The role of locality in perturbation theory

[[!redirects The Role of locality in perturbation theory]]

The page collects material related to

* [[Henri Epstein]] and [[Vladimir Glaser]],

  _The Role of locality in perturbation theory_,

  Annales Poincaré Phys. Theor. A 19 (1973) 211

  ([numdam:AIHPA_1973__19_3_211_0](http://www.numdam.org/item?id=AIHPA_1973__19_3_211_0))

on [[renormalization]] of [[quantum field theory]] compatible with [[causal locality]]. The approach came to be known as _[[causal perturbation theory]]_ (see there for more). Its generalization to [[quantum field theory on curved spacetimes]] is known as _[[locally covariant perturbative quantum field theory]]_.

This is based on earlier work in

* [[Ernst Stückelberg]], D. Rivier, Helv. Phys. Acta, 22 (1949) 215.

* [[Ernst Stückelberg]], J. Green, Helv. Phys. Acta, 24 (1951) 153.

* [[Ernst Stückelberg]], A. Peterman, , _La normalisation des constants dans la theorie des quanta_, Helv. Phys. Acta 26, 499 (1953);

* {#BogoliubovShirkov59} [[Nikolay Bogoliubov]], [[DmitryShirkov]], _Introduction to the Theory of Quantized Fields_, New York (1959)

Hence one speaks of the _Stückelberg-Bogoliubov-Epstein-Glaser method
of [[renormalization]]_, for instance in

* {#BrunettiFredenhagen00} [[Romeo Brunetti]], [[Klaus Fredenhagen]], _Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds_, Commun. Math. Phys. 208 : 623-661,2000 ([math-ph/9903028](https://arxiv.org/abs/math-ph/9903028))

category: reference


The Self-Organizing Universe

This page collects links related to the book

* [[Erich Jantsch]],

  _The Self-Organizing Universe_

  (1980)

  ([on Google Books](http://books.google.de/books/about/The_Self_organizing_Universe.html?id=1fyYNAEACAAJ&redir_esc=y))

proposing a sketch of a scientific pciture of how the structures of the [[observable universe]] ranging from fundamental [[physics]] to properties of [[life]] and [[mind]] have an explanation by principles of [[autopoiesis]] exhibited by many complex [[dissipative systems]], following ideas and results of [[Ilya Prigogine]].

## related $n$Lab entries

* [[observable universe]]

category: reference


The Stacks Project

The document

* [[Aise Johan de Jong]],  _The Stacks Project_ collaborative textbook project, $\gt 7500$ pages ([online version](https://stacks.math.columbia.edu/), [pdf](https://stacks.math.columbia.edu/download/book.pdf))

is a textbook that leads from basics of [[category theory]] and [[algebra]] to a discussion of [[scheme | schemes]], [[algebraic space | spaces]] and [[algebraic stack | algebraic stacks]].

## See also

* [[Kerodon]]

* [[bananaspace]]

category: reference

[[!redirects Stacks Project]]
[[!redirects Stacks project]]


This page compiles material related to:

* [[André Joyal]],

  \linebreak

  **The Theory of Quasi-Categories and its Applications**

  \linebreak

  lectures at: *[Advanced Course on Simplicial Methods in Higher Categories](https://lists.lehigh.edu/pipermail/algtop-l/2007q4/000017.html)*, CRM 2008

  350 pages

  [pdf at mat.uab.cat](http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf), [pdf at ncatlab.org](https://ncatlab.org/nlab/files/JoyalTheoryOfQuasiCategories.pdf)

on [[quasicategories]] -- a model for [[(infinity,1)-category theory|$(\infty,1)$-category theory]].

category: reference


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This page collects resources related to

* [[Eugene Wigner]]

  _The Unreasonable Effectiveness of Mathematics in the Natural Sciences_

  Richard Courant lecture in mathematical sciences delivered at New York University, May 11, 1959.

  Communications on Pure and Applied Mathematics. 13: 1–14 (1960)

  [doi:10.1002/cpa.3160130102](https://doi.org/10.1002/cpa.3160130102)

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

## Related references

* [[Galileo Galilei]] ([quote](Galileo+Galilei#Quotes)):

  > [[natural philosophy|Philosophy]] $[$i.e. [[physics]] $]$ is written in this grand book — I mean the universe — which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of [[mathematics]], and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering around in a dark labyrinth. ([Galilei, Il Saggiatore, 1623](http://en.wikipedia.org/wiki/The_Assayer)).

* [[David Hilbert]], _[[Naturerkennen und Logik]]_, Lecture at the _Kongress der Gesellschaft Deutscher Naturforscher und Ärtze_, 1930  ([pdf](http://math.sfsu.edu/smith/Documents/HilbertRadio/HilbertRadio.pdf), [audio](http://math.sfsu.edu/smith/Documents/HilbertRadio/HilbertRadio.mp3))

* {#AtiyahDijkgraafHitchin10} [[Michael Atiyah]], [[Robbert Dijkgraaf]], [[Nigel Hitchin]], _Geometry and physics_, Phil. Trans. R. Soc. A 13 March 2010 vol. 368 no. 1914 913-926 ([doi;10.1098/rsta.2009.0227](https://royalsocietypublishing.org/doi/10.1098/rsta.2009.0227))

  on, conversely: _[[The Unreasonable Effectiveness of Physics in the Mathematical Sciences]]_

## Related entries

* [[mathematical physics]]

category: reference


The Unreasonable Effectiveness of Physics in the Mathematical Sciences

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In view of the developments in [[theoretical physics]] from about the the 1980s, and in reference to [[Eugene Wigner]]'s famous phrase of the _[[The Unreasonable Effectiveness of Mathematics in the Natural Sciences]]_, [Atiyah-Dijkgraaf-Hitchin 10](#AtiyahDijkgraafHitchin10) wrote this:

> But over the past 30 years $[$ 1980s- 2010s $]$ a new type of interaction has taken place, probably unique, in which physicists, exploring their new and still speculative theories,have stumbled across a whole range of mathematical "discoveries". These are derived by physical intuition and heuristic arguments, which are beyond the reach, as yet, of mathematical rigour, but which have withstood the tests of time and alternative methods. There is great intellectual excitement in these mutual exchanges.

> The impact of these discoveries on mathematics has been profound and widespread. Areas of mathematics such as topology and algebraic geometry, which lie at the heart of pure mathematics and appear very distant from the physics frontier, have been dramatically affected.

> The meaning of all this is unclear and one may be tempted to invert [[The Unreasonable Effectiveness of Mathematics in the Natural Sciences|Wigner's comment]] and marvel at "the unreasonable effectiveness of physics in mathematics".

> ([Atiyah-Dijkgraaf-Hitchin 10, p. 915 (3 of 14)](#AtiyahDijkgraafHitchin10))

The phrase is picked up also in [Dijkgraaf 14](#Dijkgraaf14), [Moore 14, p. 10](#Moore14), [Tong 17](#Tong17) and elsewhere.

## References

* {#AtiyahDijkgraafHitchin10} [[Michael Atiyah]], [[Robbert Dijkgraaf]], [[Nigel Hitchin]], _Geometry and physics_, Phil. Trans. R. Soc. A 13 March 2010 vol. 368 no. 1914 913-926 ([pdf](http://rsta.royalsocietypublishing.org/content/368/1914/913.full.pdf), [doi:10.1098/rsta.2009.0227](https://royalsocietypublishing.org/doi/10.1098/rsta.2009.0227))

* {#Moore14} [[Gregory Moore]], _[[Physical Mathematics and the Future]]_, talk at [Strings 2014](http://physics.princeton.edu/strings2014/) ([talk slides](http://physics.princeton.edu/strings2014/slides/Moore.pdf), [companion text pdf](http://www.physics.rutgers.edu/~gmoore/PhysicalMathematicsAndFuture.pdf), [[MooreVisionTalk2014.pdf:file]])

* {#Dijkgraaf14} [[Robbert Dijkgraaf]], _The Unreasonable Effectiveness Of Quantum Physics in Modern Mathematics_, [Perimeter Public Lecture](http://pirsa.org/C04001) 2014 ([web](http://pirsa.org/14030094), [pdf slides](http://pirsa.org/pdf/loadpdf.php?pirsa_number=14030094)).

* {#Tong17} [[David Tong]], _The Unreasonable Effectiveness of Physics in Mathematics_, [LMS Popular Lecture](https://www.lms.ac.uk/events/popular-lectures) 2017 ([webpage](https://www.maths.cam.ac.uk/features/lms-popular-lecture-unreasonable-effectiveness-physics-mathematics-david-tong), [talk recording](https://youtu.be/UVuKyZ4pBzg))

* [[Natalie Paquette]], _The Unreasonable Effectiveness of Physics in Mathematics_, [talk](https://www.youtube.com/watch?v=XPxwabVs4Lc)

category: reference


The World in Eleven Dimensions

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This page collects material related to the book collection

* [[Mike Duff]]:

  \linebreak

  **The world in eleven dimensions:**

  **Supergravity, Supermembranes and $M$-theory**

  \linebreak

  IoP 1999

  [ISBN:9780750306720 ](https://www.crcpress.com/The-World-in-Eleven-Dimensions-Supergravity-supermembranes-and-M-theory/Duff/9780750306720)

that collects and comments key articles in the (pre-)history of [[M-theory]] ([[M-branes]] in [[11-dimensional supergravity]]) in the 1990s.

See also

* [[Edward Witten]],  _[[String Theory Dynamics In Various Dimensions]]_,  Nucl.Phys.B443:85-126,1995 ([arXiv:hep-th/9503124](http://arxiv.org/abs/hep-th/9503124))

For related texts see at _[[books about string theory]]_.

See also at _[[string theory FAQ]]_

#Contents#
* table of contents
{:toc}

## Eleven-dimensional supergravity

On [[11-dimensional supergravity]]

On classification of [[supersymmetry]]-[[super Lie algebras]]:

* {#Nahm78} [[Werner Nahm]], _[[Supersymmetries and their Representations]]_, Nucl.Phys. B135 (1978) 149 ([spire:120988](https://inspirehep.net/record/120988), doi:10.1016/0550-3213(78)90218-3)

(...)

On the [[KK-monopole]]:

* {#HanKoh85} Seung Kee Han, I. G. Koh, _$N = 4$ Remaining Supersymmetry in Kaluza-Klein Monopole Background in D=11 Supergravity Theory_, Phys.Rev. D31 (1985) 2503, in [[Michael Duff]] (ed.), _[[The World in Eleven Dimensions]]_ 57-60 ([doi:10.1103/PhysRevD.31.2503](https://doi.org/10.1103/PhysRevD.31.2503), [spire:204521](http://inspirehep.net/record/204521))

## The eleven-dimensional supermembrane

On the [[M2-brane]]

## The eleven-dimensional superfivebrane

On the [[M5-brane]]

## $M$-theory (before $M$-theory was cool)

on [[M-theory]]

## Intersecting branes and black holes

On [[black branes]], [[BPS-states]] of intersecting M-branes,

* [[Arkady Tseytlin]], _Harmonic superpositions of M-branes_, Nucl.Phys.B475:149-163,1996 ([arXiv:hep-th/9604035](https://arxiv.org/abs/hep-th/9604035))

as well as [[black holes in string theory]] and [[Bekenstein-Hawking entropy]]

## $M$-theory and duality

On [[duality in string theory]], mostly [[S-duality]] and [[dual heterotic string theory]], ending with [[AdS-CFT]]

On whether "M" is really for "[[membrane]]" after all:

> In other words, perhaps $D = 11$ supergravity together with [[M-brane|its BPS configurations]]: [[M-wave|plane wave]], [[M2-brane|membrane]], [[M5-brane|fivebrane]], [[KK monopole]] and the $D = 11$ embedding of the Type II [[D8-brane|eight brane]] are are all there is to $M$-theory and that we need look no further for new degrees of freedom. (p. 329)

Concluding with:

> future historians may judge the period 1984-95 as a time when theorists were like boys playing by the sea shore, and diverting themselves with the smoother pebbles or prettier shells of perturbative ten-dimensiorial superstrings while the great ocean of non-perturbative eleven-dimensional M-theory lay all undiscovered before them. (p.330)

