division rig

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### Context
#### Linear algebra
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[[!include homotopy - contents]]
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#### Algebra
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[[!include higher algebra - contents]]
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# Contents
* table of contents
{: toc}

## Definitions

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

A **division rig** is a [[rig]] in which every nonzero element has a multiplicative [[inverse]] and $0 \neq 1$ (which may be combined as: an element is invertible if and only if it is nonzero).
=--

### Constructive notions
{#constructive}

Division rigs are (arguably) not a purely [[algebra|algebraic]] notion in that they don\'t form an [[algebraic category]] (see [discussion below](#category)).  For this reason, it should be unsurprising that in [[constructive mathematics]] (including the [[internal logic]] of a [[topos]]) there are different inequivalent ways to define a division rig.  In this case the classical definition is not usually the best one; for instance, the non-negative real numbers do not satisfy it. There are several potential replacements with their own advantages and disadvantages.

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

If we replace "an element is invertible iff it is nonzero" in Definition \ref{classical} by "an element is invertible [[xor]] it equals zero" (which is equivalent in [[classical logic]] but stronger in [[constructive logic]]), then we obtain the notion of **discrete division rig**.  This condition means that every element is either $0$ or invertible, and it also implies that $0\neq 1$.
=--

Such a division rig $R$ is 'discrete' in that it decomposes as a coproduct $R = \{0\} \sqcup R^\times$ (where $R^\times$ is the subset of invertible elements).  An advantage is that this is a [[coherent logic|coherent theory]] and hence also a [[geometric theory]].  A disadvantage is that this axiom is not satisfied (constructively) by the rig of non-negative [[real numbers]] (however these are defined), although it is satisfied by the rig of non-negative [[rational number|rationals]].

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

If we interpret 'nonzero' in Definition \ref{classical} as a reference to a [[tight apartness relation]], thus defining the apartness relation $\#$ by $x # y$ iff $x - y$ is invertible, then we obtain the notion of **Heyting division rig**, where the rig operations become strongly extensional functions.  In addition to $0\# 1$, the condition then means that every element apart from $0$ is invertible.
=--

This is how 'practising' constructive analysts of the Bishop school usually define the simple word 'division rig'.  An advantage is that the (located Dedekind) non-negative [[real numbers]] form a Heyting division rig, although (for example) the (less located) non-negative [[MacNeille real number]]s need not form a Heyting division rig; another disadvantage is that this is not a coherent axiom and so cannot be [[internalization|internalized]] in as many categories.

Every discrete division rig is also a Heyting division rig. A Heyting division rig is a discrete division rig if and only if [[decidable equality|equality is decidable]]; it is in this sense that a discrete division rig is 'discrete'.

## Examples
 {#Examples}

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

There are the division rigs of:

* non-negative [[rational numbers]],

* non-negative [[real numbers]],

* all [[division rings]], such as [[fields]]

* for a division rig $R$, all $R$-[[division algebras]]

=--

## Related concepts

* [[rig]], [[integral cancellable rig]]

* [[division ring]]

* [[field]]

[[!redirects division rig]]
[[!redirects division rigs]]

[[!redirects commutative division rig]]
[[!redirects commutative division rigs]]


division superalgebra

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###Context###
#### Super-Algebra
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[[!include supergeometry - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The concept of a **division superalgebra** (also **super division algebra**) is the analog of that of *[[division algebras]]* in the context of [[superalgebra]]. As such, a division superalgebra is a [[superalgebra]] in which every nonzero homogeneous element is [[invertible]].

The classification of the [[associative]] real division superalgebras by a "[[ten-fold way]]" is essentially attributed to [Dyson (1962)](#Dyson62) by [Freed & Moore (2013, Appendix C)](#FreedMoore13) (in the context of the [[K-theory classification of topological phases of matter]]), further expanded on in [Moore (2013), Section 14](#Moore13) and [Geiko & Moore (2021)](#GeikoMoore21).

## Related concepts

* [[division algebra]]

## References

* {#Dyson62} [[Freeman Dyson]], *The Threefold Way. Algebraic Structure of Symmetry Groups and Ensembles in Quantum Mechanics*, J. Math. Phys. **3** 1199 (1962) [[doi:10.1063/1.1703863](https://doi.org/10.1063/1.1703863)]

* [[Todd Trimble]], The super Brauer group and super division algebras (April 27, 2005) [[webpage](https://math.ucr.edu/home/baez/trimble/superdivision.html)]

* {#FreedMoore13} [[Daniel Freed]], [[Gregory Moore]], Appendix C of: *Twisted equivariant matter*, Ann. Henri Poincaré **14** (2013)  1927-2023 [[arXiv:1208.5055](https://arxiv.org/abs/1208.5055), [doi:10.1007/s00023-013-0236-x](https://doi.org/10.1007/s00023-013-0236-x)]

* {#Moore13} [[Gregory Moore]], Section 14 of: _Quantum symmetries and compatible Hamiltonians_ (2013) [[pdf](http://www.physics.rutgers.edu/~gmoore/695Fall2013/CHAPTER1-QUANTUMSYMMETRY-OCT5.pdf)]

* {#GeikoMoore21} Roman Geiko, [[Gregory W. Moore]], _Dyson’s classification and real division superalgebras_, April 2021, Journal of High Energy Physics 2021(4):299, ([doi: 10.1007/jhep04(2021)299](http://dx.doi.org/10.1007/jhep04%282021%29299)).

[[!redirects division superalgebras]]

[[!redirects super division algebra]]
[[!redirects super division algebras]]


divisor

* [[divisor (ring theory)]]

  * [[zero divisor]]

* [[divisor (algebraic geometry)]]

  * [[Weil divisor]], [[Cartier divisor]]

  * [[theta divisor]]

category: disambiguation

[[!redirects divisors]]


divisor (algebraic geoemetry) > history

see _[[divisor (algebraic geometry)]]_


divisor (algebraic geometry)

#Contents#
* table of contents
{:toc}

## Idea

In [[algebraic geometry]] a _divisor_ (or _[[Weil divisor]]_ for definiteness) in a given [[variety]] is a [[formal linear combination]] of sub-[[varieties]] of [[codimension]] 1. Accordingly the _divisor group_ is the [[free abelian group]] on the set of subvarieties of codimension 1.

A divisor is called _effective_ if all its [[coefficients]] are [[positive numbers]].

In [[complex analytic geometry]] any [[meromorphic function]] induces a divisor whose summands are irreducible codimension-1 subvarieties weighted by the order of the [[poles]] of the function on that subvariety. Divisors arising this way are called _principal divisors_. More generally in [[algebraic geometry]] this leads to the concept of _[[Cartier divisor]]_.

Two divisors are called _linearly equivalent_ if their difference is a principal divisor.

## Properties

### Relation to line bundles and the Picard group

Under suitable conditions the [[Poincaré duality|Poincaré dual]] of a divisor may hence define the [[first Chern class]] of an [[algebraic line bundle]] and this construction yields group homomorphism from the group of divisors to the [[Picard group]]. After quotienting out linear equivalence of divisors this yields an inclusion and in complex geometry this inclusion exhibits precisely the subgroup of the [[Picard group]] of [[holomorphic line bundle]] which admit at least one global holomorphic section ([e.g. Huybrechts 04, cor. 2.3.20](#Huybrechts04)).

More abstractly in terms of [[abelian sheaf cohomology]]:

A divisor $D = \sum_i n_i D_i$ on $X$ induces a [[short exact sequence]] of [[abelian sheaves]] on $X$ ([Brylinski 94, (1-1)](#Brylinski94)) of the form

$$
  0
  \to
  \mathcal{O}^\times_X
  \longrightarrow
  \mathcal{O}_X(\ast Y)^\times
  \stackrel{v_Y}{\longrightarrow}
  (v_Y)_\ast \mathbb{Z}_{\tilde Y}
  \to 0
  \,,
$$

where $Y$ is the support of $D$ (i.e. $Y \coloneqq \underset{i | n_i \neq 0}{\cup} D_i$) and $\nu_Y \colon \tilde Y \to Y$ is its normalization in $Y$ (...). The induced [[long exact sequence in homology|long exact sequence]] in [[abelian sheaf cohomology]] has as first [[connecting homomorphism]] a map of the form

$$
  \delta \colon H^0(\tilde Y, \mathbb{Z}) \longrightarrow H^1(X,\mathcal{O}^\times_X)
  \,.
$$

Now the divisor $D = \sum_i n_i D_i$ is incarnated as the locally constant function $n$ on $\tilde Y$ whose value on $\tilde D_i$ is $n_i \in \mathbb{Z}$, hence the divisor is incarnated a class in $H^0(\tilde Y, \mathbb{Z})$. The class $\mathcal{O}(D)$ of the line bundle it induces as the image of this class under the above connecting homomorphims:

$$
  \mathcal{O}(D) \simeq \delta(n)
  \,.
$$

([Brylinski 94, prop. 1.1](#Brylinski94)).

A higher analog of the connecting homomorphism abive may be interpreted as being the [[Beilinson regulator]] from $K_1(X)$ to [[Deligne cohomology]] in degree 3 ([Brylinski 94, section 3](#Brylinski94)).

## Examples

* [[canonical divisor]]

* [[theta divisor]]

## Related concepts

* [[algebraic cycle]]

* [[exceptional divisor]] ([[resolution of singularities]])

## References

* Wikipedia, _[Divisor (algebraic geometry)](http://en.wikipedia.org/wiki/Divisor_(algebraic_geometry)_

* [[The Stacks Project]], _[53 Divisors on Algebraic Spaces](http://stacks.math.columbia.edu/chapter/53)_

In [[complex analytic geometry]]:

* {#Brylinski94} [[Jean-Luc Brylinski]], _Holomorphic gerbes and the Beilinson regulator_, Astérisque 226 (1994): 145-174 ([[Brylinski94.pdf:file]])

* {#Huybrechts04} [[Daniel Huybrechts]] section 2.3 of _Complex geometry - an introduction_. Springer (2004). Universitext. 309 pages. ([pdf](http://www.math.uh.edu/~shanyuji/2012/Geometry/Huybrechts.pdf))

Relation of [[divisors]] to [[complex cobordism cohomology]]:

* [[Burt Totaro]], _Torsion algebraic cycles and complex cobordism_, J. Amer. Math. Soc. 10 (1997), 467-493 ([doi:10.1090/S0894-0347-97-00232-4](https://doi.org/10.1090/S0894-0347-97-00232-4))

* [MO discussion](https://mathoverflow.net/a/272131/381)

[[!redirects principal divisor]]
[[!redirects principal divisors]]


divisor (ring theory)

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### Context
#### Algebra
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[[!include higher algebra - contents]]
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#### Arithmetic
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[[!include arithmetic geometry - contents]]
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# Contents
* table of contents
{: toc}

## Definitions

Let $R$ be a [[commutative ring]] (or any [[ring]]).

An element $x$ of $R$ is a __divisor__ of an element $y$ of $R$ if there exists an element $z$ of $R$ such that $z x = y$.

If $R$ is (or may be) non-commutative, then we may distinguish __left__ and __right__ divisors in the usual way.

## Related concepts

* [[zero divisor]]

* [[divisor (algebraic geometry)]]

## See also

* Wikipedia, _Divisibility (ring theory)_


Dixmier-Douady class

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###Context###
#### Cohomology
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[[!include cohomology - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

A [[bundle gerbe]] or [[circle n-bundle with connection|circle 2-bundle]] has a unique [[characteristic class]] in [[integral cohomology]] in degree 3, the higher analog of the [[Chern class]] of a [[circle group]]-[[principal bundle]] (or complex [[line bundle]]): this is called the **Dixmier-Douady class** of the [[bundle gerbe]].

## Definition

In the literature one find a universal Dixmier-Douady class defined for different entities, notably for [[projective unitary group|projective unitary]]-[[principal bundles]] and for $U(1)$-[[bundle gerbes]], as well as for [[C-star algebra]] constructions related to these. All these notions are equivalent in one sense, namely in bare [[homotopy theory]], but differ in other sense, namely in geometric homotopy theory.

