deformation quantization

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## Idea

Broadly, by *deformation quantization* one means notions of [[quantization]] of [[Poisson manifolds]] $\big(X, \{-,-\}\big)$ in terms of [[families]] of [[noncommutative algebra|non-commutative]] [[algebras]] $A_{\hbar}$ parameterized by specific admissible (formal) values of [[Planck's constant]] $\hbar$ including $\hbar = 0$, where $A_0 = C^\infty(X)$ is the ordinary commutative [[algebra of functions]] on the [[underlying]] [[smooth manifold]]. The idea is that $A_{\hbar}$ is the [[algebra of observables]] of the corresponding [[quantum system]] at that value of $\hbar$, arising from "deforming" the commutative product of $A_0$ in a way that increases with $\hbar$ and is [[infinitesimal|infinitesimally]] controlled by the given [[Poisson bracket]] $\{-,-\}$.

Deformation quantization is often taken by default to refer to the historically first notion of

* **[[formal deformation quantization]]**,

where $\hbar$ is just a formal [[variable]] (i.e. an [[infinitesimal]], but not an actual [[number]]) and the [[underlying]] [[vector space]] of all algebras in question is that of [[formal power series]] in $\hbar$.

One might naively imagine that the [[formal power series]] appearing in [[formal deformation quantization]] have a finite [[radius of convergence]] $\epsilon \in \mathbb{R}_+$ thus yielding actual (non-formal) deformation quantizations for $\hbar \lt \epsilon$, but in practice this happens rarely (see reference [here](C*+algebraic+deformation+quantization#ConvergingFormalDeformationQuantization)). Indeed, [[geometric quantization]] makes manifest that [[prequantization]] conditions typically force admissible values of [[Planck's constant|$\hbar$]] to form a [[discrete space|discrete]] [[subspace]] of $\mathbb{R}_+$ with only an [[accumulation point]] at $\hbar = 0$ (namely, in [[geometric quantization]] the [[fraction|inverse]] $1/\hbar$ is typically constrained to come in [[natural number|integral]] multiples).

Therefore, beyond formal deformation quantization there are notions of

* **[[strict deformation quantizations]]**

where $\hbar$ is allowed to take discrete [[positive number|positive]] [[real numbers|real values]] with an [[accumulation point]] at $\hbar = 0$, and where to each such value is associated an actual [[C*-algebra]]-[[algebra of observables|of observables]].

In its focus on [[algebras of observables]] the notion of deformation quantization is roughly dual to *[[geometric quantization]]*, which primarily constructs the [[spaces of quantum states]]. In special sitations both notions are compatible, but in general there is a large amount of ambiguity in [[quantization]], between but also within the different approaches.

## Related entries

* [[geometric quantization]]

* algebraic quantization

  * [[formal deformation quantization]]

  * [[strict deformation quantization]]

[[!redirects deformation quantizations]]


deformation quantization - table

**[[algebraic deformation quantization]]**

| [[dimension]] | [[classical field theory]]  | [[Lagrangian]] [[BV-BRST formalism|BV]] [[quantum field theory]]  |  [[factorization algebra of observables]] |
|---|---|---|--|
| general $n$ | [[P-n algebra]] | [[BD-n algebra]] | [[E-n algebra]] |
| $n = 0$ | [[Poisson 0-algebra]] |  [[BD-0 algebra]] = [[BD algebra]] |  [[E-0 algebra]] = [[pointed object|pointed]] space |
| $n = 1$ | [[P-1 algebra]] = [[Poisson algebra]] | [[BD-1 algebra]] | [[E-1 algebra]] = [[A-∞ algebra]] |


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## Idea

The 2-dimensional [[sphere]] naturally carries the structure of a [[Poisson manifold]], in fact of a [[symplectic manifold]], with its standard [[volume form]] serving as the [[symplectic form]]. As such one may consider the ([[strict deformation quantization|strict]]) [[deformation quantization]] of its [[Poisson algebra]] of functions.

## Details
 {#Details}

There are various notions of deformation quantization which one may consider. By default one often means [[formal deformation quantization]] with values in [[formal power series]] in [[Planck's constant]] $\hbar$. One definition of *strict* deformation quantization takes it to be such a formal deformation quantization which happens to [[convergence of a sequence|converge]] for finite [[real number]] values of $\hbar$. This is not known to exist for the 2-sphere.

However, there are other notions of strict quantization via deformation of the algebra of functions (review in [Hawkins (2007), Section 2](#Hawkins07)).

### Formal deformation quantization

(...)

### Strict deformation quantization

One kind of [[strict deformation quantization]] of the 2-sphere is obtained as follows [[Hawkins (2007), section 4](#Hawkins07)].

Take the [[volume]] of the 2-sphere to be a [[natural number]]. Then there is a [[prequantum line bundle]] $(L,\nabla)$ on $S^2$ whose [[curvature]] 2-form is the [[symplectic form]], hence the [[volume form]], and which is a [[holomorphic line bundle]] with respect to the standard [[complex manifold]] structure of the 2-sphere (the [[Riemann sphere]]).

For $N \in \mathbb{N}_+$ a positive [[natural number]], the [[geometric quantization]] of the 2-[[sphere]] for [[Planck's constant]] $\hbar = 1/N$ produced the [[space of quantum states]]

$$
  \mathcal{H}_N = \Gamma_{hol}(S^2, L^{\otimes N})
$$

which is the [[space of sections|space of holomorphic sections]] of the $N$th tensor power of the [[prequantum line bundle]]. See at _[[geometric quantization of the 2-sphere]]_.

This is a [[finite dimensional vector space|finite-dimensional]] [[complex vector space|complex]] [[Hilbert space]], hence becomes a [[module]] over the  [[matrix algebra]] $Mat_N(\mathbb{C})$.

One finds that the assignment

$$
  \left\{
    0 , \cdots, \frac{1}{N}, \cdots, \frac{1}{3}, \frac{1}{2}, 1
  \right\}
  \longrightarrow
  C^\ast Alg
$$

which sends $\frac{1}{N}$ to $Mat_N(\mathbb{C})$ is a [[strict deformation quantization]] of the 2-sphere.

## Related concepts

* [[quantization of the 2-sphere]]

  * [[geometric quantization of the 2-sphere]]

  * **deformation quantization of the 2-sphere**

## References

* {#Hawkins07} [[Eli Hawkins]], Section 4 of: *An Obstruction to Quantization of the Sphere*, Communications in Mathematical Physics **283** (2008) 675–699  [[arXiv:0706.2946](http://arxiv.org/abs/0706.2946), [doi:10.1007/s00220-008-0517-2](https://doi.org/10.1007/s00220-008-0517-2)]

following

* {#BordemannMeinrenkenSchlichenmaier93} [[Martin Bordemann]], [[Eckhard Meinrenken]], [[Martin Schlichenmaier]], _Toeplitz Quantization of Kähler Manifolds and $gl(N)$ $N\to\infty$_,  	Commun. Math. Phys. **165** (1994) 281-296 [[arXiv:hep-th/9309134](http://arxiv.org/abs/hep-th/9309134), [doi:10.1007/BF02099772](https://doi.org/10.1007/BF02099772)]

[[!redirects strict deformation quantization of the 2-sphere]]


deformation retract

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## Idea

A _deformation retract_ is a [[retract]] which is also a [[section]] up to [[homotopy]]. Equivalently, it is a [[homotopy equivalence]] one of whose two homotopies is in fact an [[identity]].

## Definition

Let $\mathcal{C}$ be a [[category]] equipped with a notion of [[homotopy]] between its [[morphisms]]. Then a _deformation retraction_ of a morphism

$$
  i : A \to X
$$

(the _deformation retract_ itself) is another morphism

$$
  r : X \to A
$$

such that

$$
  \array{
    && X
    \\
    & {}^{\mathllap{i}}\nearrow &\Downarrow^=& \searrow^{\mathrlap{r}}
    \\
    A &&\stackrel{=}{\to}&& A
  }
$$

and

$$
  \array{
    && A
    \\
    & {}^{\mathllap{r}}\nearrow &\Downarrow^{\simeq}& \searrow^{\mathrlap{i}}
    \\
    X &&\stackrel{=}{\to}&& X
  }
  \,.
$$

In particular, if "homotopy" in $\mathcal{C}$ means [[left homotopy]] with respect to an [[cylinder object]] $I \otimes X$

$$
  \array{
    X
    \\
    \downarrow^{\mathrlap{d_0}} & \searrow^{\mathrlap{Id_X}}
    \\
    I \otimes X &\stackrel{\sigma_X}{\to}& X
    \\
    \uparrow^{\mathrlap{d_1}} & \nearrow_{\mathrlap{Id_X}}
    \\
    X
  }
  \,,
$$

then a deformation retract of $i : A \to X$ is a morphism $r : X \to A$ such that $r \circ i = id_A$ and such that there exists a morphism $\eta : I \otimes X \to X$ fitting into a diagram

$$
  \array{
    X &\stackrel{r}{\to}& A
    \\
    \downarrow^{\mathrlap{d_0}} && \downarrow^{\mathrlap{i}}
    \\
    I \otimes X &\stackrel{\eta}{\to}& X
    \\
    \uparrow^{\mathrlap{d_1}} & \nearrow_{\mathrlap{Id_X}}
    \\
    X
  }
  \,.
$$

Hence a deformation retract is a (left) [[homotopy equivalence]] where one of the two homotopies occuring is in fact an [[identity]].

If the cylinder object assignment here is functorial, we say that $\eta$ is a **strong deformation retract** if moreover

$$
  \eta \circ (I \otimes i) = \sigma_X \circ (I \otimes i)
$$

(hence if the homotopy restricted to the inclusion is "constant" as seen by the chosen cylinder object).

In parts of the literature, deformation retracts are required to be strong by default.

## Examples

### In topological spaces

In the category [[Top]] of [[topological spaces]] the standard [[cylinder object]] is given by [[cartesian product]] with the [[interval]] $I \coloneqq [0,1]$.

With respect to the corresponding notion of [[left homotopy]],  if $X$ is a topological space and $A\subset X$ a [[topological subspace|subspace]], then $A$ is a strong deformation retract of $X$ if there exists a [[continuous map]] $H \colon X\times I\to X$ such that $H(a,t) = a$ for all $a\in A$, $t\in I=[0,1]$, $H(x,0) = x$ for all $x\in X$ and $H(x,1)\in A$ for all $x\in X$.

Equivalently, there are [[continuous maps]] $i \colon A\to X$ and $r \colon X\to A$ such that $r \circ i = id_A$ and $i\circ r\sim id_X (rel A)$, where $\sim (rel A)$ denotes [[homotopy]] with fixed $A$. More generally, for any continuous map $j \colon Z\to Y$ we say that it is __deformation retractable__ if there is $r \colon Y\to Z$ such that $j\circ r\sim id_Y$ and $r\circ j = id_Z$.

