specialization topology

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###Context###
#### Topology
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[[!include topology - contents]]
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# The specialisation topology
* table of contents
{: toc}

## Idea

The __specialisation topology__, also called the __Alexandroff topology__, is a natural structure of a [[topological space]] induced on the underlying [[set]] of a [[preordered set]].  This is similar to the [[Scott topology]], which is however coarser.

Spaces with this topology, called _Alexandroff spaces_ and named after [[Paul Alexandroff]] (Pavel Aleksandrov), should not be confused with _[[Alexandrov spaces]]_ (which arise in [[differential geometry]] and are named after [[Alexander Alexandrov]]).

## Definition

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

Let $P$ be a [[preordered set]].

Declare a [[subset]] $A$ of $P$ to be an _[[open subset]]_ if it is upwards-closed.  That is, if $x \leq y$ and $x \in A$, then $y \in A$. (Equivalently, closed subsets are the downwards-closed subsets.)

This defines a [[topology]] on $P$, called the **specialization topology** or **Alexandroff topology**.
=--

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

One may also use the convention that the open sets are the downwards-closed subsets; this is the specialisation topology on the [[opposite category|opposite]] $P^\op$.
=--

## Examples

* [[Sierpinski space]] is the poset of [[truth values]] with the specialisation topology.

## Properties

### General

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

Every [[finite topological space]] is an Alexandroff space.

=--

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

A [[preorder]] $P$ is a [[partial order|poset]] if and only if its specialisation topology is $T_0$.
=--

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

A [[preorder]] $P$ is a [[equivalence relation]] if and only if its specialisation topology is $R_0$.
=--

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

The specialisation topology of [[equality]] is $T_1$.
=--

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

A [[function]] between preorders is order-preserving if and only if it is a [[continuous map]] with respect to the specialisation topology.
=--

### Alexandroff topological spaces

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

An **Alexandroff space** is a [[topological space]] for which arbitrary (as opposed to just finite) [[intersections]] of [[open subsets]] are still open.

Write

$$
  AlexTop \hookrightarrow Top
$$

for the [[full subcategory]] of [[Top]] on the Alexandroff spaces.
=--

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

Every Alexandroff space is obtained by equipping its [[specialization order]] with the Alexandroff topology.
=--

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

The specialization topology embeds the category $\Proset$ of preordered sets [[full subcategory|fully-faithfully]] in the category [[Top]] of topological spaces.

$$
  Proset \hookrightarrow Top
  \,.
$$

If we restrict to a [[finite set|finite]] underlying set, then the categories $\Fin\Pros$ and $\Fin\Top$ of finite prosets and [[finite topological spaces]] are [[equivalence of categories|equivalent]] in this way.

=--

### Alexandroff locales
 {#AlexandrovLocales}

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

Write $AlexLocale$ for the non-[[full subcategory|full]] [[subcategory]] of [[Locale]] whose

* [[object]]s are __Alexandroff locales__, that is locales of the form $Alex P$ for $P\in Poset$ with $Open(Alex(P)) = UpSets(P)$.
Equivalently, Alexandroff locales can be defined as [[locales]]
that admit a [[base]] consisting of supercompact opens,
i.e., opens $a$ for which any open cover of $a$ contains $a$ itself.

* [[morphisms]] are those morphisms of locales $f\colon Alex P \to Alex Q$, for which the dual [[inverse image]] morphism of [[frames]] $f^*\colon UpSet(Q) \to UpSet(P)$ has a [[left adjoint]] $f_!\colon UpSet(P) \to UpSet(Q)$.
=--

This appears as ([Caramello, p. 55](#Caramello)).

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

By the definition of the [[2-category]] [[Locale]] (see there), this means that $AlexPoset$ consists of those morphisms which have _right_ adjoints in [[Locale]].

=--

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

The functor $Alex\colon Poset \to Locale$ factors through $AlexLocale$ and exhibits an [[equivalence of categories]]

$$
  Alex
   \colon
  Poset
    \stackrel{\simeq}{\to}
  AlexLocale
  \,.
$$

The inverse functor

$$
  AlexLocale \to Poset
$$

sends a locale $L$ to the [[poset]] of supercompact elements of $L$ (defined above)
and a morphism $f\colon L\to L'$ to the restriction of $f_!$
to supercompact elements (which are preserved by $f_!$).

=--

This appears as ([Caramello, theorem 4.2](#Caramello)). Dually, reversing the direction of arrows of Alexandroff locales,

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

The [[category]] of Alexandroff frames is [[equivalence of categories|equivalent]] to that of [[completely distributive lattice|completely distributive]] [[algebraic lattices]].

=--

This appears as ([Caramello, remark 4.3](#Caramello)).

## Related concepts

* [[order topology]]

* [[Scott topology]]

## References

* Wikipedia (English), [Alexandrov topology](http://en.wikipedia.org/wiki/Alexandrov_topology)

The original article is

* [[Paul Alexandroff]], _Diskrete Räume_ ([web](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=sm&paperid=5579&option_lang=eng)) (1973)

Details on Alexandroff spaces are in

* F. Arenas, _Alexandroff spaces_, Acta Math. Univ. Comenianae Vol. LXVIII, 1 (1999), pp. 17–25 ([pdf](http://www.emis.de/journals/AMUC/_vol-68/_no_1/_arenas/arenas.pdf))

* Timothy Speer, _A Short Study of Alexandroff Spaces_ ([arXiv:0708.2136](http://arxiv.org/abs/0708.2136))

A useful discussion of the abstract relation between posets and Alexandroff [[locales]] is in section 4.1 of

* [[Olivia Caramello]], _A topos-theoretic approach to Stone-type dualities_ ([arXiv:1103.3493](http://arxiv.org/abs/1103.3493))
 {#Caramello}

See also around page 45 in

*  [[Peter Johnstone]], _[[Stone Spaces]]_

A discussion of [[abelian sheaf cohomology]] on Alexandroff spaces is in

* Morten Brun, Winfried Bruns, Tim Römer, _Cohomology of partially ordered sets and local cohomology of section rings_ Advances in Mathematics 208 (2007) 210–235 ([pdf](http://www.home.uni-osnabrueck.de/wbruns/brunsw/pdf-article/BrunBrunsRoemer.published.pdf))

[[!redirects specialization topology]]
[[!redirects specialization topologies]]
[[!redirects specialisation topology]]
[[!redirects specialisation topologies]]
[[!redirects Alexandroff topology]]
[[!redirects Alexandroff topologies]]
[[!redirects Alexandrov topology]]
[[!redirects Alexandrov topologies]]

[[!redirects Alexandroff space]]
[[!redirects Alexandroff spaces]]

[[!redirects Alexandroff locale]]
[[!redirects Alexandroff locales]]
[[!redirects Alexandrov locale]]
[[!redirects Alexandrov locales]]

[[!redirects completely distributive algebraic lattice]]
[[!redirects completely distributive algebraic lattices]]


species

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### Context
#### Category theory
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[[!include category theory - contents]]
=--
#### Combinatorics
+-- {: .hide}
[[!include combinatorics - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A **(combinatorial) species** is a [[presheaf]] or [[higher category theory|higher categorical]] presheaf on the [[groupoid]] [[core]]([[FinSet]]), the [[permutation groupoid]].

A species is a _[[symmetric sequence]]_ by another name. Meaning: they are categorically equivalent notions.

## Definition

### 1-categorical

The [[category]] of **species**, $Species := PSh(core(FinSet))$, is the [[category of presheaves]] on the [[groupoid]] $\mathbb{P} := core(FinSet)$ of [[finite sets]] and [[bijections]], the [[core]] of [[FinSet]]:

$$
  \hat \mathbb{P} := PSh(core(FinSet))
  \,.
$$

For more, see [[structure type]].

### 2-categorical

A **2-species** (usually called a [[stuff type]]) is a [[2-presheaf]] on [[core]]([[FinSet]]), i.e. a [[pseudofunctor]]

$$
  core(FinSet) \to Grpd
$$

to the [[2-category]] [[Grpd]].

### $(\infty,1)$-categorical

Generally, an **$\infty$-species** or **homotopical species** is an [[(∞,1)-presheaf]] on $core(FinSet)$, i.e. an [[(∞,1)-functor]]

$$
  core(FinSet)^{op} \to \infty Grpd
$$

to the [[(∞,1)-category]] [[∞Grpd]] ([Gepner-Haugseng-Kock 17, Def. 3.2.9](#GHK)).

The [[(∞,1)-category]] of $\infty$-species is the [[(∞,1)-category of (∞,1)-presheaves]]

$$
  \infty Species := PSh_{(\infty,1)}(core(FinSet))
  \,,
$$

For more on this see also below the discussion _[In homotopy type theory](#InHomotopyTypeTheory)_.

### Operations on species

There are in fact 5 important monoidal structures on the category of species.

#### Sum

The **sum** $A + B$ of two species $A$, $B$ is their [[coproduct]] $A \coprod B$. Since [[colimit]]s of [[presheaves]] are computed objectwise, we have

$$
  (A + B)_n \simeq A_n \coprod B_n
  \,.
$$

The generating function of the sum of species is the sum of their generating
functions.

