zero object

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### Context
#### Category theory
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[[!include category theory - contents]]
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#### Additive and abelian categories
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[[!include additive and abelian categories - contents]]
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# Zero objects
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{: toc}

## Definition

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

In a [[category]], an [[object]] is called
a **zero object**, **null object**, or **biterminator** if it is both an [[initial object]] and a [[terminal object]].

=--

A category with a zero object is sometimes called a _[[pointed category]]_.

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

This means that $0 \in \mathcal{C}$ is a zero object precisely if for every other object $A$ there is a unique [[morphism]] $A \to 0$ to the zero object as well as a unique morphism $0 \to A$ from the zero object.

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

If $\mathcal{C}$ is a [[pointed category]], then an object $A$ of $\mathcal{C}$ is a zero object precisely when the only endomorphism of $A$ is the identity morphism.

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

There is also a notion of **zero object** in [[algebra]] which does not always coincide with the category-theoretic terminology. For example the zero [[ring]] $\{0\}$ is not an [[initial object]] in the category of unital rings (this is instead the [[integers]] $\mathbb{Z}$); but it is the
[[terminal object]].  However, the zero ring *is* the zero object in the category of [[nonunital ring]]s (although it happens to be unital).

=--

## Examples

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

* The one-point set is the zero object of the [[category of pointed sets]] (denoted $\Set_*$) and of the category of [[pointed topological spaces]] (denoted $\Top_*$), but only the [[terminal object]] of [[Set]] and [[Top]].

* The [[trivial group]] is a zero object in the category [[Grp]] of [[groups]] and in the category [[Ab]] of [[abelian groups]].

* For $R$ a [[ring]], the trivial $R$-[[module]] (that whose underlying abelian group is the [[trivial group]]) is the zero-object in $R$[[Mod]].

  In particular for $R = k$ a [[field]], the $k$-[[vector space]] of [[dimension]] 0 is the zero object in [[Vect]].

* For $R$ and $S$ [[rings]], the trivial $R$-$S$-[[bimodule]] (that whose underlying abelian group is the [[trivial group]]) is the zero-object in $R$-$S$-[[Bimod]].

* However, the zero [[ring]] is not a zero object in the category of [[ring|rings]], at least as long as rings are required to have units (and ring homomorphisms to preserve them).

* For every category $C$ with a [[terminal object]] $*$ the [[under category]] $pt \downarrow C$ of [[pointed objects]] in $C$ has a zero object: the morphism $Id_{pt}$.

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

In any category $C$ [[enriched category|enriched]] over the [[category of pointed sets]] $(Set_*, \wedge)$ with [[tensor product]] the [[smash product]], any object that is either initial _or_ terminal is automatically both and hence a zero object.

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

Write $* \in Set_*$ for the singleton pointed set. Suppose $t$ is [[terminal object|terminal]]. Then $C(x,t) = *$ for all $x$ and so in particular $C(t,t) = *$ and hence the [[identity]] morphism on $t$ is the basepoint in the pointed [[hom-set]]. By the axioms of a [[category]], every morphism $f : t \to x$ is equal to the composite

$$
  f : t \stackrel{Id}{\to} t \stackrel{f}{\to} x
  \,.
$$

By the axioms of an $(Set_*, \wedge)$-enriched category, since $Id_{t}$ is the basepoint in $C(t,t)$, also this composite is the basepoint in $C(t,x)$ and is hence the [[zero morphism]]. So $C(t,x) = *$ for all $x$ and therefore $t$ is also an [[initial object]].

Analogously from assuming $t$ to be initial it follows that it is also terminal.

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

This is a special case of an [[absolute limit]].

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

Categories enriched in $(Set_*, \wedge)$ include in particular [[Ab]]-enriched categories. So any [[additive category]], hence every [[abelian category]] has a zero object.

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* In the [[stable homotopy category]]: [[zero spectrum]].

## Properties

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

A category has a zero object precisely if it has an [[initial object]] $\emptyset$ and a [[terminal object]] $*$ and the unique morphism $\emptyset \to *$ is an [[isomorphism]].

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

In a category with a zero object 0, there is always a canonical morphism from any object $A$ to any other object $B$ called the _[[zero morphism]]_, given by the composite $A\to 0 \to B$.

Thus, such a category becomes [[enriched category|enriched]] over the [[category of pointed sets]], a partial converse to prop \ref{InCatsEnrichedInPointedSets}.

=--

## Related concepts

* [[pointed category]]

  [[pointed (∞,1)-category]], [[pointed model category]]

* [[zero morphism]]

* [[zero object in a derivator]]

## References

* [[Saunders MacLane]], §I.5 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

[[!redirects zero object]]
[[!redirects zero objects]]
[[!redirects 0 object]]
[[!redirects 0 objects]]
[[!redirects 0-object]]

[[!redirects null object]]
[[!redirects null objects]]

[[!redirects biterminator]]
[[!redirects biterminators]]


zero section

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

## Definition

For $E \to X$ [[vector bundle]], its **zero section** is the [[section]] $X \to E$ that sends every point to the 0-vector over it.

## Properties

A vector bundle minus its zero-section is a [[spherical fibration]].

## Related concepts

* [[Thom space]], [[Thom spectrum]]

* [[projective bundle]]

* [[vanishing locus]], [[critical locus]], [[derived critical locus]]

* [[Hörmander's criterion]] on [[wave front set]]

[[!redirects zero-section]]

[[!redirects zero sections]]
[[!redirects zero-sections]]


zero spectrum

## Idea

The [[zero object]] in the [[stable (infinity,1)-category of spectra]]/[[stable homotopy category]].

As a [[sequential spectrum]] this is represented by the sequence of [[pointed topological space]] which consists of the one-point space in each degree (an [[Omega-spectrum]]). This particular representative happens to be itself the [[zero object]] in the [[1-category]] of [[sequential spectra]]. But, as usual, there are other sequential spectra, not [[isomorphism|isomorphic]] to this one, which still represent the [[zero-spectrum]] in the [[stable homotopy theory]] of spectra (hence which are connected to the spectrum constant on the point by a sequence of [[weak equivalences]] in the stable [[model structure on topological sequential spectra]]).

[[!redirects zero spectra]]

[[!redirects zero-spectrum]]
[[!redirects zero-spectra]]


zero-divisor

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###Context###
#### Algebra
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[[!include higher algebra - contents]]
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#### Monoid theory
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[[!include monoid theory - contents]]
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# Zero-divisors
* table of contents
{: toc}

## Idea

A zero-divisor is something that, like [[zero]] itself, when [[multiplication|multiplied]] by something possibly nonzero still produces zero as a product.

## Definitions

Let $M$ be a [[absorption monoid]] (such as a [[commutative ring]] or any [[ring]]).

An element $x$ of $M$ is a __non-zero-divisor__ if, whenever $x \cdot y = 0$ or $y \cdot x = 0$, then $y = 0$.  An element $x$ is a __zero-divisor__ if there exists $y \ne 0$ such that $x \cdot y = 0$ or $y x = 0$.

In [[constructive mathematics]], we want $\ne$ to be a [[tight apartness relation]] on $M$ in the definition of zero-divisor.  We also say that $x$ is a __strong non-zero-divisor__ if, whenever $y \ne 0$, then $x y \ne 0$ and $y x \ne 0$.  (The notion of (weak) non-zero-divisor makes sense even without any apartness relation.)

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

## Properties

By this definition, [[zero]] itself is a zero-divisor if and only if $M$ is non-trivial.  (Some authorities will differ on this point, but if you think about it, this is clearly the correct definition, by the same principle that the trivial ring is not a field, $1$ is not a prime number, etc.  See [[too simple to be simple]].)

An [[integral domain]] is precisely a commutative ring (whose multiplicative monoid is an absorption monoid by definition) in which [[zero]] is the unique zero-divisor of the multiplicative monoid of the commutative ring (or constructively, in which the strong non-zero-divisors are precisely the strong non-zero elements in the multiplicative monoid, that is those elements $x$ such that $x \ne 0$).

The non-zero-divisors of any absorption monoid $M$ form a [[monoid]] under multiplication, which may be denoted $M^{\times}$.  Note that if $M$ happens to be a [[field]], then this $M^{\times}$ agrees with the usual notation $M^{\times}$ for the [[group]] of invertible elements of the multiplicative monoid $M$, but $M^{\times}$ is not a group in general.  (We may use $M^{\div}$ or $M^*$ for the group of invertible elements.)

## Generalisations

If $I$ is any [[ideal of a monoid|ideal]] of $M$, then we can generalise from a zero-divisor to an $I$-[[divisor]].  In a way, this is nothing new; $x$ is an $I$-divisor in $M$ if and only if $[x]$ is a zero-divisor in $M/I$.  Ultimately, this is related to the notion of [[divisor]] in [[algebraic geometry]].

## Related concepts

* [[absorption monoid]]

* [[integral domain]]

* [[cancellative element]]

[[!redirects zero-divisor]]
[[!redirects zero-divisors]]
[[!redirects zero divisor]]
[[!redirects zero divisors]]


zero-one law

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### Context
#### Measure and probability theory
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[[!include measure theory - contents]]
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## Idea

(...Work in progress...)

## See also

* [[Markov category]]

* [[probability monad]]

* [[probability theory]]

## References

* {#fritzrischel} [[Tobias Fritz]] and Eigil Fjeldgren Rischel, _Infinite products and zero-one laws in categorical probability_, Compositionality 2(3) 2020. ([arXiv:1912.02769](http://arxiv.org/abs/1912.02769))

* {#ergodic_dagger} Noé Ensarguet, [[Paolo Perrone]], Categorical probability spaces, ergodic decompositions, and transitions to equilibrium, [arXiv:2310.04267](https://arxiv.org/abs/2310.04267)

category: probability

[[!redirects zero-one laws]]
[[!redirects 0-1 law]]
[[!redirects 0-1 laws]]
[[!redirects Kolmogorov zero-one law]]
[[!redirects Hewitt-Savage zero-one law]]
[[!redirects Kolmogorov 0-1 law]]
[[!redirects Hewitt-Savage 0-1 law]]


zero-section into Thom space of universal line bundle is weak equivalence

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#### Bundles
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#### Homotopy theory
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[[!include homotopy - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The [[Thom space]] of the [[universal complex line bundle]] is [[weak homotopy equivalence|weakly homotopy equivalent]] to the base space of the line bundle, hence to the [[classifying space]] for the [[circle group]], and this is equivalence is exhibited by the [[zero section]] of the universal line bundle, followed by its inclusion into its [[Thom space]] ([Adams 74, Part I, Example 2.1](#Adams74), see Prop. \ref{TheStatement} below).

This statement plays a key role in the discussion of [[complex oriented cohomology]], as it implies that any choice of [[universal characteristic class|universal]] first [[Conner-Floyd Chern class]] $c^E_1$ is equivalent to a choice of universal [[Thom class]] on the [[universal complex line bundle]], and hence induces a Thom class, hence a "fiberwise complex orientation" on any [[complex line bundle]]. (See at _[Conner-Floyd E-Chern classes are E-Thom classes](universal+complex+orientation+on+MU#ConnerFloydChernClassesAreThomClasses)_ for more on this.)

In this context the statement is naturally stated in the form

$$
  B \mathrm{U}(1) \overset{\simeq}{\longrightarrow} M \mathrm{U}(1)
  \,,
$$

where on the right the [[Thom space]] is thought of as the component space in degree 2 (complex degree 1) of the universal [[unitary group|unitary]] [[Thom spectrum]] [[MU]].

The analogous statement is true also for the universal [[real vector bundle|real]]- and [[quaternionic line bundle|quaternionic]] [[line bundles]], and it implies the analogous consequence for [[quaternionic oriented cohomology theory]], etc., notably

$$
  B Sp(1) \overset{\simeq}{\longrightarrow} M Sp(1)
  \,,
$$

where on the right the [[Thom space]] is thought of as the component space in degree 4 (quaternionic degree 1) of the universal [[quaternionic unitary group|quaternionic unitary]] [[Thom spectrum]] [[MSp]].

