determinant line bundle

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

#Contents#
* table of contents
{:toc}

## Definition
 {#DeterminantLineBundles}

Let $V$ and $W$ be two [[vector space]]s of [[dimension]] $n = dim V = dim W$ and

$$
  T : V \to W
$$

a [[linear map]].

Write $\wedge^n V$ and $\wedge^n W$ for the _top exterior power_ of these vector spaces, the skew-symmetrized $n$th [[tensor algebra|tensor power]] of $V$ and $W$. These are 1-dimensional vector spaces, hence [[lines]] over the [[ground field]]. The linear map $T$ induces a linear map

$$
  det T \colon \wedge^n V\to \wedge^n W
$$

between these lines. This is the **[[determinant]]** of $T$. More specifically, if $V = W$ (being of the same finite dimension, both are necessarily [[isomorphic]] but not necessarily canonically so) then $det T : \wedge^n V \to \wedge^n V$ is a linear [[endomorphism]] of a 1-dimensional vector space and, by the equivalence

$$
  End(\wedge^n V) \simeq k
$$

of such endomorphisms with the [[ground field]] $k$, is identified with an element in $k$

$$
  det T \in k
  \,.
$$

This is the standard meaning of the **determinant** of a linear endomorphism.

Notice that the determinant construction:

$$
  det : (V \stackrel{T}{\to} W) \mapsto (\wedge^n V \stackrel{det T}{\to} \wedge^n W)
$$

is a [[functor]] from the [[category]] [[Vect]] to itself

$$
  det : Vect \to Vect
  \,.
$$

{#AnySuchFunctor} Any such functor $F \colon Vect \to Vect$ with certain continuity assumptions induces an endo-functor on the category of [[vector bundles]] $VectBund(X)$ over any [[manifold]] $X$ (see [there](topological+vector+bundle#FiberwiseOperations)).

Concretely, if a vector bundle $E \to X$ is given by a [[Cech cohomology|Cech cocycle]]

$$
  \array{
    C(U_i) &\stackrel{(g_{i j})}{\to}& \mathbf{B} GL(n) &\to& Vect
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    X
  }
$$

with respect to an [[open cover]] $\{U_i \to X\}$ (see [[principal bundle]] and [[associated bundle]] for details), hence by transition functions

$$
  (g_{i j} \in C(U_i \cap U_j, GL(n)))
$$

with values in the [[general linear group]], its image under $det : VectBund(X) \to VectBund(X)$ is the bundle with transition functions the determinants of these transition functions

$$
  (det g_{i j} \in C(U_i \cap U_j, GL(1) \simeq k^\times))
  \,.
$$

This are the transition functions for the bundle $\wedge^\bullet E \to X$ which is fiberwise the top exterior power of $E \to X$. This is the **[[determinant line]] bundle** of $E$.

## Properties

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

Let

$$
 E : X \stackrel{\simeq}{\leftarrow} C(U_i) \stackrel{g_{i j}}{\to} \mathbf{B} U(n)
$$

be a [[unitary group]]-[[principal bundle]] (to which is canonically [[associated bundle|associated]] a rank-$n$ [[complex vector bundle]]). Then the single [[characteristic class]]

$$
  [det E] \in H^2(X, \mathbb{Z})
$$

of its determinant [[circle bundle]]

$$
 det E
  \colon
  X \stackrel{\simeq}{\leftarrow} C(U_i) \stackrel{det g_{i j}}{\to} \mathbf{B} U(1)
$$

is the [[first Chern class]] of $E$

$$
  [det E] = c_1(E)
  \,.
$$

Moreover, if $X$ is a [[smooth manifold]] and $(g_{i j}, A_i)$ is the data of a [[connection on a bundle]] $(E, \nabla)$ on $E$ then $(det g_{i j}, tr A_i)$ (where we take the [[trace]] $tr : \mathfrak{u}(n) \to \mathfrak{u}(1)$ on the [[Lie algebra]] of the [[unitary group]]) is a [[circle n-bundle with connection|line bundle with connection]] that refines the first Chern-class to [[ordinary differential cohomology]]. In other words, this is the image under the refined [[Chern-Weil homomorphism]] of $(E, \nabla)$ induced by the canonical unary [[invariant polynomial]] on $\mathfrak{u}(n)$.

=--

An explicit version of this statement is for instance in ([GriffithsHarris, p. 414](#GriffithsHarris)).

One can now look at operators $T:E\to F$ where $E,F$ are vector bundles of rank $n$ and the induced operators $\Lambda^n T : \Lambda^n E\to \Lambda^n F$ which can be considered as elements $det T\in (\Lambda^n E)^*\otimes\Lambda^n F$.

Even more important is the case of when $X$ is replaced by an appropriate [[moduli space of connections]], instantons, holomorphic structures or some other objects related to Fredholm operators for which the determinants can be defined.

## Examples

### Quillen's determinant line bundle

There is a specific version called **Quillen's determinant line bundle** which is certain [[line bundle]] over the [[moduli space of complex structures]] on a fixed smooth vector bundle $E$ over a fixed [[Riemann surface]] $M$. A complex structure on the bundle corresponds to an operator which in local coordinates looks as $D = d\bar{z}(\partial_z+\alpha(z))$ where $\alpha(z)$ is a smooth matrix valued function. The set of such operators is an affine space $\mathcal{A}$ whose underlying vector space is the space of $(0,1)$-End-valued forms $\Omega^{0,1} (End M)$. Then again a determinant is an element of a line $\mathcal{L}_D = \lambda(Ker D)^*\otimes \lambda(Coker D)$ where $\lambda$ is taking the top exterior power. Now one has a family $\mathcal{L}_D$ depending on $D$, which determines a holomorphic line bundle over $\mathcal{A}$. This is the **determinant line bundle**.

If we had a trivialization of the Quillen's determinant line bundle, then we could identify every section with a holomorphic function on the base space, hence a holomorphic rule giving a number to a Cauchy-Riemann operator. For this one restricts first to the component consisting of the operators with the zero Fredholm index. Next, one considers the corresponding [[Laplace operator]] $D^* D$ and its [[functional determinant]] related to the [[zeta function of an elliptic differential operator]]. (This is related to the [[analytic torsion]]).

### Determinant bundle on the Grassmannian

Let $Gr_k(V)$ be the [[Grassmannian]] of $k$-dimensional subspaces of a finite dimensional vector space $V$. Let $W\subset V$ be a point in $Gr_k(V)$ and $\Lambda^k(W)$ its top exterior power; it is a fiber of the bundle $Det$ over $Gr_k(V)$. The determinant bundle $Det$ has no non-zero holomorphic global sections. Consider its dual $Det^*$ with fiber $\Lambda^k(W)^*$ over $W$. Then the space of of global holomorphic sections $\Gamma_{hol}(Det^*) \cong \Lambda^k(V^*)$. This construction can be suitably extended for the Segal Grassmannian, where $V= V_+\oplus V_-$ is a separable Hilbert space equipped with a polarization, see chapter 7 and especially 7.7 in the Pressley-Segal book listed below.

### Comparing Quillen's and Segal's determinant line bundles

The determinant line bundle of Quillen is in fact related to a variant of Segal's determinant bundle on the "semiinfinite" Grassmannian. Namely one considers instead $Gr_{cpt}(H)$ which is the set (space eventually) of closed supspaces $W\subset H$ where the projection $W\to H_+$ is [[Fredholm operator|Fredholm]] and $W\to H_-$ is [[compact operator|compact]]; then one follows the Segal's prescription to define $Det$ on $Gr_{cpt}(H)$. Notice that $Gr_{cpt}(H)$ is not a homogeneous space. Now there is a span of maps with contractible fibers

$$
Gr_{cpt}(H)\leftarrow \mathcal{B}\to Fred(H_+).
$$
The Quillen's determinant line bundle is defined in general on the whole $Fred(H_+)$ and its pullback to $\mathcal{B}$ is isomorphic to the pullback of the determinant bundle on $Gr_{cpt}(H)$; in fact the Quillen's version can be reconstructed from this pullback by certain quotienting construction.

### Pfaffian line bundle

In dimensîon $8k+2$ for $k \in \mathbb{N}$ the determinant line bundle has a canonîcal square root line bundle, the [[Pfaffian line bundle]].

### From fermionic path integrals

See at _[[fermionic path integral]]_.

### Relation to theta function

the determinant of the Dirac operator is, up to choice of isomorphism, the [[theta function]]-section of the determinant line bundle ([Freed 87, pages 30-31](#Freed87)).

### Relation to vacuum energy, partition function

See at _[[vacuum energy]]_

## Related concepts

* [[density bundle]]

* [[pseudoscalar]]

* [[theta function]]

[[!include square roots of line bundles - table]]

## References

The relation between determinant line bundles and the first Chern class is stated explicitly for instance on p. 414 of

* Griffiths and Harris, _Principles of algebraic geometry_
 {#GriffithsHarris}

Literature on determinant line bundles of infinite-dimensional bundles includes the following:

* [[Daniel Quillen|D.G. Quillen]], _Determinants of Cauchy-Riemann operators over a Riemann surface_, Funkcionalnii Analiz i ego Prilozhenija __19__ (1985), 37-41, ([pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=faa&paperid=1334&volume=19&year=1985&issue=1&fpage=37&what=fullt&option_lang=eng) of Russian version).

  reviewed e.g. in

  Arlo Caine, _Quillen's construction of Determinants of
Cauchy–Riemann operators over Riemann Surfaces_, 2005 ([pdf](http://web.archive.org/web/20150326122017/http://www.cpp.edu/~jacaine/pdf/quillen_determinant_notes.pdf))

* [[Michael Atiyah]], [[Isadore Singer]], _Dirac operators coupled to vector potentials_, Proc. Nat. Acad. Sci. USA __81__, 2597-2600 (1984) ([pdf](http://www.pnas.org/content/81/8/2597.full.pdf) at pnas site)

* {#Freed87} [[Daniel Freed]], _On determinant line bundles_, Math. aspects of [[string theory]], ed. S. T. Yau, World Sci. Publ. 1987,  ([dg-ga/9505002](http://arxiv.org/abs/dg-ga/9505002), [revised pdf](http://www.math.utexas.edu/~dafr/Index/determinants.pdf))

* [[Jean-Michel Bismut]], [[Daniel Freed]], _The analysis of elliptic families.I. Metrics and connections on determinant bundles_, Comm. Math. Phys. __106__, 1 (1986), 159-176, [euclid](http://projecteuclid.org/euclid.cmp/1104115586), _II. Dirac operators, eta invariants, and the holonomy theorem_, Comm. Math. Phys. __107__, 1 (1986), 103-163. [euclid](http://projecteuclid.org/euclid.cmp/1104115934)

* [[Jean-Michel Bismut]], _Quillen metrics and determinant bundles_, 2 conference lectures in honour of A. N. Tyurin, video at [link](http://www.mathnet.ru/php/presentation.phtml?presentid=62&option_lang=eng)

* A. Pressley, [[Graeme Segal|G. Segal]], _Loop Groups_, Oxford Math. Monographs, 1986.

* Kenro Furutani, _On the Quillen determinant_, J. Geom. Phys. __49__, 4, 366-375, [math.DG/0309127](http://arxiv.org/abs/math/0309127), [doi](http://dx.doi.org/10.1016/j.geomphys.2003.07.001)

* [[M. Kontsevich]], S. Vishik, _Geometry of determinants of elliptic operators_, in Functional Analysis on the Eve of the 21st Century.
Vol. I (S. Gindikin, et al., eds.) In honor of the 80th birthday of [[Israel Gel'fand|I.M. Gelfand]]. Birkhäuser, Progr. Math. __131__ (1993), 173-197, [pdf](http://193.51.104.7/~maxim/TEXTS/geometry_determinants_12.pdf), [hep-th/9406140](http://arxiv.org/abs/hep-th/9406140)

* [[Robbert Dijkgraaf]], [[Edward Witten|E. Witten]], _Topological gauge theories and group cohomology_, Commun. Math.Phys. __129__, 393--429 (1990), [euclid](http://projecteuclid.org/euclid.cmp/1104180750), [MR1048699](http://www.ams.org/mathscinet-getitem?mr=1048699)

Discussion in the context of the [[modular functor]] is in

* [[Graeme Segal]], section 6 and section 5 of _The definition of conformal field theory_ , preprint, 1988; also in [[Ulrike Tillmann]] (ed.) _Topology, geometry and quantum field theory_ , London Math. Soc. Lect. Note Ser., Vol. 308. Cambridge University Press, Cambridge (2004) 421-577. ([pdf](https://people.maths.ox.ac.uk/segalg/0521540496txt.pdf))

[[!redirects determinant line bundles]]


determinant of a perfect complex

## Related

* [[perfect complex]]

* [[determinant]]

* [[algebraic K-theory]]

## References

* [[Finn Faye Knudsen]], [[David Mumford]], _The projectivity of the moduli space of stable curves. I. Preliminaries on "det" and "Div"_. Math. Scand. 39 (1976), no. 1, 19–55. ([EuDML](https://eudml.org/doc/166481), [pdf](http://www.mscand.dk/article/download/11642/9658))

* [[Timo Schürg]], [[Bertrand Toën]], [[Gabriele Vezzosi]], _Derived algebraic geometry, determinants of perfect complexes, and  applications to obstruction theories for maps and complexes_, [arXiv:1102.1150](http://arxiv.org/abs/1102.1150v4).

