red-shift conjecture

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###Context###
#### Cohomology
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[[!include cohomology - contents]]
=--
#### Stable Homotopy theory
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[[!include stable homotopy theory - contents]]
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#### Higher algebra
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[[!include higher algebra - contents]]
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=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[chromatic homotopy theory]] the _redshift conjecture_ is a [[conjecture]] about the nature of the [[iterated algebraic K-theory|iterated]] [[algebraic K-theory]] spectrum $K(R)$ of a [[connective spectrum|connective]] [[E-infinity ring]] $R$. Roughly, its say that $K(R)$ has [[chromatic level]] one higher than $R$ has.

The conjecture was originally formulated by [[John Rognes]] ([Rognes 99](#Rognes99), [Rognes 00](#Rognes00)) and appeared in [Ausoni & Rognes 2008](#AusoniRognes08), review in [Rognes 2014](#Rognes14).

## Related concepts

* [[BDR 2-vector bundle]]

* [[iterated algebraic K-theory]]

[[!include chromatic tower examples - table]]

## References

Exposition in:

* {#Rognes14} [[John Rognes]], _Chromatic redshift_, lecture notes, MSRI 2014 ([arXiv:1403.4838](https://arxiv.org/abs/1403.4838))

The conjecture originates with

* {#Rognes99} [[John Rognes]], _Algebraic K-theory of finitely presented ring spectra_, lecture at Schloss Ringberg, Germany, January 1999 ([pdf](https://www.mn.uio.no/math/personer/vit/rognes/papers/red-shift.pdf), [[Rognes_AlgebraicKTheoryOfRingSpectra.pdf:file]])

* {#Rognes00} [[John Rognes]], _Algebraic K-theory of finitely presented ring spectra_, Oberwolfach talk September 2000 (OWF abstract [[Rognes_AlgebraicKTheoryOfRingSpectra_OWF.pdf:file]])

The conjecture appears published in

* {#AusoniRognes08} [[Christian Ausoni]], [[John Rognes]], _The chromatic red-shift in algebraic K-theory_, Enseign. Math. (2) 54 (2008), 9-11. ([pdf](http://www.math.univ-paris13.fr/~ausoni/papers/gmconj-AR.pdf), [[AusoniRognesRedShiftInAlgebraicKTheory.pdf:file]], [doi:10.5169/seals-109873](http://doi.org/10.5169/seals-109873))

See also

* [[Tyler Lawson]], in section 3 of: _The future_, Talbot lectures 2013 ([pdf](http://math.mit.edu/conferences/talbot/2013/19-Lawson-thefuture.pdf))

* [[Benjamin Antieau]], _Some open problems in the K-theory of ring spectra_ ([pdf](https://dantie1.people.uic.edu/files/openkt-20150709.pdf))

Previous work motivating the conjecture was the study (see also at _[[iterated algebraic K-theory]]_) of the [[algebraic K-theory]] $K(ku)$ of the [[complex K-theory spectrum]] $ku$ (also thought of as the [[classifying space]] for [[BDR 2-vector bundles]]) in

* {#BaasDundasRognes07} [[Nils Baas]], [[Bjørn Dundas]], [[Birgit Richter]], [[John Rognes]], _Ring completion of rig categories_ ([arXiv:0706.0531](http://arxiv.org/abs/0706.0531))

which was motivated by the desire to turn [[topological K-theory]] into "a form of" [[elliptic cohomology]] by a kind of [[categorification]].

For more see the references at _[[iterated algebraic K-theory]]_.

[[!redirects red shift conjecture]]
[[!redirects redshift conjecture]]


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redshift

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### Context
#### Cosmology
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[[!include cosmology -- contents]]
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#### Gravity
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[[!include gravity contents]]
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#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[cosmology]]

## Disambiguation

For the notion in chromatic homotopy theory, see [[red-shift conjecture]]

## References

See also

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

[[!redirects redshifts]]

[[!redirects red shift]]

[[!redirects red shifts]]


reduced cohomology

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

## Idea

Given any notion of _[[cohomology]]_ defined on [[pointed objects]], the corresponding _reduced cohomology_ is that part of the cohomology which vanishes on the basepoint.

Specifically for [[Whitehead-generalized cohomology theories]] the reduced cohomology is the cohomology relative to the base point, hence is the [[kernel]] of the operation of [[pullback in cohomology|pullback]] to the base point See [below](#RelationToUnreducedCohomology) and see at _[generalized cohomology -- Relation between reduced and unreduced](generalized+Eilenberg-Steenrod+cohomology#RelationBetweenReducedAndUnreduced)_ for more.

## Definition

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

A **reduced [[cohomology theory]]** is a [[functor]]

$$
  \tilde E^\bullet
   \;\colon\;
  \big(
    Top^{\ast/}_{CW}
  \big)^{op}
    \longrightarrow
  Ab^{\mathbb{Z}}
$$

from the [[opposite category|opposite]] of [[pointed topological spaces]] ([[CW-complexes]]) to $\mathbb{Z}$-[[graded abelian groups]] ("[[cohomology groups]]"), in components

$$
  \tilde E
    \;\colon\;
  \big(
    X \stackrel{f}{\longrightarrow} Y
  \big)
    \;\mapsto\;
  \big(
    \tilde E^\bullet(Y)
      \stackrel{f^\ast}{\longrightarrow}
    \tilde E^\bullet(X)
  \big)
  \,,
$$

and equipped with a [[natural isomorphism]] of degree +1, to be called the **[[suspension isomorphism]]**, of the form

$$
  \sigma
    \;\colon\;
  \tilde E^{\bullet +1}(\Sigma -)
    \overset{\simeq}{\longrightarrow}
  \tilde E^\bullet(-)
$$

such that:

1. **([[homotopy invariance]])** If $f_1,f_2 \colon X \longrightarrow Y$ are two morphisms of pointed topological spaces such that there is a (base point preserving) [[homotopy]] $f_1 \simeq f_2$ between them, then the induced [[homomorphisms]] of abelian groups are [[equality|equal]]

   $$
     f_1^\ast = f_2^\ast
     \,.
   $$

1. {#ReducedExactnessAxiom} **(exactness)** For $i \colon A \hookrightarrow X$ an inclusion of pointed topological spaces, with $j \colon X \longrightarrow Cone(i)$ the induced [[mapping cone]], then this gives an [[exact sequence]] of graded abelian groups

   $$
     \tilde E^\bullet(Cone(i))
      \overset{j^\ast}{\longrightarrow}
     \tilde E^\bullet(X)
       \overset{i^\ast}{\longrightarrow}
     \tilde E^\bullet(A)
     \,.
   $$

We say $\tilde E^\bullet$ is **additive** if in addition

* **([[wedge axiom]])** For $\{X_i\}_{i \in I} $ any set of pointed CW-complexes, then the canonical comparison morphism

  $$
    \tilde E^\bullet(\vee_{i \in I} X_i)
     \longrightarrow
    \prod_{i \in I} \tilde E^\bullet(X_i)
  $$

  is an [[isomorphism]], from the functor applied to their [[wedge sum]], example \ref{WedgeSumAsCoproduct}, to the [[product]] of its values on the wedge summands, .

We say $\tilde E^\bullet$ is **ordinary** if its value on the [[0-sphere]] $S^0$ is concentrated in degree 0:

* **(Dimension)**  $\tilde E^{\bullet\neq 0}(\mathbb{S}^0) \simeq 0$.

A [[homomorphism]] of reduced cohomology theories

$$
  \eta \;\colon\; \tilde E^\bullet \longrightarrow \tilde F^\bullet
$$

is a [[natural transformation]] between the underlying functors which is compatible with the suspension isomorphisms in that all the following [[commuting square|squares commute]]

$$
  \array{
    \tilde E^\bullet(X) &\overset{\eta_X}{\longrightarrow}&  \tilde F^\bullet(X)
    \\
    {}^{\mathllap{\sigma_E}}\downarrow && \downarrow^{\mathrlap{\sigma_F}}
    \\
    \tilde E^{\bullet + 1}(\Sigma X)
    &\overset{\eta_{\Sigma X}}{\longrightarrow}&
    \tilde F^{\bullet + 1}(\Sigma X)
  }
  \,.
$$

=--

(e.g. [AGP 02, def. 12.1.4](#AGP02))

## Properties

### Brown representability

The [[Brown representability theorem]] says that for any reduced cohomology theory $\tilde E^\bullet$ there is an [[Omega-spectrum]] $E$ which [[representable functor|represents]] $\tilde E^\bullet$ on pointed connected CW-complex $X$, in that

$$
  \tilde E^n(X) \simeq [X,E_n]_\ast
  \,.
$$

### Relation to unreduced cohomology
 {#RelationToUnreducedCohomology}

For an unreduced [[cohomology theory]] $E^\bullet$  the induced **reduced cohomology** is the [[kernel]] of operation of [[pullback in cohomology|pullback]] to the base point.

$$
  \tilde E^k(X,x_0)
    \;\coloneqq\;
  E^k(X,\{x_0\})
    \;=\;
  ker\big(
    H^k(X) \to H^k(\{x_0\})
  \big)
$$

(e.g. [AGP 02, theorem 12.1.12](#AGP02)).

