Dold-Kan correspondence

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#### Homological algebra
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#### Homotopy theory
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## Idea

The _Dold--Kan correspondence_ asserts there is an [[equivalence of categories]] between [[abelian group|abelian]] [[simplicial groups]] and [[connective chain complex|connective]] [[chain complexes]] of [[abelian groups]].

Since every [[simplicial group]] is in particular a [[Kan complex]] with [[group object|group structure]], hence an [[∞-groupoid]] with group structure, hence an [[∞-group]], the Dold-Kan correspondence says that [[connective chain complexes]] are a model for certain [[abelian ∞-groups]], thus the correspondence interpolates between [[homological algebra]] and general [[simplicial homotopy theory]]. (This is part of the [[cosmic cube]] of [[higher category theory]]). The relevance of this is that chain complexes are typically easier to handle: all the tools of [[homological algebra]] apply. In fact, the [[functor]] that identifies simplicial abelian groups with their corresponding chain complexes -- the [[Moore complex|normalized chains functor]] -- does precisely this: it _normalizes_ an abelian group by discarding irrelevant information and constructing a smaller and less redundant model for it.

There are various variants and generalizations of the Dold-Kan correspondence. These are discussed further [below](#Variants).

## Statement (abelian case)

Let $A$ be an [[abelian category]].

We say a [[chain complex]] in $A$ is _connective_ if it is concentrated in non-negative degree. The [[full subcategory]]

$$
  Ch^+_\bullet(A) \hookrightarrow Ch_\bullet(A)
$$

of [[connective chain complexes]] is naturally identified with the category of $\mathbb{N}$-graded chain complexes.

### Equivalence of categories {#EquivalenceOfCategories}

\begin{theorem}
**(Dold--Puppe)**
For $A$ an [[abelian category]]
there is an [[equivalence of categories]]

$$
  N \;\colon\; A^{\Delta^{op}}
      \stackrel{\leftarrow}{\to} Ch_\bullet^+(A) \;\colon\; \Gamma
$$

between

* the [[category of simplicial objects]] in $A$;

* the [[category of chain complexes|category of connective chain complexes]] in $A$;

where

* $N$ is the [[normalized chains complex]]/normalized [[Moore complex]] functor.

\end{theorem}

([Dold 58](#Dold58), [Kan 58](#Kan58), [Dold-Puppe 61](#DoldPuppe61)).

\begin{theorem}
**(Kan)** \linebreak
For the case that $A$ is the category [[Ab]] of [[abelian group]]s, the functors $N$ and $\Gamma$ are [[nerve and realization]] with respect to the cosimplicial chain complex

$$
  \mathbb{Z}[-]
  \;\colon\;
  \Delta \longrightarrow Ch_+(Ab)
$$

that sends the standard $n$-[[simplex]] to the normalized [[Moore complex]] of the [[free simplicial abelian group]] $F_{\mathbb{Z}}(\Delta^n)$ on the [[simplicial set]] $\Delta^n$, i.e.

$$
    \Gamma(V)
    \;\colon\;
    [k]
    \mapsto
    Hom_{Ch_\bullet^+(Ab)}
    \Big(
      N
      \big(
        \mathbb{Z}(\Delta[k])
      \big),
      \,
      V
    \Big)
    \,.
$$

\end{theorem}

This is due to ([Kan 58](#Kan58)).

More explicitly:

\begin{proposition}\label{ExplicitFormOfGamma}

* For $V \in Ch_\bullet^+$ the [[simplicial abelian group]] $\Gamma(V)$ is in degree $n  \in \mathbb{N}$ given by

  $$
    \Gamma(V)_n
    \;=\;
    \bigoplus_{[n] \twoheadrightarrow [k]} V_k
  $$

  and for $\theta : [m] \to [n]$ a morphism in $\Delta$ the corresponding map
  $\Gamma(V)_n \to \Gamma(V)_m$

  $$
    \theta^* : \bigoplus_{[n] \twoheadrightarrow [k]} V_k
     \to
     \bigoplus_{[m] \twoheadrightarrow [r]} V_r
  $$

  is given on the [[direct sum|direct summand]] indexed by some $\sigma \colon [n] \to [k]$ by the composite

  $$
    V_k
      \overset{d^*}{\to}
    V_s
      \hookrightarrow
    \bigoplus_{[m] \twoheadrightarrow [r]}
      V_r
    \,,
  $$

  where

  $$
    [m]
     \overset{t}{\twoheadrightarrow}
    [s]
      \overset{d}{\hookrightarrow}
    [k]
  $$

  is the [[epi-mono factorization]] of the [[composition|composite]] $[m] \stackrel{\theta}{\to} [n] \stackrel{\sigma}{\to} [k]$.

\end{proposition}

In these terms (Prop. \ref{ExplicitFormOfGamma}) we have:

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

* The [[natural isomorphism]] $\Gamma N \to Id$ is given on $A \in sAb^{\Delta^{op}}$ by the map

  $$
    \bigoplus_{[n] \twoheadrightarrow [k]}
     (N A)_k
    \to
    A_n
  $$

  which on the [[direct sum|direct summand]] indexed by $\sigma \colon [n] \to [k]$ is the composite

  $$
    N A_k \hookrightarrow A_k \stackrel{\sigma^*}{\to} A_n
    \,.
  $$

* The [[natural isomorphism]] $Id \to N \Gamma$ is on  a chain complex $V$ given by the composite of the projection

  $$
    V \to C(\Gamma(V)) \to C(\Gamma(C))/D(\Gamma(V))
  $$

  with the inverse

  $$
    C(\Gamma(V))/D(\Gamma(V)) \to N \Gamma(V)
  $$

  of

  $$
    N \Gamma(V) \hookrightarrow C(\Gamma(V)) \to C(\Gamma(V))/D(\Gamma(V))
  $$

  (which is indeed an [[isomorphism]], as discussed at [[Moore complex]]).

=--

This is spelled out in  [Goerss & Jardine, prop. 2.2 in section III.2](#GoerssJardine99)

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

With the explicit choice for $\Gamma N \stackrel{\simeq}{\to} Id$ as [above](#ExplicitUnitAndCounit) we have that $\Gamma$ and $N$ form an [[adjoint equivalence]] $(\Gamma \dashv N)$

=--

This is for instance ([Weibel, exercise 8.4.2](#Weilbel)).

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

It follows that with the inverse structure maps, we also have an [[adjunction]] the other way round: $(N \dashv \Gamma)$.

=--

### Quillen equivalence of model categories {#ModelCatVersion}

Both $Ch_\bullet^+(A)$ and $A^{\Delta^{op}}$ are [[categories with weak equivalences]] in an standard way:

* the weak equivalences of simplicial abelian groups are the [[weak homotopy equivalences]] of the underlying [[Kan complexes]], hence morphisms that induce [[isomorphisms]] on all [[simplicial homotopy groups]];

* the weak equivalences of chain complexes are the [[quasi-isomorphisms]]: the morphisms that induces isomorphisms on all [[chain homology]] groups.

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

These functors $N$ and $\Gamma$ both respect all weak equivalences
with respect to the [[classical model structure on simplicial sets]] [[model structure on chain complexes|and on chain complexes]] in that they induce isomorphisms between [[simplicial homotopy groups]] and [[homology group]]s.

=--

The structures of categories with weak equivalences have standard refinements to [[model category]] structures:

* the _projective_ [[model structure on chain complexes]] $Ch_\bullet$ has as fibrations the chain maps that are surjections in each positive degree;

* the _[[model structure on simplicial abelian groups]]_ has as [[fibrations]] those whose underlying morphisms in [[sSet]] are fibrations  with respect to the [[classical model structure on simplicial sets]] (hence: [[Kan fibrations]]).

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

Both

$$
  (N \dashv \Gamma)
  \;\colon\;
  Ch_\bullet^+
    \stackrel{\overset{N}{\leftarrow}}{\underset{\Gamma}{\to}}
  sAb
$$

as well as

$$
  (\Gamma \dashv N)
  \;\colon\;
  sAb
    \stackrel{\overset{\Gamma}{\leftarrow}}{\underset{N}{\to}}
  Ch_\bullet^+
$$

are [[Quillen equivalences]] with respect to these model structures.

=--

This is due to [Quillen 67, Section II.4 item 5](#Quillen67), see also [Schwede-Shipley 03, section 4.1](#SchwedeShipley03), [p.17](http://www.math.uic.edu/~bshipley/monoidalequi.final.pdf#page=17), [Jardine 03, Lemma 1.5](#Jardine03)).

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

The category [[sAb]] $ = Ab^{\Delta^{op}}$ -- being a [[category of simplicial objects]] of a category with [[colimits]] -- is naturally an [[sSet]]-[[enriched category]] and with [[model structure on simplicial abelian groups|its model structure]] this makes it a [[simplicial model category]].

Since the DK-correspondence is even an [[equivalence of categories]], this induces accordingly the structure of a simplicial model category also on $Ch_\bullet^+$. Therefore the above Quillen equivalence is even a [[simplicial Quillen adjunction]].

=--

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

The [[free functor|free]]/[[stuff, structure, property|forgetful]] [[adjunction]] $(F \dashv U) : Ab \stackrel{\leftarrow}{\to} Set$ prolongs to [[simplicial object]]s

$$
  (F\dashv U) : sAb \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}}
  sSet
$$

as an [[sSet]]-[[enriched functor|enriched]] adjunction. Moreover, by the above the right adjoint $U$ is a right Quillen functor to the standard [[model structure on simplicial sets]].

This means we have a [[simplicial Quillen adjunction]]

$$
  ( \Gamma F \dashv U N) : Ch_\bullet^+ \stackrel{\overset{}{\leftarrow}}{\underset{U N}{\to}} sSet
  \,.
$$

This manifestly [[presentable (infinity,1)-category|exhibits]] [[connective chain complexes]] as models for certain [[∞-groupoids]].

=--

## Statement (additive case)
 {#StatementAdditiveACase}

The Dold-Kan correspondence generalizes from  [[Ab]] to any [[abelian category]] ([Lurie, §1.2.3](#LurieHA)) and fruther to [[additive categories]] for which all [[retractions]] come from [[direct sums]], as is the case for [[triangulated categories]] ([Lurie, §1.2.4](#LurieHA)).

## Statement (general nonabelian case) {#StatementGeneral}

### Globular and cubical version {#GlobularAndCubical}

There are versions of the Dold-Kan correspondence for other [[geometric shapes for higher structures]] than the [[simplex]], also for the [[globe]] and the [[cube]].

\begin{theorem}
**(globular Dold-Kan correspondence)**\linebreak

Write [[Ab]] for the category of [[abelian group]]s. (Could be any [[additive category]] with [[kernel]]s for the following to be true). Then the following categories of structures [[internalization|internal to]] $Ab$ are equivalent.

1. The category of [[chain complex]]es (in non-negative degree).

1. The category of [[crossed complex]]es.

1. The category of [[cubical set]]s with [[connection on a cubical set]].

1. The category of cubical [[strict ∞-groupoid]]s.

1. The category of [[globe|globular]] [[strict ∞-groupoid]]s.

\end{theorem}

A proof with references to the rich literature can be found for instance in

* [[Ronnie Brown]], [[Philip Higgins]], [[Rafael Sivera]], _[[Nonabelian Algebraic Topology]]_

see the section  [Cubical Dold-Kan theorem](http://ncatlab.org/nlab/show/Nonabelian+Algebraic+Topology#CubicalDoldKan).

This version of the Dold-Kan theorem reproduces the simplicial Dold-Kan theorem after application of the [[omega-nerve]], i.e. the simplicial Dold-Kan correspondence factors through the globular one via the $\omega$-nerve.

### Presentation of strict groupal $\infty$-groupoids

It was mentioned above that the standard simplicial Dold-Kan correspondence $Ch_\bullet(Ab) \stackrel{\leftarrow}{\to} sAb$ may be understood as identifying strictly abelian  [[strict ∞-groupoid]]s among all [[∞-groupoid]]s. This statement is also surveyed and put into a larger context at [[cosmic cube of higher category theory]].

We now give a formal version of this statement, following an observation by [[Richard Garner]]. A different but closely analogous sequence of arguments to the same extent is also in the book

* [[Ronnie Brown]], [[Philip Higgins]], [[Rafael Sivera]],  _[[Nonabelian Algebraic Topology]]_, European Math. Soc. Tracts in Mathematics 15, 2011.

see Dold-Kan map and omega-nerve.

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

Write

$$
  (L \dashv R) : Ch_\bullet(Ab)^+ \stackrel{\leftarrow}{\to}
  Str \infty Cat(Ab)
  \stackrel{\leftarrow}{\to}
  Str \infty Cat(Set)
$$

for the [[adjunction]] obtained by composing the [globular Dold-Kan correspondence](#GlobularAndCubical) with the forgetful functor which forgets the abelian group structure on a strict $\infty$-category in the image of the globular/cubical Dold-Kan map.

=--

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

The functor

$$
  C_\bullet : \Delta \to Ch_\bullet^+
$$

which sends a [[simplex]] to its (normalized) chain complex factors as

$$
  C_\bullet : \Delta \stackrel{\mathcal{O}}{\to} Str \infty Cat
  \stackrel{L}{\to} Ch_\bullet^+
  \,,
$$

where the cosimplicial strict $\infty$-category $\mathcal{O}$ is the [[oriental]] functor.