## Preliminaries

### The universal line bundle

Let $\mathbb{K} \,\in\, \{\mathbb{R}, \mathbb{C}, \mathbb{H}\}$
be the [[real numbers]] or [[complex numbers]] or [[quaternions]].
Write

\[
  \label{GroupOfUnitNormElements}
  S(\mathbb{K})
  \;\coloneqq\;
  \big\{
    q \in \mathbb{K}
    \;\big\vert\;
    q q^\ast =1
  \big\}
  \;\;
  \in
  \;
  Groups
\]

for its [[multiplicative group]] of unit-[[norm]] elements. Specifically this is the [[cyclic group of order 2]], the [[circle group]] or the [[quaternionic unitary group]]/[[SU(2)]]:

$$
  S(\mathbb{R})
  \,\simeq\,
  \mathbb{Z}/2
  \,,
  \phantom{AA}
  S(\mathbb{C})
  \,\simeq\,
  \mathrm{U}(1)
  \,,
  \phantom{AA}
  S(\mathbb{H})
  \,\simeq\,
  Sp(1) \,\simeq\, SU(2)
$$

By either left or right multiplication in $\mathbb{K}$ this group [[action|acts]] on $\mathbb{K}$, $\mathbb{R}$-[[linear map|linearly]], making $\mathbb{K}$ a [[linear representation]]

\[
  \label{KAsSKRepresentation}
  \mathbb{K}
    \,\in\,
  S(\mathbb{K})
  Representations_{\mathbb{R}}
  \,.
\]

Hence with

$$
  E
  \big(
    S(\mathbb{K})
  \big)
  \overset{\;\;\;}{\longrightarrow}
  B
  \big(
    S(\mathbb{K})
  \big)
$$

denoting the $S(\mathbb{K})$-[[universal principal bundle]] over the [[classifying space]] for the group (eq:GroupOfUnitNormElements), the [[real vector bundle]] underlying the [[universal complex line bundle|universal K-line bundle]] is the corresponding [[associated bundle]] via the above action (eq:KAsSKRepresentation):

\[
  \label{UniversalKLineBundle}
  \array{
     E
     \big(
       S(\mathbb{K})
     \big)
     \underset{
       S(\mathbb{K})
     }{\times}
     \mathbb{K}
     \\
     \big\downarrow
     \\
     B
     \big(
       S(\mathbb{K})
     \big)
  }
\]

### Thom spaces

The [[Thom space]] of a [[real vector bundle|real]] [[topological vector bundle]] $\mathcal{V}_X$ over some base space $X$ is the [[homotopy cofiber]] of its associated [[spherical fibration]]:

\[
  \label{ThomSpaceAsHomotopyCofiber}
  S_X(\mathcal{V}_X)
  \overset{ p_{S(\mathcal{V}_X)} }{\longrightarrow}
  X
  \overset{ hocofib }{\longrightarrow}
  Th(X)
  \,.
\]

When the [[topological space]] $X$ has the [[mathematical structure|structure]] of a [[CW-complex]] then a [[cofibration]] which models the [[homotopy type]] of $p_{S(\mathcal{V}_X)}$ in the [[classical model structure on topological spaces]] is the inclusion of the unit [[sphere bundle]]  into the unit disk bundle $D_X(\mathcal{V}_X)$ of $\mathcal{V}_X$ (with respect to any choice of fiberwise [[metric]]) since this is then a [[relative cell complex]]-inclusion. Therefore the homotopy cofiber (eq:ThomSpaceAsHomotopyCofiber) is then represented by the [[1-category|1-category theoretic]] [[cofiber]]

\[
  \label{ThomSpaceAsActualCofiber}
  S_X(\mathcal{V}_X)
  \overset{ i_{S_X(\mathcal{V}_X)} }{\longrightarrow}
  D_X(\mathcal{V}_X)
  \overset{ cofib }{\longrightarrow}
  Th(X)
  \,.
\]

Finally, since the [[zero section]] of the unit disk bundle is manifestly the  [[weak homotopy equivalence]] that exibits this [[cofibrant resolution]], we may call

\[
  \label{ZeroSectionIntoTheThomSpace}
  0_X
  \;\colon\;
  X
    \underoverset
    {\simeq}
    {
      0_{D_X(\mathcal{V}_X)}
    }
    {\longrightarrow}
  D(\mathcal{V}_X)
   \overset
    {cofib\big(  i_{S_X(\mathcal{V}_X)} \big)}
    {\longrightarrow}
  Th(X)
\]

the zero-section _into_ the Thom spaces.

## Statement

\begin{prop}\label{TheStatement}
  The zero-section (eq:ZeroSectionIntoTheThomSpace) into the [[Thom space]] of the universal $\mathbb{K}$-line bundle (eq:UniversalKLineBundle)

  $$
    \array{
       Th
       \Big(
         E (S(\mathbb{K}))
         \underset{
           S(\mathbb{K})
         }{\times}
         \mathbb{K}
       \Big)
       \\
       {}^{{}_{\mathllap{
         0_{
           B \big(
             S(\mathbb{K})
           \big)
         }
       }}}
       \big\uparrow
       {}^{{}_{
         \mathrlap{\simeq}}
       }
       \\
       B \big(
         S(\mathbb{K})
       \big)
    }

  $$
  is a [[weak homotopy equivalence]].
\end{prop}

\begin{proof}
  The point is that for the universal _line_ bundle, the associated [[sphere bundle]] is [[homotopy equivalent]] to the [[universal principal bundle]] and hence [[contractible homotopy type|weakly contractible]].

One way to see it is to unwind the definition of the unit sphere bundle in the universal line bundle as follows:

\[
  \label{SphereBundleOfUniversalLineBundleIsContractible}
  S_{B \big(S(\mathbb{K})\big)}
  \Big(
    E\big(S(\mathbb{K})\big)
I       \underset{S(\mathbb{K})}{\times}
    \mathbb{K}
  \Big)
  \;=\;
  \Big(
    E\big(S(\mathbb{K})\big)
      \underset{S(\mathbb{K})}{\times}
    S(\mathbb{K})
  \Big)
   \;=\;
  E\big(S(\mathbb{K})\big)
    \;\simeq\;
  \ast
  \,.
\]

{#ViaContractibleInfiniteDimensionalSpheres} Another way to see the same is to observe that the sphere bundle associated to the universal line bundle is the [[sequential colimit]] over the [[tautological line bundle|tautological]] [[principal bundles]] over the finite-dimensional [[complex projective space]],  which themselves are [[n-spheres]] (see [there](tautological+line+bundle#eq:TautologicalPrincipalBundleOverProjectiveSpace)). With this, the statement (eq:SphereBundleOfUniversalLineBundleIsContractible) follows from the fact that [[the infinite-dimensional sphere is weakly contractible]] (see [there](infinite-dimensional+sphere#AsAnInfiniteSphericalCellComplex)):

$$
  S_{\mathbb{K}P^\infty}
  \big(
    \mathcal{L}^\ast_{\mathbb{K}P^\infty}
  \big)
  \;\simeq\;
  \underset{ \underset{n}{\longrightarrow} }{\lim}
  \;
  \big(
    S_{\mathbb{K}P^n}
    \left(
      \mathcal{L}^\ast_{\mathbb{K}P^n}
    \right)
  \big)
  \;\simeq\;
  \underset{ \underset{n}{\longrightarrow} }{\lim}
  \;
  \left(
    S^{ (n+1) \cdot dim_{{}_{\mathbb{R}}}(\mathbb{K}) - 1 }
  \right)
  \;\simeq\;
  S^\infty
  \;\simeq\;
  \ast
  \,.
$$

In any case, this means that we have the following solid [[commuting diagram]], where the solid vertical morphisms are all [[weak homotopy equivalences]]:

\begin{xymatrix@R=25pt@C=50pt}
      && &&
      B(S(\mathbb{K}))
      \ar[d]
        _-{
           0
        }
        |-{
          \hspace{3pt}
          \scalebox{.7}{
            \rotatebox{-90}{$
              \;\simeq\;
            $}
          }
        }
      \\
      \ast
      \;
      \ar@{^{(}->}[rr]_-{ \in \, \mathrm{Cofibrations} }
      \ar[d]
        |-{
          \hspace{3pt}
          \scalebox{.7}{
            \rotatebox{-90}{$
              \;\simeq\;
            $}
          }
        }
      &&
      D_{B(S(\mathbb{K}))}
      \Big(
        E\big( S(\mathbb{K}) \big)
          \underset{S(\mathbb{K})}{\times}
        \mathbb{K}
      \Big)
      \ar[rr]
        ^-{ \mbox{\tiny cofiber} }
        _-{ \mbox{\tiny $\Rightarrow$ homotopy cofiber} }
      \ar@{=}[d]
      &&
      D_{B(S(\mathbb{K}))}
      \Big(
        E\big( S(\mathbb{K}) \big)
          \underset{S(\mathbb{K})}{\times}
        \mathbb{K}
      \Big)
      \ar@{-->}[d]
      \\
      S_{B(S(\mathbb{K}))}
      \Big(
        E\big( S(\mathbb{K}) \big)
          \underset{S(\mathbb{K})}{\times}
        \mathbb{K}
      \Big)
      \;
      \ar@{^{(}->}[rr]
        _-{ \in \, \mathrm{Cofibrations} }
      &&
      D_{B(S(\mathbb{K}))}
      \Big(
        E\big( S(\mathbb{K}) \big)
          \underset{S(\mathbb{K})}{\times}
        \mathbb{K}
      \Big)
      \ar[rr]
        ^-{ \mbox{\tiny cofiber} }
        _-{ \mbox{\tiny $\Rightarrow$ homotopy cofiber} }
      &&
      \mathrm{Th}
      \Big(
        E\big( S(\mathbb{K}) \big)
          \underset{S(\mathbb{K})}{\times}
        \mathbb{K}
      \Big)
\end{xymatrix}

(Here the left vertical map picks any point of the sphere bundle. There is then a unique horizontal map on the left to make the left square [[commuting diagram|commute]].)

Now, since the [[classifying space]] $B(S(\mathbb{K}))$ does have the structure of a [[CW-complex]] (given, for instance, by its realization as infinite [[real projective space|real]]/[[complex projective space|complex]]/[[quaternionic projective space|quaternionic]] [[projective space]] $\mathbb{K}P^\infty$, via the [[cell structure of projective space]]), the bottom [[cofiber]] here represents, as in (eq:ThomSpaceAsActualCofiber), the defining [[homotopy cofiber]].

Since [[homotopy cofibers]] are preserved, up to [[weak equivalence]], by weak equivalences of their diagrams (by [this Prop.](Introduction+to+Homotopy+Theory#FiberOfFibrationIsCompatibleWithWeakEquivalences)), it follows that the dashed vertical morphism is a weak equivalence in the [[classical model structure on topological spaces]], hence a [[weak homotopy equivalence]].

This is the statement that was to be shown. Or more explicitly: By [[two-out-of-three]] also the [[composition|composite]] vertical morphism on the right is a weak homotopy equivalence, which is the desired morphism in the form (eq:ZeroSectionIntoTheThomSpace).
\end{proof}

## Related statements

* [[cell structure of projective space]]

* [Conner-Floyd E-Chern classes are E-Thom classes](universal+complex+orientation+on+MU#ConnerFloydChernClassesAreThomClasses)

See also:

* [[complex oriented cohomology]], [[universal complex orientation on MU]]

* [[quaternionic oriented cohomology]]

## References

* {#Adams74} [[Frank Adams]], Part I, Example 2.1 in: _[[Stable homotopy and generalised homology]]_, 1974

* {#Kochmann96} [[Stanley Kochman]], Lemma 2.6.5 in: _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

[[!redirects zero-section into Thom space of universal line bundle is weak homotopy equivalence]]


zero-set structure

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#### Topology
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[[!include topology - contents]]
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\tableofcontents

## Definition

Given a [[set]] $X$, a **zero-set structure** on $X$ is a [[sigma-topology|$\sigma$-topology]] $\mathcal{Z}$ such that

* For every pair of distinct points in $X$ there is an open set $Z \in \mathcal{Z}$ containing precisely one of these points.

* If $Z \in \mathcal{Z}$ then there are $Z_n \in \mathcal{Z}$ for all natural numbers $n$ such that $Z^c = \bigcup_{n \in \mathbb{N}} Z_n$.

* If $Z_1, Z_2 \in \mathcal{Z}$ and $Z_1 \cap Z_2 = \emptyset$, then there are $V_1, V_2 \in \mathcal{Z}$ with $Z_1 \subseteq V_1^c$, $Z_2 \subseteq V_2^c$, and $V_1^c \cap V_2^c = \emptyset$.

## See also

* [[zero set]]
* [[sigma-topology]]

## References

* Fedor Petrov, *"countable" topology*, MathOverflow ([web](https://mathoverflow.net/questions/173255/countable-topology))

* Hugh Gordon, *Rings of functions determined by zero-sets*. Hugh Gordon. Pacific J. Math. Volume 36, Number 1 (1971), 133-157. ([pdf](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-36/issue-1/Rings-of-functions-determined-by-zero-sets/pjm/1102971274.pdf))

[[!redirects zero-set structure]]
[[!redirects zero-set structures]]


zeta function

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#### Theta functions
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#Contents#
* table of contents
{:toc}

## Idea

The concept of _zeta function_ originates in [[number theory]], but to get an idea of what they "really are" it is helpful to proceed anachronistically:

$\zeta$-functions are [[meromorphic functions]] $s \mapsto \zeta(s)$ on the [[complex plane]], which behave like [[analytic continuations]] of [[traces]] of powers

$$
  s \mapsto Tr \left(\frac{1}{H}\right)^s
$$

of suitable [[elliptic differential operators]] $H$ (in [[physics]] these are [[regularization (physics)|regularized]] [[traces]] of [[Feynman propagators]] leading to expressions for [[vacuum amplitudes]]), which means that for sufficiently nice such $H$ these are analytic continuations in $s$ of sums of the form

$$
  s \mapsto \underset{\lambda}{\sum}  \lambda^{-s}
  \,,
$$

where the summation is over the [[eigenvalues]] $\lambda$ of $H$.