Compare to the situation 20 years later as in

* [[Greg Moore]], _[[Physical Mathematics and the Future]]_, talk at [Strings 2014](http://physics.princeton.edu/strings2014/)

with its section 12: "Keep true to the dreams of thy youth: M-theory"

For more see at _[M-theory -- The open problem](M-theory#TheOpenProblem)_

category: reference


Theo Johnson-Freyd

* [webpage](http://categorified.net/)

## Selected writings

On [[extended TQFTs]] modelling [[topological phases of matter]] in [[solid state physics]]:

* [[Davide Gaiotto]], [[Theo Johnson-Freyd]], *Condensations in higher categories* ([arXiv:1905.09566](https://arxiv.org/abs/1905.09566))

Construction of [[2d SCFTs]] from [[error-correcting codes]] and suggestion that the [[Chern-Dold character]] from [[tmf]] to [[modular forms]], sends a [[2d SCFT]] to its [[partition function]]/[[elliptic genus]]/supersymmetric index:

* [[Davide Gaiotto]], [[Theo Johnson-Freyd]], *Holomorphic SCFTs with small index*, Canadian Journal of Mathematics , **74** 2 (2022) 573-601  [[arXiv:1811.00589](https://arxiv.org/abs/1811.00589), [doi:10.4153/S0008414X2100002X](https://doi.org/10.4153/S0008414X2100002X)]

and discussion of the [[2d SCFTs]] (namely [[supersymmetry|supersymmetric]] [[SU(2)]]-[[WZW-models]]) conjecturally corresponding, under this conjectural identification, to the elements of
$\mathbb{Z}/24$
[$\simeq$](tmf#BoardmanHomomorphismInTmfIs6Connected)
$
  tmf^{-3}(\ast)
    =
  \pi_3(tmf)
$
[$\simeq$](tmf#BoardmanHomomorphismInTmfIs6Connected)
$\pi_3(\mathbb{S})$
(the [[third stable homotopy group of spheres]]):

* [[Davide Gaiotto]], [[Theo Johnson-Freyd]], [[Edward Witten]], p. 17 of: *A Note On Some Minimally Supersymmetric Models In Two Dimensions*, ([arXiv:1902.10249](https://arxiv.org/abs/1902.10249)) in S. Novikov et al. *Integrability, Quantization, and Geometry: II. Quantum Theories and Algebraic Geometry*, Proc. Symposia Pure Math., 103(2), 2021 ([ISBN: 978-1-4704-5592-7](https://bookstore.ams.org/pspum-103-2))

* [[Davide Gaiotto]], [[Theo Johnson-Freyd]], *Mock modularity and a secondary elliptic genus* ([arXiv:1904.05788](https://arxiv.org/abs/1904.05788))

* [[Theo Johnson-Freyd]], *Topological Mathieu Moonshine* ([arXiv:2006.02922](https://arxiv.org/abs/2006.02922))

On [[Lie groups]] and [[quantum groups]]:

* [[Richard Borcherds]], [[Mark Haiman]], [[Theo Johnson-Freyd]], [[Nicolai Reshetikhin]], [[Vera Serganova]], *Berkeley Lectures on Lie Groups and Quantum Groups*, 2020 ([pdf](http://categorified.net/LieQuantumGroups.pdf))

On the relation between [[functorial quantum field theory]] (axiomatizing the [[Schrödinger picture]] of [[quantum field theory]]) and [[algebraic quantum field theory]] (axiomatizing the [[Heisenberg picture]]):

* {#Johnson-Freyd21} [[Theo Johnson-Freyd]], *Heisenberg-picture quantum field theory*, in *Representation Theory, Mathematical Physics, and Integrable Systems*, Progress in Mathematics **340** (2021) [[arXiv:1508.05908](https://arxiv.org/abs/1508.05908), [doi:10.1007/978-3-030-78148-4_13](https://doi.org/10.1007/978-3-030-78148-4_13)]

On [[higher dagger categories]]:

* [[Theo Johnson-Freyd]], *Dagger Higher Categories*, workshop (2023) [[web](http://categorified.net/dagger2023.html)]

* [[Theo Johnson-Freyd]], *Higher Dagger Categories*, [talk at](Center+for+Quantum+and+Topological+Systems#JohnsonFreydNov2023) [[CQTS]] (08 Nov 2023) [slides:[pdf](http://categorified.net/NYUADtalk.pdf), [[JohnsonFreyd-HigherDaggerNYUAD.pdf:file]], video:[YT](https://youtu.be/zvtziTpl2T0)]

category: people


Theodor Bröcker

* [Wikipedia page](https://de.wikipedia.org/wiki/Theodor_Br%C3%B6cker) (deutsch)

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=18319)

## Selected writings

On [[cobordism theory]]:

* [[Theodor Bröcker]], [[Tammo tom Dieck]], *Kobordismentheorie*, Lecture Notes in Mathematics **178**, Springer (1970) [[ISBN:9783540053415](https://link.springer.com/book/9783540053415)]

On [[equivariant bordism homology theory]]:

* [[Theodor Bröcker]], [[Edward Hook]], _Stable equivariant bordism_, Math Z 129, 269–277 (1972) ([doi:10.1007/BF01187353](https://doi.org/10.1007/BF01187353))

On [[differential topology]]:

* [[Theodor Bröcker]], [[Klaus Jänich]], *Introduction to differentiable topology* (1982) [[ISBN:9780521284707](https://www.cambridge.org/ae/universitypress/subjects/mathematics/differential-and-integral-equations-dynamical-systems-and-co/introduction-differential-topology?format=PB&isbn=9780521284707)]

  > (translated from the German 1973 edition)

On the [[representation theory]] of [[compact Lie groups]]:

* [[Tammo tom Dieck]], [[Theodor Bröcker]], *Representations of compact Lie groups*, Springer 1985 ([doi:10.1007/978-3-662-12918-0](https://link.springer.com/book/10.1007/978-3-662-12918-0))

category: people

[[!redirects Theodor Broecker]]
[[!redirects Theodor Brocker]]


Theodor Kaluza

German mathematician (1885-1954)

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

## related $n$Lab entries

* [[Kaluza-Klein mechanism]]

category: people


Theodora Ioannidou

* [Institute page](http://users.auth.gr/~ti3/)

* [Google Scholar page](https://scholar.google.de/citations?user=yYZbsa0AAAAJ&hl=en)

## Selected writings

On the [[moduli space of Yang-Mills monopoles]] via [[rational maps]] ([[holomorphic maps]], really) from the [[Riemann sphere]] to a [[flag variety]]:

* {#IonnadouSutcliffe99a} [[Theodora Ioannidou]], [[Paul Sutcliffe]], *Monopoles and Harmonic Maps*, J. Math. Phys. 40:5440-5455 (1999) ([arXiv:hep-th/9903183](https://arxiv.org/abs/hep-th/9903183))

* {#IonnadouSutcliffe99b} [[Theodora Ioannidou]], [[Paul Sutcliffe]], *Monopoles from Rational Maps*, Phys. Lett. B457 (1999) 133-138 ([arXiv:hep-th/9905066](https://arxiv.org/abs/hep-th/9905066))

category: people


Theodore Courant

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

## Related entries

* [[Lie algebroid]]

* [[Courant Lie 2-algebroid]]

category: people

[[!redirects Ted COurant]]


Theodore Erler

* [webpage uscb](http://web.physics.ucsb.edu/~terler/)

* [webpage lmu](http://www.theorie.physik.uni-muenchen.de/17ls_th_cosmology_en/members/sci_mem/erler/index.html)

## Selected writings

On [[Sen's conjecture]] via [[superstring field theory]]:

* [[Theodore Erler]], _Tachyon Vacuum in Cubic Superstring Field Theory_, JHEP 0801:013,2008 ([arXiv:0707.4591](http://arxiv.org/abs/0707.4591))

* [[Theodore Erler]], _Analytic Solution for Tachyon Condensation in Berkovits' Open Superstring Field Theory_, JHEP 1311 (2013) 007 ([arXiv:1308.4400](https://arxiv.org/abs/1308.4400))

On [[string field theory]]:

* [[Theodore Erler]], *Four Lectures on Closed String Field Theory*, Physics Reports **851** (2020) 1-36 [[arXiv:1905.06785](https://arxiv.org/abs/1905.06785), [doi:10.1016/j.physrep.2020.01.003](https://doi.org/10.1016/j.physrep.2020.01.003)]

## Related $n$Lab entries

* [[string field theory]]

category: people

[[!redirects Ted Erler]]


Theodore Frankel

* [webpage](http://math.ucsd.edu/people/emeriti/Ted-Frankel/)

## Selected writings

On [[Maxwell's equations]] via [[exterior calculus]] of [[differential forms]]:

* [[Theodore Frankel]], *Maxwell's equations*, The American Mathematical Monthly **81** 4 (1974) \[doi:10.1080/00029890.1974.11993557, jstor:2318995, doi:10.2307/2318995\]

On [[mathematical physics]] via, largely, [[exterior calculus]] of [[differential forms]]:

* {#Frankel97} [[Theodore Frankel]], *[[The Geometry of Physics - An Introduction]]*, Cambridge University Press (1997, 2004, 2012) \[doi:10.1017/CBO9781139061377\]

category: people


Theodore Jacobson

[[!redirects Ted Jacobson]]

* [webpage](http://terpconnect.umd.edu/~jacobson/)

## Related entries

* [[gravity]]

* [[Bekenstein-Hawking entropy]]

category: people


Theodore Voronov

* [website](http://www.maths.manchester.ac.uk/~tv/)

category: people


theorem

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> There are good reasons why the theorems should all be easy and the definitions hard. ([[Michael Spivak]], preface to "[[Calculus on Manifolds]]" )

# Theorems
* table of contents
{: toc}

## Idea

In the traditional language of [[mathematics]], a _theorem_ is a statement which is of interest in its own right and which has been proven to be true, though the proof may not be immediately obvious.  This contrasts with a _[[lemma]]_ (which is usually of interest primarily because of its implications for other statements), a _[[conjecture]]_ (which has not yet been proved), an _[[axiom]]_ (which is obviously true or assumed to be true), a _definition_ (which becomes true by virtue of its assigning meaning to a word or phrase), a _[[proposition]]_ (which usually follows more easily from known facts than a theorem does), or a _corollary_ (which follows immediately from facts recently proven).

The discipline of [[logic]] formalizes the notion of [[proof]], but not the notions of "interest" or "immediacy".  Thus, to a logician, *any* proved statement is often called a theorem.  (Mathematicians know this meaning too, but still usually reserve the term 'theorem' for important theorems in their published work.)  The term '[[proposition]]', to a logician, means *any* statement and does *not* imply the existence of a proof.  The term '[[axiom]]' is used in a way that somewhat matches its ordinary usage, but as a logician counts trivial proofs as proofs, an axiom is also a special case of a theorem.  Logic rarely studies [[definition]]s explicitly, but in some theories they do play a role, similar to their informal usage.  The other terms appear not to be used in logic.

## Definition

In a given [[logic]], in a given [[context]], we have various [[propositions]] and various [[proofs]] of propositions.  In that context, a __theorem__ is a proposition with a proof.

Classically, a theorem is a proposition for which there *exists* a proof, but in some contexts (such as, perhaps, fully formalized [[constructive mathematics|constructive]] [[type theory]]), one may use "theorem" to mean a proposition *together with* a proof.

A theorem should be contrasted with a [[tautology]]: a proposition that is [[truth|true]] in all [[models]].  If every theorem in a given logic is a tautology in a given class of models for that logic, then we say that the class of models is __sound__ for that logic; if conversely every tautology is a theorem, then we say that the class of models is __complete__.

## Examples

... we might list famous important theorems/lemmas/etc in the $n$Lab here ...

