orthomodular lattice

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## Definition

An _orthomodular lattice_ is an [[orthocomplemented lattice]] in which in addition a weak form of [[modular lattice|modularity]] holds in that

$$
  a \leq c
$$

implies that

$$
  a \vee (a^\perp \wedge c) = c
  \,.
$$

## Examples

The canonical example are the [[Hilbert lattices]] that interpret Birkhoff-vonNeumann [[quantum logic]].

## References

* P. Pták and S. Pulmannová, _Orthomodular structures as quantum logics_, ser. Fundamental theories of physics. Kluwer Academic Publishers,
1991.

[[!redirects orthomodular lattices]]


orthosymplectic super Lie algebra

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## Idea

The _orthosymplectic supergroup_ $OSp(N|2p)$ is the sub-[[supergroup]] of the [[general linear supergroup]] $GL(N|2p)$ on those elements which preserve the canonical graded-symmetric [[bilinear form]] on $\mathbb{R}^{N|2p}$, i.e. the form which is the canonical symmetric bilinear form on the even elements $\mathbb{R}^{N}$, is the canonical [[symplectic form]] on the odd elements in $\mathbb{R}^{0|2p}$ and is zero on mixed pairs of elements.

The corresponding [[super Lie algebras]] are called the _orthosymplectic Lie algebras_ $\mathfrak{osp}(N|2p)$. Over a [[field]] of [[characteristic zero]] these constitute the infinite $B$- and $D$-series in the [classification of simple super Lie algebras](super%20Lie%20algebra#Classification).
They are closely related to [[superconformal]] symmetry (e.g. [D'Auria-Ferrara-Lledo-Varadarajan 00](#DAuriaFerrareLledoVaradarajan00)), see at _[supersymmetry -- Classification -- Superconformal symmetry](supersymmetry#ClassificationSuperconformal)_.

## Examples
 {#Examples}

* $\mathfrak{osp}(1\vert 32)$ is a limiting case of the [[M-theory super Lie algebra]] ([Fernandez-Izquierdo-Olmo 15](#FernandezIzquierdoOlmo15)).

  Further relation also to the [[type II supersymmetry algebra]] had been discussed in ([Bergshoess-Proeyen 00](#BergshoessProeyen00)).

* The [[M-theory super Lie algebra]] is actually a sub-super Lie algebra of $\mathfrak{osp}(1\vert 64)$ ([vanHolten-VanProeyen 82](#vanHoltenVanProeyen82), [Bars-Deliduma-nMinic 99, (11)-(15)](#BarsDelidumanMinic99), [West 00, section 7](#West00))

  In fact $OSp(1\vert 64)$ is the smallest simple supergroup that contains the [[M-theory super Lie algebra]]. Moreover, it is generated from its M-theory sub-algebra and the [[special conformal transformations]]  ([Bars-Deliduman-Minic 99, p. 5](#BarsDelidumanMinic99))

* The [[super anti de Sitter spacetimes]] are quotients of orthosymplectic super Lie groups:

| $\phantom{A}$$d$$\phantom{A}$ | $\phantom{A}$[[super anti de Sitter spacetime]]$\phantom{A}$ |
|-----|---|
| $\phantom{A}$4$\phantom{A}$ | $\;\;\;\;\frac{OSp(8\vert4)}{Spin(3,1) \times SO(7)}\;\;\;\;$ |
| $\phantom{A}$5$\phantom{A}$ | $\;\;\;\;\frac{SU(2,2 \vert 5)}{Spin(4,1)\times SO(5)}\;\;\;\;$ |
| $\phantom{A}$7$\phantom{A}$ | $\;\;\;\;\frac{OSp(6,2 \vert 4)}{Spin(6,1) \times SO(4)}\;\;\;\;$ |

## Related concepts

* [[orthogonal Lie algebra]]

## References

### General

* {#Kac77b} [[Victor Kac]], pages 9-10 of _A sketch of Lie superalgebra theory_, Comm. Math. Phys. Volume 53, Number 1 (1977), 31-64. ([EUCLID](https://projecteuclid.org/euclid.cmp/1103900590))

* [[Manfred Scheunert]], chapter II, 4.3.A of _The theory of Lie superalgebras. An introduction_, Lect. Notes Math. 716 (1979)

* {#Farmer84} Richard Joseph Farmer, *Orthosymplectic superalgebras in mathematics and science*, PhD Thesis (1984)
[[[Farmer-OSpSuperalgebra.pdf:file]]]

* {#CastellaniDAuriaFre91} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], section II.2.2 in _[[Supergravity and Superstrings - A Geometric Perspective]]_, World Scientific (1991)

* {#DAuriaFerrareLledoVaradarajan00} [[Riccardo D'Auria]], S. Ferrara, M. A. Lledó, [[Veeravalli Varadarajan]], _Spinor Algebras_, J.Geom.Phys. 40 (2001) 101-128 ([arXiv:hep-th/0010124](http://arxiv.org/abs/hep-th/0010124))

* N. A. Gromov, I. V. Kostyakov, V. V. Kuratov, _Cayley-Klein contractions of orthosymplectic superalgebras_ ([arXiv:hep-th/0110257](http://arxiv.org/abs/hep-th/0110257))

* Steven V Sam, _Orthosymplectic Lie superalgebras, Koszul duality, and a complete intersection analogue of the Eagon-Northcott complex_ ([arXiv:1312.2255](http://arxiv.org/abs/1312.2255))

* G.I. Lehrer, R.B. Zhang, _The second fundamental theorem of invariant theory for the orthosymplectic supergroup_ ([arXiv:1407.1058](http://arxiv.org/abs/1407.1058))

Appearance of $\mathfrak{osp}(1|2) \times $[[sl(2)]]-[[modular functor]] is found in the seemingly plain [[sl(2)]]-[[WZW model]] for *fractional* [[level (Chern-Simons theory)|level]] (see [there](Wess-Zumino-Witten+model#FractionalLevelWZWModelReferences)) in:

* [[Boris Feigin]], [[Feodor Malikov]], *Modular functor and representation theory of $\widehat{\mathfrak{sl}_2}$ at a rational level*, p. 357-405 in: Loday, Stasheff, Voronov (eds.) *Operads: Proceedings of Renaissance Conferences*,  Contemporary Mathematics **202**, AMS (1997) [[arXiv:q-alg/9511011](https://arxiv.org/abs/q-alg/9511011), [ams:conm-202](https://bookstore.ams.org/conm-202)]

### In superstring theory

* [[Igor Bandos]], Jerzy Lukierski, Christian Preitschopf, [[Dmitri Sorokin]], _OSp supergroup manifolds, superparticles and supertwistors_, Phys.Rev.D61:065009, 2000 ([arXiv:hep-th/9907113](https://arxiv.org/abs/hep-th/9907113))

* {#BergshoessProeyen00} [[Eric Bergshoeff]], [[Antoine Van Proeyen]], _The many faces of $OSp(1|32)$_, Class. Quantum Grav. 17 (2000) 3277–3303 ([arXiv:hep-th/0003261](http://arxiv.org/abs/hep-th/0003261))

* [[Horatiu Nastase]], _Towards a Chern-Simons M theory of $OSp(1\vert 32) \times OSp(1\vert 32)$_ ([arXiv:hep-th/0306269](https://arxiv.org/abs/hep-th/0306269))

* [[Itzhak Bars]], Dmitry Rychkov. *Background Independent String Field Theory* [[arXiv:1407.4699](https://arxiv.org/abs/1407.4699)]

  > (on [[string field theory]] with [[orthosymplectic super Lie algebra|$OSp(d\vert 2)$]]-symmetry)

* {#FernandezIzquierdoOlmo15} J. J. Fernandez, [[José M. Izquierdo]], M. A. del Olmo, _Contractions from $osp(1|32) \oplus osp(1|32)$ to the M-theory superalgebra extended by additional fermionic generators_, Nuclear Physics B **897** (2015) 87-97 [[arXiv:1504.05946](http://arxiv.org/abs/1504.05946)]

  > (relating to the [[M-theory super Lie algebra]])

* {#vanHoltenVanProeyen82} [[Jan-Willem van Holten]], [[Antoine Van Proeyen]], _$N=1$ Supersymmetry Algebras in $D=2$, $D=3$, $D=4$ $MOD-8$_, J.Phys. A15 (1982) 3763 ([spire:177060](http://inspirehep.net/record/177060))

* {#BarsDelidumanMinic99} [[Itzhak Bars]], C. Deliduman, D. Minic, _Lifting M-theory to Two-Time Physics_, Phys.Lett. B457 (1999) 275-284 ([arXiv:hep-th/9904063](https://arxiv.org/abs/hep-th/9904063))

* {#Bars00} [[Itzhak Bars]], _2T Physics Formulation of Superconformal Dynamics Relating to Twistors and Supertwistors_, Phys.Lett. B483 (2000) 248-256 ([arXiv:hep-th/0004090](https://arxiv.org/abs/hep-th/0004090))

* {#West00} [[Peter West]], _Hidden Superconformal Symmetry in M Theory_, JHEP 0008:007, 2000 ([arXiv:hep-th/0005270](https://arxiv.org/abs/hep-th/0005270))

### Representation theory

The [[representation theory]] ([[singleton representation|singleton representations]]) of the orthosymplectic group is discussed in

* {#NicolaiSezgin84} [[Hermann Nicolai]], [[Ergin Sezgin]], _Singleton representations of $Osp(N,4)$_,  Physics Letters B, Volume 143, Issues 4–6, 16 August 1984, Pages 389-395

[[!redirects osp]]

[[!redirects orthosymplectic super Lie algebras]]

[[!redirects orthosymplectic supergroup]]
[[!redirects orthosymplectic supergroups]]

[[!redirects orthosymplectic super group]]
[[!redirects orthosymplectic super groups]]

[[!redirects orthosymplectic super Lie group]]
[[!redirects orthosymplectic super Lie groups]]


Oscar Chacaltana

* [G+ page](https://plus.google.com/114696728745448913269/posts)

category: people


Oscar Cunningham

* Pesonal webpage and blog: [oscarcunningham.com](https://oscarcunningham.com)
* [StackExchange](https://stackexchange.com/users/497517/oscar-cunningham)
* Email: mail@oscarcunningham.com

category: people


Oscar E. Lanford

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

## Selected writings

Collected papers: [pdf](https://people.math.harvard.edu/~knill/history/lanford/papers/LanfordCollectedWork.pdf)

On [[proof assistants]] for [[proofs]] in [[analysis]]:

* [[Oscar E. Lanford]], *Computer assisted proofs*, in: *Computational Methods in Field Theory*, Lecture Notes in Physics, **409**, Springer (1992) [[doi:10.1007/3-540-55997-3_30](https://doi.org/10.1007/3-540-55997-3_30)]

category: people

[[!redirects Oscar Lanford]]


Oscar Randal-Williams

Oscar Randal-Williams is professor of pure [[mathematics]] at Cambridge.

* [webpage](https://www.dpmms.cam.ac.uk/~or257/)

## Selected writings

On the [[ordinary homology]] of [[configuration spaces of points]]:

* {#RandallWilliams13} [[Oscar Randal-Williams]], _Homological stability for unordered configuration spaces_, The Quarterly Journal of Mathematics, Volume 64, Issue 1, March 2013, Pages 303–326 ([arXiv:1105.5257](https://arxiv.org/abs/1105.5257))

On [[submanifold]] [[cobordism categories]]:

* [[Oscar Randal-Williams]], _Embedded Cobordism Categories and Spaces of Manifolds_, 	Int. Math. Res. Not. IMRN 2011, no. 3, 572-608 ([arXiv:0912.2505](https://arxiv.org/abs/0912.2505))

On [[semi-simplicial topological spaces]]:

* [[Johannes Ebert]], [[Oscar Randal-Williams]], *Semi-simplicial spaces*, Algebr. Geom. Topol. 19 (2019) 2099-2150 ([arXiv:1705.03774](https://arxiv.org/abs/1705.03774))

On [[action functionals]] for [[topological field theories]] via [[equivariant ordinary differential cohomology]] (such as [[Chern-Simons theory]]):

* Joe Davighi, [[Ben Gripaios]], [[Oscar Randal-Williams]], _Differential cohomology and topological actions in physics_ ([arXiv:2011.05768](https://arxiv.org/abs/2011.05768))

## Related $n$Lab entries

* [[moduli space of framed surfaces]]

category: people


oscillation

An __oscillation__ is a periodic motion. In other words it is a solution of [[equations of motion]] that is a [[periodic function]] of [[time]] with a fixed period $T$. One also talks about slight generalizations of this definition, like [[damping|damped]] oscillations and so on.

A [[wave]] is a transportation of oscillations in medium, in other words a disturbance of a medium periodic in time (for a fixed point) and propagation in [[physical space]].

Cf. [[quasinormal mode]].

