geometric embedding

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###Context###
#### Topos Theory
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[[!include topos theory - contents]]
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#Contents#
* automatic table of contents
{:toc}

## Idea

A **geometric embedding** is the right notion of embedding or inclusion of [[topoi]] $F \hookrightarrow E$, i.e. of [[subtoposes]].

Notably the inclusion $Sh(S) \hookrightarrow PSh(S)$ of a [[category of sheaves]] into its [[presheaf]] [[topos]] or more generally the inclusion $Sh_j E \hookrightarrow E$ of sheaves in a topos $E$ into $E$ itself, is a geometric embedding. Actually every geometric embedding is of this form, up to equivalence of [[topos|topoi]].

Another perspective is that a geometric embedding $F \hookrightarrow E$ is the [[localizations]] of $E$ at the class $W$ or morphisms that the [[left adjoint]] $E \to F$ sends to isomorphisms in $F$.

The induced geometric morphism of a topological immersion $X \hookrightarrow Y$ is a geometric embedding. The converse holds if $Y$ is a $T_0$ space. (Example A4.2.12(c) in ([Johnstone](#Johnstone)))

## Definition

For $F$ and $E$ two [[topos|topoi]], a [[geometric morphism]]

$$
  F \stackrel{f}{\to}
  E
  \;\;\;\;
  F \stackrel{\stackrel{f_*}{\longrightarrow}}{\underset{f^*}{\longleftarrow}} E
$$

is a **geometric embedding** if the following equivalent conditions are satisfied

* the [[direct image]] functor $f_*$ is [[full and faithful functor|full and faithful]] (so that $F$ is a full [[subcategory]] of $E$);

* the counit $\epsilon :  f^* f_* \to Id_{F}$ of the [[adjoint functor|adjunction]] $(f^* \dashv f_*)$ is an [[isomorphism]]

* there is a [[Lawvere-Tierney topology]] on $E$ and an [[equivalence of categories]] $e : F \stackrel{\simeq}{\to} Sh_j E$ such that the diagram of geometric morphisms $\array{ F &\stackrel{f_*}{\to}& E \\ & {}_{e}\searrow^\simeq & \uparrow^{i} \\ && Sh_j E}$ commutes up to natural isomorphism $e^* i^* \simeq f^*$

That the first two conditions are equivalent is standard, that the third one is equivalent to the first two is for instance corollary 7 in section VII, 4 of ([MacLaneMoerdijk](#MacLaneMoerdijk))

## Properties

### Relation to localization

There is a close relation between geometric embedding and
[[localization]]: [[reflective localization]].

Let $f : F \hookrightarrow E$ be a geometric embedding
and let $W \subset Mor(E)$ be the class of morphisms sent by $f^*$ to isomorphisms in $F$.

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

We have:

* $F$ is equivalent to the [[localization]] $E[W]^{-1}$;

* $F$ is equivalent to the [[full subcategory]] of $E$ on $W$-[[local objects]].

=--

This fact connects for instance the description of [[sheafification]] in terms of geometric embedding $Sh(S) \hookrightarrow PSh(S)$ as described for instance in

* MacLane-Moerdijk, [[Sheaves in Geometry and Logic]]

with that in terms of [[localization]] at [[local isomorphisms]], as described in

* Kashiwara-Schapira, [[Categories and Sheaves]].

Moreover, this is the basis on which sheafification is generalized to [[(∞,1)-sheafification]] in

* [[Jacob Lurie]], [[Higher Topos Theory]] .

The following gives a detailed proof of the above assertion.

Write $\eta : Id_E \to f_* f^*$ for the unit of the
adjunction.

Since $f_*$ is fully faithful we will identify objects and
morphism of $F$ with their images in $E$. To further trim down the notation write $\bar {(-)} := f^*$ for the left adjoint.

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

Write $W$ for the class of morphism that are sent to isomorphism under $f^*$,

$$
  W = (f^*)^{-1}\{g: c\stackrel{\simeq}{\to} d \in Mor(E)\}
  \,.
$$

=--

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

$E$ equipped with the class $W$ is a [[category with weak equivalences]], in that $W$ satisfies 2-out-of-3.

=--

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

Follows since isomorphisms satisfy 2-out-of-3.

=--

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

$W$ is a left [[calculus of fractions|multiplicative system]].

=--

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

This follows using the fact that $f^*$ is left exact and hence preserves finite limits.

In more detail:

We have already seen in the previous proposition that

* every isomorphism is in $W$;

* $W$ is closed under composition.

It remains to check the following points:

Given any

$$
  \array{
     && a
     \\
     && \downarrow^w
     \\
     b &\stackrel{h}{\to}& c
  }
$$

with $w \in W$, we have to show that there is

$$
  \array{
     d &\to& a
     \\
     \downarrow^{w'} && \downarrow^w
     \\
     b &\stackrel{h}{\to}& c
  }
$$

with $w' \in W$.

To get this, take this to be the [[pullback]] diagram,
$w' := h^* w$.
Since $f^*$ preserves pullbacks, it follows that

$$
  \array{
     \bar d &\to& \bar a
     \\
     \downarrow^{\bar w'} && \downarrow^{\bar w}
     \\
     \bar b &\stackrel{\bar h}{\to}& \bar c
  }
$$

is a pullback diagram in $F$ with $\bar w' = \bar h^* \bar w$. But by assumption $\bar w$ is an isomorphism. Therefore $\bar w'$ is an isomorphism, therefore $w'$ is in $W$.

Finally for every

$$
  a \stackrel{\stackrel{r}{\to}}{\stackrel{s}{\to}} b \stackrel{w}{\to} c
$$

with $w \in W$ such that the two composites coincide,
we need to find

$$
  d \stackrel{w'}{\to} a \stackrel{\stackrel{r}{\to}}{\stackrel{s}{\to}} b
$$

with $w' \in W$ such that the composites again coincide.

To get this, take $w'$ to be the [[equalizer]]
of the two morphisms. Sending everything with $f^*$ to $F$ we find from

$$
  \bar a \stackrel{\stackrel{\bar r}{\to}}{\stackrel{\bar s}{\to}} b \stackrel{\bar w}{\to} c
$$

that $\bar r = \bar s$, since $\bar w$ is an isomorphism. This implies that $\bar w'$ is the equalizer

$$
  \bar d \stackrel{\bar w'}{\to} a \stackrel{\stackrel{\bar r}{\to}}{\stackrel{\bar s}{\to}} b
$$

of two equal morphism, hence an identity. So $w'$ is in $W$.

=--

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

For every object $a \in E$

* the unit $\eta_a : a \to \bar a$ is in $W$;

* if $a$ is already in $F$ then the unit is already an isomorphism.

=--

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

This follows from the [[triangle identities]]
of the [[adjoint functors]].

$$
  \array{
     &
     \nearrow &\Downarrow^{\eta}& \searrow^{Id_E}
     \\
     E
       &\stackrel{\bar{(-)}}{\to}&
     F
       &\hookrightarrow&
     E
       &\stackrel{\bar{(-)}}{\to}&
     F
     \\
     &&&
     \searrow &\Downarrow^{\simeq}& \nearrow_{Id_F}
  }
  \;\;\;\;
  =
  \;\;\;\;
  \array{
     & \nearrow \searrow^{\bar{(-)}}
    \\
    E
    &\Downarrow^{Id}&
    F
    \\
    & \searrow \nearrow_{\bar{(-)}}
  }
$$

and

$$
  \array{
     &&&
     \nearrow &\Downarrow^{\eta}& \searrow^{Id_E}
     \\
     F
       &\hookrightarrow&
     E
       &\stackrel{\bar{(-)}}{\to}&
     F
       &\hookrightarrow&
     E
     \\
     &
     \searrow &\Downarrow^{\simeq}& \nearrow_{Id_F}
  }
  \;\;\;\;
  =
  \;\;\;\;
  \array{
     & \nearrow \searrow
    \\
    F
    &\Downarrow^{Id}&
    E
    \\
    & \searrow \nearrow
  }
$$

In components they say that

* for every $a \in E$ we have $(\bar a \stackrel{\bar \eta_a}{\to} \bar{\bar a} \stackrel{\simeq}{\to} \bar a) = Id_{\bar a}$

* for every $a \in F$ we have $(a \stackrel{\eta_a}{\to} \bar a \stackrel{\simeq}{\to} a)  = Id_a$

This implies the claim.

=--

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

An object $a \in E$ is **$W$-[[local object]]** if for every
$g : c \to d$ in $W$ the map

$$
  g^* : Hom_E(d,a) \stackrel{\simeq}{\to}
        Hom_E(c,a)
$$

obtained by precomposition is an isomorphism.

=--

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

Up to isomorphism, the $W$-local objects are precisely the objects of $F$ in $E$

=--

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

First assume that $a \in F$. We need to show that $a$ is $W$-local.

Notice that the existence of the required isomorphism
$
  Hom_F(d,a) \simeq Hom_F(c,a)
$
is equivalent to the statement that for every diagram

$$
  \array{
     c &\stackrel{}{\to}& d
     \\
     \downarrow^{h}
     \\
     a
  }
$$

there is a unique extension

$$
  \array{
     c &\stackrel{}{\to}& d
     \\
     \downarrow^{h} & \swarrow
     \\
     a
  }
  \,.
$$

To see the existence of this extension, hit the original diagram with $f^*$ to get

$$
  \array{
     \bar c &\stackrel{\simeq}{\to}& \bar d
     \\
     \downarrow^{\bar h}
     \\
     \bar a \simeq a
  }
  \,.
$$

By the assumption that $c \to d$ is in $W$ the morphism $\bar c \to \bar d$ here is an isomorphism. By the assumption that $a$ is already in $F$ we have $\bar a \simeq a$ since the counit is an isomorphism. Therefore this diagram clearly has a unique extension

$$
  \array{
     \bar c &\stackrel{\simeq}{\to}& \bar d
     \\
     \downarrow^{\bar h} & \swarrow_{\exists ! k}
     \\
     \bar a \simeq a
  }
  \,.
$$

By the hom-isomorphism (using full faithfullness of $f_*$ to work entirely in $E$)

$$
  Hom_E(\bar d, a) \simeq Hom_E(d,a)
$$

this defines a morphism $k : d \to a$. Chasing $k$ through the naturality diagram of the hom-isomorphism

$$
  \array{
     Hom_E(\bar d, \bar a) &\stackrel{\simeq}{\to}& Hom_E(d,\bar a)
     \\
     \downarrow && \downarrow
     \\
     Hom_E(\bar c, \bar a) &\stackrel{\simeq}{\to}& Hom_E(c,\bar a)
  }
  \,.
$$

shows that $k : d \to a$ does extend the original diagram. Again by the Hom-isomorphism, it is the unique morphism with this property.

So $a \in F$ is $W$-local.

