Thomas-Pandharipande invariants

A certain class of [[BPS invariants]]

## References

* R. Pandharipande, R. P. Thomas, _Stable pairs and BPS invariants_ ([arXiv:0711.3899](http://arxiv.org/abs/0711.3899))


Thomas-Paul Hack

* [LQP page](https://www.lqp2.org/user/339)

## writings

* _Cosmological Applications of Algebraic Quantum Field Theory in Curved Spacetimes_, Springer 2016 ([arXiv:1506.01869](https://arxiv.org/abs/1506.01869), [doi:10.1007/978-3-319-21894-6](https://doi.org/10.1007/978-3-319-21894-6))

  on applications of [[AQFT on curved spacetimes]] to [[cosmology]]

## related $n$Lab entries

* [[AQFT on curved spacetimes]]

* [[quantum gravity]]

* [[cosmology]]

* [[cosmological constant]]

* [[standard model of cosmology]]

category: people


Thomason model category

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## Definition

A __Thomason model category__ is a (Quillen) [[model category]]
such that [[functorial factorizations]] exist.

## Properties

A [[functor category]] with values in a Thomason model category is automatically again a Thomason model category and hence yields a notion of [[global model structure on functors]].

[[Mark Hovey]] later popularized this addition, including the data of functorial factorizations (and not just their existence) into his definition of a [[model category]] (he attributed this addition to Dwyer–Hirschhorn–Kan–Smith).

## References

* [[Charles Weibel]], _Homotopy Ends and Thomason Model Categories_ ([arXiv](http://arxiv.org/abs/math/0106052))

* [n-Café discussion](http://golem.ph.utexas.edu/category/2009/07/question_on_functor_categories.html#c025798)

See Weibel's Thomason obituary for some details.


Thomason model structure

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## Idea

By regarding a [[simplicial set]] as an [[object]] in the [[classical model structure on simplicial sets]], one effectively identifies it (up to [[simplicial weak homotopy equivalence|weak equivalence]]) with the [[∞-groupoid]] that it presents under [[Kan fibrant replacement]].
If the original simplicial set is the [[nerve]] of a [[category]], then the corresponding Kan fibrant replacement is something like the _$\infty$-groupoidification_ of that category: see *[[geometric realization of categories]]*.

This way each ordinary category models an [[∞-groupoid]]. The _Thomason_ [[model category]] structure on [[Cat]] exhibits this: in this [[model category]] a morphism between two categories is a [[weak equivalence]] precisely if it induces a weak equivalence of the corresponding $\infty$-groupoids.

It turns out the Thomason model structure on [[Cat]] is [[Quillen equivalence|Quillen equivalent]] to the [[classical model structure on simplicial sets]].

This is remarkable, in that it says that every [[homotopy type]], i. e. every [[equivalence in an (infinity,1)-category|equivalence]] class of [[infinity-groupoid|$\infty$-groupoids]], is obtained by $\infty$-groupoidifying just categories.

In fact, every [[cofibrant object]] in the Thomason model structure structure is a [[poset]]. Since every object in a model category is weakly equivalent to a cofibrant one, this means that even the [[nerves]] of just [[posets]] are sufficient to model all [[homotopy types]].

This is a rather curious aspect of the [[Robert Thomason|Thomason]] model on [[Cat]]: it does not really have anything intrinsically to do with [[category|categories]], but rather uses these as a way to present [[∞-groupoids]]. In particular, the class of [[weak equivalences]] is much larger than just the [[equivalence of categories|equivalences of categories]].
There is a _different_ model structure on [[Cat]] in which the weak equivalences are precisely the "true" weak [[equivalence of categories|equivalences of categories]] (not of anything constructed from them). This is called the *[[canonical model structure on Cat]]*.