### In bare homotopy-type theory

The [[classifying space]] of the [[circle n-group|circle 2-group]] $\mathbf{B}U(1)$ is an [[Eilenberg-MacLane space]] $B \mathbf{B} U(1) \simeq B^3 \mathbb{Z} \simeq K(\mathbb{Z}, 3)$.
The bare _Dixmier-Douday class_ is the [[universal characteristic class]]

$$
  DD : B B U(1) \stackrel{\simeq}{\to} K(\mathbb{Z}, 3)
$$

exhibited by this equivalence. Hence if we identify $B B U(1)$ with $K(\mathbb{Z}, 3)$, then the DD-class is the _identity_ on this space.

This is directly analogous to how the [[first Chern class]] is, as a [[universal characteristic class]], the identity on $K(\mathbb{Z},2) \simeq B U(1)$.

This means conversely that the [[equivalence class]] of a $U(1)$-[[bundle gerbe]]/[[circle n-bundle with connection|circle 2-bundle]] is entirely characterized by its Dixmier-Douady class.

### In smooth homotopy-type theory

The [[circle 2-group]] $\mathbf{B}U(1)$ naturally carries a _smooth_ structure, hence is naturally regarded not just as an [[∞-group]] in [[∞Grpd]], but as a [[smooth ∞-group]] in $\mathbf{H} \coloneqq$ [[Smooth∞Grpd]].

For each $n$, the [[central extension]] of Lie groups

$$
  U(1) \to U(n) \to PU(n)
$$

that exhibits the [[unitary group]] as a [[circle group]]-extension of the [[projective unitary group]] induces the corresponding morphism of smooth [[moduli stacks]]

$$
  \mathbf{B} U(1) \to \mathbf{B} U(n) \to \mathbf{B} PU(n)
$$

in $\mathbf{H}$.

This is part of a [[fiber sequence|long fiber sequence]] in $\mathbf{H}$ which continues to the right by a [[connecting homomorphism]] $\mathbf{dd}_n$

$$
  \mathbf{B} U(1) \to \mathbf{B} U(n) \to \mathbf{B} PU(n)
   \stackrel{\mathbf{dd}_n}{\to}
  \mathbf{B}^2 U(1)
$$

in $\mathbf{H}$. Here the last morphism is presented in
[[simplicial presheaves]] by the zig-zag/[[∞-anafunctor]] of sheaves of [[crossed modules]]

$$
  \array{
     [U(1) \to U(n)] &\to& [U(1) \to 1]
     \\
     {}^{\mathllap{\simeq}}\downarrow
     \\
     PU(n)
  }
  \,.
$$

To get rid of the dependence on the rank $n$ -- to _stabilize_ the rank --
we may form the [[directed colimit]] of smooth moduli stacks

$$
  \mathbf{B}U \coloneqq \underset{\rightarrow_n}{\lim} \mathbf{B} U(n)
$$

$$
  \mathbf{B} PU \coloneqq \underset{\rightarrow_n}{\lim} \mathbf{B} PU(n)
  \,.
$$

On these we have the smooth universal class

$$
  \mathbf{dd} : \mathbf{B} PU \to \mathbf{B}^2 U(1)
  \,.
$$

Since the [[(∞,1)-topos]] [[Smooth∞Grpd]] has [[universal colimits]], it follows that there is a [[fiber sequence]]

$$
  \array{
    \mathbf{B}U &\to& \mathbf{B} PU
    \\
    && \downarrow^{\mathbf{dd}}
    \\
    && \mathbf{B}^2 U(1)
  }
$$

exhibiting the moduli stack of smooth stable unitary bundles as the homotopy fiber of $\mathbf{dd}$.

## References
 {#References}

* [[Jacques Dixmier]], [[Adrien Douady]], _Champs continus d'espaces hilbertiens et de $C^*$-algebres_, Bull. Soc. Math. France 91 (1963), 227–-284

* [[Jacques Dixmier]], _Les $C^*$-alg\'ebres et leurs représentations, Gauthier--Villars, Paris, 1969


DLO

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### Context
#### Model theory
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[[!include model theory - contents]]
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#### Analysis
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[[!include analysis - contents]]
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# Contents
* table of contents
{: toc}

## Idea

Countable [[dense]] linear orders without endpoints are unique up to isomorphism, and are canonically modeled by the [[rational numbers]] $(\mathbb{Q}, \lt )$.

## Definition

The theory $\mathsf{DLO}$ of the _dense linear order without endpoints_ is the [[first-order theory]] of $(\mathbb{Q}, \lt)$. It is axiomatized by the usual axioms of a linear order, plus the sentences which state that the order is dense and that there is neither an upper nor lower bound on the order.

## Remarks

* $\mathsf{DLO}$ is a prototypical [[stability theory|unstable structure]].

* Cantor's theorem (the uniqueness up to isomorphism of a model of $\mathsf{DLO}$ assuming the model is countable) says precisely that $\mathsf{DLO}$ is an [[omega-categorical structure | omega-categorical]] theory.

* Since $\mathsf{DLO}$ is unstable, however, its uncountable models fall into many isomorphism classes.

* [[Dedekind cut|Dedekind cuts]] arise as [[type (in model theory)|types]] over bounded infinite parameter sets in a single variable.

* $\mathsf{DLO}$ is a [[Fraïssé limit]]; its finitely-generated substructures are precisely the finite linear orders.

* $\mathsf{DLO}$ admits [[quantifier-elimination]].

* If we view $(\mathbb{Q},<)$ as a category, the [[subobject classifier]] of the topos $\mathbf{Sets}^\mathbb{Q}$ can be identified in a canonical way with the [[Dedekind cut|Dedekind cuts]] on $\mathbb{Q}$.

* Let $(A,\lt)$ be a model of $\mathsf{DLO}$. Then $A$ has a [[frame]] of [[open subsets]] with respect to its linear order. When regarded as a [[locale]], the frame of open subsets is isomorphic to the [[locale of real numbers]].

## Related concepts

* [[dense]]

* [[omega-categorical structure]]

* [[homogeneous structure]]

* [[o-minimal structure]]

* [[locator]]

## References

- Ivan Tomasic, [Model theory](http://www.maths.qmul.ac.uk/~ivan/ModTh/lectures.pdf)

[[!redirects DLO]]
[[!redirects dense linear order]]
[[!redirects dense linear orders]]
[[!redirects countable dense linear order]]
[[!redirects countable dense linear orders]]


Dmiti Alekseevesky

* [webpage](http://www.hull.ac.uk/php/masdva/)

category: people

[[!redirects Dmitri Alekseevebsky]]


Dmitri Bykov

* [personal page](https://homepage.mi-ras.ru/~dbykov/)

* [institute page](https://itmp.msu.ru/en/institute/scientific-members/bykov-d-v)

## Selected writings

On [[flag manifold sigma-models]]:

* [[Ian Affleck]], [[Dmitri Bykov]], [[Kyle Wamer]], *Flag manifold sigma models: spin chains and integrable theories*,  Phys. Rept. **953** (2022) 1-93 [[arXiv:2101.11638](https://arxiv.org/abs/2101.11638), [doi:10.1016/j.physrep.2021.09.004](https://doi.org/10.1016/j.physrep.2021.09.004)]

On the [[Gross-Neveu model]] and [[coadjoint orbits]]:

* [[Dmitri Bykov]], *Sigma models as Gross-Neveu models. II* [[arXiv:2310.15394](https://arxiv.org/abs/2310.15394)]

category: people


Dmitri Diakonov

* [webpage](https://www.ae-info.org/ae/Member/Diakonov_Dmitri)

## Selected writings

* [[Dmitri Diakonov]], [[Victor Petrov]], _Exotic baryon resonances in the Skyrme model_ ([arXiv:0812.1212](https://arxiv.org/abs/0812.1212), [doi:10.1142/9789814704410_0004](https://doi.org/10.1142/9789814704410_0004)), Chapter 3 in: _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))

> It is astounding that [[Tony Skyrme|Skyrme]] had suggested [[Skyrmion|his model]] as early as in 1961 before it has been generally accepted that [[pions]] are (pseudo) [[Goldstone bosons]] associated with the [[spontaneous symmetry breaking|spontaneous breaking]] of [[chiral symmetry]], and of course long before [[QCD|Quantum Chromodynamics (QCD)]] has been put forward as the microscopic theory of [[strong nuclear force|strong interactions]].

> The revival of the Skyrme idea in 1983 is [due to Witten](Skyrmion#Witten83) who explained the _raison d'ˆetre_ of the [[Skyrmion|Skyrme model]] from the viewpoint of [[QCD]]. In the [[chiral perturbation theory|chiral limit]] when the light [[quark]] masses $m_u$, $m_d$, $m_s$ tend to zero, such that the octet of the pseudoscalar mesons [[pion|π]], [[kaon|K]] , η become nearly massless (pseudo) [[Goldstone bosons]], they are the lightest degrees of freedom of [[QCD]]. The [[effective field theory|effective]] chiral Lagrangian (EχL) for pseudoscalar mesons, understood as an infinite expansion in the derivatives of the pseudoscalar (or chiral) fields, encodes, in principle, full information about QCD. The famous two-term Skyrme Lagrangian can be understood as a low-energy truncation of this infinite series. Witten has added an important four-derivative [[WZW-term|Wess–Zumino term]] to the original Skyrme Lagrangian and pointed out that the overall coefficient in front of the EχL is proportional to the number of quark [[color charge|colours]] $N_c$.

> $[...]$

> Soon after Witten's work it has been realized that it is possible to bring the Skyrme model and the Skyrmion even closer to QCD and to the more customary language of constituent quarks. It has been first noticed $[$[6](#DiakonovEides83), [7a](#DharWadia84), [7b](#DarShankarWadia85), [8](#DiakonovPetrov86)$]$ that a simple chiral invariant Lagrangian for massive (constituent) quarks $Q$ interacting with the octet chiral field $\pi_A$ $(A = 1, ..., 8)$,

> $$\mathcal{L} = \overline{Q} \left( \mathrlap{\partial}/ - M e^{ \tfrac{i \pi^A \lambda^A \gamma_5}{F_\pi}  }  \right) Q$$

> induces, via a quark loop in the external pseudoscalar fields (see Fig. 3.1), the EχL whose lowest-derivative terms coincide with the Skyrme Lagrangian, including
automatically the Wess–Zumino term, with the correct coefficient!

> $[...]$

> The condition that the winding number of the trial field is unity needs to be imposed to get a deeply bound state, that is to guarantee that the baryon number is unity. $[$[9](#DiakonovPetrovPobylitsa88)$]$ The Skyrmion is, thus, nothing but the **mean chiral field binding quarks in a baryon**.

\linebreak

* {#DiakonovEides83} [[Dmitri Diakonov]], Michael I. Eides, _Chiral Lagrangian from a functional integral over quarks_, JETP Letters 38.7 (1983): 433-436 ([pdf](http://jetpletters.ac.ru/ps/1483/article_22635.pdf), [[DiakonovEides83.pdf:file]])

* {#DharWadia84} A. Dhar, Spenta R. Wadia, _Nambu—Jona-Lasinio Model: An Effective Lagrangian for Quantum Chromodynamics at Intermediate Length Scales_, Phys. Rev. Lett. 52, 959 (1984) ([doi:10.1103/PhysRevLett.52.959](https://doi.org/10.1103/PhysRevLett.52.959))

* {#DarShankarWadia85} Avinash Dhar, R. Shankar, Spenta R. Wadia, _Nambu–Jona-Lasinio–type effective Lagrangian: Anomalies and nonlinear Lagrangian of low-energy, large-N QCD_, Phys. Rev. D 31, 3256 (1985) ([doi:10.1103/PhysRevD.31.3256](https://doi.org/10.1103/PhysRevD.31.3256))

* {#DiakonovPetrov86} [[Dmitri Diakonov]], [[Victor Petrov]], _A theory of light quarks in the instanton vacuum_, Nuclear Physics B Volume 272, Issue 2, 21 July 1986, Pages 457-489 (doi:10.1016/0550-3213(86)90011-8)

* {#DiakonovPetrovPobylitsa88} [[Dmitri Diakonov]], [[Victor Petrov]], P.V. Pobylitsa, _A Chiral Theory of Nucleons_, Nucl. Phys. B306 (1988) 809 ([spire:247700](http://inspirehep.net/record/247700), doi:10.1016/0550-3213(88)90443-9)

category: people


Dmitri Faddeev

Dmitri Faddeev (Russian: Дми́трий Константи́нович Фадде́ев)
was a Russian mathematician working in [[Galois theory]], [[group cohomology]],
and numerical [[linear algebra]].
He is the father of [[Ludwig Faddeev]].