Now for the converse, assume that a given $a$ is $W$-local.

By one of the above propositions we know that the unit
$\eta_a : a \to \bar a$ is in $W$, so by the $W$-locality of
$a$ it follows that

$$
  \array{
     a &\stackrel{\eta_a}{\to}& \bar a
     \\
     \downarrow^{Id_a}
     \\
     a
  }
$$

has an extension

$$
  \array{
     a &\stackrel{\eta_a}{\to}& \bar a
     \\
     \downarrow^{Id_a} & \swarrow_{\rho_a}
     \\
     a
  }
  \,.
$$

By the 2-out-of-3 property of $W$ shown in one of the above propositions, (using that $Id_a$, being an isomorphism, is in $W$) it follows that $\rho_a : \bar a \to a$ is in $W$.

Since $\bar a$ is in $F$ and therefore $W$-local by the above, it follows that also

$$
  \array{
     \bar a &\stackrel{\rho_a}{\to}& a
     \\
     \downarrow^{Id_{\bar a}}
     \\
     \bar a
  }
$$

has an extension

$$
  \array{
     \bar a &\stackrel{\rho_a}{\to}& a
     \\
     \downarrow^{Id_{\bar a}} & \swarrow_{\lambda_a}
     \\
     \bar a
  }
  \,.
$$

So $\eta_a$ has a left inverse $\rho_a$ which itself has a left inverse $\lambda_a$. It follows that $\rho_a$ is also a right inverse to $\eta_a$, since

$$
  \begin{aligned}
    \stackrel{\rho_a}{\to}
    \stackrel{\eta_a}{\to}
    & =
    \stackrel{\rho_a}{\to}
    \stackrel{\eta_a}{\to}
    \underset{id}{\underbrace{
      \stackrel{\rho_a}{\to}
       \stackrel{\lambda_a}{\to}
    }}
   \\
    & =
    \stackrel{\rho_a}{\to}
    \underset{id}{\underbrace{
      \stackrel{\eta_a}{\to}
      \stackrel{\rho_a}{\to}
    }}
    \stackrel{\lambda_a}{\to}
    \\
    &=
    \stackrel{\rho_a}{\to}
    \stackrel{\lambda_a}{\to}
    \\
    &=
    Id
  \end{aligned}
  \,.
$$

So if $a$ is $W$-local we find that $\eta_a : a \to \bar a$ is an isomorphism, hence that $a$ is isomorphic to an object of $F$.

=--

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

$F$ is equivalent to the full [[subcategory]] $E_{W-loc}$ of $E$ on $W$-[[local objects]].

=--

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

By standard reasoning (e.g. [[Categories and Sheaves|KS lemma 1.3.11]]) there is a functor $F \to E_{W-loc}$ and a natural isomorphism

$$
  \array{
     F &&\hookrightarrow&& E
     \\
     & \searrow &\Downarrow^{\simeq}& \nearrow
     \\
     && E_{W-loc}
  }
  \,.
$$

Since $F \hookrightarrow E$ and $E_{W-loc} \hookrightarrow E$ are [[full and faithful functor|full and faithful]], so is $F \to E_{W-loc}$. Since by the above it is also essentially surjective, it establishes the equivalence $F \simeq E_{W-loc}$.

=--

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

$F$ is [[equivalence of categories|equivalent]] to the [[localization]] $E[W^{-1}]$ of $E$ at $W$.

=--

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

By one of the above propositions we know that
$W$ is a [[calculus of fractions|left multiplicative systems]].

This implies that the localization $E[W^{-1}]$ is (equivalent to) the category with the same objects as $E$, and with [[hom-sets]] given by

$$
  Hom_{E[W^{-1}]}(a,b)
  =
  \underset{a' \stackrel{p \in W}{\to}a}{colim} Hom_E(a',b)
  \,.
$$

There is an obvious candidate for a functor

$$
  F \to E[W^{-1}]
$$

given on objects by the usual embedding by $f_*$ and on morphism by the map which regards a morphism trivially as a [[span]] with left leg the identity

$$
  (a \to b)
   \;\;
   \mapsto
   \;\;
   \left(
   \array{
     a &\to& b
     \\
     \downarrow^{Id_a}
     \\
     a
  }
  \right)
  \,.
$$

For this to be an [[equivalence of categories]] we need to show that this is a [[essentially surjective functor|essentially surjective]] and [[full and faithful functor]].

To see essential surjectivity, let $a$ be any object in $E$ and let $\eta_a : a \to \bar a$ be the component of the unit of our adjunction on $a$, as above. By one of the above propositons, $\eta_a$ is in $W$. This means that the span

$$
  \array{
    a &\stackrel{Id_a}{\to}& a
    \\
    \downarrow^{\eta_a}
    \\
    \bar a
  }
$$

represents an element in $Hom_{E[W^{-1}]}(\bar a,a)$, and this element is clearly an isomorphism: the inverse is represented by

$$
  \array{
    a &\stackrel{\eta_a}{\to}& \bar a
    \\
    \downarrow^{Id_a}
    \\
    a
  }
  \,.
$$

Since every $\bar a$ is in the image of our functor, this shows that it is essentially surjective.

To see fullness and faithfulness, let $a, b\in F$ be any two objects. By one of the above propositions this means in particular that $b$ is a $W$-[[local object]]. As discussed above, this means that every span

$$
  \array{
    a' &\to& b
    \\
    \downarrow^w
    \\
    a
  }
$$

with $w \in W$ has a unique extension

$$
  \array{
    a' &\to& b
    \\
    \downarrow^w & \nearrow
    \\
    a
  }
  \,.
$$

But this implies that in the colimit that defines the [[hom-set]] of $E[W^{-1}]$ all these spans are identified
with spans whose left leg is the identiy. And these are clearly in bijection with the morphisms in $Hom_E(a,b) \simeq Hom_F(a,b)$ so that indeed

$$
  Hom_{E[W^{-1}]}(a,b)
  \simeq
  Hom_{F}(a,b)
$$

for all $a,b \in F$. Hence our functor is also full and faithful and therefore define an [[equivalence of categories]]

$$
  F \stackrel{\simeq}{\to} E[W^{-1}]
  \,.
$$

=--

### Factorizations and images

There is a [[factorization system on a 2-category|factorization system]] on the [[2-category]] [[Topos]] whose left class is the [[surjective geometric morphisms]] and whose right class is the geometric embeddings.  The factorization of a geometric morphism can be said to construct its [[image]] in the topos-theoretic sense.

See [[geometric surjection/embedding factorization]].

Moreover, each geometric embedding itself has a [[(dense,closed)-factorization]].

## Related concepts

In the more general context of [[(∞,1)-topos theory]] an **$(\infty,1)$-geometric embedding** is an [[(∞,1)-geometric morphism]]

$$
  (f^* \dashv f_*) : \mathcal{X} \stackrel{\leftarrow}{\hookrightarrow}
   \mathcal{Y}
$$

such that the [[right adjoint]] [[direct image]] $f_*$ is a [[full and faithful (∞,1)-functor]].

See [[reflective sub-(∞,1)-category]] for more details.

## References

Section VII, 4 of

* [[Saunders Mac Lane]] and [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_
{#MacLaneMoerdijk}

and section A4.2 of

* [[Peter Johnstone]], _[[Elephant|Sketches of an elephant: a topos theory compendium]]_.
{#Johnstone}

[[!redirects geometric embeddings]]

[[!redirects (∞,1)-geometric embedding]]
[[!redirects (infinity,1)-geometric embedding]]

[[!redirects sub-(∞,1)-topos]]
[[!redirects sub-(∞,1)-toposes]]
[[!redirects sub-(infinity,1)-topos]]
[[!redirects sub-(infinity,1)-toposes]]

[[!redirects sub-topos]]
[[!redirects sub-toposes]]
[[!redirects sub-topoi]]

[[!redirects subtopos]]
[[!redirects subtoposes]]
[[!redirects subtopoi]]

[[!redirects geometric inclusion]]
[[!redirects geometric inclusions]]


+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

By embedding [[quantum field theories]] into [[string theory]]  -- typically as the [[worldvolume]] theories of various [[branes]], e.g. [[super Yang-Mills theory]] on [[D-branes]], [[6d (2,0)-superconformal QFT]] on [[M5-branes]], or else at [[O-planes]] -- the various [[duality in string theory|dualities of string theory]] will relate different QFTs in way that are typically far from obvious from just looking at these QFTs themselves.

The investigation specifically of [[N=2 D=4 super Yang-Mills theory]] and [[N=1 D=4 super Yang-Mills theory]] in this fashion has come to be known as _geometric engineering of quantum field theory_ ([Katz-Klemm-Vafa 97](#KatzKlemmVafa97), [Katz-Klemm 96](#KatzKlemm96) ).

Specifically, the geometrically engineered QFTs are those on the [[worldvolume]] of [[black brane|black]] [[D-branes]] that end on (are suspended between) [[black brane|black]] [[NS5-branes]] (due to [Hanany-Witten 97](#HananyWitten97), review includes [Fazzi 17](#Fazzi17)). See at _[D-branes ending on NS5-branes](NS5-brane#DBranesEndingOnNS5Branes)_.

> graphics grabbed from [Fazzi 17, p. 25](#Fazzi17)

> graphics grabbed from [Fazzi 17, p. 32](#Fazzi17)

For more relations between QFTs found via string theory see at _[[string theory results applied elsewhere]]_.

