Giry monad

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### Context
#### Measure and probability theory
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# Contents
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## Idea

The _Giry monad_ ([Giry 80](#Giry80), following [Lawvere 62](#Lawvere62)) is the [[monad]] on a [[category]] of suitable [[spaces]] which sends each suitable space $X$ to the space of suitable [[probability measures]] on $X$.

It is one of the main examples of a [[probability monad]], and hence one of the main structures used in [[categorical probability]].

## Definition

The Giry monad is defined on the category of [[measurable spaces]], assigning to each measurable space $X$ the space of all probability measures on $X$, $G(X)$, endowed with the $\sigma$-algebra generated by the set of all the evaluation maps
$$
ev_U \colon G(X) \to [0,1]
$$
sending a probability measure $P$ to $P(U)$, where $U$ ranges over all the measurable sets of $X$. The unit of the monad sends a point $x \in X$ to the [[Dirac measure]] at $x$, $\delta_x$, while the multiplication of the monad is defined by the natural transformation
$$
  \mu_{X}: G(G(X)) \to G(X)
$$

given by

$$
 \mu_X (Q)(U) := \int_{q \in G(X)} ev_U(q) dQ.
$$
This makes the [[endofunctor]] $G$ into a [[monad]], and this is the _Giry monad_  on Measurable spaces, as originally given by Lawvere.

An alternative choice, convenient for analysis purposes, and introduced by Giry, is obtained by restricting the category of [[measurable spaces]] to the (full) subcategory which are those measurable spaces generated by [[Polish spaces]], $Pol$, which are  separable [[metric spaces]] for which a complete metric exists.

Write

$$
  P \colon Pol \to Pol
$$

for the [[endofunctor]] which sends a space, $X$, to the space of [[probability measures]] on the [[Borel subsets]] of $X$. $P(X)$ is equipped with the weakest [[topology]] which makes the [[integration]] map $\tau \mapsto \int_{X}f d\tau$ [[continuous function|continuous]] for any $f$, a bounded, continuous, real function on $X$.

There is a [[natural transformation]]

$$
  \mu_{X}: P(P(X)) \to P(X)
$$

given by

$$
 \mu_X (M)(A) := \int_{P(X)} \tau(A) M(d\tau).
$$

This makes the [[endofunctor]] $P$ into a [[monad]], and this is the _Giry monad_ on [[Polish spaces]].  The shortcoming of restricting the Giry monad to Polish spaces is that discrete spaces have no $P$-algebras (even though they do have $G$-algebras).  For example, take the discrete (measurable) space $2$, which we can also view as a topological discrete space.  The measurable map $\epsilon_{2}: G(2) \to 2$ defined by $\epsilon_{2}( (1-\alpha) \delta_0 + \alpha \delta_1) = 0$ for all $\alpha \in [0,1)$, and by $\epsilon_{2}(\delta_1)=1$ shows that $(2,\epsilon_2)$ is a $G$-algebra, yet it is not a $P$-algebra. (As Doberkat has noted,  a $P$-algebra must be connected, but that is impossible for discrete spaces.)  The map $\epsilon_{2}$ plays a critical role in the analysis of the $G$-algebras, because it is the unique affine map from the geometric (convex) space $G(2)$, which is isomorphic to the unit interval $[0,1]$ with its natural convex structure, to the discrete (combinatorial) convex space $2$.

## Properties

### Kleisli category

The [[Kleisli morphisms]] of the Giry monad on [[Meas]] (and related subcategories) are [[Markov kernels]]. Therefore its [[Kleisli category]] is the category [[Stoch]].
It is one of the most important examples of a [[Markov category]].

### Algebras over the Giry monad

([Doberkat 03](#Doberkat03)) works out the [[algebra over a monad|algebras]] for the Giry monad using the Giry monad defined on $Pol$. We want measurable maps between $P(X)$ and $X$, such that the 'fibres' are convex and closed, and such that $\delta_{x}$, the delta distribution on $x$, is in the fibre over $x$. And there's another condition which requires a compact subset of $P(X)$ to be sent to a compact subset of $X$.

Now, as ever, $P(X)$ will support an algebra, $\mu_{X}: P(P(X)) \to P(X)$. This is the analogue of a [[free construction|free]] [[group]] being an algebra of the group monad. But just as there are many interesting groups which are not free, we should want to find algebras of Giry's monad which are not of the $\mu_{X}$ form. Doberkat shows that for such an algebra $X$ must be connected, and suggests this example

$$
h: P([0, 1]) \ni \tau \mapsto \int_{0}^{1} t d\tau{(t)} \in [0, 1].
$$

which is the free algebrea $\mu_{\{0, 1\}}$ using the fact that   the probability measures on $\{0, 1\}$ are just binomial, parameterised by $t \in [0, 1]$, i.e., every $p \in P(\{0,1\})$ is represented by $p=(1-t) \delta_0 + t \delta_1$.
The other example he gives has $X$ a bounded, closed and convex subset of $\mathbb{R}^n$, and probability measures being sent to their barycentre. (This is an example of the Giry (probability) monad for compact Hausdorff spaces which was shown by Swirszcz (Monadic functors and convexity (1974).)

Doberkat has a longer article on [[Eilenberg-Moore algebra]]s of the Giry monad as item 5 [here](http://ls10-www.cs.uni-dortmund.de/index.php?id=18). (Unfortunately, the monograph 'Stochastic Relations: Foundations for Markov Transition Systems' doesn't appear to be available.) There are two monads being treated here, one which sends a Polish space to the space of all probability measures, the other to the space of all subprobability measures. The extra structure relating to these monads, is that of a (positive) convex structure. In the case of a convex structure, this intuitively captures the idea that a weighted sum of points in the space has barycentre within the space.

The fundamental question concerning algebras is the existence of algebras which are not free.  The Eilenberg-Moore category of algebras, for any monad, is descriptive in nature but not constructive, and says nothing about the existence of algebras which are not free.  For this reason it is necessary to find a
different representation of the category of $G$-algebras. (Recall that the full subcategory of the Eilenberg-Moore category whose objects are the free algebras is isomorphic to the Kleisi category.)

([VanBelle 21](#VanBelle21)) shows, in effect, that the restricted Giry monad (viewed as just a functor) $G|: \mathbf{Meas}_c \rightarrow \mathbf{Meas}$ has the codensity monad $G$, where $\mathbf{Meas}_c$ is the category of countable measurable spaces with the discrete $\sigma$-algebra (hence, in effect, one can view these simply as sets). This suggest that the natural numbers $\mathbb{N}$ are sufficient in some sense. To make this more precise, we first make the observation that  given any $G$-algebra $h:G(X) \rightarrow X$ on a (coseparated) measurable space $X$, the underlying set $X$ has the structure of a superconvex space, denoted $X_h$, defined by $\sum_{i=1}^{\infty} p_i x_i := h( \sum_{i=1}^{\infty} p_i \delta_{x_i})$, where $\sum_{i=1}^{\infty} p_i = 1$ and each $p_i \ge 0$. The proof is

\begin{equation}
\begin{array}{rcl}
 (h \mu_X)(\sum_{i \in \mathbb{N}} p_i \delta_{P_i}) &=& (h \circ \G(h))(\sum_{i \in \mathbb{N}} p_i \delta_{P_i})   \\
 h(\sum_{i \in \mathbb{N}} p_i P_i) &=& h(\sum_{i \in \mathbb{N}} p_i \delta_{h(P_i)})  \\
 h(\sum_{i \in \mathbb{N}} p_i P_i) &=& \sum_{i \in \mathbb{N}} p_i h(P_i)
\end{array}.
\end{equation}

  Moreover, given any map of $G$-algebras, $f: (X,h) \rightarrow (Y,k)$ it follows that $f$ is a countably affine map because
\begin{equation}
\begin{array}{lcl}
f( \sum_{i \in \mathbb{N}} p_i x_i) &=& f(h( \sum_{i \in \mathbb{N}} p_i \delta_{x_i} ))  \\
&=& k\big( G{(f)}( \sum_{i \in \mathbb{N}} p_i \delta_{x_i}  ) \big) \\
&=& k\big(  \sum_{i \in \mathbb{N}} p_i \delta_{f(x_i)} \big) \\
&=&  \sum_{i \in \mathbb{N}} p_i k(\delta_{f(x_i))}  \\
&=&  \sum_{i \in \mathbb{N}} p_i f(x_i)
\end{array}
\end{equation}

  For this reason we are naturally led to the category of [[super convex spaces]], denoted $\mathbf{SCvx}$, which have been referred to as strongly convex spaces in the quantum mechanics literature, e.g., See Mackey - Math. Found. Q.M..  The morphisms of $\mathbf{SCvx}$ are countably affine map (barycenter maps) which generalizes (and is consistent with) the elementary examples of probability monads on compact Hausdorff spaces, etc.. These $G$-algebras, with measurable maps, provide the connection between the continuous (geometric) spaces and discrete (combinatorial) spaces which are necessary for modeling non-determinism.

Now we can describe the two ways in which $\mathbb{N}$ is ''sufficient''.  The full subcategory of super convex spaces consisting of the single object $G(\mathbb{N})$
is dense in $\mathbf{SCvx}$. (This is almost immediate from the definition because if $A$ is any super convex space then a countably affine map $f: G(\mathbb{N}) \rightarrow A$ is completely defined by where $f$ maps each $\delta_i \in G(\mathbb{N})$.)  On the other hand, there exists a codense functor from the full subcategory of $\mathbf{SCvx}$ consisting of the single object $\mathbb{N}$ to the category of separated standard measurable spaces.  These two constructions are the content of the article ([Sturtz 22](#Sturtz22)) which attempts  to show the category of $G$-algebras on the subcategory (of $\mathbf{Meas}$) given by the category of standard measurable spaces is isomorphic to the category of super convex space.
## Related constructions

See also [[monads of probability, measures, and valuations]].

### Voevodsky's work

[[Vladimir Voevodsky]] has also worked on a category theoretic treatment of [[probability theory]], and gave few talks on this at IHES, Miami, in Moscow etc. Voevodsky had in mind applications in [[mathematical biology]], for example, population genetics:

See [Miami Talk abstract](http://www.math.miami.edu/anno/voevodsky.htm)

> ...a categorical study of probability theory where "categorical" is understood in the sense of category theory. Originally, I developed this approach to probability to get a better understanding of the constructions which I had to deal with in population genetics. Later it evolved into something which seems to be also interesting from a purely mathematical point of view. On the elementary level it gives a category which is useful for the work with probabilistic constructions involving complicated combinations of stochastic processes of different types. On a more advanced level, applying in this context the old idea of a functor as a generalized object one gets a better view of the relationship between probability and the theory of (pre-)ordered topological vector spaces.

A talk in Moscow (20 Niv 2008, in Russian) can be viewed [here](http://www.mi.ras.ru/index.php?c=obs_sem/&l=1), [wmv 223.6 Mb](http://www.mi.ras.ru/media/335_full.wmv). Abstract:

> In early 60-ies Bill Lawvere defined a category whose objects are measurable spaces and morphisms are Markov kernels. I will try to show how this category allows one to think about many of the notions of probability theory in categorical terms and to connect probabilistic objects to objects of other types through various functors.

Voevodsky's unfinished [notes on categorical probability theory](http://www.math.ias.edu/Voevodsky/voevodsky-publications_abstracts.html#markov) have been released posthumously.

### Panangaden's monad

[[Prakash Panangaden]] in [Probabilistic Relations](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.52.4840) defines the [[category]] $SRel$ (stochastic [[relation]]s) to have as [[object]]s [[set]]s equipped with a $\sigma$[[sigma-field|-field]]. [[morphism|Morphisms]] are conditional probability densities or stochastic kernels. So, a morphism from $( X, \Sigma_X)$ to $( Y, \Sigma_Y)$ is a function $h: X \times \Sigma_Y \to [0, 1]$ such that

1. $\forall B \in \Sigma_Y . \lambda x \in X . h(x, B)$ is a bounded measurable function,
1. $\forall x \in X . \lambda B \in \Sigma_Y . h(x, B)$ is a subprobability measure on $\Sigma_Y$.

If $k$ is a morphism from $Y$ to $Z$, then $k \cdot h$ from $X$ to $Z$ is defined as $(k \cdot h)(x, C) = \int_Y  k(y, C)h(x, d y)$.

Panangaden's definition differs from Giry's in the second clause where subprobability measures are allowed, rather than ordinary probability measures.

