Adjointness for 2-Categories

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#### Category theory
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#### 2-category theory
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This entry is about the book

* [[John Gray]],

  _Formal category theory: adjointness for $2$-categories_

  Lecture Notes in Mathematics, Vol. 391.

  Springer-Verlag, Berlin-New York, 1974. xii+282 pp.

  [doi:10.1007/BFb0061280](https://doi.org/10.1007/BFb0061280)

on [[formal category theory]], which is [[category theory]] formulated "formally" via the [[2-category theory]] of the [[2-category]] [[Cat]] of all categories.

This is one of the most influential and comprehensive historical books in low-dimensional [[higher category theory]], following the spirit of:

* {#Law66} [[William Lawvere]], *The Category of Categories as a Foundation for Mathematics*, pp. 1-20 in: [[John W. Gray]], *Fibred and Cofibred Categories*, in: [[Samuel Eilenberg|S. Eilenberg]], [[D. K. Harrison]], [[S. MacLane]], [[H. Röhrl]] (eds.): *[[Proceedings of the Conference on Categorical Algebra - La Jolla 1965]]*, Springer (1966) [[doi:10.1007/978-3-642-99902-4](https://doi.org/10.1007/978-3-642-99902-4)]

More recently, this approach is echoed in [Riehl & Verity 13](homotopy+2-category+of+infinity-categories#RiehlVerity13), where [[Cat]] is enhanced to the [[homotopy 2-category of (∞,1)-categories]] in order to provide [[2-category theory|2-category theoretic]] foundations for [[(∞,1)-category theory]]. At least for [[presentable (infinity,1)-categories|presentable $\infty$-categories]], this is also obtained as the [[2-localization]] of the 2-category of [[combinatorial model categories]] and [[left Quillen functors]] at the [[Quillen equivalences]]: see at *[[2Ho(CombModCat)]]]*.

The book was supposed to be the first part of a four volume work, but unfortunately later volumes/chapters never appeared. It has some parts of 2- and 3-category theory; including the treatment of the famous [[Gray tensor product]] on 2-Cat. See also [[Gray-category]].

Unfortunately, due to changes in terminology, the book may be difficult to read nowadays.  Gray uses prefixes such as 'quasi,' 'iso,' and 'weak' to indicate various levels of weakness, but his choice of terminology is not entirely consistent, can be confusing, and is completely different from the standard modern terminology which uses 'lax,' 'oplax,' and 'pseudo' with (mostly) precise and consistent meanings.

The following is a list of some of the definitions given in the book,
along with their modern names and links to nLab entries.

* Section 2

  * _2-category_ (see [[strict 2-category]])
  * _2-functor/$Cat$-functor_ (i.e. [[strict 2-functor]])
  * _Cat-natural transformation_ (see [[enriched functor category]])
  * _Quasi-natural transformation_ ([[lax natural transformation]])
  * [[modification|Modification]]
  * [[2-comma category]]
  * [[double category|Double category]]

* Section 3
  * [[bicategory|Bicategory]]
  * _Pseudo-functor_ ([[lax 2-functor]]), _homomorphic pseudo-functor_
  ([[pseudofunctor]]), _strict pseudo-functor_ (= strict 2-functor -- see [[enriched functor]])

* Section 4

  * $Fun(A,B)$ and $Pseud(A,B)$ (see [[Gray tensor product]])

* Section 6
  * _Adjoint morphisms_ in a 2-category ([[adjunction]])
  * [[Kan extension]] in a 2-category

* Section 7

  * _Transcendental quasi-adjunction_ ([[lax 2-adjunction]])

category: reference

[[!redirects Gray-adjointness-for-2-categories]]


Adjointness in Foundations

This page collects links related to the text

* [[William Lawvere]],

  _Adjointness in Foundations_,

  Dialectica 23 (1969), 281-296,

  Reprints in Theory and Applications of Categories, No. 16 (2006), pp 1-16 (revised 2006-10-30)

  ([TAC](http://www.emis.de/journals/TAC/reprints/articles/16/tr16abs.html))

on [[foundations of mathematics]] in [[categorical logic]] and the role of [[adjoints]] in the formalization of [[quantifiers]] ([[base change]]). The main point is that the logical operations on [[propositions]]

[[existential quantifier]] $\dashv$ [[context extension]] $\dashv$ [[universal quantifier]]

constitute an [[adjoint triple]]. In [[type theory]] this is lifted from operations on propositions to operations on [[types]], where it becomes

[[dependent sum]] $\dashv$ [[context extension]] $\dashv$ [[dependent product]].

In [[topos theory]] and [[geometry]] this adjoint triple is often known as _[[base change]]_.

## Related articles

* [[William Lawvere]], _[[Equality in hyperdoctrines and comprehension schema as an adjoint functor]]_, Proceedings of the AMS Symposium on Pure Mathematics XVII (1970), 1-14. ([pdf](https://ncatlab.org/nlab/files/LawvereComprehension.pdf))

* {#Lambek} [[Joachim Lambek]], _The Influence of [[Heraclitus]] on Modern Mathematics_, In _Scientific Philosophy Today: Essays in Honor of Mario Bunge_, edited by Joseph Agassi and Robert S Cohen, 111–21. Boston: D. Reidel Publishing Co. (1981)

## Related $n$Lab entries

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

* [[adjoint modality]]

category: reference

[[!redirects Adjointness in Foundations]]
[[!redirects Adjointness in foundations]]


adjoints preserve (co-)limits

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### Context
#### Category theory
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#### Limits and colimits
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#Contents#
* table of contents
{:toc}

## Idea

One of the basic facts of [[category theory]] is that [[left adjoint|left]]/[[right adjoint|right]] [[adjoint functors]] [[preserved limit|preserves]] [[colimits|co]]/[[limits]], respectively.

## Statement

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

Let $\mathcal{C}$ and $\mathcal{D}$ be two [[categories]] and let

$$
  (L \dashv R)
  \;\colon\;
  \mathcal{C}
    \underoverset
      {\underset{R}{\longrightarrow}}
      {\overset{L}{\longleftarrow}}
      {\bot}
  \mathcal{D}
$$

be a pair of [[adjoint functors]] between them.

Then

1. If $X \colon \mathcal{I} \to \mathcal{C}$ is a [[diagram]] whose [[limit]] $\underset{\longleftarrow}{\lim}_{i} X_i$ exists in $\mathcal{C}$, then this limit is preserved by the [[right adjoint]] $R$ in that there is a [[natural isomorphism]]

   $$
     R
     \left(
       \underset{\longleftarrow}{\lim}_i \left(X_i\right)
     \right)
     \;\simeq\;
     \underset{\longleftarrow}{\lim}_i \left( R(X_i) \right)
     \,,
   $$

   where on the right we have the [[limit]] in $\mathcal{D}$ over the [[diagram]] $R \circ X \colon \mathcal{I} \overset{X}{\longrightarrow} \mathcal{C} \overset{R}{\longrightarrow} \mathcal{D}$.

1. If $X \colon \mathcal{I} \to \mathcal{D}$ is a [[diagram]] whose [[colimit]] $\underset{\longrightarrow}{\lim}_{i} X_i$ exists in $\mathcal{D}$, then this colimit is preserved by the [[left adjoint]] $L$ in that there is a [[natural isomorphism]]

   $$
     L
     \left(
       \underset{\longrightarrow}{\lim}_i \left(X_i\right)
     \right)
     \;\simeq\;
     \underset{\longrightarrow}{\lim}_i \left( L(X_i) \right)
     \,,
   $$

   where on the right we have the [[colimit]] in $\mathcal{C}$ over the [[diagram]] $L \circ X \colon \mathcal{I} \overset{X}{\longrightarrow} \mathcal{D} \overset{L}{\longrightarrow} \mathcal{C}$.

=--

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

We show the first statement, the proof of the second is [[formal dual|formally dual]].

We use the following facts

1. There is a [[natural isomorphism]], $Hom_{\mathcal{C}}(L(d),c) \simeq Hom_{\mathcal{D}}(d,R(c))$; this equivalently characterizes the fact that $(L \dashv R)$ is a pair of [[adjoint functors]];

1. ([[hom-functor preserves limits]]) The [[hom-functor]] sends colimits in the first argument and limits in the second argument to limits of [[hom-sets]]

   $$
      Hom\left( X, \underset{\longleftarrow}{\lim}_i X_i \right)
     \simeq
     \underset{\longleftarrow}{\lim}_i Hom\left(X,X_i\right)
   $$

   and

   $$
     Hom\left(\underset{\longrightarrow}{\lim}_i X_i, X\right)
     \simeq
     \underset{\longleftarrow}{\lim} \left(Hom\left(X_i,X\right) \right)
     \,.
   $$

   Again, this is essentially by definition of [[limits]]/[[colimits]].

1. ([[Yoneda lemma]]) If for two objects $X$ and $Y$ in some [[category]] the [[hom-sets]] out of or into these objects (their [[representable functors]]) are [[naturally isomorphic]], then the two objects are isomorphic.

Now using the first two items, we obtain the following chain of [[natural isomorphisms]], for every [[object]] $Y \in \mathcal{D}$:

$$
  \begin{aligned}
    Hom_{\mathcal{D}}\left( Y, R \left( \underset{\longleftarrow}{\lim}_i X_i \right) \right)
    & \simeq
    Hom_{\mathcal{C}}\left( L(Y), \underset{\longleftarrow}{\lim}_i X_i\right)
    \\
    & \simeq
    \underset{\longleftarrow}{\lim}_i \left( Hom_{\mathcal{C}}\left(L\left(Y\right), X_i\right)\right)
    \\
    & \simeq
    \underset{\longleftarrow}{\lim}_i \left(Hom_{\mathcal{D}}\left(Y, R\left(X_i\right)\right)\right)
    \\
    & \simeq
    Hom_{\mathcal{D}}\left(Y, \underset{\longleftarrow}{\lim}_i \left(R\left(X_i\right) \right) \right)
  \end{aligned}
  \,.
$$

Hence the third item above, the [[Yoneda lemma]], implies the claim.