Indeed, such _[[zeta functions of elliptic differential operators]]_ constitutes one class of examples of zeta functions.  Of particular interest is the case where $H$ is a [[Laplace operator]] of a [[hyperbolic manifold]] and in particular on a hyperbolic [[Riemann surface]], for that case one obtains the _[[zeta function of a Riemann surface]]_, in particular the _[[Selberg zeta function]]_.

In modern language one also speaks of _[[L-functions]]_. Where a zeta function of some space is like the [[Feynman propagator]] of _the_ canonical [[Laplace operator]] of that space, an L-function is defined from an extra "twisting" information such as that of a [[flat bundle]]/[[local system of coefficients]] on the space (and is hence like the [[Feynman propagator]] of the corresponding twisted/coupled [[Laplace operator]]). The major properties satisfied by anything that qualifies as a zeta function or [[L-functions]] are: these are [[meromorphic functions]] $s \mapsto L(s)$ on the [[complex plane]] such that

1. for $\Re(s) \gt 1$ they have a [[convergence|converging]] [[series]] expansion of the above form, and/or a [[infinite product|multiplicative series]] expression, the _[[Euler product]]_;

1. such that [[analytic continuation]] of the series expression exists to a meromorphic function $L(-)$ on the complex plane;

1. and such that the result satisfies a _[[functional equation]]_ which says that the product $\hat L$ of $L$ with some correcion functions satisfies $\hat L(1-s) = \hat L(s)$.

Proceeding from the above class of examples in [[complex analytic geometry]] one may wonder if there are [[analogy|analogs]] also in [[arithmetic geometry]]. Indeed, by the [[function field analogy]] there are. All the way down "on [[Spec(Z)]]" the analog of the [[Selberg zeta function]] is the [[Riemann zeta function]], which _historically_ is the first of all zeta functions, defined by [[analytic continuation]] of the [[series]]

$$
   s \mapsto \underoverset{n = 1}{\infty}{\sum} n^{-s}
  \,.
$$

The _[[Riemann hypothesis]]_ [[conjecture|conjectures]] a characterization of the [[roots]] of this zeta function and is regarded as one of the outstanding problems in [[mathematics]]. It has evident analogs for all other zeta functions (for some of which it has been proven).

More generally, over [[arithmetic curves]] which are [[spectrum of a commutative ring|spectra]] of [[rings of integers]] of more general [[number fields]], the Riemann zeta function has generalization to the _[[Artin L-functions]]_ defined intrinsically in terms of [[characteristic polynomials]] of [[Galois representations]]. When the Galois representation is 1-dimensional, then the Artin L-function may be expressed (by "[[Artin reciprocity]]") in terms of "more arithmetic" data by [[Dirichlet L-functions]] and [[Hecke L-functions]]. When the Galois representation is higher dimensional, then the [[Langlands correspondence]] [[conjecture]] asserts that the Artin L-function may be expressed "arithmetically" as the [[automorphic L-function]] of an [[automorphic form]].

Similarly on [[arithmetic curves]] given by [[function fields]] there is the [[Goss zeta function]] and in [[higher dimensional arithmetic geometry]] the [[Weil zeta function]], famous from the _[[Weil conjectures]]_. The

When interpreting the [[Frobenius morphisms]] that appear in the [[Artin L-functions]] geometrically as flows (as discussed at _[Borger's arithmetic geometry -- Motivation](Borger's+absolute+geometry#Motivation)_) then this induces an evident analog of [[zeta function of a dynamical system]]. This in turn has strong analogies with [[Alexander polynomials]] in [[knot theory]] (see at _[[arithmetic topology]]_).

[[!include zeta-functions and eta-functions and theta-functions and L-functions -- table]]

## Properties

### Function field analogy
 {#FunctionFieldAnalogy}

One way to understand the plethora of different zeta functions is to see them as the incarnation of the same general concept in different flavors of [[geometry]]. This is expressed at least in parts by the

[[!include function field analogy -- table]]

## Related concepts

* [[multiple zeta values]], [[motivic multiple zeta values]], [[motivic integration]], [[motive]]

* [[Weil conjecture]]

* [[Riemann hypothesis]]

* there are attempts to understand the Riemann zeta function as the spectrum of a [[Hamiltonian]] of a [[quantum mechanical system]]. See at _[[Riemann hypothesis and physics]]_.

* [[Beilinson regulator]]

* [[zeta function of a Riemann surface]]

## References

### General

A useful survey of the zoo of zeta functions is in

* [[Jeffrey Lagarias]], _Number theory zeta functions and Dynamical zeta functions_ ([pdf](http://www.math.lsa.umich.edu/~lagarias/doc/numberthzeta.pdf))

Further general review includes

* {#Kowalski} E. Kowalski, first part of _Automorphic forms, L-functions and number theory (March 12–16) Three Introductory lectures_ ([pdf](http://www.math.ethz.ch/~kowalski/lectures.pdf))

* [[Alain Connes]], [[Matilde Marcolli]], chapter II of _[[Noncommutative Geometry, Quantum Fields and Motives]]_

Discussion in the more general context of [[higher dimensional arithmetic geometry]] is in

* {#Fesenko08} [[Ivan Fesenko]], _Adelic approch to the zeta function of arithmetic schemes in dimension two_, Moscow Math. J. 8 (2008), 273–317 ([pdf](https://www.maths.nottingham.ac.uk/personal/ibf/ada.pdf))

### In algebraic geometry

* [[Yuri Manin]], _Lectures on zeta functions and motives (according to Deninger and Kurokawa)_, Astérisque __228__:4 (1995) 121--163, and preprint MPIM1992-50 [pdf](http://www.mpim-bonn.mpg.de/preblob/4793)

* Nobushige Kurokawa, _Zeta functions over $F_1$_,  Proc. Japan Acad. Ser. A Math. Sci. __81__:10 (2005) 180-184 [euclid](http://projecteuclid.org/euclid.pja/1135791771)

* [[Bruno Kahn]], _Fonctions zêta et $L$ de variétés et de motifs_, [arXiv:1512.09250](http://arxiv.org/abs/1512.09250v1).

### Categorical approaches

* [[Michael Larsen|M. Larsen]], [[Valery Lunts|V. A. Lunts]], _Motivic measures and stable birational geometry_, Mosc. Math. J. __3__, 1 (2003) 85--95; _Rationality criteria for motivic zeta functions_, Compos. Math. __140__:6 (2004) 1537–1560
* [[Vladimir Guletskii]], _Zeta functions in triangulated categories_, Mathematical Notes __87__, 3 (2010) 369--381, [math/0605040](http://arxiv.org/abs/math/0605040)
* [[Maxim Kontsevich|M. Kontsevich]], _Notes on motives in finite characteristics_, [math.AG/0702206](http://arxiv.org/abs/math.AG/0702206)

* [[John Baez]], _[[johnbaez:Zeta functions]]_

* [[Sergey Galkin]], [[Evgeny Shinder]], _On a zeta-function of a dg-category_, [arXiv:1506.05831](http://arxiv.org/abs/1506.05831).

[[!redirects zeta functions]]


zeta function of a dynamical system

#Contents#
* table of contents
{:toc}

## Idea

When interpreting the [[Frobenius morphisms]] that appear in the [[Artin L-functions]] geometrically as flows (as discussed at _[Borger's arithmetic geometry -- Motivation](Borger's+absolute+geometry#Motivation)_) then this induces an evident analog of _[[zeta function]] of a [[dynamical system]]_.

The archetypal example is the _[[Ruelle zeta function]]_.

## Definition

## References

Lecture notes include

* [[Mark Pollicott]], _Dynamical zeta functions_ ([pdf](https://homepages.warwick.ac.uk/~masdbl/grenoble-16july.pdf))

* _Dynamical zeta functions_ ([pdf](http://homepages.warwick.ac.uk/~masdbl/leiden-lectures.pdf))

* [[Jeffrey Lagarias]], _Number theory zeta functions and Dynamical zeta functions_ ([pdf](http://www.math.lsa.umich.edu/~lagarias/doc/numberthzeta.pdf))

[[!redirects zeta functions of dynamical systems]]

[[!redirects dynamical zeta function]]
[[!redirects dynamical zeta functions]]


zeta function of a Riemann surface

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
#### Functional analysis
+--{: .hide}
[[!include functional analysis - contents]]
=--
#### Operator algebra
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[zeta function]] naturally associated to a [[Riemann surface]]/[[complex curve]], hence the [[zeta function of an elliptic differential operator]] for the [[Laplace operator]] on the Riemann surface (and hence hence essentially the [[Feynman propagator]] for the [[scalar fields]] on that surface) is directly analogous to the zeta functions associated with [[arithmetic curves]], notably the [[Artin L-functions]].

([Minakshisundaram-Pleijel 49](#MinakshisundaramPleijel49)) considered the [[zeta function of an elliptic differential operator]] for the [[Laplace operator]] on a [[Riemann surface]].

Motivated by the resemblance of the [[Selberg trace formula]] to Weil's formula for the sum of zeros of the [[Riemann zeta function]], ([Selberg 56](#Selberg56)) defined for any compact hyperbolic [[Riemann surface]] a [[zeta function]]-like expression, the _[[Selberg zeta function]] of a Riemann surface_. (e.g. [Bump, below theorem 19](#Bump)).

Much of this is more generally defined/considered on higher dimensional [[hyperbolic manifolds]].

That the Selberg zeta function is indeed proportional to the [[zeta function of an elliptic differential operator|zeta function]] of a [[Laplace operator]] is due to ([D'Hoker-Phong 86](#DHokerPhong86), [Sarnak 87](#Sarnak87)), and that it is similarly related to the [[eta function of a self-adjoint operator|eta function]] of a [[Dirac operator]] on the given Riemann surface/hyperbolic manifold goes back to ([Milson 78](#Milson78)), with further development including ([Park 01](#Park01)). For review of the literature on this relation see also the beginning of ([Friedman 06](#Friedman06)).

## Examples

### For a complex torus / complex elliptic curve

For $\mathbb{C}/(\mathbb{Z}\oplus \tau \mathbb{Z})$
a [[complex torus]] (complex [[elliptic curve]]) equipped with its
standard flat [[Riemannian metric]], then the [[zeta function of an elliptic differential operator|zeta function]] of the corresponding [[Laplace operator]] $\Delta$ is

$$
  \zeta_{\Delta} = (2\pi)^{-2 s} E(s)
   \coloneqq
   (2\pi)^{-2 s} \underset{(k,l)\in \mathbb{Z}\times\mathbb{Z}-(0,0)}{\sum}
  \frac{1}{{\vert k +\tau l\vert}^{2s}}
  \,.
$$

The corresponding [[functional determinant]] is

$$
  \exp(
    E^\prime_{\Delta}(0)
  )
  = (Im \tau)^2 {\vert \eta(\tau)\vert}^4
  \,,
$$

where $\eta$ is the [[Dedekind eta function]].

(recalled e.g. in [Todorov 03, page 3](#Todorov03))

### Of Dirac operators twisted by a flat connection
 {#OfDiracOperatorTwistedByFlatConnection}

For $A$ a [[flat connection]] on a [[Riemannian manifold]], write $D_A$ for the [[Dirac operator]] twisted by this connection.

On a suitable [[hyperbolic manifold]], the [[partition function]]/[[theta function]] for $D_A$ appears in ([Bunke-Olbrich 94, prop. 6.3](#BunkeOlbrich94)) (and [Bunke-Olbrich 94a, def. 3.1](#BunkeOlbrich94a)) for the odd dimensional case). The corresponding Selberg zeta formula is ([Bunke-Olbrich 94a, def. 4.1](#BunkeOlbrich94a)). This has a form analogous to that of [[Artin L-functions]] with the flat connection replaced by a [[Galois representation]].

## Properties

### Analogy with Artin L-function
 {#AnalogyWithArtinLFunction}

That the Selberg/Ruelle zeta function is equivalently an [[Euler product]] of [[characteristic polynomials]] is due to ([Gangolli 77, (2.72)](#Gangolli77) [Fried 86, prop. 5](#Fried86)).

That it is in particular the Euler product of characteristic polynomials of the [[monodromies]]/[[holonomies]] of the [[flat connection]] corresponding to the given [[group representation]] is ([Bunke-Olbrich 94, prop. 6.3](#BunkeOlbrich94)) for the even-dimensional case and ([Bunke-Olbrich 94a](#BunkeOlbrich94a)) for the odd-dimensional case.

Notice that this is analogous to the standard definition of an [[Artin L-function]] if one interprets a [[Frobenius map]] $Frob_p$ (as discussed there) as an element of the arithmetic fundamental group of an [[arithmetic curve]] and a [[Galois representation]] as a [[flat connection]].