* [[Alexander Beilinson]], _Topological $\epsilon$-factors_, [arXiv:math/0610055](http://arxiv.org/abs/math/0610055).

* [[Bhargav Bhatt]], [[Peter Scholze]], _Projectivity of the Witt vector affine Grassmannian_, [arXiv:1507.06490](http://arxiv.org/abs/1507.06490).

* [[Fernando Muro]], _Picard categories, determinant functors and K-theory_, talk notes, [pdf](http://personales.us.es/fmuro/files/slides/pontevedra.pdf)

* appendix A of "Discriminants, Resultants, and Multidimensional Determinants" by Gelfand, Kapranov and Zelevinsky

* [MO/7124](http://mathoverflow.net/questions/7124/determinant-of-a-perfect-complex), _Determinant of a perfect complex_.


determinantal gerbe

## Definition

[Kapranov-Vasserot IV, (1.7)](#KapranovVasserotIV)

## References

* [[Vladimir Drinfeld]], _Infinite-dimensional vector bundles in algebraic geometry_, in [[Pavel Etingof]], V. Retakh, [[Isadore Singer]], _The unity of mathematics_, Progr. Math., 244, Birkhäuser Boston (2006)

* [[Mikhail Kapranov]], E. Vasserot, _Formal loops IV: Chiral differential operators_ ([arXiv:math/0612371](http://arxiv.org/abs/math/0612371))
 {#KapranovVasserotIV}

[[!redirects determinantal gerbes]]


determinantal point process

Related entries include [[random matrix]], [[Fredholm determinant]]

* wikipedia [determinantal point process](http://en.wikipedia.org/wiki/Determinantal_point_process)
* A. Soshnikov, Determinantal random point fields. Russian Math. Surveys, 2000, 55 (5), 923–975 Original: А. Б. Сошников, _Детерминантные случайные точечные поля_, УМН, 55:5(335) (2000), 107–160 [doi](http://dx.doi.org/10.4213/rm321) [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=rm&paperid=321&what=fullt&option_lang=rus)
* R. Lyons, _Determinantal probability measures_, Publ. Math. Inst. Hautes Études Sci., 2003, no. 98, 167–212 [MR2031202](http://www.ams.org/mathscinet-getitem?mr=2031202)
* T. Shirai, Y. Takahashi, _Random point fields associated with certain Fredholm determinants. I. Fermion, Poisson and boson point processes_, J. Funct. Anal., 205:2 (2003), 414–463 [MR2018415](http://www.ams.org/mathscinet-getitem?mr=2018415)
* A. Borodin, G. Olshanski, _Representation theory and random point processes_, European Congress of Math. 2005, 73–94 [MR2185737](http://www.ams.org/mathscinet-getitem?mr=2185737)
* A. Borodin, A. Okounkov, G. Olshanski, _On asymptotics of Plancherel measures for symmetric groups_, [arxiv:math/9905032](http://arxiv.org/abs/math/9905032)
* M. Bertola, M. Cafasso, _Fredholm determinants and pole-free solutions to the noncommutative Painlevé II equation_, Comm. Math. Phys. __309__ (2012), no. 3, 793–833 [MR2885610](http://www.ams.org/mathscinet-getitem?mr=2885610) [doi](http://dx.doi.org/10.1007/s00220-011-1383-x)

category: probability
[[!redirects determinantal random point process]]
[[!redirects determinantal point processes]]


deterministic automaton

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

## Definitions

(For the moment, mostly as an example of a [[coalgebra for an endofunctor]].)

This will give the classical definition as a state-based system and then show how to turn that form into the coalgebraic form.  The point is that other coalgebras should then be easier to interpret.

+-- {: .un_defn}
###### Definition
A **deterministic automaton** consists of the following data:

* a set of _states_, $Q$;

* a set, $\Sigma$, of _inputs_ ;

* a function, $\delta : Q\times \Sigma \to Q$, called the _next state function_,

and

* a predicate, $final : Q \to bool = \{\bot, \top\}$, the [[boolean domain]].
=--

In the usual interpretation if the automaton is in state $q$, and is 'given' the input $\sigma$, then it changes to being in state $\delta(q,\sigma)$.

The 'final' predicate, of course, returns $\top$ if a state is a _final state_ and $\bot$ otherwise. (There will, in general, be a set of final or 'accept' states.)

(For the moment we will not look in any detail at the links between automata and languages.)

## Currying $\delta$

The first step in transforming this to the coalgebraic form is to [[currying|curry]] $\delta$, so as to obtain it in the form $\delta : Q\to Q^\Sigma$. We thus have for a state, $q\in Q$, $\delta(q,) : \Sigma \to Q$.  We then also have a product function

$$\alpha = \delta \times (final) : Q \to  Q^\Sigma\times bool.$$

If we now write $HQ = Q^\Sigma\times bool$, we get a functor (for you to check) $H : Set \to Set$ and the deterministic automaton corresponded precisely to a [[coalgebra for an endofunctor|coalgebra]], $(Q,\alpha)$, for $H$.

## ... and the morphisms?

Rather than start with a classically defined morphism of automata, we will take a morphism $f : (Q,\alpha)\to (Q',\alpha')$ of coalgebras. We thus have that $f : Q\to Q'$ is a mapping of states, and that $\alpha'\circ f = H(f)\circ \alpha$. Now let $q\in Q$ and $\sigma\in \Sigma$, then the equation tells us that $\delta'(f(q),\sigma) = f(\delta(q,\sigma)$ and that the final states of $Q$ are mapped onto those of $Q'$.  In other words the concept of morphism of these $H$-coalgebras is precisely that of functional simulation of deterministic automata.

## Terminal coalgebra

The [[terminal coalgebra]], $(T,\alpha_T)$, for this endofunctor is neat. It has state space, $T = \mathcal{P}(\Sigma^*)$, the power set of the [[free monoid]]/ Kleene star of the set of 'symbol labels'. This power set is precisely the set of all languages over the alphabet $\Sigma$, as a [[language]] is defined to be a subset of $\Sigma^*$.

The next state function for $(T,\alpha_T)$ is defined by $\delta_T(L,s) = \{w\in \Sigma^* |sw \in L\}$.

## References

For a summary of automata theory , look at the [Wikipedia](http://en.wikipedia.org/wiki/Automata_theory).

For a thorough treatment, see

*  [[M. V. Lawson]], _Finite Automata_, [Google books](http://books.google.co.uk/books?id=MDQ_K7-z2AMC&dq=automata+theory+lawson&source=bl&ots=0w-Xp8AaDa&sig=6dXKjG0tVY6aUlvlAc5UaT-ObX8&hl=en&ei=Ywy6TPmtCJSOjAfzgYXYDg&sa=X&oi=book_result&ct=result&resnum=1&sqi=2&ved=0CBoQ6AEwAA)

or other texts on the subject.

For the coalgebraic treatment, this is discussed in:

*  [[A. Kurz]]:  _Coalgebras and Modal Logic._ Course Notes for ESSLLI 2001, Version of October 2001. Appeared on the CD-Rom ESSLLI'01, Department of Philosophy, University of Helsinki, Finland, available from [site](http://www.cs.le.ac.uk/people/akurz/works.html).

category: computer science
[[!redirects deterministic automaton]]
[[!redirects deterministic automata]]


deterministic random variable

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

#Contents#
* table of contents
{:toc}

## Idea

A [[random variable]] is called *deterministic* if it happens to be non-random ("[[red herring principle]]") in that it does not actually depend on the given [[event (in probability theory)|events]], hence if it is a [[constant function]] on the underlying [[measure space]]/[[probability space]] (or at least takes a fixed value [[almost surely]]).

## Properties

\begin{remark}
  The [[law of a random variable|law]] of a deterministic random variable is a [[Dirac measure]].
\end{remark}

## Related concepts

* [[Dirac measure]]
* [[constant function]] /  [[locally constant function]]

* [[constant sheaf]] / [[locally constant sheaf]]

* [[constant stack]] / [[locally constant stack]]

* [[constant ∞-stack]] / [[locally constant ∞-stack]]

* [[local system]]

* [[steady function]]

[[!redirects deterministic random variables]]


Detlef Gromoll

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

## Selected writings

On the [[Cheeger-Gromoll splitting theorem]]:

* {#CheegerGromoll} [[Jeff Cheeger]], [[Detlef Gromoll]], _The splitting theorem for manifolds of nonnegative Ricci curvature_, J. Differential Geom. Volume 6, Number 1 (1971), 119-128 ([euclid:1214430220](https://projecteuclid.org/euclid.jdg/1214430220))

On [[Gromoll-Meyer spheres]]:

* {#GromollMeyer74} [[Detlef Gromoll]], Wolfgang Meyer, _An Exotic Sphere With Nonnegative Sectional Curvature_, Annals of Mathematics Second Series, Vol. 100, No. 2 (Sep., 1974), pp. 401-406 ([jstor:1971078](https://www.jstor.org/stable/1971078))

category: people


Detlev Buchholz

* [webpage](http://www.theorie.physik.uni-goe.de/~buchholz/)

## Selected writings

On the [[Gauss law]] in relation to [[infraparticles]]:

* {#Buchholz86} [[Detlev Buchholz]], _Gauss’ law and the infraparticle problem_, Phys.  Lett.  B **174** (1986) 331-334 \[doi:10.1016/0370-2693(86)91110-X\]

On the [[Gauss law]] in relation to non-[[local field theory|locality]] of charged states in [[quantum field theory]]:

* [[Giovanni Morchio]], [[Franco Strocchi]], *Localization and symmetries*, J. Phys. A **40** (2007) 3173-3188 [[arXiv:math-ph/0607015](https://arxiv.org/abs/math-ph/0607015), [doi:10.1088/1751-8113/40/12/S17](https://doi.org/10.1088/1751-8113/40/12/S17)]

* [[Detlev Buchholz]], Fabio Ciolli, [[Giuseppe Ruzzi]], Ezio Vasselli, *Gauss's law, the manifestations of gauge fields, and their impact on local observables* [[arXiv:2212.11009](https://arxiv.org/abs/2212.11009)]

Examples of [[non-perturbative quantum field theory|non-perturbative]] [[interacting quantum field theory|interacting]] [[scalar field theory]] in _any_ [[spacetime]] [[dimension]] (in particular in $d \geq 4$):

* [[Detlev Buchholz]], [[Klaus Fredenhagen]], _A $C^\ast$-algebraic approach to interacting quantum field theories_,  Commun. Math. Phys. **377** (2020) 947–969  [[arXiv:1902.06062](https://arxiv.org/abs/1902.06062), [doi:10.1007/s00220-020-03700-9](https://doi.org/10.1007/s00220-020-03700-9)]

## Related entries

* [[AQFT]]

category: people


Deutsch-Jozsa algorithm

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An early example of an [[algorithm]] for which [[quantum computers]] are expected to drastically outperform classical computers.