For more see at _[generalized cohomology -- Relation between reduced and unreduced](generalized+Eilenberg-Steenrod+cohomology#RelationBetweenReducedAndUnreduced)_.

## Related entries

* [[reduced homology]]

* [[vanishing at infinity]]

## References

See the references at _[[generalized (Eilenberg-Steenrod) cohomology]]_.

* {#AGP02} Marcelo Aguilar, [[Samuel Gitler]], Carlos Prieto, section 12 of: _Algebraic topology from a homotopical viewpoint_, Springer (2002) ([toc pdf](http://tocs.ulb.tu-darmstadt.de/106999419.pdf))

* [[Jacob Lurie]], _[[A Survey of Elliptic Cohomology - cohomology theories]]_

[[!redirects reduced cohomology theory]]


reduced cylinder

#Contents#
* table of contents
{:toc}

## Idea

The concept of _reduced cylinder_ is the analog for [[pointed objects]] of [[cylinder]] constructions.

The [[mapping cone]] of $X \to \ast$ formed with the standard reduced cylinder is the [[reduced suspension]] of $X$.

Applying the reduced cylinder construction degreewise to a [[sequential spectrum]] yields the standard [[cylinder spectrum]] construction.

## For pointed topological spaces

Specifically in [[topological spaces]], with $I = [0,1] \subset \mathbb{R} \in $ [[Top]] the [[closed interval]] with its [[Euclidean space|Euclidean]] [[metric topology]] and $I_+ \in Top^{\ast/}$ its [[pointed topological space|pointed]] version with a basepoint freely adjoined, then for $X$ a [[pointed topological space]], the standard **reduced cylinder** over it is the [[smash product]]

$$
  X \wedge (I_+) \;\; \in Top^{\ast/}
  \,.
$$

This is obtained from the ordinary standard cylinder $X \times I$ by passing to the [[quotient space]] ([this example](quotient+space#QuotientBySubspace)) given by collapsing the copy of $I$ that sits over the basepoint $x$ of $X$:

$$
  X \wedge (I_+) \simeq (X \times I)/(\{x\} \times I)
  \,.
$$

For the purposes of [[generalized (Eilenberg-Steenrod) cohomology]] theory typically it does not matter whether one evaluates on the standard cylinder or the reduced cylinder. For example for [[topological K-theory]] since since $\{x\} \times I$ is a [[contractible topological space|contractible]] [[closed subspace]], then [this prop.](topological+vector+bundle#VectorBundlesOverQuotientByContractibleSubspaceAreEquivalentToVectorBundlesOnTotalSpace) says that [[topological vector bundles]] do not see a difference as long as $X$ is a [[compact Hausdorff space]].

## References

Early lecture notes include

* [[Frank Adams]], part III, section 2 _[[Stable homotopy and generalised homology]]_, 1974

[[!redirects reduced cylinders]]

[[!redirects reduced cylinder object]]
[[!redirects reduced cylinder objects]]


reduced homology

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###Context###
#### Homological algebra
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[[!include homological algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In general, the [[homology]] of a [[point]] is not trivial but is concentrated in degree 0 on the given [[coefficient]] object. For some applications, though, it is convenient to divide out that contribution such as to have the homology of the point be entirely trivial. This is called _reduced homology_.

## Definition

### Reduced singular homology

We discuss the reduced version of [[singular homology]].

Let $X$ be a [[topological space]]. Write $C_\bullet(X)$ for its [[singular chain complex]].

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

The **augmentation map** is the homomorphism of abelian groups

$$
  \epsilon \colon C_0(X) \to \mathbb{Z}
$$

which adds up all the coefficients of all 0-chains:

$$
  \epsilon \colon \colon \sum_{i} n_i \sigma_i \mapsto \sum_i n_i
  \,.
$$

Since the boundary of a 1-chain is in the [[kernel]] of this map, it constitutes a [[chain map]]

$$
  \epsilon \colon C_\bullet(X) \to \mathbb{Z}
  \,,
$$

where now $\mathbb{Z}$ is regarded as a chain complex concentrated in degree 0.

=--

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

The **reduced singular chain complex** $\tilde C_\bullet(X)$ of $X$ is the [[kernel]] of the augmentation map, the chain complex sitting in the [[short exact sequence]]

$$
  0 \to \tilde C_\bullet(X) \to C_\bullet(X) \stackrel{\epsilon}{\to} \mathbb{Z} \to 0
  \,.
$$

The **reduced singular homology** $\tilde H_\bullet(X)$ of $X$ is the [[chain homology]] of the reduced singular chain complex

$$
  \tilde H_\bullet(X) \coloneqq H_\bullet(\tilde C_\bullet(X))
  \,.
$$

=--

Equivalently:

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

The **reduced singular homology** of $X$, denoted $\tilde H_\bullet(X)$, is the [[chain homology]] of the [[augmentation|augmented]] chain complex

$$
  \cdots \to C_2(X) \stackrel{\partial_2}{\to} C_1(X) \stackrel{\partial_1}{\to} C_0(X) \stackrel{\epsilon}{\to}
  \mathbb{Z} \to 0
  \,.
$$

=--

## Properties

### Relation to ordinary homology
 {#RelationToOrdinaryHomology}

Let $X$ be a [[topological space]], $H_\bullet(X)$ its [[singular homology]] and $\tilde H_\bullet(X)$ its reduced singular homology, def. \ref{ReducedSingularChainComplex}.

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

For $n \in \mathbb{N}$ there is an [[isomorphism]]

$$
  H_n(X)
  \simeq
  \left\{
    \array{
      \tilde H_n(X) & for \; n \geq 1
      \\
      \tilde H_0(X) \oplus \mathbb{Z} & for\; n = 0
    }
  \right.
$$

=--

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

The [[homology long exact sequence]] of the defining short exact sequence $\tilde C_\bullet(C) \to C_\bullet(X) \stackrel{\epsilon}{\to} \mathbb{Z}$ is, since $\mathbb{Z}$ here is concentrated in degree 0, of the form

$$
  \cdots \to \tilde H_n(X) \to H_n(X) \to 0 \to \cdots \to
  0 \to
  \cdots \to \tilde H_1(X) \to H_1(X) \to 0 \to
  \tilde H_0(X) \to H_0(X) \stackrel{\epsilon}{\to}  \mathbb{Z} \to 0
  \,.
$$

Here [[exact sequence|exactness]] says that all the morphisms $\tilde H_n(X) \to H_n(X)$ for positive $n$ are [[isomorphisms]]. Moreover, since $\mathbb{Z}$ is a [[free abelian group]], hence a [[projective object]], the remaining [[short exact sequence]]

$$
  0 \to \tilde H_0(X) \to H_0(X) \to \mathbb{Z} \to 0
$$

is [[split exact sequence|split]] (as discussed there) and hence $H_0(X) \simeq \tilde H_0(X) \oplus \mathbb{Z}$.

=--

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

For $X = *$ the [[point]], the morphism

$$
  H_0(\epsilon) \colon H_0(X) \to \mathbb{Z}
$$

is an [[isomorphism]]. Accordingly the reduced homology of the point vanishes in every degree:

$$
  \tilde H_\bullet(*) \simeq 0
  \,.
$$

=--

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

By the discussion at _[Singular homology -- Relation to homotopy groups](singular%20homology#RelationToHomotopyGroups)_ we have that

$$
  H_n(*) \simeq
  \left\{
    \array{
       \mathbb{Z} & for \; n = 0
       \\
       0 & otherwise
    }
  \right.
  \,.
$$

Moreover, it is clear that $\epsilon \colon C_0(*) \to \mathbb{Z}$ is the [[identity]] map.

