Barr's theorem

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## Idea

**Barr's theorem** was originally conjectured by [[William Lawvere]] as an infinitary generalization of the [[Deligne completeness theorem]] for [[coherent toposes]] which can be expressed as the existence of a surjection $\mathcal{S}/K\to\mathcal{E}$ for a coherent topos $\mathcal{E}$ with set of points $K$. General toposes $\mathcal{E}$ may fail to have [[enough points]] but [[Michael Barr]] showed that a surjection from a suitable [[Boolean topos]] still exists.

## Statement

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

If $\mathcal{E}$ is a [[Grothendieck topos]], then there is a  [[surjective geometric morphism]]

$$
  \mathcal{F} \to \mathcal{E}
$$

where $\mathcal{F}$ satisfies the [[axiom of choice]].

=--

## Remark

[[Deligne completeness theorem|Deligne's completeness theorem]] says that a coherent Grothendieck topos has enough points in $Set$ and this corresponds to the Gödel-Henkin completeness theorem for first-order theories. Similarly, Barr's theorem can interpreted as saying that a Grothendieck topos has sufficient _Boolean-valued_ points and is in turn closely related to Mansfield's **Boolean-valued completeness theorem** for infinitary first-order theories.

## Constructive proof and classical detour

Let $\mathbb{T}$ be a [[geometric theory]] and $U_\mathbb{T}$ its [[classifying topos|universal model]]. Recall that $U_\mathbb{T}$ represents deducibility in geometric logic in the sense that a geometric sequent $ \sigma$ is deducible from $\mathbb{T}$ precisely iff $U_\mathbb{T}\models \sigma $.

Suppose that $M=f^*(U_\mathbb{T})$ is a $\mathbb{T}$-model in a topos $\mathcal{E}$ where $f:\mathcal{E}\to Set[U_\mathbb{T}]$ is a [[surjective geometric morphism]] to the [[classifying topos]] of $\mathbb{T}$ and let $ (\varphi \vdash_{\vec{x}} \psi) $ be a geometric sequent such that $M\models (\varphi \vdash_{\vec{x}} \psi) $. Now by the definition of the satisfaction relation, this is the same as to say that the monomorphism $\{\vec{x}.\varphi\wedge\psi\}_M\hookrightarrow\{\vec{x}.\varphi\}_M$ is an isomorphism but, $f$ being surjective, $f^*$ is [[conservative functor|conservative]] whence $U_\mathbb{T}\models (\varphi \vdash_{\vec{x}} \psi) $ and, accordingly, $(\varphi \vdash_{\vec{x}} \psi)$ is deducible from $\mathbb{T}$ in geometric logic.

This together with the existence of a Boolean cover assured by Barr's theorem implies the following important

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

Let $\mathbb{T}$ be a [[geometric theory]] and $\sigma$ a geometric sequent that holds in all $\mathbb{T}$-models in Boolean toposes. Then $\sigma$ is deducible from $\mathbb{T}$ in geometric logic.

=--

In other words, if a statement in [[geometric logic]] is deducible from a [[geometric theory]] using classical [[logic]] and the [[axiom of choice]], then it is also deducible from it in [[constructive mathematics]]. Unfortunately, the proof of Barr's theorem itself is highly non-constructive whence is of no direct help in finding such constructive replacements for classical proofs of geometric statements.

## Related entries

* [[Deligne completeness theorem]]

* [[Boolean topos]]

## Link

* MO discussion on topos without points: _[Topos Without point, from the point of view of logic.](http://mathoverflow.net/questions/98729/topos-without-point-from-the-point-of-view-of-logic)_

## References

* [[Michael Barr|M. Barr]], _Toposes without points_ , JPAA **5** (1974) pp.265-280. ([preprint](http://www.math.mcgill.ca/barr/papers/top.no.pt.pdf)) doi:[10.1016/0022-4049(74)90037-1](http://dx.doi.org/10.1016/0022-4049(74%2990037-1)

Extensive discussion of the context of Barr's theorem is in chapter 7 of:

* [[P. T. Johnstone]], _Topos Theory_ , Academic Press New York 1977 (Dover reprint 2014).

A proof sketch and a survey of its model-theoretic context is in

* [[Gonzalo Reyes|Gonzalo E. Reyes]], _Sheaves and concepts: A model-theoretic interpretation of Grothendieck topoi_ , Cah. Top. Diff. Géo. **Cat. XVIII** no.2 (1977) pp.405-437. ([numdam](http://www.numdam.org/item?id=CTGDC_1977__18_2_105_0))

For a discussion of the importance of the corollary in constructive algebra see also

* [[Gavin Wraith]], _Intuitionistic algebra: some recent developments in topos theory_  In: Proceedings of the International Congress of Mathematicians (Helsinki, 1978), pages 331–337, Helsinki, 1980. Acad. Sci. Fennica. ([pdf](http://www.mathunion.org/ICM/ICM1978.1/Main/icm1978.1.0331.0338.ocr.pdf))
 {#Wraith}

For proof-theoretic approaches to Barr's theorem see

* Sara Negri, _Contraction-free sequent calculi for geometric theories with an
application to Barr's theorem_ , Archive for Mathematical Logic **42** (2003) pp.389–401.

For the connection with the Boolean-valued completeness theorem see also

* R. Goldblatt, _Topoi - The Categorical Analysis of Logic_ , North-Holland 1982². (Dover reprint New York 2006; [project euclid](http://projecteuclid.org/euclid.bia/1403013939))

* R. Mansfield, _The Completeness Theorem for Infinitary Logic_ , JSL **37** no.1 (1972) pp.31-34.

* [[Michael Makkai]], [[Gonzalo E. Reyes]], _First-order Categorical Logic_ , LNM **611** Springer Heidelberg 1977.


Barr-exact category > history

< [[Barr-exact category]]
[[!redirects Barr-exact category -- history]]


Barratt-Eccles operad

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## Idea

The _Barratt-Eccles operad_ $\mathcal{E}$ is a specific realization of an [[E-infinity operad]], a cofibrant [[resolution]] of the [[commutative operad]] (in the [[model structure on operads]]).

As a [[topological operad]] it is given by $\mathcal{E}_n := E \Sigma_n$, the [[universal principal bundle]] for the [[symmetric group]] $\Sigma_n$. As an [[sSet]]-operad it has $\mathcal{E}_n = N(\Sigma_n // \Sigma_n)$, the [[nerve]] of the [[action groupoid]] of $\Sigma_n$ acting on itself.

## Definition

### As a simplicial operad

We give the definition of the Barratt-Eccles operad as an object of the category of multi-colored [[symmetric operad|symmetric]] [[simplicial operads]] ([[sSet]]-[[enriched category|enriched]] [[symmetric multicategories]]). (See [Berger-Fress, section 1.1.5 ](#BerFre01).)

The **Barratt-Eccles operad** $\mathcal{E}$ is the operad defined as follows.

It has a single color.

For $n  \in \mathbb{N}$, its [[simplicial set]] $\mathcal{E}(n)$ of $n$-ary operations is the [[nerve]] $N(\Sigma_n//\Sigma_n)$ of the [[action groupoid]] $\Sigma_n//\Sigma_n$ of the [[symmetric group]] $\Sigma_n$ permuting $n$ elements acting by right multiplication on itself.

$$
  \mathcal{E}(n) := N(\Sigma_n // \Sigma_n)
  \,.
$$

Explicitly, this is the [[simplicial set]] whose $k$-cells are $(k+1)$-tuples of group elements

$$
  \mathcal{E}(n)_k =_{iso} (\Sigma_n)^{\times k+1}
  \,.
$$

Regarded as the nerve of the action groupoid, the face maps on $\mathcal{E}(n)$ are given by multiplication in $\Sigma_n$

$$
  d_i (g_0, g_1, \cdots, g_k)
  =
  g_0, \cdots, g_{i-1}, g_i \cdot g_{i+1}, g_{i+2}, \cdots, g_k
  \;\;\;\;\;
  0 \leq i \leq k
  \,.
$$

But, alternatively, we can parameterize $\mathcal{E}(n)_k$ by the tuples

$$
  (w_0, w_1, \cdots, w_k) :=
  (g_0, g_0 \cdot g_1, g_0 \cdot g_1 \cdot g_2,
  \cdots, \prod_{i=0}^{k} g_i)
  \,.
$$

In terms of this the $i$th face map is given simply by omitting the $i$th entry

$$
  d_i(w_0, \cdots, w_k) = (w_0, \cdots, \hat w_i, \cdots, w_k)
  \,.
$$

The $i$th degeneracy map is given by repeating the $i$th entry

$$
  s_i (w_0, \cdots, w_n)
  :=
  (w_0, \cdots, w_{i-1}, w_i, w_i, w_{i+1}, \cdots, w_n)
  \;\;\;
  0 \leq i \leq n
  \,.
$$

In terms of this, the $\Sigma_n$-[[action]] on $\mathcal{E}(n)$ (giving the structure of a [[symmetric operad]]) is then the [[diagonal]] action

$$
  \sigma \cdot (w_0, w_1, \cdots, w_k)
  :=
  (\sigma \cdot w_0, \sigma \cdot w_1, \cdots, \sigma \cdot w_k)
  \,.
$$

The composition operations in the operad

$$
  \mathcal{E}(r) \times (\mathcal{E}(n_1) \times \cdots \times \mathcal{E}(n_r))
  \to
  \mathcal{E}(n_1 + \cdots + n_r)
$$

are the morphisms of simplicial sets which in degree $k$ are maps on tuples, which in each degree $i$ are given by the natural function

$$
  \Sigma_r \times (\Sigma_{n_1} \times \cdots \times \Sigma_{n_r})
  \to
  \Sigma_{n_1 + \cdots + n_r}
$$

that composes $r$ permutations with a permutation of $r$ elements to a permutation of $\sum_{i = 0}^r n_r$ elements.

(This function is in fact that which gives the composition in [[Assoc]] when regarded as a [[symmetric operad]], $Assoc := Symm(*)$.)

### As a dg-operad

(...)

See ([Berger-Fresse](#BerFre01)).

## Properties

### As a simplicial operad

Each of the simplicial sets $\mathcal{E}(n)$ for $n \in \mathbb{N}$ is [[contractible]]. One way to see this is to observe that $\Sigma_n // \Sigma_n$ is (the [[nerve]] of) the [[pullback]]

$$
  \array{
     \Sigma_n // \Sigma_n
     &\to&
     *
     \\
     \downarrow && \downarrow
     \\
     (\mathbf{B} \Sigma_n)^I &\to& \mathbf{B}\Sigma_n
  }
  \,,
$$

where $\mathbf{B}\Sigma_n$ is one-object groupoid with $\Sigma_n$ as its morphisms, $(\mathbf{B}\Sigma_n)^I$ is its [[arrow category]] and the bottom vertical map is evaluation at the source. Since this is an acyclic fibration, so is the top vertical morphism.

It follows that the canonical morphism of simplicial operads

$$
  \mathcal{E} \to Comm
$$

to the [[commutative operad]] (which has $Comm(n) = *$ for all $n \in \mathbb{N}$) is a weak equivalence (in the [[model structure on operads]]). In fact, it is a cofibrant resolution.

### As a dg-operad

(...)

## Related concepts

* [[E-infinity algebra]]

* [[(2,1)-algebraic theory of E-infinity algebras]]

## References

As a [[simplicial operad]], the Barratt-Eccles operad was introduced in

* [[M. Barratt]], [[P. Eccles]], _On $\Gamma^+$-structures I. A free group functor for stable homotopy theory_, Topology 13 (1974), 25-45.