## Related concepts

* [[quiver gauge theory]]

* [[(p,q)5-brane webs]]

* [[AdS-QCD correspondence]]

* [[M-theory on G2-manifolds]]

## References

### General

The original articles are

* {#KatzKlemmVafa97} [[Sheldon Katz]], [[Albrecht Klemm]], [[Cumrun Vafa]], _Geometric Engineering of Quantum Field Theories_, Nucl.Phys. B497 (1997) 173-195 ([arXiv:hep-th/9609239](http://arxiv.org/abs/hep-th/9609239))

* {#KatzKlemm96} [[Sheldon Katz]], [[Cumrun Vafa]], _Geometric Engineering of $N=1$ Quantum Field Theories_, Nucl.Phys. B497 (1997) 196-204 ([arXiv:hep-th/9611090](http://arxiv.org/abs/hep-th/9611090))

* {#HananyWitten97} [[Amihay Hanany]], [[Edward Witten]], _Type IIB Superstrings, BPS Monopoles, And Three-Dimensional Gauge Dynamics_, Nucl. Phys. B492:152-190, 1997 ([arXiv:hep-th/9611230](https://arxiv.org/abs/hep-th/9611230))

Review:

* [[Amit Giveon]], [[David Kutasov]], _Brane Dynamics and Gauge Theory_, Rev. Mod. Phys. 71:983-1084, 1999 ([arXiv:hep-th/9802067](https://arxiv.org/abs/hep-th/9802067) [doi](https://doi.org/10.1103/RevModPhys.71.983))

  (emphasis on the [[s-rule]])

* [[Andreas Karch]], _Field Theory Dynamics from Branes in String Theory_, PhD thesis (1998) ([doi:10.18452/14371](https://doi.org/10.18452/14371))

* [[David Morrison]], _Limitations of Geometric Engineering: Implications for Model Building_, talk (2008) [pdf slides](http://hans.math.upenn.edu/StringPhenom2008/slides/d.morrison.pdf)

* {#Fazzi17} [[Marco Fazzi]], _Higher-dimensional field theories from type II supergravity_ ([arXiv:1712.04447](https://arxiv.org/abs/1712.04447))

Further developments are in

* [[Balázs Szendrői]], _Nekrasov's Partition Function and Refined Donaldson-Thomas Theory: the Rank One Case_,  SIGMA, 2012, Volume 8, 088  ([doi](https://doi.org/10.3842/SIGMA.2012.088) [web](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=sigma&paperid=765&option_lang=eng))

### Examples
 {#Examples}

#### Geometric engineering of $D = 6$ $\mathcal{N} = (2,0)$ SCFT

For [[geometric engineering]] of the [[D=6 N=(2,0) SCFT]], see at _[duality between M-theory on Z2-orbifolds and type IIB string theory on K3-fibrations -- Geometric engineering of 6d (2,0)-SCFT](duality+between+M-theory+on+Z2-orbifolds+and+type+IIB+string+theory+on+K3-fibrations#GeometricEngineeringOfThe6d2SuperconformalQFT)_.

#### Geometric engineering of $D = 6$ $\mathcal{N} = (1,0)$ SCFT

On [[D=6 N=(1,0) SCFTs]] via [[geometric engineering of QFT|geometric engineering]] on [[M5-branes]]/[[NS5-branes]] at D-, E-type [[ADE-singularities]], notably from [[M-theory on S1/G_HW times H/G_ADE]], hence from [[orbifolds]] of [[type I' string theory]] (see at [half NS5-brane](NS5-brane#NSHalfBranes)):

* Michele Del Zotto, [[Jonathan Heckman]], [[Alessandro Tomasiello]], [[Cumrun Vafa]], _6d Conformal Matter_, JHEP02(2015)054 ([arXiv:1407.6359](https://arxiv.org/abs/1407.6359))

* {#GaiottoTomasiello14} [[Davide Gaiotto]], [[Alessandro Tomasiello]], _Holography for $(1,0)$ theories in six dimensions_, JHEP12(2014)003 ([arXiv:1404.0711](https://arxiv.org/abs/1404.0711))

* [[Kantaro Ohmori]], [[Hiroyuki Shimizu]], _$S^1/T^2$ Compactifications of 6d $\mathcal{N} = (1,0)$ Theories and Brane Webs_, J. High Energ. Phys. (2016) 2016: 24 ([arXiv:1509.03195](https://arxiv.org/abs/1509.03195))

* {#HKLY15} Hirotaka Hayashi, Sung-Soo Kim, Kimyeong Lee, Futoshi Yagi, _6d SCFTs, 5d Dualities and Tao Web Diagrams_, JHEP05 (2019)203 ([arXiv:1509.03300](https://arxiv.org/abs/1509.03300))

* Ibrahima Bah, Achilleas Passias, [[Alessandro Tomasiello]], _$AdS_5$ compactifications with punctures in massive IIA supergravity_, JHEP11 (2017)050 ([arXiv:1704.07389](https://arxiv.org/abs/1704.07389))

[[!redirects geometric engineering]]

[[!redirects geometric engineering of quantum field theories]]

[[!redirects geometric engineering of gauge theories]]

[[!redirects geometric engineering of QFT]]
[[!redirects geometric engineering of QFTs]]

[[!redirects Hanany-Witten theory]]
[[!redirects Hanany-Witten rule]]
[[!redirects Hanany-Witten rules]]


geometric fibre

#Contents#
* table of contents
{:toc}

##Idea

In [[algebraic geometry]] a _geometric fiber_ is a [[fiber]] of a [[bundle]] over a [[geometric point]].

For a [[bundle]] $p: E\to X$ of [[topological spaces]], the [[fibre]] over a [[point]], $x\in X$ may be thought of as the [[preimage]] $E_x = p^{-1}(x)$ equipped with its [[subspace topology]]. More abstract, this is the [[pullback]] ${x} \times_X E$ of $p$ along the map sending a singleton space $*$ to $x \in X$, the object which is [[universal property|universal]] with the property of making this [[commuting diagram|diagram commute]]:

$$
  \array{
    E_x &\to& E
    \\
    \downarrow && \downarrow^{\mathrlap{p}}
    \\
    {*} &\stackrel{x}{\to}& X
  }
  \,.
$$

Adapting this to the algebraic context we get the following definition.

## Definition

Let $X$ be a [[scheme]] over some base field $k$.  Fix an [[algebraic closure]] $\overline{k}$ of $k$ and let $\overline{\xi} : Spec(\overline{k}) \to X$ be a [[geometric point]] in $X$.

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

For a morphism, $p: E\to X$,  the **geometric fibre** over the [[geometric point]] $x$ is the [[pullback]]

$$
  E_x \coloneqq E \times_X Spec (\overline{k})
  \,.
$$

=--

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

In [[EGA]], the case is also considered in which the field is replaced by a [[local ring]], (see EGA p. 112), in which case the word 'geometric' is dropped. (This needs checking.)

=--

##Example and application

An important case is where $p$ is a finite [[étale cover]], and then the geometric fibre is just a [[finite set]].

##References

* [[EGA]] I 3.4.5, p. 112.

[[!redirects geometric fibrs]]
[[!redirects geometric fiber]]
[[!redirects geometric fibers]]


geometric fixed point spectrum

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

#Contents#
* table of contents
{:toc}

## Idea

In the context of [[equivariant stable homotopy theory]] and $G$ one distinguishes, for a [[G-spectrum]] $E$, between the plain  _[[fixed point spectrum]]_ $F^G(E)$ and its _geometric fixed point spectrum_ $\Phi^G(E)$.

Here the terminology "geometric" is in the sense of _[[point-set topology]]_, as opposed to [[homotopy theory]]: If $X$ is a ([[pointed topological space|pointed]]) [[topological space]] equipped with a [[continuous function]] $G$-[[action]] (a [[topological G-space]]), so that one may consider its $G$-[[equivariant suspension spectrum]] $\Sigma^\infty_G X \in G Spectra$, then the _geometric fixed point spectrum_ $\Phi^G(\Sigma^\infty_G X)$ of the latter is equivalent to the plain [[suspension spectrum]] of the plain [[fixed point]]-space $X^G$:

$$
  \Phi^G\big(  \Sigma^\infty_G X \big)
     \;\simeq\;
  \Sigma^\infty\big( X^G \big)
  \,,
$$

see the characterization in Prop. \ref{AbstractCharacterizationOfGeometricFixedPointSpectra}, below.

In general this is different from (not [[equivalence in an (infinity,1)-category|equivalent]] to) both of the following other notions of fixed point spectra:

1. the plain (really: [[homotopy theory|homotopy theoretic]]) [[fixed point spectrum]] $F^G(\Sigma^\infty_G X)$, which is instead the [[derived functor]] of forming topological fixed points $X \mapsto X^G$, hence which applies this construction only after [[fibrant resolution]]; the difference between the two is described by the _[[tom Dieck splitting theorem]]_, see Prop. \ref{AsWedgeSummandInTomDieckSplitting} below.

1. the [[categorical fixed point spectrum]]...

## Definition

A concrete definition of geometric fixed points of an equivariant spectrum is in ([Schwede 15, 7.3](#Schwede15)). A conceptual characterization in terms of [[localization of spectra]] is in ([Mathew-Naumann-Noel 15, def. 6.12](#MathewNaumannNoel15)).

## Properties

### For equivariant suspension spectra
 {#ForEquivariantSuspensionSpectra}

+-- {: .num_prop #AsWedgeSummandInTomDieckSplitting}
###### Proposition
**(as a [[wedge sum|wedge summand]] in the [[tom Dieck splitting]])**

For $X$ a [[topological G-space]] and $\Sigma^\infty_G X$ its [[equivariant suspension spectrum]], there is a canonical comparison morphism (...)

$$
  \Phi^G(\Sigma^\infty_G X)
  \;\simeq\;
  \Sigma^\infty( X^G )
  \hookrightarrow
  F^G(\Sigma^\infty_G X)
$$

which exhibits its geometric fixed point spectrum as precisely the first summand in the [[tom Dieck splitting]] of the plain [[fixed point spectrum]]

$$
  F^G(\Sigma^\infty_G X)
  \simeq
  \Sigma^\infty( X^G )
    \vee
  \left(
    \underset{{[H\subset G]} \atop {1 \neq H \neq G}}{\vee}
    \Sigma^\infty( E (W_G H)_+ \wedge_{W_G H} X^H )
  \right)
    \vee
  \Sigma^\infty( E G_+ \wedge_{G} X )
  \,.
$$

=--

([Schwede 15, Example 7.7](#Schwede15))

In fact:

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

The construction of geometric fixed point spectra is essentially uniquely characterized by the  property

$$
  \Phi^G\big(  \Sigma^\infty_G X \big)
     \;\simeq\;
  \Sigma^\infty\big( X^G \big)
$$

and the property of being [[left derived functor|left derived]] [[strong monoidal functor|strong monoidal]] and preserving [[homotopy colimits]].

=--

([Schwede 15, remark 7.15](#Schwede15), [Blumberg 17, around Def. 2.5.16](#Blumberg17))

More generally:

+-- {: .num_prop #PartialGeometricFixedPoint}
###### Proposition/Remark
**(partial geometric fixed point spectra)**

There is a "partial" geometric fixed point functor, which for a given [[subgroup]] $H \subset G$ sends

$$
  \Phi^H \;\colon\; G Spectra \longrightarrow W_G H Spectra
$$

(for $W_G/H$ the [[Weyl group]], which is the [[quotient group]] $G/H$ in the case that $H$ is a [[normal subgroup]]) and satisfies for a $G$-[[equivariant suspension spectrum]] $\Sigma^\infty_G X$ the relation

\[
  \label{PartialGeometricFixedPointsOfEqui}
  \Phi^N \big(  \Sigma^\infty_G X \big)
  \;\simeq\;
  \Sigma^\infty_{W_G H} X^H
  \,,
\]

hence, if $H = N \subset G$ already happens to be a [[normal subgroup]]:

\[
  \Phi^N \big(  \Sigma^\infty_G X \big)
  \;\simeq\;
  \Sigma^\infty_{G/N} X^H
  \,.
\]

=--

([Lewis-May-Steinberger 86, II.9, Def. 9.7, Cor. 9.9](#LewisMaySteinberger86), [Lewis 00, Scholium 10.2](#Lewis00))

$\,$

### In terms of smashing localization
 {#InTermsOfSmashingLocalization}

We collect some facts from [Lewis-May-Steinberger 86, section II.9](#LewisMaySteinberger86).