=--

## Related entries

* [[hom-functor preserves limits]]

* [[limits preserve limits]]

* [[limits of presheaves are computed objectwise]]

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

* [[limits and colimits by example]]

[[!redirects left adjoints preserve colimits and right adjoints preserve limits]]
[[!redirects right adjoints preserve limits and left adjoints preserve colimits]]

[[!redirects left adjoints preserve colimits]]
[[!redirects right adjoints preserve limits]]


adjunct

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#Contents#
* table of contents
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## Definition

A pair

$$
  (L \dashv R) : C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D
$$

of [[adjoint functor]]s between [[category|categories]] $C$ and $D$, is characterized by a [[natural isomorphism]]

$$
  C(L X,Y) \cong D(X,R Y)
$$

of [[hom-set]]s for [[object]]s $X\in D$ and $Y\in C$.  Two [[morphism]]s
$f:L X \to Y$ and $g : X \to R Y$ which correspond under this bijection are said to be **adjuncts** of each other.  That is, $g$ is the (right-)adjunct of $f$, and $f$ is the (left-)adjunct of $g$.  Sometimes one writes $g = f^\sharp$ and $f = g^\flat$, as in musical notation.

Sometimes people call $\tilde f$ the "adjoint" of $f$, and vice versa, but this is potentially confusing because it is the _functors_ $F$ and $G$ which are adjoint.  Other possible terms are _conjugate_, _transpose_, and [[mate]].

## Properties

+-- {: .num_prop #AdjunctionCoUnitGiveesAllAdjuncts}
###### Proposition
**(adjuncts in terms of adjunction (co-)unit)**

Let $\eta_X \colon X \to R L X$ be the [[unit of an adjunction|unit of the adjunction]] and $\epsilon_X \colon L R X \to X$ the [[counit of an adjunction|counit]].

Then

* the adjunct of $f \colon X \to R Y$ in $D$ is the composite

  $$
    \tilde f
      \colon
    L X
     \overset{L f}{\longrightarrow}
    L R Y
     \overset{\epsilon_Y}{\longrightarrow}
    Y
  $$

* the adjunct of $g \colon L X \to Y$ in $C$ is the composite

  $$
    \tilde g
      \colon
    X
      \stackrel{\eta_X}{\longrightarrow}
    R L X
      \overset{R g}{\longrightarrow}
    R Y
    \,.
  $$

=--

For proof see [this Prop.](adjoint+functor#GeneralAdjunctsInTermsOfAdjunctionUnitCounit) at *[[adjoint functor]]*.

## Examples

* The process of [[currying]] is an instance of passage to adjuncts, specialized to the tensor-hom adjunction of a [[closed monoidal category]].

## Related concepts

* [[adjunction]]

* [[zig-zag law]]/[[triangle identity]]

* [[unit of an adjunction]]

* [[mate]]

* **adjunct**

## References

* [[Categories Work]], second edition, p. 81

* [[Category Theory in Context]], p. 116, 124.

[[!redirects adjuncts]]


adjunction

> This page is about adjunctions in general [[2-categories]]. Specifically for the common case of adjunctions in [[Cat]] see at *[[adjoint functors]]*. For the notion of "adjunction of a set to a field" in [[field theory]], see [[field extension]].

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#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

A [[pair]] of [[1-morphisms]] in a [[2-category]] form an **adjunction** if they are **dual** to each other ([Lambek (1982)](#Lambek82), cf. [here](geometry+of+physics+--+categories+and+toposes#CategoryTheoryIsTheoryOfDuality)) in a precise sense.

There are two archetypical classes of examples:

* If $A$ is a [[monoidal category]] and $\mathbf{B}A$ denotes the [[singleton|one-]][[object]] 2-category whose single [[hom-category]] is $A$ (the [[delooping]] of $A$), then the notion of adjoint morphisms in $\mathbf{B}A$ coincides precisely with the notion of [[dualizable object|dual objects]] in $A$, subsuming, in turn, classical examples such as [[dual vector spaces]] in the case that $A =  $ [[FinDimVect]].

* Adjunctions in the [[2-category]] [[Cat]] of [[categories]] are  _[[adjoint functors]]_.

  * Notice that essentially everything that makes [[category theory]] nontrivial and interesting, beyond [[groupoid]]-theory, is governed by the concept of [[adjoint functors]]. In particular [[universal constructions]] such as [[limit|limits and colimits]], [[Kan extensions]], ([[coend|co]])[[ends]] are examples of adjunctions in [[Cat]].

  * Similarly, for $V$ any [[cosmos for enrichment]], adjunctions in the 2-category [[VCat]] of $V$-[[enriched categories]] are equivalently [[enriched adjoint functors]]. Already in simple cases such $V = $ [[truth values]] this subsumes classical concepts such as that of [[Galois connections]].

  * Remarkably, even adjunctions in the [[homotopy 2-category]] of [[(infinity,1)-categories|$(\infty,1)$-categories]] are equivalent to [[adjoint (infinity,1)-functors|adjoint $\infty$-functors]], see the examples [below](#AdjointInfinityOneFunctors).

  These classes of examples make adjunctions a key notion in *[[formal category theory]]*.

Finally, the notion of *adjunction* may usefully be thought of as a generalization of the notion of *[[equivalence in a 2-category]]*: an adjoint [[1-morphism]] need not be [[invertible morphism|invertible]] (even up to [[2-isomorphism]]) but it does have, in a precise sense, _a left inverse from below_ or _a right inverse from above_.

If an adjoint 1-morphisms happens to be a genuine [[equivalence in a 2-category]], then the adjunction is called an *[[adjoint equivalence]]*.

## Definition
 {#Definition}

\begin{defn} \label{DefinitionAdjunction}
An _adjunction_ in a [[2-category]] is

* a [[pair]] of [[objects]] $C$ and $D$

* a [[pair]] of [[1-morphisms]]

  $L \colon C \longrightarrow D$ (the *[[left adjoint]]*)

  $R \colon D \longrightarrow C$ (the *[[right adjoint]]*)

* a [[pair]] of [[2-morphisms]]

  $\eta \colon 1_C \longrightarrow R \circ L$ (the *[[adjunction unit]]*)

  $\epsilon \colon L \circ R \longrightarrow 1_D$ (the *[[adjunction counit]]*)

such that the following equivalent conditions hold:

* **[[triangle identity]]** the following [[commuting diagram|diagrams commute]] in the [[hom-categories]] (where "$\cdot$" denotes [[whiskering]]):

\begin{centre}
  \begin{tikzcd}
    L \ar[r, "L \cdot \eta"]
    \ar[dr, swap, "\mathrm{id}"]
    & L \circ R \circ L \ar[d, "\epsilon \cdot L"]
    \\ & L
  \end{tikzcd}
\end{centre}

\begin{centre}
  \begin{tikzcd}
    R \ar[r, "\eta \cdot R"] \ar[dr, swap, "id"] & R \circ L \circ R \ar[d, "R \cdot \epsilon"]
   \\ & R
  \end{tikzcd}
\end{centre}

* **[[zig-zag law]]** the following [[equality]] of [[2-morphisms]]:

\begin{tikzcd}[row sep=10pt]
  C
  \ar[rr, bend left=40, "{\mathrm{id}}",
      "{\ }"{name=s1, swap}
  ]
  \ar[r, "L"{description}, "{\ }"{name=t1, pos=.9}]
  \ar[
    from=s1,
    to=t1,
    Rightarrow,
    "{ \eta }"{swap}
  ]
  &
  D
  \ar[rr, bend right=40, "{\mathrm{id}}"{swap},
    "{\ }"{name=t2}
  ]
  \ar[r, "R"{description}]
  &
  C
  \ar[r, "L"{description}, "{\ }"{swap, name=s2, pos=.1}]
  \ar[
    from=s2,
    to=t2,
    Rightarrow,
    "{ \epsilon }"
  ]
  &
  D
  &=&
  C
  \ar[r, bend left=40, "{L}", "{\ }"{name=s3, swap}]
  \ar[r, bend right=40, "{L}"{swap}, "{\ }"{name=t3}]
  \ar[from=s3, to=t3, Rightarrow, "{\mathrm{id}}"]
  &
  D
\end{tikzcd}

\begin{tikzcd}[row sep=10pt]
  D
  \ar[rr, bend right=40, "{\mathrm{id}}"{swap},
      "{\ }"{name=s1}
  ]
  \ar[r, "R"{description}, "{\ }"{name=t1, pos=.9, swap}]
  \ar[
    from=t1,
    to=s1,
    Rightarrow,
    "{ \epsilon }"{swap, pos=.3}
  ]
  &
  C
  \ar[rr, bend left=40, "{\mathrm{id}}",
    "{\ }"{name=t2, swap}
  ]
  \ar[r, "L"{description}]
  &
  D
  \ar[r, "R"{description}, "{\ }"{name=s2, pos=.1}]
  \ar[
    from=t2,
    to=s2,
    Rightarrow,
    "{ \eta }"{pos=.8}
  ]
  &
  C
  &=&
  D
  \ar[r, bend left=40, "{R}", "{\ }"{name=s3, swap}]
  \ar[r, bend right=40, "{R}"{swap}, "{\ }"{name=t3}]
  \ar[from=s3, to=t3, Rightarrow, "{\mathrm{id}}"]
  &
  C
\end{tikzcd}

{#InTermsOfStringDiagrams} In terms of [[string diagrams]] the above data entering the definition looks like

[[adjunction-L.png:pic]]          [[adjunction-R.png:pic]]          [[adjunction-unit.png:pic]]          [[adjunction-co-unit.png:pic]]

(where 1-cells read from right to left and 2-cells from bottom to top), and the [[zig-zag identities]] appear as moves "pulling zigzags straight" (hence the name):

[[adjunction-up-string.png:pic]]          [[adjunction-down-string.png:pic]]

Often, arrows on strings are used to distinguish $L$ and $R$, and most or all other labels are left implicit; so the zigzag identities, for instance, become:

[[adjunction-up-string-minimal.png:pic]]          [[adjunction-down-string-minimal.png:pic]]

\end{defn}

## Properties

### Relation to monads

See at _[monad -- Relation between adjunctions and monads](monad#RelationBetweenAdjunctionsAndMonads)_.