#### Cauchy product

The category $core(FinSet)$ becomes a [[monoidal category]] under disjoint union of finite sets. This monoidal structure $(core(FinSet), \coprod)$ induces canonically the [[Day convolution]] monoidal structure on $Species := PSh(core(FinSet))$.  This is often called the **Cauchy product** of species, or sometimes (confusingly) just the "product", since the generating function of the Cauchy product of species is the product of their generating functions.

Explicitly, the Cauchy product of species $A$ and $B$ is given by the [[coend]]

$$
  (A \otimes B)_n
  \simeq
  \int^{n \in FinSet}  A_k \times B_l \times Hom_{core(FinSet)}(n,k+l)
  \simeq
  \coprod_{k+l = n} \prod_{\frac{(k+l)!}{k! + l!}} (A_k \times B_l)
  \,.
$$

#### Hadamard product

The category of species also has cartesian [[product]]s.  Since products of [[presheaves]] are computed objectwise, we have

$$
  (A \times B)_n \simeq A_n \times B_n
  \,.
$$

This operation is often called the **Hadamard product** of species.

#### Dirichlet product

The category $core(FinSet)$ also becomes a [[monoidal category]] under cartesian product of finite sets. This monoidal structure induces another [[Day convolution]] monoidal structure on $Species := PSh(core(FinSet))$.  This monoidal structure on species seems to be what Maia and Méndez call the **arithmetic product**, but here we call it the **Dirichlet product** of species, since the Dirichlet series of this product of species is the product of their Dirichlet series.  (See [Dirichlet series and the Hasse--Weil zeta function](https://ncatlab.org/johnbaez/show/Dirichlet+species+and+the+Hasse-Weil+zeta+function).)

#### Composition product

Finally, there is a monoidal structure on the category of species known as **plethysm** or **composition**.  The generating function of a composite of two species is the composite of their generating functions (in the sense of formal power series).    A monoid with respect to the plethysm tensor product is called an [[operad]].

## In Homotopy Type Theory
 {#InHomotopyTypeTheory}

The following discusses formalization of the concept of species in [[homotopy type theory]].

Let [[FinSet]] be the [[type]] of [[finite sets]] (see at _[[hSet]]_). Notice that in [[homotopy type theory]] this automatically comes out as the [[groupoid]] of finite sets (see also at _[[universe in type theory]]_), so we may (and do) suppress writing "[[core]]" here.

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

A _species_ is a type $X \colon Type$ equipped with a [[function]] into $FinSet$. Hence the type of all species is the [[dependent sum]]

$$
  Species \coloneqq \sum_{X : Type} (X \to FinSet)
$$

of all [[function types]] $X \to FinSet$ as $X$ ranges over the [[type of types]].

=--

Write

$$
  FinSet \longrightarrow {\|FinSet\|_{0}} \simeq \mathbb{N}
$$

for the [[unit of a monad|unit]] of the [[n-truncation modality|0-truncation modality]] on [[FinSet]], going into the [[type of natural numbers]].

Given a species $f \colon X \to FinSet$ as in def. \ref{HoTTSpecies}, its "[[decategorification]]" is the composite

$$
  f \circ card \colon X \stackrel{f}{\longrightarrow} FinSet \to {\|FinSet\|_{0}} \simeq \mathbb{N}
$$

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

The _[[generating function]]_ of a species $f \colon X \to FinSet$, def. \ref{HoTTSpecies}, is

$$
  {|X|}(z)
  \coloneqq
    \sum_{n=0}^{\infty} {\big| fib_{f \circ card}(n) \big|}\,  z^{n}
  =
  \sum_{n=0}^{\infty} {\big| fib_{f}(Fin(n)) \big|}\, \frac{z^{n}}{n!}
  \,,
$$

where $fib(-)$ denotes the [[homotopy fiber]] and $Fin(n)$ denotes a [[0-type]] with exactly $n$ elements.

=--

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

If $\Phi : FinSet \to Type$ is a family of types, then we obtain the species

$$
  pr_{1} : \bigg(\sum_{A : FinSet} \Phi(A)\bigg) \to FinSet
$$

with generating function

$$
  {|\Phi|}(z)
  =
  \sum_{n=0}^{\infty} {|\Phi(Fin(n))|} \frac{z^{n}}{n!}
$$

=--

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

If $\Phi$ is a [[stuff, structure, property|stuff, structure, or property]]
on finite sets, then we immediately know something about its generating function, considered as an exponential generating function:

* If $\Phi(A)$ is [[contractible type|contractible]] for all $A$, then $\Phi(Fin(n)) = 1$, so its generating function is the [[exponential function]] $e^{z}$.

* If $\Phi(A)$ is a family of [[mere propositions]], then every coefficient is either $0$ or $1$, according to whether $A$ can be equipped with a $\Phi$-structure or not.

* If $\Phi(A)$ is a family of sets (i.e., a [[structure type]]), then every coefficient is an element of the [[type of natural numbers]] $\mathbb{N}$.

* If $\Phi(A)$ has a higher homotopy level (i.e., is a [[stuff type]]), then the coefficients are in $[0, \infty)$.

=--

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

Since any species $f : X \to Type$ is equal to

$$
  pr_{1} : \bigg(\sum_{x:X}fib_{f}(x)\bigg) \to Type
$$

every species is equal to one obtained from a [[stuff type]].

=--

### Operations on species

The five monoidal structures mentioned above under [Operations on species](#operations_on_species)
can be represented in HoTT, along with a couple other useful operations.  We
give four of the five monoidal structures here.
For more operations, see ([Dougherty15](#Dougherty15)).  We assume throughout this subsection that $f : X \to \Finset$ and $g : Y \to Finset$ are two species, and $\Phi, \Psi : FinSet \to Type$ are two stuff types.

#### Coproduct

The recursion principle for the coproduct gives the species

$$ \array{
  (f + g) &:& X + Y \to Finset \\
  (f + g) &:& rec_{X + Y}(FinSet, f, g) = z \mapsto \left\{\array{ f(x) & \text{if}\; z \equiv inl x \\ g(y) & \text {if}\; z \equiv inr y .}\right.
}$$

If $X$ and $Y$ are obtained from $\Phi$ and $\Psi$, respectively, then the coproduct is equal to

$$
  \sum_{A : FinSet} (\Phi(A) + \Psi(A))
$$

The generating function for the coproduct is

$$
  {|X + Y|}(z) = {|X|}(z) + {|Y|}(z)
$$

#### Hadamard product

The Hadamard product of species is

$$ \array{
  \langle X, Y \rangle &:& X \times_{FinSet} Y \to FinSet \\
  \langle X, Y \rangle &:& (x, y, p) \mapsto f(x)
}$$

which for stuff types is equal to

$$ \array{
  \sum_{A : FinSet} (\Phi(A) \times \Psi(A))
}$$

The generating functions are related by

$$
  {|\langle X, Y \rangle|}(z)
  =
  \sum_{n=0}^{\infty} n! X_{n} Y_{n} z^{n}
$$

where $X_{n}$ and $Y_{n}$ are the $n$th coefficients of the functions ${|X|}(z)$ and ${|Y|}(z)$, respectively.

The name "Hadamard product" is used in ([Aguiar-Mahajan10](#AguiarMahajan10)) and ([Bergeron-Labelle-Leroux08](#Bergeron08)).  In ([Baez-Dolan01](#Baez01)) ${|\langle X, Y \rangle|}(1)$ is called the "inner product" of stuff types, because equipping the category of stuff types with this operation makes it a categorified version of the Hilbert space of a quantum harmonic oscillator.

#### Cauchy product
 {#HoTTCauchyProduct}

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

The _Cauchy product_ of species is given by sending two species
$f,g$ as in def. \ref{HoTTSpecies} to the species given by

$$ \array{
  (f \cdot g) &:& X \times Y \to FinSet \\
  (f \cdot g) &:& (x, y) \mapsto f(x) + g(y)
}
$$

=--

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

The Cauchy product, def. \ref{HoTTCauchyProduct}, is eqivalently the [phased tensor product](%28infinity,n%29-category+of+correspondences#PhasedTensorProduct) on the slice over the [[symmetric monoidal category|symmetric monoidal]] groupoid $(FinSet, \coprod)$.

=--

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

For [[stuff types]] $\Phi$, $\Psi$, their Cauchy product, def. \ref{HoTTCauchyProduct}, is equal to

$$
  \sum_{A, U, V : FinSet} \sum_{p : U + V = A} (\Phi(U) \times \Psi(V))
$$

In this form the Cauchy product is discussed in ([Aguiar-Mahajan 10, def. 8.5](#AguiarMahajan10), [Aguiar-Mahajan 12, section 2.2](#AguiarMahajan12)).
[[monoids|Monoids]] and [[comonoid]] in the resulting [[monoidal category]] of species are discussed in ([Aguiar-Mahajan 10, section 8.2](#AguiarMahajan10)) and [[Hopf monoids]] in this category in ([Aguiar-Mahajan 12, section 2.2](#AguiarMahajan12)).

This means that $\Phi\Psi$ stuff on a finite set is the stuff of "being chopped in two, with $\Phi$ stuff on one part and $\Psi$ stuff on the other".