## Definition
 {#Definition}

Recall the [[subdivision]] functor $Sd$ and its [[right adjoint]] [Ex-functor](Kan+fibrant+replacement#ExFunctor).

\begin{definition}
\label{ThomasonModelStructure}
Let $C$ and $D$ be [[small categories]]. A functor $f \colon C \to D$ is called :

* a *Thomason fibration* iff $Ex^2 N(f)$ is a Kan fibration,

* *Thomason weak equivalence* iff its [[nerve]] $N(f)$ is a [[weak homotopy equivalence]],

* *Thomason-cofibration* iff it has the [[left lifting property]] against the morphisms that are both Thomason-fibrations as well as Thomason-equivalences.

\end{definition}
\begin{proposition}
Equipped with these classes of morphisms, the [[1-category]] of [[small categories]] is a [[proper model category|proper]] [[model category]].
\end{proposition}
This is [Thomason (1980), corollary 5.5](#Thomason1980).

## Properties

### (Co)Fibrant objects

\begin{proposition}
\label{EveryThomasocCofibrantCategoryIsAPoset}
 Every [[cofibrant object]] in the Thomas model structure (Def. \ref{ThomasonModelStructure}) is a [[poset]] ([regarded](partial+order#AsACategoryWithExtraProperties) as a [[small category]]).
\end{proposition}
This is [Thomason (1980), proposition 5.7](#Thomason1980).

### Equivalence with the classical model structure on simplicial sets

The Thomason model structure on Cat is [[Quillen equivalent]] to the [[classical model structure on simplicial sets]] by the adjunction

$$ h Sd^2 : sSet \rightleftarrows Cat : Ex^2 N $$

We can assert more: this is also an adjoint weak equivalence. These are relative functors between the [[relative categories]] $(sSet, Kan)$ and $(Cat, Thomason)$, and both the adjunction unit and counit are natural weak equivalences.

### Relation to ∞Gpd

The localization $L : Cat \to \infty Gpd$ sending a category to its homotopy type has a direct interpretation in terms of ∞-category theory:

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###### Proposition
$L$ is the ∞-groupoidification functor functor $C \mapsto C[C^{-1}]$.
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### Homotopy colimits

Let $f : C \to Cat$ be a functor. Its [[homotopy colimit]] in the Thomason model
structure can be computed using the [[Grothendieck construction]]:

### The nerve is a homotopy colimit of simplicial sets

By [Hirschhorn proposition 18.1.6](#Hir), the nerve of any category is the homotopy colimit of the constant point-valued diagram:

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###### Proposition
For any category $C$, $N(C) \simeq hocolim_{c \in C} \Delta^0$
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In fact, when Hirschhorn's definition of the homotopy colimit for functors valued in simplicial model categories:
$$
  hocolim(f) = N(- \downarrow C)^{op} \otimes_{C} f
$$
we get an isomorphism $N(C)^{op} \cong hocolim(1)$.

## Related concepts

* [[basic localizer]]

* [[Dwyer map]]

* [[homotopy hypothesis]]

* [[model structure on simplicial sets]], [[model structure on topological spaces]]

* [[Strøm model structure]]

## References

The original reference:

*  {#Thomason1980} [[Robert W. Thomason]], *Cat as a closed model category*, Cahiers Topologie Géom. Différentielle **21** 3 (1980) 305-324  [[numdam:CTGDC_1980__21_3_305_0](http://www.numdam.org/item/?id=CTGDC_1980__21_3_305_0), MR0591388 (82b:18005)]

A correction to this article was made in

* [[Denis-Charles Cisinski]], _La classe des morphismes de Dwyer n'est pas stable par rétractes_, Cahiers Topologie et Géom. Différentielle Catégoriques, **40** no. 3 (1999), pp. 227--231. ([Numdam](http://www.numdam.org/item?id=CTGDC_1999__40_3_227_0))

Review and generalization to [[n-fold category|$n$-fold categories]]:

* [[Thomas M. Fiore]], [[Simona Paoli]], *A Thomason model structure on the category of small $n$-fold categories*, Algebr. Geom. Topol. **10** 4 (2010) 1933-2008 [[arXiv:0808.4108](http://arxiv.org/abs/0808.4108), [doi:10.2140/agt.2010.10.1933](https://projecteuclid.org/journals/algebraic-and-geometric-topology/volume-10/issue-4/A-Thomason-model-structure-on-the-category-of-small-nfold/10.2140/agt.2010.10.1933.full)]

