Tom Banks

* [wikipedia entry](http://en.wikipedia.org/wiki/Tom_Banks_%28physicist%29)

## Selected writings

On [[string phenomenology]] via [[D=2 N=(2,0) SCFT]]:

* {#BDFF88} [[Tom Banks]], [[Lance Dixon]], [[Dan Friedan]], [[Emil Martinec]], _Phenomenology and Conformal Field Theory or Can String Theory Predict the Weak Mixing Angle?_, Nucl. Phys. B299 (1988) 613.  ([pdf](http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-pub-4377.pdf))

On the [[AdS/CFT correspondence]]:

* [[Tom Banks]], [[Michael Douglas]], [[Gary Horowitz]], [[Emil Martinec]], _AdS Dynamics from Conformal Field Theory_ ([arXiv:hep-th/9808016](https://arxiv.org/abs/hep-th/9808016), [spire:474214](http://inspirehep.net/record/474214))

On the [[BFSS matrix model]] ([[M-theory]] as [[D0-brane]] [[quantum mechanics]] in [[non-perturbative effect|non-perturbative]] [[type IIA string theory]]):

* {#BanksFischlerShenkerSusskind96} [[Tom Banks]], [[Willy Fischler]], [[Stephen Shenker]], [[Leonard Susskind]], _M Theory As A Matrix Model: A Conjecture_  Phys. Rev. D55 (1997). ([arXiv:hep-th/9610043](http://arxiv.org/abs/hep-th/9610043))

* [[Tom Banks]], _Matrix Theory_, Nucl.Phys.Proc.Suppl. 67 (1998) 180-224 ([arXiv:hep-th/9710231](https://arxiv.org/abs/hep-th/9710231))

Discussion of longitudinal [[M5-branes]] in the [[BFSS matrix model]]:

* [[Tom Banks]], [[Nathan Seiberg]], [[Stephen Shenker]], _Branes from Matrices_, Nucl. Phys. B490:91-106, 1997 ([arXiv:hep-th/9612157](https://arxiv.org/abs/hep-th/9612157))

On [[black holes in string theory]] via the [[BFSS matrix model]]:

* [[Tom Banks]], [[Willy Fischler]], [[Igor Klebanov]], [[Leonard Susskind]],  _Schwarzschild Black Holes from Matrix Theory_, Phys.Rev.Lett.80:226-229,1998 ([arXiv:hep-th/9709091](https://arxiv.org/abs/hep-th/9709091))

* [[Tom Banks]], [[Willy Fischler]], [[Igor Klebanov]], [[Leonard Susskind]], _Schwarzchild Black Holes in Matrix Theory II_, JHEP 9801:008,1998 ([arXiv:hep-th/9711005](https://arxiv.org/abs/hep-th/9711005))

On [[massive type IIA string theory]]:

* [[Tom Banks]], K. van den Broek, _Massive IIA flux compactifications and U-dualities_, JHEP 0703:068, 2007 ([arXiv:hep-th/0611185](https://arxiv.org/abs/hep-th/0611185))

On the [[firewall problem]] via the [[BFSS matrix model]]:

* [[Tom Banks]], [[Willy Fischler]], _No Firewalls in Holographic Space-Time or Matrix Theory_ 2013 ([arXiv:1305.3923](http://arxiv.org/abs/1305.3923))

On [[D=4 N=8 supergravity]]:

 * [[Tom Banks]], _Why I don't Believe N=8 SUGRA is Finite_, talk at _[Workshop "Supergravity versus Superstring Theory in the Ultraviolet"](http://www.gravity.psu.edu/events/supergravity/program.shtml)_, PennState Univ, PA USA, August 27-30 2009.

On [[quantum field theory]]:

* [[Tom Banks]], _Modern quantum field theory, A concise introduction_ ([pdf](http://web.phys.ntnu.no/~mika/banks.pdf))

On the [[landscape of string theory vacua]] (or not):

* [[Tom Banks]], _Landskepticism: or Why Effective Potentials Don't Count String Models_ ([arXiv:hep-th/0412129](https://arxiv.org/abs/hep-th/0412129))

* [[Tom Banks]], _The Top $10^{500}$ Reasons Not to Believe in the Landscape_ ([arXiv:1208.5715](https://arxiv.org/abs/1208.5715))

Specifically on the [[swampland conjectures]]/[[landscape of string theory vacua]] in view of [[non-perturbative effects]] in [[string theory]] (i.e. in [[M-theory]]):

* [[Tom Banks]], _The Top $10^{500}$ Reasons Not to Believe in the Landscape_ ([arXiv:1208.5715](https://arxiv.org/abs/1208.5715))

* {#Banks19} [[Tom Banks]], _On the Limits of Effective Quantum Field Theory: Eternal Inflation, Landscapes, and Other Mythical Beasts_ [[arxiv:1910.12817](https://arxiv.org/abs/1910.12817)]

  from pages [14](https://arxiv.org/pdf/1910.12817.pdf#PAGE=14)-22:

  > these considerations lead to conclusions at odds with the seemingly similar arguments of [the [[swampland conjectures]]]. $[\cdots]$  Perturbative moduli space completely distorts the true nature of the class of consistent models.

  > It’s important to realize that the entire procedure just outlined for finding (meta) stable AdS minima of a non-perturbative effective potential is purely hypothetical and has no basis in well founded string theory calculations. $[\cdots].$

  > The hypothesis of the String Landscape is entirely based on low energy effective field theory ideas about finding "vacua" by minimizing an effective potential.  Everything that’s been said above indicates that this idea has no validity in genuine models of quantum gravity. $[\cdots]$

  > The most serious issue,  in  my  opinion,  is  the  contention  that  one  can  make  the  AdS  radius  much  larger  than the size of the compact manifold.  All well established examples of large radius AdS/CFT havea compact manifold of dimension 2 or greater whose radius is comparable to that of the AdSspace.  In Appendix A we’ll present an argument based on the properties of AdS black holes,that this is in fact necessary.

  > The next step in the construction of "realistic" models involves "adding an anti-brane to break supersymmetry and make the c.c.  positive".  This is supposed to be a small modification of the model, calculable in low energy effective field theory, and that seems manifestly incorrect. $[\cdots]$.

  > even if one believes that the construction of meta-stable dS models is reliable, there is no clear argument about  what  the  proper  observables  of  the  model  are  nor  that different  dS  constructions are part of the same model.  Neither is there an interpretation of these correlators as transition amplitudes in a quantum mechanical model. $[\cdots]$

  > The conclusion  that  effective  field  theorists  should  draw  from  this is that unlike super-symmetric string models in flat or AdS space-time, many of which have at least perturbative definitions as mathematical models obeying the axioms of quantum mechanics, all literature on the String Landscape is speculation based on the unfounded notion that all string models with a given amount of SUSY are part of one single model and that it makes sense to define an effective action that encompasses all string models. Every single non-perturbative construction of string models contradicts this claim $[\cdots]$

Musings on [[holography]], [[quantum gravity]] and [[quantum information]] (such as through [[holographic entanglement entropy]]):

* [[Tom Banks]], *Holographic Space-time and Quantum Information*, Front. Phys. **8** (2020) [[arXiv:2001.08205](https://arxiv.org/abs/2001.08205), [doi:10.3389/fphy.2020.00111](https://doi.org/10.3389/fphy.2020.00111)]

* [[Tom Banks]], *Hilbert Bundles and Holographic Space-time Models* [[arXiv:2306.07038](https://arxiv.org/abs/2306.07038)]

category: people


Tom Braden

* [webpage](http://www.math.umass.edu/~braden/)

category: people


Tom Bridgeland

* [webpage](http://www.all-souls.ox.ac.uk/people.php?personid=2079)

Related $n$Lab pages include [[Bridgeland stability condition]].

## Selected works

* Tom Bridgeland, _Stability conditions on triangulated categories_, Ann. of Math. __166__ (2007) 317--345 [math.AG/0212237](https://arxiv.org/abs/math/0212237)
* Tom Bridgeland, _An introduction to motivic Hall algebra_, [arXiv:1002.4372](https://arxiv.org/abs/1002.4372)
A realization of (Drinfeld-Jimbo) [[quantum group]]s via a "Bridgeland" version of [[twisted Hall algebra]],

* [[Tom Bridgeland]], _Quantum groups via Hall algebras of complexes_, Ann of Math __177__ (2) (2013) 739--759

category: people


Tom de Jong

* [webpage](https://www.cs.bham.ac.uk/~txd880/)

## Selected writings and talks

On [[domain theory]] in [[constructive mathematics|constructive]] [[set theory]]:

* [[Tom de Jong]], *Sharp Elements and Apartness in Domains*, in: Proceedings MFPS 2021, EPTCS 351, 2021, pp. 134-151 ([doi:10.4204/EPTCS.351.9](https://doi.org/10.4204/EPTCS.351.9), [arXiv:2112.14052](https://arxiv.org/abs/2112.14052))

On [[domain theory]] in [[homotopy type theory]]/[[univalent foundations]] ([[mathematics presented in HoTT]]):

* [[Tom de Jong]], [[Martín Hötzel Escardó]], _Domain Theory in Constructive and Predicative Univalent Foundations_, in: _29th EACSL Annual Conference on Computer Science Logic_, [CSL 2021](https://csl2021.fmf.uni-lj.si/), LIPIcs proceedings 183, 2021 ([doi:10.4230/LIPIcs.CSL.2021.28](https://doi.org/10.4230/LIPIcs.CSL.2021.28), [arXiv:2008.01422](https://arxiv.org/abs/2008.01422))

On [[ordinals]] in [[type theory]] and [[marked extensional well-founded orders]]:

* {#deJongEtAl23} [[Tom de Jong]], [[Nicolai Kraus]], [[Fredrik Nordvall Forsberg]], [[Chuangjie Xu]], *Set-Theoretic and Type-Theoretic Ordinals Coincide* [[arXiv:2301.10696](https://arxiv.org/abs/2301.10696)]

On [[epimorphisms]] in [[homotopy type theory]]:

* [[Tom de Jong]], *Acyclic types and epimorphisms in HoTT*, [[Homotopy Type Theory Electronic Seminar Talks]], 17 November 2022 ([slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Jong-2022-11-17-HoTTEST.pdf), [video](https://www.youtube.com/watch?v=vohAYPAAWxs))

* [[Ulrik Buchholtz]], [[Tom de Jong]], [[Egbert Rijke]], *Epimorphisms and Acyclic Types in Univalent Mathematics*, ([arXiv:2401.14106](https://arxiv.org/abs/2401.14106))

category: people


tom Dieck splitting

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

#Contents#
* table of contents
{:toc}

## Idea

The objects of [[equivariant stable homotopy theory]] -- genuine [[G-spectra]] -- are very rich: Already the [[fixed point spectra]] of [[equivariant suspension spectra]] contain considerably more information than just the [[suspension spectra]] of the plain underlying [[fixed point spaces]] -- the latter are just the _[[geometric fixed point spectra]].

The _tom Dieck splitting_  ([tom Dieck 75](#tomDieck75), [Lewis-May-Steinberger 86, V.11](#LewisMaySteinberger86)) gives an explicit description of all the [[wedge sum|wedge summands]] appearing in [[fixed point spectra]] of [[equivariant suspension spectra]]. These wedge sums start out with the [[geometric fixed point spectra]] and then have one summand for each [[conjugacy class]] of [[subgroups]] $H \subset G$, given by the plain [[suspension spectra]] of the [[homotopy quotient]] of the $H$-[[fixed point spaces]] by the corresponding [[Weyl group]]-[[action]].

Induced from this [[wedge sum]] splitting formula for the spectra themselves is a corresponding [[direct sum]]-formula of the [[equivariant stable homotopy groups]] in terms of plain [[stable homotopy groups]].

The richness of this splitting, hence of [[G-spectra]], is witnessed by its simplest non-trivial example, which is the [[equivariant stable homotopy groups]] of the [[equivariant sphere spectrum]]: This yields the [[abelian group]] underlying the [[Burnside ring]], which is [[free abelian group|freely generated]] from the [[conjugacy classes]] of [[subgroups]] of $G$ (see [below](#ForTheEquivariantSphereSpectrum)).