## Quotations

*  A mathematician is a device for turning coffee into theorems.  ---Alfréd Rényi

*  Lemmas do the work in mathematics: Theorems, like management, just take the credit.  ---[[Paul Taylor]]

## Related concepts

[[!include mathematical statements --- contents]]

## References

* [[Thomas Hales]], _Formal proof_ ([pdf](http://www.ams.org/notices/200811/tx081101370p.pdf))

* John Harrison, _Formal proof -- theory and practice_ ([pdf](http://www.ams.org/notices/200811/tx081101395p.pdf))

[[!redirects theorem]]
[[!redirects theorems]]
[[!redirects Theorem]]

[[!redirects lemma]]
[[!redirects lemmas]]
[[!redirects lemmata]]

[[!redirects corollary]]
[[!redirects corollaries]]


theorem of the heart

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#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[heart of a t-structure]]

* [[exact (∞,1)-category]]

* [[Waldhausen K-theory]]

## References

* [[Amnon Neeman]], *K-Theory for Triangulated Categories III(A): The Theorem of the Heart*, Asian Journal of Mathematics **2** 3 (1998) 495-589 [[arXiv:math/9901091](https://arxiv.org/abs/math/9901091), [doi:10.4310/AJM.1998.v2.n3.a4](https://doi.org/10.4310/AJM.1998.v2.n3.a4)]

* [[Amnon Neeman]], *K-theory for triangulated categories III(B): The theorem of the heart*, Asian Journal of Mathematics **3** 3 (1999) 557–608 [[doi:10.4310/AJM.1999.v3.n3.a2](https://doi.org/10.4310/AJM.1999.v3.n3.a2), [pdf](https://www.intlpress.com/site/pub/files/_fulltext/journals/ajm/1999/0003/0003/AJM-1999-0003-0003-a002.pdf)]

* [[Amnon Neeman]], *K-Theory for Triangulated Categories $3\frac{3}{4}$: A Direct Proof of the Theorem of the Heart*, K-Theory **22** 1-2 (2001) 1-144 [[doi:10.1023/A:1011172502978](http://dx.doi.org/10.1023/A:1011172502978)]

* [[Clark Barwick]], *On exact infinity-categories and the Theorem of the Heart*, Compositio Mathematica **151** (2015) 2160-2186 [[arXiv:1212.5232](https://arxiv.org/abs/1212.5232), [doi:10.1112/S0010437X15007447](https://doi.org/10.1112/S0010437X15007447)]


theory

> This entry is about the notion in [[logic]]. For the notion of the same name in [[physics]] see at _[[theory (physics)]]_.

***

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#Contents#
* table of contents
{:toc}

## Idea

In [[mathematical logic]], a _theory_ is a formal [[language]] used to precisely [[axiom|axiomatize]] a certain class of [[models]].

In principle also all other notions of _theory_, such as in the sense of [[physics]] should be special cases of this, but in practice of course there are many systems called "theories" which are not (yet) as fully formalized as in mathematical logic.

## Definition

There are several different viewpoint on theories:

* The

  [syntactic view](#SyntacticView)

  is that the theory itself consists of a set of [[formulas]] in the first order language $Lang(\Sigma)$ of a [[signature (in logic)|signature]] $\Sigma$.  Classically, these formulas are assumed to have no [[free variables]] (i.e. to be "sentences"), but in weaker logics that lack [[universal quantification]] it is better to take them to be formulas-in-[[context]].  One also sometimes considers the theory to include all logical consequences (aka theorems) of the axioms in $A$, relative to (some specified) fragment of [[first-order logic]] --- that is, to be "saturated" with respect to provability.

* The

  [semantic view](#SemanticView)

  is that the theory is given by the class of its [[model]]s appropriate to that fragment of logic. Gödel's [[completeness theorem]] is that a sentence in $\mathcal{T}$ is a theorem iff it is satisfied in every model.

* The

  [categorical view](#CategoricalView)

  is that the logical formalism of a theory $\mathcal{T}$ can frequently be embodied in a [[syntactic category]] of terms $C_{\mathcal{T}}$, so that models of a theory $\mathcal{T}$ are identified with [[functor]]s

  $$C_{\mathcal{T}} \to Set$$

  that preserve some (typically [[stuff, structure, property|property-like]]) [[stuff, structure, property|structures]] on $C_{\mathcal{T}}$, such as certain classes of [[colimit]]s or of [[limits]], pertinent to the fragment of logic at hand. Then a completeness theorem would be the statement that the canonical map

  $$C_{\mathcal{T}} \to \prod_{models F in Set} Set$$

  is a [[full and faithful functor|full faithful embedding]] (one that preserves all relevant logical structure). For this reason, completeness theorems are also known as _embedding theorems_.

+--{.query}
Hm, is that the way it should be said?
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  In fact, the notion of model can be generalized away from $Set$ to more general  [[categories]], namely those that have enough structure to "internalize" the fragment of logic at hand. From this very general point of view on model, the syntactic category $C_{\mathcal{T}}$ is the generic or universal model for $\mathcal{T}$, and if we simply call $C_{\mathcal{T}}$ the theory, then models and theories are placed on the same footing.

In this article we mostly consider the categorical view on "theory".

### Syntactic view
 {#SyntacticView}

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

In first-order [[logic]], a **theory** $\mathcal{T}$ is presented by

1. a [[signature (in logic)|signature]] $\Sigma$, specifying the allowed [[type]]s of variables and the [[function]]s and [[relation]]s between these;

1. a set $A$ of [[sequent]]s of [[formula]]s in this signature, called the **axioms** of the theory:  expressed in the first-order language $Lang(\Sigma)$ with equality generated by $\Sigma$

=--

For instance ([Johnstone, def. D1.1.6](#Johnstone)).

### Semantic view
{#SemanticView}

(...)

### Categorical view
 {#CategoricalView}

(...)

* [[categorical semantics]]

(...)

## Examples

### Classes of theories

There are many different kinds of "theory" depending on the strength of the "logic": a by-no-means complete list includes

* [[equational logic]],

* [[first-order theory]]

* [[Horn theory]],

* [[essentially algebraic theory]],

* [[geometric logic]],

* [[regular logic]],

* [[exact logic]],

* [[extensive logic]],

* [[coherent logic]],

* [[first-order logic]] (aka pretopos logic),

and corresponding theories for these logics.

* [[essentially algebraic theory]]

* [[generalized algebraic theory]]

* [[algebraic theory]]

  * [[commutative algebraic theory]]

  * finitary algebraic theory: [[Lawvere theory]]

    * [[Fermat theory]]

#### Hierarchy of theories: cartesian, regular, coherent, geometric

There is a hierarchy of theories that can be interpreted in the [[internal logic]] of a hierarchy of types of [[categories]].  Since [[predicates]] in the internal logic are represented by [[subobjects]], in order to interpret any connective or quantifier in the internal logic, one needs a corresponding operation on subobjects to exist in the category in question, and be well-behaved.  For instance:

* **cartesian theories** Since $\wedge$ and $\top$ are represented by [[intersections]] and identities (top elements), these can be interpreted in any [[lex category]].  Theories involving only these are [[cartesian theory|cartesian theories]].

* **regular theories** Since $\exists$ is represented by the [[image]] of a subobject, it can be interpreted in any [[regular category]].  Theories involving only $\wedge$, $\top$, and $\exists$ are [[regular theory|regular theories]].

* **coherent theories** Since $\vee$ and $\bot$ are represented by [[union]] and [[bottom elements]], these can be interpreted in any [[coherent category]].  Theories which add these to regular logic are called [[coherent theory|coherent theories]].

* **geometric theories** Finally, theories which also involve infinitary $\bigvee$, which is again represented by an infinitary union, can be interpreted in [[infinitary coherent categories]], aka *[[geometric categories]]*.  These are **[[geometric theories]]**.

Note that the axioms of one of these theories are actually of the form
$$ \varphi \vdash_{\vec{x}} \psi $$
where $\varphi$ and $\psi$ are formulas involving only the specified connectives and quantifiers, $\vdash$ means entailment, and $\vec{x}$ is a [[context]].  Such an axiom can also be written as
$$ \forall \vec{x}. (\varphi \Rightarrow \psi) $$
so that although $\Rightarrow$ and $\forall$ are not strictly part of any of the above logics, they can be applied "once at top level."  In an axiom of this form for geometric logic, the formulas $\varphi$ and $\psi$ which must be built out of $\top$, $\wedge$, $\bot$, $\bigvee$, and $\exists$ are sometimes called *positive* formulas.

### Abelian theories

Interestingly, one form of logic which made an early appearance but is not ordinarily thought of as logic at all is the logic of [[abelian categories]], which is characterized by certain exactness properties. Here a small abelian category $A$ can be thought of as a syntactic site for some "abelian theory"; models of the theory are exact additive functors with domain $A$. The classical models would in fact be exact additive functors $A \to Ab$, or exact additive functors to a category of modules. A "Freyd-Heron-Lubkin-Mitchell" embedding theorem is then a completeness theorem with respect to the classical models, and assures us that a statement in the language of abelian category theory is provable if and only if it is true when interpreted in any module category.

### Specific examples
 {#SpecificExamples}

The simplest nontrivial theory is the

* [[theory of objects]]

A still pretty simple but very nontrivial theory is

* [[elementary function arithmetic]]

* [[second order arithmetic]]

The theories

* $Th(Cat)$ of [[categories]]

* $Th(Lex)$ of [[finitely complete categories]]

* $Th(Topos)$ of [[elementary toposes]]

* $Th(ETCS)$ of [[sets]] ([[ETCS]])

are discussed at _[[fully formal ETCS]]_.

## Models for a theory

### Set-theoretic models for a first-order theory in syntactic approach

The basic concept is of a structure for a first-order language $L$: a set $M$ together with an interpretation of $L$ in $M$. A theory is specified by a language and a set of sentences in $L$. An $L$-structure $M$ is a __model__ of $T$ if for every sentence $\phi$ in $T$, its interpretation in $M$, $\phi^M$ is true ("$\phi$ holds in $M$"). We say that $T$ is __consistent__ or satisfiable (relative to the universe in which we do [[model theory]]) if there exist at least one model for $T$ (in our universe). Two theories, $T_1$, $T_2$ are said to be __equivalent__ if they have the same models.

Given a class $K$ of structures for $L$, there is a theory $Th(K)$ consisting of all sentences in $L$ which hold in every structure from $K$. Two structures $M$ and $N$ are __elementary equivalent__ (sometimes written by equality $M=N$, sometimes said "elementarily equivalent") if $Th(M)=Th(N)$, i.e. if they satisfy the same sentences in $L$. Any set of sentences which is equivalent to $Th(K)$ is called a __set of axioms__ of $K$. A theory is said to be __finitely axiomatizable__ if there exist a finite set of axioms for $K$.

A theory is said to be __complete__ if it is equivalent to $Th(M)$ for some structure $M$.

#### Categorical point of view and models in topoi

From the categorical point of view, for every theory $T$ there exists a [[category]] $C_T$ -- the  **[[syntactic category]]** $C_T$ -- such that a model for $T$ is a [[functor]] $C_T \to \mathcal{T}$ into some [[topos]] $\mathcal{T}$, satisfying certain conditions.

For instance the syntactic categories of [[Lawvere theory|Lawvere theories]] are precisely those categories that have finite cartesian [[product]]s and in which every object is [[isomorphic]] to a finite cartesian power $x^n$ of a distinguished object $x$. A model for a Lawvere theory is precisely a finite product preserving functor $C_T \to \mathcal{T}$.

We say a functor $\mathcal{T}_1 \to \mathcal{T}_2$ of toposes (for instance a [[logical morphism]] or a [[geometric morphism]]) _preserves a theory_ $T$ if for every model $C_T \to \mathcal{T}_1$ of $T$ in $\mathcal{T}_1$, the composite $C_T \to \mathcal{T}_1 \to \mathcal{T}_2$ is a model of $T$ in $\mathcal{T}_2$.

For instance, every [[geometric morphism]] preserves every [[Lawvere theory]] since, being a [[right adjoint]], it preserves [[limit]]s, hence finite products.

## Related concepts

* [[algebraic theory]], [[globular theory]]

* [[type theory]]

* [[theory of presheaf type]]

* [[ordinal analysis]]

* [[2-theory]], [[3-theory]]

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[[!include mathematical statements --- contents]]

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## References

* H. Jerome Keisler, _Fundamentals of model theory_, A2 in JOhn Barwise, Handbook of mathematical logic (pp.47-103), North Holland 1977

A standard textbook reference for the [[categorical semantics]] is section D of

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

A discussion of the relation between theories and their [[syntactic categories]] is at

* n-category cafe 2010: [what is a theory](http://golem.ph.utexas.edu/category/2010/07/what_is_a_theory.html)

Other references include

* [[Mamuka Jibladze]], [[Teimuraz Pirashvili]], _Cohomology of algebraic theories_, J. Algebra  __137__ (1991),  no. 2, 253--296, doi

* wikipedia: [list of first-order theories](http://en.wikipedia.org/wiki/List_of_first-order_theories), [Morley's categoricity theorem](http://en.wikipedia.org/wiki/Morley%27s_categoricity_theorem), [Decidability (logic)](http://en.wikipedia.org/wiki/Decidability_%28logic%29), [signature (logic)](http://en.wikipedia.org/wiki/Signature_%28logic%29)

In [[Coq]] theories are specified with the

* _[[Gallina specification language]]_.