## Related concepts

* [[Shor's algorithm]]

* [[Grover's algorithm]]

* [[quantum complexity theory]]

* [[quantum supremacy]]

## References

Due to:

* [[David Deutsch]], [[Richard Jozsa]], *Rapid solution of problems by quantum computation*. Proc. R. Soc. Lond. A **439** 1907 (1992) 553-558 [[pdf](https://www.isical.ac.in/~rcbose/internship/lectures2016/rt08deutschjozsa.pdf), [doi:10.1098/rspa.1992.0167](https://doi.org/10.1098/rspa.1992.0167)]

Review:

* Renato Portugal, *Basic Quantum Algorithms* [[arXiv:2201.10574](https://arxiv.org/abs/2201.10574)]

Implementation in [[Quipper]]:

* Safat Siddiqui, Mohammed Jahirul Islam, Omar Shehab, §4.1-2 in: *Five Quantum Algorithms Using Quipper* [[arXiv:1406.4481](https://arxiv.org/abs/1406.4481)]

See also:

* Wikipedia, *[Deutsch-Jozsa algorithm](https://en.wikipedia.org/wiki/Deutsch%E2%80%93Jozsa_algorithm)*


Dev Sandbox

<div class="rightHandSide toc">
[[!include differential graded objects - contents]]
</div>

##Test

##Test two


Dev Sinha

* [webpage](https://math.uoregon.edu/profile/dps)

## Selected writings

On [[Fulton-MacPherson compactifications]] of [[configuration spaces of points]]:

* {#Sinha03} [[Dev Sinha]], _Manifold theoretic compactifications of configuration spaces_, Selecta Math. (N.S.) 10(3):391-428, 2004 ([arXiv:math/0306385](https://arxiv.org/abs/math/0306385))

On the [[homology]] of the [[little disks operad]]

* [[Dev Sinha]], _The homology of the little disks operad_ ([arXiv:math/0610236](http://arxiv.org/abs/math/0610236))

On [[Lie coalgebras]], [[rational homotopy theory]] and [[Hopf invariants]] via [[homotopy Whitehead integrals]] ("[[functional cup products]]"):

* {#SinhaWalter13} [[Dev Sinha]], [[Ben Walter]], _Lie coalgebras and rational homotopy theory II: Hopf invariants_, Trans. Amer. Math. Soc. 365 (2013), 861-883  ([arXiv:0809.5084](https://arxiv.org/abs/0809.5084), [doi:10.1090/S0002-9947-2012-05654-6](https://doi.org/10.1090/S0002-9947-2012-05654-6))

On models of [[knot spaces]] via ([[cosimplicial object|co]]-)[[simplicial object|simplicial]] systems of [[configuration spaces of points]]:

* [[Dev Sinha]], _The Topology of Spaces of Knots: Cosimplicial Models_, American Journal of Mathematics, Vol. 131, No. 4 (Aug., 2009), pp. 945-980 (36 pages) ([arXiv:math/0202287](https://arxiv.org/abs/math/0202287), [jstor:40263760 ](https://www.jstor.org/stable/40263760))

On the [[bar and cobar construction]] and [[Koszul duality]]:

* [[Dev Sinha]], _Koszul duality in algebraic topology - an historical perspective_, J. Homotopy Relat. Struct. (2013) 8: 1 ([arXiv:1001.2032](https://arxiv.org/abs/1001.2032))

category: people


# The development and current state of mathematics
* table of contents
{: toc}

This is not part of [[philosophy]] in a narrow sense, but the opinions on what is good [[mathematics]], how mathematics could develop better, and how it should be stimulated to develop, as well as overviews of the current state of mathematics and its important points, is an important part of reflection, but also of the practical action of a mathematician.

## On the relation between conjectural and proved mathematics styles

This is a topic of a well-known discussion series in 1990s.

* Arthur Jaffe, Frank Quinn, _"Theoretical mathematics": Toward a cultural synthesis of mathematics and theoretical physics_ , Bull. Amer. Math. Soc. (N.S.) __29__ (1993), no. 1, 1--13; [MR94h:00007](http://www.ams.org/mathscinet-getitem?mr=1202292),[math.HO/9307227](http://arxiv.org/abs/math.HO/9307227), [doi](http://dx.doi.org/10.1090/S0273-0979-1993-00413-0)

* [[Michael Atiyah]], [[Armand Borel]], G. J. Chaitin, Daniel Friedan, James Glimm, Jeremy J. Gray, Morris W. Hirsch, [[Saunders MacLane]], Benoit B. Mandelbrot, David Ruelle, [[Albert Schwarz]], Karen Uhlenbeck, René Thom, [[Edward Witten]], Christopher Zeeman, _Responses to "Theoretical Mathematics: Toward a cultural synthesis of mathematics and theoretical physics", by A. Jaffe and F. Quinn_, Bull. Am. Math. Soc. __30__:178-207, 1994, [math.HO/9404229](http://arxiv.org/abs/math/9404229)

* [[Michael Atiyah]], _Reflections on geometry and physics_, Surveys in differential geometry, Vol. II (Cambridge, MA, 1993), 1–6, Int. Press, Cambridge, MA, 1995.

* [[William P. Thurston]],  _On proof and progress in mathematics_, Bull. Amer. Math. Soc. (N.S.) __30__ (1994), no. 2, 161–177, [MR94m:00006](http://www.ams.org/mathscinet-getitem?mr=1249357), [doi](http://dx.doi.org/10.1090/S0273-0979-1994-00502-6), [arxiv/math.HO](http://arxiv.org/abs/math/9307227)

* [[David Corfield]] at el. _Wittgenstein and Thurston on understanding_,  [nCafe](http://golem.ph.utexas.edu/category/2006/10/wittgenstein_and_thurston_on_u.html)

Somewhat later and with different scope, but related to this series, is also a long article

* Frank Quinn, _Contributions to a science of contemporary mathematics_, [pdf](https://personal.math.vt.edu/fquinn/history_nature/nature0.pdf)

## Some other issue

* The [[big picture]] in [[mathematics]].

[[!redirects opinions on the development of mathematics]]
[[!redirects opinions on development of mathematics]]
[[!redirects development of mathematics]]
[[!redirects development and current state of mathematics]]


DevTest2

Test.


DevTest3

    Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Dolor sed viverra ipsum nunc aliquet bibendum enim. In massa tempor nec feugiat. Nunc aliquet bibendum enim facilisis gravida. Nisl nunc mi ipsum faucibus vitae aliquet nec ullamcorper. Amet luctus venenatis lectus magna fringilla. Volutpat maecenas volutpat blandit aliquam etiam erat velit scelerisque in. Egestas egestas fringilla phasellus faucibus scelerisque eleifend. Sagittis orci a scelerisque purus semper eget duis. Nulla pharetra diam sit amet nisl suscipit. Sed adipiscing diam donec adipiscing tristique risus nec feugiat in. Fusce ut placerat orci nulla. Pharetra vel turpis nunc eget lorem dolor. Tristique senectus et netus et malesuada.

    Etiam tempor orci eu lobortis elementum nibh tellus molestie. Neque egestas congue quisque egestas. Egestas integer eget aliquet nibh praesent tristique. Vulputate mi sit amet mauris. Sodales neque sodales ut etiam sit. Dignissim suspendisse in est ante in. Volutpat commodo sed egestas egestas. Felis donec et odio pellentesque diam. Pharetra vel turpis nunc eget lorem dolor sed viverra. Porta nibh venenatis cras sed felis eget. Aliquam ultrices sagittis orci a. Dignissim diam quis enim lobortis. Aliquet porttitor lacus luctus accumsan. Dignissim convallis aenean et tortor at risus viverra adipiscing at.


DevTest4

Test.


Dexter Chua

This page exists so that I don't see a question mark by my name all the time.

 * Here is my [website](https://dec41.user.srcf.net/home).
 * Here are the [lecture notes](https://dec41.user.srcf.net/) I took as a student.

category : people


DF space

{:myproof: .proof style="margin-left:2em;margin-right:1em;background:#fef;-moz-border-radius:10px;padding:1ex;"}
{:mynumdef: .num_defn style="border:solid #cccccc;border-width:2px 1px;padding:0 1em;margin:0 1em;-moz-border-radius:10px;background:#ffe;"}
{:mynumprop: .num_prop style="border:solid #cccccc;border-width:2px 1px;padding:0 1em;margin:0 1em;-moz-border-radius:10px;background:#eff;"}

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

#Contents#
* table of contents
{:toc}

## Idea

A **DF space** is a type of [[locally convex topological vector space]].  The basic idea is that a DF space is _morally_ the [[strong dual]] of a [[Fréchet space]].  That is, it has all the nice structure that such a dual would have but without the bother of actually having to be a dual space.

## Definition

+-- {: mynumdef #DFspace}
###### Definition
A [[locally convex topological vector space]] is a **DF space** if it possesses a [[fundamental system|fundamental sequence]] of [[subsets of lctvs|bounded sets]] and if every [[subsets of lctvs|strongly bounded]] countable union of [[subsets of lctvs|equicontinuous]] subsets of the dual is again equicontinuous.
=--

## Properties

1. The [[strong dual]] of a [[Fréchet space|metrisable]] locally convex topological vector space is a DF space.
2. Every [[normable space]] is a DF space.
3. Every [[infrabarrelled space]] that has a [[fundamental system|fundamental sequence]] of [[subsets of lctvs|bounded sets]] is a DF space.
4. In particular, every [[LB space]] is a DF space, since any bounded set in an LB space is contained in one of the factors.

+-- {: mynumprop #MetDF}
###### Proposition
A locally convex topological vector space that is [[Fréchet space|metrisable]] and is a DF space is [[normable space|normable]].
=--

+-- {: myproof}
###### Proof
Let $E$ be a metrisable DF space.

To prove the result, we shall use a proposition recorded in [[functional analysis bibliography|Schaefer (IV.6.7)]].  That says that a [[subsets of lctvs|convex]], [[subsets of lctvs|circled]] subset $V$ of $E$ is a [[neighbourhood]] of $0$ if (and only if) for every [[subsets of lctvs|convex]], [[subsets of lctvs|circled]] [[subsets of lctvs|bounded]] subset $B \subseteq E$, $B \cap V$ is a $0$-neighbourhood in $B$.

As $E$ is metrisable, it has a countable $0$-neighbourhood base, say $(V_n)$.  As $E$ is a DF space, it has a countable fundamental family of bounded sets, say $(B_n)$.  Let us assume, without loss of generality, that this family is increasing.

For each $k \in \mathbb{N}$, as $B_k$ is bounded, there is some $\lambda_k \gt 0$ such that $B_k \subseteq \lambda_k V_k$.  Since, by assumption, the $B_k$ are an increasing family, we have $B_k \subseteq \lambda_l V_l$ for all $l \ge k$.  Let $B \coloneqq \bigcap \lambda_k V_k$.  This is a bounded set since it is contained in $\lambda_k V_k$ for each $k$.  Let $l \in \mathbb{N}$ and consider $B_l \cap B$.  We can write this as

$$
B_l \cap \big(\bigcap_{k=1}^{l-1} \lambda_k V_k\big) \cap B_l \cap \big( \bigcap_{k \ge l} \lambda_k V_k \big).
$$

Since $B_l \subseteq \lambda_k V_k$ for $k \ge l$, the last part is unnecessary and so we see that

$$
B_l \cap B = B_l \cap \big(\bigcap_{k=1}^{l-1} \lambda_k V_k \big).
$$

This is then a finite intersection of open sets in $B_l$ and so is open in $B_l$.

As this holds for any of the $B_l$s, it holds for any bounded set, whence the proposition from Schaefer applies to show that $B$ is a $0$-neighbourhood.  Thus $E$ possesses a bounded $0$-neighbourhood, whence is normable.
=--

## References

See [[functional analysis bibliography]].


dg-algebra > history

< [[dg-algebra]]


dg-category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
#### Enriched category theory
+--{: .hide}
[[!include enriched category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Differential graded categories_ or _dg-categories_ are linear analogues of [[spectral categories]]. In other words they are [[linear (infinity,1)-category|linear]] [[stable (infinity,1)-categories]]. It is common and useful to view them as [[enhanced triangulated categories]].

## Definition

A _dg-category_ over a [[commutative ring]] $k$ is an [[(infinity,1)-category]] [[enriched (infinity,1)-category|enriched]] in the [[(infinity,1)-category of chain complexes]] of $k$-[[modules]].  Equivalently, it is an ordinary category [[enriched category|strictly enriched]] in [[chain complexes]] (see [Haugseng 13](#Haugseng13)).

Hence a dg-category is a category with [[mapping complexes]] of morphisms between any two objects. By taking the [[homologies]] of these [[chain complexes]] in degree zero, one gets an ordinary category, called the [[homotopy category of a dg-category]].
Notice that a dg-category with a single object is the same thing as a [[dg-algebra]].

## Properties

### The (infinity,1)-category of dg-categories

The [[Dwyer-Kan model structure on dg-categories]] presents the [[(infinity,1)-category of dg-categories]].

### Relation to stable $\infty$-categories

By the [[stable Dold-Kan correspondence]], the [[(infinity,1)-category]] of dg-categories is equivalent to the [[(infinity,1)-category]] of [[(infinity,1)-categories]] [[enriched (infinity,1)-category|enriched]] in the [[symmetric monoidal (infinity,1)-category]] of [[module spectra|modules]] over the [[Eilenberg-Mac Lane spectrum]] $H k$.
The latter is equivalent, at least morally, to the [[(infinity,1)-category]] of $k$-[[linear (infinity,1)-category|linear]] [[stable (infinity,1)-categories]].

More precisely, it is shown in [Cohn 13](#Cohn13) that the [[Morita model structure on dg-categories]] presents the [[(infinity,1)-category]] of [[idempotent complete (infinity,1)-category|idempotent complete]] [[linear (infinity,1)-category|linear]] [[stable (infinity,1)-categories]].

