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

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

The collection of [[functors]] from ([[pointed topological space|pointed]]) [[topological spaces]] to [[abelian groups]] which assign [[cohomology groups]] of [[ordinary cohomology]] (e.g. [[singular cohomology]]) may be [[axiom|axiomatized]] by a small set of natural conditions, called the _Eilenberg-Steenrod axioms_ ([Eilenberg-Steenrod 52, I.3](#EilenbergSteenrod52)), see [below](#TheEilenbergSteenrodAxioms). One of these conditions, the "dimension axiom" ([Eilenberg-Steenrod 52, I.3 Axiom 7](#EilenbergSteenrod52)) says that the (co)homology groups assigned to the point are concentrated in degree 0. The class of functors obtained by discarding this "dimension axiom" came to be known as _generalized (co)homology theories_ ([Whitehead 62](#Whitehead62)) or _extraordinary (co)homology theories_.

Examples include [[topological K-theory]] ([Atiyah-Hirzebruch 61, 1.8](https://ncatlab.org/nlab/show/topological+K-theory#AtiyahHirzebruch61)), [[elliptic cohomology]] and [[cobordism cohomology theory]]. Dually one speaks of _[[generalized homology]]_.

Notice that, while the terminology "generalized cohomology" is standard in [[algebraic topology]] with an eye towards [[stable homotopy theory]], it is somewhat unfortunate in that there are various _other_ and _further_ generalizations of the axioms that all still deserve to be and are called "[[cohomology]]". For instance dropping the [[suspension isomorphism|suspension]] axiom leads to [[nonabelian cohomology]] and dropping the "homotopy axiom" (and taking the domain spaces to be [[smooth manifolds]]) leads to the further generality of [[differential cohomology]].
This entry here is concerned with the generalization obtained from the Eilenberg-Steenrod axioms by _just_ discarding the dimension axiom. For lack of a better term, we say "generalized (Eilenberg-Steenrod) cohomology" here.

In ([Whitehead 62](#Whitehead62)) it was observed that every [[spectrum]] induces a generalized homology theory.  The _[[Brown representability theorem]]_ ([Brown 62](#Brown62)) asserts that every generalized (co)homology arises this way, being [[representable functor|represented]] by [[mapping spectra]] into/[[smash product]] with a [[spectrum]].  But beware that the cohomology theory represented by a spectrum in general contains strictly less information than the spectrum, due to the existence of "[[phantom maps]]".

On the other hand, if one refines the concept of a generalized homology theory from taking values in [[graded abelian]] groups to taking values in [[homotopy types]] then it does become equivalent to the concept of _spectrum_, this is the statement at _[excisive functor -- Examples -- Spectrum objects](excisive+%28%E2%88%9E%2C1%29-functor#SpectrumObjects)_.

This means that from a perspective of [[higher category theory]], generalized Eilenberg-Steenrod cohomology is the intrinsic [[cohomology]] of the [[(∞,1)-category of spectra]], or better: [[twisted cohomology|twisted]] generalized Eilenberg-Steenrod cohomology is the intrinsic cohomology of the [[tangent (∞,1)-topos]] of [[parameterized spectra]].

+-- {: .standout}

Generalized Eilenberg-Steenrod cohomology is
[[cohomology]] $E(X)= H(X,E)$ with [[coefficients]] $E$ a [[spectrum object]].

=--

## Definition
 {#TheEilenbergSteenrodAxioms}

This sections states the classical formulation of the Eilenberg-Steenrod axioms due to ([Eilenberg-Steenrod 52, I.3](#EilenbergSteenrod52)) in terms of concepts from classical [[algebraic topology]], such as [[CW-pairs]] and [[mapping cones]].

More abstractly, via the [[classical model structure on topological spaces]], these structures are seen to serve as presentations for certain [[homotopy pushouts]]. In terms of "abstract homotopy theory" ([[(infinity,1)-category theory]]) one obtains a more streamlined formulation, which we turn to [below](#AbstractFormulation).

$\,$

There are two versions of the statement of the axioms:

* _[Reduced cohomology](#ReducedCohomology)_;

* _[Unreduced cohomology](#UnreducedCohomology)_.

There are functors taking any reduced cohomology theory to an unreduced one, and vice versa. When some fine detail in the axioms is suitably set up, then this establishes an [[equivalence of categories|equivalence]] between reduced and unreduced generalized cohomology:

* _[Relation between reduced and unreduced cohomology](#RelationBetweenReducedAndUnreduced)_

The fine detail in the axioms that makes this work is such as to ensure that a cohomology theory is a functor on the [[opposite category|opposite]] of the (pointed/pairwise) [[classical homotopy category]]. Since this has different presentations, there are corresponding different versions of suitable axioms:

1. On the one hand, $Ho(Top_{Quillen})$ may be presented by topological spaces homeomorphic to [[CW-complexes]] and with [[homotopy equivalence]]-classes of continuous functions between them, and accordingly a generalized cohomology theory may be taken to be a funtor on (pointed/pairs of) CW-complexes invariant under [[homotopy equivalence]].

1. On the other hand, $Ho(Top_{Quillen})$ may be presented by all topological spaces with [[weak homotopy equivalences]] [[localization|inverted]], and accordingly a generalized cohomology theory may be taken to be a functor on all (pointed/pairs of) topological spaces that sends weak homotopy equivalences to isomorphisms.

Notice however that "[[classical homotopy category]]" is already ambiguous. Pre Quillen this was the category of _all_ topological spaces with homotopy equivalence classes of maps between them, and often generalized cohomology functors are defined on this larger category and only restricted to CW-complexes or required to preserve weak homotopy equivalences when need be (e.g. [Switzer 75, p.117](#Switzer75)), such as for establishing the equivalence between reduced and unreduced theories.

Moreover, historically, these conditions have been decomposed in several numbers of ways. Notably ([Eilenberg-Steenrod 52](#EilenbergSteenrod52)) originally listed 7 axioms for unreduced cohomology, more than typically counted today, but their axioms 1 and 2 jointly just said that we have a functor on topological spaces, axiom 3 was the condition for the connecting homomorphism to be a [[natural transformation]], conditions which later ([Switzer 75, p. 99,100](#Switzer75)) were absorbed in the underlying structure.

Finally, following the historical development it is common to state the exactness properties of cohomology functors in terms of [[mapping cone]] constructions. These are models for [[homotopy cofibers]], but in general only when some technical conditions are met, such that the underlying topological spaces are CW-complexes.

For these reasons, in the following we stick to two points of views: where we discuss cohomology theories as functors on topological spaces we restrict attention to those homeomorphic to CW-complexes. In a second description we speak fully abstractly about functors on the homotopy category of a given model category of $\infty$-category.

### Reduced cohomology
 {#ReducedCohomology}

Throughout, write [[Top]]${}_{CW}$ for the category of [[topological spaces]] [[homeomorphism|homeomorphic]] to [[CW-complexes]]. Write $Top^{\ast/}_{CW}$ for the corresponding category of [[pointed topological spaces]].

Recall ([here](final+functor#FiberProductsInASliceCategory)) that [[colimits]] in $Top^{\ast/}$ are computed as colimits in [[Top]] ([here](Top#UniversalConstructions)) after adjoining the base point and its inclusion maps to the given diagram.

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

The [[coproduct]] in [[pointed topological spaces]] is the _[[wedge sum]]_, denoted $\vee_{i \in I} X_i$.

=--

Write

$$
  \Sigma \coloneqq S^1 \wedge (-)
  \;\colon\;
  Top^{\ast/}_{CW} \longrightarrow Top^{\ast/}_{CW}
$$

for the [[reduced suspension]] functor.

Write $Ab^{\mathbb{Z}}$ for the category of integer-[[graded abelian groups]].

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

A **reduced [[cohomology theory]]** is a [[functor]] ("[[pullback in cohomology]]")

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

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

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

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

* {#WedgeAxiom} **([[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](#AguilarGitlerPrieto02))

We may rephrase this more intrinsically and more generally:

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

Let $\mathcal{C}$ be an [[(∞,1)-category]] with [[(∞,1)-pushouts]], and with a [[zero object]] $0 \in \mathcal{C}$. Write $\Sigma \colon \mathcal{C} \to \mathcal{C}\colon X\mapsto 0 \underset{X}{\sqcup} 0$ for the corresponding [[suspension]] [[(∞,1)-functor]].

A **reduced generalized cohomology theory** on $\mathcal{C}$ is

1. a [[functor]]

   $$
     E^\bullet \;\colon \; Ho(\mathcal{C})^{op} \longrightarrow Ab^{\mathbb{Z}}
   $$

   (from the [[opposite category|opposite]] of the [[homotopy category of an (infinity,1)-category|homotopy category]] of $\mathcal{C}$ into $\mathbb{Z}$-[[graded abelian groups]]);

1. a [[natural isomorphisms]] ("[[suspension isomorphisms]]") of degree +1

   $$
     \sigma \; \colon \; H^\bullet \longrightarrow H^{\bullet+1} \circ \Sigma
   $$

such that $H^\bullet$

1. takes small [[coproducts]] to [[products]];

1. takes [[homotopy cofiber sequences]] to [[exact sequences]].

=--

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

Given a generalized cohomology theory $(H^\bullet,\sigma)$ on some $\mathcal{C}$ as in def. \ref{GeneralizedCohomologyOnGeneralInfinityCategory}, and given a [[homotopy cofiber sequence]] in $\mathcal{C}$

$$
  X \stackrel{f}{\longrightarrow} Y \stackrel{g}{\longrightarrow} Z
  \stackrel{coker(g)}{\longrightarrow}
  \Sigma X
  \,,
$$

then the corresponding **[[connecting homomorphism]]** is the composite

$$
  \partial
    \;\colon\;
  \widetilde  E^\bullet(X)
   \stackrel{\sigma}{\longrightarrow}
  \widetilde  E^{\bullet+1}(\Sigma X)
   \stackrel{coker(g)^\ast}{\longrightarrow}
  \widetilde E^{\bullet+1}(Z)
  \,.
$$

=--

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

The [[connecting homomorphisms]] of def. \ref{ConnectinHomomorphismForCohomologyTheoryOnInfinityCategory} are part of [[long exact sequences]]

$$
  \cdots
   \stackrel{\partial}{\longrightarrow}
  \widetilde E^{\bullet}(Z)
    \longrightarrow
  \widetilde E^\bullet(Y)
    \longrightarrow
  \widetilde  E^\bullet(X)
    \stackrel{\partial}{\longrightarrow}
  \widetilde  E^{\bullet+1}(Z)
   \to \cdots
  \,.
$$

=--

See at _[[long exact sequence in generalized cohomology]]_.

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

By the defining exactness of $E^\bullet$, def. \ref{GeneralizedCohomologyOnGeneralInfinityCategory}, and the way this appears in def. \ref{ConnectinHomomorphismForCohomologyTheoryOnInfinityCategory}, using that $\sigma$ is by definition an isomorphism.

=--

### Unreduced cohomology
 {#UnreducedCohomology}

In the following a _pair_ $(X,A)$ refers to a [[subspace]] inclusion of [[topological spaces]] ([[CW-complexes]])  $A \hookrightarrow X$.  Whenever only one space is mentioned, the subspace is assumed to be the [[empty set]] $(X, \emptyset)$. Write $Top_{CW}^{\hookrightarrow}$ for the category of such pairs (the [[full subcategory]] of the [[arrow category]] of $Top_{CW}$ on the inclusions). We identify $Top_{CW} \hookrightarrow Top_{CW}^{\hookrightarrow}$ by $X \mapsto (X,\emptyset)$.

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

A _[[cohomology theory]]_ (unreduced, [[relative cohomology|relative]]) is a [[functor]]

$$
  E^\bullet : (Top_{CW}^{\hookrightarrow})^{op} \to Ab^{\mathbb{Z}}
$$

to the category of $\mathbb{Z}$-[[graded abelian groups]], as well as a [[natural transformation]] of degree +1, to be called the **[[connecting homomorphism]]**, of the form

$$
  \delta_{(X,A)}
    \;\colon\;
  E^\bullet(A, \emptyset) \to E^{\bullet + 1}(X, A)
  \,.
$$

such that:

1. **(homotopy invariance)** For $f \colon (X_1,A_1) \to (X_2,A_2)$ a [[homotopy equivalence]] of pairs, then

   $$
     E^\bullet(f)
      \;\colon\;
     E^\bullet(X_2,A_2) \stackrel{\simeq}{\longrightarrow} E^\bullet(X_1,A_1)
   $$

   is an [[isomorphism]];

1. {#ExactnessUnreduced} **(exactness)** For $A \hookrightarrow X$ the induced sequence

   $$
     \cdots
       \to
     E^n(X, A)
       \longrightarrow
     E^n(X)
       \longrightarrow
     E^n(A)
       \stackrel{\delta}{\longrightarrow}
     E^{n+1}(X, A)
       \to
     \cdots
   $$

   is a [[long exact sequence]] of [[abelian groups]].