[[!redirects theories]]

[[!redirects logical theory]]
[[!redirects logical theories]]

[[!redirects theory (logic)]]


theory (physics)

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

## Idea

### General idea

In [[physics]] the term _theory_ or _physical theory_ traditionally refers, somewhat vaguely, to a given set of notions and rules, usually formulated in the language of [[mathematics]], that describe how some [[physical system]] or class of physical systems behaves. Typically these systems are highly idealized, in that the theories describe only certain aspects.

Often a given such theory depends on many free parameters. When a choice of such parameters is made or the range of the parameters is being restricted one tends to call the result a _[[model (in theoretical physics)]]_. For more on this see _[Theories and their Models](#TheoriesAndTheirModels)_ below.

The most accurate general theory of fundamental physics known is _[[Einstein gravity]]_ and _[[quantum field theory]]_. The best available choices of parameters in this general theory that make it fit the specifics of the observed world ([[phenomenology]]) are [[model (in theoretical physics)|models]] which accordingly are called the _standard models_: there is the _[[standard model of particle physics]]_ and the _[[standard model of cosmology]]_.

### Formalization

Beware that, therefore, the use of the terms _theory_ and _model_ in [[physics]] is _different_ from the same terms as used in [[logic]] (see at [[theory (logic)]] and [[model (logic)]]).

But most theories of fundamental physics (and many theories of effective physics such as [[solid state physics]])  fit into a pattern that can be [[axiom|axiomatized]] at least to some extent:

these physical theories are specified by a ([[local Lagrangian|local]]/[[extended Lagrangian|extended]]) [[Lagrangian]] on a space of [[field (physics)|fields]] over a given [[spacetime]]/[[worldvolume]] [[manifold]] $X$, hence by an [[action functional]]

$$
  S \;\colon\; [X, \mathbf{Fields}]_{\mathbf{H}} \to U(1)
  \,.
$$

In particular the corresponding [[classical field theory]] has as its "space of models" the [[critical locus]]

$$
  \underset{\phi \in [X, \mathbf{Fields}]_{\mathbf{H}}}{\sum}
  ( \mathbf{d} S_\phi \simeq 0 )
$$

of such an [[action functional]], the space of solutions of the [[Euler-Lagrange equations]]. A point in this space is a single "physically realizable" configuration of [[field (physics)|fields]] in this theory, disregarding [[quantum field theory]]-corrections, and a small-parameter subspace is often referred to as "a model" of the theory.

In this perspective of classical field theory, two different action functionals on different spaces of fields but with [[equivalence|equivalent]] [[critical loci]] are regarded as "equivalent physical theories". One often sees the term "classically equivalent" for this notion used in the literature.

But the full [[quantum field theory]] determined by a [[Lagrangian]]/[[action functional]] depends on more than just the [[critical locus]], which is just something like the lowest order approximation to the quantum theory (in a sense that can be made precise, for instance in [[deformation quantization]] in terms of [[power series]] developments in [[Planck's constant]].)

## Examples

### General principles

One broad way of classifying physical theories is by the extent to which they take [[quantum physics]] into account. We have

* [[classical field theory]]

* [[prequantum field theory]]

* [[quantum field theory]]

Here [[quantum field theory]] is the most refined framework, which underlies the [[standard model of particle physics]].

The notion of quantum field theory, fundamental as it is, is quite flexible and in particular it naturally captures the concept that a given quantum field theory only describes phenomena that occur below a certain [[energy]] range and treats all phenomena at higher energy as the average over an unspecified more refined theory. This is the notion of

* [[effective quantum field theory]].

Crucially, [[Einstein gravity]] is not known to have a formulation as a _fundamental_ quantum field theory with finitely many unspecified parameters ([[renormalization|renormalizable]]).  But it may well be a [[effective quantum field theory]], the approximation to a more refined physical theory valid at higher energies. (This is the issue of [[quantum gravity]].) A proposal for a physical theory that achieves this is called _[[string theory]]_.

### Types of theories

* [[free field theory]]

* [[gauge theory]]

### The fundamental phenomenological theories

* [[gravity]], [[Yang-Mills theory]]

* [[Einstein-Maxwell theory]]

* [[Einstein-Yang-Mills theory]]

* [[Einstein-Maxwell-Yang-Mills-Dirac-Higgs theory]]

[[!include standard model of fundamental physics - table]]

### Theories and their models
 {#TheoriesAndTheirModels}

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

The [[classical field theory]] of [[gravity]] is a physical theory which asserts that [[spacetime]] is modeled by a [[pseudo-Riemannian manifold]] equipped with certain further [[force]] and [[matter]] [[field (physics)|fields]], such that this data satisfies [[Einstein equations]]. But if one furthermore specifies a particular such pseudo-Riemannian manifold etc. one may call this a _model_ of [[gravity]]/[[cosmology]]. The _[[FRW model]]_ is an example: here one specifies that the given pseudo-Riemannian metric and the [[matter]] [[field (physics)|field]] content is _homogenous_ and _isotropic_. This is highly restrictive but still does not single out a unique solution. The remaining parameter is $k \in \{-1,0,1\}$, determining whether in this solution space has negative, positive or vanishing constant curvature.

=--

## Related concepts

* [[model (physics)]], [[experiment]], [[coordination]]

* [[theory of everything]]

* [[computable physics]]

* [[mathematical physics]]

* [string theory FAQ -- How do physical theories generally make predictions, anyway?](http://ncatlab.org/nlab/show/string+theory+FAQ#AsideHowToPhysicalTheorieyGenerallyMakePredictionsAnyway)

## References

* [[James Wells]], _Lexicon of Theory Qualities_ ([pdf](http://www-personal.umich.edu/~jwells/prms/prm8.pdf))

[[!redirects theories (physics)]]

[[!redirects physical theory]]
[[!redirects physical theories]]

[[!redirects theory in physics]]
[[!redirects theories in physics]]

[[!redirects theory of physics]]
[[!redirects theories of physics]]


Theory and Applications of Categories

_Theory and Applications of Categories_ (or just _TAC_) is an online journal about [[category theory]].  In addition to new material, they also make available important classics that have been hard to find. The journal was launched in 1995 on the [[categories mailing list]], with [this email](https://www.mta.ca/~cat-dist/catlist/1999/elec-journal).

The website is <http://www.tac.mta.ca/tac/>, from which you have access to all articles in all issues.

## Reprints in Theory and Applications of Categories

This subpublication republishes old, but important works in category theory.
Some publications are original.
In their own words:

> Reprints in Theory and Applications of Categories will disseminate articles or other works from the body of important literature in Category Theory and closely related subjects which has never been published in journal form, or which has been published in journals whose narrow circulation makes access very difficult.

* <http://www.tac.mta.ca/tac/reprints/>

## Expositions in Theory and Applications of Categories

This subpublication publishes expository articles.
In their own words:

> Expositions in Theory and Applications of Categories ("TAC Expositions") is a new series specifically designed for publication of well-written and novel expository articles on topics of current research interest in the theory and/or applications of categories.

* <http://www.tac.mta.ca/tac/expositions/>

category: reference

[[!redirects TAC]]


theory of abelian groups

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#### Type theory
#### Topos Theory
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[[!redirects theory of divisible abelian groups]]
[[!redirects theory of torsion-free abelian groups]]
[[!redirects theory of torsion abelian groups]]
[[!redirects theory of ordered abelian groups]]

##Idea##
The _theory of abelian groups_ $T$ is the logical [[theory]] whose models in a [[cartesian category]] are the [[abelian group]] objects.

##Definition##
The _theory of abelian groups_ $T$ is the theory over the signature with one sort $X$, one constant $0$, two function symbols $+$ (of sort $X\times X\to X$) and $-$ (of sort $X\to X$) and equality with axioms:

* $\top\vdash_x x+0=x \quad$
* $\top\vdash_{x,y,z} x+(y+z)=(x+y)+z\quad$,
* $\top\vdash_x  x+(-x)=0\quad$.
* $\top\vdash_{x,y} x+y=y+x\quad$.

The theory of abelian groups is [[algebraic theory|algebraic]]. In the following we list some extensions that use increasingly stronger fragments of geometric logic.

## Extensions of the theory of abelian groups

The **theory of torsion-free abelian groups** $T^\infty$ results from $T$ by addition of the following axioms:

* For all $n\geq 2$: $((nx=0))\vdash_x (x=0))\quad$.

The resulting theory is a Horn theory.

The **theory of divisible abelian groups** $T^\backslash$ results from $T$ by addition of the following regular axioms:

* For all $n\geq 2$: $\top\vdash_x (\exists y) (ny=x)\quad$.

The **theory of divisible torsion-free abelian groups** $T^{\backslash\infty}$ results from $T^\backslash$ by adding the above axioms for torsion-freeness. The resulting theory is cartesian.

The **theory of torsion abelian groups** $T^~$ is an infinitary geometric theory resulting from $T$ by addition of:

* $\top\vdash_x \bigvee_{n\geq 2} (nx=0)\quad$.

##Related concepts##

* [[abelian group]]

* [[torsion]]

* [[geometric theory]]

* [[theory of objects]]

* [[theory of decidable objects]]

##References##

* [[Peter Johnstone]], _Sketches of an Elephant II_, Oxford UP 2002. pp.812-815.


theory of algebraically closed fields

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#Contents#
* table of contents
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## Idea
Fields are finitely [[first-order theory|first-order]] axiomatizable. An [[algebraically closed field]] is further axiomatized by infinitely many sentences which state that all non-constant polynomials have a root. Once we additionally specify a characteristic $p$, $\mathsf{ACF}_p$ turns out to be complete, [[elimination of imaginaries|eliminates imaginaries]], is [[stability theory|stable]], and admits [[quantifier elimination]].

## Definition
$\mathsf{ACF}$ is the countable collection of sentences in the language $\mathcal{L}_{\operatorname{ring}}$ of rings given by:

$$\{\text{field axioms}\} + \left\{ (\forall a_0, \ldots, a_{n-1}) (\exists x)[x^n + \sum_{i=0}^{n-1} a_i x_i^i = 0] \right\},$$

where $n = 1, 2, \ldots$.

We can additionally specify a characteristic $p$ to obtain $\mathsf{ACF}_p$ by either adding the axioms $\{ 1 \ne 0 , 1+1 \ne 0 , \cdots \}$ to get $\mathsf{ACF}_0$ or adding the axiom $\underset{p\; terms}{\underbrace{1 + \cdots + 1}} = 0$ to get $\mathsf{ACF}_p$ (where $p$ is prime).

## Properties
* $\mathsf{ACF}$ has quantifier elimination. This amounts to a special case of Chevalley's direct image theorem from algebraic geometry.

* $\mathsf{ACF}$ is [[stability theory|stable]].

* $\mathsf{ACF}$ is _totally transcendental_: [[Morley rank]] is defined everywhere. In this setting Morley rank subsumes the usual Krull dimension of an algebraic variety.

* $\mathsf{ACF}$ is _uncountably categorical_: for each uncountable $\kappa$, models of $\mathsf{ACF}$ of size $\kappa$ must all be isomorphic. This fails at $\aleph_0$: $\mathbb{Q}^{\operatorname{alg}}$ has transcendence degree zero while there exist countable algebraically closed overfields of $\mathbb{Q}$ with infinite transcendence degree.

* $\mathsf{ACF}$ [[elimination of imaginaries|eliminates imaginaries]]. This means that its [[syntactic category]] $\mathbf{Def}(\mathsf{ACF})$ has effective [[internal congruence|internal congruences]], and has a good [[model-theoretic Galois theory|Galois theory]].