## Statement

For $H\subset G$ a [[subgroup]], write

* $[H \subset G]$ for its [[conjugacy class]];

* $W_G H \coloneqq (N_G H)/H$ for its [[Weyl group]], the [[quotient group]] of the [[normalizer subgroup]] (of $H$ in $G$) by $H$;

* $E W_G H$ for the [[universal principal bundle]] of the [[Weyl group]].

### Of fixed point spectra of equivariant suspension spectra
 {#OfFixedPointSpectraOfEquivariantSuspensionSpectra}

The [[fixed point spectrum]] of an [[equivariant suspension spectrum]] is given by the [[wedge sum]] formula

$$
  F^G(\Sigma^\infty_G X)
  \simeq
  \underset{[H\subset G]}{\bigvee}
  \Sigma^\infty( E (W_G H)_+ \wedge_{W_G H} X^H )
$$

where $\Sigma^\infty$ is the plain [[suspension spectrum]] construction.

(e.g. [Guillou-May 12, theorem 5.3](#GuillouMay12), [Schwede 15, example 7.7](#Schwede15))

In particular, since $W_G G = 1$ and $W_G 1 = G$, the extremal summands for $H = G$ and $H = 1$ are just the [[suspension spectrum]] of the plain [[fixed point space]] $X^G$ and of the [[homotopy quotient]] $X\sslash G$ (equivalently the  [[Borel construction]] $X \sslash G \simeq E G_+ \times_G X$ ) of the full space, respectively:

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

Here the first summand is the _[[geometric fixed point spectrum]]_ inside the full [[fixed point spectrum]]

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

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

### For equivariant homotopy groups

It follows that for $X$ a pointed [[topological G-space]], its [[equivariant homotopy groups]] are

$$
  \begin{aligned}
    \pi_\bullet^G(\Sigma^\infty X)
    &
    \simeq
    \underset{[H \subset G]}{\bigoplus}
    \pi_\bullet^{W_G H}(\Sigma^\infty (E (W_G H)_+ \wedge X^H))
    \\
    &\simeq
    \pi_\bullet(\Sigma^\infty X^G)
    \oplus
    \underset{{[H \subset G]} \atop {1 \neq H \neq G}}{\bigoplus}
    \pi_\bullet^{W_G H}(\Sigma^\infty (E (W_G H)_+ \wedge X^H))
    \oplus
    \pi_\bullet^{G}(\Sigma^\infty (E G_+ \wedge X))
  \end{aligned}
$$

where the [[direct sum]] is over [[conjugacy classes]] of [[subgroups]] $H$ of $G$.

(e.g. [Schwede 15, theorem 6.12](#Schwede15))

### For rational equivariant homotopy theory

For $G$ finite and for [[rational equivariant stable homotopy theory]] this becomes ([Greenlees, 6.2](#Greenlees))

$$
  G RationalSpectra \simeq \underset{[H \subset G] }{\prod} \mathbb{Q}(W_G H) Mod
$$

where the product is over [[conjugacy classes]] of [[subgroups]] $H$ of $G$ and $W_G H$ denotes the [[Weyl group]] of $H$ in $G$

## Examples

### For the equivariant sphere spectrum
 {#ForTheEquivariantSphereSpectrum}

For the [[equivariant sphere spectrum]] $\mathbb{S} = \Sigma^\infty_G S^0$ the tom Dieck splitting says that its 0th [[equivariant homotopy group]]
is the [[free abelian group]] on the set of [[conjugacy classes]] of [[subgroups]] of $G$:

$$
  \pi_0^G(\mathbb{S})
  \simeq
  \underset{[H \subset G]}{\oplus} \pi_0^{W H}(\Sigma_+^\infty E W H)
  \simeq
  \mathbb{Z}\big[\text{conjugacy classes of subgroups}\big]
$$

(e.g. [Schwede 15, p. 64](#Schwede15))

This is the group underlying the [[Burnside ring]].

## Related concepts

* [[equivariant Whitehead theorem]]

* [[Elmendorf's theorem]]

* [[G-CW approximation]]

## References

The theorem at the level of [[stable homotopy groups]] is due to

* {#tomDieck75} [[Tammo tom Dieck]], Satz 2 of _Orbittypen und  äquivariante Homologie. II._, Arch. Math. (Basel) 26 (1975), no. 6, 650–662

The refinement to [[spectra]] is achieved in section V.11 of

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

An alternative proof is in

* {#GuillouMay12} [[Bert Guillou]], [[Peter May]], section 6.2 of _Permutative $G$-categories in equivariant infinite loop space theory_, Algebr. Geom. Topol. 17 (2017) 3259-3339 ([arXiv:1207.3459](http://arxiv.org/abs/1207.3459))

Detailed lecture notes are in

* {#Schwede15} [[Stefan Schwede]], section 6 of _[[Lectures on Equivariant Stable Homotopy Theory]]_, 2015 ([pdf](http://www.math.uni-bonn.de/people/schwede/equivariant.pdf))

* {#Blumberg17} [[Andrew Blumberg]], section 2.7 of _The Burnside category_, 2017 ([pdf](https://www.ma.utexas.edu/users/a.debray/lecture_notes/m392c_EHT_notes.pdf), [GitHub](https://github.com/adebray/equivariant_homotopy_theory))

A brief mentioning appears in this survey of [[rational equivariant stable homotopy theory]]:

* {#Greenlees} [[John Greenlees]], p. 3 of _Triangulated categories of rational equivariant cohomology theories_ ([pdf](https://pdfs.semanticscholar.org/eda8/ee6effb2ae487085a2e7615a4a7e1964c493.pdf), [[GreenleesRationalEquivariant.pdf:file]])

[[!redirects tom  Dieck splittings]]

[[!redirects tom Dieck splitting theorem]]
[[!redirects tom Dieck splitting theorems]]


Tom Hirschowitz

* [home page](http://www.lama.univ-savoie.fr/~hirschowitz/)

category: people


Tom Kibble

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

* [[Michael Duff]], [[Kellogg Stelle]], _Sir Thomas Walter Bannerman Kibble CBE_ ([arXiv:2011.13257](https://arxiv.org/abs/2011.13257))

## Related $n$Lab entries

* [[cosmic string]]

* [[standard model of particle physics]]

category: people


Tom Lada

* [website](http://www4.ncsu.edu/~lada/)

category: people


Tom LaGatta

* [webpage](http://www.cims.nyu.edu/~lagatta/)

category: people


Tom Leinster

__Tom Leinster__ is a mathematician at the University of Edinburgh.

* [web](http://www.maths.ed.ac.uk/~tl/)

## Selected writings

* _Rethinking set theory_, e-print [arXiv:1212.6543 [math.LO]](https://arxiv.org/abs/1212.6543), 2012; see also [discussion on nCafé](https://golem.ph.utexas.edu/category/2012/12/rethinking_set_theory.html)

  This is an expository article for a general mathematical readership. It describes [[ETCS]], but without assuming any knowledge of category theory or ever defining a category.

* _A survey of definitions of n-category_, e-print [math.CT/0107188](http://arxiv.org/abs/math.CT/0107188),
2001; also [Theory and Applications of Categories 10 (2002), no. 1, 1–70.](http://www.tac.mta.ca/tac/volumes/10/1/10-01abs.html)

* _An informal introduction to topos theory_, Publications of the nLab vol. 1 no. **_1_** (2011).

  At https://ncatlab.org/publications/published/Leinster2011 , but as of 2022 the live version is not entirely readable: the diagrams and some other typesetting are broken.

  A [readable version](https://web.archive.org/web/20131212071554/http://ncatlab.org:80/publications/published/Leinster2011) (give the scripts a few seconds to load after opening the page) can be found in the Internet Archive.

On [[commutativity of limits with colimits]]:

* [[Marie Bjerrum]], [[Peter Johnstone]], [[Tom Leinster]], [[William F. Sawin]], *Notes on commutation of limits and colimits*, Theory and Applications of Categories **30** (2015), 527-532 [[arXiv:1409.7860](http://arxiv.org/abs/1409.7860), [tac:3015](http://www.tac.mta.ca/tac/volumes/30/15/30-15abs.html)]

On [[Isbell duality]] and [[reflexive completion]]:

* [[Tom Avery]], [[Tom Leinster]]. _Isbell conjugacy and the reflexive completion_. Theory and Applications of Categories, **36** 12 (2021) 306-347 [[tac:36-12](http://www.tac.mta.ca/tac/volumes/36/12/36-12abs.html), [pdf](http://www.tac.mta.ca/tac/volumes/36/12/36-12.pdf)]

## Books

* _[Basic Category Theory](http://www.maths.ed.ac.uk/~tl/bct/)_, 2014

  Short description from author's web page for book:

  > Basic Category Theory is an introductory category theory textbook. Features:
  >
  > * It doesn't assume much, either in terms of background or mathematical maturity.
  >
  > * It sticks to the basics.
  >
  > * It's short.
  >
  > Advanced topics are omitted, leaving more space for careful explanations of the core concepts. I have used versions of this text to teach final-year undergraduate and master's-level courses at the Universities of Glasgow and Edinburgh.

* {#Leinster04} [[Tom Leinster]], *Higher operads, higher categories*, London Math. Soc. Lec. Note Series **298**, Cambridge University Press (2004) [[math.CT/0305049](http://arxiv.org/abs/math.CT/0305049), [doi:10.1017/CBO9780511525896](https://doi.org/10.1017/CBO9780511525896)]

* _[Entropy and Diversity: The Axiomatic Approach](https://www.cambridge.org/us/academic/subjects/mathematics/logic-categories-and-sets/entropy-and-diversity-axiomatic-approach?isbn=9781108832700)_, Cambridge UP, 2021

category: people

[[!redirects T. Leinster]]
[[!redirects Leinster]]


Tom Mainiero

* [personal page](https://tommainiero.com/)

* [GitHub page](https://github.com/tmainiero)

## Selected writings

On [[homological algebra]]-tools for characterizing [[quantum entanglement]] in [[quantum field theory]]:

* {#Mainiero2019} [[Tom Mainiero]], *Homological Tools for the Quantum Mechanic* (2019) [[arXiv:1901.02011](https://arxiv.org/abs/1901.02011), [10.10.11.6/handle/1/1339](http://10.10.11.6/handle/1/1339)]

## Related entries

* [[functorial aspects of the GNS representation]]

category: people


Tom Pugh

* [webpage](https://www.theorie.physik.uni-muenchen.de/MATH/members/former_mem/former_mpi/thomas_george_pugh/index.html)

## Selected writings

On [[M-theory on Spin(7)-manifolds]], [[F-theory on Spin(7)-manifolds]] and [[Witten's Dark Fantasy]]:

* [[Tom Pugh]], _M-theory on $Spin(7)$-manifolds and their F-theory duals_ ([[PughMTheoryOnSpin7.pdf:file]])

* {#BonettiGrimmPugh13} [[Federico Bonetti]], [[Thomas Grimm]], [[Tom Pugh]], _Non-Supersymmetric F-Theory Compactifications on $Spin(7)$ Manifolds_, JHEP 01 (2014) 112 ([arXiv:1307.5858](https://arxiv.org/abs/1307.5858))

* {#BonettiGrimmPugh13b} [[Federico Bonetti]], [[Thomas Grimm]], [[Eran Palti]], [[Tom Pugh]], _F-Theory on $Spin(7)$ Manifolds: Weak-Coupling Limit_, JHEP 02 (2014) 076 ([arXiv:1309.2287](https://arxiv.org/abs/1309.2287))

category: people


Tom Rudelius

* [webpage](https://www.ias.edu/scholars/thomas-rudelius)

## Selected writings

On [[little string theory]] from [[F-theory]]:

* Lakshya Bhardwaj, Michele Del Zotto, [[Jonathan Heckman]], [[David Morrison]], [[Tom Rudelius]], [[Cumrun Vafa]], _F-theory and the Classification of Little Strings_ ([arXiv:1511.05565](http://arxiv.org/abs/1511.05565))

Comprehensive review of expected [[top-down model building|top-down constructions]] of [[D=6 N=(2,0) SCFT|$D=6$ $\mathcal{N}=(2,0)$ SCFTs]] and [[D=6 N=(1,0) SCFT|$D=6$ $\mathcal{N}=(1,0)$ SCFTs]]

* [[Jonathan J. Heckman]], [[Tom Rudelius]], *Top Down Approach to 6D SCFTs*, J. Phys. A: Math. Theor. **52** 9 (2019) 093001 [[arXiv:1805.06467](https://arxiv.org/abs/1805.06467), [doi:10.1088/1751-8121/aafc81](https://doi.org/10.1088/1751-8121/aafc81)]

On the [[weak gravity conjecture]]:

* [[Ben Heidenreich]], [[Matthew Reece]], [[Tom Rudelius]], _Sharpening the Weak Gravity Conjecture with Dimensional Reduction_, JHEP02(2016)140 ([arXiv:1509.06374](https://arxiv.org/abs/1509.06374))

* [[Ben Heidenreich]], [[Matthew Reece]], [[Tom Rudelius]], _Repulsive Forces and the Weak Gravity Conjecture_ ([arXiv:1906.02206](https://arxiv.org/abs/1906.02206))

On [[swampland conjectures]]:

* [[Ben Heidenreich]], [[Matthew Reece]], [[Tom Rudelius]], _Emergence and the Swampland Conjectures_ ([arXiv:1802.08698](https://arxiv.org/abs/1802.08698))

category: people


Tom W. B. Kibble

* [Wikipedia page](https://en.wikipedia.org/wiki/Tom_Kibble)

* [InSpire page](https://inspirehep.net/authors/1003042)

## Selected writings

On [[classical mechanics]]:

* [[Tom W. B. Kibble]], [[Frank Berkshire]], *Classical mechanics*, McGraw-Hill (1966, 1973, 2004) [[pdf](https://physicaeducator.files.wordpress.com/2018/02/classical-mechanics-by-kibble-and-berkshire.pdf), Wikipedia entry]

Introducing the [[symplectic formulation of quantum mechanics]]:

* {#Kibble79} [[Tom W. B. Kibble]], *Geometrization of quantum mechanics*, Communications in Mathematical Physics **65** (1979) 189–201 [[doi:10.1007/BF01225149](https://doi.org/10.1007/BF01225149)]

category: people

[[!redirects Thomas W. B. Kibble]]
[[!redirects Tom Kibble]]
[[!redirects Thomas Walter Bannerman Kibble]]


Tom Weston

* [webpage](http://people.math.umass.edu/~weston/)

## related $n$Lab entries

* [[cobordism theory]]

category: people


Tom Zlosnik

* [website](https://ceico.cz/team/postdocs/tom-zlosnik)

## Selected writings

On a [[relativistic field theory]] version of [[MOND]]:

* [[Constantinos Skordis]], [[Tom Zlosnik]], *A new relativistic theory for Modified Newtonian Dynamics*. Physical Review Letters. Volume 127, Issue 16. Published 15 October 2021 ([doi:10.1103/PhysRevLett.127.161302](https://doi.org/10.1103/PhysRevLett.127.161302), [arXiv:2007.00082](https://arxiv.org/abs/2007.00082))

* [[Constantinos Skordis]], [[Tom Zlosnik]], *Aether scalar tensor theory: Linear stability on Minkowski space*. Physical Review D. Volume 106, Issue 10. Published 15 November 2022. ([doi:10.1103/PhysRevD.106.104041](https://doi.org/10.1103/PhysRevD.106.104041), [arXiv:2109.13287](https://arxiv.org/abs/2109.13287))

On [[cosmic structure formation]] in view of [[MOND]]:

* [[Daniel Thomas]], [[Ali Mozaffari]], [[Tom Zlosnik]], *Consistent cosmological structure formation on all scales in relativistic extensions of MOND* ([arXiv:2303.00038](https://arxiv.org/abs/2303.00038))

category: people


Tomas Everaert

[[!redirects T. Everaert]]
Tomas Everaert works on homological and homotopical algebra in semi-abelian categories.  He is based at the _Vrije Universiteit Brussel_.

* [university research page](https://researchportal.vub.be/en/persons/tomas-everaert)

* [Orcid link](https://orcid.org/0000-0002-3096-4105)

## Selected publications

On the [[canonical model structure]] for [[internal categories]] in [[finitely complete categories]]:

* [[Tomas Everaert]], [[Rudger W. Kieboom]], [[Tim Van der Linden]], _Model structures for homotopy of internal categories_, Theory and Applications of Categories **15** 3 (CT2004) 66-94 [[tac:15-03](http://www.tac.mta.ca/tac/volumes/15/3/15-03abs.html)]

category:people


Tomas Ortin

## Selected writings

On [[double dimensional reduction]] of the [[M5-brane]] to the [[D4-brane]]:

* [[Eric Bergshoeff]], [[Mees de Roo]], [[Tomas Ortin]], *The Eleven-dimensional Five-brane*, Phys. Lett. B **386** (1996) 85-90 [[arXiv:hep-th/9606118](https://arxiv.org/abs/hep-th/9606118), doi:10.1016/0370-2693(96)00913-6]

On [[massive type IIA string theory]] and its embedding in [[M-theory]] via [[M9-branes]]:

* [[Eric Bergshoeff]], [[Yolanda Lozano]], [[Tomas Ortin]], *Massive Branes*, Nucl. Phys. B **518** (1998) 363-423 [[arXiv:hep-th/9712115](https://arxiv.org/abs/hep-th/9712115), doi:10.1016/S0550-3213(98)00045-5]

On [[D8-branes]] as [[black branes]] in [[massive type IIA string theory]]:

* {#JanssenMeessenOrtin99} [[Bert Janssen]], [[Patrick Meessen]], [[Tomas Ortin]], _The D8-brane tied up: String and brane solutions in massive type IIA supergravity_, Phys. Lett. B453 (1999) 229-236 ([spire:494174](http://inspirehep.net/record/494174), doi:10.1016/S0370-2693(99)00315-9)

Formulation of ([[Lagrangian densities]] for) [[type II supergravity]] with "democratic"/"[[pregeometric RR-field|pregeometric]]" [[RR-fields]] subject to [[self-dual higher gauge theory|self-duality]]:

* [[Eric Bergshoeff]], [[Renata Kallosh]], [[Tomas Ortin]], [[Diederik Roest]], [[Antoine Van Proeyen]], *New Formulations of D=10 Supersymmetry and D8-O8 Domain Walls*, Class. Quant. Grav. **18** (2001) 3359-3382 [[arXiv:hep-th/0103233](https://arxiv.org/abs/hep-th/0103233), [doi:10.1088/0264-9381/18/17/303](https://doi.org/10.1088/0264-9381/18/17/303)]

On [[defect branes]]:

* {#BershoeffOrtinRiccioni11} [[Eric Bergshoeff]], [[Tomas Ortin]], [[Fabio Riccioni]], _Defect Branes_, Nuclear Physics B **856** 2 (2012) 210-227 [[arXiv:1109.4484](https://arxiv.org/abs/1109.4484), [doi:10.1016/j.nuclphysb.2011.10.037](https://doi.org/10.1016/j.nuclphysb.2011.10.037)]

category: people


Tomas Petricek

* [personal page](https://tomasp.net/)

## Selected writings

On [[comonads]] and [[graded modalities]] [[comonads in computer science|in computer science]];

* {#POM13} [[Tomas Petricek]], [[Dominic Orchard]], [[Alan Mycroft]], *Coeffects: Unified Static Analysis of Context-Dependence*, in: *Automata, Languages, and Programming. ICALP 2013*, Lecture Notes in Computer Science **7966** Springer (2013) [[doi:10.1007/978-3-642-39212-2_35](https://doi.org/10.1007/978-3-642-39212-2_35)]

category: people


Tomasz Brzezinski

_Tomasz Brzeziński_ is a Polish-UK mathematician and [[mathematical physicist]] with a position at the University of Swansea, Wales and University of Białystok, Poland. Large part of his work is concerned with [[noncommutative geometry]], especially [[quantum principal bundles]], coalgebraic and Hopf-Galois theories and the aspects of the [[descent theory]] in algebraic context. He has played a large role in the revival of the theory of [[corings]]. With [[Shahn Majid]] he rediscovered the mixed [[distributive laws]] under the name of [[entwining structures]], motivated by the quantum principal bundle theory.

* Tomasz's Swansea pages: [homepage](https://www.swansea.ac.uk/staff/science/maths/brzezinski-t/).

## Selected publications

On [[coalgebra]] ([[corings]], [[comodules]])

* T. Brzeziński and [[S. Majid]], _Coalgebra bundles_, Commun. Math. Phys, 191, (1998), 467 – 492

* T. Brzeziński, R. Wisbauer, _Corings and comodules_, London Math. Soc. Lec. Note Series 309, Cambridge 2003.

On associative [[bialgebroids]] and in particalar about those arising as smash products of a [[Hopf algebra]] $H$ with a braided commutative Yetter-Drinfeld $H$-module algebra $A$ ([[scalar extension bialgebroid|scalar extension bialgebroids]]):

* [[Tomasz Brzeziński]], [[Gigel Militaru]], *Bialgebroids, $\times_A$-bialgebras and duality*,  J. Algebra __251__ (2002) 279-294 [[math.QA/0012164](https://arxiv.org/abs/math.QA/0012164), [doi:10.1006/jabr.2001.9101](https://doi.org/10.1006/jabr.2001.9101)]

See also:

* T. Brzeziński, A. Vazquez Marquez and [[J. Vercruysse]], _The Eilenberg-Moore category and a Beck-type theorem for a Morita context_, Applied Categorical Structures, 19, (2011), 821–858.

[[!redirects Tomasz Brzeziński]]
[[!redirects T. Brzeziński]]
[[!redirects T. Brzezinski]]
[[!reidrects Tomasz Brzeziński]]


Tomasz Maszczyk

__Tomasz Maszczyk__ is a Polish mathematician (Warsaw University and [IMPAN](http://www.impan.pl//EN)) with main interests in [[algebraic geometry]], noncommutative algebra and [[noncommutative geometry]] (both of algebraic and of operator-algebraic flavour). With [[Piotr Hajac]], he coordinates a weakly active seminar in Warsaw on noncommutative geometry at IMPAN, frequently featuring international leading experts in noncommutative geometry.

Maszczyk created an original viewpoint and approach to noncommutative geometry based on [[monoidal category|monoidal]] [[abelian category|abelian]] categories viewed as categories of quasicoherent sheaves. Galois and [[reconstruction theorems]] of various sort (Tannakian, Morita-type...), and remarkable and consistent application of categorical thinking (neglected by most of the mainstream schools in noncommutative geometry), play role at many places in his work. His works gave light to a number of problems related to cyclic homology, [[coring]]s, Hopf-Galois extensions, descent theory and mathematics of [[regular differential operator]]s in commutative and [[noncommutative algebraic geometry]].

Most of Maszczyk's main program is still unpublished (even on arXiv). Among his articles on the arXiv cf.

* _Noncommutative geometry through monoidal categories I_, [math.QA/0611806] (http://arxiv.org/abs/math.QA/0611806)

For an anouncement of some unpublished interesting results see the following abstract

T. Maszczyk, [NCG Seminar IMPAN](http://www.impan.gov.pl/~pmh/seminar/sem07.html) 18 Feb 2008

NONCOMMUTATIVE CORRESPONDENCES AND GALOIS-TANNAKA RECONSTRUCTION

> According to Grothendieck-Galois theory, there is a close relation between splittings of commutative rings by an appropriate base change and (groupoid) actions. The reconstruction of the action from a given splitting is called the Galois reconstruction. According to Grothendieck-Deligne-Saavedra Rivano-Tannaka theory, there is another close relation between representations of a given groupoid and the groupoid itself. The reconstruction of the groupoid from its representations is called the Tannaka reconstruction. We show that both reconstructions are particular cases of our theorem about splittings of flat covers in the bicategory of monoidal categories.