He discovered [[group cohomology]] independently of [[Eilenberg]] and [[MacLane]];
the publication of his first paper on this subject was delayed till 1947 because of World War II.

[MacTutor biography](http://www-history.mcs.st-andrews.ac.uk/Biographies/Faddeev.html).

## Related concepts

* [[group cohomology]]

## Selected writings

On [[group cohomology]]:

* [[Dmitri Faddeev]], _On factor-systems in Abelian groups with operators_. (Russian), Doklady Akad. Nauk SSSR (N. S.) 58, (1947). 361–364.


Dmitri Nikshych

__Dmitri Nikshych__ is a mathematician at University of New Hampshire ([web](https://ceps.unh.edu/person/dmitri-nikshych)). He has been one of the key figures in the development of the theory of [[fusion categories]].

## Selected writings

On [[tensor categories]]:

* [[Pavel Etingof]], [[Shlomo Gelaki]], [[Dmitri Nikshych]], [[Victor Ostrik]], _Topics in Lie theory and Tensor categories -- 9 Tensor categories_, Lecture notes (spring 2009) ([pdf](http://ocw.mit.edu/courses/mathematics/18-769-topics-in-lie-theory-tensor-categories-spring-2009/lecture-notes/MIT18_769S09_lec09.pdf) [web](http://ocw.mit.edu/courses/mathematics/18-769-topics-in-lie-theory-tensor-categories-spring-2009/lecture-notes/))

* [[Pavel Etingof]], [[Shlomo Gelaki]], [[Dmitri Nikshych]], [[Victor Ostrik]], *Tensor Categories*, AMS Mathematical Surveys and Monographs **205** (2015) $[$[ISBN:978-1-4704-3441-0](https://bookstore.ams.org/surv-205)$]$

On [[braided monoidal category|braided]] [[fusion categories]]:

* {#EtingofNikshychOstrik05} [[Pavel Etingof]], [[Dmitri Nikshych]], [[Victor Ostrik]], *On fusion categories*, Annals of Mathematics Second Series, Vol. 162, No. 2 (Sep., 2005), pp. 581-642 ([arXiv:math/0203060](http://arxiv.org/abs/math/0203060), [jstor:20159926](https://www.jstor.org/stable/20159926))

* {#DrinfeldGelakiNikshychOstrik10} [[Vladimir Drinfeld]], [[Shlomo Gelaki]], [[Dmitri Nikshych]], [[Victor Ostrik]], *On braided fusion categories I*, Selecta Mathematica. New Series 16 (2010), no. 1, 1–119 ([doi:10.1007/s00029-010-0017-z](https://doi.org/10.1007/s00029-010-0017-z))

category: people


Dmitri Orlov

**Dmitri Orlov** (Дмитрий Олегович Орлов) is a Russian algebraic geometer.

* [math-net.ru](http://www.mathnet.ru/php/person.phtml?&personid=11896&option_lang=eng) page
* personal page at Russian Academy: [orlov](http://www.mi.ras.ru/~orlov)

Orlov is one of the pioneers of the modern emerging categorical framework which unites the commutative and [[noncommutative algebraic geometry]], via the study of [[enhanced triangulated categories]] of [[quasicoherent sheaves]].

## Related

* [[triangulated categories of sheaves]]
* [[enhanced triangulated categories]]
* [[Bondal-Orlov reconstruction theorem]]
* [[derived category of singularities]]

## Publications

* D. Orlov, _Quasi-coherent sheaves in commutative and non-commutative geometry_, Izv. RAN. Ser. Mat. __67__:3 ( 2003)119--138 (see also preprint version [dvi](http://www.mpim-bonn.mpg.de/preprints/send?bid=57), [ps](http://www.mpim-bonn.mpg.de/preprints/send?bid=56))

* D. O. Orlov, _Derived categories of coherent sheaves and equivalences between them_, Uspekhi Mat. Nauk, 2003,  Vol. __58__,  issue 3(351), pp. 89–172, [Russian pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=rm&paperid=629&volume=58&year=2003&issue=3&fpage=89&what=fullt&option_lang=eng), transl. as Russian Mathematical Surveys (2003),58(3):511, [doi link](http://dx.doi.org/10.1070/RM2003v058n03ABEH000629) ([pdf](http://www.mi.ras.ru/~orlov/papers/Uspekhi2003.pdf) at Orlov's webpage, not on arXiv!)

* 5 lectures on derived categories and D-branes in Bonn, slides: [djvu](http://www.irb.hr/korisnici/zskoda/orlovMPIslides.djvu)

* D. Orlov, _Formal completions and idempotent completions of triangulated categories of singularities_, [arxiv/0901.1859](http://arxiv.org/abs/0901.1859)

* A. Bondal, D. Orlov, _Semi-orthogonal decomposition for algebraic varieties_, PreprintMPI/95–15, [alg-geom/9506006](http://arxiv.org/abs/alg-geom/9506012)

* A. Bondal, D. Orlov, _Reconstruction of a variety from the derived category and groups of autoequivalences_, Compositio Math. __125__ (2001), 327–344 [doi](http://dx.doi.org/10.1023/A:1002470302976) ; see also [[Bondal-Orlov reconstruction theorem]]

* V. A. Lunts, D. O. Orlov, _Uniqueness of enhancement for triangulated categories_, [arXiv:0908.4187](http://arxiv.org/abs/0908.4187).

* 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);
   * _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)

* D. Orlov, _Equivalences of derived categories of coherent sheaves_,  [MSRI lectures](http://www.msri.org/publications/ln/msri/2002/hodgetheory/orlov/1/index.html)

* A. I. Bondal, D. O. Orlov, _Derived categories of coherent sheaves_, Proc. Internat. Congress of Mathematicians (Beijing, 2002)

* A. N. Kapustin, D. O. Orlov, _Lectures on mirror symmetry, derived categories, and D-branes_,  Uspehi Mat. Nauk __59__  (2004),  no. 5(359), 101--134;  translation in  Russian Math. Surveys __59__  (2004), no. 5, 907--940, [math.AG/0308173](http://arxiv.org/abs/math/0308173)

category: people

[[!redirects Дмитрий Олегович Орлов]]
[[!redirects Дмитрий Орлов]]
[[!redirects Dmitri Olegovich Orlov]]
[[!redirects Dmitri Orlov]]
[[!redirects D. Orlov]]
[[!redirects D. O. Orlov]]
[[!redirects Orlov]]


Dmitri Pavlov

* [website](http://math.berkeley.edu/~pavlov/)

## Selected wrints

On [[rectification]] of [[symmetric operad|symmetric]] [[colored operads]]:

* {#PavlovScholbach18} [[Dmitri Pavlov]], [[Jakob Scholbach]], *Admissibility and rectification of colored symmetric operads*, Journal of Topology **11** 3 (2018) 559-601 $[$[doi:10.1112/topo.12008](https://doi.org/10.1112/topo.12008), [arXiv:1410.5675](https://arxiv.org/abs/1410.5675)$]$

On identifying the [[shape modality]] on the [[cohesive (∞,1)-topos]] of [[smooth ∞-groupoids]] with the [[path ∞-groupoid]]-construction ([[shape via cohesive path ∞-groupoid]]):

* {#BEBP} [[Daniel Berwick-Evans]], [[Pedro Boavida de Brito]], [[Dmitri Pavlov]], _Classifying spaces of infinity-sheaves_ ([arXiv:1912.10544](https://arxiv.org/abs/1912.10544))

On [[local field theory|locality]] of [[extended functorial field theory|extended]] [[functorial field theory]]:

* [[Daniel Grady]], [[Dmitri Pavlov]], _Extended field theories are local_ ([arXiv:2011.01208](https://arxiv.org/abs/2011.01208))

On the [[cobordism hypothesis]] for [[extended functorial field theory|extended]] [[functorial field theory]] equipped with geometric structure (such as [[conformal field theory]] etc.):

* [[Daniel Grady]], [[Dmitri Pavlov]], *The geometric cobordism hypothesis* ([arXiv:2111.01095](https://arxiv.org/abs/2111.01095))

Review:

* [[Dmitri Pavlov]], *The geometric cobordism hypothesis*, talk at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023) [[pdf](https://dmitripavlov.org/nyuad.pdf), [[Pavlov-GCHatCQTS.pdf:file]], [web](Center+for+Quantum+and+Topological+Systems#PavlovMar2023), video: [YT](https://www.youtube.com/watch?v=n6Eog_z82eA)]

On the [[Brown-Gersten property]] for [[simplicial presheaves]] over [[topological spaces]] and [[smooth manifolds]] ([[smooth infinity-groupoids|smooth $\infty$-groupoids]]):

* {#Pavlov22} [[Dmitri Pavlov]], *Numerable open covers and representability of topological stacks* [[arXiv:2203.03120](https://arxiv.org/abs/2203.03120)]

category: people

[[!redirects Dmitri Pavlov]]
[[!redirects Dmitry Pavlov]]


Dmitri Raikov

* [Wikipedia entru](https://en.wikipedia.org/wiki/Dmitrii_Abramovich_Raikov)

* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=107048)

## Selected writings

The characterization of [[state on a star-algebra|states]] on [[group algebras]] and what came to be known as the [[Gelfand-Raikov theorem]]:

* {#GelfandRaikov43} И. М. Гельфанд, Д. А. Райков, *Неприводимые унитарные представления локально бикомпактных групп*, Матем. сб., 13(55):2–3 (1943)  301–316 [[mathnet pdf](https://www.mathnet.ru/links/728d630fdf2220b1ea8bdb2a5f6a5488/sm6181.pdf)]

  \linebreak

  [[Israel Gelfand]], [[Dmitri Raikov]], *Irreducible unitary representations of locally bicompact groups*, Recueil Mathématique. N.S., 13(55) 2–3 (1943) 301–316 [[mathnet:eng/sm6181](https://www.mathnet.ru/eng/sm6181)]

category: people

[[!redirects Dmitri A. Raikov]]


Dmitri Sorokin

* [webpage](http://www.perimeterinstitute.ca/people/dmitri-sorokin)

## Selected writings

On [[self-dual higher gauge fields]]:

* [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], _On Lorentz Invariant Actions for Chiral P-Forms_, Phys.Rev. D55 (1997) 6292-6298 ([arXiv:hep-th/9611100](https://arxiv.org/abs/hep-th/9611100))

On [[superstrings]] and [[super membranes]] formulated in [[superspace]]:

* [[Igor Bandos]], [[Dmitrij Sorokin]], [[Dmitrij Volkov]], _On the generalized action principle for superstrings and supermembranes_, Phys. Lett. B352:269-275, 1995 ([arXiv:hep-th/9502141](https://arxiv.org/abs/hep-th/9502141))

On the [[superembedding approach]] for [[super p-branes|super $p$-branes]]/[[Green-Schwarz sigma-models]]:

* {#BPSTV95} [[Igor Bandos]], [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], [[Dmitry Volkov]], _Superstrings and supermembranes in the doubly supersymmetric geometrical approach_, Nucl. Phys. B **446** (1995) 79-118 [[arXiv:hep-th/9501113](https://arxiv.org/abs/hep-th/9501113)]

* {#Sorokin99} [[Dmitri Sorokin]], _Superbranes and Superembeddings_, Phys.Rept.329:1-101, 2000 ([arXiv:hep-th/9906142](http://arxiv.org/abs/hep-th/9906142))

* {#Sorokin01} [[Dmitri Sorokin]], _Introduction to the Superembedding Description of Superbranes_, AIP Conference Proceedings 589, 98 (2001) ([arXiv:hep-th/0105102](https://arxiv.org/abs/hep-th/0105102))

* [[Igor A. Bandos]], [[Dmitri P. Sorokin]], *Superembedding approach to superstrings and super-$p$-branes*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2301.10668](https://arxiv.org/abs/2301.10668)]

On [[kappa-symmetry]] explained via the [[superembedding approach]]:

* {#SorokinTkachVolkov89} [[Dmitri Sorokin]], V. Tkach and [[Dmitrij Volkov]], _Superparticles, twistors and Siegel symmetry_, Mod. Phys. Lett. A4 (1989) 901-908 ([spire:271923](http://inspirehep.net/record/271923), [doi:10.1142/S0217732389001064](https://doi.org/10.1142/S0217732389001064))

On the [[Green-Schwarz action functional]] for the [[M5-brane]]:

* [[Igor Bandos]], [[Kurt Lechner]], [[Alexei Nurmagambetov]], [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], _On the equivalence of different formulations of the M Theory five--brane_, Phys. Lett. B408 (1997) 135-141 ([arXiv:hep-th/9703127](http://arxiv.org/abs/hep-th/9703127))

* {#PST97} [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], _Covariant Action for a D=11 Five-Brane with the Chiral Field_, Phys. Lett. B398 (1997) 41 ([arXiv:hep-th/9701037](https://arxiv.org/abs/hep-th/9701037))

* Sheng-Lan Ko, [[Dmitri Sorokin]], Pichet Vanichchapongjaroen, _The M5-brane action revisited_, JHEP11(2013)072 ([arXiv:1308.2231](https://arxiv.org/abs/1308.2231))

On [[type II supergravity]]:

* {#BNS04} [[Igor Bandos]], [[Alexei Nurmagambetov]], [[Dmitri Sorokin]], _Various Faces of Type IIA Supergravity_, Nucl. Phys. B676 (2004) 189-228 ([arXiv:hep-th/0307153](https://arxiv.org/abs/hep-th/0307153))

Proposal for a non-abelian and [[supersymmetry|supersymmetric]] [[DBI-action]] for [[coincident brane|coincident]] [[D0-branes]]:

* Sudhakar Panda, [[Dmitri Sorokin]], *Supersymmetric and Kappa-invariant Coincident D0-Branes*, 	JHEP 0302 (2003) 055 [[arXivLhep-th/0301065](https://arxiv.org/abs/hep-th/0301065), [doi:10.1088/1126-6708/2003/02/055](https://doi.org/10.1088/1126-6708/2003/02/055)]

On [[membranes]] in [[D=4 supergravity]] ([[super 2-brane in 4d]]):

* [[Igor Bandos]], [[Fotis Farakos]] Stefano Lanza, L. Martucci,  [[Dmitri Sorokin]], _Higher Forms and Membranes in 4D Supergravities_, in: Proceedings of _[[Higher Structures in M-Theory 2018]]_, Fortschritte der Physik, Special Issue Volume 67, Issue 8-9 ([arXiv:1903.02841](https://arxiv.org/abs/1903.02841), [doi:10.1002/prop.201910020](https://doi.org/10.1002/prop.201910020))

## Related $n$Lab entries

* [[super p-brane]]

* [[kappa-symmetry]]

* [[superembedding approach]]

* [[M5-brane]]

category: people

[[!redirects Dmitrij Sorokin]]

[[!redirects Dmitri P. Sorokin]]


Dmitri Vey

* [GoogleScholar page](https://scholar.google.fr/citations?user=Ylpu_RgAAAAJ&hl=fr)

## Selected writings

On [[n-plectic geometry|$n$-plectic]] formulation of [[gravity]] and [[Maxwell theory]]:

* [[Dmitri Vey]], _$n$-Plectic Maxwell Theory_ [[arxiv:1303.2192](https://arxiv.org/abs/1303.2192)]

* [[Dmitri Vey]], *Multisymplectic formulation of vielbein gravity. De Donder-Weyl formulation, Hamiltonian $(n-1)$-forms*,  Classical and Quantum Gravity **32** 9 (2015) [[arxiv:1404.3546] (https://arxiv.org/abs/1404.3546), [doi:10.1088/0264-9381/32/9/095005](https://iopscience.iop.org/article/10.1088/0264-9381/32/9/095005)[

category: people


Dmitrij Volkov

* [Memorial Page](https://ivv5hpp.uni-muenster.de/u/douplii/volkov/)

* [obituary](http://www.kipt.kharkov.ua/conferences/itp/DV2000/dvbiogr.html)

* Interview with Dmitry Volkov by Yuri Ranyuk, Imperial College, London March 1995 ([aip oral histories:4392](https://www.aip.org/history-programs/niels-bohr-library/oral-histories/4392))

* S. I. Volkova, A. A. Zheltukhin, _Glimpses of Dmitry Volkov's life and work_, Nuclear Physics B - Proceedings Supplements
Volume 101, Issues 1–3, August 2001, Pages 20-25 (doi:10.1016/S0920-5632(01)01489-X)

* A.S. Bakai, S. V. Peletminskii, N.F. Shul'ga, Yu.V. Slyusarenko, D.V. Uvarov, A.A. Zheltukhin, _To the 90-th anniversary of Dmitry Vasilievich Volkov's birthday_, Ukrainian Journal of Physics v.60 N7 (2015) 675-677 ([arXiv:1506.03003](https://arxiv.org/abs/1506.03003))

* [[Steven Duplij]], _Supergravity was discovered by D.V. Volkov and V.A. Soroka in 1973, wasn't it?_, East Eur. J. Phys. p. 81-82 (2019) ([arXiv:1910.03259](https://arxiv.org/abs/1910.03259))

## Selected writings:

The idea of [[supergravity]] was proposed by

* [[Dmitry Volkov]], V. P. Akulov, _Possible universal neutrino interaction_, ZhETF Pis. Red. (JETP Letters, AIP translation), 16, n.11 (1972) 621 ([pdf](http://www.jetpletters.ac.ru/ps/1766/article_26864.shtml))

followed up by the first model of supergravity (in nonlinear realization) constructed in

* [[Dmitry Volkov]] and [[Vyacheslav Soroka]], _Higgs effect for Goldstone particles with spin 1/2_, ZhETF Pis. Red. (JETP Letters, AIP translation), 18, n.8 (1973) 529 ([pdf](https://www.jetpletters.ac.ru/ps/1568/article_24038.shtml))

On [[D=4 supergravity]] formulated in [[superspace]]:

* V. Akulov, [[Dmitry Volkov]] and [[Vyacheslav Soroka]], _Generally covariant theories of gauge fields on superspace_, Theor. Math. Phys. 31 (1977) 285 ([doi:10.1007/BF01041233](https://doi.org/10.1007/BF01041233))

On the history of [[supergravity]]:

* [[Dmitry Volkov]], _Supergravity before and after 1976. The story of goldstonions_, in  [Concise Encyclopedia of Supersymmetry,
S. Duplij, J. Bagger, W. Siegel (Eds.), Springer, 2004](http://www.springer.com/gp/book/9781402013386) or [arXiv:hep-th/9404153](http://arxiv.org/abs/hep-th/9404153)

On [[kappa-symmetry]] explained via the [[superembedding approach]]:

* {#SorokinTkachVolkov89} [[Dmitri Sorokin]], V. Tkach and [[Dmitrij Volkov]], _Superparticles, twistors and Siegel symmetry_, Mod. Phys. Lett. A4 (1989) 901-908 ([spire:271923](http://inspirehep.net/record/271923), [doi:10.1142/S0217732389001064](https://doi.org/10.1142/S0217732389001064))

On [[superstrings]] and [[super membranes]] formulated in [[superspace]]:

* [[Igor Bandos]], [[Dmitrij Sorokin]], [[Dmitrij Volkov]], _On the generalized action principle for superstrings and supermembranes_, Phys. Lett. B352:269-275, 1995 ([arXiv:hep-th/9502141](https://arxiv.org/abs/hep-th/9502141))

On the derivation of the [[equations of motion]] for the [[superstring]] and the [[M2-brane]] via the [[superembedding approach]]:

* {#BPSTV95} [[Igor Bandos]], [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], [[Dmitry Volkov]], _Superstrings and supermembranes in the doubly supersymmetric geometrical approach_, Nucl. Phys. B446:79-118, 1995 ([arXiv:hep-th/9501113](https://arxiv.org/abs/hep-th/9501113))

## Related $n$Lab entries

* [[kappa-symmetry]]

* [[superembedding formalism]]

category: people

[[!redirects Dmitry Volkov]]

[[!redirects Dmitrij Vasilievich Volkov]]


Dmitriy Morozov

* [personal page](https://mrzv.org/)

## Selected writings

Introducing [[well groups]]:

* {#EMP11} [[Herbert Edelsbrunner]], [[Dmitriy Morozov]], [[Amit Patel]], *Quantifying Transversality by Measuring the Robustness of Intersections*, Foundations of Computational Mathematics, **11** 3  (2011) 345–361 $[$[arXiv:0911.2142](https://arxiv.org/abs/0911.2142)$]$

* [[Paul Bendich]], [[Herbert Edelsbrunner]], [[Dmitriy Morozov]], [[Amit Patel]], *The Robustness of Level Sets*, In: M. de Berg, U. Meyer (eds.) _Algorithms – ESA 2010_. ESA 2010. Lecture Notes in Computer Science **6346** Springer (2010)  $[$[doi:10.1007/978-3-642-15775-2_1](https://doi.org/10.1007/978-3-642-15775-2_1)$]$

* [[Paul Bendich]], [[Herbert Edelsbrunner]], [[Dmitriy Morozov]], [[Amit Patel]], *Homology and Robustness of Level and Interlevel Sets*, Homology, Homotopy and Applications, **15** (2013) 51-72 $[$[euclid:1383943667](https://projecteuclid.org/euclid.hha/1383943667)$]$

On [[persistent homology]]:

* [[Herbert Edelsbrunner]], [[Dmitriy Morozov]], *Persistent homology: theory and practice*, in: *European Congress of Mathematics Kraków, 2–7 July, 2012* EMS $[$[doi:10.4171/120-1/3](https://www.ems-ph.org/books/show_abstract.php?proj_nr=170&vol=1&rank=3), [pdf](http://mrzv.org/publications/persistent-homology-theory-practice/ecm)$]$

On [[zigzag persistence]], connecting [[persistent homology]] to [[ADE classification|A-type]] [[quiver representation]]-theory, applied to [[level sets]]:

* [[Gunnar Carlsson]], [[Vin de Silva]], [[Dmitriy Morozov]], *Zigzag persistent homology and real-valued functions*, in: *SCG '09: Proceedings of the twenty-fifth annual symposium on Computational geometry (2009) 247–256 $[$[doi:10.1145/1542362.1542408](https://doi.org/10.1145/1542362.1542408)$]$

* [[Gunnar Carlsson]], [[Vin de Silva]], [[Sara Kališnik]], [[Dmitriy Morozov]], *Parametrized Homology via Zigzag Persistence*, Algebr. Geom. Topol. **19** (2019) 657-700 $[$[arXiv:1604.03596](https://arxiv.org/abs/1604.03596), [doi:10.2140/agt.2019.19.657](https://doi.org/10.2140/agt.2019.19.657)$]$

[[!redirects Sara Kalisnik]]


Dmitry Feichtner-Kozlov

* [Wikipedia entry](https://en.wikipedia.org/wiki/Dmitry_Feichtner-Kozlov)

* [Institute page](https://www.alta.uni-bremen.de/members/dfk/)

* *[Applied Topology Unit](https://groups.oist.jp/atu)*

* [GoogleScholar page](https://scholar.google.co.il/citations?user=ZMc1jGQAAAAJ&hl=en)

## Selected writings

On [[combinatorics|combinatorial]] [[algebraic topology]] (cf. [[applied topology]]):

* [[Dmitry N. Kozlov]], *Trends in Topological Combinatorics* [[arXiv:math/0507390](https://arxiv.org/abs/math/0507390)]

* [[Dmitry Kozlov]], *Combinatorial Algebraic Topology*,  Algorithms and Computation in Mathematics **21**, Springer (2008) [[doi:10.1007/978-3-540-71962-5](https://doi.org/10.1007/978-3-540-71962-5)]

category: people

[[!redirects Dmitry Kozlov]]

[[!redirects Dmitry N. Kozlov]]


Dmitry Kaledin

**Dmitry Kaledin** (sometimes transliterated as Dmitri) is a Russian mathematician.

Kaledin has proved some cases of the [[degeneration conjecture]] of [[Maxim Kontsevich]].

* [homepage at Russian Academy of Sciences](http://www.mi.ras.ru/~kaledin), [arXiv papers](http://arxiv.org/find/math/1/au:+Kaledin_D/0/1/0/all/0/1)

* D. Kaledin, _Tokyo lectures "Homological methods in non-commutative geometry"_, [pdf](http://imperium.lenin.ru/~kaledin/tokyo/final.pdf), [TeX](http://imperium.lenin.ru/~kaledin/tokyo/final.tex); and related but different [Seoul lectures](http://imperium.lenin.ru/~kaledin/seoul)

* D. Kaledin, _Cartier isomorphism and Hodge theory in the non-commutative case_,  Arithmetic geometry,  537--562, Clay Math. Proc. __8__, Amer. Math. Soc. 2009, [arxiv/0708.1574](http://arxiv.org/abs/0708.1574)

* D. Kaledin, _Cyclic homology with coefficients_, [math.KT/0702068](http://arxiv.org/abs/math.KT/0702068), to appear in Yu. Manin's 70th anniversary volume.