Panangaden emphasises that the mechanism is similar to the way that the category of relations can be constructed from the [[power set]] [[functor]]. Just as the category of relations is the [[Kleisli category]] of the powerset functor over the category of sets [[Set]], $SRel$ is the Kleisli category of the functor over the category of [[measurable space]]s and [[measurable function]]s which sends a measurable space, $X$, to the measurable space of sub[[probability measure]]s on $X$. This functor gives rise to a [[monad]].

What is gained by the move from probability measures to subprobability measures? One motivation seems to be to model probabilistic processes from $X$ to a [[coproduct]] $X + Y$. This you can iterate to form a process which looks to see where in $Y$ you eventually end up. This relates to $SRel$ being traced.

There is a [[monad]] on $MeasureSpaces$, $1 + -: Meas \to Meas$. A probability measure on $1 + X$ is a subprobability measure on $X$. Panangaden's monad is a composite of Giry's and $1 + -$.

## Related concepts

* [[monads of probability, measures, and valuations]]

* [[measure]], [[probability measure]], [[pushforward measure]]

* [[measure theory]], [[probability theory]], [[random variable]]

* [[Radon monad]], [[distribution monad]], [[probabilistic powerdomain]], [[extended probabilistic powerdomain]]

* [[categorical probability]], [[Markov category]]

* [[Markov kernel]], [[Stoch]]

## History

The [[adjoint functor|adjunction]] underlying the Giry monad was originally developed [by Lawvere in 1962](#Lawvere62), prior to the full recognition of the relationship between [[monads]] and [[adjunctions]]. Although P. Huber had already shown in 1961 that every [[adjoint functor|adjoint pair]] gives rise to a [[monad]], it wasn't until 1965 that the constructions of Eilenberg-Moore, and Kleisli, made the essential equivalence of both concepts manifest.

Lawvere's construction was written up as an [appendix to a proposal](#Lawvere62) to the Arms Control and Disarmament Agency,  set up by President Kennedy as part of the State Department to handle planning and execution of certain treaties with the Soviet Union. This appendix was intended to provide a reasonable framework for arms control verification protocols ([Lawvere 20](#Lawvere20)).

At that time, Lawvere was working for a "think tank" in California, and the purpose of the proposal was to provide a means for verifying compliance with limitations on nuclear weapons.   In the 1980's, [[Michèle Giry]] was collaborating with another French mathematician at that time who was also working with the French intelligence agency, and was able to obtain a copy of the appendix. Giry then developed and extended some of the ideas in the appendix ([Giry 80](#Giry80))

[[Gian-Carlo Rota]] had also (somehow) obtained a copy of [the appendix](#Lawvere62), which ended up in the library at The American Institute of Mathematics, and only became publicly available in 2012.

From [Lawvere 20](#Lawvere20):

> I'd like to say that the idea of the category of probabilistic mappings, the document corresponding to that was not part of a seminar, as some of the circulations say, essentially it was the document submitted to the arms control and disarmament agency after suitable checking that the Pentagon didn't disagree with it. Because of the fact that for arms control agencies as a side responsibility the forming of arms control agreements and part of these agreements must involve agreed upon protocols of verification. So the idea of that paper did not provide such protocols, but it purported to provide reasonable framework within which such protocol can be formulated.

## References

The idea originates with

* {#Lawvere62} [[W. Lawvere]], _The category of probabilistic mappings_, ms. 12 pages, 1962  ([[lawvereprobability1962.pdf:file]])

  (notice that the statement of origin on p.1 is wrong)

and was picked up and published in:

* {#Giry80} [[Michèle Giry]], _A categorical approach to probability theory_, Categorical aspects of topology and analysis (Ottawa, Ont., 1980), pp. 68–85, Lecture Notes in Math. **915** Springer 1982 ([doi:10.1007/BFb0092872](https://doi.org/10.1007/BFb0092872))

  (there are allegedly a few minor [[analysis|analytically]] incorrect points and gaps in proofs, observed by later authors).

Historical comments on the appearance of [Lawvere 62](#Lawvere62) are made in

* {#Lawvere20} [[Bill Lawvere]], video discussion, 6 June 2020 (in the talk "The algebra of statistical theories and models" by Evan Patterson, at [Categorical Probability and Statistics](http://perimeterinstitute.ca/personal/tfritz/2019/cps_workshop/), 5–8 June 2020,
([YouTube video, timestamp 1:00:20](http://www.youtube.com/watch?v=BhKaHAY8Ec8&t=60m21s))

According to [E. Burroni (2009)](#Burroni09), the Giry monad appears also in

* O. de la Tullaye, _L'intégration considérée comme l'algèbre d'un triple_. Rapport de Stage de D.E.A. manuscrit 1971.

The article

* {#VanBelle21} [[Ruben Van Belle]], *Probability monads as codensity monads*  (2021) $[$[arXiv:2111.01250](https://arxiv.org/abs/2111.01250)$]$

shows, in effect, that the Giry monad restricted to countable measurable spaces (with the discrete $\sigma$-algebra) yields the restricted Giry functor $G|: \mathbf{Meas}_c \rightarrow \mathbf{Meas}$ which has the codensity monad $G$.  This suggest that the natural numbers $\mathbb{N}$ are ''sufficient'' in the sense characterized above which suggest a factorization of the Giry monad, defined on the category of standard measurable spaces, through the category of super convex spaces, spelled out in

* {#Sturtz22} [[Kirk Sturtz]], _Giry algebras for standard measurable spaces_ $[$[arXiv:2202:10819](https://arxiv.org/abs/2202.10819)$]$

which has evolved from the preliminary work

* [[Kirk Sturtz]], _Categorical Probability Theory_, $[$[arXiv:1406.6030](http://arxiv.org/abs/1406.6030)$]$

which views probability measures via double dualization, restricted to weakly averaging affine maps which preserves limits.  (A more satisfactory description of probability measures arises from recognizing the need for viewing them as weakly-averaging countably affine maps, rather than just finite affine maps, as discussed in that article.)

Some corrections from an earlier version of the article, were pointed out in

* {#Avery14} [[Tom Avery]], _Codensity and the Giry monad_, Journal of Pure and Applied Algebra **220** 3 (2016) 1229-1251 [[arXiv:1410.4432](https://arxiv.org/abs/1410.4432), [doi:10.1016/j.jpaa.2015.08.017](https://doi.org/10.1016/j.jpaa.2015.08.017)]

Apart from these papers, there are similar developments in

* Franck van Breugel, _The metric monad for probabilistic nondeterminism_, features both the Lawvere/Giry monad and Panangaden's monad.

* {#Doberkat03} [[Ernst-Erich Doberkat]], _Characterizing the Eilenberg-Moore algebras for a monad of stochastic relations_ ([pdf](https://eldorado.tu-dortmund.de/bitstream/2003/2717/1/147.pdf))

* [[Ernst-Erich Doberkat]], _Kleisli morphisms and randomized congruences_, Journal of Pure and Applied Algebra Volume 211, Issue 3, December 2007, Pages 638-664 <https://doi.org/10.1016/j.jpaa.2007.03.003>

* N. N. Cencov, _Statistical decisions rules and optimal Inference_, Translations of Math. Monographs __53__, Amer. Math. Society 1982

(blog comment) Cencov's "category of statistical decisions" coincides with Giry's (Lawvere's) category.  I ($\leftarrow$ somebody) have the sense that Cencov discovered this category independently of Lawvere although years later.

* category cafe related to Giry monad: [category theoretic probability](http://golem.ph.utexas.edu/category/2007/02/category_theoretic_probability.html), [coalgebraic modal logic](http://golem.ph.utexas.edu/category/2009/09/coalgebraic_modal_logic.html)

* Samson Abramsky et al. _Nuclear and trace ideals in tensored ∗-Categories_,[arxiv:math/9805102](http://arxiv.org/abs/math/9805102), on the representation of probability theory through monads, which looks to work Giry's monad into a context even more closely resembling the category of relations.

There is also relation with work of Jacobs et al.

* Robert Furber, [[Bart Jacobs]], _Towards a categorical account of conditional probability_, [arxiv:1306.0831](http://arxiv.org/abs/1306.0831)

* [[Bart Jacobs]], _Probabilities, distribution monads and convex categories_, Theoretical
Computer Science 412(28) (2011) pp.3323–3336. <https://doi.org/10.1016/j.tcs.2011.04.005>, ([preprint](http://www.cs.ru.nl/B.Jacobs/PAPERS/triangle.pdf))

J. Culbertson and K. Sturtz use the Giry monad in their categorical approach to [[Bayesian reasoning]] and inference (both articles contain further references to the categorical approach to probability theory):

* Jared Culbertson and [[Kirk Sturtz]], _A categorical foundation for Bayesian probability_, Applied Cat. Struc. 2013 (preprint as [arXiv:1205.1488](http://arxiv.org/abs/1205.1488))

* Jared Culbertson and [[Kirk Sturtz]], _Bayesian machine learning via category theory_, 2013 ([arxiv:1312.1445](http://arxiv.org/abs/1312.1445))

* {#Burroni09} Elisabeth Burroni, _Lois distributives. Applications aux automates stochastiques_, TAC 22, 2009 pp.199-221 ([journal page](http://www.tac.mta.ca/tac/volumes/22/7/22-07abs.html))

where she derives stochastic [[automata]] as algebras for a suitable [[distributive law]]
 on the monoid and Giry monads.

B. Fong has a section on the Giry monad in his paper on Bayesian networks:

* Fong: _Causal Theories - A Categorical Perspective on Bayesian Networks_, (2013) arXiv:[1301.6201](http://arxiv.org/abs/1301.6201)

To do:

* Bring in material from [Progic posts](http://golem.ph.utexas.edu/category/2007/09/progic.html)

category: probability

[[!redirects Giry monad]]
[[!redirects Giry's monad]]
[[!redirects Giry's monad]]
[[!redirects Giry\'s monad]]
[[!redirects Giry's monad]]


GIT-stable point

#Contents#
* table of contents
{:toc}

## Idea

Generally, given an [[action]] of a [[group]] on some [[space]], then a _stable point_ of the action is one whose [[stabilizer subgroup]] is a [[finite group]]. This means that the [[quotient stack]] restricted to the stable points is "tame" in that it is just an [[orbifold]].

More specifically, in [[geometric invariant theory]] one gives criteria for (semi-)stability of points in [[projective varieties]] under the [[action]] of [[reductive groups]]. A central application here is to the construction of [[moduli stacks]] of (semi-)[[stable vector bundles]], which are, roughly, the (semi-)stable points under the [[action]] of the [[gauge group]].

A review of the definition of GIT-(semi-)stable points is in ([Saiz 09, pp. 27-28](#Saiz09)). Review of (semi-)[[stable vector bundles]] as GIT-(semi-)stable points is in ([Saiz 09, section 2.3](#Saiz09)).

## References

Review is in

* {#Saiz09} Alfonso Zamora Saiz, _On the stability of vector bundles_, Master thesis 2009 ([[SaizStableBundles.pdf:file]])

* Wikipedia, _[Geometric invariant theory#Stability](http://en.wikipedia.org/wiki/Geometric_invariant_theory#Stability)_

For original references see at _[[geometric invariant theory]]_.

[[!redirects GIT-stable points]]

[[!redirects stable point]]
[[!redirects stable points]]

[[!redirects GIT-semistable point]]
[[!redirects GIT-semistable points]]

[[!redirects GIT-semi-stable point]]
[[!redirects GIT-semi-stable points]]

[[!redirects semi-stable point]]
[[!redirects semi-stable points]]


Giuliano Benenti

* [GoogleScholar entry](https://scholar.google.com/citations?user=gBECZGAAAAAJ&hl=en)

## Selected writings

On [[quantum computation]] and [[quantum information]]:

* [[Giuliano Benenti]], [[Giulio Casati]], [[Davide Rossini]], *Principles of Quantum Computation and Information*, World Scientific 2018  ([doi:10.1142/10909](https://doi.org/10.1142/10909), [2004 pdf](http://mmrc.amss.cas.cn/tlb/201702/W020170224608149307696.pdf))

category: people


Giulio Casati

* [Wikipedia entry](https://en.wikipedia.org/wiki/Giulio_Casati)

## Selected writings

On [[quantum computation]] and [[quantum information]]:

* [[Giuliano Benenti]], [[Giulio Casati]], [[Davide Rossini]], *Principles of Quantum Computation and Information*, World Scientific 2018  ([doi:10.1142/10909](https://doi.org/10.1142/10909), [2004 pdf](http://mmrc.amss.cas.cn/tlb/201702/W020170224608149307696.pdf))

category: people


Giuseppe

A wannabe differential geometer


Giuseppe Dibitetto

* webpage ?