=--

This is a remark by [[Richard Garner]] posted here.

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

Use that $\mathcal{O}(n)$ is the [[free construction|free]] strict $\infty$-category on a [[computad]].

Observe that $L$ sends a strict $\omega$-category $X$ to the chain complex obtained from the abelian reflexive globular set $X \times \mathb{Z}$. In particular the value on the $n$-[[globe]] is the chain complex

$$
  \mathbb{Z} \to \mathbb{Z}\oplus\mathbb{Z} \to
   \mathbb{Z}\oplus\mathbb{Z} \to \cdots \to
   \mathbb{Z}\oplus\mathbb{Z}
$$

with $(n+1)$ terms and differential given by $x \mapsto (x, -x)$ in each dimension.

Moreover, the value of $L$ on the boundary of the $n$-globe is the chain complex obtained from this by removing the uppermost copy of $\mathbb{Z}$.

Given a [[computad]] $C$, the associated abelian chain complex $L C$ has for $(L C)_n$ the free abelian group on the set of generating $n$-cells of $C$, and differential given by $\partial x = \sum_j t_j - \sum_i s_i$, where $\{s_i\}$ and $\{t_i\}$ are the sets of source- and or target-cells, respectively. A glance at [[Ross Street]]'s presentation of the [[oriental]]s shows that $L(\mathcal{O}(n)) = C_\bullet(\Delta[n])$.

=--

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

The simplicial Dold-Kan map

$$
  Hom(C_\bullet \Delta[n], -) : Ch_\bullet + \to sSet
$$

factors as the identification of chain complexes with strictly abelian strict $\infty$-groupoids, followed by the functor that forgets the abelian structure and then followed by the [[omega-nerve]] operation that embeds strict $\infty$-groupoids into all $\infty$-groupoids.

=--

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

Use the above adjunction and proposition to write for $K_\bullet$ a chain complex

$$
  Hom_{Ch_\bullet}(C_\bullet \Delta[n], K)
  =
  Hom_{Ch_\bullet}(L \mathcal{O}(n), K)
  =
  Hom_{Str \infty Cat}(\mathcal{O}(n), R K)
  =
  N (R K)_n
  \,.
$$

=--

**Remark** The alternative construction in [[Nonabelian Algebraic Topology]] factors also versions of the nonabelian Dold-Kan correspondence through the $\omega$-nerve.

##Non-abelian forms of the Dold-Kan correspondence.

Perhaps the 'ultimate' form of a 'classical' Dold--Kan result is by [[Pilar Carrasco]], who identified the extra structure on chain complexes of groups in order that they be [[Moore complex|Moore complexes]] of simplicial groups.  [[Dominique Bourn]] has a general form of this result for his [[semi-abelian category|semi-abelian categories]]. His results provide a neat categorical gloss on the theorem.

Dominique Bourn's formulation is very pretty. The Moore complex functor is [[monad|monadic]] when the basic category is semi-Abelian (Th. 1.4. p.113 in _Bourn2007_ below). Of course for simplicial _groups_, the  monad on chain complexes of groups gives the [[hypercrossed complex]]es of [[Carrasco]] and [[Cegarra]], but here they fall out from the theory.  On the down side there is apparently no
full analysis as yet of the actual form of this monad.

Gary Nan Tie showed that a certain full subcategory of filtered systems of hypergroupoids of rank n is equivalent to the following three categories, n-groupoids, n-simplicial T-complexes and n-crossed complexes, and in doing so proved a non-abelian Dold-Kan theorem for crossed complexes:

A Dold-Kan theorem for crossed complexes,
Journal of Pure and Applied Algebra,
Volume 56, Issue 2, 1989, Pages 177-194,
ISSN 0022-4049,
https://doi.org/10.1016/0022-4049(89)90143-6

Iterated W̄ and T-groupoids,
Journal of Pure and Applied Algebra,
Volume 56, Issue 2, 1989, Pages 195-209,
ISSN 0022-4049,
https://doi.org/10.1016/0022-4049(89)90144-8

## Stable Dold-Kan correspondence
 {#StableDoldKanCorrespondence}

The Dold-Kan correspondence [[stabilization|stabilizes]]
to identify _unbounded_ [[chain complexes]]
with the category of stably simplicial abelian groups.
The latter are closely related to [[combinatorial spectrum|combinatorial spectra]] of [[Daniel Kan]] and can be defined as stably simplicial objects in
the category of abelian groups.
More precisely, we have the following definitions.

+-- {: .num_defn}
###### Definition
The category of _stable simplices_ has integer numbers as objects.
Given two objects $k$ and $l$, the set of morphisms
from $k$ to $l$ is the set of order-preserving maps $h$
from the set of natural numbers to itself
such that $h(n)=n+l-k$ for all but a finite number of $n$.
Morphisms are composed by composing the corresponding maps.
=--

+-- {: .num_defn}
###### Definition
A _stably simplicial abelian group_ is a presheaf $F$ of abelian groups
on the category of stable simplices
such that for any integer $k$ every element $x$ of $F(k)$
belongs to the kernels of all but a finite number of degeneracy maps.
Morphisms of stably simplicial abelian groups are morphisms
of presheaves.
=--

The following theorem was established in 1963 by [[Daniel Kan]]
in his paper "Semisimplicial spectra" (see Proposition 5.8):

+-- {: .num_theorem}
###### Theorem
The [[category of unbounded chain complexes]] is [[equivalence of categories|equivalent]] to the
category of stably simplicial abelian groups,
with equivalences being given by the same functors

as in the unstable Dold-Kan correspondence, but appropriately
extended to the above categories.
=--

Similarly to the unstable case,
the above categories, when interpreted as ∞-categories,
are also equivalent to the ∞-category of module spectra over
the Eilenberg-MacLane ring spectrum of the integers.
For more see at _[[stable Dold-Kan correspondence]]_.

## Generalizations and variants {#Variants}

There are various variants, generalizations and enhancements of the
Dold--Kan correspondence.

* The [[monoidal Dold-Kan correspondence]] relates [[simplicial algebra]]s with [[dg-algebra]]s.

* In [[rational homotopy theory]], Quillen proved and used an analogous statement for [[Lie algebra]]s: a Quillen equivalence between the reduced rational [[dg-Lie algebra]]s and reduced rational simplicial Lie algebras:

  D. Quillen, _Rational homotopy theory_ , Ann. Math. 90 (1969), 204--265.

* The statement of the Dold--Kan correspondence generalizes to [[sheaf|sheaves]] with values in the respective categories and this way from [[∞Grpd]] to more general $(\infty,1)$-[[(infinity,1)-topos|topoi]]:

  For $X$ be a [[site]], let $Sh(X, sAb)$ be the category of  _simplicial abelian sheaves_ -- i.e. [[simplicial presheaf|simplicial sheaves]] which take values in simplicial abelian groups -- and let $Sh(X, Ch_+(Ab))$ be the category of [[sheaf|sheaves]] on $S$ with values in non-negatively graded [[chain complexes]] of abelian groups. The normalized chain complex extends objectwise to a functor
  $$
   Sh(X,sAb) \stackrel{\simeq}{\to} Sh(X, Ch_+(Ab))
  $$
  which is an [[equivalence]] of categories. Moreover, both these categories are naturally  [[category with weak equivalences|categories with weak equivalences]]: the weak equivalences in $Sh(X, sAb)$ are the stalkwise [[model structure on simplicial sets|weak equivalences of simplicial sets]]
and the weak equivalences in $Sh(X, Ch_+(Ab))$ are the [[quasi-isomorphisms]]. The normalized chain complex functor preserves these weak equivalences.
This sheaf version of the Dold--Kan correspondence  allows to understand [[abelian sheaf cohomology]] as a special case of [[nonabelian cohomology]].

  See page 9,10 of

  * K. Brown, [[BrownAHT|Abstract Homotopy Theory and Generalized Sheaf Cohomology]]

* There is a version of the Dold--Kan correspondence in the context of $(\infty,1)$-[[(infinity,1)-category|categories]]:

  let $C$ be a [[stable (∞,1)-category]]. Then the $(\infty,1)$-categories of non-negatively graded [[complexes]] in $C$ is equivalent to the $(\infty,1)$-category of [[simplicial objects]] in $C$

  $$
    Fun(N(\mathbb{Z}_{\geq 0}), C)
    \simeq
    Fun(N(\Delta)^{op}, C)
    \,.
  $$

  This is [[infinity-Dold-Kan correspondence]] is [theorem 12.8, p. 50](http://www.math.harvard.edu/~lurie/topoibook/DAGI.pdf) of

  * [[Jacob Lurie]], [[Stable ∞-Categories]]

* There is a version of the Dold--Kan correspondence with [[simplicial sets]] generalized to [[dendroidal sets]]. This is described in

  * [[Javier Gutierrez|Javier Gutiérrez]], [[Andor Lukacs]], [[Ittay Weiss]], _Dold-Kan correspondence for dendroidal abelian groups_ ([arXiv](http://arxiv.org/abs/0909.3995))

* Various functor categories of interest in stable homotopy theory and homological stability are involved in generalized Dold-Kan equivalences. These equivalences have been studied independently by several authors, including [Pirashvili](#Piraashvili)
, [Słomińska](#S?omi?ska), [Helmstutler](#Helmstutler), and [Lack and Street](#LackStreet)

* There is a [[categorification]] of the correspondence, [[categorified Dold-Kan correspondence]] ([Dyckerhoff17](#Dyckerhoff17))

## Applications
  {#Applications}

### Eilenberg-MacLane objects
  {#EilenbergMacLaneObjects}

The Dold-Kan correspondence gives a convenient construction of [[Eilenberg-MacLane object]]s in [[simplicial group]]s.

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

For $A$ an [[abelian group]] write $A[-n]$ for the [[chain complex]] concentrated on $A$ in degree $n$.

The simplicial abelian group $\Gamma (A[-n])$ is an [[Eilenberg-MacLane object]] $K(A,n)$.

And conversely, every such Eilenberg-MacLane object in simplicial abelian groups is related by an [[∞-anafunctor]]-equivalence to a $\Gamma(A[-n])$.

=--

### Looping and delooping
 {#LoopingAndDelooping}

The Dold-Kan correspondence provides a convenient way to describe formation of [[loop space object]]s and [[delooping]] for anything in the image of $\Xi : Ch_\bullet \to sSet$:

by the basic fact that the homotopy groups of $\Xi(V_\bullet)$ are the homology groups of $V_\bullet$, looping and delooping simply corresponds to shifting chain complexes up or down in degree.

But the relation is also strongly coherent: it respects the standard delooping functor $\bar W : sGrp \to sSet$ for [[simplicial group]]s (see there and at [[looping and delooping]]) (notice that restricted to simplicial abelian groups this produces simplicial abelian groups $\bar W : sAbGrp \to sAbGrp$):

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

There is a [[natural isomorphism]]
$$N \bar W G \simeq (N G)[-1]$$
natural in $G \in sAbGrpd$.

=--

This appears for instance as
([Goerss & Jardine, remark III.5.6](#GoerssJardine99)) or around ([Jardine, theorem 4.57](#Jardine)).

### Abelian sheaf cohomology in nonabelian cohomology

Composed with the [[forgetful functor]] $sAb \to sSet$ the Dold-Kan correspondence presents certain [[simplicial set]]s by chain complexes. Since this is entirely functorial, it prolongs to a functor from chain complexes of (pre)[[sheaves]] on any [[site]] $S$, to [[simplicial presheaves]]
$$\Gamma : [S^{op}, Ch_\bullet^+(ab)] \to [S^{op}, sSet]$$

If $[S^{op}, sSet]$ is equipped with the projective [[model structure on simplicial presheaves]] it models the [[(∞,1)-sheaf (∞,1)-topos]] on $S$. The [[derived hom-space]]s compute general [[nonabelian cohomology]].

If the coefficient objects come from sheaves of chain complexes along $\Gamma$, this cohomology restricts to ordinary [[abelian sheaf cohomology]]. See there for more details.

### Computational aspects

One may view the (monoidal) Dold-Kan correspondence as a relation between a well-behaved theory (simplicial/higher methods) that work in any characteristic but is very abstract and mainly suited to the proof of abstract theorems, and a more computational theory (strict structures in dg-modules) that are particularly well adapted to computations. The relation between these two (symmetric monoidal) theories may only be properly used with characteristic 0 coefficients. This remark is very naive and basic, but certainly at the center of computational implementations of abstract homotopical methods.