## Examples

### Adjoint functors
  {#AdjointFunctors}

\begin{proposition}\label{AdjunctionsInCatAreAdjointFunctors}
An adjunction in the 2-category [[Cat]] of [[categories]], [[functors]] and [[natural transformations]] is equivalently a pair of [[adjoint functors]].
\end{proposition}
See also the proof [here](adjoint+functor#AdjointnessInTermsOfHomIsomorphismEquivalentToAdjunctionInCat) at *[[adjoint functor]]*.
\begin{proof}
Suppose given functors $L \,\colon\, C \to D$, $R: D \to C$ and the structure of a pair of [[adjoint functors]] in the form of a natural isomorphism of [[hom-sets]] ([here](adjoint+functor#InTermsOfHomIsomorphism))

$$
  \Psi_{c, d}
  \;\colon\;
  \hom_D(L(c), d) \cong \hom_C(c, R(d))
$$

Now the idea is that, in the spirit of the (proof of the) [[Yoneda lemma]], we would like $\Psi$ to be determined by what it does to [[identity morphisms]]. With that in mind, define the [[adjunction unit]] $\eta \colon 1_C \to R L$ by the formula $\eta_c = \Psi_{c, L(c)}(1_{L(c)})$. Dually, define the counit $\varepsilon \,\colon\, L R \to 1_D$ by the formula
$$
  \varepsilon_d
  \,\coloneqq\, \Psi^{-1}_{R(d), d}(1_{R(d)})
  \,.
$$

Then given $g \,\colon\, L(c) \to d$, the claim is that

$$
  \Psi_{c, d}(g)
    \,=\,
  (c \stackrel{\eta_c}{\to} R(L(c)) \stackrel{R(g)}{\to} R(d))
  \,.
$$

This may be left as an exercise in the yoga of the Yoneda lemma, applied to $\hom_D(L(c), -) \to \hom_C(c, R(-))$. By [[formal duality]], given $f \,\colon\, c \to R(d)$,

$$
  \Psi^{-1}_{c, d}(f) = (L(c) \stackrel{L(f)}{\to} L(R(d)) \stackrel{\varepsilon_d}{\to} d)
  \,.
$$

(We spell out the Yoneda-lemma proof of this dual form [below](#YonedaLemmaArgument).)

Finally, these operations should obviously be mutually inverse, but that can again be entirely encapsulated Yoneda-wise in terms of the effect on identity maps. Thus, if $\eta_c \coloneqq \Psi_{c, L(c)}(1_{L(c)})$, via the recipe just given for $\Psi^{-1}$ we recover

$$1_{L(c)} = (L(c) \stackrel{L(\eta_c)}{\to} L R L(c) \stackrel{\varepsilon_{L(c)}}{\to} L(c))$$

and this is one of the famous [[triangle identities]]: $1_L = (L \stackrel{L \eta}{\to} L R L \stackrel{\varepsilon L}{\to} L)$.  Here, juxtaposition of functors and natural transformations denotes neither functor application, nor vertical composition, nor horizontal composition, but [[whiskering]]. By duality, we have the other [[triangle identity]] $1_R = (R \stackrel{\eta R}{\to} R L R \stackrel{R \varepsilon}{\to} R)$. These two triangular equations are enough to guarantee that the recipes for $\Psi$ and $\Psi^{-1}$ indeed yield mutual inverses.

In conclusion it is perfectly sufficient to define an adjoint pair of functors in $Cat$ as given by unit and counit transformations $\eta: 1_C \to R L$, $\varepsilon: L R \to 1_D$, satisfying the triangle identities above.
\end{proof}

\begin{remark}
The definition of adjunctions via [[unit of an adjunction|units]] and [[counit of an adjunction|counits]] is an "elementary" definition (so that by implication, the formulation in terms of [hom-isomorphisms](adjoint functor#InTermsOfHomIsomorphism) is not elementary) in the sense that while the [[hom-functor]] formulation relies on some notion of [[hom-set|hom-*set*]], the formulation in terms of units and counits is purely in the [[first-order logic|first-order]] [[language]] of [[categories]] and makes no reference to a model of [[set theory]]. The definition via (co)units therefore gives us a definition of adjunctions even if an assumption of  [[locally small|local smallness]] is not made.
\end{remark}

\begin{proof}\label{YonedaLemmaArgument}
**(Yoneda-lemma argument)**
\linebreak
We claim that $\Psi^{-1}_{c, d} \,\colon\, \hom_C(c, R(d)) \to \hom_D(L(c), d)$ can be defined by the formula

$$\Psi^{-1}_{c, d}(f: c \to R(d)) = (L(c) \stackrel{L(f)}{\to} L(R(d)) \stackrel{\varepsilon_d}{\to} d)$$

where $\varepsilon_d \coloneqq \Psi^{-1}_{R(d), d}(1_{R(d)})$. This is by appeal to the proof of the Yoneda lemma applied to the transformation

$$\Psi^{-1}_{-, d}: \hom_C(-, R(d)) \to \hom_D(L(-), d)$$

For the naturality of $\Psi^{-1}$ in the argument $(-)$ would imply that given $f: c \to R(d)$, we have a commutative square

$$\array{
  \hom_C(R(d), R(d))
  &
  \stackrel{\Psi^{-1}_{R(d), d}}{\to}
  & \hom_D(L(R(d)), d)
  \\
  \hom_C(f, R(d))
  \big\downarrow
  & &
  \big\downarrow \hom_D(L(f), d)
  \\
  \hom_C(c, R(d))
  &
  \underset{\Psi^{-1}_{c, d}}{\to}
  &
  \hom_D(L(c), d)
}$$

Chasing the element $1_{R(d)}$ down and then across, we get $f: c \to R(d)$ and then $\Psi^{-1}_{c, d}(f)$. Chasing across and then down, we get $\varepsilon_d$ and then $\varepsilon_d \circ L(f)$. This completes the verification of the claim.
\end{proof}

\begin{corollary}
An adjunction in its [[core]] [[2-groupoid]] $Core(Cat)$ is an [[adjoint equivalence]].
\end{corollary}

### Enriched adjoint functors

Similarly one sees:
\begin{example}\label{EnrichedAdjointFunctors}
  For $V$ a [[cosmos for enrichment]], an adjunction in the [[2-category]] [[VCat]] of $V$-[[enriched categories]] is equivalently a pair $V$-[[enriched functors]].
\end{example}

\begin{example}
Let $U \,\colon\, Grp \to Set$ from [[Grp]] to [[Set]] denote the usual [[forgetful functor]] from [[Grp]] to [[Set]]. When we say "$F(X)$ is the [[free group]] generated by a set $X$", we mean there is a function $\eta_X \,\colon\, X \to U(F(X))$ which is universal among functions from $X$ to the underlying set of a group, which means in turn that given a function $f: X \to U(G)$, there is a unique group homomorphism $g \colon F(X) \to G$ such that

$$
  f = (X \stackrel{\eta_X}{\to} U(F(X)) \stackrel{U(g)}{\to} U(G))
$$

Here $\eta_X$ is a component of what we call the [[unit of an adjunction|unit of the adjunction]] $F \dashv U$, and the equation above is a recipe for the relationship between the map $g: F(X) \to G$ and the map $f: X \to U(G)$ in terms of the unit.
\end{example}

### Adjoint $(\infty,1)$-functors
 {#AdjointInfinityOneFunctors}

\begin{proposition}
\label{AdjointInfinityFunctorsInHomotopy2Category}
  An adjunction in the [[homotopy 2-category of (infinity,1)-categories|homotopy 2-category of $(\infty,1)$-categories]] is equivalently a pair of [[adjoint (infinity,1)-functors|adjoint $(\infty,1)$-functors]].
\end{proposition}
([Riehl & Verity 2015, Rem. 4.4.5](adjoint+infinity1-functor#RiehlVerity15); [Riehl & Verity 2022, Prop. F.5.6](adjoint+infinity1-functor#RVElements); see [there](adjoint+infinity1-functor#InTheHomotopy2Category) for more).

\begin{remark}
  In view of Prop. \ref{AdjunctionsInCatAreAdjointFunctors},
  the remarkable aspect of Prop. \ref{AdjointInfinityFunctorsInHomotopy2Category} is that the [[homotopy 2-category of (infinity,1)-categories|homotopy 2-category of $\infty$-categories]] is sufficient to detect adjointness of [[(infinity,1)-functors|$\infty$-functors]], which would, *a priori*, be defined as a kind of [[higher homotopy]]-[[coherence law|coherent]] adjointness in the full [[(infinity,2)-category|$(\infty,2)$-category]] [[(infinity,1)Cat|$Cat_{(\infty,1)}$]]. For more on this reduction of homotopy-coherent adjunctions to plain adjunctions see [Riehl & Verity 2016, Thm. 4.3.11, 4.4.11](adjoint+infinity1-functor#RiehlVerity16).
\end{remark}

## Related concepts

* [[adjoint functor]], [[adjoint (∞,1)-functor]]

* [[left adjoint]], [[right adjoint]]

* [[duality]]

* [[adjunct]]

* [[dual adjunction]], [[Galois connection]]

* [[monoidal adjunction]]

* [[enriched adjunction]]

* [[derived adjunction]]

* [[adjoint monad]]

* [[heteromorphism]]

* [[adjoint string]]

  * [[adjoint triple]], [[adjoint quadruple]]

* [[ambidextrous adjunction]]

* [[adjoint cylinder]]

* [[adjoint logic]]

## References

Adjunctions in 2-categories were introduced (together with the notion of [[strict 2-categories]] itself) in:

* [[Jean-Marie Maranda]], pp. 762 in: *Formal categories*, Canadian Journal of Mathematics **17** (1965) 758-801 [[doi:10.4153/CJM-1965-076-0](https://doi.org/10.4153/CJM-1965-076-0), [pdf](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/A7C463460EB8CAC64C2CA340F870CF80/S0008414X00039729a.pdf/formal-categories.pdf)]

Though see also the following, which uses more modern terminology:

* [[Max Kelly]], §2 in: *Adjunction for enriched categories*, in: *Reports of the Midwest Category Seminar III*, Lecture Notes in Mathematics **106**, Springer (1969) [[doi:10.1007/BFb0059145](https://doi.org/10.1007/BFb0059145)]

A thorough 2-categorical account is contained in:

* [[Claude Auderset]], _Adjonctions et monades au niveau des $2 $-catégories_, Cahiers de topologie et géométrie différentielle 15.1 (1974): 3-20.