[[!redirects oscillation]]
[[!redirects oscillations]]


Oskar Becker

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

* Wikipedia entry (de)

## Selected writings

Early disucussion of [[modal logic]]:

* [[Oskar Becker]], *Zur Logik der Modalitäten*, Jahrbuch für Philosophie und phänomenologische Forschung **11** (1930)  497-548 [[digizeit](http://www.digizeitschriften.de/dms/img/?PID=PPN827944462_0011%7CLOG_0010), [ophen:101138](https://ophen.org/pub-101138), [[Becker-LogikModalitaeten.pdf:file]]]

translated to English in:

*  Stefania Centrone, Pierluigi Minari, *Oskar Becker, On the Logic of Modalities (1930): Translation, Commentary and Analysis*,  Synthese Library **444**, Springer (2022) [[doi:10.1007/978-3-030-87548-0](https://doi.org/10.1007/978-3-030-87548-0)]

category: people


Oskar Klein

Swedish physicist, 1894-1977

* [Wikipedia](http://en.wikipedia.org/wiki/Oskar_Klein)

## related $n$Lab entries

* [[Kaluza-Klein mechanism]]
* [[Klein-Gordon equation]]

category: people


Oskar Pelc

* [InSpire page](https://inspirehep.net/authors/993923)

## Selected writings

On the [[Hanany-Witten effect]]:

* {#Pelc08} [[Oskar Pelc]], , *On the Quantization Constraints for a D3 Brane in the Geometry of NS5 Branes*,  	JHEP 0008 (2000) 030 [[arXiv:hep-th/0007100](https://arxiv.org/abs/hep-th/0007100), [doi:10.1088/1126-6708/2000/08/030](https://doi.org/10.1088/1126-6708/2000/08/030)]

On [[D2-D6 brane bound states]]:

* [[Oskar Pelc]], Ruud Siebelink, *The D2-D6 System and a Fibered AdS Geometry*, Nucl. Phys. B **558** (1999) 127-158 [[arXiv:hep-th/9902045](https://arxiv.org/abs/hep-th/9902045), doi:10.1016/S0550-3213(99)00332-6]

category: people


Osterwalder-Schrader theorem

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## Idea

The _Osterwalder-Schrader theorem_ ([Osterwalder-Schrader 73](#OsterwalderSchrader73)) states precise conditions under which [[Wick rotation]] between [[relativistic field theory]] and [[Euclidean field theory]] works.

Rough idea: The [[Wightman axioms]] describe how the algebra of observables of a [[quantum field theory]] on [[Minkowski spacetime]] is generated by quantum fields. The Wightman reconstruction theorem asserts that knowing all correlation functions of all fields in the vacuum state is equivalent to knowing the quantum fields. The _Osterwalder--Schrader theorem_ states conditions that correlation functions on Euclidean spacetime have to satisfy to be equivalent to the correlation functions of a Wightman QFT on Minkowski spacetime.

In this sense the Osterwalder--Schrader theorem states and proves conditions that assure that the [[Wick rotation]] is a well defined isomorphism of quantum field theories on Minkowski and on Euclidean spacetime.

## Axioms of euclidean field theory

The axioms of euclidean field theory are the euclidean analogue of the [[Wightman axioms]] on Minkowski spacetime.
The axioms may be formulated for tempered distributions, but we follow the lines of Glimm and Jaffe and define them for $\mathcal{D}'(\mathbb{R}^d)$, the space of distributions that is dual to the space of all smooth functions with compact support, $\mathcal{D}(\mathbb{R}^d)$. In the original paper of Osterwalder and Schrader the axioms are given in terms of the Schwinger functions. Here the axioms are given in a form more directly related to the measure on field space and its characteristic function, rather than the Schwinger functions themselves. This form was first presented by Fröhlich.
We define the generating functional on $\mathcal{D}(\mathbb{R}^d)$
$$
S(f) := \integral e^{i \phi(f)} d\mu
$$
as the inverse Fourier transform of a Borel probability measure $d\mu$ on $\mathcal{D}'(\mathbb{R}^d)$.

* **OS0 (analyticity)**: For every finite set of test functions $f_1, f_2,...f_n$ and complex numbers $z:= (z_1, z_2, ...z_n)$ the function
$$
        z \mapsto S(\sum_{k=1}^n z_k f_k)
$$
is entire analytic on $\mathbb{C}^n$.

* **OS1 (regularity)**: For some p with $1 \le p \le 2$ and some constant c the following inequality holds for all test functions f:
$$
     | S(f) | \le exp(c \| f \|_{L_1} + \| f\|^p_{L_p})
$$

* **OS2 (invariance)**: S is invariant under euclidean symmetries E of $\mathcal{R}^d$ (translations, rotations, reflections), that is S(f) = S(Ef) for all symmetries E and test functions f.

* **OS3 ([[reflection positivity]])**
We define exponential functionals on $\mathcal{D}'(\mathbb{R}^d)$ via
$$
A(\phi) := \sum_{k=1}^n c_k exp(\phi(f_k))
$$
Let $\mathcal{A}$ be the set of all these functionals, by axiom OS0 this is a subset of $L_2(d\mu)$. Euclidean symmetries act on $\mathcal{D}'(\mathbb{R}^d)$ via duality, that is $E\phi(f) = \phi(Ef)$, and thus define an unitary continuous action on $L_2(d\mu)$. Let $\mathcal{A}^+ \subset \mathcal{A}$ be the set of functionals with $supp(f_i) \subset \mathbb{R}^d_+$ where $\mathbb{R}^d_+ := \{(x, t): t \gt 0 \}$. Let $\theta: (x, t) \mapsto (x, -t)$ be the time reflection. Then the content of the axiom is:
$$
   0 \le \langle \theta A, A\rangle_{L_2}
$$

* **OS4 (ergodicity)**: the time translation subgroup acts ergodically on the measure space $(\mathcal{D}'(\mathbb{R}^d), d\mu)$.

* **theorem (Schwinger functions)**: A measure that satisfies OS0 has moments of all order, the nth moment has a density $S \in \mathcal{D}'(\mathbb{R}^{nd})$. These distributions are called Schwinger functions.

## The theorem
One possible formulation: To every measure satisfying the axioms stated above there is a Wightman field such that the Schwinger and Wightman functions are related by:
$$
\integral \phi_E(x_1, t_1) \cdots \phi_E(x_n, t_n) = \langle \Omega, \phi_M(x_1, i t_1) \cdots \phi_M(x_n, i t_n) \Omega \rangle
$$
$\phi_E$ is a Schwinger function, $\phi_M$ is a Wightman field and $\Omega$ is the vacuum vector of the Wightman fields.
See theorem 6.15 in the book by Glimm and Jaffe (see references).

## Related theorems

* [[Reeh-Schlieder theorem]]

* [[spin-statistics theorem]]

## References

The original article is

* {#OsterwalderSchrader73} [[Konrad Osterwalder]], [[Robert Schrader]], _Axioms for Euclidean Green's functions_, Comm. Math. Phys. Volume 31, Number 2 (1973), 83-112 ([Euclid:1103858969](https://projecteuclid.org/euclid.cmp/1103858969))

Discussion for compact/periodic Euclidean time, as needed for [[thermal quantum field theory]] is in

* {#KleinLandau81} Abel Klein, Lawrence Landau, _Periodic Gaussian Osterwalder-Schrader positive processes and the two-sided Markov property on the circle_, Pacific Journal of Mathematics, Vol. 94, No. 2, 1981 ([
DOI: 10.2140/pjm.1981.94.341](https://msp.org/pjm/1981/94-2/p12.xhtml), [pdf](https://msp.org/pjm/1981/94-2/pjm-v94-n2-p12-s.pdf))

Exposition is in

* Dirk Schlingemann, _From euclidean field theory to quantum field theory_ ([arXiv:hep-th/9802035](http://arxiv.org/abs/hep-th/9802035))

A textbook account is in

* [[James Glimm]], [[Arthur Jaffe]], _[[Glimm-Jaffe|Quantum physics: a functional integral point of view]]_

[[!redirects Osterwalder-Schrader theorem]]
[[!redirects Osterwalder–Schrader theorem]]
[[!redirects Osterwalder--Schrader theorem]]
[[!redirects Osterwalder-Schrader axioms]]
[[!redirects Osterwalder--Schrader axioms]]


Ostrowski's theorem

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## Statement

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###### Theorem
**([[Alexander Ostrowski]])**

Any non-trivial [[absolute value]] on the [[rational numbers]] is equivalent either to the standard [[real numbers|real]] absolute value, or to the [[p-adic numbers|p-adic]] absolute value.

=--

+-- {: .num_theorem}
###### Theorem
**(Ostrowski)**

Any [[complete field|complete]] [[archimedean valued field]] is, as an [[associative algebra]] and as a [[topological space]], [[isomorphism|isomorphic]] to either the [[real numbers]] or the [[complex numbers]].

=--

## Related entries

* [[absolute value]], [[archimedean valued field]]

* [[p-adic numbers]]

* [[analytic geometry]]

## References

* [[Frédéric Paugam]], p. 23 _Global analytic geometry and the functional equation_ (2010) ([pdf](http://www.math.jussieu.fr/~fpaugam/documents/enseignement/master-global-analytic-geometry.pdf))

Discussion in [[point-free topology]]:

* [[Ming Ng]], [[Steven Vickers]], *A Point-Free Look at Ostrowski's Theorem and Absolute Values* [[arXiv:2308.14758](https://arxiv.org/abs/2308.14758)]

[[!redirects Ostrowski theorem]]

[[!redirects Ostrowski's theorems]]
[[!redirects Ostrowski theorems]]


Oswald Teichmüller

* [Wikipedia entry](http://en.wikipedia.org/wiki/Oswald_Teichmüller)

## related $n$Lab entries

* [[Teichmüller theory]]

category: people


Oswald Wyler

* [Mathematics Genealogy page](https://www.mathgenealogy.org/id.php?id=34270)

## Selected writings

On [[convenient categories of topological spaces]]:

* {#Wyler73} [[Oswald Wyler]], *Convenient categories for topology*, General Topology and its Applications **3** 3 (1973) 225-242 (doi:10.1016/0016-660X(72)90014-1)

On [[toposes]] and [[quasitoposes]]:

* {#Wyler} [[Oswald Wyler]], _Lecture Notes on Topoi and Quasitopoi_ , World Scientific Singapore 1991 ([doi:10.1142/1047](https://doi.org/10.1142/1047))

category: people


Otgonbayar Uuye

Otgonbayar Uuye is a researcher working in Noncommutative Geometry – (index theory, K-theory and KK-theory, cyclic theory), and
Noncommutative Algebraic Topology –( noncommutative homotopy theory, connective K-theory).  He is currently based at the National University of Mongolia.
Previously he worked in Centre for Symmetry and Deformation in Copenhagen and in Cardiff in the Mathematical Physics group.

* [personal website](http://www.cardiff.ac.uk/maths/subsites/uuyeo/index.html)

* [Cardiff University maths. website](http://www.cardiff.ac.uk/maths/contactsandpeople/profiles/uuyeo.html)

## Related pages

* [[homotopical structure on C*-algebras]]

category: people


Othmar Steinmann

* [Mathematics Genealgy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=121837)

## related $n$Lab entries

* [[anomalous magnetic moment of the electron]]

category: people


Otto Schreier

__Otto Schreier__ (1901--1929) was a German algebraist born in Vienna famous for his work in [[combinatorial group theory]]
([[Nielsen-Schreier theorem|Nielsen--Schreier method]] and HNN-extensions). His 1926 article on [[group extensions|extensions of groups]] contains ideas of [[nonabelian cohomology]] and [[pseudofunctors]]; nowadays this is called _Schreier's theory of (nonabelian) extensions_. The work has been completed by [[Sammy Eilenberg|Eilenberg]] and [[Saunders Mac Lane|Mac Lane]] in a series of three papers in the late 1940s. Higher analogues of Schreier's theory were studied by L. Breen. Schreier's theory can be also extended to some other algebraic structures, see e.g. [[nonabelian Lie algebra cohomology]].

See biographical and some mathematical data at [the MacTutor biography](http://www-history.mcs.st-and.ac.uk/history/Biographies/Schreier.html). For more on the mathematical side see

* [[W. Magnus]], The history of combinatorial group theory: a case study in the history of ideas

## Selected writings

Proving the [[Nielsen-Schreier theorem]] on [[subgroups]] of [[free groups]] and introducing the notion of what came to be known as the [[amalgamated free product of groups]]:

* [[Otto Schreier]], *Die Untergruppen der freien Gruppen*, Abh. Math. Semin. Univ. Hambg. **5** (1927) 161–183 [[doi:10.1007/BF02952517](https://doi.org/10.1007/BF02952517)]

## Related entries

* [[Schreier theory]]


outer automorphism

#Contents#
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## Definition

For $G$ a [[group]], the **outer automorphism group** $Out(G)$ is the [[quotient]] of the [[automorphism group]] by the [[normal subgroup]] of [[inner autormorphism]]s:

$$
  Out(G) := Aut(G)/Inn(G)
  \,.
$$

## Properties

### As a truncation of the automorphism 2-group

Write

$$
  AUT(G) := Aut_{Grpd}(\mathbf{B}G)
$$

for the [[automorphism 2-group]] of $G$. This is the [[strict 2-group]] coming from the [[crossed module]]

$$
  G \stackrel{Ad}{\to} Aut(G)
  \,.
$$

Therefore the [[truncated|0-truncation]] of $AUT(G)$ is $Out(G)$:

$$
  Out(G) \simeq \tau_0 AUT(G)
  \,.
$$

This perspective generalizes to the notion of [[outer automorphism ∞-group]].

## Examples

The connected components of the [[subgroup]] of outer automorphisms of the a [[super Poincaré group]] which fixes the underlying [[Poincaré group]] is known as the _[[R-symmetry group]]_ in [[supersymmetry]].