## Related concepts

* **Dold-Kan correspondence**

* [[monoidal Dold-Kan correspondence]], [[Eilenberg-Zilber/Alexander-Whitney deformation retraction]]

* [[operadic Dold-Kan correspondence]]

* [[infinity-Dold-Kan correspondence]]

* [[stable Dold-Kan correspondence]]

* [[categorified Dold-Kan correspondence]]

## References

The original articles:

* {#Dold58} [[Albrecht Dold]], *Homology of symmetric products and other functors of complexes*, Annals of Mathematics Second Series **68** 1 (Jul., 1958) 54-80  ([jstor:1970043](http://www.jstor.org/stable/1970043))

considers the correspondence for categories of [[modules]], and

* {#DoldPuppe61} [[Albrecht Dold]], [[Dieter Puppe]], _Homologie nicht-additiver Funktoren. Anwendungen_, Annales de l'institut Fourier **11** (1961) 201-312  $[$[eudml:73776](https://eudml.org/doc/73776)$]$

generalizes the result to arbitrary [[abelian category|abelian categories]].

The general expression of the correspondence in terms of [[nerve and realization]] is due to

* {#Kan58} [[Daniel Kan]], _Functors involving c.s.s complexes_, Transactions of the American Mathematical Society, **87** 2 (March 1958) 330-346 ([jstor:1993103](http://www.jstor.org/stable/1993103)).

  > This remarkable article introduces not just the abstract [[nerve and realization]] form of the Dold-Kan correspondence, but introduces the general notion of nerve and realization and in fact the general notion of what is now called *[[Kan extension]]*.

The Dold-Kan correspondence as a [[Quillen equivalence]] between the [[model structure on simplicial abelian groups]] and the projective [[model structure on chain complexes]] is due to

* {#Quillen67} [[Daniel Quillen]], Section II.4 item 5 in: _[[Homotopical Algebra]]_, Lecture Notes in Mathematics 43, Springer 1967([doi:10.1007/BFb0097438](https://doi.org/10.1007/BFb0097438))

A textbook reference for the ordinary Dold-Kan correspondence:

* {#GoerssJardine99} [[Paul Goerss]], [[J. F. Jardine]], Section III.2 of: _[[Simplicial homotopy theory]]_, Progress in Mathematics, Birkhäuser (1999) Modern Birkhäuser Classics (2009) [[doi:10.1007/978-3-0346-0189-4](https://link.springer.com/book/10.1007/978-3-0346-0189-4), [webpage](http://web.archive.org/web/19990208220238/http://www.math.uwo.ca/~jardine/papers/simp-sets/)]

Similar material is in section 4.6 of

* {#Jardine} [[Rick Jardine]], _Generalized etale cohomology theories_ Modern Birkhäuser Classics, (1991)

Interpretation via [[model categories]]:

* {#SchwedeShipley03} [[Stefan Schwede]], [[Brooke Shipley]], _Equivalences of monoidal model categories_, Algebr. Geom. Topol. **3** (2003) 287-334 [[arXiv:math.AT/0209342](http://arxiv.org/abs/math.AT/0209342), [euclid:euclid.agt/1513882376](https://projecteuclid.org/euclid.agt/1513882376)]

* {#Jardine03} [[J. F. Jardine]], Lemma 1.5 in: _Presheaves of chain complexes_, K-theory 30.4 (2003): 365-420 ([pdf](https://pdfs.semanticscholar.org/360c/fd4eb76da956e0c8dd897377f6d6f342e44e.pdf), [[JardinePresheavesOfChainComplexes.pdf:file]])

Further review:

* [[Jacob Lurie]], *The Dold-Kan correspondence*, Section 2.5.6 in *[[Kerodon]]* [[tag 00QQ](https://kerodon.net/tag/00QQ)]

Discussion in the generality of [[abelian categories]] (and further generalization, in [section 1.2.4](#LurieHA) to [[additive categories]] in which [[retractions]] come from [[direct sums]]):

* {#LurieHA} [[Jacob Lurie]], section 1.2.3 of: _[[Higher Algebra]]_

The relation between [[strict ∞-groupoids]] and [[crossed complexes]] is in

* [[Ronnie Brown|R. Brown]] and P.J. Higgins, _The equivalence of $\infty$-groupoids and crossed complexes_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 22 no. 4 (1981), p. 371--386  ([pdf](http://archive.numdam.org/article/CTGDC_1981__22_4_371_0.pdf))

P. 59 of

* R. Brown, _Groupoids and crossed objects in algebraic topology_, Homology, Homotopy and Applications, 1 (1999) 1-78.

gives seven equivalent categories with the equivalences all expressing  nonabelian versions of the Dold--Kan correspondence. One of these is given in

* Ashley, N., _Simplicial T-complexes and crossed complexes: a  nonabelian version of a theorem of Dold and Kan_. University of Wales PhD Thesis, (1978);  Dissertationes Math. (Rozprawy Mat.) 265 (1988) 1--61.

The relation of these with the abelian version is given in

* Brown, R. and Higgins, P. J., _Cubical abelian groups with connections are equivalent to   chain complexes_. Homology Homotopy Appl. 5 (1) (2003) 49--52.

The paper

* [[G. J. Ellis]] and [[R. Steiner]] _Higher-dimensional crossed modules and the homotopy groups  of $(n+1)$}-ads. _J. Pure Appl. Algebra_ **46** (2-3) (1987) 117--136.

should also be seen as of Dold-Kan type. The homotopical applications  considerably generalise results on the [[Blakers-Massey theorem]].

See also

* Brown, R. _Modelling and computing homotopy types: I_, Indagationes Math: Special Issue in honor of L.E.J. Brouwer, (2017)
 ([pdf](http://groupoids.org.uk/pdffiles/brouwer-honor.pdf))

The discussion of Dold--Kan in the general context of [[semi-abelian category|semi-abelian categories]] is in

* [[Dominique Bourn]], _Moore normalisation and Dold--Kan theorem for semi-Abelian categories_, in
Categories in algebra, geometry and mathematical physics , volume 431 of Contemp. Math.,
105--124, Amer. Math. Soc., Providence, RI. (2007)

The classical Dold-Kan theorem occurs as a special case among others from [[combinatorics]] and [[representation theory]], and in particular from homological stability, in:

* {#Slomiiska} [[Jolanta Słomińska]], _Dold-Kan type theorems and Morita equivalences of functor categories_, Journal of Algebra **274** 1 (2004) 118-137 $[$[doi:10.1016/j.jalgebra.2003.10.025](https://doi.org/10.1016/j.jalgebra.2003.10.025)$]$

A similar framework was independently rediscovered in:

* [[Stephen Lack]], [[Ross Street]], _Combinatorial Categorical Equivalences_, arxiv:1402.7151 (2014).  ([link](http://arxiv.org/abs/1402.7151))

A stable homotopical version of these general correspondences was developed in:

* {#Helmstutler} Randall Helmstutler, _Model category extensions of the Pirashvili-Słomińska theorems_ $[$[arXiv:0806.1540](https://arxiv.org/abs/0806.1540))$]$

Among the correspondences "of Dold-Kan type" included in this theory are an equivalence between [[FI-modules]] and linear [[combinatorial species]]:

* [[Thomas Church]], [[Jordan S. Ellenberg]], [[Benson Farb]], *FI-modules and stability for representations of symmetric groups*, Duke Math. J. **164** 9 (2015) 1833-1910 [[arXiv:1204.4533](https://arxiv.org/abs/1204.4533), [doi:10.1215/00127094-3120274](https://doi.org/10.1215/00127094-3120274)]

A Dold-Kan theorem for $\Gamma$-groups:

* {#Pirashvili} Teimuraz Pirashvili. _Dold-Kan type theorem for ∞-groups_, Mathematische Annalen 318.2 (2000): 277-298. ([link](https://doi.org/10.1007/s002080000120))

An equivalence between representations of the category of finite-dimensional $\mathbb{F}_q$-vector spaces and representations of its underlying groupoid:

* Nicholas Kuhn, _Generic representation theory of finite fields in nondescribing characteristic_, arxiv:1405.0318 ([link](https://arxiv.org/abs/1405.0318))

A [[categorification]] to a [[categorified Dold-Kan correspondence]] is discussed here:

* {#Dyckerhoff17} [[Tobias Dyckerhoff]], _A categorified Dold-Kan correspondence_ ([arXiv:1710.08356](https://arxiv.org/abs/1710.08356))

[[!redirects Dold--Kan correspondence]]
[[!redirects Dold–Kan correspondence]]
[[!redirects Dold-Kan theorem]]
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Dold-Kan theorem > history

< [[Dold-Kan theorem]]
[[!redirects Dold-Kan theorem -- history]]


Dold-Thom theorem

+-- {: .un_defn}
###### Definition
**(Symmetric Product)**

There is a natural forgetful functor from topological commutative monoids to [[pointed space|pointed topological spaces]], where the basepoint of a commutative monoid is taken to be its unit element. The induced monad is the _infinite symmetric product_ construction:

$$
  Sym(X,e) = \mathrm{colim}_N Sym^N(X) \qquad
  Sym^N(X) = (X \times \dots \times X)/\Sigma_N
$$

Here the colimit is taken over the maps $Sym^N(X) \to Sym^{N+1}(X)$, $(x_1,\dots, x_n) \mapsto (x_1,\dots, x_n, e)$ where $e$ is the basepoint.

Thus $Sym(X,e)$ is the free topological commutative monoid on $X$ with the basepoint $e$ acting as the unit.

###### Theorem
**(Dold-Thom)**

The [[ordinary homology]],  hence the standard $n$-th [[homology group]] of a [[pointed connected CW-complex]] $(X,e)$ is [[isomorphism|isomorphic]] to the $n$-th [[homotopy group]] of the infinite symmetric product of $(X,e)$

$$
  H_i(X) = \pi_i (\mathrm{colim}_N\, \mathrm{Sym}^N X)
  \,.
$$

=--

The [[Mayer-Vietoris sequence]] for homology is a consequence of applying $\pi_*(-)$ to the [[homotopy pullback]] square resulting from the application of $\mathrm{Sym}^\infty$ to the [[homotopy pushout]] square formed by the inclusions of the intersection, $A \cap B$, of two subspaces $A$ and $B$ of a space $X$ into $A$ and $B$.

## References

The original article is

* [[Albrecht Dold|A. Dold]], [[René Thom|R. Thom]], _Quasifaserungen und unendliche symmetrische Produkte_ , Ann. Math. (2) 69 (1959), 239--281. ([pdf](http://www.maths.ed.ac.uk/~aar/papers/doldthom.pdf))

For a textbook account, see [Hatcher](https://pi.math.cornell.edu/~hatcher/AT/AT.pdf#page=484), Sec. 4.K p. 475.


domain

## Disambiguation

There are multiple notions of domain of mathematics. These include:

* in [[ring theory]]:

  [[domain (ring theory)]] and [[integral domain]]

* in [[domain theory]]:

  [[Scott domains]]: [[pointed directed-complete partial orders]]

* in [[category theory]]:

  [[source]] of a [[morphism]]

* [[domain of discourse]]

* The [[subset]] $A\subseteq X$ for which there exists a [[relation]] $R\colon X \to Y$.

* in [[differential geometry]] and especially in [[complex analysis]]:

  an [[open subset]] of a [[manifold]] (cf. *[[Stein domain]]*)

category: disambiguation

[[!redirects domain]]
[[!redirects domains]]


domain (ring theory)

> This entry is about domains in [[ring theory]]. For other uses, see at _[[domain]]_.

***

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## Definition

Some authors call a domain an [[integral domain]]. However, we maintain a distinction between domains and integral domains, and reserve "integral domain" for the commutative domains.

### By the cancellative property

A unital [[ring]] $R$ is a __domain__ if it is nontrivial and the multiplicative submonoid $R \backslash \{0\}$ is a [[cancellative monoid]] (i.e., $1 \neq 0$ and left and right multiplication by $c$ is injective if $c \neq 0$, which may be combined as left and right multiplication by $c$ is injective if and only if $c \neq 0$)

### By zero divisors

A unital [[ring]] $R$ is an __domain__ if it is nontrivial and has no non-zero [[zero divisors]] (i.e., $1 \ne 0$ and $a b = 0$ implies $a = 0$ or $b = 0$).

In this definition, the [[trivial ring]] is [[too simple to be simple|too simple]] to be an integral domain. You can see this by phrasing this definition without [[bias]] as: any product of (finitely many) nonzero elements of $R$ (which includes the empty product $1$) must be nonzero.

## Properties

A domain $R$ is an __[[Ore domain]]__ if the set of all nonzero elements is an [[Ore set]] in $R$. In that case the Ore localized ring is called the _[[Ore quotient ring]]_ of $R$.

## Examples

For example, the ring of [[integers]], any [[skewfield]], the ring of global sections of the structure sheaf of any [[integral scheme]], an [[Ore extension]] of any other domain.

## Generalizations

In principle, one could just as easily consider a [[rig]] or [[semiring]] $R$.  In that case, however, only the definition involving the cancellative property extends to rigs and semirings. Furthermore, we should add the additional requirement that addition in $R$ is cancellable (that is, addition by any element is injective), to make the analogue of the previous paragraph correct. These could be called a *domain rig* or *domain semiring*. One could also relax the requirement that the domain be associative or unital, this could be called a *domain $\mathbb{Z}$-algebra*, in the context where $R$-algebras are usually not assumed to be [[associative unital algebras]].