1. {#excision} **([[excision]])** For $U \hookrightarrow A \hookrightarrow X$ such that $\overline{U} \subset Int(A)$, then the natural inclusion of the pair $i \colon (X-U, A-U) \hookrightarrow (X, A)$ induces an isomorphism

   $$
     E^\bullet(i)
      \;\colon\;
     E^n(X, A)
      \overset{\simeq}{\longrightarrow}
     E^n(X-U, A-U)
   $$

We say $E^\bullet$ is **additive** if it takes [[coproducts]] to [[products]]:

* **(additivity)** If $(X, A) = \coprod_i (X_i, A_i)$ is a [[coproduct]], then the canonical comparison morphism

  $$
    E^n(X, A) \overset{\simeq}{\longrightarrow} \prod_i E^n(X_i, A_i)
  $$

  is an [[isomorphism]] from the value on $(X,A)$ to the [[product]] of values on the summands.

We say $E^\bullet$ is **ordinary** if its value on the point is concentrated in degree 0

* **(Dimension)**: $E^{\bullet \neq 0}(\ast,\emptyset) = 0$.

A [[homomorphism]] of unreduced cohomology theories

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

is a [[natural transformation]] of the underlying functors that is compatible with the connecting homomorphisms, hence such that all these [[commuting square|squares commute]]:

$$
  \array{
     E^\bullet(A,\emptyset) &\overset{\eta_{(A,\emptyset)}}{\longrightarrow}& F^\bullet(A,\emptyset)
     \\
     {}^{\mathllap{\delta_E}}\downarrow && \downarrow^{\mathrlap{\delta_F}}
     \\
     E^{\bullet +1}(X,A)
       &\overset{\eta_{(X,A)}}{\longrightarrow}&
     F^{\bullet +1}(X,A)
  }
  \,.
$$

=--

e.g. ([AGP 02, def. 12.1.1 ](#AguilarGitlerPrieto02)).

+-- {: .num_defn #AlternativeFormulationOfExcisionAxiom}
###### Lemma

The excision axiom in def. \ref{GeneralizedCohomologyTheory} is equivalent to the following statement:

For all $A,B \hookrightarrow X$ with $X = Int(A) \cup Int(B)$, then the inclusion

$$
  i \colon (A, A \cap B) \longrightarrow (X,B)
$$

induces an isomorphism,

$$
  i^\ast
    \;\colon\;
  E^\bullet(X, B)
    \overset{\simeq}{\longrightarrow}
  E^\bullet(A, A \cap B)
$$

=--

(e.g [Switzer 75, 7.2](#Switzer75))

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

In one direction, suppose that $E^\bullet$ satisfies the original excision axiom. Given $A,B$ with $X = \Int(A) \cup Int(B)$, set $U \coloneqq X-A$ and observe that

$$
  \begin{aligned}
     \overline{U}
        & = \overline{X-A}
     \\ & = X- Int(A)
     \\ & \subset Int(B)
  \end{aligned}
$$

and that

$$
  (X-U, B-U) = (A, A \cap B)
  \,.
$$

Hence the excision axiom implies $  E^\bullet(X, B) \overset{\simeq}{\longrightarrow} E^\bullet(A, A \cap B)$.

Conversely, suppose $E^\bullet$ satisfies the alternative condition. Given $U \hookrightarrow A \hookrightarrow X$ with $\overline{U} \subset Int(A)$, observe that we have a cover

$$
  \begin{aligned}
          Int(X-U) \cup Int(A)
      & = (X - \overline{U}) \cup \Int(A)
   \\ & \supset (X - Int(A)) \cup Int(A)
   \\ & = X
  \end{aligned}
$$

and that

$$
  (X-U, (X-U) \cap A) = (X-U, A - U)
  \,.
$$

Hence

$$
  E^\bullet(X-U,A-U)
  \simeq
  E^\bullet(X-U, (X-U)\cap A)
  \simeq
  E^\bullet(X,A)
  \,.
$$

=--

The following lemma shows that the dependence in pairs of spaces in a generalized cohomology theory is really a stand-in for evaluation on [[homotopy cofibers]] of inclusions.

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

Let $E^\bullet$ be an cohomology theory, def. \ref{GeneralizedCohomologyTheory}, and let $A \hookrightarrow X$. Then there is an isomorphism

$$
  E^\bullet(X,A)
    \stackrel{\simeq}{\longrightarrow}
  E^\bullet(X \cup Cone(A), \ast)
$$

between the value of $E^\bullet$ on the pair $(X,A)$ and its value on the [[mapping cone]] of the inclusion, relative to a basepoint.

If moreover $A \hookrightarrow X$ is (the [[retract]] of) a [[relative cell complex]] inclusion, then also the morphism in cohomology induced from the [[quotient]] map $p \;\colon\; (X,A)\longrightarrow (X/A, \ast)$ is an [[isomorphism]]:

$$
  E^\bullet(p)
    \;\colon\;
  E^\bullet(X/A,\ast)
    \longrightarrow
  E^\bullet(X,A)
  \,.
$$

=--

(e.g [AGP 02, corollary 12.1.10](#AguilarGitlerPrieto02))

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

Consider $U \coloneqq (Cone(A)-A \times \{0\}) \hookrightarrow Cone(A)$, the cone on $A$ minus the base $A$. We have

$$
  ( X\cup Cone(A)-U, Cone(A)-U) \simeq (X,A)
$$

and hence the first isomorphism in the statement is given by the excision axiom followed by homotopy invariance (along the contraction of the cone to the point).

Next consider the quotient of the [[mapping cone]] of the inclusion:

$$
  ( X\cup Cone(A), Cone(A) ) \longrightarrow (X/A,\ast)
  \,.
$$

If $A \hookrightarrow X$ is a cofibration, then this is a [[homotopy equivalence]] since $Cone(A)$ is contractible and since by the dual [[factorization lemma]] $X \cup Cone(A)\to X/A$ is a weak homotopy equivalence, hence a homotopy equivalence on CW-complexes.

Hence now we get a composite isomorphism

$$
  E^\bullet(X/A,\ast)
    \overset{\simeq}{\longrightarrow}
  E^\bullet( X\cup Cone(A), Cone(A) )
    \overset{\simeq}{\longrightarrow}
  E^\bullet(X,A)
  \,.
$$

=--

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

As an important special case of : Let $(X,x)$ be a [[pointed topological space|pointed]] [[CW-complex]]. For $p\colon (Cone(X), X) \to (\Sigma X,\{x\})$ the quotient map from the reduced cone on $X$ to the [[reduced suspension]], then

$$
  E^\bullet(p)
  \;\colon\;
  E^\bullet(Cone(X),X)
   \overset{\simeq}{\longrightarrow}
  E^\bullet(\Sigma X, \{x\})
$$

is an isomorphism.

=--

+-- {: .num_prop #ExactSequenceOfATriple}
###### Proposition
**(exact sequence of a triple)**

For $E^\bullet$ an unreduced generalized cohomology theory, def. \ref{GeneralizedCohomologyTheory}, then every inclusion of two consecutive subspaces

$$
  Z \hookrightarrow Y \hookrightarrow X
$$

induces a [[long exact sequence]] of cohomology groups of the form

$$
  \cdots
   \to
  E^{q-1}(Y,Z)
    \stackrel{\bar \delta}{\longrightarrow}
  E^q(X,Y)
    \stackrel{}{\longrightarrow}
  E^q(X,Z)
    \stackrel{}{\longrightarrow}
  E^q(Y,Z)
    \to
  \cdots
$$

where

$$
  \bar \delta
    \;\colon \;
  E^{q-1}(Y,Z)
   \longrightarrow
  E^{q-1}(Y)
   \stackrel{\delta}{\longrightarrow}
  E^{q}(X,Y)
  \,.
$$

=--

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

Apply the [[braid lemma]] to the interlocking long exact sequences of the three pairs $(X,Y)$, $(X,Z)$, $(Y,Z)$. See [here](braid+lemma#ExactSequenceForTripleInGeneralizedHomology) for details.

The dual braid diagram for [[generalized homology]] is this:

<img src="http://www.ncatlab.org/nlab/files/BraidDiagramForHomologyOnTripled.jpg" width="500">

(graphics from [this Maths.SE comment](http://math.stackexchange.com/a/1180681/58526))

=--

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

The exact sequence of a triple in prop. \ref{ExactSequenceOfATriple} is what gives rise to the [[Cartan-Eilenberg spectral sequence]] for $E$-cohomology of a [[CW-complex]] $X$.

=--

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

For $(X,x)$ a [[pointed topological space]] and $Cone(X) = (X \wedge (I_+))/X$ its reduced [[cone]], the long exact sequence of the triple $(\{x\}, X, Cone(X))$, prop. \ref{ExactSequenceOfATriple},

$$
   0
   \simeq
   E^q(Cone(X), \{x\})
     \longrightarrow
   E^q(X,\{x\})
     \overset{\bar \delta}{\longrightarrow}
   E^{q+1}(Cone(X),X)
     \longrightarrow
   E^{q+1}(Cone(X), \{x\})
   \simeq 0
$$

exhibits the [[connecting homomorphism]] $\bar \delta$ here as an [[isomorphism]]

$$
   \bar \delta
     \;\colon\;
   E^q(X,\{x\})
     \overset{\simeq}{\longrightarrow}
   E^{q+1}(Cone(X),X)
  \,.
$$

This is the _[[suspension isomorphism]]_ extracted from the unreduced cohomology theory, see def. \ref{FromUnreducedToReducedCohomology} below.

=--

+-- {: .num_prop #MayerVietorisSequenceInGeneralizedCohomology}
###### Proposition
**([[Mayer-Vietoris sequence]])**

Given $E^\bullet$ an unreduced cohomology theory, def. \ref{GeneralizedCohomologyTheory}. Given a topological space covered by the [[interior]] of two spaces as $X = Int(A) \cup Int(B)$, then for each $C \subset A \cap B$ there is a [[long exact sequence]] of cohomology groups of the form

$$
  \cdots
  \to
  E^{n-1}(A \cap B , C)
  \overset{\bar \delta}{\longrightarrow}
  E^n(X,C)
  \longrightarrow
  E^n(A,C) \oplus E^n(B,C)
  \longrightarrow
  E^n(A \cap B, C)
  \to
  \cdots
  \,.
$$

=--

e.g. ([Switzer 75, theorem 7.19](#Switzer75), [Aguilar-Gitler-Prieto 02, theorem 12.1.22](#AguilarGitlerPrieto02), see also at *[[Brown-Gersten property]]*)

## Relation between reduced and unreduced cohomology
 {#RelationBetweenReducedAndUnreduced}

+-- {: .num_defn #FromUnreducedToReducedCohomology}
###### Definition
**(unreduced to reduced cohomology)**

Let $E^\bullet$ be an unreduced cohomology theory, def. \ref{GeneralizedCohomologyTheory}. Define a reduced cohomology theory, def. \ref{ReducedGeneralizedCohomology} $(\tilde E^\bullet, \sigma)$ as follows.

For $x \colon \ast \to X$ a [[pointed topological space]], set

$$
  \tilde E^\bullet(X,x) \coloneqq E^\bullet(X,\{x\})
  \,.
$$

This is clearly [[functor|functorial]]. Take the [[suspension isomorphism]] to be the composite

$$
  \sigma
   \;\colon\;
  \tilde E^{\bullet+1}(\Sigma X)
   =
  E^{\bullet+1}(\Sigma X, \{x\})
   \overset{E^\bullet(p)}{\longrightarrow}
  E^{\bullet+1}(Cone(X),X)
    \overset{\bar \delta^{-1}}{\longrightarrow}
  E^\bullet(X,\{x\})
   =
  \tilde E^{\bullet}(X)
$$

of the isomorphism $E^\bullet(p)$ from example \ref{GeneralizedCohomologyOnHomotopyQuotientMaps} and the [[inverse]] of the isomorphism $\bar \delta$ from example \ref{ExtractingSuspensionIsomorphismFromUnreducedCohomology}.

=--

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

The construction in def. \ref{FromUnreducedToReducedCohomology} indeed gives a reduced cohomology theory.