* _Hochschild cohomology, Entropy and information theory.

Tomasz Maszczyk has also developed a cohomological theory of Entropy based on Hochschild cohomology. It is related to the information cohomology of Baudot-Bennequin.

The link to the abstract of this talk is below.

[Hochschild cohomology and Shannon entropy](https://www.mimuw.edu.pl/aktualnosci/seminaria/hochschild-cohomology-abstract-convexity-and-shannon-entropy)

Here is the link to Tomasz Maszczyk's [homepage](http://www.mimuw.edu.pl/~maszczyk).

category: people


Tomer Schlank

[[!redirects T. M. Schlank]]

Tomer Schlank is a Professor at The Hebrew University of Jerusalem working in Homotopy Theory and Arithmetic Geometry. Previously, he was a Simons Postdoc in the Mathematics Department at MIT.

## Selected writings

* [[I. Barnea]] and T. M. Schlank, 2011, _A Projective Model Structure on Pro Simplicial Sheaves, and the Relative Etale Homotopy Type_, [arXiv:1109.5477](http://arxiv.org/abs/1109.5477)

*  [[I. Barnea]] and T. M. Schlank, 2013, _Functorial Factorizations in Pro Categories_, [arXiv:1305.4607](http://arxiv.org/abs/1305.4607).

* [[I. Barnea]] and T. M. Schlank, (2014), _Model Structures on Ind Categories and the Accessibility Rank of Weak Equivalences_, [arXiv:1407.1817](http://arxiv.org/abs/1407.1817)

* Yonatan Harpaz, Tomer M. Schlank, _Homotopy Obstructions to Rational Points_, [arXiv:1110.0164](http://arxiv.org/abs/1110.0164)

On the [[Eckmann-Hilton argument]] for [[(∞,1)-operads]]:

* [[Tomer Schlank]], [[Lior Yanovski]], *The ∞-Categorical Eckmann-Hilton Argument*, Algebr. Geom. Topol. 19 (2019) 3119-3170 ([arXiv:1808.06006](https://arxiv.org/abs/1808.06006), [doi:10.2140/agt.2019.19.3119](https://doi.org/10.2140/agt.2019.19.3119))

category:people


Tommaso Dorigo

* [webpage](https://dorigo.wordpress.com/who-am-i/)

## selected writings

* {#Dorigo15} [[Tommaso Dorigo]], _Extraordinary claims: the $0.000029\%$ solution_, EPJ Web of Conferences 95, 02003 (2015) ([doi:10.1051/epjconf/20159502003](https://doi.org/10.1051/epjconf/20159502003))

## related $n$Lab entries

* [[statistical significance]]

* [[flavour anomaly]]

category: people


Tommaso Macrelli

* [Institute page](https://www.surrey.ac.uk/people/tommaso-macrelli)

## Selected writings

On [[quantum L-infinity algebras]] for the [[BV-BRST formalism]]:

* [[Branislav Jurčo]], [[Tommaso Macrelli]], [[Christian Sämann]], [[Martin Wolf]], *Loop Amplitudes and Quantum Homotopy Algebras*, Journal of High Energy Physics **2020** 3 (2020) [doi:10.1007/JHEP07(2020)003, [arXiv:1912.06695](https://arxiv.org/abs/1912.06695)]

* [[Branislav Jurčo]], [[Hyungrok Kim]], [[Tommaso Macrelli]], [[Christian Saemann]], [[Martin Wolf]], *Perturbative Quantum Field Theory and Homotopy Algebras*, PoS (CORFU2019) 199 [[arXiv:2002.11168](https://arxiv.org/abs/2002.11168), [doi:10.22323/1.376.0199](https://doi.org/10.22323/1.376.0199)]

On the [[classical double copy]] ([[KLT relations]], relating [[gravity]] to [[Yang-Mills theory]]) via [[L-infinity algebras|$L_\infty$-algebras]]:

* [[Leron Borsten]], [[Branislav Jurčo]], [[Hyungrok Kim]], [[Tommaso Macrelli]], [[Christian Saemann]], [[Martin Wolf]], _Double copy from homotopy algebras_, Fortschritte der Physik, **69** 8-9 (2021) 2100075 $[$[arXiv:2102.11390](https://arxiv.org/abs/2102.11390), [doi:10.1002/prop.202100075](https://doi.org/10.1002/prop.202100075)$]$

Extending double copy construction from tree level to loop level:

* [[Leron Borsten]], [[Branislav Jurčo]], [[Hyungrok Kim]], [[Tommaso Macrelli]], [[Christian Saemann]], [[Martin Wolf]], _Becchi-Rouet-Stora-Tyutin-Lagrangian double copy of Yang-Mills theory_, Phys. Rev. Lett. **126** (2021) 191601 [[doi:10.1103/PhysRevLett.126.191601](https://doi.org/10.1103/PhysRevLett.126.191601)]

> We show that the double copy of gauge theory amplitudes to N=0 supergravity amplitudes extends from tree level to loop level. We first explain that color-kinematics duality is a condition for the Becchi-Rouet-Stora-Tyutin operator and the action of a field theory with cubic interaction terms to double copy to a consistent gauge theory. We then apply this argument to Yang-Mills theory, where color-kinematics duality is known to be satisfied on shell at the tree level. Finally, we show that the latter restriction can only lead to terms that can be absorbed in a sequence of field redefinitions, rendering the double copied action equivalent to N=0 supergravity.

category: people


Tomonaga-Schwinger equation

#Contents#
* table of contents
{:toc}

## Idea

The generalization of the [[Schrödinger equation]] from [[quantum mechanics]] to [[quantum field theory]]:

from [Torre-Varadarajan 98 p.2](#TorreVaradarajan98):

The idea of evolving a quantum field from any [[Cauchy surface]] to any other seems to have originated in the mid 1940's with the work of Tomonaga [1] and
Schwinger [2] on relativisticquantum field theory. Tomonaga and Schwinger wanted an invariant generalization of the Schrödinger equation, which describes time evolution of the state of a quantum field relative to a fixed inertial reference frame. By allowing for all possible Cauchy surfaces in the
description of dynamical evolution one easily accommodates all possible notions of time for all possible inertial observers. Thus a dynamical formalism incorporating arbitrary Cauchy surfaces does allow for an invariant generalization of the Schrödinger equation. Since, the space of Cauchy surfaces is infinite-dimensional, it is impossible to describe time evolution along arbitrary surfaces by using a single time parameter.  In essence, one needs a distinct time parameter for every possible foliation of spacet
ime. As shown by Tomonaga and Schwinger, if one formulates dynamics in terms of general Cauchy surfaces, the resulting dynamical evolution equation is, formally, a functional differential equation, which is usually called the "Tomonaga-Schwinger equation".

## Related concepts

* [[Schrödinger picture]]

* [[Wheeler superspace]]

## References

Named after [[Shin'ichirō Tomonaga]] and [[Julian Schwinger]].

* {#TorreVaradarajan98} [[Charles Torre]], M. Varadarajan, _Functional Evolution of Free Quantum Fields_, Class.Quant.Grav. 16 (1999) 2651-2668 ([arXiv:hep-th/9811222](https://arxiv.org/abs/hep-th/9811222))


Tomoyuki Arakawa

Tomoyuki Arakawa is a Japanese mathematician and mathematical physicist at RIMS, Kyoto with research interests in [[representation theory]] and [[vertex operator algebras]].

* personal [webpage](https://www.kurims.kyoto-u.ac.jp/~arakawa/index-e.html)

* [wikipedia](https://en.wikipedia.org/wiki/Tomoyuki_Arakawa)

## Selected works

On [[class S]] [[chiral algebras]]:

* Tomoyuki Arakawa, _Chiral algebras of class S and Moore-Tachikawa symplectic varieties_, [arXiv:1811.01577](https://arxiv.org/abs/1811.01577)

An analogue of [[Borel-Weil-Bott theorem]] for [[chiral differential operators]]

* Tomoyuki Arakawa, [[Fyodor Malikov]], _A chiral Borel--Weil--Bott theorem_, Advances in Mathematics __229__:5 (2012) 2908--2949 [doi](https://doi.org/10.1016/j.aim.2011.11.002)

category: people


Tomáš Brauner

* [webpage](http://gemma.ujf.cas.cz/~brauner/)

category: people

[[!redirects Tomas Brauner]]


Tomáš Bzdušek

* [personal page](https://www.tomasbzdusek.com/)

* [institute page](https://www.physik.uzh.ch/en/groups/neupert/team/bzdusek.html)

## Selected writings

On [[topological semimetals]]:

* [[Tomáš Bzdušek]], *Symmetry and topology of nodal semimetals*, ETH (2017) [[doi:10.3929/ethz-b-000216959](https://doi.org/10.3929/ethz-b-000216959)]

On non-trivial [[braid group statistics|braiding]] of [[semi-metal|nodal points]] in "momentum space" (i.e. in the [[Brillouin torus]]):

* QuanSheng Wu, Alexey A. Soluyanov, [[Tomáš Bzdušek]], *Non-Abelian band topology in noninteracting metals*, Science **365** (2019) 1273-1277 ([arXiv:1808.07469](https://arxiv.org/abs/1808.07469) ,[doi:10.1126/science.aau8740](https://doi.org/10.1126/science.aau8740))

* Apoorv Tiwari, [[Tomáš Bzdušek]], *Non-Abelian topology of nodal-line rings in PT-symmetric systems*, Phys. Rev. B **101** (2020) 195130 ([doi:10.1103/PhysRevB.101.195130](https://doi.org/10.1103/PhysRevB.101.195130))

category: people

[[!redirects Tomas Bzdusek]]


Tonny Springer

* [Wikipedia entry](https://en.wikipedia.org/wiki/T._A._Springer)

## selected writings

* {#SpringerVeltkamp00} [[Tonny Springer]], [[Ferdinand Veldkamp]], _Octonions, Jordan Algebras, and Exceptional Groups_, Springer Monographs in Mathematics, 2000

## related $n$Lab entries

* [[octonions]]

* [[triality]]

* [[exceptional Lie group]]

category: people


Tony Pantev

* [website](http://www.math.upenn.edu/~tpantev)

## Selected writings

* Tony Pantev, [[Bertrand Toen]], M. Vaquie, [[G. Vezzosi]], _Shifted symplectic structures_, [arxiv/1111.3209](http://arxiv.org/abs/1111.3209)
* [[Carlos Simpson]], Tony Pantev, [[Ludmil Katzarkov]], _Nonabelian mixed Hodge structures_, [arXiv:math/0006213](http://arxiv.org/abs/math/0006213)
* [[Ludmil Katzarkov]], [[Maxim Kontsevich]], Tony Pantev, _Hodge theoretic aspects of mirror symmetry_, [arxiv/0806.0107](http://arxiv.org/abs/0806.0107)
* [[Ludmil Katzarkov]], [[Tony Pantev]], [[Bertrand Toën]], _Schematic homotopy types and non-abelian Hodge theory_, [math.AG/0107129](http://arxiv.org/abs/math/0107129)

On [[string phenomenology]] in [[heterotic M-theory]]:

* {#DOPW99} [[Ron Donagi]], [[Burt Ovrut]], [[Tony Pantev]], [[Daniel Waldram]], _Standard Models from Heterotic M-theory_, Adv. Theor. Math. Phys. 5 (2002) 93-137 ([arXiv:hep-th/9912208](https://arxiv.org/abs/hep-th/9912208))

* {#DOPW00} [[Ron Donagi]], [[Burt Ovrut]], [[Tony Pantev]], [[Daniel Waldram]], _Standard Model Vacua in Heterotic M-Theory_, talk at [Strings '99](http://strings99.aei.mpg.de/), Potsdam, Germany, 19 - 24 Jul 1999 ([arXiv:hep-th/0001101](https://arxiv.org/abs/hep-th/0001101))

## Related $n$Lab entries

* [[geometric Langlands]]
* [[nonabelian Hodge theory]]
* [[deformation quantization]]
* [[Stokes phenomenon]]
* [[homological mirror symmetry]]
* [[deformation quantization]]
* [[schematic homotopy type]]

category: people
[[!redirects T. Pantev]]


Tony Skyrme

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

Regarding the introduction of [[skyrmions]], [Ranada-Trueba 01, p. 200](On+Vortex+Atoms#RanadaTrueba01) know that

> [[Tony Skyrme|Skyrme]] had studied with attention [[On Vortex Atoms|Kelvin's ideas on vortex atoms]].