* [[Stephen Schanuel]] and [[Ross Street]], _The free adjunction_, Cahiers de topologie et géométrie différentielle catégoriques 27.1 (1986): 81-8

Review:

* [[Saunders MacLane]], §XII.4 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

* {#Lack10} [[Steve Lack]], §2.1 in: *A 2-categories companion*,  in: *[[Towards Higher Categories]]*, The IMA Volumes in Mathematics and its Applications **152** Springer (2010) [[arXiv:math.CT/0702535](http://arxiv.org/abs/math.CT/0702535), [doi:10.1007/978-1-4419-1524-5_4](https://doi.org/10.1007/978-1-4419-1524-5_4)]

* {#JohnsonYau20} [[Niles Johnson]], [[Donald Yau]], Chapter 6 of: _2-Dimensional Categories_, Oxford University Press 2021 ([arXiv:2002.06055](http://arxiv.org/abs/2002.06055), [doi:10.1093/oso/9780198871378.001.0001](https://oxford.universitypressscholarship.com/view/10.1093/oso/9780198871378.001.0001/oso-9780198871378))

Fr the special case of [[adjoint functors]] see any text on [[category theory]] (and see the references at _[[adjoint functor]]_), for instance:

* {#Borceux94} [[Francis Borceux]], Vol 1, Section 3 of _[[Handbook of Categorical Algebra]]_

* _[[geometry of physics -- categories and toposes]] -- [Adjunctions](https://ncatlab.org/nlab/show/geometry+of+physics+--+categories+and+toposes#Adjunctions)_

For some early history and illustrative examples see

* {#Lambek82} [[Joachim Lambek]], _The Influence of Heraclitus on Modern Mathematics_, In _Scientific Philosophy Today: Essays in Honor of Mario Bunge_, edited by Joseph Agassi and Robert S Cohen, 111–21. Boston: D. Reidel Publishing Co. (1982) ([doi:10.1007/978-94-009-8462-2_6](https://link.springer.com/chapter/10.1007/978-94-009-8462-2_6))

  (more along these lines at _[[objective logic]]_).

The fundamental role of [[adjoint functors]] in [[logic]]/[[type theory]] originates with the observaiton that [[substitution]] forms an [[adjoint triple]] with [[existential quantification]] and [[universal quantification]]:

* {#Lawvere69} [[William Lawvere]], _Adjointness in Foundations_, ([tac:16](http://www.emis.de/journals/TAC/reprints/articles/16/tr16abs.html)), Dialectica 23 (1969), 281-296

* [[William Lawvere]], _Quantifiers and sheaves_, Actes, Congrès intern, math., 1970. Tome 1, p. 329 à 334 ([pdf](http://www.mathunion.org/ICM/ICM1970.1/Main/icm1970.1.0329.0334.ocr.pdf))

Adjunctions in [[programming languages]] (though mainly again just [[adjoint functors]]):

* {#Hinze12} Ralf Hinze, _Generic Programming with Adjunctions_,  In: J. Gibbons  (ed.) _Generic and Indexed Programming_ Lecture Notes in Computer Science, vol 7470. Springer 2012 ([pdf](http://www.cs.ox.ac.uk/ralf.hinze/LN.pdf), [slides](http://www.cs.ox.ac.uk/ralf.hinze/SSGIP10/Slides.pdf) [doi:10.1007/978-3-642-32202-0_2](https://doi.org/10.1007/978-3-642-32202-0_2))

* Jeremy Gibbons, Fritz Henglein, Ralf Hinze, Nicolas Wu, _Relational Algebra by Way of Adjunctions_, Proceedings of the ACM on Programming Languages archive Volume 2 Issue ICFP, September 2018 Article No. 86  ([pdf](https://www.cs.ox.ac.uk/jeremy.gibbons/publications/reladj.pdf), [doi:10.1145/3236781](https://dl.acm.org/citation.cfm?doid=3243631.3236781))

See also

* Wikipedia, *[Adjoint Functors](http://en.wikipedia.org/wiki/Adjoint_functors)*

* Catsters, _Adjunctions_ ([YouTube](http://www.youtube.com/watch?v=loOJxIOmShE&feature=channel_page))

-

[[!redirects adjoint pair]]
[[!redirects adjoint pairs]]
[[!redirects adjunctions]]


adjunction > zigzagepsilon

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adjunction > zigzageta

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adjunction > zigzageta > history

< [[adjunction > zigzageta]]
[[!redirects adjunction/zigzag]]
[[!redirects adjunction/zigzageta]]
[[!redirects adjunction/zigzag -- history]]


+-- {: .num_prop #AdjunctionBetweenTopologicalSpacesAndDiffeologicalSpaces}
###### Proposition
**([[adjunction between topological spaces and diffeological spaces]])**

There is a pair of [[adjoint functors]]

\[
  \label{AdjointFunctorsBetweenTopSpAndDifflgSp}
  TopSp
  \underoverset{
    \underset{
      Cdfflg
    }{\longrightarrow}
  }{
    \overset{
      Dtplg
    }{\longleftarrow}
  }{\phantom{AA}\bot\phantom{AA}}
  DifflgSp
\]

between the [[categories]] of [[Top|TopologicalSpaces]] and of [[DiffeologicalSpaces]], where

* $Cdfflg$ takes a [[topological space]] $X$ to the **continuous diffeology**, namely the diffeological space on the same underlying set $X_s$ whose plots $U_s \to X_s$ are the [[continuous functions]] (from the underlying [[topological space]] of the [[domain]] $U$).

* $Dtplg$ takes a [[diffeological space]] to the **diffeological topology** ([[D-topology]]), namely the [[topological space]] with the same underlying set $X_s$ and with the [[final topology]] that makes all its plots $U_{s} \to X_{s}$ into [[continuous functions]]: called the _[[D-topology]]_.

  Hence a [[subset]] $O \subset \flat X$ is an [[open subset]] in the [[D-topology]] precisely if for each plot $f \colon U \to X$ the [[preimage]] $f^{-1}(O) \subset U$ is an [[open subset]] in the [[Cartesian space]] $U$.

Moreover:

1. the [[fixed point of an adjunction|fixed points of this adjunction]] $X \in$[[Top|TopologicalSpaces]] (those for which the [[counit of an adjunction|counit]] is an [[isomorphism]], hence here: a [[homeomorphism]]) are precisely the [[Delta-generated topological spaces]] ([i.e.](Delta-generated+topological+space#AsDTopologicalSpaces) [[D-topological spaces]]):

   $$
     X \;\,\text{is}\;\Delta\text{-generated}
     \;\;\;\;\;
     \Leftrightarrow
     \;\;\;\;\;
     Dtplg(Cdfflg(X))
     \underoverset{\simeq}{\;\;\epsilon_X\;\;}{\longrightarrow}
     X
   $$

1. this is an [[idempotent adjunction]], which exhibits $\Delta$-generated/[[D-topological spaces]] as a [[reflective subcategory]] inside [[diffeological spaces]] and a [[coreflective subcategory]] inside all [[topological spaces]]:

\[
  \label{DeltaGeneratedSpacesInIdempotentAdjunction}
  TopologicalSpaces
  \underoverset
    {
      \underset{
       Cdfflg
      }{\longrightarrow}
    }
    {
      \overset{
      }{\hookleftarrow}
    }
    {\phantom{AA}\bot\phantom{AA}}
  DTopologicalSpaces
  \underoverset
    {
      \underset{
      }{\hookrightarrow}
    }
    {
      \overset{
       Dtplg
      }{\longleftarrow}
    }
    {\phantom{AA}\bot\phantom{AA}}
  DiffeologicalSpaces
\]

Finally, these adjunctions are a sequence of [[Quillen equivalences]] with respect to the:

|  |  |  |
|--|--|--|
| [[classical model structure on topological spaces]] | [[model structure on D-topological spaces]] | [[model structure on diffeological spaces]] |

> Caution: There was a gap in the original proof that $DTopologicalSpaces \simeq_{Quillen} DiffeologicalSpaces$. The gap is claimed to be filled now, see the commented references [here](model+structure+on+diffeological+spaces#References).

=--

Essentially these adjunctions and their properties are observed in [Shimakawa, Yoshida & Haraguchi 2010, Prop. 3.1, Prop. 3.2, Lem. 3.3](diffeological+space#SYH10), see also [Christensen, Sinnamon & Wu 2014, Sec. 3.2](D-topology#CSW13). The model structures and Quillen equivalences are due to [Haraguchi 13, Thm. 3.3](#model+structure+on+Delta-generated+topological+spaces#Haraguchi13) (on the left) and [Haraguchi-Shimakawa 13, Sec. 7](model+structure+on+diffeological+spaces#HaraguchiShimakawa13) (on the right).