=--

### Relation to relative homology
 {#RelationToRelativeHomology}

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

For $X$ an [[inhabited set|inhabited]] [[topological space]], its reduced singular homology, def. \ref{ReducedSingularChainComplex}, coincides with its [[relative singular homology]] relative to any base point $x \colon * \to X$:

$$
  \tilde H_\bullet(X)
  \simeq
   H_\bullet(X,*)
  \,.
$$

=--

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

Consider the sequence of [[topological subspace]] inclusions

$$
  \emptyset \hookrightarrow * \stackrel{x}{\hookrightarrow} X
  \,.
$$

By the discussion at _[Relative homology - long exact sequences](relative%20homology#LongExactSequences)_ this induces a [[long exact sequence]] of the form

$$
  \cdots
  \to
  H_{n+1}(*) \to H_{n+1}(X) \to H_{n+1}(X,*)
  \to
  H_n(*) \to H_n(X) \to H_n(X,*)
  \to
  \cdots
  \to
  H_1(X) \to H_1(X,*) \to  H_0(*) \stackrel{H_0(x)}{\to}  H_0(X) \to H_n(X,*)
  \to 0
  \,.
$$

Here in positive degrees we have $H_n(*) \simeq 0$ and therefore [[exact sequence|exactness]] gives [[isomorphisms]]

$$
  H_n(X) \stackrel{\simeq}{\to} H_n(X,*)\;\; \forall_{n \geq 1}
$$

and hence with prop. \ref{RelationBetweenReducedSingularAndSingular} isomorphisms

$$
  \tilde H_n(X) \stackrel{\simeq}{\to} H_n(X,*)\;\; \forall_{n \geq 1}
  \,.
$$

It remains to deal with the case in degree 0. To that end, observe that $H_0(x) \colon H_0(*) \to H_0(X)$ is a [[monomorphism]]: for this notice that we have a [[commuting diagram]]

$$
  \array{
     H_0(*) &\stackrel{id}{\to}& H_0(*)
     \\
     {}^{\mathllap{H_0(x)}}\downarrow &{}^{\mathllap{H_0(f)}}\nearrow& \downarrow^{\mathrlap{H_0(\epsilon)}}_\simeq
     \\
     H_0(X) &\stackrel{H_0(\epsilon)}{\to}& \mathbb{Z}
  }
  \,,
$$

where $f \colon X \to *$ is the terminal map.
That the outer square commutes means that $H_0(\epsilon) \circ H_0(x) = H_0(\epsilon)$ and hence the composite on the left is an [[isomorphism]]. This implies that $H_0(x)$ is an injection.

Therefore we have a [[short exact sequence]] as shown in the top of this diagram

$$
  \array{
    0 &\to& H_0(*) &\stackrel{H_0(x)}{\hookrightarrow}&
    H_0(X) &\stackrel{}{\to}& H_0(X,*)
    &\to&
    0
    \\
    && & {}_{\mathllap{\simeq}}\searrow & \downarrow^{\mathrlap{H_0(\epsilon)}} &
   \\
   && && \mathbb{Z}
  }
  \,.
$$

Using this we finally compute

$$
  \begin{aligned}
    \tilde H_0(X)
    & \coloneqq
    ker H_0(\epsilon)
    \\
    & \simeq coker( H_0(x) )
    \\
    & \simeq H_0(X,*)
  \end{aligned}
  \,.
$$

=--

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

Moreover, for "good" inclusions $A \hookrightarrow X$ of topological space, the reduced singular homology of the quotient $X/A$ is isomorphic to the $A$-[[relative singular homology]] of $X$.

See at _[Relative homology - Relation to reduced homology of quotient topological spaces](relative+homology#RelationToQuotientTopologicalSpaces)_.

=--

### Relation to wedge sums
 {#RelationToWedgeSums}

Let $\{* \to X_i\}_i$ be a set of [[pointed object|pointed]] [[topological spaces]]. Write $\vee_i X_i \in Top$ for their [[wedge sum]] and write
$\iota_i \colon X_i \to \vee_i X_i$ for the canonical inclusion functions.

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

For each $n \in \mathbb{N}$ the homomorphism

$$
  (\tilde H_n(\iota_i))_i \colon \oplus_i \tilde H_n(X_i) \to \tilde H_n(\vee_i X_i)
$$

is an [[isomorphism]].

=--

For instance ([Hatcher, cor. 2.25](#Hatcher)).

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

This follows with _[this proposition](relative+homology#HomologyOfQuotientSpace)_ at _[[relative homology]]_.

=--

## Examples

### For singular homology

For $X$ a [[topological space]], write $H_n(X)$ for its [[singular homology]] with integer coefficients.

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

If $X$ is a [[contractible topological space]], then for all $n \in \mathbb{N}$

$$
  \tilde H_n(X) \simeq 0
  \,.
$$

=--

+-- {: .num_example #ReducedHomologyOf0Sphere}
###### Example

The reduced singular homology of the 0-[[sphere]] $S^0 \simeq {*} \coprod {*}$ is

$$
  \tilde H_n(S^0) \simeq
  \left\{
    \array{
       \mathbb{Z} & if \; n = 0
       \\
       0 & otherwise
    }
  \right.
  \,.
$$

=--

## Related concepts

* [[reduced cohomology]]

## References

Reduced singular homology is discussed for instance around p. 119 of

* [[Alan Hatcher]], _[Algebraic Topology](http://www.math.cornell.edu/~hatcher/AT/ATpage.html)_

[[!redirects reduced singular homology]]


reduced K-theory

## Idea

Reduced K-theory is the [[reduced cohomology theory]] of [[K-theory]].

For the moment see [here](topological+K-theory#ReducedKGroup) at _[[topological K-theory]]_

## Related concepts

* [[relative K-theory]]

[[!redirects reduced K-group]]
[[!redirects reduced K-groups]]


reduced local ring

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

\tableofcontents

## Definition

A **reduced local ring** is a [[commutative ring]] which is both a [[local ring]] and a [[reduced ring]], hence a commutative ring such that:

* The ring is nontrivial $0 \neq 1$;

* if the sum of two elements $x + y$ is equal to $1$, then $x$ is [[invertible]] or $y$ is invertible;

* every [[nilpotent element]] is equal to [[zero]]; equivalently, every element which squares to zero is equal to zero.

## Examples

* Every [[Heyting field]] is a reduced local ring which is also a [[Artinian ring]].

* Given a [[discrete field]] $K$, the ring of [[power series]] $K[[x]]$ is a reduced local ring which is not a [[Heyting field]].

## Properties

The theory of reduced local rings is a [[coherent theory]].

## See also

* [[ordered reduced local ring]]

## References

For a division algorithm for [[polynomials]] over reduced local rings:

* [[Fred Richman]], *A division algorithm*. Journal of Algebra and Its Applications, Vol. 04, No. 04, pp. 441-449 (2005) [[doi:10.1142/S0219498805001289](https://doi.org/10.1142/S0219498805001289)]

| [[commutative ring]] | [[reduced ring]] | [[integral domain]] |
---------------------|-----------------|-----------------|
| [[local ring]] | [[reduced local ring]] | [[local integral domain]] |
| [[Artinian ring]] | [[semisimple ring]] | [[field]] |
| [[Weil ring]] | [[field]] | [[field]] |

[[!redirects reduced local ring]]
[[!redirects reduced local rings]]


reduced object

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{:toc}

## Idea
 {#Idea}

In a context of [[synthetic differential geometry]]/[[differential cohesion]] a _reduced object_ is one "without purely infinitesimal extension".

For instance in the context of [[formal schemes]]/[[formal smooth manifolds]] the reduced objects are the genuine [[schemes]] and the genuine [[smooth manifold]], those without formal extension.

Accordingly, an [[anti-reduced object]] is one consisting entirely of infinitesimal extension, hence is an [[infinitesimally thickened point]].

Beware that reduced objects in general do "contain infinitesimals in between their classical points", in that not every map from an [[anti-reduced object]] into them is necessarily [[constant map|constant]]. The objects "without any infinitesimals" in the sense that all such maps are constant are instead the [[coreduced objects]].

## Definition

A context of [[differential cohesion]] is determined by the existence of an [[adjoint triple]] of  [[modalities]]

$$
  \Re \dashv \Im \dashv \&
  \,,
$$

where $\Re$ and $\&$ are [[idempotent monad|idempotent]] [[comonads]] and $\Im$ is an [[idempotent monad]].

A **reduced object** or **reduced type** is one in the [[full subcategory]] defined by the leftmost modality $\Re$.

## Examples
 {#Examples}

### In smooth differential geometry

Consider the [[Cahiers topos]]

$$
  \mathbf{H} = Sh(\{\mathbb{R}^n \times Spec(W)\}_{n,W})
$$

as a [[categorical semantics|model]] of [[synthetic differential geometry]]/[[differential cohesion]] (where $W$ denotes [[infinitesimally thickened point|Weil algebras]]/[[Artin algebras]]). The sub-topos of reduced obects

$$
  \mathbf{H}_{reduced} \hookrightarrow \mathbf{H}
$$

is the topos of [[smooth spaces]]

$$
  \mathbf{H}_{reduced} = Sh(\{\mathbb{R}^n\})
  \,.
$$

An object $D \in \mathbf{H}$ with is an [[anti-reduced object]], hence whose reduction coreflection is the [[terminal object]], $\Re(D) \simeq   \ast $ is an _[[infinitesimally thickened point]]_.