## Selected writings

Introducing [[Skyrmions]]:

* {#Skyrme62} [[Tony Skyrme]], _A unified field theory of mesons and baryons_, Nuclear Physics Volume 31, March–April 1962, Pages 556-569 (doi:10.1016/0029-5582(62)90775-7)

category: people


too simple to be simple

# Too simple to be simple
* table of contents
{: toc}

## Idea

There is a general principle in mathematics that
+-- {: .standout}
A trivial object is too simple to be simple.
=--

Quite often, classical references will define 'simple' (or an analogous term) in naïve way, so that a 'trivial' object *is* simple, but later it will become clear that more sophisticated theorems (especially classification theorems) work better if the definition is changed so that the trivial object is *not* simple.  Usually this can be done by changing 'if' to 'iff' (or sometimes changing 'or' to 'xor') in the classical definition.

The general pattern is a progression of definitions (of 'simple') from more to less naïve:

1.  Imprecise summary: there are no foos.  (Example: in a field, every element is invertible.)
2.  Classical definition: there are no nontrivial foos.  (Example: in a field, every element other than $0$ is invertible.)
3.  Modern definition: there are no nontrivial foos, but there is the trivial foo.  (Example: in a field, every element other than $0$ is invertible, and $0$ is non-invertible.)

This should be distinguished from barring the trivial object entirely.  Historically, many people wanted to say that the [[empty set]] is not a [[set]], and some people still say that the [[trivial ring]] is not a [[ring]] or that the [[improper filter]] is not a [[filter]].  This allows one to state later definitions (as of [[field]] or [[ultrafilter]]) without having to exclude the trivial object, but this does the exclusion too early.

## Examples

Examples include:

*  [[one|1]] is not a [[prime number]].
*  The [[trivial ring]] is not a [[field]] (or even an [[integral domain]]).
*  The [[trivial group]] is not a [[simple group]] (which is the trope-namer).
*  A [[zero object]] is not a [[simple object]] (generalizing the previous example).
*  The [[empty space]] is not a [[path-connected space]] (or even a [[connected space]]).
*  The [[improper ideal]] is not a [[maximal ideal]] (or even a [[prime ideal]]).
*  The [[improper filter]] is not an [[ultrafilter]].
*  An [[empty function]] is not a [[constant function]].
*  A [[bottom element]] is not an [[atom]].

But of course one may still find definitions used that disagree (see discussion at [[connected space]] and [[empty space]], for example).

Perhaps the basic example is that

*  [[false|False]] is not [[true]].

Nobody would be naïve enough to believe otherwise in this case, of course.  The point with this example is that, in the other cases, the naïve definition imposes only a uniqueness requirement (so that some set of possibilities ---such as the set of proper divisors of a prime number, or the set of non-invertible elements of a field--- must be a [[subsingleton]]) when it should in fact impose an *existence and uniqueness* requirement (so that the set of possibilities must be a [[singleton]]).  With [[truth values]], uniqueness is automatic, so existence is easier to notice.  More abstractly, the naïve definition is about $(-1)$-[[(-1)-truncation|truncation]], while the more sophisticated definition is about $(-2)$-[[(-2)-truncation|truncation]], which is more often relevant.

## Relationship to biased definitions

In many of the above examples one can obtain the sophisticated definition from the naïve definition by replacing a $2$-ary function by a function of arbitrary (finite) arity. For example we would replace

* $n$ is prime if whenever $n = a b$ we have $n = a$ or $n = b$

with

* $n$ is prime if whenever $n = \prod_{i=1}^k a_i$ we have $n = a_i$ for some $i$

Then $1$ is not a prime because it is equal to the empty product ($k = 0$) but not equal to any of the $a_i$ (because there aren't any)! Similarly we have:

* The [[empty space]] is the empty union, and therefore not [[connected space|connected]].
* In the [[trivial ring]] the empty product is equal to zero,
and so this ring is not [[integral domain|integral]].
* [[false|False]] is the empty disjunction, and hence is not [[true]].

This illustrates one advantage of using unbiased rather than [[biased definition|biased]] definitions: if one has replaced "for all $n$" with "for both $0$ and $2$" then it is very easy to forget the $0$ case and end up with a definition that fails for the trivial case.  (Unfortunately, it seems to be easy to forget $0$ even when using unbiased definitions, but hopefully less easy.)

In a similar vein we can define [[path-connected space|path connected]] by

* A space is path connected if for each ([[Kuratowski-finite|Kuratowski]])-finite subset there is a [[path]] (a [[continuous map]] from the [[unit interval]]) passing through every point of that subset.

Then the empty space is not path connected because it has no paths at all and hence no path through the empty subset.

[[!redirects too simple to be simple]]
[[!redirects too simple]]
[[!redirects trivial]]
[[!redirects trivial object]]
[[!redirects trivial objects]]


This page collects material related to the text

* [[William Lawvere]],

  _Tools for the advancement of objective logic: closed categories and toposes_,

  in J. Macnamara, [[Gonzalo Reyes]] (Eds.),

  _The Logical Foundations of Cognition_,

  Oxford University Press 1993

  (Proceedings of the Febr. 1991 Vancouver Conference "Logic and Cognition"),
pages 43-56, 1994.

  ([pdf](https://github.com/mattearnshaw/lawvere/blob/master/pdfs/1994-tools-for-the-advancement-of-objective-logic-closed-categories-and-toposes.pdf))

which argues that "[[objective logic]]" in the sense of [[Georg Hegel]]'s _[[Science of Logic]]_ should be understood as being captured by [[universal constructions]] of [[category theory]]/[[topos theory]], whereas the "[[subjective logic]]" is to be understood as ordinary [[deduction]]/inference about objects such constructed.

The text explicitly discusses formalization in [[categorical logic]] of Hegelian concepts such as

* [[abstract general]], [[concrete general]] and [[concrete particular|particular]] by categorical [[theories]] ([[Lawvere theories]]) and their [[models]];

* [[unity of opposites]] by [[adjoint modalities]].

Various related suggestions are implicit in other texts by Lawvere, see at _[William Lawvere -- Mathematics related to philosophy](William+Lawvere#RelationToPhilosophy)_

### Contents

#### 1. Categorical Refinement of a Hegelian Principle

#### 2. Deepening, Metrics, Unity and Identity of Adjoint Opposites

#### 3. Realization of Plans

#### 4. Detailed Models of Becoming

category: reference


Top

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

#Contents#
* table of contents
{:toc}

## Definition

**Top** denotes the [[category]] whose [[objects]] are [[topological spaces]] and whose [[morphisms]] are [[continuous functions]] between them. Its [[isomorphisms]] are the [[homeomorphisms]].

For exposition see _[[Introduction to Topology -- 1|Introduction to point-set topology]]_.

Often one considers (sometimes by default) [[subcategories]] of [[nice topological spaces]] such as [[compactly generated topological spaces]], notably because these are [[cartesian closed category|cartesian closed]]. There other other [[convenient categories of topological spaces]]. With any one such choice understood, it is often useful to regard it as "the" category of topological spaces.

The [[homotopy category]] of $Top$ given by its [[localization]] at the [[weak homotopy equivalences]] is the [[classical homotopy category]] [[Ho(Top)]]. This is the central object of study in [[homotopy theory]], see also at _[[classical model structure on topological spaces]]_.  The [[simplicial localization]] of [[Top]] at the [[weak homotopy equivalences]] is the archetypical [[(∞,1)-category]], [[equivalence of (infinity,1)-categories|equivalent]] to [[∞Grpd]] (see at _[[homotopy hypothesis]]_).

## Properties

### Universal constructions
 {#UniversalConstructions}

We discuss [[universal constructions]] in [[Top]], such as [[limits]]/[[colimits]], etc. The following definition suggests that universal constructions be seen in the context of $Top$ as a [[topological concrete category]] (see Proposition \ref{topcat} below).

$\,$

[[!include universal constructions of topological spaces -- table]]

$\,$

+-- {: .num_defn #InitialAndFinalTopologies}
###### Definition
**([[weak topology]] and [[strong topology]])**

Let $\{X_i = (S_i,\tau_i) \in Top\}_{i \in I}$ be a [[class]] of [[topological spaces]], and let $S \in Set$ be a bare [[set]]. Then

* For $\{S \stackrel{f_i}{\to} S_i \}_{i \in I}$ a set of [[functions]] out of $S$, the _[[initial topology]]_ $\tau_{initial}(\{f_i\}_{i \in I})$ is the topology on $S$ with the [[minimum]] collection of [[open subsets]] such that all $f_i \colon (S,\tau_{initial}(\{f_i\}_{i \in I}))\to X_i$ are [[continuous function|continuous]].

* For $\{S_i \stackrel{f_i}{\to} S\}_{i \in I}$ a set of [[functions]] into $S$, the _[[final topology]]_ $\tau_{final}(\{f_i\}_{i \in I})$ is the topology on $S$ with the [[maximum]] collection of [[open subsets]] such that all $f_i \colon X_i \to (S,\tau_{final}(\{f_i\}_{i \in I}))$ are [[continuous function|continuous]].

=--

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

For $X$ a single topological space, and $\iota_S \colon S \hookrightarrow U(X)$ a subset of its underlying set, then the initial topology $\tau_{intial}(\iota_S)$, def. \ref{InitialAndFinalTopologies}, is the [[subspace topology]], making

$$
  \iota_S
  \;\colon\;
  (S, \tau_{initial}(\iota_S))
  \hookrightarrow
  X
$$

a [[topological subspace]] inclusion.

=--

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

Conversely, for $p_S \colon U(X) \longrightarrow S$ an [[epimorphism]], then the final topology $\tau_{final}(p_S)$ on $S$ is the _[[quotient topology]]_.

=--

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

Let $I$ be a [[small category]] and let $X_\bullet \colon I \longrightarrow Top$ be an $I$-[[diagram]] in [[Top]] (a [[functor]] from $I$ to $Top$), with components denoted $X_i = (S_i, \tau_i)$, where $S_i \in Set$ and $\tau_i$ a topology on $S_i$. Then:

1. The [[limit]] of $X_\bullet$ exists and is given by [[generalized the|the]] topological space whose underlying set is [[generalized the|the]] limit in [[Set]] of the underlying sets in the diagram, and whose topology is the [[initial topology]], def. \ref{InitialAndFinalTopologies}, for the functions $p_i$ which are the limiting [[cone]] components:

   $$
     \array{
       && \underset{\longleftarrow}{\lim}_{i \in I} S_i
       \\
       & {}^{\mathllap{p_i}}\swarrow && \searrow^{\mathrlap{p_j}}
       \\
       S_i && \underset{}{\longrightarrow} && S_j
     }
     \,.
   $$

   Hence

   $$
     \underset{\longleftarrow}{\lim}_{i \in I} X_i
     \simeq
     \left(\underset{\longleftarrow}{\lim}_{i \in I} S_i,\; \tau_{initial}(\{p_i\}_{i \in I})\right)
   $$

1. The [[colimit]] of $X_\bullet$ exists and is the topological space whose underlying set is the colimit in [[Set]] of the underlying diagram of sets, and whose topology is the [[final topology]], def. \ref{InitialAndFinalTopologies} for the component maps $\iota_i$ of the colimiting [[cocone]]

   $$
     \array{
       S_i && \longrightarrow && S_j
       \\
       & {}_{\mathllap{\iota_i}}\searrow && \swarrow_{\mathrlap{\iota_j}}
       \\
       && \underset{\longrightarrow}{\lim}_{i \in I} S_i
     }
     \,.
   $$

   Hence

   $$
     \underset{\longrightarrow}{\lim}_{i \in I} X_i
     \simeq
     \left(\underset{\longrightarrow}{\lim}_{i \in I} S_i,\; \tau_{final}(\{\iota_i\}_{i \in I})\right)
   $$

=--

(e.g. [Bourbaki 71, section I.4](#Bourbaki71))

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

The required [[universal property]] of $\left(\underset{\longleftarrow}{\lim}_{i \in I} S_i,\; \tau_{initial}(\{p_i\}_{i \in I})\right)$ is immediate: for

$$
  \array{
    && (S,\tau)
    \\
    & {}^{\mathllap{f_i}}\swarrow && \searrow^{\mathrlap{f_j}}
    \\
    X_i && \underset{}{\longrightarrow} && X_j
  }
$$

any [[cone]] over the diagram, then by construction there is a unique function of underlying sets $S \longrightarrow \underset{\longleftarrow}{\lim}_{i \in I} S_i$ making the required diagrams commute, and so all that is required is that this unique function is always [[continuous function|continuous]]. But this is precisely what the initial topology ensures.