## Selected writings

On [[D=8 supergravity]]:

* [[Mees de Roo]], [[Giuseppe Dibitetto]], Yihao Yin, _Critical points of maximal $D=8$ gauged supergravities_, JHEP 1201:029 (2012) [[arXiv:1110.2886](http://arxiv.org/abs/1110.2886)]

On [[black brane|black]]$\;$[[D6-D8-brane bound states]] in [[massive type IIA string theory]], with [[defect QFT|defect]] [[D2-D4-brane bound states]] inside them realizing [[AdS3-CFT2]] "inside" [[AdS7-CFT6]]:

* [[Giuseppe Dibitetto]], [[Nicolò Petri]], _6d surface defects from massive type IIA_, JHEP 01 (2018) 039 ([arxiv:1707.06154](https://arxiv.org/abs/1707.06154))

* [[Giuseppe Dibitetto]], [[Nicolò Petri]], _$AdS_3$ vacua and surface defects in massive IIA_ ([arxiv:1904.02455](https://arxiv.org/abs/1904.02455))

On [[black brane|black]]$\;$[[D4-D8-brane bound states]] in [[massive type IIA string theory]], with [[defect QFT|defect]] [[D2-D6-brane bound states]] inside them realizing [[AdS3-CFT2]] "inside" [[AdS7-CFT6]]:

* [[Giuseppe Dibitetto]], [[Nicolò Petri]], _Surface defects in the D4 − D8 brane system_, JHEP 01 (2019) 193 ([arxiv:1807.07768](https://arxiv.org/abs/1807.07768))

* [[Giuseppe Dibitetto]], [[Nicolò Petri]], _$AdS_3$ vacua and surface defects in massive IIA_ ([arxiv:1904.02455](https://arxiv.org/abs/1904.02455))

On [[D6-branes]]/[[KK-monopoles]] in [[M-theory]] discussed via [[G-structure]]:

* [[Ulf Danielsson]], [[Giuseppe Dibitetto]], Adolfo Guarino, _KK-monopoles and $G$-structures in M-theory/type IIA reductions_, JHEP 1502 (2015) 096 ([arXiv:1411.0575](https://arxiv.org/abs/1411.0575))

On [[de Sitter spacetime]] [[cosmology]] [[brane world models]] in ambient $\sim$[[anti de Sitter spacetime|AdS]]-[[bulk]] [[spacetime]]:

* {#BDDGS18} Souvik Banerjee, [[Ulf Danielsson]], [[Giuseppe Dibitetto]], Suvendu Giri, Marjorie Schillo, _Emergent de Sitter cosmology from decaying AdS_ ([arXiv:1807.01570](https://arxiv.org/abs/1807.01570))

* Souvik Banerjee, [[Ulf Danielsson]], [[Giuseppe Dibitetto]], Suvendu Giri, Marjorie Schillo, _de Sitter Cosmology on an expanding bubble_ ([arXiv:1907.04268](https://arxiv.org/abs/1907.04268))

On [[AdS2/CFT1]] in [[massive type IIA string theory]]:

* [[Giuseppe Dibitetto]], [[Nicolò Petri]], _$AdS_2$ solutions and their massive IIA origin_, JHEP 05 (2019) 107 ([arXiv:1811.11572](https://arxiv.org/abs/1811.11572))

On [[AdS2/CFT1]] on [[D1-D3 brane intersections]]:

* [[Giuseppe Dibitetto]], [[Yolanda Lozano]], [[Nicolò Petri]], Anayeli Ramirez, _Holographic Description of M-branes via $AdS_2$_ ([arXiv:1912.09932](https://arxiv.org/abs/1912.09932))

category: people


Giuseppe Longo

Giuseppe Longo is a scientific polymath whose principal fields of interest are: [[logic]] and theory of computation; [[denotational semantics]] and [[lambda-calculus]]; [[type theory]], [[category theory]] and their applications to Computer Science, cognitive foundations of Mathematics; interfaces of Mathematics, Physics, Biology. He is Research Director of the Centre Cavaillès at Ecole Normale Supérieure, and Adjunct Professor at the Department of Integrative Physiology and Pathobiology, Tufts University School of Medicine in Boston.

* [website](http://www.di.ens.fr/users/longo/)

###References

* [downloadable publications](http://www.di.ens.fr/users/longo/download.html)
* Giuseppe Longo, Andrea Asperti, _[Categories, Types, and Structures: An Introduction to Category Theory for the working Computer Scientist](http://www.di.ens.fr/users/longo/files/CategTypesStructures/book.pdf)_, MIT Press (1991)
* Giuseppe Longo, Francis Bailly, _Mathematics and the Natural Sciences. The Physical Singularity of Life_, Imperial College Press (2011)
* Giuseppe Longo, Maël Montévil, _Perspectives on Organisms: Biological Time, Symmetries and Singularities_, Springer (2014)

category: people


Giuseppe Peano

* [Wikipedia entry](https://en.wikipedia.org/wiki/Giuseppe_Peano)

## Selected writings

Introducing the modern notion of [[vector spaces]]:

* [[Giuseppe Peano]], *Calcolo Geometrico secondo l'Ausdehnungslehre di H. Grassmann preceduto dalle Operazioni della Logica Deduttiva*, Fratelli Bocca Editori, Torino, 1888, pp. XI, 171 $[$[Mathematica Italiana opere 138](http://mathematica.sns.it/opere/138), [pdf](http://mathematica.sns.it/media/volumi/138/Calcolo%20geometrico%20secondo%20l'Ausdehnungslhere%20di%20H.%20Grassmann_bw.pdf)$]$

  > (referring to [[Hermann Grassmann]]'s *[[Ausdehnungslehre]]*)

Introducing the [[Peano curve]]:

* {#Peano1890} [[Giuseppe Peano]], *Sur une courbe, qui remplit toute une aire plane*, Mathematische Annalen, **36** 1 (1890) 157–160 [[doi:10.1007/BF01199438](https://doi.org/10.1007/BF01199438)]

## Related entries

* [[Peano arithmetic]]

* [[Peano curve]]

* [[Borel's theorem on power series]]

[[!redirects Peano]]

[[!redirects Guiseppe Peano]]

category: people


Giuseppe Policastro

* [webpage](http://www.phys.ens.fr/~policast/)

## related $n$Lab entries

* [[AdS-CFT]], [[AdS-CFT in condensed matter physics]]

* [[TCFT]]

category: people


Giuseppe Primiero

[website](http://www.philosophy.ugent.be/giuseppeprimiero/personal)

category: people


Giuseppe Rosolini

**Giuseppe Rosolini** is an Italian theoretical computer scientist.

 * [homepage](http://www.disi.unige.it/person/RosoliniG/)

## Selected writings

On [[fibered category|fibrations]] of [[enriched groupoids]] as [[categorical semantics]] for [[identity types]] in [[homotopy type theory]]:

* [[Jacopo Emmenegger]], [[Fabio Pasquali]], [[Giuseppe Rosolini]], §3 in: *Elementary fibrations of enriched groupoids*, Mathematical Structures in Computer Science **31** 9 (2021) 958-978 [[doi:10.1017/S096012952100030X](https://doi.org/10.1017/S096012952100030X)]

A [[2-comonad]] characterizing [[Grothendieck fibrations]]:

* [[Jacopo Emmenegger]], [[Luca Mesiti]], [[Giuseppe Rosolini]], [[Thomas Streicher]], *A comonad for Grothendieck fibrations* [[arXiv:2305.01474](https://arxiv.org/abs/2305.01474)]

## Related entries

* [[synthetic domain theory]]

* [[realizability topos]]

* [[recursive topos]]

category: people

[[!redirects Pino Rosolini]]
[[!redirects G. Rosolini]]


Giuseppe Ruzzi

* [publications](http://inspirehep.net/search?p=find+a+ruzzi,+g)

category: people


GL(2,3)

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## Idea

The [[finite group|finite]] [[general linear group]] $GL(2,\mathbb{3})$ for [[coefficients]] the [[prime field]] $\mathbb{F}_3$.

## Properties

### Character table

The [[character table]] for $GL(2,\mathbb{F}_3)$ over the [[complex numbers]] looks like that of the [[binary octahedral group]]. The only difference is the [[Schur indices]]. Hence the character tables over the [[real numbers]] do differ.

[[!include character table of GL(2,3)]]

## Related concepts

* [[SL(2,3)]]

## References

* [[GroupNames]], _GL(2,3)_


Gleason's theorem

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## Idea

### General

The classical _Gleason theorem_ says that a [[state on a star-algebra|state]] on the [[C*-algebra]] $\mathcal{B}(\mathcal{H})$ of all [[bounded operators]] on a [[Hilbert space]] is uniquely described by the values it takes on the orthogonal [[projections]] $\mathcal{P}$, if the dimension of the [[Hilbert space]] $\mathcal{H}$ is not 2.

In other words: every [[quasi-state]] is already a state if  $dim(H) \gt 2$.

It is possible to extend the theorem to certain types of [[von Neumann algebras]] (e.g. obviously [[von Neumann algebra factor|factors]] of type $I_2$ have to be excluded).

Gleason's theorem is also valid for real and quaternionic Hilbert spaces as proved by Varadarajan in 1968. A gap of that proof has been fixed in 2018 by V.Moretti and M.Oppio.

### Implications for Quantum Logic

Roughly, Gleason's theorem says that "a quantum state is completely determined by only knowing the answers to all of the possible yes/no questions".

## Definitions

+-- {: .num_defn}
###### Definition
Let $\rho: \mathcal{P} \to [0, 1]$ such that for every finite family $\{ P_1, ..., P_n: P_i \in \mathcal{P} \}$ of pairwise orthogonal projections we have $\rho(\sum_{i=1}^n P_i) = \sum_{i=1}^n \rho(P_i) $, then $\rho$ is a **finitely additive measure** on $\mathcal{P}$.

If the family is not finite, but countable, then $\rho$ is a **sigma-finite measure**.
=--

## The Theorem

### Classical Gleason's Theorem

+-- {: .num_theorem}
###### Theorem
If $dim(\mathcal{H}) \neq 2$ then each finitely additive measure on $\mathcal{P}$ can be uniquely extended to a state on $\mathcal{B}(\mathcal{H})$. Conversly the restriction of every state to $\mathcal{P}$ is a finitley additive measure on $\mathcal{P}$.

The same holds for sigma-finite measures and [[normal state|normal states]]: Every sigma-finite measure can be extended to a normal state and every normal state restricts to a sigma-finite measure.
=--

### Extension to Certain Types of von Neumann Algebras

...

### Gleason\'s Theorem for POVMs

In [[quantum information theory]], one often considers positive operator-valued measures ([[POVM]]s) instead of [[Hermitian operators]] as [[observables]].  While a Hermitian operator is given by a family of [[projection operator]]s $P_i$ such that $\sum_i P_i = 1$, a POVM is given more generally by any family of positive-semidefinite operators $E_i$ such that $\sum_i E_i = 1$.

In the analog of Gleason's Theorem for POVMs, therefore, we start with $\rho\colon \mathcal{E} \to [0,1]$, where $\mathcal{E}$ is the space of all positive-semidefinite operators.  Then if $\sum_i \rho(E_i) = 1$ whenever $\rho(\sum_i E_i) = 1$, the theorem states that $\rho$ has a unique extension to a mixed [[quantum state]].

As a theorem, Gleason\'s Theorem for POVMs is much weaker than the classical Gleason\'s Theorem, since we must begin with $\rho$ defined on a much larger space of operators.  However, some content does remain, since we have not assumed any continuity properties of $\rho$.  Also, Gleason\'s Theorem for POVMs has a much simpler proof, which works regardless of the dimension.

## Examples

### Counterexample For Dimension Two

See example 8.1 in the book by Parthasarathy (see references).
Our Hilbert space is $\mathbb{R}^2$. Projections $P$ on it are either identical zero, the identity, or projections on a one dimensional subspace, so that these $P$ can be written in the [[bra-ket notation]] as
$$
 P = {|u \rangle} {\langle u|}
$$
with a unit vector $u$, i.e. $u \in \mathbb{R}^2, {\|u\|} = 1$. In this finite dimensional case sigma-finite and finite are equivalent, and a finite probability measure is equivalent to a (complex valued) function such that
$$
f(c u) = f(u)
$$

$$
\sum_i f(u_i) = 1
$$
for every scalar $c$ of modulus one, every unit vector $u$ and every orthonormal basis $\{u_1, u_2\}$. If there is a state that extends such a measure and therefore restricts to such a measure on projections, there would be a linear operator $T$ such that
$$
f(u) = {\langle u | T u \rangle}
$$
for all unit vectors $u$.