The generating function is

$$
  {|X \cdot Y|}(z) = {|X|}(z) \cdot {|Y|}(z)
  \,.
$$

=--

#### Composition

The composition of two species is

$$ \array{
  (f \tilde{\circ} g) &:& \bigg(\sum_{x:X}(Fin(card(f(x)) \to Y)\bigg) \to FinSet \\
  (f \tilde{\circ} g) &:& (x, P, p) \mapsto \bigoplus_{i=1}^{card(f(x))}g(P(i))
}$$

where $\bigoplus_{i=1}^{n}g(P(i))$ is the $n$-ary direct sum of the finite sets $g(P(i))$.  For stuff types, this amounts to

$$
  \sum_{(A, C : FinSet)}\sum_{(B : Fin(card(C)) \to FinSet)} \bigg(
  (B\vdash_{card(C)}A) \times \Phi(C) \times \prod_{k : Fin(card(C))} \Psi(B(k))
  \bigg)
$$

That is, $\Phi \tilde{\circ} \Psi$ stuff on a finite set is a partition of the set into $card(C)$ subsets, with $\Phi$ stuff on the partition and $\Psi$ stuff on each of the subsets.

The generating function of the composition is

$$
{|X \tilde{\circ} Y|}(z) = {|X|}({|Y|}(z))
$$

This species is also called the [[plethysm]] product.

## Properties

### Cardinality

Under [[groupoid cardinality]]

$$
  {|-|} : \infty Grpd_{tame} \to \mathbb{R}
$$

every (tame) [[∞-groupoid]] is mapped to a [[real number]]

$$
  X \mapsto {|X|} := \sum_{[x] \in \pi_0(X)}\prod_{i = 1}^{\infty} (\pi_i(X,x)^{(-1)^{i}})
  \,.
$$

A species $\mathbf{X}$ assigns an [[∞-groupoid]] $\mathbf{X}_n$ to each [[natural number]] $n \in \mathbb{Z}$. Therefore under [[groupoid cardinality]] we may naturally think of a tame species as mapping to a [[power series]]

$$
  \mathbf{X} \mapsto {|\mathbf{X}|}
  :=
  \sum_{n = 0}^{\infty} \frac{1}{n!} {|\mathbf{X}_n|} z^n
  \in
  \mathbb{R}[ [ z ] ]
  \,.
$$

This cardinality operation maps the above addition and multiplication of combinatorial species to addition and multiplication of [[power series]].

That [[coproduct]] of species maps to sum of their cardinalities is trivial. That [[Day convolution]] of species maps under cardinality to the product of their cardinality series depends a little bit more subtly on the combinatorial prefactors:

$$
  {| A \otimes B |}
  =
  {|A|} \cdot {|B|}
  =
  \sum_{n=0}^\infty
   \frac{1}{n!}
   \sum_{k+l = n} \frac{n!}{k! l!}
   {|A_k|} \cdot {|B_l|}
  \,.
$$

## Variants

* If in the definition of combinatorial species the domain [[core]]([[FinSet]]) is replaced with [[FinDimVect]] and also the [[presheaves]] are take with values in [[FinDimVect]] then one obtains the notion of  **[[Schur functor]]**.

* Instead of $core(FinSet)$, we may choose as domain $\mathbf{B} A$ for a small category $A$, with objects $\langle a_i \rangle_{i \in I}$, for $I$ in $core(FinSet)$, i.e., finite families of objects of $A$. Also, instead of $Set$ valued presheaves, we may consider those valued in presheaves on a small category $B$. This joint generalisation yields what are called **generalised species**. These generalised species can be collected in a [[cartesian closed 2-category]] ([FGHW](#FGHW)).

## References

The notion of species goes back to

* {#Joyal81} [[André Joyal]], _Une théorie combinatoire des séries formelles_ , Advances in Mathematics 42:1-82 (1981) doi, [MR84d:05025](http://www.ams.org/mathscinet-getitem?mr=633783)

An expository discussion can be found at

* [[Todd Trimble]], _Exponential Generating Function and Introduction to Species_ ([blog](http://topologicalmusings.wordpress.com/2009/03/28/number-of-idempotent-endofunctions/)) (scroll down a bit).

See also Wikipedia: [combinatorial species](http://en.wikipedia.org/wiki/Combinatorial_species) and

* François Bergeron, Gilbert Labelle, Pierre Leroux, _Théorie des espèces et combinatoire des structures arborescentes_ , LaCIM, Montréal (1994). English version: [[Combinatorial species and tree-like structures]], Cambridge University Press (1998).

* {#Bergeron08} F. Bergeron, G. Labelle, P. Leroux, _Introduction to the Theory of
Species of Structures_, 2008, [pdf](http://bergeron.math.uqam.ca/Site/bergeron_anglais_files/livre_combinatoire.pdf)

* François Bergeron, _Species and variations on the theme of species_, invited talk at [Category Theory and Computer Science '04](http://www.itu.dk/research/theory/ctcs2004/), Copenhagen (2004). Slides ([pdf](http://bergeron.math.uqam.ca/Especes_trans.pdf)).

* G. Labelle, [video](http://www.sms.cam.ac.uk/media/937;jsessionid=9540827CB40AC9F1E61BF944127EBAF4) intro into combinatorial species at Newton Institute, Cambridge 2008

* {#AguiarMahajan10} [[Marcelo Aguiar]], [[Swapneel Mahajan]], _[[Monoidal Functors, Species and Hopf Algebras]]_, With forewords by [[Kenneth Brown]], [[Stephen Chase]], [[André Joyal]]. CRM Monograph Series __29__ Amer. Math. Soc. 2010. lii+784 pp. ([author pdf](http://www.math.cornell.edu/~maguiar/a.pdf))

* {#AguiarMahajan12} [[Marcelo Aguiar]], [[Swapneel Mahajan]], _Hopf monoids in the category of species_ ([arXiv:1210.3120](http://arxiv.org/abs/1210.3120))

The Dirichlet product of species seems to match what Maia and Méndez call the **arithmetic product**:

* Manuel Maia and Miguel Méndez, On the arithmetic product of combinatorial species, _Discrete Mathematics_ **308**, (2008), 5407--5427.   ([web](https://www.sciencedirect.com/science/article/pii/S0012365X07007960))

An application in computer science:

* [[Brent Yorgey]], _Random testing and beyond! with combinatorial species_, slides [pdf](http://www.cis.upenn.edu/~byorgey/talks/combtesting-ku-20091124.pdf); _Species and functors and types, oh my_, article, [pdf](http://www.cis.upenn.edu/~byorgey/papers/species-pearl.pdf)

Homotopical species are defined in (Def. 3.2.9):

* {#GHK} [[David Gepner]], [[Rune Haugseng]], [[Joachim Kock]], _∞-Operads as Analytic Monads_, ([arXiv:1712.06469](https://arxiv.org/abs/1712.06469))

Formalization in [[homotopy type theory]]:

* [[Brent Yorgey]], _Combinatorial Species and labelled structures._ ([pdf](http://ozark.hendrix.edu/~yorgey/pub/thesis.pdf))

* {#Dougherty15} [[John Dougherty]], _Species in HoTT_ ([pdf](https://github.com/jdoughertyii/hott-species/blob/master/species.pdf?raw=true)) ([formalization in Coq](https://github.com/jdoughertyii/hott-species))

An application in statistical mechanics:

* W. Faris, _Combinatorial species and cluster expansions_, Mosc. Math. J. __10__:4 (2010), 713–727 [pdf](http://www.ams.org/distribution/mmj/vol10-4-2010/faris.pdf), [MR2791054](http://www.ams.org/mathscinet-getitem?mr=2791054)

Discussion in relation to [[Feynman diagrams]]:

* {#Baez01} [[John Baez]] and [[James Dolan]], From finite sets to Feynman diagrams, in _Mathematics Unlimited - 2001 and Beyond_, vol. 1, eds. Björn Engquist and Wilfried Schmid, Springer, Berlin, 2001, pp. 29-50.  ([arXiv](http://arxiv.org/abs/math.QA/0004133))

Discussion of generalised species:

* {#FGHW} M. Fiore, N. Gambino, M. Hyland, G. Winskel, _The cartesian closed bicategory of generalised species of structures_, ([pdf](http://www1.maths.leeds.ac.uk/~pmtng/Publications/generalised-species.pdf))

[[!redirects homotopical species]]
[[!redirects (infinity,1) species]]
[[!redirects ∞-species]]
[[!redirects combinatorial species]]
[[!redirects species of structure]]


Specker sequence

# Specker sequences
* table of contents
{: toc}

## Definition

A __Specker sequence__ is a [[bounded function|bounded]] [[computable function|computable]] [[increasing function|increasing]] [[infinite sequence]] of [[rational numbers]] with no [[computable number|computable]] [[supremum]].

## Examples

Since there is a sequence of all [[Turing machine|Turing machines]], define a sequence $(S_n)_n$ as
$$ S_n = \sum_{i + j = n} 2^{-i} \{i\}_j ,$$
where $\{i\}_j$ is the [[bit]] ($0$ or $1$) giving whether the $i$th Turing machine halts before $j$ steps.  The theoretical limit of this sequence is
$$ \sum_i 2^{-i} \{i\} ,$$
where $\{i\}$ is the bit giving whether the $i$th Turing machine halts at all, but this is uncomputable (on pain of solving the [[halting problem]]).

Many famous non-computable numbers may be expressed as limits of Specker sequences.  For example, a [[Chaitin constant]] is defined as the sum of an [[infinite series]] with non-negative computable terms (each of which is a [[dyadic rational]]), and so is the limit of a Specker sequence.