## Related concepts

* [[group]], [[∞-group]],

* [[automorphism group]], [[automorphism ∞-group]],

* [[center]], [[center of an ∞-group]]

* **outer automorphism group**, [[outer automorphism ∞-group]]

* [[Outer space]]

[[!redirects outer automorphism group]]
[[!redirects outer automorphism groups]]

[[!redirects outer automorphisms]]


outer automorphism infinity-group

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## Definition

Let $\mathcal{X}$ be an [[(∞,1)-topos]] and $G \in \infty Grpd(\mathcal{X})$ an [[n-truncated]] [[∞-group]] object, for some $n \in \mathbb{N}$ (an [[n-group]] in $\mathcal{X}$).

Write

$$
  AUT(G) := \underline{Aut}(\mathbf{B}G) \hookrightarrow [\mathbf{B}G, \mathbf{B}G]
  \in
  \mathcal{X}
$$

for the [[internalization|internal]] [[automorphism ∞-group]].

Then the [[truncated|n-truncation]]

$$
  Out(G) := \tau_n AUT(G) \in \infty Grp(\mathcal{X})
$$

is the **outer automorphism $\infty$-group** of $G$.

## Examples

* For $\mathcal{X} = $ [[∞Grpd]] and $n = 0$, $G$ is an ordinary [[discrete group]], and $AUT(G)$ is its [[automorphism 2-group]]. Then $Out(G)$ is the ordinary group of ordinary [[outer automorphism]]s.

## Applications

* Outer automorphism $\infty$-groups control part of the [[nonabelian cohomology]] of [[∞-gerbe]]s. See there for more details.

## Related concepts

* [[group]], [[∞-group]],

* [[automorphism group]], [[automorphism ∞-group]],

* [[center]], [[center of an ∞-group]]

* [[outer automorphism group]], **outer automorphism $\infty$-group**

[[!redirects outer automorphism ∞-group]]
[[!redirects outer automorphism ∞-groups]]
[[!redirects outer automorphism infinity-groups]]


outer horn

An **outer horn** is a [[horn]] $\Lambda[n]_i$ with $i = 0$ or $i = n$.

[[!redirects outer horns]]


Outer space

#Contents#
* table of contents
{:toc}

## Idea

The Culler-Vogtmann __Outer space__ (capitalized for [[outer automorphism group|Out]]) is a free group analog of the [[Teichmüller space]] of a Riemann surface. It is of importance in the study of the [[outer automorphism group]] of the [[free group]] on $n$ letters, $Out(F_n)$.

## Related entries

* [[free group]], [[graph complex]], [[Teichmüller theory]], [[moduli space of curves]]

## References

* wikipedia, [Karen Vogtmann](http://en.wikipedia.org/wiki/Karen_Vogtmann)
* Mark Culler, [[Karen Vogtmann]], _Moduli of graphs and automorphisms of free groups__, Inventiones Mathematicae 84 (1): 91–119 (1986) [doi](http://dx.doi.org/10.1007/BF01388734), [MR 0830040](http://www.ams.org/mathscinet-getitem?mr=0830040)
* J. Conant, K. Vogtmann, _On a theorem of Kontsevich_, Algebraic and Geometric Topology 3 (2003), pp. 1167–1224 [pdf](http://www.emis.ams.org/journals/UW/agt/ftp/main/2003/agt-3-42.pdf)
* Vincent Guirardel, _Dynamics of $Out(F_n)$ on the boundary of Outer space_. Annales Scientifiques de l'École Normale Supérieure (4), vol. 33 (2000), no. 4, 433–465.

[[!redirects outer space]]


over category

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

#Contents#
* table of contents
{:toc}

## Definition

The **slice category** or **over category** $\mathbf{C}/c$ of a [[category]] $\mathbf{C}$ over an object $c \in \mathbf{C}$ has

* objects that are all arrows $f \in \mathbf{C}$ such that $cod(f) = c$, and

* morphisms $g: X \to X' \in \mathbf{C}$ from $f:X \to c$ to $f': X' \to c$ such that $f' \circ g = f$.

$$
  C/c
  =
  \left\lbrace
    \array{
       X &&\stackrel{g}{\to}&& X'
       \\
       & {}_f \searrow && \swarrow_{f'}
       \\
       && c
    }
  \right\rbrace
$$

The slice category is a special case of a [[comma category]].

There is a [[forgetful functor]] $U_c: \mathbf{C}/c \to \mathbf{C}$ which maps an object $f:X \to c$ to its domain $X$ and a morphism $g: X \to X' \in \mathbf{C}/c$ (from $f:X \to c$ to $f': X' \to c$ such that $f' \circ g = f$) to the morphism $g: X \to X'$.

The [[duality|dual]] notion is an [[under category]].

## Examples

* If $\mathbf{C} = \mathbf{P}$ is a [[partial order|poset]] and $p \in \mathbf{P}$, then the slice category $\mathbf{P}/p$ is the [[down set]] $\downarrow (p)$ of elements $q \in \mathbf{P}$ with $q \leq p$.

* If $1$ is a [[terminal object]] in $\mathbf{C}$, then $\mathbf{C}/1$ is isomorphic to $\mathbf{C}$.

* For $X$ a [[topological space]] then the [[category of covering spaces]] over $X$ is a [[full subcategory]] of the slice category $Top_{/X}$ of the [[category of topological spaces]].

* The [[fundamental theorem of topos theory]] states that the slice category over any object in a topos is itself a topos.

* For a [[monoidal category]] the slice category over any [[monoid object]] is monoidal.

  {#MonoidalTopos} For instance, the [[slice topos]] of a given [[topos]] over any [[monoid object]] is canonically a *[[monoidal topos]]* (see the Example [there](monoidal+topos#SliceToposOverAMonoidObject)).

## Properties

### Relation to codomain fibration

The assignment of overcategories $C/c$ to objects $c \in C$ extends to a [[functor]]

$$
  C/(-) : C \to Cat
$$

Under the [[Grothendieck construction]] this functor corresponds to the [[codomain fibration]]

$$
  cod : [I,C] \to C
$$

from the [[arrow category]] of $C$.  (Note that unless $C$ has [[pullbacks]], this functor is not actually a [[Grothendieck fibration|fibration]], though it is always an opfibration.)

[[!include sliced adjoint functors -- section]]

### Presheaves on over-categories and over-categories of presheaves {#RelWithPresheaves}

Let $C$ be a [[category]], $c$ an [[object]] of $C$ and let $C/c$ be the [[over category]] of $C$ over $c$. Write
$PSh(C/c) = [(C/c)^{op}, Set]$ for the [[category of presheaves]] on $C/c$ and write
$PSh(C)/Y(c)$ for the [[over category]] of [[presheaf|presheaves]] on $C$ over the presheaf $Y(c)$, where $Y : C \to PSh(c)$ is the [[Yoneda embedding]].

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

There is an [[equivalence]] of categories

$$
  e : PSh(C/c) \stackrel{\simeq}{\to} PSh(C)/Y(c)
  \,.
$$

=--

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

The functor $e$ takes $F \in PSh(C/c)$ to the presheaf
$F' : d \mapsto \sqcup_{f \in C(d,c)} F(f)$ which is equipped with the natural transformation $\eta : F' \to Y(c)$ with component map $\eta_d: \sqcup_{f \in C(d,c)} F(f) \to C(d,c)$.

A weak inverse of $e$ is given by the functor
$$
  \bar e : PSh(C)/Y(c) \to PSh(C/c)
$$
which sends
$
  \eta : F' \to Y(C))
$ to $F \in PSh(C/c)$ given by
$$
  F : (f : d \to c) \mapsto F'(d)|_c
  \,,
$$
where $F'(d)|_c$ is the [[pullback]]
$$
  \array{
     F'(d)|_c &\to& F'(d)
     \\
     \downarrow && \downarrow^{\eta_d}
     \\
     pt &\stackrel{f}{\to}& C(d,c)
  }
  \,.
$$

=--

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

Suppose the presheaf $F \in PSh(C/c)$ does not actually depend on the morphisms to $C$, i.e. suppose that it factors through the forgetful functor from the [[over category]] to $C$:
$$
  F : (C/c)^{op} \to C^{op} \to Set
  \,.
$$

Then
$
  F'(d) = \sqcup_{f \in C(d,c)} F(f)
  = \sqcup_{f \in C(d,c)} F(d)
  \simeq
  C(d,c) \times F(d)
$
and hence $F ' = Y(c) \times F$ with respect to the [[closed monoidal structure on presheaves]].

=--

See also [[functors and comma categories]].

For the analogous statement in [[(∞,1)-category theory]]  see at
*[[infinity1-category+of+infinity1-presheaves#WithOvercategories|(∞,1)-category of (∞,1)-presheaves -- Interaction with overcategories]]*.

### Limits and colimits
 {#LimitsAndColimits}

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

A  [[colimit]] in an [[over category]] is computed as a colimit in the underlying category.

Precisely: let $\mathcal{C}$ be a [[category]], $t \in \mathcal{C}$ an [[object]], and $\mathcal{C}/t$ the corresponding [[overcategory]], and $p \colon \mathcal{C}/t \to \mathcal{C}$ the obvious projection.

Let $F \colon D \to \mathcal{C}/t$ be any [[functor]]. Then, if it exists, the [[colimit]] of $p \circ F$ in $\mathcal{C}$ is the image under $p$ of the colimit over $F$:

$$
  p
  \big(
    \underset{\longrightarrow}{\lim}
    F
  \big)
  \;\simeq\;
  \underset{\longrightarrow}{\lim} (p \circ F)
$$

and $\underset{\longrightarrow}{\lim} F$ is uniquely characterized by $\underset{\longrightarrow}{\lim} (p \circ F)$ this way.

=--

This statement, and its proof, is the [[formal dual]] to the corresponding statement for [[undercategories]], see [there](under+category#LimitsAndColimits).

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

For $\mathcal{C}$ a [[category]], $X \;\colon\; \mathcal{D} \longrightarrow \mathcal{C}$ a [[diagram]], $\mathcal{C}_{/X}$ the [[comma category]] (the over-category if $\mathcal{D}$ is the point) and $F \;\colon\; K \to \mathcal{C}_{/X}$ a [[diagram]] in the [[comma category]], then the [[limit]] $\underset{\leftarrow}{\lim} F$ in $\mathcal{C}_{/X}$ coincides with the limit $\underset{\leftarrow}{\lim} F/X$ in $\mathcal{C}$.

=--

For a proof see at [[(∞,1)-limit]]  _[here](limit+in+a+quasi-category#InOvercategories)_.

### Initial and terminal objects
 {#InitialAndTerminalObjects}

As a special case of the above discussion of limits and colimits in a
slice $\mathcal{C}_{/X}$ we obtain the following statement, which of
course is also immediately checked explicitly.

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

* If $\mathcal{C}$ has an initial object $\emptyset$, then $\mathcal{C}_{/X}$ has an [[initial object]], given by $\langle \emptyset \to X\rangle$.

* The [[terminal object]] of $\mathcal{C}_{/X}$ is $\mathrm{id}_X$.

=--

## Related concepts

* **over-category**

  * [[enriched over category]]

  * [[under category]]

  * [[over topos]]

* [[slice 2-category]]

* [[over (∞,1)-category]],

  * [[model structure on an over category]]

  * [[over-(∞,1)-topos]]

[[!redirects overcategory]]
[[!redirects over categories]]
[[!redirects over-category]]
[[!redirects over-categories]]
[[!redirects overcategories]]
[[!redirects slice category]]
[[!redirects slice categories]]


over-(infinity,1)-category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For $C$ an [[(∞,1)-category]] and $X \in C$ an [[object]], the _over-$(\infty,1)$-category_ or _slice $(\infty,1)$-category_ $C_{/X}$ is the $(\infty,1)$-category whose [[objects]] are [[morphism]] $p : Y \to X$ in $C$, whose morphisms $\eta : p_1 \to p_2$ are [[2-morphisms]] in $C$ of the form

$$
  \array{
     Y_1 &&\stackrel{f}{\to}&& Y_2
     \\
     & {}_{\mathllap{p_1}}\searrow &\swArrow_{\simeq}^{\eta}& \swarrow_{\mathrlap{p_2}}
     \\
     && X
  }
  \,,
$$

hence 1-morphisms $f$ as indicated together with a [[homotopy]] $\eta \colon p_2 \circ f \stackrel{\simeq}{\to} p_1$; and generally whose [[n-morphisms]] are [[diagrams]]

$$
  \Delta[n+1] = \Delta[n] \star \Delta[0]\to C
$$

in $C$ of the shape of the cocone under the $n$-[[simplex]] that take the tip of the [[cocone]] to the object $X$.

This is the generalization of the notion of [[over-category]] in ordinary [[category theory]].

## Definition

We give the definition in terms of the model of [[(∞,1)-categories]] in terms of [[quasi-categories]].

Recall from [[join of quasi-categories]] that there are two different but quasi-categorically equivalent definitions of _join_, denoted $\star$ and $\diamondsuit$. Accordingly we have the following two different but quasi-categoricaly equivalent definitions of over/under quasi-categories.

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

Let $C$ be a [[quasi-category]]. let $K$ be any [[simplicial set]] and let

$$
  F : K \to C
$$

be an [[(∞,1)-functor]] -- a morphism of simplicial sets.

1. The **under-quasi-category** $C_{F/}$ is the
   simplicial set characterized by the property
   that for any other simplicial set $S$ there is
   a natural bijection of hom-sets

   $$
    Hom_{sSet}(S, C_{F/})
     \cong
     (Hom_{(K\downarrow SSet)})(i_{K,S} , F)
     \,,
   $$

   where $i_{K,S} : K \to K \star S$ is the canonical inclusion of $K$ into its [[join of simplicial sets]] with $S$.

