}{\to} & T Y
}$$
Since $X = (0 \times X) + T X$, the two obvious arrows $0 \times X \to Y$, $T X \to Y$ combine to give an arrow $(f, g): X \to Y$. It is straightforward to check that $0 \times (f, g) = f$ (because the functor $0 \times -$ kills the type T summands) and that $T(f, g) = g$.
=--

+-- {: .num_theorem}
###### Theorem
The functor $F$ is an equivalence.
=--

+-- {: .proof}
###### Proof
All that is left to check is that $F$ is essentially surjective, but this is clear because given a type A object $X$ and a type T object $Y$, we have $0 \times (X + Y) \cong X$ (use axiom 4 plus the fact that $0 \times Y \cong 0$), and
$$T(X + Y) \cong T X + T Y \cong 0 + T Y \cong Y$$
which completes the proof.
=--

This completes Freyd's "second task".

## Topos-theoretic considerations

In order to beef up the type T objects to a topos, Freyd imposes some extra structure on top of the AT axioms. Since it is now the topos axioms that are in the ascendant, I will christen these categories _TA categories_.

Thus, a **TA category** is an AT category $C$ together with

* Functions $P, E\colon Ob(C) \to Ob(C)$,

* Functions $l, r\colon Ob(C) \to Mor(C)$, of the form $l X\colon E X \to P X$, $r X\colon E X \to X$,

* An operation $\Lambda$ which assigns to each pair of morphisms $f\colon R \to Y$, $g\colon R \to X$ (where $R$ and $Y$ are assumed to be type T), a morphism
$\Lambda(f, g)\colon Y \to P X$.

This data is to satisfy the following axioms:

1. $\langle l X, r X \rangle: E X \to P X \times X$ is monic and $P X$, $E X$ are of type T,

2. The composite of relations
$$Y \stackrel{\Lambda(f, g)}{\to} P X \stackrel{(l X)^{op}}{\to} E X \stackrel{r X}{\to} X$$
is equal to
$$Y \stackrel{f^{op}}{\to} R \stackrel{g}{\to} X,$$

3. If
$$\array{
R & \stackrel{g}{\to} & E X \\
f \downarrow & & \downarrow l X \\
Y & \underset{h}{\to} & T X
}$$
is a pullback and $R$ is of type T, then $\Lambda(f, (r X) \circ g) = h$.

Freyd: "Note that in the full category of type-T objects, $P$ yields
power-objects with $E$, $l$, $r$  naming the universal relations. (The third axiom provides the uniqueness condition.)"

And: "In any abelian category the only type T object is the zero
object, which forces $P = E = 0$ for abelian categories."

And finally: "It's routine that both $0 \times {-}$ and $T$ preserve the new structure. We can now remove the existential from AE. Define $\xi\colon T X \to X$  as the image of $r X$. The third task is finished with:

1. For every $X$, $0 \times X \to X \stackrel{\xi}{\leftarrow} T X$ is a coproduct diagram."

_This section is likely to be rewritten and cleaned up._

## References

* {#Freyd99} [[Peter Freyd]] in reply to [[Vaughan Pratt]], *Abelian-topos (AT) categories*, mailing list comments (1999) [[catlist:1999/atcat](http://www.mta.ca/~cat-dist/catlist/1999/atcat), [[Freyd_AT-categories.txt:file]]]

[[!redirects AT category]]
[[!redirects AT categories]]
[[!redirects AT-category]]
[[!redirects AT-categories]]

[[!redirects TA category]]
[[!redirects TA categories]]
[[!redirects TA-category]]
[[!redirects TA-categories]]


Athanasios Chatzistavrakidis

Athanasios Chatzistavrakidis is a Greek theoretical and mathematical physicist working at the Ruđer Bošković Institute in Croatia.

* [webpage](https://www.irb.hr/eng/Divisions/Division-of-Theoretical-Physics/Quantum-Gravity-and-Mathematical-Physics-Group/Employees/Athanasios-Chatzistavrakidis)

category: people


Atihya-Bott-Shapiro orientation > history

see at _[[K-orientation]]_


Atish Dabholkar

* [webpage](https://en.wikipedia.org/wiki/Atish_Dabholkar)

## Selected writings

On [[orientifolds]] and [[discrete torsion]]:

* [[Atish Dabholkar]], [[Jaemo Park]], _Strings on Orientifolds_, Nucl. Phys. B477 (1996) 701-714 ([arXiv:hep-th/9604178](https://arxiv.org/abs/hep-th/9604178))

On [[orientifolds]] and [[duality in string theory]]:

* [[Atish Dabholkar]], _Lectures on Orientifolds and Duality_, In *Trieste 1997, High energy physics and cosmology* 128-191 ([arXiv:hep-th/9804208](https://arxiv.org/abs/hep-th/9804208), [spire:454332](https://inspirehep.net/literature/454332))

Introducing the [[generalized Scherk-Schwarz mechanism]]:

* [[Atish Dabholkar]], [[Chris Hull]], _Duality Twists, Orbifolds, and Fluxes_, JHEP 0309:054, 2003 ([arXiv:hep-th/0210209](https://arxiv.org/abs/hep-th/0210209))

category: people


Atiyah 2-framing

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

# Contents
* table of contents
{: toc}

## Definition

In the general terminology of _$n$-[[framing]]_ then a _2-framing_ of a [[manifold]] $\Sigma$ of [[dimension]] $d \leq 2$ is a trivialization of $T \Sigma \oplus \mathbb{R}^{2-d}$.

In ([Atiyah 90](#Atiyah1990)) the term "2-framing" is instead used for a trivialization of the double of the tangent bundle of a 3-manifold. So this is a different concept, but it turns out to be closely related to the _3-framing_ (in the previous sense) of surfaces.

For $X$ a [[compact space|compact]], [[connected]], [[orientation|oriented]] 3-[[dimension|dimensional]] [[manifold]], write

$$
  2 T X := T X \oplus T X
$$

for the [[direct sum of vector bundles|fiberwise direct sum]] of the [[tangent bundle]] with itself. Via the [[diagonal]] embedding

$$
  SO(3) \to SO(3) \times SO(3) \hookrightarrow SO(6)
$$

this naturally induces a [[special orthogonal group|SO(6)]]-[[principal bundle]].

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

The underlying $SO(6)$-principal bundle of $2 T X$ always admits a [[lift of structure group|lift]] to a [[spin group|spin(6)]]-[[principal bundle]].

=--

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

By the sum-rule for [[Stiefel-Whitney classes]] (see at [SW class -- Axiomatic definition](Stiefel-Whitney+class#AxiomaticDefinition)) we have that

$$
  w_2(2 T X) = 2 w_0(T X) \cup w_2(T X) + w_1(T X) w_1(T X)
  \,.
$$

Since $T X$ is assumed [[orientation|oriented]], $w_1(T X) = 0$ (since this is the [[obstruction]] to having an [[orientation]]). So $w_2(2 T X) = 0 \in H^2(X,\mathbb{Z}_2)$ and since this in turn is the further [[obstruction]] to having a [[spin structure]], this does exist.

=--

Therefore the following definition makes sense

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

A **2-framing** in the sense of ([Atiyah 90](#Atiyah1990)) on a [[compact space|compact]], [[connected]], [[orientation|oriented]] 3-[[dimension|dimensional]] [[manifold]] $X$ is the [[homotopy class]] of a trivializations of the [[spin-group]]-[[principal bundle]] underlying [[Whitney sum|twice]] its [[tangent bundle]].

=--

More in detail, we may also remember the [[groupoid]] of 2-framings and the smooth structure on collections of them:

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

The [[moduli stack]] $At2\mathbf{Frame}$ is the [[homotopy pullback]] in

$$
  \array{
    At2\mathbf{Frame} &\to& *
    \\
    \downarrow && \downarrow
    \\
    \mathbf{B}SO(3) &\stackrel{}{\to}&
    \mathbf{B} Spin(6)
  }
$$

in [[Smooth∞Grpd]].

=--

In terms of this a 2-framing on $X$ with [[orientation]] $\mathbf{o} \colon X \to \mathbf{B}SO(3)$ is a lift $\hat {\mathbf{o}}$ in

$$
  \array{
    && At 2 \mathbf{Frame}
    \\
    & {}^{\mathllap{\hat {\mathbf{o}}}}\nearrow & \downarrow
    \\
    X &\stackrel{\mathbf{o}}{\to}& \mathbf{B}SO(3)
  }
  \,.
$$

## Properties

### Relation to bounding 4-manifolds
 {#RelationToBounding4Manifolds}

In ([Atiyah](#Atiyah1990)) it is shown how a framing on a compact connected oriented 3-manifold $X$ is induced by a 4-manifold $Z$ with [[boundary]] $\partial Z \simeq X$. In fact, a framing is equivalently a choice of [[cobordism]] class of bounding 4-manifolds ([Kerler](#Kerler)).

Discussion of 2-framing entirely in terms of bounding 4-manifolds is for instance in ([Sawin](#Sawin)).