### Function field analogy

[[!include function field analogy -- table]]

## Related concepts

[[!include zeta-functions and eta-functions and theta-functions and L-functions -- table]]

## References

Original articles include

* {#MinakshisundaramPleijel49} S. Minakshisundaram, ; Å Pleijel, _Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds_ (1949), Canadian Journal of Mathematics 1: 242–256, doi:10.4153/CJM-1949-021-5, ISSN 0008-414X, MR 0031145 ([web](http://cms.math.ca/10.4153/CJM-1949-021-5))

* {#Selberg56} [[Atle Selberg]], _Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series_, Journal of the Indian Mathematical Society 20 (1956) 47-87.

* {#Milson78} [[John Milson]], _Closed geodesic and the $\eta$-invariant_, Ann. of Math., 108, (1978) 1-39 ([](http://www.jstor.org/stable/1970928))

Review includes

* Wikipedia, _[Selberg zeta function](http://en.wikipedia.org/wiki/Selberg_zeta_function)_

* Wikipedia, _[Minakshisundaram–Pleijel zeta function](http://en.wikipedia.org/wiki/Minakshisundaram–Pleijel_zeta_function)_

* {#Watkins} [[Matthew Watkins]], citation collection on _[Selberg trace formula and zeta functions](http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/physics4.htm)_

* {#Bump} Bump, below theorem 19 in _Spectral theory of $\Gamma \backslash SL(2,\mathbb{R})$_ ([[BumpSpectralTheory.pdf:file]])

Expression of the Selberg/Ruelle zeta function as an [[Euler product]] of [[characteristic polynomials]] is due to

* {#Gangolli77} Ramesh Gangolli, _Zeta functions of Selberg's type for compact space forms of symmetric spaces of rank one_, Illinois J. Math. Volume 21, Issue 1 (1977), 1-41. ([Euclid](http://projecteuclid.org/euclid.ijm/1256049498))

* {#Fried86} [[David Fried]], _The zeta functions of Ruelle and Selberg. I_, Annales scientifiques de l'École Normale Supérieure, Sér. 4, 19 no. 4 (1986), p. 491-517 ([Numdam](http://www.numdam.org/item?id=ASENS_1986_4_19_4_491_0))

Discussion of the relation between, on the one hand, [[zeta function of an elliptic differential operator|zeta function]] of [[Laplace operators]]/[[eta function of a self-adjoint operator|eta funcstions]] of [[Dirac operators]] and, on the other hand, Selberg zeta functions includes

* {#DHokerPhong86} [[Eric D'Hoker]] [[Duong Phong]], _Communications in Mathematical Physics_, Volume 104, Number 4 (1986), 537-545 ([Euclid](http://projecteuclid.org/euclid.cmp/1104115166))

* {#Sarnak87} [[Peter Sarnak]], _Determinants of Laplacians_, Communications in Mathematical Physics, Volume 110, Number 1 (1987), 113-120. ([Euclid](http://projecteuclid.org/euclid.cmp/1104159171))

* [[Ulrich Bunke]], [[Martin Olbrich]], Andreas Juhl, _The wave kernel for the Laplacian on the classical locally symmetric spaces of rank one, theta functions, trace formulas and the Selberg zeta function_, Annals of Global Analysis and Geometry February 1994, Volume 12, Issue 1, pp 357-405

* {#BunkeOlbrich94} [[Ulrich Bunke]], Martin Olbrich, _Theta and zeta functions for locally symmetric spaces of rank one_ ([arXiv:dg-ga/9407013](http://arxiv.org/abs/dg-ga/9407013))

and for odd-dimensional spaces also in

* {#BunkeOlbrich94a} [[Ulrich Bunke]], [[Martin Olbrich]], _Theta and zeta functions for odd-dimensional locally symmetric spaces of rank one_ ([arXiv:dg-ga/9407012](http://arxiv.org/abs/dg-ga/9407012))

* {#BunkeOlbrich94b} [[Ulrich Bunke]], [[Martin Olbrich]] _$\Gamma$-Cohomology and the Selbeg zeta function_ ([arXiv:dg-ga/9411004](http://arxiv.org/abs/dg-ga/9411004))

* [[Ulrich Bunke]], [[Martin Olbrich]], _Group cohomology and the singularities of the Selberg zeta function associated to a Kleinian group_ ([arXiv:dg-ga/9603003](http://arxiv.org/abs/dg-ga/9603003))

* {#BunkeOlbrich95} [[Ulrich Bunke]], [[Martin Olbrich]], _Selberg zeta and theta functions: a differential operator approach_, Akademie Verlag 1995

* {#Park01} Jinsung Park, _Eta invariants and regularized determinants for odd dimensional hyperbolic manifolds with cusps_ ([arXiv:0111175](http://arxiv.org/abs/math/0111175))

* {#Friedman04} [[Joshua Friedman]], _The Selberg trace formula and Selberg zeta-function for cofinite Kleinian groups with finite-dimensional unitary representations_ ([arXiv:math/0410067](http://arxiv.org/abs/math/0410067))

* {#Friedman06} [[Joshua Friedman]], _Regularized determinants of the Laplacian for cofinite Kleinian groups with finite-dimensional unitary representations_, Communications in Mathematical Physics ([arXiv:math/0605288](http://arxiv.org/abs/math/0605288))

See also

* {#Todorov03} [[Andrey Todorov]], _The analogue of the Dedekind eta function for CY threefolds_, 2003 [pdf](http://www.ma.huji.ac.il/conf/crelle.pdf)

[[!redirects Selberg zeta function of a Riemann surface]]


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

#Contents#
* table of contents
{:toc}

## Idea

For a suitable [[linear operator]] $H$ (say on [[section]] of a [[line bundle]] over a [[Riemann surface]]), its _zeta function_ is the [[analytic continuation]] of the [[trace]]

$$
  \zeta_H(s) \coloneqq Tr(H^{-s})
$$

of the $-s$ power of $H$, which, if $H$ is suitably self-adjoint, is
the [[sum]] of the $-s$-powers of all its [[eigenvalues]], as a function of $s$. This is analogous to the _[[Riemann zeta function]]_ and the [[Dedekind zeta function]] (or would be if there were something like a [[Laplace operator]] on [[Spec(Z)]] or more generally on an [[arithmetic curve]], see at _[[function field analogy]]_).

The exponential of the derivative of the zeta function at $n = 0$ also encodes the [[functional determinant]] of $H$, a [[regularization (physics)|regularized]] version ("[[zeta function regularization]]") of the naive and generally ill-defined product of all eigenvalues. As such, zeta functions play a central role in [[quantum field theory]].

Generally, the values of $\zeta_H(s)$ of interest in [[physics]] (when regarding $H$ as a [[Hamilton operator]]) are those for (low) integral $s$. These are just the _[[special values of L-functions]]_.

## Definition

### The zeta function

Given an [[elliptic differential operator]] with positive lower bound $c$,  write $H$ for its [[self-adjoint extension]] and write

$$
  0 \lt \lambda_1 \leq \lambda_2 \leq \cdots
$$

for its [[eigenvalues]].

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

The _zeta function_ of $H$ is the [[holomorphic function]] defined by the [[series]]

$$
  \begin{aligned}
    \zeta_H(s)
    & \coloneqq
    Tr( H^{-s} )
    \\
    & \coloneqq
    \underoverset{n = 1}{\infty}{\sum} \frac{1}{(\lambda_n)^s}
  \end{aligned}
  \,.
$$

where this [[convergence|converges]] and then extended by [[analytic continuation]].

=--

(e.g. ([Duistermaat-Guillemin 75 (2.13)](#DuistermaatGuillemin75), [Berline-Getzler-Vergne 04, section 9.6](#BerlineGetzlerVergne04) ) ).

### Functional determinant and zeta-function regularization
 {#FunctionalDeterminant}

Notice that the first [[derivative]] $\zeta^\prime_H$ of this zeta function is, where the original series converges, given by

$$
  \zeta_H^\prime(s)
  =
  \sum_{n = 1}^\infty \frac{- \ln \lambda_n}{ (\lambda_n)^s}
  \,.
$$

Therefore one says ([Ray-Singer 71](#RaySinger71)) that the _[[functional determinant]]_ of $H$ is the exponential of the derivative of zeta function of $H$ at 0:

$$
  det_{reg} H
  \coloneqq
  \exp(- \zeta_H^\prime(0))
  \,.
$$

Via the [[analytic continuation]] involved in defining $\zeta_H(0)$ in the first place, this may be thought of as a _[[regularization (physics)|regularization]]_ of the ill-defined  naive definition "$\prod_n \lambda_n$" of the [[determinant]] of $H$. As such functional determinants often appear in [[quantum field theory]] as what is called _[[zeta function regularization]]_.

Conversely, the [[logarithm]]

$$
  Z \coloneqq - \frac{1}{2}\zeta_H^\prime(0) = \tfrac{1}{2} log\,det_{reg} H
$$

is what is called the _[[vacuum energy]]_ in [[quantum field theory]] (for $H^{-1}$ the [[Feynman propagator]]).

If $H = D^2$ has a square root $D$ (a [[Dirac operator]]-type square root as in [[supersymmetric quantum mechanics]]) then under some conditions on the growth of the eigenvalues, then the functional determinant may also be expressed in terms of the [[eta function]] of $D$ as

$$
  det H
  =
  det (D^2)
   =
  \exp( \frac{\partial}{\partial s}\frac{\partial}{\partial c} \eta_{D}(0))
  \,.
$$

See at _[eta invariant -- Relation to zeta function](eta+invariant#RelationToTheZetaFunction)_ for more on this.

### Relation to partition functions and number-theoretic zeta/theta functions
 {#AnalogyWithNumberTheoreticZetaFunctions}

By basic [[integration]] identities we have that

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

The series expression in def. \ref{ZetaBySeries} is equal to the [[Mellin transform]] of the [[partition function]]

$$
    \zeta_H(s)
     =
     \int_{(0,\infty)}
     t^{s-1}
     \left(
        \underset{\lambda_k \neq 0}{\sum}
        \exp(-t \lambda_k)
     \right)
     d t
  \,.
$$

=--

(see e.g. [Quine-Heydari-Song 93 (8)](#QuineHeydariSong93), [Richardson, pages 8-9](#Richardson), [BCEMZ 03, section A.2](#BCEMZ03), [Connes-Marcolli 06, theorem 13.11](#ConnesMarcolli06)).

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

If one thinks of the operation $H$ as a [[Hamiltonian]] of a [[quantum mechanical system]], then the term

$$
  Tr(\exp(-\beta H))
  \coloneqq
   \underset{\lambda_k \neq 0}{\sum}
        \exp(-\beta \lambda_k)
   + 1
$$

is the _[[partition function]]_ of this system. Accordingly, prop. \ref{IntegralKernelExpression} says that the zeta function of $H$ is obtained from its partition function by

$$
  \zeta_H(s)
  =
   \int_{(0,\infty)}
     \beta^{s-1}
     \;
     \left(Tr(\exp(-\beta H)) - 1 \right)
     \;
   d \beta
   \,.
$$

=--

Further, by a change of integration variable $t\coloneqq x^2$
in the expression in prop. \ref{IntegralKernelExpression}
one obtains

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

The series expression in def. \ref{ZetaBySeries} is equal to

$$
  \begin{aligned}
     \zeta_H(s)
     & =
     2
     \int_{(0,\infty)}
     x^{2s-1}
     \left(
        \underset{\lambda_k \neq 0}{\sum}
        \exp(- x^2 \lambda_k)
     \right)
      d x
  \end{aligned}
 \,.
$$

In particular if $H = D^2$ is the square of a [[Dirac operator]]/[[supersymmetric quantum mechanics]]-type square root operator $D$ with [[eigenvalues]] $\pm \alpha_k$,then $\lambda_k = \alpha_k^2$ and hence in this case the series is

$$
  \begin{aligned}
     \zeta_H(s)
     & =
     2
     \int_{(0,\infty)}
     x^{2s-1}
     \left(
        \underset{\alpha_k \neq 0}{\sum}
        \exp(- (x \alpha_k)^2)
     \right)
      d x
  \end{aligned}
 \,.
$$

=--

By comparison one observes:

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

The integral expression in prop. \ref{IntegralKernelExpressionInSquares}
is analogous to the expression of [[zeta functions]] in [[number theory]]/[[arithmetic geometry]] as integrals of a [[theta function]] (for instance discussed [here](Riemann%20zeta%20function#RelationToThetaFunctions) for the [[Riemann zeta function]])

$$
  \hat\zeta_f(2 s)
  =
  \int_{(0,\infty)} (\theta(x^2) - 1) x^{2s-1} d x
  \,.
$$

Under this analogy the [[theta function]] in the case of the differential operator $H$ is

$$
  \theta_H(x)
   \coloneqq
   \underset{\lambda_k \neq 0}{\sum}
        \exp(- x \lambda_l)
   \,.
$$

This is formally the same definition as that of adelic theta functions (e.g.[Garrett 11, section 1.8](Iwasawa-Tate%20theory#Garrett11))

=--

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

The [[determinant line bundle]] of the [[functional determinant]] of the [[Dirac operator]] on a [[complex torus]] is a complex-analytic [[theta function]] as above, quotiented by the [[Dedekind eta function]].