For instance the [[Isbell duality|formal dual]] $D^1 = Spec(\mathbb{R}[\epsilon](\epsilon^2))$ of the [[ring of dual numbers]] is such that its reduction is the point $\Re(D) \simeq \ast$.

Under [[Yoneda embedding]] every [[smooth manifold]] is in $\mathbf{H}_{reduced}$ and is hence a reduced object in $\mathbf{H}$. More generally there are [[formal smooth manifolds]] in $\mathbf{H}$ and they are generally not reduced.

For example for $\Sigma \in SmoothMfd \hookrightarrow \mathbf{H}_{reduced}\hookrightarrow \mathbf{H}$ an ordinary smooth manifold (hence reduced) the object

$$
  \Sigma \times D^1 \in \mathbf{H}
$$

is a [[formal smooth manifold]] which is not reduced. It reduction is

$$
  \Re(\Sigma\times D^1) \simeq \Sigma
  \,.
$$

In particular the [[real line]] which is the smooth line object of the [[smooth topos]] $\mathbf{H}$

$$
  \mathbb{R}^1 \in \mathbf{H}_{reduced}\hookrightarrow \mathbf{H}
$$

is reduced, $\Re(\mathbb{R}) \simeq \mathbb{R}$.

Observe from these examples that reduced objects do "contain infinitesimal points in between their classical points", which just means that there are non-[[constant map|constant]] morphisms of the form

$$
  D \longrightarrow \Sigma
  \,.
$$

###Contrast between reduced and coreduced objects

The objects $X \in \mathbf{H}$ for which all maps out of anti-reduced objects $D$ are [[constant maps]] are instead the [[coreduced objects]].

The [[coreduced objects]] are the ones with "no infinitesimal behavior", and the reduced objects are the ones "whose infinitesimal behavior is determined by their non-infinitesimal behavior".  A reduced object does contain infinitesimal points; what it lacks are "purely infinitesimal directions" while a coreduced object has no infinitesimal points.

### In algebraic geometry

A [[reduced scheme]] is one all whose local [[rings]] of functions have no non-zero nilpotent elements.

## Related concepts

* [[infinitesimal space]], [[infinitesimally thickened point]]

[[!include cohesion - table]]

* [[formally smooth morphism]], [[formally etale morphism]], [[formally unramified morphism]]

[[!redirects reduced objects]]

[[!redirects reduced type]]
[[!redirects reduced types]]

[[!redirects reduced space]]
[[!redirects reduced spaces]]


reduced phase space

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#Contents#
* table of contents
{:toc}

## Idea

In [[physics]], a [[local Lagrangian]] induces a [[covariant phase space]] equipped with a canonical [[presymplectic form]]. The [[quotient]] of this by [[symmetries]] that, in good cases, make the pre-symplectic form a genuine [[symplectic form]], is called the _reduced phase space_.

Generally, given a [[symplectic manifold]] or [[presymplectic manifold]] or [[Poisson manifold]] regarded as a [[phase space]] equipped with a suitable ([[Hamiltonian action|Hamiltonian-]]) [[action]] by a [[Lie group]], the corresponding [[symplectic reduction]] or [[presymplectic reduction]] or [[Poisson reduction]] is, if it exists, the corresponding reduced phase space.

[[!include symplectic reduction - table]]

## Details

### For local Lagrangians

Given a [[local Lagrangian]] (we display it in [[codimension]] 1 [[mechanics]] for simplicity of notation)

$$
  L \;\colon\; \Omega^{0,1}\left(\mathbf{Fields}(I) \times I\right)
$$

the corresponding [[covariant phase space]] is the space of solutions of the [[Euler-Lagrange equations]]

$$
  \{EL = 0\}
$$

equipped with the [[presymplectic form]]

$$
  \omega = \delta \frac{\delta L}{\delta \dot \phi} \wedge \delta \phi
$$

which is exact, with [[potential]]

$$
  \theta = \frac{\delta L}{\delta \dot \phi} \wedge \delta \phi
$$

as discussed in detail at _[[covariant phase space]]_.

Together, this is a [[prequantum bundle]], hence a [[circle bundle with connection]] whose [[curvature]] is $\omega$. It so happens that the underlying U(1)-[[principal bundle]] of this is trivial, and hence the [[connection on a bundle|connection]] is given by the globally defined [[differential 1-form]] $\theta$.

But this trivialility is only superficial: the [[symmetry]] [[group]] $G$ of the Lagrangian is supposed to act by [[Hamiltonian flows]] and the prequantum connection $\theta$ is to be equipped with $G$-[[equivariant connection]] structure for it to count as a connection on the [[reduced phase space]].

Another way to say this, using the [[higher differential geometry]] of [[smooth groupoids]]: the above [[prequantum bundle]] is modulated by a map $\theta \;\colon\; \{EL = 0\} \longrightarrow \mathbf{B}U(1)_{conn}$ to the [[smooth groupoid|smooth]] [[moduli stack]] of [[circle n-bundles with connection|circle bundles with connection]], and  the $G$-[[Hamiltonian action]] induced the [[action groupoid]] $\{EL = 0\} \longrightarrow \{EL = 0\}//G$; and the construction of the reduced [[prequantization]] is the construction of the diagonal [[morphism]] $\nabla_{red}$ in the following [[diagram]] (of [[smooth groupoids]])

$$
  \array{
    \{EL = 0\} &\stackrel{\theta}{\longrightarrow}& \mathbf{B}U(1)_{conn}
    \\
    \downarrow & \nearrow_{\nabla_{red}}
    \\
    \{EL = 0\}//G
  }
  \,.
$$

### For extended local Lagrangians

For $n$-dimensional [[field theory]], the [[local Lagrangian]] of the above dsicussion arises as the [[transgression]] of an $n$-form Lagrangian down in [[codimension]] $n$, a refinement discussed in more detail at _[[local prequantum field theory]]_. Such a Lagrangian may analogously be understood as being the [[principal infinity-connection|higher connection]] on a [[prequantum n-bundle]] which, in turn, maps to the ordinary [[prequantum bundle]] under [[transgression]]. Hence one may ask for a _universal_ or _extended_ [[equivariant structure]] already on this [[prequantum n-bundle]] which is such that under [[transgression]] (to any [[Cauchy surface]]) it induces an equivariant structure and hence a reduced phase space as above.

Examples of such extended reduced phase space structures include:

* the [[universal Chern-Simons circle 3-connection]]

* the [[universal Chern-Simons circle 7-connection]]

and other examples discussed at _[[local prequantum field theory]]_.
## Related concepts

* [[symplectic reduction]]

* [[quantization commutes with reduction]]

* [[equivariant cohomology]]

* [[BV-BRST formalism]]

## References

Textbook account:

* [[Marc Henneaux]], [[Claudio Teitelboim]], §2.2.3 in: _[[Quantization of Gauge Systems]]_, Princeton University Press (1992) [[ISBN:9780691037691](https://press.princeton.edu/books/paperback/9780691037691/quantization-of-gauge-systems), [jstor:j.ctv10crg0r](https://www.jstor.org/stable/j.ctv10crg0r)]

See also the references at _[[quantization commutes with reduction]]_.

[[!redirects reduced phase spaces]]


reduced ring

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\tableofcontents

## Definition

Given a [[commutative ring]] $R$, $R$ is **reduced** or has a **trivial nilradical** if $x \cdot x = 0$ implies that $x = 0$ for all $x \in R$.

\begin{theorem}
For every natural number $n$, $x^{n + 1} = 0$ implies that $x = 0$ for all $x \in R$.
\end{theorem}

\begin{proof}
Let the function $f:\mathbb{N} \to \mathbb{N}$ be defined as the [[ceiling]] of half of $n$, $f(n) \coloneqq \lceil n/2 \rceil$. Then $x^{n + 1} = 0$ implies that $x^{f(n + 1)} = 0$, and for every natural number $n$, the $(n + 1)$-th iteration of the function $f$ evaluated at $n + 1$ is always equal to $1$, $f^{n + 1}(n + 1) = 1$, thus resulting in $x^{f^{n + 1}(n + 1)} = x = 0$. Thus, the nilradical of $R$ is trivial.
\end{proof}

As a result, the theory of a reduced ring is a [[coherent theory]].

## Properties

* Every [[integral domain]] is a reduced ring. Thus, every field is a reduced ring.

* An example of a reduced ring which is not an [[integral domain]] is the quotient ring $R[x, y]/(x \cdot y)$.

* Given a [[square-free integer]] $n$, the [[integers modulo n]] $\mathbb{Z}/n\mathbb{Z}$ is a reduced ring. Since for every integer $n$,  $\mathbb{Z}/n\mathbb{Z}$ is a [[prefield ring]], $\mathbb{Z}/n\mathbb{Z}$ is an [[integral domain]] and thus a [[field]] if and only if $n$ is a [[prime number]].