=--

(e.g. [Switzer 75, 7.34](#Switzer75))

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

We need to check the exactness axiom given any $A\hookrightarrow X$. By lemma \ref{EvaluationOfCohomologyTheoryOnGoodPairIsEvaluationOnQuotient} we have an isomorphism

$$
  \tilde E^\bullet(X \cup Cone(A))
  =
  E^\bullet(X \cup Cone(A), \{\ast\})
    \overset{\simeq}{\longrightarrow}
  E^\bullet(X,A)
  \,.
$$

Unwinding the constructions shows that this makes the following [[commuting diagram|diagram commute]]:

$$
  \array{
     \tilde E^\bullet(X\cup Cone(A))
        &\overset{\simeq}{\longrightarrow}&
     E^\bullet(X,A)
     \\
     \downarrow && \downarrow
     \\
     \tilde E^\bullet(X) &=& E^\bullet(X,\{x\})
     \\
     \downarrow && \downarrow
     \\
     \tilde E^\bullet(A) &=& E^\bullet(A,\{a\})
  }
  \,,
$$

where the vertical sequence on the right is exact by prop. \ref{ExactSequenceOfATriple}. Hence the left vertical sequence is exact.

=--

+-- {: .num_defn #ReducedToUnreducedGeneralizedCohomology}
###### Definition
**(reduced to unreduced cohomology)**

Let $(\tilde E^\bullet, \sigma)$ be a reduced cohomology theory, def. \ref{ReducedGeneralizedCohomology}. Define an unreduced cohomolog theory $E^\bullet$, def. \ref{GeneralizedCohomologyTheory}, by

$$
  E^\bullet(X,A)
  \coloneqq
  \tilde E^\bullet( X_+ \cup Cone(A_+))
$$

and let the connecting homomorphism be as in def. \ref{ConnectinHomomorphismForCohomologyTheoryOnInfinityCategory}.

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

The construction in def. \ref{ReducedToUnreducedGeneralizedCohomology} indeed yields an unreduced cohomology theory.

=--

e.g. ([Switzer 75, 7.35](#Switzer75))

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

Exactness holds by prop. \ref{LongExactSequenceOfACohomologyTheoryOnAnInfinityCategory}. For excision, it is sufficient to consider the alternative formulation of lemma \ref{AlternativeFormulationOfExcisionAxiom}.  For CW-inclusions, this follows immediately with lemma \ref{EvaluationOfCohomologyTheoryOnGoodPairIsEvaluationOnQuotient}.

=--

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

The constructions of def. \ref{ReducedToUnreducedGeneralizedCohomology} and def. \ref{FromUnreducedToReducedCohomology} constitute a pair of [[functors]] between then [[categories]] of reduced cohomology theories, def. \ref{ReducedGeneralizedCohomology} and unreduced cohomology theories, def. \ref{GeneralizedCohomologyTheory} which exhbit an [[equivalence of categories]].

=--

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

(...careful with checking the respect for suspension iso and connecting homomorphism..)

To see that there are [[natural isomorphisms]] relating the two composites of these two functors to the identity:

One composite is

$$
  \begin{aligned}
    E^\bullet
    & \mapsto
    (\tilde E^\bullet \colon (X,x) \mapsto E^\bullet(X,\{x\}))
    \\
    & \mapsto
    ((E')^\bullet \colon (X,A) \mapsto E^\bullet( X_+ \cup Cone(A_+) ), \ast)
  \end{aligned}
  \,,
$$

where on the right we have, from the construction, the reduced mapping cone of the original inclusion $A \hookrightarrow X$ with a base point adjoined. That however is isomorphic to the unreduced mapping cone of the original inclusion. With this the natural isomorphism is given by lemma \ref{EvaluationOfCohomologyTheoryOnGoodPairIsEvaluationOnQuotient}.

The other composite is

$$
  \begin{aligned}
    \tilde E^\bullet
     & \mapsto
    (E^\bullet \colon (X,A) \mapsto \tilde E^\bullet(X_+ \cup Cone(A_+)))
    \\
    & \mapsto
    ((\tilde E')^\bullet \colon X \mapsto \tilde E^\bullet(X_+ \cup Cone(*_+)))
  \end{aligned}
$$

where on the right we have the reduced mapping cone of the point inclusion with a point adoined. As before, this is isomorphic to the unreduced mapping cone of the point inclusion. That finally is clearly homotopy equivalent to $X$, and so now the natural isomorphism follows with homotopy invariance.

=--

Finally we record the following basic relation between reduced and unreduced cohomology:

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

Let $E^\bullet$ be an unreduced cohomology theory, and $\tilde E^\bullet$ its reduced cohomology theory from def. \ref{FromUnreducedToReducedCohomology}. For $(X,\ast)$ a pointed topological space, then there is an identification

$$
  E^\bullet(X) \simeq \tilde E^\bullet(X) \oplus E^\bullet(\ast)
$$

of the unreduced cohomology of $X$ with the [[direct sum]] of the reduced cohomology of $X$ and the unreduced cohomology of the base point.

=--

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

The pair $\ast \hookrightarrow X$ induces the sequence

$$
  \cdots
   \to
  E^{\bullet-1}(\ast)
    \stackrel{\delta}{\longrightarrow}
  \tilde E^\bullet(X)
    \stackrel{}{\longrightarrow}
  E^\bullet(X)
    \stackrel{}{\longrightarrow}
  E^\bullet(\ast)
    \stackrel{\delta}{\longrightarrow}
  \tilde E^{\bullet+1}(X)
    \to
  \cdots
$$

which by the exactness clause in def. \ref{GeneralizedCohomologyTheory} is [[exact sequence|exact]].

Now since the composite $\ast \to X \to \ast$ is the identity, the morphism
$E^\bullet(X) \to E^\bullet(\ast)$ has a [[section]] and so is in particular an [[epimorphism]]. Therefore, by exactness, the [[connecting homomorphism]] vanishes, $\delta = 0$ and we have a [[short exact sequence]]

$$
  0
    \to
  \tilde E^\bullet(X)
    \stackrel{}{\longrightarrow}
  E^\bullet(X)
    \stackrel{}{\longrightarrow}
  E^\bullet(\ast)
    \to
  0
$$

with the right map an epimorphism. Hence this is a [[split exact sequence]] and the statement follows.

=--

## Brown functoriality

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

Given a generalized cohomology functor $E^\bullet \colon Ho(\mathcal{C})^{op}\to Ab^{\mathbb{Z}}$, def. \ref{GeneralizedCohomologyOnGeneralInfinityCategory}, its underlying [[Set]]-valued functors $H^n \colon Ho(\mathcal{C})^{op}\to Ab\to Set$ are [[Brown functors]], def. \ref{BrownFunctorOnInfinityCategory}.

=--

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

The first condition on a [[Brown functor]] holds by definition of $H^\bullet$. For the second condition, given a homotopy pushout square

$$
  \array{
    X_1 &\stackrel{f_1}{\longrightarrow}& Y_1
    \\
    \downarrow^{} && \downarrow
    \\
    X_2 &\stackrel{f_2}{\longrightarrow}& Y_2
  }
$$

in $\mathcal{C}$, consider the induced morphism of the [[long exact sequences]] given by prop. \ref{LongExactSequenceOfACohomologyTheoryOnAnInfinityCategory}

$$
  \array{
    H^\bullet(coker(f_2))
      &\longrightarrow&
    H^\bullet(Y_2)
      &\stackrel{f^\ast_2}{\longrightarrow}&
    H^\bullet(X_2)
      &\stackrel{}{\longrightarrow}&
    H^{\bullet+1}(\Sigma coker(f_2))
    \\
    {}^{\mathllap{\simeq}}\downarrow && \downarrow && \downarrow && \downarrow^{\mathrlap{\simeq}}
    \\
    H^\bullet(coker(f_1))
      &\longrightarrow&
    H^\bullet(Y_1)
      &\stackrel{f^\ast_1}{\longrightarrow}&
    H^\bullet(X_1)
      &\stackrel{}{\longrightarrow}&
    H^{\bullet+1}(\Sigma coker(f_1))
  }
$$

Here the outer vertical morphisms are [[isomorphisms]], as shown, due to the [[pasting law]] (see also at _[fiberwise recognition of stable homotopy pushouts](homotopy+pullback#FiberwiseRecognitionInStableCase)_).

This means that the [[four lemma]] applies to this diagram. Inspection shows that this implies the claim.

=--

## Examples

* [[ordinary cohomology]]: [[Eilenberg-Mac Lane spectrum]]

* [[K-theory]]: [[K-theory spectrum]]
* [[tmf]]

* [[cobordism cohomology theory]]

* [[Morava K-theory]], [[integral Morava K-theory]]

* [[Morava E-theory]]

* [[stable cohomotopy]]

## Properties

### Expression by ordinary cohomology via Atiyah-Hirzebruch spectral sequence

The [[Atiyah-Hirzebruch spectral sequence]] serves to express generalized cohomology $E^\bullet$ in terms of ordinary cohomology with coefficients in $E^\bullet(\ast)$.

### Whitehead theorem
 {#WhiteheadTheorem}

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

Let $\phi \colon E \longrightarrow F$ be a morphism of reduced generalized (co-)homology functors, def. \ref{ReducedGeneralizedCohomology} (a [[natural transformation]]) such that its component

$$
  \phi(S^0) \colon E(S^0) \longrightarrow F(S^0)
$$

on the [[0-sphere]] is an [[isomorphism]]. Then $\phi(X)\colon E(X)\to F(X)$ is an [[isomorphism]] for $X$ any [[CW-complex]] with a [[finite number]] of cells. If both $E$ and $F$ satisfy the [[wedge axiom]], then $\phi(X)$ is an isomorphism for $X$ any [[CW-complex]], not necessarily finite.

=--

For $E$ and $F$ [[ordinary cohomology]]/[[ordinary homology]] functors a proof of this is in ([Eilenberg-Steenrod 52, section III.10](#EilenbergSteenrod52)). From this the general statement follows (e.g. [Kochman 96, theorem 3.4.3, corollary 4.2.8](#Kochman96)) via the [[natural transformation|naturality]] of the [[Atiyah-Hirzebruch spectral sequence]] (the classical result gives that $\phi$ induces an isomorphism between the second pages of the AHSSs for $E$ and $F$). A complete proof of the general result is also given as ([Switzer 75, theorem 7.55, theorem 7.67](#Switzer75))

## Related concepts

* [[Brown representability theorem]]

* [[generalized homology theory]]

* [[Kronecker pairing]]

* [[bivariant cohomology theory]]

* [[multiplicative cohomology theory]]

* [[Chern-Dold character]]

[[!include twisted generalized cohomology in linear homotopy type theory -- table]]

## References

The original article on the Eilenberg–Steenrod axioms:

* [[Samuel Eilenberg]], [[Norman E. Steenrod]], _Axiomatic Approach to Homology Theory_, Proceedings of the National Academy of Sciences **31** 4 (1945) 117–120  [[doi:10.1073/pnas.31.4.117](http://dx.doi.org/10.1073/pnas.31.4.117)]

Further development and an expository account:

* {#EilenbergSteenrod52} [[Samuel Eilenberg]], [[Norman Steenrod]], _Foundations of algebraic topology_, Princeton 1952 ([pdf](http://www.maths.ed.ac.uk/~aar/papers/eilestee.pdf), [ISBN:9780691653297](https://press.princeton.edu/books/hardcover/9780691653297/foundations-of-algebraic-topology))

The concept of generalized homology obtained by discarding the dimension axiom and the observation that every [[spectrum]] induces an example is due to

* {#Whitehead62} [[George Whitehead]], _Generalized homology theories_, Transactions of the American Mathematical Society, 102 (1962) 227-283 ([pdf](http://www.ams.org/journals/tran/1962-102-02/S0002-9947-1962-0137117-6/S0002-9947-1962-0137117-6.pdf), [jstor:1993676](https://www.jstor.org/stable/1993676))

The proof that every generalized (co)homology theory arises this way ([[Brown representability theorem]]) is due to

* {#Brown62} [[Edgar Brown]], _Cohomology theories_, Annals of Mathematics, Second Series 75: 467–484 (1962) ([jstor:1970209](https://www.jstor.org/stable/1970209))

* {#Brown65} [[Edgar Brown]], _Abstract homotopy theory_, Trans. AMS 119 no. 1 (1965) ([doi:10.1090/S0002-9947-1965-0182970-6](https://doi.org/10.1090/S0002-9947-1965-0182970-6))

An early lecture note account is in

* {#Adams74} [[Frank Adams]], part III, sections 2 and 6 of _[[Stable homotopy and generalised homology]]_, 1974

Textbook accounts:

* {#Switzer75} [[Robert Switzer]], chapter 7 (and 8-12) of _Algebraic Topology - Homotopy and Homology_, Die  Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.