## See also

* [[integral domain]]

## External links

* Wikipedia, *Domain (ring theory)*

[[!redirects domain ring]]
[[!redirects domain rings]]

[[!redirects domain rig]]
[[!redirects domain rigs]]
[[!redirects domain semiring]]
[[!redirects domain semirings]]

[[!redirects domain Z-algebra]]
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domain of holomorphy

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## Related concepts

* [[Stein manifold]]

## References

* Wikipedia, _[Domain of holomorphy](http://en.wikipedia.org/wiki/Domain_of_holomorphy)_

[[!redirects domains of holomorphy]]


domain opfibration

# Contents
* toc
{: toc}

## Idea

The notion of _domain opfibration_ is [[duality|dual]] to that of [[codomain fibration]]. See there for more details.

## Definition

Let $C$ be a [[category]] and $Arr(C)= C^2$ the corresponding [[arrow category]]: the objects in $Arr(C)$ are morphisms in $C$ and the morphisms $(f:x\to x')\to (g:y\to y')$ in $Arr(C)$ are the [[commutative squares]] of the form

$$\array{
x &\stackrel{u}\to& y\\
\downarrow\mathrlap{f} &&\downarrow\mathrlap{g}\\
x' &\stackrel{v}\to& y'
}$$

with the obvious composition.

There is a [[functor]] $dom:Arr(C)\to C$ given on objects by the domain (= source) map, and on morphisms it gives the upper arrow of the commutative square.  If $C$ has [[pushouts]], then this functor is in fact an [[Grothendieck opfibration|opfibered (cofibered) category]] in the sense of Grothendieck, whose pushforward functor $u_*$ amounts to the usual pushout of $f$ along $u$ in $C$.  The fiber over an object $c$ in $C$ is the [[undercategory]] $c\downarrow C$. This opfibered category is called the **domain opfibration** over $C$ (some say the *basic* opfibration). This notion is dual to the notion of [[codomain fibration]].

## Remarks on notation {#notation}

Although the pushforward functor in an opfibration is usually written $u_!$, in the case of the domain opfibration we usually write it as $u_*$ instead, following the notation of [[algebraic geometry]].  Each such functor also has a [[right adjoint]], given by precomposition (just as in the [[codomain fibration]] the pullback functors have left adjoints given by postcomposition).  Thus, the domain opfibration is in fact a [[bifibration]], though traditionally its opfibered aspect is emphasised (and it even motivates the notion of cocartesianess for [[categories over categories]]).  And while the right adjoints in a bifibration are usually written as $u^*$, for the domain opfibration we write them as $u^!$, again to conform to usage in algebraic geometry, where the standard string of adjoints is $u_! \dashv u^* \dashv u_* \dashv u^!$.

[[!redirects domain cofibration]]


domain semantics

1. aaa
1. aaaa

1.

1.


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## Idea

A _[[domain specific programming language]]_ is one designed for  a specialized kind ("domain") of applications. A _domain specific embedded programming language_ (DSEL) is a domain specific language realized "inside" a general-purpose high level ([[type theory|typed]]) [[programming language]].

## Relation to synthetic mathematics
 {#RelationToSyntheticMathematics}

There is at least some similarity between DSELs and [[synthetic mathematics]], see for instance ([Hudak 98, section 3.2](#Hudak98)). In ([Hudak 98, figure 2](#Hudak98)) this shows aspects of a real-world DSL for "geometric region analysis" embedded in [[Haskell]] which under the [[relation between type theory and category theory]]/[[computational trinitarianism]] one immediately recognizes as a fragment of [[synthetic geometry]].

## Relation to monadic effects
 {#RelationToMonadicEffects}

There is a close relation between embedding a domain-specific language and declaring [do-notation](monad+in+computer+science#DoNotation) for [[monadic effects]], see [Benton, Hughes & Moggi 2002, §5.3](#BentonHughesMoggi02) who write:

> Every time a functional programmer designs a combinator library, then, we might as well say that he or she designs a domain specific programming language [...]. This is a useful perspective, since it encourages programmers to produce a modular design, with a clean separation between the semantics of the DSL and the program that uses it, rather than mixing combinators and 'raw' semantics willy-nilly. And since monads appear so often in programming language semantics, it is hardly surprising that they appear often
in combinator libraries also!

## Examples
 {#Examples}

### Quantum programming
 {#QuantumProgramming}

Many existing [[quantum programming languages]] are actually domain-specific languages for the description of [[quantum circuits]], and as such many are embedded in ambient [[type theories]].

For instance:

* *[[Quipper]]* is embedded into into [[Haskell]] [[Green, Lumsdaine, Ross, Selinger & Valiron (2013)](Quipper#GLRSV13)];

* *[[QWIRE]]* has been embedded into [[Coq]] ([Rand, Paykin & Zdancewic (2018)](QWIRE#RandPaykinZdancewic18)) and [[homotopy type theory]] [[Paykin & Zdancewic (2019)](QWIRE#PaykinZdancewic19)].

* [[CoqQ]] is another quantum programming language embedded in [[Coq]] [Ying et al. (2022)]

See also [Rennela & Staton (2020)](#RennelaStaton17) for more general discussion.

## References

* [[Paul Hudak]], *Building Domain-Specific Embedded Languages*, ACM Computing Surveys **28** 4e (1996) 196–es [[doi:10.1145/242224.242477](https://doi.org/10.1145/242224.242477), [web](https://dl.acm.org/doi/fullHtml/10.1145/242224.242477)]

* {#Hudak98} [[Paul Hudak]], _Modular Domain Specific Languages and Tools_, in: _Proceedings of Fifth International Conference on Software Reuse_, IEEE Computer Society Press  (1998)  [[pdf](http://staff.um.edu.mt/afra1/seminar/ModularDSL.pdf), [doi:10.5555/551789.853532](https://dl.acm.org/doi/10.5555/551789.853532)]

With emphasis on the relation to [[monads in computer science]]:

* {#BentonHughesMoggi02} [[Nick Benton]], [[John Hughes]], [[Eugenio Moggi]], §5.3 in: *Monads and Effects*, in: *Applied Semantics*, Lecture Notes in Computer Science **2395**, Springer (2002) 42-122 [[doi:10.1007/3-540-45699-6_2](https://doi.org/10.1007/3-540-45699-6_2)]

Discussion for embedding in [[Haskell]]:

* HaskellWiki, _[Embedded domain specific language](http://www.haskell.org/haskellwiki/Embedded_domain_specific_language)_

A list of literature is at

* Haskell Wiki, _[Research papers/Domain specific languages](http://www.haskell.org/haskellwiki/Research_papers/Domain_specific_languages)_

Discussion specifically of [[quantum programming languages]] for [[quantum circuits]] as domain specific embedded languages:

* {#RennelaStaton17} [[Mathys Rennela]], [[Sam Staton]], *Classical Control, Quantum Circuits and Linear Logic in Enriched Category Theory*, Logical Methods in Computer Science **16** 1 (2020) [[arXiv:1711.05159](https://arxiv.org/abs/1711.05159), doi:10.23638/LMCS-16(1:30)2020]

[[!redirects domain specific embedded programming languages]]

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domain specific programming language

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## Idea

A _[[programming language]]_ is called _domain specific_ if it is designed for a specific kind (domain) of applications, in contrast to general purpose programming  languages.

If a domain specific language is realized inside a general-purpose language, one speaks of _[[domain specific embedded programming language]]_.

## Examples
 {#Examples}

* Many existing [[quantum programming languages]] are actually domain-specific languages for the description of [[quantum circuits]], and as such many are [[domain specific embedded programming language|embedded]] (see there for more).

## References

### General

* Wikipedia, _[Domain specific language](http://en.wikipedia.org/wiki/Domain-specific_language)_

* Haskell Wiki, _[Research papers/Domain specific languages](http://www.haskell.org/haskellwiki/Research_papers/Domain_specific_languages)_

### Examples

For [[differential equations]]:

* {#ALORW12} Martin S. Alnaes, Anders Logg, Kristian B. Oelgaard, Marie E. Rognes, Garth N. Wells, _Unified Form Language: A domain-specific language for weak formulations of partial differential equations_ ([arXiv:1211.4047](http://arxiv.org/abs/1211.4047))

## References

* [[Paul Hudak]], *Building Domain-Specific Embedded Languages*, ACM Computing Surveys **28** 4e (1996) 196–es [[doi:10.1145/242224.242477](https://doi.org/10.1145/242224.242477), [web](https://dl.acm.org/doi/fullHtml/10.1145/242224.242477)]

* [[Paul Hudak]], *Domain Specific Languages*, Chapter 3 (pp. 39–60) in:  Peter H. Salas (ed), *Handbook of Programming Languages*, Vol. III: *Little Languages and Tools*, MacMillan  (1998) [[pdf](http://cs448h.stanford.edu/DSEL-Little.pdf), [[Hudak-DSLs.pdf:file]]]

* [[Paul Hudak]], _Modular Domain Specific Languages and Tools_, in: _Proceedings of Fifth International Conference on Software Reuse_, IEEE Computer Society Press  (1998)  [[pdf](https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=685738), [doi:10.5555/551789.853532](https://dl.acm.org/doi/10.5555/551789.853532)]

[[!redirects domain specific programming languages]]

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[[!redirects DSL]]


domain theory

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## Idea
 {#Idea}

*Domain theory* has its origin in

1. topological algebra concerned with Lawson semilattices,

1. [[computer science]], concerned with the problem of finding a viable [[denotational semantics]] for certain theories of [[computability]] (such as the untyped [[lambda calculus]]): this resists straightforward interpretations in terms of plain [[functions]] between sets but does have interpretation in terms of [[monotone functions]] between [[partially ordered sets]] (certain [[lattices]]) [[Scott (1970)](#Scott70), [Scott & Strachey (1971)](#ScottStrachey71)].

The theory has since grown into an area which weaves together diverse strands in [[logic]], [[computability]], [[lattice]] theory, [[general topology]], and [[category theory]].

Domain theory can be said to have come into existence with the insight from [Scott (1970)](#Scott70) of interpreting untyped [[lambda calculus]] in terms of [[monotone functions]] between [[continuous lattices]]. In rough terms, the problem can be set out as follows:

* [[lambda calculus|Lambda calculus]] is a [[syntax]] of [[functions]] and [[functional]] application, whose basic constituents are [[types]], [[terms]] of types, and type and term [[type formation|formation schemes]] with which one can speak of [[product types]] $A \times B$ and [[function types]] $A \Rightarrow B$.

* One problem is to construct a *meaning* or *[[semantics]]* for such [[syntax]] in terms of "actual" [[sets]] and [[functions]]. The [[categorical semantics]] of [[lambda calculus]] is by [[cartesian closed categories]], in which [[types]] are interpreted as [[objects]] $A$ and [[terms]] are interpreted as ([[generalized element|generalized]]) [[elements]] or "functions" $X \to A$.

* In so-called untyped lambda calculus (whose [[syntax]] is closely connected with the theory of [[computability]] and [[recursive functions]]), all [[terms]] may be regarded as being of the same [[type]] $D$ (which therefore need not be mentioned, hence "untyped"), so that, intuitively speaking, elements of $D$ and functions on $D$ are treated on one and the same footing.

* The problem Scott solved is to give faithful [[categorical semantics]] for untyped lambda calculus; where the basic problem is to construct a [[cartesian closed category]] with essentially just one [[object]] $D$ (really two objects: $D$ and a [[terminal object]] $1$), so that $D$ is closed under formation of [[products]] and [[internal homs]]: $D \cong D \times D$ and $D \cong D \Rightarrow D$. Notice that in the [[category of sets]], the only solution is to take $D \cong 1$, so that all terms are then equal ("algorithmic inconsistency"). This is not a faithful modeling of untyped lambda calculus, which has provably unequal terms.

In 1969, Dana Scott solved this problem topologically: the terms were interpreted as continuous functions on a suitable space $D$ isomorphic to its own function space. This $D$ is called a _domain_. Decades later, we now know many techniques for constructing such domains as suitable objects in cartesian closed categories, but Scott's basic insight, that computability could be interpreted as continuity,
continues to exert a decisive influence today.

## Terminology

[[Martín Escardó]] writes on the [nForum](https://nforum.ncatlab.org/discussion/10150/dcpo/?Focus=90736#Comment_90736):

> The usual practice of [[domain theory]] papers and talks is to start by defining “domain” for the purposes of the paper or talk. It is a reusable word. Usually only compound words using the word “domain” have a fixed meaning, such as [[Scott domain]] (bounded complete [[algebraic dcpo]]), [[continuous Scott domain]] (bounded complete [[continuous poset|continuous]] [[dcpo]]), [[FS domain]], [[SFP domain]], [[L-domain]]. I wouldn’t use the plain word “domain” in a paper without first explicitly defining it, as it has been used with so many different meanings in the domain theory literature. Also the terminological conventions vary a bit depending on whether domain theory is used for the purposes of computation (e.g., [[denotational semantics|programming language semantics]], and [[computability|theory of computability]]) or [[topology]] and [[algebra]]. For example, in programming language semantics papers one often encounters [[posets]] that have sups of ascending sequences, rather than [[directed sets]], and a least element, because all the authors are interested in is the existence of (least) fixed point of continuous endo-functions, and these assumptions are enough for this purpose. Such [[posets]] are often called domains in such papers. But for applications of [[domain theory]] to [[topology]], [[dcpo|directed completeness]] is what one needs in general, and often we have more (even all sups — for example, a topological space is [[exponentiable object|exponentiable]] if its [[lattice]] of open sets is a [[continuous poset|continuous]] [[dcpo]] — but this [[dcpo]] has all sups and moreover is a [[distributive lattice]], and the continuous distributive lattices are precisely the topologies of [[exponentiable object|exponentiable spaces]], up to isomorphism). Because these applications and communities are so diverse, it is natural to see a divergence of terminology, even if many of the techniques are the same.