A pair $(X,A)$ is an __[[NDR-pair]]__ if there is a [[pair]] of continuous maps, $u \colon X\to I,\; H \colon X\times I\to X$ such that $H(a,t)=a$ for all $a\in A$ and all $t$, $H(x,0)=x$ for all $x\in X$, $u^{-1}(0)=A$ and $H(x,1)\in A$ for all $x$ such that $u(x)\lt 1$.
If $(X,A)$ is an NDR-pair, then the inclusion has a left [[homotopy inverse]] iff $A$ is also a [[retract]] of $X$ (in [[Top]], in the standard [[category theory|category-theoretic]] sense).

The pair $(X,A)$ is a __DR-pair__ if it is a deformation retract and there is a function $u \colon X\to I$ such that $A=u^{-1}(0)$ (i.e. it gives simultaneously a deformation retract and a NDR-pair). If $(X,A)$ is an NDR-pair then the inclusion $A\hookrightarrow X$ is a homotopy equivalence iff $A$ is a deformation retract of $X$. Any map $f:X\to Y$ is a [[homotopy equivalence]] iff $X$ is the deformation retract of the mapping cylinder of $f$. If $(X,A)$ is an NDR-pair and $A$ is [[contractible space|contractible]], then the quotient map $X\to X/A$ is a homotopy equivalence.

### In chain complexes
 {#ExamplesInChainComplexes}

\begin{proposition}\label{EZAWDeformationRetract}
**([[Eilenberg-Zilber/Alexander-Whitney deformation retraction]])** \linebreak

Let

* $A, B \,\in\, sAb = $ [[SimplicialAbelianGroups]]

and denote

* by $N(A), N(B) \,\in\, Ch^+_\bullet = $ [[ConnectiveChainComplexes]] their [[normalized chain complexes]],

* by $A \otimes B \,\in\, sAb$ the degreewise [[tensor product of abelian groups]],

* by $N(A) \otimes N(B)$ the [[tensor product of chain complexes]].

Then there is a [[deformation retraction]]

\begin{tikzcd}
  N(A) \otimes N(B)
  \ar[
    rr,
    bend right=20,
    "\mathrm{id}"{below},
    "\ "{above, name=t}
  ]
  \ar[
    rr,
    phantom,
    "\ "{name=s, yshift=-6pt}
  ]
  \ar[
    r,
    "\nabla_{A,B}"
  ]
  &
  N( A \otimes B )
  \ar[
    r,
    "\Delta_{A,B}"
  ]
  &
  N(A) \otimes N(B)
  \ar[
    from=s,
    to=t,
    -,
   shift left=1pt
  ]
  \ar[
    from=s,
    to=t,
    -,
   shift right=1pt
  ]
\end{tikzcd}

\begin{tikzcd}
  N( A \otimes B )
  \ar[
    rr,
    bend right=20,
    "\mathrm{id}"{below},
    "\ "{above, name=t}
  ]
  \ar[
    rr,
    phantom,
    "\ "{name=s, yshift=-6pt}
  ]
  \ar[
    r,
    "\Delta_{A,B}"
  ]
  &
  N(A) \otimes N(B)
  \ar[
    r,
    "\nabla_{A,B}"
  ]
  &
  N( A \otimes B )
  \ar[
    from=s,
    to=t,
    Rightarrow
  ]
\end{tikzcd}

where

* $\nabla_{A,B}$ is the [[Eilenberg-Zilber map]];

* $\Delta_{A,B}$ is the [[Alexander-Whitney map]].

\end{proposition}

For unnormalized chain complexes, where we have a [[homotopy equivalence]], this is the original [[Eilenberg-Zilber theorem]] ([Eilenberg & Zilber 1953](EZAW+deformation+retraction#EilenbergZilber53), [Eilenberg & MacLane 1954, Thm. 2.1](EZAW+deformation+retraction#EilenbergMacLane54)). The above [[deformation retraction]] for normalized chain complexes is [Eilenberg & MacLane 1954, Thm. 2.1a](EZAW+deformation+retraction#EilenbergMacLane54). Both are reviewed in [May 1967, Cor. 29.10](EZAW+deformation+retraction#May67). Explicit description of the [[homotopy operator]] is given in [Gonzalez-Diaz & Real 1999](EZAW+deformation+retraction#GonzalezDiazReal99).

## Related concepts

* [[retract]], [[neighbourhood retract]]

* [[section]]

* There is also the notion of a [[deformation retract of a homotopical category]], which has a similar feel in some ways but is not closely related.  (It should not be confused with the idea of a deformation retract *in* a [[model category]], which is a direct generalization of the notion described above for [[Top]].)

## References ##

Textbook accounts

* [[George Whitehead]], chapter 1 of: *Elements of Homotopy Theory*, Springer 1978 ([doi:10.1007/978-1-4612-6318-0](https://link.springer.com/book/10.1007/978-1-4612-6318-0))

* [[Peter May]], Section 6.4 of: _[[A concise course in algebraic topology]]_, University of Chicago Press 1999 ([ISBN: 9780226511832](https://www.press.uchicago.edu/ucp/books/book/chicago/C/bo3777031.html), [pdf](http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf))

[[!redirects deformation retracts]]

[[!redirects deformation retraction]]
[[!redirects deformation retractions]]

[[!redirects strong deformation retract]]
[[!redirects strong deformation retracts]]


deformation retract for the enrichment

#Definition#

A _deformation retract for the enrichment_ of a category  $C$ [[enriched category|enriched over]] a [[closed monoidal homotopical category]] $V$ is a [[deformation retract of a homotopical category]] for the [[two-variable adjunction]]

([[power]], [[copower]], [[closed category|internal hom]]) : $V \times C \to C$

satisfying a bunch of extra conditions.

#References#

This is [definition 16.1, p. 46] in Shulman: [Homotopy limits and colimits and enriched homotopy theory](http://arxiv.org/abs/math.AT/0610194v1)

[[!redirects deformation retract for the enrichement]]


deformation retract of a homotopical category

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## Idea ##

_Deformations_ and _deformation retracts_ are tools in [[homotopy theory]] for constructing the [[homotopy category]] of a [[model category]] or more general [[homotopical category]].

The idea of a deformation retract is to find a [[full subcategory]] of a given [[homotopical category]] $C$ such that

* a given functor $F$ from $C$ to another [[homotopical category]] becomes a [[homotopical functor]] on this subcategory;

* every object in the category is naturally weakly equivalent to an object in the subcategory.

Deformations are a generalizations of _cofibrant replacement functors_ in a [[model category]].

##Definition##

Let $C$ be a [[homotopical category]].

* A **left deformation** of $C$ is a functor $Q : C \to C$ equipped with a natural weak equivalence
$$
  \array{
    & \nearrow  \searrow^{Q}&
    \\
    C
    &\Downarrow^{q}_\simeq&
    C
    \\
    & \searrow  \nearrow_{Id}
  }
$$
(it follows from [[category with weak equivalences|2-out-of-3]] that $Q$ is a [[homotopical functor]]).

* A **left deformation retract** is a [[full subcategory]] $C_Q$ containing the image of a left deformation $(Q,q)$.

Now let $F : C \to D$ be a functor between [[homotopical category|homotopical categories]].

* A **left deformation retract for $F$**  is a left deformation retract $C_Q$ of $C$ such that $F$ becomes a [[homotopical functor]] when restricted to $C_Q$.

Right deformations are defined analogously.

##Generalization##

There are pretty obvious generalizations of deformation retracts for functors of more than one variable.

* A **deformation retract** for a [[two-variable adjunction]] $(\otimes , hom_l, hom_r) : C \times D \to E$ consists of left deformation retracts $C_Q$, $D_Q$ for $C$ and $D$, respectively, and a right deformation retract $E_Q$ of $E$, such that

  * $\otimes$ is [[homotopical functor|homotopical]] on $C_Q \times D_Q$;

  * $hom_l$ is [[homotopical functor|homotopical]] on $C_Q^{op} \times E_Q$;

  * $hom_r$ is [[homotopical functor|homotopical]] on $D_Q^{op} \times E_Q$.

##References##

The definition of deformation and deformation retract is in paragraph 40 of

* William G. Dwyer, Philip S. Hirschhorn, Daniel M. Kan, and Jeffrey H. Smith. Homotopy Limit Functors on Model Categories and Homotopical Categories, volume 113 of Mathematical Surveys and Monographs. American Mathematical Society, 2004.

The notion of deformation retract of a [[two-variable adjunction]] is [definition 15.1, p. 43](http://arxiv.org/PS_cache/math/pdf/0610/0610194v1.pdf#page=43)
in

* Michael Shulman, _Homotopy limits and colimits and enriched homotopy theory_ ([arXiv](http://arxiv.org/abs/math.AT/0610194v1))


deformation retraction > history

see _[[deformation retract]]_


Deformation Theory

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This entry is about the article

* [[Jacob Lurie]], _Deformation theory_ ([arXiv:0709.3091](http://arxiv.org/abs/0709.3091))

on the notions of [[deformation theory]] -- such as [[Kähler differentials]] and [[cotangent complex]]es -- generalized from the ordinary context of [[algebra]] to that of [[higher algebra]]. In particular it studies the case where ordinary [[commutative rings]] are refined to  by $E_\infty$-[[E-∞-ring|rings]]. See also at _[[E-∞ geometry]]_.

Much of this has meanwhile be absorbed in section 8.3 of

* [[Jacob Lurie]], _[[Higher Algebra]]_

## Keywords

* [[deformation theory]]

  * [[derivation]]

  * [[Kähler differential]]

* [[tangent (∞,1)-category]]

  * [[cotangent complex]]

...

category: reference


deformation theory

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## Idea {#Idea}

Deformation theory studies problems of extending structures to  extensions of their domains. __Formal deformation theory__, is the part of the deformation theory where the extensions are _[[infinitesimal object|infinitesimal]]_.

A typical problem in formal deformation theory has the structure that

* a [[morphism]] $f : X \to Y$ of certain [[space]]s is given,

* and [[infinitesimal object|infinitesimal]] thickenings $\tilde X$ and $\tilde Y$ of $X$ and $Y$ are prescribed, with injection morphisms $X \to \tilde X$ and $Y \to \tilde Y$

and asks whether a bottom horizontal morphism $\tilde f$ in the diagram

$$
  \array{
    X &\stackrel{f}{\to}& Y
    \\
    \downarrow && \downarrow
    \\
    \tilde X &\stackrel{\tilde f}{\to}& \tilde Y
  }
$$

may be found. This morphism $\tilde f$ would be called an _infinitesimal deformation_ of $f$.

In other words:

+-- {: .standout}

Formal deformation theory studies the **[[obstruction|obstruction theory]]** of extensions to **infinitesimal** thickenings.

=--

A typical example of an _infinitesimal thickening_ is a _square-0-extension_ of a [[ring]]:

let $R$ be a [[ring]], to be thought of as the ring of functions on the [[space]] $X$ in the above diagram. Let furthermore $N$ be an $R$-module, to be thought of as the $R$-module of [[section]]s of a [[vector bundle]] over $X$.