## Aspects of dg-categories

* [[homotopy category of a dg-category]].
* [[equivalence of dg-categories]]
* [[dg-modules]], [[perfect dg-modules]]
* [[derived dg-category]]
* [[dg-Yoneda embedding]]
* [[pretriangulated dg-category]]
* [[dg-localization]], [[dg-quotient]]
* [[dg-nerve]]
* [[Waldhausen K-theory of a dg-category]]
* [[semi-topological K-theory of a dg-category]]
* [[derived moduli stack of objects in a dg-category]]

## Related concepts

* [[enhanced triangulated category]]
* [[A-infinity category]]
* [[spectral category]]
* [[stable (infinity,1)-category]]

## References

Historically, dg-categories were introduced in

* [[G. M. Kelly]], _Chain maps inducing zero homology maps_, Proc. Cambridge Philos. Soc. **61** (1965), 847–854, doi:[10.1017/S0305004100039207](https://doi.org/10.1017/S0305004100039207)

whilst their modern development can be traced to

*  [[A. I. Bondal]], [[Mikhail Kapranov]], _Enhanced triangulated categories_, Матем. Сборник, Том 181 (1990), No.5, 669--683 (Russian); transl. in USSR Math. USSR Sbornik, vol. 70 (1991), No. 1, pp. 93--107, (MR91g:18010) ([[bondalKaprEnhTRiangCat.pdf:file]])

### Overviews

For concise reviews of the theory, see section 1 of

* [[A. Beilinson]], [[V. Vologodsky]], _DG guide to Voevodsky's motives_.

as well as the introduction and appendices to

* [[V. Drinfeld]], _DG quotients of DG categories_, [pdf](http://www.math.harvard.edu/~gaitsgde/grad_2009/DGquotients.pdf).

For longer surveys, see

* [[Bernhard Keller]], _On differential graded categories_, International Congress of Mathematicians. **II** (2006) 151-190, Eur. Math. Soc., Zürich [[arXiv:math/0601185](http://arxiv.org/abs/math/0601185)]

and

* [[Bertrand Toën]], _Lectures on dg-categories_ ([pdf](https://perso.math.univ-toulouse.fr/btoen/files/2012/04/swisk.pdf)).

### Homotopy theory of dg-categories

The [[homotopy theory]] of [[dg-categories]] is studied in

* [[Gonçalo Tabuada]], _Théorie homotopique des DG-caté́gories_, Thesis, Paris, 2007, [pdf](https://arxiv.org/pdf/0710.4303.pdf).

* [[Gonçalo Tabuada]], _Une structure de catégorie de modèles de Quillen sur la catégorie des dg-catégories_,  C. R. Math. Acad. Sci. Paris  340  (2005),  no. 1, 15--19.

The equivalence with the [[homotopy theory]] of [[stable (infinity,1)-categories]] is discussed in

* {#Cohn13} [[Lee Cohn]], _Differential Graded Categories are k-linear Stable Infinity Categories_ ([arXiv:1308.2587](http://arxiv.org/abs/1308.2587))

(Note that the proof works over any ring, even though it is stated there for [[characteristic zero]].)

In the following it is shown that the [[homotopy theory]] of [[(infinity,1)-categories]] enriched in the [[(infinity,1)-category]] of [[chain complexes]] is equivalent to the [[homotopy theory]] of ordinary categories strictly enriched in [[chain complexes]].

* {#Haugseng13} [[Rune Haugseng]], _Rectification of enriched infinity-categories_, [arXiv:1312.3881](http://arxiv.org/abs/1312.3881v2).

### Derived noncommutative geometry

The following references discuss the use of dg-categories in [[derived noncommutative algebraic geometry]] and [[noncommutative motives]].

* [[Gonçalo Tabuada]], _Invariants additifs de DG-catégories_, Int. Math. Res. Not.  2005,  no. 53, 3309--3339; Addendum in Int. Math. Res. Not.  2006, Art. ID 75853, 3 pp. ; Erratum in Int. Math. Res. Not. IMRN  2007,  no. 24, Art. ID rnm149, 17 pp.

* [[Marco Robalo]], _Théorie homotopique motivique des espaces noncommutatifs_, [pdf](http://webusers.imj-prg.fr/~marco.robalo/these.pdf).

* S. Mahanta, _Noncommutative geometry in the framework of differential graded categories_, (arXiv:0805.1628)

* D. Orlov, _Smooth and proper noncommutative schemes and gluing of DG categories_, Adv. Math. 302 (2014) [doi](https://dx.doi.org/10.1016/j.aim.2016.07.014)

### Other aspects

* [[Bernhard Keller]], _Deriving DG categories_,  Ann. Sci. École Norm. Sup. (4)  27  (1994),  no. 1, 63--102 (numdam)

* [[Dmitry Tamarkin]], _What do dg-categories form?_,
Compos. Math. 143 (2007), no. 5, 1335--1358.

* [[Michael Batanin|M. Batanin]], _What do dg-categories form_ (after Tamarkin), talks at Paris 7 and Australian category seminar (abstract), [math.CT/0606553](http://arxiv.org/abs/math.CT/0606553)

* [[Oren Ben-Bassat]], [[Jonathan Block]], _Cohesive DG categories I: Milnor descent_, [arxiv/1201.6118](http://arxiv.org/abs/1201.6118)

[[!redirects dg-category]]
[[!redirects dg-categories]]
[[!redirects dg category]]
[[!redirects dg categories]]
[[!redirects DG-category]]
[[!redirects DG-categories]]
[[!redirects DG category]]
[[!redirects DG categories]]
[[!redirects differential graded category]]
[[!redirects differential graded categories]]


dg-category presented by a dg-model category

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

## Idea

Any [[dg-model category]] $M$ presents an associated [[dg-category]], whose objects are the [[cofibrant object|cofibrant]] [[fibrant object|fibrant]] objects of $M$, and whose [[mapping complexes]] are given by the dg-enrichment of $M$. This [[dg-category]] has the property of being [[fibrant object|fibrant]] in the [[Dwyer-Kan model structure on dg-categories]].

This is the analogue of the [[simplicially enriched category]] presented by a [[simplicial model category]].

## Definition

## Related concepts

* [[dg-model category]]
* [[dg-localization]]

[[!redirects dg-category presented by dg-model category]]
[[!redirects dg-category presented by a model category]]
[[!redirects dg-category presented by model category]]


dg-coalgebra > history

< [[dg-coalgebra]]


dg-comodule

#Contents#
* table of contents
{:toc}

## Idea

A _dg-comodule_ is a [[comodule]] in [[chain complexes]] over a [[dg-coalgebra]]. This is the [[duality|dual]] concept to that of a _[[dg-module]]_ over a [[dg-algebra]].

## Related concepts

* [[model structure on dg-comodules]]

* [[comodule spectrum]]

[[!redirects dg-comodules]]


dg-geometry

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

#Contents#
* table of contents
{:toc}

## Idea

This entry describes the [[higher geometry]]/[[derived geometry]] modeled on [[(∞,1)-site]]s of formal duals of [[dg-algebra]]s, bounded or unbounded, over a [[field]] of [[characteristic]] 0.

The corresponding [[(∞,1)-topos]] is the context for classical [[rational homotopy theory]], which arises by forming [[function algebras on ∞-stacks]] over [[constant ∞-stack]]s. It is also the context in which classical and higher order [[Hochschild homology]] of algebras and dg-algebras arises naturally as the function $\infty$-algebra on [[free loop space object]]s.

## The $(\infty,1)$-toposes

We discuss some basic aspects of the [[(∞,1)-topos]]es over [[(∞,1)-site]]s of formal duals of cdg-algebras and of cdg-algebras of functions on its objects.

### Over formal duals of non-positively graded cdg-algebras

Let $k$ be a [[field]] of [[characteristic]] 0, or more generally a commutative $\mathbb{Q}$-[[associative algebra|algebra]].

#### Definition

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

Write

* $cdgAlg_k$ for the category of graded-commutative cochain [[dg-algebra]]s (meaning: with [[differential]] of degree +1) in arbitrary degree;

* $cdgAlk_k^-$ for the full [[subcategory]] on objects with vanishing [[cochain cohomology]] in positive degree, $H^{\bullet \geq 1}(-) = 0$.

There are the standard projective [[model structures on dg-algebras]] on these categories, whose weak equivalences are the [[quasi-isomorphism]]s and whose fibrations are the degreewise surjections.

=--

This is considered in ([Toën-Vezzosi, 2.3.1](#ToenVezzosiStacks))

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

Let

$$
  C \hookrightarrow ((cdgAlg_k^-)^{op})^\circ
$$

be a [[small (∞,1)-category|small]] full [[sub-(∞,1)-category]] of the [[(∞,1)-category]] [[presentable (∞,1)-category|presented]] by this model structure, and let $C$ be equipped with the structure of a [[subcanonical coverage|subcanonical]] [[(∞,1)-site]].

Write

$$
  \mathbf{H} := Sh_{(\infty,1)}(C)
$$

for the [[(∞,1)-category of (∞,1)-sheaves]] on $C$. We have a derived [[Isbell duality]]

$$
  (\mathcal{O} \dashv j)
    :
  (cdgAlg_k^{op})^\circ
   \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{j}{\to}}
  \mathbf{H}
$$

where the [[left adjoint|left]] [[adjoint (∞,1)-functor]] $\mathcal{O}$ is the [[Yoneda extension]] of the inclusion $cdgAlg^+_k \hookrightarrow cdgAlg_k$.

=--

This is considered in ([Ben-ZviNadler](#Ben-ZviNadler)). See [[function algebras on ∞-stacks]] for details.

#### Properties

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

The inclusion

$$
  cdgAlg^-_k \hookrightarrow cdgAlg_k
$$

is a _homotopical context_ in the sense of ([ToënVezzosi, def. 1.1.0.11](#ToenVezzosiStacks)).

=--

This is ([ToënVezzosi, lemma 2.3.11](#ToenVezzosiStacks)). We record the following implications of this statement

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

$(cdgAlg_k, \otimes_k)$ is a [[symmetric monoidal category|symmetric]] [[monoidal model category]].

=--

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

For $B \in (dgcAlg_k)_{proj}$ a cofibrant object, the [[tensor product]] with $B$ preserves weak equivalences.

=--

This follows from ([ToënVezzosi, assumption 1.1.0.4](#ToenVezzosiStacks)).

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

The inclusion

$$
  (cdgAlg_k^-)^{op} \hookrightarrow (cdgAlg_k)^{op}
$$

preserves [[homotopy limit]]s, hence the induced inclusion

$$
  ((cdgAlg_k^-)^{op})^\circ \hookrightarrow ((cdgAlg_k)^{op})^\circ
$$

preserves [[(∞,1)-limit]]s.

=--

This follows from ([ToënVezzosi, assumption 1.1.0.6](#ToenVezzosiStacks)).

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

For $A \in cdgAlg_k$ a [[nLab:dg-algebra]], write

* $A Mod$ for its [[nLab:model structure on dg-modules|category of dg-modules]];

  This is naturally a [[nLab:symmetric monoidal category]].

* $cdgAlg_A := CMon(A Mod)$ for the [[category of commutative monoids]] in $A Mod$, the category of **cdg-$A$-algebras**.

=--

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

For any $A\in cdgAlg_k$ say a morphism in $cdgAlg_A$ is

* a weak equivalence precisely if it is a [[nLab:quasi-isomorphism]];

* a fibration precisely if it is degreewise surjective.

This makes $cdgAlg_A$ into a [[model category]] that is

* [[combinatorial model category|combinatorial]];

* [[proper model category|proper]].

There is an [[equivalence of categories]] with the [[under category]] of cdg-algebras under $A$

$$
  cdgAlg_A \simeq A/cdgAlg_k
$$

which is a [[Quillen equivalence]] with respect to the standard [[model structure on an under category]] on the right.

=--

This is ([ToënVezzosi, assumption 1.1.0.4, remark on p. 18](#ToenVezzosiStacks)).

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

For $B \in cdgAlg_A$ cofibrant with respect to the model structure in
cor \ref{ModelStructureOndgAAlgebras}, the [[tensor product]] ([[base change]]) functor

$$
  B \otimes_A (-) : A Mod \to B Mod
$$

preserves weak equivalences.

=--

This is ([ToënVezzosi, assumption 1.1.0.4](#ToenVezzosiStacks)).

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

The [[monoidal Dold-Kan correspondence]] provides a [[Quillen equivalence]]

$$
  (\Gamma^{cmon} \dashv N_\bullet)
  :
  cAlg_k^{\Delta^{op}}
    \stackrel{\overset{\Gamma^{cmon}}{\leftarrow}}{\underset{N_\bullet}{\to}}
  cdgAlg_k^+
$$

(since $k$ is assumed to be of characteristic 0). Under this equivalence we have that $U \in cAlg_k \hookrightarrow cAlg_k^{\Delta^{op}} \hookrightarrow \mathbf{H}$ is $\mathcal{O}$-perfect:

$$
  \mathcal{O} (X^{K}) \simeq K \cdot \mathcal{O}(X)
$$

and this recovers the constructions discussed above in [The Hochschild chain complex of an associative algebra](#HochschildChainComplex).