* Given a [[discrete field]] $F$, let $\overline{F}$ denote its [[algebraic closure]]. Given a [[square-free polynomial]] $q \in \overline{F}[x]$, the [[quotient ring]] $\overline{F}[x]/q\overline{F}[x]$ is a reduced ring. Since for every polynomial $q \in \overline{F}[x]$, $\overline{F}[x]/q\overline{F}[x]$ is a [[prefield ring]], $\overline{F}[x]/q\overline{F}[x]$ is an [[integral domain]] and thus a [[field]] if and only if $q$ is a prime polynomial in $\overline{F}[x]$, a [[monic polynomial]] of degree one; the resulting quotient ring is equivalent to $\overline{F}$.

## See also

* [[radical]]

* [[reduction modality]]

| [[commutative ring]] | [[reduced ring]] | [[integral domain]] |
---------------------|-----------------|-----------------|
| [[local ring]] | [[reduced local ring]] | [[local integral domain]] |
| [[Artinian ring]] | [[semisimple ring]] | [[field]] |
| [[Weil ring]] | [[field]] | [[field]] |

## References

* Wikipedia, [Reduced ring](https://en.wikipedia.org/wiki/Reduced_ring)

[[!redirects reduced ring]]
[[!redirects reduced rings]]

[[!redirects reduced algebra]]
[[!redirects reduced algebras]]

[[!redirects trivial nilradical]]
[[!redirects trivial nilradicals]]


reduced scheme

#Contents#
* table of contents
{:toc}

## Idea

A [[scheme]] is _reduced_ if it has no "purely [[infinitesimal object|infinitesimal]] directions".

## Definition

An algebraic [[scheme]] is __reduced__ if all [[stalks]] of the [[structure sheaf]] are [[local rings]] without nonzero nilpotent elements.

This means that it is a [[reduced object]] in the sense of [[synthetic differential geometry]].

Equivalently a scheme $X$ is reduced iff it can be covered by a family $\{U_\alpha\}_\alpha$ of [[open sets]] such that $\mathcal{O}_X(U)$ has no nonzero nilpotent elements. Equivalently, every open $U\subset X$ has the property that $\mathcal{O}_X(U)$ has no nonzero nilpotent elements.

Reducedness can also be characterized by the [[internal language]] of the sheaf topos $\mathrm{Sh}(X)$: $X$ is reduced iff $\mathcal{O}_X$ is a reduced ring in $\mathrm{Sh}(X)$ iff $\mathcal{O}_X$ is a [[field|residue field]] in $\mathrm{Sh}(X)$ (i.e. every non-unit is zero). This can be used to give an internal proof of the fact that $\mathcal{O}_X$-modules of [[coherent sheaf|finite type]] over a reduced scheme are locally free iff their [[rank]] is constant (by internalizing the constructive proof of the linear algebra fact that [[finitely generated modules]] over residue fields are [[free module|free]], provided the minimal number of elements needed to generate the module exists as a natural number).

## Examples

For instance the dual $Spec k[\epsilon](\epsilon^2)$ of the [[ring of dual numbers]] is *not* reduced, since it is the point with an infinitesimal extension added to it. Its reduction is the point itself. Generally, [[formal schemes]] are not reduced.

## Related concepts

* [[formal scheme]]

* [[reduced object]]

* [[de Rham space]]

* [[formally smooth scheme]]

category: algebraic geometry
[[!redirects reduced schemes]]


reduced Segal space

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# Reduced Segal space
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{: toc}

## Idea

A _reduced Segal space_ is a [[category object in an (∞,1)-category|category object]] in [[∞Grpd]] equipped with an (essentially) surjective functor from the terminal such. So it is the [[delooping]] of a [[monoid in an (∞,1)-category]].

Accordingly, if the reduced Segal space happens to be actually a [[groupoid object in an (∞,1)-category|groupoid object]], then it is the [[delooping]] of a [[group object in an (∞,1)-category|group object]] in [[∞Grpd]]: an [[∞-group]].

The notion was first introduced by [[G. Segal]] (who named it "special $\Delta$-space") as a means for the characterization of an [[infinite loop space]] via the notion of [[Gamma-space]].

When a reduced Segal space is group-like, it becomes a model for an [[infinity-group]] aka a loop space. Group-like reduced Segal spaces characterize loop spaces by means of finite products and weak equivalences and as such transparently show that the property of being a loop space is  invariant under any product-preserving endofunctor of topological spaces.

If $M$ is a topological monoid, its classifying space $BM$ has a model as a simplicial space in which the 0th level is a point and the nth level is n-times the product of the 1st level. The idea of a reduced Segal space is requiring the above two properties of the simplicial space $BM$ to hold up to homotopy and capture in this way all $\infty$-monoids.

The passage from reduced Segal spaces to group-like reduced Segal spaces is the step of adding "inverses up to coherent homotopy". It turns out to be equivalent to simply requiring that the monoid of path components of the 1st level admits a group structure.

## Definition

A simplicial space $X:\Delta^{op}\to Top$ is called a reduced Segal space if:

(1) the space $X_0$ is weakly [[contractible]];

(2) for each $n\geq 1$, the Segal map $X_n\to X_1\times ...\times X_1$ is a weak equivalence.

$X$ is called group-like reduced Segal space if in addition:

(3) the monoid structure on $\pi_0 X_1$, induced from the $H$-space structure on $X_1$, admits inverses (i.e. it is a group).

## Properties

Proposition([[G. Segal]]): if $X$ is a group-like reduced Segal space, the map $X_1\to \Omega |X|$ is a weak equivalence.

## Generalizations

* The idea of reduced Segal spaces can be used for the characterization of n-fold loop spaces by n-simplicial spaces $X:(\Delta^{op})^n\to Top$ in which the corresponding Segal maps are weak equivalences.
This is explained in "Iterated
Monoidal Categories" (below).

## Related concepts

* [[Segal space]], [[complete Segal space]], [[Gamma space]]

## References

* [[G. Segal]], "Categories and Cohomology Theories", Topology 13 (1974).

*  C. Balteanu, Z. Fiedorowicz, R. Schwanzl and [[R. Vogt]], "Iterated
Monoidal Categories", Advances in Mathematics (2003).

[[!redirects reduced Segal spaces]]

[[!redirects special Delta-space]]
[[!redirects special Delta-spaces]]


reduced simplicial set

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#Contents#
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{:toc}

## Definition

A [[simplicial set]] $X$ is (sometimes) called _reduced_ if it has a single vertex, $X_0 \simeq *$.

More generally, for $n \in \mathbb{N}$ a simplicial set is _$n$-reduced_ if its $n$-[[skeleton]] is the point, $sk_n X = \Delta[0]$.

## Properties

### Reflection

Write $sSet_0 \hookrightarrow$ [[sSet]] for the [[full subcategory]] inclusion of the reduced simplicial sets into all of them.

This is a [[reflective subcategory]]. The reflector

$$
  red : sSet \to sSet_0
$$

identifies all vertices of a simplicial set.

Write $sSet^{*/}$ for the category of [[pointed object|pointed]] simplicial sets. There is also a full inclusion $sSet_0 \hookrightarrow sSet^{*/}$. This has a right adjoint $red : sSet^{*/} \to sSet_0$ which sends a pointed simplicial set to the subobject all whose $n$-cells have as 0-faces the given point.

### Coreflection

The inclusion $sSet_0 \hookrightarrow sSet^{*/}$ into [[pointed object|pointed]] simplicial sets is [[coreflective subcategory|coreflective]]. The coreflector is the [[Eilenberg subcomplex]] construction in degree 1.

### Model structure

There is a [[model structure on reduced simplicial sets]] (see there) which serves as a [[presentable (∞,1)-category|presentation]] of the [[(∞,1)-category]] of [[pointed object|pointed]] [[connected]] [[∞-groupoids]].

### As a model for $\infty$-groups

There is a [[Quillen equivalence]]

$$
  (G \dashv \bar W)
   \;\colon\;
  sGrp
   \stackrel{\overset{\Omega}{\leftarrow}}{\underset{\bar W}{\to}}
  sSet_0
$$

between the [[model structure on simplicial groups]] and the [[model structure on reduced simplicial sets]] (thus exhibiting both of these as models for [[infinity-groups]]). Its [[left adjoint]] $G$, the _[[simplicial loop space]] construction_, is a concrete model for the loop space construction with values in [[simplicial groups]].