It turns out however that the conditions imposed on $f$ are not enough in two dimensions to enforce this kind of linearity of $f$. Heuristically, in three dimensions there are more rotations than in two, therefore the "rotational invariance" of (the conditions imposed on) $f$ is more restrictive in three dimensions than it is in two dimensions.

In two dimensions, choose a function $g$ on $[0, \frac{\pi}{2})$ such that $0 \le g(\theta) \le 1$ everywhere. There are no further restrictions, that is $g$ need not be continuous, for example. Now we can define a probability measure on the projections by
$$
f(u) =
\begin{cases}

  g(\theta) \; \; \text{for} \; \; 0 \le \theta \lt \frac{\pi}{2} \\
  1 - g(\theta -  \frac{\pi}{2}) \; \; \text{for} \; \; \frac{\pi}{2} \le \theta \lt \pi \\
  f(-u) \; \; \text{as defined in the first two items, else}
\end{cases}
$$
This probability measure will in general not extend to a state.

## Related theorems

Other theorems about the foundations and [[interpretation of quantum mechanics]] include:

* [[order-theoretic structure in quantum mechanics]]

  * [[Kochen-Specker theorem]]

  * [[Alfsen-Shultz theorem]]

  * [[Harding-Döring-Hamhalter theorem]]

* [[Fell's theorem]]

* [[Wigner theorem]]

* [[Bell's theorem]]

* [[Bub-Clifton theorem]]

* [[Kadison-Singer problem]]

## References

Gleason's original paper outlining the theorem:

* {#Gleason57} [[Andrew Gleason]], _Measures on the closed subspaces of a Hilbert space_, Journal of Mathematics and Mechanics,  Indiana Univ. Math. J. **6** 4 (1957), 885-893 [[IUMJ:56050](http://www.iumj.indiana.edu/IUMJ/FULLTEXT/1957/6/56050), [pdf](http://www.iumj.indiana.edu/IUMJ/FTDLOAD/1957/6/56050/pdf)]

Textbook accounts:

* [[Huzihiro Araki]], Theorem 2.3 (without proof) in: _[[Mathematical Theory of Quantum Fields]]_ ,

* [[Roland Omnès]], §3 of: *[[The Interpretation of Quantum Mechanics]]*, Princeton University Press (1994) [[ISBN:9780691036694](http://press.princeton.edu/titles/5525.html)]

A monograph stating and proving both the classical theorem and extensions to [[von Neumann algebras]]:

* Jan Hamhalter, _Quantum measure theory_ ([ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1038.81003&format=complete))

The classical theorem is proved also in this monograph:

* K. R. Parthasarathy, _An Introduction to Quantum Stochastic Calculus_ ([ZMATH] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0751.60046&format=complete))

Gleason\'s Theorem for POVMs is proved here:

* Paul Busch, _Quantum states and generalized observables: a simple proof of Gleason's theorem_; (1999) ([arXiv](http://arxiv.org/abs/quant-ph/9909073))

The failure of Gleason's theorem for _classical_ states (on [[Poisson algebra]]s) is discussed in

* Michael Entov, Leonid Polterovich, _Symplectic quasi-states and semi-simplicity of quantum homology_ ([arXiv](http://arxiv.org/abs/0705.3735)).

Gleason\'s Theorem proved for real, complex and quaternionic Hilbert spaces using the notion of real trace.

* [[Valter Moretti]], Marco Oppio, _The correct formulation of Gleason's theorem in quaternionic  Hilbert spaces_, Ann. Henri Poincaré 19 (2018), 3321-3355  ([arXiv:1803.06882](http://arxiv.org/abs/1803.06882))

See also

* Wikipedia on [Gleason's theorem] (http://en.wikipedia.org/wiki/Gleason%27s_theorem)

[[!redirects Gleason's theorem]]
[[!redirects Gleason's theorem]]
[[!redirects Gleason theorem]]
[[!redirects Gleason's Theorem]]
[[!redirects Gleason's Theorem]]
[[!redirects Gleason Theorem]]


Gleb Arutyunov

* [webpage](http://www.staff.science.uu.nl/~aruty101/)

## Selected writings

On [[multi-trace operators]]:

* [[Gleb Arutyunov]], [[Sergey Frolov]], A. C. Petkou, _Perturbative and instanton corrections to the OPE of CPOs in $\mathcal{N}=4$ $SYM_4$_, Nucl. Phys. B602:238-260, 2001; Erratum-ibid. B609:540, 2001 ([arXiv:hep-th/0010137](https://arxiv.org/abs/hep-th/0010137))

## Related $n$Lab entries

* [[Green-Schwarz action functional]]

category: people


Glen Bredon

* [Wikipedia entry](https://en.wikipedia.org/wiki/Glen_Bredon)

## Selected writings

On [[equivariant cohomology]] ([[Bredon cohomology]]):

* Glen Bredon, _[[Equivariant cohomology theories]]_, Springer Lecture Notes in Mathematics Vol. 34. 1967

On [[transformation groups]] ([[topological G-spaces]]):

* Glen Bredon, _[[Introduction to compact transformation groups]], Academic Press 1972

On [[topology]], [[differential geometry]], [[algebraic topology]] and [[homotopy theory]]:

* {#Bredon93} [[Glen Bredon]], _Topology and Geometry_, Graduate texts in mathematics **139**, Springer 1993 ([doi:10.1007/978-1-4757-6848-0](https://link.springer.com/book/10.1007/978-1-4757-6848-0),  [pdf](http://virtualmath1.stanford.edu/~ralph/math215b/Bredon.pdf))

## Related $n$Lab entries

* [[Bredon cohomology]], [[equivariant cohomology]]

* [[equivariant homotopy theory]]

* [[orbit category]]

category: people


Glenn Alexander Magee

* [webpage](http://www.liu.edu/CWPost/Academics/Faculty/Faculty/M/Glenn-Alexander-Magee)

## related entries

* [[hermeticism]]

* [[Science of Logic]]

category: people


Glenn Barnich

* [web](http://homepages.ulb.ac.be/~gbarnich)

## Selected writings

On [[local BRST cohomology]] and [[BV-formalism]]:

* {#BarnichBrandtHenneaux94} [[Glenn Barnich]], [[Friedemann Brandt]], [[Marc Henneaux]], _Local BRST cohomology in the antifield formalism: I. General theorems_, Commun. Math. Phys. **174** (1995) 57-92 [[arXiv:hep-th/9405109](https://arxiv.org/abs/hep-th/9405109), [doi:10.1007/BF02099464](https://doi.org/10.1007/BF02099464)]

* {#BarnichHenneaux96} [[Glenn Barnich]], [[Marc Henneaux]], _Isomorphisms between the Batalin-Vilkovisky antibracket and the Poisson bracket_, J. Math. Phys. **37** (1996) 5273-5296 [[arXiv:hep-th/9601124](https://arxiv.org/abs/hep-th/9601124), [doi:10.1063/1.531726](https://doi.org/10.1063/1.531726)]

* {#BarnichBrandtHenneaux00} [[Glenn Barnich]], [[Friedemann Brandt]], [[Marc Henneaux]], _Local BRST cohomology in gauge theories_, Phys. Rept. **338** (2000) 439-569 [doi:10.1016/S0370-1573(00)00049-1[arXiv:hep-th/0002245](https://arxiv.org/abs/hep-th/0002245)]

and application to [[gravity]] and/or [[Yang-Mills theory]] ([[Einstein-Yang-Mills theory]]):

* [[Glenn Barnich]], [[Friedemann Brandt]], [[Marc Henneaux]], _Local BRST cohomology in Einstein-Yang-Mills theory_, Nucl. Phys. B **455** (1995) 357-408 [doi:10.1016/0550-3213(95)00471-4[arXiv:hep-th/9505173](https://arxiv.org/abs/hep-th/9505173)]

On [[discrete light cone quantization]]:

* [[Glenn Barnich]], Sucheta Majumdar, [[Simone Speziale]], Wen-Di Tan, *Lessons from discrete light-cone quantization for physics at null infinity: Bosons in two dimensions* [[arXiv:2401.14873](https://arxiv.org/abs/2401.14873)]

## Related entries

* [[local BRST cohomology]]

category: people


Glenys Luke

* [institute page](https://www.maths.ox.ac.uk/people/glenys.luke)

## Selected writings

On [[vector bundles]]:

* [[Glenys Luke]], [[Alexandr S. Mishchenko]], *Vector bundles and their applications*, Math. and its Appl. **447** Kluwer (1998) [[doi:10.1007/978-1-4757-6923-4](https://doi.org/10.1007/978-1-4757-6923-4), [MR99m:55019](http://www.ams.org/mathscinet-getitem?mr=99m:55019)]

category:  people


Global analytic geometry

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

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## Idea

Global [[analytic geometry]] is a developing subject that gives an alternative/complementary approach to scheme theory in [[arithmetic geometry]] and [[analytic number theory]]. The starting point of this theory is in Vladimir Berkovich's book about spectral theory and non-archimedean analytic geometry. It was then developped further by [[Jérôme Poineau]].

Many interesting results on polynomial equations can be proved using the mysterious interactions between algebraic, complex analytic and p-adic analytic geometry. The aim of global analytic geometry is to construct a category of spaces which contains these three geometries.

## Motivation
 {#Motivation}

Global analytic geometry

* combines [[non-archimedean analytic geometry|non-archimedean]] and archimedean [[analytic geometry]];

* contains [[algebraic geometry]] as a sub-theory;

* treats all [[places]] on equal footing, contrary to scheme theory. This implies that it is closer in spirit to the automorphic philosophy and [[Langlands program]], and more historically to Tate's proof of the [[functional equation]].

One aim of the theory is to define, using global analytic tools, a good [[Hodge theory]] for arithmetic varieties.

## Possible set of initial constraints
For a [relaxed](http://en.wikipedia.org/wiki/Relaxation_%28approximation%29) approach to global Hodge theory: it is not an easy task to find a good set of constraints on such a [[global Hodge theory]], but they are useful to  understand better the motivations underlying the construction of global analytic spaces.

1. having a good theory of linear coefficients on global analytic spaces,
with the Grothendieck six operations (this should be done by the use of the étale [[sub-analytic]] topology in characteristic $0$, and by a "probably quite hard to develop in characteristic p, but easy to develop in characteristic 0"
model theoretical description of [[definable]] sets for the étale G-topology on strict and non-strict overconvergent analytic spaces). It seems that global analytic motivic
spectral coefficients (given by imposing homotopy invariance with respect to the unit
disc are not so well adapted to the study of torsion phenomena in the characteristic p situation).
2. having a good theory of higher and [[derived global analytic geometry]], with a well-behaved notion of de Rham type cohomology theory and a Chern character. The constraints on such a theory would be:
* get back (or be isomorphic to) the usual algebraic de Rham Chern character when one works with usual schemes.
* get back the p-adic analytic de Rham Chern character (on Ayoub's motivic cohomology) of dagger spaces when one works with dagger p-adic spaces.
* get back the usual de Rham Chern character when one works over $\C$.

This first set of constraints is worked out in the theory of [[overconvergent global analytic geometry]].

The problems related to the torsion phenomena for coefficients in characteristic $p$ may be perhaps be solved by the development of a really non-linear analytic homotopy theory, meaning a proper theory of [[spectral global analytic geometry]].

## A more optimistic set of constraints

1. having a theory of derived analytic microlocalization of sheaves and differential equations, allowing the proper settlement of a [[global analytic index theory]].

1. being able to prove the functional equation of [[zeta functions]] of arbitrary [[arithmetic varieties]];

2. being able to settle down an [[analytic langlands program]], giving a correspondence between general (non-algebraic) [[automorphic representations]] and a sort of global [[analytic motives]]. The p-adic Langlands program should be a particular case of this general construction when the base Banach ring is $\mathbb{Z}_p$.

1. being able to devise a robust and simple enough [[arithmetic cryptography]] protocol based on a [[discrete logarithm problem]] or on a [[cohomological product problem]] on a given [[geometric cohomology]] theory for global analytic spaces.

Argument in favor of its use are:

* the fact that archimedean factors are deeply related to (real and complex analytic)
[[Hodge theory]];

* the fact that all [[proofs]] of parts of [[local Langlands program]] use deeply non-archimedean analytic spaces that are out of the scope of classical algebraic scheme theory.