## Related concepts

* [[topological domain theory]]

* [[synthetic domain theory]]

* [[synthetic guarded domain theory]]

* [[denotational semantics]]

## References

### In computation theory

Origin of domain theory in [[denotational semantics]] for [[programming languages]]:

* {#Scott70} [[Dana S. Scott]], *Outline of a mathematical theory of computation*, in: Proceedings of the *Fourth Annual Princeton Conference on Information Sciences and Systems* (1970) 169–176. [[pdf](https://ropas.snu.ac.kr/~kwang/520/readings/sco70.pdf), [[Scott-TheoryOfComputation.pdf:file]]]

* {#ScottStrachey71} [[Dana S. Scott]], [[Christopher Strachey]], *Toward a Mathematical Semantics for Computer Languages*, Oxford University Computing Laboratory, Technical Monograph PRG-6 (1971) [[pdf](https://www.cs.ox.ac.uk/files/3228/PRG06.pdf), [[ScottStrachey-MathematicalSemantics.pdf:file]]]

* {#Scott76} [[Dana Scott]], *Data types as lattices*. SIAM Journal of Computing **5** 3 (1976) 522--587 [[doi:10.1137/0205037](https://doi.org/10.1137/0205037), [pdf](https://www.cs.ox.ac.uk/files/3287/PRG05.pdf)]

Discussion [[internalization|internal]] to [[cartesian closed categories]]:

* [[Michael Barr]], *Fixed points in cartesian closed categories*, Theoretical Comp. Sci. **70** (1990) 65–72 [doi:10.1016/0304-3975(90)90152-8, [pdf](https://www.math.mcgill.ca/barr/papers/dom.pdf), [[Barr-FixedPointsInCCCs.pdf:file]]]

Introductions and lecture notes:

* [[Robert D. Tennent]], *The denotational semantics of programming languages*, Communications of the ACM **19** 8 (1976) 437–453 [[doi:10.1145/360303.360308](https://doi.org/10.1145/360303.360308), [pdf](https://dl.acm.org/doi/pdf/10.1145/360303.360308)]

* [[Andrew M. Pitts]], *Lecture Notes on Denotational Semantics* (2012) [[pdf](https://www.cl.cam.ac.uk/teaching/1112/DenotSem/dens-notes-bw.pdf), [[Pitts-DenotationalSemantics.pdf:file]]]

Textbook accounts:

* [[Glynn Winskel]], §8 of: *The Formal Semantics of Programming Languages*, MIT Press (1993) [[ISBN:9780262731034](https://mitpress.mit.edu/9780262731034/the-formal-semantics-of-programming-languages/), [pdf](https://www.cin.ufpe.br/~if721/intranet/TheFormalSemanticsofProgrammingLanguages.pdf)]

* [[David A. Schmidt]], *Denotational Semantics -- A methodology for language development*, Allyn and Bacon (1986)  [[pdf](https://people.cs.ksu.edu/~schmidt/text/DenSem-full-book.pdf), [webpage](https://people.cs.ksu.edu/~schmidt/text/densem.html)]

* {#GierzHofmannKeimelLawsonLisloveScott03} G. Gierz, [[Karl H. Hofmann]], K. Keimel, J. D. Lawson, [[Michael W. Mislove]], [[Dana S. Scott]], *Continuous Lattices and Domains*, in: *Encyclopedia of Mathematics and its Applications* **93**, Cambridge University Press (2003)  [[doi:10.1017/CBO9780511542725](https://doi.org/10.1017/CBO9780511542725)]

* [[Thomas Streicher]], *Domain-Theoretic Foundations of Functional Programming*, World Scientific (2006) [[pdf](https://doc.lagout.org/programmation/Functional%20Programming/Domain-Theoretic%20Foundations%20of%20Functional%20Programming%20%5BStreicher%202006-12-04%5D.pdf), [doi:10.1142/6284](https://www.worldscientific.com/worldscibooks/10.1142/6284)]

See also:

* Wikipedia, *[Domain theory](http://en.wikipedia.org/wiki/Domain_theory)*

Discussion in [[homotopy type theory]]/[[univalent foundations]]:

* {#deJongEscardo21} [[Tom de Jong]], [[Martín Hötzel Escardó]], _Domain Theory in Constructive and Predicative Univalent Foundations_, in: _29th EACSL Annual Conference on Computer Science Logic_, [CSL 2021](https://csl2021.fmf.uni-lj.si/), LIPIcs proceedings 183, 2021 ([doi:10.4230/LIPIcs.CSL.2021.28](https://doi.org/10.4230/LIPIcs.CSL.2021.28), [arXiv:2008.01422](https://arxiv.org/abs/2008.01422))

Review:

* [[Samson Abramsky]], [[Achim Jung]], *Domain Theory*, in: *[[Handbook of Logic in Computer Science]]* **3**, Oxford University Press (1995) [[ISBN:9780198537625](https://global.oup.com/academic/product/handbook-of-logic-in-computer-science-9780198537625?cc=de&lang=en&), [pdf](https://www.cs.bham.ac.uk/~axj/pub/papers/handy1.pdf)]

* Gordon Plotkin, *Pisa Notes on Domains* [[ps](http://homepages.inf.ed.ac.uk/gdp/publications/Domains_a4.ps)]

### Relation to causets
 {#ReferencesRelationToCausets}

Possible relation to [[spacetime]] [[causality]] (cf. *[[causets]]*):

* [Domain theory and general relativity](http://www.cs.mcgill.ca/~prakash/Pubs/dom_gr_review.pdf), Keye Martin and [[Prakash Panangaden]]

>**Summary**. We discuss the current state of investigations into the domain theoretic structure of spacetime, including recent developments which explain the connection between measurement, the Newtonian concept of time and the Lorentz distance.

* [A Domain of Spacetime Intervals in General Relativity](http://www.cs.mcgill.ca/~prakash/Pubs/cmp.pdf), Keye Martin and [[Prakash Panangaden]]

> **Abstract**: We prove that a globally hyperbolic [[spacetime]] with its causality relation is a bicontinuous [[poset]] whose interval topology is the manifold topology. From this one can show that from only a countable dense set of events and the causality relation, it is possible to reconstruct a globally hyperbolic spacetime in a purely order theoretic manner. The ultimate reason for this is that globally hyperbolic spacetimes belong to a category that is equivalent to a special category of domains called interval domains. We obtain a mathematical setting in which one can study causality independently of geometry and differentiable structure, and which also suggests that spacetime emerges from something discrete.

### Categorical notions of domain
 {#ReferencesCategoricalNotionsOfDomain}

In [[category theory|category theoretic]] generalizations of  the notion, the informal idea is to replace [[partial orders]] with more general [[categories]], so that there is more space for intensionality. An early reference, which proposes  the notion of *[[Scott-complete categories]]*:

* [[Jiří Adámek]], *A categorical generalization of Scott domains*, Mathematical Structures in Computer Science **7** 5  (1997) 419 - 443 [[doi:10.1017/S0960129597002351](https://doi.org/10.1017/S0960129597002351)]

The following reference proposes using [[presheaf categories]] as a generalization of prime [[algebraic lattices]]. This also connects to work on [[species#variants|generalized species]]:

* [[Mikkel Nygaard]], [[Glynn Winskel]], *Domain theory for concurrency*, Theoretical Computer Science
**316** 1–3 (2004) 153-190 [[doi:10.1016/j.tcs.2004.01.029](https://doi.org/10.1016/j.tcs.2004.01.029), [pdf](https://www.cl.cam.ac.uk/~gw104/DomThy.pdf)]

category: computer science


domain wall

#Contents#
* table of contents
{:toc}

## Idea

In general, a _domain wall_ is a [[QFT with defects|defect]] of [[codimension]] 1.

More specifically, in a [[gauge theory]] with a degenerate [[vacuum]] (such as when a [[Higgs mechanism]] applies), the [[moduli space]] of [[vacua]] is the [[quotient]] $G/H$ (the [[coset]]) of the [[gauge group]] $G$ by the [[stabilizer subgroup]] $H \hookrightarrow G$ of any of these vacua ([[spontaneous symmetry breaking]]).

This means that gauge equivalence classes of vacuum configurations on a [[spacetime]] $X$ are given by [[homotopy]] classes of maps $X \to \Pi(G/H)$ (where the notation on the right denotes the underlying [[homotopy type]] of the [[coset]] space, $\Pi$ is the [[shape modality]]).

If [[spacetime]] is locally to be taken of the form $\mathbb{R} \times (\mathbb{R}^3 - D^1 \times \mathbb{R}^2)$, hence with a 1-dimensional ("wall"-like) piece taken out, them homotopy classes of maps $X \to \Pi(G/H)$ are classified by the 0-th [[homotopy group]] $\pi_0(G/H)$. For a given nontrivial element here the corresponding [[vacuum]] is said to contain a **domain wall defect**.

For more see at _[[QFT with defects]]_ the section _[Topological defects from spontaneously broken symmetry](http://ncatlab.org/nlab/show/QFT+with+defects#DefectsFromBrokenSymmetry)_.

## Related concepts

* [[cosmic string]], [[monopole]]

[[!include field theory with boundaries and defects - table]]

* [[wall crossing]]

* [[defect brane]]

## References

* {#VilenkinShellard94} [[Alexander Vilenkin]], E. P. S. Shellard, *Cosmic strings and other topological defects*, Cambridge University Press (1994) [[ISBN:9780521654760](https://www.cambridge.org/ae/universitypress/subjects/physics/theoretical-physics-and-mathematical-physics/cosmic-strings-and-other-topological-defects?format=PB&isbn=9780521654760), [spire:1384873](https://inspirehep.net/literature/1384873)]

See also:

* Wikipedia, _[Domain wall](http://en.wikipedia.org/wiki/Domain_wall)_

[[!redirects domain walls]]


Domenico Fiorenza

\begin{imagefromfile}
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    "width": 280,
    "unit": "px",
    "margin": {
        "top": -50,
        "right": 10,
        "bottom": -10,
        "left": 20
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* [website](http://www.mat.uniroma1.it/~fiorenza/)

* [contact](#contact)

* [my position](#myposition)

* [my research](#myresearch)

* [exchange of thoughts](#exchangeofthoughts)

* [beta versions](#betaversions)

## contact {#contact}

The best way to contact me is by sending me an email at <fiorenza@mat.uniroma1.it>

## my position {#myposition}

I have my PhD in [[mathematics|math]] from the University of Pisa. Have been a postdoc in mathematics for three years (in Roma Tor Vergata and Sapienza). Since 2005 I am assistant professor at the Sapienza - Università di Roma

## my research {#myresearch}

I'm principally interested in categorical constructions arising from theoretical [[physics]]. See _[[Hypothesis H]]_.

Something like a personal research wiki with more information is beginning to develop at

* [[domenicofiorenza:HomePage|my personal web within the nLab]].

## exchange of thoughts {#exchangeofthoughts}

I enjoy exchanging thoughts on topics of interest for my research and always felt it would be a pity not to use the web for that purpose. To satisfy that need I use to work on

*  the wiki [The nLab](http://ncatlab.org/nlab/show/HomePage) (this wiki here).

* [[domenicofiorenza:HomePage|my personal web within the nLab]]

## beta versions {#betaversions}

Here is where I develop revisions of nLab entries which (from my point of view) deserve consistent revision. By this I do not mean these entries  are wrong (in any sense you can give to this word); what I mean is that there are a few entries which have been edited severla times and have suffered an excess of stratification (add a line today, add another tomorrow...). So it could be not a bad idea to clean up thema a bit. As revision is complete I'll post them on the nLab. Everything you find here is work in progress: please, edit it!