Throughout, consider a [[finite group]] $G$ and a [[normal subgroup]] $N \subset G$.

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

We write

$$
  \mathcal{F}[N]
  \;\coloneqq\;
  \big\{
    N ⊄ H\; \subset G
  \big\}
$$

for the set of [[subgroups]] of $G$ that do not contain $N$, and

$$
  \mathcal{F}[N]^'
  \;\coloneqq\;
  \big\{
    N \subset H \subset G
  \big\}
$$

for the subset of [[subgroups]] of $G$ that do contain $N$.

=--

([LMS86, p. 107 & bottom of p. 109](#LewisMaySteinberger86))

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

There is a [[pointed topological space|pointed]] [[G-space]]

$$
  \widetilde E \mathcal{F}[N]
  \;\in\;
  G Spaces
$$

whose [[fixed point spaces]] for [[subgroups]] $H \subset G$ are

$$
  \big(
    \widetilde E \mathcal{F}[N]
  \big)^H
  \;\simeq\;
  \left\{
    \array{
      \ast
        &\vert&
      H \in \mathcal{F}[N]
        \;\Leftrightarrow\;
      N ⊄ H
      \\
      S^0
        &\vert&
      H \in \mathcal{F}[N]^'
        \;\Leftrightarrow\;
      N \subset H
    }
  \right.
$$

=--

([LMS86, beginning of II.9](#LewisMaySteinberger86))

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

We say that a morphism $f \colon X \to Y$ of [[G-spectra]] is an _$\mathcal{F}[N]^'$-equivalence_ if its [[smash product]] with $\tilde E \mathcal{F}[N]$ (Def. \ref{tildeE})

$$
  f \wedge id_{\tilde E \mathcal{F}[N]}
  \;\colon\;
  X \wedge \tilde E \mathcal{F}[N]
   \longrightarrow
  Y \wedge \tilde E \mathcal{F}[N]
$$

is an equivalence of [[G-spectra]].

=--

[LMS 86, bottom of p. 107](#LewisMaySteinberger86)

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

The [[localization]] of $G Spectra$ at the $\mathcal{F}[N]'$-equivalences (Def. \ref{FPrimeEquivalences}) is a [[smashing localization]], given by smashing with the [[equivariant suspension spectrum]] of $\tilde E \mathcal{F}[N]$ (Def. \ref{tildeE})

$$
  \Sigma^\infty_G \tilde E \mathcal{F}[N]
  \;\in\;
  G Spectra
$$

In particular, we have

$$
  Ho_{G Spectra}\left(
    \tilde E \mathcal{F}[N] \wedge X,
    \tilde E \mathcal{F}[N] \wedge Y
  \right)
  \;\simeq\;
  Ho_{G Spectra}\left(
    X,
    \tilde E \mathcal{F}[N] \wedge Y
  \right)
$$

=--

([Lewis-May-Steinberger 86, Prop. II 9.1, 9.2 & top of p. 109](#LewisMaySteinberger86))

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

Hence

$$
  (L_{\mathcal{F}[N]'})_{X,Y}
  \coloneqq
  Ho_{G Spectra}(X, (S^0 \to \tilde E \mathcal{F}) \wedge Y)
  \;\colon\;
  Ho_{G Spectra}(X,Y)
    \longrightarrow
  Ho_{L_{\mathcal{F}[N]'} G Spectra}(X,Y)
$$

is $\mathcal{F}'[N]$-localization on [[hom-objects]].

=--

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

For $X$ and $Y$ [[G-CW-complexes]], the following are [[bijections]] of [[hom-sets]]:

$$
  \array{
  Ho_{G Spaces)}
  \big(
    X, \widetilde E \mathcal{F}[N] \wedge Y
  \big)
  &
  \underoverset{\simeq}{
    Ho_{G Spaces}
    \big(
      X^N \hookrightarrow X, \widetilde E \mathcal{F}[N]
    \big)
  }{\longrightarrow}
   &
  \mathrm{Hom}_{Ho(G Spaces)}
  \big(
    X^N, \widetilde E \mathcal{F}[N] \wedge Y
  \big)
  \\
  && =
  \\
  Ho_{G Spaces}
  \big(
    X^N, Y
  \big)
  &
  \underoverset{\simeq}{
    Ho_{G Spaces}
    \big(
      X^N, (S^0 \to \tilde E \mathcal{F}[N]) \wedge Y
    \big)
    }{\longrightarrow}
  &
  Ho_{G Spaces}
  \big(
    X^N, \widetilde E \mathcal{F}[N] \wedge Y
  \big)
  \\
  =
  \\
  Ho_{G Spaces}
  \big(
    X^N, Y^N
  \big)
  }
$$

=--

([LMS 86, prop. II 9.3 with remark below the proof](#LewisMaySteinberger86))

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

On [[hom-sets]] of [[G-spaces]] $Ho_{G Spaces}(X,Y)$, [[postcomposition|postcomposing]] with the [[smash product|smashing]] $(S^0 \to \tilde E \mathcal{F}[N]) \wedge Y$ is isomorphic to restricting along $X^N \hookrightarrow X$: The following is a [[commuting square]] (by nature of the [[hom-functor]]) and the right and bottom morphisms are bijections by Lemma \ref{SmashingWithTildeOnSpaces}:

$$
  \array{
    Ho_{G Spaces}
    \big(
      X, Y
    \big)
    &
    \overset{
      Ho_{G Spaces}
      \big(
        X^N \hookrightarrow N, Y
      \big)
    }{ \longrightarrow }
    &
    Ho_{G Spaces}\big( X^N, Y \big)
    &\simeq&
    Ho_{G Spaces}\big( X^N, Y^N \big)
    \\
    {}^{
      \mathllap{
        Ho_{G Spaces}( X, (S^0 \to \tilde E \mathcal{F}[N]) \wedge Y )
      }
    }
    \big\downarrow
    &&
    {}^{\mathllap{\simeq}}\big\downarrow
    {}^{
      \mathrlap{
        Ho_{G Spaces}( X^N, (S^0 \to \tilde E \mathcal{F}[N]) \wedge Y )
      }
    }
    \\
    Ho_{G Spaces}\big( X, \tilde E \mathcal{F}[N] \wedge Y  \big)
    &
      \underoverset
        {\simeq}
        {
          Ho_{G Spaces}
          \big(
            X^N \hookrightarrow X, \tilde E \mathcal{F}[N] \wedge Y
          \big)
        }
        {\longrightarrow}
    &
    Ho_{G Spaces}\big( X^N, \tilde E \mathcal{F}[N] \wedge Y  \big)
  }
$$

=--

+-- {: .num_lemma #GeometricFixedPointsInTermsOfPlainFixedPoints}
###### Lemma
**(geometric fixed point spectra in terms of homotopy fix point spectra)**

The partial geometric fixed point functor (Prop. \ref{PartialGeometricFixedPoint})

$$
  \Phi^N
  \;\colon\;
  G Spectra
    \longrightarrow
  G/N Spectra
$$

is given on [[equivariant suspension spectra]] $\Sigma^\infty_G X$ equivalently by first smashing with $\tilde E \mathcal{F}[N]$ (Def. \ref{tildeE}) followed by forming the partial plain [[fixed point spectrum]]:

$$
  \Phi^N \Sigma^\infty_G X \;\simeq\;
  \big(
    \tilde E \mathcal{F}[N]
    \;\wedge\;
    \Sigma^\infty_G X
  \big)^N
  \,.
$$

=--

([Lewis-May-Steinberger 86, Cor. 9.9](#LewisMaySteinberger86))

We will also need this here:

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

For $X$ a [[G-CW-complex]] $E$ a [[G-spectrum|G-]] [[CW-spectrum]] and $N \subset G$ a [[normal subgroup]], the partial $N$-[[fixed point spectrum]] functor on spectra and the plain fixed point functor on spaces are compatible with smash product:

$$
  \left(
     \tilde E \mathcal{F}[N]
     \wedge E
     \wedge X
  \right)^N
  \;\simeq\;
  \left(
     \tilde E \mathcal{F}[N]
     \wedge E
  \right)
     \wedge X^N
$$

=--

([Lewis-May-Steinberger 86, prop. II 9.12](#LewisMaySteinberger86))

### Via inversion of Euler classes
 {#AsInversionOfEulerClasses}

We discuss an explicit formula (Prop. \ref{GeometricFixedPointCohomologtIsColimitOverSmashWithEulerClasses} below, due to [Lewis-May-Steinberger 86](#LewisMaySteinberger86)) that expresses [[equivariant cohomology|equivariant]] [[cohomology groups]] with [[coefficients]] in partial geometric fixed point spectra (Prop. \ref{PartialGeometricFixedPoint}) as the [[equivariant cohomology|equivariant]] [[cohomology groups]] with [[coefficients]] in the original spectrum, but with certain "[[Euler classes]] [[localization|inverted]]".

As an application, we show (Example \ref{EquivariantStableCohomotopyOfPointsInNontrivalROGDegree} below) that the [[equivariant stable cohomotopy]] of the point in certain non-trivial [[RO(G)-degrees]] $V$ surjects onto the corresponding partially equivariant stable cohomotopy in degree 0 (the latter being well-understood: given by the [[Burnside ring]], by [[Burnside ring is equivariant stable cohomotopy of the point|this Prop]]).

$\,$

A key role in this discussion is played by those [[RO(G)-degrees]] which trivialize when passing to partial fixed points:

+-- {: .num_defn #ROGDegreeWithoutNonTrivialHFixedPoints}
###### Definition
**([[RO(G)-degrees]] without non-trivial $H$-fixed points)**

For $H \subset G$ a [[subgroup]], say that a $G$-[[representation]] $V$ has _no non-trivial $H$-fixed points_ if the [[fixed point space]] of $V$ with respect to the $H$-[[action]] is the origin (which is necessarily fixed), hence is the [[zero object|zero]]-[[representation]]:

$$
  V^H \;=\; 0
$$

=--

We also use the following notation, following [Lewis-May-Steinberger 86](#LewisMaySteinberger86):

+-- {: .num_defn}
###### Definition
**([[base change]] along [[normal subgroup]]-inclusions of [[equivariant homotopy theory|equivariance]]-groups)**

Given a [[normal subgroup]]-inclusion

$$
  N \overset{\phantom{AA}}{\hookrightarrow} G
  \overset{ \phantom{A} \epsilon \phantom{A} }{\longrightarrow}
  G/N
$$

with induced [[projection]] $\epsilon$ to the [[quotient group]] $G/N$
this induces various [[base change]] [[adjunctions]] (on [[homotopy categories]], say), such as on [[topological G-spaces]], to be denoted

\[
  \label{PullbackGSpace}
  G/N Spaces
  \overset{ \epsilon^\sharp }{\longrightarrow}
  G Spaces
\]

and on $G$-[[representations]], to be denoted

\[
  \label{PullbackRepresentation}
  G/N Rep
      { \overset{ \epsilon^\ast }{\longrightarrow} }
  G Rep
\]

and on [[G-spectra]], to be denoted

\[
  \label{PullbackSpectrum}
  G/N Spectra
    \underoverset
      { \underset{(-)^N}{\longleftarrow} }
      { \overset{ \epsilon^\sharp }{\longrightarrow} }
      {\phantom{AA}\bot\phantom{AA}}
  G Spectra
\]

where the [[right adjoint]] $(-)^N$ is the partial [[fixed point spectrum]]-functor (in contrast to the _geometric_ fixed point functor).