The case of the colimit is [[formal dual|formally dual]].

=--

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

The limit over the empty diagram in $Top$ is the [[point]] $\ast$ with its unique topology.

=--

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

For $\{X_i\}_{i \in I}$ a set of topological spaces, their [[coproduct]] $\underset{i \in I}{\sqcup} X_i \in Top$ is their _[[disjoint union]]_.

=--

In particular:

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

For $S \in Set$, the $S$-indexed [[coproduct]] of the point, $\underset{s \in S}{\coprod}\ast $,  is the set $S$ itself equipped with the [[final topology]], hence is the [[discrete topological space]] on $S$.

=--

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

For $\{X_i\}_{i \in I}$ a set of topological spaces, their [[product]] $\underset{i \in I}{\prod} X_i \in Top$ is the [[Cartesian product]] of the underlying sets equipped with the _[[product topology]]_, also called the _[[Tychonoff product]]_.

In the case that $S$ is a [[finite set]], such as for binary product spaces $X \times Y$, then a [[basis for a topology|sub-basis]] for the product topology is given by the [[Cartesian products]] of the open subsets of (a basis for) each factor space.

=--

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

The [[equalizer]] of two [[continuous functions]] $f, g \colon X \stackrel{\longrightarrow}{\longrightarrow} Y$ in $Top$ is the equalizer of the underlying functions of sets

$$
  eq(f,g)
   \hookrightarrow
  S_X
    \stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}}
  S_Y
$$

(hence the largets subset of $S_X$ on which both functions coincide) and equipped with the [[subspace topology]], example \ref{TopologicalSubspace}.

=--

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

The [[coequalizer]] of two [[continuous functions]] $f, g \colon X \stackrel{\longrightarrow}{\longrightarrow} Y$ in $Top$ is the coequalizer of the underlying functions of sets

$$
  S_X
    \stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}}
  S_Y
    \longrightarrow
  coeq(f,g)
$$

(hence the [[quotient set]] by the [[equivalence relation]] generated by $f(x) \sim g(x)$ for all $x \in X$)  and equipped with the [[quotient topology]], example \ref{QuotientTopology}.

=--

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

For

$$
  \array{
     A &\overset{g}{\longrightarrow}& Y
     \\
     {}^{\mathllap{f}}\downarrow
     \\
     X
  }
$$

two [[continuous functions]] out of the same [[domain]], then the [[colimit]] under this diagram is also called the _[[pushout]]_, denoted

$$
  \array{
     A &\overset{g}{\longrightarrow}& Y
     \\
     {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{g_\ast f}}
     \\
     X &\longrightarrow& X \sqcup_A Y
     \,.
  }
  \,.
$$

(Here $g_\ast f$ is also called the pushout of $f$, or the _[[base change|cobase change]]_ of $f$ along $g$.) If $g$ is an inclusion, one also write $X \cup_f Y$ and calls this the _[[attaching space]]_.

By example \ref{CoequalizerInTop} the [[pushout]]/[[attaching space]] is the [[quotient topological space]]

$$
  X \sqcup_A Y \simeq (X\sqcup Y)/\sim
$$

of the [[disjoint union]] of $X$ and $Y$ subject to the [[equivalence relation]] which identifies a point in $X$ with a point in $Y$ if they have the same pre-image in $A$.

(graphics from [Aguilar-Gitler-Prieto 02](#AguilarGitlerPrieto02))

=--

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

As an important special case of example \ref{PushoutInTop},  let

$$
  i_n \colon S^{n-1}\longrightarrow D^n
$$

be the canonical inclusion of the standard [[n-sphere|(n-1)-sphere]] as the [[boundary]] of the standard [[n-disk]] (both regarded as [[topological spaces]] with their [[subspace topology]] as subspaces of the [[Cartesian space]] $\mathbb{R}^n$).

Then the colimit in [[Top]] under the diagram, i.e. the [[pushout]] of $i_n$ along itself,

$$
  \left\{
     D^n \overset{i_n}{\longleftarrow} S^{n-1} \overset{i_n}{\longrightarrow}
     D^n
  \right\}
  \,,
$$

is the [[n-sphere]] $S^n$:

$$
  \array{
    S^{n-1} &\overset{i_n}{\longrightarrow}& D^n
    \\
    {}^{\mathllap{i_n}}\downarrow &(po)& \downarrow
    \\
    D^n &\longrightarrow& S^n
  }
  \,.
$$

(graphics from Ueno-Shiga-Morita 95)

=--

+-- {: .num_example #ClosedSubspacesGluing}
###### Example
**([[union]] of two [[open subset|open]] or two [[closed subset|closed]] [[subspaces]] is [[pushout]])**

Let $X$ be a [[topological space]] and let $A,B \subset X$ be [[subspaces]] such that

1. $A,B \subset X$ are both [[open subsets]] or are both [[closed subsets]];

1. they constitute a [[cover]]: $X = A \cup B$

Write $i_A \colon A \to X$ and $i_B \colon B \to X$ for the corresponding inclusion [[continuous functions]].

Then the [[commuting square]]

$$
  \array{
     A \cap B &\longrightarrow& A
     \\
     \downarrow && \downarrow^{\mathrlap{i_A}}
     \\
     B &\underset{i_B}{\longrightarrow}& X
  }
$$

is a [[pushout]] square in $Top$ (example \ref{PushoutInTop}).

By the [[universal property]] of the [[pushout]] this means in particular that for $Y$ any [[topological space]] then a function of underlying sets

$$
  f \;\colon\; X \longrightarrow Y
$$

is a [[continuous function]] as soon as its two restrictions

$$
  f\vert_A \;\colon\; A \longrightarrow Y
  \phantom{AAAA}
  f\vert_A \;\colon\; B \longrightarrow Y
$$

are continuous.

=--

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

Clearly the underlying diagram of underlying [[sets]] is a pushout in [[Set]]. Therefore by prop. \ref{DescriptionOfLimitsAndColimitsInTop} we need to show that the [[topological space|topology]] on $X$ is the [[final topology]] induced by the set of functions $\{i_A, i_B\}$, hence that
a [[subset]] $S \subset X$ is an [[open subset]] precisely if the [[pre-images]] (restrictions)

$$
  i_A^{-1}(S) = S \cap A
   \phantom{AAA}
    \text{and}
   \phantom{AAA}
  i_B^{-1}(S) = S \cap B
$$

are open subsets of $A$ and $B$, respectively.

Now by definition of the [[subspace topology]], if $S \subset X$ is open, then the intersections $A \cap S \subset A$ and $B \cap S \subset B$ are open in these subspaces.

Conversely, assume that $A \cap S \subset A$ and $B \cap S \subset B$ are open. We need to show that then $S \subset X$ is open.

Consider now first the case that $A;B \subset X$ are both open open. Then by the nature of the [[subspace topology]], that $A \cap S$ is open in $A$ means that there is an open subset $S_A \subset X$ such that $A \cap S = A \cap S_A$. Since the intersection of two open subsets is open, this implies that $A \cap S_A$ and hence $A \cap S$ is open.  Similarly $B \cap S$. Therefore

$$
  \begin{aligned}
    S
     & =
    S \cap X
    \\
    & = S \cap (A \cup B)
    \\
    & =  (S \cap A) \cup (S \cap B)
  \end{aligned}
$$

is the union of two open subsets and therefore open.

Now consider the case that $A,B \subset X$ are both closed subsets.

Again by the nature of the subspace topology, that $A \cap S \subset A$ and $B \cap S \subset B$ are open means that there exist open subsets $S_A, S_B \subset X$ such that $A \cap S = A \cap S_A$ and $B \cap S = B \cap S_B$. Since $A,B \subset X$ are closed by assumption, this means that $A \setminus S, B \setminus S \subset X$ are still closed, hence that $X \setminus (A \setminus S), X \setminus (B \setminus S) \subset X$ are open.

Now observe that (by [[de Morgan duality]])

$$
  \begin{aligned}
    S
      & = X \setminus (X \setminus S)
    \\
      & =
    X \setminus ( (A \cup B) \setminus S )
    \\
      & =
    X \setminus ( (A \setminus S) \cup (B \setminus S) )
    \\
     & =
    (X \setminus (A \setminus S)) \cap (X \setminus (B \setminus S))
    \,.
  \end{aligned}
$$

This exhibits $S$ as the intersection of two open subsets, hence as open.

=--

+-- {: .num_example #attach}
###### Example
If $X, Y, Z$ are [[normal topological spaces]] and $h: X \to Z$ is a [[closed embedding of topological spaces]] and $f: X \to Y$ is a [[continuous function]], then in the [[pushout]] diagram in $Top$ (example \ref{PushoutInTop})

$$\array{
X & \stackrel{h}{\to} & Z \\
\mathllap{f} \downarrow & & \downarrow \mathrlap{g} \\
Y & \underset{k}{\to} & W,
}$$

the space $W$ is normal and $k: Y \to W$ is a closed embedding.

=--

For **proof** of this and related statements see at _[[colimits of normal spaces]]_.

### Relation with $Set$

Write [[Set]] for the [[category]] of [[sets]].

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

Write

$$
  U \colon Top \longrightarrow Set
$$

for the [[forgetful functor]] that sends a topological space $X = (S,\tau)$ to its underlying set $U(X) = S \in Set$ and which regards a [[continuous function]] as a plain [[function]] on the underlying sets.

=--

Prop. \ref{DescriptionOfLimitsAndColimitsInTop} means in particular that:

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

The category [[Top]] has all small [[limits]] and [[colimits]]. The [[forgetful functor]] $U \colon Top \to Set$ from def. \ref{ForgetfulFunctorFromTopToSet} [[preserved limit|preserves]] and [[lifted limit|lifts]] limits and colimits.

=--

(But it does not [[created limit|create]] or [[reflected limit|reflect]] them.)

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

The [[forgetful functor]] $U$ from def. \ref{ForgetfulFunctorFromTopToSet} has a [[left adjoint]] $disc$, given by sending a [[set]] $S$ to the corresponding [[discrete topological space]], example \ref{DiscreteTopologicalSpaceAsCoproduct}

$$
  Top
  \stackrel{\overset{disc}{\longleftarrow}}{\underset{U}{\longrightarrow}}
  Set
  \,.
$$

=--

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

The [[forgetful functor]] $U$ from def. \ref{ForgetfulFunctorFromTopToSet} exhibits $Top$ as

* a [[concrete category]]

* a [[topological concrete category]].