## Related concepts

* [[Berkovich space]], [[Huber space]], [[perfectoid space]]

* [[Arakelov geometry]]

* [[function field analogy]]

* [[Arithmetic cryptography]]

* [[global analytic index theory]]

* [[overconvergent global analytic geometry]]

* [[spectral global analytic geometry]]

## References

A short introduction for large audience is in

* {#Poineau07} [[Jérôme Poineau]], _Global analytic geometry_, pages 20-23 in EMS newsletter September 2007 ([pdf](http://www.ems-ph.org/journals/newsletter/pdf/2007-09-65.pdf))

For more see

* [[Jérôme Poineau]], _La droite de Berkovich sur Z_, Asterisque 334 (2010).

  (on fundamental properties of the [[affine line]])

* [[Jérôme Poineau]], _Espaces de Berkovich sur Z: etude locale_  [arXiv:1202.0799](http://arxiv.org/abs/1202.0799)

  (on the coherence of the sheaf of analytic functions on higher dimensional affine spaces)

* {#Paugam08} [[Frédéric Paugam]], _Overconvergent global analytic geometry_
([preprint](http://webusers.imj-prg.fr/~frederic.paugam/documents/Overconvergent-Global-Analytic-Geometry.pdf))

* {#Paugam08} [[Frédéric Paugam]], _Global analytic geometry_ ([arXiv:0803.0148](http://arxiv.org/abs/0803.0148))

* [[Frédéric Paugam]], _Global analytic geometry and the functional equation_, lecture notes ([pdf](http://www.math.jussieu.fr/~fpaugam/documents/enseignement/master-global-analytic-geometry.pdf))

* {#BassatKremnitzer13} [[Oren Ben-Bassat]], [[Kobi Kremnizer]], section 7 of _Non-Archimedean analytic geometry as relative algebraic geometry_ ([arXiv:1312.0338](http://arxiv.org/abs/1312.0338))

[[!redirects global analytic geometries]]


global analytic index theory

[[!redirects Global analytic index theory]]
[[!redirects global analytic index theory]]

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## A new formulation for classical index theory

Classical real or [[complex analytic geometry|complex analytic]] [[index theory]] may be described as a setting that contains:

* [[Riemann-Roch theorem|Riemann-Roch]] type theorems, such as Bismut's Riemann-Roch theorem for a general smooth proper morphism of complex analytic spaces, Grothendieck's Riemann-Roch theorem for a morphism of schemes, or Riou's motivic Riemann-Roch theorem for a projective morphism of schemes, and

* Index theorems for elliptic operators or elliptic complexes on real analytic manifolds (that actually also give Riemann-Roch results, when the elliptic complex is induced by the $\bar{\partial}$-operator on differential forms).

* Equivariant [[Atiyah-Singer index theorems]] and noncommutative Connes-Moscovisci local index theorems.

One may formulate many of these results in an essentially unified setting by using index theory on derived analytic stacks over $\mathbb{R}$ or $\mathbb{C}$.

Recall also that one may solve a linear partial differential equation
$$P(\partial)=0$$
when $P$ has constant coefficients by using the analytic continuation of the complex-valued function $(x,s)\mapsto P(x)^s$. This method is called the Bernstein-Sato method for finding the fundamental solution. There are also other methods, but a natural constraint on the given system is to try to do everything in the analytic setting, and it is thus natural to use complex powers to try to solve it. This will be discussed in more details later.

## The challenge of global analytic index theory

Global analytic index theory should give a setting for index theory on [[global analytic spaces]] that gives back usual index theorems when one works over $\mathbb{C}$ or $\mathbb{R}$, but that gives new interesting theorems when one works with analytic spaces or even analytic stacks over $\mathbb{Z}$ equipped with its archimedean norm. It should also give back the p-adic index theorems of Christol-Mebkhout when one works in a Frobenius-equivariant setting over $\mathbb{Z}_p$ or $\mathbb{Q}_p$.

The starting point of this approach should be given by the settlement of a global analytic derived version of Kashiwara and Schapira's microlocal formulation of various index theorems. It is essential, for this approach to work, to have a viewpoint on the microlocalization techniques that is very close to Toen and Vezzosi's loop space approach to the Chern character, through Hochschild and negative cyclic homology of rigid symmetric monoidal stacks of $\infty$-categories over $\infty$-stacks, with an additional microlocal fashion. One also needs a version of [[Grothendieck duality]] adapted to global analytic derived geometry, because it is necessary to define microlocal Hochschild homology in Kashiwara and Schapira's approach to index theory.

## An important difficulty

A difficulty to be overcome in the seek for a global analytic index theory is
that the theory of $D$-modules without denominators, or in characteristic $p$,
lacks of a good notion of de Rham cohomology: the usual definition of it gives
etale cohomology with $k$-coefficients on an algebraically closed field $k$ of characteristic $p$ (Ogus' article on the infinitesimal site). There is also not yet a
clear idea of what the analog of $\mathbb{R}$-constructible coefficients could
be on a general global analytic space.

For example, if one mimicks the approach of Kashiwara and Schapira in the characteristic $p$ setting, one has to replace
$\mathbb{C}/\mathbb{R}$ by $\bar{\mathbb{F}}_p/\mathbb{F}_p$, which is an infinite extension. The natural analog in this situation of the elliptic pair
$(\mathcal{O}_X,\mathbb{C}_M)$ associated to $M/\mathbb{R}$, for $M/\mathbb{F}_p$ a projective smooth variety, has de Rham cohomology an $\bar{\mathbb{F}}_p$-module that doesn't seem to be finite dimensional.

Two approaches to $\mathcal{D}$-modules in characteristic $p$ were developed by Berthelot and Mebkhout-Narvaez-Macarro. One may try to extend this last approach (using weak formal schemes over $\mathbb{Z}_p$) to a global base, but one will face the problem of dealing with (naive) $\mathcal{D}$-modules in characteristic $p$, that are not well behaved.

Another (maybe more promising) approach is to get inspiration from Buium's differential calculus over the integers and Madsen/Hesselholt's results on topological cyclic homology: making a topological version of $\mathcal{D}$-modules in the setting of [[spectral global analytic geometry]].

## Pursuing global analytic trace kernels

Recall that a trace kernel in Kashiwara and Schapira's sense
is a $k_{M\times M}$-module $K$ together with a sequence of morphisms
$$k_\Delta\to K\to \omega_\Delta$$
and the corresponding class is a section of $\mu hom(k_\Delta,\omega_\Delta)$.

In the coherent setting, the analog of a trace kernel associated to a coherent $\mathcal{O}$-module $\mathcal{F}$ is given by the sequence
$$
\mathcal{O}_\Delta\to \mathcal{F}\boxtimes D\mathcal{F}\to \omega^{hol}_\Delta.
$$

In the $\mathcal{D}$-module setting, the analog of a trace kernel of a coherent
$\mathcal{D}$-module $\mathcal{M}$ is given by the sequence
$$
\mathcal{B}_\Delta\to \mathcal{M}\boxtimes D_\mathcal{D}\mathcal{M}\to \mathcal{B}^\vee_\Delta
$$
and one sends it to the trace kernel given by taking the de Rham functor (tensor product operation $\omega_{X\times X}\otimes_\mathcal{D}-$. The corresponding microlocal euler class is given by the microlocalization of the above sequence
$$
\mathcal{B}_\Delta^E\to \mathcal{M}^E\boxtimes D_\mathcal{E}\mathcal{M}^E\to \mathcal{B}^{E\vee}_\Delta
$$
because one has a canonical isomorphism
$$
\mathbb{R}Hom_{\mathcal{E}}(\mathcal{B}_\Delta^E,\mathcal{B}_\Delta^{E\vee})
\overset{\sim}{\longrightarrow}
\mu hom(\mathbb{C}_\Delta,\omega_\Delta).
$$

The problem is now the following: given an $R$-constructible sheaf $F$ on $M$,
one may define a trace kernel by using the sequence
$$\mathbb{C}_\Delta\to F\boxtimes DF\to \omega_\Delta.$$
How can one formulate this construction in terms of $\mathcal{E}$-modules (maybe with an additional structure, such as a kind of filtration, if one takes inspiration from $p$-adic Hodge theory, with filtered modules with connections, or $p$-adic Riemann-Hilbert, with $p$-adic differential equations with Frobenius structure)?

The easiest thing to try is to simply use the (awful) sheaf of rings $\mathcal{E}^\infty$ of infinite order microdifferential operators. Indeed, if $F$ is an $R$-constructible sheaf, then $\mu Hom(F,\mathcal{O})$ is naturally an $\mathcal{E}^\infty$-module. Moreover, any coherent $\mathcal{D}$-module gives rise to an $\mathcal{E}$-module, and thus an $\mathcal{E}^\infty$-module. It is thus quite tempting to try to define a global version of $\mathcal{E}^\infty$ and to find a nice class of $\mathcal{E}^\infty$-modules, containing both (non-torsion) constructible sheaves and the images of coherent $\mathcal{D}$-modules.

A first step in this global direction is to develop an index theory that works in a uniform way over $\mathbb{Q}_p$ and $\mathbb{R}$. This seems to be an almost done theory (because there is no lack of denominators): the notion of constructible sheaf is replaced by its de Rham version. More precisely, a constructible sheaf is constructible for the, say, sub-analytic overconvergent etale topology, i.e., sub-analytically etaly isomorphic to $\mathrm{DR}(\mathcal{O})$, in the sense of $\mathcal{D}^\infty$-modules, and not as usual to a locally constant sheaf: this only works in the complex setting because smooth analytic spaces are locally isomorphic for the semi-analytic topology in the complex case/semi-analytic etale topology in the p-adic case to discs only in the complex situation/to rational domains in the $p$-adic situation, and one may not try to solve $p$-adic differential equations on a rational domain too naively because the exponential doesn't converge so nicely as in the complex case.

One should not try to develop a theory of $p$-adic locally constant coefficients
because the $p$-adic etale cohomology of the unit disc is infinite dimensional:
only de Rham $p$-adic coefficients make sense in (overconvergent) $p$-adic analytic geometry. This is also a good reason to try to develop a global version of de Rham coefficients through the use of topological methods (to encode the de Rham Witt vector constructions in a canonical way).

Once the theory of $p$-adic constructible coefficients is developed, it becomes easy to copy Kashiwara and Schapira's index theory.

## A possible set of constraints for a coherent theory

One should be able to define global analytic analogs of:

1. The derived duality functor on coherent $\mathcal{D}$-modules and in particular coherent $\mathcal{O}$-modules (carefully defined as modules on the underlying dagger algebra, locally for the $G$-topology, even in the affinoid case).

1. The Grothendieck six operations on:
* coherent $\mathcal{O}$-modules,
* coherent $\mathcal{O}^t$-modules, where $\mathcal{O}^t$ (definition due to P. Schapira) is the sheaf for the $G$-topology given on a rational domain $R$ with Berkovich interior $\overset{\circ}{R}$ by the subsheaf of $\mathcal{O}_{\overset{\circ}{R}}$ given by functions $f$ with temperate growth, that one may define as the $\mathcal{D}$-module generated by the ---G-topological--- sheaf $\mathcal{O}^b$ of bounded holomorphic functions.
* sub-analytic torsion étale (i.e., Artin-Verdier) or more generally motivic analytic sheaves (using the sheaf-cosheaf formalism of Lurie's book higher algebra, generalized to the $1$-localic topos of étale sheaves on global analytic spaces; one must use the associated etale Berkovich topos to define properly the notion of properness, that is needed for the definition of direct images with proper support; in the strict or generalized Arakelov compactified case, one may need to restrict to morphism of proper logarithmic spaces to get the six operations; this is in some sense analogous to the fact that schemes always have a stacky compactification, i.e., a proper smooth hypercovering, following Deligne), and
* global analytic holonomic $\mathcal{D}$-modules and global analytic exponential motivic sheaves; the $\mathcal{D}$-module situation should involve also global microlocalization techniques (to get natural conditions for the definition of inverse images) and maybe also the Ran space approach with the $*$-tensor product $\overset{*}{\otimes}$.
* One may also define the Grothendieck six operations on general $\mathcal{D}^\infty$-modules.

1. The Grothendieck six operations on sub-analytic Hodge structures (given by a loop space construction of the Hodge filtration, without denominators, together with a Betti structure, given by topological $K$-theory with the $\Lambda$-structure on the Chern character; this may involve the use of topological/spectral methods to take care of Witt vector constructions on an integral or characteristic $p$ base).