   Similarly, the **over quasi-category** over $F$ is the simplicial set
   characterized by the property that

   $$
    Hom_{sSet}(S, C_{/F})
     \simeq
     Hom_{(K\downarrow SSet)}(
       j_{K,S} , F
     )
   $$

   naturally in $S$, where $j_{K,S}$ is the canonical inclusion map $K\to S\star K$.

1. The functor

   $$
      sSet \to sSet_{K/}
   $$

   $$
     S \mapsto K \diamondsuit S
   $$

   with $\diamondsuit$ denoting the other definition of [[join of quasi-categories]] (as described there)

   has a [[right adjoint]]

   $$
      sSet_{K/} \to sSet
   $$

   $$
     (F : K \to C) \mapsto C^{F/}
   $$

   and its image $C^{F/}$ is another definition of
   the quasi-category under $F$.

=--

The first definition in terms of the the mapping property is due to [[Andre Joyal]]. Together with the discussion of the concrete realization it appears as [[Higher Topos Theory|HTT, prop 1.2.9.2]]. The second definition appears in [[Higher Topos Theory|HTT]] above prop. 4.2.1.5.

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

The simplicial sets $C_{/F}$ and $C_{F/}$ are indeed themselves again [[quasi-categories]].

=--

This appears as [[Higher Topos Theory|HTT, prop. 1.2.9.3]]

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

The two definitions yield equivalent results in that the canonical morphism

$$
  C_{/F} \to C^{/F}
  \,.
$$

is an equivalence of quasi-categories.

=--

This is [[Higher Topos Theory|HTT, prop. 4.2.1.5]]

From the formula

$$
  (C_{/F})_n = (Hom_{sSet})_F(\Delta^n \star K , C)
$$

we see that

* an object in the over quasi-category $C_{/F}$ is a **[[cone]]** over $F$;.

  For instance if $K = \Delta[1]$ then an object in $C_{/F}$ is a 2-cell

  $$
    \array{
       && v
       \\
       & \swarrow &\swArrow& \searrow
       \\
       F(0) &&\to&& F(1)
    }
  $$

  in $C$.

* a morphism in $C_{/F}$ is a morphism of cones,

* etc:.

So we may think of the overcategory $C_{/F}$ as the **quasi-category of cones over $F$**. Accordingly we have that

* the [[terminal object in a quasi-category|terminal object]] in $C_{/F}$ is (if it exists) the [[limit in a quasi-category|limit in]] $C$ over $F$;

* the [[terminal object in a quasi-category|initial object]] in $C_{F/}$ is (if it exists) the [[limit in a quasi-category|colimit of]] $F$ in $C$.

## Properties

### Relation to over-1-categories

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

For $C = N(\mathcal{C})$ (the [[nerve]] of) an ordinary [[category]] $\mathcal{C}$ and $K = *$, this construction coincides with the ordinary notion of [[overcategory]] $\mathcal{C}/F$ in that there is a canonical [[isomorphism]] of simplicial sets

$$
  N(\mathcal{C}/F) \simeq N(\mathcal{C})/F
  \,.
$$

=--

This appears as [[Higher Topos Theory|HTT, remark 1.2.9.6]].

### Functoriality of the slicing

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

If $q : C \to D$ is a [[model structure for quasi-categories|categorical equivalence]] then so is the induced morphism $C_{/F} \to D_{/q F}$.

=--

This appears as [[Higher Topos Theory|HTT, prop 1.2.9.3]].

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

For $C$ a [[quasi-category]] and $p : X \to C$ any morphism of simplicial sets, the canonical morphisms

$$
  C_{p/} \to C
$$

and

$$
  C^{p/} \to C
$$

are both [[left Kan fibrations]].

=--

This is a special case of [[Higher Topos Theory|HTT, prop 2.1.2.1]] and [[Higher Topos Theory|prop. 4.2.1.6]].

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

Let $v \colon K \to \tilde K$ be a map of small [[(∞,1)-categories]], $\mathcal{C}$ an $(\infty,1)$-category, $\tilde{p}: \tilde{K} \to \mathcal{C}$ and $p = \tilde{p}v$. The induced [[(∞,1)-functor]] between slice $(\infty,1)$-categories

$$
  \mathcal{C}_{/ \tilde{p}} \to \mathcal{C}_{/p}
$$

 is an [[equivalence of (∞,1)-categories]] for each diagram $\tilde{p}$ precisely if $v$ is an op-[[final (∞,1)-functor]] (hence if $v^{op}$ is final).

=--

This is ([Lurie, prop. 4.1.1.8](#Lurie)).

### Hom-spaces in a slice
 {#HamSpacesInASlice}

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

For $C$ an [[(∞,1)-category]] and $X \in C$ an [[object]] in $C$ and $f : A \to X$ and $g : B \to X$ two objects in $C/X$, the [[derived hom-space|hom-∞-groupoid]] $C/X(f,g)$ is equivalent to the [[homotopy fiber]] of
$C(A,B) \stackrel{g_*}{\to} C(A,X)$ over the morphism $f$: we have an [[(∞,1)-pullback]] [[diagram]]

$$
  \array{
     C/X(f,g) & \longrightarrow & C(A,B)
     \\
     \downarrow && \downarrow^{\mathrlap{g_*}}
     \\
     {*} & \stackrel{\vdash f}{\longrightarrow} & C(A,X)
  }
  \,.
$$

=--

This is [[Higher Topos Theory|HTT, prop. 5.5.5.12]].

### Limits and Colimits in a slice
 {#LimitsAndColimits}

The forgetful functor $\mathcal{C}_{/X} \to \mathcal{C}$ out of a slice ([[dependent sum]]) [[reflected limit|reflects]] [[(∞,1)-colimits]]:

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

Let $f \colon K \to \mathcal{C}_{/X}$ be a [[diagram]] in the slice of an [[(∞,1)-category]] $\mathcal{C}$ over an object $X \in \mathcal{C}$. Then if the composite $K \stackrel{f}{\to} \mathcal{C}_{/X} \to \mathcal{C}$ has an [[(∞,1)-colimit]], then so does $f$ itself and the projection $\mathcal{C}_{/q} \to \mathcal{C}$ takes the latter to the former. Conversely, a cocone $K \star \Delta^0  \to \mathcal{C}_{/X}$ under $f$ is an [[(∞,1)-colimit]] of $f$ precisely if the composite $K \star \Delta^0 \to \mathcal{C}_{/X} \to \mathcal{C}$ is an $(\infty,1)$-colimit of the projection of $f$.

=--

This appears as ([Lurie, prop. 1.2.13.8](#Lurie)).

On the other hand [[(∞,1)-limits]] in the slice are computed as limits over the diagram with the slice-cocone adjoined:

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

For $\mathcal{C}$ an [[(∞,1)-category]], $X \;\colon\; \mathcal{D} \longrightarrow \mathcal{C}$ a [[diagram]], $\mathcal{C}_{/X}$ the [[comma category]] (the over-$\infty$-category if $\mathcal{D}$ is the point) and $F \;\colon\; K \to \mathcal{C}_{/X}$ a [[diagram]] in the [[comma category]], then the [[(∞,1)-limit]] $\underset{\leftarrow}{\lim} F$ in $\mathcal{C}_{/X}$ coincides with the limit $\underset{\leftarrow}{\lim} F/X$ in $\mathcal{C}$.

=--

For a proof see at [[(∞,1)-limit]]  _[here](limit+in+a+quasi-category#InOvercategories)_.

[[!include sliced adjoint functors -- section]]

## Examples

* [[pointed homotopy types]]

## Related concepts

* [[arrow category]]

* [[over-category]]

  * [[slice category]]

  * [[under category]]

  * [[over topos]]

* **over-(∞,1)-category**

  * [[model structure on an over category]]

  * [[over-(∞,1)-topos]], [[etale geometric morphism]]

* [[arrow (∞,1)-category]], [[arrow (∞,1)-topos]]

* [[slice 2-category]]

## References

Section 1.2.9 of

* [[Jacob Lurie]], _[[Higher Topos Theory]]_
 {#Lurie}

[[!redirects over-category in quasi-categories]]
[[!redirects over quasi-categories]]
[[!redirects over quasicategory]]
[[!redirects over-quasi-category]]
[[!redirects over-quasicategory]]
[[!redirects overquasicategory]]
[[!redirects slice quasi-category]]
[[!redirects slice quasicategory]]

[[!redirects over-(∞,1)-category]]
[[!redirects over-(infinity,1)-category]]

[[!redirects slice infinity-category]]
[[!redirects slice infinity-categories]]

[[!redirects over-(∞,1)-categories]]
[[!redirects over-(infinity,1)-categories]]

[[!redirects under-(∞,1)-category]]
[[!redirects under-(infinity,1)-category]]

[[!redirects under-(∞,1)-categories]]
[[!redirects under-(infinity,1)-categories]]

[[!redirects over (∞,1)-category]]
[[!redirects over (infinity,1)-category]]

[[!redirects over (∞,1)-categories]]
[[!redirects over (infinity,1)-categories]]

[[!redirects over quasi-category]]

[[!redirects slice-(∞,1)-category]]
[[!redirects slice-(∞,1)-categories]]

[[!redirects slice (∞,1)-category]]
[[!redirects slice (∞,1)-categories]]
[[!redirects slice (infinity,1)-category]]
[[!redirects slice (infinity,1)-categories]]

[[!redirects coslice-(∞,1)-category]]
[[!redirects coslice-(∞,1)-categories]]

[[!redirects coslice (∞,1)-category]]
[[!redirects coslice (∞,1)-categories]]
[[!redirects coslice (infinity,1)-category]]
[[!redirects coslice (infinity,1)-categories]]

[[!redirects over-infinity1-category]]


over-(infinity,1)-topos

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For $\mathbf{H}$ an [[(∞,1)-topos]] and $X \in \mathbf{H}$ any [[object]], the [[over-(∞,1)-category]] $\mathbf{H}/X$ is itself an $(\infty,1)$-topos: the _over-$(\infty,1)$-topos_ (the *[[fundamental theorem of (infinity,1)-topos theory|fundamental theorem of $\infty$-topos theory]]*).

If we think of $\mathbf{H}$ as a [[big topos]], then for $X \in \mathbf{H}$ we may think of $\mathbf{H}/X \in $ [[(∞,1)-topos]] as the [[little topos]]-incarnation of $X$. The [[object]]s of $\mathbf{H}/X$ we may think of as [[(∞,1)-sheaves]] on $X$.

This correspondence between objects of $X$ and their little-topos incarnation is entirly natural: $\mathbf{H}$ is equivalently recovered as the [[(∞,1)-category]] whose objects are over-$(\infty,1)$-toposes $\mathbf{H}/X$ and whose morphisms are [[(∞,1)-geometric morphism]]s _over_ $\mathbf{H}$.

## Definition

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

For $\mathbf{H}$ an [[(∞,1)-topos]] and $X \in \mathbf{H}$ an [[object]] also the [[over-(∞,1)-category]] $\mathbf{H}/X$ is an $(\infty,1)$-topos. This is the **over-$(\infty,1)$-topos** of $\mathbf{H}$ over $X$.

=--

This is [[Higher Topos Theory|HTT, prop 6.3.5.1 (1)]].

## Properties

### Base change to the point

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

There is a canonical [[(∞,1)-geometric morphism]]

$$
  \mathbf{H}/X
    \stackrel{\overset{X_!}{\longrightarrow}}{\stackrel{\overset{X^*}{\longleftarrow}}{\underset{X_*}{\longrightarrow}}}
  \mathbf{H}
$$

where the extra [[left adjoint]] $X_!$ is the obvious projection $X_! : (Y \to X) \mapsto X$, and $X_*$ is given by forming the [[(∞,1)-limit|(∞,1)-product]] with $X$.

=--

This is called an [[etale geometric morphism]]. See there for more details.

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

The fact that $(X_! \dashv X^*)$ follows from the universal property of the products. The fact that $X^*$ preserves [[(∞,1)-colimit]]s and hence has a further [[right adjoint]] $X_*$ by the [[adjoint (∞,1)-functor theorem]] follows from that fact that $\mathbf{H}$ has [[universal colimits]].

=--

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

If $\mathbf{H}$ is a [[locally ∞-connected (∞,1)-topos]] then for all $X \in \mathbf{H}$ also the over-$(\infty,1)$-topos $\mathbf{H}/X$ is locally $\infty$-connected.

=--

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

The composite of [[(∞,1)-geometric morphism]]s

$$
  \mathbf{H}/X
    \stackrel{\overset{X_!}{\longrightarrow}}{\stackrel{\overset{X^*}{\longleftarrow}}{\underset{X_*}{\longrightarrow}}}
  \mathbf{H}
  \stackrel{\overset{\Pi_{\mathbf{H}}}{\longrightarrow}}{\stackrel{\overset{LConst_{\mathbf{H}}}{\longleftarrow}}{\underset{\Gamma_{\mathbf{H}}}{\longrightarrow}}}
  \infty Grpd
$$

is itself a geometric morphism. Since [[∞Grpd]] is the [[terminal object]] in [[(∞,1)Topos]] this must be the [[global section]] geometric morphism for $\mathbf{H}/X$. Since it has the extra left adjoint $\Pi \circ X_!$ it is locally $\infty$-connected.