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

We spell out the existence of the [[idempotent adjunction]] (eq:DeltaGeneratedSpacesInIdempotentAdjunction):

First, to see we have an [[adjunction]] $Dtplg \dashv Cdfflg$, we check the hom-isomorphism ([here](adjoint+functor#eq:HomIsomorphismForAdjointFunctors)).

Let $X \in DiffeologicalSpaces$ and $Y \in TopologicalSpaces$. Write $(-)_s$ for the underlying sets. Then a [[morphism]], hence a [[continuous function]] of the form

$$
  f \;\colon\; Dtplg(X) \longrightarrow Y
  \,,
$$

is a [[function]] $f_s \colon X_s \to Y_s$ of the underlying [[sets]] such that for every [[open subset]] $A \subset Y_s$ and every [[smooth function]] of the form $\phi \colon \mathbb{R}^n \to X$ the [[preimage]] $(f_s \circ \phi_s)^{-1}(A) \subset \mathbb{R}^n$ is open. But this means equivalently that for every such $\phi$, $f \circ \phi$ is [[continuous function|continuous]].  This, in turn, means equivalently that the same underlying function $f_s$ constitutes a [[smooth function]] $\widetilde f \;\colon\; X \longrightarrow Cdfflg(Y)$.

In summary, we thus have a [[bijection]] of [[hom-sets]]

$$
  \array{
    Hom( Dtplg(X), Y )
    &\simeq&
    Hom(X, Cdfflg(Y))
    \\
    f_s &\mapsto& (\widetilde f)_s = f_s
  }
$$

given simply as the [[identity function|identity]] on the underlying [[functions]] of underlying sets.  This makes it immediate that this hom-isomorphism is [[natural bijection|natural]] in $X$ and $Y$ and this establishes the [[adjunction]].

Next, to see that the [[D-topological spaces]] are the [[fixed point of an adjunction|fixed points]] of this adjunction,
we apply the above [[natural bijection]] on hom-sets to the case

$$
  \array{
    Hom( Dtplg(Cdfflg(Z)), Y )
    &\simeq&
    Hom(Cdfflg(Z), Cdfflg(Y))
    \\
    (\epsilon_Z)_s &\mapsto& (\mathrm{id})_s
  }
$$

to find that the [[counit of the adjunction]]

$$
  Dtplg(Cdfflg(X))
  \overset{\epsilon_X}{\longrightarrow}
  X
$$

is given by the [[identity function]] on the underlying sets $(\epsilon_X)_s = id_{(X_s)}$.

Therefore $\eta_X$ is an [[isomorphism]], namely a [[homeomorphism]], precisely if the open subsets of $X_s$ with respect to the topology on $X$ are precisely those with respect to the topology on $Dtplg(Cdfflg(X))$, which means equivalently that the open subsets of $X$ coincide with those whose pre-images under all continuous functions $\phi \colon \mathbb{R}^n \to X$ are open. This means equivalently that $X$ is a D-topological space.

Finally, to see that we have an [[idempotent adjunction]], it is sufficient to check (by [this Prop.](idempotent+adjunction#EquivalentConditionsForIdempotency)) that the [[comonad]]

$$
  Dtplg \circ Cdfflg \;\colon\; TopologicalSpaces \to TopologicalSpaces
$$

is an [[idempotent comonad]], hence that

$$
  Dtplg \circ Cdfflg
  \overset{
    Dtplg \cdot \eta \cdot Cdfflg
  }{\longrightarrow}
  Dtplg \circ Cdfflg \circ Dtplg \circ Cdfflg
$$

is a [[natural isomorphism]]. But, as before for the adjunction counit $\epsilon$, we have that also the [[adjunction unit]] $\eta$ is the [[identity function]] on the underlying sets. Therefore, this being a natural isomorphism is equivalent to the operation of passing to the D-topological refinement of the topology of a topological space being an idempotent operation, which is clearly the case.

=--


adjunction of dg-categories

## Idea

...

## References

See section 2.1 in

* [[B. Toen]], _Derived Azumaya algebras and generators for twisted derived categories_, [arXiv:1002.2599](http://arxiv.org/abs/1002.2599).

[[!redirects adjunctions of dg-categories]]


adjusted connection

## Idea

A notion of connection for [[principal ∞-bundles]] that does not impose the fake flatness condition.

## Related concepts

* [[adjusted Weil algebra]]

## References

The construction of non-fake flat [[principal ∞-connections]] originates with

* {#FSS12} [[Domenico Fiorenza]], [[Urs Schreiber]], [[Jim Stasheff]], [p. 77](https://arxiv.org/pdf/1011.4735.pdf#page=77) and [p. 80](https://arxiv.org/pdf/1011.4735.pdf#page=80) of: _[[schreiber:Cech cocycles for differential characteristic classes|Čech cocycles for differential characteristic classes]]_, Advances in Theoretical and Mathematical Physics, **16** 1 (2012) 149-250 [[arXiv:1011.4735](https://arxiv.org/abs/1011.4735), [doi:10.1007/BF02104916](https://doi.org/10.1007/BF02104916)]

based on the [[adjusted Weil algebras]] discussed earlier in

* {#SSS09} [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], Def. 23 on  [p. 47](https://arxiv.org/pdf/0801.3480.pdf#page=47) with Prop. 21 on [p. 48](https://arxiv.org/pdf/0801.3480.pdf#page=48) in:  *[[schreiber:L-infinity algebra connections|$L_{\infty}$ algebra connections and applications to String- and Chern-Simons $n$-transport]]*, in *Quantum Field Theory*, Birkhäuser (2009) 303-424 [[arXiv:0801.3480](https://arxiv.org/abs/0801.3480), [doi:10.1007/978-3-7643-8736-5_17](https://doi.org/10.1007/978-3-7643-8736-5_17)]

* {#SSS12} [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]]:  middle boxes on [p. 28](https://arxiv.org/pdf/0910.4001.pdf#page=28) and [p. 32](https://arxiv.org/pdf/0910.4001.pdf#page=32) of: *[[schreiber:Twisted Differential String and Fivebrane Structures]]*, Communications in Mathematical Physics **315** 1 (2012)  169-213 [[arXiv:0910.4001](https://arxiv.org/abs/0910.4001), [doi:10.1007/s00220-012-1510-3)]([arXiv:](https://link.springer.com/article/10.1007/s00220-012-1510-3))]

As the title of [FSS 12](#FSS12) indicates, this procedure constructs [[Čech cohomology]] [[cocycles]] for non-fake flat higher connections in the style of the cocycles in

* {#BrylinskiMcLaughlin96} [[Jean-Luc Brylinski]], D. A. McLaughlin. *Čech cocycles for characteristic classes*, Comm. Math. Phys., **178** 1 (1996) 225–236 [[doi:10.1007/BF02104916](https://doi.org/10.1007/BF02104916)]

for the underlying bundles.

This construction is based on the [[Lie integration]] of [[L-infinity algebras]] by the "[path method](Lie+integration#SmoothIntegration)" and as such works generally but produces very "large" cocycle data, in a sense.

A variant construction tailored towards "smaller" cocycles for low-degree [[Lie n-algebras]] was later proposed in:

* [[Dominik Rist]], [[Christian Saemann]], [[Martin Wolf]], _Explicit Non-Abelian Gerbes with Connections_, [arXiv:2203.00092](https://arxiv.org/abs/2203.00092)

Examples for T-duality:

* [[Hyungrok Kim]], [[Christian Saemann]], _T-duality as Correspondences of Categorified Principal Bundles with Adjusted Connections_, [arXiv:2303.16162](https://arxiv.org/abs/2303.16162)

Further references for T-duality:

* [[Konrad Waldorf]], _Geometric T-duality: Buscher rules in general topology_, [arXiv:2207.11799](https://arxiv.org/abs/2207.11799).

* [[Thomas Nikolaus]], [[Konrad Waldorf]], _Higher geometry for non-geometric T-duals_, [arXiv:1804.00677](https://arxiv.org/abs/1804.00677).

[[!redirects adjusted connections]]


adjusted Weil algebra

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The traditional notion of *[[connection on a principal bundle]]* with given [[structure group|structure]] [[Lie algebra]] $\mathfrak{g}$ has a slick [[dg-algebra|dg-algebraic]]-formulation in terms of the [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{g})$ and the [[Weil algebra]] $W(\mathfrak{g})$ of $\mathfrak{g}$ (due to [Henri Cartan 1950](Weil+algebra#Cartan50)).

These concepts of *[[Cartan connection]]* may be generalized to [[L-infinity algebras|$L_\infty$-algebras]] (originally so in [SSS09](#SSS09), [SSS12](#SSS12), [FSS12](#FSS12), which was the starting point for the more comprehensive development in [dcct12](#dcct12)) -- see at *[[connection on a smooth principal infinity-bundle|connection on a smooth principal $\infty$-bundle]]*. In fact, noticing (see [here](L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra)) that the [[Chevalley-Eilenberg algebra]]-construction $CE(-)$ constitutes a [[full subcategory]]-embedding of the 1-category of [[finite type]] [[L-infinity algebras|$L_\infty$-algebras]] into the [[opposite category|opposite 1-category]] of [[dgc-algebras]], there is an immediate $L_\infty$-analog of the notion of [[Weil algebra]] ([SSS09, Def. 16](#SSS09)).

However, this is a [[1-category|1-]][[category theory|category theoretic]] analog, hence is a particularly strict model, defined up to [[isomorphism]], of what the would-be $\infty$-Weil algebra of an $L_\infty$-algebra should be. The latter will only be well-defined up to compatible [[quasi-isomorphism]].