=--

### Mono-/Epimorphisms
 {#MonoEpiMorphisms}

+-- {: .num_prop #SubspaceInclusionsAreRegularMonos}
###### Proposition
**([[regular monomorphisms]] of [[topological spaces]])**

In the [[category]] [[Top]] of [[topological spaces]],

1. the [[monomorphisms]] are those [[continuous functions]] which are [[injective functions]];

1. the [[regular monomorphisms]] are the [[topological embeddings]] (i.e. those continuous functions which are [[homeomorphisms]] onto their [[images]] equipped with the [[subspace topology]]).

=--

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

Regarding the first statement: An injective continuous function $f \colon X \to Y$ clearly has the cancellation property that defines monomorphisms: for parallel continuous functions $g_1,g_2 \colon Z \to X$, if $f \circ g_1 = f \circ g_2$, then $g_1 = g_2$, because continuous functions are equal precisely if their underlying functions of sets are equal. Conversely, if $f$ has the cancellation property, then testing on points $g_1, g_2 \colon \ast \to X$ gives that $f$ is injective.

Regarding the second statement: from the construction of [[equalizers]] in [[Top]] (example \ref{EqualizerInTop}) we have that these are topological subspace inclusions.

Conversely, let  $i \colon X \to Y$ be a [[topological subspace embedding]]. We need to show that this is the equalizer of some pair of parallel morphisms.

To that end, form the [[cokernel pair]] $(i_1, i_2)$ by taking the [[pushout]] of $i$ against itself (in the category of sets, and using the [[quotient topology]] on a [[disjoint union space]]). By [this prop.](regular+monomorphism#RegEquEff), the equalizer of that pair is the set-theoretic equalizer of that pair of functions endowed with the [[subspace topology]]. Since monomorphisms in [[Set]] are regular, we get the function $i$ back, and again by example \ref{EqualizerInTop}, it gets equipped with the subspace topology. This completes the proof.

=--

### Intersections and quotients

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

The [[pushout]] in [[Top]] of any (closed/open) [[topological subspace]] inclusion $i \colon A \hookrightarrow B$, example \ref{TopologicalSubspace}, along any [[continuous function]] $f \colon A \to C$ is itself an a (closed/open) subspace $j \colon C \hookrightarrow D$.

=--

For proof see [there](subspace+topology#pushout).

## Related concepts

* [[topological concrete category]]

* [[Ho(Top)]], [[∞Grpd]]

* [[convenient category of topological spaces]]

* [[TopGrp]]

## References

For general references see those listed at _[[topology]]_, such as

* {#Bourbaki71} [[Nicolas Bourbaki]], chapter 1 _Topological Structures_ of _Elements of Mathematics III: General topology_, Springer 1971, 1990

See also

* {#AguilarGitlerPrieto02} Marcelo Aguilar, [[Samuel Gitler]], Carlos Prieto, section 12 of _Algebraic topology from a homotopical viewpoint_, Springer (2002) ([toc pdf](http://tocs.ulb.tu-darmstadt.de/106999419.pdf))

An axiomatic desciption of $Top$ along the lines of [[ETCS]] for [[Set]] is discussed in

* Dana Schlomiuk, _An elementary theory of the category of topological space_, Transactions of the AMS, volume 149 (1970)

category: category

[[!redirects TopologicalSpaces]]
[[!redirects category of topological spaces]]

[[!redirects TopSp]]


top

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

#Contents#
* table of contents
{:toc}

## Idea

In a [[partial order|poset]] $P$, a __top__ of $P$ is a __greatest__ element: an element $\top$ of $P$ such that $a \leq \top$ for every element $a$.  Such a top may not exist; if it does, then it is unique.

In a [[preorder|proset]], a top may be defined similarly, but it need not be unique.  (However, it is still unique up the natural [[equivalence]] in the proset.)

A top of $P$ can also be understood as a [[meet]] of zero elements in $P$.

A poset that has both top and [[bottom]] is called _bounded_.

As a poset is a special kind of [[category]], a top is simply a [[terminal object]] in that category.

The top of the poset of [[subsets]] or [[subobject]]s of a given set or object $A$ is always $A$ itself.

## Related concepts

[[!include logic symbols -- table]]

[[!redirects top]]
[[!redirects tops]]
[[!redirects top element]]
[[!redirects top elements]]
[[!redirects greatest element]]
[[!redirects greatest elements]]


top Chern class

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic topology
+--{: .hide}
[[!include algebraic topology - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

For $\mathcal{V}_X$ a [[complex vector bundle]] of complex [[rank]] $n$, the highest degree [[Chern class]] that may generally be non-vanishing is $c_n$. This is hence often called the _[[top Chern class]]_ of the vector bundle.

## Properties

### Relation to Euler- and Thom-class

\begin{prop}
  The [[top Chern class]] of a [[complex vector bundle]] $\mathcal{V}_X$ equals the [[Euler class]] $e$ of the underlying [[real vector bundle]] $\mathcal{V}^{\mathbb{R}}_X$:

$$
  \mathcal{V}_X
  \;
  \text{has complex rank}\;n
  \;\;\;\;\;
  \Rightarrow
  \;\;\;\;\;
  c_n
  \big(
    \mathcal{V}_X
  \big)
  \;\;
    =
  \;\;
  e
  \big(
    \mathcal{V}^{\mathbb{R}}_X
  \big)
  \;\;\;\;
  \in
  H^{2n}
  \big(
   X;
   \,
   \mathbb{Z}
  \big)
  \,.
$$
\end{prop}

(e.g. [Bott-Tu 82 (20.10.6)](#BottTu82))

\begin{prop}
  The [[top Chern class]] of a [[complex vector bundle]] $\mathcal{V}_X$ equals the [[pullback in cohomology|pullback]] of any [[Thom class]]
$th \;\in\; H^{2n}\big( \mathcal{V}_X; \mathbb{Z} \big)$ on $\mathcal{V}_X$ along the [[zero-section]]:

$$
  \mathcal{V}_X
  \;
  \text{has complex rank}\;n
  \;\;\;\;\;
  \Rightarrow
  \;\;\;\;\;
  c_n
  \big(
    \mathcal{V}_X
  \big)
  \;\;\;
    =
  \;\;\;
  (0_X)^\ast
  (th)
  \;\;
  \in
  \;
  H^{2n}
  \big(
    X ;
    \,
    \mathbb{Z}
  \big)
$$
\end{prop}

(e.g. [Bott-Tu 82, Prop. 12.4](#BottTu82))

\begin{remark}
  This relation to [[Thom classes]] generalizes to [[Conner-Floyd-Chern classes]] in [[complex oriented cohomology|complex oriented]] [[Whitehead-generalized cohomology]]. See at _[[universal complex orientation on MU]]_.
\end{remark}

## Related concepts

* [[first Chern class]]

* [[Euler class]], [[Thom class]]

## References

* {#BottTu82} [[Raoul Bott]], [[Loring Tu]], _[[Differential Forms in Algebraic Topology]]_, Graduate Texts in Mathematics 82, Springer 1982 ([doi:10.1007/BFb0063500](https://doi.org/10.1007/BFb0063500))

For more references see at _[[Chern class]]_ and at _[[characteristic class]]_.

[[!redirects top Chern classes]]


top quark

+-- {: .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]] the _top quark_ is the up-type [[quark]] in the third [[generation of fermions]], a [[heavy quark]].

It is the [[fundamental particle]] with by far the largest [[rest mass]] (after [[Higgs mechanism]]). This implies that top quark [[virtual particles]] in [[Feynman diagrams]] typically gives the dominant contribution to [[loop order|loop effects]] in [[perturbative quantum field theory|perturbative]] [[quantum observables]], notably in relation to the [[Higgs field]].

## Related concepts

[[!include flavors of fundamental particles -- table]]

## References

The top quark discovery was announced jointly in

* F. Abe et al. (CDF Collaboration), _Observation of Top Quark Production in $\bar p p$ Collisions with the Collider Detector at Fermilab_, Phys. Rev. Lett. 74, 2626 – Published 3 April 1995 ([doi:10.1103/PhysRevLett.74.2626](https://doi.org/10.1103/PhysRevLett.74.2626))

* S. Abachi et al. (D0 Collaboration), _Observation of the Top Quark_, Phys. Rev. Lett. 74, 2632 – Published 3 April 1995 ([doi:10.1103/PhysRevLett.74.2632](https://doi.org/10.1103/PhysRevLett.74.2632))

See also

* Wikipedia, _[Top quark](https://en.wikipedia.org/wiki/Top_quark)_

[[!redirects top quarks]]


TopGrp

#Contents#
* table of contents
{:toc}

## Definition

$TopGrp$ is the [[category]] whose [[objects]] are [[topological groups]] and whose [[morphisms]] are [[continuous function|continuous]] [[group homomorphisms]] between these.

## Properties

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

The [[category]] [[TopGrp]] of topological groups and [[continuous function|continuous]] [[group homomorphisms]] between them is a [[protomodular category]].

=--

See at _[topological group -- Protomodularity](http://ncatlab.org/nlab/show/topological+group#Protomodularity)_.

## Related entries

* [[Top]], [[Grp]]

category: category


TopMfd

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

#Contents#
* table of contents
{:toc}

## Definition

The [[category]] $TopMfd$ is that whose

* [[object]]s are [[topological manifold]]s;

* [[morphism]]s are [[continuous function]]s between these.

## Properties

### As a site

The category $TopMfd$ becomes a [[large site]] equipped with the [[open cover]] [[coverage]]. This is a [[small-generated site]]: a [[dense sub-site]] is [[CartSp]]${}_{top}$.

The [[hypercompletion|hypercompleted]] [[(∞,1)-sheaf (∞,1)-topos]] over $TopMfd$ is that of [[Euclidean-topological ∞-groupoid]]s.

$$
  ETop\infty Grpd \simeq \hat Sh_{(\infty,1)}(TopMfd)
  \,.
$$

See there for details.

## Related concepts

* **TopMfd**

* [[Diff|DiffMfd]]

category: category


topological abelian group

# Topological abelian groups
* table of contents
{: toc}

## Definition

A __topological abelian group__ (or __TAG__) is a [[topological group]] whose underlying [[group]] is [[abelian group|abelian]].

Topological abelian groups and [[continuous map|continuous]] group [[homomorphisms]] form a [[category]] $Ab Top$.

## Remarks

Often TAGs are easier to understand than arbitrary topological groups, because no distinction need be made between left and right.  For example, a TAG has only one [[uniform space|uniform structure]], a [[locally compact space|locally compact]] TAG has only one [[Haar measure]] (up to scale), etc.

## Applications

TAGs are important in [[analysis]].  For example, every [[topological vector space]] (TVS) is a TAG, and much of the theory of TVSs can be generalised to TAGs.  See also [[G-norm]].

## References

*  [[Jiri Adamek]], [[Horst Herrlich]], and [[George Strecker]], _Abstract and concrete categories: the joy of cats_. [free online](http://katmat.math.uni-bremen.de/acc/acc.pdf)

*  _[[HAF]]_ (for applications to analysis)

[[!redirects topological abelian group]]
[[!redirects topological abelian groups]]
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topological algebra

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topology
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[[!include topology - contents]]
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#### Algebra
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[[!include higher algebra - contents]]
=--
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#Contents#
* table of contents
{:toc}

## Definition

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

A _topological algebra_ is equivalently

* a [[topological ring]] which is also an [[associative algebra]] over some base [[topological ring]];

* an [[associative algebra]] [[internalization|internal]] to the [[category]] [[Top]] of [[topological spaces]] and [[continuous functions]] between them

* an [[associative algebra]] structure on a [[topological space]] such that the algebra operations are [[continuous functions]].

=--

## Examples

A [[Banach algebra]] is in particular a topological algebra.