=--

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

Since the [[(∞,1)-Yoneda embedding]] $y$ commutes with [[(∞,1)-limit]]s we have that the powering $(y(U))^{K} \simeq y(U^K)$ is still representable. Therefore

$$
  \mathcal{O} ((y(U))^K) \simeq \mathcal{O}(U^K) \;\;
  \in (cdgAlg_k^-)^{op} \hookrightarrow (cdgAlg_k)^{op}
$$

is simply the formal dual of $U^K$, which is $K \cdot \mathcal{O}(U)$ formed in $cdgAlg_k$  by formal duality. By the [above proposition](cdgAlgInclusionPreservesLimits) the inclusion
$cdgAlg_k^- \hookrightarrow cdgAlg_k$ preserves this $(\infty,1)$-colimit.

=--

### Over formal duals of general cdg-algebras

(...)

## Applications

* [[dg-manifold]], [[dg-scheme]]

* [[Hochschild cohomology]]: section Higher order HH modeled on cdg-algebras;

* [[rational homotopy theory]]: [[function algebras on ∞-stacks]]

* [[perfect ∞-stack]]s and their [[geometric ∞-function theory]]

* [[BRST-BV complex]]

## References

Various [[model category]] presentations of dg-geometry are presented in

* [[Bertrand Toën]], [[Gabriele Vezzosi]], _Homotopical Algebraic Geometry II: geometric stacks and applications _ ([arXiv](http://arxiv.org/abs/math/0404373))
{#ToenVezzosiStacks}

The [[geometric ∞-function theory]] of [[perfect ∞-stack]]s in dg-geometry, and the corresponding [[Hochschild cohomology]] is considered in

* [[David Ben-Zvi|Ben-Zvi]], [[John Francis]],  [[David Nadler]], _Integral transforms and Drinfeld centers in derived algebraic geometry_ ([arXiv](http://arxiv.org/abs/0805.0157))
{#Ben-ZviFrancisNadler}

The $(\mathcal{O} \dashv Spec)$-adjunction for dg-geometry is studied in

* [[David Ben-Zvi]], [[David Nadler]], _Loop Spaces and Connections_ ([arXiv:1002.3636](http://arxiv.org/abs/1002.3636))
{#Ben-ZviNadler}

The basic reference for the [[model structure on dg-algebras]] (see there for more details) for the commutative case over a field of characteristic 0 is

* [[Vladimir Hinich]],  _Homological algebra of homotopy algebras_ Communications in algebra, 25(10). 3291-3323 (1997)([arXiv:q-alg/9702015](http://arxiv.org/abs/q-alg/9702015), _Erratum_ ([arXiv:math/0309453](http://arxiv.org/abs/math/0309453)))
{#Hinich}

Details on the use of this [[model category]] structure for modelling dg-spaces are in

* [[Kai Behrend]], _Differential graded schemes I: prefect resolving algebras_ ([arXiv:0212225](http://arxiv.org/abs/math/0212225))

* [[Kai Behrend]], _Differential Graded Schemes II: The 2-category of Differential Graded Schemes_ ([arXiv:0212226](http://arxiv.org/abs/math/0212226))


dg-ideal

A **dg-ideal** $J \subset A$ is an alternative name for an [[ideal]] in a [[dg-algebra]] $A$, i.e. a $A$-$A$-dg-subbimodule of $A$. Of course, one can also talk about one-sided ideals.

This means that $J$ is a graded sub-vector space that is closed under product by element in $A$ from either side, and closed under acting with the differential.


dg-localization

## Idea

_dg-localization_ is the analogue in the world of [[dg-categories]] to the notion of [[simplicial localization]]. In good cases it is [[dg-category presented by a model category|presented]] by a [[dg-model category]].

## Definition

+-- {: .un_defn}
###### Definition
The **dg-localization** of a [[dg-category]] $T$ at a subset of morphisms $S$ is the data of a morphism
  $$ \gamma : T \longrightarrow T[S^{-1}] $$
in the [[(infinity,1)-category of dg-categories]] $dg-cat$ such that for any [[dg-category]] $T' \in dg-cat$ the induced morphism
  $$ \gamma^* : Map(T[S^{-1}], T')
       \longrightarrow Map(T, T') $$
of [[mapping spaces]] is [[injective]] on [[connected components]] and has [[image]] the subset of morphisms $T \to T'$ in $dgcat$ that send morphisms of $S$ to [[equivalences]] in $T'$.
=--

+-- {: .un_proposition}
###### Proposition (Toen)
The dg-localization of any [[dg-category]] at a set of [[morphisms]] exists.
=--

## Related concepts

* [[dg-quotient]]
* [[simplicial localization]]

## References

See section 8.2 of

* [[B. Toen]], _The homotopy theory of dg-categories and derived Morita theory_, [arXiv:math/0408337](http://arxiv.org/abs/math/0408337).

and section 4.3 of

* [[B. Toen]], _Lectures on dg-categories_, [pdf](http://www.hamilton.tcd.ie/events/swisk.pdf).


dg-model category

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

## Idea

A _dg-model category is the analogue of a [[simplicial model category]] in the [[dg-category|dg]]-setting. Any dg-model category [[dg-category presented by a dg-model category|presents]] an associated [[dg-category]], as described at [[dg-category presented by a dg-model category]].

## Definition

A **dg-model structure** is a $dgmod_k$-[[enriched model structure]], where $dgmod_k$ denotes the category of [[dg-modules]] over the base [[commutative ring]] $k$.

[[!redirects dg-model categories]]

[[!redirects dg-model structure]]
[[!redirects dg-model structures]]


dg-module

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

#Contents#
* table of contents
{:toc}

## Idea

The notion of _dg-module_ over a [[dg-algebra]] is the specialization of [[module over a monoid]] to the category of [[chain complexes]]. More generally one can consider dg-modules over [[dg-algebras]] with many objects, i.e. [[dg-categories]].

## Definition

### Over dg-algebras

Let $A$ be a [[dg-algebra]] in an [[abelian category]], hence a [[monoid]] in the [[symmetric monoidal category|symmetric monoidal]] [[category of chain complexes]] with respect to the [[tensor product of chain complexes]].

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

A left (resp. right) **dg-module** over $A$ is a left (resp. right) [[module over a monoid|module]] over the [[monoid]] $A$.

Spelling this out in components, it means the following:

A _[[dg-module]]_ over $A$ is a [[chain complex]] $V$ equipped with a [[chain map]]

$$
  \rho \;\colon\; A \otimes V \longrightarrow V
$$

out of the [[tensor product of chain complexes]], which satisfies the [[action]] property. Explicitly this means that

1. $\rho$ preserves degrees: for all $a \in A$ and $v \in V$ of homogeneous degrees ${\vert a \vert}$ and ${\vert v \vert}$, respectively, then $\vert \rho(a,v)\vert = {\vert a \vert} + {\vert v \vert}$;

1. for all $a \in A$, $v \in V$ with $a$ in degree $\vert a \vert$, then

   $$
     d_V (\rho(a,v)) = \rho(d_A a, v) + (-1)^{\vert a\vert} \rho(a,d_V v)
   $$

1. for all $a,b \in A$, $v \in V$ then

   $$
     \rho(a,\rho(b,v)) = \rho(a \cdot b,v)
     \,.
   $$

For $(V_i, \rho_i)$ two [[dg-modules]] over $A$, then a [[homomorphism]] between them is a [[chain map]]

$$
  \phi \;\colon\; V_1 \longrightarrow V_2
$$

such that for all $a \in A$ and $v \in V$ then

$$
  \rho_2(a, \phi(v)) = \phi(\rho_1(a,v))
  \,.
$$

This defines a [[category]] $A Mod$ of [[dg-modules]] over $A$.

We say that a morphism of dg-modules is a _[[quasi-isomorphism]]_ if its underlying [[chain map]] is, hence if it induces an isomorphism on [[chain homology]]. This makes $A Mod$ a [[category with weak equivalences]].

=--

### Over dg-categories

More generally let $T$ be a [[dg-category]].

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

A left (resp. right) **dg-module** over $T$ is a [[module over an enriched category|module]] over $T$ in the sense of [[enriched category theory]]. That is, it is a functor of [[dg-categories]]
  $$ M : T^{op} \longrightarrow dg-mod_k $$
(resp. $M : T \longrightarrow dg-mod_k$) where $k$ is the base [[commutative ring]].
=--

Note that when $T$ is a [[dg-category]] with a single object, a dg-module over $T$ is the same thing as a dg-module over the [[dg-algebra]] of [[endomorphisms]] of the unique object.

+-- {: .num_defn}
###### Definition
The **shift** of a dg-module $M$ over $T$ by an [[integer]] $n$ is the dg-module $M[n]$ defined by composing $M$ with the shift [[endofunctor]] $-[n]$ on $dg-mod_k$.
=--

## Properties

By general [[enriched category theory]], dg-modules themselves form a [[dg-category]] which we denote $dg-mod_T$.

+-- {: .num_defn}
###### Definition
A dg-module $M \in dg-mod_T$ is called **representable** if it is in the [[essential image]] of the [[dg-Yoneda embedding]].

$M$ is called **free** if it is equivalent to a [[direct sum]] of shifts of representable dg-modules.

$M$ is called **semi-free** if it admits a [[filtration]] whose [[associated graded objects]] are free dg-modules.
=--

## Related concepts

* [[minimal dg-module]]

* [[dg-comodule]]

## References

* [[Bertrand Toen]], section 3 in _The homotopy theory of dg-categories and derived Morita theory_, [arXiv:math/0408337](http://arxiv.org/abs/math/0408337).

Paragraph 2.2 of

* [[Dmitri Orlov]], _Smooth and proper noncommutative schemes and gluing of DG categories_, [arXiv:1402.7364](http://arxiv.org/abs/1402.7364).

[[!redirects dg-modules]]

[[!redirects dg-presheaf]]
[[!redirects dg-presheaves]]


dg-nerve

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
#### Enriched category theory
+--{: .hide}
[[!include enriched category theory contents]]
=--
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

> This section is no good...

Any pretriangulated [[dg-category]] $\mathcal{C}$ presents a [[stable (infinity,1)-category]]. A plain dg-category only presents a spectrally enriched (infinity,1)-category. One way to construct this is to apply the [[Dold-Kan correspondence]] on each [[hom-object]] to produce a fibrant [[sSet-enriched category]] and then, if desired, form the [[homotopy coherent nerve]] of that to obtain a [[quasi-category]].

On the other hand, the _dg-nerve_ of $\mathcal{C}$ is a more direct construction that directly sends the dg-category to a [[simplicial set]] which is the [[quasi-category]] incarnation of the corresponding [[stable (∞,1)-category]].

To the extent that one may think of $\mathcal{C}$ as analogous to a [[category of chain complexes]], the dg-nerve may be thought of producing the [[simplicial set]] whose $k$-[[simplices]] are the [[local systems]] on $\Delta^k$ with [[coefficients]] in $\mathcal{C}$ ([[flat ∞-connections]] with coefficients in $\mathcal{C}$). The formula is just as for [[Lie integration]] of [[L-infinity algebroids]].

## Properties

### Compatibility with the simplicial nerve
 {#CompatibilityWithSimplicialNerve}

Recall:

\begin{remark}\label{QuasiCategoricalLocalization}
From a [[relative category]] $(\mathcal{C}, \mathrm{W})$ (a [[category with weak equivalences]] $\mathrm{W} \subset Mor(\mathcal{C})$) one obtains:

1. the [[sSet-enriched category]] $\mathcal{C}[\mathrm{W}^{-1}]$ which is its [[simplicial localization]],

1. the [[quasi-category]] which is the [[homotopy coherent nerve]] $N \mathcal{C}[\mathrm{W}^{-1}]$ of that.

or alternatively, up to [[equivalence of quasi-categories]]:

1. the [[simplicial nerve]] $N \mathcal{C}$,

1. its [[localization of a quasi-category|localization as a quasi-category]] $N\mathcal{C}[\mathrm{W}^{-1}]$.

\end{remark}

Now:

\begin{proposition}
For $\mathcal{A}$ an [[additive category]], write

1. $Ch_\bullet(\mathcal{A})$ for the plain [[category of chain complexes]] in $\mathcal{A}$,

1. $\mathbf{Ch}_\bullet(\mathcal{A})$ for the [[dg-category]] obtained form the [[enriched category|self-enrichment]] of $Ch_\bullet(\mathcal{A})$

   (via its [[symmetric monoidal category|symmetric monoidal]] structure given by the [[tensor product of chain complexes]]),

1. $\mathrm{W}_{che} \,\subset\, Mor\big(Ch_\bullet(\mathcal{A})\big)$ for the [[class]] of [[chain homotopy equivalences]]

   (which for $\mathcal{A} = k$ [[Vect]] coincide with the [[quasi-isomorphisms]], but not otherwise).