## In constructive mathematics

In [[Russian constructivism]], all [[real numbers]] are computable, so a Specker sequence has no (located) supremum, giving a counterexample to the [[classical mathematics|classical]] [[least upper bound principle]] ($LUP$).

In many other varieties of [[constructive mathematics]], the computability of all real numbers can be neither proved nor refuted, but Specker sequences still provide [[weak counterexample|weak counterexamples]].

For a non-computable number that can be expressed as the limit of a Specker sequence, the sequence itself is a way to work with the number in constructive mathematics without assuming that it exists as a real number.  (This amounts to treating it as a computable [[lower real number]].)

[[!redirects Specker sequence]]
[[!redirects Specker sequences]]


spectral action

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## Idea

The _spectral action_ is a natural [[functional]] on the space of [[spectral triples]]. It is essentially a regularized [[heat kernel]] expansion of the [[Dirac operator]]-like operator in the spectral triple.

Since a spectral triple encodes [[Riemannian geometry]] in a generalized context of [[noncommutative geometry]], a functional on a space of spectral triples is comparable to the [[Einstein-Hilbert action]] functional on the space of ordinary Riemennian manifolds. And indeed, on spectral triples corresponding to ordinary Riemannian geometry the spectral action reduces to the Einstein-Hilbert action plus a series of integrals over higher [[curvature]] invariants. Moreover, by an incarnation of the [[Kaluza-Klein mechanism]], on spectral triples corresponding to a Riemannian manifold times a non-classical space of classical dimension 0, the spectral action to first order is an [[Einstein-Yang-Mills-Dirac theory]] and in fact [[Einstein-Maxwell-Yang-Mills-Dirac-Higgs theory]], hence also includes [[gauge theory]] ([[Yang-Mills theory]]).

## Application to particle phenomenology

The spectral action has been proposed as an [[action functional]] for describing fundamental [[physics]] (the [[standard model of particle physics]]). See at

* [[higher category theory and physics]] the section _[The standard model and gravity](higher+category+theory+and+physics#SpecStandModAndGravity)_.

Notice that a [[spectral triple]] which describes the [[worldline]] [[quantum mechanics]] of a point [[particle]] may be regarded as the point-particle degeneration limit of a [[2d SCFT]] (a "[[2-spectral triple]]", [Roggenkamp-Wendland 03](#RoggenkampWendland03), [Soibelman 11](#Soibelman11)) which describes the [[worldsheet]] [[quantum field theory]] of a [[superstring]]. It has been argued ([Chamseddine 97](#Chamseddine97)) that the spectral action of the spectral triple correspondingly reproduces that of the [[effective field theories]] of these [[string theories]].

## Related concepts

* [[Connes-Lott-Chamseddine model]]

## References

### General On spectral triples

The notion of [[spectral triple]] and of spectral action was introduced in

* [[Alain Connes]], _Gravity coupled with matter and foundation of non-commutative geometry_ ([arXiv:hep-th/9603053](http://arxiv.org/abs/hep-th/9603053))

A discussion specifically of the spectral action is in

* [[Ali Chamseddine]], [[Alain Connes]], _The Spectral Action Principle_ [arXiv:hep-th/9606001](http://arxiv.org/abs/hep-th/9606001),

Earlier articles on this include

* [[Ali Chamseddine]], _Remarks on the Spectral Action Principle_ ([arxiv:hep-th/9806046](http://arxiv.org/abs/hep-th/9806046))

### Phenomenological models

In the article

* {#Connes08} [[Alain Connes]], _Noncommutative Geometry and the standard model with neutrino mixing_, JHEP0611:081,2006 ([arXiv:hep-th/0608226](http://arxiv.org/abs/hep-th/0608226))

a previous version of [[Connes-Lott-Chamseddine model]] was refined in some technical fine print, and it was found that the [[KO-dimension]] of the classically 0-dimensional compactification spaces has to be 6 (just as in [[superstring theory]]). Also a concise [[bimodule]]-characterization of the particle content of the standard model was claimed (prop. 2.2).

### Lift to 2-spectral triples

A claim that the spectral action for something like a [[2-spectral triple]] does reproduce the effective background action of [[string theory]] is in

* {#Chamseddine97} [[Ali Chamseddine]], _An Effective Superstring Spectral Action_ ([arXiv:hep-th/9705153](http://arxiv.org/abs/hep-th/9705153))

A detailed derivation of how ordinary [[spectral triples]] arise as point particle limits of [[vertex operator algebra]]s for 2d [[SCFTs]] is in

* {#RoggenkampWendland03} [[Daniel Roggenkamp]], [[Katrin Wendland]], _Limits and Degenerations of Unitary Conformal Field Theories_ ([arXiv:hep-th/0308143](http://arxiv.org/abs/hep-th/0308143))

A summary of this is in

* [[Daniel Roggenkamp]], [[Katrin Wendland]], _Decoding the geometry of conformal field theories_ ([arXiv:0803.0657](http://arxiv.org/abs/0803.0657))

Also

* Sebastiano Carpi, Robin Hillier, [[Yasuyuki Kawahigashi]], [[Roberto Longo]], _Spectral Triples and the Super-Virasoro Algebra_ Communications in Mathematical Physics, Volume 295, Number 1 / April 2010 ([Springer pdf](http://www.springerlink.com/content/316wx4r4x885172w/fulltext.pdf))

A brief indication of some ideas of [[Yan Soibelman]] and [[Maxim Kontsevich]] on this matter is at

* [[Urs Schreiber]],  [_Spectral triples and graph field theory_](http://golem.ph.utexas.edu/category/2007/06/had_the_pleasure_of_talking.html)

Details are in

* {#Soibelman11} [[Yan Soibelman]], _Collapsing CFTs, spaces with non-negative Ricci curvature and nc-geometry_ , in [[Hisham Sati]], [[Urs Schreiber]] (eds.), _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_, Proceedings of Symposia in Pure Mathematics, AMS (2001)

[[!redirects spectral action principle]]


spectral affine line

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## Idea

The concept of _spectral affine line_ is the generalization of the concept of _[[affine line]]_ to [[spectral geometry]], in the sense of [[E-∞ geometry]].

Given a ([[connective spectrum|connective]]) [[E-∞ ring]] $R$, then the spectral affine line over $R$ is the affine [[spectral scheme]] given by the [[spectral symmetric algebra]] on a single generator:

$$
  \mathbb{A}^1_R = Spec( Sym_R(R) )
  \,.
$$

(e.g. [Lurie Schemes, below prop. 2.20](#LurieSchemes))

For $R = \mathbb{S}$ the [[sphere spectrum]], then this may be called the _absolute spectral affine line_. This is discussed in [Strickland-Turner 97](#StricklandTurner97).

## References

Some general comments are in

* {#LurieSchemes} [[Jacob Lurie]], below prop. 2.20 _[[Spectral Schemes]]_.

The absolute spectral affine line over $ R = \mathbb{S}$ the [[sphere spectrum]] is discussed in

* {#StricklandTurner97} [[Neil Strickland]], [[Paul Turner]], _Rational Morava $E$-theory and $D S^0$_, Topology Volume 36, Issue 1, January 1997, Pages 137-151 ([pdf](http://hopf.math.purdue.edu/Strickland-PTurner/rme.pdf))


Spectral Algebraic Geometry

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This page is to record material related to:

* [[Jacob Lurie]],

  _Spectral Algebraic Geometry_

  ([pdf]( http://math.harvard.edu/~lurie/papers/SAG-rootfile.pdf))

on [[spectral algebraic geometry]].

#### Contents
* 0 Introduction

#### I Fundamentals of Spectral Algebraic Geometry
* 1 Schemes and Deligne-Mumford Stacks
* 2 Quasi-Coherent Sheaves
* 3 Spectral Algebraic Spaces

(see also _[[Structured Spaces]]_)

#### II Proper Morphisms
* 4 Morphisms of Finite Presentation
* 5 Proper Morphisms in Spectral Algebraic Geometry
* 6 Grothendieck Duality
* 7 Nilpotent, Local, and Complete Modules
* 8 Formal Spectral Algebraic Geometry

#### III Tannaka Reconstruction and Quasi-Coherent Stacks
* 9 Tannaka Duality
* 10 Quasi-Coherent Stacks
* 11 Smooth and Proper Linear $\infty$-Categories

#### IV Formal Moduli Problems
* 12 Deformation Theories: Axiomatic Approach
* 13 Moduli Problems for Commutative Algebras
* 14 Moduli Problems for Associative Algebras
* 15 Moduli Problems for En-Algebras
* 16 Examples of Formal Moduli Problems

#### V Representability Theorems
* 17 Deformation Theory and the Cotangent Complex
* 18 Artin's Representability Theorem
* 19 Applications of Artin Representability

#### VI Structured Spaces

* 20 Fractured $\infty$-Topoi

  [[big and little toposes]]

  [[fractured (infinity,1)-topos|fractured $\infty$-topos]]

* 21 Structured sheaves

#### VII Variants of Spectral Algebraic Geometry

#### VIII Higher Algebraic Stacks

#### IX Rational and $p$-adic Homotopy Theory

#### X Appendix
* A Coherent $\infty$-Topoi
* B Grothendieck Topologies in Commutative Algebra
* C Prestable $\infty$-Categories
* D Descent for Modules and Linear $\infty$-Categories
* E Profinite Homotopy Theory

category: reference


spectral algebraic geometry

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## Idea

**Spectral algebraic geometry** (or maybe [[E-∞ geometry]]) is the theory of [[homotopical algebraic geometry]] specialized to the [[(infinity,1)-category of spectra]].
Hence it is a generalization of ordinary [[algebraic geometry]] where instead of [[commutative rings]], [[spectral schemes]] are locally modelled on [[commutative ring spectra]].