* It's easy to see that $\mathsf{ACF}$ [[elimination of imaginaries|codes]] all finite sets: if $R$ is a finite set of points inside a monster $\mathbb{M}$, its code is the tuple $c$ of coefficients for the polynomial

$$f(X) \overset{\operatorname{df}}{=} \displaystyle \prod_{r \in R} (X - r),$$

so that $c$ is fixed by an automorphism of $\mathbb{M}$ if and only if that automorphism permutes $R$.

## Related concepts
* [[theory of differentially closed fields]]

* [[quantifier elimination]]

* [[stable theory]]

* [[Morley rank]]

## References
* Dave Marker, _Model Theory: An Introduction_.

[[!redirects ACF]]


theory of categories

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#### Topos Theory
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[[!redirects  geometric theory of categories]]
[[!redirects essentially algebraic theory of categories]]
[[!redirects cartesian theory of categories]]

#Contents#
* table of contents
{:toc}

##Idea
The [[geometric theory|geometric]] **theory of categories** $\mathbb{K}$ is a [[first-order logic|first-order]] axiomatisation of the concept of a  [[category]]. $\mathbb{K}$ is (morally) an [[essentially algebraic theory]], and literally a cartesian theory i.e. _grosso modo_ a [[regular theory]] where existential quantification is provably unique.

Historically, the first axiomatisation was given by [[William Lawvere| F. William Lawvere]] in his [[Functorial Semantics of Algebraic Theories|dissertation]] in 1963 as part of a formalisation of the [[category of categories]] (cf. Lawvere ([1963](#Law63},[1966](#Law66)) or [[ETCC]] for further information).

## Definition

The [[signature]] $\Sigma_\mathbb{K}$ has two sorts $O$ and $A$ for objects and arrows, respectively, three function symbols $d_0:A\to O$ , $d_1:A\to O$ and $id:O\to A$, assiging to morphisms their domain, respectively, codomain and to the objects their [[identity|identity morphisms]], as well as a relation symbol $C\rightarrowtail A\times A\times A$ expressing composition of morphisms.

The **theory of categories** is the theory $\mathbb{K}$ over the signature $\Sigma_\mathbb{K}$ with the following sequents:[^var]

[^var]: Here and in the following $x,y$ are variables ranging over O, whereas the $f_1,f_2,\dots$ range over A; the contexts are assumed to be canonical i.e. consisting of all free distinct variables occurring in a sequent listed in order of first appearance.

* $\top\vdash d_0(id(x))=x\wedge d_1(id(x))=x\quad$ ,

* $C(f_1,f_2,f_3)\vdash d_0(f_1)=d_0(f_3)\wedge d_1(f_1)=d_0(f_2)\wedge d_1(f_2)=d_1(f_3)\quad $ ,

* $C(f_1,f_2,f_3)\wedge C(f_1,f_2,f_4)\vdash f_3=f_4\quad $ ,

* $d_1(f_1)=d_0(f_2)\vdash\exists f_3 C(f_1,f_2,f_3)\quad$ ,

* $\top \vdash C(f,id(d_1(f)),f)\wedge C(id(d_0(f)),f,f)\quad $ ,

* $C(f_1,f_2,f_4)\wedge C(f_2,f_3,f_5)\wedge C(f_4,f_3,f_6)\wedge C(f_1,f_5,f_7)\vdash f_6=f_7\quad $.

### Remark

The third axiom implies that the existential quantification occurring in the fourth axioms is unique thus $\mathbb{K}$ is indeed _cartesian_. Since the chosen signature uses a composition relation instead of a partial composition operation the theory as given here is not literally [[essentially algebraic theory|essentially algebraic]] in the syntactic sense, though, of course, it is [[Morita equivalence|Morita equivalent]] to an essentially algebraic axiomatisation. Other signatures, like a single sorted "arrows only" one, are also possible, the one used here (taken from Johnstone [1977](#J77)) leans on the so called **elementary theory of abstract categories** as presented in Lawvere ([1966](#Law66)).

## Properties

Foremost, we have

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

Let $\mathcal{E}$ be a [[Grothendieck topos]]. Then $Mod_{\mathbb{K}}(\mathcal{E})=cat(\mathcal{E})$ with $cat(\mathcal{E})$ the category of (small) [[internal category|internal categories]] in $\mathcal{E}$ with internal functors as morphisms.$\qed$
=--

+-- {: .num_prop}
###### Proposition
The [[classifying topos]] $Set[\mathbb{K}]$ is the presheaf topos on the opposite of the category of finitely presentable categories in $Set$.
=--

This follows from the fact that the classifying toposes of [[cartesian theory|cartesian theories]] are generally given as the presheaf toposes on the opposite of the category of finitely presentable models in $Set$ (cf. Johnstone [2002](#J02), p.891).[^diss] The proposition holds more generally with $Set$ replaced by an arbitrary topos $\mathcal{E}$ with a [[natural numbers object]] (cf. Johnstone-Wraith [1978](#JW78), p.226).$\qed$

[^diss]: Apparently $Set[\mathbb{K}]$ made its first appearance in Johnstone's 1974 Cambridge dissertation on "Internal category Theory" (cf. Tierney [1976](#Tierney76)).

## Some related theories

* The [[theory of model homomorphisms|theory $\mathbb{K}^2$ of $\mathbb{K}$-model homomorphisms]] is the **theory of functors**.

* Recall that a _discrete opfibration_ is an internal functor $F\,:\,\mathbf{F}\to \mathbf{C}$ with the property that the following diagram
$$
\array{
   & & \\
F_1 &\overset{d_0}{\longrightarrow}& F_0
    \\
    {}^{\mathllap{\gamma_1}}
    \downarrow
    & &
    \downarrow^{\mathrlap{\gamma_0}}
    \\
    C_1
      &\underset{d_0}{\longrightarrow}&
    C_0
    \\  & &
  }
$$
  is pullback. Whence by adding the appropriate geometric axiom to the theory $\mathbb{K}^2$ of functors one obtains as a quotient the **theory of discrete opfibrations** $\mathbb{K}^2_\downarrow$.

  Let $\mathbf{C}\in cat(\mathcal{E})$ be an internal category. By pulling back its classifying morphism $\overline{\mathbf{C}}\,:\,\mathcal{E}\to \mathcal{E}[\mathbb{K}]$ along the geometric morphism $d_1\,:\,\mathcal{E}[\mathbb{K}^2_\downarrow]\to \mathcal{E}[\mathbb{K}]$ that classifies the codomain of the generic discrete opfibration as an internal category
$$
\array{ \mathcal{E}[\mathbb{K}^\downarrow_\mathbf{C}] &\longrightarrow &\mathcal{E}[\mathbb{K}^2_{\downarrow}]
    \\
    \downarrow
    & &
    \downarrow^{d_1}
    \\
    \mathcal{E}
      &\underset{\overline{\mathbf{C}}}{\longrightarrow}&
    \mathcal{E}[\mathbb{K}]
    \\ & &
  }
$$
  one obtains the classifying topos $\mathcal{E}[\mathbb{K}^{\downarrow}_\mathbf{C}]$ for discrete opfibrations on $\mathbf{C}$, or, equivalently, internal diagrams (=copresheaves) on $\mathbf{C}$. Since an internal [[profunctor]] $\mathbf{C} ⇸ \mathbf{D}$ is by definition an internal diagram on $\mathbf{C}^{op}\times\mathbf{D}\,$, one can use this construction to obtain the _classifying topos for internal profunctors_ from $\mathbf{C}$ to $\mathbf{D}$ by pulling back along the classifying morphism for $\mathbf{C}^{op}\times\mathbf{D}$ (cf. Johnstone [1977](#J77), p.209).

* By adding the following sequents to the theory $\mathbb{K}$ of categories one obtains the **theory of filtered categories** $\mathbb{K}^\gt$ with models the internal [[filtered category|filtered categories]] (cf. Johnstone [1977](#J77), p.203):

  $\array{\\
\top\vdash \exists x (x=x)
\\
\\
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\top\vdash \exists f_1\exists f_2 (d_0(f_1)=x\wedge d_0(f_2)=y\wedge d_1(f_1)=d_1(f_2))
\\
d_0(f_1)=d_0(f_2)\wedge d_1(f_1)=d_1(f_2)\vdash \exists f_3\exists f_4\big (C(f_1,f_3,f_4)\wedge C(f_2,f_3,f_4)\big )\quad}$

## Related entries

* [[category]]

* [[category theory]]

* [[geometric theory]]

* [[theory of objects]]

* [[theory of model homomorphisms]]

* [[theory of flat functors]]

* [[ETCC]]

## References

* [[Peter Freyd|P. J. Freyd]], A. Scedrov, _[[Categories, Allegories]]_ , North-Holland Amsterdam 1990.

* {#J77}[[Peter Johnstone]], _Topos Theory_ , Academic Press New York (1977). (Also available as Dover Reprint, Mineola 2014, p.202f)

* {#J02}[[Peter Johnstone]], _Sketches of an Elephant vol.2_ , Oxford UP 2002. (D1.1.7(e), p.813, D1.3.4, p.833)

* {#JW78}[[Peter Johnstone]], [[Gavin Wraith]], _Algebraic Theories in Toposes_ , pp.141-242 in Johnstone, Paré (eds.), _Indexed Categories and Their Applications_ , Springer LNM **661** Heidelberg 1978.

* {#Law63}F. W. Lawvere, _[[Functorial Semantics of Algebraic Theories]]_ , Ph.D. thesis, Columbia University New York 1963. (With an author's comment and a supplement in: Reprints in Theory and Applications of Categories, no.5 (2004) pp.1-121 ([pdf](http://www.tac.mta.ca/tac/reprints/articles/5/tr5.pdf)))

* {#Law66} *[[William Lawvere]], *The Category of Categories as a Foundation for Mathematics*, pp. 1-20 in: [[John W. Gray]], *Fibred and Cofibred Categories*, in: [[Samuel Eilenberg|S. Eilenberg]], [[D. K. Harrison]], [[S. MacLane]], [[H. Röhrl]] (eds.): *[[Proceedings of the Conference on Categorical Algebra - La Jolla 1965]]*, Springer (1966) [[doi:10.1007/978-3-642-99902-4](https://doi.org/10.1007/978-3-642-99902-4)]

* {#Tierney76}[[Myles Tierney]], _Forcing Topologies and Classifying Toposes_ , pp.211-219 in Heller, Tierney (eds.), _Algebra, Topology and Category Theory_  , Academic Press New York 1976.


theory of decidable objects

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[[!redirects theory of infinite decidable object]]
[[!redirects theory of infinite decidable objects]]

## Idea
The _theory of decidable objects_ is the logical [[theory]] $\mathbb{D}$ whose models in a [[coherent category]] are precisely the [[decidable object|decidable objects]].

##Definition##
The _theory of decidable objects_ is the theory $\mathbb{D}$ over the signature with one sort and one binary relation $#$ besides equality with axioms $(x#x)\vdash_x\perp$ and $\top\vdash_{x,y} ((x#y)\vee(x=y))$.

## Properties

* For a [[topos]] $\mathcal{E}$ the category of models $Mod_{\mathbb{D}}(\mathcal{E})$ is the category of [[decidable object|decidable objects]] in $\mathcal{E}$. The _[[classifying topos]]_ $Set[\mathbb{D}]$ for the theory of decidable objects is the functor category $[FinSet_{mono},Set]$ where $FinSet_{mono}$ is the category of finite sets and monomorphisms. $Set[\mathbb{D}]$ is a [[locally decidable topos]].

## Infinite decidable objects

* The theory of _infinite decidable objects_ $\mathbb{D}_\infty$ adds to $\mathbb{D}$ the axioms $\top\vdash_{x_1,\dots,x_n} (\exists y)\bigwedge_{i=1}^{n}(y#x_i)$ for all $n$ with $\top\vdash(\exists y)\top$ for $n=0$. The models of $\mathbb{D}_\infty$ are precisely the infinite decidable objects and its classifying topos $Set[\mathbb{D}_\infty]$ is the [[Schanuel topos]].

## Related entries

* [[theory of objects]]

* [[decidable object]]

* [[locally decidable topos]]

* [[Schanuel topos]]

##References##

* [[Peter Johnstone]], _Sketches of an [[Elephant]] II_, Oxford UP 2002. (pp.906,925)


theory of everything

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## Idea

In _[[quantum physics|quantum]] [[physics]]_, specifically in fundamental physics/[[high energy physics]], the term "theory of everything" is used to refer to a (hypothetical) [[theory (physics)|theory]] which coherently subsumes "all" of fundamental physics, in some sense.