Then consider the new ring, whose underlying group is the [[direct sum]] $R \oplus N$, equipped with the product structure

$$
  (r_1, n_1) \cdot (r_2, n_2) = (r_1 r_2, r_1 n_2 + n_1 r_2)
  \,.
$$

This is the **square 0-extension** of $R$ by $N$. It should be thought of as the algebra of functions that consists of elements of $R$ and  $N$, where the elements in $N$ are thought of as functions with values in [[infinitesimal object|infinitesimal quantities]], so that their would-be product "$n_1 \cdot n_2$" vanishes.

So the ring $R \oplus N$ may be thought of as the ring of functions on the infinitesimal extension $\tilde X$ of $X$, which is the space obtained by adding to $X$ all the _vectors of infinitesimal length_ in the vector bundle over $X$.

There is a canonical ring homomorphism $R\oplus N \to R$ that is the identity on $R$ and $0$ on $N$. This is to be thought of as the pullback of functions on spaces along the inclusion of spaces $X \to \tilde X$ (which in turn may be thought of as the 0-[[section]] of the vector bundle on $X$).

Similarly, let $R_2$ be another ring with module $N_2$ and square-0 extension $R_2 \oplus N_2$, thought of, respectively, as the ring of functions on a space $Y$, the module of sections of a vector bundle on $Y$ and the ring of functions on the space of infinitesimal vectors of this vector bundle.

In terms of these function rings, a morphism $f : X \to Y$ of spaces corresponds to a ring homomorphism $R_1 \leftarrow R_2 : f^*$. Hence we have a situation

$$
  \array{
    R_1 &\stackrel{f^*}{\leftarrow}& R_2
    \\
    \uparrow && \uparrow
    \\
    R_1 \oplus N_1 && R_2 \oplus N_2
  }
  \,.
$$

The obvious obstruction problem now is whether we can **deform** $f^*$ to a morphism $R_1 \oplus N_1 \leftarrow R_2 \oplus N_2 : \tilde f^*$ of rings, such that we get a commuting diagram

$$
  \array{
    R_1 &\stackrel{f^*}{\leftarrow}& R_2
    \\
    \uparrow && \uparrow
    \\
    R_1 \oplus N_1 &\stackrel{\tilde f^*}{\leftarrow}& R_2 \oplus N_2
  }
  \,.
$$

The obstruction to the existence of such lifts is measured by [[cohomology]] with coefficients in the [[cotangent complex]] of $R_1$.

This is the archetypical problem that deformation theory deals with. As always, after studying this a bit it turns out that in order to obtain a good theory, one needs to adopt the [[nPOV]]. Problems as above may be stated in the [[category]] [[Ring]] of rings, but they may have good answers only in [[vertical categorification|categorifications]] of this for instance to the [[(∞,1)-category]] of [[E-∞-ring]]s.

### Modules, derivations and Kähler differentials {#Modules}

In order to better see the structure of the above archetypical problem of deformation theory, we describe some aspects of the canonical [[bifibration]] of ring modules in a way that nicely organizes all the concepts [[module]], [[derivation]], [[Kähler differential]] in a single picture that lends itself to [[vertical categorification]]. (Following [[Deformation Theory|DefTheory]].)

With [[Ring]] denoting the [[category]] of (commutative, unital) [[rings]], write

$$
  p : Mod \to Ring
$$

for the [[bifibration]] of [[modules]] over [[rings]]: objects of $Mod$ are pairs consisting of a ring $R$ an an $R$-[[module]] $N$, and morphism $(R_1,N_1) \to (R_2, N_2)$ are pairs consisting of a ring homomorphism $f : R_1 \to R_2$ and a morphism $F : N_1 \to N_2 \otimes_f R_2$ of $R_2$-modules.

(Recall for instance from the discussion at [[Sweedler coring]]
that this bifibration is a way to think of the [[stack]] of algebraic [[vector bundles]].)

But there is also another functor $G : Mod \to Ring$ of interest: for $N$ any $R$-module, we may form the ring $G(N) := R \oplus N$ called the **square 0-extension** of $R$, in which multiplication is given by

$$
  (r_1,n_1) \cdot (r_2, n_2) := (r_1 r_2, n_1 r_2 + n_2 r_1)
  \,.
$$

Moreover, there is a [[natural transformation|natural]] morphism of rings $G(N) \to R$ given by sending $(r,n) \mapsto r$. A [[section]] $v : R \to G(n)$ of this morphism is precisely a [[derivation]] of $R$ with values in the module $N$.

This may be organized into a single functor

$$
   Mod \to [I,Ring]
$$

into the [[arrow category]] of [[Ring]], that sends to the $R$-module $N$ to the morphism $G(N) \to R$. The original bifibration factors through this morphism by the right endpoint evaluation

$$
  \array{
    Mod &&\stackrel{p}{\to}&& Ring
    \\
    & \searrow && \nearrow_{\mathrlap{d_1}}
    \\
    && [I,Ring]
  }
  \,.
$$

Finally notice that the [[functor]] $G$ has a [[left adjoint|left]] [[adjoint functor]]

$$
  \Omega : Ring \to Mod
$$

that sends a ring $R$ to the $R$-module $\Omega_K(R)$ of [[Kähler differentials]], i.e. to the module that encodes the [[cotangent bundle]].

### Cotangent complex

Using the module of Kähler differentials is not appropriate in general; instead we need to take its derived version. To talk about the nonabelian derived functors, Quillen introduced a model category structure on the category of simplicial commutative rings. Given a morphism $f: A\to B$ of rings, which makes $B$ an $A$-algebra, the category $AbGr(A-Alg/B)$ of abelian group objects in the slice category $A$-$Alg/B$ of $A$-algebras over $B$ is equivalent both to the category of $B$-modules and the trivial (= square zero) extensions of $A$ by $B$-modules. In particular we can consider the forgetful functor $AbGr(A-Alg/B)\to A-Alg/B$ which has a left adjoint $Ab_{B/A} : A-Alg/B\to AbGr(A-Alg/B)\cong {}_B Mod$. All said is true for simplicial commutative rings as well. Now the **relative cotangent complex** $L_{B/A}$ is the value on $B$ of the left [[derived functor]] $\mathbb{L} Ab_{B/A}(B)$. Regarding that the left adjoint at the nonderived level (and for usual rings) can be expressed via Kähler differentials, this explains the phrase "derived version of the module Kähler differentials".

The above situation generalizes from the category [[Ring]] to an arbitrary presentable [[(∞,1)-category]] $C$ by replacing the [[bifibration]] $Mod \to Ring$ by the [[stabilization]] $T_C \to C$ of the [[codomain fibration]] of $C$: the [[tangent (∞,1)-category]] of $C$.

The projection $p : T_C \to C$ still has a [[left adjoint]]

$$
  \Omega : C \to T_C
$$

for which a representative which is also a section (in a strict sense) of $p$ may be taken; any such representative is called the [[cotangent complex]] functor for $C$. The special property section property, like in the motivating example above, says that the composition

$$
  C \stackrel{\Omega}{\to} T_C \stackrel{p}{\to} C
$$

is the identity [[(∞,1)-functor]].

### Further categorification

...

## Deformation theory via differential graded Lie algebras

Over a field of characteristic zero, there is an approach to deformation theory via [[differential graded Lie algebras]] (or more generally [[L-infinity algebras]]). One can find some exposition about this approach in the Kontsevich and Lurie references below. See also discussion at MathOverflow: [def theory and dgla-s](http://mathoverflow.net/questions/385/deformation-theory-and-differential-graded-lie-algebras).

In this approach, one begins with an object $X$ (for example a scheme, or a complex manifold, or a vector bundle, or an associative algebra, or a dg category, or ...) that one would like to deform. Then the general principle is that there exists a dgLa $L_X$ with the property that the functor $Def_{L_X} : Art \to Set$, which sends a local Artin algebra $(A,m)$ to the set of [[Maurer-Cartan equation|Maurer-Cartan solutions]] in $(L_X \otimes m)^1$ modulo the gauge action of $(L_X \otimes m)^0$, is isomorphic to the functor which sends a local Artin algebra $(A,m)$ to the set of isomorphism classes of deformations of $X$ over $\operatorname{Spec} A$. Note the similarity with Schlessinger's theory: both here and in Schlessinger's work, we deal with functors from Artin algebras to sets.

In the case of a compact complex manifold, the dgLa in question is given by the so-called [[Kodaira-Spencer theory|Kodaira-Spencer]] dgLa: holomorphic vector fields tensor $(0,q)$-forms (this is just the [[Dolbeault cohomology|Dolbeault resolution]] of the sheaf of holomorphic vector fields). In the case of an associative algebra (or a dg algebra, or an A-infinity algebra, or a dg category, or an A-infinity category), the appropriate dgLa is the Hochschild complex with the Hochschild differential and the Gerstenhaber bracket.

In this language, the [[Tian-Todorov theorem]] on the unobstructedness of deformations of [[Calabi-Yau manifold]]s translates to the statement that the [[Kodaira-Spencer theory|Kodaira-Spencer]] dgLa of a Calabi-Yau manifold is homotopy abelian --- that is, it is quasi-isomorphic to an abelian dg Lie algebra. Barannikov-Kontsevich proved more generally that the dgLa given by holomorphic polyvector fields tensor $(0,q)$-forms on a Calabi-Yau manifold is homotopy abelian. The deformation-theoretic consequence is that the "extended deformations" of Calabi-Yau manifolds are unobstructed. These "extended deformations" should be realized by certain $A_\infty$ deformations of (a dg enhancement of) the derived category of coherent sheaves on the Calabi-Yau.

The following paper is a good introduction to these ideas:

* Marco Manetti, _Deformation theory via differential graded Lie algebras_ , °([arXiv:0507284](http://arxiv.org/abs/math/0507284))

The Kontsevich and Soibelman references below are also good.

## Related entries

* [[Kuranishi deformation theory]]

* [[Kodaira-Spencer theory]]

* [[Artin-Mazur formal group]]

* [[deformation context]], [[formal moduli problem]], [[Lie differentiation]]

* [[tangent complex]], [[André-Quillen cohomology]], [[Hochschild cohomology]]

* [[cotangent complex]], [[André-Quillen homology]], [[Hochschild homology]]

* [[derived deformation theory]], [[derived algebraic geometry]]

* [[formal scheme]], [[formal smoothness]]

* [[deformation quantization]]

* [[Lubin-Tate theory]]

* [[Galois deformation ring]]

## References

* {#SchlessingerStasheff85} [[Michael Schlessinger]], [[Jim Stasheff]], _The Lie algebra structure of tangent cohomology and deformation theory_, Journal of pure and applied algebra 38 (1985) 313-322

* {#SchlessingerStasheff12} [[Michael Schlessinger]], [[Jim Stasheff]], _Deformation theory and rational homotopy type_ ([arXiv:1211.1647](https://arxiv.org/abs/1211.1647))

* [[Charles Doran]], S. Wong, *Deformation Theory: An Historical Annotated Bibliography*, Chapter 2 in Deformation of Galois Representations, to appear in the AMS-IP Studies in Advanced Mathematics Series.