Early references explaining this include [Alvarez-Gaumé & Moore & Vafa 86](#AlvaresGaumeMooreVafa86), [Alvarez-Gaumé & Bost & Moore & Nelson & Vafa 87](#AlvaresGaumeBostMooreNelsonVafa87). In a bigger perspective, this relation plays a central role in the general discussion of [[self-dual higher gauge theory]] ([Witten 96](self-dual%20higher%20gauge%20theory#Witten96)).

=--

## Examples

### Of Laplace operator on complex torus and Dedekind eta function

For $\mathbb{C}/(\mathbb{Z}\oplus \tau \mathbb{Z})$
a [[complex torus]] (complex [[elliptic curve]]) equipped with its
standard flat [[Riemannian metric]], then the [[zeta function of an elliptic differential operator|zeta function]] of the corresponding [[Laplace operator]] $\Delta$ is

$$
  \zeta_{\Delta} = (2\pi)^{-2 s} E(s)
   \coloneqq
   (2\pi)^{-2 s} \underset{(k,l)\in \mathbb{Z}\times\mathbb{Z}-(0,0)}{\sum}
  \frac{1}{{\vert k +\tau l\vert}^{2s}}
  \,.
$$

The corresponding [[functional determinant]] is

$$
  \exp(
    E^\prime_{\Delta}(0)
  )
  = (Im \tau)^2 {\vert \eta(\tau)\vert}^4
  \,,
$$

where $\eta$ is the [[Dedekind eta function]].

(recalled e.g. in [Todorov 03, page 3](#Todorov03))

For more see also at _[[zeta function of a Riemann surface]]_.

### Analytic torsion

The functional determinant of a [[Laplace operator]] of a [[Riemannian manifold]] acting on [[differential n-forms]] is up to a sign in the exponent a factor in what is called the _[[analytic torsion]]_ of the manifold.

## Related concepts

* [[partition function]]

[[!include zeta-functions and eta-functions and theta-functions and L-functions -- table]]

## References

### General

An early reference is

* {#DuistermaatGuillemin75} [[Hans Duistermaat]], [[Victor Guillemin]], _The Spectrum of Positive Elliptic Operators and Periodic Bicharacteristics_,Inventiones mathematicae (1975) Volume: 29, page 39-80 ([EuDML](https://eudml.org/doc/142329))

 (see also at _[[Duistermaat-Guillemin trace formula]]_)

Textbook accounts include

* {#BerlineGetzlerVergne04} [[Nicole Berline]], [[Ezra Getzler]], [[Michèle Vergne]], section 9.6 of _Heat Kernels and Dirac Operators_

Review includes

* {#Richardson} [[Ken Richardson]], section 3 of _Introduction to the Eta invariant_ ([pdf](http://faculty.tcu.edu/richardson/Seminars/etaInvariant.pdf))

* {#QuineHeydariSong93} J. R. Quine, S. H. Heydari, R. Y. Song, _Zeta regularized products_, Transactions of the AMS volume 338, number 1, 1993 ([[QuineZetaRegularization.pdf:file]])

* Wikipedia, _[Functional determinant -- Zeta function version](http://en.wikipedia.org/wiki/Functional_determinant#Zeta_function_version)_

* Wikipedia, _[Zeta function regularization](http://en.wikipedia.org/wiki/Zeta_function_regularization)_

* {#ConnesMarcolli06} [[Alain Connes]], [[Matilde Marcolli]], _A walk in the noncommutative garden_ ([arXiv:0601054](http://arxiv.org/abs/math/0601054))

### Zeta function regularization

* {#Speer71} [[Eugene Speer]], _On the structure of Analytic Renormalization_, Comm. math. Phys. 23, 23-36 (1971) ([Euclid](http://projecteuclid.org/euclid.cmp/1103857549))

* {#BCEMZ03} A. Bytsenko, G. Cognola, [[Emilio Elizalde]], [[Valter Moretti]], S. Zerbini, section 2 of _Analytic Aspects of Quantum Fields_, World Scientific Publishing, 2003, ISBN 981-238-364-6

### Functional determinant

The definition of a _[[functional determinant]]_ via the exponential of the derivative of the zeta function at 0 originates in

* {#RaySinger71} D. Ray, [[Isadore Singer]], _R-torsion and the Laplacian on Riemannian manifolds_, Advances in Math. 7: 145–210, (1971) doi:10.1016/0001-8708(71)90045-4, MR 0295381

Discussion in the special case of [[2d CFT]] ([[worldsheet]] [[string theory]]) is in

* {#AlvaresGaumeMooreVafa86} [[Luis Alvarez-Gaumé]], [[Gregory Moore]], [[Cumrun Vafa]], _Theta functions, modular invariance, and strings_, Communications in Mathematical Physics Volume 106, Number 1 (1986), 1-4 ([Euclid](http://projecteuclid.org/euclid.cmp/1104115581))

* {#AlvaresGaumeBostMooreNelsonVafa87} [[Luis Alvarez-Gaumé]], [[Jean-Benoit Bost]], [[Gregory Moore]], Philip Nelson, [[Cumrun Vafa]], _Bosonization on higher genus Riemann surfaces_, Communications in Mathematical Physics, Volume 112, Number 3 (1987), 503-552 ([Euclid](http://projecteuclid.org/euclid.cmp/1104159982))

* {#Todorov03} [[Andrey Todorov]], _The analogue of the Dedekind eta function for CY threefolds_, 2003 [pdf](http://www.ma.huji.ac.il/conf/crelle.pdf)

[[!redirects zeta functions of elliptic differential operators]]


zeta function regularization

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#### Physics
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[[!include physicscontents]]
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#### Theta functions
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[[!include theta functions - contents]]
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#### Complex geometry
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[[!include complex geometry - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

In the context of [[regularization (physics)|regularization in physics]], _zeta function regularization_ is a method/prescription for extracing finite values for [[traces]] of powers of [[Laplace operators]]/[[Dirac operators]] by

1. considering $s$-powers for all values of $s$ in the [[complex plane]] where the naive trace does make sense and then

1. using [[analytic continuation]] to obtain the desired [[special values of L-functions|special value]] at $s = 1$ -- as for [[zeta functions]].

### Analytic regularization of propagators

One speaks of _analytic regularization_ ([Speer 71](#Speer71)) or _zeta function regularization_ (e.g. [M 99](#M99), [BCEMZ 03, section 2](#BCEMZ03)) if a [[Feynman propagator]]/[[Green's function]] for a [[boson|bosonic]] [[field (physics)|field]], which is naively given by the expression
"$Tr\left(\frac{1}{H}\right)$" (for $H$ the given [[wave operator]]/[[Laplace operator]]) is made well defined by interpreting it as the [[principal value]] of the [[special values of L-functions|special value]] at $s= 1$

$$
  Tr_{reg}
  \left(\frac{1}{H}\right)
  \coloneqq
  pv\, \zeta_H(1)
$$

of the [[zeta function of an elliptic differential operator|zeta function]] which is given by the expression

$$
  \zeta_H(s)
    \coloneqq
  Tr\left(
    \frac{1}{H}
  \right)^s
$$

for all values of $s \in \mathbb{C}$ for which the right hand side exists, and is defined by [[analytic continuation]] elsewhere.

Analogously the zeta function regularization of the [[Dirac propagator]] for a [[fermion]] [[field (physics)|field]] with [[Dirac operator]] $D$ is defined by

$$
  Tr_{reg}
  \left(\frac{D}{D^2} \right)
  \coloneqq
  pv\, \eta_D(1)
$$

where $\eta$ is the [[eta function]] of $D$.

### Functional determinants

Notice that the first [[derivative]] $\zeta^\prime_H$ of this [[zeta function of an elliptic differential operator|zeta function]] is, where the original series converges, given by

$$
  \zeta_H^\prime(s)
  =
  \sum_{n = 1}^\infty \frac{- \ln \lambda_n}{ (\lambda_n)^s}
  \,.
$$

Therefore the _[[functional determinant]]_ of $H$ ([Ray-Singer 71](#RaySinger71)) is the exponential of the zeta function of $H$ at 0:

$$
  Det_{reg} H
  \coloneqq
  \exp(- \zeta_H^\prime(0))
  \,.
$$

(see also [BCEMZ 03, section 2.3](#BCEMZ03))

Via the [[analytic continuation]] involved in defining $\zeta_H(0)$ in the first place, this may be thought of as a _[[regularization (physics)|regularization]]_ of the ill-defined  naive definition "$\prod_n \lambda_n$" of the [[determinant]] of $H$. As such functional determinants often appear in [[quantum field theory]] as what is called _[[zeta function regularization]]_.

### Higher amplitudes

Accordingly, more general [[scattering amplitudes]] are controled by [[multiple zeta functions]] (...).

## Examples

### Of Laplace operator on complex torus and Dedekind eta function

For $\mathbb{C}/(\mathbb{Z}\oplus \tau \mathbb{Z})$
a [[complex torus]] (complex [[elliptic curve]]) equipped with its
standard flat [[Riemannian metric]], then the [[zeta function of an elliptic differential operator|zeta function]] of the corresponding [[Laplace operator]] $\Delta$ is

$$
  \zeta_{\Delta} = (2\pi)^{-2 s} E(s)
   \coloneqq
   (2\pi)^{-2 s} \underset{(k,l)\in \mathbb{Z}\times\mathbb{Z}-(0,0)}{\sum}
  \frac{1}{{\vert k +\tau l\vert}^{2s}}
  \,.
$$

The corresponding [[functional determinant]] is

$$
  \exp(
    E^\prime_{\Delta}(0)
  )
  = (Im \tau)^2 {\vert \eta(\tau)\vert}^4
  \,,
$$

where $\eta$ is the [[Dedekind eta function]].

(recalled e.g. in [Todorov 03, page 3](#Todorov03))

### Zeta regularization for divergent integrals ###

the zeta regularizatio method can be extended to include also a regularization for the divergent integrals $ \int_{a}^{\infty}x^{m}dx $ which appears in QFT, this is made by means of the identity

$$\begin{array}{l}
\int_{a}^{\infty }x^{m-s} dx =\frac{m-s}{2} \int_{a}^{\infty }x^{m-1-s} dx +\zeta (s-m)-\sum_{i=1}^{a}i^{m-s}  +a^{m-s}  \\
-\sum_{r=1}^{\infty }\frac{B_{2r} \Gamma (m-s+1)}{(2r)!\Gamma (m-2r+2-s)}  (m-2r+1-s)\int_{a}^{\infty }x^{m-2r-s} dx \end{array}  $$

for the case of $ m=-1$ although the harmonic series has a pole we can regularize by the 2 possibilities

$ \sum_{n=0}^{\infty} \frac{1}{n+a} = -\Psi (a) $ or $ \sum_{n=0}^{\infty} \frac{1}{n+a} = -\Psi (a)+log(a) $ in particular
$ \sum_{n=1}^{\infty} \frac{1}{n} = \gamma $  Euler-Mascheroni constant, and $ \Psi(a)= -\frac{\Gamma '(a)}{\Gamma (a)} $

So within this reuglarization there wouldn't be any UV ultraviolet divergence
### Analytic torsion

The functional determinant of a [[Laplace operator]] of a [[Riemannian manifold]] acting on [[differential n-forms]] is up to a sign in the exponent a factor in what is called the _[[analytic torsion]]_ of the manifold.

## Related concepts

[[!include zeta-functions and eta-functions and theta-functions and L-functions -- table]]

## References

Original articles include

* {#Speer71} [[Eugene Speer]], _On the structure of Analytic Renormalization_, Comm. Math. Phys. 23, 23-36 (1971) ([Euclid](http://projecteuclid.org/euclid.cmp/1103857549))

* {#RaySinger71} D. Ray, [[Isadore Singer]], _R-torsion and the Laplacian on Riemannian manifolds_, Advances in Math. 7: 145–210, (1971) doi:10.1016/0001-8708(71)90045-4, MR 0295381

Modern accounts and reviews include

* {#Freed87} [[Daniel Freed]], page 8 of _On determinant line bundles_, Math. aspects of [[string theory]], ed. S. T. Yau, World Sci. Publ. 1987,  (revised [pdf](http://www.math.utexas.edu/~dafr/Index/determinants.pdf), [dg-ga/9505002](http://arxiv.org/abs/dg-ga/9505002))

* {#Elizalde95} [[Emilio Elizalde]], _Ten Physical Applications of Spectral Zeta Functions_ (1995)

* {#M99}  [[Valter Moretti]], _Local z-function techniques vs point-splitting procedures: a few rigorous results_
Commun. Math. Phys. 201, 327 (1999).

* {#BCEMZ03} A. Bytsenko, G. Cognola, [[Emilio Elizalde]], [[Valter Moretti]], S. Zerbini, section 2 of _Analytic Aspects of Quantum Fields_, World Scientific Publishing, 2003, ISBN 981-238-364-6

* {#Robles09} [[Nicolas Robles]], _Zeta function regularization_, 2009 ([[RoblesZetaRegularization.pdf:file]])

See also

* {#Todorov03} [[Andrey Todorov]], _The analogue of the Dedekind eta function for CY threefolds_, 2003 [pdf](http://www.ma.huji.ac.il/conf/crelle.pdf)

[[!redirects zeta function regularizations]]

[[!redirects functional determinant]]
[[!redirects functional determinants]]

[[!redirects zeta-function regularization]]
[[!redirects zeta-function regularizations]]


zeta polynomial

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#Contents#
* table of contents
{:toc}

## Idea

The **zeta polynomial** $Z_P(n)$ of a finite [[partially ordered set]] $P$ counts the number of multichains (also known as "weakly increasing sequences") of length $n$ in $P$.