* [[John Michael Boardman]], Section 3 of: _Stable Operations in Generalized Cohomology_ [[pdf](https://math.jhu.edu/~wsw/papers2/math/28a-boardman-stable.pdf), [[Boardman-StableOperations.pdf:file]]] in: [[Ioan Mackenzie James]] (ed.) _[[Handbook of Algebraic Topology]]_ Oxford 1995 ([doi:10.1016/B978-0-444-81779-2.X5000-7](https://doi.org/10.1016/B978-0-444-81779-2.X5000-7))

* {#Kochman96} [[Stanley Kochman]], section 3.4 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* {#May99} [[Peter May]] chapter 18 of _A Concise Course on Algebraic Topology_, Chicago Lecture Notes  1999 ([pdf](http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf))

* {#AguilarGitlerPrieto02} Marcelo Aguilar, [[Samuel Gitler]], Carlos Prieto, section 12.1 of _Algebraic topology from a homotopical viewpoint_, Springer (2002) ([toc pdf](http://tocs.ulb.tu-darmstadt.de/106999419.pdf))

* {#KonoTamaki02} [[Akira Kono]], [[Dai Tamaki]], _Generalized cohomology_, AMS 2002, esp. chapter 2 $[$[[GeneralizedCohomology.pdf:file]], [ISBN: 978-0-8218-3514-2](https://bookstore.ams.org/mmono-230)$]$

* [[Stefan Schwede]], chapter II, section 6 of  _[[Symmetric spectra]]_, 2012  ([pdf](http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf))

Discussion in the further generality of [[equivariant cohomology]] is in

* {#tomDieck79} [[Tammo tom Dieck]], section 7 of _[[Transformation Groups and Representation Theory]]_, Lecture Notes in Mathematics 766, Springer 1979

A pedagogical introduction to [[spectrum|spectra]] and generalized (Eilenberg-Steenrod) cohomology is in

* [[John Baez]], [TWF 149](http://math.ucr.edu/home/baez/twf_ascii/week149)

Formulation in [[(infinity,1)-category theory]] is in

* {#LurieHigherAlgebra} [[Jacob Lurie]], section 1.4.1 of _[[Higher Algebra]]_

More references relating to the [[nPOV]] on cohomology include:

* [[Mike Hopkins]], _[[Complex oriented cohomology theories and the language of stacks]]_ course notes ([pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/coctalos.pdf))

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

Formulation in [[homotopy type theory]] (cf. *[[cohomology in homotopy type theory]]*):

* [[Evan Cavallo]], Section 3.2 of: *Synthetic Cohomology in Homotopy Type Theory* (2015) [[pdf](https://staff.math.su.se/evan.cavallo/works/thesis15.pdf), [[Cavallo-CohomologyInHoTT.pdf:file]]]

* [[Floris van Doorn]], around Def. 5.4.2 in: _On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory_ (2018) [[arXiv:1808.10690](https://arxiv.org/abs/1808.10690), [pdf](http://florisvandoorn.com/papers/dissertation.pdf)]

[[!redirects generalized Eilenberg-Steenrod cohomology]]
[[!redirects generalized Eilenberg?Steenrod cohomology]]
[[!redirects generalized (Eilenberg?Steenrod) cohomology]]
[[!redirects generalized (Eilenberg--Steenrod) cohomology]]
[[!redirects generalized (Eilenberg-Steenrod) cohomology theory]]
[[!redirects generalized (Eilenberg?Steenrod) cohomology theory]]

[[!redirects generalized (Eilenberg--Steenrod) cohomology theory]]
[[!redirects generalized (Eilenberg--Steenrod) cohomology theories]]
[[!redirects generalized (Eilenberg-Steenrod) cohomology theory]]
[[!redirects generalized (Eilenberg-Steenrod) cohomology theories]]

[[!redirects generalised (Eilenberg-Steenrod) cohomology]]
[[!redirects generalised (Eilenberg?Steenrod) cohomology]]
[[!redirects generalised (Eilenberg--Steenrod) cohomology]]
[[!redirects generalised (Eilenberg-Steenrod) cohomology theory]]
[[!redirects generalised (Eilenberg?Steenrod) cohomology theory]]
[[!redirects generalised (Eilenberg--Steenrod) cohomology theory]]
[[!redirects Eilenberg-Steenrod cohomology]]
[[!redirects Eilenberg?Steenrod cohomology]]
[[!redirects Eilenberg--Steenrod cohomology]]
[[!redirects Eilenberg-Steenrod cohomology theory]]
[[!redirects Eilenberg?Steenrod cohomology theory]]
[[!redirects Eilenberg--Steenrod cohomology theory]]

[[!redirects Eilenberg-Steenrod axioms]]

[[!redirects generalized (Eilenberg-Steenrod) cohomology theory]]
[[!redirects generalized (Eilenberg-Steenrod) cohomology theories]]

[[!redirects generalized (Eilenberg-Steenrod) cohomlogy theory]]
[[!redirects generalized (Eilenberg-Steenrod) cohomlogy theories]]

[[!redirects extraordinary cohomology theory]]
[[!redirects extraordinary cohomology theories]]

[[!redirects Eilenberg–Steenrod cohomology theory]]
[[!redirects Eilenberg–Steenrod cohomology theories]]

[[!redirects Whitehead generalized cohomology]]
[[!redirects Whitehead generalized cohomology theory]]
[[!redirects Whitehead generalized cohomology theories]]

[[!redirects Whitehead-generalized cohomology]]
[[!redirects Whitehead-generalized cohomology theory]]
[[!redirects Whitehead-generalized cohomology theories]]


The ordinary [[homotopy groups]] of a space $X$ are
$$
\pi_n(X) = \pi^{S^0}_n(X) = [\Sigma^n S^0, X] = [S^n, X],
$$
where $S^0$ is the 0-[[sphere]]. We can choose another [[pointed space|based space]], say $A$. Thus,
$$
\pi^{A}_n(X) = [\Sigma^n A, X],
$$
are the generalized homotopy groups of $X$ with (co)-coefficients in $A$.

+-- {: .query}
But should this page, mentioning Eilenberg-Steenrod, be about generalized _stable_ homotopy? I.e., should we focus on $\Sigma^n A$ as a spectrum? Mind you, in [[spectrum]] it requires $E_n \cong \Omega E_{n + 1}$, where $\Omega$ denotes the based loop space. Don't we want the requirement $E_{n + 1} \cong \Sigma E_n$? Need to check whether adjunction means this makes no difference.

[[Tim]]: To my mind, there should be a spectrum based _generalised stable cohomotopy_ of $X$ as well perhaps, but the paradigm we have been using has been that it is the spaces that are the first importance here so I would stick with homotopy as $[\Sigma^n A,X]$ but also would ask about not using pointed spaces.  The free case is possibly more fun and useful.
=--

##Reference##

* Hans Baues, Algebraic Homotopy, Cambridge University Press, 1989, p. 117

[[!redirects generalized (Eilenberg--Steenrod) homotopy]]
[[!redirects generalized (Eilenberg–Steenrod) homotopy]]
[[!redirects generalized (Eilenberg-Steenrod) homotopy theory]]
[[!redirects generalized (Eilenberg--Steenrod) homotopy theory]]
[[!redirects generalized (Eilenberg–Steenrod) homotopy theory]]
[[!redirects generalised (Eilenberg-Steenrod) homotopy]]
[[!redirects generalised (Eilenberg--Steenrod) homotopy]]
[[!redirects generalised (Eilenberg–Steenrod) homotopy]]
[[!redirects generalised (Eilenberg-Steenrod) homotopy theory]]
[[!redirects generalised (Eilenberg--Steenrod) homotopy theory]]
[[!redirects generalised (Eilenberg–Steenrod) homotopy theory]]
[[!redirects Eilenberg-Steenrod homotopy]]
[[!redirects Eilenberg--Steenrod homotopy]]
[[!redirects Eilenberg–Steenrod homotopy]]
[[!redirects Eilenberg-Steenrod homotopy theory]]
[[!redirects Eilenberg--Steenrod homotopy theory]]
[[!redirects Eilenberg–Steenrod homotopy theory]]


generalized (infinity,1)-operad

[[!redirects generalized (∞,1)-operad]]

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

#Contents#
* table of contents
{:toc}

## Idea

An ordinary [[(∞,1)-operad]] $\mathcal{O}$ is equivalently incarnated as its [[(∞,1)-category of operators]] $\mathcal{O}^\otimes \to FinSet_*$, which in particular has the property that $\mathcal{O}^{\otimes}_{\langle 0\rangle} \simeq *$.

A _generalized $(\infty,1)$-operad_ is like an [[(∞,1)-category of operators]] only that this condition is dropped and

$$
  \mathcal{C} \coloneqq \mathcal{O}^{\otimes}_{\langle 0\rangle}
$$

is allowed to be any [[(∞,1)-category]].

It is essentially the same kind of data as a [[family of (∞,1)-operads]] (see there for more) parameterized over $\mathcal{C}$.

## References

Section 2.3.2 of

* [[Jacob Lurie]], _[[Higher Algebra]]_

[[!redirects generalized (infinity,1)-operads]]

[[!redirects generalized (∞,1)-operads]]


generalized algebraic theory

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

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

A *generalized algebraic theory* (GAT, [Cartmell 1978](#Cartmell78)/[86](#Cartmell86)) is a [[dependent type theory]] in the [[syntax|syntactic]] sense: There are [[judgments]] declaring [[types]] and [[terms]] of types, where, critically, a type $A$ declared in [[context]] $\Gamma$ is allowed to [[dependent type|depend]] on terms of $\Gamma$. The only other judgments allowed are of [[equality]] ([[judgemental equality]]) of pairs of types or terms.

[Cartmell 1978](#Cartmell78)/[86](#Cartmell86) introduced the notion of [[contextual category]] alongside that of generalized algebraic theories (GATs), specifically to provide the latter with a [[functorial semantics]]. He proved that the [[contextual category]] generated by a GAT (its [[syntactic category]]) is its [[initiality conjecture|initial model]], taking an early step in understanding the [[categorical semantics of dependent type theory]].

The latter program has more generally dealt with complicated [[coherence]] issues that arise for richer kinds of [[contextual categories]], corresponding to type theories admitting further [[type formation]] [[inference rules]] such as for [[dependent product types]] ("$\Pi$-types"), for [[identification types]] ("Id-types"), and so on. The key distinction is that a plain GAT cannot use a judgment of the form "$A$ is a [[type]] in [[context]] $\Gamma$" as a [[premise]], and hence cannot express general [[dependent type]]-[[type formation|formation]] [[inference rule|rules]].

The basic target for [[categorical semantics]] of a GAT is the contextual category of [[families]] of (families of families of...) sets, which avoids some [[coherence]] issues via stratifying its objects by their number of layers of families. Much of the difficulty in providing [[categorical semantics for dependent type theory]] is in understanding how to transfer such models to the ordinary category of sets, other [[locally cartesian closed category|locally cartesian closed categories]] and beyond.

### Examples ###

The theory of categories is a GAT:

1. Its type judgments are that $\mathrm{Ob}$ is a type and that, in the context of two terms $a,b:\mathrm{Ob},$ we have that $\mathrm{Hom}(a,b)$ is a type.

2. Its term judgments are that $\mathrm{id}(x):\mathrm{Hom}(x,x)$ in context $x:\mathrm{Ob}$ and that $g\circ f:\mathrm{Hom}(x,z)$ in context $x,y,z:\mathrm{Ob},f:\mathrm{Hom}(x,y),g:\mathrm{Hom}(y,z).$

3. Its equations are those imposing associativity and unitality, as usual.

One can also directly axiomatize the theory of [[monoidal categories]] by
adding:

1. A term $I$ of sort $\mathrm{Ob},$ in the empty context.

2. Terms $a\otimes b:\mathrm{Ob}$ in context $a,b:\mathrm{Ob}.$

3. Similarly, terms $f\otimes g:\mathrm{Hom}(x\otimes y,z\otimes w)$ in
the appropriate context, as well as for the unitors and the associators.

4. Additional equations corresponding to the pentagon and triangle diagrams, functoriality of $\otimes$, and inverse+naturality squares for remaining operations.

The axiomatization of [[strict monoidal categories]] is similar, but briefer.

The theory of 2-categories is also a GAT, and adds types $2-\mathrm{Hom}(f,g)$ in context $x,y:\mathrm{Ob},f,g:\mathrm{Hom}(x,y).$ This and further higher-categorical examples illustrate the need for *iterated* families of sets in the models.

Finally, [[multicategories]] form a GAT with infinitely many type and term formation rules, namely that we have a type $\mathrm{Ob}$ and, for every $n,$ a type $\mathrm{Hom}(x_1,\ldots,x_n;y)$ in context $x_1,\ldots,x_n:\mathrm{Ob}.$ We must, similarly, axiomatize all the composition operations of different arities separately.