=--

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

Let $((\infty,1)Topos/\mathbf{H})_{et} \subset (\infty,1)Topos/\mathbf{H}$ be the full [[sub-(∞,1)-category]] on the [[etale geometric morphism]]s $\mathbf{H}/X \to \mathbf{H}$. Then there is an [[equivalence in an (∞,1)-category|equivalence]]

$$
  ((\infty,1)Topos/\mathbf{H})_{et}
  \simeq
  \mathbf{H}
$$

=--

See [[etale geometric morphism]] for more details.

### General base change

See [[base change geometric morphism]].

### As $(\infty,1)$-sheaves on the big $(\infty,1)$-site of an object {#SheavesOnBigSite}

We spell out how $\mathbf{H}/X$ is the [[(∞,1)-category of (∞,1)-sheaves]] over the [[big site]] of $X$.

The following may be seen as the presheaf version of the [[fundamental theorem of (infinity,1)-topos theory|fundamental theorem of $\infty$-topos theory]].

+-- {: .num_prop #SlicingCommutesWithPassingToPresheaves}
###### Proposition
**(forming overcategories commutes with passing to presheaves)**

Let $C$ be a [[small (∞,1)-category]] and $X \colon K \longrightarrow C$ a [[diagram]]. Write $C_{/X}$
and $PSh(C)_{/X}$ for the corresponding [[over (∞,1)-categories]], where
-- notationally implicitly -- we use the [[(∞,1)-Yoneda embedding]] $C \hookrightarrow PSh(C)$.

Then we have an [[equivalence of (∞,1)-categories]]

$$
  PSh\big(
    C_{/X}
  \big)
  \stackrel{\simeq}{\longrightarrow}
  PSh(C)_{/X}
  \,.
$$

=--

This appears as [[Higher Topos Theory|HTT, 5.1.6.12]]. For more on this see [[(∞,1)-category of (∞,1)-presheaves]].

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

Here we may think of $C/X$ as the [[big site]] of the object $c \in PSh(C)$, hence of $PSh(C/X)$ as presheaves on $X$.

=--

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

Let $C$ be equipped with a [[subcanonical coverage]], let $X \in C$ and regard $C/X$ as an [[(∞,1)-site]] with the [[big site]]-[[coverage]]. Then we have

$$
  Sh(C/X) \simeq Sh(C)/X
  \,.
$$

=--

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

By the discussion of adjoint (∞,1)-functors on over-(∞,1)-categories [[adjoint (∞,1)-functor]]s we have that the adjunction

$$
  (F \dashv i) : Sh(C) \stackrel{\overset{F}{\leftarrow}}{\hookrightarrow}
  PSh(C)
$$

passes to an adjunction on the [[over-(∞,1)-categories]]

$$
  (F/X \dashv i/X) : Sh(C)/X \stackrel{\overset{F}{\leftarrow}}{\hookrightarrow}
  PSh(C)/X
  \,,
$$

(where we are using that $F  i X \simeq X$ by the assumption that the coverage is subcanonical, so that the representable $X$ is a [[(∞,1)-sheaf]]), such that $i/X$ is still a [[full and faithful (∞,1)-functor]] (where we are using that the unit $X \to F i X$ is an equivalence, since $X$ is a sheaf).

Since moreover the [[(∞,1)-limits]] in $Sh(C)/X$ are computed as limits in $Sh(C)$ over diagrams with a bottom element adjoined (as discussed at limits in over-(∞,1)-categories) it follows that with $F$ preserving all finite limits, so does $F/X$.

In summary we have that $(F/X \dashv i/X)$ is a [[reflective sub-(∞,1)-category]] of $PSh(C/X)$ hence is the [[(∞,1)-category of (∞,1)-sheaves]] on the category $C/X$ for _some_ [[(∞,1)-site]]-structure. But since $F/X$ inverts precisely those morphisms that are inverted by $F$ under the projection $PSh(C)/X \to PSh(C)$, it follows that this is the [[big site]] structure on $C/X$ (this is the defining property of the big site).

=--

Specifically for the $(\infty,1)$-topos $\mathbf{H} = $ [[∞Grpd]] we also have the following characterization.

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

For $\mathbf{H} = $ [[∞Grpd]] we have that for $X \in \infty Grp$ any [[∞-groupoid]] the corresponding over-$(\infty,1)$-topos is equivalent to the [[(∞,1)-category of (∞,1)-presheaves]] on $X$:

$$
  \infty Grpd/X \simeq PSh(X) \simeq [X, \infty Grpd]
  \,.
$$

=--

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

This is a special case of the [[(∞,1)-Grothendieck construction]]. See the section (∞,0)-fibrations over ∞-groupoids.

=--

The following proposition asserts that the over-$(\infty,1)$-topos over an $n$-[[truncated]] object indeed behaves like a generalized [[n-groupoid]]

\begin{proposition}
\label{SliceOfNLocalicToposOverNTruncatedObject}
For $n \in \mathbb{N}$ and $\mathcal{X}$ an [[n-localic (∞,1)-topos]], then the over-$(\infty,1)$-topos $\mathcal{X}/U$ is $n$-localic precisely if the object $U$ is $n$-[[truncated]].
\end{proposition}

This is ([StrSp, lemma 2.3.16](#StrSp)).

### Object classifier
 {#ObjectClassifier}

If $Obj_\kappa \in \mathbf{H}$ is an [[object classifier]] for $\kappa$-[[small objects]], then the [[projection]] $Obj_\Kappa \times X \to X$ regarded as an object in the slice is a $\kappa$-small object classifier in $\mathbf{H}_{/X}$.

### In homotopy type theory
 {#InHomotopyTypeTheory}

If a [[homotopy type theory]] is the [[internal language]] of $\mathbf{H}$, then then theory in [[context]] $x : X \vdash \cdots$ is the internal language of $\mathbf{H}_{/X}$.

## Related concepts

* [[fundamental theorem of topos theory]]

* [[arrow category]]

* [[over-category]]

  * [[slice category]]

  * [[under category]]

  * [[over topos]]

* [[over (∞,1)-category]],

  * [[model structure on an over category]]

  * **over-(∞,1)-topos**

* [[arrow (∞,1)-category]], [[arrow (∞,1)-topos]]

## References

* [[Jacob Lurie]], Section 6.3.5 of: *[[Higher Topos Theory]]*, Annals of Mathematics Studies 170, Princeton University Press 2009 ([pup:8957](https://press.princeton.edu/titles/8957.html), [pdf](https://www.math.ias.edu/~lurie/papers/HTT.pdf))

Some related remarks are in:

* {#StrSp} [[Jacob Lurie]], _[[Structured Spaces]]_

[[!redirects over-(∞,1)-topos]]

[[!redirects over-(∞,1)-toposes]]
[[!redirects over-(∞,1)-topoi]]
[[!redirects over-(infinity,1)-toposes]]
[[!redirects over-(infinity,1)-topoi]]

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

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

[[!redirects slice homotopy-topos]]
[[!redirects slice homotopy-toposes]]
[[!redirects slice homotopy-topoi]]

[[!redirects slice infinity1-topos]]


over-category in quasi-categories > history

< [[over-category in quasi-categories]]
[[!redirects over-category in quasi-categories -- history]]


over-topos

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

#Contents#
* table of contents
{:toc}

## Idea

For $\mathcal{T}$ a [[topos]] and $X \in \mathcal{T}$ any [[object]] the [[over category]] $\mathcal{T}/X$ -- the **slice topos** or **over-topos** -- is itself a topos: the "big [[little topos]]" incarnation of $X$.  This fact is has been called the *[[fundamental theorem of topos theory]]* ([McLarty 1992, Thm. 17.4](#McLarty92)).

More generally, given a functor $u : E \to F$ between [[toposes]] that preserves [[pullbacks]], the [[comma category]] $(id_F/u)$ is again a [[topos]], called the [[Artin gluing]].

## Definition / Existence
 {#Definition}

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

For $\mathcal{T}$ a [[topos]] and $X \in \mathcal{T}$ any [[object]], the [[slice category]] $\mathcal{T}{/X}$ is itself again a [[topos]].

=--

A proof is spelled out for instance in [MacLane-Moerdijk, IV.7 theorem 1](#MacLaneMoerdijk). In particular we have

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

If $\Omega \in \mathcal{T}$ is the [[subobject classifier]] in $\mathcal{T}$, then the [[projection]] $\Omega \times X \to X$ regarded as an object in the slice over $X$ is the subobject classifier of $\mathcal{T}{/X}$.

=--

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

The [[power object]] of a map $f: A \to X$ is given by the equalizer of the maps $p, t$:
$$
  \array{
    P_X f && \dashrightarrow && P A \times X && \underoverset{t}{p}{\rightrightarrows} && P A
    \\
    &&&&\downarrow^{\mathrlap{1 \times \{\cdot\}_X}} &&&& \uparrow^{\mathrlap{\wedge}}
    \\
    &&&& P A \times P X && \underset{1 \times P f}{\rightarrow} && P A \times P A
  },
$$
where $p$ is the projection map and $t$ is the composition $\wedge \circ (1 \times P f) \circ (1 \times \{\cdot\}_X)$. In the internal language, this says
$$
  P_X f = \{(B, x) \in P A \times X: (\forall b \in B) f(b) = x\}.
$$
The map to $X$ is given by projection onto the second factor.
=--

+-- {: .num_remark #AsExampleOfToposOfCoalgebrasOverAComonad}
###### Remark
The fact that the slice $\mathcal{T}/X$ is a topos, and particularly the construction of power objects above, can be deduced from a more general result: that the category of coalgebras of a pullback-preserving comonad $G: \mathcal{T} \to \mathcal{T}$ is a topos. See at _[[topos of coalgebras over a comonad]]_. In the case of a slice topos, the comonad would be $X \times -: \mathcal{T} \to \mathcal{T}$ (with comultiplication induced by the diagonal $X \to X \times X$, and counit induced by the projection $!: X \to 1$). This result also subsumes the weaker result where $G$ is assumed to preserve finite limits. See the [[Elephant]], Section A, Remark 4.2.3. A proof of a still more general result may be found [here](https://ncatlab.org/toddtrimble/published/Three+topos+theorems+in+one).
=--

## Properties

### Étale geometric morphism {#EtaleGeometricMorphism}

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

For $\mathcal{T}$ a [[Grothendieck topos]] and $X \in \mathcal{T}$ any object, the canonical projection functor $X_! : \mathcal{T}/X \to \mathcal{T}$ is part of an [[essential geometric morphism]]

$$
  (X_! \dashv X^* \dashv X_*)
  :
  \mathcal{T}/X \stackrel{\overset{X_!}{\to}}{\stackrel{\overset{X^*}{\leftarrow}}{\underset{X_*}{\to}}}
  \mathcal{T}
  \,.
$$

=--

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

The functor $X^*$ is given by taking the [[product]] with $X$:

$$
  X^* : K \mapsto (p_2 : K \times X \to X)
  \,,
$$

since [[commuting diagram]]s

$$
  \array{
    A &&\to&& K \times X
    \\
    & \searrow && \swarrow_{\mathrlap{p_2}}
    \\
    && X
  }
$$

are evidently uniquely specified by their components $A \to K$.

Moreover, since in the Grothendieck topos $\mathcal{T}$ we have [[universal colimits]], it follows that $(-) \times X$ preserves all [[colimit]]s. Therefore by the [[adjoint functor theorem]] a further [[right adjoint]] $X_*$ exists.

=--

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

One also says that $X_!$ is the _[[dependent sum]]_ operation and $X_*$ the [[dependent product]] operation. As discussed there, this can be seen to compute spaces of [[section]]s of [[bundles]] over $X$.

Moreover, in terms of the [[internal logic]] of $\mathcal{T}$ the functor $X_!$ is the _[[existential quantifier]]_ $\exists$ and $X_*$ is the _[[universal quantifier]]_ $\forall$.

=--

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

A [[geometric morphism]] $\mathcal{E} \to \mathcal{T}$ equivalent to one of the form $(X_! \dashv X^* \dashv X_*)$ is called an
**[[etale geometric morphism]]**.

=--

More generally:

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

For $\mathcal{E}$ a [[Grothendieck topos]] and $f : X \to Y$ a [[morphism]] in $\mathcal{E}$,
there is an induced [[essential geometric morphism]]

$$
  (f_! \dashv  f^* \dashv f_*)
  :
  \mathcal{E}/X
    \stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}}
  \mathcal{E}/Y
  \,,
$$

where $f_!$ is given by postcomposition with $f$ and $f^*$ by [[pullback]] along $X$.

=--

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

By [[universal colimit]]s in $\mathcal{E}$ the [[pullback]] functor $f^*$ preserves both [[limit]]s and [[colimit]]s. By the [[adjoint functor theorem]] and using that the over-toposes are [[locally presentable categories]], this already implies that it has a [[left adjoint]] and a [[right adjoint]]. That the left adjoint is given by postcomposition with $f$ follows from the universality of the pullback: for $(a : A \to X)$ in $\mathcal{E}/X$ and $(b : B \to Y)$ in $\mathcal{E}/Y$ we have unique factorizations

$$
  \array{
    A &\to& X \times_X B &\to& B
    \\
    &{}_{\mathllap{a}}\searrow& \downarrow^{\mathrlap{f^*(b)}} && \downarrow^{\mathrlap{b}}
    \\
    && X &\stackrel{f}{\to}& Y
  }
$$

in $\mathcal{E}$, hence an isomorphism

$$
  \mathcal{E}/Y(f_!(A \to X), (B \to Y)) \simeq
  \mathcal{E}/X((A \to X), f^*(B \to Y))
  \,.
$$

=--

### In terms of sheaves on a slice site
  {#PresheafOverTopos}

Generally, for $C$ a [[site]], $c \in C$ an [[object]], and $L(y(c)) \in Sh(C)$ the [[sheafification]] of its image under the [[Yoneda embedding]], there is an [[equivalence of categories]]

$$
  Sh\big(
    C /c
  \big)
  \;\simeq\;
  Sh(C)/(L(y(c)))
$$

between the [[category of sheaves]] on the [[slice category]] $C/c$ with its evident induced [[structure]] of a [[site]], and the [[slice topos]] of the category of sheaves on $C$, sliced over $L(y(c))$.