Its realization as a [[dg-operad]] is discussed in detail in

* C. Berger, _Combinatorial models for real configuration spaces and $E_n$-operads_, [pdf](http://math.unice.fr/~cberger/config.pdf)
* [[Clemens Berger]], [[Benoit Fresse]] _Combinatorial operad actions on cochains_, Math. Proc. Cambridge Philos. Soc. 137 (2004), 135-174, [arXiv:math.AT/0109158](http://arxiv.org/abs/math/0109158), [doi](http://10.1017/S0305004103007138)
{#BerFre01}

* [[Clemens Berger]], [[Benoit Fresse]], _A prismatic decomposition of the Barratt-Eccles operad_, [math.AT/0204326](http://arxiv.org/abs/math/0204326)

Elmendorf and Mandell show that the Barratt-Eccles operad $E\Sigma_*$ is obtained by applying functor $E$ to certain operad $\Sigma_*$ in

* A.D. Elmendorf, M.A. Mandell, _Rings, modules, and algebras in infinite loop space theory_, Advances in Mathematics __205__, n.1, 2006, pp 163–228, [arxiv](http://front.math.ucdavis.edu/0403.5403), [doi](http://dx.doi.org/10.1016/j.aim.2005.07.007)

[[!redirects Barratt-Eccles operads]]


barreled topological vector space

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## Idea

Barreled spaces are [[topological vector spaces]] for which the theorem of [[Banach-Steinhaus theorem|Banach–Steinhaus]] is valid. This theorem says, roughly, that for a set of continuous linear maps $L(E, F)$ from a barreled space $E$ to a [[locally convex space|locally convex]] [[TVS]] boundedness in the topology of pointwise convergence implies boundedness in the topology of bounded convergence.

## Definition

A subset $T \subset E$ of a [[TVS]] E is a **barrel** if it is

* absorbing

* balanced

* closed

* convex

A [[TVS]] $E$ is **barreled** (or __barrelled__) if every barrel is a neighborhood of zero.

Sometimes [[locally convex]] is included in the definition, this is not implied by barreled as defined above, i.e. there are barreled spaces that are not locally convex.

In the definition of **quasibarreled** or **infrabarreled** the barrels are replaced by sets that are barrels and which absorb all bounded sets (sets with the latter property are also called **bornivorous**).

## Properties

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

A [[locally convex space|locally convex]] [[TVS]] which is a [[Baire space]] is barreled.
=--

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

A [[locally convex space|locally convex]] [[TVS]] is barreled iff its topology is the [[strong topology]].
=--

## Examples

Since all [[locally convex space|locally convex]] [[TVS]]es that are [[Baire space|Baire spaces]] are barreled, the examples naturally include [[Fréchet spaces]], [[Banach spaces]] and [[Hilbert spaces]].

## References

See the [[functional analysis bibliography]].

The definition of quasibarreled is from

* S.M. Khaleelulla: _Counterexamples in Topological Vector Spaces_.

It is called _infrabarreled_ in

* H.H. Schaefer: _Topological vector spaces_.

[[!redirects barreled]]
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[[!redirects Barrelled spaces]]

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[[!redirects barrelled topological vector space]]
[[!redirects Barrelled topological vector space]]
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[[!redirects Barrelled topological vector spaces]]

[[!redirects quasibarreled space]]
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[[!redirects Quasibarrelled spaces]]

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[[!redirects Quasibarreled topological vector space]]
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[[!redirects Quasibarreled topological vector spaces]]
[[!redirects quasibarrelled topological vector space]]
[[!redirects Quasibarrelled topological vector space]]
[[!redirects quasibarrelled topological vector spaces]]
[[!redirects Quasibarrelled topological vector spaces]]

[[!redirects infrabarreled space]]
[[!redirects Infrabarreled space]]
[[!redirects infrabarreled spaces]]
[[!redirects Infrabarreled spaces]]
[[!redirects infrabarrelled space]]
[[!redirects Infrabarrelled space]]
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[[!redirects Infrabarreled topological vector space]]
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Barry Fawcett

* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=37865)

## Selected writings

On [[completely distributive lattices]]:

* {#WoodFawcett} [[Barry Fawcett]], [[Richard J. Wood]], *Constructive complete distributivity I* , Math. Proc. Camb. Phil. Soc. **107** (1990) 81-89 [[doi:10.1017/S0305004100068377](https://doi.org/10.1017/S0305004100068377)]

category: people

[[!redirects B. Fawcett]]


Barry Kurtz

* [personal page](https://cs.appstate.edu/~blk/)

## Selected writings

On [[syntax]] and [[semantics]] of [[programming languages]]:

* [[Kenneth Slonneger]], [[Barry Kurtz]], *Formal Syntax and Semantics of Programming Languages*, Addison-Wesley (1995) [[webpage](https://homepage.divms.uiowa.edu/~slonnegr/plf/Book/), [pdf](https://doc.lagout.org/science/Artificial%20Intelligence/Natural%20Language%20Processing/Formal%20Syntax%20and%20Semantics%20of%20Programming%20Languages%20-%20Kenneth%20Slonneger.pdf)]

## Related entries

* [[denotational semantics]]

category: people


Barry Mazur

Barry Charles Mazur is a mathematician at Harvard University.

* [website](http://www.math.harvard.edu/~mazur/)

## Selected writings

Early occurence of the argument that later came to be known as the *[[Eilenberg swindle]]*:

* {#Mazur61} [[Barry C. Mazur]], _On embeddings of spheres_,  Acta Mathematica **105** 1–2 (1961) 1–17   [[doi:10.1007/BF02559532](https://doi.org/10.1007%2FBF02559532)]

Introducing [[Artin-Mazur formal groups]]:

* {#ArtinMazur77} [[Michael Artin]], [[Barry Mazur]], _Formal Groups Arising from Algebraic Varieties_, Annales scientifiques de l'École Normale Supérieure, Sér. 4, 10 no. 1 (1977), p. 87-131  ([numdam:ASENS_1977_4_10_1_87_0](http://www.numdam.org/item?id=ASENS_1977_4_10_1_87_0), [MR56:15663](http://www.ams.org/mathscinet-getitem?mr=56:15663))

On [[elliptic curves]] over general [[commutative rings]] (in [[arithmetic geometry]]):

* [[Nicholas Katz]], [[Barry Mazur]], _Arithmetic moduli of elliptic curves, Annals of Mathematics Studies_, vol. 108, Princeton University Press, 1985 ([jstor:j.ctt1b9s05p](https://www.jstor.org/stable/j.ctt1b9s05p))

[[!redirects B. Mazur]]
[[!redirects Barry C. Mazur]]
[[!redirects Barry Charles Mazur]]

category: people


Barry Mitchell

Barry Mitchell (1933-2021) was an American mathematician who wrote one of the earliest books on category theory (1965).

He proved what is now known as the *[[Freyd-Mitchell embedding theorem]]*, which shows that any [[abelian category]] has an exact embedding into a [[category of modules]].

* [Local newspaper obituary](https://www.legacy.com/obituaries/dailyfreeman/obituary.aspx?n=barry-m-mitchell&pid=197672237&fhid=3910)

## Selected writings

Introducing the [[Mitchell embedding theorem]]:

* [[Barry Mitchell]], *The Full Imbedding Theorem*, American Journal of Mathematics **86** 3 (1964) 619-637 [[doi:10.2307/2373027](https://doi.org/10.2307/2373027)]

On [[category theory]]:

* [[Barry Mitchell]], *Theory of categories*, Pure and Applied Mathematics **17**, Academic Press (1965) [[ISBN:978-0-12-499250-4](https://www.elsevier.com/books/theory-of-categories/mitchell/978-0-12-499250-4)]

See also:

* [[Barry Mitchell]], *The dominion of Isbell*, Transactions of the American Mathematical Society **167** (1972) 319-331 [[doi:10.2307/1996142](https://doi.org/10.2307/1996142)]

## Related entries

* [[Mitchell embedding theorem]]

* [[semicategory]]


Barry Simon

* [webpage](http://www.math.caltech.edu/people/simon.html)

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

## Selected writings

On [[mathematical physics]]:

* [[Michael Reed]] and [[Barry Simon]]: *[[Methods of Modern Mathematical Physics]]*, Academic Press (1978)

On the [[Berry phases]] as the [[holonomy]] of [[connections]] on [[fiber bundles]]:

* {#Simon83} [[Barry Simon]], *Holonomy, the Quantum Adiabatic Theorem, and Berry's Phase*, Phys. Rev. Lett. **51** (1983) 2167 ([doi:10.1103/PhysRevLett.51.2167](https://doi.org/10.1103/PhysRevLett.51.2167))

category: people


Bart Jacobs

* [papers](http://www.cs.ru.nl/B.Jacobs/PAPERS/index.html)

## Selected writings

On [[dependent type theory]] and its [[categorical semantics]] ([[categorical semantics of dependent types]]):

* [[Bart Jacobs]], *Comprehension categories and the semantics of type dependency*, Theoret. Comput. Sci. **107** 2 (1993) 169-207 [[MR1201808](http://www.ams.org/mathscinet-getitem?mr=1201808), doi:10.1016/0304-3975(93)90169-T]

* [[Bart Jacobs]], *Categorical Logic and Type Theory*, Studies in Logic and the Foundations of Mathematics **141**, Elsevier (1998)  [[ISBN:978-0-444-50170-7](https://www.sciencedirect.com/bookseries/studies-in-logic-and-the-foundations-of-mathematics/vol/141), [pdf](https://people.mpi-sws.org/~dreyer/courses/catlogic/jacobs.pdf), [webpage](http://www.cs.ru.nl/B.Jacobs/CLT/bookinfo.html)]

On [[category theory|category theoretic]] [[probability theory]] (cf. *[[convex powerset of distributions monad]]*):

* [[Bart Jacobs]], *Coalgebraic trace semantics for combined possibilitistic and probabilistic systems*, Proceedings of CMCS 2008, ENTCS **203** 5 (2008) 131-152 [[doi:10.1016/j.entcs.2008.05.023](https://doi.org/10.1016/j.entcs.2008.05.023)]

On [[CD-categories]] ([[gs-monoidal categories]]):

* Kenta Cho, [[Bart Jacobs]], _Disintegration and Bayesian inversion via string diagrams_, Mathematical Structures in Computer Science **29** 7 (2019) 938-971 [[arXiv:1709.00322](https://arxiv.org/abs/1709.00322), [doi:10.1017/S0960129518000488](https://doi.org/10.1017/S0960129518000488)]

## Related pages

* [[definability (fibred category theory)]]

category: people


Bartek

I am a PhD student at the University of Auckland, working on [[Cartan geometry]], and more specifically [[Parabolic geometry]].


Bartel Leendert van der Waerden

Bartel Leendert van der Waerden was a mathematician at the universities of Groningen, Leipzig, and Zürich.

He got his PhD degree in 1926 at the University of Amsterdam, advised by Hendrick de Vries.

## Selected works

* [[Moderne Algebra]]. Teil I, Die Grundlehren der mathematischen Wissenschaften, vol. 33, Berlin, New York: Springer-Verlag, 1930, ISBN 978-3-540-56799-8

* [[Moderne Algebra]]. Teil II, Die Grundlehren der mathematischen Wissenschaften, vol. 34, Springer-Verlag, 1931, ISBN 978-3-540-56801-8

Based on lectures by [[Emil Artin]] and [[Emmy Noether]],
it was the first textbook using the modern approach to [[algebra]].

category: reference


Bartlomiej Czech

* [spire page](https://inspirehep.net/authors/1024822)

* [institute page](https://www.ias.tsinghua.edu.cn/en/info/1059/1176.htm)

## Selected writings

On [[quantum information theory]] in view of [[AdS-CFT duality]] ([[holographic entanglement entropy]]):

* [[Bowen Chen]], [[Bartlomiej Czech]], [[Zi-zhi Wang]], *Quantum Information in Holographic Duality*, Rept. Prog. Phys. **85** (2022) 4, 046001 [[arXiv:2108.09188](https://arxiv.org/abs/2108.09188), [doi:10.1088/1361-6633/ac51b5](https://doi.org/10.1088/1361-6633/ac51b5)]

category: people

[[!redirects Bart Czech]]


Bartomeu Monserrat

* [institute page](https://www.msm.cam.ac.uk/people/monserrat)

* [Monserrat Lab](https://www.tcm.phy.cam.ac.uk/~bm418/)

## Selected writings

On [[braid group statistics|anyonic braiding]] of nodal points in the [[Brillouin zone]] of [[semi-metals]] ("braiding in momentum space"):

* Siyu Chen, [[Adrien Bouhon]], [[Robert-Jan Slager]], [[Bartomeu Monserrat]], *Non-Abelian braiding of Weyl nodes via symmetry-constrained phase transitions* (formerly: *Manipulation and braiding of Weyl nodes using symmetry-constrained phase transitions*),
Phys. Rev. B **105** (2022) L081117 $[$[arXiv:2108.10330](https://arxiv.org/abs/2108.10330), [doi:10.1103/PhysRevB.105.L081117](https://doi.org/10.1103/PhysRevB.105.L081117)$]$