## Applications

### Elliptic cohomology

Historically, the first application of spectral algebraic geometry was in the study of [[elliptic cohomology]] and [[topological modular forms]].  In particular it allowed the construction and study of the [[tmf]] [[spectrum]] as a certain [[derived moduli stack]] of [[derived elliptic curves]]. This construction is based on the [[Artin-Lurie representability theorem]].  See

* [[Jacob Lurie]], [[A Survey of Elliptic Cohomology]]
* [[Jacob Lurie]], [[Elliptic Cohomology I]]

## See also

* [[derived algebraic geometry]]
* [[brave new algebra]]
* [[E-infinity ring]]

## References

The foundations of the theory are developed in

* [[Jacob Lurie]], _DAG VII: Spectral schemes_, [pdf](http://www.math.harvard.edu/~lurie/papers/DAG-VII.pdf).

* [[Jacob Lurie]], _DAG VIII: Quasi-coherent sheaves and Tannaka duality theorems_, [pdf](http://www.math.harvard.edu/~lurie/papers/DAG-VIII.pdf).

* [[Jacob Lurie]], _DAG IX: Closed immersions_, [pdf](http://www.math.harvard.edu/~lurie/papers/DAG-IX.pdf).

* [[Jacob Lurie]], _DAG XI: Descent theorems_, [pdf](http://www.math.harvard.edu/~lurie/papers/DAG-XI.pdf).

* [[Jacob Lurie]], _DAG XII: Proper morphisms, completions, and the Grothendieck existence theorem_, [pdf](http://www.math.harvard.edu/~lurie/papers/DAG-XII.pdf).

* [[Jacob Lurie]], _DAG XIV: Representability theorems_, [pdf](http://www.math.harvard.edu/~lurie/papers/DAG-XIV.pdf).

A textbook account is developing in

* [[Jacob Lurie]], _[[Spectral Algebraic Geometry]]_ ([pdf]( http://math.harvard.edu/~lurie/papers/SAG-rootfile.pdf))

See also

* [[Clark Barwick]], _Applications of derived algebraic geometry to homotopy theory_, lecture notes, mini-course in Salamanca, 2009, [pdf](https://www.maths.ed.ac.uk/~cbarwick/papers/salamanca.pdf).

* [[Paul Goerss]], [[Topological Algebraic Geometry - A Workshop]]

For the 2014 installment of UOregon's Moursund Lecture Series, Jacob Lurie gave three (video recorded) lectures on spectral algebraic geometry

* [[Jacob Lurie]], [Theory of 'Spectral' Algebraic Geometry](http://media.uoregon.edu/channel/2014/05/21/moursund-mathematics-lecture-series-dr-jacob-lurie-lecture-1-theory-of-%E2%80%98spectral-algebraic-geometry/)

* [[Jacob Lurie]], [Ambidexterity](http://media.uoregon.edu/channel/2014/05/23/moursund-mathematics-lecture-series-dr-jacob-lurie-lecture-2-ambidexterity/)

* [[Jacob Lurie]], [Roots of Unity in Stable Homotopy Theory](http://media.uoregon.edu/channel/2014/05/28/moursund-mathematics-lecture-series-dr-jacob-lurie-lecture-3-roots-of-unity-in-stable-homotopy-theory/)

More review:

* [[Charles Rezk]], *Spectral algebraic geometry* ([pdf](https://faculty.math.illinois.edu/~rezk/sag-chapter-web.pdf)) in:  [[Andrew J. Blumberg]], [[Teena Gerhardt]], [[Michael A. Hill]] (eds.), *[[Stable categories and structured ring spectra]]*, MSRI Book Series, Cambridge University Press

[[!redirects spectral algebraic space]]
[[!redirects spectral algebraic spaces]]

[[!redirects E-∞ algebraic space]]
[[!redirects E-∞ algebraic spaces]]


spectral category

__Spectral category__ is a [[enriched category|category enriched]] in the [[symmetric spectrum|symmetric monoidal category of spectra]]. Hence they are a nonlinear version of [[dg-categories]]. More precisely, pretriangulated idempotent complete spectral categories are a model of [[stable (infinity,1)-category|stable]] [[idempotent complete (infinity,1)-category|idempotent complete]] [[(infinity,1)-categories]]. This equivalence is recorded in

* [[Andrew J. Blumberg]], [[David Gepner]], [[Goncalo Tabuada]], _A universal characterization of higher algebraic K-theory_, [arXiv:1001.2282](http://arxiv.org/abs/1001.2282v4).

See also

* [[Stefan Schwede]], _Enhancements of triangulated categories_, [pdf](http://www.math.uni-bonn.de/people/schwede/EnhancedSeminar.pdf) (broken link)

* [[Stefan Schwede]], [[Brooke Shipley]], _Stable model categories are categories of modules_, Topology __42__:1 (2003) 103--153, MR2003g:55034

Background material on symmetric spectra is in

* [[Stefan Schwede]], _Symmetric spectra_, book project in progress, beta version [pdf](http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf)

[[!redirects spectral categories]]


spectral cookbook

## Idea

Typically spectra of categories involve utilization of some preordering and looking for "almost final" objects in that preordering and declaring them points of spectra. For example, one can look at some class of topologizing subcategories and equip that class with a preordering. A deeper inspection shows that an additional functor may be involved.

#### Historical motivation

While the spectrum of a commutative ring $R$ is obtained just from studying the ideal in a ring, that is $R$-submodules in $R$, the structure of various sets of ideals in a noncommutative ring usually is too small and otherwise inadequate for geometric purposes. One needs to consider not only ideals, but the whole category of (say left) $R$-modules, thus not necessarily submodules of $R$. A similar thing is with the reconstruction of commutative schemes: the whole abelian category of [[quasicoherent sheaves]] of $\mathcal{O}$-modules, not only the quasicoherent $\mathcal{O}$-submodules of $\mathcal{O}$, is needed for the reconstruction.

#### The development of spectral machinery for categories

[[Pierre Gabriel]] introduced his spectrum of indecomposable injectives to reconstruct Noetherian separated schemes from their categories of qausicoherent sheaves; now it is often called the Gabriel spectrum. Later many other spectra of abelian categories were invented, including the 1980-s [[A. L. Rosenberg]]'s spectrum used for the reconstruction of quasicompact quasiseparated schemes. Around 2000, Rosenberg noticed that the zoo of many spectra has their common feature and that all the spectra can be produced in analogous way. This pattern for producing spectra is introduced as a __spectral cookbook__ in

* (RosenbergSpectraNSp) A. L. Rosenberg, _Spectra of noncommutative spaces_, MPIM2003-110 [ps](http://www.mpim-bonn.mpg.de/preblob/1946) [dvi](http://www.mpim-bonn.mpg.de/preblob/1945) (2003)

## The cookbook

#### Local categories and local spectrum

(RosenbergSpectraNSp 1.1) A category $C$ is __local__ if the full subcategory generated by all objects which are not initial, has itself an initial object. In particular, every local category has initial objects.

(RosenbergSpectraNSp 1.2) The __local spectrum__ of an arbitrary small category $C$ is the full subcategory $\mathfrak{Spec}^1(C)\hookrightarrow C$ whose objects are all $x$ in $Ob C$ such that the [[undercategory]] $x\backslash C$ is local.

#### Support of an object

(RosenbergSpectraNSp 1.3) If $x$ is an object in $C$, its support in $C$ is the full subcategory $\mathfrak{Supp}_C(x)\subset C$ whose objects are all objects $y$ in  $C$ such that $C(x,y)=\emptyset$.

#### $\mathfrak{Spec}^0$

(RosenbergSpectraNSp 1.4) $\mathfrak{Spec}^0(C)\subset C$ is the full subcategory of $C$ generated by those objects $x$ in $C$ whose support $\mathfrak{Supp}_C(x)$ has a final object, $\hat{x}$ (in particular the support is nonempty).

(RosenbergSpectraNSp 1.4.4) A choice of a final object $\hat{x}$ for every object $x$ in $\mathfrak{Spec}^0(C)$ extends to a functor $\theta_c : \mathfrak{Spec}^0(C)\to C$.

#### When $C$ is a preorder

(RosenbergSpectraNSp 1.4.6) Now specialize to the case where $C$ is a [[preorder]] category having finite coproducts (equivalently supremum for every pair of objects). Then the functor $\theta_c : \mathfrak{Spec}^0(C)\to C$ factors through the embedding $\mathfrak{Spec}^1(C)\to C$. Consequently, it corestricts to a functor
$\mathfrak{Spec}^0(C)\to \mathfrak{Spec}^1(C)$ which may also be denoted as $\theta_C$.