1. The Chern character in various situations, using either the formalism of Toen-Vezzosi or the formalism of Kashiwara-Schapira, applied to the following stacks of rigid (maybe non) symmetric monoidal $\infty$-categories on the $\infty$-topos of derived dagger analytic stacks with its pro-etale, Nisnevich or usual analytic topology:
* the stack $X\mapsto Perf(X)$ of perfect $\mathcal{O}$-modules (derived analog of coherent modules),
* the stack $X\mapsto Const_{proet}(X,\mathcal{C})$ of constructible pro-etale sub-analytic sheaves with values in a symmetric monoidal stable presentable $\infty$-category, with their natural tensor product and duality; be careful, the pro-etale constructibility is meant in the above de Rham sense: one should not use locally constant coefficients because they don't work well even over $\mathbb{Q}_p$. The problem with this stack is that it doesn't take care of the derived (infinitesimal) information, so its treatment should be given in a Kashiwara-Schapira approach, using loop space microlocalization and some additional structure that will probably be necessary to encode an R-constructible sheaf in a microdifferential structure.
* one would also like to treat the case of (at least holonomic) $\mathcal{D}$-modules, either using the Ran space or another construction. It seems that Toen-Vezzosi's approach has to be modified to treat this case properly because of the compound tensor structure $(\otimes^!,\otimes^*)$ on $\mathcal{D}$-modules on the Ran space.

1. The treatment of $\mathcal{D}$-modules and $\mathcal{D}_q$-modules in a uniform setting may involve the following generalization of the Kashiwara and Schapira approach (in a Toen-Vezzosi type of setting) due to F. Petit: one must work with trace kernels, and compute the trace of their action on the Hochschild or cyclic homology of ---non-symmetric--- monoidal categories of (typically) bimodules on an associative algebra. The proper treatment of this theory will be discussed in the section [[non-commutative global analytic index theory]].

## Derived index theory

Using the methods of [[derived microlocalization]] and Kashiwara and Schapira's [[microlocal formulation of index theory]], one may develop a derived index theory.

## References

[[Masaki Kashiwara]] and [[Pierre Schapira]] _Microlocal Euler classes and Hochschild homology_ [arXiv](http://arxiv.org/abs/1203.4869)

One may be very tempted to use the following reference, to get a better way of looking at Deligne (or global analytic motivic) cohomology, in use for the Grothendieck-Riemann-Roch theorem:

* J. P. Pridham, _A K-theoretic interpretation of real Deligne cohomology_ [arXiv](http://arxiv.org/pdf/1411.5705v1.pdf)


global dimension theorem

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

#Contents#
* table of contents
{:toc}

## Idea

The global dimension theorem relates various different notions of [[dimension]] of a [[ring]].

## References

The global dimension theorem appears for instance as theorem 4.1.2 in

* [[Charles Weibel]], _[[An Introduction to Homological Algebra]]_

See also the notes

* Daniel Katz, _Global dimension theorem_ ([pdf](http://www.math.ku.edu/~dlk/Lecture08.pdf))


global element

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

# Global elements
* table of contents
{: toc}

## Idea

One striking difference between [[set theory]] and [[category theory]] is that, while [[objects]] of a [[category]] need not have any other structure, a [[set]] comes equipped with the notion of _element_, identifying other sets which belong to it.  Sometimes, a category turns out to possess a similar structure, designating certain [[morphisms]] as __global elements__ (or __global points__ in geometric contexts) of an object.

## Definition

If a category $C$ has a [[terminal object]] $1$, a __global element__ of another object $x$ is a morphism $1 \to x$.

So a global element is a [[generalized element]] at "stage of definition" $1$.

For example:

* In [[Set]], global elements are just elements: a function from a one-element set into $x$ picks out a single element of $x$.

* In [[Cat]], global elements are objects: the terminal category $1$ is the [[discrete category]] with one object, and a [[functor]] from $1$ into a category $C$ singles out an object of $C$.

* In a [[topos]], a global element of the [[subobject classifier]] is called a [[truth value]].

* Working in a [[over category|slice category]] $C/b$, a global element of the object $\pi: e \to b$ is a map into it from the terminal object $1_b: b \to b$; i.e., a [[right inverse]] for $\pi$.  In the context of [[bundles]], a global element of a bundle is called a *[[global section]]*.

If $C$ does not have a terminal object, we can still define a global element of $x\in C$ to be a global element of the [[represented functor|represented]] [[presheaf]] $C(-,x) \in [C^{op},Set]$.  Since the [[Yoneda embedding]] $x \mapsto C(-,x)$ is fully faithful and preserves any [[limits]] that exist in $C$, including terminal objects, if $C$ does have a terminal object then this definition coincides with the more naive one.

If we unravel the general definition more explicitly, it says that a global element of $x$ consists of, for every object $i\in C$, a morphism $e_x : i\to x$ (i.e. a [[generalized element]] of $x$ at stage $i$), such that for any morphism $f:i\to j$ we have $e_j \circ f = e_i$.

## Variations

Many (but not all) of the examples above are [[cartesian closed categories]].  In a more general [[closed category]], a morphism from the unit object to $x$ can be called an _element_ of $x$. For example, an element of an [[abelian group]] $x$ is a morphism from the group $\mathbf{Z}$ of integers to $x$, and of course this is equivalent to the usual notion of element of $x$. Here the adjective 'global' would not conform to the usage above since $\mathbf{Z}$ is not terminal, although we warn that some authors may call a map $\mathbf{Z} \to A$ a "global element" of $A$.

Thus generally, when $C$ is cartesian closed or even [[semicartesian monoidal category|semicartesian monoidal]] closed, the monoidal unit $I$ is terminal and such elements $I \to A$ are global elements in the sense of this article. But again, even in the general monoidal case, some authors call maps of the form $I \to A$ "global elements", even though this conflicts with our usage.

As an important special case, there is[^1] for [[closed monoidal categories]] a notion of "name of a morphism", as follows. Let $\mathcal{C}$ be closed monoidal, with external ($Set$-valued) homs denoted by $\mathcal{C}(A, B)$, the monoidal product by $\otimes$ and the monoidal unit by $I$, and internal homs by $[A, B]$. Then for each pair $(A, B)$, the evident composite map

$$\mathcal{C}(A, B) \stackrel{\mathcal{C}(\lambda_A, 1_B)}{\to} \mathcal{C}(I \otimes A, B) \cong \mathcal{C}(I, [A, B])$$

($\lambda_A: I \otimes A \to A$ the left unit isomorphism) defines a map which we denote as $name_{A, B}: \mathcal{C}(A,B)\rightarrow\mathcal{C}(I, [A,B])$. Notice this is the component at $(A, B)$ of a natural bijection $name$; it takes a map $f: A \to B$ in $\mathcal{C}$ to its name, typically denoted as $\text{"}f\text{"}: I \to [A, B]$, and which is an element of the internal hom $[A, B]$.

[^1]: See, e.g., John Baez, Quantum Gravity Seminar, University of California, Riverside, Fall 2006, notes taken by Derek Wise, lecture of 2 November 2006.

Finally, in contrast to a global element, a morphism to $x$ from _any_ object $i$ whatsoever may be seen as a [[generalized element]] of $x$. For example, if $i$ is the [[unit interval]] (in topology, chain complexes, etc), then a map from $i$ to $x$ is a *path* (rather than a point) in $x$. Or in a slice category $C/b$, if $\rho: a \to b$ is an [[embedding]], then a morphism from $\rho$ to $\pi$ is a _local_ section of $\pi$.

[[!redirects global element]]
[[!redirects global elements]]
[[!redirects global point]]
[[!redirects global points]]


global equivariant bordism homology theory

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

#Contents#
* table of contents
{:toc}

## Idea

The generalization of [[equivariant bordism homology theory]] (hence of [[bordism homology theory]]) to [[global equivariant homotopy theory]].

## Related concepts

* [[ultracommutative ring spectrum]]

[[!include flavours of cobordism cohomology theories -- table]]

## References

* [[Stefan Schwede]], Section 6 of: _[[Global homotopy theory]]_, New Mathematical Monographs 34, Cambridge University Press, 2018 ([doi:10.1017/9781108349161](https://doi.org/10.1017/9781108349161), [arXiv:1802.09382](https://arxiv.org/abs/1802.09382))

Survey:

* {#Schwede15} [[Stefan Schwede]], _Equivariant bordism from the global perspective_, 2015 ([pdf](http://www.math.uni-bonn.de/people/schwede/glasgow-handout.pdf), [[SchwedeGlobalEquivariantBordism.pdf:file]])

[[!redirects mO]]
[[!redirects equivariant mO]]
[[!redirects global equivariant mO]]

[[!redirects mU]]
[[!redirects equivariant mU]]
[[!redirects global equivariant mU]]


global equivariant homotopy theory

> under construction (some more harmonization needed)

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

#Contents#
* table of contents
{:toc}

## Idea

What is called _global equivariant homotopy theory_ is a variant of [[equivariant cohomology]] in [[homotopy theory]] where [[pointed object|pointed]] [[topological spaces]]/[[homotopy types]] are equipped with $G$-[[infinity-actions]] "for all [[compact Lie groups]] $G$ at once", or more generally for a [[global family]].

Sometimes this is referred to just as "global homotopy theory", leaving the equivariance implicit. There is also a [[stabilization|stable]] version involving [[spectra]] equipped with [[infinity-actions]], see at _[[global equivariant stable homotopy theory]]_.

More precisely, the _global equivariant homotopy category_ is the [[(∞,1)-category]] (or else its [[homotopy category of an (∞,1)-category|homotopy category]]) of [[(∞,1)-presheaves]] $PSh_\infty(Orb)$ on the  [[global orbit category]] $Orb$ ([Henriques-Gepner 07, section 1.3](#HenriquesGepner07)), regarded as an [[(∞,1)-category]].

Here $Orb$ has as [[objects]] [[compact Lie groups]] and the [[(∞,1)-categorical hom-spaces]] $Orb(G,H) \coloneqq \Pi [\mathbf{B}G, \mathbf{B}H] $, where on the right we have the [[geometric realization of cohesive infinity-groupoids|fundamental (∞,1)-groupoid]] of the [[topological groupoid]] of [[group homomorphisms]] and [[conjugation|conjugations]].

## Definition
 {#Definition}

We follow ([Rezk 14](#Rezk14)). Beware that the terminology there differs slightly but crucially in some places from ([Henriques-Gepner 07](#HenriquesGepner07)). Whatever terminology one uses, the following are the key definitions.

The following is the _[[global equivariant indexing category]]_.

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

Write $Glo$  for the [[(∞,1)-category]] whose

* [[objects]] are [[compact Lie groups]];

* [[(∞,1)-categorical hom-spaces]] $Glo(G,H)$ are the [[geometric realizations]] of the [[Lie groupoid]] of smooth functors and smooth [[natural transformations]] $Top\infty Grpd(\mathbf{B}G, \mathbf{B}H)$.

=--

([Rezk 14, 2.1](#Rezk14))

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

Equivalent models for the global indexing category, def. \ref{GlobalIndexingCategory} include the category "$O_{gl}$" of ([May 90](#May90)).
Another variant is $\mathbf{O}_{gl}$ of ([Schwede 13](#Schwede13)).

=--

([Rezk 14, 2.4, 2.5](#Rezk14))

The following is the _[[global orbit category]]_.

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

Write

$$
  Orb \longrightarrow Glo
$$

for the non-[[full sub-(∞,1)-category|full]] [[sub-(∞,1)-category]] of the global indexing category, def. \ref{GlobalIndexingCategory}, on the [[injection|injective]] group homomorphisms.

=--

([Rezk14, 4.5](#Rezk14))

The following defines the _global equivariant homotopy theory_ $PSh_\infty(Glo)$.

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

Write

$$
  Top_{Glo} \coloneqq PSh_\infty(Glo)
$$

for the [[(∞,1)-category of (∞,1)-presheaves]] (an [[(∞,1)-topos]]) on the global indexing category $Glo$ of def. \ref{GlobalIndexingCategory}, and write
$$
  \mathbb{B} \;\colon\; Glo \longrightarrow PSh_\infty(Glo)
$$

for the [[(∞,1)-Yoneda embedding]].

Similarly write

$$
  Top_{Orb} \coloneqq PSh_\infty(Orb)
$$

for the [[(∞,1)-category of (∞,1)-presheaves]] on the  [[global orbit category]] $Orb$ of def. \ref{GlobalOrbitCategory}, and write again

$$
  \mathbb{B} \;\colon\; Orb \longrightarrow PSh_\infty(Orb)
$$

for its [[(∞,1)-Yoneda embedding]].