=--

(e.g. [Lewis-May-Steinberger 86, above theorem 9.5](#LewisMaySteinberger86))

+-- {: .num_prop #GeometricFixedPointCohomologtIsColimitOverSmashWithEulerClasses}
###### Proposition
**(partial geometric fixed point cohomology via inversion of Euler classes)**

Let $E \;\in\; G Spectra$ be a [[G-spectrum]] with partial geometric fixed point spectrum $\Phi^N E \;\in\; G/N Spectra$ (Prop. \ref{PartialGeometricFixedPoint})
and let
$
  X
  \;\in\;
  G/N Spectra^{fin}
  \overset{\epsilon^\sharp}{\longrightarrow}
  G Spectra
$
be [[finite spectrum|finite]] $G$-[[CW-spectrum]].

Then the $G/N$-[[equivariant cohomology|equivariant]] [[cohomology groups]] in [[RO(G)-degree|RO(G/N)-degree]] $\alpha$ of $X$ with [[coefficients]] in the partial geometric fixed point spectrum $\Phi^N E$ are equivalently the [[colimit]] over the $G$-[[equivariant cohomology|equivariant]] [[cohomology groups]] of $\epsilon^\sharp X$ (eq:PullbackGSpace) with [[coefficients]] in $E$, but in [[RO(G)-degree]] $\epsilon^\ast \alpha$ (eq:PullbackRepresentation) plus a shift by all those [[representations]] $V$ that have no nontrivial $N$-fixed points (Def. \ref{ROGDegreeWithoutNonTrivialHFixedPoints}):

\[
\label{ColimitConstructionForCohomologyWithCoeffsInPartialGeometricFixedPointSpectra}
  (\Phi^N E)^\bullet_{G/N}(X)
  \;\simeq\;
  \underset{\underset{\{V \vert V^N = 0\}}{\longrightarrow}}{\lim}
  E_G^{\epsilon^\ast \bullet + V}(\epsilon^\sharp X)
  \,,
\]

where the [[colimit]] is over the [[diagram]] that has precisely one morphism for every inclusion  $V_1 \subset V_2$ of $G$-representations without non-trivial $N$-fixed points (Def. \ref{ROGDegreeWithoutNonTrivialHFixedPoints})

$$
  E_G^{\epsilon^\ast \bullet + V_1}(\epsilon^\sharp X)
  \overset{ (-) \wedge \chi_{V_2 - V_1} }{\longrightarrow}
  E_G^{\epsilon^\ast \bullet + V_2}(\epsilon^\sharp X)
$$

given by [[smash product]] with the [[Euler class]]

$$
  \chi_{V}
  \;\coloneqq\;
  1
  \in
  E_G^V(S^V)
  \overset{ (S^0 \to S^V)^\ast }{\longrightarrow}
  E_G^V(S^0)
$$

of $V \coloneqq V_2 - V_1$.

=--

([Lewis-May-Steinberger 86, chapter II, prop. 9.13](#LewisMaySteinberger86))

+-- {: .num_prop #CanonicalComparisonMap}
###### Proposition
**(comparison map to partial geometric fixed point cohomology)**

Prop. \ref{GeometricFixedPointCohomologtIsColimitOverSmashWithEulerClasses} provides a canonical comparison morphism, to be denoted

\[
  \label{ComparisonMap}
  E_G^{\epsilon^\ast \bullet}(\epsilon^\sharp X)
  \overset{
    \phantom{AA}
      p_E^N(X)
    \phantom{AA}
  }{\longrightarrow}
  (\Phi^N E)^\bullet_{G/N}(X)
\]

from the $G$-[[equivariant cohomology|equivariant]] [[cohomology groups]] with [[coefficients]] in $E$ to those with [[coefficients]] in the partial geometric fixed point spectrum: Namely the component of the [[colimit|colimiting]] [[cocone]](eq:ColimitConstructionForCohomologyWithCoeffsInPartialGeometricFixedPointSpectra) at stage $V = 0$:

$$
  \array{
    E^{\epsilon^\ast \bullet }(\epsilon^\sharp X)
    && \overset{ (-) \wedge \chi_{V} }{\longrightarrow} &&
    E^{\epsilon^\ast \bullet + V}(\epsilon^\sharp X)
    &\to&
    \cdots
    \\
    & {}_{ \mathllap{ p_E^N(X) } }\searrow && \swarrow & \cdots
    \\
    &&
    (\Phi^N E)^\bullet_{G/N}(X)
  }
$$

This component is equal to the following composite of isomorphisms with $\mathcal{F}[N]'$-localization $L_{\mathcal{F}[N]'}$ (Def. \ref{FPrimeEquivalences}):

\[
  \label{LocalizationIsComparisonMorphism}
  \begin{aligned}
  p_E^N(X)
  \;\colon\;
  E^{\epsilon^\ast \alpha}(\epsilon^\sharp X)
  & =
  Ho_{G Spectra}\left(
    \epsilon^\sharp \Sigma^\infty_{G/N} X
    \;,\;
    \Sigma^\infty_G S^{\epsilon \alpha} \wedge E
  \right)
  \\
  & \simeq
  Ho_{G Spectra}\left(
    \epsilon^\sharp \Sigma^\infty_{G/N} X
    \;,\;
    \Sigma^\infty_G S^{\epsilon \alpha} \wedge S^0 \wedge E
  \right)
  \\
  &
  \overset{ L_{\mathcal{F}[N]'} }{\longrightarrow}
  Ho_{G Spectra}\left(
    \epsilon^\sharp \Sigma^\infty_{G/N} X
    \;,\;
    \Sigma^\infty_G S^{\epsilon \alpha} \wedge \tilde E \mathcal{F}[N]\wedge E
  \right)
  \\
  & \simeq
  Ho_{G/N Spectra}\left(
     \Sigma^\infty_{G/N} X
    \;,\;
    \left(
       \Sigma^\infty_G S^{\epsilon^\ast \alpha}
         \wedge
       \tilde E \mathcal{F}[N] \wedge E
    \right)^N
  \right)
  \\
  & \simeq
  Ho_{G/N Spectra}\left(
     \Sigma^\infty_{G/N} X
    \;,\;
    \left(
      S^{\epsilon^\ast \alpha}
    \right)^N
         \wedge
    \left(
       \tilde E \mathcal{F}[N] \wedge E
    \right)^N
  \right)
  \\
  & \simeq
  Ho_{G/N Spectra}\left(
     \Sigma^\infty_{G/N} X
    \;,\;
    S^{\alpha}
    \wedge
    \Phi^N E
  \right)
  \\
  & =
  (\Phi^N E)^{\alpha}(X)
  \end{aligned}
\]

=--

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

This follows from the proof of ([Lewis-May-Steinberger 86, chapter II, prop. 9.13](#LewisMaySteinberger86)). We make this explicit: The proof there says that the comparison map is given by the [[smashing localization|smashing]] with $S^0 \to \tilde E \mathcal{F}$, up to re-identifications:

1. The first equality in (eq:LocalizationIsComparisonMorphism) is the definition of cohomology classes;

1. the second step  is the [[unitor]] isomorphism for the [[tensor unit]] being the [[sphere spectrum]];

1. the third step is smashing with $S^0 \to \tilde E \mathcal{F}[N]$, which is $\mathcal{F}'$-localization by Prop. \ref{FPrimeLocalizationIsSmashingLocalization};

1. the fourth step is the hom-isomorphism for the [[adjoint functor|adjunction]] $( \epsilon^\sharp \dashv (-)^N )$ from (eq:PullbackSpectrum);

1. the fifth step is application of Lemma \ref{FixedPointSpectrumOfSmashProduct};

1. the sixth step is the evident identification $(S^{\epsilon^\ast \alpha})^N = S^\alpha$ in the first smash factor, and is Lemma \ref{GeometricFixedPointsInTermsOfPlainFixedPoints} in the second factor.

1. the seventh step is again the definition of cohomology.

=--

$\,$

## Examples

+-- {: .num_example #EquivariantStableCohomotopyOfPointsInNontrivalROGDegree}
###### Example
**([[equivariant stable cohomotopy]] of the point in non-trivial [[RO(G)-degree]])**

Let $G$ be a [[finite group]]. Then the canonical comparison morphism (eq:ComparisonMap) from Def. \ref{CanonicalComparisonMap} exhibits the $G$-[[equivariant stable cohomotopy]] [[cohomology groups|group]] of the point in any [[RO(G)-degree]] $V$ that has trivial $N$-fixed points ($V^N = 0$, Def. \ref{ROGDegreeWithoutNonTrivialHFixedPoints}) as a [[group extension]] of the $G/N$-[[equivariant stable cohomotopy]] of the point in [[RO(G)-degree|RO(G/N)-degree]] zero, hence of the group underlying
the [[Burnside ring]] $A(G/N)$ ([[Burnside ring is equivariant stable cohomotopy of the point|this Prop.]]):

\[
  \label{SurjectionFromEquivariantStableCohomotopyInDegreeVToDegreeZero}
  \mathbb{S}_G^{V}(\ast)
  \overset
    { \phantom{A} \text{epi} \phantom{A} }
    {\longrightarrow}
  \mathbb{S}_{G/N}^0(\ast)
  \simeq
  A(G/N)
  \,.
\]

=--

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

First observe that, in the given situation, the comparison morphism $p_{\mathbb{S}}^N(\ast)$ (eq:ComparisonMap) is indeed of the form shown, up to isomorphism: We are in the situation of Prop. \ref{GeometricFixedPointCohomologtIsColimitOverSmashWithEulerClasses} for

1. $X \coloneqq \Sigma^\infty_{G/N}(\ast_+) = \Sigma^\infty_{G/N} S^0$, which is clearly a [[finite spectrum|finite]] $G/N$-[[CW-spectrum]];

1. $E \coloneqq \Sigma^V_G\mathbb{S}_G \coloneqq \Sigma^\infty_G S^V$ the $V$-shifted $G$-[[equivariant sphere spectrum]], being the [[G-spectrum]] [[Brown representability theorem|representing]] $G$-[[equivariant stable cohomotopy]], by definition;

1. $\Phi^N E \simeq \Sigma^\infty_{G/N} (S^V)^N \simeq \Sigma^\infty_{G/N} S^0 \simeq \mathbb{S}_{G/N}$ the unshifted $G/N$-[[equivariant sphere spectrum]], by (eq:PartialGeometricFixedPointsOfEqui) and by assumption on $V$.