#### Relative spectra

(RosenbergSpectraNSp 1.6) Given a functor between small categories $F: C\to D$ the relative spectra $\mathfrak{Spec}^0(C,F), \mathfrak{Spec}^1(C,F)$ are defined as isocomma objects ("2-categorical pullbacks")

$$\array{
\mathfrak{Spec}^0(C,F)&\stackrel{\theta_F}\longrightarrow&C\\
{\pi_F^0}_\mathrlap{}\downarrow&&\downarrow{}_\mathrlap{F}\\
\mathfrak{Spec}^0(C)&\underset{\theta_C}\longrightarrow&D
}
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\array{
\mathfrak{Spec}^1(C,F)&\stackrel{\theta^1_F}\longrightarrow&C\\
{\pi_F^1}_\mathrlap{}\downarrow&&\downarrow{}_\mathrlap{F}\\
\mathfrak{Spec}^1(C)&\underset{\theta^1_C}\longrightarrow&D
}
$$

It appears that, for a fixed target category $D$, the constructions $(C,F)\mapsto \mathfrak{Spec}^0(C,F)$ and $(C,F)\mapsto \mathfrak{Spec}^1(C,F)$ extend to pseudofunctors $Cat/D\to Cat$. If $D$ is a preorder with finite coproducts then every functor $F: C\to D$
induces a canonical functor

$$
\theta_{C,F} : \mathfrak{Spec}^0(C,F)\longrightarrow \mathfrak{Spec}^1(C,F)
$$

which are the components of a natural transformation of pseudofunctors $Cat/D\to Cat$

$$
\theta : \mathfrak{Spec}^0\longrightarrow \mathfrak{Spec}^1
$$

## Applications to spectra of Abelian categories

#### Idea

Starting with an Abelian category $A$ we proceed in three steps. In the first step we construct some preorder $C_A$ from $A$, or of a functor $F_A: C_A\to D_A$ with a preorder category $D_A$ and then apply the spectral cookbook. The objects of $C_A$ may be some special subcategories of $A$ (e.g. some class of [[topologizing subcategories]]), of multiplicative systems of morphisms, as used in the localization theory and so on. In a third step, one often passes to the equivalence classes of objects in the obtained spectra.

Additionally, one may add construction of some sort of topology or additional "structure stack" to obtained spectrum using possibly supplemental knowledge about the input data.

#### Remark on monoidal case

Some reconstruction theorems (like the theorems of Balmer and of Garkusha) consider abelian symmetric monoidal categories instead; their spectra are very analogous but the subcategories used to construct the intermediate preorder category are monoidal subcategories as well and various constructions respect the monoidal structure. The pattern is still the same.

category: noncommutative geometry


spectral curve

#Contents#
* table of contents
{:toc}

## Idea

Roughly speaking, a __spectral curve__ is a locus of zeros of the [[characteristic polynomial]] of the [[Lax pair|Lax matrix]] of a classical [[integrable system]].

[[Hitchin]] has taken a deeper geometric point of view in terms of the [[moduli space]] of [[stable vector bundles]] of fixed rank $r$ and degree $d$ over a fixed projective nonsingular curve $\Sigma$ of [[genus]] $g$. A point in the [[cotangent bundle]] of this bundle is a stable vector bundle equipped with a 1-form valued endomorphism; to each such datum one associates a spectral curve.

## Related concepts

* [[topological recursion]]

## References

* [[Nigel Hitchin|N.J. Hitchin]], _The self-duality equations on a Riemann surface_, Proc. London Math. Soc. (3) 55 (1987), no. 1, 59–126, [MR887284](http://www.ams.org/mathscinet-getitem?mr=887284) [doi](http://dx.doi.org/10.1112/plms/s3-55.1.59); _Stable bundles and integrable systems_, Duke Math. J. , 54 (1987) pp. 91–114 [MR89a:32021](http://www.ams.org/mathscinet-getitem?mr=89a:32021) [doi](http://dx.doi.org/10.1215/S0012-7094-87-05408-1) [euclid](http://projecteuclid.org/euclid.dmj/1077305506)

* O. Babelon, D. Bernard, M. Talon, _Introduction to classical integrable systems_, Cambridge Univ. Press 2003.

* Eyal Markman, _Spectral curves and integrable systems_ Compositio Math. __93__ no. 3 (1994), p. 255-290, [numdam](http://www.numdam.org/item?id=CM_1994__93_3_255_0)

* [[Ron Donagi]], Eyal Markman, _Spectral covers, algebraically completely integrable, Hamiltonian systems, and moduli of bundles_, pp. 1-119 in Integrable systems and quantum groups (expanded CIME lectures, Montecatini Terme, 1993), Lecture Notes in Math. __1620__ (1996) [arXiv:alg-geom/9507017](http://arxiv.org/abs/alg-geom/9507017), [MR97h:14017](http://www.ams.org/mathscinet-getitem?mr=1397273)

* [[Ron Donagi]], _Spectral covers_, MSRI series 28, [pdf](http://www.msri.org/publications/books/Book28/files/donagi.pdf)

* A. Beauville, _Jacobiennes des courbes spectrales et systèmes Hamiltoniens complètement intégrables_, Acta Math. __164__ (1990), 211-235,
[MR86c:32030](http://www.ams.org/mathscinet-getitem?mr=86c:32030), [doi](http://dx.doi.org/10.1007/BF02392754)

* A. Beauville, M. S. Narasimhan, S. Ramanan, _Spectral curves and the generalized theta divisor_, J. Reine Angew. Math. 398 (1989), 169-179.

* [[Carlos Simpson|Carlos T. Simpson]], _Higgs bundles and local systems_, Publ. Mathématiques de l'IHÉS, 75 (1992), p. 5-95, [numdam](http://www.numdam.org/item?id=PMIHES_1992__75__5_0)

* J. C. Hurtubise, _Integrable systems and algebraic surfaces_, Duke Math. J. __83__:1 (1996), 19-50, [euclid](http://projecteuclid.org/euclid.dmj/1077244044); erratum  Duke Math. J. __84__:3 (1996), 815, [euclid](http://projecteuclid.org/euclid.dmj/1077244246)

* Anton Izosimov, _Singularities of integrable systems and algebraic curves_, [arxiv/1509.08996](http://arxiv.org/abs/1509.08996)

Quantum spectral curve:

* V. Talalaev, _The quantum Gaudin system_, Funct. Anal. Appl. __40__, 73–77 (2006) ( _Квантовая система Годена_, Функц. анализ и его прил. __40__, вып. 1, 2006, с. 86–91, [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=faa&paperid=25&what=fullt&option_lang=rus))

* Д. Талалаев, А. Червов, _Уравнение КЗ, G-оперы, квантовая редукция Дринфельда–Соколова и квантовое тождество Гамильтона–Кэли_, Zapiski Naučnyh Seminarov POMI __360__, 2008, 246–259, [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=znsl&paperid=2168&what=fullt&option_lang=rus); Engl, transl. in J. Math, Sci. (New York), 2009, 158:6, 904–911, [doi](http://dx.doi.org/10.1007/s10958-009-9415-1)

[[!redirects spectral curves]]


spectral geometry

#Contents#
* table of contents
{:toc}

## Idea

Since there are several meanings of _[[spectrum]]_ there are accordingly several things one may mean by "spectral geometry". Most of those belong
either to the geometry as seen either by point spectra of [[spectral theory]] (of operators, families of operators, operator algebras, rings, associative algebras, abelian categories etc.), or by spectra in the sense of [[stable homotopy theory]] like [[symmetric spectra]], $E_\infty$-spectra, ring spectra...

### Geometry seen by operator spectra

For the notion of [[spectrum of an operator]] spectral geometry is geometry as seen by the spectra of operators ("[[hearing the shape of a drum]]"), which is geometry as seen by [[spectral triples]] (often inaccurately referred to as "[[noncommutative geometry]]").

See also

* [[spectral triple]]

* [[spectral action]]

### Geometry modeled on duals of $E_\infty$-rings

For the notion of [[spectrum]] as in [[stabilization]] in [[stable homotopy theory]], spectral geometry is geometry over [[Isbell duality|formal duals]] of [[E-∞ rings]] (the full version of [[derived algebraic geometry]]).

See at _[[E-∞ geometry]]_.

See also

* [[E-∞ scheme]]

[[!redirects spectral geometries]]

category: disambiguation


spectral global analytic geometry

[[!redirects spectral analytic geometry]]
[[!redirects spectral analytic spaces]]
[[!redirects spectral analytic space]]

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###Context###
#### Analytic geometry
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[[!include analytic geometry -- contents]]
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#Contents#
* table of contents
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## Idea

If one thinks of [[global analytic geometry]] as a theory based on the study of monoids in the monoidal category of Banach modules over a base Banach ring (e.g., $(\mathbb{Z},|\cdot|_\infty)$), it is quite tempting to look for a geometric theory of the associated cohomology theories, that one may call spectral global analytic geometry.

A first step in this direction of a good homotopy theory for global analytic spaces (that would be useful for the study of their cohomological invariants) would be to work with spectral rings in the symmetric monoidal category of Banach modules (over a given fixed Banach ring).