## Related concepts

* [[dense subalgebra]]

* [[C*-algebra]]

## References

* {#DubucPorta71} [[Eduardo Dubuc]], [[Horacio Porta]], _Convenient categories of topological algebras_, Bull. Amer. Math. Soc., Volume 77, Number 6 (1971), 975-979 ([Euclid:1183533170](https://projecteuclid.org/euclid.bams/1183533170))

* [[eom]], _[Topological algebra](http://www.encyclopediaofmath.org/index.php/Topological_algebra)_

[[!redirects topological algebras]]


Topological Algebraic Geometry - A Workshop

This entry is about the slides:

* [[Paul Goerss]], _Topological Algebraic Geometry: A workshop at the University of Copenhagen_ ([pdf](http://www.math.northwestern.edu/~pgoerss/papers/tag.pdf))

on [[derived algebraic geometry]] in terms of [[structure sheaf|structure sheaves]] with values in [[spectrum|spectra]] and [[E-infinity ring]]s.

category: reference


topological André-Quillen homology

#Contents#
* table of contents
{:toc}

## Idea

Topological André-Quillen homology is to [[A-∞ algebras]] as [[Kähler differentials]] are to [[associative algebra]].

## Related concepts

* [[André-Quillen homology]]

* [[topological Hochschild homology]]

## References

The definition is recalled for instance in section 2 of

* {#McCarthyManasian03} Randy McCarthy, Vahagn Minasian, _HKR Theorem for Smooth $S$-algebras_, [arxiv:math/0306243](http://arxiv.org/pdf/math/0306243v1.pdf)

which otherwise proves the analog of the [[Hochschild-Kostant-Rosenberg theorem]] in this context.

[[!redirects topological André-Quillen cohomology]]

[[!redirects topological Andre-Quillen homology]]
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topological automorphic form

#Contents#
* table of contents
{:toc}

## Idea

Topological automorphic forms are a generalization of [[topological modular forms]]: where the latter come with the [[moduli space]] of [[elliptic curves]], topological automorphic forms are associated to a given [[Shimura variety]].
Moreover, just as topological modular forms refine to the [[tmf]]-[[spectrum]] representing the corresponding [[cohomology theory]], so every Shimura variety induces a cohomology theory $taf$.

## Properties

### Chromatic pattern
 {#ChromaticPattern}

| [[chromatic level]] $n = $ | 1 | 2 | $\geq 3$ |
|--|--|--|--|
| [[cohomology theory]]/[[spectrum]] $E = $ | [[topological K-theory|KO]] | [[TMF]] | [[TAF]] |
| [[algebraic group]] | $GL_1$ | $GL_2$ | $U(1,n-1)$ |
| geometric object | [[multiplicative group]] | [[elliptic curve]] | [[Shimura variety]] |
| [[FQFT]] | [[(1,1)-dimensional Euclidean field theories and K-theory|superparticle]] |  [[(2,1)-dimensional Euclidean field theories and tmf|heterotic superstring]] | ?? |

## Related concepts

* [[topological modular forms]]
* [[automorphic form]]

## References

The definition is due to

* {#BehrensLawson07} [[Mark Behrens]], [[Tyler Lawson]], _Topological automorphic forms_, Mem. Amer. Math. Soc., 204 (2010), no. 958 ([arXiv:math/0702719](http://arxiv.org/abs/math/0702719))

An introductory survey is in

* {#Lawson08} [[Tyler Lawson]], _An overview of abelian varieties in homotopy theory_ ([arXiv:0810.0507](http://arxiv.org/abs/0810.0507))

Lecture notes include

* [[Mark Behrens]], _Topological Automorphic Forms_, Lecture series (2011) ([lecture notes](http://www.math.ist.utl.pt/~ggranja/SummerLect11_files/abstractmark.html))

* [[Doug Ravenel]], _Seminar on topological automorphic forms_ (2009-11) ([web](http://www.math.rochester.edu/u/faculty/doug/taf.html))

* [[Dylan Wilson]] and [[Paul VanKoughnett]], _TAF reading course_, 2013 ([web](http://math.northwestern.edu/~pvankoug/taf/taf.html))

* [[Chermaine Sia]], _Topological automorphic forms_, talk at [Talbot Workshop 2013](http://math.mit.edu/conferences/talbot/index.php?year=2013&sub=talks) ([pdf](http://www.math.harvard.edu/~sia/notes/talbot_taf.pdf))

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topological base

> This page discusses bases for the topology on [[topological spaces]]. For the concept of topological [[linear basis]] see at _[[basis in functional analysis]]_. For bases on [[sites]], that is for [[Grothendieck topologies]], see at _[[Grothendieck pretopology]]_.

+-- {: .rightHandSide}
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### Context
#### Topology
+-- {: .hide}
[[!include topology - contents]]
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# Contents
* table of contents
{: toc}

## Idea

A _base_ and a _subbase_ for a [[topological space]] are ways of generating its topology from something simpler.  This is the application to topology of the general concept of [[base]].

## Definition {#ForTopologicalSpaces}

Let $X$ be a [[topological space]], and let $\tau$ be its collection of [[open subsets]] (its 'topology').

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

A __base__ or __basis__ for (or "of") $X$ (or $\tau$) is a [[subset|collection]] $B \subset \tau$ -- whose members are called __basic open subsets__ or __generating open subsets__ -- such that every open subset is a [[union]] of basic ones.
=--

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

A __subbase__ for (or "of") $X$ (or $\tau$) is a subcollection $S \subset \tau$ -- whose members are called __subbasic open subsets__ -- such that every open subset is a [[union]] of [[finitary intersections]] of subbasic ones.
=--

If only the [[underlying set]] of $X$ is given, then a __base__ or __subbase__ on this [[set]] is any collection of [[subsets]] of $X$ that is a base or subbase for *some* topology on $X$.  See [below](#generation) for a characterisation of which collections these can be.

Now fix a [[point]] $a$ in $X$.

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

A __local base__ or __base of neighborhoods__ or __fundamental system of neighborhoods__ for (or "of") $X$ (or $\tau$) at $a$ is a subcollection $B \subset \tau$ -- whose members are called __basic neighborhoods__ or __generating neighborhoods__ of $a$ -- such that every basic neighborhood of $a$ is a [[neighborhood]] and every neighborhood of $a$ is a [[superset]] of some basic neighborhood.
=--

We may also allow basic neighborhoods to be non-open, but this really doesn\'t make any difference; any local base may be refined to a local base of open neighborhoods, and most local bases in practices already come that way.

A __local subbase__ at $a$ is a family of neighbourhoods $a$ such that each neighbourhood of $a$ contains a finite [[intersection]] of elements of the family.

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

The minimum [[cardinality]] of a base of $X$ is the __weight__ of $X$.  The minimum cardinality of a base of neighborhoods at $a$ is the __character__ of $X$ at $a$.  The [[supremum]] of the characters at all points of $X$ is the __character__ of $X$.
=--

We have assumed the [[axiom of choice]] to simplify the description of this concept; but in general one must speak of classes of cardinalities rather than individual cardinalities.

If the character of $X$ is [[countable set|countable]], we say that $X$ satisfies the [[first-countable space|first axiom of countability]]; if the weight is countable, we say that $X$ satisfies the [[second-countable space|second axiom of countability]].

## Examples

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

For the [[discrete topology]] on a set $X$, the collection of all [[singleton]] subsets is a base, and the singleton $\{x\}$ is a local base at $x$.  Thus every discrete space is first-countable, but only [[countable set|countable]] discrete spaces are second-countable.
=--

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

For every [[metric space]], in particular every [[paracompact space|paracompact]] [[Riemannian manifold]], the collection of [[open subsets]] that are [[open balls]] forms a base for the topology.  (For instance, a base for the topology on the [[real line]] is given by the collection of open intervals $(a,b) \subset \mathbb{R}$.)  Similarly, the collection of open balls containing a given point is a local basis at that point.
=--

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

This means that [[covering]] families consisting of such _basic_ open subsets are [[good open covers]].
=--

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

Refining the previous example, every [[metric space]] has a basis consisting of the [[open balls]] with *[[rational number|rational]]* radius.  (For instance, a base for the topology on the [[real line]] is given by the collection of open intervals $(a,b) \subset \mathbb{R}$ where $b - a$ is rational.)  Similarly, the collection of open balls with rational radius containing a given point is a local base at that point.  Therefore, every metric space is first-countable.
=--

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

Now consider a [[separable space|separable]] metric space; that is, we have a [[dense subset]] $D$ which is [[countable set|countable]].  Now the space has a basis consisting of the [[open balls]] with [[rational number|rational]] radius and centres in $D$.  (For instance, a base for the topology on the [[real line]] is given by the collection of open intervals $(a,b) \subset \mathbb{R}$ where $a$ and $b$ are rational.)  Therefore, every separable metric space is second-countable.
=--

## Properties

### Generating topologies
 {#generation}

Let $X$ be simply a [[set]].

+-- {: .num_prop #Recognition}
###### Proposition
**(recognition of topological bases)**

A collection $B$ of [[subsets]] of $X$ is a base for *some* topology on $X$ iff these conditions are met:

*  The elements of $B$ cover $X$;
*  For any $U, V \in B$ and any point $x \in U \cap V$ there is a $W \in B$ such that $W \subseteq U \cap V$ and $x \in W$.
=--

These conditions amount to saying that for each $x\in X$, the subcollection of those $U\in B$ such that $x\in U$ is a base for a [[filter]] on $X$ (which is then the [[neighborhood]] filter of $x$) --- in other words, that these subcollections are "colaxly closed" under finite intersections.

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

*Every* collection $S$ of subsets of $X$ is a subbase for some topology on $X$.
=--

A subbase naturally generates a base (for the same topology) by [[Moore closure|closing]] it under finitary intersections.  (The resulting base will actually be closed under intersection.)

### Relation to Grothendieck topologies and coverages
 {#ForSites}

If one thinks of the topology on $X$ as being encoded in the standard [[Grothendieck topology]] that it induces on its [[category of open subsets]] $Op(X)$, then a base for the topology induces a _[[coverage]]_ on $Op(X)$, whose covering families are the open covers by basic open subsets, which generates this Grothendieck topology.

This coverage is *not* in general a [[basis for the Grothendieck topology]], because a base for a topological space is in general not closed under [[intersection]] with arbitrary [[open subsets]]; a coverage is only a basis if is stable under [[pullback]] (here, closed under these intersections) and transitive.  Unfortunately the established terminology "basis" in [[topology]] and [[topos theory]] is not quite consistent with the inclusion of topological spaces into topos theory: "basis" in topology corresponds to "coverage" in topos theory, not to "basis" in topos theory.

## Related concepts

* [[neighbourhood base]]

## Literature

* [[Ryszard Engelking]], _General Topology_, PWN; Warszawa 1977; Revised and completed edition in: Sigma series in pure mathematics __6__, Heldermann 1989 [ISBN 388538-006-4](https://www.heldermann.de/SSPM/SSPM06/sspm06.htm)

* Parimal, Proofs (?), Lecture 13: [Basis for a topology](https://ece.iisc.ac.in/~parimal/proofs/lecture-13.pdf

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topological chiral homology

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

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

## Idea ##

_Topological chiral homology_ is a generalization of [[Hochschild homology]]. Where Hochschild homology is given by [[(∞,1)-colimit]]s of functors constant on an $\infty$-algebra over a diagram that is an [[∞-groupoid]], topological chiral homology is given by colimits of constant functors over [[(∞,1)-categories]] of open subsets of a [[manifold]].

This generalizes the concept of _chiral homology_ by Beilinson-Drinfeld.

## Definition

For the moment see the section Topological chiral homology at the entry on [[Hochschild homology]].

## Related concepts

The notion of topological chiral homology should be closely related to that of

* [[blob homology]]

* [[factorization algebra]]

* [[factorization homology]]

and be related to concepts in

* [[AQFT]].

Other related concepts

* [[nonabelian Poincaré duality]]

## References

A quick definition and comments on its relation to [[FQFT]] are in section 4.1 of

* [[Jacob Lurie]], _[[On the Classification of Topological Field Theories]]_

Technical details are in section 3 of

* [[Jacob Lurie]], _[[Ek-Algebras]]_

which meanwhile has becomes part of section 5 of

* [[Jacob Lurie]], _[[Higher Algebra]]_ .