Then the dg-nerve of $\mathbf{Ch}_\bullet(\mathcal{A})$ is [[equivalence of quasi-categories|equivalent as a quasi-category]] to the (homotopy coherent) nerve of the (simplicial) localization of $Ch_\bullet(\mathcal{A})$ at $\mathrm{W}_{che}$ in the sense of Rem. \ref{QuasiCategoricalLocalization}:

$$
  N_{dg}
  \big(
    \mathbf{Ch}_\bullet(\mathcal{A})
  \big)
  \;\;
  \simeq
  \;\;
  N Ch_\bullet(\mathcal{A})[\mathrm{W}^{-1}_{che}]
  \,.
$$
\end{proposition}
[[Lurie (2017), Prop. 1.3.4.5](#Lurie17)]

Similarly:
\begin{proposition}
  Given $\mathbf{C}$ a [[category of chain complexes|$Ch_\bullet(R Mod)$]]-[[enriched model category]] (over some [[ring]] $R$) all whose [[objects]] are [[cofibrant object|cofibrant]], then
$$
  N_{dg}(\mathbf{C})
  \;\;
  \simeq
  \;\;
  N C[\mathrm{W}_{che}^{-1}]
$$
where $C$ denotes the [[underlying]] ordinary category and $\mathrm{W}_{che} \subset Mor(C)$ again the class of [[chain homotopy equivalences]].
\end{proposition}
[[Gwilliam & Pavlov (2018), Prop. 5.17](#GwilliamPavlov18)]

## References

The definition originates with:

* {#BlockSmith14} [[Jonathan Block]], [[Aaron M. Smith]], Def. 2.3 (2.4 in the preprint) of: *The higher Riemann--Hilbert correspondence*, Advances in Mathematics **252** (2014) 382-405 [[arXiv:0908.2843](https://arxiv.org/abs/0908.2843), [doi:10.1016/j.aim.2013.11.001](https://doi.org/10.1016/j.aim.2013.11.001)]

* {#Lurie17} [[Jacob Lurie]], Construction 1.3.1.6 in: *[[Higher Algebra]]* (2017)

Extension to [[A-infinity categories|$A_\infty$-categories]], proof that the dg-nerve maps [[pretriangulated dg-categories]] to [[stable (∞,1)-categories]].

* {#Faonte13} [[Giovanni Faonte]], *Simplicial nerve of an $\mathcal{A}_\infty$-category*, Theory and Applications of Categories **32** 2 (2017) 31-52 [[arXiv:1312.2127](http://arxiv.org/abs/1312.2127), [tac:32-02](http://www.tac.mta.ca/tac/volumes/32/2/32-02abs.html)]

Further discussion:

* {#GwilliamPavlov18} [[Owen Gwilliam]], [[Dmitri Pavlov]], *Enhancing the filtered derived category*, Journal of Pure and Applied Algebra **222** 11 (2018) 3621-3674 [[arXiv:1602.01515](https://arxiv.org/abs/1602.01515), [doi:10.1016/j.jpaa.2018.01.004](https://doi.org/10.1016/j.jpaa.2018.01.004)]

[[!redirects dg-nerves]]
[[!redirects nerve of a dg-category]]
[[!redirects nerves of dg-categories]]


dg-operad

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

#Contens#
* table of contents
{:toc}

## Definition

A **dg-operad** is an [[operad]] in a [[category of chain complexes]].

## Properties

There is a [[model structure on dg-operads]].

## Related concepts

* [[topological operad]]

* **dg-operad**

[[!redirects dg-operads]]


dg-quiver

##Idea##

As explained in that entry, '[[quiver]]' is often used to refer to a [[directed graph]], but, as is argued at that entry, this is almost always then used to derive the free category on the that directed graph. The notion of a differential graded quiver, as studied by Lyubashenko and Manzyuk (see reference below), is an enriched analogue of directed graph, enriched over the category of differential graded modules over a commutative ring $k$. (It only uses the underlying category in fact, not the tensor structure). It is introduced _precisely_ to define the analogue of the [[path category]] of a directed graph in this enriched context *but* the structure they define is in fact a [[A-infinity category|linear A-infinity category]].

##Definition##
A **differential graded $k$-quiver** (or simply **dg-quiver**) $\mathcal{Q}$ is specified by

* a small set of _objects_, denoted $Ob(\mathcal{Q})$;

* for each pair $(x,y)$ of objects of $\mathcal{Q}$, a [[chain complex]] (= graded $k$-module with differential of degree +1) denoted $\mathcal{Q}(x,y)$.

(More work to be done here.)
###References###

*  V. Lyubashenko and O. Manzyuk, _Free $A_\infty$-categories_,  Theory and Applications of Categories, Vol. 16, 2006, No. 9, pp 174-205.([pdf](http://www.tac.mta.ca/tac/volumes/16/9/16-09.pdf))

*  J. P. May, _Operadic categories, $A_\infty$-categories, and $n$-categories_, Talk given in Morelia, Mexico, May 25, 2001 available at: ([pdf](http://www.math.uchicago.edu/~may/NCATS/PostMexico.pdf))

[[Note on Formatting|✄]]


dg-scheme

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

#Contents#
* table of contents
{:toc}

## Idea

The notion of _dg-schemes_ was introduced by [[Maxim Kontsevich]] as
the first approach to [[derived algebraic geometry]], and was further
developed by [[Mikhail Kapranov]] and [[Ionut Ciocan-Fontanine]].

## Definition

A **differential graded scheme** (**dg-scheme**) is a [[scheme]] $(X,
O_X)$ together with a [[sheaf]] $O_X^\bullet$ of nonnegatively graded
commutative [[dg-algebra|differential graded]] $O_X$-[[dg-algebra|algebras]], such that $O_X \to H^0(O_X^\bullet)$ is
[[surjective]].

## Examples

[[Ciocan-Fontanine]] and [[Kapranov]] construct $RQuot(X, F)$, a
derived enhancement of the classical [[Quot scheme]] parametrizing
subsheaves of a given [[coherent sheaf]] $F$ on a smooth projective
[[variety]] $X$ [(1999)](#CiKa1).  Similarly they also construct a
dg-scheme $RHilb_h(X)$, a derived enhancement of the [[Hilbert
scheme]] parametrizing [[subschemes]] of a given [[projective scheme]]
$X$ with [[Hilbert polynomial]] $h$ [(2000)](#CiKa2).  As an
application they construct the [[derived moduli stack]] of [[stable
maps]] of [[curves]] to a given [[projective variety]].

## Relation with derived stacks

There is a [[functor]] from the [[category]] of dg-schemes to the
category of [[derived stacks]] of [[Bertrand Toen]] and [[Gabriele
Vezzosi]].  It takes values in the [[full subcategory]] of 1-geometric
derived stacks, but is not known (or expected) to be [[fully
faithful]].

In particular, the dg-schemes $RQuot(X, F)$ and $RHilb_h(X)$,
discussed above, also induce [[derived stacks]] in the modern sense.

## Related concepts

* [[dg-manifold]]
* [[dg-geometry]]
* [[derived stack]]

## References

A prediction of [[derived moduli spaces]] is spelled out (in a bit different language) in

* M. Kontsevich, _Enumeration of rational curves via torus actions_. The moduli space of curves (Texel Island, 1994), 335--368, Progr. Math. __129__, Birkhäuser 1995. MR1363062 (97d:14077), [hep-th/9405035](http://arxiv.org/abs/hep-th/9405035).
{#KontsevichEnum}

The first examples of [[derived moduli spaces]], using dg-schemes, are constructed in

* [[Ionut Ciocan-Fontanine]], [[Mikhail Kapranov]], _Derived Quot
schemes_, 1999,
[arXiv:math/9905174](http://arxiv.org/abs/math/9905174).
{#CiKa1

* [[Ionut Ciocan-Fontanine]], [[Mikhail Kapranov]], _Derived Hilbert
schemes_, 2000,
[arXiv:math/0005155](http://arxiv.org/abs/math/0005155).
{#CiKa2}

* M. Kapranov, _Injective resolutions of BG and derived moduli spaces of local systems_,  J. Pure Appl. Algebra  155  (2001),  no. 2-3, 167--179; [math/alg-geom/9710027](http://arxiv.org/abs/alg-geom/9710027), MR1801413 (2002b:18017)
 {#Kapranov}

* [[Kai Behrend]], _Differential graded schemes I: prefect resolving
algebras_ ([arXiv:0212225](http://arxiv.org/abs/math/0212225))

* [[Kai Behrend]], _Differential Graded Schemes II: The 2-category of
Differential Graded Schemes_
([arXiv:0212226](http://arxiv.org/abs/math/0212226))

[[!redirects dg-schemes]]
[[!redirects dg scheme]]
[[!redirects dg schemes]]
[[!redirects differential graded scheme]]
[[!redirects differential graded schemes]]


dg-Yoneda embedding

## Idea

The _dg-Yoneda embedding_ is the analogue of the [[Yoneda embedding]] in the setting of [[dg-categories]].

## Definition

Let $dg-mod_T$ denote the [[dg-category]] of [[dg-modules]] over $T$. The [[enriched Yoneda lemma|enriched Yoneda embedding]] is a [[fully faithful]] functor of [[dg-categories]]
  $$ h : T \hookrightarrow dg-mod_T. $$

This induces the [[dg-Yoneda embedding]]
  $$ h : T \hookrightarrow D(T) $$
into the [[derived dg-category of a dg-category|derived dg-category]] of $T$, which is a full sub-[[dg-category]] of $dg-mod_T$.

## References

See the references at [[dg-module]].


dgAlg

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

$dg Alg$ is the [[category]] whose [[objects]] are [[dg-algebra]]s and whose [[morphisms]] are dg-algebra [[homomorphisms]].

category: category


dgc-coalgebra

## Idea

A [[differential graded coalgebra]] which is graded-co-commutative.

The linear dual of the concept of _[[dgc-algebras]]_

## Examples

The chains [[Chevalley-Eilenberg algebra]] of any [[L-infinity algebra]].

[[!redirects dgc-coalgebras]]

[[!redirects differential graded-cocommutative coalgebra]]
[[!redirects differential graded-cocommutative coalgebras]]


dgcAlg

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

#Contents#
* table of contents
{:toc}

## Idea

The [[category]] $dgcAlg$ whose [[objects]] are [[differential graded-commutative algebras]] and whose [[morphisms]] are [[homomorphisms]] between these (i.e. [[algebra homomorphisms]] that respect the [[graded object|grading]] and the [[differential]]).

## Properties

### Model structure

* [[model structure on dgc-algebras]]

### Full subcategories

Important [[full subcategories]] include thos of

* [[connective]] dgc-algebras $dgcAlg_{\geq 0}$ which are concentrated in non-positive degree;

* [[semifree dgc-algebras]], equivalently [[Chevalley-Eilenberg algebras]] of [[L-infinity algebras]]

## Related concepts

* [[Alg]], [[Mod]]

* [[rational homotopy theory]]

* [[monoidal Dold-Kan correspondence]]

* [[Lie integration]]

## References

See the references at _[[model structure on dgc-algebras]]_.

category: category

[[!redirects dgcAlgebras]]

[[!redirects category of dgc-algebras]]
[[!redirects categories of dgc-algebras]]

[[!redirects category of differential graded-commutative algebras]]
[[!redirects categories of differential graded-commutative algebras]]

[[!redirects category of connective dgc-algebras]]
[[!redirects categories of connective dgc-algebras]]

[[!redirects category of connective differential graded-commutative algebras]]
[[!redirects categories of connective differential graded-commutative algebras]]


DHR category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### AQFT
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}

## Idea ##

Given a [[quantum field theory]] presented by a [[local net of observables]] ([[AQFT]])

$$
  \mathcal{A} : Open(X) \to Algebras
$$

a _local endomorphism_ is a [[natural transformation|natural]] [[associative algebra|algebra]] [[homomorphism]] $\rho : \mathcal{A} \to \mathcal{A}$ which is supported (nontrivial) on a [[compact space|compact]] region of [[spacetime]] $X$.

These _local endomorphism_ are physically interpreted as local [[charges]]. By the locality of the local net, one finds that local endomorphisms naturally form a [[braided monoidal category]]. This is called the _DHR category_.

The DHR category is thus built from data used in [[DHR superselection theory]] and is used to provide a simplified proof of the [[Doplicher-Roberts reconstruction theorem]].

## Abstract ##

After the definition of objects and arrows we show several structures that the DHR category has.

## Definition ##

See [[DHR superselection theory]] and [[Haag-Kastler vacuum representation]] for the terminology used here.

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

The transportable endomorphisms are the **[[objects]]** of the DHR category $\Delta$.

=--

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

For two transportable endomorphisms the **set of [[intertwiners]]** are the **[[morphisms]]**.