However, one would like the theory not to depend on the choice of a particular Banach ring, because it is also desirable to be invariant by the natural power action of $\mathbb{R}_+^*$ on [[proxi-Banach ring|Banach proxi-norms]], that are multiplicative maps that fulfill the weak triangular inequality
$$
|a+b|\leq \max(|2|,|1|)\cdot \max(|a|,|b|).
$$

A possible research direction is to look at Gromov's ideas about metric homotopy theory: one may try to define a category of geometric objects whose homotopy groups are naturally equipped with Banach pseudo-norms. If one looks for non-linear versions of Banach spaces, it is natural to look at metric spaces with Lipschitz maps between them (because their sets of maps are equipped with a natural "norm"). So one may try to develop global analytic homotopy theory (a way to look for interesting cohomological invariants for global analytic spaces, such as [[topological cyclic homology]] or [[topological derived de Rham cohomology]]) by developing the stable homotopy theory of metric spaces.

## Constraints

A natural very small set of constraints for a relaxed approach to the homotopy theory of metric spaces is the following:

* A metrized ring spectrum should be "something like" a commutative ring spectrum $A$ together with norms on its homotopy groups such that $\pi_0(A)$
is a seminormed ring and $\pi_i(A)$ is a seminormed module over $\pi_0(A)$ for all $i$.

* Having a metric ring sphere spectrum $\mathbb{S}$ such that $\pi_0(\mathbb{S})=(\mathbb{Z},|\cdot|_\infty)$.

* Having a natural action of $\mathbb{R}_+^*$ on the homotopy category (this is not contradictory with the previous constraint: one actually needs a family of sphere ring spectra, or its limit, that would have $\pi_0(\mathbb{S})=(\mathbb{Z},|\cdot|_\infty^\infty)$, in the sense of pro-objects).

* Getting back the homotopy theory of spectral Banach modules when one works over a fixed base Banach ring.

* To each spectrum, one associated a cohomology theory with values in complete ind-Banach modules over a given Banach ring.

## Objectives

One would like to be able to define "de Rham-like coefficients" over arbitrary global analytic spaces, even in mixed characteristic. One may hope (following Hesselholt, Connes and Bhatt/Scholze/Morrow) that topological coefficients (such as topological cyclic homology and topological derived de Rham/André-Quillen cohomology) may be a first step in this very optimistic direction.
Indeed, they seem to take care of something like Buium's differential calculus along the integers, and thus also of Witt vector constructions, in a completely canonical way.

## Constructions

To develop a homotopy theory for metric spaces, we first need a category of "metric spaces" that has all limits and colimits (usual metric spaces don't even have finite coproducts). One may then develop the associated stable homotopy theory by the usual methods, once fixed a metric on the interval $[0,1]$. To get a theory that is $\mathbb{R}_+^*$-invariant, one actually needs to equip $[0,1]$ with the limit of the family of [[proxi-metric space|proxi-metrics]]
$$d_\infty^t(x,y):=|x-y|_\infty^t$$
as $t$ goes to $\infty$.
This formal limit makes sense in a convenient category of metric spaces, where one keeps some track of the finite level $t$ information.

## References

This nlab page.


spectral group scheme

#Contents#
* table of contents
{:toc}

## Idea

A **spectral group scheme** is a [[group scheme]] in [[E-∞ geometry]]. In the affine case then the [[formal dual]] notion is a [[Hopf E-∞ algebra]].

(Integrate comment [here](https://nforum.ncatlab.org/discussion/7553/spectral-group-scheme/#Item_9) about smooth and flat versions.)

## Example

* $GL$: Consider $R$ an [[E-∞ ring]], $CAlg_R$, the ∞-category of $R$-algebras, $Mod_R$ an ∞-category of $R$-modules.
Let $Sym_R$ be [[left adjoint]] to the forgetful functor $CAlg_R \to Mod_R$. Set $M_{n,R} = Spec Sym_R End_R(R^{\oplus n})$, a spectral monoid scheme which represents $R \mapsto End_R(R^{\oplus n})$. Let $GL_{n,R}$ be the spectral group scheme obtained by inverting the determinant element in $\pi_o M_{n,R}$ ([Ohara](#Ohara)).

## References

* {#Ohara} Mariko Ohara, _On the K-theory of finitely generated
projective modules over a spectral ring_, slide 27 of [slides](http://userpage.fu-berlin.de/jcirici/posters/Ohara.pdf).


spectral interpretation

###Context###
#### Arithmetic geometry
#### Automorphic representation theory

* table of contents
{:toc}

## Overview

Many results on arithmetic or automorphic [[zeta functions]] and [[L-functions]] are obtained by using the fact that their zeroes and poles may be seen as the spectrum of an operator acting either on a given [[geometric cohomology]] theory (in arithmetic geometry) or on a given [[topological vector space]] (in the automorphic theory).

For example, the functional equation is often proved by using an equivariant pairing on the space that gives the spectral interpretation (Poincaré duality in arithmetic geometry and Poisson summation formula in the automorphic theory).

The finite characteristic analog of the Riemann hypothesis was also proved, following the [[Weil conjectures]], by the use of a kind of geometric Fourier duality result, in Laumon's approach.

## Challenges

The main challenge is to find a [[geometric cohomology]] theory for strict [[global analytic spaces]] over $\mathb{Z}$ (with its archimedean norm) that will allow the spectral interpretation of zeroes and poles of motivic L-functions. It may also be interesting to propose a similar definition for non-strict [[global analytic spaces]], in order to get a unified treatment of the finite characteristic case (seen as a theory of derived overconvergent analytic spaces over $\mathbb{Z}_p$) and the global case (of flat regular schemes over $\mathbb{Z}$).

## References

* Connes on the spectral interpretation of Riemann's zeta function (Hilbert space approach).

* Meyer on the spectral interpretation of Dedekind zeta functions (Schwartz space approach).

* Soulé on the spectral interpretation for Godement-Jacquet's automorphic L-functions (Hilbert space approach).

* Grothendieck-Deligne: proof of the Weil conjectures ($\ell$-adic cohomology), using monodromy. The Riemann hypothesis is proved using a Rankin-Selberg type of idea.

* Laumon: proof of the Weil conjectures using Fourier-Deligne transform and ideas similar to those used by Witten for his super-symmetric proof of the Morse inequalities.

* Caro-Kedlaya: p-adic proof of the full Weil conjectures.

* Deninger: system of constraints for a cohomological theory of global arithmetic L-functions, i.e., a global analog of the Weil conjectures (for flat schemes over Z).

* Katz-Rudnick-Sarnak: proposition of a global analog of the notion of monodromy used by Deligne in his Rankin-Selberg type proof of the Riemann hypothesis, based on the conjectured Montgomery link between zeroes of L-functions and random matrix theory.


spectral measure

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

# Idea
Spectral measures are an essential tool of [[functional analysis]] on [[Hilbert space]]s. Spectral measures are projection-valued measures and are used to state various forms of [[spectral theorem]]s.

In the following, let $\mathbb{H}$ be a [[Hilbert space]] and $\mathcal{B}(\mathcal{H})$ be the algebra of bounded linear operators on $\mathbb{H}$ and $\mathcal{P}(\mathcal{H})$ the orthogonal projections.

##real spectral measure##
The following paragraphs explain the concept of a spectral measure in the real case, sufficient for [[spectral theorem]]s of [[selfadjoint operator]]s.

###resolution of identity###
Do not confuse this concept with the [[partition of unity]] in [[differential geometry]].

definition: A __resolution of the identity operator__ is a map $E: \mathbb{R} \to \mathcal{P}(\mathcal{H})$ satisfying the following conditions:

1. (**monotony**): For $\lambda_1, \lambda_2 \in \mathbb{R}$ with $\lambda_1 \leq \lambda_2$ we have $E(\lambda_1) \leq E(\lambda_2)$.

2. (**continuity from above**): for all $\lambda \in \mathbb{R}$ we have $s-\lim_{\epsilon \to 0, \epsilon \gt 0} E(\lambda + \epsilon) = E(\lambda)$.

3. (**boundary condition**): $s-\lim_{\epsilon \to -\infty} E(\lambda) = 0$ and $s-\lim_{\epsilon \to \infty} E(\lambda) = \mathbb{1}$.

If there is a finite $\mu \in \mathbb{R}$ such that $E_{\lambda} = 0$ for all $\lambda \leq \mu$ and $E_{\lambda} = \mathbb{1}$ for all $\lambda \geq \mu$, than the resolution is called **bounded**, otherwise **unbounded**.

###spectral measure and spectral integral###
Let E be a spectral resolution and $I$ be a bounded interval in $\mathbb{R}$. The spectral measure of $I$ with respect to $E$ is given by
$$
E(J):= \begin{cases}

           E(y-) - E(x)  & \text{for }\quad I=(x,y) \\
           E(y-) - E(x-) & \text{for }\quad I=[x,y) \\
           E(y)  - E(x)  & \text{for }\quad I=(x,y] \\
           E(y)  - E(x-) & \text{for }\quad I=[x,y] \\

        \end{cases}
$$

This allows us to define the integral of a step function $u = \sum_{k=1}^{n} \alpha_k \chi_{I_k} $ with respect to E as

$$
\integral u(\lambda) dE(\lambda) := \sum_{k=1}^{n} \alpha_k E(I_k)
$$

The value of this integral is a bounded operator.