## Definition

By a _multichain_ of length $n$ in $P$, we mean a sequence of elements $x_0 \le x_1 \le \cdots \le x_n$, which can be identified with an [[order-preserving function]] from the [[linear order]] $[n] = \{ 0 \lt 1 \lt \cdots \lt n \}$ into $P$.
To see that
$$Z_P(n) = |Hom([n],P)|$$
defines a [[polynomial]] in $n$,[^Note] first observe that any function $[n] \to P$ factors as a [[surjection]] from $[n]$ onto some $[k] = \{ 0 \lt 1 \lt \cdots \lt k \}$ (where $k \le n$), followed by an [[injection]] from $[k]$ to $P$. The total number of order-preserving functions from $[n]$ to $P$ can therefore be calculated explicitly as
$$Z_P(n) = \sum_{k=0}^{d} b_k \binom{n}{k}$$
where $b_k$ is the number of _chains_ $x_0 \lt x_1 \lt \cdots \lt x_k$ in $P$ (i.e., injective order-preserving functions from $[k]$ to $P$), and where $d$ is the length of the longest chain. Hence $Z_P(n)$ is a polynomial of degree equal to the length of the longest chain in $P$.

[^Note]: Note that the definition we use here for $Z_P(n)$ has an index shift from the definition that seems to be more standard in combinatorics. For example, the definition in [(Stanley, 3.12)](#StanleyEC1) counts multichains of length $n-2$ rather than of length $n$. Accordingly, one should apply a substitution to get some of the properties stated here to match equivalent results in the literature.

## Examples

The zeta polynomial of $[2] = \{ 0 \lt 1 \lt 2 \}$ is
$$3 + 3n + \binom{n}{2} = \frac{n^2 + 5n + 6}{2}$$
For example, evaluating the polynomial at $n=0$ and $n=1$ confirms that $[2]$ contains 3 points and 6 [[intervals]], while evaluating it at $n=2$ confirms that there are 10 order-preserving functions from $[2]$ to itself.

The zeta polynomial of the 5-element poset
$$P = \array{&&v&& \\ &&\uparrow&& \\ &&u&& \\ &\nearrow& &\nwarrow& \\ y &&&& z \\ &\nwarrow& &\nearrow& \\ &&x&&}$$
is $5 + 9n + 7\binom{n}{2} + 2\binom{n}{3} = \frac{2n^3 + 15n^2 + 37n + 30}{6}$.
Evaluating at $n=1$, we compute that $P$ contains 14 distinct intervals.

## Properties

### Relation to order polynomial

The [[order polynomial]] is related to the  zeta polynomial by the equation
$$
\Omega_P(n+2) = Z_{P^\downarrow}(n)
$$
where $P^\downarrow \cong (2)^P$ is the lattice of [[lower sets]] in $P$.
This can be seen as a consequence of the [[currying]] isomorphisms
$$Hom([n], P^\downarrow) \cong Hom([n] \times P, (2)) \cong Hom(P, [n]^\downarrow)$$
together with the isomorphisms $[n]^\downarrow \cong [n+1] \cong (n+2)$.

### Relation to zeta function

Using the formalism of [[incidence algebras]], the zeta polynomial has a simple expression in terms of the _zeta function_ of $P$ (defined by $\zeta_P(x,y) = 1$ if $x\le y$ and $\zeta_P(x,y) = 0$ otherwise):
$$Z_P(n) = \sum_{x,y\in P} \zeta_P^n(x,y)$$
where $\zeta_P^n$ is the $n$-fold convolution product of $\zeta_P$. (In other words, if we view the zeta function as a square matrix, then the zeta polynomial is the sum of the entries in its $n$-fold matrix product.) This follows immediately from the definition of the convolution product,
$$(f\cdot g)(x,y) = \sum_{x \le z \le y} f(x,z) \cdot g(z,y)$$
since $\zeta_P^n(x,y)$ computes the number of multichains of length $n$ in $P$ from $x$ to $y$.

As a special case, if $P$ has both a [[bottom]] element 0 and a [[top]] element 1, then
$$Z_P(n) = \zeta_P^{n+2}(0,1)$$
since an arbitrary multichain $x_0 \le x_1 \le \cdots \le x_n$ of length $n$ can be extended to a multichain $0 \le x_0 \le x_1 \le \cdots \le x_n \le 1$ of length $n+2$ between 0 and 1.

## Related concepts

* [[order polynomial]]
* [[incidence algebra]]
* [[Möbius inversion]]
* [[nerve]]

## References

* Paul H. Edeleman. Zeta Polynomials and the Möbius Function. European Journal of Combinatorics 1(4), 1980.
{#Edelman80}

* Richard P. Stanley, _Enumerative combinatorics_, vol.1 ([pdf](http://www-math.mit.edu/~rstan/ec/ec1.pdf))
{#StanleyEC1}

* Joseph P. S. Kung, [[Gian-Carlo Rota]], Catherine H. Yan. Combinatorics: The Rota Way. Cambridge, 2009.
{#KungRotaYan}

category: combinatorics


zeta-functions and eta-functions and theta-functions and L-functions -- table

| context/[[function field analogy]]  | [[theta function]] $\theta$ | [[zeta function]] $\zeta$ (= [[Mellin transform]] of $\theta(0,-)$) | [[L-function]] $L_{\mathbf{z}}$ (= [[Mellin transform]] of $\theta(\mathbf{z},-)$) | [[eta function]] $\eta$  |  [[special values of L-functions]] |
|---|-----|-----------|---|-----|----|
| [[physics]]/[[2d CFT]] | [[partition function]] $\theta(\mathbf{z},\mathbf{\tau}) = Tr(\exp(-\mathbf{\tau} \cdot (D_\mathbf{z})^2))$ as function of [[complex structure]] $\mathbf{\tau}$ of [[worldsheet]] $\Sigma$ (hence [[polarization]] of [[phase space]])  and [[background field|background]] [[gauge field]]/[[source]] $\mathbf{z}$  | analytically continued [[trace]] of [[Feynman propagator]] $\zeta(s) = Tr_{reg}\left(\frac{1}{(D_{0})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(0,\tau)\, d\tau$  | analytically continued [[trace]] of [[Feynman propagator]] in [[background field|background]] [[gauge field]] $\mathbf{z}$: $L_{\mathbf{z}}(s) \coloneqq Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(\mathbf{z},\tau)\, d\tau$  | analytically continued [[trace]] of [[Dirac propagator]] in [[background field|background]] [[gauge field]] $\mathbf{z}$ $\eta_{\mathbf{z}}(s) = Tr_{reg} \left(\frac{sgn(D_{\mathbf{z}})}{ { \vert D_{\mathbf{z}} } \vert }\right)^s $  |  [[zeta function regularization|regularized]] [[1-loop vacuum amplitude]] $pv\, L_{\mathbf{z}}(1) = Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)$ / [[zeta function regularization|regularized]] fermionic [[1-loop vacuum amplitude]] $pv\, \eta_{\mathbf{z}}(1)= Tr_{reg} \left( \frac{D_{\mathbf{z}}}{(D_{\mathbf{z}})^2} \right)$ / [[vacuum energy]] $-\frac{1}{2}L_{\mathbf{z}}^\prime(0) = Z_H = \frac{1}{2}\ln\;det_{reg}(D_{\mathbf{z}}^2)$ |
| [[Riemannian geometry]] ([[analysis]]) |   | [[zeta function of an elliptic differential operator]] | [[zeta function of an elliptic differential operator]] | [[eta function of a self-adjoint operator]] |  [[functional determinant]], [[analytic torsion]] |
| [[complex analytic geometry]] | [[section]] $\theta(\mathbf{z},\mathbf{\tau})$  of [[line bundle]] over [[Jacobian variety]] $J(\Sigma_{\mathbf{\tau}})$ in terms of covering coordinates $\mathbf{z}$ on $\mathbb{C}^g \to J(\Sigma_{\mathbf{\tau}})$ | [[zeta function of a Riemann surface]] |  [[Selberg zeta function]] |   |  [[Dedekind eta function]] |
| [[arithmetic geometry]] for a [[function field]] |   |  [[Goss zeta function]] (for [[arithmetic curves]])  and [[Weil zeta function]] (in [[higher dimensional arithmetic geometry]]) |  |  |  |
| [[arithmetic geometry]] for a [[number field]] | [[Hecke theta function]], [[automorphic form]] | [[Dedekind zeta function]] ([being](Artin+L-function#RelationToDedekindZeta) the [[Artin L-function]] $L_{\mathbf{z}}$ for $\mathbf{z} = 0$ the [[trivial representation|trivial]] [[Galois representation]]) |   [[Artin L-function]] $L_{\mathbf{z}}$ of a [[Galois representation]] $\mathbf{z}$, expressible "in coordinates" (by [[Artin reciprocity]]) as a finite-order [[Hecke L-function]] (for 1-dimensional representations) and generally (via [[Langlands correspondence]]) by an [[automorphic L-function]]  (for higher dimensional reps) |  | [[class number]] $\cdot$ [[regulator]]  |
| [[arithmetic geometry]] for $\mathbb{Q}$ | [[Jacobi theta function]] ($\mathbf{z} = 0$)/ [[Dirichlet theta function]] ($\mathbf{z} = \chi$ a [[Dirichlet character]]) | [[Riemann zeta function]] (being the [[Dirichlet L-function]] $L_{\mathbf{z}}$ for [[Dirichlet character]] $\mathbf{z} = 0$) | [[Artin L-function]] of a [[Galois representation]] $\mathbf{z}$ , expressible "in coordinates" (via [[Artin reciprocity]]) as a [[Dirichlet L-function]] (for 1-dimensional Galois representations) and generally (via [[Langlands correspondence]]) as an [[automorphic L-function]] |  |  |

[[!redirects zeta-functions and eta-functions and L-functions -- table]]
[[!redirects zeta functions table]]


ZFA

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# Zermelo--Fraenkel set theory with atoms
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## Idea

**ZFA** is a variant of the [[material set theory]] [[ZF]] which allows for objects, called _atoms_ or _[[urelements]]_ (hence the alternative name ZFU), which may be members of [[sets]], but are not made up of other elements. ZFA featured in early independence proofs, notably [[Fraenkel-Mostowski permutation models]], for example showing AC is independent of the rest of the axioms of ZFA.

[[Zermelo|Zermelo's]] original 1908 axiomatisation of set theory included atoms, but they were soon discarded as a foundational approach as they could be modeled inside of atomless set theory.

## Definition

There are two possible approaches to formulating ZFA. In both cases, we can further require that the [[axiom of choice]] is satisfied, and obtain ZFCA.

### Empty atoms
In this approach, atoms are empty. We start by adding an additional unary predicate $A$, where we interpret $A(x)$ as saying "$x$ is an atom". We write $(\forall set x)$ to mean $\forall x, \neg A(x) \Rightarrow$, and similarly write $(\forall atom x)$ to mean $\forall x, A(x) \Rightarrow$.

Then the [[axiom of extensionality]] says
$$
  (\forall set x) (\forall set y) (\forall z, z \in x \Leftrightarrow z \in y)  \Rightarrow x = y,
$$
and the [[axiom of empty set]] says
$$
  (\exists set x) (\forall y) (y \notin x).
$$
We also add the axiom that says atoms are empty:
$$
  (\forall atom a) (\forall x) x \notin a.
$$
Sometimes it is also convenient to assume that we have a set of atoms:
$$
  (\exists X) (\forall x) (A(x) \Leftrightarrow x \in X),
$$
but in some cases, we might also like to consider models with a proper class of atoms.

### Reflexive/Quine atoms
We can give up on the [[axiom of foundation]], and introduce the urelements as [[reflexive sets]], ie. sets $x$ such that $x = \{x\}$. In place of the axiom of foundation, we can have an axiom of _weak_ foundation, where we require the existence of a set A such that every element of A is reflexive, and the [[cumulative hierarchy]] built up from $A$ is the whole universe. In other words, if we define

 * $R(0) = A$,

 * $R(\alpha + 1) = P(R(\alpha))$ for any ordinal $\alpha$,

 * $R(\lambda) = \bigcup_{\gamma \lt \lambda} R(\gamma)$ for $\lambda$ a limit ordinal,

then $V = \bigcup_\alpha R(\alpha)$.

## Models

### Fraenkel--Mostowski models
By allowing atoms in our models, we lend ourselves to the method of [[Fraenkel-Mostowski models]], where we can obtain models in which the [[axiom of choice]] fails by imposing some symmetry on the atoms (so that we cannot uniformly pick an atom out of many). Such models are closely related to [[categories of G-sets]].