* [[Bo Peng]], [[Adrien Bouhon]], [[Robert-Jan Slager]], [[Bartomeu Monserrat]], *Multi-gap topology and non-Abelian braiding of phonons from first principles*, Phys. Rev. B **105** (2022) 085115 ([arXiv:2111.05872](https://arxiv.org/abs/2111.05872), [doi;10.1103/PhysRevB.105.085115](https://doi.org/10.1103/PhysRevB.105.085115))

* [[Bo Peng]], [[Adrien Bouhon]], [[Bartomeu Monserrat]], [[Robert-Jan Slager]], *Phonons as a platform for non-Abelian braiding and its manifestation in layered silicates*, Nature Communications volume 13, Article number: 423 (2022) ([doi:10.1038/s41467-022-28046-9](https://doi.org/10.1038/s41467-022-28046-9))

category: people


Barton Zwiebach

* [website](http://web.mit.edu/physics/people/faculty/zwiebach_barton.html)

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

## Selected writings

On [[bosonic string|bosonic]] [[closed string|closed]] [[string field theory]] and introducing its relation to [[L-infinity algebras|$L_\infty$-algebras]]:

* [[Barton Zwiebach]], *Issues In Covariant Closed String Theory*, pp 192-200 in: *Proceedings of [10th and Final Workshop on Grand Unification](http://inspirehep.net/record/966930)*, 20-22 Apr 1989. Chapel Hill, North Carolina [[spire:282685](http://inspirehep.net/record/282685)]

* [[Barton Zwiebach]], *Closed string field theory: Quantum action and the Batalin-Vilkovisky master equation*, Nucl. Phys. B **390** 1 (1993) 33-152 [[arXiv:hep-th/9206084](http://arxiv.org/abs/hep-th/9206084), doi:10.1016/0550-3213(93)90388-6]

On [[string junctions]] in relation to [[Lie algebras]] and [[Lie algebra representations]]:

* Oliver DeWolfe, [[Barton Zwiebach]], _String Junctions for Arbitrary Lie Algebra Representations_,  Nucl. Phys. B541 (1999) 509-565 ([arXiv:hep-th/9804210](https://arxiv.org/abs/hep-th/9804210))

* Oliver DeWolfe, Tamas Hauer, [[Amer Iqbal]], [[Barton Zwiebach]], _Uncovering the Symmetries on $[p,q]$ 7-branes: Beyond the Kodaira Classification_, Adv. Theor. Math. Phys. 3 (1999) 1785-1833 ([arXiv:hep-th/9812028](https://arxiv.org/abs/hep-th/9812028))

## Related entries

* [[string field theory]]

category: people


Bartosz Milewski

### Contact ###
bartosz@relisoft.com

### Publications ###
* [Bartosz Milewski's Programming Cafe](http://bartoszmilewski.com).
* [Profunctor optics: a categorical update](https://arxiv.org/abs/2001.07488)
* [Category Theory for Programmers](https://github.com/hmemcpy/milewski-ctfp-pdf)

### Selected writings

On [[categorical semantics]] of [[functional programming languages]] (such as for [[monads in computer science]]):

* [[Bartosz Milewski]] (compiled by Igal Tabachnik), *Category Theory for Programmers*, Blurb (2019) [[pdf](https://github.com/hmemcpy/milewski-ctfp-pdf/releases/download/v1.3.0/category-theory-for-programmers.pdf), [github](https://github.com/hmemcpy/milewski-ctfp-pdf), [webpage](https://bartoszmilewski.com/2014/10/28/category-theory-for-programmers-the-preface/), [ISBN:9780464243878](https://www.blurb.com/b/9621951-category-theory-for-programmers-new-edition-hardco)]

* [[Bartosz Milewski]], *The Dao of Functional Programming* (2023) [[pdf](https://github.com/BartoszMilewski/Publications/blob/master/TheDaoOfFP/DaoFP.pdf), [github](https://github.com/BartoszMilewski/Publications/tree/master/TheDaoOfFP)]

On [[optics (in computer science)]]:

* [[Bryce Clarke]], Derek Elkins, [[Jeremy Gibbons]], [[Fosco Loregian]], [[Bartosz Milewski]], Emily Pillmore, [[Mario Román]], _Profunctor optics, a categorical update_, 2020. ([arXiv:2001.07488](https://arxiv.org/abs/2001.07488))

category: people


Baruch Spinoza

+-- {: .rightHandSide}
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### Context
#### Philosophy
+-- {: .hide}
[[!include philosophy - contents]]
=--
=--
=--

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

* [Stanford Encycl. of Phil. entry](http://plato.stanford.edu/entries/spinoza/)

## Hegel on Spinoza

From [[Georg Hegel]], _[Lectures on the history of Philosophy --  Spinoza](https://www.marxists.org/reference/archive/hegel/works/hp/hpspinoz.htm)_:

> The philosophy of Descartes underwent a great variety of unspeculative developments, but in Benedict Spinoza a direct successor to this philosopher may be found, and one who carried on the Cartesian principle to its furthest logical conclusions. For him soul and body, thought and Being, cease to have separate independent existence. The dualism of the Cartesian system Spinoza, as a Jew, altogether set aside. For the profound unity of his philosophy as it found expression in Europe, his manifestation of Spirit as the identity of the finite and the infinite in God, instead of God's appearing related to these as a Third — all this is an echo from Eastern lands. The Oriental theory of absolute identity was brought by Spinoza much more directly into line, firstly with the current of European thought, and then with the European and Cartesian philosophy, in which it soon found a place.

> First of all we must, however, glance at the circumstances of Spinoza's life. He was by descent a Portuguese Jew, and was born at Amsterdam in the year 1632; the name he received was Baruch, but he altered it to Benedict. In his youth he was instructed by the Rabbis of the synagogue to which he belonged, but he soon fell out with them, their wrath having been kindled by the criticisms which he passed on the fantastic doctrines of the Talmud. He was not, therefore, long in absenting himself from the synagogue, and as the Rabbis were in dread lest his example should have evil consequences, they offered him a yearly allowance of a thousand gulden if he would keep away from the place and hold his tongue. This offer he declined; and the Rabbis thereafter carried their persecution of him to such a pitch that they were even minded to rid themselves of him by assassination. After having made a narrow escape from the dagger, he formally withdrew from the Jewish communion, without, however, going over to the Christian Church. He now applied himself particularly to the Latin language, and made a special study of the Cartesian philosophy. Later on he went to Rhynsburg, near Leyden, and from the year 1664 he lived in retirement, first at Voorburg, a village near the Hague, and then at the Hague itself, highly respected by numerous friends: he gained a livelihood for himself by grinding optical glasses. It was no arbitrary choice that led him to occupy himself with light, for it represents in the material sphere the absolute identity which forms the foundation of the Oriental view of things. Although he had rich friends and mighty protectors, among whom even generals were numbered, he lived in humble poverty, declining the handsome gifts offered to him time after time. Nor would he permit Simon von Vries to make him his heir; he only accepted from him an annual pension of three hundred florins; in the same way he gave up to his sisters his share of their father's estate. From the Elector Palatine, Carl Ludwig, a man of most noble character and raised above the prejudices of his time, he received the offer of a professor's chair at Heidelberg, with the assurance that he would have liberty to teach and to write, because "the Prince believed he would not put that liberty to a bad use by interfering with the religion publicly established." Spinoza (in his published letters) very wisely declined this offer, however, because "he did not know within what limits that philosophic liberty would have to be confined, in order that he might not appear to be interfering with the publicly established religion." He remained in Holland, a country highly interesting in the history of general culture, as it was the first in Europe to show the, example of universal toleration, and afforded to many a place of refuge where they might enjoy liberty of thought; for fierce as was the rage of the theologians there against Bekker, for example (Bruck. Hist. crit. phil. T. IV. P. 2, pp. 719, 720), and furious as were the attacks of Voetius on the Cartesian philosophy, these had not the consequences which they would have had in another land. Spinoza died on the 21st of February, 1677, in the forty-fourth year of his age. The cause of his death was consumption, from which he had long been a sufferer; this was in harmony with his system of philosophy, according to which all particularity and individuality pass away in the one substance. A Protestant divine, Colerus by name, who published a biography of Spinoza, inveighs strongly against him, it is true, but gives nevertheless a most minute and kindly description of his circumstances and surroundings — telling how he left only about two hundred thalers, what debts he had, and so on. A bill included in the inventory, in which the barber requests payment due him by M. Spinoza of blessed memory, scandalizes the parson very much, and regarding it he makes the observation: "Had the barber but known what sort of a creature Spinoza was, he certainly would not have spoken of his blessed memory." The German translator of this biography writes under the portrait of Spinoza: characterem reprobationis in vultu gerens, applying this description to a countenance which doubtless expresses the melancholy of a profound thinker, but is otherwise wild and benevolent. The reprobatio is certainly correct; but it is not a reprobation in the passive sense; it is an active disapprobation on Spinoza's part of the opinions, errors and thoughtless passions of mankind.(1)

> Spinoza used the terminology of Descartes, and also published an account of his system. For we find the first of Spinoza's works entitled "An Exposition according to the geometrical method of the principles of the Cartesian philosophy." Some time after this he wrote his Tractatus theologico-politicus, and by it gained considerable reputation. Great as was the hatred which Spinoza roused amongst his Rabbis, it was more than equalled by the odium which he brought upon himself amongst Christian, and especially amongst Protestant theologians — chiefly through the medium of this essay. It contains his views on inspiration, a critical treatment of the books of Moses and the like chiefly from the point of view that the laws therein contained are limited in their application to the Jews. Later Christian theologians have written critically on this subject, usually making it their object to show that these books were compiled at a later time, and that they date in part from a period subsequent to the Babylonian captivity; this has become a crucial point with Protestant theologians, and one by which the modern school distinguishes itself from the older, greatly pluming itself thereon. All this, however, is already to be found in the above-mentioned work of Spinoza. But Spinoza drew the greatest odium upon himself by his philosophy proper, which we must now consider as it is given to us in his Ethics. While Descartes published no writings on this subject, the Ethics of Spinoza is undoubtedly his greatest work; it was published after his death by Ludwig Mayer, a physician, who had been Spinoza's most intimate friend. It consists of five parts; the first deals with God (De Deo). General metaphysical ideas are contained in it, which include the knowledge of God and nature. The second part deals with the nature and origin of mind (De natura et origine mentis). We see thus that Spinoza does not treat of the subject of natural philosophy, extension and motion at all, for he passes immediately from God to the philosophy of mind, to the ethical point of view; and what refers to knowledge, intelligent mind, is brought forward in the first part, under the head of the principles of human knowledge. The third book of the Ethics deals with the origin and nature of the passions (De oriqine et natura affectuum); the fourth with the powers of the same, or human slavery (De servitute humana seu de affectuum viribus); the fifth, lastly, with the power of the understanding, with thought, or with human liberty (De potentia intellectus seu de libertate humana). (2) Kirchenrath Professor Paulus published Spinoza's works in Jena; I had a share in the bringing out of this edition, having been entrusted with the collation of French translations.

## related entries

* [[Spinoza's system]]

category: people

[[!redirects Spinoza]]


barycenter

##Barycenter of a simplex
+--{: .un_defn}
######Definition
If $\sigma = \{ v_0, \ldots, v_q\} \in K_q$, the set of $q$-simplices of a [[simplicial complex]], $K$, then its **barycentre**, $b(\sigma)$, is the point

$$b(\sigma) = \sum_{0\leq i \leq q}\frac{1}{q + 1} v_i \in |K|.$$
=--
For the use of barycenters in the **barycentric subdivision**, see [[classical triangulation]] or

* wikipedia: [barycentric subdivision](http://en.wikipedia.org/wiki/Barycentric_subdivision), [barycentric coordinates](http://en.wikipedia.org/wiki/Barycentric_coordinates_%28mathematics%29)

* [barycentric coordinates homepage](http://www.inf.usi.ch/hormann/barycentric)

* Wolframworld: [[barycentric coordinates](http://mathworld.wolfram.com/BarycentricCoordinates.html)


baryogenesis

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

#Contents#
* table of contents
{:toc}
## Idea

In [[cosmology]] and [[particle physics]],  by _baryogenesis_ one refers to the process by which an abundance of [[baryons]] is produced in the early [[observable universe]], leading to the baryonic [[matter]] seen all around. Since in the [[standard model of particle physics]] [[matter]] and [[antimatter]] are _essentially_ always produced symmetrically ("conservation of baryon number"), the topic of baryogenesis is to a large extent the study of how the matter/antimatter symmetry may be [[broken symmetry|broken]].