* _Non-commutative Hodge-to-de Rham degeneration via the method of
Deligne-Illusie_, Pure Appl. Math. Quat. __4__ (2008), 785–875.
* _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)

* [[Dmitri Kaledin]], [[Misha Verbitsky]], _Hyperkahler manifolds_, International Press of Boston 2000

* [[Dmitry Kaledin]], _Geometry and topology of symplectic resolutions_, in: Algebraic Geometry: 2005 Summer Research Institute, edited by Dan Abramovich, pp. 595--628, arXiv:[math.AG/0608143](https://arxiv.org/abs/math/0608143)

category:people

[[!redirects D. Kaledin]]
[[!redirects Dmitry Kaledin]]
[[!redirects Dmitri Kaledin]]
[[!redirects Dmitriy Kaledin]]
[[!redirects Dmitrij Kaledin]]
[[!redirects Dmitriĭ Kaledin]]
[[!redirects Дмитрий Каледин]]
[[!redirects Дмитрий Борисович Каледин]]


Dmitry Melnikov

* [inSpire page](https://inspirehep.net/authors/1274266)

* [iip page](https://iip.ufrn.br/eventslecturer?inf===QTq10d)

## Selected writings

On [[topological quantum computation]] with [[Chern-Simons theory]] (e.g. with [[su(2)-anyons]]):

* [[Dmitry Melnikov]], [[Andrei Mironov]], [[Sergey Mironov]], [[Alexei Morozov]], [[Andrey Morozov]], _Towards topological quantum computer_, Nucl. Phys. B926 (2018) 491-508 ([arXiv:1703.00431](https://arxiv.org/abs/1703.00431), [doi:10.1016/j.nuclphysb.2017.11.016](https://doi.org/10.1016/j.nuclphysb.2017.11.016))

category: people


Dmitry Roytenberg

* [personal page](https://sites.google.com/site/dmitryroytenberg/)

* [arXiv articles](http://front.math.ucdavis.edu/author/D.Roytenberg)

* [GoogleScholar page](https://scholar.google.com/citations?user=ygCqV38AAAAJ&hl=en)

## Selected writings

On [[symplectic Lie n-algebroids]]:

* [[Dmitry Roytenberg]], _Courant algebroids, derived brackets and even symplectic supermanifolds_, PhD thesis [[arXiv:9910078](http://arxiv.org/abs/math/9910078)]

* [[Dmitry Roytenberg]], _On the structure of graded symplectic supermanifolds and Courant algebroids_, in: [[Theodore Voronov]] (ed.).  _Quantization, Poisson Brackets and Beyond_, Contemp. Math. **315**, Amer. Math. Soc. (2002) [[arXiv:0203110](http://arxiv.org/abs/math/0203110), [ISBN:978-0-8218-7905-4](https://bookstore.ams.org/conm-315)]

Introducing [[hemistrict Lie 2-algebras]]:

* {#Roytenberg07} [[Dmitry Roytenberg]]: *On weak Lie 2-algebras*, in: *XXVI Workshop on Geometrical Methods in Physics*, AIP Conference Proceedings **956**, American Institute of Physics, (2007) [[arXiv:0712.3461](https://arxiv.org/abs/0712.3461), [hdl:10993/10465](http://hdl.handle.net/10993/10465)]

On [[smooth superalgebras]]:

* [[David Carchedi]], [[Dmitry Roytenberg]], *On Theories of Superalgebras of Differentiable Functions* [[arXiv:1211.6134](http://arxiv.org/abs/1211.6134)]

On equivalence of dg manifolds

* [[Dmitry Roytenberg]], _Equivalences of differential graded manifolds_, talk at Global Poisson Webinar [yt](https://www.youtube.com/watch?v=N-kLG1JuiTA)

## Related entries

* [[n-symplectic manifold]]

category: people


Dmitry Rudinsky

* [personal page](https://itmp.msu.ru/en/institute/phd-students/rudinsky-dmitry)

## Selected writings

On [[L-infinity algebras|$L_\infty$-algebras]] from [[BV-formalism]] in [[local field theory|local]] [[gauge field|gauge]] [[field theory]]:

* [[Maxim Grigoriev]], [[Dmitry Rudinsky]], *Notes on the $L_\infty$-approach to local gauge field theories*, Journal of Geometry and Physics **190** (2023) 104863 [[arXiv2303.08990](https://arxiv.org/abs/2303.08990), [doi:10.1016/j.geomphys.2023.104863](https://doi.org/10.1016/j.geomphys.2023.104863)]

category: people


Dmitry Shirkov

[[!redirects DmitryShirkov]]

* [Wikipedia entry](https://en.wikipedia.org/wiki/Dmitry_Shirkov)

## Selected writings

On [[non-perturbative effects]] in [[quantum field theory]]:

* {#BakulevShirkov10} Alexander P. Bakulev, [[Dmitry Shirkov]], _Inevitability and Importance of Non-Perturbative Elements in Quantum Field Theory_, Proceedings of the 6th Mathematical Physics Meeting, Sept. 14--23, 2010, Belgrade, Serbia, pp. 27--54 ([arXiv:1102.2380](https://arxiv.org/abs/1102.2380), [ISBN 978-86-82441-30-4](http://www.proceedings.com/22646.html), [book webpage](http://www.mphys6.ipb.ac.rs/proceedings6.htm), [book TOC pdf](http://toc.proceedings.com/22646webtoc.pdf))

## Related $n$Lab entries

* [[causal perturbation theory]]

* [[Bogoliubov's formula]]

category: people


Dmitry Tamarkin

**Dmitry Tamarkin** is a mathematician (with some training in theoretical physics as well) at Northwestern University. His expertise includes  deformation theory (with his famous proof of Kontsevich's formality theorem), [[D-modules]], [[homological algebra]], symplectic geometry, microlocal analysis and various strong homotopy algebras ($A_\infty$, $L_\infty$, $G_\infty$...). His advisor was [[Boris Tsygan]]. See also [[noncommutative differential calculus]] and [[microlocal category]].

* list of [arxiv articles](http://arxiv.org/find/math/1/au:+Tamarkin_D/0/1/0/all/0/1)

* homepage: [dept](http://www.math.northwestern.edu/people/facultyProfiles/dmitry.tamarkin.html), [personal](http://www.math.northwestern.edu/~tamarkin/)

* Dmitry Tamarkin, _What do dg-categories form?_,
Compos. Math. __143__ (2007), no. 5, 1335--1358.

* Dmitry Tamarkin, _Deformations of chiral algebras_, Proceedings of the ICM, Beijing 2002, vol. 2, 105--118

* D. Tamarkin, [[Boris Tsygan]], _The ring of differential operators on forms in noncommutative calculus_, Graphs
and patterns in mathematics and theoretical physics, Proc. Sympos. Pure Math. 73, Amer. Math. Soc. 2005, pp. 105--131  [doi](https://doi.org/10.1090/pspum/073/2131013) [MR2131013](https://mathscinet.ams.org/mathscinet-getitem?mr=2131013)

* V. Dolgushev, D. Tamarkin, B. Tsygan, _Noncommutative calculus and the Gauss-Manin connection_, in: Higher structures in geometry and physics, 139–158, Progr. Math., 287, Birkhäuser/Springer, New York, 2011 , [arXiv:0902.2202](http://arxiv.org/abs/0902.2202)

* Dmitry Tamarkin, _Microlocal category for a closed symplectic manifold_, talk at IAS [youtube](https://www.youtube.com/watch?v=GxdwyLt-i-k)
* Dmitry Tamarkin, _Microlocal category_, [arXiv:1511.08961](https://arxiv.org/abs/1511.08961)

category: people
[[!redirects Dmitri Tamarkin]]


Dmitry Vaintrob

* [webpage](http://math.mit.edu/people/profile.php?pid=1348)

category: people


Dmitry Vladimirovich Alekseevsky

* [Webpage](http://iitp.ru/en/users/2420.htm)

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=79852)

## Selected writings

On [[G-structure]]:

* {#Alekseevskii} [[Dmitry Alekseevsky]], _$G$-structure on a manifold_ in M. Hazewinkel (ed.) _Encyclopedia of Mathematics, Volume 4_

On [[Sp(n).Sp(1)]]-[[special holonomy]]:

* {#Alekseevskii68} [[Dmitri Alekseevskii]], [_Riemannian spaces with exceptional holonomy groups_](https://link.springer.com/article/10.1007%2FBF01075943), Functional Analalysis and its Applications (1968) 2: 97.

On [[quaternionic manifolds]] and [[quaternion-Kähler manifolds]] in terms of [[G-structure]]:

* [[Dmitri V. Alekseevsky]], [[Stefano Marchiafava]], _Quaternionic-like structures on a manifold: Note I. 1-integrability and integrability conditions_, Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1993) Volume: 4, Issue: 1, page 43-52 ([dml:244082](https://eudml.org/doc/244082))

* [[Dmitri V. Alekseevsky]], [[Stefano Marchiafava]], _Quaternionic-like structures on a manifold: Note II. Automorphism groups and their interrelations_, Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1993) Volume: 4, Issue: 1, page 53-61 ([dml:244299](https://eudml.org/doc/244299))

* [[Dmitry V. Alekseevsky]], [[Stefano Marchiafava]], _Quaternionic structures on a manifold and subordinated structures_, Annali di Matematica pura ed applicata 171, 205–273 (1996) ([doi:10.1007/BF01759388](https://doi.org/10.1007/BF01759388))

On [[Killing spinors]]:

* [[Dmitri Alekseevsky]], [[Vicente Cortés]], [[Chandrashekar Devchand]], [[Uwe Semmelmann]], _Killing spinors are Killing vector fields in Riemannian Supergeometry_ ([arXiv:dg-ga/9704002](http://arxiv.org/abs/dg-ga/9704002))

On [[Lie algebra extension]], [[nonabelian Lie algebra cohomology]]:

* {#ACDP} [[Dmitry Vladimirovich Alekseevsky]], [[Vicente Cortés]], [[Chandrashekar Devchand]], [[Antoine Van Proeyen]], _Polyvector Super-Poincaré Algebras_ Commun.Math.Phys. 253 (2004) 385-422 ([arXiv:hep-th/0311107](http://arxiv.org/abs/hep-th/0311107))

* [[Dmitri Alekseevsky]], [[Peter Michor]], Wolfgang Ruppert, Extensions of Lie algebras ([math.DG/0005042](http://arxiv.org/abs/math/0005042))

On [[super Lie algebra]] extensions:

* [[Dmitri Alekseevsky]], [[Peter Michor]], Wolfgang Ruppert, _Extensions of super Lie algebras_, J. Lie Theory 15 (2005) No. 1, 125--134 ([arXiv:math/0101190](http://arxiv.org/abs/math/0101190))

On [[special geometry]]

* D. V. Alekseevsky, V. Cortés, C. Devchand, _Special complex manifolds_,  Journal of Geometry and Physics __42__:1-2 (2002) 85--105 doi

## Related $n$Lab entries

* [[Sp(n).Sp(1)]]

* [[Killing spinor]]

* [[super Poincaré Lie algebra]]

* [[charge conjugation matrix]]

* [[supergeometry]]

* [[extended supersymmetry]]

category: people

[[!redirects Dmitry Alekseevsky]]

[[!redirects Dmitri Alekseevskii]]

[[!redirects Dmiti Alekseevsky]]

[[!redirects Dmitri Alekseevsky]]

[[!redirects Dmitry V. Alekseevsky]]

[[!redirects Dmitri V. Alekseevsky]]


Dmytro Shklyarov

Dmytro Shklyarov worked on the subject of quantum groups with late L. Vaksman n Ukraine. After that he got Ph. D. in mathematics from Kansas State University where he studied noncommutative algebraic geometry under [[Yan Soibelman]]. He is now a postdoc in Augbusrg, Germany.