## Related concepts

 * [[ZFC]]
 * [[Fraenkel-Mostowski models]]
 * [[permutation model]]
 * [[urelements]]

[[!redirects ZFA]]
[[!redirects ZFU]]
[[!redirects ZFCA]]
[[!redirects ZFAC]]


ZFC

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# Contents
* table of contents
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## Idea

The most commonly accepted standard [[foundation of mathematics]] today is a [[material set theory]] commonly known as _Zermelo--Fraenkel set theory with the [[axiom of choice]]_ or $ZFC$ for short.  There are many variations on that theory (including [[constructive mathematics|constructive]] and [[class]]-based versions, which are also discussed here).

Accompanying ZFC, especially taking into account the [[axiom of foundation]], is a picture (or 'ontology') of material sets forming a [[cumulative hierarchy]] organized by an [[ordinal number|ordinal]]-valued rank function.  This picture (sometimes referred to as the 'iterative conception') considers sets as generated by starting at bottom with the [[empty set]] and building to higher ranks by applying a [[power set]] operation to get to a next successor ordinal rank, and taking [[unions]] to get to limit ordinal ranks.  This iterative conception finds alternative expression in [[algebraic set theory]].

## History

The first version was developed by [[Ernst Zermelo]] in 1908; in 1922, [[Abraham Fraenkel]] and [[Thoralf Skolem]] independently proposed a precise [[first order logic|first-order]] version with the [[axiom of replacement]]; von Neumann added the [[axiom of foundation]] in 1925.  All of these versions included the [[axiom of choice]], but this was considered controversial for some time; one has merely $ZF$ if it is taken out.

$ZFC$ is similar to the [[class]] theories [[NBG]] (due to [[John von Neumann]], [[Paul Bernays]], and [[Kurt Gödel]]) and [[Morse–Kelley set theory|MK]] (due to [[Anthony Morse]] and [[John Kelley]]).  The former is a conservative, finitely axiomatisable extension of $ZFC$, while the latter is stronger and cannot be finitely axiomatised (although a conservative extension involving meta-classes could be).

Contemporary set theorists often accept additional [[large cardinal]] [[axioms]], which can greatly increase the strength of $ZFC$, far beyond even $MK$.  Other additional axioms which are sometimes added are the [[axiom of determinacy]] (or various weaker versions of it) or the [[axiom of constructibility]].  There are also weaker variants of $ZFC$, especially for [[constructive mathematics|constructive]] and [[predicative mathematics|predicative]] mathematics.  Then there are alternatives on a different basis, notably [[New Foundations|NFU]] (a very [[predicative mathematics|impredicative]] [[material set theory]] with a set of all sets) and [[ETCS]] (a [[structural set theory]]).

(The source for this history, especially the dates, is mostly [the English Wikipedia](http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory).)

## Axioms

$ZFC$ is a [[simple type theory|simply typed]] [[first-order theory|first-order]] [[material set theory]], with a single type $V$ called the __[[cumulative hierarchy]]__ and a binary __membership__ [[predicate]] $\in$ on $V$. Every term of $V$ in standard $ZFC$ is a [[pure set]], which we will call simply a set.
A set $a$ is said to belong to, be in, be a __member__ of, or be an __element__ of a set $b$ if $a \in b$, and set $b$ is said to have the member $a$.

There are also variations of ZFC with non-set terms such as [[urelements]] and [[classes]]. Urelements may be distinguished from sets and classes since they have no elements (although the [[empty set]] also has no elements); sets are usually those classes that are themselves elements (members) of sets. Urelements are also called *atoms*, and $ZF$ with atoms included is sometimes called [[ZFA]] or $ZFU$.

1.  [[axiom of extensionality|Extensionality]]:  If two sets have the same members, then they are __[[equality|equal]]__ and themselves members of the same sets.  See [[axiom of extensionality]] for variations, such as whether this is taken as a definition or an axiomatisation of equality of sets, and how the condition might be strengthened if (10) is left out.

2.  [[axiom of the null set|Null Set]]:  There is an __[[empty set]]__: a set $\empty$ with no elements.  By (1), it follows that this set is unique; by even the weakest version of (5), it is enough to state the existence of some set.  (Analogous remarks apply to axioms (3), (4), (6), (7), and (8), except when (5) is omitted and (4) and (6) are needed to derive it.)

3.  [[axiom of pairing|Pairing]]:  If $a$ and $b$ are sets, then there is a set $\{a,b\}$, the __[[unordered pairing]]__ of $a$ and $b$, whose elements are precisely those sets equal to $a$ or $b$.  Between them, (2) and (3) form a nullary/binary pair; the unary version follows from (3), since $\{a\} = \{a,a\}$ by (1).  The higher finitary versions rely also on (4) (with $\{a,b,c\} = \{a\} \cup \{b,c\}$, etc), so they should probably be stated explicitly as an [[axiom scheme]] if (4) is omitted (but nobody seems to do that).

4.  [[axiom of union|Union]]:  If $\mathcal{C}$ is a set, then there is a set $\bigcup \mathcal{C}$, the __[[union]]__ of $\mathcal{C}$, whose elements are precisely the elements of the elements of $\mathcal{C}$.  It is normal to write $A \cup B$ for $\bigcup \{A,B\}$, etc, $A \cup B \cup C$ for $\bigcup \{A,B,C\}$, etc.  Note that $\bigcup \{A\} = A$ and $\bigcup \empty = \empty$ (using (1) to prove these results), so no special notation is needed for these.

5.  [[axiom of separation|Separation/Specification/Comprehension]]:  Given any [[predicate]] $\phi[x]$ in the language of set theory with the chosen [[free variable]] $x$ of type $V$, if $U$ is a set, then there is a set $\{x \in U \;|\; \phi[x]\}$, the __[[subset]]__ of $U$ given by $\phi$, whose elements are precisely those elements $x$ of $U$ such that $\phi[x]$ holds.  There are many variations, from Bounded Separation to Full Comprehension, which we should probably describe at [[axiom of separation]]; the version listed here is Full Separation.  This is an [[axiom scheme]], but it can be made a single axiom in $NBG$ (but not completely in $MK$).  Note that (5) follows from (4) and (6) using [[classical logic]], so it is often left out, but it must be included in intuitionistic variations, variations where (6) is omitted or weakened (or where (4) is omitted or weakened, in principle, although that never happens in practice), and variations where (5) itself is strengthened.

6.  [[axiom of replacement|Replacement/Collection]]:  Given a [[predicate]] $\psi[x,Y]$ with the chosen [[free variables]] $x$ and $Y$ of type $V$, if $U$ is a set and if for every $x$ in $U$ there is a unique $Y$ such that $\psi[x,Y]$ holds, then there is set $\{\iota\,Y.\;\psi[x,Y] \;|\; x \in U\}$, the __[[image]]__ of $U$ under $\psi$, whose elements are precisely those sets $Y$ such that there is an element $x$ of $U$ such that $\psi[x,Y]$ holds; if $\Psi[x]$ is a defined term, then we write $\{\Psi[x] \;|\; x \in U\}$ for $\{\iota\,Y.\; Y = \Psi[x] \;|\; x \in U\}$.  Again there are many variations, from Weak Replacement to Strong Collection, which we should probably describe at [[axiom of replacement]]; the one described here is Replacement.  This is also an [[axiom scheme]], but it can be made into a single axiom in both $NBG$ and $MK$.  One could combine this with (5) to produce $\{\iota\,Y.\;\psi[x,Y] \;|\; x \in U \;|\; \phi[x]\}$, but nobody seems to do this.

7.  [[axiom of power sets|Power Sets]]:  If $U$ is a set, then there is a set $\mathcal{P}U$, the __[[power set]]__ of $U$, whose elements are precisely the [[subsets]] of $U$, that is the sets $A$ whose elements are all elements of $U$.  When using [[intuitionistic logic]], it is possible to accept only a weak version of this, such as [[subset collection|Subset Collection]] or (even weaker) [[function sets|Exponentiation]].  But in classical logic, Power Sets follows from Exponentiation and the weakest form of (5).

8.  [[axiom of infinity|Infinity]]:  There is a set $\omega$ of __[[ordinal number|finite ordinals]]__ as pure sets.  Normally one states that $\empty \in \omega$ and $a \cup \{a\} \in \omega$ whenever $a \in \omega$, although variations are possible.  Using any but the weakest version of (6), it is enough to state that there is a set satisfying [[Peano arithmetic|Peano's axioms]] of [[natural numbers]], or even any [[Dedekind-infinite set]].  It seems to be uncommon to incorporate (2) into (8), but in principle (8) implies (2).

9.  [[axiom of choice|Choice]]:  If $\mathcal{C}$ is a set of pairwise disjoint sets, each of which has an element, then there is a set with exactly one element from each element of $\mathcal{C}$.  Note that this set is *not* unique, nor can we construct a canonical version which is, so we do not give it any name or notation.  This version is the simplest to state in the language of $ZFC$; see [[axiom of choice]] for further discussion and weak versions.  It is possible to incorporate (9) into (5) or (6), but this seems to be rare.

10. [[axiom of foundation|Foundation/Regularity/Induction]]:  Given a formula $\phi$ with a chosen [[free variable]] $X$ of type $V$, if $\phi$ holds whenever $\phi[a/X]$ holds for every $a \in X$, then $\phi$ holds absolutely.  For variations (including the axiom of anti-foundation), see [[axiom of foundation]].  This [[axiom scheme]] can be made into a single axiom even in $ZFC$ itself (although not in versions with intuitionistic logic; in that case it can be made a single axiom only in a class theory).

## Variations

Zermelo\'s original version consists of axioms (1--5) and (7--9), in a somewhat imprecise form (which affects the interpretation of 5) of [[higher-order logic|higher-order]] [[classical logic]].

The modern $ZF$ consists of (1--8) and (10), using [[first-order logic|first-order]] [[classical logic]], the strongest form of (6) (that is, Strong Collection, although the standard Replacement is sufficient with [[classical logic]]), and the strongest form of (5) possible using only sets and not classes (Full Separation).  Since Full Separation follows from Replacement with [[classical logic]], it is often omitted from the list of axioms.

$ZFC$ adds (9) and is thus the strongest version without classes or additional axioms.  The version originally formulated by Fraenkel and Skolem did not include (10), although the three founders all eventually accepted it.

It is common to take __Zermelo set theory__ ($\mathrm{Z}$) to be $ZF$ without (6), although Zermelo never accepted the first-order formulation, and his axioms did not include the Axiom of Foundation (10). Note that the weakest form (Weak Replacement) of (6) follows from (7) and (5), so it holds even in $\mathrm{Z}$.

Another variant is __bounded Zermelo set theory__ ($BZ$), which is like $\mathrm{Z}$ but with only Bounded Separation; this is of interest to category theorists because $BZC$ is equivalent to [[ETCS]].

### Constructive versions

See also [[constructive set theory]].

The most well-known foundations for [[constructive mathematics]] through material set theory are [[Peter Aczel]]\'s __constructive Zermelo--Fraenkel set theory__ ($CZF$) and [[John Myhill]]\'s __intuitionistic Zermelo--Fraenkel set theory__ ($IZF$).

$CZF$ uses axioms (1--8) and (10), usually weak forms, in [[intuitionistic logic]]; specifically, it uses Bounded Separation for (5), Strong Collection for (6), and an intermediate (Subset Collection) form of (7).  $IZF$ is simliar, but it uses Full Separation for (5) and the full strength of (7); Myhill\'s original version uses only Replacement for (6), but Collection (equivalent to Strong Collection using Full Separation) is standard now.

Note that adding (9) to $IZF$ implies [[excluded middle]] and so makes $ZFC$.  However, some authors like to include a weak form of (9), such as [[dependent choice]] or [[COSHEP]].

[[Mike Shulman]]\'s survey of material and structural set theories ([Shulman 2018](#Shulman2018)) takes $CPZ^{\circlearrowleft-}$ as the most basic form; it consists of (1--4) and the weakest versions (Bounded Separation and Weak Replacement) of (5&6) in intuitionistic logic.  Adding (10) gives $CPZ^{-}$, adding (8) gives $CPZ^{\circlearrowleft}$, and adding both gives $CPZ$, __constructive pre-Zermelo set theory__.  Shulman gives systematic notation for other versions, which includes those (constructive and classical) listed above.

Myhill has another version, __constructive set theory__ ($CST$); this consists of (1--4), Bounded Separation for (5), Replacement for (6), the weakest (Exponentiation) form of (7), (8), and a weak version (Dependent Choice) of (9).  It also uses a variation of the language, with [[urelements]] for natural numbers; note that the existence of $\omega$ still follows using (6).  This classifies $CST$ as $\mathrm{C}{\Pi}ZF^{\circlearrowleft} + DC$ in Shulman\'s system if one ignores the use of urelements and strengthens Replacement to Strong Collection.

### Class theories

__[[Morse--Kelley class theory]]__ ($MK$) features both [[sets]] and [[proper classes]].  This allows it to strengthen (5) to Full Comprehension, since $\phi$ can include [[quantification]] over classes; the same holds in (6) and (10), although this does not add any additional strength.