([Sakharov 67](#Sakharov67)) formulated three conditions jointly necessary for matter/antimatter asymmetry leading to baryogenesis in the early universe, now known as the _Sakharov conditions_

1. violation of baryon number conservation;

1. [[CP problem|C and CP violation]]

1. departure from thermal equilibrium.

Indeed, by a [[quantum anomaly]] in the [[standard model of particle physics]], the _[[axial anomaly]]_ (a [[chiral anomaly]]) (see e.g. [Jackiw 08](#Jackiw08)),  the [[divergence]] of the [[baryon current]] (the chiral version of the [[Dirac current]]) does not quite vanish but is ([t'Hooft 76](#Hooft76)) in traditional physics notation

$$
  \partial_\mu j_b^\mu
   =
  \partial_\mu j_L^\mu
   =
  n_f
  \left(
    \frac{g^2}{32 \pi^2}
    W_{\mu \nu}^\alpha \tilde W^{\alpha \mu \nu}
    -
    \frac{{g'}^2}{32\pi^2} \epsilon_{\mu \nu \rho \sigma} F^{\mu \nu} F^{\rho \sigma}
  \right)
 \,,
$$

where $F_\nabla$ is the [[field strength]] of the [[electroweak field]] (the [[curvature]] of a $SU(2)$-[[principal connection]] $\nabla$).

(see  e.g. [Dine 90, (4)](#Dine90) [Trodden-Carroll, 4.9.3](#TroddenCarroll), [Shu 08, (7.6)](#Shu08)).

In more intrinsic notation and ignoring numerical constants, the statement is that

$$
  \mathbf{d} \star j_b = \langle F_\nabla \wedge F_\nabla \rangle
  \,.
$$

Properly interpreted (see at _[Yang-Mills instanton -- from the correct maths to the traditional physics story](Yang-Mills+instanton#FromTheMathsToThePhysicsStory)_) this is the _local_ expression for the [[connection on a 3-bundle|3-connection]] on the [[Chern-Simons line 3-bundle]] associated with the gauge field. If the latter is globally nontrivial in that it is in an _[[instanton]]_-sector (has nontrivial [[second Chern class]]), then the [[integral]] of $\langle F\wedge F\rangle$ over closed manifolds is an integer -- the "[[instanton]] number" -- and in conclusion there is baryon generation proportional to this number (physics lingo also: _sphaleron transitions_).

Notice that this would mean that baryons "are" a kind of topological twist, different from, but not entirely unlike to the old idea of _[[On Vortex Atoms]]_. See there at _[Similarlity to concepts of modern particle physics](On+Vortex+Atoms#SimilarityToParticlePhysics)_

That this process was at least one source of baryogenesis in the early universe is plausible but not certain. The process is heavily suppressed by inverse energy/temperature (e.g. [Dine 90, around (9)](#Dine90)) and out of reach of conceivable experiments in the present age of the universe, but plausibly may have occurred in the very early universe -- just as it should be for realistic baryogenesis.

## Exposition
 {#Exposition}

One needs to be careful that the concept of "particle" itself is only well defined on Minkowski spacetime, hence not on cosmological scales. But nevertheless there is a fermion current 3-form $J$, such that its integral

$$
  Q_\Sigma \coloneqq   \int_\Sigma J \in \mathbb{R}
$$

has the interpretation of the total net _charge_ carried by the stuff measured by $J$, hence counts the net  "fermion number" in all of space" as seen by $\Sigma$.

This is familiar from, say, Maxwell's equations, where such a $J$ appears on the right hand side as the electromagnetic source form

$$
  d  \star F = J
  \,.
$$

Now given any such current 3-form, then its de Rham differential $d J$ measures how the charges $Q$ changes as time progresses. Because let $\Sigma_1$ and $\Sigma_2$ be two spatial slices of spacetimes, and $X_4$ a piece of spacetime connecting them, i.e. such that

$$
  \partial X_4 = \Sigma_2 - \Sigma_1
  \,.
$$

Then [[Stokes's theorem]] gives that $d J$ is the local change of charge, because:

$$
  \begin{aligned}
    Q_{\Sigma_2} - Q_{\Sigma_1}
    &=
    \int_{\Sigma_2} J - \int_{\Sigma_1} J
    \\
    &=
    \int_{\partial X_4} J
    \\
    & =
     \int_{X_4} d J
  \end{aligned}
$$

So if $J$ is closed, then total charge is conserved, while if $J$ is not closed, then $d J $ is the local measure for how charge is being "created" or how it disappears.

Now the remarkable fact is that in the standard model of particle physics, $J$ comes out non-closed. Its differetial instead comes out proportial to the 4-form which measures instanton number

$$
  d J \propto \langle F_\nabla \wedge F_\nabla \rangle
  \,.
$$

This is the famous _[[chiral anomaly]]_.

So far this is highlighted in every textbook. But the following further crucial subtlety tends not to be recognized for what it is.

Namely on cosmological spacetimes that carry [[instantons]], then (as discussed at _[Yang-Mills instanton -- from the correct maths to the traditional physics story](Yang-Mills+instanton#FromTheMathsToThePhysicsStory)_) the 4-form $\langle F_\nabla \wedge F_\nabla\rangle$ is not in fact globally exact. The above formulas hold only locally, on a chart of spacetime. But on intersections two such pieces of data need to be glued by a gauge transformation. If we do make the usual simple assumptions (for simplicity of the discussion), then this gauge transformation is that famous "Chern-Simons winding number" $S^3 \to SU(2)$, which they keep handing the physics students without properly explaining it (such as the one who I am reacting to [here](http://physicsoverflow.org/38243/topological-number-integral-theory-boundary-volume-forms)).

As a result, in the presence of instantons, the integral of $\langle F_{\nabla} \wedge F_{\nabla}\rangle$ may be non-zero on a _closed_ cosmological spacetime.

The usual picture (which you see displayed in many popular accounts) is: imagine a 4d cup $D^4$ (a unit disk thought of as a "cup" cobordism from nothing to $S^3$) where the big bang expands spacetime from nothing. Then glue on something like $S^3 \times LongInterval$ and think of the result as being a simple model for the universe. Then assume some boundary condition saying that far in the future from the big bang the gauge fields $\nabla$ that carry those instantons decay away, hence are [[vanishing at infinity]]. So then for computing the total fermion charge in this universe, we are effectively dealing with its one-point compactification. In the present simple example this is the 4-sphere $S^4$.

So since the four form vanishes "at infinity", we may just as well assume that it is supported on that cup $D^4$ in our model, the neighbourhood of the big bang.

We learn that that:

1. The total net fermion charge in the universe is $\int_{S^4} \langle F_\nabla \wedge F_\nabla\rangle \in \mathbb{Z}$;

1. this fermion number is picked up incrementally increasing from zero (at the origin of our $D^4$-cup, the "big bang singularity) and reaching at "comoving time $t \lt 1$" the value

$$
   Q_t
   =
   \int_{D^4_{t}} \langle F_\nabla \wedge F_\nabla\rangle
   =
   \int_{S^3} J_t
$$

(where $D^4_t$ is the disk of radius $t \lt 1$).

Then at $t = 1$ (in this paramneterization) the four form goes to zero (and in this fashion eventually extends to a global 4-form on our one-point compactified cosmological spacetime).

So after the comoving time $t = 1$ there is no net particle creation anymore. Moreover, the net particle number picked up until then is equal to the second Chern class of the cosmological gauge field, hence equal to the number that masure the "knottedness" of the cosmological gauge field (which we here think of as all being concentrated around the big big).

One particle per knottedness of the cosmological gauge field.

## Related concepts

* [[leptogenesis]]

* [[nucleosynthesis]]

* [[On Vortex Atoms]]

* [[comoving time]]

## References

The original article stating the _Sakharov conditions_ is

* {#Sakharov67} [[Andrei Sakharov]], _Violation of CP invariance, C asymmetry, and baryon asymmetry of the universe_. Journal of Experimental and Theoretical Physics 5: 24–27. 1967, republished as A. D. Sakharov (1991). _Violation of CP invariance, C asymmetry, and baryon asymmetry of the universe_. Soviet Physics Uspekhi 34 (5): 392–393. Bibcode:1991SvPhU..34..392S. doi:[10.1070/PU1991v034n05ABEH002497](http://dx.doi.org/10.1070/PU1991v034n05ABEH002497). 1967

reviewed in

* [[Dennis Perepelitsa]], _Sakharov Conditions for Baryogenesis_, 2008 ([pdf](http://phys.columbia.edu/~dvp/dvp-sakharov.pdf))

The derivation that [[instantons]] lead to baryon number violation is due to

* {#Hooft76} [[Gerard 't Hooft]], _Symmetry Breaking through Bell-Jackiw Anomalies_ Phys. Rev. Lett. 37 (1976) ([pdf](http://www.staff.science.uu.nl/~hooft101/gthpub/symm_br_bell_jackiw.pdf))

* [[Gerard 't Hooft]] _Computation of the quantum effects due to a four-dimensional pseudoparticle_, Phys. Rev. D14:3432-3450 (1976)

A good review of the axial anomaly is in

* {#Jackiw08} [[Roman Jackiw]], _Axial anomaly_, (2008), Scholarpedia, 3(10):7302. ([web](http://www.scholarpedia.org/article/Axial_anomaly))

and of the resulting [[phenomenology]] of baryon number non-conservation is in

* {#Dine90} [[Michael Dine]], _Baryon number violation at high-energy in the standard model: Fact or fiction?_ Sep 1990, ([spire](https://inspirehep.net/record/299206?ln=de), [pdf](http://www.slac.stanford.edu/cgi-wrap/getdoc/ssi90-020.pdf))

Review:

* Heidi Kuismanen, section 2 of _Leptogenesis as the origin of matter-antimatter asymmetry in extra dimensionaL and supersymmetric modeLs_ ([[KuismanenBaryogenesis.pdf:file]])

* V. A. Rubakov, [[Mikhail Shaposhnikov]], *Electroweak Baryon Number Non-Conservation in the Early Universe and in High Energy Collisions*, Usp. Fiz. Nauk **166** (1996) 493-537 Phys. Usp. 39  (1996) 461-502 7lbrack;[arXiv:hep-ph/9603208](https://arxiv.org/abs/hep-ph/9603208)]

* Antonio Riotto, Mark Trodden, _Recent Progress in Baryogenesis_, Ann.Rev.Nucl.Part.Sci.49:35-75,1999 ([arXiv:hep-ph/9901362](http://arxiv.org/abs/hep-ph/9901362))

* Laurent Canetti, Marco Drewes, [[Mikhail Shaposhnikov]], _Matter and Antimatter in the Universe_, New J. Phys. 14 (2012) 095012 ([arXiv:1204.4186](https://arxiv.org/abs/1204.4186))

* Yanagida, _The origin of matter_ ([[YanagidaBaryogenesis.pdf:file]])

* {#TroddenCarroll} Mark Trodden, Sean Carroll, TASI Lectures: Introduction to Cosmology _[4.9 Baryon Number violation](http://ned.ipac.caltech.edu/level5/Sept03/Trodden/Trodden4_9.html)_

* James M. Cline, _Baryogenesis_, lectures at Les Houches Summer School, Session 86: Particle Physics and Cosmology: the Fabric of Spacetime, 7-11 Aug. 2006 ([arXiv:hep-ph/0609145](https://arxiv.org/abs/hep-ph/0609145))

* {#Shu08} Jing Shu, section 7 of _Connecting LHC Signals with Deep Physics at the TeV Scale and Baryogenesis_ 2008

Discussion with focus on [[QCD]]-effects:

* Dominik J. Schwarz, _The first second of the Universe_, Annalen Phys.12:220-270, 2003 ([arXiv:astro-ph/0303574](https://arxiv.org/abs/astro-ph/0303574))


baryon

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In the [[standard model of particle physics]] (specifically in [[QCD]]), a _baryon_ is [[bound state]] of three [[quarks]] via the [[strong nuclear force]]. Baryons are the "heavy" types of [[hadrons]], the other being the [[mesons]].