As in conventional measure and integration theory, the integral can be extended from step functions to Borel-measurable functions. In this case one often used notation is
$$
E(u) = \integral u(\lambda) dE(\lambda)
$$
For general function $u, E(u)$ need not be a bounded operator of course, the domain of $E(u)$ is (theorem):
$$
D(E(u)) = \{ f \in \mathcal{H} : \int |u(\lambda)|^2 d\langle E(\lambda)f, f\rangle \lt \infty \}
$$

# Spectrum of Representations of Groups, the SNAG Theorem
The SNAG theorem is necessary to explain the spectrum condition of the [[Haag-Kastler axioms]].

Let $\mathcal{G}$ be a locally compact, abelian topological group, $\hat \mathcal{G}$ the character group of $\mathcal{G}$, $\mathcal{H}$ a  Hilbert space and $\mathcal{U}$ an unitary representation of $\mathcal{G}$ in the algebra of bounded operators of $\mathcal{H}$.
The following theorem is sometimes called (classical) **SNAG** theorem (SNAG = Stone-Naimark-Ambrose-Godement):

+-- {: .num_theorem #specmeasure}
* **Theorem**: There is a unique regular spectral measure $\mathcal{P}$ on $\hat \mathcal{G}$ such that:

$$
     \mathcal{U}(g) = \int_{\chi\in\hat \mathcal{G}} \langle g, \chi\rangle \mathcal{P}(d\chi) \qquad \forall g \in \mathcal{G}
$$
 =--
The equality holds in the weak sense, i.e. the integral converges in the weak operator topology.
The **spectrum** of $\mathcal{U}(\mathcal{G})$, denoted by $spec\mathcal{U}(\mathcal{G})$, is defined to be the support of this spectral measure $\mathcal{P}$.

## The Case of the Translation Group
The groups of translations $\mathcal{T}$ on $\R^n$ is both isomorph to $\R^n$ and to it's own character group, every character is of the form $a \mapsto exp(i \langle a, k\rangle)$ for a fixed $k \in \R^n$. So in this case theorem \ref{specmeasure} becomes:
$$
\mathcal{U}(t) = \int_{k\in \R^n} e^{i \langle t, k\rangle} \mathcal{P}(k) \qquad \forall t \in \mathcal{T}
$$
This allows us to talk about the support of the spectral measure, i.e. the spectrum of $\mathcal{U}(\mathcal{T})$, as a subset of $\R^n$.

## Related entries

* [[projection measure]]

## References

The theorem \ref{specmeasure} is theorem 4.44 in the following classic book:

* [[Gerald B. Folland]], *A course in abstract harmonic analysis*, Studies in Advanced Mathematics. CRC Press (1995) [[pdf](https://sv.20file.org/up1/1415_0.pdf), [gBooks](http://books.google.com/books?hl=en&lr=&id=0VwYZI1DypUC)]

See also:

* A. A. Kirillov, A. D. Gvišiani, Теоремы и задачи функционального анализа (theorems and exercises in functional analysis), Moskva, Nauka 1979, 1988

[[!redirects spectral measures]]


spectral network

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[N=2 D=4 super Yang-Mills theory]]

## References

The notion of _spectral networks_ was introduced in:

* [[Davide Gaiotto]], [[Gregory Moore]], [[Andrew Neitzke]], _Spectral networks_, Annales Henri Poincaré **14** (2013) 1643–1731, [[arxiv/1204.4824](http://arxiv.org/abs/1204.4824), [doi:10.1007/s00023-013-0239-7](https://doi.org/10.1007/s00023-013-0239-7)]

Review:

* [[Gregory Moore]], _Spectral networks and their applications_, 2012 ([talk slides pdf](http://hep.caltech.edu/ym35/presentations/Moore.pdf))

Further developments:

* [[Carlos Simpson]], _Pre-buildings as sheaves on the site of enclosures_ talk at _Topos a l'IHES_ 2015 ([recording](https://www.youtube.com/watch?v=iWv_LoXcFQA&index=15&list=PLx5f8IelFRgFjhhrWWl96sRSClcG5YIx6))

Relation to [[Chern-Simons theory with complex gauge group]]:

* [[Daniel S. Freed]], [[Andrew Neitzke]], *3d spectral networks and classical Chern-Simons theory* [[arXiv:2208.07420](https://arxiv.org/abs/2208.07420)]

[[!redirects spectral networks]]


spectral presheaf

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### Context
#### Topos Theory
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[[!include topos theory - contents]]
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#### Operator algebra
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[[!include AQFT and operator algebra contents]]
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#### Noncommutative geometry
+--{: .hide}
[[!include noncommutative geometry - contents]]
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#### Physics
+-- {: .hide}
[[!include physicscontents]]
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#Contents#
* table of contents
{:toc}

## Idea

Given a [[C*-algebra]] $A$, not necessarily [[commutative C*-algebra|commutative]], write $ComSub(A)$ for its [[poset of commutative subalgebras]], whose [[morphisms]] are the inclusion maps. By [[Gelfand duality]], the [[presheaf topos]] $PSh(ComSub(A))$ over this contains a canonical [[object]], namely the [[presheaf]]

$$
  \Sigma \;\colon\; C \mapsto \Sigma_C
$$

which maps a [[commutative C*-algebra]] $C \hookrightarrow A$ to (the point set underlying) its [[Gelfand spectrum]] $\Sigma_C$. This is called the _spectral presheaf_ of $A$ ([Isham-Döring 07](#IshamDoering07))

## Properties

The [[Kochen-Specker theorem]] of [[quantum mechanics]] is equivalent to the statement that for $H$ a (complex) [[Hilbert space]] of [[dimension]] greater than 2, then the spectral presheaf of the [[algebra of bounded operators]] $\mathcal{B}(H)$ (the [[quantum observables]]) has no [[global element]] ([Butterfield-Hamilton-Isham 98](#ButterfieldHamiltonIsham)). (This observation motivates the topos-theoretic development in ([Isham-Döring 07](#IshamDoering07))).

See at _[[Kochen-Specker theorem]]_ and at _[[Bohr topos]]_ for more on this.

## Related concepts

* [[Bohr topos]]

* [[daseinisation]]

## References

The term "spectral presheaf" was introduced in

* [[Jeremy Butterfield]], John Hamilton, [[Chris Isham]], _A topos perspective on the Kochen-Specker theorem_, _I. quantum states as generalized valuations_, Internat. J. Theoret. Phys. 37(11):2669--2733, 1998, [MR2000c:81027](http://www.ams.org/mathscinet-getitem?mr=1669557), [doi](http://dx.doi.org/10.1023/A:1026680806775); _II. conceptual aspects and classical analogues_ Int. J. of Theor. Phys. 38(3):827--859, 1999, [MR2000f:81012](http://www.ams.org/mathscinet-getitem?mr=1697983), [doi](http://dx.doi.org/10.1023/A:1026652817988); _III. Von Neumann algebras as the base category_, Int. J. of Theor. Phys. 39(6):1413--1436, 2000, [arXiv:quant-ph/9911020](http://arxiv.org/abs/quant-ph/9911020), [MR2001k:81016](http://www.ams.org/mathscinet-getitem?mr=1788498),[doi](http://dx.doi.org/10.1023/A:1003667607842); _IV. Interval valuations_, Internat. J. Theoret. Phys. __41__ (2002), no. 4, 613–639, [MR2003g:81009](http://www.ams.org/mathscinet-getitem?mr=1902067), [doi](http://dx.doi.org/10.1023/A:1015276209768)    {#ButterfieldHamiltonIsham}

* [[Chris Isham]], [[Andreas Döring]], _A Topos Foundation for Theories of Physics_, ([arXiv:quant-ph/0703060](http://arxiv.org/abs/quant-ph/0703060), [arXiv:quant-ph/0703062](http://arxiv.org/abs/quant-ph/0703062), [arXiv:quant-ph/0703066](http://arxiv.org/abs/quant-ph/0703066))
 {#IshamDoering07}

[[!redirects spectral presheaves]]


Spectral Schemes

+-- {: .rightHandSide}
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###Context###
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

This page collects some links related to the article:

* [[Jacob Lurie]],

  _Spectral Schemes_ ([pdf](http://www.math.harvard.edu/~lurie/papers/DAG-VII.pdf))

This studies the theory of [[generalized scheme]] whose [[structure sheaf]] takes values in [[E-∞ ring]]-spectra. Hence [[structured (∞,1)-toposes]] in [[E-∞ geometry]].

See at _[[E-∞ scheme]]_.

See also

* [[Paul Goerss]], [[Topological Algebraic Geometry - A Workshop]]

#Contents#
* table of contents
{:toc}

## Sheaves of spectra

* [[sheaf of spectra]]

* [stabilization via sheaves of spectra](stable+%28infinity%2C1%29-category#StabilizationBySheavesOfSpectra)

* [inclusion of sheaves of connective spectra](connective+spectrum#InclusionIntoAllSpectra)

## Spectral schemes

* [[E-infinity scheme]]

## Coherent $\infty$-Topoi

* [[coherent (∞,1)-topos]]

* [[compact topos]]

* [[n-coherent object]]

## Deligne's completeness theorem

* [[Deligne-Lurie completeness theorem]]

* [[Deligne completeness theorem]]

##  Flat descent

## Flat descent for modules

## Digression: Henselian rings

## Spectral Deligne-Mumford stacks

* [[spectral Deligne-Mumford stack]]

## Comparison results

category: reference