Currently revising:

* [[domenicofiorenza:twisted cohomology|twisted cohomology]]

* [[domenicofiorenza:over quasi-category|over quasi-category]]

* [[domenicofiorenza:limit|limit]]

## Selected writings

On [[t-structures]] and [[reflective factorization systems]]:

* [[Domenico Fiorenza]], [[Fosco Loregian]], *$t$-Structures are normal torsion theories*, Appl Categor Struct **24** (2016) 181–208 [[arxiv:1408.7003](http://arxiv.org/abs/1408.7003), [doi:10.1007/s10485-015-9393-z](https://doi.org/10.1007/s10485-015-9393-z)]

* [[Domenico Fiorenza]], [[Fosco Loregian]], [[Giovanni Marchetti]], *Hearts and towers in stable $\infty$-categories*,  J. Homotopy Relat. Struct. **14**  (2019) 993–1042 [[arXiv:1501.04658](https://arxiv.org/abs/1501.04658), [doi:10.1007/s40062-019-00237-0](https://doi.org/10.1007/s40062-019-00237-0)]

category: people

[[!redirects domenico_fiorenza]]
[[!redirects D. Fiorenza]]


Domenico Giulini

* [webpage](https://www.itp.uni-hannover.de/~giulini/)

## Selected writings

On [[geometrodynamics]]:

* [[Domenico Giulini]], *Matter from Space*, in: *Beyond Einstein* Einstein Studies **14** Birkhäuser (2018) [[doi:10.1007/978-1-4939-7708-6_12](https://doi.org/10.1007/978-1-4939-7708-6_12), [arXiv:0910.2574](https://arxiv.org/abs/0910.2574)]

## Related entries

* [[Wheeler superspace]]

category: people


Domenico Seminara

* [webpage](http://florencestringtheorygroup.weebly.com/domenico-seminara.html)

## related $n$Lab entries

* [[11-dimensional supergravity]]

category: people


dominance

# Dominance

* table of contents
{: toc}

## Idea

In [[synthetic topology]] done as a branch of [[constructive mathematics]], a *dominance* is a set $\Sigma$ that functions as an analogue of the [[Sierpinski space]].  In particular, it allows us to define synthetically a notion of [[open subset]]: $\Sigma$ is a [[subset]] of the set of [[truth values]] $\Omega$, and a subset of a set $A$ is called "open" if its classifying map $A\to \Omega$ lands in $\Sigma$.

The name *dominance* is meant to evoke that the set is used to define the domains of a class of [[partial functions]]. I.e., in synthetic topology the partial functions whose domain is an open set and in [[synthetic computability theory]] the domains of [[computable functions|partial computable functions]].

## Definition

Let $\Omega$ be the set of [[truth values]].  A **dominance** is a subset $\Sigma\subseteq \Omega$ such that

1. $\top \in \Sigma$, and
1. If $U\in\Sigma$ and $P\in\Omega$ and $U\Rightarrow (P\in \Sigma)$, then $U\wedge P \in \Sigma$.

The second condition implies that $\Sigma$ is closed under binary [[meets]] $\wedge$, and hence is a sub-meet-[[semilattice]] of $\Omega$.  In [[type theory|type-theoretic]] language, the second condition says that $\Sigma$ is closed under [[dependent sums]].

The elements of $\Sigma$ are called **open** truth values.

### In homotopy type theory

In [[homotopy type theory]], every [[univalent]] [[type of propositions]] is a [[subtype]] of the [[type of all propositions]] $\Omega$. A **dominance** is a univalent type of propositions $(\Sigma, \mathrm{El}_\Sigma)$ with an element $\top:\Sigma$ such that

* $\mathrm{El}_\Sigma(\top)$ is a [[contractible type]]
* given a [[mere proposition]] $P$, for all elements $U:\Sigma$ with a function from $\mathrm{El}_\Sigma(U)$ to the type that $P$ is essentially $\Sigma$-small, the [[product type]] $\mathrm{El}_\Sigma(U) \times P$ is essentially $\Sigma$-small.
$$\frac{\Gamma \vdash P \; \mathrm{type} \quad \Gamma \vdash p:\mathrm{isProp}(P)}{\Gamma, U:\Sigma \vdash q:\left(\mathrm{El}_\Sigma(U) \to \sum_{B:\Sigma} \mathrm{El}_\Sigma(B) \simeq P\right) \to \left(\sum_{C:\Sigma} \mathrm{El}_\Sigma(C) \simeq \mathrm{El}_\Sigma(U) \times P\right)}$$

## Open subsets

We define a subset $U\subseteq A$ of an arbitrary set $A$ to be **open** if for each $x\in A$, the proposition "$x\in U$" is an open truth value.  The second condition above is equivalent to saying that if $U\subseteq A$ is open and also $V\subseteq U$ is open, then $V\subseteq A$ is open.

Note that for any function $f:A\to B$, the preimage of any open set is open, since $(x\in f^{-1}(U))\iff (f(x) \in U)$.  Thus, any function is "[[continuous function|continuous]]" with respect to this "intrinsic topology."

## Overt sets

It is hard to get very far without an additional assumption that $\Sigma$ is closed under some [[joins]] as well.  However, if it were closed under *all* joins, then it would be all of $\Omega$, since any $P\in \Omega$ is the join $\bigvee_{i\in \{\star | P\}} \top$.

Given a dominance $\Sigma$, we say that a set $I$ is **overt** if $\Sigma$ is closed under $I$-indexed joins.  (This is related to, but not identical to, the notion of [[overt space]].)  In general it is reasonable to expect [[discrete object|discrete]] sets to be overt in this sense.  In some frameworks such as [[spatial type theory]] there is a formal notion of "discrete" and we can actually assert that all discrete sets are overt.   Otherwise we can assume that specific sets that we expect to be discrete are overt.  For instance, we might assume that:

* The [[empty set]] is overt, i.e. $\bot\in\Sigma$.
* [[finite sets|Finite sets]] are overt, i.e. $\Sigma$ is also a sub-join-semilattice of $\Omega$.
* The [[natural numbers]] are overt, i.e. $\Sigma$ is closed under countable joins in $\Omega$.

## Examples

* The singleton $\{\top\}$ is a dominance, for which only singletons are overt.

* The set $\{\bot,\top\}$ is a dominance.  This is the smallest dominance such that the empty set is overt.  (In [[classical mathematics]], of course, this and the previous example are the only two dominances, and the theory trivializes.)

* The set of all truth values of the form $\exists n, f(n) = 1$ for some function $f:\mathbb{N}\to \mathbf{2}$ is often a dominance, though this may not be provable without further assumptions.  For instance, this is the case if we assume [[countable choice]] or (perhaps) the [[propositional axiom of choice]].  When it is a dominance, this is the smallest dominance such that $\mathbb{N}$ is overt; it is called the **Rosolini dominance**.  Equivalently, it is the set of truth values of the form $x\gt 0$ for some [[Cauchy real number]] $x$.

* The set of all truth values of the form $x\gt 0$ for some [[Dedekind real number]] $x$ is also often a dominance, though this also may not be provable without further assumptions.

## See also

* [[streak]]

## References

*  G. Rosolini. *Continuity  and  Effectiveness  in  Topoi*.  PhD thesis, University of Oxford, 1986. [pdf](https://www.researchgate.net/publication/35103849_Continuity_and_effectiveness_in_topoi)

* [[Martin Escardo]], *Topology via higher-order intuitionistic logic.*, unfinished paper, [pdf](http://www.cs.bham.ac.uk/~mhe/papers/pittsburgh.pdf)

* [[Martin Escardo]], *Synthetic  topology  of  data  types  and  classical  spaces.*  Electronic Notes in Theoretical Computer Science, 87:21--156, 2004.  [pdf](http://www.cs.bham.ac.uk/~mhe/papers/entcs87.pdf)

* [[Andrej Bauer]] and Davorin Lesnik, *Metric Spaces in Synthetic Topology*, [pdf](http://math.andrej.com/wp-content/uploads/2010/01/csms_in_synthtop.pdf)

[[!redirects dominance]]
[[!redirects dominances]]
[[!redirects Rosolini dominance]]
[[!redirects open truth value]]
[[!redirects open truth values]]
[[!redirects open proposition]]
[[!redirects open propositions]]


dominant functor

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

# Dominant functors
* table of contents
{: toc}

## Idea

A [[functor]] $F\colon C \to D$ is **dominant** if it is "surjective on objects up to retracts".

## Definition

$F\colon C \to D$ is **dominant** if for every [[object]] $y$ of $D$, there exists an object $x$ of $C$ for which $y$ is a [[retract]] of $F x$, i.e. there is a pair of morphisms $y \to F x \to y$ composing to the identity.

## Examples

* Every [[essentially surjective on objects functor]] is dominant.

## Properties

* Let $L \dashv R I$ be an [[adjunction]]. If $I$ is dominant and [[full]], then $I L \dashv R$. In this case, the two adjunctions induce the same [[monad]]. This is Proposition 1.5 of [DFH75](#DFH75). (For a converse, see [[fully faithful functor]].)

* Let $L \dashv R$. If $L$ is dominant, then $R$ is [[faithful]]. $R$ is [[full]] if it is full on the image of $L$. This is Proposition 1.7 of [DFH75](#DFH75).

* Let $G : A \to B$ be a functor between small categories. The induced functor $[G^{op}, Set] : [B^{op}, Set] \to [A^{op}, Set]$ between [[presheaf categories]] is a [[surjective geometric morphism]] if and only if $G$ is dominant, if and only if $[G^{op}, Set]$ is [[monadic]]. This is Example A4.2.7(b) of [[Sketches of an Elephant]].

## Related concepts

[[!include properties of functors -- contents]]

## References

* {#DFH75} [[Aristide Deleanu]], [[Armin Frei]], [[Peter Hilton]], _Idempotent triples and completion_, Mathematische Zeitschrift **143** (1975) 91-104 [[doi:10.1007/BF01173053](https://doi.org/10.1007/BF01173053)]

Called **quasi-surjective on objects functors** in:

* [[Gabriella Böhm]], [[Steve Lack]], [[Ross Street]], _Idempotent splittings, colimit completion, and weak aspects of the theory of monads_, Journal of Pure and Applied Algebra __216__:2 (2012) 385--403 [doi](https://doi.org/10.1016/j.jpaa.2011.07.003)

[[!redirects dominant functors]]


dominant geometric morphism

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

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

## Idea

The concept of **dominance** is a weakening of the concept of [[surjective geometric morphism|surjectivity]] for [[geometric morphism|geometric morphisms]]. It generalizes the concept of a continuous function whose image is a [[dense subspace]] of its codomain in topology.

## Definition

A geometric morphism $f:\mathcal{F}\to\mathcal{E}$ is called _dominant_ if the following equivalent conditions hold:

* The [[direct image]] satisfies: $f_\ast(\emptyset_\mathcal{F})\cong\emptyset_\mathcal{E}$.

* The [[inverse image]] satisfies: from $f^\ast(Z)\cong\emptyset_\mathcal{F}$ follows $Z\cong\emptyset_\mathcal{E}$.

* The [[inverse image]] satisfies: from $f^\ast(Z)\cong\emptyset_\mathcal{F}$ follows $Z\cong\emptyset_\mathcal{E}$ for $Z$ a [[subterminal object]].

## Examples

* A geometric morphism $f:Sh(X)\to Sh(Y)$  between the toposes of sheaves on two topological spaces $X, Y$  is dominant iff the corresponding continuous map $f:X\to Y$  has a [[dense subspace|dense]] image $f(X)$ in $Y$.

* A geometric embedding $i:\mathcal{E}\hookrightarrow\mathcal{F}$ is dominant precisely when it exhibits $\mathcal{E}$ as a [[dense subtopos]].

## Properties

+-- {: .num_prop #dominant_surjection}
###### Proposition
If $f:\mathcal{F}\to\mathcal{E}$ is a surjection (i.e. $f^\ast$ is [[faithful]]) then $f$  is dominant.
=--

**Proof**:
Suppose $f^\ast(Z)\cong \emptyset_\mathcal{F}$ is initial, then $Hom_\mathcal{F}(\emptyset_\mathcal{F},f^\ast(Y))=Hom_\mathcal{F}(f^\ast(Z),f^\ast(Y))$ is a singleton for all $Y\in\mathcal{E}$ , but by faithfulness of $f^\ast$ this implies that $Hom_\mathcal{E}(Z,Y)$ is a singleton for all $Y\in\mathcal{E}$ , which says that $Z$ is initial in $\mathcal{E}$. $\qed$

+-- {: .num_prop #dominant_dense}
###### Proposition
Let $f:\mathcal{F}\to\mathcal{E}$ be a geometric morphism and $i\circ s$ its [[(geometric surjection, embedding) factorization system|surjection-inclusion factorization]]. $f:\mathcal{F}\to\mathcal{E}$ is dominant iff $i:Im(f)\hookrightarrow\mathcal{E}$ is a [[dense subtopos|dense inclusion]].
=--

**Proof**: Suppose $i$ is dense, hence dominant. $s$ as a surjection is dominant as well, and so is their composition $f$.

Conversely, suppose $f:\mathcal{F}\to\mathcal{E}$ is dominant and $i^\ast(Z)\cong \emptyset_{Im(f)}$. Since $s^\ast$ preserves colimits, $\emptyset_\mathcal{F}\cong s^\ast (\emptyset_{Im(f)})\cong s^\ast\circ i^\ast (Z)=f^\ast(Z)$ but $f$ is dominant by assumption, therefore $Z\cong \emptyset_\mathcal{E}$, hence $i$ is dense. $\qed$

The following is a slight generalization of the [[(dense,closed)-factorization]] employing dominant geometric morphisms:

+-- {: .num_prop #dominant-closed}
###### Proposition
Let $f:\mathcal{F}\to\mathcal{E}$ be a geometric morphism. Then $f$ factors as a [[dominant geometric morphism]] $d$ followed by a [[closed subtopos|closed inclusion]] $c$.
=--

**Proof**: Let $i\circ d_1$ be the [[(geometric surjection, embedding) factorization system|surjection-inclusion factorization]] of $f$. Since $d_1$ is surjective, it is dominant (cf. [above](#dominant_surjection)). Then we use the [[(dense,closed)-factorization]] to factor $i$ into $c\circ d_2$. Since both $d_i$ are dominant, so is $d:=d_2\circ d_1$ and $c\circ d$ yields the demanded factorization of $f$. $\qed$

## Remark

The concept can be generalized to morphisms _in_ a topos $\mathcal{E}$ by calling $f:X\to Y$ _dominant_ if the induced geometric morphism $f^\ast\dashv\prod_f:\mathcal{E}/X\to\mathcal{E}/Y$ is dominant, where $f^\ast$ denotes the pullback functor.