Examples of baryons are the [[nucleons]]: [[protons]] and [[neutrons]]. Other examples are [[Lambda baryons]].

Also [[pentaquarks]] are counted as baryons.

## Properties

### Conceptualization and computation in AdS/QCD
 {#ConceptualizationAndComputationInAdSQCD}

In the [[Witten-Sakai-Sugimoto model]] for [[non-perturbative effect|strongly coupled]] [[QCD]] via an [[intersecting D-brane model]], the [[hadrons]] in [[QCD]] correspond to [[string theory|string-theoretic]]-phenomena in an ambient [[bulk field theory]] on an approximately [[anti de Sitter spacetime]]:

1. the [[mesons]] ([[bound states]] of 2 [[quarks]]) correspond to [[open strings]] in the bulk, whose two endpoints on the [[asymptotic boundary]] correspond to the two [[quarks]];

1. [[baryons]] ([[bound states]] of $N_c$ [[quarks]]) appear in two different but equivalent ([Sugimoto 16, 15.4.1](#Sugimoto16)) guises:

   1. as [[wrapped brane|wrapped]] [[D4-branes]] with $N_c$ [[open strings]] connecting them to the [[D8-brane]]

      ([Witten 98b](#Witten98b), [Gross-Ooguri 98](#GrossOoguri98))

   1. as [[skyrmions]]

      ([Sakai-Sugimoto 04, section 5.2](#SakaiSugimoto04), [Sakai-Sugimoto 05, section 3.3](#SakaiSugimoto05), see [Bartolini 17](#Bartolini17)).

For review see [Sugimoto 16](#Sugimoto16), also [Rebhan 14, around (18)](#Rebhan14).

> graphics grabbed from [Sugimoto 16](#Sugimoto16)

This produces [[baryon]] [[mass]] spectra with moderate quantitative agreement with [[experiment]] ([HSSY 07](#HSSY07)):

> graphics grabbed from [Sugimoto 16](#Sugimoto16)

## Related concepts

* [[constituent quark]]

* [[diquark]]

* [[quantum hadrodynamics]]

* [[hadron supersymmetry]], [[hadron Kaluza-Klein theory]]

  * [[flavor brane]]

* [[Walecka model]]

* [[baryogenesis]]

* [[skyrmion]]

* [[baryon-lepton symmetry]]

## References

### General

Introduction and review:

* [[Franz Gross]], [[Eberhard Klempt]] et al., Chapter 9 of: *50 Years of Quantum Chromodynamics*, EJPC [[arXiv:2212.11107](https://arxiv.org/abs/2212.11107)]

Baryons as 3-[[constituent quark]] [[bound states]]:

* Gernot Eichmann, Helios Sanchis-Alepuz, Richard Williams, Reinhard Alkofer, Christian S. Fischer, _Baryons as relativistic three-quark bound states_, Progress in Particle and Nuclear Physics Volume 91, November 2016, Pages 1-100 ([arXiv:1606.09602](https://arxiv.org/abs/1606.09602), [doi:10.1016/j.ppnp.2016.07.001](https://doi.org/10.1016/j.ppnp.2016.07.001))

Baryons as [[quark]]/[[diquark]] [[bound states]]:

* Martin Oettel, _Baryons as Relativistic Bound States of Quark and Diquark_ ([arXiv:nucl-th/0012067](https://arxiv.org/abs/nucl-th/0012067), [spire:538966](https://inspirehep.net/literature/538966))

Experiment:

* Eberhard Klempt, Jean-Marc Richard, _Baryon spectroscopy_, Rev. Mod. Phys. 82:1095-1153, 2010 ([arXiv:0901.2055](https://arxiv.org/abs/0901.2055)

See also

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

* Aarts, _Baryons at finite temperature_ ([[AartsBaryonsAtFiniteTemperature.pdf:file]])

[[!include baryon chiral perturbation theory -- references]]

### Baryons as Skyrmions

The [[Skyrmion]]-model for baryons (see there for more references):

* {#Weigel96} [[Herbert Weigel]], _Baryons as Three Flavor Solitons_,  	Int. J. Mod. Phys. A11:2419-2544, 1996 ([arXiv:hep-ph/9509398](https://arxiv.org/abs/hep-ph/9509398), [cds:288541](http://cds.cern.ch/record/288541), [doi:10.1142/S0217751X96001218](https://doi.org/10.1142/S0217751X96001218))

* {#Weigel08} [[Herbert Weigel]], _Chiral Soliton Models for Baryons_,  Lecture Notes in Physics book series, volume 743, Springer 2008 ([doi:10.1007/978-3-540-75436-7](https://doi.org/10.1007/978-3-540-75436-7))

* {#RhoZahed16} [[Mannque Rho]], [[Ismail Zahed]] (eds.) _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))

[[!include hadrons as KK-modes of 5d Yang-Mills theory -- references]]

### In the large $N$ limit

In the [[large N limit]]:

* {#Witten79} [[Edward Witten]], _Baryons in the $1/n$ Expansion_, Nucl. Phys. B160 (1979) 57-115 ([spire:140391](http://inspirehep.net/record/140391), doi:10.1016/0550-3213(79)90232-3)

[[!redirects baryons]]


baryon chiral perturbation theory -- references

### Baryon chiral perturbation theory
 {#BaryonChiralPerturbationTheoryReferences}

Discussion of [[baryon chiral perturbation theory]], i.e of [[chiral perturbation theory]] with explicit [[effective field theory|effective]] (as opposed to or in addition to implicit [[Skyrmion|skyrmionic]]) [[baryon]] [[field (physics)|fields]] included (see also _[[Walecka model]]_ and _[[quantum hadrodynamics]]_):

Review:

* [[Ulf-G. Meissner]], _Chiral QCD: Baryon dynamics_, in:  At The Frontier of Particle Physics, pp. 417-505 (2001) ([arxiv:hep-ph/0007092](https://arxiv.org/abs/hep-ph/0007092))

* [[Véronique Bernard]], _Chiral Perturbation Theory and Baryon Properties_, Prog. Part. Nucl. Phys. 60:82-160, 2008 ([arXiv:0706.0312](https://arxiv.org/abs/0706.0312))

* [[Stefan Scherer]], _Baryon chiral perturbation theory_, PoS CD09:075, 2009 ([arXiv:0910.0331](https://arxiv.org/abs/0910.0331))

Original articles:

* Elizabeth Jenkins, Aneesh V. Manohar, _Baryon chiral perturbation theory using a heavy fermion lagrangian_, Physics Letters B Volume 255, Issue 4, 21 February 1991, Pages 558-562 (doi:10.1016/0370-2693(91)90266-S)

* Robert Baur, Joachim Kambor, _Generalized Heavy Baryon Chiral Perturbation Theory_, Eur. Phys. J. C7:507-524, 1999 ([arXiv:hep-ph/9803311](https://arxiv.org/abs/hep-ph/9803311))

Higher order terms:

* José Antonio Oller, Michela Verbeni, Joaquim Prades, _Meson-baryon effective chiral Lagrangians to $\mathcal{O}(q^3)$_,  Journal of High Energy Physics, Volume 2006, JHEP09(2006) ([arXiv:hep-ph/0608204](https://arxiv.org/abs/hep-ph/0608204), [doi:10.1088/1126-6708/2006/09/079](https://iopscience.iop.org/article/10.1088/1126-6708/2006/09/079))

* Matthias Frink, [[Ulf-G. Meissner]], _On the chiral effective meson-baryon Lagrangian at third order_, Eur. Phys. J. A29:255-260, 2006 ([arXiv:hep-ph/0609256](https://arxiv.org/abs/hep-ph/0609256))

* Jose Antonio Oller, Joaquim Prades, Michela Verbeni, _Meson-Baryon Effective Chiral Lagrangians at $\mathcal{O}(q^3)$ Revisited_ ([arXiv:hep-ph/0701096](https://arxiv.org/abs/hep-ph/0701096), [spire:742291](https://inspirehep.net/literature/742291))

See also:

* Lisheng Geng, _Recent developments in $SU(3)$ covariant baryon chiral perturbation theory_, Front. Phys., 2013, 8(3): 328-348 ([arXiv:1301.6815](https://arxiv.org/abs/1301.6815))


baryon current

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[conserved current]] whose [[charge]] is the [[number]] of [[baryons]].

(Not actually conserved, generally, in the [[standard model of particle physics]], due to the [[chiral anomaly]]. See also st _[[baryogenesis]]_.)

## Related concepts

* [[Dirac current]]

## References

### General

(...)

[[!include WZW term of QCD chiral perturbation theory -- references]]

[[!redirects baryon currents]]


baryon number

## References

See also:

* Wikipedia, _[Baryon number](https://en.m.wikipedia.org/wiki/Baryon_number)


baryon-lepton symmetry

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[nuclear physics]], the term _baryon-lepton symmetry_ ([Gamba-Marshak-Okubo 59](#GambaMarshakOkubo59)) refers to the observation that [[interactions]] involving the [[weak nuclear force]] are close to invariant under joint exchange of the [[light baryons|light]] [[baryons]] ([[protons]], [[neutrons]]) with the "[[light leptons|light]] [[leptons]]" ([[electron]] and its [[neutrino]]), i.e. within the [[first generation of fermions]].

## Related concepts

* [[hadron supersymmetry]]

## References

The original article:

* {#GambaMarshakOkubo59} A. Gamba, [[Robert Marshak]], S. Okubo, _On a symmetry in weak interactions_, PNAS June 1, 1959 45 (6) 881-885 ([doi:10.1073/pnas.45.6.881](https://doi.org/10.1073/pnas.45.6.881))

Review:

* [[Robert Marshak]], _Yukawa Meson, Sakata Model and Baryon-Lepton Symmetry Revisited_, Progress of Theoretical Physics Supplement, Volume 85, May 1985, Pages 61–74 ([dpi:10.1143/PTP.85.61](https://doi.org/10.1143/PTP.85.61))

Further developments:

* Rabindra N. Mohapatra, _From Old Symmetries to New Symmetries: Quarks, Leptons and B-L_, International Journal of Modern Physics AVol. 29, No. 29, 1430066 (2014) ([arXiv:1409.7557](https://arxiv.org/abs/1409.7557))


Bas Spitters

Bas Spitters is professor for [[formal logic]] and [[computer science]] at Aarhus University and leader of the *Concordium Blockchain Research Center*

* [website](http://www.cs.au.dk/~spitters/)

* [Concordium page](https://cs.au.dk/research/centers/concordium/about)

## Selected writings

On [[constructive mathematics|constructive]] [[integration]] theory:

* [[Bas Spitters]], *Constructive algebraic integration theory*, Annals of Pure and Applied Logic, Volume 137, Issues 1–3, January 2006, Pages 380-390 ([pdf](https://www.cs.au.dk/~spitters/ftop2.pdf), [doi:10.1016/j.apal.2005.05.031](https://doi.org/10.1016/j.apal.2005.05.031))

* [[Bas Spitters]], *Constructive algebraic integration theory without choice*. In [[Thierry Coquand]], [[Henri Lombardi]], Marie-Francoise Roy, Mathematics, Algorithms, Proofs, 2005, Dagstuhl Seminar Proceedings 05021, Internationales Begegnungs- und Forschungszentrum (IBFI), Schloss Dagstuhl, Germany. ([pdf](https://www.cs.au.dk/~spitters/obs.pdf))

On [[constructive analysis]] with exact [[real numbers]] via [[type theory]]:

* [[Herman Geuvers]], [[Milad Niqui]], [[Bas Spitters]], [[Freek Wiedijk]], _Constructive analysis, types and exact real numbers_, Mathematical Structures in Computer Science **17** 01 (2007) 3-36 [[doi:10.1017/S0960129506005834](https://doi.org/10.1017/S0960129506005834)]

and specifically with [[ufias2012:Type classes]] in [[Coq]]:

* {#KrebbersSpitters11} [[Robbert Krebbers]], [[Bas Spitters]], *Type classes for efficient exact real arithmetic in Coq*, Logical Methods in Computer Science, **9** 1 (2013) lmcs:958 [[arXiv:1106.3448](http://arxiv.org/abs/1106.3448/), doi:10.2168/LMCS-9(1:1)2013]