* [webpage](http://www.math.uni-augsburg.de/geo/mitarbeiter/shklyarov/index_en.html)
* _Hirzebruch-Riemann-Roch theorem for differential graded algebras_, PhD Thesis, Kansas State University (2009). Available at <http://krex.k-state.edu/dspace/handle/2097/1381>, with preprint version _Hirzebruch-Riemann-Roch theorem for DG algebras_, [arxiv/0710.1937](http://arxiv.org/abs/0710.1937)
* _2-Representations and Equivariant 2D Topological Field Theories_, [arxiv/0812.4482](http://arxiv.org/abs/0812.4482)
* _Non-commutative Hodge structures: Towards matching categorical and geometric examples_, [arxiv/1107.3156](http://arxiv.org/abs/1107.3156)

category: people

[[!redirects D. Shklyarov]]


doctrinal adjunction

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###Context###
#### 2-Category theory
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#### Higher algebra
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[[!include higher algebra - contents]]
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# Contents
* table of contents
{:toc}

## Idea

_Doctrinal Adjunction_ is the title of a 1974 paper ([Kelly](#Kelly)) that gives conditions under which [[adjunction|adjoint morphisms]] $f \dashv u$ in a [[2-category]] $K$, and additionally the [[unit of an adjunction|unit]] and [[counit of an adjunction|counit]], may be lifted to the category $T$-$Alg_l$ for some [[2-monad]] $T$ on $K$.

Here $T$-$Alg_l$ is the 2-category of strict $T$-algebras, [[lax morphism|lax T-morphisms]], and $T$-transformations, but the result works as well for pseudo algebras.

The term 'doctrinal' refers to the concept [[doctrine]], in particular to its definition as a [2-monad](doctrine#as_2monads).

## Statement

+-- {: .num_theorem}
###### Theorem

Let $f \dashv u$ be an [[adjunction]] in some [[2-category]] $K$ and let $T$ be a [[2-monad]] on $K$.

There is a [[bijection]] between [[2-morphism]]s $\bar u$ making $(u,\bar u)$ a lax $T$-morphism and 2-morphisms $\tilde f$ making $(f,\tilde f)$ a colax $T$-morphism; it is given by taking [[mates]] with respect to the adjunctions $T f \dashv T u$ and $f \dashv u$.

=--

The proof ([Kelly](#Kelly))
relies solely on the properties of the mate correspondence.

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

For the unit and counit of the adjunction $f \dashv u$ to be $T$-transformations, and hence for the adjunction to live in $T$-$Alg_l$, it is necessary and sufficient that $\tilde f$ have an inverse $\bar f$ that makes $(f,\bar f)$ into a lax $T$-morphism, and hence $(f,\bar f)$ into a strong $T$-morphism.

=--

Again, the proof hinges on the properties of mates:  we take the conditions for the unit and counit to be $T$-transformations and pass to mates wrt $T f \dashv T u$ and $1 \dashv 1$.  Noting that $\tilde f$ is the mate of $\bar u$, the conditions are seen to be equivalent to requiring that $\bar f$ and $\tilde f$ are mutually inverse.

It follows that

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

$(f, \bar f) \dashv (u, \bar u)$ in $T$-$Alg_l$ if and only if $f \dashv u$ in $K$ and $\bar f$ has inverse $\tilde f$ = the mate of $\bar u$.

=--

## In terms of double categories

Doctrinal adjunction can be stated cleanly in terms of [[double categories]].  Namely, for any 2-monad $T$ there is a double category  $T$-**Alg** whose objects are $T$-algebras, whose horizontal arrows are lax $T$-morphisms, whose vertical arrows are colax $T$-morphisms, and whose 2-cells are 2-cells in the base 2-category $K$ that make a certain cube commute; see [[double category of algebras]].  The horizontal 2-category of this double category is $T$-$Alg_l$, and its vertical 2-category is $T$-$Alg_c$.  There is an obvious forgetful double functor $T \mathbf{Alg} \to \mathbf{Sq}(K)$, where $\mathbf{Sq}(K)$ is the double category of squares or [[quintets]] in $K$.

It is straightforward to verify that a [[conjunction]] in the double category $T$-**Alg** is precisely an adjunction in $K$ between $T$-algebras whose left adjoint is colax, whose right adjoint is lax, and for which the lax and colax structure maps are mates under the adjunction -- i.e. a "doctrinal adjunction" in the above sense.  Furthermore, an arrow in $T$-**Alg** has a [[companion]] precisely when it is a *strong* (= pseudo) $T$-morphism.  The two central results of Kelly's paper can then be stated as:

1. The forgetful double functor $U\colon T \mathbf{Alg} \to \mathbf{Sq}(K)$ creates conjunctions.  I.e. given a horizontal arrow $u$ in $T \mathbf{Alg}$ and a left conjoint $f$ of $U(u)$ (i.e. a left adjoint of $u$ in $K$), there is a unique left conjoint of $u$ in $T\mathbf{Alg}$ lying over $f$.

1. Let $f\colon A\to B$ be a vertical arrow in $T \mathbf{Alg}$ (i.e. a colax $T$-morphism) and let $f'\colon A\to B$ and $u\colon B\to A$ be horizontal arrows (i.e. lax $T$-morphisms).  Then from any two of the following three data we can uniquely construct the third.
   1. Data making $f'$ a horizontal companion of $f$;
   1. Data making $u$ a right conjoint of $f$;
   1. Data making $f'$ a left adjoint of $u$ in the horizontal 2-category.

Of these, the second is actually a general statement about companions and conjoints in any double category.  Of course, the first is a special property of the forgetful double functor from the double category of $T$-algebras.

## Examples

### Monoidal categories

Let $K = $ [[Cat]] and $T$ the 2-monad whose 2-algebras are [[monoidal categories]]. Then

* a lax $T$-morphism  is a [[lax monoidal functor]];

* an oplax $T$-morphism is an [[oplax monoidal functor]].

The above theorem then asserts

+-- {: .un_corollary}
###### Corollary

For two [[adjoint functors]] $(L \dashv R)$ between monoidal categories, $L$ is oplax monoidal precisely if $R$ is lax monoidal.

=--

See at _[[oplax monoidal functor]]_ and at _[[monoidal adjunction]]_ for more details.

## Related concepts

* [[monoidal adjunction]]

## References

* {#Kelly} [[Max Kelly]], _Doctrinal Adjunction_ Lecture Notes in Mathematics, (1974), Volume 420/1974, doi:[10.1007/BFb0063105](https://dx.doi.org/10.1007/BFb0063105)

The following article explains the double category perspective:

* [[Mike Shulman]], _Comparing composites of left and right derived functors_, New York Journal of Mathematics, Volume 17 (2011) pp 75-125, ([arXiv:0706.2868](https://arxiv.org/abs/0706.2868), [journal abstract page](http://nyjm.albany.edu/j/2011/17-5.html))

[[!redirects doctrinal adjunctions]]


doctrine

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[[!include higher algebra - contents]]
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# Doctrines
* table of contents
{: toc}

## Idea

The concept of **doctrine**, as the word was originally used by [[Jon Beck]], is a [[vertical categorification|categorification]] of the concept of "[[theory]]".  Thus, a doctrine could also reasonably be called a "2-theory."

Here we are using the word "theory" in a vague non-precise sense.  For instance, there is a theory of [[groups]], a theory of [[abelian group]]s, a theory of [[commutative ring]]s, and so on.  Likewise, there is a doctrine of [[monoidal categories]], a doctrine of [[cartesian monoidal category|categories with finite product]]s, a doctrine of [[rig category|rig categories]], and so on.

Moreover, just as categories are objects of a 2-category (the categorification of a category), ordinary _theories_ are objects of some _doctrine_.  The doctrine in which a theory lives specifies the structure on categories in which models of that theory can be [[internalization|internalized]].  For instance, the theory of groups lives naturally in the doctrine of categories with finite products, since a [[group object]] can be defined in any such category.  Likewise, the theory of [[monoid]]s lives in the doctrine of monoidal categories, and so on.

Moreover, in general there is a "free" category containing a model (the "[[walking structure|walking model]]") of some given theory: for instance there is a category with finite products freely generated by a group object (namely, the [[Lawvere theory]] of a group), and a monoidal category freely generated by a monoid object (namely, the [[augmented simplex category]]).  In this way, a theory whose _models_ are categories in some doctrine can itself be regarded as a category in that doctrine, with the property that the category of models of a theory $T$ in a category $K$ is the hom-category $hom(T,K)$ in the relevant doctrine.  In fact, we can fruitfully think of a doctrine _as_ the collection of all theories in that doctrine, since (for example) any category with finite products can be regarded as a "theory" that can be interpreted in any other such category, via finite-product-preserving functors.

For this to make sense, whatever a doctrine is, it must in particular induce a 2-category of its models.  Or, at least, of its models in [[Cat]], which is so far all that we have talked about.  But one can also imagine considering 2-theories as objects of some [[3-theory]], and so on.  (Unfortunately this approach involves us in an infinite regress when we look for a formal definition!)

## Definitions

### As 2-monads

A number of people, starting with Lawvere, have defined "doctrine" to mean [[2-monad]] (particularly a 2-monad on $Cat$).  This may seem reasonable, since all the examples of doctrines we considered above are, in fact, (the algebras for) some 2-monad on $Cat$.  However, it is not really correct.

At a lower level, the "theories" considered above of which a doctrine is supposed to be a categorification are also _not_ the same as [[monad]]s.  In particular, the morphisms between such theories are not all representable by morphisms of monads.  For instance, there is a morphism of theories from the theory of commutative rings to the theory of abelian groups which sends a ring to its multiplicative group of units, but this is not induced by any morphism of monads because it does not preserve the underlying set.

Likewise, not all "morphisms of doctrines" are representable by morphisms of 2-monads.

### As 2-categories

Another possible definition of "doctrine" is just "a [[2-category]]".  In this approach, we identify a doctrine with its category of models (in [[Cat]]).  This has the advantage of simplicity.  Moreover, we can say easily what a model of a theory in a doctrine is: it is just a morphism in the 2-category (of models of that doctrine).  It is also easy to generalize to [[Higher doctrines]] (given a notion of higher category).

However, while useful, this approach has the disadvantage of being too broad: there are many 2-categories that we would not really want to consider as being the category of models of some doctrine.

See also at _[[2-algebraic theory]]_.

### As logics

The word "doctrine" can also be used to design a particular setting for categorical logic (e.g., categories with finite products, [[geometric logic]] etc...; see [[internal logic]]).  This can also be formalized in a more general way using weak higher categories.  This generalization is necessary to treat complicated algebraic structures like symmetric operads or higher operads in a categorical logical way.

### As a class of limits

Especially in the theory of [[accessible categories]] and [[commutativity of limits and colimits]], a "doctrine" is sometimes taken to be simply a class of shapes (or [[weighted limit|weights]]) for [[limits]].  For more about this, see [[sound doctrine]].

## Related concepts

* [[algebraic theory]] / [[Lawvere theory]] / [[(∞,1)-algebraic theory]] / [[2-algebraic theory]]

* [[monad]] / [[2-monad]]/ **doctrine** / [[3-theory]] / [[(∞,1)-monad]]

* [[operad]] / [[(∞,1)-operad]]

## References

[[William Lawvere]], [23 Mar 2006](http://www.acsu.buffalo.edu/~wlawvere/Beck%20posting.htm):

>  The word "doctrine" itself is entirely due to [[Jon Beck]] and signifies something which is like a theory, except appropriate to be interpreted in the category of categories, rather than, for example, in the category of sets; of course, an important example of a doctrine is a 2-monad, and among 2-monads there are key examples whose category of "algebras" is actually a category of theories in the set-interpretable sense. Among such "theories of theories", there is a special kind whose study I proposed in that paper. This kind has come to be known as "Kock-Zoeberlein" doctrine in honor of those who first worked out some of the basic properties and ramifications, but the recognition of its probable importance had emerged from those discussions with Jon.

The concept of doctrine is apparently first appears in print in

* {#Lawvere69} [[William Lawvere]],  _Ordinal sums and equational doctrines_, Lecture Notes in Math., Vol. 80 (Springer, Berlin, 1969).