On [[deformation theory]] in terms of [[L-infinity algebra|$L_\infty$-algebras]] and their [[Maurer-Cartan equations]]:

* [[Martin Doubek]], [[Martin Markl]], [[Petr Zima]], equation (31) in: *Deformation Theory (lecture notes)*, Archivum mathematicum 43(5), 2007, 333-371 ([arXiv:0705.3719](https://arxiv.org/abs/0705.3719))

* [[Martin Markl]], _Deformation theory of algebras and their diagrams_, 129 pp, CBMS __116__, AMS 2012 ([ISBN:978-0-8218-8979-4](http://www.ams.org/bookstore-getitem/item=CBMS-116), [toc pdf](https://www.gbv.de/dms/goettingen/721467989.pdf))

* Wikipedia: [deformation theory](http://en.wikipedia.org/wiki/Deformation_theory), [cotangent complex] (http://en.wikipedia.org/wiki/Cotangent_complex)
* E. Sernesi, _An overview of classical deformation theory_, [pdf](http://www.mat.uniroma3.it/users/sernesi/sernesioverviewdefth.pdf)

* [[Alexander Grothendieck]], _Catégories cofibrées additives et complexe cotangent relatif_, Lecture Notes in Mathematics 79

* [[Luc Illusie]], _Complexe cotangent et déformations I_, Lec. Notes Math. __239__, Springer 1971, xv+355 pp.; _II_, LNM __283__, Springer 1972. vii+304 xv+355 pp. (errata: [pdf](https://www.imo.universite-paris-saclay.fr/~luc.illusie/ErrSLN239.pdf))

* [[Maxim Kontsevich]], [[Yan Soibelman]], _Deformation theory I_ ([ps](http://www.math.ksu.edu/~soibel/Book-vol1.ps)); _Notes on A-infinity algebras, A-infinity categories and non-commutative geometry. I_, [math.AG/0606241](http://arxiv.org/abs/math/0606241) -- two parts of large unfinished books on the subject; [[Yan Soibelman]], _Lectures on deformation theory and mirror symmetry_ ([ps](http://www.math.ksu.edu/~soibel/ipam-final.ps))

* [[Maxim Kontsevich]], _Topics in deformation theory_ (A rough write up of a Berkeley course, early 90-s), [ps](http://www.math.uchicago.edu/~mitya/langlands/kontsdef.ps)
* [[Jacob Lurie]], [[Deformation Theory]] ([arXiv:0709.3091](http://arxiv.org/abs/0709.3091)) -- describes a very setup for deformation theory over any [[(∞,1)-category]] is described. Then as an application the deformation theory of [[E-∞-ring]]s is developed. An application: J. Lurie, _Moduli problems for ring spectra_, [moduli.pdf](http://www.math.harvard.edu/~lurie/papers/moduli.pdf).

* E. Sernesi, _Deformations of algebraic schemes_ (monograph)
Grundlehren der Math. Wiss. __334__, Springer 2006. xii+339 pp. [MR2008e:14011](http://www.ams.org/mathscinet-getitem?mr=2008e:14011)

* Alexander I. Efimov, [[Valery Lunts|Valery A. Lunts]], [[Dmitri Orlov|Dmitri O. Orlov]], _Deformation theory of objects in homotopy and derived categories_
   * _I: general theory_ [arXiv:math/0702838](http://arxiv.org/abs/math/0702838);
   * _II: pro-representability of the deformation functor_ [arXiv:math/0702839](http://arxiv.org/abs/math/0702839);
   * _III: abelian categories_ [arXiv:math/0702840](http://arxiv.org/abs/math/0702840)

* [[Martin C. Olsson]], _Deformation theory of representable morphisms of algebraic stacks_, Mathematische Zeitschrift__253__, n. 1, 25--62 (2006) [doi](http://dx.doi.org/10.1007/s00209-005-0875-9); _Tangent spaces and obstructon theories_, lectures, [MSRISummer07.pdf](http://math.berkeley.edu/~molsson/MSRISummer07.pdf)

* B. Fantechi, M. Manetti, _Obstruction calculus for functors of Artin rings I_, J. Algebra __202__ (1998), no. 2, 541--576, [MR99f:14004](http://www.ams.org/mathscinet-getitem?mr=99f:14004).

* Domenico Fiorenza, Marco Manetti, Elena Martinengo, _Semicosimplicial DGLAs in deformation theory_, [arxiv/0803.0399](http://arxiv.org/abs/0803.0399)

* S. Merkulov, B. Vallette, _Deformation theory of properads_, [arXiv:0707.0889](http://arxiv.org/abs/0707.0889)

* [[Vladimir Hinich]], _Deformations of homotopy algebras_,  Communication in Algebra, 32 (2004), 473-494, [MR2005m:18009](http://www.ams.org/mathscinet-getitem?mr=2101417), [doi](http://dx.doi.org/10.1081/AGB-120027907), [pdf](http://math.haifa.ac.il/hinich/WEB/mypapers/dha.pdf); _Deformations of sheaves of algebras_, Adv. Math. __195__ (2005), no. 1, 102–164, [MR2007d:13021](http://www.ams.org/mathscinet-getitem?mr=2145794), [doi](http://dx.doi.org/10.1016/j.aim.2004.07.007), [pdf](http://math.haifa.ac.il/hinich/WEB/mypapers/dsa.pdf); _DG coalgebras as formal stacks_, J. Pure Appl. Algebra __162__ (2001), no. 2-3, 209--250 (doi), [math.AG/9812034](http://arxiv.org/abs/math.AG/9812034); _Formal deformations of sheaves of algebras_, video of a talk at MSRI 2002, [link](http://www.msri.org/publications/ln/msri/2002/hodgetheory/hinich/1/index.html)

* [[Vladimir Hinich]], [[Vadim Schechtman]], _Deformation theory and Lie algebra homology_ I. Algebra Colloq. 4 (1997), no. 2, 213–240, [MR2000c:14006a](http://www.ams.org/mathscinet-getitem?mr=1682729); part II., Algebra Colloq. 4 (1997), no. 3, 291–316, [MR2000c:14006b](http://www.ams.org/mathscinet-getitem?mr=1681547); preprint version [alg-geom/9405013](http://arxiv.org/abs/alg-geom/9405013);_Homotopy Lie algebras_, I.M. Gelfand Seminar, Adv. in Sov. Math., 16 (1993), Part 2, 1-28.

* W. Lowen, M. Van den Bergh, _Deformation theory of abelian categories_, Trans. Amer. Math. Soc. __358__  (2006),  no. 12, 5441--5483; [arXiv:math.CT/0405226](http://arxiv.org/abs/math/0405226).

* M. Van den Bergh, _Notes on formal deformations of abelian categories_, [arXiv:1002.0259](http://arxiv.org/abs/1002.0259)

* M. Talpo, [[Angelo Vistoli|A. Vistoli]], _Deformation theory from the point of view of fibered categories_, [arxiv/1006.0497](http://arxiv.org/abs/1006.0497)

* B. Mazur, _Perturbations, deformations, and variations (and "Near-misses") in Geometry, Physics, and Number Theory_, [BAMS 41(3), 307-336](http://www.ams.org/bull/2004-41-03/S0273-0979-04-01024-9/S0273-0979-04-01024-9.pdf)

* M. Artin, _Deformations of singularities_, TATA Lecture Notes vol. 54.

* M. Artin, _Versal deformations and algebraic stacks_, Invent. Math. 1974

* K. Kodaira, L. Nirenberg, D. C. Spencer, _On the existence of deformation of complex analytic structures_, Ann. Math. __68__, 450-459 (1958).

* K. Kodaira, D. C. Spencer, _On deformation of complex analytic structures_, I II, Ann. Math. __67__, 328-466 (1958).

* M. Schlessinger, _Functors of Artin rings_, Trans. AMS 130, 208-222 (1968) -- this was a groundbreaking article at the time, still much cited.

* B. Osserman, _Deformation theory and moduli in algebraic geometry_, [pdf](http://www.msri.org/people/members/defthy07/lectures/brian.pdf)

* [[Robin Hartshorne]], _Deformation theory_, Grad. texts in math. Springer 2010,  viii+234 pp. (draft of the chap. I-IV: [pdf](http://math.berkeley.edu/~robin/math274root.pdf)), [gBooks](http://books.google.hr/books?id=bwhEX01JlXkC&lpg=PP1&ots=2zkSvT3Bvz&dq=Robin%20Hartshorne%20deformation%20theory&pg=PP1#v=onepage&q&f=false), [doi](http://dx.doi.org/10.1007/978-1-4419-1596-2)

* [homepage](http://math.stanford.edu/~vakil/727) of Ravi Vakil's graduate Stanford class on deformation theory and moduli spaces

* [[Kai Behrend]], [[Barbara Fantechi|B. Fantechi]], _The intrinsic normal cone_, Invent. Math. __128__ (1997), no. 1, 45--88, [MR98e:14022](http://www.ams.org/mathscinet-getitem?mr=98e:14022) [arXiv:alg-geom/9601010](http://arxiv.org/abs/alg-geom/9601010)

* [[Dennis Gaitsgory]], _Lie theory from the point of view of derived algebraic geometry_, Lecture notes from a mini-course, Nantes, 2014, [web](http://www.math.harvard.edu/~gaitsgde/Nantes14/).

* Vladimir Dotsenko, Sergey Shadrin, [[Bruno Vallette]], _Pre-Lie deformation theory_, [arxiv/1502.03280](http://arxiv.org/abs/1502.03280)

* M. Gerstenhaber, S. D. Schack, _Algebras, bialgebras, quantum groups, and algebraic deformations_, in: Deformation theory and quantum groups with applications to mathematical physics (Amherst, MA, 1990), 51-92, Contemp. Math. __134__, Amer. Math. Soc. 1992.

* Gregory Ginot, Sinan Yalin, _Deformation theory of bialgebras, higher Hochschild cohomology and formality_, [arxiv/1606.01504](http://arxiv.org/abs/1606.01504)
[[!redirects deformation]]
[[!redirects deformations]]


deformation theory of sheaves

This article collects some references about the deformation theory of sheaves on [[schemes]] and [[derived schemes]].

## Related

* [[deformation theory]]

* [[perfect complexes]], [[triangulated categories of sheaves]]

* [[derived moduli stack of objects in a dg-category]]

## References

* [[Max Lieblich]], _Moduli of complexes on a proper morphism_, [arXiv:math/0502198](http://arxiv.org/abs/math/0502198).

* [[Wendy T. Lowen]], _Obstruction theory for objects in abelian and derived categories_, [arXiv:math/0407019](http://arxiv.org/abs/math/0407019).

* [[D. Huybrechts]], [[R. P. Thomas]], _Deformation-obstruction theory for complexes via Atiyah and  Kodaira--Spencer classes_, [arXiv:0805.3527](http://arxiv.org/abs/0805.3527).