Exposition:

* [[Bas Spitters]], *Verified Implementation of Exact Real Arithmetic in Type Theory*, talk at *Computable Analysis and Rigorous Numeric* (2013) [[pdf](https://users-cs.au.dk/spitters/CARN.pdf), [[Spitters-ExactRealArithmetic.pdf:file]]]

On [[Bohr toposes]]:

* {#HeunenLandsmanSpitters09} [[Chris Heunen]], [[Klaas Landsman]], [[Bas Spitters]], *Bohrification of operator algebras and quantum logic*, Synthese **186** 3 (2012) 719-752 [[arXiv:0905.2275](http://arxiv.org/abs/0905.2275), [doi;10.1007/s11229-011-9918-4](https://doi.org/10.1007/s11229-011-9918-4)]

On [[mathematical structures]] formulated in [[dependent type theory]], specifically via [[ufias2012:Type classes]] in [[Coq]]:

* [[Bas Spitters]], Eelis van der Wegen *Type classes for mathematics in type theory*, Mathematical Structures in Computer Science **21** 4 "Interactive Theorem Proving and the Formalisation of Mathematics" (2011) 795-825 [[doi:10.1017/S0960129511000119](https://doi.org/10.1017/S0960129511000119)]

On [[modalities]] in [[homotopy type theory]] ([[modal homotopy type theory]]):

* {#RSS} [[Egbert Rijke]], [[Mike Shulman]], [[Bas Spitters]], *Modalities in homotopy type theory*,  Logical Methods in Computer Science, **16** 1 (2020) [[arXiv:1706.07526](https://arxiv.org/abs/1706.07526), [episciences:6015](https://lmcs.episciences.org/6015), doi:10.23638/LMCS-16(1:2)2020]

On [[software verification]] for [[blockchain]]-technology:

* [[Bas Spitters]], *[Formal verificaton](https://cs.au.dk/research/centers/concordium/research-areas/formal-verification)* at *[Concordium Blockchain Research Center](https://cs.au.dk/research/centers/concordium/)*

* {#ThomsenSpitters20} Søren Eller Thomsen, [[Bas Spitters]], *Formalizing Nakamoto-Style Proof of Stake* [[eprint:2020/917](https://eprint.iacr.org/2020/917)]

category: people

[[!redirects Bas Spitters]]
[[!redirects BasSpitters]]


Bas van Fraassen

**Bastiaan Cornelis van Fraassen** is a distinguished [[philosophy of science|philosopher of science]].

* [homepage](http://www.basvanfraassen.org/)

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

## References

* {#vanFraasen80} [[Bas van Fraassen]], _The Scientific Image_ (1980) ([philpapers:VANTSI](https://philpapers.org/rec/VANTSI),\linebreak [doi:10.1093/0198244274.001.0001/acprof-9780198244271](https://www.oxfordscholarship.com/view/10.1093/0198244274.001.0001/acprof-9780198244271))

  (on [[constructive empiricism]])

* B. van Fraassen, _Identity in Intensional Logic: Subjective Semantics_ , pp.201-220 in Eco, Santambrogio, Violi (eds.), _Meaning and Mental Representations_ , Indiana UP Bloomington 1988. ([pdf](http://www.princeton.edu/~fraassen/abstract/SubjSemantics.pdf))

* B. C. van Fraassen, _Laws and Symmetry_ , Oxford UP 1989.

category:people


base

# Bases and subbases
* table of contents
{: toc}

## Idea

For many notions of [[structure]], particularly for [[topological concrete category|topological categories]], one can specify a structure by a _base_ or _subbase_ that generate the structure.  Besides being convenient ways to specify a structure, they may even be necessary when using weak [[foundations]].

+-- {: .num_remark #basevsbasis}
###### Warning

Sometimes one says '[[basis]]' instead of 'base', but I ([[Toby Bartels]]) think that it\'s safest to save the former term for the generating set of a [[free object]] (or an analogous situation), especially in an [[algebraic category]].  Although a basis and a base can both generate something, they tend to do so in very different ways.  (It doesn\'t help that 'bases' is the plural of both, although the pronunciation is different.)
=--

Typically, every structure of an appropriate type is both a base and a subbase for itself, while every base is a subbase.  Bases and subbases can also be characterised independently; every subbase generates a base (which tends to be _saturated_ in some way), while every base (saturated or not) generates a complete structure.

## Examples

### Bases for filters

Recall that a __[[filter]]__ on a [[poset]] $L$ is a [[subset]] $F$ of $L$ such that:

1.  If $x \leq y$ and $x \in F$, then $y \in F$.
2.  Some $x \in F$.
3.  If $x, y \in F$, then for some $z \in F$, $z \leq x, y$.

More generally, any subset $F$ satisfying (2,3) is a __filter base__.

Given a filter base $F$ in a poset, we generate a filter $\overline{F}$ by closing under (1); that is, if $F$ is a filter base on a poset $L$, then
$$ \overline{F} \coloneqq \{ y \;|\; \exists x \in F,\; x \leq y \} $$
is a filter on $L$.

If $L$ is a [[meet]]-[[semilattice]], then we can equivalently define a filter as a subset $F$ such that:

1.  If $x \leq y$ and $x \in F$, then $y \in F$ (same as before).
2.  The [[top element]] $\top \in F$.
3.  If $x, y \in F$, then $x \wedge y \in F$.

Now any subset satisfying (2,3) is a __saturated filter base__, and any subset whatsoever is a __filter subbase__.

Given a filter subbase $F$ in a semilattice, we generate a base $\vec{F}$ by closing under (2,3) in the second list; that is, if $F$ is a filter subbase on a semilattice $L$, then
$$ \vec{F} \coloneqq \{ \bigwedge_{i=1}^n x_i \;|\; x_i \in F \} $$
is a filter base on $L$, which in fact is saturated.  (Note that $\top \in \vec{F}$ follows when $n = 0$.)

Given a filter subbase $F$ in a semilattice, we can generate a filter by first generating a base $\vec{F}$ and then generating a filter $\overline{\vec{F}}$.  Alternatively, we can generate the same filter by closing under (1,2,3) in the first list all at once.  That is, if $F$ is a filter subbase on a semilattice $L$, then
$$ \overline{F} \coloneqq \{ y \;|\; \exists x_1,\ldots,x_n \in F,\; \forall z \leq x_1,\ldots,x_n,\; z \leq y \} $$
is a filter on $L$.  Furthermore, this is the same filter as $\overline{\vec{F}}$.

The [[intersection]] of any family of filters on a semilattice $L$ is a filter; that is, being a filter is a [[Moore closure]] property on subsets of $L$.  The filter generated by a filter subbase $F$ (which is an arbitrary subset of $L$, remember) is the same as the Moore closure of $F$ under this property, that is the intersection of all filters on $L$ that contain $F$.

Unlike filters and filter bases, the concept of filter subbase does not seem to make sense on an arbitrary poset, but only on a semilattice.

### Bases for topologies

Recall that a [[topological space|topology]] on a [[set]] $X$ is a collection $\mathcal{O}$ of [[subsets]] of $X$ such that:

1.  Any [[union]] of elements of $\mathcal{O}$ belongs to $\mathcal{O}$.
2.  $X$ itself belongs to $\mathcal{O}$.
3.  If $U,V \in \mathcal{O}$, then $U \cap V \in \mathcal{O}$.

More generally, any collection $\mathcal{O}$ satisfying (2,3) is a __saturated topological base__, and any collection whatsoever is a __topological subbase__.

A slightly more complicated but equivalent definition of topology is this:

1.  Again, any union of elements of $\mathcal{O}$ belongs to $\mathcal{O}$.
2.  $X$ itself is a union of elements of $\mathcal{O}$.
3.  If $U,V \in \mathcal{O}$, then $U \cap V$ is contained in a union of elements of $\mathcal{O}$.

Now any collection satisfying (2,3) is a __[[basis for a topology|topological base]]__ (not necessarily saturated).

Given a topological subbase $\mathcal{O}$, we generate a base $\vec{\mathcal{O}}$ by closing under (2,3) in the first list; that is, if $\mathcal{O}$ is a topological subbase on a set $X$, then
$$ \vec{\mathcal{O}} \coloneqq \{ \bigcap_{i=1}^n U_i \;|\; U_i \in \mathcal{O} \} $$
is a topological base on $X$, which in fact is saturated.  (Note that $X \in \vec{\mathcal{O}}$ follows when $n = 0$.)

Given a topological base $\mathcal{O}$, we generate a topology $\overline{\mathcal{O}}$ by closing under (1); that is, if $\mathcal{O}$ is a topological base on a set $X$, then
$$ \overline{\mathcal{O}} \coloneqq \{ V \;|\; \forall p \in V,\; \exists U \in \mathcal{O},\; p \in U \subseteq V \} $$
is a topology on $X$.

Given a topological subbase $\mathcal{O}$, we can generate a topology by first generating a base $\vec{\mathcal{O}}$ and then generating a topology $\overline{\vec{\mathcal{O}}}$.  Alternatively, we can generate the same topology by closing under (1,2,3) in the second list all at once.  That is, if $\mathcal{O}$ is a topological subbase on a set $X$, then
$$ \overline{\mathcal{O}} \coloneqq \{ V \;|\; \forall p \in V,\; \exists U_1, \ldots, U_n \in \mathcal{O},\; p \in \bigcap_{i=1}^n U_i \subseteq V \} $$
is a topology on $\mathcal{O}$.  Furthermore, this is the same topology as $\overline{\vec{\mathcal{O}}}$.

As with filters, being a topology is a [[Moore closure]] property, this time on subsets of the [[power set]] $\mathcal{P}X$, and the topology generated by a topological subbase $\mathcal{O}$ is the intersection of all topologies on $X$ that contain $\mathcal{O}$.

Very analogous considerations apply to [[local bases]] for a topology and bases for [[pretopological space|pretopologies]], [[convergence space|convergence structures]], [[gauge space|gauge structures]], [[Cauchy space|Cauchy structures]], etc.

### Bases for uniformities

Uniformities are a little trickier than topologies, at least in the case of subbases.  For now, see the page [[uniform space]] for definitions of base and subbase for a uniformity.

### Bases for $\sigma$-algebras

Recall that a $\sigma$-[[sigma-algebra|algebra]] on a [[set]] $X$ is ...

...

### Bases for Grothendieck coverages

See  [[basis for a Grothendieck topology]].

## General theory

Is there a general theory of bases?  That\'s a good question.  I don\'t know!

Obviously this has something to do with [[Moore closures]] (and hence [[monads]]); generating a structure from a subbase is (often) taking a Moore closure.  But there\'s some particular property of some closure operators that makes the intermediate concept of base work out.

## References

This paper discusses bases in the generality of algebras over a monad:

* [[Stefan Zetzsche]], [[Alexandra Silva]], [[Matteo Sammartino]]: _Generators and bases for algebras over a monad_ (2020), ([arXiv:010.10223](https://arxiv.org/abs/2010.10223))

[[!redirects base]]
[[!redirects subbase]]
[[!redirects saturated base]]


base (infinity,1)-topos

[[!redirects base (∞,1)-topos]]

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

#Contents#
* table of contents
{:toc}

## Idea

In ordinary [[topos theory]] it is common to "work over a fixed [[base topos]]" which may or may not be the canonical choice [[Set]].

Similarily, in [[(∞,1)-topos theory]] one may choose to work over a fixed _base $(\infty,1)$-topos_ $\mathbf{B}$ other than [[∞Grpd]].

Basically this amounts to working not with the [[(∞,1)-category]] [[(∞,1)Topos]], but instead in the [[over-(∞,1)-topos]] $(\infty,1)Topos/\mathbf{B}$.

## Related concepts

* [[base topos]]

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


base change

> This entry is about base change of [[slice categories]]. For base change in [[enriched category theory]] see at [[change of enriching category]].