Hence with all identifications made explicit, the morphism (eq:SurjectionFromEquivariantStableCohomotopyInDegreeVToDegreeZero) in question is the composite

\[
  \label{ProjectionFromEquivariantCohomotopyOfPpintInRODegreeToBurnside}
  \mathbb{S}_G^V(\ast)
  \simeq
  (\Sigma^\infty_G S^V)^0_{G}(\ast)
  \overset{
    \phantom{AA}
      p_{ \Sigma^V_G \mathbb{S}_G }^N(\ast)
    \phantom{AA}
   }{\longrightarrow}
  (\Phi^N \Sigma^\infty_G S^V)^0_{G/N}(\ast)
  \simeq
  (\Sigma^\infty_{G/N} S^0)^0_{G/N}(\ast)
  \simeq
  \mathbb{S}^0_{G/N}(\ast)
  \simeq
  A(G/N)
\]

of $p_{ \Sigma^V_G \mathbb{S}_G }^N(\ast)$ with a sequence of [[isomorphisms]], and hence our task is to prove that $p_{ \Sigma^V_G \mathbb{S}_G }^N(\ast)$ is a surjection.

We first prove this for the case that $V = 0$. In this case the identification with the [[Burnside ring]] (via [[Burnside ring is equivariant stable cohomotopy of the point|this Prop.]]) applies also to the [[domain]] cohomology group:

$$
  \mathbb{S}_G^V(\ast)
  \;\simeq\;
  \underset{\underset{V \in G Rep}{\longrightarrow}}{\lim}
  Ho_{G Spaces}\big(S^V, S^V \big)
  \simeq A(G)
  \,,
$$

By Prop. \ref{CanonicalComparisonMap}
the comparison morphism acts on this by smashing the [[codomain]] of the [[hom-sets]] with $(S^0 \to \tilde E \mathcal{F}[N])$. But by Corollary \ref{SmashingWithTildeEOnspacesEquivalentToRestriction} this is equivalent to restricting to $N$-[[fixed point spaces]] so that (eq:SurjectionFromEquivariantStableCohomotopyInDegreeVToDegreeZero) becomes simply the projection of [[Burnside rings]]

$$
  \array{
    A(G)
      &\simeq&
    \underset{\underset{ V \in G Rep }{\longrightarrow}}{\lim}
    &
    Ho_{G Spaces}\big( S^V, S^V \big)
      &\simeq&
    \mathbb{S}^0_G(\ast)
    \\
    {}^{\mathllap{ (-)^N }}\big\downarrow
    &&
    \big\downarrow
    {}^{{}_{
    \mathrlap{
      \array{
        \underset{\underset{V \in G Rep}{\longrightarrow}}{\lim}
        Ho_{G Spaces}\big(
          S^V,
          S^V \wedge (S^0 \to \tilde E\mathcal{F}[E])
        \big)
        \\
        =
        \\
        \underset{\underset{V \in G Rep}{\longrightarrow}}{\lim}
        Ho_{G Spaces}\big(
          (S^V)^N \hookrightarrow  S^V,
          S^V
        \big)
      }
      }
    }
    }
    &
    &&
    \big\downarrow{}^{
      \mathrlap{
        p_\mathbb{S}^N(\ast)
      }
    }
    \\
    A(G/N)
      &\simeq&
    \underset{\underset{ W \in G/N Rep }{\longrightarrow}}{\lim}
    &
    Ho_{G/N Spaces}\big( S^W, S^W \big)
      &\simeq&
    \mathbb{S}^0_{G/N}(\ast)
  }
$$

sending any [[G-set]] $K$ to its [[subset]] $K^N$ of $N$-[[fixed points]] regarded with its residual $G/N$-[[action]].

This is clearly surjective. (The irreducible elements on the right are the isomorphism classes of the  [[transitive action|transitive]] $G/N$-[[actions]]  $(G/N)/H$ for $H \subset G/H$, which are canonically also [[G-sets]], hence have a pre-image on the left.)

In order to deduce the general statement from this special case, we now make use of the fact that Prop. \ref{GeometricFixedPointCohomologtIsColimitOverSmashWithEulerClasses} says that the comparison map for $V = 0$ is one coprojection map of a [[colimit|colimiting]] [[cocone]]-[[diagram]], which for each $G$-[[representation]] $V$ without non-trivial $N$-fixed points (Def. \ref{ROGDegreeWithoutNonTrivialHFixedPoints}) contains a [[cocone]] component of the following form:

\[
  \label{ColimitingCoconeForComparisonAtTrivialROG}
  \array{
    A(G)
    &=&
   (\Sigma^\infty_G S^0)^0_{G}(\ast)
   &\overset{ (-) \wedge \chi_V }{\longrightarrow}&
   (\Sigma^\infty_G S^V)^0_{G}(\ast)
   &\to& \cdots
   \\
   {}^{(-)^N}\big\downarrow && {}^{\mathllap{epi}}\big\downarrow{}^{p_0} &
   \swarrow_{\mathrlap{p_V}}
   \\
   A(G/N)
   &=&
   (\Sigma^\infty_{G/N} S^0)^0_{G/N}(\ast)
  }
\]

Since we know, as just argued, that the map $(-)^N$ on the left is a surjection, the [[commutative diagram|commutativity]] of this diagram implies that also the component projection $p_V$ is surjective. (Every element $c \in A(G/N)$ has a lift to $\widehat c \in A(G)$, but then the commutativity of the triangle means that $\widehat c \wedge \chi_V$ is a pre-image of $c$ under $p_V$.)

This we may use to deduce the statement for the general case, where the codomain of (eq:ProjectionFromEquivariantCohomotopyOfPpintInRODegreeToBurnside) is in degree $V$:

By the assumption that the [[RO(G)-degree]] $V$ has no non-trivial $N$-fixed points, Prop. \ref{GeometricFixedPointCohomologtIsColimitOverSmashWithEulerClasses} says that the [[colimit|colimiting]] [[cocone]] in which the map $p_{ \Sigma^V_G \mathbb{S}_G }^N(\ast)$ appears, by Def. \ref{CanonicalComparisonMap}, looks just like the one above, except that it "starts" not in degree 0, but in degree $V$:

\[
  \label{ColimitingCoconeForComparisonAtNonTrivialROG}
  \array{
   &&
   (\Sigma^\infty_G S^V)^0_{G}(\ast)
   &\to& \cdots
   \\
   &
   \swarrow_{\mathrlap{p_V}}
   \\
   (\Sigma^\infty_G S^0)^0_{G/N}(\ast)
  }
\]

In particular the [[cocone]] in (eq:ColimitingCoconeForComparisonAtTrivialROG) restricts to a [[cocone]] over this sub-diagram in (eq:ColimitingCoconeForComparisonAtNonTrivialROG), so that the [[universal property]] of the cocone in (eq:ColimitingCoconeForComparisonAtNonTrivialROG) implies an [[endomorphism]] $\phi$ of the [[abelian group]] underlying the [[Burnside ring]] $(\Sigma^\infty_G S^0)^0_{G/N}(\ast) = A(G/n)$ such that

\[
  \label{Factorization}
  p_V
  \;=\;
  \phi \;\circ\; p_{ \Sigma^V_G \mathbb{S}_G }^N(\ast)
  \,.
\]

Since $p_V$ is surjective, it is now sufficient to prove that this $\phi$ is in fact an [[isomorphism]].

To see this, observe that, since $G$ is a [[finite group]] by assumption, the [[abelian group]] underlying the [[Burnside ring]] $A(G/N)$ is a [[finitely generated module|finitely generated]] [[free abelian group]] (spanned by the cosets $(G/N)/H$ as $H$ ranges over the [[finite set]] of [[conjugacy classes]] of [[subgroups]] of $G/N$ ).
By the [structure theory of free abelian groups](principal+ideal+domain#StructureTheoryOfModules), this means that $\phi$ may be represented by a [[matrix]] in [[Smith normal form]]. Specifically, since $\phi$ is an [[endomorphism]], it is represented by a square matrix in [[Smith normal form]]. Since $\phi \circ p_{ \Sigma^V_G \mathbb{S}_G }^N(\ast)$ is surjective, by (eq:Factorization) and the surjectivity of $p_V$ established before, this implies that $\phi$ is represented by a [[diagonal matrix]] all whose diagonal entries are non-vanishing and invertible, hence that $\phi$ is in fact an [[isomorphism]].

With this, (eq:Factorization) says that with $p_V$ also $p_{ \Sigma^V_G \mathbb{S}_G }^N(\ast)$ is surjective.

=--

## Related concepts

* [[fixed point space]]

* [[fixed point spectrum]]

* [[categorical fixed point spectrum]]

* [[homotopy fixed points]]

## References

* {#LewisMaySteinberger86} [[L. Gaunce Lewis]], [[Peter May]], and Mark Steinberger (with contributions by J.E. McClure), _Equivariant stable homotopy theory_, Springer Lecture Notes in Mathematics Vol.1213. 1986 ([pdf](http://www.math.uchicago.edu/~may/BOOKS/equi.pdf))

* {#Lewis00} [[L. Gaunce Lewis, Jr.]], section 10 of _Splitting theorems for certain equivariant spectra_, Memoirs of the AMS, number 686, March 2000, Volume 144 ([pdf](http://hopf.math.purdue.edu/LewisG/spltspec.pdf))

* {#Schwede15} [[Stefan Schwede]], section 7.3 of _[[Lectures on Equivariant Stable Homotopy Theory]]_, 2015 ([pdf](http://www.math.uni-bonn.de/people/schwede/equivariant.pdf))

* {#Schwede18} [[Stefan Schwede]], section 3.3. of _[[Global homotopy theory]]_ ([arXiv:1802.09382](https://arxiv.org/abs/1802.09382))

* {#Blumberg17} [[Andrew Blumberg]], Def. 2.5.16 in _Equivariant homotopy theory_, 2017 ([pdf](https://www.ma.utexas.edu/users/a.debray/lecture_notes/m392c_EHT_notes.pdf), [GitHub](https://github.com/adebray/equivariant_homotopy_theory))

* {#MathewNaumannNoel15} [[Akhil Mathew]], [[Niko Naumann]], [[Justin Noel]], _Nilpotence and descent in equivariant stable homotopy theory_ ([arXiv:1507.06869](http://arxiv.org/abs/1507.06869))

* [[Tom Bachmann]], [[Marc Hoyois]], remark 9.9 in _Norms in motivic homotopy theory_ ([arxiv:1711.03061](https://arxiv.org/abs/1711.03061))

Relation to [[spectral Mackey functors]]:

* {#Glasman15} [[Saul Glasman]], _Stratified categories, geometric fixed points and a generalized Arone-Ching theorem_ ([arXiv:1507.01976](https://arxiv.org/abs/1507.01976), [talk notes pdf](http://www-users.math.umn.edu/~sglasman/strattalk.pdf))

[[!redirects geometric fixed point spectra]]

[[!redirects geometric fixed point]]
[[!redirects geometric fixed points]]

[[!redirects geometric fixed point functor]]
[[!redirects geometric fixed point functors]]


geometric function object

#Content#

* automatic table of contents goes here
{:toc}

# Idea #

This entry list details on concrete constructions for examples of [[geometric function theory|geometric function theories]], or closely related structures.