=--

## Properties ##

### DHR is a C-star-category with a direct product ###

It is straightforward to see that $\Delta$ is a [[category]]:

The [[identity]] morphism for each object in $\Delta$ is given by the identiy in $\mathcal{A}$. The [[composition]] of arrows is simply the composition of [[intertwiners]]:

From

$$
  R \rho_1 = \rho_2 R
$$

$$
  T \rho_2 = \rho_3 T
$$

follows

$$
  T R \rho_1 = \rho_3 T R
$$

Several structural properties follow immediatly from the definition:

+-- {: .num_lemma }
###### Lemma

$\Delta$ is a $\mathbb{C}-$[[algebroid]].

=--

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

$\Delta$ is a [[dagger-category]] since, if $R$ is an [[intertwiner]] of the pair $(\rho_1, \rho_2)$, then $R^*$ is obviously an [[intertwiner]] of the pair $(\rho_2, \rho_1)$.

=--

Combining these two structures we get that $\Delta$ is a [[star-category]].

Since the arrows inherit a norm, we actually get

+-- {: .num_lemma }
###### Lemma

 $\Delta$ is a [[C-star-category]].

=--

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

It is possible to introduce a finite [[direct product]] in $\Delta$, if the net satisfies the [[Borchers property]].

=--

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

The [[Haag-Kastler vacuum representation]] that we talk about here satisfies the [[Borchers property]].

=--

+-- {: .proof}
###### Sketch of the Proof

Let $\pi_1, \pi_2$ be admissible representations and $\rho_1, \rho_2$ be their transportable [[localized endomorphism|endomorphisms localized]] in $K_1, K_2$ respectively. Choose a double cone $K_0 \in \mathcal{J}_0$ that contains $K_1$ and $K_2$. Since the local von Neumann algebra $\mathcal{M}(K_0)$ is not trivial, it contains a nontrivial projection $E$, that is $0 \lt E \lt \mathbb{1}$.

Thanks to the Borchers property there is a double cone $K$ containing the closure of $K_0$, and partial isometries $W_1, W_2 \in \mathcal{M}(K)$ such that $W_1 W_1^* = E, W_2 W_2^* = \mathbb{1} - E$.

Now we set
$$
\rho := W_1 \rho_1 W_1^* + W_2 \rho_2 W_2^*
$$
It is possible to show that $\pi_0 \rho$ is unitarily equivalent to $\pi_1 \oplus \pi_2$ and that $\rho$ is a transportable (and therefore in particular a localized) endomorphism. So we will call $\rho$ a **direct sum** of $\rho_1$ and $\rho_2$.
=--

### DHR is a symmetric monoidal category ###

We first define the "[[tensor product]]":

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

For endomorphisms we set $\rho_1 \otimes \rho_2 := \rho_1 \rho_2$.

For intertwiners $S \in Hom(\rho, \rho^{\prime})$ and $T \in Hom(\sigma, \sigma^{\prime})$ we define the tensor product via $S \otimes T := S \rho(T)$.

=--

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

In the [[AQFT]] literature the tensor product of arrows is sometimes called the _crossed product_ of intertwiners.

=--

+-- {: .num_lemma }
###### Lemma

The tensor product as defined above turns $\Delta$ into a [[monoidal category]].

=--

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

First: The tensor product of arrows is well defined, for any $A \in \mathcal{A}$ we have:

$$
S \otimes T (\rho_1 \otimes \rho_2) (A)

= (S \rho(T)) \rho (\sigma(A))

= S \rho(T \sigma(A))

= \rho^{\prime} (T \sigma(A)) S

= \rho^{\prime} (\sigma^{\prime}(A) T) S

= \rho^{\prime} \sigma^{\prime}(A) \rho^{\prime}(T) S

= \rho^{\prime} \sigma^{\prime}(A) S \rho(T)

$$

which shows that the tensor product of intertwiners is an intertwiner for the tensor product of endomorphisms. The unit object is the identity endomorphism $\mathbb{1} \in \mathcal{A}$, left and right unitor and the associator are the identities, that is, $\Delta$ is **strict**.

=--

Now to the braiding. The braiding is [[symmetric monoidal category|symmetric]] in $d \ge 3$ dimensions only, this has a topological reason: The causal complement of a double cone is pathwise connected in $d \ge 3$ dimensions only, but not in $d \le 2$ dimensions.

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

When we talk about $d = 1$ dimensions we mean one space dimension and zero time dimensions, so that a double cone in this context is simply an open interval, and spacelike separated double cones are simply disjoint open intervals.

To define the braiding we will need the following concepts:

=--

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

For transportable endomorphisms $\rho, \sigma$ choose causally separated double cones $K_1 \perp K_2$ and $\rho_0 \in \hat \rho$ localized in $K_1$ and $\sigma_0 \in \hat \sigma$ localized in $K_2$. These endomorphisms $\rho_0, \sigma_0$ are then called **spectator endomorphisms** of $\rho$ and $\sigma$.

=--

Note that it is a theorem that causally disjoint transportable endomorphisms commute, therefore spectator endomorphisms always commute.

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

For transportable endomorphisms $\rho, \sigma$ and spectator endomorphisms $\rho_0, \sigma_0$ choose unitary interwiners $U \in Hom(\rho, \rho_0)$ and $V \in \Hom(\sigma, \sigma_0)$. Such unitaries are called **transporters**.

=--

Obviously both spectator endomorphisms and transporters are not unique, in general.

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

For transportable endomorphisms $\rho, \sigma$, spectator endomorphisms $\rho_0, \sigma_0$ and transporters U, V define the **permutator** or **permutation symmetry** via
$$
\epsilon(\rho, \sigma) := (V^* \otimes U^*) (U \otimes V)
$$

=--

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

The permutators are well defined and independent of the choice of spectator endomorphisms and transporters.

=--

## References ##

See at _[[DHR superselection theory]]_.

[[!redirects DHR categories]]


DHR superselection theory

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

#Contents#
* the following line creates the automatic table of contents
{:toc}

## Idea ##
The DHR superselection theory is about [[superselection sectors]] in the [[Haag-Kastler approach]] to [[AQFT]].
As such, it has to state one or more conditions on representations of the given [[Haag-Kastler net]] that specify the representations of the quasi-local algebra that are deemed physically admissible.

It is named after [[Sergio Doplicher]], [[Rudolf Haag]] and [[John Roberts]].

In the following we consider the theory starting with a [[Haag-Kastler vacuum representation]] on [[Minkowski spacetime]].

The DHR condition says that **all expectation values (of all observables) should approach the vacuum expectation values, uniformly, when the region of measurement is moved away from the origin.** This is one possible abstraction of the situation in a high energy particle collider, where the registered events are localized both in time and in space.

The DHR condition excludes long range forces like electromagnetism from consideration, because, by [[Stokes' theorem]], the electric charge in a finite region can be measured by the flux of the field strength through a sphere of arbitrary large radius. The DHR analysis is of interest nevertheless, because it has reached a certain maturity and is therefore an excellent object to study: see for example the [[Doplicher-Roberts reconstruction theorem]].

## Abstract ##

After the definition of admissible representations we collect some notions that will enable us to state a description of all admissible representations using intrinsic properties of the quasi-local algebra. The central concept that we define is that of a **transportable endomorphism**, these form the objects of a [[category]], the [[DHR category]].

## Definition ##
We start with a [[Haag-Kastler vacuum representation]] that we assume to be irreducible and [[Haag dual]].

Let $\pi_0$ be the vacuum representation from now on, $K$ denote double cones and $\mathcal{J}_0$ be the [[causal index set]] of double cones, just as $\mathcal{O}$ are bounded open sets and $\mathcal{J}$ the [[causal index set]] of bounded open sets.

Recall that the [[C-star algebra]]

$$
\mathcal{A} := clo_{\| \cdot \|} \bigl( \bigcup_{\mathcal{O}\in\mathcal{J}}\mathcal{M}(\mathcal{O}) \bigr)
$$

is called the quasi-local algebra of the given net. We will in the same way associate a [[C-star algebra]] to every open region $\mathcal{O}_0$ via

$$
\mathcal{A}(\mathcal{O}_0) := clo_{\| \cdot \|} \bigl( \bigcup_{\mathcal{O}_0 \supseteq \mathcal{O}\in\mathcal{J}}\mathcal{M}(\mathcal{O}) \bigr)
$$

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

A representation $\pi$ of the local algebra $\mathcal{A}$ is called (DHR) **admissible** if $\pi | \mathcal{A}(\mathcal{K}^{\perp})$ is unitarily equivalent to $\pi_0 | \mathcal{A}(\mathcal{K}^{\perp})$ for all $K \in \mathcal{J}_0$, that is, for all double cones $K$.
=--

Note that the equivalence of the representations is required for the causal complement of _all_ double cones, not a special one, so that the characterization that "the representations can not be distinguished at space-like infinity" is misleading.

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

Recall that a mapping
$$
\rho: \mathcal{A} \to \mathcal{A}
$$
is called a **unital endomorphism** if $\rho$ is linear, multiplicative ($\rho(AB) = \rho(A) \rho(B)$) and $\rho(\mathbb{1})=1$.
=--

We will drop "unitarily" from now on.

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

For any representation $\pi$ and endomorphism $\rho$ the composition $\pi \circ \rho$ is another representation.

Conversely, under the assumption of [[Haag duality]] every DHR admissible representation $\rho$ comes from an endomorphism this way. For let $A \in \mathcal{A}(K)$ and $B \in \mathcal{A}(K^\pert)$ be any two causally unrelated localized obserbales. Then for $\rho : \mathcal{A} \to \mathcal{B}(\mathcal{H})$ a DHR representation, we have on the one hand

$$
  \rho(A B) = \rho(A) \pi_0(B)
$$

and on the other, due to the causal structure

$$
  \rho(A B) = \rho(B A) = \pi_0(B) \rho(A)
  \,.
$$

This says that $\rho(A) \in \pi_0(\mathcal{A}(K'))'$ and hence by [[Haag duality]] $\cdots \in \pi_0(\mathcal{A}(K))''$. Under the assumption that the [[local net of observables]] takes values in [[von Neumann algebras]] this in turn in $\cdots = \pi_0(\mathcal{A}(K))$ and so $\rho$ indeed factors as an endomrphism of $\mathcal{A}$ followed by the vaccum representation.

=--

For this reason one can hope to gain some insights into the representations by studying the endomorphisms, while the set of endomorphisms has certainly more structure than that of representations: For example, endomorphisms may have inverses, and endomorphisms form a [[monoid]] by composition.

First we define "unitarily equivalent" and "intertwiner" analog to the definition for [[representations of C-star algebras]]:

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

Two endomorphisms $\rho_1, \rho_2$ are **unitarily equivalent** if there is a [[unitary operator]] $U \in \mathcal{A}$ such that $\rho_1 = ad(U) \rho_2 = U \rho_2 U^{-1}$. The endomorphisms $ad(U)$ with $U \in \mathcal{A}$ are called **inner automorphisms** (of $\mathcal{A}$).
An element $R \in \mathcal{A}$ such that $R \rho_1 = \rho_2 R$, not necessarily unitary, is called an **intertwiner** or **intertwining operator**.
=--

The DHR selection criterion selects representations that "look like the vaccum" on the causal complement of elements of our index set (in this case the double cones). The analog for endomorphims is this:

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

An endomorphim $\rho$ is **[[localized endomorphism|localized]]** or **localizable** if there is a bounded open set $\mathcal{O} \in \mathcal{J}$ such that $\rho$ is the identity on the algebra of the causal complement $\mathcal{A}(\mathcal{O}^{\perp})$. Such an endomorphism is **localized in $\mathcal{O}$**.
=--

We have to restrict the notion of "unitarily equivalent" to a "localized" version accordingly:

+-- {: .num_defn}
###### Definition
An element $A \in \mathcal{A}$ is a **local operator** if there is a double cone K such that $A \in \mathcal{M}(K)$. In particular, a local unitary operator $U$ is a **local unitary**, and two endomorphisms are **locally unitarily equivalent** if there is a unitary as in the definition of unitarily equivalent that is a local operator.
=--

For an endomorphism $\rho$ let $\hat \rho$ be the equivalence class with respect to locally unitarily equivalence. Then we can define:

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

A [[localized endomorphism]] is **transportable** if for every bounded open set $\mathcal{O} \in \mathcal{J}$ there is a $\rho_0 \in \hat \rho$ that is localized in  $\mathcal{O}$.
=--

Transportable endomorphisms naturally form the objects of a category, the [[DHR category]].

## Properties ##

### General

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

**transportable endomorphisms are compatible with the net structure**

Let $\rho$ be a transportable endomorphism that is localized in the double cone $K_0$, then for all double cones $K \supset K_0$, $\rho$ maps $\mathcal{M}(K)$ to itself.
=--

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

**intrinsic characterization of admissible representations**

A representation $\pi$ of $\mathcal{A}$ is admissible iff there is a transportable endomorphism $\rho$ such that $\pi$ is unitarily equivalent to $\pi_0 \circ \rho$.
=--

Reference: This is theorem 2.1.3 in the book by Baumgärtel.