* [[Alexander I. Efimov]], [[Valery A. Lunts]], [[Dmitri O. Orlov]], _Deformation theory of objects in homotopy and derived categories I:  general theory_, [arXiv:math/0702838](http://arxiv.org/abs/math/0702838).

* [[Alexander I. Efimov]], [[Valery A. Lunts]], [[Dmitri O. Orlov]], _Deformation theory of objects in homotopy and derived categories II:  pro-representability of the deformation functor_, [arXiv:math/0702839](http://arxiv.org/abs/math/0702839).

* [[Alexander I. Efimov]], [[Valery A. Lunts]], [[Dmitri O. Orlov]], _Deformation theory of objects in homotopy and derived categories III:  abelian categories_, [arXiv:math/0702840](http://arxiv.org/abs/math/0702840).

* [[B. Toen]], [[M. Vaquie]], _Moduli of objects in dg-categories_, [arXiv:math/0503269](http://arxiv.org/abs/math/0503269).

* [[J. P. Pridham]], Derived moduli of schemes and sheaves, [arXiv:1011.2742](http://arxiv.org/abs/1011.2742).


defunctionalization

# Defunctionalization
* table of contents
{: toc}

## Idea

_Defunctionalization_ is a method of converting a [[functional programming|functional program]] into one that involves no higher order functions nor lambda expressions. It is a whole program transformation.

## Program transformation

The idea is it to consider a type $Defun(A,B)$ for every pair of types $A,B$ as a proxy for the true [[function type]] $A\to B$.
This type $Defun(A,B)$ is a [[coproduct]]
$$
 Defun(A,B)=\coprod_{x_1:C_1,\dots,x_n:C_n\, \vdash \,\lambda y.\,t\,  :\,  A\to B}C_1\times \dots \times C_n
$$
over all [[lambda abstractions]] $\lambda y.\,t$ that appear syntactically in the program. Notice that lambda abstractions may have free variables $x_1:C_1,\dots,x_n:C_n$, because they may appear deep inside the program.
An inhabitant of the type $Defun(A,B)$ is sometimes called a _closure_, because it is a syntactic expression together with an environment: a valuation for its free variables.

The type $Defun(A,B)$ has a canonical evaluation function
$eval:Defun(A,B)\times A\to B$
roughly given by
$$
 eval(((x_1:C_1\dots x_n:C_n\vdash \lambda y.\,t:A\to B),(x_1,\dots x_n)),y)=t
$$
and so it behaves like a function space.

Since $A$, $B$ and the $C_i$’s can also be defunctionalized, this is a method to transform a whole functional program into one with no lambda expressions nor any true function types.

The coproduct type $Defun(A,B)$ is typically not a true function space in the sense of [[cartesian closed categories]] for two reasons.

* it does not admit arbitrary lambda abstractions -- just the lambda abstractions that happen to be included in the coproduct.

* it does not equate lambda abstractions that are semantically equal but syntactically different.

(Thus it violates both existence and uniqueness in the universal property.)

## Connection to adjoint functor theorems

Recall that a cartesian [[internal hom]] $B^A$, if it exists, is a [[representable functor|representation]] of $Hom(-\times A,B)$.
As such, in many concrete situations, we can deduce its existence from an [[adjoint functor theorem]].
Recall that the [[solution set condition]] in this instance requires a set $I$ and an $I$-indexed family of objects $C_i$
together with morphisms
$$
 f_i:C_i\times A\to B
$$
such that every morphism $f:C\times A\to B$ factors as
$$
 C\times A \stackrel{g\times A}\to C_i\times A\stackrel{f_i}\to B
$$
for some $i\in I$ and $g\colon C\to C_i$.
Then, as is usually observed, the coproduct
$$
 \coprod_{i\in I}C_i
$$
is a weak internal hom, from which the internal hom is recovered by taking a [[coequalizer]]. This coproduct is strongly reminiscent of the coproduct type $Defun(A,B)$ used in defunctionalization.

## Example
Consider the [[factorial]] function in [[continuation-passing style]], written in [[Haskell]]:

    factcps :: Integer -> (Integer -> Integer) -> Integer
    factcps x k = if x == 0 then k 1 else factcps (x-1) (\y -> k (x * y))

    fact :: int -> int
    fact x = factcps x (\x -> x)

The idea is that `factcps x k` calculates the factorial of `x` but does not return the result, instead, it calls the "continuation" `k` with the result. Since `fact 5` calls `factcps 5` with the identity function, it will eventually return `120`.

If this is the whole program, then there are only two [[lambda abstractions]] we need to deal with, both of type `Integer -> Integer`. The first one `\y -> k (x * y)` has free variables `x` and `k`; we'll call it `Multiply`. The second `\x -> x` has no free variables; we'll call it `Identity`.

To defunctionalize this we define a type `DefunIntInt`, which is $Defun(Integer,Integer)$ above:

    data DefunIntInt = Multiply Integer DefunIntInt | Identity
    eval :: DefunIntInt -> Integer -> Integer
    eval (Multiply x k) y = eval k (x * y)
    eval (Identity) x = x

And now we can write the program without [[lambda abstractions]]:

    factcps :: Integer -> DefunIntInt -> Integer
    factcps x k = if x == 0 then eval k 1 else factcps (x-1) (Multiply x k)

    fact :: Integer -> Integer
    fact x = factcps x Identity

In fact, in this example, the type `DefunIntInt` is isomorphic to the type of integer lists, and squinting slightly it can be seen as a call stack.

## References

Defunctionalization was probably introduced by John Reynolds in

* [[John C. Reynolds]], Definitional Interpreters for Higher-Order Programming Languages. In Proceedings of the Annual ACM Conference. 1972.  [pdf]( https://homepages.inf.ed.ac.uk/wadler/papers/papers-we-love/reynolds-definitional-interpreters-1998.pdf)

There and elsewhere a common theme is to first convert programs to [[continuation-passing style]], which makes the control stack explicit, and then to defunctionalize, which makes the control stack look like a stack; the net result is a kind-of abstract machine.


degeneracy pressure

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#### Solid state physics
+-- {: .hide}
[[!include solid state physics -- contents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An intrinsic outward [[pressure]] felt by all [[fermion|fermionic]] [[matter]] due to the _[[Pauli exclusion principle]]_, responsible for the formation of [[solid state physics|solid bodies]] (also [[neutron stars]] and other phenomena).

## References

* Wikipediam _[Electron degeneracy pressure](http://en.wikipedia.org/wiki/Electron_degeneracy_pressure)_

[[!redirects electron degeneracy pressure]]
[[!redirects fermion degeneracy pressure]]


#Contents#
* table of contents
{:toc}

## Idea

In [[algebraic geometry]], there is a __[[Hodge to de Rham spectral sequence]]__ and the statement of the sufficient conditions when it degenerates. [[Maxim Kontsevich]] has conjectured an extension of this to [[noncommutative algebraic geometry]] based on $A_\infty$-[[A-infinity category|categories]]. A somewhat weaker case, in the framework of [[dg-category|dg-categories]] has been proved by [[Dmitri Kaledin]]. Although the conjecture is in [[characteristic zero]], Kaledin has used a method in positive characteristic, combining the cyclic homology with ideas from the one of the proofs of the classical Hodge-dR ss degeneration with positive characteristic methods due [[Pierre Deligne]] and [[Luc Illusie]], involving Frobenius automorphism and so-called [[Cartier operator]]. This is one of the most nontrivial facts in [[noncommutative geometry]].

## References

* _Degeneration of the Hodge-de-Rham spectral sequence_ ([pdf](http://math.bu.edu/people/potthars/writings/HdRSS.pdf))
* [[D. Kaledin]], Non-commutative Hodge-to-de Rham degeneration via the method of
Deligne-Illusie, Pure Appl. Math. Quat. __4__ (2008), 785–875.
* D. Kaledin, Spectral sequences for cyclic homology, in Algebra, Geometry and
Physics in the 21st Century (Kontsevich Festschrift), Birkhäuser, Progress in Math.
__324__ (2017), 99–129.
* D. Kaledin, A. Konovalov, K. Magidson, _Spectral algebras and non-commutative Hodge-to-de Rham degeneration_, [arxiv/1906.09518](https://arxiv.org/abs/1906.09518)

[[!redirects Hodge to de Rham spectral sequence degeneration]]
[[!redirects degeneration of Hodge to de Rham spectral sequence]]
[[!redirects degeneration of Hodge-to-de-Rham spectral sequence]] [[!redirects degeneration conjecture]]

[[!redirects degeneration conjecture]]

[[!redirects degeneration of the Hodge-to-de Rham spectral sequence]]


degree

* [[degree of a continuous map]]

* [[degree of a coherent sheaf]]

* [[degree of a polynomial]]

category: disambiguation


degree of a coherent sheaf

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#Contents#
* table of contents
{:toc}

## Idea

The concept of _degree_ of a [[coherent sheaf]] is a generalization of the concept of [[first Chern class]] of a [[line bundle]], to which it reduces for coherent sheaves of [[sections]] of [[holomorphic line bundles]] over [[Riemann surfaces]] in [[complex analytic geometry]] (see at _[first Chern class -- In complex analytic geometry](first+Chern+class#InComplexAnalyticGeometry)_).

## Definition

Discussion of degrees of coherent sheaves over [[complex curves]] is  in
e.g. ([Huybrechts-Lehn 96, def. 1.2.11](#HuybrechtsLehn96)).

On a [[compact topological space|compact]] [[Kähler manifold]] $(X,\omega)$ of complex [[dimension]] $n$, then the degree of a [[torsion subgroup|torsion]]-free [[holomorphic vector bundle]] $E$ is the [[integral of differential forms]]

$$
  deg(E)
  =
  \int_X c_1(E) \wedge \omega^{\wedge_{n-1}}
$$

of the [[de Rham cohomology|de Rham representative]] of the actual [[first Chern class]] of $E$ wedge with copies of the given [[symplectic form]] (e.g. [Scheinost-Schottenloher 96, def. 1.14](#ScheinostSchottenloher96)).

## In complex analytic geometry

## Related concepts

* [[rank of a coherent sheaf]]

* [[slope of a coherent sheaf]]

* [[stable coherent sheaf]], [[Bridgeland stability condition]]

## References

* {#HuybrechtsLehn96} [[Daniel Huybrechts]], [[Manfred Lehn]], _The Geometry of the Moduli Spaces of Sheaves_, 1996 ([[HuybrechtsLehn.pdf:file]])

* {#ScheinostSchottenloher96} Peter Scheinost, [[Martin Schottenloher]], _Metaplectic quantization of the moduli spaces of flat and parabolic bundles_, J. reine angew. Mathematik, 466 (1996) ([web](https://eudml.org/doc/153753))

[[!redirects degrees of a coherent sheaf]]

[[!redirects degrees of coherent sheaves]]

[[!redirects degree of a vector bundle]]
[[!redirects degrees of a vector bundle]]
[[!redirects degrees of vector bundles]]


degree of a continuous function

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###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given a [[continuous function]] between two [[connected topological space|connected]] [[closed manifold|closed]] [[orientation|oriented]] [[topological manifolds]] of the same [[dimension]], its _degree_ is a measure for how often the function "wraps its domain around its codomain".