The need to pass to compatible deformations of $L_\infty$-Weil algebras for a satisfactory definition of [[string 2-connections]] ([[1-brane]] connections) and analogous higher notions ([[fivebrane structure|5-brane]]-connection, [[ninebrane structure|9-brane]], etc.) was first discussed around [SSS09, Prop. 21](#SSS09) (and used to exhibit the [[higher gauge theory|higher gauge-theoretic]] nature of the [[Green-Schwarz mechanism]] in [SSS12, §3.2](higher+gauge+theory+of+the+Green-Schwarz+mechanism+--+references#SatiSchreiberStasheff12)), and was justified there by the condition that the canonical [[invariant polynomial]] in that situation does lift as expected.

A more general discussion of the necessary *adjustments* of $L_\infty$-Weil algebras was then given in [Saemann & Schmidt 20](#SaemannSchmidt20), who observe that a good general condition to impose is that the induced [[BRST complex|BRST complexes]] of $L_\infty$-connections ([SSS09, §9.3](#SSS09)) remain well-defined when the [[antifields]] and [[auxiliary fields]] are required to strictly vanish.
These authors introduce the terminology *adjusted Weil algebras* ([Saemann & Schmidt 20, Def. 4.2](#SaemannSchmidt20)) and the resulting *adjusted $L_\infty$-connections* and *adjusted [[higher parallel transport]]* ([Kim & Saemann 2020](KimSaemann20)).

Concretely, the choice of compatible deformation of the Weil algebra determines the [[Bianchi identities]] on the [[curvature characteristic forms]] of corresponding [[L-infinity algebra valued differential forms|$L_\infty$-algebra valued differential forms]] (these [[Bianchi identities]] are embodied by the [[differential]] in the Weil algebra restricted to shifted generators) and without adjustment the Bianchi identities are stronger than (i.e. just special cases of what) they ought to be.

Notice that, while the [[Weil algebra]] by itself is [[contractible homotopy type]], its *adjustments* along [[quasi-isomorphisms]] must satisfy [[horizontal differential form|horizontality]] constraints (such as the vanishing of those [[antifields]]) which makes this a non-trivial procedure.

While the characterization of adjusted Weil algebras for $L_\infty$-algebra in [Saemann & Schmidt 20, Def. 4.2](#SaemannSchmidt20) clearly (generalizes and) conceptually improves on [SSS09, Prop. 21](#SSS09), and while in applications it clearly gives the right answers (the correct higher [[Bianchi identities]]), what is still missing is a purely [[homotopy theory|homotopy theoretic]] justification. Since in practice these adjusted Weil algebras behave and are used much like [[resolutions]] by [[minimal fibrations]] in [[model category]]-theory, it is natural to wonder if there is the structure of a [[homotopical category|homotopical]] [[fibration category]] on the Weil algebra choices for $L_\infty$-algebras, such that this is indeed the case. This question is open.

On the other hand, [Borsten, Kim & Saemann 2021](#BorstenKimSaemann21) argue that adjustment is naturally understood after embedding [[L-infinity algebras|$L_\infty$-algebras]] within [[EL-infinity algebra|$E L_\infty$-algebras]] (and that this is what exhibits [[tensor hierarchies]] as a [[higher gauge theory]]-phenomenon).

## Related entries

* [[principal infinity-connection]]

* [[nonabelian differential cohomology]]

* [[supergravity Lie 6-algebra]]

## References

The original discussion for the special case of [[string 2-connections]] and their higher analogs (such as [[Fivebrane structure|fivebrane]] 6-connections, [[Ninebrane structure|Ninebrane]] 10-connections etc.):

* {#Schreiber07} [[Urs Schreiber]], *[Obstructions to $n$-Bundle Lifts Part II](https://golem.ph.utexas.edu/category/2007/10/obstructions_to_nbundle_lifts.html)* (Oct 2007) [bottom right corner in this hand-drawn diagram: [[Schreiber-PrincipalInfinityConnections-2007.pdf:file]]]

* {#SSS09} [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], Def. 23 on  [p. 47](https://arxiv.org/pdf/0801.3480.pdf#page=47) with Prop. 21 on [p. 48](https://arxiv.org/pdf/0801.3480.pdf#page=48) in:  *[[schreiber:L-infinity algebra connections|$L_{\infty}$ algebra connections and applications to String- and Chern-Simons $n$-transport]]*, in *Quantum Field Theory*, Birkhäuser (2009) 303-424 [[arXiv:0801.3480](https://arxiv.org/abs/0801.3480), [doi:10.1007/978-3-7643-8736-5_17](https://doi.org/10.1007/978-3-7643-8736-5_17)]

* {#SSS12} [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]]:  middle boxes on [p. 28](https://arxiv.org/pdf/0910.4001.pdf#page=28) and [p. 32](https://arxiv.org/pdf/0910.4001.pdf#page=32) of: *[[schreiber:Twisted Differential String and Fivebrane Structures]]*, Communications in Mathematical Physics **315** 1 (2012)  169-213 [[arXiv:0910.4001](https://arxiv.org/abs/0910.4001), [doi:10.1007/s00220-012-1510-3)]([arXiv:](https://link.springer.com/article/10.1007/s00220-012-1510-3))]

* {#FSS12} [[Domenico Fiorenza]], [[Urs Schreiber]], [[Jim Stasheff]], [p. 77](https://arxiv.org/pdf/1011.4735.pdf#page=77) and [p. 80](https://arxiv.org/pdf/1011.4735.pdf#page=80) of: _[[schreiber:Cech cocycles for differential characteristic classes|Čech cocycles for differential characteristic classes]]_, Advances in Theoretical and Mathematical Physics, **16** 1 (2012) 149-250 [[arXiv:1011.4735](https://arxiv.org/abs/1011.4735), [doi:10.1007/BF02104916](https://doi.org/10.1007/BF02104916)]

* {#dcct13} [[Urs Schreiber]], Def. 5.2.91 on [p. 645](https://arxiv.org/pdf/1310.7930v1.pdf#page=645) in: *[[schreiber:dcct|differential cohomology in a cohesive topos]]* (2013-)

{#WithHindsight} With hindsight, an early example of such adjustment (namely for the  "[[supergravity Lie 6-algebra]]", see [there](supergravity+Lie+6-algebra#ModifiedWeilAlgebra)), is given in:

* {#CastellaniDAuriaFre} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], vol 2, III (8.24) of: *[[Supergravity and Superstrings - A Geometric Perspective]]*, World Scientific (1991) [[doi:10.1142/0224](https://doi.org/10.1142/0224), [epdf](https://epdf.pub/supergravity-and-superstrings-a-geometric-perspective-vol-2-supergravity.html)]

(recognized as a "modified Weil algebra" in [revision 8, 2011](https://ncatlab.org/nlab/revision/diff/supergravity+Lie+6-algebra/8), long before the "adjusted"-terminology was proposed).

Characterization of adjusted $\mathfrak{g}$-Weil algebras by requiring well-behaved [[BRST-complexes]] for [[L-infinity algebra valued differential forms|$L_\infty$-algebra valued differential forms]]:

* {#SaemannSchmidt20} [[Christian Saemann]], [[Lennart Schmidt]], *Towards an M5-Brane Model II: Metric String Structures*, Fortschr. Phys. **68** (2020) 2000051 [[arXiv:1908.08086](https://arxiv.org/abs/1908.08086), [doi:10.1002/prop.202000051](https://doi.org/10.1002/prop.202000051)]

Revisiting the special case of the [[string Lie 2-algebra]]:

* {#Schmidt19} [[Lennart Schmidt]], _Twisted Weil Algebras for the String Lie 2-Algebra_, in [[Christian Saemann]], [[Urs Schreiber]], [[Martin Wolf]] (eds.) _[Higher Structures in M-Theory](http://www.maths.dur.ac.uk/lms/109/index.html)_ [Durham Symposium](http://www.maths.dur.ac.uk/lms/) 2018, Fortschritte der Physik 2019 [[arXiv:1903.02873](https://arxiv.org/abs/1903.02873), [doi:10.1002/prop.201910016](https://doi.org/10.1002/prop.201910016)]

Application to [[higher parallel transport]]:

* {#KimSaemann20} [[Hyungrok Kim]], [[Christian Saemann]], *Adjusted Parallel Transport for Higher Gauge Theories*, J. Phys. A **52** (2020) 445206 [[arXiv:1911.06390](https://arxiv.org/abs/1911.06390), [doi:10.1088/1751-8121/ab8ef2](https://doi.org/10.1088/1751-8121/ab8ef2)]

Relation to [[EL-infinity algebras|$E L_\infty$-algebras]] and [[tensor hierarchies]]:

* {#BorstenKimSaemann21} [[Leron Borsten]], [[Hyungrok Kim]], [[Christian Saemann]].  *$E L_\infty$-algebras, Generalized Geometry, and Tensor Hierarchies* [[arXiv:2106.00108](https://arxiv.org/abs/2106.00108)]

Survey and review:

* [[Christian Saemann]], *Adjusted Higher Gauge Theory: Connections and Parallel Transport* Lisbon (2021) [[pdf](https://math.tecnico.ulisboa.pt/seminars/download.php?fid=1485), [[Saemann-AdjustedHGT.pdf:file]]]

* [[Christian Saemann]], *$E L_\infty$-algebras, Generalized Geometry, and Tensor Hierarchies*, talk at [SFT@Cloud 2021](https://indico.cern.ch/event/1042834/) (2021) [[pdf](https://indico.cern.ch/event/1042834/contributions/4487418/attachments/2313514/3937748/saemann_generalized_geometry.pdf), [[Saemann-ELInfinityAndTensorHierarchies.pdf:file]]]

* [[Christian Saemann]], *Atiyah Algebroids for Higher and Groupoid Gauge Theories*, [talk at](M-Theory+and+Mathematics#Saemann2024) *[[M-Theory and Mathematics]]*, [[CQTS]] (2024) [[[Saemann-AtiyahAlgebroids.pdf:file]]]

Application to geometric refinement of [[topological T-duality]]:

* [[Hyungrok Kim]], [[Christian Saemann]], *T-duality as Correspondences of Categorified Principal Bundles with Adjusted Connections* [[arXiv:2303.16162](https://arxiv.org/abs/2303.16162)]

[[!redirects adjusted Weil algebras]]


ADM mass

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

## Idea

In [[general relativity]] the _ADM mass_ is a notion of total [[mass]] contained in [[asymptotically flat spacetimes]].