Taken at face value this terminology has its evident problems, but its usage is to be understood in the context of the situation of the field of theoretical physics at the turn to the 21st century, where it has the evident and justified restricted meaning of a theory which, somehow, coherently subsumes and possibly explains from more fundamental principles both:

* the [[nLab:standard model of particle physics]], based on [[nLab:Yang-Mills theory]];

* the [[nLab:standard model of cosmology]], based on [[nLab:Einstein gravity]].

One issue here is to unify these at the level of [[quantum field theory]] proper (which seems to be accomplished by [[string theory]] at least in broad outline, for more see at *[[quantum gravity]]*), the next issue is to go beyond [[perturbative quantum field theory]] and account for still largely elusive [[non-perturbative effects]] (working title: "[[M-theory]]") such as [[confinement]] via the [[mass gap problem|mass gap]] of [[QCD]]. Yet another issue is to possibly find an explanation for some of the random-looking structures in these standard models, for more on this see also at *[[grand unified field theory]]* and *[[exceptional naturalism]]*; or else to understand these as just random artefacts (in a "[[multiverse]]" of physically allowed structures).

Notice here that in their colloquial meaning the terms _theory of everything_ and _grand unified field theory_ largely overlap, but that as technical terms they are understood quite differently: The term _[[grand unified field theory]]_ is used almost exclusively for [[nLab:gauge group]]-unification, which is not even part of all proposals for a "theory of everything".

## Related concepts

* [[high energy physics]]

* [[quantum gravity]]

* [[grand unified theory]]

* [[mass gap problem]]

* [[mathematical physics]]

* [[string theory]], [[M-theory]]

* [[exceptional naturalism]]

* [[gauge coupling unification]]

* [[multiverse]]

## References

Back in the day, [[David Hilbert]] would speak of *Weltgesetze* ("world laws"), see starting on page 396 (esp. 417, 423) in:

* {#SauerMajer} [[Tilman Sauer]], [[Ulrich Majer]] with  Arne Schirrmacher, Heinz-Jürgen Schmidt (eds.): *David Hilbert's Lectures on the foundations of physics 1915-1927 --- Relativity, Quantum Theory and Epistemology*, Springer (2009) [[doi:10.1007/b12915](https://doi.org/10.1007/b12915)]

See also:

* Wikipedia, _[Theory of everything](http://en.wikipedia.org/wiki/Theory_of_everything)_

[[!redirects theories of everything]]

[[!redirects theories of everything]]


theory of flat functors

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[[!redirects theory of flat diagrams]]
[[!redirects Theory of flat diagrams]]
[[!redirects Theory of flat functors]]
[[!redirects Theory of J-continuous flat diagrams]]
[[!redirects theory of continuous flat functors]]
[[!redirects theory of J-continuous flat functors]]

## Idea

The **theory of J-continuous flat functors** on a [[site]] $(\mathcal{C},J)$ is the [[geometric theory]] $\mathbb{T}_J^{\mathcal{C}}$ whose models in a Grothendieck topos $\mathcal{E}$ correpond precisely to J-continuous flat functors $\mathcal{C}\to\mathcal{E}$ and, by well known representation theorems, to [[geometric morphisms]] $\mathcal{E}\to Sh(\mathcal{C},J)$ whence $\mathbb{T}_J^{\mathcal{C}}$ provides a uniform albeit (in general) uneconomical axiomatization of the [[geometric theory]] classified by $Sh(\mathcal{C},J)$.

## Definition

Let $(\mathcal{C},J)$ be a small site. The **theory of J-continuous flat functors** is given by the following geometric theory $\mathbb{T}_J^{\mathcal{C}}$ over the signature $\Sigma$ which has a sort $\lceil A\rceil$ for every object $A$ of $\mathcal{C}$ and function symbols $\lceil f\rceil :\lceil A\rceil\to\lceil B\rceil$ for every function $f:A\to B$:

* axioms $\top\vdash_x \lceil i\rceil(x)=x$ for all identity morphisms $i$.

* axioms $\top\vdash_x\lceil f\rceil(x)=\lceil h\rceil(\lceil g\rceil (x))$ for all triples $f,g,h$ with $f=g\circ h$.

* an axiom $\top\vdash\underset{A\in\mathcal{C}}{\bigvee} (\exists x:\lceil A\rceil) \top$.

* axioms $\top\vdash_{x:A,y:B}\underset{A\overset{f}{\leftarrow} C\overset{g}{\rightarrow B}}{\bigvee}\exists (z:\lceil C\rceil)\big(\lceil f\rceil (z:\lceil C\rceil)=x:\lceil A\rceil\wedge \lceil g\rceil (z:\lceil C\rceil)=y:\lceil B\rceil\big)$ where the disjunction ranges over all cones on the discrete diagram consisting of $A$ and $B$.

* axioms $\lceil f\rceil (x:\lceil A\rceil)=\lceil g\rceil (x:\lceil A\rceil)\vdash_{x:\lceil A\rceil}\bigvee_{h:C\to A|f\circ h=g\circ h} \exists(z:\lceil C\rceil)\big(\lceil h\rceil(z:\lceil C\rceil)=x:\lceil A\rceil\big)$ for any pair $f,g:A\to B$, the disjunction ranging over all $h$ equalizing $f,g$.

* axioms $\top\vdash_{x:\lceil A\rceil}\bigvee_{i\in I}\exists (y_i:\lceil B_i\rceil) \big(\lceil f_i\rceil(y_i:\lceil B_i\rceil)=x:\lceil A\rceil\big)$ for each J-covering $\{f_i:B_i\to A|i\in I\}$.

### Remark

The first two axiom schemata correspond to functoriality, the third to fifth to [[filtered category|filteredness]] (this being a notion equivalent to flatness), and the last to J-continuity (turning J-covers into epimorphic families).

Given a small category $\mathcal{C}$, $Set^{\mathcal{C}^{op}}\simeq Sh(\mathcal{C},J_0)$ where $J_0$ is the [[trivial topology]]. The theory of flat functors on $\mathcal{C}$ coincides with $\mathbb{T}_{J_0}^{\mathcal{C}}$ since the last axiom schema becomes redundant for the maximal sieves whence the first five axiom schemata axiomatize the notion of a flat functor on $\mathcal{C}$ and the resulting **theory of flat functors** (also called the _theory of flat diagrams_) is seen to be of [[theory of presheaf type|presheaf type]].

## Examples

* Let $(\emptyset,\emptyset)$ be the empty site. Then the signature $\Sigma$ is empty and only the third condition takes hold, yielding $\mathbb{T}_\emptyset^{\emptyset}=\{\top\vdash\bot\}$ i.e. the empty topos $Sh(\emptyset,\emptyset)=\mathbf{1}$ classifies the **inconsistent theory**.

## Related entries

* [[flat functor]]

* [[geometric theory]]

* [[classifying topos]]

* [[Diaconescu's theorem]]

* [[theory of categories]]

* [[theory of objects]]

## References

* [[Olivia Caramello]], _Lattices of theories_ , arXiv:0905.0299 (2009). ([abstract](https://arxiv.org/abs/0905.0299), p.11)

* [[Olivia Caramello]], _Theories, Sites, Toposes_ , Oxford UP 2018. (p.58f)

* [[Peter Johnstone]], _Sketches of an Elephant vol.II_ , Oxford UP 2002. (p.897)


theory of G-torsors

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## Idea

Given a group $G\;$, the **theory of G-torsors** is the logical theory whose models in a category $\mathcal{C}$ are the [[torsor|G-torsors]].

## Definition

Let $G$ be a [[group]]. The _theory of G-torsors_ $\mathbb{T}_G$ is the [[geometric theory]] over the signature with one sort $U$, a set of unary functions symbols ${g\in G}$ and the following axiom scheme:

$$
\begin{aligned}
\top & \vdash (g_i g_j)u=g_i(g_j u)\quad\text{for all}\; g_i,g_j
\\
\top&\vdash (\exists u)\; \top \quad\text{(U is inhabited)}
\\
g_{i}u = g_{j}u &\vdash \bot\quad \text{for all pairs}\;g_i\neq g_j\quad\text{(G acts freely)}
\\
\top &\vdash \bigvee_{g\in G} gx=y\quad \text{(G acts transitively)}\quad .
\end{aligned}
$$

## Properties

Let $G$ be a group in $Set$ and $\mathcal{E}$ be a [[Grothendieck topos]] with $\Delta\dashv \Gamma:\mathcal{E}\to Set$. Recall that a _G-torsor_ over $\mathcal{E}$ is defined as an object $T$ of $\mathcal{E}$ together with a left action $\mu:\Delta(G)\times T\to T$ such that $T\to 1$ is epi and $\mu$ induces an isomorphism $(\mu,\pi_2):\Delta(G)\times T\to T \times T$.

The category of models of $\mathbb{T}_G$ in $\mathcal{E}$ is the category $Tor(\mathcal{E},G)$ of G-torsors over $\mathcal{E}$ and, furthermore, $Tor(\mathcal{E},G)\cong Hom_{Top}(\mathcal{E},Set^{G^{op}})$ i.e. $Set^{G^{op}}$ is the [[classifying topos]] $Set[\mathbb{T}_G]$ of $\mathbb{T}_G$.

It is a standard fact that Boolean presheaf toposes correspond precisely to toposes $Set^{\mathcal{G}^{op}}$ of actions of groupoids $\mathcal{G}$ whereas toposes $Set^{G^{op}}$ of group actions correspond to the connected toposes among these whence the toposes of the form $Set^{G^{op}}$ can be characterized as the connected Boolean presheaf toposes.

This affords a logical characterization of $\mathbb{T}_G$ as a **complete Boolean theory of presheaf type** since Boolean presheaf toposes are atomic and atomic toposes are connected precisely when they are [[two-valued topos|two-valued]] which is equivalent to the completeness of the classified theory (See at [[two-valued topos]] and [[theory of presheaf type]] for the logical terminology). To sum up: toposes $Set^{G^{op}}$ of group actions are up to equivalence precisely the classifying toposes of complete Boolean theories of presheaf type.[^finite]

[^finite]: Looking at $\mathbb{T}_G$ it seems reasonable to conjecture that the toposes of _finite_ group actions correspond precisely to the classifying toposes of the finitely presentable complete Boolean theories of presheaf type i.e. those with finitely many function symbols and a finite list of axioms.

## Related entries

* [[torsor]]

* [[theory of objects]]

* [[theory of decidable objects]]

* [[classifying topos]]

## References

The syntactic theory is explicitly stated p.42 in

* {#Lawvere75}[[F. William Lawvere]], _Variable sets etendu and variable structure in topoi_ , Lecture notes University of Chicago 1975.

The "semantic" side was well known to the Grothendieck school and its elements are nicely exposed e.g. in section VIII.2 of

* {#MM94} [[Saunders Mac Lane|S. Mac Lane]], [[Ieke Moerdijk|I. Moerdijk]], _[[Sheaves in Geometry and Logic]]_ , Springer Heidelberg 1994.

[[!redirects Theory of G-torsors]]
[[!redirects theory of torsors]]


theory of model homomorphisms

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[[!redirects theory of $\mathbb{T}-model homomorphisms]]
[[!redirects theory of T-model homomorphisms]]
[[!redirects theory of homomorphism]]

#Contents#
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##Idea
For any [[geometric theory]] $\mathbb{T}$ there exists a geometric theory $\mathbb{T}^2$, the **theory of $\mathbb{T}$-model homomorphisms**, whose models in a [[Grothendieck topos]] $\mathcal{E}$ are precisely the homomorphisms between $\mathbb{T}$-models in $\mathcal{E}$.

## Definition

Let $\mathbb{T}$ be a [[geometric theory]] over the [[signature]] $\Sigma$.

Let $\Sigma^2$ be the signature containing a pair of new sort symbols $X^0$, $X^1$ and a new function symbol $f_X:X^0\to X^1$ for any sort symbol $X$ in $\Sigma$, as well, as pairs $g^0:X_1^0\times\dots\times X_n^0\to Y^0$, $g^1:X_1^1\times\dots\times X_n^1\to Y^1$ of new functions symbols (resp. pairs $R^0\rightarrowtail X_1^0\times\dots\times X_n^0$, $R^1\rightarrowtail X_1^1\times\dots\times X_n^1$ of new relation symbols) for any function symbol $g:X_1\times\dots\times X_n\to Y$ (resp. relation symbol $R\rightarrowtail X_1\times\dots\times X_n$) in $\Sigma$.