This is for instance in Verdier's exposé III.5 prop.5.4 ([SGA4](#SGA4), p.295).

\linebreak

We now discuss this in more detail for the special case of over-[[presheaf toposes]].

Let $C$ be a [[small category]], $c$ an [[object]] of $C$ and let $C/c$ be the [[slice category]] of $C$ over $c$.

Write

* $PSh(C/c) = [(C/c)^{op}, Set]$ for the [[category of presheaves]] on $C/c$

* and write $PSh(C)/Y(c)$ for the [[over category]] of [[presheaf|presheaves]] on $C$ over the presheaf $Y(c)$, where $Y : C \to PSh(C)$ is the [[Yoneda embedding]].

+-- {: .num_prop #SliceOfPresheavesIsPresheavesOnSlice}
###### Proposition
**([[slice of presheaves is presheaves on slice]])**

There is an [[equivalence of categories]]

$$
  e \;\colon\;
  PSh(C/c)
  \xrightarrow{\;\;\simeq\;\;}
  PSh(C)/Y(c)
  \,.
$$

=--

([SGA4 I, Ex. 1 Prop. 5.11](#SGA41), [p. 27](http://www.cmls.polytechnique.fr/perso/laszlo/sga4/SGA4-1/sga41.pdf#page=27); [Kashiwara-Schapira 2006, Lemma 1.4.12](#KS06), [p. 26](https://www.maths.ed.ac.uk/~v1ranick/papers/kashiwara2.pdf#page=26))

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

The functor $e$ takes $F \in PSh(C/c)$ to the presheaf

$$
  F'
    \,\colon\,
  d
    \;\mapsto\;
  \underset{
    f \in C(d,c)
  }{\sqcup}
  F(f)
$$

which is equipped with the natural transformation $\eta : F' \to Y(c)$ with component map

$$
  \array{
    \eta_d
    &
    \colon
    &
    \underset
      {f \in C(d,c)}
    {\sqcup }
      F(f)
    &
    \longrightarrow
    &
    C(d,c)
    \\
    &&
    \theta \in F(f)
    &\mapsto&
    f
    \,.
  }
$$

One readily checks (for more details see [[slice of presheaves is presheaves on slice|here]]) that a weak inverse of $e$ is given by the functor
$$
  \bar e \;\colon\; PSh(C)/Y(c) \to PSh(C/c)
$$
which sends $\eta \,\colon\, F' \to Y(c)$ to $F \in PSh(C/c)$ given by
$$
  F \;\colon\; (f \,\colon\, d \to c) \mapsto F'(d)|_c
  \,,
$$
where $F'(d)|_c$ is the [[pullback]]
$$
  \array{
     F'(d)|_c &\to& F'(d)
     \\
     \downarrow && \downarrow^{\eta_d}
     \\
     pt &\stackrel{f}{\to}& C(d,c)
  }
  \,.
$$

=--

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

Suppose the presheaf $F \in PSh(C/c)$ does not actually depend on the morphisms to $c$, i.e. suppose that it factors through the forgetful functor from the [[over category]] to $C$:
$$
  F : (C/c)^{op} \to C^{op} \to Set
  \,.
$$

Then
$
  F'(d) = \sqcup_{f \in C(d,c)} F(f)
  = \sqcup_{f \in C(d,c)} F(d)
  \simeq
  C(d,c) \times F(d)
$
and hence $F ' = Y(c) \times F$ with respect to the [[closed monoidal structure on presheaves]].

=--

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

Consider $\int_C Y(c)$ , the [[category of elements]] of $Y(c):C^{op}\to Set$. This has objects $(d_1,p_1)$ with $p_1\in Y(c)(d_1)$, hence $p_1$ is just an arrow $d_1\to c$ in $C$. A map from $(d_1, p_1)$ to $(d_2, p_2)$ is just a map $u:d_1\to d_2$ such that $p_2\circ u =p_1$ but this is just a morphism from $p_1$ to $p_2$ in $C/c$.

Hence, the above Prop. \ref{SliceOfPresheavesIsPresheavesOnSlice} can be rephrased as $PSh(\int_C Y(c))\simeq PSh(C)/Y(c)$ which is an instance of the following formula:

=--

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

Let $P:C^{op}\to Set$ be a presheaf. Then there is an [[equivalence of categories]]

$$
  PSh(\int_C P) \simeq PSh(C)/P
  \,.
$$

=--

On objects this takes $F : (\int_C P)^{op} \to Set$ to
$$i(F)(A \in C) = \{ (p,a) | p \in P(A), a \in F(A,p) \} = \Sigma_{p \in P(A)} F(A,p)$$
with obvious projection to $P$. The inverse takes $f : Q \to P$ to
$$i^{-1}(f)(A, p \in P(A)) = f_A^{-1}(p)\;.$$

[Kashiwara-Schapira (2006, Lemma 1.4.12, p. 26)](#KS06).
For a more general statement involving slices of Grothendieck toposes see [Mac Lane-Moerdijk (1994, p.157)](#MacLaneMoerdijk).

In particular, this equivalence shows that _slices of presheaf toposes are presheaf toposes_.

### Geometric morphisms by slicing {#GeometricMorphismBySlicing}

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

For $(f^* \dashv f_*) : \mathcal{T} \to \mathcal{E}$ a [[geometric morphism]] of toposes and $X \in \mathcal{E}$ any [[object]], there is an induced geometric morphism between the slice-toposes

$$
  (f^*/X \dashv f_*) : \mathcal{T}/f^*X \to \mathcal{E}/X
  \,,
$$

where the [[inverse image]] $f^*/X$ is the evident application of $f^*$ to diagrams in $\mathcal{E}$.

=--

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

The slice adjunction $(f^*/X \dashv f_*/X)$ is discussed  here: the [[left adjoint]] $f^*/X$ is the evident induced functor. Since limits in an over-category $\mathcal{E}/X$ are computed as limits in $\mathcal{E}$ of diagrams with a single bottom element $X$ adjoined, $f^*/X$ preserves finite limits, since $f^*$ does, so that $(f^*/X \dashv f_*/X)$ is indeed a [[geometric morphism]].

=--

### Topos points {#Points}

We discuss [[point of a topos|topos points]] of over-toposes.

\begin{lemma}\label{points-from-pairs}
Let $\mathcal{E}$ be a [[topos]], $X \in \mathcal{E}$ an [[object]] and

$$
  (e^* \dashv e_*)  : Set \to \mathcal{E}
$$

a [[point of a topos|point]] of $\mathcal{E}$. Then for every element $x \in e^*(X)$ there is a point of the slice topos $\mathcal{E}/X$ given by the composite

$$
  (e,x)
  \;\colon\;
  Set \stackrel{\overset{x^*}{\leftarrow}}{\underset{x_*}{\to}}
  Set/e^*(X)
  \stackrel{\overset{e^*/X}{\leftarrow}}{\underset{e_*/X}{\to}}
  \mathcal{E}/X
  \,.
$$
\end{lemma}

Here $(e^*/X \dashv e_*/X)$ is the slice geometric morphism of $e$ over $X$ discussed [above](#SliceGeometricMorphism) and $(x^* \dashv x_*)$ is the étale geometric morphism discussed [above](#GeneralEtaleGeometricMorphism) induced from the morphism
$* \stackrel{x}{\to} e^*(X)$.

Hence the [[inverse image]] of $(e,x)$ sends $A \to X$ to the [[fiber]] of $e^*(A) \to e^*(X)$ over $x$.

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

If $\mathcal{E}$ has [[point of a topos|enough points]] then so does the slice topos $\mathcal{E}/X$ for every $X \in \mathcal{E}$.

=--

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

That $\mathcal{E}$ has enough points means that a morphism $f : A \to B$ in $\mathcal{E}$ is an [[isomorphism]] precisely if for every [[point of a topos|point]] $e : Set \to \mathcal{E}$ the function $e^*(f) : e^*(A) \to e^*(B)$ is an isomorphism.

A morphism in the slice topos, given by a diagram

$$
  \array{
     A &&\stackrel{f}{\to}&& B
     \\
     & \searrow && \swarrow
     \\
     && X
  }
$$

in $\mathcal{E}$ is an isomorphism precisely if $f$ is. By the above observation we have that under the [[inverse image]]s of the slice topos points $(e,x \in e^*(X))$ this maps to the fibers of

$$
  \array{
     e^*(A) &&\stackrel{e^*(f)}{\to}&& e^*(B)
     \\
     & \searrow && \swarrow
     \\
     && e^*(X)
  }
$$

over all points $* \stackrel{x}{\to} e^*(X)$. Since in [[Set]] every object $S$ is a [[coproduct]] of the point indexed over $S$, $S \simeq \coprod_S *$ and using [[universal colimits]] in $S$, we have that if $x^* e^*(f)$ is an isomorphism for all $x \in e^*(X)$ then $e^*(f)$ was already an isomorphism.

The claim then follows with the assumption that $\mathcal{E}$ has enough points.

=--

It turns out that all points of $\mathcal{E}/X$ correspond to pairs $(e,x)$ as above, with $e$ a [[point of a topos|point]] of $\mathcal{E}$ and $x \in e^*(X)$ an element. More precisely:

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

Let $\mathcal{E}$ be a [[topos]] and $X$ an [[object]] in $\mathcal{E}$. Then the category of [[point | points]] of the over-topos $\mathcal{E}/X$ is equivalent to the category with: as objects the pairs $(e,x)$ with $e$ a [[point]] of $\mathcal{E}$ and $x \in e^*(X)$ an element; and as morphisms $(e,x) \to (e',x')$ the [[natural transformation | natural transformations]] $\eta : e^* \to (e')^*$ such that $\eta_X(x) = x'$.

=--

This is [SGA4 (1972, Exposé IV, Proposition 5.12, p. 382)](#SGA41), in the special case where $E' = Set$. In the statement of the proposition, we used the (now standard) convention that a morphism of points (or [[geometric transformation]]) $e \to e'$ is a natural transformation $e^* \to (e')^*$. Note however that SGA4 uses the opposite convention, see [SGA4 (1972, Exposé IV, 3.2, p. 328)](#SGA41).

The point corresponding the pair $(e,x)$ is the one constructed in Observation \ref{points-from-pairs}.

## Related concepts

* [[over category]], [[under category]]

* [[Artin gluing]]

* [[fundamental theorem of topos theory]]

* [[over (∞,1)-category]],

  * [[model structure on an over category]]

  * [[over-(∞,1)-topos]]

## References

* {#SGA41}[[Michael Artin]], [[Alexander Grothendieck]], [[Jean-Louis Verdier]], _Théorie des Topos et Cohomologie Etale des Schémas_ ([[SGA4]]) Tome 1: *Théorie des Topos* Springer **LNM** **269** (1972) ([doi:10.1007/BFb0081551](https://link.springer.com/book/10.1007/BFb0081551), [pdf](http://www.cmls.polytechnique.fr/perso/laszlo/sga4/SGA4-1/sga41.pdf))

  (In particular, exposé III.5 and exposé IV.5 on the "induced topos" - _topos induit_ = slice topos)

* {#McLarty92} [[Colin McLarty]], *[[Elementary Categories, Elementary Toposes]]*, Oxford University Press 1992 ([ISBN:9780198514732](https://global.oup.com/academic/product/elementary-categories-elementary-toposes-9780198514732?cc=ae&lang=en&))

* {#KS06} [[Masaki Kashiwara]], [[Pierre Schapira]], _[[Categories and Sheaves]]_, Springer 2006 (  [doi:10.1007/3-540-27950-4](https://link.springer.com/book/10.1007/3-540-27950-4), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/kashiwara2.pdf))

* {#MacLaneMoerdijk} [[Saunders Mac Lane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_ , Springer Heidelberg 1994. (Especially section IV.7)

[[!redirects over topos]]
[[!redirects over toposes]]
[[!redirects over-toposes]]
[[!redirects over topoi]]
[[!redirects over-topoi]]
[[!redirects slice topos]]
[[!redirects slice toposes]]
[[!redirects slice topoi]]


overconvergent derived loop spaces

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Overconvergent derived loop spaces are analogs of [[derived loop spaces]] adapted to the setting of [[overconvergent global analytic geometry]]. Their aim is to give a take at overconvergent analogs of $\mathcal{D}$-modules.

## Overconvergent de Rham space

Let $X$ be a global analytic space over a Banach ring $R$. The [[de Rham space]] $X_{dR}$ is usually defined as the quotient of $X$ by the formal groupoid obtained by formal completion of the groupoid of pairs $X\times X$ acting on $X$ along its unit section (given by the diagonal map). If $X$ is separated, then this diagonal map is closed, and we may (in the affine case), see this formal completion as the spectrum of the projective limit of the quotients $\mathcal{O}_{X\times X}/\mathcal{I}_{\Delta}^n$.
To get an overconvergent version of this formal neighborhood of the diagonal in $X\times X$, we may simply take the ind-Ring of germs of functions on $X\times X$ around the diagonal. It may be an interesting question to try to relate sheaves on the corresponding quotient $X^\dagger_{dR}$ of $X$ by the corresponding groupoid to modules over the ring $\mathcal{D}^\infty$ of differential operators of infinite order.