## Related concepts

* [[Eilenberg subcomplex]]

* [[simplicial group]]

[[!redirects reduced simplicial sets]]


reduced space

#Contents#
* table of contents
{:toc}

## Idea

In a context of [[infinitesimal cohesion]] the **reduced type** $Red(X)$ of a type $X$ is the result of removing all infinitesimal thickening in $X$.

## Definition

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

Given [[differential cohesion]]

$$
  (\mathbf{H} \coloneqq CohesiveType)
    \stackrel{i}{\hookrightarrow}
  (\mathbf{H}_{th} \coloneqq InfThickenedCohesiveType)
$$

define the [[monad]]/[[comonad]] [[adjunction]]

$$
  (Red \dashv \Pi_{inf})
  \colon
  \mathbf{H}_{th}
  \stackrel{\overset{i_*}{\leftarrow}}{\underset{i^*}{\to}}
  \mathbf{H}
  \stackrel{\overset{i_!}{\leftarrow}}{\underset{i_*}{\to}}
  \mathbf{H}_{th}
  \,.
$$

We call $Red(X)$ the **reduced type** of $X$ and $\Pi_{inf}(X)$ the [[infinitesimal path ∞-groupoid]] of $X$.

The $(i_* \dashv i^*)$-[[unit of an adjunction|counit]]

$$
  Red X \to X
$$

we call the **inclusion of the reduced part** of $X$.

For the $(i_* \dashv i^*)$-[[unit of an adjunction|unit]] we write

$$
  InfinitesimalPathInclusion_X \colon X \to \Pi_{inf}(X)
$$

and call it the **constant infinitesimal path inclusion** on $X$.

=--

## Examples

(...)

[[!redirects reduced spaces]]
[[!redirects reduced cohesive ∞-groupoid]]
[[!redirects reduced cohesive ∞-groupoids]]

[[!redirects reduced type]]
[[!redirects reduced types]]


reduced suspension

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# Reduced suspension
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{: toc}

## Idea
 {#Idea}

For a [[pointed topological space]] $(X, x)$, its *reduced suspension* $\Sigma (X,x) $ is obtained from the plain [[suspension]]

$$
  \mathrm{S}X
  \;\coloneqq\;
  \frac{
    X \,\times\, [-1,\, +1]
  }{
    \left(
    \begin{array}{c}
      X \times \{-1\}\mathrlap{,\,}
      \\
      X \times \{+1\}
    \end{array}
    \right)
  }
$$

of the [[underlying]] [[topological space]] $X$ by [[quotient space|collapsing]] the [[meridian]] through the [[basepoint]] $x$ itself to a point --- this making $\Sigma(X,x)$ itself a [[pointed topological space]] with basepoint the [[equivalence class]] of that meridian $mer(x)$:

$$
  \Sigma (X,x)
    \;\coloneqq\;
  \frac{
    \mathrm{S}X
  }{ \{x\} \times [-1,1] }
  \;\;\;
  \in
  \;\;
  Top^{\ast/}
  \,.
$$

(Notice that this identifies in particular also the two [[antipode|antipodal]] "poles" of the plain suspension.)

If $X$ admits the structure of a [[CW-complex]] then, under passage to the [[classical homotopy category]] of [[pointed topological spaces]] (cf.  [here](classical+model+structure+on+topological+spaces#ModelstructureOnPointedTopologicalSpaces)) this construction models the [[homotopy pushout]] of the [[terminal object|terminal]] map $(X,x) \to (\ast,pt)$ along itself, which explains its prevalence in [[homotopy theory]] (especially in [[stable homotopy theory]], see also at *[[suspension spectrum]]*).

Moreover, in this case of [[CW-complexes]] the [[underlying]] space of $\Sigma (X,x)$ (i.e. forgetting its [[basepoint]]) is [[weak homotopy equivalence|weakly homotopy equivalent]] to the plain [[suspension]] $\mathrm{S} X$ of the [[underlying]] space $X$ of $(X,x)$. In this sense, reduced suspension in the context of [[homotopy theory]] may be understood as just being plain suspension but with basepoints taken into account.

## Definition
 {#Definition}

For $(X,x)$ a [[pointed topological space]], then its *reduced suspension* $\Sigma X$ is equivalently the following:

* obtained from the standard [[cylinder]] $I\times X$ ([[product topological space]] with the [[closed interval]] $I = [0,1]$) by identifying the [[subspace]] $(\{0,1\}\times X) \cup (I\times \{x\})$ to a point, i.e. the [[quotient space]] ([this example](quotient+space#QuotientBySubspace))

  $$
     (X \times [0,1])/ ( ( X \times \{0,1\} ) \cup ([0,1] \times \{x\}) )
  $$

  (Think of crushing the two ends of the cylinder and the line through the base point to a point.)

* obtained from the plain [[suspension]]

  $$
    S X = (X \times [0,1])/( X \times \{0\}, X \times \{1\})
  $$

  of $X$ by passing to the [[quotient space]] which collapses $\{x\} \times I$ to a point ([this example](quotient+space#QuotientBySubspace))

  $$
    \Sigma X \simeq S X / (  \{x\} \times I )
  $$

  For the purposes of [[generalized (Eilenberg-Steenrod) cohomology]] theory typically it does not matter whether one evaluates on the standard suspension or the reduced suspension. For example for [[topological K-theory]] since $\{x\} \times I$ is a [[contractible topological space|contractible]] [[closed subspace]], then [this prop.](topological+vector+bundle#VectorBundlesOverQuotientByContractibleSubspaceAreEquivalentToVectorBundlesOnTotalSpace) says that [[topological vector bundles]] do not see a difference as long as $X$ is a [[compact Hausdorff space]].

* obtained from the [[reduced cylinder]] by collapsing the two ends, i.e. the [[cofiber]]

  $$
    \Sigma X \simeq cofib(X \vee X \to X \wedge (I_+))
  $$

* the [[mapping cone]] in [[pointed topological spaces]] formed with respect to the [[reduced cylinder]] $X \wedge (I_+)$ of the map $X \to \ast$;

* the [[smash product]] $S^1\wedge X$, of $X$ with the [[circle]] (based at some point) with $X$.

  $$
    \Sigma X \simeq S^1 \wedge X
    \,.
  $$

## Properties

### Relation to suspension

For [[CW-complexes]] $X$ that are also [[pointed topological space|pointed]], with the point identified with a 0-cell, then their reduced suspension is [[weak homotopy equivalence|weakly homotopy equivalent]] to the ordinary suspension: $\Sigma X \simeq S X$.

### Cogroup structure

[[suspensions are H-cogroup objects]]

## Example

### Spheres

Up to [[homeomorphism]], the reduced suspension of the $n$-[[sphere]] is the $(n+1)$-sphere

$$
  \Sigma S^n \simeq S^{n+1}
  \,.
$$

See at _[one-point compactification -- Examples -- Spheres](one-point+compactification#ExamplesSpheres)_ for details.

## Related concepts

* [[loop space object]], [[free loop space object]],

  * [[delooping]]

  * [[loop space]], [[free loop space]], [[derived loop space]]

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

* [[suspension object]]

  * [[suspension type]], [[reduced suspension type]]

  * [[suspension]], **reduced suspension**

    * [[Freudenthal suspension theorem]]

    * [[Spanier-Whitehead category]]

    * [[suspension isomorphism]]

    * [[suspension of a chain complex]]

    * [[Sullivan model of a suspension]]

## References
 {#References}

Discussion of (reduced) [[suspension]] may be found in most introductions to [[homotopy theory]] (for discussion of unreduced suspension see also [there](suspension#References)).

For instance:

* [[Marcelo Aguilar]], [[Samuel Gitler]], [[Carlos Prieto]], §2.10 in: *Algebraic topology from a homotopical viewpoint*, Springer (2008) [[doi:10.1007/b97586](https://link.springer.com/book/10.1007/b97586)]

* [[Jeffrey Strom]], §3.8 and §17 in: *Modern classical homotopy theory*, Graduate Studies in Mathematics **127**, American Mathematical Society (2011) [[doi:10.1090/gsm/127](http://www.ams.org/books/gsm/127/)]

* [[Martin Arkowitz]], *Loop spaces and suspensions*, §2.3 in: *Introduction to Homotopy Theory*, Springer (2011) [[doi:10.1007/978-1-4419-7329-0](https://doi.org/10.1007/978-1-4419-7329-0)]

Review in the context of [[stable homotopy theory]]:

* [[Michael Hopkins]] (notes by [[Akhil Mathew]]), Lecture 2 of: _Spectra and stable homotopy theory_ (2012) [[pdf](http://math.uchicago.edu/~amathew/256y.pdf), [[HopkinsMathewStableHomotopyTheory.pdf:file]]]

[[!redirects reduced suspensions]]

[[!redirects reduced suspension object]]
[[!redirects reduced suspension objects]]


reduced suspension type

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

\tableofcontents

## Idea

In [[homotopy type theory]], the notion of _reduced suspension type_ is an [[axiom|axiomatization]] of the [[reduced suspension]] in [[pointed topological space|pointed]] [[homotopy theory]].