The **theory of $\mathbb{T}$-model homomorphisms** is the theory $\mathbb{T}^2$ over the signature $\Sigma^2$ with the following sequents:

 A pair of sequents $\varphi^0\vdash \psi^0$, $\varphi^1\vdash \psi^1$ for any sequent $\varphi\vdash \psi$ in $\mathbb{T}$ where the new sequents result from replacing the function and relation symbols in $\varphi\vdash \psi$ with their 0-indexed, resp. 1-indexed pendants where necessary[^prop]. Plus a sequent

:$\top\vdash f_Y(g^0(x_1,\dots,x_n))=g^1(f_{X_1}(x_1),\dots,f_{X_n}(x_n))$

resp.

:$R^0(x_1,\dots,x_n)\vdash R^1(f_{X_1}(x_1),\dots,f_{X_n}(x_n))$

for any pair $g^0$, $g^1$ of function symbols in $\Sigma^2$ corresponding to $g:X_1\times\dots\times X_n\to Y$ in $\Sigma$ (in case $n=0\,$, the sequent reads $\top\vdash f_Y(g^0)=g^1\,$), resp. any pair $R^0$, $R^1$ of relation symbols in $\Sigma^2$ corresponding to $R\rightarrowtail X_1\times\dots\times X_n$ in $\Sigma$ (in case $n=0\,$, the sequent reads $R^0\vdash R^1\,$).

[^prop]:  It is understood that variables and contexts are updated with the new sorts e.g. the sequent $\top\vdash_{x:S} x=x$ yields two sequents $\top\vdash_{x:S^0} x=x$ and $\top\vdash_{x:S^1} x=x$ and $\top\vdash \big (\forall x:S\big ) x=x$ yields $\top\vdash \big (\forall x:S^0\big ) x=x$ and $\top\vdash \big (\forall x:S^1\big ) x=x$ etc. In case, the sequent $\varphi\vdash\psi$ contains nothing to update (e.g. the sequent $\top\vdash\bot$) one just takes the old sequent as the "new pair".

## Properties

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

Let $\mathcal{E}$ be a [[Grothendieck topos]]. Then

$${\mathbb{T}^2}\text{-}Mod(\mathcal{E})=\mathbb{T}\text{-}Mod(\mathcal{E})^2=\mathbb{T}\text{-}Mod(\mathcal{E}^2)\quad.$$
=--

Cf. Johnstone ([1977](#J77), p. 203; [2002](#J02), p.425), Mac Lane-Moerdijk ([1994](#MM94), ex.X.5 p.572).

Note that the second equality determines the class of $\mathbb{T}$-model homomorphisms once the $\mathbb{T}$-models in the arrow categories $\mathcal{E}^2$ are known.

**Remark**:
This e.g. precludes the existence of geometric theories $\mathbb{T}^{op}$ or $\mathbb{T}^\times$ with the property that, given a geometric theory $\mathbb{T}$, the category of models ${\mathbb{T}^{op}}\text{-}Mod(\mathcal{E})$ and ${\mathbb{T}^{\times}}\text{-}Mod(\mathcal{E})$ coincide with the [[opposite category]] ${\mathbb{T}}\text{-}Mod(\mathcal{E})^{op}$ or the [[twisted arrow category]] ${\mathbb{T}}\text{-}Mod(\mathcal{E})^\times$ for any [[Grothendieck topos]], respectively. In particular, since the (geometric) theory of morphisms $\mathbb{O}^2$ (see the examples below) assigns the [[arrow category]] $\mathcal{E}^2$ as category of models to a Grothendieck topos $\mathcal{E}$ no geometric theory can assign all Grothendieck toposes $\mathcal{E}$ their [[twisted arrow category]] $\mathcal{E}^\times$ as category of models because the class of models coincides with the class of models for the theory of morphisms. $\qed$

Since $\mathcal{E}^2=\mathcal{E}\times Set^2$ in general and the [[Sierpinski topos]] $Set^2$ is [[exponentiable topos|exponentiable]], one gets

$${\mathbb{T}^2}\text{-}Mod(\mathcal{E})=\mathbb{T}\text{-}Mod(\mathcal{E}^2)=Hom(\mathcal{E}^2,Set[\mathbb{T}])=Hom(\mathcal{E}\times Set^2, Set[\mathbb{T}])=Hom(\mathcal{E},Set[\mathbb{T}]^{(Set^2)})$$

or in other words

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

The [[classifying topos]] for the theory $\mathbb{T}^2$ of $\mathbb{T}$-model homorphisms is $Set[\mathbb{T}^2]= Set[\mathbb{T}]^{(Set^2)}$.
=--

**Remark**:
It is worthwhile to muse a bit how "squaring" fares as a theory operator with respect to the various subdoctrines of [[geometric logic]]:
since it does not introduce logical operators like infinitary disjunctions or quantifiers unless the operators were already present in $\mathbb{T}$, one sees that the formats of the more expressive subdoctrines like Horn, [[regular logic|regular]], and [[coherent logic]] are respected by the passage to $\mathbb{T}^2$.
Also, the square of a [[localic topos|propositional theory]] i.e. one over a signature lacking sort symbols, is again propositional - this implies e.g. via the above corollary that the exponentiation of a [[localic topos]] by $Set^2$ is again a localic topos.
The order sensitive subdoctrines like [[essentially algebraic theory|cartesian]] or [[disjunctive logic]] are also respected.
One case where squaring does not preserve the subdoctrine is [[Lawvere theory|1-sorted algebraic logic]] since then $\mathbb{T}^2$ is necessarily 2-sorted. $\qed$

By an observation of Hébert ([2010](#H10), p.1), the finitely presentable objects in arrow categories $\mathcal{K}^2$, are precisely the morphisms between finitely presentable objects in $\mathcal{K}$ whence, denoting the subcategory of finitely presentable models of a theory $\mathbb{T}$ in a topos $\mathcal{E}$ by $\mathbb{T}\text{-}Mod_{fp}(\mathcal{E})$ one has

$$\mathbb{T}^2\text{-}Mod_{fp}(\mathcal{E})=\mathbb{T}\text{-}Mod_{fp}(\mathcal{E})^2\;.$$

But since classifying toposes of cartesian theories are given by the presheaf toposes on the opposite of the category of their finitely presentable models in $Set$ (cf. Johnstone [2002b](J02b), p. 891) one has

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

Let $\mathbb{T}$ be a [[essentially algebraic theory|cartesian theory]]. Then the theory $\mathbb{T}^2$ of $\mathbb{T}$-model homorphisms is cartesian and its [[classifying topos]] is $Set[\mathbb{T}^2]=Set^{\mathbb{T}^2\text{-}Mod_{fp}(Set)}=Set^{(\mathbb{T}\text{-}Mod_{fp}(Set)^2)}=\big (Set^{\mathbb{T}\text{-}Mod_{fp}(Set)}\big)^{(Set^2)}=Set[\mathbb{T}]^{(Set^2)}$.
=--

For an easy application of this result see at the theory $\mathbb{O}$ of objects in the examples below!

## Examples

* Let $\mathbb{T}^\emptyset_\emptyset$ and $\mathbb{T}^\emptyset_1$ be the [[empty theory]], resp., the inconsistent theory, over the empty signature. Then $\mathbb{T}^{\emptyset 2}_\emptyset=\mathbb{T}^\emptyset_\emptyset$ and $\mathbb{T}^{\emptyset 2}_1=\mathbb{T}^\emptyset_1$ in accordance with the fact that these theories have only empty models whence all model homomorphisms are [[identity|identity morphisms]] of empty models. In other words, squaring theories obeys the laws of arithmetic at 0 and 1.

* Let $\mathbb{O}$ be the [[theory of objects]] i.e. the theory with no sequents over the signature with one sort symbol $O$. Then $\mathbb{O}^2$ is the theory with no sequents over the signature with two sort symbols $O^0$, $O^1$ and a function symbol $f_O:O^0\to O^1$. Clearly, its models in a Grothendieck topos $\mathcal{E}$ are the morphisms of $\mathcal{E}$ and, accordingly, $\mathbb{O}^2$ is called the **theory of morphisms**.

  It is the [[exponentiable topos|dual theory]] of the theory classified by the [[Sierpinski topos]] $Set^2$, or in other words, its [[classifying topos]] called the _morphism classifier_ (cf. Johnstone [1977](#J77), p.184; [2002](#J02), p.426) is
$$ Set[\mathbb{O}^2]=Set[\mathbb{O}]^{Set^2}=(Set^{FinSet})^{Set^2}=Set^{(FinSet^2)}\quad .$$

  The _generic morphism_ is given by the [[natural transformation]] $\eta\,:\,\mathbf{O}\circ dom\to\mathbf{O}\circ cod$ with $\mathbf{O}$ the _generic object_ i.e. the inclusion functor $FinSet\hookrightarrow Set$, and $dom\,:\,FinSet^2\to FinSet\, ,\, (X\to Y)\mapsto X$ the domain projection functor, and $cod\,:\,FinSet^2\to FinSet\,,\, (X\to Y)\mapsto Y$ the codomain projection functor, with components $\eta_{(X\to Y)}=\mathbf{O}(X\to Y)$.

* Let $\mathbb{K}$ be the [[theory of categories]] (e.g. Johnstone [1977](#J77), p.202) whose models in a [[Grothendieck topos]] $\mathcal{E}$ are the [[internal category|internal categories]] $\mathbf{C}\in cat(\mathcal{E})$. Then $\mathbb{K}^2$ is the **theory of functors**. $\mathbb{K}$ likely being the most famous cartesian theory, one has $Set[\mathbb{K}^2]=Set^{(\mathbb{K}\text{-}Mod_{fp}(Set)^2)}$.

## The connection to natural homotopy

The [[Sierpinski topos]] $Set^2$ is a connected, locally connected and local topos. As a result of [[Artin gluing]] along $Set\overset{id}{\to}Set$ it has two [[point of a topos|topos points]] (corresponding to the two points of the underlying [[Sierpinski space]]). Whence one can take $Set^2$ as an abstract [[interval object]] for a homotopy theory of Grothendieck toposes correlated with their nature as "generalized spaces" and view, accordingly, the exponential $\mathcal{E}^{Set^2}$ as a path space for $\mathcal{E}$.

(Cf. Beke [2000](#B00), p.11f)

## Related entries

* [[geometric theory]]

* [[theory of objects]]

* [[theory of decidable objects]]

* [[natural homotopy]]

* [[exponentiable topos]]

* [[Sierpinski topos]]

## References

* {#B00}[[Tibor Beke]], _Homotopoi_ , ms. University of Massachusetts Lowell (2000). ([dvi](http://faculty.uml.edu/tbeke/topos.dvi))

* {#H10}[[Michel Hébert]], _Finitely presentable morphisms in exact sequences_ , TAC **24** no.9 (2010) pp.209-220. ([abstract](http://tac.mta.ca/tac/volumes/24/9/24-09abs.html))

* {#J77}[[Peter Johnstone]], _Topos Theory_ , Academic Press New York (1977). (Also available as Dover Reprint, Mineola 2014)

* {#J02a}[[Peter Johnstone]], _Sketches of an Elephant vol.1_ , Oxford UP 2002.

* {#J02b}[[Peter Johnstone]], _Sketches of an Elephant vol.2_ , Oxford UP 2002.

* {#JW84}[[André Joyal]], [[Gavin Wraith]], _Eilenberg-Mac Lane Toposes and Cohomology_ , pp.117-131 in Cont. Math. **92** AMS 1984.

* {#MM94} [[Saunders Mac Lane]], [[Ieke Moerdijk]], _Sheaves in Geometry and Logic_ , Springer Heidelberg 1994.


theory of objects

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#### Topos Theory
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[[!redirects theory of pointed objects]]
[[!redirects theory of inhabited objects]]

#Contents#
* table of contents
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##Idea
The _theory of objects_ is the logical theory whose models in a category $\mathcal{C}$ are precisely the objects of $\mathcal{C}$. From an algebraic perspective one can think of it as the theory of an algebra with no operations.