=--

([Rezk 14, 3.1 and 4.5](#Rezk14))

The following recovers the ordinary ("local") [[equivariant homotopy theory]] of a given [[topological group]] $G$ ("of $G$-spaces").

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

For $G$ a [[topological group]], write

$$
  G Top \coloneqq PSh_\infty(Orb)/\mathbb{B}G
$$

for the [[slice (∞,1)-topos]] of $PSh_\infty(Orb)$ over the image of $G$ under the [[(∞,1)-Yoneda embedding]], as in def. \ref{GlobalEquivariantHomotopyTopos}.

=--

This is ([Rezk 14, 1.5](#Rezk14)). Depending on axiomatization this is either a definition or [[Elmendorf's theorem]], see at _[[equivariant homotopy theory]]_ for more on this.

## Properties

### Cohesion

For more see at *[[cohesion of global- over G-equivariant homotopy theory]]*.

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

The global equivariant homotopy theory $PSh_\infty(Glo)$ of def. \ref{GlobalEquivariantHomotopyTopos} is a [[cohesive (∞,1)-topos]] over the canonical [[base (∞,1)-topos]] [[∞Grpd]]:

the [[global section geometric morphism]]

$$
  (\Delta \dashv \Gamma)
  \;\colon\;
  PSh_\infty(Glo)
  \longrightarrow
  \infty Grpd
$$

is given (as for all (∞,1)-presheaf (∞,1)-toposes) by the [[direct image]]/[[global section]] functor being the [[homotopy limit]] over the opposite [[(∞,1)-site]]

$$
  \Gamma X \simeq \underset{\leftarrow}{\lim}(Glo^{op}\stackrel{X}{\to} \infty Grpd)
$$

and the [[inverse image]]/[[constant ∞-stack]] functor literally assigning constant presheaves:

$$
  \Delta S \colon G \mapsto S
 \,.
$$

This is a [[full and faithful (∞,1)-functor]].

Moreover,  $\Delta$ has a further [[left adjoint]] $\Pi$ which preserves [[finite products]], and $\Gamma$ has a further [[right adjoint]] $\nabla$.

=--

([Rezk 14, 5.1](#Rezk14))

More in detail, the [[shape modality]], [[flat modality]] and [[sharp modality]] of this [[cohesion]] of the global equivariant homotopy theory has the following description.

### Relation between global and local equivariant homotopy theory

Some aspects of the [[cohesion of global- over G-equivariant homotopy theory]]:

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

For $G$ a [[compact Lie group]] define an [[(∞,1)-functor]]

$$
  \delta_G \;\colon\; G Top \longrightarrow PSh_\infty(Glo)
$$

sending a [[topological G-space]] to the  he presheaf which sends a group $H$ to the [[geometric realization]] of the [[topological groupoid]] of maps from $\mathbf{B}H$ to the [[action groupoid]] $X//G$:

$$
  \delta_G(X)\;\colon\; H \mapsto \Pi( [\mathbf{B}H, X//G] )
  \,.
$$

Observe that by def. \ref{GlobalEquivariantHomotopyTopos} this gives $\delta_G(\ast) \simeq \mathbb{B}G$ and so $\delta_G$ induces a functor

$$
  \Delta_G \;\colon\; G Top \simeq G Top/\ast \simeq PSh_\infty(Orb)/\mathbb{B}G \stackrel{\delta_G}{\longrightarrow} PSh_\infty(Glo)/\mathbb{B}G
 \,.
$$

=--

([Rezk 14, 3.2](#Rezk14))

+-- {: .num_prop #OrdinaryQuotientAndHomotopyQuotientViaCohesion}
###### Proposition
**(ordinary [[quotient]] and [[homotopy quotient]] via equivariant cohesion)**

On a $G$-space $X \in G Top$ included via def. \ref{InclusionOfGSpacesInTheGlobalTheory} into the global equivariant homotopy theory,

* the [[shape modality]] of def. \ref{CohesionOfGlobalEquivariantHomotopyTheory} produces the [[homotopy type]] of the ordinary [[quotient]] of the $G$-[[action]]

  $$
    \Pi(\delta_G(X)) \simeq \vert X/G \vert
    \,,
  $$

* the [[flat modality]] of def. \ref{CohesionOfGlobalEquivariantHomotopyTheory} produces the [[homotopy type]] of the [[homotopy quotient]]/[[homotopy coinvariants]] of the $G$-[[action]] ([[∞-action]])

  $$
    \Gamma(\delta_G(X)) \simeq \vert X//G \vert
    \,,
  $$

In particular then the [[points-to-pieces transform]] of general [[cohesion]] yields the comparison map

$$
  \vert X//G \vert
  \longrightarrow
  \vert X/G \vert
   \,.
$$

=--

([Rezk 14, 5.1](#Rezk14))

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

For $G$ any [[compact Lie group]], the [[cohesion]] of the global equivariant homotopy theory, prop. \ref{CohesionOfGlobalEquivariantHomotopyTheory}, descends to the [[slice (∞,1)-toposes]]

$$
  PSh_\infty(Glo)/\mathbb{B}G
  \longrightarrow
  PSh_\infty(Orb)/\mathbb{B}G \simeq G Top
  \,,
$$

hence to cohesion over the "local" $G$-[[equivariant homotopy theory]].

The inclusion $\Delta_G$ is that of def. \ref{InclusionOfGSpacesInTheGlobalTheory}.

=--

([Rezk 14, 5.3](#Rezk14))

### Relation to topological stacks and orbispaces

> under construction

By the main theorem of ([Henriques-Gepner 07](#HenriquesGepner07)) the [[(∞,1)-presheaves]] on the  [[global orbit category]] are equivalently "cellular" [[topological stacks]]/[[topological groupoids]] ("[[orbispaces]]"), we might write this as

$$
  ETopGrpd^{cell} = PSh_\infty(Orb)
  \,.
$$

(As such the global equivariant homotopy theory should be similar to [[ETop∞Grpd]]. Observe that this is a [[cohesive (∞,1)-topos]] with $\Pi$ such that it sends a topological [[action groupoid]] of a [[topological group]] $G$ acting on a [[topological space]] $X$ to the [[homotopy quotient]] $\Pi(X)//\Pi(G)$.)

The central theorem of ([Rezk 14](#Rezk14)) (using a slightly different definition than [Henriques-Gepner 07](#HenriquesGepner07)) is that $PSh_\infty(Orb)$ is a [[cohesive (∞,1)-topos]] with $\Gamma$ producing homotopy quotients.

## Related concepts

* [[cohesion of global- over G-equivariant homotopy theory]]

* [[orbifold cohomology]]

* [[global equivariant stable homotopy theory]]

* [[proper equivariant homotopy theory]]

[[!include equivariant homotopy theory -- table]]

[[!include homotopy type representation theory -- table]]

## References

The [[global orbit category]] $Orb$ is considered in

* {#HenriquesGepner07} [[André Henriques]], [[David Gepner]], _Homotopy Theory of Orbispaces_ ([arXiv:math/0701916](http://arxiv.org/abs/math/0701916))

* {#Lurie} [[Jacob Lurie]], Section 3 of _Elliptic cohomology III: Tempered Cohomology_ ([pdf](http://www.math.harvard.edu/~lurie/papers/Elliptic-III-Tempered.pdf))

Global unstable equivariant homotopy theory is discussed as a [[localization]] of the category of "orthogonal spaces" (the unstable version of [[orthogonal spectra]]) in

* {#Schwede17} [[Stefan Schwede]], _Orbispaces, orthogonal spaces, and the universal compact Lie group_ ([arXiv:1711.06019](https://arxiv.org/abs/1711.06019))

* {#Schwede18} [[Stefan Schwede]], _[[Global homotopy theory]]_, New Mathematical Monographs **34**, Cambridge University Press, 2018 ([doi:10.1017/9781108349161](https://doi.org/10.1017/9781108349161), [arXiv:1802.09382](https://arxiv.org/abs/1802.09382))

see also

* {#May90} [[Peter May]], _Some remarks on equivariant bundles and classifying spaces_, Asterisque 191 (1990), 7, 239-253. International Conference on Homotopy Theory (Marseille-Luminy, 1988).

Discussion of the global equivariant homotopy theory as a [[cohesive (∞,1)-topos]] is in

* {#Rezk14} [[Charles Rezk]], _[[Global Homotopy Theory and Cohesion]]_ (2014)

Discussion of a model structure for global equivariance with respect to [[geometrically discrete ∞-groupoid|geometrically discrete]] [[simplicial groups]]/[[∞-group]] (globalizing the [[Borel model structure]] for [[∞-actions]]) is in

* [[Yonatan Harpaz]], [[Matan Prasma]], _The Grothendieck construction for model categories_ ([arXiv:1404.1852](http://arxiv.org/abs/1404.1852))

Discussion from a perspective of [[homotopy type theory]] is in

* {#Shulman15} [[Mike Shulman]], _Univalence for inverse EI diagrams_ ([arXiv:1508.02410](http://arxiv.org/abs/1508.02410))

The example of global equivariant [[algebraic K-theory]]:

* [[Stefan Schwede]], _Global algebraic K-theory_ ([arXiv:1912.08872](https://arxiv.org/abs/1912.08872))

On [[orbifold cohomology]] seen in [[global equivariant homotopy theory]]:

* {#Juran20} [[Branko Juran]], _Orbifolds, Orbispaces and Global Homotopy Theory_ ([arXiv:2006.12374](https://arxiv.org/abs/2006.12374))

following the suggestion in [Schwede 17, Intro](#Schwede17), [Schwede 18, p. ix-x](#Schwede18).

[[!redirects global homotopy theory]]

[[!redirects global equivariant homotopy category]]

[[!redirects globally equivariant homotopy theory]]


global equivariant indexing category

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#### Representation theory
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#### Cohomology
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#Contents#
* table of contents
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## Idea

The _global equivariant indexing category_ is the full subcategory of [[topological ∞-groupoids]] on those which are [[deloopings]] of [[compact Lie groups]], with hom-spaces being the [[geometric realization]] of the [[internal homs]] there.

The [[(∞,1)-presheaves]] over the global equivariant indexing category is the [[global equivariant homotopy theory]]. This is a [[cohesive (∞,1)-topos]] over [[∞Grpd]] ([Rezk 14](#Rezk14)).

## Definition

The following defines the _global equivariant indexing category_ $Glo$.

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

Write $Glo$  for the [[(∞,1)-category]] whose

* [[objects]] are [[compact Lie groups]];

* [[(∞,1)-categorical hom-spaces]] $Glo(G,H)$ are the [[geometric realizations]] of the [[Lie groupoid]] of smooth functors and smooth [[natural transformations]] $Top\infty Grpd(\mathbf{B}G, \mathbf{B}H)$.

=--

([Rezk 14, 2.1](#Rezk14))

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

Equivalent models for the global indexing category, def. \ref{GlobalIndexingCategory} include the category "$O_{gl}$" of ([May 90](#May90)).
Another variant is $\mathbf{O}_{gl}$ of ([Schwede 13](#Schwede13)).

=--

([Rezk 14, 2.4, 2.5](#Rezk14))

The following is the _[[global orbit category]]_.

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

Write

$$
  Orb \longrightarrow Glo
$$

for the non-full [[sub-(∞,1)-category]] of the global indexing category, def. \ref{GlobalIndexingCategory}, on the [[injection|injective]] group homomorphisms.

=--

([Rezk 14, 4.5](#Rezk14))

## Properties

### Relation to the local orbit category

The [[slice (∞,1)-category]] of the global orbit category over $\mathbf{B}G$ is the local [[orbit category]] of $G$

$$
  Glo_{/\mathbf{B}G}  \simeq Orb_G
 \,.
$$

### Relation to orbispaces and $G$-spaces

The [[(∞,1)-category of (∞,1)-presheaves]] over the global orbit category is that of [[orbispaces]].

Accordingly, by the discussion [here](over-%28infinity%2C1%29-topos#SheavesOnBigSite), the [[slice (∞,1)-topos]] of orbispaces over $\mathbf{B}G$ is that of [[G-spaces]]

$$
  PSh_\infty(Orb)_{/\mathbf{B}G}
  \simeq
  PSh_\infty(Orb_{/\mathbf{B}G})
  \simeq
  PSh_\infty(Orb_G)
  \simeq
  G Space
$$

(where the last step is [[Elmendorf's theorem]]).