## Related entries

* [[dense subtopos]]

* [[skeletal geometric morphism]]

* [[open geometric morphism]]

* [[(dense,closed)-factorization]]

* [[dense functor]]

## References

* {#SGA4}[[M. Artin]], [[A. Grothendieck]], [[J. L. Verdier]], _Théorie des Topos et Cohomologie Etale des Schémas ([[SGA4]])_, LNM **269** Springer Heidelberg 1972. (pp.429-430, 462)


Dominic Culver

* [personal page](https://dlculver.github.io/)

## Selected writings

On [[fully dualizable object|fully]] [[dualizable objects]]:

* [[Dominic Culver]], [[Mitchell Faulk]], *Duality Notes*, lecture notes for *West Coast Algebraic Topology Summer School* (2014) [[pdf](https://mathtube.org/sites/default/files/lecture-notes/Duality%20Notes.pdf), [[CulverFaulk-DualityNotes.pdf:file]] [mathtube](https://www.mathtube.org/lecture/notes/duality-notes)]

category: people


Dominic G. B. Edelen

* [InSpire page](https://inspirehep.net/authors/2307214)

## Selected writings

On [[exterior calculus]] with [[differential forms]]:

* [[Dominic G. B. Edelen]], *Applied exterior calculus*, Wiley (1985) \[GoogleBooks\]

## Related pages

* [[Maxwell equations]]

category: people

[[!redirects Dominic Edelen]]


Dominic Hughes

* [webpage](http://boole.stanford.edu/~dominic/)

category: people


Dominic Joyce

Dominic Joyce is professor for pure [[mathematics]] at Oxford

* [website](http://people.maths.ox.ac.uk/joyce), [publications](http://people.maths.ox.ac.uk/joyce/publ.html)

Warning: distinguish from mathematician David Joyce who worked on knot theory and [[quandles]] (and invented the terminology "quandle")

## Selected writings

On [[hypercomplex manifolds]] in terms of [[G-structure]]:

* [[Dominic Joyce]], _Manifolds with many complex structures_, The Quarterly Journal of Mathematics, Volume 46, Issue 2, June 1995, Pages 169–184 ([doi:10.1093/qmath/46.2.169](https://doi.org/10.1093/qmath/46.2.169))

On [[Calabi-Yau orbifolds]]:

* [[Dominic Joyce]], _Deforming Calabi-Yau orbifolds_, Asian Journal of Mathematics 3.4 (1999): 853-868 ([pdf](https://www.intlpress.com/site/pub/files/_fulltext/journals/ajm/1999/0003/0004/AJM-1999-0003-0004-a007.pdf))

Construction of [[compact topological space|compact]] [[G2-manifolds]]:

* {#Joyce96}[[Dominic Joyce]], _Compact Riemannian 7-manifolds with holonomy $G_2$_, Journal of Differential Geometry vol 43, no 2, 1996 ([Euclid:jdg/1214458109](https://projecteuclid.org/euclid.jdg/1214458109))

* {#Joyce00} [[Dominic Joyce]], _Compact Manifolds with Special Holonomy_, Oxford Mathematical Monographs, Oxford University Press (2000)

* [[Dominic Joyce]], _Compact Riemannian manifolds with exceptional holonomy_, Surveys in Differential Geometry
Volume 6 (2001) ([doi:10.4310/SDG.2001.v6.n1.a3](https://dx.doi.org/10.4310/SDG.2001.v6.n1.a3))

Construction of [[compact topological space|compact]] [[Spin(7)-manifolds]];

* [[Dominic Joyce]], _A new construction of compact 8-manifolds with holonomy $Spin(7)$_, J. Differential Geom. Volume 53, Number 1 (1999), 89-130 ([euclid:jdg/1214425448](https://projecteuclid.org/euclid.jdg/1214425448))

On [[Donaldson-Thomas invariants]]:

* Dominic Joyce, Yinan Song, _A theory of generalized Donaldson-Thomas invariants_ ([arxiv/0810.5645](http://arxiv.org/abs/0810.5645

On [[derived smooth manifolds]] (derived [[orbifolds]]):

Something similar (derived [[orbifolds]]):

* [[Dominic Joyce]], _D-orbifolds, Kuranishi spaces, and polyfolds_ talk notes (Jan 2010) [[pdf](http://people.maths.ox.ac.uk/joyce/dmtalk.pdf), [[Joyce-DOrbifolds.pdf:file]]]

## Related entries

* [[special holonomy]]

* [[G2-manifolds]]

* [[C-infinity-ring]],

* [[derived geometry]]

* [[special Lagrangian submanifold]],

* [[Donaldson-Thomas invariant]]

category: people


Dominic Orchard

* [website](https://www.cs.kent.ac.uk/people/staff/dao7/)

## Selected writings

On [[comonads]] and [[graded modalities]] [[comonads in computer science|in computer science]];

* {#POM13} [[Tomas Petricek]], [[Dominic Orchard]], [[Alan Mycroft]], *Coeffects: Unified Static Analysis of Context-Dependence*, in: *Automata, Languages, and Programming. ICALP 2013*, Lecture Notes in Computer Science **7966** Springer (2013) [[doi:10.1007/978-3-642-39212-2_35](https://doi.org/10.1007/978-3-642-39212-2_35)]

* [[Dominic Orchard]], [[Alan Mycroft]], *A Notation for Comonads*, in: *Implementation and Application of Functional Languages. IFL 2012*, Lecture Notes in Computer Science **8241** [[doi:10.1007/978-3-642-41582-1_1](https://doi.org/10.1007/978-3-642-41582-1_1)]

  > (on [[codo-notation]])

* {#GKOBU16} Marco Gaboardi, [[Shin-ya Katsumata]], [[Dominic Orchard]], Flavien Breuvart, [[Tarmo Uustalu]], *Combining effects and coeffects via grading*, ICFP 2016: Proceedings of the 21st ACM SIGPLAN International Conference on Functional Programming (2016) 476–489 [[doi:10.1145/2951913.2951939](https://doi.org/10.1145/2951913.2951939), [talk abstract](https://icfp16.sigplan.org/details/icfp-2016-papers/31/Combining-Effects-and-Coeffects-via-Grading), [video rec](https://www.youtube.com/watch?v=l1ZNMT3fQCo)]

On [[dependent linear types]] and [[graded modalities]]:

* [[Benjamin Moon]], [[Harley Eades III]], [[Dominic Orchard]], *Graded Modal Dependent Type Theory*. In: N. Yoshida (ed.) *Programming Languages and Systems ESOP 2021*, Lecture Notes in Computer Science **12648**, Springer (2021) 462-490 [[doi:10.1007/978-3-030-72019-3_17](https://doi.org/10.1007/978-3-030-72019-3_17), [arxiv:2010.13163](https://arxiv.org/abs/2010.13163)]

## Related entries

* [[graded modality]]

category: people


Dominic Verity

Dominic Verity is a British category theorist, based in Australia. He is an Emeritus Professor at Macquarie University.

He has worked on the theory of [[complicial sets]] and their [[weak complicial sets|weak analogues]], which followed up on ideas of [[John Roberts]] on cohomology and, effectively, [[omega-category]] theory.

More recently he has worked with [[Emily Riehl]] on foundations of [[(infinity,1)-category theory|$(\infty, 1)$-category theory]] seen through their [[homotopy 2-category]], and using the concept of [[∞-cosmoi]] to capture common structure of different presentations of $(\infty, 1)$-categories.

* [Departmental webpage](https://researchers.mq.edu.au/en/persons/dominic-verity)

* [Mathematics Genealogy page](https://www.mathgenealogy.org/id.php?id=61019)

## Selected writings

On [[enriched category theory]] and [[internal categories]]:

* _Enriched categories, internal categories and change of base_ Ph.D. thesis, Cambridge University (1992), reprinted as Reprints in Theory and Applications of Categories, No. 20 (2011) pp 1-266 ([TAC](http://www.tac.mta.ca/tac/reprints/articles/20/tr20abs.html))

Introducing the notion of [[traced monoidal categories]]:

* {#JoyalStreetVerity96} [[André Joyal]], [[Ross Street]], [[Dominic Verity]], _Traced monoidal categories_, Math. Proc. Camb. Phil. Soc. **119** (1996) 447-468 [[pdf](http://sci-prew.inf.ua/v119/3/S0305004100074338.pdf), [doi:10.1017/S0305004100074338](https://doi.org/10.1017/S0305004100074338)]

On [[weak complicial sets]]:

*   D. R. Verity, 2005, Complicial Sets , available from : [arXiv:math.CT/0410412](http://arxiv.org/abs/math.CT/0410412).

*   D. Verity, 2006, Weak complicial sets I: basic homotopy theory , available from : [arXiv:math/0604414](http://arxiv.org/abs/math/0604414).

*    D. R. Verity, 2006, Weak complicial sets. III. Enriched and internal quasi-category theory , (in preparation).

*    D. R. Verity, 2007, Weak complicial sets. II. Nerves of complicial Gray-categories , in Categories in algebra, geometry and mathematical physics , volume 431 of Contemp. Math., 441--467, Amer. Math. Soc., Providence, RI.

On [[comprehensive factorization systems]] and [[torsors]]:

* [[Ross Street]], [[Dominic Verity]], *The comprehensive factorization and torsors*, [[TAC]] **23** (2010)  42-75 [[tac:23-03](http://www.tac.mta.ca/tac/volumes/23/3/23-03abs.html)]

On [[(∞,1)-category theory]] via the [[homotopy 2-category of (∞,1)-categories]] of [[∞-cosmoi]] ([[formal (infinity,1)-category theory|formal $(\infty,1)$-category theory]]):

* [[Emily Riehl]], [[Dominic Verity]], _The 2-category theory of quasi-categories_, Advances in Mathematics Volume 280, 6 August 2015, Pages 549-642 ([arXiv:1306.5144](http://arxiv.org/abs/1306.5144), [doi:10.1016/j.aim.2015.04.021](https://doi.org/10.1016/j.aim.2015.04.021))

* {#RiehlVerity16} [[Emily Riehl]], [[Dominic Verity]], _Infinity category theory from scratch_, Higher Structures Vol 4, No 1 (2020) ([arXiv:1608.05314](https://arxiv.org/abs/1608.05314), [pdf](http://www.math.jhu.edu/~eriehl/scratch.pdf))

* {#RiehlVerity22} [[Emily Riehl]], [[Dominic Verity]], _[[Elements of ∞-Category Theory]]_, Cambridge studies in advanced mathematics **194**, Cambridge University Press (2022) $[$[doi:10.1017/9781108936880](https://doi.org/10.1017/9781108936880), ISBN:978-1-108-83798-9, [pdf](https://emilyriehl.github.io/files/elements.pdf)$]$

On [[(∞,1)-functors]] and [[(∞,1)-monads]]:

* {#RiehlVerity16} [[Emily Riehl]], [[Dominic Verity]], _Homotopy coherent adjunctions and the formal theory of monads_, Advances in Mathematics, Volume 286, 2 January 2016, Pages 802-888 ([arXiv:1310.8279](http://arxiv.org/abs/1310.8279), [doi:10.1016/j.aim.2015.09.011](https://doi.org/10.1016/j.aim.2015.09.011))

On [[Reedy model structures]] via [[weighted colimits]]:

* [[Emily Riehl]], [[Dominic Verity]], *The theory and practice of Reedy categories*, Theory and Applications of Categories **29** 9 (2014) 256-301 [[tac:29-09](http://www.tac.mta.ca/tac/volumes/29/9/29-09abs.html),[arXiv:1304.6871](https://arxiv.org/abs/1304.6871)]

On the [[Yoneda lemma for (∞,1)-categories]]:

* [[Emily Riehl]], [[Dominic Verity]], _Fibrations and Yoneda's lemma in an $\infty$-cosmos_, Journal of Pure and Applied Algebra Volume 221, Issue 3, March 2017, Pages 499-564 ([arXiv:1506.05500](https://arxiv.org/abs/1506.05500), [doi:10.1016/j.jpaa.2016.07.003](https://doi.org/10.1016/j.jpaa.2016.07.003))

## Related entries

* [[simplicial model for weak omega-categories]]

* [[stratified simplicial set]]

* [[complicial set]]

* [[Verity on descent for strict omega-groupoid valued presheave|Verity on descent for strict omega-groupoid valued presheaves]]

* [[infinity-cosmos]]

[[!redirects D. Verity]]
[[!redirects Verity]]

category: people


Dominik B. Bucher

* [institute page](https://www.ch.tum.de/en/ch/faculty/staff/active-members/b/dr-dominik-bucher/)