The [[underlying]] type of a [[pointed type|pointed]] reduced suspension type is [[equivalence in type theory|equivalent]] to the plain [[suspension type]] of the given underlying type, so that, up to [[equivalence in type theory|equivalence]] one may simply take the reduced suspension to be the plain suspension equipped with any one of its terms as basepoint (e.g, [UFP13, Rem. 8.6.3](#UFP13)).

However, in practice it is sometimes convenient to use a different (even if equivalent) construction of the reduced suspension:

## Definition

### As smash product with the circle type

For a given [[type]] $X$, we follow the conventions of the [[reduced suspension]] article and write $\mathrm{S} X$ for the plain [[suspension type]] of $X$ and $\Sigma X$ for the reduced suspension type of $A$.

The reduced suspension type of a [[pointed type]] $(X,x)$ with [[basepoint]] [[term]]  $x \colon X$ may be defined as the [[smash product type]] of the [[pointed type|pointed]] [[circle type]] $(S^1, s)$ with $X$:

$$
  \Sigma X
    \;\coloneqq\;
  (S^1,s) \wedge (X,x)
  \,.
$$

### In cohesive homotopy type theory

In [[cohesive homotopy type theory]], let $(X, x)$ be a pointed cohesive type, and let $\mathbf{\Sigma} X$ be the [[reduced suspension]] of $X$, as a cohesive type. Then the reduced suspension type of $X$, as a [[discrete space|discrete]] [[homotopy type]], is the [[shape modality|shape]] of the reduced suspension of $X$: $\Sigma X \coloneqq \esh(\mathbf{\Sigma} X)$.

In particular, if we take $(X, x)$ to be a [[pointed topological space]] in [[real-cohesive homotopy type theory]], this allows a construction of reduced suspension types directly from reduced suspensions in classical [[algebraic topology]].

## Properties

The reduced suspension type of the $n$-sphere type $S^n$ is [[equivalence of types|equivalent to]] the $(n + 1)$-sphere type $S^{n + 1}$

$$\Sigma S^n \simeq S^{n + 1}$$

## See also

* [[suspension object]]

  * [[suspension type]], **reduced suspension type**

  * [[suspension]], [[reduced suspension]]

## References

* {#UFP13} [[Univalent Foundations Project]], Rem. 8.6.3 in: *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]* (2013) [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)]

[[!redirects reduced suspension type]]
[[!redirects reduced suspension types]]


reducible representation

#Contents#
* table of contents
{:toc}

## Idea

A [[representation]] is _reducible_ if it is a [[direct sum]] of [[irreducible representations]].

## Related concepts

* [[reductive group]]

* [[reductive Lie algebra]]

[[!redirects reducible representations]]


reduction and lift of structure groups

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

#Contents#
* table of contents
{:toc}

## Idea

For $G \to K$ a [[monomorphism]] of groups, a $G$-structure on a $K$-[[principal bundle]] is a _reduction_ of the [[structure group]] from $K$ to $G$.

Alternatively, for $G \to K$ an [[epimorphism]] of groups, a $G$-structure on a $K$-[[principal bundle]] is a _lift_ of the structure group from $K$ to $G$.

A $G$-reduction of the [[frame bundle]] of a [[smooth manifold]] is called a _[[G-structure]]_.

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

As one passes to [[higher differential geometry]], the [[(epi, mono) factorization system]] dissolves into the infinite tower of [[(n-epi, n-mono) factorization systems]], and hence the distinction between reduction and lift of structure groups blurs. One may just consider generally for $G\to K$ a homomorphism of [[∞-groups]] the problem of factoring a [[modulating morphism]] $X\to \mathbf{B}K$ through this morphism, up to a chosen [[homotopy]].

=--

## Definition

We spell out three equivalent definitions.

Let $\mathbf{H}$ be the ambient [[(∞,1)-topos]], let $G,K \in Grp(\mathbf{H})$ be two [[∞-groups]] and let $\phi : G \to K$ be a homomorphism, hence $\mathbf{B}\phi : \mathbf{B}G \to \mathbf{B}K$ the morphism in $\mathbf{H}$ between their [[deloopings]]. Write

$$
  \array{
    K\sslash G &\to& \mathbf{B}G
    \\
    && \downarrow^{\mathrlap{\mathbf{B}\phi}}
    \\
    && \mathbf{B}K
  }
$$

for the corresponding [[fiber sequence]], with $K \sslash G$ the [[homotopy fiber]] of the given morphism. By the discussion at _[[∞-action]]_ this exhibits the canonical $K$-[[∞-action]] on the [[coset]] object $K\sslash G$.

Let furthermore $P \to X$ be a $K$-[[principal ∞-bundle]] in $\mathbf{H}$. By the discussion there this is modulated essentially uniquely by a [[cocycle]] morphism $k : X \to \mathbf{B}K$ such that there is a [[fiber sequence]]

$$
  \array{
    P &\to& X
    \\
    && \downarrow
    \\
    && \mathbf{B}K
  }
  \,.
$$

### Reduction of the cocycle

The reduction of the structure of the cocycle $k$ is a diagram

$$
  \array{
     X &&\stackrel{\sigma}{\to}&& \mathbf{B}G
     \\
     & {}_{\mathllap{k}}\searrow &\swArrow_{\tilde\sigma}& \swarrow
     \\
     && \mathbf{B}K
  }
$$

in $\mathbf{H}$, hence a morphism

$$
  \sigma : k \to \mathbf{B}\phi
$$

in the [[slice (∞,1)-topos]] $\mathbf{B}_{/\mathbf{B}K}$.

### Section of the associated coset-bundle

By the discussion at [[associated ∞-bundle]] such a diagram is equivalently a section

$$
  \sigma \in \Gamma_X(P \times_{K} K\sslash G)
$$

of the [[associated ∞-bundle|associated]] $K \sslash G$ [[fiber ∞-bundle]].

### Equivariant map to the coset

The above is the [[categorical semantics]] of what in the [[homotopy type theory]] [[internal language]] of $\mathbf{H}$ is given by the [[syntax]]

$$
  \vdash (\prod_{x : \mathbf{B}K} P \to K\sslash G) : Type
  \,.
$$

See the discussion at _[[∞-action]]_.

This expresses the fact that the reduction of the structure group along $\phi$ is equivalently a $K$-equivariant map $P \to K\sslash G$.

## Examples
 {#Examples}

* reduction of [[tangent bundle]] along [[orthogonal group]] inclusion $O(n) \hookrightarrow GL(n)$:  [[vielbein]], [[orthogonal structure]],

* reduction of [[tangent bundle]] along [[symplectic group]] inclusion $Sp(2n) \to GL(2n)$: [[almost symplectic structure]];

  subsequent lift to the [[metaplectic group]] $Mp(2n) \to Sp(2n)$: [[metaplectic structure]]

  induced lift over [[Lagrangian submanifolds]] to the [[metalinear group]] $Ml(n) \to GL(n)$: [[metalinear structure]];

* reduction of [[tangent bundle]] along inclusion of [[complex numbers|complex]] [[general linear group]] $GL(n, \mathbb{C}) \hookrightarrow GL(2n, \mathbb{R})$: [[almost complex structure]];

  further reduction to the [[unitary group]] $U(n) \hookrightarrow GL(n,\mathbb{C})$: [[almost Hermitian structure]];

* reduction of [[generalized tangent bundle]] along $U(n,n) \hookrightarrow O(2n,2n)$: [[generalized complex geometry]],

  * further reduction along $SU(n,n) \hookrightarrow O(2n,2n)$: [[generalized Calabi-Yau manifold]] ;

    * for $n = 3$: further reduction along $SU(3) \times SU(3) \hookrightarrow SU(3,3)$, [N=2 type II sugra compactification](exceptional+generalized+geometry#HigherSupersymmetry)

* reduction of [[generalized tangent bundle]] along $G_2 \times G_2 \hookrightarrow SO(7,7)$: [[G2-structure]];

* reduction of [[generalized tangent bundle]] along $O(n) \times O(n) \hookrightarrow O(n,n)$: [[generalized vielbein]], [[type II geometry]];

* reduction of [[exceptional tangent bundle]] along [[maximal compact subgroup]] of [[exceptional Lie group]] $H_n \hookrightarrow E_{n(n)}$: [[exceptional generalized geometry]]

* reduction of [[exceptional tangent bundle]] along $SU(7) \hookrightarrow E_{7(7)}$: [N=1 11d sugra compactification on ](exceptional+generalized+geometry#HigherSupersymmetry)

## References

In the generality of [[principal infinity-bundles]], reductions/lifts of structure groups are discused in section 4.3 of

* {#NSS} [[Thomas Nikolaus]], [[Urs Schreiber]], [[Danny Stevenson]], _[[schreiber:Principal ∞-bundles -- theory, presentations and applications|Principal ∞-bundles -- General theory]]_ ([arXiv:1207.0248](http://arxiv.org/abs/1207.0248))

[[!redirects lift of structure groups]]
[[!redirects lift of structure group]]
[[!redirects lift of the structure group]]

[[!redirects reduction of structure group]]
[[!redirects reduction of the structure group]]

[[!redirects extension of principal bundles]]
[[!redirects extension of a principal bundle]]

[[!redirects reduction of structure groups]]

[[!redirects reduction of a structure group]]
[[!redirects reductions of a structure group]]


reduction modality

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohesion
+--{: .hide}
[[!include cohesive infinity-toposes - contents]]
=--
#### Modalities, Closure and Reflection
+-- {: .hide}
[[!include modalities - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In a context of [[synthetic differential geometry]]/[[differential cohesion]] the _reduction modality_ characterizes [[reduced objects]]. It forms itself the [[left adjoint]] in an [[adjoint modality]] with the [[infinitesimal shape modality]].