General discussions of the "doctrine" concept:

* [Algebraic Geometry for Category Theorists (blog post)](http://golem.ph.utexas.edu/category/2009/06/algebraic_geometry_for_categor.html)

* [Doctrines by Dolan and Baez](http://ncatlab.org/johnbaez/show/Doctrines)

* [A duality relative to a limit doctrine](http://www.emis.de/journals/TAC/volumes/10/20/10-20abs.html) by C. Centazzo and E.M. Vitale

A pedadogically aimed study of higher doctrines, using the simple definition of doctrines as (weak) [[n-categories]] described by [[generators and relations]]:

* [[Frédéric Paugam]]: Chapter 1 of
[Towards the mathematics of quantum field theory](http://people.math.jussieu.fr/~fpaugam/documents/enseignement/master-mathematical-physics.pdf).

* [[Higher doctrines]].

[[!redirects doctrines]]
[[!redirects 2-theory]]
[[!redirects 2-theories]]


doctrine of ideas

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### Context
#### Philosophy
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[[!include philosophy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

What has come to be called the _doctrine of ideas_ or the _theory of forms_ is  a central part of the [[philosophy]] of [[Plato]], going back mostly to the [[Parmenides dialogue]] and the allegory of 'The Cave' in the [[Republic dialogue]].

[[Aristotle]] expands on this. In the 19th century it becomes _[[idealism]]_. With  [[Hegel]] there is "The Idea" in an absolute way ([WdL -- Die Idee](http://ncatlab.org/nlab/show/Science%20of%20Logic#TheIdea)).

## Hegel's perspective
 {#HegelPerspective}

> For Hegel identifies the idea not with Plato's archetype but with [[Aristotle]]'s formal–final cause. Hegel saw Aristotle, not Plato, as the proper founder of [[absolute idealism]]. ([Beiser 05, p. 66](#Beiser05))

> Hegel contrasts Plato's Idea with "Aristotle's Idea" (sic), ... in contrast to Plato's it makes the end effective ([Ferrarin 04, p. 106](#Ferrarin04))

* [[Georg Hegel]], _[[Science of Logic]]_:

> [WdL §55](Science+of+Logic#55) Um dies in die Vorstellung wenigstens aufzunehmen, ist die Meinung auf die Seite zu legen, als ob die Wahrheit etwas Handgreifliches sein müsse. Solche Handgreiflichkeit wird zum Beispiel selbst noch in die Platonischen Ideen, die in dem Denken Gottes sind, hineingetragen, als ob sie gleichsam existierende Dinge, aber in einer anderen Welt oder Region seien, außerhalb welcher die Welt der Wirklichkeit sich befinde und eine von jenen Ideen verschiedene, erst durch diese Verschiedenheit reale Substantialität habe. Die Platonische Idee ist nichts anderes als das Allgemeine oder bestimmter der Begriff des Gegenstandes; nur in seinem Begriffe hat etwas Wirklichkeit; insofern es von seinem Begriffe verschieden ist, hört es auf, wirklich zu sein, und ist ein Nichtiges; die Seite der Handgreiflichkeit und des sinnlichen Außersichseins gehört dieser nichtigen Seite an.

> §55 To get some idea of this one must discard the prejudice that truth must be something tangible. Such tangibility is, for example, imported even into the Platonic Ideas which are in God's thinking, as if they are, as it were, existing things but in another world or region; while the world of actuality exists outside that region and has a substantial existence distinct from those Ideas and only through this distinction is a substantial reality. The Platonic Idea is the universal, or more definitely the Notion of an object; only in its Notion does something possess actuality and to the extent that it is distinct from its Notion it ceases to be actual and is a non-entity; the side of tangibility and sensuous self-externality belongs to this null aspect. But on the other side, one can appeal to the conceptions of ordinary logic itself; for it is assumed, for example, that the determinations contained in definitions do not belong only to the knower, but are determinations of the object, constituting its innermost essence and its very own nature. Or, if from given determinations others are inferred, it is assumed that what is inferred is not something external and alien to the object, but rather that it belongs to the object itself, that to the thought there is a correspondent being.

* [[Georg Hegel]],  _[[Lectures on the History of Philosophy]]_

> Chapter III: First Period, Third Division

> The third division, which deals with Plato and Aristotle, is found in Greek science where objective thought, the Idea, forms itself into a whole. The concrete, in itself determining Thought, is, with Plato, the still abstract Idea, but in the form of universality; while with Aristotle that Idea was conceived of as the self-determining, or in the determination of its efficacy or activity. ([source](https://www.marxists.org/reference/archive/hegel/works/hp/hpconten.htm))

* [[Georg Hegel]], _[[Lectures on the History of Philosophy]] -- [Aristotle -- 1. Metaphyiscs](Lectures+on+the+History+of+Philosophy#AristotleMetaphysics)_ ([English translation](https://www.marxists.org/reference/archive/hegel/works/hp/hpconten.htm)):

> From this definition we now see clearly the sort of opposition in which the Idea of Aristotle stands to that of Plato, for although the Idea of Plato is in itself essentially concrete and determined, Aristotle goes further. In so far, namely, as the Idea is determined in itself, the relation of the moments in it can be more closely specified, and this relation of the moments to each other is to be conceived of as nothing other than activity.

> The Platonic Idea, on the other hand, is rather that abrogation of opposites, where one of the opposites is itself unity. While, therefore, with Plato the main consideration is the affirmative principle, the Idea as only abstractly identical with itself, in Aristotle there is added and made conspicuous the moment of negativity, not as change, nor yet as nullity, but as difference or determination. The principle of individualization, not in the sense of a casual and merely particular subjectivity, but in that of pure subjectivity, is peculiar to Aristotle.

## Related concepts

* [[idealism]]

  [[objective idealism]]

  [[absolute idealism]]

## References

* Wikipedia, _[Theory of Forms](http://en.wikipedia.org/wiki/Theory_of_Forms)_

* Wikipedia, _[Platonic idealism](http://en.wikipedia.org/wiki/Platonic_idealism)_

* {#WdL} [[Georg Hegel]], section _[The Idea](Science+of+Logic#TheIdea)_ in _[[Science of Logic]]_

*  {#Ferrarin04} Alfredo Ferrarin, _Hegel and Aristotle_, 2004 ([web](https://books.google.it/books?id=H_SjGi51DrEC&pg=PA106#v=onepage&q&f=false))

* {#Beiser05} Frederick Beiser, _Hegel_ (2005), passim and page 66

[[!redirects Platonic idealism]]


dodecahedron

[[!include ADE -- table]]

## References

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


Dolbeault cohomology

> under construction

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###Context###
#### Cohomology
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[[!include cohomology - contents]]
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#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
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=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Dolbeault cohomology_ of a [[complex manifold]] $X$ is the [[chain cohomology]] of the [[Dolbeault complex]] of $X$ (see there for more).

## Properties

### Dolbeault theorem

By the _[[Dolbeault theorem]]_ (see there) Dolbeault cohomology is equivalently the [[abelian sheaf cohomology]] $H^q(X;\Omega_X^p)$, of the [[abelian sheaf]] $\Omega_X^p$ which is the [[Dolbeault complex]] of [[holomorphic p-forms]].

### Hodge isomorphism

For $X$ a [[Hermitian manifold]] write $\mathcal{H}^{p,q}(X)$ for the space of $(p,q)$-[[harmonic differential forms]] and write $H^{p,q}$ for its [[Dolbeault cohomology]] in the bidegree.

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

There is a canonical [[homomorphism]]

$$
  \mathcal{H}^{p,q}(X) \longrightarrow H^{p,q}(X)
  \,.
$$

If $X$ is [[compact topological space|compact]], then this is an [[isomorphism]], the _[[Hodge isomorphism]]_

=--

Also

$$
  \mathcal{H}^k(X,\mathbb{C}) \longrightarrow H^k_{dR}(X,\mathbb{C})
$$

is an isomorphism

(e.g. [Maddock,  prop. 4.2.7](#Maddock))

### Serre duality

[[Serre duality]]: On a [[Hermitian manifold]] $X$ of complex dimension $dim_{\mathbb{C}}(X) = n$ the [[Hodge star operator]] induces [[isomorphisms]]

$$
  \mathcal{H}^{p,q}(X)\simeq \mathcal{H}^{n-p, n-q}(X)
  \,.
$$

(e.g. [Maddock,  prop. 4.2.8](#Maddock))

### Hodge decomposition

If $X$ is a [[compact topological space|compact]] [[Kähler manifold]] then

$$
  \mathcal{H}^k(X) \simeq \underset{p+q = k}{\oplus} \mathcal{H}^{p,q}(X)
$$

(e.g. [Maddock,  prop. 4.2.9](#Maddock))

This is called the _[[Hodge decomposition]]_.

### Hodge symmetry

Over [[complex manifolds]] $X$, _[[Hodge symmetry]]_ is the property that the [[Dolbeault cohomology]] groups $H^{p,q}(X)$ are taken into each other under [[complex conjugation]] followed by switching the bidegree:

$$
  H^{p,q}(X) \simeq \overline{H^{q,p}(X)}
  \,.
$$

In particular this means that the [[dimension]] of the cohomology groups in degree $(p,q)$ coincides with that in bidegree $(q,p)$.

### Dolbeault resolution

Given any [[holomorphic vector bundle]] $E$, one can form the Dolbeault resolution $E \otimes \Omega^{0,q}$, where $\Omega^{0,q}$ is the sheaf of $C^\infty$ $(0,q)$-forms. This is an acyclic resolution of $E$ and hence computes its [[sheaf cohomology]].

(...)

### Double complex and Frölicher spectral sequence

The Dolbeault complexes naturall fit into a [[double complex]]

$$
  \array{
    \Omega^{p,0} &\stackrel{\bar \partial}{\to}& \Omega^{p-1,1} &\stackrel{\bar \partial}{\to}& \cdots
    \\
    \downarrow^{\mathrlap{\partial }}
    &&
    \downarrow^{\mathrlap{\partial }}
    \\
    \Omega^{p+1,0} &\stackrel{\bar \partial}{\to}& \Omega^{p,1} &\stackrel{\bar \partial}{\to}& \cdots
    \\
    \downarrow^{\mathrlap{\partial }}
    &&
    \downarrow^{\mathrlap{\partial }}
    \\
    \vdots && \vdots
  }
  \,.
$$

The corresponding [[spectral sequence of a double complex]] is called the _[[Frölicher spectral sequence]]_. On a [[Kähler manifold]] it exhibits the [[Hodge filtration]].

## Related concepts

* [[Hodge number]]

* [[Kodaira vanishing theorem]]

* [[de Rham cohomology]]

* [[Bott-Chern cohomology]]

## References

* {#Maddock} Zachary Maddock, _Dolbeault cohomology_ ([pdf](http://www.math.columbia.edu/~maddockz/notes/dolbeault.pdf))

[[!redirects Dolbeault resolution]]


Dolbeault complex

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

#Contents#
* table of contents
{:toc}

## Idea

The _Dolbeault complex_ is the analog of the [[de Rham complex]] in [[complex geometry]].

## Definition

### Dolbeault complex

On a [[complex manifold]] $X$ the [[de Rham complex]] $\Omega^\bullet(X)$ refines to a [[bigraded object|bigraded]] complex $\Omega^{\bullet, \bullet}(X)$, where a [[differential form]] of bidegree $(p,q)$ has holomorphic degree $p$ and antiholomorphic degree $q$, hence is given on a local [[coordinate chart]] by an expression of the form

$$
  \omega = \sum f_{I J} d z_{i_1} \wedge \cdots \wedge d z_{i_p} \wedge d \bar z_{j_1} \wedge \cdots \wedge d \bar z_{j_q}
 \,.
$$

Moreover, the [[de Rham differential]] $\mathbf{d}$ decomposes as

$$
  \mathbf{d} = \partial + \bar \partial
  \,,
$$

where $\partial \colon \Omega^{\bullet, \bullet}\to \Omega^{\bullet + 1, \bullet}$ and $\bar \partial \colon \Omega^{\bullet, \bullet}\to \Omega^{\bullet, \bullet + 1}$.

The _Dolbeault complex_ of $X$ is the [[chain complex]] $(\Omega^{\bullet, \bullet}(X), \bar \partial)$. The _[[Dolbeault cohomology]]_ of $X$ is the [[cochain cohomology]] of this complex.

### Holomorphic differential forms

Here $\Omega^{p,0}(X)$ defines a [[holomorphic vector bundle]] and a [[holomorphic section]] is a differential form with local expression as above, such that the coefficient functions $f_{I J}$ are [[holomorphic functions]]. This is called a _holomorphic differential form_.

For $p \lt dim_{\mathbb{C}}(X)$ equivalently this is a differential form in the [[kernel]] of the antiholomorphic Dolbeault operator $\bar \partial$.