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

**stability of admissible representations**

(i)  finite direct sums of admissible representations are admissible.

(ii) subrepresentations of admissible representations are admissible.
=--

For transportable endomorphisms we get even more:

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

**products**
The product, i.e. the concatenation, of transportable endomorphisms is a transportable endomorphism.
=--

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

**product of causally disjoint localized endomorphisms is commutative**

Let $\rho_1, \rho_2$ be transportable endomorphisms localized in $K_1, K_2$ respectivley with $K_1 \perp K_2$, then $\rho_1 \rho_2 = \rho_2 \rho_1$.
=--

Note that the product of equivalenc classes is well defined, and $\hat \rho_1 \hat \rho_2 = \widehat{\rho_1\rho_2}$. Therefore the theorem above implies that the product of equivalence classes is commutative.

### In terms of local net cohomology

The DHR superselection theory has a reformulation in terms of [[cohomology of local net of observables|cohomology of local nets of observables]]. For the moment, see there for more.

## References ##

see [[AQFT]] ff.

Of particular relevance (besides the original work of Doplicher and Roberts) are

* Hellmut Baumgärtel: _Operator algebraic Methods in Quantum Field Theory. A series of lectures._ Akademie Verlag 1995 ([ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0839.46063&format=complete))

* [[Hans Halvorson]], [[Michael Müger]], _Algebraic Quantum Field Theory_ ([arXiv:math-ph/0602036](http://arxiv.org/abs/math-ph/0602036))

Discussion in the context of [[holographic entanglement entropy]]:

* {#CHMP19} Horacio Casini, Marina Huerta, Javier M. Magan, Diego Pontello, _Entanglement entropy and superselection sectors I. Global symmetries_ ([arXiv:1905.10487](https://arxiv.org/abs/1905.10487))

[[!redirects DHR analysis]]
[[!redirects Doplicher-Haag-Roberts superselection theory]]
[[!redirects Doplicher-Haag-Roberts superselection sector]]
[[!redirects Doplicher-Haag-Roberts superselection sectors]]


Diaconescu's theorem

> This entry is about the theorem in [[topos theory]]. For the theorem in [[logic]] that often goes by the same name, see at _[[Diaconescu-Goodman–Myhill theorem]]_.

***

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

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

## Idea

_Diaconescu's theorem_ asserts that any [[presheaf topos]] is the [[classifying topos]] for [[internally flat functors]] on its [[site]].

Often a special case of this is considered, which asserts that for every [[topological space]] $X$ and [[discrete group]] $G$ there is an [[equivalence of categories]]

$$
  Topos(Sh(X),[\mathbf{B}G, Set]) \simeq G Tors(X)
$$

between the [[geometric morphism]]s from the [[sheaf topos]] over $X$ to the category of [[permutation representation]]s of $G$ and the category of $G$-[[torsor]]s on $X$.

## Statement

For $C$ a [[category]], write

$$
  PSh(C) := [C^{op}, Set]
$$

for its [[presheaf topos]].

For $\mathcal{E}$ any [[topos]], write

$$
  FlatFunc(C, \mathcal{E}) \hookrightarrow [C, \mathcal{E}]
$$

for the [[full subcategory]] of the [[functor category]] on the [[internally flat functors]].

+-- {: .num_theorem}
###### Theorem
**(Diaconescu's theorem)**

There is an [[equivalence of categories]]

$$
  Topos(\mathcal{E}, PSh(C))
   \simeq
  FlatFunc(C, \mathcal{E})
$$

between the category of [[geometric morphism]]s $f : \mathcal{E} \to PSh(C)$ and the category of [[internally flat functors]] $C \to \mathcal{E}$.

This equivalence takes $f$ to the composite

$$
  C \stackrel{j}{\to} PSh(C) \stackrel{f^*}{\to} \mathcal{E}
  \,,
$$

where $j$ is the [[Yoneda embedding]] and $f^*$ is the [[inverse image]] of $f$.

=--

See for instance ([Johnstone, theorem B3.2.7](#Johnstone)).

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

If $C$ is a [[finitely complete category]] we may think of it as the [[syntactic category]] and in fact the [[syntactic site]] of an [[essentially algebraic theory]] $\mathbb{T}_C$.  An [[internally flat functor]] $C \to \mathcal{E}$ is then precisely a [[finite limit]] preserving functor, hence is precisely a $\mathbb{T}$-[[model]] in $\mathcal{E}$.

Therefore the above theorem says in this case that there is an [[equivalence of categories]]

$$
  Topos(\mathcal{E}, PSh(C))
   \simeq
  \mathbb{T}_C Mod(\mathcal{E})
$$

between the [[geometric morphism]]s and the $\mathbb{T}$-models in $\mathcal{E}$.

This says that $PSh(C)$ is the [[classifying topos]] for $\mathbb{T}_C$.

=--

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

If $G$ is a [[discrete group]] and $C = \mathbf{B}G$ is its [[delooping]] [[groupoid]], $PSh(C) \simeq [\mathbf{B}G, Set]$ is the category of [[permutation representation]]s of $G$, also called the [[classifying topos]] of $G$.

In this case an internally flat functor $ C = \mathbf{B}G \to \mathcal{E}$ may be identified with a $G$-[[torsor]] object in $\mathcal{E}$.

For this reason one sees in the literature sometimes the term "torsor" for internally flat functors out of any category $C$. It is however not so clear in which sense this terminology is helpful in cases where $C$ is not a delooping groupoid or at least _some_ groupoid.

=--

## References

A standard reference is section B3.2 in

* {#Johnstone} [[Peter Johnstone]], _[[Sketches of an Elephant]]_

The first proof of this result can be found in:

* [[Radu Diaconescu]], _Change of base for toposes with generators_ , J. Pure Appl. Algebra __6__ (1975), no. 3, 191-218 (doi:10.1016/0022-4049(75)90015-8)

Another proof is in

* [[Ieke Moerdijk]], _Classifying spaces and classifying topoi_, Lecture Notes in Mathematics __1616__, Springer 1995. vi+94 pp. ISBN: 3-540-60319-0

[[!redirects Diaconescu's theorem]]
[[!redirects Diaconescu's theorem]]
[[!redirects Diaconescu theorem]]
[[!redirects Disconescu's theorem]]
[[!redirects Diaconescu's Theorem]]
[[!redirects Diaconescu's Theorem]]
[[!redirects Diaconescu Theorem]]
[[!redirects Disconescu's Theorem]]


Diaconescu-Goodman-Myhill theorem

> This entry is about the theorem in [[logic]] which is often just called _Diaconescu's theorem_, but not to be confused with [[Diaconescu's theorem]] in [[topos theory]].

***

#Contents#
* table of contents
{:toc}

## Idea

The Diaconescu-Goodman–Myhill theorem ([Diaconescu 75](#Diaconescu75), [Goodman-Myhill 78](#GoodmanMyhill78)) states that the [[law of excluded middle]] may be regarded as a very weak form of the [[axiom of choice]].

## Statement

The following are equivalent:

1. The principle of [[excluded middle]].
1. [[finite set|Finitely indexed]] sets are [[projective object|projective]] (in fact, it suffices 2-indexed sets to be projective).
1. [[finite set|Finite sets]] are [[choice object|choice]] (in fact, it suffices for 2 to be choice).

(Here, a [[set]] $A$ is **[[finite set|finite]]** or **finitely-indexed** (respectively) if, for some natural number $n$, there is a [[bijection]] or [[surjection]] (respectively) $\{0, \ldots, n - 1\} \to A$.)

## Proof

The [[proof]] is as follows.  If $p$ is a truth value, then divide $2 = \{0,1\}$ by the equivalence relation where $0 \equiv 1$ iff $p$ holds.  Then we have a surjection $2\to A$, whose domain is $2$ (and in particular, finite), and whose codomain $A$ is finitely-indexed.  But this surjection splits iff $p$ is true or false, so if either $2$ is choice or $2$-indexed sets are projective, then PEM holds.

On the other hand, if PEM holds, then we can show by induction that if $A$ and $B$ are choice, so is $A\sqcup B$ (add details).  Thus, all finite sets are choice.  Now if $n\to A$ is a surjection, exhibiting $A$ as finitely indexed, it has a section $A\to n$.  Since a finite set is always projective, and any retract of a projective object is projective, this shows that $A$ is projective.

In particular, the axiom of choice implies PEM.  This argument, due originally to [Diaconescu 75](#Diaconescu75), can be [[internalization|internalized]] in any [[topos]]. However, other weak versions of choice such as [[countable choice]] (any surjection to a countable set (which for this purpose is any set isomorphic to the set of natural numbers) has a section), [[dependent choice]], or even [[COSHEP]] do not imply PEM. In fact, it is often claimed that axiom of choice is *true* in constructive mathematics (by the BHK or [[Brouwer-Heyting-Kolmogorov interpretation]] of predicate logic), leading to much argument about exactly what that means.

## References

* {#Diaconescu75} [[Radu Diaconescu]], _Axiom of choice and complementation_, Proceedings of the American Mathematical Society 51:176-178 (1975) ([doi:10.1090/S0002-9939-1975-0373893-X](https://doi.org/10.1090/S0002-9939-1975-0373893-X))

* {#GoodmanMyhill78} N. D. Goodman J. Myhill, _Choice Implies Excluded Middle_, Zeitschrift fuer Mathematische Logik und Grundlagen der Mathematik 24:461 (1978)

Review includes

* {#McLarty96} [[Colin McLarty]], _Elementary Categories, Elementary Toposes_, Oxford University Press, 1996

See also

* Wikipedia, _[Diaconescu' theorem](https://en.wikipedia.org/wiki/Diaconescu%27s_theorem)_

[[!redirects Diaconescu-Goodman–Myhill theorem]]


Diaconescu-Moore-Witten anomaly

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A potential [[quantum anomaly]] in [[M-theory]]/[[type IIA string theory]].

## References
 {#References}

### General

The anomaly was found in

* [[Duiliu-Emanuel Diaconescu]], [[Gregory Moore]], [[Edward Witten]], _$E_8$ Gauge Theory, and a Derivation of K-Theory from M-Theory_, Adv.Theor.Math.Phys.6:1031-1134 (2003) [[arXiv:hep-th/0005090](http://arxiv.org/abs/hep-th/0005090)]

while mostly assuming vanishing NS-fields (the [[B-field]]). The non-vanishing B-field in this context was considered in

* [[Gregory Moore]], [[Natalia Saulina]], _T-Duality, and the K-Theoretic Partition Function of TypeIIA Superstring Theory_, Nucl.Phys. B670 (2003) 27-89 ([arXiv:hep-th/0206092](http://arxiv.org/abs/hep-th/0206092))

where however the anomaly as such is not discussed. This was then put together in

* [[Varghese Mathai]], [[Hisham Sati]], _Some Relations between Twisted K-theory and E8 Gauge Theory_, JHEP0403:016,2004 ([arXiv:hep-th/0312033](http://arxiv.org/abs/hep-th/0312033))

which hence discusses the anomaly for [[twisted K-theory]].

### Interpretaton as Orientation in second integral Morava K-theory
 {#ReferencesInterpretationInSecondMoravaKTheory}

An interpretation of the phenomenon in terms of [[orientation in generalized cohomology|orientation]] in second [[integral Morava K-theory]] is in

* [[Igor Kriz]], [[Hisham Sati]], _M-theory, type IIA superstrings, and elliptic cohomology_, Adv. Theor. Math. Phys. 8 (2004), no. 2, 345–394 ([Euclid](http://projecteuclid.org/euclid.atmp/1091543172), [arXiv:hep-th/0404013](http://arxiv.org/abs/hep-th/0404013))
 {#SatiKriz04}

* {#Buhne11} [[Lukas Buhné]], _[[Properties of Integral Morava K-Theory and the Asserted Application to the Diaconescu-Moore-Witten Anomaly]]_, Diploma thesis Hamburg (2011) ([spire:899170](http://inspirehep.net/record/899170))

## Related concepts

[[!include orientations in higher quantization - table]]


diagonal action

#Contents#
* table of contents
{:toc}

## Idea

For $G$ a [[group]] with [[action]] on two [[objects]] $X$ and $Y$, there is a canonically induced action on the [[product]] $X \times Y$. This is the _diagonal action_.

## Definition
For every $g\in G$, $g:X\rightarrow X$, $x\mapsto gx$,  and $g:Y\rightarrow Y$, $y\mapsto gy$ are both automorphisms, and
the maps $g:X\times Y$, $g(x,y)=(gx, gy)$ define the *diagonal action* of $G$ on $X\times Y$.

## Related concepts

* [[diagonal]]

* [[Atiyah Lie groupoid]]

[[!redirects diagonal actions]]