__Von Neumann--Bernays--Gödel class theory__ ($NBG$) uses the same language as $MK$, but it still uses only Full Separation for (5).  This makes it conservative over $ZFC$ and also allows for a finite axiomatisation; we replace the formulas in (5) and (6) with classes, and add some special cases of (5) for subclasses, one for each logical connective.  (It is provable that plain $ZF$, if consistent, cannot be finitely axiomatized in its own first-order language; $NBG$ escapes this conclusion by extending the language with the notion of classes.)

One can also rework all of the weak versions of set theory above into a class theory like $NBG$, which is conservative over the original set theory.  One can also use a class theory like $MK$, although this destroys any attempt to use a weak version of (5).

### Large cardinals

One often adds axioms for [[large cardinals]] to $ZFC$.  Even (8) can be seen as a large cardinal axiom, stating that $\aleph_0$ exists.  These additional axioms are most commonly studied in the context of a material set theory, but they work just as well in a structural set theory.

Note that adding an [[inaccessible cardinal]] (commonly considered the smallest sort of large cardinal) to $ZFC$ is already stronger than $MK$: given an inaccessible cardinal $\kappa$, one can interpret the sets and classes in $MK$ as the sets in $V_\kappa$ and $V_{\kappa+1}$, respectively.  Of course, one can add a large cardinal to $MK$ to get something even stronger.

It is often convenient to assume that one always has more large cardinals when necessary.  You cannot say this in an absolute sense, but you can adopt the axiom that every set belongs to some [[Grothendieck universe]].  Adding this axiom to $ZFC$ makes __[[Tarski-Grothendieck set theory]]__ ($TG$).  This is not the last word, however; you can make it stronger by adding classes in the style of $MK$, or even adding a cardinal which is inaccessible from $TG$.  In fact, we have barely begun the large cardinals known to modern set theory!

### Miscellaneous variations

The [[axiom of constructibility]], usually notated "$V = L$", is a very strong axiom that can be added to $ZF$; it asserts that all sets belong to the [[constructible universe]] $L$, which can be "constructed" in a definable way through a transfinite procedure.  This notion of "constructible" should not be confused with [[constructive mathematics]]; for instance, $V = L$ implies the [[axiom of choice]] and thus also [[excluded middle]] even with intuitionistic logic.  $V = L$ also implies the [[generalized continuum hypothesis]] ($GCH$), which is how Gödel originally proved that $GCH$ was consistent with $ZFC$.  However, it is incompatible with the sufficiently large cardinals: the existence of a [[measurable cardinal]] implies that $V \neq L$.  Most contemporary set theorists do not regard $V = L$ as potentially "true."

The [[axiom of determinacy]] ($AD$) is another axiom that can be added to $ZF$; it asserts that a certain class of infinite [[game]]s is determined (one player or the other has a winning strategy).  $AD$ is inconsistent with the full axiom of choice, although it is consistent with dependent choice.  A weaker form of $AD$ called "projective determinacy" is consistent with $AC$ and is equiconsistent with certain large cardinal assertions.

The $GCH$ itself, or its negation, could also be regarded as an additional axiom that can be added to $ZF$.  Many set theorists would prefer to find a more "natural" axiom, such as a large cardinal axiom, which implies either $GCH$ or its negation.  The equiconsistency of projective determinacy with a large cardinal assertion can be regarded as a step in this direction.

### Relation to structural set theories

The structural set theory **[[ETCS]]** is equivalent to $BZC$ in that the [[category of sets]] in that theory satisfies $ETCS$ while the well-founded [[pure sets]] in $ETCS$ satisfy $BZC$.  This uses (1--4), Bounded Separation for (5), and (7--10), with Weak Replacement following from (5) and (7).

[[Mike Shulman]]\'s **[[SEAR|SEARC]]** is equivalent to $ZFC$ in the same way.  $SEAR$, which lacks the axiom of choice, is equivalent to $ZF^{\circlearrowleft}$, which is $ZF$ without (10), in a weaker sense of equivalence.

[Shulman 2018](#Shulman2018) is an extensive survey of variations of $ZFC$ and variations of $ETCS$ (mostly weak ones), showing how these correspond.

## Related entries

* [[material set theory]]

* [[algebraic set theory]]

* [[cumulative hierarchy]]

* [[structural ZFC]]

## References

Zermelo's axiomatisation grew out of his reflections on his proofs of the well-ordering theorem (1904/08) and was published in

* [[Ernst Zermelo]], _Untersuchungen über die Grundlagen der Mengenlehre I_ , Math. Ann. **65** (1908) pp.261-81. ([gdz](http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002262002))

English versions of the early key texts on set theory by Zermelo, Fraenkel, Skolem, von Neumann et al. can be found in

* J. van Heijenoort, _From Frege to Gödel - A Sourcebook in Mathematical Logic 1879-1931_ , Harvard UP 1967.

There are many texts which discuss ZFC and the [[cumulative hierarchy]] from a traditional (material) set-theoretic perspective. A good example is

* Kenneth Kunen, _Set Theory: An Introduction to Independence Proofs_, Studies in Logic and the Foundations of Mathematics Vol. 102 (2006), Elsevier.

A classification of axioms of variants of $ZFC$, with an eye towards corresponding [[structural set theories]], is

* [[Michael Shulman]] (2018). Comparing material and structural set theories. [arXiv:1808.05204](https://arxiv.org/abs/1808.05204).
{#Shulman2018}

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Zhaohui Luo

* [webpage](http://www.cs.rhul.ac.uk/~zhaohui/)

## Selected writings

On [[type theory]] in [[computer science]] (via the [[calculus of constructions]]):

* [[Zhaohui Luo]], *Computation and Reasoning -- A Type Theory for Computer Science*, Clarendon Press (1994) $[$[ISBN:9780198538356](https://global.oup.com/academic/product/computation-and-reasoning-9780198538356), [pdf](https://www.researchgate.net/profile/Zhaohui-Luo-4/publication/243770570_Computation_and_Reasoning_A_Type_Theory_for_Computer_Science/links/54d0dc6b0cf29ca81103f26a/Computation-and-Reasoning-A-Type-Theory-for-Computer-Science.pdf)$]$

## Related entries

* [[type universe]]

* [[coercion]]

* [[dependent type theoretic methods in natural language semantics]]

category: people


Zhen Huan

* [webpage](https://huanzhen84.github.io/zhenhuan/)

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=233130)

## Selected writings

On [[equivariant elliptic cohomology]] in the form of [[orbifold K-theory]] on [[Huan's inertia orbifolds]] (a form of [[twisted ad-equivariant Tate K-theory]]):

* [[Zhen Huan]], _Quasi-elliptic cohomology_, 2017 ([hdl](http://hdl.handle.net/2142/97268))

* [[Zhen Huan]], *Quasi-Elliptic Cohomology and its Power Operations*, J. Homotopy and Related Structures **13** (2018) 715–767 [[arXiv:1612.00930](https://arxiv.org/abs/1612.00930), [doi:10.1007/s40062-018-0201-y](https://doi.org/10.1007/s40062-018-0201-y)]

* [[Zhen Huan]], *Quasi-elliptic cohomology and its Spectrum* ([arXiv:1703.06562](https://arxiv.org/abs/1703.06562))

* {#Huan18} [[Zhen Huan]], _Quasi-Elliptic Cohomology I_, Advances in Mathematics, Volume 337, 15 October 2018, Pages 107-138 ([arXiv:1805.06305](https://arxiv.org/abs/1805.06305), [doi:10.1016/j.aim.2018.08.007](https://doi.org/10.1016/j.aim.2018.08.007))

* [[Zhen Huan]], *Quasi-theories* [[arXiv:1809.06651](https://arxiv.org/abs/1809.06651)]

* [[Zhen Huan]], *Quasi-theories and their equivariant orthogonal spectra* [[arXiv:1809.07622](https://arxiv.org/abs/1809.07622)]

  > (via [[orthogonal spectra]])

* [[Zhen Huan]], [[Matthew Spong]], *Twisted Quasi-elliptic cohomology and twisted equivariant elliptic cohomology* ([arXiv:2006.00554](https://arxiv.org/abs/2006.00554))

* [[Zhen Huan]], [[Nathaniel Stapleton]], *Level structures on p-divisible groups from the Morava E-theory of abelian groups* ([arXiv:2001.10075](https://arxiv.org/abs/2001.10075))

Review:

* [[Zhen Huan]], *Quasi-elliptic cohomology theory and the twisted, twisted Real theories*, talk at Perimeter Institute (2020) [video: [doi:10.48660/20050059](https://doi.org/10.48660/20050059)]

Generalization to [[twisted equivariant K-theory|twisted equivariant]] [[KR-theory]]:

* [[Zhen Huan]], [[Matthew B. Young]], *Twisted Real quasi-elliptic cohomology* [[arXiv:2210.07511](https://arxiv.org/abs/2210.07511)]

On [[2-representations]]:

* [[Zhen Huan]], _2-Representations of Lie 2-groups and 2-Vector Bundles_ ([arXiv:2208.10042](https://arxiv.org/abs/2208.10042))

## Related entries

* [[Tate K-theory]]

* [[orbifold K-theory]]

* [[equivariant elliptic cohomology]]

* [[Huan's inertia orbifold]], [[cyclic loop space]]

category: people


Zhen Lin > history

see *[[Zhen Lin Low]]*


Zhen Lin Low

* [webpage](https://zll22.user.srcf.net/dpmms/)

* [blog](http://semmibol.wordpress.com/)

## Selected writings

On [[Grothendieck universes]] for [[category theory]]:

* [[Zhen Lin Low]], _Universes for category theory_ [[arxiv/1304.5227](http://arxiv.org/abs/1304.5227)]

On [[geometric homotopy theory|geometric]] [[homotopy theory]] via [[simplicial objects]] [[internalization|internal]] to a [[Grothendieck topos]] ([[simplicial sheaves]]):

* {#Low14} [[Zhen Lin Low]], _Internal and local homotopy theory_ (2014) $[$[arXiv:1404.7788](https://arxiv.org/abs/1404.7788)$]$

On [[simplicial homotopy theory]], [[(∞,1)-category theory]] and [[(∞,1)-topos theory]]:

* {#Low15} [[Zhen Lin Low]], *[[Notes on homotopical algebra]]*, 2015 ([pdf](http://zll22.user.srcf.net/writing/homotopical-algebra/2015-11-10-Main.pdf))

On generalization of the notion of [[schemes]] (to something like [[Structured Spaces]]):

* {#Low16} [[Zhen Lin Low]], *Categories of spaces built from local models*, 2016 ([pdf](https://www.repository.cam.ac.uk/bitstream/handle/1810/256998/Low-2016-PhD.pdf;jsessionid=76AD1098F39D92EDDFE705324F14BBA5?sequence=1), [[LowCategoriesOfSpaces.pdf:file]])

category: people

[[!redirects Zhen Lin]]


Zhenbin Yang

* [webpage](https://profiles.stanford.edu/zhenbin-yang?releaseVersion=7.16.0)

## Selected writings

General aspects of [[nearly AdS2-CFT1]] ([[JT-gravity]]/matrix models):

* [[Juan Maldacena]], [[Douglas Stanford]], [[Zhenbin Yang]], _Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space_ ([arXiv:1606.01857](https://arxiv.org/abs/1606.01857))

Argument that the image of *[[wormhole]] traversal* under [[AdS-CFT duality]] is *[[quantum teleportation]]*:

* [[Juan Maldacena]], [[Douglas Stanford]], [[Zhenbin Yang]], *Diving into traversable wormholes*,Fortsch. Phys. **65** 5 (2017) 1700034 [[arXiv:1704.05333](https://arxiv.org/abs/1704.05333), [doi:10.1002/prop.201700034](https://doi.org/10.1002/prop.201700034)]

On the [[SYK model]]:

* Biao Lian, S. L. Sondhi, [[Zhenbin Yang]], _The chiral SYK model_ ([arXiv:1906.03308](https://arxiv.org/abs/1906.03308))

Claim that the proper application of [[holographic entanglement entropy]] to the discussion of [[Bekenstein-Hawking entropy]] resolves the apparent [[black hole information paradox]]:

* [[Geoff Penington]], [[Stephen Shenker]], [[Douglas Stanford]], [[Zhenbin Yang]], _Replica wormholes and the black hole interior_ ([arXiv:1911.11977](https://arxiv.org/abs/1911.11977))

category: people


Zheng Yin

* [SemanticScholar page](https://www.semanticscholar.org/author/Zheng-Yin/2048400828)

## Selected writings

On a [[matrix model]] for the [[3-brane in 6d]] arising as the [[brane intersection|intersection]] of two [[M5-branes]]:

* {#KachruOzYin98} [[Shamit Kachru]], [[Yaron Oz]], [[Zheng Yin]], *Matrix Description of Intersecting M5 Branes* JHEP 9811:004, (1998) ([arXiv:hep-th/9803050](https://arxiv.org/abs/hep-th/9803050))

category: people