***

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
#### Topos theory
+--{: .hide}
[[!include topos theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For $f : X \to Y$ a [[morphism]] in a [[category]] $C$ with [[pullbacks]], there is an induced [[functor]]

$$
  f^* : C/Y \to C/X
$$

of [[over-categories]]. This is the _base change_ morphism. If $C$ is a [[topos]], then this refines to an [[essential geometric morphism]]

$$
  (f_! \dashv f^* \dashv f_*) : C/X \to C/Y
  \,.
$$

More generally, such a triple adjunction holds whenever $C$ is [[locally cartesian closed category|locally cartesian closed]], and indeed this [[locally cartesian closed category#DependentProductImpliesLocalCartesinClosure|characterises]] locally cartesian closed categories. The [[duality|dual]] concept is [[cobase change]].

## Definition

### Pullback

For $f : X \to Y$ a [[morphism]] in a [[category]] $C$ with [[pullback]]s, there is an induced [[functor]]

$$
  f^* : C/Y \to C/X
$$

of [[over-categories]].  It is on objects given by [[pullback]]/[[fiber product]] along $f$

$$
  (p : K \to Y)
  \mapsto
  \left(
  \array{
    X \times_Y K &\to & K
    \\
    {}^{\mathllap{p^*}}\downarrow && \downarrow
    \\
    X &\stackrel{f}{\to}& Y
  }
  \right)
  \,.
$$

On morphisms, it follows from the universal property of pullback

$$
  \left\lbrace
    \array{
       K &&\stackrel{g}{\to}&& K'
       \\
       & {}_p \searrow && \swarrow_{p'}
       \\
       && Y
    }
  \right\rbrace
  \mapsto
  \left\lbrace
    \array{
       X \times_Y K &&\stackrel{g^*}{\to}&& X \times_Y K'
       \\
       & {}_{p^*} \searrow && \swarrow_{p'^*}
       \\
       && X
    }
  \right\rbrace
$$

by observing that this square commutes

$$
  \array{
     &&&& X \times_Y K
     \\&
    &&
      {}^{p^*}\swarrow
      && \searrow^{g \circ p_K}
    \\
     && X &&&& K'
      \\
      &
      && {}_f\searrow
       &
       & \swarrow_{p'}
      &&
     \\
     &&&&
     Y
     &&&&
  }
  \,.
$$

### In a fibered category

The concept of base change generalises from this case to other [[fibered category|fibred categories]].

### Base change geometric morphisms
  {#GeometricMorphism}

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

For $\mathbf{H}$ a [[topos]] (or [[(∞,1)-topos]], etc.) $f : X \to Y$ a [[morphism]] in $\mathbf{H}$, then base change induces an [[essential geometric morphism]] between over-toposes/[[over-(∞,1)-topos]]es

$$
  (\sum_f \dashv f^* \dashv \prod_f) : \mathbf{H}/X
   \stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}}
 \mathbf{H}/Y
$$

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

=--

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

That we have [[adjoint functor]]s/[[adjoint (∞,1)-functor]]s $(f_! \dashv f^*)$ follows directly from the universal property of the pullback. The fact that $f^*$ has a further [[right adjoint]] is due to the fact that it preserves all small [[colimit]]s/[[(∞,1)-colimit]]s by the fact that in a topos we have [[universal colimits]] and then by the [[adjoint functor theorem]]/[[adjoint (∞,1)-functor theorem]].

=--

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

The ([[comonad|co-]])[[monads]] induced by the [[adjoint triple]] in prop. \ref{BaseChangeIsEssentialGeometricMorphism} have special names in some contexts:

* $f_\ast f^\ast$ is also called the [[function monad]] (or "[[reader monad]]", see at _[[monad (in computer science)]]_).

* $f_! f^\ast$ is also called the "[[writer comonad]]" (in computer science)

* in [[modal type theory]] $f^\ast f_\ast$ is _[[necessity]]_ while $f^\ast f_!$ is _[[possibility]]_.

=--

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

Here $f^\ast$ is a [[cartesian closed functor]], hence base change of toposes constitutes a cartesian [[Wirthmüller context]].

=--

See at _[[cartesian closed functor]]_ for the proof.

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

$f^*$ is a [[logical functor]]. Hence $(f^* \dashv f_*)$ is also an [[atomic geometric morphism]].

=--

This appears for instance as ([MacLaneMoerdijk, theorem IV.7.2](#MacLaneMoerdijk)).

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

By prop. \ref{BaseChangeIsEssentialGeometricMorphism} $f^*$ is a [[right adjoint]] and hence preserves all [[limit]]s, in particular [[finite limit]]s.

Notice that the [[subobject classifier]] of an [[over topos]] $\mathbf{H}/X$ is $(p_2 : \Omega_{\mathbf{H}} \times X \to X)$. This [[product]] is preserved by the [[pullback]] by which $f^*$ acts, hence $f^*$ preserves the subobject classifier.

To show that $f^*$ is logical it therefore remains to show that it also preserves [[exponential object]]s.  (...)

=--

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

A (necessarily essential and atomic) geometric morphism of the form $(f^* \dashv \prod_f)$ is called the **base change geometric morphism** along $f$.

The [[right adjoint]] $f_* = \prod_f$ is also called the [[dependent product]] relative to $f$.

The [[left adjoint]] $f_! = \sum_f$ is also called the [[dependent sum]] relative to $f$.

In the case $Y = *$ is the  [[terminal object]], the base change geometric morphism is also called an **[[etale geometric morphism]]**. See there for more details

=--

## Properties

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

If $\mathcal{C}$ is a [[locally cartesian closed category]] then for every morphism $f \colon X \to Y$ in $\mathcal{C}$ the [[inverse image]] $f^* \colon \mathcal{C}_{/Y} \to \mathcal{C}_{/X}$ of the base change is a [[cartesian closed functor]].

=--

See at _[cartesian closed functor -- Examples](cartesian%20closed%20functor#Examples)_ for a proof.

## Examples

### Along $\mathbf{B}H \to \mathbf{B}G$
 {#AlongDeloopingsOfGroupHomomorphisms}

For $\mathbf{H}$ an [[(∞,1)-topos]] and $G$ an group object in $\mathbf{H}$ (an [[∞-group]]), then the [[slice (∞,1)-topos]] over its [[delooping]] may be identified with the [[(∞,1)-category]] of $G$-[[∞-actions]] (see there for more):

$$
  Act_G(\mathbf{H}) \simeq \mathbf{H}_{/\mathbf{B}G}
  \,.
$$

Under this identification, then left and right base change long a morphism of the form $\mathbf{B}H \to \mathbf{B}G$ (corresponding to an [[∞-group]] homomorphism $H \to G$) corresponds to forming [[induced representations]] and [[coinduced representations]], respectively.

### Along $\ast \to \mathbf{B}G$
 {#AlongPointInclusionIntoBG}

As the special case of the [above](#AlongDeloopingsOfGroupHomomorphisms) for $H = 1$ the trivial group we obtain the following:

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

Let $\mathbf{H}$ be any  [[(∞,1)-topos]] and let $G$ be a group object in $\mathbf{H}$ (an [[∞-group]]). Then the base change along the canonical point inclusion

$$
  i \;\colon\; \ast \to \mathbf{B}G
$$

into the [[delooping]] of $G$ takes the following form:

There is a pair of [[adjoint ∞-functors]] of the form

$$
  \mathbf{H}
    \underoverset
      { \underset{i_\ast \simeq [G,-]/G}{\longrightarrow}}
      { \overset{i^\ast \simeq hofib}{\longleftarrow}}
      {\bot}
  \mathbf{H}_{/\mathbf{B}G}
  \,,
$$

where

* $hofib$ denotes the operation of taking the [[homotopy fiber]] of a map to $\mathbf{B}G$ over the canonical basepoint;

* $[G,-]$ denotes the [[internal hom]] in $\mathbf{H}$;

* $[G,-]/G$ denotes the [[homotopy quotient]] by the [[conjugation action|conjugation]] [[∞-action]] for $G$ equipped with its canonical [[∞-action]] by left multiplication and the argument
regarded as equipped with its trivial $G$-$\infty$-action

  (for $G = S^1$ then this is the [[cyclic loop space]] construction).

Hence for

* $\hat X \to X$ a $G$-[[principal ∞-bundle]]

* $A$ a [[coefficient]] object, such as for some [[differential cohomology|differential]] [[generalized cohomology theory]]

then there is a [[natural equivalence]]

$$
  \underset{
    \text{original} \atop \text{fluxes}
  }{
  \underbrace{
    \mathbf{H}(\hat X\;,\; A)
  }
  }
    \;\;
    \underoverset
      {\underset{oxidation}{\longleftarrow}}
      {\overset{reduction}{\longrightarrow}}
      {\simeq}
    \;\;
  \underset{
     \text{doubly} \atop { \text{dimensionally reduced} \atop \text{fluxes} }
  }{
  \underbrace{
    \mathbf{H}(X \;,\; [G,A]/G)
  }
  }
$$

given by

$$
  \left(
     \hat X \longrightarrow A
  \right)
  \;\;\;
    \leftrightarrow
  \;\;\;
  \left(
    \array{
       X && \longrightarrow && [G,A]/G
       \\
       & \searrow && \swarrow
       \\
       && \mathbf{B}G
    }
  \right)
$$

=--

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

The statement that $i^\ast \simeq hofib$ follows immediately by the definitions. What we need to see is that the [[dependent product]] along $i$ is given as claimed.

To that end, first observe that the [[conjugation action]] on $[G,X]$ is the [[internal hom]] in
the [[(∞,1)-category]] of $G$-[[∞-actions]] $Act_G(\mathbf{H})$.
Under the [[equivalence of (∞,1)-categories]]

$$
  Act_G(\mathbf{H}) \simeq \mathbf{H}_{/\mathbf{B}G}
$$

(from [NSS 12](https://ncatlab.org/schreiber/show/Principal+%E2%88%9E-bundles+--+theory%2C+presentations+and+applications)) then $G$ with its canonical [[∞-action]] is $(\ast \to \mathbf{B}G)$
and $X$ with the trivial action is $(X \times \mathbf{B}G \to \mathbf{B}G)$.

Hence

$$
  [G,X]/G
    \simeq
  [\ast, X \times \mathbf{B}G]_{\mathbf{B}G}
  \;\;\;\;\;
  \in
   \mathbf{H}_{/\mathbf{B}G}
  \,.
$$

So far this is the very definition of what $[G,X]/G \in \mathbf{H}_{/\mathbf{B}G}$ is to mean in the first place.

But now since the [[slice (∞,1)-topos]] $\mathbf{H}_{/\mathbf{B}G}$ is itself [[cartesian closed (infinity,1)-category|cartesian closed]], via

$$
  E \times_{\mathbf{B}G}(-)
  \;\;\;
   \dashv
  \;\;\;
  [E,-]_{\mathbf{B}G}
$$

it is immediate that
there is the following sequence of [[natural equivalences]]

$$
  \begin{aligned}
   \mathbf{H}_{/\mathbf{B}G}(Y, [G,X]/G)
   & \simeq
   \mathbf{H}_{/\mathbf{B}G}(Y, [\ast, X \times \mathbf{B}G]_{\mathbf{B}G})
   \\
   & \simeq
   \mathbf{H}_{/\mathbf{B}G}(
     Y \times_{\mathbf{B}G} \ast,
     \underset{p^\ast X}{\underbrace{X \times \mathbf{B}G }}
  )
  \\
  & \simeq
  \mathbf{H}(
     \underset{hofib(Y)}{\underbrace{p_!(Y \times_{\mathbf{B}G} \ast)}},
     X
  )
  \\
  & \simeq
  \mathbf{H}(hofib(Y),X)
  \end{aligned}
$$

Here $p \colon \mathbf{B}G \to \ast$ denotes the terminal morphism and $p_! \dashv p^\ast$ denotes the [[base change]] along it.

=--

See also at _[[double dimensional reduction]]_ for more on this.

### Along $V/G \to \mathbf{B}G$

More generally:

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

Let $\mathbf{H}$ be an [[(∞,1)-topos]] and $G \in Grp(\mathbf{H})$ an [[∞-group]].