## Definition

For $X$ is a [[connected topological space|connected]] [[closed manifold|closed]] [[orientation|oriented]] [[manifold]] of [[dimension]] $n$, its top [[singular homology|homology group]] $H_n(X) = H_n(X; \mathbb{Z})$ is isomorphic to $\mathbb{Z}$, where the generator $1 \in \mathbb{Z}$ is identified with the orientation class $[\omega_X]$ of $X$, the _[[fundamental class]]_ of $X$.

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

Given a [[continuous map]] $f \colon X \to Y$ between two such [[manifolds]], the [[homomorphism]] $f_\ast = H_n(f) \colon H_n(X) \to H_n(Y)$ is therefore specified by the [[integer]] $n$ such that $f_\ast [\omega_X] = n [\omega_Y]$. This integer is called the **degree** of $f$.

=--

## Computing the degree

We suppose throughout that $X$ and $Y$ are connected closed oriented manifolds of the same dimension $n$. The degree of a continuous function $g \colon X \to Y$ is frequently computed according to the following considerations:

* The space of continuous functions $g \colon X \to Y$ has a [[dense subspace]] consisting of [[smooth functions]] $f \colon X \to Y$, and in particular every continuous function $g$ is [[homotopy|homotopic]] to a smooth function $f$. It therefore suffices to compute the degree of $f$.

* By [[Sard's theorem]], the set of [[singular values]] of a smooth function $f$ has [[measure]] zero (using for example the orientation on $Y$ to define a volume form and hence a measure). Accordingly, we may choose a [[regular value]] $y \in Y$.

* The [[inverse image]] $f^{-1}(y)$ is a compact $0$-dimensional manifold, hence consists of finitely many (possibly zero) points $x_1, \ldots, x_r \in X$. Since these are regular points, $f$ restricts to a diffeomorphism
$$f_i \colon U_i \to V$$
where $U_i$ is a small neighborhood of $x_i$ and $V$ is a small neighborhood of $y$. The diffeomorphism $f_i$ either preserves or reverses the orientation of $U_i$, i.e., the sign of the [[determinant]] as a mapping between [[differential n-forms]]
$$\Omega^n(U_i) \to \Omega^n(V)$$
is either $+1$ or $-1$.

* By a straightforward application of the [[excision axiom]] in [[homology]], it follows that the degree of $f$ is the sum of these signs:
$$\deg(f) = \sum_{i=1}^r sign(\Omega^n(f_i))$$
and this quantity is independent of the choice of regular value $y$.

## Properties

### Hopf degree theorem

+-- {: .num_prop}
###### Proposition
**(Hopf degree theorem)**

Let $n \in \mathbb{N}$ be a [[natural number]] and $X \in Mfd$ be a [[connected topological space|connected]] [[orientation|orientable]] [[closed manifold]] of [[dimension]] $n$. Then the $n$th [[cohomotopy]] classes $\left[X \overset{c}{\to} S^n\right] \in \pi^n(X)$ of $X$ are in [[bijection]] to the [[degree of a continuous function|degree]] $deg(c) \in \mathbb{Z}$ of the representing functions, hence the canonical function

$$
  \pi^n(X)
    \underoverset{\simeq}{S^n \to K(\mathbb{Z},n)}{\longrightarrow}
   H^n(X,\mathbb{Z})
     \;\simeq\;
   \mathbb{Z}
$$

from $n$th [[cohomotopy]] to $n$th [[integral cohomology]] is a [[bijection]].

=--

(e.g. [Kosinski 93, IX (5.8)](Hopf+degree+theorem#Kosinski93), [Kobin 16, 7.5](Hopf+degree+theorem#Kobin16))

### Poincaré–Hopf theorem

See at _[[Poincaré–Hopf theorem]]_.

### Generalization to the Adams d-invariant

The Hopf degree of a map is a special case of its _[[Adams d-invariant]]_; see [there](d-invariant#DInvariantSpecializesToDegreeOfAContinuousFunction) for more.

## Examples

* [[winding number]]

## Related concepts

* [[Poincaré–Hopf theorem]]

## References

Texbook accounts:

* {#DubrovinNovikovFomenko85} B. A. Dubrovin, [[S. P. Novikov]], A. T. Fomenko, section 13 of: _Modern Geometry — Methods and Applications: Part II: The Geometry and Topology of Manifolds_, Graduate Texts in Mathematics 104, Springer-Verlag New York, 1985

See also:

* Wikipedia, _[Degree of a continuous mapping](http://en.wikipedia.org/wiki/Degree_of_a_continuous_mapping)_

[[!redirects degree of a map]]
[[!redirects degree of a continuous map]]
[[!redirects degree of a continuous mapping]]


degree of a polynomial

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

\tableofcontents

## Definition
 {#Definition}

The following definition of the degree of a polynomial is formulated vis the *formal [[derivative]]* or *left shift* of a polynomial.

The idea behind this lies in the traditional conception of polynomials as [[sums]] of [[monomials]], namely of [[multiplication|products]] of non-zero [[scalars]] with powers of an "indeterminate" -- which have an associated exponent because commutative rings are [[power-associative]] in [[multiplication]].

Both the derivative and the left shift are functions on polynomials which reduce the exponent of the associated indeterminate by one and take polynomials, relative to the given indeterminate, to zero. Since both operations are also [[linear functions]], they act this way on each monomial summand, which means that after repeatedly taking derivatives or left shifts of polynomials, the result  will eventually become [[zero]].

This allows us to define the degree of a polynomial using [[induction]] on the [[natural numbers]] and on [[lists]] of natural numbers, as well as [[composition]] of left shifts, without having to write out long formulae using indices and ellipses everywhere in the definition, and without having to include additional structure with each polynomial $p$ in the form of the [[scalar]] [[coefficients]] of $p$.

### Polynomials in one indeterminate

Given a [[commutative ring]] $R$, the commutative ring of [[polynomials]] $R[x]$ in one indeterminate $x$ is the [[initial object|initial]] commutative ring $R[x]$ with an element $x \in R[x]$ and a [[ring homomorphism]] $h:R \to R[x]$.

#### Using derivatives

The **formal derivative** $\frac{d (-)}{d x}:R[x] \to R[x]$ is a function which is inductively defined on $R[x]$ by

* Constants go to zero: $\frac{d h(r)}{d x} = 0$ for all $r \in R$
* Indeterminate goes to one: $\frac{d x}{d x} = h(1)$
* Addition preservation: $\frac{d (p + q)}{d x} = \frac{d p}{d x} + \frac{d q}{d x}$ for all $p \in R[x]$ and $q \in R[x]$.
* Leibniz rule: $\frac{d (p \cdot q)}{d x} = \frac{d q}{d x} \cdot q + p \cdot \frac{d q}{d x}$ for all $p \in R[x]$ and $q \in R[x]$.

By the universal property of $R[x]$, these constructors are enough to define the derivative on all polynomials $p \in R[x]$.

Let $\frac{d^{(-)} (-)}{d x^{(-)}}:\mathbb{N} \to (R[X] \to R[X])$ be the function which takes a natural number $n \in \mathbb{N}$ to the $n$-th derivative in the [[function set]] $R[X] \to R[X]$. This is inductively defined on the natural numbers by

* $\frac{d^{0} (-)}{d x^{0}} = \mathrm{id}_{R[X]}$
* $\frac{d^{s(n)} (-)}{d x^{s(n)}} = \frac{d (-)}{d x} \circ \frac{d^{n} (-)}{d x^{n}}$ for all $n \in \mathbb{N}$

If $R$ has [[characteristic zero]], given a non-zero polynomial $p \in R[x]$, the **degree** of $p$ is the maximum [[natural number]] $n$ where the $n$-th iteration of the formal [[derivative]] of $p$ is non-zero:
$$\mathrm{deg}(p) \coloneqq \max_{n \in \mathbb{N}, \frac{d^{n} p}{d x^{n}} \neq 0}(n)$$

The degree of the zero polynomial is undefined, since the formal derivative of the zero polynomial is never non-zero.

The above definition involving formal derivatives works for commutative rings $R$ with [[characteristic zero]]. However, if $R$ has [[positive characteristic]], the formal derivative is problematic for the degree, because given characteristic $n$, the derivative of the $n$-th power of the indeterminate $x$ is zero. Instead, we have to use something more general, the left shift.

#### Using left shifts

For every polynomial $p \in R[x]$, $p$ could be represented as the sum of a constant polynomial $c_p$ and the product of the indeterminate $x$ and a polynomial $s_L(p)$ called the **left shift** of $p$: $p = c_p + x \cdot s_L(p)$. Given any two constant polynomials $p = c_p$ and $q = c_q$, by definition, $c_{p \cdot q} = c_p \cdot c_q$. Now, given any two general polynomials $p = c_p + x \cdot s_L(p)$ and $q = c_q + x \cdot s_L(q)$, one then has the following, through the axioms of a [[commutative ring]]:
$$p = x \cdot s_L(p) + c_p, q = x \cdot s_L(q) + c_q$$
$$p \cdot q = x \cdot s_L(p \cdot q) + c_{p \cdot q} = x s_L(p \cdot q) + c_{p} \cdot c_{q}$$
$$p \cdot q = (x \cdot s_L(p) + c_p) \cdot (x \cdot  s_L(q) + c_q) = x^2 \cdot s_L(p) \cdot s_L(q) + x \cdot s_L(p) \cdot c_q + x \cdot s_L(q) \cdot c_p + c_p \cdot c_q$$
$$x \cdot s_L(p \cdot q) = x^2 \cdot s_L(p) \cdot s_L(q) + x \cdot s_L(p) \cdot c_q + x \cdot s_L(q) \cdot c_p$$
$$s_L(p \cdot q) = x \cdot s_L(p) \cdot s_L(q) + s_L(p) \cdot c_q + s_L(q) \cdot c_p$$
$$p - x \cdot s_L(p) = c_p, q - x \cdot s_L(q) = c_q$$
$$s_L(p \cdot q) = x \cdot s_L(p) \cdot s_L(q) + s_L(p) \cdot (q - x \cdot s_L(q)) + s_L(q) \cdot (p - x \cdot  s_L(p)) = s_L(p) \cdot q + s_L(q) \cdot p - x \cdot s_L(p) \cdot s_L(q)$$
Thus, we have the formula for the left shift of the product $p \cdot q$ for any two polynomials $p \in R[x]$ and $q \in R[x]$:
$$s_L(p \cdot q) = s_L(p) \cdot q + p \cdot s_L(q) - x \cdot s_L(p) \cdot s_L(q)$$
The only difference from the [[Leibniz rule]] for the formal [[derivative]] is the extra term $-x \cdot s_L(p) \cdot s_L(q)$ at the end of the formula.