## Properties

### Dolbeault theorem
 {#DolbeaultTheorem}

The complex analog of the [[de Rham theorem]] is the [[Dolbeault theorem]]:

for $X$ a [[complex manifold]] then its [[Dolbeault cohomology]] in bi-degree $(p,q)$ is [[natural isomorphism|naturally isomorphic]] to the [[abelian sheaf cohomology]] in degree $q$ of the [[abelian sheaf]] $\Omega^p \coloneqq \Omega^{p,0}$ of [[holomorphic p-forms]]

$$
  H^{p,q}(X)\simeq H^q(X,\Omega^p)
  \,.
$$

(...)

Let $Disk_{compl}$ be the [[category]] of complex [[polydiscs]] in $\mathbb{C}^n$ and [[holomorphic functions]] between them.

For $p \in \mathbb{N}$ write $\Omega^p \colon Disk_{complex}^{op} \to Set$ for the [[sheaf]] of holomorphic [[differential n-form|differential p-forms]].

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

For $X$ a [[complex manifold]], let $\{U_i \to X\}$ be a holomorphic good open cover. Then the [[Cech cohomology]] of this cover with [[coefficients]] in $\Omega^p$ in degree $q$ is the [[Dolbeault cohomology]] in bidegree $(p,q)$

$$
  H^{p,q}(X) \simeq \pi_0 sPSh(Disk_{comp})(C(\{U_i\}, \Omega^p[q]))
  \,.
$$

=--

For instance ([Maddock, theorem 1.0.1](#Maddock)).

### On Stein manifolds

+-- {: .num_prop}
###### Proposition
**([[Cartan theorem B]])**

For $X$ a [[Stein manifold]],

$$
  H^k(\Omega^{p,\bullet}(X), \bar \partial)
  =
  \left\{
    \array{
      0 & k \neq 0
      \\
      \Omega^p_{hol}(X) & k = 0
    }
  \right.
  \,.
$$

=--

For instance ([Gunning-Rossi](#GunningRossi)).

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

For $X$ a [[Stein manifold]] of complex [[dimension]] $n$, the [[compactly supported cohomology|compactly supported]] Dolbeault [[cohomology]] is

$$
  H^k(\Omega_c^{p, \bullet}(X), \bar \partial)
  =
  \left\{
    \array{
      0 , & k \neq n
      \\
      (\Omega_{hol}^{n-p}(X))^\ast
    }
  \right.
  \,,
$$

where on the right $(-)^\ast$ denotes the continuous linear dual.

=--

First noticed in ([Serre](#Serre)).

### Todd genus

By the [[Hirzebruch-Riemann-Roch theorem]] the [[index]] of the Dolbeault operator is the [[Todd genus]].

### Relation to $Spin^c$-structures

A [[complex manifold]], being in particular an [[almost complex manifold]], carries a canonical [[spin^c structure]]. The corresponding [[Spin^c Dirac operator]] identifies with the Dolbeault operator under the identification of the [[spinor bundle]] with that of [[holomorphic differential forms]]

$$
  S(X) \simeq \wedge^{0,\bullet} T^\ast X
  \,.
$$

## Related concepts

* [[Dolbeault cohomology]]

* [[Dolbeault-Dirac operator]]

* [[chiral Dolbeault complex]]

* [[holomorphic de Rham complex]]

## References

* {#Voisin02} [[Claire Voisin]], section 2.3 of _[[Hodge theory and Complex algebraic geometry]] I,II_, Cambridge Stud. in Adv. Math. __76, 77__, 2002/3

* {#Maddock} Zachary Maddock, _Dolbeault cohomology_ ([pdf](http://www.math.columbia.edu/~maddockz/notes/dolbeault.pdf))

* {#GunningRossi} [[Robert C. Gunning]], [[Hugo Rossi]], *Analytic functions of several complex variables*, Prentice-Hall Inc., Englewood Cliffs (1965)

* {#Serre} [[Jean-Pierre Serre]], _Quelques problèmes globaux relatifs aux variétés de Stein_, Colloque sur les fonctions de plusieurs variables, tenu à Bruxelles, 1953, Georges Thone, Liège, 1953, pp. 57–68. MR 0064155 (16,235b)

A [[formal geometry]] version:

* Shilin Yu, _Dolbeault dga of a formal neighborhood_, [arxiv/1206.5155](http://arxiv.org/abs/1206.5155); _The Dolbeault dga of the formal neighborhood of a diagonal_, [arxiv/1211.1567](http://arxiv.org/abs/1211.1567)

[[!redirects Dolbeault complexes]]

[[!redirects Dolbeault cohomology]]

[[!redirects holomorphic smooth function]]
[[!redirects holomorphic smooth functions]]

[[!redirects holomorphic differential 1-form]]
[[!redirects holomorphic 1-form]]
[[!redirects holomorphic differential form]]

[[!redirects holomorphic differential 1-forms]]
[[!redirects holomorphic 1-forms]]
[[!redirects holomorphic differential forms]]

[[!redirects holomorphic form]]
[[!redirects holomorphic forms]]

[[!redirects holomorphic differential p-form]]
[[!redirects holomorphic differential p-forms]]
[[!redirects holomorphic p-form]]
[[!redirects holomorphic p-forms]]

[[!redirects Dolbeault differential]]
[[!redirects Dolbeault differentials]]

[[!redirects Dolbeault operator]]
[[!redirects Dolbeault operators]]


Dolbeault theorem

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

#Contents#
* table of contents
{:toc}

## Idea

The [[complex geometry]]-analog of the [[de Rham theorem]] is the _Dolbeault theorem_:

For $X$ a [[complex manifold]] then its [[Dolbeault cohomology]] in bi-degree $(p,q)$ is [[natural isomorphism|naturally isomorphic]] to the [[abelian sheaf cohomology]] in degree $q$ of the [[abelian sheaf]] $\Omega^p \coloneqq \Omega^{p,0}$ of [[holomorphic p-forms]]

$$
  H^{p,q}(X)\simeq H^q(X,\Omega^p)
  \,.
$$

## Related concepts

* [[de Rham theorem]]

* [[Hodge theorem]]

## References

Review includes

* {#Maddock} Zachary Maddock, _Dolbeault cohomology_ ([pdf](http://www.math.columbia.edu/~maddockz/notes/dolbeault.pdf))

[[!redirects Dolbeault isomorphism]]


Dolbeault-Dirac operator

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
#### Spin geometry
+-- {: .hide}
[[!include higher spin geometry - contents]]
=--
#### Index theory
+-- {: .hide}
[[!include index theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

On a [[Kähler manifold]] $(X,\omega)$ with a choice of [[spin structure]] given by a [[Theta characteristic]] $\sqrt{\Omega^{0,n}}$, the sum of the [[Dolbeault operator]] $\overline{\partial}$ with its adjoint $\overline{\partial}^\ast$ are identified with the corresponding [[Dirac operator]]

$$
  \overline{\partial}
  +
  \overline{\partial}^\ast
  \;\colon\;
  S_X\simeq \Omega^{0,\bullet}_X \otimes \sqrt{\Omega^{0,n}}
  \to
  \Omega^{0,\bullet}_X \otimes \sqrt{\Omega^{0,n}}
  \simeq
  S_X
  \,.
$$

Here the action of $\overline{\partial} + \overline{\partial}^\ast$ on $\Omega^{0,\bullet}$ is the canonical one. For the action on $\sqrt{\Omega^{n,0}}$ choose any [[line bundle with connection|connection]] $\nabla$ on this line bundle. Then on a local [[coordinate patch]] the action is given by differentiating along a coordinate vector, multiplying with the corresponding Clifford element and projecting on the antiholomorphic part, then summing this over all coordinate vectors (e.g. [Friedrich 97, p. 79](#Friedrich97)).

## Properties

### Index and genus

The [[index]] is the [[Todd genus]] (see there).

## Related concepts

* [[Kähler-Dirac operator]]

* [[spin^c Dirac operator]]

## References

Textbook accounts include

section 3.4 (specifically around p. 79) of

* [[Thomas Friedrich]], _Dirac operators in Riemannian geometry_, Graduate studies in mathematics 25, AMS (1997)
 {#Friedrich97}

[[!redirects Dirac-Dolbeault operator]]


Dold fibration

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

[[Serre fibration]]  $\Leftarrow$ [[Hurewicz fibration]] $\Rightarrow$ **Dold fibration**$\Leftarrow$ [[shrinkable map]]

#Contents#
* automatic table of contents goes here
{:toc}

## Idea ##

A **Dold fibration** is a continuous map of [[topological space]]s that allows [[right lifting property|lifting]] of [[homotopies]], with initial condition agreeing with a given map up to a vertical homotopy.

## Definition ##

The morphism $f:E \to B$ of [[topological space]]s is said to have the **weak covering homotopy property** (WCHP) for the space $Y$ if for all squares

$$
\begin{matrix}
  Y\times\{0\}& \stackrel{g_0}{\to} & E \\
  \downarrow&&\, \downarrow f\\
  Y\times I &\underset{g}{\to} & B
\end{matrix}
$$
there is a map $\hat{g}:Y\times I \to E$ such that $f\circ \hat{g} = g$ and there is a vertical homotopy between $\hat{g}(-,0):Y\to E$ and $g_0$. The synonymous expression _weak homotopy lifting property_ (WHLP) is also used.

A continuous map is a **Dold fibration** if it has the WCHP for all spaces. Somewhat surprisingly, there is an equivalent condition in terms of [[delayed homotopies]]. A __[[delayed homotopy]]__ is a homotopy $H:Y\times I\to Z$ such that $H(y,t)=H(y,0)$ for $0\leq t\leq t_0$ for some $t_0\gt 0$.
A continuous map is a Dold fibration iff in the diagram above in which $g$ is a delayed homotopy, can be filled with a diagonal map $\hat{g}:Y\times I \to E$ such that the diagram is strictly commutative. It is of course not required that $\hat{g}$ be delayed (one can require, but then one allows $t_0$ for $\hat{g}$ to be possibly smaller than $t_0$ for $g$). This is sometimes called the __delayed homotopy lifting property__.

## Relation to other fibrations

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

Every [[Hurewicz fibration]] is a Dold fibration.

=--

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

Not all Serre fibrations are Dold fibrations.

=--

+--{: .query}
[[David Roberts]]:  Can we come up with a counterexample of a Serre fibration that isn't a Dold fibration? I'll ask on MathOverflow.
=--

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

Not all Dold fibrations are [[Serre fibration]]s.

=--

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

Here is a very simple counter-example due to Dold. Consider the union of line segments
$$
E:= [-1,0]\times\{2\} \cup \{0\}\times [1,2] \cup [0,1]\times\{1\}
$$
in $\mathbf{R}^2$, and the map projecting on to the first coordinate, $pr_1:E \to [-1,1]$. Then this map is a Dold fibration but not a Serre fibration.

=--

+--{: .query}
[[Chris Schommer-Pries]]: I believe this is an example of a quasi-fibration which is not a Serre fibration, but it is not a Dold fibration either.
=--

One could consider maps that have the WCHP for just cubes -- these are a sort of hybrid Dold--Serre fibration (warning! nonstandard terminology. I just made it up, suggestions appreciated). For these maps there exists a [[long exact sequence]] in homotopy once basepoints are chosen. For classes of maps determined by (homotopy) lifting properties, this is about the minimum one needs to define such a long exact sequence. On the other hand, [[quasifibration|quasifibrations]] give rise to a long exact sequence in homotopy, but are defined by homotopy properties of the fibres.

## Examples

+-- {: .un_lemma}
###### Observation

Every [[shrinkable]] map is a Dold fibration.

=--

This result follows from a theorem of Dold about [[locally homotopy trivial map]]s being the same as Dold fibrations. It should be obvious that a shrinkable map is globally homotopy trivial, with trivial fibre.

+-- {: .un_cor}
###### Corollary

Let $\{U_i \to X\}$ be a [[numerable open cover]]. Let $C(\{U_i\})$ the [[Cech nerve]] (a [[simplicial object]] in [[topological space]]s) and $|C(\{U_i\})| \in Top$ its [[geometric realization]]. Then the canonical map

$$
  |C(\{U_i\})| \to X
$$

is shrinkable, hence a Dold fibration.

=--

This observation is due to Segal. See [[shrinkable map]].

[[!redirects Dold fibrations]]