Let moreover $V \in \mathbf{H}$ be an object equipped with a $G$-[[∞-action]] $\rho$, equivalently (by the discussion there) a [[homotopy fiber sequence]] of the form

$$
  \array{
    V
    \\
    \downarrow
    \\
    V/G & \overset{p_\rho}{\longrightarrow}& \mathbf{B}G
  }
$$

Then

1. pullback along $p_\rho$ is the operation that assigns to a morphism $c \colon X \to \mathbf{B}G$ the $V$-[[fiber ∞-bundle]] which is [[associated ∞-bundle|associated]] via $\rho$ to the $G$-[[principal ∞-bundle]] $P_c$ classified by $c$:

   $$
     (p_\rho)^\ast \;\colon\; c \mapsto P_c \times_G V
   $$

1. the right base change along $p_\rho$ is given on objects of the form $X \times (V/G)$ by

   $$
     (p_\rho)_\ast \;\colon\; X \times (V/G) \;\mapsto\;  [V,X]/G
   $$

=--

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

The first statement is [NSS 12, prop. 4.6](https://ncatlab.org/schreiber/show/Principal+%E2%88%9E-bundles+--+theory%2C+presentations+and+applications).

The second statement follows as in the proof of prop. \ref{CyclicLoopSpace}: Let

$$
  \left(
    \array{
      Y
      \\
      \downarrow^{\mathrlap{c}}
      \\
      \mathbf{B}G
    }
  \right)
  \;\in\;
  \mathbf{H}_{/\mathbf{B}G}
$$

be any object, then there is the following sequence of [[natural equivalences]]

$$
  \begin{aligned}
   \mathbf{H}_{/\mathbf{B}G}(Y, [V,X]/G)
   & \simeq
   \mathbf{H}_{/\mathbf{B}G}(Y, [V/G, X \times \mathbf{B}G]_{\mathbf{B}G})
   \\
   & \simeq
   \mathbf{H}_{/\mathbf{B}G}(
     Y \times_{\mathbf{B}G} (V/G),
     \underset{p^\ast X}{\underbrace{X \times \mathbf{B}G }}
  )
  \\
  & \simeq
  \mathbf{H}_{/\mathbf{B}G}
  (
    (p_\rho)_!( P_c \times_G (V/G) ),
    p^\ast X
  )
  \\
  & \simeq
     \mathbf{H}_{/(V/G)}
     (
       P_c \times_G V, (p_\rho)^\ast p^\ast X
     )
  \\
  &
  \simeq
    \mathbf{H}_{/(V/G)}(P_c \times_G V, X \times (V/G))
  \end{aligned}
$$

where again

$$
  p \colon \mathbf{B}G \to \ast
  \,.
$$

=--

+-- {: .num_example #SymmetricPowers}
###### Example
**(symmetric powers)**

Let

$$
  G = \Sigma(n) \in Grp(Set) \hookrightarrow Grp(\infty Grpd) \overset{LConst}{\longrightarrow} \mathbf{H}
$$

be the [[symmetric group]] on $n$ elements, and

$$
  V = \{1, \cdots, n\} \in Set \hookrightarrow \infty Grpd \overset{LConst}{\longrightarrow} \mathbf{H}
$$

the $n$-element [[set]] ([[h-set]]) equipped with the canonical $\Sigma(n)$-[[action]]. Then prop. \ref{RightBaseChangeAlongUniversalFiberBundleProjection} says that right base change of any $p_\rho^\ast p^\ast X$ along

$$
  \{1, \cdots, n\}/\Sigma(n)
    \longrightarrow
  \mathbf{B}\Sigma(n)
$$

is equivalently the $n$th [[symmetric power]] of $X$

$$
 [\{1,\cdots, n\},X]/\Sigma(n)
  \simeq
  (X^n)/\Sigma(n)
  \,.
$$

=--

## Related concepts

* **base change**

  * [[dependent sum]], [[dependent product]]

  * [[dependent sum type]], [[dependent product type]]

  * [[necessity]], [[possibility]], [[reader monad]], [[writer comonad]]

* [[proper base change theorem]]

* Base change geometric morphisms may be interpreted in terms of [[fiber integration]]. See [[integral transforms on sheaves]] for more on this.

* [[change of enriching category]]

[[!include notions of pullback -- contents]]

## References

A general discussion that applies (also) to [[enriched categories]] and [[internal categories]] is in

* [[Dominic Verity]], _Enriched categories, internal categories and change of base_ Ph.D. thesis, Cambridge University (1992), reprinted as Reprints in Theory and Applications of Categories, No. 20 (2011) pp 1-266 ([TAC](http://www.tac.mta.ca/tac/reprints/articles/20/tr20abs.html))

Discussion in the context of [[topos theory]] is around example A.4.1.2 of

* {#Johnstone} [[Peter Johnstone]],  _[[Sketches of an Elephant]]_

and around theorem IV.7.2 in

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

Discussion in the context of [[(infinity,1)-topos theory]] is in section 6.3.5 of

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

See also

* A. Carboni, G. Kelly, R. Wood, _A 2-categorical approach to change of base and geometric morphisms I_ ([numdam](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1991__32_1/CTGDC_1991__32_1_47_0/CTGDC_1991__32_1_47_0.pdf))

[[!redirects change of base]]
[[!redirects changes of base]]
[[!redirects base changes]]

[[!redirects base change geometric morphism]]
[[!redirects base change geometric morphisms]]


base change spectral sequence

For $R$ a [[ring]] write $R$[[Mod]] for its category of [[modules]].
Given a [[homomorphism]] of [[ring]] $f : R_1 \to R_2$ and an $R_2$-[[module]] $N$ there are composites of [[base change]] along $f$ with the [[hom-functor]] and the [[tensor product]] functor

$$
  R_1 Mod \stackrel{\otimes_{R_1} R_2}{\to} R_2 Mod \stackrel{\otimes_{R_2} N}{\to} Ab
$$

$$
  R_1 Mod \stackrel{Hom_{R_1 Mod}(-,R_2)}{\to}
  R_2 Mod
  \stackrel{Hom_{R_2}(-,N)}{\to}
  Ab
  \,.
$$

The [[derived functors]] of $Hom_{R_2}(-,N)$ and $\otimes_{R_2} N$ are the [[Ext]]- and the [[Tor]]-functors, respectively, so the [[Grothendieck spectral sequence]] applied to these composites is the _base change spectral sequence_ for these.


base of enrichment

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

#Contents#
* table of contents
{:toc}

## Idea

In [[enriched category theory]], a *base of enrichment* is some kind of [[category theory|category theoretic]] [[structure]] $V$ -- most often a [[monoidal category]], but potentially something more general like a [[bicategory]], [[virtual double category]], or [[skew-monoidal category]] -- over which one intends to consider $V$-[[enriched categories]].

Frequently, but not always, $V$ will be a [[Bénabou cosmos]], which provides sufficient infrastructure to carry out enriched versions of most of the standard category theoretic constructions, for example of [[enriched functor categories]], [[tensor products of enriched categories]], [[enriched presheaves|enriched]] [[presheaf categories]], [[Eilenberg-Moore categories]], specific [[weighted limits]] and [[weighted colimits]], and so on.

In this context, "change of base" or "change of base of enrichment", refers to a [[2-functor]] $V\text{-}Cat \longrightarrow W\text{-}Cat$ between ([[very large category|very large]]) [[categories of enriched categories]] that is induced by a [[lax monoidal functor]] $V \to W$ between bases of enrichment. See also the section *[Base change](enriched+category#BaseChange)* at *[[enriched category]]*.

## References

* {#Kelly82} [[Max Kelly]], p. 9 of: _Basic concepts of enriched category theory_, London Math. Soc. Lec. Note Series __64__, Cambridge Univ. Press (1982), Reprints in Theory and Applications of Categories **10** (2005) 1-136  [[ISBN:9780521287029](https://www.cambridge.org/de/academic/subjects/mathematics/logic-categories-and-sets/basic-concepts-enriched-category-theory?format=PB&isbn=9780521287029), [tac:tr10](http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf)]

* [[Francis Borceux]], §6.4 of: _[[Handbook of Categorical Algebra]] Vol 2: Categories and Structures_, Encyclopedia of Mathematics and its Applications **50**, Cambridge University Press (1994) [[doi:10.1017/CBO9780511525865](https://doi.org/10.1017/CBO9780511525865)]

[[!redirects base of enrichment]]
[[!redirects bases of enrichment]]

[[!redirects base for enrichment]]
[[!redirects bases for enrichment]]


base topos

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

#Contents#
* table of contents
{:toc}

## Idea

Every [[sheaf topos]] $\mathcal{E}$ of sheaves with values in [[Set]] is canonically and essentially uniquely equipped with its [[global section]] [[geometric morphism]] $\Gamma : \mathcal{E} \to Set$. So in particular for $\mathcal{E} \to \mathcal{F}$ any other [[geometric morphism]], we have necessarily a [[diagram]]

$$
  \array{
    \mathcal{E} &&\to&& \mathcal{F}
    \\
    & \searrow &\swArrow_{\simeq}& \swarrow
    \\
    && Set
  }
$$

in the [[2-category]] [[Topos]].

Accordingly, if $\mathcal{E}$ and $\mathcal{F}$ are both equipped with [[geometric morphism]] to some other topos $\mathcal{S}$, it makes sense to restrict attention to those geometric morphisms between them that do form commuting triangles as before

$$
  \array{
    \mathcal{E} &&\to&& \mathcal{F}
    \\
    & \searrow &\swArrow_{\simeq}& \swarrow
    \\
    && \mathcal{S}
  }
$$

but now over the new _base topos_ $\mathcal{S}$. This is a [[morphism]] in the [[slice 2-category]] [[Topos]]$/\mathcal{S}$.

One can develop essentially all of [[topos theory]] in $Topos/\mathcal{S}$ instead of in $Topos$ itself.

To some extent it is also possible to speak of a base topos entirely [[internal logic|internally]] to a given topos. See for instance ([AwodeyKishida](#AwodeyKishida)).

## Constructions

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

To $\mathcal{S}$ itself we associate the $\mathcal{S}$-[[indexed category]] (the canonical self-indexing) $\mathbb{S}$ given by

$$
  \mathbb{S}^I = \mathcal{S}/I
  \,.
$$

To $p : \mathcal{E} \to \mathcal{S}$ a topos over a base $\mathcal{S}$, we associate the $\mathcal{S}$-[[indexed category]]

$$
  \mathbb{E} : \mathcal{S}^{op} \to Cat
$$

which sends an object $I \in \mathcal{S}$ to the [[over-topos]] of $\mathcal{E}$ over the [[inverse image]] of $I$ under the [[geometric morphism]] $p$

$$
  \mathcal{E}^I := \mathcal{E}/p^*(I)
  \,.
$$

=--

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

The [[geometric morphism]] $p : \mathcal{E} \to \mathcal{S}$ induces an $\mathcal{S}$-indexed geometric morphism (hence a geometric morphism [[internalization|internal to]] the [[slice 2-category]] [[Topos]]$/\mathcal{S}$)

$$
  \mathbb{p} : \mathbb{E} \to \mathbb{S}
  \,.
$$

=--

By the discussion at [[indexed category]].

## Related concepts

* [[base (∞,1)-topos]]
* [[internal site]]
* [[internal sheaves]]
* [[bounded geometric morphism]]
* [[unbounded topos]]

## References

The general notion of base toposes is the topic of section B3 of

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

An internal description of base toposes in the context of [[modal logic]] appears in

* {#AwodeyKishida} [[Steve Awodey]], Kohei Kishida, _Topology and modality: the topological interpretation of first-order modal logic_ ([pdf](http://www.andrew.cmu.edu/user/awodey/preprints/FoS4.phil.pdf))

[[!redirects base toposes]]

[[!redirects topos over a base]]
[[!redirects toposes over a base]]


This page collects material related to the textbook

* [[Dale Husemoeller]], [[Michael Joachim]], [[Branislav Jurčo]], [[Martin Schottenloher]],

  _Basic Bundle Theory and K-Cohomology Invariants_,

  Lecture Notes in Physics, Springer 2008

  [doi:10.1007/978-3-540-74956-1](https://doi.org/10.1007/978-3-540-74956-1)

  [pdf](http://www.mathematik.uni-muenchen.de/~schotten/Texte/978-3-540-74955-4_Book_LNP726corr1.pdf)

on [[fiber bundles]] ([[principal bundles]], [[vector bundles]], [[classifying space]], [[characteristic classes]]), [[topological K-theory]], [[twisted K-theory]], [[equivariant K-theory]], [[twisted ad-equivariant K-theory]] and applications to the [[K-theory classification of D-brane charge]].

category: reference