## Related concepts

* [[positive energy theorem]]

## References

* Dan N. Vollick, _On the Meaning of Various Mass Definitions for Asymptotically Flat Spacetimes_ ([arXiv:2101.12570](https://arxiv.org/abs/2101.12570))

See also

* Wikipedia, _[ADM formalism](https://en.m.wikipedia.org/wiki/ADM_formalism)


admin > history


Admir Greljo

* [institute page](https://theory.cern/roster/greljo-admir)

## Selected writings

On [[flavour anomalies]] and [[leptoquarks]]:

* {#GST21} [[Admir Greljo]], [[Peter Stangl]], Anders Eller Thomsen, _A Model of Muon Anomalies_ ([arXiv:2103.13991](https://arxiv.org/abs/2103.13991))

category: people


admissible rule

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### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Deduction and Induction
+-- {: .hide}
[[!include deduction and induction - contents]]
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\tableofcontents

## Idea

An [[inference rule]] in a [[deductive system]], such as a [[logic]] or [[type theory]], is *admissible* if there is some algorithm for constructing a derivation of the conclusion from derivations of the premises.  It is *not* one of the "specified" or [[primitive rules]] of the deductive system, but its admissibility means that it could be added as a primitive rule without changing the set of derivable [[judgments]].

Compare to a *[[derivable rule]]*, which is an admissible rule that is given by a "parametric" proof, or more formally by simply applying a finite number of primitive rules.  In contrast, the admissibility algorithm for an admissible rule is allowed to inspect and arbitrarily deconstruct the given derivations of the premises.

Unlike primitive and derivable rules, an admissible rule need *not* be satisfied by all semantics.  However, for most deductive systems there is a specified set of "important" admissible rules, and one considers only semantics that do satisfy these rules.

## Examples

* the [[cut rule]] and the identity in [[sequent calculus]]

* [[substitution]] and [[weakening rule|weakening]] in [[natural deduction]]

* [[negation]] in [[classical logic|classical]] [[linear logic]]

## Related Concepts

* [[focusing]]

* [[structural rule]]

[[!redirects admissible rules]]


admissible subcategory

A **right-admissible subcategory** (resp. left admissible subcategory) of a [[triangulated category]] $A$ is a strictly [[full]] [[triangulated subcategory]] $B$ such that the inclusion functor $i : B \hookrightarrow A$ admits a right adjoint $i_*$ (resp. a left adjoint $i^*$).  It is called simply **admissible** if it is both right- and left-admissible.

## See also

* [[semi-orthogonal decomposition]]
* [[Verdier quotient]]


Adolf Hurwitz

* [Wikipedia entry](http://en.wikipedia.org/wiki/Adolf_Hurwitz)

## Selected writings

On [[branched cover|branched]]  [[Riemann surfaces]] and [[braid groups]] as [[fundamental groups]] of [[configuration spaces of points]]:

* [[Adolf Hurwitz]], *Über Riemann'sche Flächen mit gegebenen Verzweigungspunkten*, Mathematische Annalen **39** (1891) 1–60 [[doi:10.1007/BF01199469](https://doi.org/10.1007/BF01199469)]

Proving the [[Hurwitz theorem]]:

* [[Adolf Hurwitz]], _Über die Composition der quadratischen Formen von beliebig vielen Variabeln_, Nachr. Ges. Wiss. Göttingen (1898) pp 309--316 ([GDZ pdf](http://gdz.sub.uni-goettingen.de/download/PPN252457811_1898/LOG_0034.pdf), [EuDML entry](https://eudml.org/doc/58420))

* [[Adolf Hurwitz]],  _Über die Komposition der quadratischen Formen_, Math. Ann. 88 (1923) pp 1--25, doi:[10.1007/bf01448439](http://dx.doi.org/10.1007/bf01448439) ([GDZ pdf](http://gdz.sub.uni-goettingen.de/download/PPN235181684_0088/LOG_0005.pdf), [EuDML entry](https://eudml.org/doc/158975))

## Related entries

* [[Hurwitz theorem]]

* [[composition algebra]]

category: people


Adrian Clingher

* [webpage](http://math.stanford.edu/~clingher/)

## related $n$Lab entries

* [[Sen limit]]

category: people


Adrian Clough

* member of [[CQTS]]

* [institute page](https://nyuad.nyu.edu/en/research/faculty-labs-and-projects/cqts/christopher-adrian-clough.html) at NYU Abu Dhabi

* [personal webpage](https://adrianclough.github.io/)

## Selected writings

On pro-algebraic resolutions of [[regular schemes]]:

* [[Adrian Clough]], *Pro-algebraic Resolutions of Regular Schemes*, MSc thesis, ETC Zürich (2014) [[pdf](https://web.ma.utexas.edu/users/adrian.clough/Master_thesis.pdf)]

On [[smooth ∞-groupoids]] (called "[[differentiable stacks]]") and the [[smooth Oka principle]]:

* {#Clough21} [[Adrian Clough]], *A Convenient Category for Geometric Topology*, PhD thesis, UT Austin (2021) [[pdf](https://repositories.lib.utexas.edu/bitstream/handle/2152/114981/CLOUGH-DISSERTATION-2021.pdf), [[Clough-ConvenientCategory.pdf:file]]]

* *The homotopy theory of differentiable sheaves* [[arXiv:2309.01757](https://arxiv.org/abs/2309.01757)]

* [[Adrian Clough]], *The Homotopy Theory of Differentiable Sheaves*, [talk at](CQTS#CloughOct2023) *[Workshop on Homotopy Theory and Applications](CQTS#WorkshopOnHomotopyTheory2023)*, [[CQTS]] (Oct 2023) [video:[YT](https://youtu.be/5NrKo-fPk2A)]

category: people


Adrian Norbert Schellekens

* [webpage](http://www.nikhef.nl/~t58/Site/Home.html)

* [articles on arXiv](https://arxiv.org/search/?searchtype=author&query=Schellekens%2C+A)

Bert Schellekens has been working on [[2d CFT]] and specifically [[SCFT]] in relation to [[model building]] in [[string phenomenology]]. In particular he studied [[rational conformal field theory|RCFT]]-[[Gepner model]]-[[orbifold]]-[[KK-compactification|compactifications]] of the [[heterotic string]] and of [[intersecting D-brane models]] (via [[boundary states]]) in the [[type II superstring]].

Recognizing the abundance of semi-[[standard model of particle physics|realistic  models]] ([Lerche-Lüst-Schellekens 86](#LercheLustSchellekens86), [Lerche-Lüst-Schellekens 87](#LercheLustSchellekens87)), Schellekens was maybe the first to recognize and definitely the first to make explicit the existence and relevance of the _[[landscape of string theory vacua]]_ and to coin that term in the first place ([Schellekens 98](#Schellekens98), see [Schellekens 06](#Schellekens06), [Schellekens 08](#Schellekens08), [Schellekens 16](#Schellekens16)) long before this became more widely appreciated.

Schellekens' scans of the "[[landscape of string theory vacua|landscape]]" of [[vertex algebra|vertex-algebraically]] well-defined [[Gepner model]] [[KK-compactification]] (in the sense of _[Spectral Standard Models and String Compactifications](https://www.physicsforums.com/insights/spectral-standard-model-string-compactifications/)_) of the [[heterotic string]] and of [[intersecting D-brane models]] (via [[boundary states]]) of the [[type II superstring]] ([Dijkstra-Huiszoon-Schellekens 04a](#DijkstraHuiszoonSchellekens04a), [Dijkstra-Huiszoon-Schellekens 04b](#DijkstraHuiszoonSchellekens04b)) showed that solid and substantial investigation into the [[landscape of string theory vacua]] is possible, worthwhile and probably necessary.

As an example, from [Dijkstra-Huiszoon-Schellekens 04b](#DijkstraHuiszoonSchellekens04b), the following is a plot of [[standard model of particle physics|standard model]]-like [[coupling constants]] in a computer scan of [[Gepner model]]-[[KK-compactification]] of [[intersecting D-brane models]]. The blue dot indicates the couplings in $SU(5)$-[[GUT]] theory. The faint lines are NOT drawn by hand, but reflect increased density of Gepner models as seen by the computer scan.