This allows us to inductively define on $R[x]$ the formal left shift operator $s_L:R[x] \to R[x]$ in the same manner as the formal derivative, by

* Constants go to zero: $s_L(h(r)) = 0$ for all $r \in R$
* Indeterminate goes to one: $s_L(x) = h(1)$
* Addition preservation: $s_L(p + q) = s_L(p) + s_L(q)$ for all $p:R[x]$ and $q:R[x]$.
* Left shift product rule: $s_L(p \cdot q) = s_L(p) \cdot q + p \cdot s_L(q) - x \cdot s_L(p) \cdot s_L(q)$ for all $p \in R[x]$ and $q \in R[x]$.

By the universal property of $R[x]$, these constructors are enough to define the left shift on all polynomials $p \in R[x]$.

Let $s_L^n:\mathbb{N} \to (R[X] \to R[X])$ be the function which takes a natural number $n \in \mathbb{N}$ to the $n$-th left shift in the [[function set]] $R[X] \to R[X]$. This is inductively defined on the natural numbers by the following constructors

* $s_L^0 = \mathrm{id}_{R[X]}$
* $s_L^{s(n)} = s_L \circ s_L^n$ for all $n \in \mathbb{N}$

In this defintion, $R$ is not required to have [[characteristic zero]]. Given a non-zero polynomial $p \in R[x]$, the **degree** of $p$ is the [[maximum]] [[natural number]] $n$ where the $n$-th iteration of the formal left shift operator of $p$ is non-zero:
$$\mathrm{deg}(p) \coloneqq \max_{n \in \mathbb{N}, s_L^n(p) \neq 0}(n)$$

The degree of the zero polynomial is undefined, since the left shift of the zero polynomial is never non-zero.

### Polynomials in a finite number of indeterminates

Let us define the standard [[finite set]] $\mathrm{Fin}(n)$ as the set of all [[natural numbers]] less than $n$:

$$\mathrm{Fin}(n) \coloneqq \{i \in \mathbb{N} \vert i \lt n\}$$

Given a natural number less than $n$, $i:\mathrm{Fin}(n)$, let $\mathrm{Fin}(n) \setminus \{i\}$ be the set of natural numbers less than $n$ which are not equal to $i$.

Given a [[commutative ring]] $R$ with [[characteristic zero]] and a natural number $n$, the commutative ring of [[polynomials]] $R[X]$ is the [[initial object|initial]] [[commutative ring]] $R[X]$ with a function $X:\mathrm{Fin}(n) \to R[X]$ and a [[ring homomorphism]] $h:R \to R[X]$. We write $X_i$ for $X(i)$ throughout. Given a natural number less than $n$, $i \in \mathrm{Fin}(n)$, let $R[X \setminus \{X_i\}]$ be the polynomial [[subring]] of $R[X]$ whose indeterminates do not include $X_i$, with monomorphism $m:R[X \setminus \{X_i\}] \hookrightarrow R[X]$.

Let $\mathrm{Fin}(n)^*$ be the [[free monoid]] on the set of natural numbers less than $n$. Every [[free monoid]] $\mathrm{Fin}(n)^*$ has a length function $\mathrm{len}:\mathrm{Fin}(n)^* \to \mathbb{N}$ which returns the number of elements in a list $a \in \mathrm{Fin}(n)^*$, inductively defined by the following constructors

* Monoidal unit preservation: $\mathrm{len}(\epsilon) = 0$
* Monoidal product preservation: $\mathrm{len}(a b) = \mathrm{len}(a) + \mathrm{len}(b)$ for all $a \in \mathrm{Fin}(n)^*$ and $b \in \mathrm{Fin}(n)^*$
* Generators have length 1: $\mathrm{len}(i) = 1$ for all $i \in \mathrm{Fin}(n)$

#### With partial derivatives

The **formal partial derivative** $\frac{\partial(-)}{\partial X_{(-)}}:\mathrm{Fin}(n) \times R[X] \to R[X]$ is inductively defined by

* Constants relative to the indeterminant go to zero: $\frac{\partial h(r)}{\partial X_i} = 0$ for all $i \in \mathrm{Fin}(n)$ and $r \in R[X \setminus \{X_i\}]$
* Indeterminates go to one: $\frac{\partial X_i}{\partial X_i} = 1$ for all $i \in \mathrm{Fin}(n)$
* Addition preservation: $\frac{\partial (P + Q)}{\partial X_i} = \frac{\partial P}{\partial X_i} + \frac{\partial Q}{\partial X_i}$ for all $i \in \mathrm{Fin}(n)$, $P \in R[X]$ and $Q \in R[X]$.
* Leibniz rule: $\frac{\partial (P \cdot Q)}{\partial X_i} = \frac{\partial P}{\partial X_i} \cdot Q + P \cdot \frac{\partial Q}{\partial X_i}$ for all $i \in \mathrm{Fin}(n)$, $P \in R[X]$ and $Q \in R[X]$.

By the universal property of $R[X]$, these constructors are enough to define the partial derivatives on all polynomials $P \in R[X]$.

Let $d:\mathrm{Fin}(n)^* \to (R[X] \to R[X])$ be the [[monoid homomorphism]] which takes a list $a \in \mathrm{Fin}(n)^*$ of natural numbers less than $n$, to the composition of formal partial derivatives
$$d(a) \coloneqq \frac{\partial^{\mathrm{len}(a)} (-)}{\partial X_{a_0} \partial X_{a_1} \partial X_{a_2} \ldots} \coloneqq \frac{\partial (-)}{\partial X_{a_0}} \circ \frac{\partial (-)}{\partial X_{a_1}} \circ \frac{\partial (-)}{\partial X_{a_2}} \circ \ldots$$
in the [[function set]] $R[X] \to R[X]$. This is inductively defined by the following constructors

* Monoidal unit preservation: $d(\epsilon) = \mathrm{id}_{R[X]}$
* Monoidal product preservation: $d(a b) = d(a) \circ d(b)$ for all $a \in \mathrm{Fin}(n)^*$ and $b \in \mathrm{Fin}(n)^*$
* Generators to partial derivatives: $d(i) = \frac{\partial (-)}{\partial X_i}$ for all $i \in \mathrm{Fin}(n)$

If $R$ has [[characteristic zero]], given a non-zero polynomial $P \in R[x]$, the **degree** of $P$ is the maximum length of all lists $a \in \mathrm{Fin}(n)^*$ of natural numbers less than $n$ such that the evaluation of $d(a)$ at $P$ is non-zero
$$\mathrm{deg}(P) \coloneqq \max_{a \in \mathrm{Fin}(n)^*, d(a)(P) \neq 0}(\mathrm{len}(a))$$
The degree of the zero polynomial is undefined, since any composition of partial derivatives evaluated at the zero polynomial is never non-zero.

The above definition involving formal partial derivatives works for commutative rings $R$ with [[characteristic zero]]. However, if $R$ has [[positive characteristic]], the formal derivative is problematic for the degree, because given characteristic $p$, the derivative of the $p$-th power of any indeterminate $X_i$ for natural number less than $n$, $i \in \mathrm{Fin}(n)$, is zero. Instead, we have to use something more general, partial left shifts.

#### With partial left shifts

The **formal partial left shift operator** $s_{\partial L}:\mathrm{Fin}(n) \times R[X] \to R[X]$ is inductively defined by

* Constants relative to the indeterminant go to zero: $s_{\partial L}(i, r) = 0$ for all $i \in \mathrm{Fin}(n)$ and $r \in R[X \setminus \{X_i\}]$
* Indeterminates go to one: $s_{\partial L}(i, X_i) = 1$ for all $i \in \mathrm{Fin}(n)$
* Addition preservation: $s_{\partial L}(i, P + Q) = s_{\partial L}(i, P) + s_{\partial L}(i, Q)$ for all $i \in \mathrm{Fin}(n)$, $P \in R[X]$ and $Q \in R[X]$.
* Left shift product rule: $s_{\partial L}(i, P \cdot Q) = s_{\partial L}(i, P) \cdot Q + P \cdot s_{\partial L}(i, Q) - X_i \cdot s_{\partial L}(i, P) \cdot s_{\partial L}(i, Q)$ for all $i \in \mathrm{Fin}(n)$, $p \in R[x]$ and $q \in R[x]$.

By the universal property of $R[X]$, these constructors are enough to define the partial left shifts on all polynomials $P \in R[X]$.

Let $S:\mathrm{Fin}(n)^* \to (R[X] \to R[X])$ be the [[monoid homomorphism]] which takes a list $a\in \mathrm{Fin}(n)^*$ of natural numbers less than $n$, to the composition of formal partial left shifts
$$S(a) \coloneqq s_{\partial L}(a_0, -) \circ s_{\partial L}(a_1, -) \circ s_{\partial L}(a_2, -) \circ \ldots$$
in the [[function set]] $R[X] \to R[X]$. This is inductively defined by the following constructors

* Monoidal unit preservation: $S(\epsilon) = \mathrm{id}_{R[X]}$
* Monoidal product preservation: $S(a b) = S(a) \circ S(b)$ for all $a \in \mathrm{Fin}(n)^*$ and $b \in \mathrm{Fin}(n)^*$
* Generators to partial left shifts $S(i) = s_{\partial L}(i, -)$ for all $i \in \mathrm{Fin}(n)$

Given a non-zero polynomial $P \in R[x]$, the **degree** of $P$ is the maximum length of all lists $a \in \mathrm{Fin}(n)^*$ of natural numbers less than $n$ such that the evaluation of $S(a)$ at $P$ is non-zero
$$\mathrm{deg}(P) \coloneqq \max_{a \in \mathrm{Fin}(n)^*, S(a)(P) \neq 0}(\mathrm{len}(a))$$
The degree of the zero polynomial is undefined, since any composition of partial derivatives evaluated at the zero polynomial is never non-zero.

## Homogeneous polynomials

...

Given a polynomial $p \in R[x]$ in one indeterminate, it is said to be an **homogeneous polynomial** of degree $n$ if $\frac{d p}{d x}.x = n.p$ (see the [[homogeneous polynomial|Euler identity]]).

Given a polynomial $p \in R[X]$ in a finite number of indeterminates, it is said to be an homogeneous polynomial of degree $n$ if $\underset{i \in \mathrm{Fin}(n)}{\sum} \frac{\partial p}{\partial X_i}.X_i = n.p$.

One could also define a polynomial $p \in R[X]$ to be homogeneous of degree $n$ with respect to $i \in \mathrm{Fin}(n)$ if $\frac{\partial p}{\partial X_i}.X_i = n.p$. It is equivalent to the fact that $p$ can be written under the form $p=X_i^{n}.q$ where $q \in R[X]$ is such that $\frac{\partial p}{\partial X_i} = 0$.

## In constructive mathematics

In [[constructive mathematics]], where [[excluded middle]] does not hold, the above definition is correct only if the ring $R$ has [[decidable equality]]. In more general circumstances, one has to assume that the ring $R$ and thus the polynomial ring $R[X]$ is an [[inequality space]] with a [[tight apartness relation]] $\#$, and replace all instances of "non-zero" $p \neq 0$ with instances of "apart from zero" $p \# 0$. (That every set $R$ is an inequality space in classical mathematics follows from the [[stable equality|stability]] of decidable equality.) For example, the degree function on the [[Dedekind real numbers]] is only defined on polynomials with at least one coefficient whose [[absolute value]] is greater than zero.