### Relationship to Many-Sorted Algebraic Theories ###

A [many-sorted algebraic theory](algebraic+theory#multisorted_algebraic_theories) is a GAT with no nontrivially dependent types.

### Relationship to Essentially Algebraic Theories ###

Cartmell's paper explains how for every GAT there is an [[essentially algebraic theory]] (EAT) with the same models and for every EAT there is a GAT with the same models in the category of sets, up to the coherence issues in moving between the categories of sets and of families of sets mentioned above.

## Related concepts

* [[initiality conjecture]]
* [[Martin-Löf dependent type theory]]

## References

* {#Cartmell78} [[John Cartmell]], *Generalised Algebraic Theories and Contextual Categories*, PhD thesis, Oxford University (1978) [[[Cartmell-Thesis.pdf:file]]]

* {#Cartmell86} [[John Cartmell]], *Generalised Algebraic Theories and Contextual Categories*, Annals of Pure and Applied Logic, **32** (1986) 209-243  [[doi:10.1016/0168-0072(86)90053-9](http://dx.doi.org/10.1016/0168-0072%2886%2990053-9)]

[[!redirects generalized algebraic theories]]
[[!redirects generalised algebraic theories]]
[[!redirects generalised algebraic theory]]


generalized Calabi-Yau manifold

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A generalization of the notion of [[Calabi-Yau manifold]] in the context of [[generalized complex geometry]].

## Definition

### In terms of $G$-structure
 {#InTermsOfGStructure}

For $X$ a $2n$-[[dimension|dimensional]] [[smooth manifold]], a _[[generalized complex geometry|generalized complex structure]]_ on $X$ is a [[reduction of the structure group]] of the [[generalized tangent bundle]] $T X \oplus T^* X$ along the inclusion

$$
  U(n,n) \hookrightarrow O(2n,2n)
$$

into the [[Narain group]].

Recall that for $X$ an ordinary [[compact topological space|compact]] [[complex manifold]] of real [[dimension]] $2n$, a [[Calabi-Yau manifold]] structure on $X$ is a [[reduction of the structure group]] along the inclusion $SU(n) \hookrightarrow U(n)$ of the [[special unitary group]] into the [[unitary group]].

A **generalized Calabi-Yau structure** on a generalized complex manifold $X$ is a further [[reduction of the structure group]] along

$$
  SU(n,n) \hookrightarrow U(n,n) \hookrightarrow O(2n,2n)
  \,.
$$

([Hitchin, section 4.5](#Hitchin))

[[!include Spin(8)-subgroups and reductions -- table]]

## Related concepts

* [[Calabi-Yau object]]

  * [[Calabi-Yau algebra]], [[Calabi-Yau manifold]], **generalized Calabi-Yau manifold**

* [[supersymmetry and Calabi-Yau manifolds]]

* [[generalized G2-manifold]]

## References

The notion was introduced in

* {#Hitchin} [[Nigel Hitchin]], _Generalised Calabi-Yau Manifolds_, Quart. J. Math. Oxford Ser. 54:281-308, (2003) ([math.DG/0209099](http://arxiv.org/abs/math.dg/0209099))

Relation to (non-[[integrable G-structure|integrable]]) [[G-structure]] for $G = $ [[SU(n)]] (see also at _[[MSU]]_):

* {#Prins16} [[Daniël Prins]], _On flux vacua, $SU(n)$-structures and generalised complex geometry_, Université Claude Bernard -- Lyon I, 2015.  ([arXiv:1602.05415](https://arxiv.org/abs/1602.05415), [tel:01280717](https://tel.archives-ouvertes.fr/tel-01280717))

The role of generalized CY-manifolds as (factors of) [[target spaces]] in [[string theory]] is discussed for instance in

* [[Mariana Graña]], [[Ruben Minasian]], Michela Petrini, [[Alessandro Tomasiello]], _Type II Strings and Generalized Calabi-Yau Manifolds_ ([arXiv:hep-th/0409176](http://arxiv.org/abs/hep-th/0409176))

* [[Mariana Grana]], [[Ruben Minasian]], Michela Petrini, [[Alessandro Tomasiello]], _Generalized structures of N=1 vacua_, [hep-th/0505212](http://arxiv.org/abs/hep-th/0505212)

* [[Jan Louis]], _Generalized Calabi-Yau compactifications with D-branes and fluxes_, Forthschr. Phys. 53, no 7-8 (2005) ([pdf](http://www.desy.de/~jlouis/talks/770_a.pdf))

[[!redirects generalized Calabi-Yau manifolds]]

[[!redirects generalized Calabi-Yau space]]
[[!redirects generalized Calabi-Yau spaces]]

[[!redirects generalized Calabi-Yau structure]]
[[!redirects generalized Calabi-Yau structures]]


generalized Cauchy real number

[[!redirects generalized Cauchy real numbers]]
[[!redirects generalised Cauchy real number]]
[[!redirects generalised Cauchy real numbers]]

[[!redirects Cauchy complete real number]]
[[!redirects Cauchy complete real numbers]]
[[!redirects Cauchy-complete real number]]
[[!redirects Cauchy-complete real numbers]]

[[!redirects Cauchy net real number]]
[[!redirects Cauchy net real numbers]]
[[!redirects Cauchy-net real number]]
[[!redirects Cauchy-net real numbers]]

+-- {: .rightHandSide}
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### Context
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea ##

The Cauchy completion of the rational numbers, in the sense that all [[Cauchy nets]] or [[Cauchy filters]] in the rational numbers [[converge]].

## Definition ##

### Using filters

...

### Using nets

Let $\mathbb{Q}$ be the [[rational numbers]] and let

$$\mathbb{Q}_{+} \coloneqq \{x \in \mathbb{Q} \vert 0 \lt x\}$$

be the set of positive rational numbers. Let $I$ be a [[directed set]] and

$$C(I, \mathbb{Q}) \coloneqq \{x \in \mathbb{Q}^I \vert \forall \epsilon \in \mathbb{Q}_{+}. \exists N \in I. \forall i \in I. \forall j \in I. (i \geq N) \wedge (j \geq N) \wedge (\vert x_i - x_j \vert \lt \epsilon)\}$$

is the set of all [[Cauchy nets]] with index set $I$ and values in $\mathbb{Q}$. Let

$$CauchyNetsIn(\mathbb{Q}) \coloneqq \bigcup_{I \in Ob(DirectedSet_\mathcal{U})} C(I, \mathbb{Q})$$

be the (large) set of all Cauchy nets in $\mathbb{Q}$ in a [[universe]] $\mathcal{U}$, where $DirectedSet_\mathcal{U}$ is the [[category]] of all directed sets in $\mathcal{U}$.

Let the [[relation]] $\equiv_{I, J}$ in the Cartesian product $C(I, \mathbb{Q}) \times C(J, \mathbb{Q})$ for directed sets $I$ and $J$ be defined as

$$a \equiv_{I, J} b \coloneqq \forall \epsilon \in \mathbb{Q}_{+}, \exists M \in I. \forall i \in I. \exists N \in J. \forall j \in J. (i \geq M) \wedge (j \geq N) \wedge (\vert a_i - b_j \vert \lt \epsilon)$$

Let a *generalized Cauchy algebra* be defined as a [[set]] $A$ with a function $\iota \in {A}^{CauchyNetsIn(\mathbb{Q})}$ such that

$$\forall I \in Ob(DirectedSet_\mathcal{U}). \forall J \in Ob(DirectedSet_\mathcal{U}). \forall a \in C(I, \mathbb{Q}). \forall b \in C(J, \mathbb{Q}). (a \equiv_{I, J} b) \implies (\iota(a) = \iota(b))$$

A *generalized Cauchy algebra homomorphism* is a function $f:B^A$ between Cauchy algebras $A$ and $B$ such that

$$\forall a \in C(\mathbb{Q}). f(\iota_A(a)) = \iota_B(a)$$

The *category of generalized Cauchy algebras* is the category $GCAlg$ whose objects $Ob(GCAlg)$ are generalized Cauchy algebras and whose morphisms $Mor(GCAlg)$ are generalized Cauchy algebra homomorphisms. The set of **generalized Cauchy real numbers**, denoted $\mathbb{R}$, is defined as the initial object in the category of Cauchy algebras.

## See also ##

* [[HoTT book real numbers]]
* [[Cauchy real numbers]]


generalized cohomology

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###Context###
#### Cohomology
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[[!include cohomology - contents]]
=--
=--
=--

By a **generalized cohomology theory** is usually meant a contravariant functor on a [[homotopy category]] satisfying all abstract properties of [[ordinary cohomology]], except possibly for the dimension axiom. For more on this see at

* **[[Whitehead-generalized cohomology]]**.

  (and for the dual concept see at _[[generalized homology]]_).

But there are more general generalizations of the concept of ordinary cohomology, too. For instance there is also

* [[nonabelian cohomology]]

* [[sheaf cohomology]]

* [[differential cohomology]]

etc.

For a fully general concept of generalized cohomology, see at

* [[cohomology]].

[[!include homotopy-homology-cohomology]]

[[!redirects generalized cohomology theory]]
[[!redirects generalized cohomology theories]]
[[!redirects generalised cohomology theory]]
[[!redirects generalised cohomology theories]]
[[!redirects cohomology theories]]
[[!redirects cohomology theory]]


generalized complex geometry

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#### Complex geometry
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#### Differential geometry
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[[!include synthetic differential geometry - contents]]
=--
=--
=--

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

## Idea

_Generalized complex geometry_ is the study of the geometry of [[symplectic Lie n-algebroid|symplectic Lie 2-algebroid]] called [[standard Courant algebroid]]s $\mathfrak{c}(X)$ (over a [[smooth manifold]] $X$).

This geometry of symplectic Lie 2-algebroids turns out to unify, among other things, [[complex geometry]] with [[symplectic geometry]]. This unification notably captures central aspects of [[T-duality]].

## Definition
 {#Definition}

### On a vector space

Let $V$ be a [[dimension|finite dimensional]] [[vector space]] over the [[real numbers]].

Recall that a _[[complex structure]]_ on $V$ is a [[linear map]]

$$
  J : V \to V
$$

such that $J \circ J = - id_V$. And a [[symplectic structure]] on $V$ is equivalently a linear [[isomorphism]]

$$
  \omega : V \to V^*
$$

such that

$$
  \omega^* = - \omega
  \,,
$$

where $V^*$ denotes the [[dual vector space]] and $\omega^*$ the dual linear map.

The following definition may be thought of as combining these two concepts.

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

A **generalized complex structure** on $V$ is a [[linear map]]

$$
  \mathcal{J} : V \oplus V^* \to V \oplus V^*
$$

(an [[endomorphism]] of the [[direct sum]] of $V$ with its [[dual vector space]])

such that it is both

1. a [[complex structure]] on $V \oplus V^*$ in that $\mathcal{J}^2 = - id$;

1. a [[symplectic structure]] on $V \oplus V^*$ in that $\mathcal{J}^* = - \mathcal{J}$.

=--

The following shows that this is indeed a joint generalization of complex and symplectic structures.

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

Let $J : V \to V$ be an ordinary complex structure on $V$. Then the linear endomorphism of $V \oplus V^*$ defined by [[matrix calculus]] as

$$
  \mathcal{J}_j :=
  \left(
    \array{
       -J & 0
       \\
       0 & J^*
    }
  \right)
$$

is a generalized complex structure on $V$.

Similarly, let $\omega : V \to V^*$ be an ordinary symplectic structure on $V$. then the endomorphism

$$
  \mathcal{J}_\omega :=
  \left(
    \array{
      0 & - \omega^{-1}
      \\
      \omega & 0
    }
  \right)
$$

is a generalized complex structure on $V$.

=--

### On a manifold

A generalized complex structure on a manifold is a generalized complex structure on the fibers of the [[generalized tangent bundle]].

(...)

### In terms of reduction of the structure group

A generalized complex structure on $V \oplus V^*$ is equivalently a [[reduction of the structure group]] along the inclusion

$$
  U(n,n) \hookrightarrow O(2n,2n)
  \,,
$$

where the left hand is identified as $U(n,n) = O(2n,2n) \cap GL(2n, \mathbb{C})$ ([Gualtieri, prop. 4.6](#Gualtieri)).

The analog of this with the unitary group replaced by the orthogonal group yields _[[type II geometry]]_.

## Properties

One finds (as described at [[standard Courant algebroid]]) that

* choices of sub-[[Lie algebroid]]s of $\mathfrak{c}(X)$ encode (almost) **[[Dirac structure]]s** and -- after [[complexification]] -- **generalized [[complex structure]]s**;

* choices of [[section]]s of the canonical morphism $\mathfrak{c}(X) \to T X$ to the [[tangent Lie algebroid]]s encode **generalized Riemannian metrics**: pairs consisting of a (possibly [[pseudo-Riemannian metric|pseudo]]-)[[Riemannian metric]] and a [[differential form|2-form]].