## Overconvergent derived loop spaces

The basic idea of overconvergent derived loop spaces is to give a loop space version of the above construction of $X^\dagger_{dR}$. Recall that if $X$ is an analytic Artin derived stack, we may define the associated formal loop space as the formal completion $\hat{L} X$ of the loop space groupoid $LX=Hom_{dSt}(S^1,X)$ along the constant loop unit section $c:X\to LX,\;x\mapsto c_x$.

Recall that more concretely, one will have $LX\cong X\times^h_{X\times X} X$. In some sense, this construction corresponds to the intuitive idea of using a kind of (derived) tubular neighborhood of the diagonal to make its auto-intersection interesting (of course, the classical auto-intersection is equal to itself).
One may be tempted to replace both $X$'s that appear in the above formula by
the corresponding germs neighborhood of the diagonal $X$ inside $X\times X$, denoted $X^dagger_{X\times X}$, and computing the associated homotopical fiber product
$$
L^\dagger X=X^\dagger_{X\times X}\times^h_{X\times X} X^\dagger_{X\times X}.
$$
To get a sensible dagger generalization of the formal completion of $L^dagger X$ along the constant loop unit section $c:X\to LX$, we will (be careful, It is not so clear that such a construction will fulfill smooth descent, so that we may need to restrict to usual analytic spaces to get a sensible result) take the analytic germs completion $\hat{L}^\dagger X$ of $L^\dagger X$ along $c:X\to LX$. This means that we consider the filtered colimit (aka intersection) of all _Berkovich_ open subspaces of $L^\dagger X$ that contain $c(X)$. Trying to work everywhere with convergent power series seems like a very natural thing to do if we are doing analytic geometry.


overconvergent global analytic geometry

[[!redirects Overconvergent global analytic geometry]]
[[!redirects overconvergent global analytic geometry]]

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## Idea

Overconvergent [[global analytic geometry]] is a setting for global analytic geometry that allows the definition of strict and non-strict analytic spaces over an arbitrary [[Banach rings]], [[ind-object|ind-]]Banach ring, or more [[ind-object|ind-]][[pseudo-Banach ring]]. It also gives a notion of [[analytic motivic homotopy theory]] and [[derived global analytic geometry]].

## Basic ideas
 {#BasicIdeas}

The basic building blocks for overconvergent global analytic geometry over a given [[Banach ring]] $(R,{\vert\cdot\vert})$ are given by [[polydiscs]] of [[radius]] $1$ (in the strict situation) or arbitrary real radius (in the non-strict situation), and more generally, by strict (or non-strict) rational domains in them. This gives two categories $RatAlg_R^s$ and $RatAlg_R$ of rational domain algebras that actually form [pre-geometries](geometry+%28for+structured+%28infinity%2C1%29-toposes%29#Pregeometry) in the sense of Lurie.

One then defines analytic (resp. derived analytic) algebras as functors (resp. homotopical functors) of functions on the categories of rational domain algebras. The various types of finitely presented analytic algebras define various types of [[geometry (for structured (infinity,1)-toposes)|geometries]] in Lurie's sense. One may then define analytic (resp. derived analytic) stacks as functors (resp. homotopical functors) of points on analytic (resp. derived analytic) algebras.

This gives in particular four categories $An_R^\dagger$, $An_R^{\dagger,s}$, $DAn_R^\dagger$ and $DAn_R^{\dagger,s}$ of strict and non-strict overconvergent derived and non-derived analytic spaces.

## Examples

### Projective line
 {#ProjectiveLine}

One may define the strict [[projective line]] $\mathbb{P}^1_R$ over $R$ by pasting the overconvergent unit disc $D^1=\mathbb{M}(R\{T\}^\dagger)$ with itself along its boundary $U(1)=\mathbb{M}(R\{T,S\}^\dagger/(ST-1))$ by the map $T\mapsto 1/T$. This gives a strict analytic space over $R$. This strict construction of the projective line is particularly interesting when the base Banach ring is the ring $\mathbb{Z}$ of [[integers]] equipped with its usual [[archimedean absolute value]]. This strict structure is intuitively very close to what people call an archimedean compactification, e.g., in [[Arakelov geometry]].

### Unitary group
 {#UnitaryGroup}

One may try to use a similar construction to define higher dimensional global analogs of [[unitary groups]]: one may simply start from the strict [[polydisc]]

$$
  M_n^{\leq 1}
  := \{(a_{ij})\in M_n,\;\max_{i,j}|a_{ij}|\leq 1\}
$$
in the (non-strict analytic) [[affine space]] $M_n=\mathbb{A}^{n^2}$ of [[matrices]].
Since matrix multiplication on $M_n$ does not restrict to a morphism
$$M_n^{\leq 1}\times_R M_n^{\leq 1}\to M_n^{\leq 1},$$
this matrix polydisc is not a monoid. This implies that the naive definition of a global analytic version of $U(n)$, given by
$$
U(n)
\coloneqq
\{(A,B)\in M_n^{\leq 1},\;A B = B A = I\},
$$
only gives a strict analytic space that is not equipped with an analytic group structure
(except if one works in a non-archimedean setting).

One may however associate to $M_n^{\leq 1}$ a [[simplicial object|simplicial]] [[analytic space]] (a kind of [[nerve]]) given by forcing the multiplication to be defined:
$$
BM_n^{\leq 1}_{[k]}\coloneqq
\{(A_i)\in (M_n^{\leq 1})^{k},\;
\prod_{j\in [s]} A_j\in (M_n^{\leq 1})^{s}\;\forall f:[s]\hookrightarrow [k]\}.
$$
One may then define the $n$-dimensional global unitary [[infinity-groupoid]] $BU(n)$ to be given by the analytic sub-groupoid of $B\GL_n$ defined by
$$
  BU(n)
  \coloneqq
  B\GL_n\cap BM_n^{\leq 1}
  \,.
$$

Remark now that $BM_n^{\leq 1}$ is a strict simplicial analytic space, because it is a subspace of the strict simplicial space $[k]\mapsto (M_n^{\leq 1})^k$ defined
by strict inequalities. One may also see $BU(n)$ as a closed strict simplicial analytic
subspace of $BM_n^{\leq 1}\times BM_n^{\leq 1}$ of pairs $(A,B)$ such that
$AB=BA=I$.

We may now look at what we get over various standard Banach rings:

* Over $(\mathbb{Z},|\cdot|_0)$, the scheme (i.e., strict analytic space over a trivially normed ring) $U(n)$  is isomorphic to $GL_{n,\mathbb{Z}}$.

* Over $(\mathbb{Q}_p,|\cdot|_p)$, the rigid analytic space $U(n)$ is such that
$U(n)(\mathbb{Q}_p)=\GL_n(\mathbb{Z}_p)$.

* Over $(\mathbb{Z},|\cdot|_\infty)$, the space $U(n)$ is not an analytic group, and $U(n)(\mathbb{Z},|\cdot|_\infty)$ is closer to what people usually denote $\GL_n(\mathbb{F}_{\{\pm 1\}})$, where $\mathbb{F}_{\{\pm 1\}}$ denotes the [[field with one element]] in Durov's sense.

* Over $\mathbb{C}$, the space $U(n)$ contains the classical unitary group $U(n)(\mathbb{C})$, since one has, for every complex matrix $A=(a_{ij})$, a natural inequality
$$\|A\|_{max}:=\max(|a_{ij}|)\leq \|A\|_2,$$
where $\|A\|_2$ is the operator norm for the hermitian norm on $\mathbb{C}^n$.
This thus gives a natural morphism
$$BU_{n}(\mathbb{C})\to BU(n)(\mathbb{C})$$
from the simplicial classifying space of the classical unitary group $U_{n}(\C)$ to the complex points of the simplicial "classifying space" $BU(n)$.

It is also clearly compact (contained in a product of polydiscs) and contained in the non-strict analytic group $\GL_{n}(\mathbb{C})$. Since $U(n)(\mathbb{C})$ is a maximal compact subgroup, we have that $U(n)$ is indeed the complex unitary group over $\mathbb{C}$.

###  Arithmetic principal unitary bundles

The good properties of the global analytic unitary group **may** imply that there is a natural bijection
$$
\pi_0(BU(n)(\mathbb{Z},|\cdot|_\infty))\cong
K\backslash \GL_n(\mathbb{A})/\GL_n(\mathbb{Q}),$$
where $K$ is the maximal compact subgroup $O(n,\mathbb{R})\times \GL_n(\hat{\mathbb{Z}})$ of $\GL_n(\mathbb{A})$.
It is thus quite a tempting idea to try to formulate the arithmetic [[Langlands program]] in geometric terms (similar to the ones used in the [[geometric Langlands correspondence]] over a function field) using the classifying stack $BU(n)$ of principal $U(n)$-bundles on strict global analytic spaces.

## Related entries

[[derived analytic geometry]]

[[global Hodge theory]]

[[global analytic index theory]]

[[generalized global analytic geometry]]

## Reference

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


overlap algebra

\tableofcontents

## Idea

Overlap algebras are one possible constructive version
of [[complete Boolean algebras]].
In the presence of [[excluded middle]],
the category of overlap algebras
is equivalent to the category of [[complete Boolean algebras]]
and continuous homomorphisms.

However, in the absence of [[excluded middle]],
the [[powerset]] of a [[set]] is no longer a [[complete Boolean algebra]],
but it is an overlap algebra.

## Definition

(See Definition 2.1 in [OA](#OA).)

An __overlap algebra__ is an [[overt]] [[frame]] $L$
such that if for all $z\in L$ the positivity of $z\wedge x$
implies the positivity of $z\wedge y$,
then we have $x\le y$.

Morphisms of overlap algebras are precisely
[[open]] morphisms of [[frames]].

Thus, in the presence of [[excluded middle]]
the category of overlap algebras is equivalent
to the category of [[complete Boolean algebras]],
see Proposition 5.2 in [OA](#OA).

## Properties

In the presence of [[excluded middle]],
overlap algebras coincide with [[complete Boolean algebras]].
(Proposition 2.2 and Proposition 5.2 in [OA](#OA).)

## References

[[Francesco Ciraulo]], [[Michele Contente]], _Overlap Algebras: a Constructive Look at Complete Boolean Algebras_, [arXiv:1904.13320](https://arxiv.org/abs/1904.13320)
{#OA}

[[!redirects overlap algebra]]
[[!redirects overlap algebras]]


overt space

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# Overt spaces
* table of contents
{: toc}

## Idea

An _overt_ space is a [[topological space]] that satisfies a condition [[de Morgan duality|logically dual]] to that satisfied by a *[[compact space]]*.  An overt space is also called _open_ (in French, there is only one word, 'ouvert').

## Motivation

Notice (see [here](closed-projection+characterization+of+compactness#StillOtherCleverConstructions)) that a [[topological space]] $X$ is _compact_ if and only if, for every other space $Y$ and any open subspace $U$ of $X \times Y$, the subspace
$$ \forall_X U = \{ b\colon Y \;|\; \forall\; a\colon X,\; (a, b) \in U \} $$
is open in $Y$.

Dually, a space is _overt_ if and only if, for every other space $Y$ and any open subspace $U$ of $X \times Y$, the subspace
$$ \exists_X U = \{ b\colon Y \;|\; \exists\; a\colon X,\; (a, b) \in U \} $$
is open in $Y$.  (Note that the duality here is only in the logical connectives, not within the category of spaces.)

Of course, *every* topological space satisfies this condition; $\exists_X U$ is the [[union]] over $a$ of the open subspaces $\{ b \;|\; (a,b) \in U \}$.  However, the condition is not trivial in some frameworks, such as [[constructive mathematics|constructive]] [[locale]] theory, [[formal topology]], and [[Abstract Stone Duality]].

## Definition

To remove it from dependence on points, we write the definition like this:

A [[space]] (topological space, locale, etc) $X$ is __overt__ (or __open__) if, given any space $Y$ and any [[open subspace]] $U$ of the [[product space]] $X \times Y$, there exists an open $\exists_X U$ in $Y$ that satisfies the [[universal property]] of [[existential quantification]]:
$$ \exists_X U \subseteq V \;\Leftrightarrow\; U \subseteq X \times V $$
for every open $V$ in $Y$.

Note that, since we quantify over all spaces $Y$ and opens $U\subseteq X\times Y$ in this definition, whether a given space is overt may depend on precisely what 'space' and 'open' mean (even if $X$ is an example for both meanings).  For example, if $X$ is a set with the [[discrete topology]], then it is always overt if 'space' means 'topological space' or 'locale' with the usual meaning of 'open'.  On the other hand, if 'space' means simply 'set', but 'open' refers to the synthetic notion induced by a [[dominance]], then overtness of $X$ is a nontrivial condition (and in fact, if all sets are overt in this sense then the dominance is trivial).

## Overt locales

In the case of [[locales]], it is sufficient to require the above definition only for the case $Y \coloneqq 1$ the [[point]] (unlike the "dual" case of compactness).  In this case, when the map $\exists_X\colon Op(X) \to Op(1) = TV$ (the set or class of [[truth values]]) exists, it is a [[positivity predicate]] on $Op(X)$.  The behavior of this predicate depends on the foundational assumptions:

*  In [[classical mathematics]], it always exists; thus classically every locale is overt.