### Relation to equivariant homotopy theory

The [[(∞,1)-category of (∞,1)-presheaves]] on the [[global equivariant indexing category]] is the [[global equivariant homotopy theory]] and under the canonical projection is a [[cohesive (∞,1)-topos]] over [[∞Grpd]]. Its [[slice (∞,1)-topos]] over the terminal orbispace is cohesive over [[orbispaces]]

$$
  PSh_\infty(Glo)_{/\mathcal{N}} \to PSh_\infty(Orb)
 \,.
$$

## References

* {#HenriquesGepner07} [[André Henriques]], [[David Gepner]], _Homotopy Theory of Orbispaces_ ([arXiv:math/0701916](http://arxiv.org/abs/math/0701916))

* {#Rezk14} [[Charles Rezk]], _[[Global Homotopy Theory and Cohesion]]_ (2014)

See also

* {#Schwede13} [[Stefan Schwede]], _Global homotopy theory_, 2013 ([pdf](http://www.math.uni-bonn.de/~schwede/global.pdf))

* {#May90} [[Peter May]], _Some remarks on equivariant bundles and classifying spaces_, Asterisque 191 (1990), 7, 239-253. International Conference on Homotopy Theory (Marseille-Luminy, 1988).


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

## Idea

What is called _global equivariant stable homotopy theory_ is a variant of [[equivariant stable homotopy theory]] where  [[spectra]] are equipped with $G$-[[infinity-actions]] "for all [[compact Lie groups]] $G$ at once".

Often this is referred to just as "global stable homotopy theory" or even just "global homotopy theory". But there is also _unstable_ _[[global equivariant homotopy theory]]_.

More precisely, given an [[orthogonal spectrum]] $X$, then every [[representation]] $\rho \colon G \to O(n)$ of a [[compact Lie group]] on the [[Cartesian space]] $\mathbb{R}^n$ by [[orthogonal group]] [[actions]] induces a $G$-[[equivariant spectrum]] and hence a notion of $G$-[[equivariant homotopy groups]].

One says that a morphism of [[orthogonal spectra]] is a **global equivariant equivalence** if it induces [[isomorphisms]] on all $G$- [[equivariant homotopy groups]], for all $G$, this way. (This definition appears for instance as ([Schwede 13, def. 2.9](#Schwede13)), there referred to just as "global equivalence". See also at _[[equivariant Whitehead theorem]]_.)

The **global equivariant stable homotopy category** $\mathcal{GH}$ is the ([[simplicial localization|simplicial]]) [[localization]] of the [[category]] of [[orthogonal spectra]] at these global equivariant equivalences (this is a [[stable (infinity,1)-category]]/[[triangulated category]].)

Since a global equivariant equivalence is in particular an ordinary [[weak homotopy equivalence]] of [[spectra]], there is a canonical functor

$$
  U \;\colon\; \mathcal{GH} \longrightarrow \mathcal{SH}
$$

from the global equivariant to the ordinary [[stable homotopy category]].

This functor has a ([[derived functor|derived]]) [[left adjoint]] and [[right adjoint]], which are both [[full and faithful functors]], hence which exhibit [[discrete object]]/[[codiscrete object]] structure

$$
  \array{
    \mathcal{GH}
    &\stackrel{\overset{L}{\leftarrow}}{\stackrel{\overset{U}{\longrightarrow}}{\underset{R}{\leftarrow}}}&
    \mathcal{SH}
  }
$$

on stable/triangulated categories ([Schwede 13, theorem IV 5.2](#Schwede13))

Global Borel-type [[equivariant cohomology]] is in the image of the [[right adjoint]] $R$ ([Schwede 13, example IV 5.12](#Schwede13))

## Properties

### Relation to plain stable homotopy theory
 {#RelationToPlainStableHomotopyTheory}

The [[forgetful functor]] from global stable homotopy theory to plain [[stable homotopy theory]] exhibits a [[recollement]].

(...)

## Examples

* [[global equivariant sphere spectrum]]

## Related concepts

* [[global family]]

## References

A comprehensive textbook account is in

* {#Schwede13} [[Stefan Schwede]], _[[Global homotopy theory]]_ ([arXiv:1802.09382](https://arxiv.org/abs/1802.09382))

Survey (with emphasis on [[global equivariant bordism homology theory]]):

* {#Schwede15} [[Stefan Schwede]], _Equivariant bordism from the global perspective_, 2015 ([pdf](http://www.math.uni-bonn.de/people/schwede/glasgow-handout.pdf), [[SchwedeGlobalEquivariantBordism.pdf:file]])

Original articles are

* [[L. Gaunce Lewis]], Jr., [[Peter May]], M. Steinberger, chapter II of _Equivariant stable homotopy theory_. Lecture Notes in Mathematics, 1213, Springer-Verlag, 1986

* [[John Greenlees]], [[Peter May]], section 5 of _Localization and completion theorems for $MU$-module spectra_ Ann. of Math. (2) 146 (1997), 509-544.

Discussion specifically in terms of equivariant [[orthogonal spectra]] is in

* [[Anna Marie Bohmann]], _Global Orthogonal Spectra_ ([arXiv:1208.4997](http://arxiv.org/abs/1208.4997))

Discussion for collections of [[finite group|finite]] [[subgroups]] includes

* [[Wolfgang Lueck]], _The Burnside Ring and Equivariant Stable Cohomotopy for Infinite Groups_ ([arXiv:math/0504051](http://arxiv.org/abs/math/0504051))

* [[Irakli Patchkoria]], _Proper equivariant stable homotopy and virtual cohomological dimension_, in _Oberwolfach Report_ No. 14/2015 ([pdf](https://www.mfo.de/document/1511/preliminary_OWR_2015_14.pdf))

* [[Markus Hausmann]], _Symmetric spectra model global homotopy theory of finite groups_ ([arXiv:1509.09270](http://arxiv.org/abs/1509.09270))

The example of [[algebraic K-theory]]:

* [[Stefan Schwede]], _Global algebraic K-theory_ ([arXiv:1912.08872](https://arxiv.org/abs/1912.08872))

[[!redirects global stable homotopy theory]]
[[!redirects global equivariant homotopy theory]]


global family

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#Contents#
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## Idea

In the context of [[global equivariant homotopy theory]] and [[global equivariant stable homotopy theory]] one considers [[equivariant spectra]] with equivariance not necessarily with respect to one fixed group, but with respect to a suitable family of groups at once. These "suitable families of groups for defining global equivariant (stable) homotopy theory" are called _global families_, for short.

## Definition:

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

A non-empty [[class]] of [[compact Lie groups]] is called a **global family** if it is closed under

1. [[isomorphism]];

1. [[closed subset|closed ]]  [[subgroups]];

1. [[quotient groups]]

=--

(e.g. [Schwede 18, def. 1.4.1](#Schwede18))

A global family is said to be **reflexive** if and only if the inclusion into the category of all compact Lie groups has a left adjoint. [Schwede 18, def 4.5.7](#Schwede18).

## Examples

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

The degenerate case of a global family (def. \ref{GlobalFamily}) contains only the [[trivial group]] $1$. The ([[global equivariant stable homotopy theory|stable]]) [[global equivariant homotopy theory]] with respect to this choice is simply ordinary non-equivariant [[homotopy theory]] ([[stable homotopy theory]]).

=--

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

The following are examples of global families (def. \ref{GlobalFamily}): the classes of

1. all [[compact Lie groups]];

1. all [[abelian group|abelian]] [[compact Lie groups]];

1. all [[quotient groups]] of [[closed subset|closed]] [[subgroups]] of a fixed [[compact Lie group]].

1. all [[finite groups]].

1. all finite abelian groups.

1. all [[topologically cyclic groups]].

1. all finite [[cyclic groups]].

1. all finite [[p-primary groups]].

1. all finite [[solvable group|solvable]] p-groups.

=--

## References

* {#Schwede18} [[Stefan Schwede]], section 1.4 of _[[Global homotopy theory]]_


global field

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#### Arithmetic geometry
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#Contents#
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## Definition

A _global field_ is either

* a [[number field]]

* or a [[function field]] over a [[finite field]];

hence is a [[finite number|finite]]-[[dimension|dimensional]] [[field extension]] of either

* the [[rational numbers]] $\mathbb{Q}$;

* the [[rational functions]] $\mathbb{F}_q(t)$ with [[coefficients]] in the [[finite field]] $\mathbb{F}_q$.

The [[function field analogy]] says how both these kinds of global fields are indeed analogous in that they behave as rings of [[rational functions]] on "[[arithmetic curves]] over the would-be [[field with one element]] $\mathbb{F}_1$" and $\mathbb{F}_q$, respectively. In particular for both types there is a concept of [[ring of adeles]], [[group of ideles]], etc.

The [[Langlands conjectures]] concern the [[arithmetic geometry]] of global fields.

The "local" version of a global field in the sense of [[formal geometry]] is a [[local field]]. The corresponding version of the Langlands program are the _[[local Langlands correspondences]]_.

## Artin-Whaples characterization

Global fields can also be described axiomatically, as those fields which satisfy a suitable [product formula](/nlab/show/group+of+ideles#ProductFormula). For example, if $k$ is a number field and $\mathbb{I}_k$ is its [[group of ideles]], then there is an idele norm map

$$N: \mathbb{I}_k \to \mathbb{C}^\times$$

$$(x_v)_{places\; v} \mapsto \prod_v {|x_v|}_v$$

and $k^\times$ is retrieved as the [[kernel]] of this map, i.e., is the subgroup of elements $(x_v)$ for which the product formula $\prod_v {|x_v|}_v = 1$ holds.

For the following theorem, assume that $k$ is a field for which there is at least one [[valuation]] that is either archimedean, or is discrete and with finite residue class field for the corresponding [[local ring]]. By a *place* we understand an equivalence class of valuations.

+-- {: .num_theorem}
###### Theorem
**(Artin-Whaples)**
Suppose $k$ is a field that has a set of valuations ${|(-)|}_v$, one for every place of $k$, such that for every $x \in k^\times$ the product formula $\prod_v {|x|}_v = 1$ is satisfied. Then $k$ is a global field (and conversely, every global field has this property).
=--

## References

* Wikipedia, _[Global field](http://en.wikipedia.org/wiki/Global_field)_

* Emil Artin and George Whaples, _Axiomatic characterization of fields by the product formula for valuations_, Bull. Amer. Math. Soc. 51 (1945), 469-492. ([link to article](http://www.ams.org/journals/bull/1945-51-07/S0002-9904-1945-08383-9/home.html))

[[!redirects global fields]]


global Hodge theory

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## Idea

Global Hodge theory is the study of Hodge theoretical invariants in
[[global analytic geometry]] and [[overconvergent global analytic geometry]].
The basic idea is to try to generalize/combine
the results of classical (complex analytic) [[Hodge theory]] and of
[[p-adic Hodge theory]] in a unique setting.

A first (non-archimedean) step in this direction was given by Bhatt in his
paper on the derived approach to p-adic Hodge theory (originally due to Beilinson), where he proposed a Hodge theory for (say semi-stable) varieties over $\mathbb{Z}$, using Illusie's [[derived de Rham cohomology]] to define a convenient [[period ring]]. From the point of view of global analytic geometry, this gives a Hodge theory for strict analytic spaces over $\mathbb{Z}$, equipped with its trivial norm. One may try to adapt these ideas to extend
them to a Hodge theory for strict analytic spaces over $\mathbb{Z}$, equipped with
its archimedean norm, but the use of a derived completion in the process prevents one from a straightforward generalization. Remark that the treatment of quasi-projective varieties should certainly involve a [[global mixed Hodge theory]] for pre-proper [[logarithmic geometry|logarithmic analytic varieties]].

The advantage of including the archimedean norm
in global Hodge theory is that this seems to give a better setting for the
study of completed [[arithmetic L-functions]] and [[automorphic L-functions]],
whose [[functional equation]] may not be proved without using the archimedean
information given by a convenient Gamma-factor. This extension is very easy
to make if one works with rational coefficients, like in Beilinson's original
paper on p-adic Hodge theory (i.e., with the `de Rham` period ring).
It remains a challenging problem to combine the non-archimedean and archimedean
information on Hodge theory without forgetting to take care of torsion phenomena in
cohomology, that are very important for the study of special values of L-functions. This may be doable if one follows the line
of ideas used in Arakelov geometry, but formulated in a global analytic way. A very natural setting for this seems to be global [[strict analytic geometry]] over $\mathbb{Z}$ equipped with its archimedean norm.

## Related concepts

* [[Berkovich space]]

* [[global analytic geometry]]

* [[overconvergent global analytic geometry]]

* [[arithmetic cryptography]]

## References