## See also

* [[polynomial]]
* [[degree]]

[[!redirects polynomial degree]]


degree of freedom

## References

* Wikipedia, [Degrees of freedom](https://en.wikipedia.org/wiki/Degrees_of_freedom)

[[!redirects degrees of freedom]]


Dehn surgery

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

# Contents
* table of contents
{:toc}

## Idea

Dehn surgery is a method for constructing one [[manifold]] from another, especially one [[3-manifold]] from another, by a kind of cut-and-paste procedure.

## Method

A standard application of Dehn surgery is [[surgery]] along a [[link]] $L$ in a [[3-sphere]] $S^3$. This works in two steps (whose description makes it sound like *Zahn* surgery[^1]):

[^1]: Yes, that's supposed to be a little joke. 'Zahn' here is the German word.

* Form the [[complement]] in the [[3-sphere]] of a [[tubular neighborhood]] of the [[embedding of topological spaces|embedded]] [[link]] $L \hookrightarrow S^3$, of the form $L \times int(D^2)$. This is called *Dehn drilling*. The result is a 3-[[manifold with boundary]] $M$, whose [[boundary]] $\partial M \cong L \times S^1$ can be viewed as the boundary of a [[disjoint union]] of [[solid tori]] $L \times D^2$.

* To each of the [[connected components]] $C_1, \ldots, C_n$ of $\partial M$, apply an (orientation-preserving) [[homeomorphism]], say $\phi_1, \ldots, \phi_n$. The [[union]] $\phi_1 \cup \ldots \cup \phi_n$ is a homeomorphism $\phi: \partial M \to \partial M$. Then perform a *Dehn filling* by constructing the [[pushout]] of an evident [[span]]:
$$
  \array{
    \partial M
    &
    \overset{\phi}{\longrightarrow}
    & \partial M
    \\
    \mathllap{incl}
    \downarrow
    & &
    \downarrow
    \mathrlap{incl}
    \\
    M
    &
    &
    L \times D^2,
  }
$$
thus refilling the drilled portion, but in a new way (along $\phi$). This gives a new [[3-manifold]] $N$.

Some further notes: the surgery can be done one [[solid torus]] at a time. A homeomorphism on a boundary torus $S^1 \times S^1$ sends a meridian $S_1 \times \{1\}$ to some simple closed curve that is homotopic to a curve of rational slope $p/q$ (the curve which is the image of the line $y = (p/q)x$ in $\mathbb{R}^2$ under the quotient map $\mathbb{R}^2 \to \mathbb{R}^2/\mathbb{Z}^2 \cong S^1 \times S^1$). It turns out that the result of the surgery depends, up to homeomorphism, only on the quantity $p/q$, called a *surgery coefficient*. If all the surgery coefficients are [[integers]], we speak of an *integral surgery*.

Put a bit different: given a [[framed link]] in an [[orientation|oriented]] 3-manifold like $S^3$, an integral surgery drills out a solid torus, [[Dehn twist|twists]] it an integral number of times according to the framing, and then reattaches it.

## Results

+-- {: .num_theorem}
###### Theorem
**(Lickorish-Wallace)**
Every [[connected space|connected]] [[orientation|oriented]] [[closed manifold|closed]] [[3-manifold]] $N$ arises by performing an integral Dehn surgery along a [[link]] $L \hookrightarrow S^3$ (i.e., surgery along a framed link).
=--

## Related concepts

* [[Kirby calculus]]

* [[surgery theory]]

## References

### General

See also

* Wikipedia, _[Dehn surgery](https://en.wikipedia.org/wiki/Dehn_surgery)_

### In Chern-Simons theory

Relation between [[Dehn surgery]] and [[Wilson loop observables]] in [[Chern-Simons theory]]:

* E. Guadagnini, _Surgery rules in quantum Chern-Simons field theory_, Nuclear Physics B
Volume 375, Issue 2, 18 May 1992, Pages 381-398 (doi:10.1016/0550-3213(92)90037-C)

* Boguslaw Broda, _Chern-Simons theory on an arbitrary manifold via surgery_ ([arXiv:hep-th/9305051](https://arxiv.org/abs/hep-th/9305051))

[[!redirects Dehn surgeries]]


Dehn twist

# Contents
* table of contents
{:toc}

## Idea

A Dehn twist is a type of self-[[homeomorphism]] on a surface (a 2-dimensional [[topological manifold]]), especially an [[orientation|orientable]] [[closed manifold]] (of dimension 2).

## Definition

Let $\Sigma$ be an orientable surface of [[genus]] $g$, and consider an embedded circle $i: S^1 \hookrightarrow \Sigma$. (For example, one may imagine a circle representing one of $2 g$ generators of $H^1(\Sigma, \mathbb{Z})$.) The embedded circle has a [[tubular neighborhood]] in $\Sigma$, homeomorphic to an annulus $S^1 \times [0, 1]$. Representing elements of $S^1$ by [[complex numbers]] $z$ of [[norm]] 1, a *twist* on the annulus may be defined by sending $(z, t) \mapsto (\exp(2\pi i t) z, t)$, which equals the identity on the boundary circles where $t = 0, t = 1$. This defines a self-homeomorphism (mod boundary) on the annulus.

The corresponding Dehn twist on $\Sigma$ is obtained by extending the self-homeomorphism on the annulus to the entire surface, by taking a point $p$ to itself if $p$ is outside the annulus. (If we are working in the category of smooth manifolds, one may modify the twist on the annulus by taking $(z, t) \mapsto (\exp(2\pi i f(t))z, t)$ where $f$ is a smooth [[bump function]] such that $f(t) = 0$ on a small neighborhood of $t = 0$ and $f(t) = 1$ on a small neighborhood of $t=1$. This modification ensures that we get a self-[[diffeomorphism]] on the surface $\Sigma$ as smooth manifold.)

## Properties

The following theorem was first proved by MaxDehn, with later simplifications by Lickorish:

+-- {: .num_theorem}
###### Theorem
**("Lickorish twist")**
The Dehn twists generate the [[mapping class group]] of $\Sigma$, of orientation-preserving homeomorphisms considered modulo [[isotopy]]. (In fact Lickorish described $3g-1$ explicit embedded circles for a surface $\Sigma$ of genus $g$ whose corresponding twists give the generators.)
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One example that is easily visualized is the mapping class group $MCG(\Sigma)$ of the [[torus]] $\Sigma = S^1 \times S^1$. Here the canonical map $MCG(\Sigma) \to Aut(\pi_1(\Sigma)) \cong Aut(\mathbb{Z} \times \mathbb{Z})$ is an isomorphism (where automorphisms of $\mathbb{Z}$ are required to preserve orientation, i.e., are elements of the group $SL_2(\mathbb{Z})$; cf. [[modular group]]). Thus in this case the Dehn twists can be visualized in terms of their action on loops representing elements of $\pi_1(\Sigma)$; the [Wikipedia article](#WP) has some decent pictures of this action.

## Related concepts

* [[mapping class group]]

* [[Dehn surgery]]

## References

* Wikipedia, _[Dehn twist](http://en.wikipedia.org/wiki/Dehn_twist)_
 {#WP}

[[!redirects Dehn twists]]


del Pezzo surface

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##Idea

A (complex) **del Pezzo surface** is a smooth projective [[complex surface]] with [[ample line bundle|ample]] anticanonical line bundle. Such a surface has a degree defined as the self-intersection of the [[canonical divisor]]. The possible degrees run between $d=1$ and $d=9$. Topologically, del Pezzo surfaces are determined by their degree except for $d=8$.

If $d$ is different from 8, then a del Pezzo surface of degree $d$, $\mathbb{B}_k$, is a generic [[blow up]] of the [[complex projective plane]] $\mathbb{P}^2$ in $9-d$ points. But if $d=8$, there are two choices: $\mathbb{P}^2$ blown up in one point, $\mathbb{B}_1$, and $\mathbb{P}^1 \times \mathbb{P}^1$, .

For surfaces formed by blowing up $9-d$ points, no three may be collinear, no six lie on a conic, and no eight of them lie on a cubic having a node at one of them. Conversely any blowup of the plane in points satisfying these conditions is a del Pezzo surface.

## Properties

These surfaces admit [[metrics]] of positive [[scalar curvature]].

Homology in degree 2: $dim H_2(\mathbb{B}_k, \mathbb{R}) = k+ 1$; a natural basis to choose is $\{H, E_1, \ldots, E_k\}$, where the $E_a$ are the "exceptional curves" obtained by blow-ups and $H$ represents the pullback of the generator of $dim H_2(\mathbb{P}^2, \mathbb{R})$ under the projection $\pi: \mathbb{B}_k \to\mathbb{B}_0  = \mathbb{P}^2$ which simply collapses each exceptional curve $E_a$ to the corresponding point $p_a \in \mathbb{P}$.

This has an integral statement as well. The orthogonal complement to the canonical class in the integral $H_2$ is isomorphic to a root lattice $R_r$. These are given as $E_8$, $E_7$, $E_6$, $D_5$, $A_4$ or $A_2 \times A_1$ depending on the value of $k$.

## Related concepts

* [[mysterious duality]]

## References

See also:

* Wikipedia, _[Del Pezzo surface](https://en.m.wikipedia.org/wiki/Del_Pezzo_surface)

On [[exceptional structures]] ([[exceptional Lie algebras]] and [[sporadic finite simple groups]]) via del Pezzo surfaces:

* [[Yang-Hui He]], [[John McKay]], _Sporadic and Exceptional_ [[arXiv:1505.06742](https://arxiv.org/abs/1505.06742)]

[[!redirects del Pezzo surfaces]]
[[!redirects Del Pezzo surface]]
[[!redirects Del Pezzo surfaces]]


delayed homotopy

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## Idea

A [[homotopy]] between [[continuous function]]s between [[topological space]]s is called _delayed_ if it starts out being constant near one boundary of the interval.

(If it is constant near both boundaries we say it has [[sitting instant]]s).

## Definition

For $I = [0,1]$ the [[unit interval]] and $X$ and $Y$ any [[topological space]]s, a [[continuous map]] $F: X\times I\to Y$ is a **delayed homotopy** (between $F(-,0)$ and $F(-1))$  if there exist $t_0\gt 0$ such that $F(x,t)=F(x,0)$ for all $0\leq t\leq t_0$.

## Properties

### In Dold-fibrations

Delayed homotopies appear in an alternative characterization of [[Dold fibration]]s.  See there for details.

### Smoothing of delayed homotopies

If a continuous homotopy between two [[smooth function]]s is delayed at both ends of the inerval it may be approximated by a _smooth homotopy_ . See [[Steenrod-Wockel approximation theorem]].

[[!redirects delayed homotopies]]