*  In impredicative [[constructive mathematics]] (such as the [[internal logic]] of a [[topos]]), the positivity predicate can be defined, but it may not satisfy the requisite univeral property of [[adjunction|adjointness]].  Thus, constructively, not every locale is overt.  However, even constructively, every [[topological locale]] is overt (so a [[sober space]] is overt regardless of whether it is viewed as a topological space or as a locale).

*  In constructive [[predicative mathematics]], a positivity predicate cannot be defined and so must be given as a structure on (predicative data that generate) the locale, as is done with a [[formal topology]].  Once this structure is assumed, one can then ask whether a formal topology is overt, i.e. whether the axiomatic positivity predicate satisfies the requisite adjointness.

In impredicative constructive mathematics, [[overt locales]] can be characterized by the *positive covering lemma*: $X$ is overt iff every open $U\in O(X)$ can be covered by positive opens, and iff every covering of an open $U$ can be refined to a covering by positive elements.

## Examples

* At least in impredicative constructive mathematics, the [[topological locale|locale induced by a topological space]] is overt. An open subspace $U$ is positive if and only if it is inhabited.

* At least in impredicative constructive mathematics, the [[spectrum of a commutative ring]] $A$ (defined as the locale whose frame is the frame of radical ideals) is overt if and only if any element of $A$ is nilpotent or not nilpotent.

## Remarks

As compact spaces go with [[proper maps]], so overt spaces go with [[open maps]].  Indeed, $X$ is compact if and only if the unique map $X \to pt$ to the [[point]] is proper, while $X$ is overt if and only if the unique map $X \to pt$ is open.  Similarly, if $X\to pt$ is instead [[closed map|closed]], then $X$ is [[covert space|covert]].

Note that overtness is expressed only in terms of a [[left adjoint]], whereas open maps of locales must additionally satisfy a [[Frobenius reciprocity]] condition.  In the case of locale maps to the point, this latter condition is automatic.

An overt space may also be called **locally $(-1)$-connected**, since this condition is the [[(0,1)-topos]]-theoretic version of the notion of [[locally connected topos]] and [[locally n-connected (n,1)-topos]].  A similar thing happens for higher local connectivity involving the Frobenius reciprocity condition, which must be imposed on general [[geometric morphisms]] to make them locally $n$-connected, but is automatic for morphisms to the point.

## In synthetic topology

In [[synthetic topology]], we interpret 'space' to mean simply 'set' (or [[type]], i.e. the basic objects of our foundational system).  If the notion of 'open' is specified by a [[dominance]], then there is an induced nontrivial notion of "overt set", defined essentially as above.  For instance, the [[Rosolini dominance]] is the smallest dominance such that $\mathbb{N}$ is overt, whereas if all sets are overt then the dominance is trivial.

On the other hand, if by 'open' we mean an open subset [in the sense of Penon](/nlab/show/open+subspace#SynTop), then all sets are overt.

## References

The notion of an open [[locale]] was originally introduced by Joyal and Tierney (and developed by Johnstone in [[Stone Spaces]]):

* [[Andre Joyal]], [[Myles Tierney]], _An extension of the Galois theory of Grothendieck_,  Mem. Amer. Math. Soc. 51 (1984), no. 309, vii+71 pp.

The term "overt" is due to [[Paul Taylor]].  For example, it appears in:

* [[Paul Taylor]], _A lambda calculus for real analysis_, Journal of Logic and Analysis 2(5), pp.1-115, 2010. ([web](http://www.paultaylor.eu/ASD/lamcra/))

Some of the history is described in the introduction to:

* [[Bas Spitters]], _Locatedness and Overt Sublocales_, Annals of Pure and Applied Logic, 162:1, October 2010, Pages 36–54. ([arxiv](http://arxiv.org/abs/math/0703561))

[[!redirects overt space]]
[[!redirects overt spaces]]
[[!redirects open space]]
[[!redirects open spaces]]

[[!redirects locally (-1)-connected space]]
[[!redirects locally (-1)-connected spaces]]
[[!redirects locally (-1)-connected locale]]
[[!redirects locally (-1)-connected locales]]

[[!redirects overt]]
[[!redirects overtness]]


overtone series

> This entry is about a notion in [[physics]] and [[music]]. For the related notion in [[mathematics]] see at *[[harmonic series]]*.

***

\tableofcontents

\section{Idea}

In [[acoustics]], [[signal processing]], and [[music]], an **overtone series** or **harmonic series** is a [[sequence]] of all [[harmonics]] of a [[periodic function|periodic]] [[waveform]].

\section{See also}

* [[pure tone]]

* [[harmonic]]

* [[sequence]]

\section{References}

See also:

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


Owen Biesel

category: people


Owen Gwilliam

* [website](http://people.mpim-bonn.mpg.de/gwilliam/)

## Selected writings

On [[factorization algebras]] and [[BV-quantization]]:

* [[Kevin Costello]], Owen Gwilliam, _[[Factorization algebras in perturbative quantum field theory]]_

* Owen Gwilliam, _Factorization algebras and free field theories_, PhD thesis ([[GwilliamThesis.pdf:file]])

On a kind of [[BV-quantization]] of the [[Loday-Quillen-Tsygan theorem]] and relating to the [[large N limit|large $N$-limit]] of [[Chern-Simons theory]]:

* [[Grégory Ginot]], [[Owen Gwilliam]], [[Alastair Hamilton]], [[Mahmoud Zeinalian]], _Large $N$ phenomena and quantization of the Loday-Quillen-Tsygan theorem_, Adv. Math. __409A__ (2022) 108631 [[arXiv:2108.12109](https://arxiv.org/abs/2108.12109), [doi:10.1016/j.aim.2022.108631](https://doi.org/10.1016/j.aim.2022.108631)]

On the [[CS-WZW correspondence]] via [[factorization algebras]]:

* {#GwilliamRabinovichWilliams2022} [[Owen Gwilliam]], [[Eugene Rabinovich]], [[Brian R. Williams]], *Factorization algebras and abelian CS/WZW-type correspondences*, Pure and Applied Mathematics Quarterly **18** 4 (2022) 1485–1553  [[arXiv:2001.07888](https://arxiv.org/abs/2001.07888), [doi:10.4310/PAMQ.2022.v18.n4.a7](https://dx.doi.org/10.4310/PAMQ.2022.v18.n4.a7)]

## Related entries

* [[factorization algebra of observables]]

* [[BV-quantization]]

* [[Feynman diagram]]

category: people


p-adic AdS-CFT > history

> see at _[[p-adic AdS/CFT correspondence]]_


p-adic AdS/CFT correspondence

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## Idea

A version of the [[AdS/CFT correspondence]] (specifically: [[AdS3/CFT2 duality]]) for [[p-adic physics]], obtained by passing on the [[conformal field theory]]-side from the [[complex geometry]], hence (via [[GAGA]])[[algebraic geometry]] of [[2d CFT]] to [[arithmetic geometry]] over [[p-adic numbers]], and on the [[string theory]]-side to [[p-adic string theory]] (?)

Here the [[arithmetic geometry]]-version of the [[bulk]] [[AdS spacetime]] is identified with the [[Bruhat-Tits tree]] for the [[projective general linear group]] $PGL(2,\mathbb{Q}_p)$:

> graphics from [Casselman 14](Bruhat-Tits+tree#Casselman14)

This may be regarded (at some finite depth truncation) as a [[tensor network state]]:

> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

(As in [HMSS 16](#HMSS16), [HLM 19](#HLM19). But  maybe one wants the Poincaré-dual networks, instead, as in [HMPS 18](#HMPS18)?)

There is the suggestion that [[BTZ black holes]] are encoded by networks that look like [[Bruhat-Tits trees]] towards the boundaries and like [[matrix product states]] towards the interior:

(As in [ESZ 19](p-adic+AdS/CFT+correspondence#ESZ19).)

## Related concepts

* [[holographic entanglement entropy]]

* [[tensor network]]

* [[number theory and physics]]

  * [[p-adic physics]]

  * [[p-adic string theory]]

  * [[arithmetic Chern-Simons theory]]

## References

Suggestion to identify the [[Bruhat-Tits tree]] $T_p$ with [[anti de Sitter spacetime]] in the [[p-adic AdS/CFT correspondence]]:

* [[Steven Gubser]], Johannes Knaute, Sarthak Parikh, Andreas Samberg, Przemek Witaszczyk, _$p$-adic AdS/CFT_, Communications in Mathematical Physics volume 352, pages 1019–1059 (2017) ([arXiv:1605.01061](https://arxiv.org/abs/1605.01061))

* [[Steven Gubser]], Sarthak Parikh, _Geodesic bulk diagrams on the Bruhat-Tits tree_ ([arXiv:1704.01149](https://arxiv.org/abs/1704.01149))

Relation to [[tensor networks]]:

* {#HMSS16} [[Matthew Heydeman]], [[Matilde Marcolli]], [[Ingmar Saberi]], [[Bogdan Stoica]], _Tensor networks, $p$-adic fields, and algebraic curves: arithmetic and the $AdS_3/CFT_2$ correspondence_ ([arXiv:1605.07639](https://arxiv.org/abs/1605.07639))

* {#BHLL17} Arpan Bhattacharyya, Ling-Yan Hung, Yang Lei, Wei Li, _Tensor network and ($p$-adic) AdS/CFT_, JHEP 1801 (2018) 139 ([arXiv:1703.05445](https://arxiv.org/abs/1703.05445))

* {#HLM19} Ling-Yan Hung, Wei Li, Charles M. Melby-Thompson, _$p$-adic CFT is a holographic tensor network_ ([arXiv:1902.01411](https://arxiv.org/abs/1902.01411))

* Lin Chen, Xirong Liu, Ling-Yan Hung, _Emergent Einstein Equation in p-adic CFT Tensor Networks_ ([arXiv:2102.12022](https://arxiv.org/abs/2102.12022))

* Lin Chen, Xirong Liu, Ling-Yan Hung, _Bending the Bruhat-Tits Tree I:Tensor Network and Emergent Einstein Equations_ ([arXiv:2102.12023](https://arxiv.org/abs/2102.12023))

* Lin Chen, Xirong Liu, Ling-Yan Hung, _Bending the Bruhat-Tits Tree II: the p-adic BTZ Black hole and Local Diffeomorephism on the Bruhat-Tits Tree_ ([arXiv:2102.12024](https://arxiv.org/abs/2102.12024))

* Feng Qu, _The effective field theory on a finite boundary of the Bruhat-Tits tree_ ([arXiv:2103.02882](https://arxiv.org/abs/2103.02882))

Including [[spinors]]:

* [[Steven Gubser]], Christian Jepsen, Brian Trundy, _Spin in $p$-adic AdS/CFT_, Journal of Physics A: Mathematical and Theoretical, Volume 52, Number 14 ([arXiv:1811.02538](https://arxiv.org/abs/1811.02538))

Including [[BTZ black holes]]:

* {#HMPS18} [[Matthew Heydeman]], [[Matilde Marcolli]], Sarthak Parikh, [[Ingmar Saberi]], _Nonarchimedean Holographic Entropy from Networks of Perfect Tensors_ ([arXiv:1812.04057](https://arxiv.org/abs/1812.04057))

* {#ESZ19} Stephen Ebert, Hao-Yu Sun, Meng-Yang Zhang, _Probing holography in $p$-adic CFT_ ([arXiv:1911.06313](https://arxiv.org/abs/1911.06313))

An expository account is given in

* [[Ingmar Saberi]], _Holography and local fields_, ([arXiv:1801.04942](https://arxiv.org/abs/1801.04942))

Proposed realization of aspects of [[p-adic AdS/CFT correspondence]] in [[solid-state physics]]:

* Gregory Bentsen, Tomohiro Hashizume, Anton S. Buyskikh, Emily J. Davis, Andrew J. Daley, [[Steven Gubser]], Monika Schleier-Smith, _Treelike interactions and fast scrambling with cold atoms_, Phys. Rev. Lett. 123, 130601 (2019) ([arXiv:1905.11430](https://arxiv.org/abs/1905.11430))

See also:

* Feng Qu, *$P$-adic AdS/CFT on subspaces of the Bruhat-Tits tree* [[arXiv:2402.03730](https://arxiv.org/abs/2402.03730)]

[[!redirects p-adic AdS/CFT duality]]

[[!redirects p-adic AdS-CFT]]


p-adic cohomology

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#### Cohomology
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## Idea

The term _$p$-adic cohomology_ would broadly refer to [[cohomology]] over [[spaces]] in [[p-adic geometry]].

More specifically it is used for [[crystalline cohomology]] in positive [[characteristic]],
based on the [[comparison theorem (crystalline cohomology)|comparison theorem]] which says that this is equivalent to [[de Rham cohomology]] in [[p-adic geometry]].

Notice that by "$p$-adic cohomology" one could also mean [[cohomology]] with [[coefficients]] in [[p-adic numbers]]. An example of this is [[ℓ-adic cohomology]] which has [[p-adic number|ℓ-adic]] [[coefficients]], genuinely so over the [[pro-étale site]]). But not in all cases does this give the expected answers.

## References

* [[Kiran Kedlaya]], _$p$-adic cohomology_ ([arXiv:math/0601507](http://arxiv.org/abs/math/0601507))