Recall the notion of _geometric function object_ from [[geometric function theory]]:

Given an [[(∞,1)-topos]] $\mathbf{H}$ of [[∞-stack]]s  -- in the simplest case just [[Top]] or [[∞-Grpd]] -- a _geometric function theory_ is some kind of assignment

$$
  C : \mathbf{H} \to (\infty,1)Cat
$$

such that for $X \in \mathbf{H}$ the object $C(X)$ behaves to some useful extent like a collection of "functions on $X$".

More concretely, this will usually be taken to mean that $C$ satisfies properties of the following kind:

* **existence of pull-push** -- For every morphism $f : A \to B$ in $\mathbf{H}$ there is naturally (functorially) an [[adjunction]] $f_* : C(A) \stackrel{\leftarrow}{\to} C(B) : f^*$ with $f_*$ playing the role of push-forward of functions along $f$ and $f^*$ playing the role of pullback of functions along $f$;

* **respect for composition of spans** -- Pull-push through [[span]]s should be functorial: if
  $$
    \array{
      &&&& Y_1 \times_{X_2} Y_2
      \\
      &&& {}^{p_1}\swarrow && \searrow^{p_2}
      \\
      && Y_1 &&&& Y_2
      \\
      & {}^t\swarrow && \searrow^u && {}^v\swarrow && \searrow^w
      \\
      X_1 &&&& X_2 &&&& X_4
    }
  $$
  is a composite of two [[span]]s, then the pull-push through both spans seperately should be equivalent to that through the total span

  $$
    w_* v^* u_* t^*
    \simeq
    {w}_* {p_2}_* p_1^* t^*
    \,.
  $$

  Of course this just means that the two ways to pull-push through the pullback diamond

  $$
    \array{
      &&&& Y_1 \times_{X_2} Y_2
      \\
      &&& {}^{p_1}\swarrow && \searrow^{p_2}
      \\
      && Y_1 &&&& Y_2
      \\
      & && \searrow^u && {}^v\swarrow &&
      \\
      &&&& X_2 &&&&
    }
  $$

  should coincide.

* **respect for fiber products** -- With respect to some suitable [[tensor product]] of geometric functions one has for each ([[homotopy pullback|homotopy]]) [[fiber product]] $X \times_Z Y$ in $\mathbf{H}$ that
  $$
   C(X \times_Z Y) \simeq C(X) \otimes_{C(Z)} C(Y)
   \,.
  $$

## over-categories and groupoidification ##

This first example is rather minimalistic and may feel a bit tautological, as compared to more involved constructions as discussed below. It does nevertheless have interesting applications and, due to its structural simplicity, should serve as a good model on which to study the structural aspects of geometric function theory.

So consider here the assignment

$$
  C := \mathbf{H}/(-) : \mathbf{H} \to (\infty,1)Cat
$$

that sends each object $X \in \mathbf{H}$ to its [[over category]] $\mathbf{H}/X$.

Checking that this assignment does satisfy a good deal of the properties of a geometric function object amounts to recalling the properties of  [[over category|over categories]].

So an [[object]] in $C(X)$ is a morphism $\psi : \Psi \to X$ in $\mathbf{H}$. A [[morphism]] $(\psi,\Psi) \to (\psi',\Psi')$ is a diagram

$$
  \array{
    \Psi &&\to&& \Psi'
    \\
    & {}_\psi\searrow && \swarrow_{\psi'}
    \\
    && X
  }
$$

in $\mathbf{H}$.

For $f : X \to Y$ a morphism in $\mathbf{H}$ the push-forward functor

$$
  f_* : C(X) \to C(Y)
$$

is simply given by postcomposition with $f$:

$$
  f_* \;\;:\;\;
  \left(
    \array{
      \Psi
      \\
      \downarrow^\psi
      \\
      X
    }
  \right)
  \;\;
  \mapsto
  \;\;
  \left(
    \array{
      \Psi
      \\
      \downarrow^\psi
      \\
      X
      \\
      \downarrow^f
      \\
      Y
    }
  \right)
  \,.
$$

The pullback functor

$$
  f^* : C(Y) \to C(X)
$$

is literally given by the ([[homotopy pullback|homotopy]]) [[pullback]]

$$
  \array{
    f^* \Psi &\to& \Psi
    \\
    \downarrow^{f^* \psi} && \downarrow^\psi
    \\
    X &\stackrel{f}{\to}& Y
  }
$$

of a morphism $\psi : \Psi \to Y$ along $f$.

A quick way to check that pushforward $f_*$ and pullback $f^*$ defined this form a pair of  [[adjoint functor]]s is to notice the hom-isomorphism

$$
  Hom_{\mathbf{H}/X}(\Psi, f^* \Phi)
  \simeq
  Hom_{\mathbf{H}/X}(f_* \Psi, \Phi)
$$

which is established by the essential uniqueness of the universal morphism into the pullback

$$
  \array{
    \Psi &&\to&&
    \\
    & \searrow^{\exists ! \bar k} & && \downarrow^{k}
    \\
    \downarrow && f^* \Phi &\to& \Phi
    \\
    &\searrow& \downarrow^{f^* \psi} && \downarrow^\psi
    \\
    && X &\stackrel{f}{\to}& Y
  }
$$

Here the outer diagram exhibits a morphism $k : f_* \Psi \to \Phi$. The universal property of the [[pullback]] says that this essentially uniquely corresponds to the [[adjunct]]  morphism $\bar k : \Psi \to f^* \Phi$.

The fact that the pull-push respects composition of spans is a direct consequence of the way [[pullback]] diagrams compose under pasting: recall that in a diagram

$$
  \array{
    A &\to& B &\to& C
    \\
    \downarrow && \downarrow && \downarrow
    \\
    D &\to& E &\to& F
  }
$$

for which the left square is a pullback, the total rectangle is a pullback precisely if the right square is, too.

Apply this to the pull-push of an object $\left(\array{ \Psi \\ \downarrow^{\psi} \\ Y_1}\right) \in C(Y_1)$ through a pullback diamond (see the introduction above)

  $$
    \array{
      &&&& Y_1 \times_{X_2} Y_2
      \\
      &&& {}^{p_1}\swarrow && \searrow^{p_2}
      \\
      && Y_1 &&&& Y_2
      \\
      & && \searrow^u && {}^v\swarrow &&
      \\
      &&&& X_2 &&&&
    }
    \,.
  $$

This is described by the diagram

  $$
    \array{
      && p_1^* \Psi
      \\
      & {}^{q_1}\swarrow && \searrow^{p_1^* \psi}
      \\
      \Psi&&&& Y_1 \times_{X_2} Y_2
      \\
      &\searrow^f && {}^{p_1}\swarrow && \searrow^{p_2}
      \\
      && Y_1 &&&& Y_2
      \\
      & && \searrow^u && {}^v\swarrow &&
      \\
      &&&& X_2 &&&&
    }
    \,.
  $$

By the above definitions, the push-pull operation $v^* u_*$ is encoded in the pullback property of the total outer rectangle. On the other hand, the pull-push operation ${p_2}_* p_1^*$ is determined by the pullback property of the upper square. By the above fact both properties are equivalent. This means that indeed

$$
  v^* u_* \simeq {p_2}_* p_1^*
$$

and hence that the pull-push operations defined by over-categories are compatible with composition of [[span]]s.

Finally, there is a simple observation on the [[cartesian product]] on [[over category|over categories]]:

for

$$
  \array{
     && Y
     \\
     && \downarrow^g
     \\
     X &\stackrel{f}{\to}& Z
  }
$$

a diagram in $\mathbf{H}$, notice that the objects in the [[fiber product]] of [[over category|over categories]]

$$
  (\mathbf{H}/X) \times_{\mathbf{H}/Y} \mathbf{H}/Y
$$

are those pairs $\psi : \Psi \to X$ and $\phi : \Phi \to Y$ such that we get a (homotopy) commutative diagram

$$
  \array{
     \Psi \simeq \Phi &\stackrel{\phi}{\to}& Y
     \\
     \downarrow^\psi && \downarrow^g
     \\
     X &\stackrel{f}{\to}& Z
  }
  \,.
$$

Again by the universal property of the pullback this is the same as maps

$$
  (\Psi \simeq \Phi) \to X \times_Z Y
$$

which are precisely the objects of $C(X \times_Z Y)$. So we get

$$
  C(X \times_Z Y) \simeq C(X) \times_{C(Z)} C(Y)
$$

### remarks ###

 * This is -- more or less implicitly -- the notion of geometric ∞-functions that underlies [[John Baez]]' notion of [[groupoidification]] as well as the generalized sections that appear at [these sigma-model notes](http://ncatlab.org/schreiber/show/Nonabelian+cocycles+and+their+sigma+model+QFTs).

* The definition seems to be disturbingly non-linearized, but this should be viewed in light of the possible nature of the $X$s considered here. If $X = E$ is, for instance, the [[groupoid]] incarnation of the total space of the [[vector bundle]] associated to a $G$-[[principal bundle]], then a choice of groupoid over $E$ picks a bunch of vectors in that bundle, hence picks a "distributional section" of that bundle.

## under-categories of $\infty$-quantities ##

By essentially simply applying [[Isbell duality]] for the case that the underlying [[site]] is [[CartSp]] to the above example one obtains the following example.

+-- {: .standout}

Tentative.

=--

Recall the notion of [[∞-quantity]]. Notice that by the discussion at [[models for ∞-stack (∞,1)-toposes]] every object $A \in \mathbf{H}$ may be modeled as a [[simplicial presheaf]]. Let $C^\infty(-)$ be the map that sends simplicial presheaves to cosimplicial copresheaves as described at [[∞-quantity]].