  In applications in [[string theory]], this encodes the field of [[gravity]] and the [[Kalb–Ramond field]], respectively. (There are also proposals for how the [[dilaton]] field appears in this context.)

In components these are structures found on the [[vector bundle]]

$$
  T X \oplus T^* X
  \,,
$$

the [[direct sum]] of the [[tangent bundle]] with the cotangent bundle of $X$.

Generalized complex geometry thus generalizes and unifies

* [[complex geometry]],
* [[symplectic geometry]], and
* [[Riemannian geometry]].

It was in particular motivated by the observation that this provides a natural formalism for describing [[T-duality]].

## Examples

* [[generalized Calabi-Yau manifold]]

## Related concepts

* [[generalized tangent bundle]], [[generalized vielbein]]

* [[type II geometry]]

* [[exceptional generalized geometry]]

* [[Bn-geometry]]

* [[generalized contact geometry]]

## References

#### General

Generalized complex geometry was proposed by [[Nigel Hitchin]] as a formalism in [[differential geometry]] that would be suited to capture the phenomena that physicists encountered in the study of [[T-duality]].
It was later and is still developed by his students, notably Gualtieri and Cavalcanti.

A standard reference is the PhD thesis

* [[Marco Gualtieri]], _Generalized complex geometry_ ([arXiv:math/0401221](http://arxiv.org/abs/math/0401221))
 {#Gualtieri}

A survey set of slides with an eye towards  the description of the [[Kalb-Ramond field]] and [[bundle gerbe]]s is

* [[Nigel Hitchin]], _B-Fields, gerbes and generalized geometry_ Oxford Durham Symposium 2005 ([pdf](http://www.maths.dur.ac.uk/events/Meetings/LMS/2005/GCFTST/Talks/hitchin1.pdf))

* [[Nigel Hitchin]], _Lectures on generalized geometry_ ([arxiv/1008.0973](http://arxiv.org/abs/1008.0973))

With emphasis on the role of [[G-structures]]:

* [[Paul Koerber]], _Lectures on Generalized Complex Geometry for Physicists_, Fortsch. Phys. 59: 169-242, 2011 ([arXiv:1006.1536](https://arxiv.org/abs/1006.1536))

#### As targets for $\sigma$-models

Generalized complex structures may serve as [[target spaces]] for [[sigma-models]]. Relations to the [[Poisson sigma-model]] and the [[Courant sigma-model]] are discussed in

* [[Alberto Cattaneo]], Jian Qiub, [[Maxim Zabzine]], _2D and 3D topological field theories for generalized complex geometry_, [euclid](http://projecteuclid.org/getRecord?id=euclid.atmp/1288619156) [MR2012a:81255](http://www.ams.org/mathscinet-getitem?mr=2721659) [pdf](http://www.math.uzh.ch/fileadmin/math/preprints/GC_BV.pdf)

#### Mirror symmetry

* Oren Ben-Bassat, _Mirror symmetry and generalized complex manifolds. I. The transform on vector bundles, spinors, and branes_,
J. Geom. Phys. __56__ (2006), no. 4, 533–558 [math.AG/0405303](http://arxiv.org/abs/math/0405303) [MR2006k:53067](http://www.ams.org/mathscinet-getitem?mr=2199280) [doi](http://dx.doi.org/10.1016/j.geomphys.2005.03.004)

#### Geometry of supergravity

Generalized complex geometry and variants of [[exceptional generalized complex geometry]] are natural for describing [[supergravity]] background compactifications in [[string theory]] with their [[T-duality]] and [[U-duality]] symmetries ([[non-geometric vacua]]).

* Ian Ellwood, _NS-NS fluxes in Hitchin's generalized geometry_ ([arXiv:hep-th/0612100](http://arxiv.org/abs/hep-th/0612100))

* Paulo Pires Pacheco, [[Daniel Waldram]], _M-theory, exceptional generalised geometry and superpotentials_ ([arXiv:0804.1362](http://arxiv.org/abs/0804.1362))

* [[Mariana Graña]], [[Ruben Minasian]], Michela Petrini, [[Daniel Waldram]], _T-duality, generalized geometry and non-geometric backgrounds_,  J. High Energy Phys. 2009, no. 4, 075, 39 pp. [arXiv:0807.4527](http://xxx.lanl.gov/abs/0807.4527) [MR2010i:81323](http://www.ams.org/mathscinet-getitem?mr=2505954) [doi](http://dx.doi.org/10.1088/1126-6708/2009/04/075)

* David Andriot, [[Ruben Minasian]], Michela Petrini, _Flux backgrounds from twists_,  J. High Energy Phys. 2009, no. 12, 028 [arXiv:0903.0633](http://arxiv.org/abs/0903.0633) [MR2011c:81201](http://www.ams.org/mathscinet-getitem?mr=2593014) [doi](http://dx.doi.org/10.1088/1126-6708/2009/12/028)

[[!redirects generalized complex geometries]]

[[!redirects generalised complex geometry]]
[[!redirects generalised complex geometries]]

[[!redirects generalized complex structure]]
[[!redirects generalized complex structures]]


generalized contact geometry

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###Context###
#### Complex geometry
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[[!include complex geometry - contents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

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

## Idea

_Generalized contact geometry_ is the odd dimensional analogue of [[generalized complex geometry]]. One proposal for an odd dimensional analogue of a generalized complex manifold is called a **generalized contact bundle** ([VitaAis15]({#VitaAis15)). This concept encompasses both not necessarily coorientable contact manifolds and line bundles equipped with an integrable [[complex structure]] on their [[Atiyah algebroid]].

## Related concepts

* [[contact geometry]]

* [[generalized complex geometry]]

## References

* {#VitaAis15} Luca Vitagliano, Aïssa Wade, _Generalized Contact Bundles_, ([arXiv:1507.03973](https://arxiv.org/abs/1507.03973))

* {#SchnVita17} Jonas Schnitzer, Luca Vitagliano, _The Local Structure of Generalized Contact Bundles_, ([arXiv:1711.08310](https://arxiv.org/abs/1711.08310))

[[!redirects generalized contact manifold]]
[[!redirects generalized contact manifold]]


generalized continuity > history

< [[generalized continuity]]


generalized cup product

#Contents#
* table of contents
{:toc}

## Idea

Steenrod's _generalized cup products_ generalize the usual simplicial [[cup products]] on [[simplicial cochains]].

Taken together, all generalized cup products organize into the [[sequence operad]].
This operad is an [[E-infinity operad]] and it acts on [[simplicial cochains]],
turning them into an [[E-infinity algebra]].

## Definition

First, we define an operation on [[simplicial chains]]
that decomposes simplices in a manner similar
to how the usual simplicial [[cup product]] decomposes
a simplex into the tensor product of a simplex
consisting of the initial $(p+1)$ vertices
and a simplex consisting of the last $(q+1)$ vertices,
with $p+q$ being the dimension of the original simplex.

\begin{definition}
(Definition 2.10(a) in [McClure-Smith 01](#Cochain).)
Given a surjection $f\colon\{1,\ldots,m\}\to\{1,\ldots,k\}$
and a simplex $\sigma\colon\Delta^p\to X$,
define
$$\sigma[f]\in(C(X))^{\otimes k}$$
as
$$\sigma[f]=\sum_A (-1)^{\epsilon(f,A)}\bigotimes_{1\le i\le k}\sigma\left(\coprod_{f(j)=i}A_j\right).$$
The variable $A$ is indexed over all overlapping partitions of $\{0,\ldots,p\}$ with $m$ parts.
\end{definition}

Here
$$\epsilon(f,A)=\sum_{j\lt j',f(j)\gt f(j')}dim(A_j)dim(A_{j'})+\sum_{1\le j\le m}dim(A_j)(\tau_f(j)-f(j)),$$
(Definition 2.9 in [Cochain](#Cochain))
where
$$\tau_f(j)=card\{j'\in\{1,\ldots,m\}\mid f(j')\lt f(j) or f(j')=f(j) and j'\le j\}$$
(Definition 2.7 in [Cochain](#Cochain)).
Also, $dim(B)=card(B)-1$ denotes the dimension of the simplex with the set of vertices $B$
(Notation 2.6 in [Cochain](#Cochain)).

We are now ready to define generalized cup products.

\begin{definition}
(See Definition 2.10(b) in [Cochain](#Cochain).)
Given a surjection $f\colon \{1,\ldots,m\}\to\{1,\ldots,k\}$,
the natural transformation
$$\langle f\rangle=\cup_f\colon (C^*(X))^{\otimes k}\to C^*(X)$$
is defined by
$$\langle f\rangle(x_1\otimes\cdots\otimes x_k)(\sigma)=(-1)^{m-k}(x_1\otimes\cdots\otimes x_k)(\sigma[f]).$$
\end{definition}

\begin{remark}
(See Lemma 2.12, Definition 2.13, and Definition 2.14 in [Cochain](#Cochain).)
If $f(l)=f(l+1)$ for some $1\le l\lt m$, then $\langle f\rangle=0$.
Thus, in the [[sequence operad]] such degenerate operations must be modded out.
The remaining operations form a basis of the [[sequence operad]].
\end{remark}

\begin{remark}
(See Remark 2.11 in [Cochain](#Cochain).)
If $k=2$, we recover the traditional cup-$i$ products $\cup_i$ defined by Steenrod as follows.
Define $f\colon\{1,\ldots,m\}\to\{1,2\}$
to be the function with alternating values 1, 2, 1, 2, …
Now $\langle f\rangle=\cup_{m-2}$.
In particular for $m=2$ (the sequence 12) we recover the [[cup product]]
and for $m=3$ (the sequence 121) we recover $\cup_1$.
\end{remark}

## Related concepts

* [[cup product]]

* [[sequence operad]]

## References

* {#Cochain} [[James E. McClure]], [[Jeffrey H. Smith]],
_Multivariable cochain operations and little $n$-cubes_
([arXiv:math/0106024](https://arxiv.org/abs/math/0106024))

[[!redirects generalized cup products]]


generalized Eilenberg-Mac Lane space

[[!redirects generalized Eilenberg-MacLane space]]

#Contents#
* table of contents
{:toc}

###Disambigation

This entry is about 'generalized Eilenberg-Mac Lane spaces' as used in Stable Homotopy Theory. The same term is used for a related concept, namely the representing fibration for [[ twisted cohomology|cohomology with local coefficients]]. That cohomology is also called [[twisted cohomology]] so we have used the term [[twisted Eilenberg - Mac Lane space]] for this second use.

## Idea

A _generalized Eilenberg-Mac Lane space_ is a [[topological space]] with the [[homotopy type]] of a [[Cartesian product]] of (finitely many) [[Eilenberg-MacLane spaces]].

Accordingly a _generalized Eilenberg-Mac Lane spectrum_ is a [[spectrum]] equivalent to a [[wedge sum]] of [[Eilenberg-Mac Lane spectra]].

> to be expanded... please add if you have the time

## Applications

* Generalized Eilenberg-Mac Lane spectra appear as [[coefficients]] in an [[Adams resolution]] used in the [[Adams spectral sequence]].

*Theorem.* Let $R$ be a commutative ring with a unit and let $fMod_R$ be the category of finitely generated free $R$-modules. If $H\colon fMod_R\to Spaces$ is a functor such that

* $H(0)$ is a contractible space,
* $H(R)$ is connected and for every $ n$ the projections $pr_k \colon R^n\to R$ induce a weak equivalence $ H(R^n) \stackrel{\simeq}{\to}  H(R)^n$

then the space $ H(R)$ is weakly equivalent to a product $\prod_{i=1}^{\infty} K(M_i , i)}$ where $M_i$ is an $ R$-module.

**Proof.** This is the main theorem of [Badzioch]

## References

{#Bdzch01} Badzioch, Bernard. "Recognition principle for generalized Eilenberg-Mac Lane spaces." Cohomological methods in homotopy theory. Birkhäuser, Basel, 2001. 21-26.

[[!redirects generalized Eilenberg-Mac Lane spaces]]
[[!redirects generalised Eilenberg-Mac Lane space]]
[[!redirects generalized Eilenberg-MacLane spectrum]]
[[!redirects generalized Eilenberg-MacLane spectra]]


generalized element

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

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

## Idea and definition
In a broad, non-technical sense, an "element" is a "building block", "component", or "basic part" of a more substantial whole. Ordinary or [[global elements|global]] elements of a [[set]] are simply the [[points]] of that set, and hence sufficiently capture this broad notion of "element" in [[Set]], since by definition sets are no more than collections of points.