### Relation to String-structures
 {#RelationToStringStructures}

By ([Atiyah 2.1](#Atiyah1990)) an Atiyah 2-framing of a 3-manifold $X$ is equivalently a
$p_1$-[[twisted differential c-structures|structure]], where $p_1$ is the first [[Pontryagin class]], hence is a homotopy class of a trivialization of

$$
  p_1(X) \colon X \to B SO(3) \stackrel{p_1}{\to} K(\mathbb{Z},4)
  \,.
$$

This perspective on Atiyah 2-framings is made explicit in ([Bunke-Naumann, section 2.3](#BunkeNaumann)). It is mentioned for instance also in ([Freed, page 6, slide 5](#Freed08)).

## Related concepts

* [[framed manifold]]

* [[p1-structure]], [[string structure]].

## References

The notion of "2-framing" in the sense of framing of the double of the tangent bundle is due to

* {#Atiyah1990} [[Michael Atiyah]], _On framings of 3-manifolds_ , Topology, Vol. 29, No 1, pp. 1-7 (1990) ([pdf](http://www.maths.ed.ac.uk/~aar/papers/atiyahfr.pdf))

making explicit a structure which slightly implicit in the discussion of the [[perturbation theory|perturbative]] [[path integral]] [[quantization of 3d Chern-Simons theory]] in

* [[Edward Witten]], _Quantum field theory and the Jones Polynomial_ , Comm. Math. Phys.  121 (1989)

reviewed for instance in

* {#Young} M. B. Young, section 2 of _Chern-Simons theory, knots and moduli spaces of connections_ ([pdf](http://www.math.sunysb.edu/~myoung/CS.pdf))

(see [Atiyah, page 6](#Atiyah)). For more on the role of 2-framings in [[Chern-Simons theory]] see also

* [[Daniel Freed]], Robert Gompf, _Computer calculation of Witten's 3-Manifold invariant_, Commun. Math. Phys. 141,79-117 (1991) ([pdf](http://www.maths.ed.ac.uk/~aar/papers/freedgompf.pdf))

* [[Gregor Masbaum]], section 2 of _Spin TQFT and the Birman-Craggs Homomorphism_, Tr. J. of Mathematics 19 (1995)  [pdf](http://journals.tubitak.gov.tr/math/issues/mat-95-19-2/pp-189-199.pdf)

* {#Freed08} [[Daniel Freed]], _Remarks on Chern-Simons theory_ ([arXiv:0808.2507](http://arxiv.org/abs/0808.2507), [pdf slides](http://www.ma.utexas.edu/users/dafr/MSRI_25.pdf))

and for discussion in the context of the [[M2-brane]] from p. 7 on in

* [[Hisham Sati]], _[[Geometric and topological structures related to M-branes]] II: Twisted $String$ and $String^c$-structures_  ([arXiv:1007.5419](http://arxiv.org/abs/1007.5419)).

The relation to $p_1$-structure is made explicit in

* {#BunkeNaumann} [[Ulrich Bunke]], [[Niko Naumann]], section 2.3 of _Secondary Invariants for String Bordism and tmf_, Bull. Sci. Math. 138 (2014), no. 8, 912–970 ([arXiv:0912.4875](http://arxiv.org/abs/0912.4875))

* C. Blanchet, N. Habegger, [[Gregor Masbaum]], [[Pierre Vogel]], _Topological quantum field theories derived from the Kauffman bracket_, Topology Vol 34, No. 4, pp. 883-927 (1995) ([pdf](http://www.maths.ed.ac.uk/~aar/papers/bhmv.pdf))

More discussion in terms of bounding 4-manifolds is in

* {#Kerler}Thomas Kerler, _Bridged links and tangle presentations of cobordism categories_. Adv. Math., 141(2):207–281, (1999) ([arXiv:math/9806114](http://arxiv.org/abs/math/9806114))

* {#Sawin} Stephen F. Sawin, _Three-dimensional 2-framed TQFTS and surgery_ (2004) ([pdf](http://digitalcommons.fairfield.edu/cgi/viewcontent.cgi?article=1020&context=mathandcomputerscience-facultypubs))

and page 9 of

* Stephen Sawin, _Invariants of Spin Three-Manifolds From Chern-Simons Theory and Finite-Dimensional Hopf Algebras_ ([arXiv:math/9910106](http://arxiv.org/abs/math/9910106)).

and more discussion for 3-manifolds with boundary includes

* {#KerlerLyubashenko01} Thomas Kerler, [[Volodymyr Lyubashenko]], section 1.6.1 of _Non-semisimple topological quantum field theories for 3-manifolds with corners_, Lecture notes in mathematics 2001

See also

* [[Greg Kuperberg]], _[MO comment](http://mathoverflow.net/a/4389/381)_

[[!redirects 2-framing]]
[[!redirects 2-framings]]