## Definition

A context of [[differential cohesion]] is determined by the existence of an [[adjoint triple]] of  [[modalities]]

$$
  \Re \dashv \Im \dashv \&
  \,,
$$

where $\Re$ and $\&$ are [[idempotent monad|idempotent]] [[comonads]] and $\Im$ is an [[idempotent monad]], furthermore $\Re$ preserves finite products.

Here $\Re$ is the **reduction modality**. The [[reflective subcategory]] that it defines is that of [[reduced objects]].

## Related concepts

[[!include cohesion - table]]

[[!redirects reduction modalities]]

[[!redirects reduction]]
[[!redirects reductions]]


reductionism

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

\tableofcontents

## Idea

Reductionism is a philosophical position which states that all of that field can be reduced to a common foundation.

This contrasts with [[emergentism]], which says that the entire field cannot be reduced to a common foundation.

Note that this is different from the [[pluralism]]/[[monism]] distinction: monist reductionists believe there is only one way to reduce a field down to a common foundation, while pluralist reductionists would believe that there are multiple ways to reduce a field down to a common foundation.

## Reductionism in the foundations of mathematics

Reductionism is inherent in the foundations of mathematics, because it posits that everything in mathematics can be reduced down to the foundations of mathematics.

## See also

* [[philosophy]]

* **reductionism**, [[emergentism]]

* [[pluralism]], [[monism]]

## References

On how ontological reductionism is problematic in the philosophy of science:

* Daniel C. Dennett (1987) “True Believers." The Intentional Stance. Cambridge, MA: MIT Press.

* Jimmy Aames (2019) Patternhood and Generality: A Peircean Approach to Emergence. European Journal of Pragmatism and American Philosophy, XI (2). ([pdf](https://philsci-archive.pitt.edu/16878/1/Patternhood%20and%20Generality.pdf), [web](https://philsci-archive.pitt.edu/16878/))

## External links

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


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

+--{: .query}
The SVG figures are still not displaying completely properly. The text labels seem to be almost all left aligned, while some of them should be right or center aligned. In particular, the (?) in the last figure should be centered. --Igor
=--

# Reductions, deformations, and resolutions in the service of physics
* table of contents
{:toc}

## Outline ##

These aim of these notes is to summarize the various mathematical
structures and constructions employed in physics to build models of
[[classical field theory|classical]] and [[quantum field theories]]. To avoid the (yet non-existent)
rigorous construction of non-linear (interacting) field theories,
non-linearities are treated as formal perturbations. Theories with [[gauge theory|gauge invariance]] ([[first class constraints]]) are treated within the [[BV-BRST formalism]].

### Reductions, deformations, resolutions ###

There are multiple operations on classical and quantum field theories
that produce new ones. I roughly classify them into three kinds. These
terms are not meant to be rigorously defined or taken literally. They
mostly reflect how these operations are viewed in the physics
literature.

* **reductions**
:	These are easy/natural operations that are essentially uniquely
:	defined. The uniqueness may not always be mathematically precise, but
:	precision can be achieved by considering extra physical input.
:	Suggestive examples are _differentiation_ and _computing cohomology_.
* **deformations**
:	These are generally difficult operations inverse to a reduction, that
:	are themselves essentially uniquely defined. Uniqueness is considered
:	in the same sense as above. A suggestive examples is _integration_.
* **resolutions**
:	These are operations inverse to a reduction that are not uniquely
:	defined and involve making significant choices in their application.
:	Suggestive examples are _solving underdetermined equations_ and
:	_choosing a resolution_ in homological algebra.

The relevant examples that will appear in these notes are the following.
A solid arrow represents a reduction, while a dashed arrow represents
the inverse deformation or resolution, in the senses described above.

+--{: style="text-align:center"}
[[!include reductions deformations resolutions in physics > cq-arr]]
=--

The transition to the _[[classical limit]]_ corresponds to taking the
limit $\hbar\to0$, which is a parameter explicitly appearing in all
physically relevant quantum systems. The inverse operation is
_quantization_, which corresponds to the mathematical field
technically called [[deformation quantization]].

+--{: style="text-align:center"}
[[!include reductions deformations resolutions in physics > ln-arr]]
=--

Explicit solutions or other kinds of information is readily available
for linear field theories. Thus, often, the first step to understanding a
non-linear field theory is to consider its _linearization_. The
inverse operation of deforming the linear theory with a _non-linear
perturbation_ is often very difficult and thus considered in the physics
literature mostly at the level of formal power series in the
perturbation parameter.

+--{: style="text-align:center"}
[[!include reductions deformations resolutions in physics > pg-arr]]
=--

Many physical theories are difficult to describe without introducing
extra/non-physical/redundant so-called [[gauge theory|gauge degrees of freedom]] and at
the same time so-called [[gauge symmetry]] acting on them. However, given such a description, the physical theory must be recovered from the
_quotient by gauge transformations_. The description involving
gauge degrees of freedom is highly non-unique, but a particularly
convenient (though still not unique) one is provided by the
_BV-BRST_ resolution.

### Summary of construction ###

The operations described above can be used as the axes of a
three-dimensional space, inhabited by what can be called the
resolution-deformation-reduction cube of [[quantization]]. Note that [[gauge fixing]] can be considered as one of the (non-uniquely defined) steps of the BV-BRST construction.

+--{: style="text-align:center"}
[[!include reductions deformations resolutions in physics > bv-axes]]
=--

One corner of the cube corresponds to some classical, non-linear,
physical theory. The goal of quantization is to jump to the corner with
the quantum version of this theory. This is difficult to do directly
because of the non-linearity and when gauge degrees of freedom are
present. The usual procedure is then to construct the remaining
auxiliary vertices of the cube and to take the indirect path, indicated
in pale blue. The meaning of the solid and dashed lines is the same as
above.

+--{: style="text-align:center"}
[[!include reductions deformations resolutions in physics > bv-diag]]
=--

### Scope and contents ###

* Classical and quantum mechanics
:	* algebraic structure of [[classical mechanics]]
:	* algebraic structure of [[quantum mechanics]]
:	* [[classical limit]] and ([[deformation quantization|deformation]]) [[quantization]]
* Classical and quantum field theory
:	* geometry of [[space-time]]
:	* algebraic and geometric structure of [[classical field theory]]
:	* algebraic and geometric structure of [[quantum field theory]]
* Construction of field theories (simplified)
:	* space-time, field content, dynamics
:	* [[phase space|(covariant) phase space]]
:	* algebra of [[observables]]
:	* (deformation) [[quantization]]
* Examples of field theories
:	* [[scalar field]] (Klein-Gordon)
:	* vector field ([[electromagnetism|Maxwell]], [[Yang-Mills theory|Yang-Mills]])
:	* [[gravity]] (general relatvity)
:	* [[spinor]] field (Dirac)
* [[fermion|Fermi fields]] and [[supergeometry]]
* [[gauge invariance|Gauge invariance]]/degeneracy
:	* [[Noether's theorem|Nöther's second theorem]]
:	* algebra and geometry of [[gauge transformation]]s
:	* physical theory from quotient by gauge
* [[BV-BRST formalism|BV-BRST construction]] of gauge theories
:	* BV-BRST (Koszul-Tate) resolution of physical theory
:	* [[gauge fixing]], well-posedness, [[Poisson algebra|Poisson structure]]
:	* [[cohomology]]
* BV-BRST, (deformation) quantization and [[quantum anomaly|gauge anomalies]]


reductions deformations resolutions in physics > bv-axes

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