However, in general, knowing about the points of an object is insufficient to count as knowing its elements (construed broadly). From the point of view of the category [[Set]], most things that can be said and done about elements of a set $X$, can more generally be said and done for morphisms $x \colon U\to X$, for _any_ other set $U$.  The point is just that many constructions can be performed "elementwise".  For instance, the fact that elements of $X\times Y$ are exactly pairs $(x,y)$ of an element of $X$ and an element of $Y$, when performed "elementwise" for morphisms out of $U$, expresses the universal property of a [[product]].  In [[structural set theory]] such as [[ETCS]], one sometimes (but not necessarily) takes this point of view for axiomatizing the structure of $Set$.

On the other hand, once elements of objects are regarded as morphisms into these objects, the same reasoning applies to _every_ [[category]] $C$.  Accordingly, for $C$ any [[category]] and $X$ an [[object]] of $C$, one may refer to a [[morphism]] $x \colon U \to X$ a **generalized element** of $X$. One says this is a generalized element with **stage of definition** given by $U$, or a **figure** of shape $U$ in $X$.

The perspective of generalized elements of objects of a category $C$ is related to regarding $C$ as its image under the [[Yoneda embedding]]

$$
  Y : C \hookrightarrow [C^{op}, Set]
$$

into its [[presheaf category]]. Under this embedding, every object $X$ of $C$ is mapped to the [[functor]] -- the [[representable functor]] represented by it --

$$
  GenEl(X) : C^{op} \to Set
$$

that sends each object $U$ of $C$ to the set of generalized elements of $X$ at stage $U$.

It is also worth noting that the [[internal logic]] or [[type theory]] of a category provides us a way to go backwards formally.  By reasoning about "abstract elements" in a set-theoretic style like ordinary elements, the interpretation then "compiles" such proofs to category-theoretic ones which actually apply to all generalized elements.

## Examples

### In $Set$

The primordial example is when $C$ is the category [[Set]] of sets and $I$ is a [[terminal object]] in $Set$ --- that is, a set with one element.  Then elements of any set $c$ are in one-to-one correspondence with functions $f: I \to c$.  This correspondence works as follows: given any element of $c$ there is a unique function $f: I \to c$ with this element in its image, and conversely each function $f: I \to c$ has a unique element of $c$ in its image.

### In concrete categories

In the same way, in a [[concrete category]] whose underlying-set functor is represented by $I$, the $I$-elements of an object are the same as the elements of its underlying set.  (The category of sets is actually a special case of this, since it is concrete, with the identity functor represented by a terminal object.)

### Global elements

Generalizing from $Set$ in another way, in any category with a terminal object $I$, we call a morphism $f : I \to c$ a [[global element]] of the object $c$.

### In toposes

The stability or [[fundamental theorem of topos theory|slice theorem]] for [[toposes]] says that if $\mathbf{E}$ is a topos, then also the [[slice category]] $\mathbf{E}/U$ is a topos for any object $U$, and the [[pullback#PullbackFunctor|pullback functor]] $U^\ast: \mathbf{E} \to \mathbf{E}/U$ is a [[logical functor]].

Observe that in $\mathbf{E}/U$, an ordinary (i.e. global) point of an object $U^\ast X$, a section $U^\ast 1 \to U^\ast X$, corresponds to a generalized element $U \to X$ in $\mathbf{E}$. Thus the slice theorem guarantees that generalized points with domain $U$ may be treated exactly as ordinary points, just in a more variable topos $\mathbf{E}/U$.

### In monoidal categories

On the other hand, it is common to take $I$ to be the unit object whenever $C$ is a [[monoidal category]].  The generalized elements defined over this $I$ are important in [[enriched category|enriched category theory]]).

### For generators

Arguably, the most general case where generalized elements defined at only one stage $I$ are "sufficient" when $I$ is some sort of [[generator]] of the category.  However, not every category has a single object as any sort of generator!  Instead, in arbitrary categories, generalized elements of *all* possible stages of definition must often be used to replace global elements.  Thus while a set is determined by its global elements, an object of an arbitrary category is determined by all of its generalized elements (this is one way to state the [[Yoneda lemma]]).

### In presheaf categories

For $C = [D^{op}, Set]$  a [[presheaf]] category and for $I = \Delta_{pt} = (d \mapsto \{\bullet\})$ the presheaf constant at the singleton set, the generalized elements of a presheaf $F$ are the _[[global section]]s_ of this presheaf, equivalently these are the elements in the [[limit]] set over $F$.

On the other hand, if $I=D(-,d)$ is a [[representable functor|representable]] presheaf, then the generalized elements of $F$ at stage $d$ are precisely the elements of the set $F(d)$, by the [[Yoneda lemma]].

### In abelian categories

An [[element in an abelian category]] is an equivalence class of generalised elements.

## Relationship to type theory

In the [[internal language|internal]] [[type theory]] of a category $C$, the generalized elements of $X$ at stage $U$ can be identified with [[terms]] of type $X$ in [[context]] $u\colon U$:
$$
   u\colon U \vdash x(u) \colon X
  \,.
$$
See the references [below](#ReferencesInTypeTheory).

The fact that all type-theoretic constructions can be performed in any context implies that we can manipulate ordinary elements, and end up speaking also about generalized elements defined at arbitrary stages.

## References

### In type theory
 {#ReferencesInTypeTheory}

The interpretation of [[terms]] in [[type theory]] as generalized elements of objects in a category is discussed for instance on p. 8 of

* [[Steve Awodey]], [[Andrej Bauer]], _Propositions as $[$Types$]$_, Journal of Logic and Computation. Volume 14, Issue 4, August 2004, pp. 447-471 ([pdf](http://math.andrej.com/data/bracket_types.pdf))

[[!redirects generalised element]]
[[!redirects generalised elements]]
[[!redirects generalized elements]]


generalized elliptic operator

#Contents#
* table of contents
{:toc}

## Idea

In the context of [[noncommutative geometry]]/[[KK-theory]], a _generalized elliptic operator_ is a [[linear operator]] of a [[Hilbert bimodule]] over two [[C*-algebras]] $A, B$ in [[C*Alg]], such that in the special case that $A = C_0(X)$ and $B = C_0(Y)$ for $X,Y$ [[smooth manifolds]], this reduces to a $Y$-parameterized collection of ordinary [[elliptic operators]] on $X$.

## Related concepts

* [[elliptic complex]]

[[!redirects generalized elliptic operators]]


generalized fiber integration synonyms - table

The following terms all refer to essentially the same concept:

* [[fiber integration in generalized cohomology]]

* [[push-forward in generalized cohomology]]

* [[Umkehr map]], [[twisted Umkehr map]]

* [[Gysin map]]

* [[Pontrjagin-Thom collapse map]]


generalized Fitting subgroup

One definition of the _generalized Fitting subgroup_  $F^*(G)$ of a [[group]] $G$ is that it is the [[subgroup]] generated by all [[normal subgroup]]s $N$ of $G$ possessing subgroups $N_1,N_2,\dots, N_r$ for some integer $r$ such that $N=N_1N_2\cdots N_r$; $x_i x_j=x_j x_i$ for all $x_i\in N_i$, $x_j\in N_j$, and distinct subscripts $i$ and $j$; and each $N_i$ either has prime power order or is a [[quasisimple group]]. [[Helmut Bender]] proved that $F^*(G)$ itself enjoys these properties.

Named after [[Hans Fitting]].


generalized function

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### Context
#### Functional analysis
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[[!include functional analysis - contents]]
=--
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#Contents#
* table of contents
{:toc}

## Idea

In [[functional analysis]], [[distributional densities]] are usefully thought of as _generalized smooth functions_ in that every [[distribution]] is the [[limit of a sequence|limit]] of a [[sequence]] of [[smooth functions]], thought of as [[non-singular distributions]] ([this prop.](non-singular+distribution#NonSingularDistributionsAreDenseInAllDistributions)).

More in detail, regarding a [[smooth function]] $f \in C^\infty(X)$ as a [[distributional density]] means to regard it as the [[integral kernel]]

$$
  u_f
  \;\colon\;
  b \mapsto \underset{X}{\textstyle{\int}} f(x) b(x) \, dvol_X(x)
$$

on [[bump functions]] $b \in C^\infty_{cp}(X)$. Therefore a [[limit of a sequence|limit]] over a sequence $\{f_n\}_{n \in \mathbb{N}}$ of smooth functions, which may not really exists as a smooth function but just as a distribition, is nevertheless usefully thought of as the integral against the "generalized function" which would be "$\underset{\underset{n}{\longrightarrow}}{\lim} f_n$" if that limit existed in $C^\infty(X)$:

$$
  \underset{X}{\textstyle{\int}}
    \left(\underset{\underset{n}{\longrightarrow}}{\lim} f_n\right)(x)
    b(x) \, dvol_X(x)
  \;\coloneqq\;
  \underset{\underset{n}{\longrightarrow}}{\lim}
  \underset{X}{\textstyle{\int}}
    f_n(x) b(x), dvol_X(x)
  \,.
$$

It is common and convenient to write evaluation of [[distributions]] on test functions in this way, and the name of many distributions reflects this habit, such as "[[Dirac delta function]]" for the [[delta distribution]] or "[[commutator function]]" for a [[causal propagator]].

In particular [[distributions]] that arise from [[integral]] expressions appearing in [[Fourier analysis]] are often conveniently written this way: For $f \in \mathbb{R}^n$ a [[smooth function]] (or [[distribution]]) in a [[variable]] $k$, its [[Fourier transform of distributions]] $\hat f$ generally is given as a generalized function by the integral expression

$$
  \hat f(x)
    \;=\;
  \underset{\mathbb{R}^n}{\textstyle{\int}} f(k) e^{i k \cdot x} \, d^n k
$$

which however in general is not an actual _function_ of $x$, in that this [[integral]] need not [[convergence of a sequence|converge]] for fixed $x \in \mathbb{R}^n$, but is a generalized function in that for appropriate test functions $b$ the further integral

$$
  \underset{\mathbb{R}^n}{\textstyle{\int}} \underset{\mathbb{R}^n}{\textstyle{\int}} f(k) e^{i k \cdot x}  b(x) \, d^n x \, d^n k
 \;\in\;
 \mathbb{C}
$$

does make sense.

For example it is in this sense that the [[delta-distribution]], regarded as a generalized function, is given by the common [[integral]] expression

$$
  \delta(x) \;=\; \underset{k \in \mathbb{R}^n}{\textstyle{\int}} e^{2 \pi i k x} d^n k
$$

(see at _[[delta distribution]]_ [this example](Dirac+distribution#FourierTransformOfDeltaDistribution)).

Thinking of distributions as generalized functions this way motivates and streamlines the extension of a variety of concepts from functions to distributions, such as the concepts of _[[product of distributions]]_, _[[derivative of distributions]]_, _[[support of distributions]]_, [[distributional solutions to a PDE]] and so on.

## Related concepts

* [[distribution]]

* [[hyperfunction]]

* [[current (distribution theory)]]

[[!redirects generalized functions]]


generalized G2-manifold

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###Context###
#### Differential geometry
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[[!include synthetic differential geometry - contents]]
=--
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
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#Contents#
* table of contents
{:toc}

## Definition

[[reduction of the structure group]] of the [[generalized tangent bundle]] of a 7-dimensional [[manifold]] along the inclusion

$$
  G_2 \times G_2 \to SO(7,7)
$$

of two copies of [[G2]].

## Properties

### As exceptional geometry

[[!include Spin(8)-subgroups and reductions -- table]]

## Related concepts

* [[G2-manifold]]

* [[exceptional generalized geometry]]

* [[generalized Calabi-Yau manifold]], [[type II geometry]]

* [[generalized Spin(7)-manifold]]

## References

* Frederik Witt, _Generalised $G_2$-structures_, ([math.dg/0411642](http://arxiv.org/abs/math/0411642))

* Claus Jeschek, Frederik Witt, _Generalised $G_2$-structures and type IIB superstrings_ ([arXiv:hep-th/0412280](http://arxiv.org/abs/hep-th/0412280))

* _Generalised $G_2$-manifolds_ (2005) ([pdf](http://geometry.ma.ic.ac.uk/juniorgeometry/IC_London_10-01-05.pdf))

* Anna Fino, Adriano Tomoassini, _Generalized $G_2$-manifolds and $SU(3)$-structure_, Int. J. Math. 19, 1147 (2008) ([web](http://www.worldscientific.com/doi/abs/10.1142/S0129167X08005072))

* J. de Boer, P de Medeiros, S El-Showk and A Sinkovics, _$G_2$ Hitchin functionals at one loop_, Class. Quantum Grav. 25 ([web](http://iopscience.iop.org/0264-9381/25/7/075006/))

[[!redirects generalized G2-manifolds]]

[[!redirects generalised G2-manifold]]
[[!redirects generalised G2-manifolds]]

[[!redirects generalized G2-structure]]
[[!redirects generalized G2-structures]]

[[!redirects generalised G2-structure]]
[[!redirects generalised G2-structures]]


generalized geometry

* [[generalized complex geometry]]

* [[type II geometry]]

* [[exceptional generalized geometry]]

* [[higher geometry]]

[[!redirects generalized geometries]]


generalized global analytic geometry

#Contents#
* table of contents
{:toc}

## Ideas and objectives

The motivation of generalized global analytic geometry is to define analytic spaces associated to rig categories, and to use their relations with classical [[global analytic geometry|global analytic spaces]] in order to properly define [[Arakelov geometry|Arakelov models]] of [[arithmetic varieties]] in an analytic setting.