* [research lab](https://www.bucherlab.org/)

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

## Selected writings

On [[quantum sensing]] via [[spin resonance]] on the [[nitrogen-vacancy center in diamond]]:

* Nick R. Neuling, Robin D. Allert, [[Dominik B. Bucher]], *Prospects of Single-Cell NMR Spectroscopy with Quantum Sensors*, Current Opinion in Biotechnology **83** (2023) 102975 [[arXiv:2306.07593](https://arxiv.org/abs/2306.07593), [doi:10.1016/j.copbio.2023.102975](https://doi.org/10.1016/j.copbio.2023.102975)]

* [[Dominik Bucher]], *Quantum Sensing with Spin Defects in Diamond*, [talk at](CQTS#BucherOct2023) [[CQTS]] (Oct 2023) [video:[YT](https://youtu.be/ArN3Ed78sSc)]

category: people

[[!redirects Dominik Bucher]]


Dominik Rist

* [inspire page](https://inspirehep.net/authors/2014022)

## Selected writings

On [[adjusted connections]] in [[higher gauge theory]]:

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

On [[higher gauge theory]]:

* [[Leron Borsten]], [[Mehran Jalali Farahani]], [[Branislav Jurčo]], [[Hyungrok Kim]], [[Jiří Nárožný]], [[Dominik Rist]], [[Christian Saemann]], [[Martin Wolf]], *Higher Gauge Theory*, in *[[Encyclopedia of Mathematical Physics 2nd ed]]*, Elsevier (2024) [[arXiv:2401.05275](https://arxiv.org/abs/2401.05275)]

category: people


Dominique Arlettaz

* [webpage](http://www.unil.ch/central/page2885_en.html)

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=24778)

## Selected writings

On bounds on [[kernel]]/[[cokernel]] of the [[Boardman homomorphism]] from [[stable Cohomotopy]] to any [[Whitehead-generalized cohomology theory]]:

* {#Arlettaz04} [[Dominique Arlettaz]], _The generalized Boardman homomorphisms_, Central European Journal of Mathematics March 2004, Volume 2, Issue 1, pp 50-56 ([doi:10.2478/BF02475949](https://doi.org/10.2478/BF02475949))

category: people


Dominique Bourn

Dominique Bourn is a [[category theory|category theorist]] who is Professeur at the  Université du Littoral, Laboratoire LMPA, BP 699, 62228 Calais CEDEX, FRANCE.

* [personal webpage](http://www-lmpa.univ-littoral.fr/~bourn/)

## Selected writings

On [[profunctors]] with application to [[shape theory]]:

* [[Dominique Bourn]], [[Jean-Marc Cordier]], _Distributeurs et théorie de la forme_, Cah. Top. Géom. Diff. Cat. **21** no.2 (1980) pp.161-189. ([numdam:CTGDC_1980__21_2_161_0](http://www.numdam.org/item/CTGDC_1980__21_2_161_0), [pdf](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1980__21_2/CTGDC_1980__21_2_161_0/CTGDC_1980__21_2_161_0.pdf))

On [[regular categories]] and [[protomodular categories]]:

* [[Dominique Bourn]], [[Marino Gran]], _Regular, Protomodular, and Abelian Categories_, Chapter IV, pp.165-211 in: Maria Pedicchio, [[Walter Tholen]] (eds.), _Categorical Foundations_ , Cambridge University Press 2004 ([doi:10.1017/CBO9781107340985.007](https://doi.org/10.1017/CBO9781107340985.007))

On [[nonabelian homological algebra]], such as its [[diagram chasing lemmas]], in [[Mal'cev categories]], [[protomodular categories]], [[semi-abelian categories]] and [[homological categories]]:

* [[Francis Borceux]], [[Dominique Bourn]], *[[Mal'cev, protomodular, homological and semi-abelian categories]]*, Mathematics and Its Applications __566__,  Kluwer 2004 ([doi:10.1007/978-1-4020-1962-3](https://link.springer.com/book/10.1007/978-1-4020-1962-3))

On [[2-categories]]

* D. Bourn, J. Penon, _2-Catégories réductibles_, [tac reprints](http://www.tac.mta.ca/tac/reprints/articles/19/tr19abs.html)
* Dominique Bourn, [_From Groups to Categorial Algebra : Introduction to Protomodular and Mal’tsev Categories_](https://doi.org/10.1007/978-3-319-57219-2), Compact Textbooks in Mathematics, Birkhäuser 2017

category: people
[[!redirects D. Bourn]]


Dominique Gouyou-Beauchamps

* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=117023)

## Selected writings

Expressing the number of [[standard Young tableaux]] with at most 4 rows through [[Catalan numbers]]:

* [[Dominique Gouyou-Beauchamps]], *Standard Young Tableaux of Height 4 and  5*, Europ. J. Combinatorics (1989) 10, 69-82 (doi:10.1016/S0195-6698(89)80034-4, [pdf](https://core.ac.uk/download/pdf/82423471.pdf))

category: people


Don H. Shimamoto

* [Mathematics Genealogy page](https://www.mathgenealogy.org/id.php?id=24062)

## Selected writings

Identifying the [[homotopy of rational maps|homotopy type of the space of rational maps]] from the [[Riemann sphere]] to a [[complex projective space]] with that of a [[configuration space of points]]:

* {#CohenShimamoto91} [[Ralph L. Cohen]], [[Don H. Shimamoto]], *Rational Functions, Labelled Configurations, and Hilbert Schemes*, Journal of the London Mathematical Socienty **43** 2 (1991) 509-528 ([doi:10.1112/jlms/s2-43.3.509](https://doi.org/10.1112/jlms/s2-43.3.509))

category: people

[[!redirects Don Shimamoto]]


Don Lichtenberg

* [webpage](https://web.physics.indiana.edu/faculty/lichten.shtml)

## Selected writings

On [[diquarks]] (with a section IX on [[hadron supersymmetry]]):

* Mauro Anselmino, Enrico Predazzi, Svante Ekelin, Sverker Fredriksson, [[Don B. Lichtenberg]], *Diquarks*, Rev. Mod. Phys. **65** (1993) 1199 ([doi:10.1103/RevModPhys.65.1199](https://doi.org/10.1103/RevModPhys.65.1199))

On [[hadron supersymmetry]]:

* [[Don Lichtenberg]], _Whither Hadron Supersymmetry?_,
Contribution to: _International Conference on Orbis Scientiae 1999: Quantum Gravity, Generalized Theory of Gravitation and Superstring Theory Based Unification_ (28th Conference on High-Energy Physics and Cosmology Since 1964), 203-208 ([arXiv:hep-ph/9912280](https://arxiv.org/abs/hep-ph/9912280), [spire:511339](https://inspirehep.net/literature/511339))

* [[Don Lichtenberg]], _Hadron Supersymmetry and Relations between Meson and Baryon Masses_, In: Kursunoglu B.N., Mintz S.L., Perlmutter A. (eds.) _Neutrino Mass, Dark Matter, Gravitational Waves, Monopole Condensation, and Light Cone Quantization_, Springer 1996 ([doi:10.1007/978-1-4899-1564-1_29](https://doi.org/10.1007/978-1-4899-1564-1_29))

Discussion of [[exotic hadron]] [[mass|masses]] via [[hadron supersymmetry]]:

* [[Don Lichtenberg]],  Renato Roncaglia, Enrico Predazzi, *Predicting exotic hadron masses from supersymmetry and a quark -- diquark model*, J. Phys. G: Nucl. Part. Phys. **23** (1997) 865 ([doi:10.1088/0954-3899/23/8/001](https://iopscience.iop.org/article/10.1088/0954-3899/23/8/001))

[[!redirects Don B. Lichtenberg]]

[[!redirects D. B. Lichtenberg]]
[[!redirects D.B. Lichtenberg]]

category: people


Don Marolf > history

see _[[Donald Marolf]]_


Don Page

* [webpage](http://www.physics.ualberta.ca/Faculty/DonPage)

* [wikipedia entry](http://en.wikipedia.org/wiki/Don_Page_%28physicist%29)

## Selected writings

On [[Freund-Rubin compactification]] of [[D=11 supergravity]] on [[round sphere|round]] and [[squashed sphere|squashed]] [[7-spheres]] (and introducing the _[[Page charge]]_ of [[black brane|black]] [[M2-branes]], see at _[[shifted C-field flux quantization]]_):

* {#Page83} [[Don Page]], _Classical stability of round and squashed seven-spheres in eleven-dimensional supergravity_, Phys. Rev. D 28, 2976 (1983) ([spire:14480](http://inspirehep.net/record/14480) [doi:10.1103/PhysRevD.28.2976](https://doi.org/10.1103/PhysRevD.28.2976))

On [[G2-conifolds]]:

* [[Gary Gibbons]], [[Don Page]], [[Christopher Pope]], _Einstein metrics on $S^3$, $\mathbb{R}^3$ and $\mathbb{R}^4$ bundles_, Comm. Math. Phys. Volume 127, Number 3 (1990), 529-553 ([euclid:cmp/1104180218](https://projecteuclid.org/euclid.cmp/1104180218))

category: people


Don Zagier

* [webpage](http://people.mpim-bonn.mpg.de/zagier/)

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

*  C. S. Aravinda, Sudhir Rao: *Speaking the language of mathematics*, interview with Don Zagier, Bhāvanā mathematics magazine **7** 2 (2023) [[web](https://bhavana.org.in/speaking-the-language-of-mathematics/)]

## Selected writings

Introducing [[modified Bernoulli numbers]]:

* {#Zagier98} [[Don Zagier]], _A modified Bernoulli number_, Nieuw Archief voor Wiskunde, 16:63–72, 1998 ([[ZagierModifiedBernoulli.pdf:file]])

* [[Don Zagier]], _Hecke operators and periods of modular forms_, Israel Math. Conf. Proc., 3 (1990), 321–336 ([hdl:21.11116/0000-0004-3936-0](http://hdl.handle.net/21.11116/0000-0004-3936-0))

On [[Bernoulli numbers]] and [[modified Bernoulli numbers]]:

* {#ZagierProperties} [[Don Zagier]], _Curious and Exotic Identities for Bernoulli
Numbers_ ([[ZagierPropertiesBernoulli.pdf:file]])

On [[perturbative quantum field theory|perturbative]] [[Chern-Simons theory with complex gauge group]]:

* {#DGLZ09} [[Tudor Dimofte]], [[Sergei Gukov]], [[Jonatan Lenells]], [[Don Zagier]], _Exact Results for Perturbative Chern-Simons Theory with Complex Gauge Group_, Commun. Num. Theor. Phys. **3** (2009) 363-443 [[arXiv:0903.2472](http://arxiv.org/abs/0903.2472), [doi:10.4310/CNTP.2009.v3.n2.a4](https://dx.doi.org/10.4310/CNTP.2009.v3.n2.a4)]

category: people


Donald Davis

* [webpage](http://www.lehigh.edu/~dmd1/)

category: people

[[!redirects Don Davis]]


Donald Knuth

__Donald E. Knuth__ is an American computer scientist and a mathematician, a professor emeritus at Stanford University. Knuth is the author of  TeX,  Metafont, the cweb computer language, of many important algorithms in formal languages, parsing, combinatorics, numerics, [[rewriting system]]s, of the concept of literate programming, of the series of books "The Art of Computer Programming", a book on surreal numbers etc.

* [website](https://www-cs-faculty.stanford.edu/~knuth/)

[[!redirects Donald E. Knuth]]
[[!redirects Don Knuth]]

category: people


Donald Knutson

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=37441)

## Selected writings

On [[algebraic spaces]]:

* [[Donald Knutson]], *Algebraic spaces*, Lecture Notes in Mathematics, **203**, Springer (1971) [[doi:10.1007/BFb0059750](https://link.springer.com/book/10.1007/BFb0059750)]

## Related entries

* [[locally affine space]]

category: people


Donald Marolf

* [webpage](http://web.physics.ucsb.edu/~marolf/)

## Selected writings

On the [[Hanany-Witten effect]] and [[T-duality]]:

* {#Marolf01} [[Donald Marolf]], §2 of: *T-duality and the case of the disappearing brane*, JHEP 0106:036 (2001) [[arXiv:hep-th/0103098](https://arxiv.org/abs/hep-th/0103098), [doi:10.1088/1126-6708/2001/06/036](https://doi.org/10.1088/1126-6708/2001/06/036)]

More on [[holographic entanglement entropy]]:

* Eugenia Colafranceschi, [[Xi Dong]], [[Donald Marolf]], Zhencheng Wang, *Algebras and Hilbert spaces from gravitational path integrals: Understanding Ryu-Takayanagi/HRT as entropy without invoking holography* [[arXiv:2310.02189](https://arxiv.org/abs/2310.02189)]

category: people

[[!redirects Don Marolf]]


Donald Pinckney

* [personal page](https://donaldpinckney.com/)

## Selected writings

On [[topological data analysis]] and [[persistent homology]]:

* [[Donald Pinckney]], *[Topological Data Analysis and Persistent Homology](https://donaldpinckney.com/machine%20learning/2019/05/02/tda.html)* (2019)

category: people