## Definition

The _theory of objects_ $\mathbb{O}$ is the [[theory]] with no axioms over the [[signature (in logic)|signature]] with a single [[type]] and no primitive symbols except [[equality]].

## Properties

Of course, $\mathbb{O}$ is a [[geometric theory]], and as [[models]] for $\mathbb{O}$ in a [[topos]] $\mathcal{E}$ correspond to [[objects]] of $\mathcal{E}$, we can use its [[classifying topos]] to get a representation of the objects of $\mathcal{E}$.

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

The [[classifying topos]] for the theory of objects is the [[presheaf topos]] $[FinSet, Set]$ over the [[opposite category]] of the category [[FinSet]] of [[finite sets]].
=--

This follows from the fact that $\mathbb{O}$ is the [[algebraic theory]] of a 1-sorted algebra without operations whence its algebras are merely sets. The classifying topos for algebraic theories is quite generally given by the presheaf topos on the opposite of the category of finitely presentable models in our case just the finite sets.

For a direct proof see at _[[classifying topos for the theory of objects]]_.
For base toposes $\mathcal{S}$ other than $Set$, it is a theorem due to [[Andreas Blass]] that the theory of objects has a classifying topos precisely if $\mathcal{S}$ has a [[natural numbers object]] ([Blass 1989](#blass)).

In the syntax-free approach to geometric theories of Johnstone (2002, I B4.2) the **theory of objects** corresponds to the forgetful functor sending an $\mathcal{S}$-topos to its underlying category. (See at [[geometric theory]] the section on the functorial definition.)

## Quotient and other related theories

* A step up on the ladder of logical complexity is the **theory of inhabited objects** $\mathbb{O}_\exists$ that adds to $\mathbb{O}$ the existential axiom $\top\vdash(\exists x)\top$. Its classifying topos $Set[\mathbb{O}_\exists]$ is the functor category $[FinSet_\exists, Set]$ with $FinSet_\exists$ the _category of finite nonempty sets_. It has the property that every topos $\mathcal{E}$ admits a [[localic topos|localic morphism]] to $Set[\mathbb{O}_\exists]$.[^fine]

[^fine]: cf. Johnstone (2002 II, p.773) and [[Andre Joyal|Joyal-Tierney (1984)]]. For some further information on $FinSet_\exists$ see the references at [[generic interval]].

* When viewed as algebraic theory the initial algebra (in $Set$) is the empty set which satisfies $(\exists x)\top\vdash\bot$. Adding the sequent to the theory of objects one obtains the **theory of empty objects** $\mathbb{O}_\emptyset$, its models are the [[initial object|initial objects]]. Its classifying topos is $Set$ with $\emptyset$ as generic object. Whence the duality between quotient theories and subtoposes implies that $Set[\mathbb{O}]$ contains $Set[\mathbb{O}_\emptyset]$. The pattern that the classifying topos of an algebraic theory contains a copy of $Set$ as a [[subtopos]] classifying $Th(\mathfrak{A}_0)$ i.e. the theory of all geometric sequents holding at the initial algebra $\mathfrak{A}_0$ in Set is valid more generally (see at [[geometric type theory]] for another example).

* If instead of an additional axiom one adds a single constant symbol to the signature of $\mathbb{O}$ one obtains the **theory of pointed objects** $\mathbb{O}_\ast$ i.e. the empty theory relative to the signature with a single sort and a single constant. Its models are pointed objects and its classifying topos is $[FinSet_\ast,Set]$. (See the discussion&references at [[classifying topos for the theory of objects]].)

* The **theory of morphisms** $\mathbb{O}^2$ is the theory with no sequents over the signature with two sort symbols $O^0$, $O^1$ and a function symbol $f_O:O^0\to O^1$. It is the [[theory of model homomorphisms|theory of $\mathbb{O}$-model homomorphisms]] and its models in a Grothendieck topos $\mathcal{E}$ are the morphisms of $\mathcal{E}$. It is the [[exponentiable topos|dual theory]] of the theory classified by the [[Sierpinski topos]] $Set^2$, or in other words, its [[classifying topos]] is $Set[\mathbb{O}^2]=Set[\mathbb{O}]^{Set^2}$.

## Related Concepts

* [[theory of decidable objects]]

* [[theory of model homomorphisms]]

## References

Sections B4.2, D3.2 of

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

For the equivalence of having an NNO to having a classifier for objects:

* {#blass} [[Andreas Blass]], _Classifying topoi and the axiom of infinity_ , Algebra Universalis **26** (1989) 341-345 ([doi:10.1007/BF01211840](https://doi.org/10.1007/BF01211840))


theory of presheaf type

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#### Topos Theory
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#Contents#
* table of contents
{:toc}

## Idea
**Theories of presheaf type** though being [[geometric theory|geometric theories]] of a particular "simple" and tractable type are yet ubiquitous in the sense that every geometric theory is a [[quotient theory]] of some theory of presheaf type.

## Definition

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

A [[geometric theory]] $\mathbb{T}$ is **of presheaf type** if its [[classifying topos]] $Set[\mathbb{T}]$ is [[equivalence of categories|equivalent]] to a [[presheaf topos]].

=--

## Examples

* Any [[cartesian theory]] $\mathbb{T}$ being (modulo neglectable size issues[^iss]) classified by $Set^{\mathbb{T}-Mod_{fp}(Set)}$ is of presheaf type with $\mathbb{T}\text{-}Mod_{fp}(Set)$ the category of finitely presentable $\mathbb{T}$-models in $Set$.

* More concretely, e.g. the [[theory of objects]] or the [[classifying topos#ForIntervals|theory of intervals]] are of presheaf type being classified by the [[classifying topos for the theory of objects|object classifier]] respectively by the [[simplicial sets|topos of simplicial sets]].

* The _inconsistent theory_ with axiom $\top\vdash\bot$ is of presheaf type since it is classified by the initial Grothendieck topos $\mathbf{1}\simeq Pr(\emptyset)$, the presheaf topos on the empty category.

## Properties

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

For a [[category]] $\mathcal{M}$ the following are equivalent:

* $\mathcal{M}$ is finitely [[accessible category|accessible]].

* $\mathcal{M}\simeq Pts(Set^{\mathcal{C}})$ the [[point of a topos|category of points]] of some presheaf topos.

* $\mathcal{M}\simeq \mathbb{T}\text{-}Mod(Set)$ for some theory $\mathbb{T}$ of presheaf type.

* $\mathcal{M}\simeq Ind \text{-}\mathcal{C}$ for some small category $\mathcal{C}$.

=--

Cf. Beke ([2004](#Beke04), p.923) and the references given there. In fact, these equivalences are mostly (direct consequences of) classical results in the theory of accessible categories or Grothendieck toposes.

Note that despite the above equivalences the finite accessibility of $\mathbb{T}\text{-}Mod(Set)$ does _not_ imply that $\mathbb{T}$ itself is of presheaf type! One sees this already in case $\mathbb{T}\text{-}Mod(Set)=\emptyset$ since there famously are non trivial (Boolean sheaf) toposes lacking points ("non empty generalized spaces without points") yet up to [[Morita equivalence]] the only theory of presheaf type corresponding to the finite accessibility of the empty category is the inconsistent theory.

The following proposition shows in which sense theories of presheaf type are still determined by their (finitely presentable) models in $Set$:

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

A [[geometric theory]] $\mathbb{T}$ is of presheaf type iff (modulo neglectable size issues[^iss])
$$Set[\mathbb{T}]\simeq  [\mathbb{T}\text{-}Mod_{fp}(Set),Set]\, .$$

=--

**Proof**. "$\Rightarrow$":

(Cf. [Caramello 2018](#Cara18), pp.198f)

By assumption $Set[\mathbb{T}]\simeq [\mathcal{C}, Set]$. Since $[\mathcal{C},Set]\simeq [\hat{\mathcal{C}}, Set]$ (by [Johnstone 2002](#JT02), p.10) we can assume that $\mathcal{C}$ is [[Cauchy complete category|Cauchy complete]].

We have:

* $Ind\text{-}\mathcal{C}\simeq Flat(\mathcal{C}^{op}, Set)$ (by [Johnstone 2002](#JT02), p.723, or [Caramello 2018](#Cara18), p.198)

* $(Ind\text{-}\mathcal{C})_{fp}\simeq \mathcal{C}$ (by [Johnstone 2002](#JT02) 4.2.2.(iii), p.724)

* $\mathbb{T}\text{-}Mod(Set)\simeq Flat(\mathcal{C}^{op}, Set)$ (from [[Diaconescu theorem|Diaconescu’s theorem]]).

Whence $(Ind\text{-}\mathcal{C})_{fp}\simeq (\mathbb{T}\text{-}Mod (Set))_{fp} =\mathbb{T}\text{-}Mod_{fp}(Set)\simeq \mathcal{C}$ and, accordingly, $[\mathcal{C},Set]]\simeq  [\mathbb{T}\text{-}Mod_{fp}(Set),Set]$. $\qed$

In other words, theories of presheaf type are precisely those [[geometric theory|geometric theories]] $\mathbb{T}$ such that their classifying toposes $Set[\mathbb{T}]$ can be represented as presheaf toposes $Set^{\mathbb{T}\text{-}Mod_{fp}(Set)}$.

This implies e.g. that any consistent theory of presheaf type has models in $Set$.

[^iss]: This means here (and in the following) that the essentially small category $\mathbb{T}\text{-}Mod_{fp}(Set)$ has to be replaced by a skeleton.

## Related entries

* [[Gabriel-Ulmer duality]]

* [[Diaconescu theorem]]

* [[natural homotopy]]

## References

* {#Beke04}[[Tibor Beke]], _Theories of presheaf type_ , JSL **69** no.3 (2004) pp.923-934. ([pdf](http://faculty.uml.edu/tbeke/jsl.pdf))

* {#Cara18}[[Olivia Caramello]], _Theories, Sites, Toposes_ , Oxford UP 2018. (chapters 6-9)

* [[Matthias Hutzler]], _Syntactic presentations for glued toposes and for crystalline toposes_, Phd. diss. Universität Augsburg 2021. ([arXiv:2206.11244](https://arxiv.org/abs/2206.11244))

* {#JT02}[[Peter Johnstone]], _[[Sketches of an Elephant]] vols.1-2_ , Oxford UP 2002.

[[!redirects theories of presheaf type]]
[[!redirects Theory of presheaf type]]


theory of primes > history

< [[theory of primes]]


thermodynamic limit

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## Idea

In [[statistical physics]] the _thermodynamic limit_ denotes the limiting behaviour of a [[physical system]] that consists of many components ([[particles]]) as

* the [[volume]] $V$ and the number $N$ of particles tends to [[infinity]];

* the density ratio $\rho \coloneqq N/V$ approaches a constant value.

Many characteristic properties of macroscopic physical systems only appear in this limit, notably [[phase transitions]], [[universality classes]] and other [[critical phenomena]].

## Related concepts

* [[universality class]]

* [[conformal bootstrap]]

* [[quantum lattice system]]

* [[mesoscopic physics]]

## References

Reviews and introductions include

* _The theory of the thermodynamic limit_ ([pdf](http://igitur-archive.library.uu.nl/dissertations/1957294/c8.pdf))

* Daniel F. Styer, _What good is the thermodynamic limit?_ American Journal of Physics -- January 2004 -- Volume 72, Issue 1, pp. 25

  **Abstract** Statistical mechanics applies to large systems: technically, its results are exact only for infinitely large systems in "the thermodynamic limit." The importance of this proviso is often minimized in undergraduate courses. This paper presents six paradoxes in statistical mechanics that can be resolved only by acknowledging the thermodynamic limit. For example, it demonstrates that the widely used microcanonical "thin phase space limit" must be taken after taking the thermodynamic limit.

* Ben Simons, _Phase Transitions and Collective Phenomena_ ([web](http://www.tcm.phy.cam.ac.uk/~bds10/phase.html))

* C. N. Yang and T. D. Lee , _Statistical Theory of Equations of State and Phase Transitions. I. Theory of Condensation_,  Phys. Rev. 87, 404–409 (1952)

[[!redirects thermodynamics limits]]

[[!redirects continuum limit]]
[[!redirects continuum limits]]