## Selected writings

On [[heterotic string theory]]:

* [[Wolfgang Lerche]], [[Dieter Lüst]], [[Adrian Norbert Schellekens]], *Ten-dimensional heterotic strings from Niemeier lattices*, Physics Letters B **181** 1–2 (1986) 71-75 [doi:10.1016/0370-2693(86)91257-8, [inspire:233203](https://inspirehep.net/literature/233203)]

  > (relation to [[Niemeier lattices]])

* {#LercheLustSchellekens87} [[Wolfgang Lerche]], [[Dieter Lüst]], [[Bert Schellekens]], _Chiral Four-dimensional Heterotic Strings from Self-dual Lattices_  Nucl. Phys. B 287, 477, 1987 ([pdf](http://lerche.web.cern.ch/lerche/papers/4dhetstrings.pdf))

* {#Schellekens91} [[Bert Schellekens]], _Classification of Ten-Dimensional Heterotic Strings_, Phys.Lett. B277 (1992) 277-284 ([arXiv:hep-th/9112006](http://arxiv.org/abs/hep-th/9112006))

On the [[partition function]] of the [[superstring]] ([[heterotic string theory|heterotic string]] and [[type II string theory|type II string]]) as a [[modular form]] with values in the [[Chern character]] of the [[background field|background]] [[field strengths]] ("character-valued partition function", a little later called the *[[elliptic genus]]*/*[[Witten genus]]*) and relation to [[Green-Schwarz anomaly cancellation]]:

* [[A. N. Schellekens]], [[Nicholas P. Warner]], *Anomalies, characters and strings*, Nuclear Physics B Volume 287, 1987, Pages 317-361 (doi:10.1016/0550-3213(87)90108-8)

* [[A. N. Schellekens]], [[Nicholas P. Warner]], *Anomalies and modular invariance in string theory*, Physics Letters B 177 (3-4), 317-323, 1986 (doi:10.1016/0370-2693(86)90760-4)

* [[Wolfgang Lerche]], [[Bengt Nilsson]], [[A. N. Schellekens]], *Heterotic string-loop calculation of the anomaly cancelling term*, Nuclear Physics B Volume 289, 1987, Pages 609-627 (doi:10.1016/0550-3213(87)90397-X)

* [[Wolfgang Lerche]], [[Bengt Nilsson]], [[A. N. Schellekens]],  [[Nicholas P. Warner]], *Anomaly cancelling terms from the elliptic genus*, Nuclear Physics B Volume 299, Issue 1, 28 March 1988, Pages 91-116 (doi:10.1016/0550-3213(88)90468-3)

On classification of strongly rational holomorphic [[vertex operator algebras]] of [[central charge]] $\leq 24$ (of relevance in [[heterotic string theory]] and [[monstrous moonshine]]):

* [[Adrian Norbert Schellekens]], *Meromorphic $c = 24$ Conformal Field Theories*, Commun. Math. Phys. **153** (1993) 159-186 [[doi:10.1007/BF02099044](https://doi.org/10.1007/BF02099044)]

On [[orientifold]] [[Gepner model]] [[string phenomenology]]:

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

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

On the [[landscape of string theory vacua]]:

* {#Schellekens98} [[Bert Schellekens]], _Naar een waardig slot_, inauguration speech ar University of Nijmegen, September 1998, ISBN 90-9012073-4 ([ubn:2066/18631](https://mobile.repository.ubn.ru.nl/handle/2066/18631))

* {#Schellekens06} [[Bert Schellekens]], _The Landscape "avant la lettre"_ ([arXiv:physics/0604134](http://arxiv.org/abs/physics/0604134))

* {#Schellekens08} [[Bert Schellekens]], _The Emperor's Last Clothes?_, Rept. Prog. Phys.71:072201, 2008 ([arXiv:0807.3249](https://arxiv.org/abs/0807.3249))

* {#Schellekens16} [[Bert Schellekens]], _Big Numbers in String Theory_ ([arXiv:1601.02462](http://arxiv.org/abs/1601.02462))

On [[string phenomenology]] with [[orbifold]]/[[orientifold]] [[intersecting D-brane models]] and [[Gepner models]]:

* [[Fernando Marchesano]], [[Bert Schellekens]], [[Timo Weigand]], *D-brane and F-theory Model Building*, in: *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2212.07443](https://arxiv.org/abs/2212.07443)]

## related $n$Lab entries

* [[Gepner model]]

* [[intersecting D-brane model]]

* [[string phenomenology]], [[GUT]]

* [[landscape of string theory vacua]]

* [[discrete torsion]]

category: people

[[!redirects A. N. Schellekens]]
[[!redirects Bert Schellekens]]


Adrian Ocneanu

Adrian Ocneanu is a Romanian origin mathematician working in Penn State.  His research is in Operator Algebras.

* [Wikipedia page](http://en.wikipedia.org/wiki/Adrian_Ocneanu)

* [Genealogy project entry](http://genealogy.math.ndsu.nodak.edu/id.php?id=3326)

category: people

[[!redirects A. Ocneanu]]


Adriano Barenco

## Selected writings

Influential early discussion of ([[controlled quantum gate|controlled]]) [[quantum logic gates]]:

* {#BBCDMSSSW95} [[Adriano Barenco]], [[Charles H. Bennett]], [[Richard Cleve]], [[David P. DiVincenzo]], [[Norman Margolus]], [[Peter Shor]], [[Tycho Sleator]], [[John A. Smolin]], [[Harald Weinfurter]], *Elementary gates for quantum computation*, Phys. Rev. A**52** (1995) 3457 [[arXiv:quant-ph/9503016](https://arxiv.org/abs/quant-ph/9503016), [doi:10.1103/PhysRevA.52.3457](https://doi.org/10.1103/PhysRevA.52.3457)]

On [[quantum computation]]:

* [[Adriano Barenco]], *Quantum Physics and Computers*,  Contemp. Phys. **37** (1996) 375-389 [[arXiv:quant-ph/9612014](https://arxiv.org/abs/quant-ph/9612014), [doi:10.1080/00107519608217543](https://doi.org/10.1080/00107519608217543)]

category: people


Adrien Bouhon

* [GoogleScholar page](https://scholar.google.com/citations?user=me2JMkUAAAAJ&hl=en)

* [institute page](https://www.su.se/english/profiles/adbo4636-1.383815)

## Selected writings

On [[braid group statistics|anyonic braiding]] of nodal points in the [[Brillouin zone]] of [[semi-metals]] ("braiding in momentum space"):

* [[Adrien Bouhon]], QuanSheng Wu, [[Robert-Jan Slager]], Hongming Weng, Oleg V. Yazyev, [[Tomáš Bzdušek]], *Non-Abelian reciprocal braiding of Weyl points and its manifestation in ZrTe*, Nature Physics **16** (2020)  1137--1143 ([arXiv:1907.10611](https://arxiv.org/abs/1907.10611), [doi:10.1038/s41567-020-0967-9](https://doi.org/10.1038/s41567-020-0967-9))

* Bin Jiang, [[Adrien Bouhon]], Zhi-Kang Lin, Xiaoxi Zhou, Bo Hou, Feng Li, [[Robert-Jan Slager]], Jian-Hua Jiang *Observation of non-Abelian topological semimetals and their phase transitions*,  	Nature Physics **17** (2021) 1239-1246 [[arXiv:2104.13397](https://arxiv.org/abs/2104.13397), [doi:10.1038/s41567-021-01340-x](https://doi.org/10.1038/s41567-021-01340-x)]

* Siyu Chen, [[Adrien Bouhon]], [[Robert-Jan Slager]], [[Bartomeu Monserrat]], *Non-Abelian braiding of Weyl nodes via symmetry-constrained phase transitions* (formerly: *Manipulation and braiding of Weyl nodes using symmetry-constrained phase transitions*),
Phys. Rev. B **105** (2022) L081117 $[$[arXiv:2108.10330](https://arxiv.org/abs/2108.10330), [doi:10.1103/PhysRevB.105.L081117](https://doi.org/10.1103/PhysRevB.105.L081117)$]$

* [[Bo Peng]], [[Adrien Bouhon]], [[Robert-Jan Slager]], [[Bartomeu Monserrat]], *Multi-gap topology and non-Abelian braiding of phonons from first principles*, Phys. Rev. B **105** (2022) 085115 ([arXiv:2111.05872](https://arxiv.org/abs/2111.05872), [doi:10.1103/PhysRevB.105.085115](https://doi.org/10.1103/PhysRevB.105.085115))

*  *[[Bo Peng]], [[Adrien Bouhon]], [[Bartomeu Monserrat]], [[Robert-Jan Slager]], *Phonons as a platform for non-Abelian braiding and its manifestation in layered silicates*, Nature Communications volume 13, Article number: 423 (2022) ([doi:10.1038/s41467-022-28046-9](https://doi.org/10.1038/s41467-022-28046-9))

* [[Adrien Bouhon]], [[Robert-Jan Slager]], *Multi-gap topological conversion of Euler class via band-node braiding: minimal models, PT-linked nodal rings, and chiral heirs* $[$[arXiv:2203.16741](https://arxiv.org/abs/2203.16741)$]$

* [[Robert-Jan Slager]], [[Adrien Bouhon]], [[Fatma Nur Ünal]], *Floquet multi-gap topology: Non-Abelian braiding and anomalous Dirac string phase* [[arXiv:2208.12824](https://arxiv.org/abs/2208.12824)]

* Wojciech J. Jankowski, Mohammedreza Noormandipour, [[Adrien Bouhon]], [[Robert-Jan Slager]], *Disorder-induced topological quantum phase transitions in Euler semimetals* [[arXiv:2306.13084](https://arxiv.org/abs/2306.13084)]

category: people


Adrien Brochier

* [webpage](http://abrochier.org/index.php)

## Selected writings

On [[horizontal chord diagrams]] and [[Vassiliev braid invariants]]:

* [[Adrien Brochier]], _Cyclotomic associators and finite type invariants for tangles in the solid torus_, Algebr. Geom. Topol. 13 (2013) 3365-3409 ([arXiv:1209.0417](https://arxiv.org/abs/1209.0417))

On construction of [[4d TQFT]] via [[factorization homology]] from [[braided monoidal category|braided]] [[tensor categories]] (with relation to [[double affine Hecke algebras]]):

* [[David Ben-Zvi]], [[Adrien Brochier]], [[David Jordan]], _Integrating quantum groups over surfaces: quantum character varieties and topological field theory_ ([arXiv:1501.04652](http://arxiv.org/abs/1501.04652))

On [[4d TQFTs]] from [[fully dualizable object|fully dualizable]] [[braided monoidal category|braided]] [[tensor categories]], via the [[cobordism hypothesis]]:

* [[Adrien Brochier]], [[David Jordan]], [[Pavel Safronov]], [[Noah Snyder]], _Invertible braided tensor categories_ ([arXiv:2003.13812](https://arxiv.org/abs/2003.13812))

category: people


Adrien Douady

* [Wikipedia entry](http://en.wikipedia.org/wiki/Adrien_Douady)

category: people


Adrien-Marie Legendre

* [St. Andrews History of mathematics entry](http://www-history.mcs.st-andrews.ac.uk/history/Biographies/Legendre.html)

## related $n$Lab entries

* [[Legendre transform]]

category:people

[[!redirects Legendre]]