Then consider the assignment

$$
  C(-) : \mathbf{H} \to (\infty,1)Cat
$$

that sends every $X$ to the $(\infty,1)$-category of cosimplicial copresheaves  to the [[under category]]

$$
  C(X) = C^\infty(X)/CoSCoSh
$$

or

$$
  C(X) = C^\infty_{loc}(X)/CoSCoSh
  \,.
$$

From the discussion at [[∞-quantity]] and [[Lie-∞ algebroid representation]] we see that we can think of objects in $C(X)$ defines this way as representations of the [[Lie-∞ algebroid]] of $X$.

Now pullback is left adjoint and push-forward is right adjoint.

## quasicoherent sheaves ##

The choice $C(X) =$ the [[stable (∞,1)-category]] of [[quasicoherent sheaf|quasicoherent sheaves]] on a [[derived stack]] $X$ is discussed at

* [[geometric ∞-function theory]].

[[!redirects examples for geometric function objects]]
[[!redirects example for geometric function objects]]
[[!redirects geometric function objects]]


geometric function theory

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

## Idea

**Geometric function theory**
considers notions of higher generalized functions on higher generalized spaces (such as on [[groupoid]]s, on [[Lie groupoid|orbifolds]] and more generally on [[infinity-stack homotopically|infinity-stacks]]) such that all suitably generalized linear maps between the monoidal $\infty$-structures of functions on two spaces arise from a higher analog of plain matrix multiplication, namely from a _pull-tensor-push_ operation in the given $\infty$-context.

### Motivating toy example: matrices

The motivating toy example is that where all spaces in question are just _finite sets_ and functions are just maps from finite sets to a given [[ground field]]: for $X$ a finite set the collection of functions $C(X)$ on $X$ is canonically identified with the vector space spanned by the elements of $X$. Moreover, for two finite sets $X_1$ and $X_2$ functions on the [[product]] set $X_1 \times X_2$ are canonically identified with $|X_1| \times |X_2|$-matrices
$$
  C(X_1 \times X_2) \simeq Mat(n= |X_1|, m = |X_2|)
  \,.
$$
The point is that from this perspective on matrices the action of a matrix on a vector is given by the pull-tensor-push operation of functions through a [[span]]

$$
  \array{
     && X_1 \times X_2
     \\
     & {}^{pr_1}\swarrow && \searrow^{pr_2}
     \\
     X_1
     &&&&
     X_2
  }
$$

Given $v \in C(X_1)$ and $M \in C(X_1 \times X_2)$ the matrix product $M \cdot v \in C(X_2)$ can be understood as

$$
  M \cdot v =  \int_{pr_2}(M \cdot (pr_1^* (v)))
  \,,
$$
where

* $pr_1^* : C(X_1) \to C(X_1 \times X_2)$ is the pullback of functions along the projection map $pr_1$;

* $M \cdot (\cdot)$ is the ordinary product of functions;

* $\int_{pr_2}$ is the push-forward of field-valued functions along $pr_2$, i.e. the operation which integrates (a finite sum in this toy example) a function on $X_1 \times X_2$ over $X_1$ to a function on $X_2$.

### Groupoidification

A closely related situation is considered in the context of [[groupoidification]], where the aim is to encode not only the operation of matrix multiplication geometrically in the above sense, but also the very notion of a function itself.

In [[groupoidification]] functions with values in fields are essentially replaced by functions with values in (finite) [[groupoid]]s, and the notion of [[groupoid cardinality]] is used to interpret such a gadget as a function with values in rational numbers.

More precisely, [[groupoidification]] in the sense of [[John Baez]] can be understood as geometric function theory for the case that collections of geometric functions are modeled as [[over category|over categories]]. This is described in more detail at [[examples for geometric function objects]].

Related toy examples of the general pattern are [[group]] algebras regarded as [[category algebra|category algebras]] (see the end of that entry for the span-description of category algebras).

### General $\infty$-categorical / homotopical setup

In full generality the idea of _geometric function theory_ is to replace in the above toy example all finite sets with generalized higher spaces in the form of [[∞-stack]]s and to realize a useful notion of higher "functions" $C(X)$, such that $C(X)$ is a [[monoidal (∞,1)-category]] in a suitable sense and behaves as expected under [[homotopy pullback]] and push-forward operations.

The **main two conditions** that one wants to have in _geometric function theory_ are

1. for $X_1 \to Y \leftarrow X_2$ two generalized spaces sitting over a third one, we have an [[equivalence]] between the generalized functions on the [[pullback|fiber product]] $X_1 \times_Y X_2$ and the [[tensor product]] of functions on $X_1$ with functions on $X_2$ over functions on $Y$:
$$
  C(X_1 \times_Y X_2)
  \simeq
  C(X_1) \otimes_{C(Y)} C(X_2)
  \,,
$$
where all operations are in the
suitable $\infty$-context (for instance the fiber product is a [[homotopy limit]]).

2. again for given $X_1 \to Y \leftarrow X_2$ the generalized functions on the (homotopy/$\infty$-)fiber product are supposed to induce via (homotopy/$\infty$-) pull-tensor-push operation through the [[span]] $X_1 \leftarrow X_1 \times_{Y} X_2 \to X_2$ all $C(Y)$-linear morphism between $C(X_1)$ and $C(X_2)$:
$$
  C(X_1 \times_Y X_2)
  \simeq
  Maps_{C(Y)}(C(X_1), C(X_2))
  \,.
$$

This is to be read as a vast generalization of ordinary matrix multiplication, to which it reduces in the case that all spaces here are finite sets and all functions are ordinary functions on finite sets with values in some field.  One step higher this gives [[Fourier-Mukai transformation|Fourier?Mukai transformations]] etc.

### Relation to Hochschild (co)homology

A particularly interesting application of items 1) and 2) above is to the case where the pullback diagram of generalized spaces in question is simply

$$
  X \stackrel{Id \times Id}{\to}
  X \times X
  \stackrel{Id \times Id}{\leftarrow}
  X
  \,.
$$

As discussed at [[span trace]] the [[homotopy pullback]] of this is the [[loop space object]] $\Lambda X$ of $X$.

So in this case statement 2) relates

* generalized functions $C(\Lambda X)$ on the [[loop space]] of $X$

with

* the homotopy tensor product of $C(X)$ with itself over $C(X)\otimes C(X)$

$$
  C(\Lambda X) \simeq C(X) \otimes_{C(X) \otimes C(X)} C(X)
  \,.
$$

The expression on the right is known as the higher [[trace]] of the [[monoidal (∞,1)-category]] $C(X)$, which is the higher [[Hochschild homology]] of $C(X)$. (See next subsection for concrete realizations).

#### Realization in terms of derived $\infty$-stacks on the algebraic site

Aspects of geometric function theory have a long history in the context of [[sheaf|sheaves]] and [[stack|stacks]] over algebraic [[site]]s. The [[Fourier-Mukai transformation|Fourier?Mukai transform]] is a classical example of a pull-tensor-push operation in a context where the object $C(X)$ of generalized functions on a space $X$ is taken to be $D(X)$, the [[derived category]] of [[coherent sheaf|coherent sheaves]] on $X$.

A general proof of the above item 2 in the context of generalized functions modeled by the [[derived category]] $D(X)$ has been given in

* B. Toën and G. Vezzosi, _Homotopical Geometry I: Topos theory._, Adv.
Math. 193 (2005), no. 2, 257–372 ([arXiv](http://arxiv.org/abs/math/0207028)).

As any [[homotopy category]], the [[derived category]] $D(X)$ of [[coherent sheaf|coherent sheaves]] is naturally understood as the $1$-categorical shadow of a [[stable (∞,1)-category]]. In a suitable context of derived $\infty$-stacks there is for each such $\infty$-stack $X$ an $(\infty,1)$-category $QC(X)$ whose homotopy category is the ordinary derived category $D(X)$ of quasi-coherent sheaves on $X$.

In this context geometric function theory with functions $C(X)$ taken to be given by $QC(X)$ was discussed in

* David Ben-Zvi, John Francis, David Nadler, _Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry_ ([arXiv](http://arxiv.org/abs/0805.0157))

where the above two items appear as items 1) and 2) in [theorem 1.2, p. 4,5](http://arxiv.org/PS_cache/arxiv/pdf/0805/0805.0157v4.pdf#page=4)

A detailed entry on this is at

* [[geometric ∞-function theory]]

### Further examples

Further discussion of realizations of geometric function objects is at

* [[examples for geometric function objects]]

## Remarks

Closely related to the spans appearing in geometric function theory is the notion of [[bi-brane]].

## Further references

[[David Ben-Zvi]] kindly wrote an expositional piece on geometric function theory for the $n$-Category Café ([blog entry](http://golem.ph.utexas.edu/category/2009/01/benzvi_on_geometric_function_t.html)):

* David Ben-Zvi, [[BenZviGeometricFunctionTheory.pdf:file]] (Jan 2009).


geometric functor

## Definition

## Related concepts

[[!include categories and logic - table]]

## References

Remark C2.2.10

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_

[[!redirects geometric functors]]


geometric group theory

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### Context
#### Group Theory
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[[!include group theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

Within [[group theory]], *geometric group theory* studies [[groups]] by equipping them with a [[metric]] and thus regarding them as [[metric spaces]].

For [[finitely generated group|finitely generated groups]] this is typically done by regarding the group's [[elements]] as the [[vertices]] of a [[graph]] -- the *[[Cayley graph]]* -- whose [[edges]] connect elements related by multiplication with a generator. The corresponding [[graph distance]] then equips the group with a [[metric]].

The methods of geometric group theory overlap with [[kernel methods]] in [[machine learning]].

## Related entries

* [[Cayley graph]]

* [[Cayley distance]], [[Kendall distance]]

## References

Textbooks accounts:

* [[Persi Diaconis]], Chapter 6 of: *Group Representations in Probability  and Statistics*, Lecture Notes - Monographs Series, Institute of Mathematical Statistics 1988 ([pdf](https://jdc.math.uwo.ca/M9140a-2012-summer/Diaconis.pdf))

* [[Cornelia Druţu]], [[Michael Kapovich]] (appendix by [[Bogdan Nica]]), *Geometric group theory*, Colloquium Publications **63**, AMS 2018 ([ISBN:978-1-4704-1104-6](https://bookstore.ams.org/coll-63/), [pdf](https://courses.maths.ox.ac.uk/node/view_material/35649))

* [[Clara Löh]], *Geometric Group Theory*, Springer 2017 ([doi:10.1007/978-3-319-72254-2](https://link.springer.com/book/10.1007/978-3-319-72254-2), [pdf](http://www.mathematik.uni-regensburg.de/loeh/teaching/ggt_ws1011/lecture_notes_old.pdf))

See also:

* Wikipedia, _[Geometric group theory](http://en.wikipedia.org/wiki/Geometric_group_theory)_