Since rig categories are categorical analogs of associative and commutative semi-rings, a proper theory for them can only be given in the setting of $\infty$-categories.

One will thus need to define [[overconvergent global analytic geometry|overconvergent global analytic spaces]] over a given rig $\infty$-category, rational domains in them, and compare the associated $G$-topology with the usual $G$-topology of overconvergent analytic geometry. Generalizing directly Haran's definition of the rig of integers of $(\mathbb{R},|\cdot|_\infty)$, we will then give examples of rig $\infty$-category with a well defined notion of rig $\infty$-category of analytic functions, and use them to define natural Arakelov-type compactifications of strict overconvergent global analytic spaces over $(\mathbb{Z},|\cdot|_\infty)$.

To be able to complete non-additive rig categories, we will have to replace seminorms by rig metrics (metrics on morphism spaces that are compatible with the rig category structure). There are rig categories associated to rings, but also to Banach rings, and for the last ones, there may be various natural choices, based on the type of modules in use. The notion of morphism will be imposed by the necessity of having a natural morphism
$$
(\Z_\eta,\|\cdot \|_2)\to (\Z_\eta,\|\cdot\|_2)^t
$$
for every $t\geq 1$.
We will then define a category of rational domain rig categories that will be the building blocks of analytic ring categories and analytic ringed spaces. The definitions proposed in the coming Sections is not clearly adapted to the setting of Banach rings, because the metric enrichment should be done on the full infinity groupoid of morphisms in a rig category (and this last point is yet a problematic issue; one should probably use simplicial Cauchy spaces or metric spaces with Cauchy maps, to make an $\infty$-category adapted to the enrichment). All these constructions will be compatible with the standard flow on metrics and absolute value given by taking $t$-th power for $t\in \mathbb{R}_+^*$.

## Seminormed and metrized ring $\infty$-categories

A ring $\infty$-category is a model of the Lawvere theory of rings (given by the category with finite products opposite to the category of finitely generated polynomial semirings over the initial ring $\mathbb{Z}$) with values in the category with products of $\infty$-categories. If we give such a rig $\infty$-category
$$\mathcal{A}:(T_{Rings},\times)\longrightarrow ({}^\infty Cat,\times),$$
we may associate to it a classical ring-category $h(\mathcal{A})$. A multiplicative seminorm on $\mathcal{A}$ will be a multiplicative seminorm on $h(\mathcal{A})$ in the sense of the article cited on this page. If $\mathcal{A}$ is a rig category in the classical sense, one may extend it by a kind of nerve construction to a rig $\infty$-category. Indeed, the nerve $N(\mathcal{A})$ of $\mathcal{A}$ gives an $\infty$-category. By sending $\mathbb{A}^n$ to $N(\mathcal{A})^n$, one defines a functor
$$N_{\oplus,\otimes}(\mathcal{A}):T_{SRings}\to {}^\infty Cat$$
given by sending a polynomial map
$f=\sum a_\alpha X^\alpha:\mathbb{A}^n\to \mathbb{A}^1$
to the corresponding functor
$$
N_{\oplus,\otimes}(f):=
(M_1,\dots,M_n)\mapsto \bigoplus_\alpha\oplus_{a_\alpha}(M_1,\dots,M_n)^{\otimes \alpha}.
$$

There is a natural functor from the category of (commutative) rings to the category of ring $\infty$-categories given by sending a ring $A$ to its category $FMod(A)$ of free finitely generated modules, together with their usual direct sum and tensor product. This functor is fully faithful.

Recall that the Berkovich spectrum of $FMod(\mathbb{Z})$ projects on the space of "Artin" seminorms, that fulfill a weak triangle inequality of the form
$$|a+b|\leq C\,\max(|a|,|b|).$$

A stable symmetric monoidal $\infty$-category is also a natural example of a ring $\infty$-category, with additive homotopy category (since the direct sum monoidal structure is given by the product, or equivalently coproduct, of the given $\infty$-category).

One may define the Berkovich spectrum of a ring $\infty$-category by setting
$$\mathcal{M}(\mathcal{A})=\mathcal{M}(h(\mathcal{A})).$$
This means that the Berkovich spectrum doesn't really encode the homotopy information. However, it is quite practical to work with rig $\infty$-categories because they have nice categorical properties such as (at least finite) limits and colimits.

We will also work with the notion of a ring metric on a ring category,
given by a ring norm on its homotopy category.

## Valuation rig $\infty$-categories

Given a multiplicative seminorm $|\cdot|$ on a rig $\infty$-category $\mathcal{A}$, one may define the associated generalized valuation ring as the rig $\infty$-category $O_\mathcal{A}$ whose homotopy category is given by
$$h(O_\mathcal{A}):=\{f\in h(\mathcal{A}),\;|f|\leq 1\}.$$
For example, Haran's definition of the ring of integers of $\mathbb{R}$ is given by
$$\mathbb{Z}_\eta:=O_{(FMod(\mathbb{R}),\|\cdot\|_2)}.$$
One may also define an ind-seminormed rig $\infty$-category
$$
\mathbb{Z}^\dagger_\eta:=\colim_{\rho\to 1}
O_{(FMod(\mathbb{R}),\|\cdot\|^\rho_2)}.
$$

Remark that the $L^2$ norm $\|\cdot\|_2$ on the rig category $FMod(\mathbb{R})$ may be extended to a family of norms on the rig category $FMod(\mathbb{R}\{X\}^\dagger)$, by putting on the free module $(\mathbb{R}\{X\}^\dagger)^n$ the norms given by
$$
\|(\sum a_{i,1} X^i,\dots,\sum a_{i,n} X^i)\|_{\rho,2}:=
\|(\sum |a_{i,1}|_\infty\rho^i,\dots,\sum |a_{i,n}|_\infty\rho^i)\|_2.
$$
This seminorm is only sub-multiplicative for the tensor product operation.
One may then give a construction of the ind-seminormed rig $\infty$-category
$$
\mathbb{Z}_\eta\{X\}^\dagger:=O_{FMod(\mathbb{R}\{X\}^\dagger)}.
$$

## Classical approach: the metrized spectrum

If $(\mathcal{A},d_{\mathcal{A}})$ is a metrized pointed ring category,
we may define its Berkovich spectrum as the set $X=\mathcal{M}(\mathcal{A},d_{\mathcal{A}}(0,\cdot))$ of multiplicative seminorms such that
$|\cdot|\leq d_\mathcal{A}(0,\cdot)$. One may define the presheaf of uniform limits (i.e., the completion) of rational functions without poles, and then germs $\mathcal{O}_x$ at a point $x\in X$ as equivalence classes of pairs $(V,(f_n))$ composed of an open neighborhood $V$ of $x$ and a Cauchy sequence of rational functions without poles $f_n\in \mathcal{K}(V)$.

We then define analytic functions as maps
$$f:U\to \coprod_{x\in U} \Hom_{\mathcal{O}_x}(M,N)$$
($M$ and $N$ are objects of $\mathcal{A}$, seen as objects of $\mathcal{O}_x$)
such that for every point $x\in U$, there is a neighborhood $V$ of $x$ such
that $f$ comes from the completion of rational functions without poles on $V$. The obtained sheaf $\mathcal{O}_X$ has a map from rational functions $\mathcal{K}_X$ by definition.

Morphisms of analytic spaces will be those induced by morphisms of
metrized ring categories.

Remark that the point $d_\infty^\infty$ doesn't even make sense in $\mathcal{D}(\mathbb{Z}_\eta,d_\infty)$, because $d_\infty^t(-1,1)$ tends to infinity.
However, the associated norm makes sense. Inspired by this remark,
we define the rig category of analytic functions on $\mathcal{M}(\mathcal{A},d_\mathcal{A}(0,-))$
as the direct image along
$$\phi:\mathcal{D}(\mathcal{A},d_\mathcal{A})\to \mathcal{M}(\mathcal{A},d_\mathcal{A}(0,-))$$
of the structural sheaf we just defined on the space of metrics.

## Functorial approach: derived stacks over metrized ringed categories

One may define a natural class of ind-metrized ring categories
by the following process: first, one defines
the category of metrized power series ring categories over
$\mathbb{Z}$ in the usual way, using the $L^2$-norm and its real powers.
Then one defines power series ring categories over $\mathbb{Z}_\eta$ as
valuation ring categories of power series ring categories over $\mathbb{R}$.
Then we define rational domain algebras over both types of
power series rings, as coequalizers of morphisms such as $fX$ and $g$
with $fX,g\in \mathrm{End}(\mathbb{1})$. This is where we need the metric
structure (because the norm will not allow completion) on morphism spaces.
This gives a subcategory of ind-metrized ring categories called
rational domain algebra categories.

One then gets inspiration from Dubuc's work on synthetic analytic geometry and Lurie's work on general pre-geometries to define general analytic ring categories
as arbitrary colimits (in the category of ind-metrized ring categories) of rational domain algebra categories, or maybe, as functors of functions that commute with products and standard pullbacks along rational domain embeddings. One may then define a Grothendieck topology on analytic ring categories and use it to define sheaves (functors of points) that will extend (in a sense to be explained) global analytic spaces (in the sense of the paper cited down this page) to a wider class of spaces.

In this new category $\An^\dagger_{\mathbb{F}_{\{\pm 1\}}}$, we may define,
for example
$$
\overline{\mathcal{M}(\mathbb{Z})}:=
\{|2|\geq 3/2\}\coprod_{\{2\geq |2|\geq 3/2\}}\{2\geq |2|\},
$$
where the left domain is from $\mathcal{M}(\mathbb{Z}_\eta)$ and the two other domains are from $\mathcal{M}(\mathbb{Z})$.

The gives a category $An^{\dagger,s}_\overline{\mathcal{M}(\mathbb{Z})}$ of Arakelov-type varieties.

## Adelic description of principal $\GL_n$-bundles on $\overline{\mathcal{M}(\mathbb{Z})}$

The main motivation for compactifying global analytic spaces using the ring categories approach is to get a geometric interpretation of the standard
adelic quotient
$$
\GL_n(\mathbb{Q})\backslash\GL_n(\mathbb{A})/K,
$$
where $K$ is the maximal compact subgroup of the adelic group,
as a moduli space of bundles.

An $F$-rig category is a rig category whose objects are all isomorphic
to objects of the form $\mathbb{1}^{\oplus n}$ for $n\in \N$, with
the convention that $\mathbb{1}^{\oplus 0}:=0$.

Let $\GL_n$ be the functor on rig categories that sends
$(\mathcal{A},\oplus,\otimes)$ to
$$\GL_n(\mathcal{A}):=\Isom_\mathcal{A}(\mathbb{1}^{\oplus n},\mathbb{1}^{\oplus n}).$$
A principal $\GL_n$-bundle on a generalized analytic space $X$ is a space
$F\to X$ that is isomorphic, locally on $X$, to the space $\GL_n$.
One may actually define an $\infty$-stack $B\GL_n$ by sheafifying the nerve
of the groupoid $B\GL_n$.

We denote $\Bun_n(\cdot):=\Hom_{St}(\cdot,B\GL_n)$
the $\infty$-stack of $\GL_n$-bundles on generalized analytic spaces.

By construction, we will have
$$
\pi_0(\Bun_n(\overline{\mathcal{M}(\mathbb{Z})}))\cong \GL_n(\mathbb{Q})\backslash\GL_n(\mathbb{A})/K,
$$
where $K$ is the maximal compact subgroup of the adelic group.
## References

[[Frédéric Paugam]] _Analytic spectrum of rig categories_ [TAC](http://www.tac.mta.ca/tac/volumes/29/6/29-06abs.html)

[[Frédéric Paugam]] _Overconvergent global analytic geometry_ [arXiv](http://arxiv.org/abs/1410.7971)