Restriction of the Thomason model structure to the category of just [[posets]]:

* [[George Raptis]], *Homotopy theory of posets*, Homology Homotopy Appl. **12** 2 (2010) 211-230 [[euclid:hha/1296223882](https://projecteuclid.org/journals/homology-homotopy-and-applications/volume-12/issue-2/Homotopy-theory-of-posets/hha/1296223882.full)]

and to the category of just (small) acyclic categories:

* [[Roman Bruckner]], *A Model Structure On The Category Of Small Acyclic Categories* [[arXiv:1508.00992](https://arxiv.org/abs/1508.00992)]

Discussion of [[fibrant objects]] in the Thomason model structure:

* [[Lennart Meier]], [[Viktoriya Ozornova]], _Fibrancy of partial model categories_,  Homology, Homotopy and Applications, Volume 17 (2015) Number 2.  ([arXiv:1408.2743](https://arxiv.org/abs/1408.2743), [doi:10.4310/HHA.2015.v17.n2.a5](http://dx.doi.org/10.4310/HHA.2015.v17.n2.a5))

* [[Lennart Meier]], _Fibrancy of (Relative) Categories_, talk at _Young Topologists Meeting 2014_ ([slides pdf](http://www.staff.science.uu.nl/~meier007/Kopenhagen2014.pdf))

Discussion of [[cofibrant objects]] (Thomason-cofibrant posets):

* [[Roman Bruckner]], [[Christoph Pegel]], _Cofibrant objects in the Thomason Model Structure_, [[arXiv:1603.05448](http://arxiv.org/abs/1603.05448)]

* [[Roman Bruckner]], *Abstract Homotopy Theory and the Thomason Model Structure*, PhD thesis, Bremen 2016 [[gbv:46-00105527-15](http://nbn-resolving.de/urn:nbn:de:gbv:46-00105527-15), [pdf](https://media.suub.uni-bremen.de/bitstream/elib/1120/1/00105527-1.pdf)]

On the analog of the Thomason model structure for the _stable_ case -- an equivalence between connective [[stable homotopy theory]] and [[nerves]] of [[symmetric monoidal categories]]:

* [[Robert W. Thomason]], _Symmetric monoidal categories model all connective spectra_, Theory and Applications of Categories **1** 5 (1995) 78-118

Some other references

* {#Hir} [[Philip Hirschhorn]], _[[Model Categories and Their Localizations]]_, AMS Math. Survey and Monographs **99** (2002) [[ISBN:978-0-8218-4917-0](https://bookstore.ams.org/surv-99-s/), [pdf toc](http://www.gbv.de/dms/goettingen/360115845.pdf), [pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/pshmain.pdf), [pdf](http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf)]

[[!redirects Thomason weak equivalence]]
[[!redirects Thomason weak equivalences]]


Thomason-type model structure

## Idea

Thomason-type model categories provide simple 1-categorical models
for (∞,1)-categorical objects.

The provide a particularly convenient setting for results
like Quillen's [[Theorem A]] and [[Theorem B]].

## Examples

|(∞,1)-categorical structure|1-categorical structure|model structure|
|-|-|-|
|[[∞-groupoid]]|[[category]]|[[Thomason model structure]]|
|[[∞-groupoid]]|[[poset]]|[[model structure on posets]]|
|[[∞-groupoid]]|[[2-category]]|[[Thomason model structure on 2-categories]]|
|[[(∞,1)-category]]|[[relative category]]|[[Barwick–Kan model structure]]|
|[[connective spectra]]|[[symmetric multicategory]]|[[Fuentes-Keuthan model structure]]|
|[[connective spectra]]|[[symmetric monoidal groupoid]]|(only a [[relative category]])|
|[[connective spectra]]|[[symmetric monoidal category]]|(only a [[relative category]])|

## Related concepts

* [[Thomason model structure]]

## References

[…]


Thompson group

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[[!redirects Thompson Group]]
[[!redirects Thompson groups]]
[[!redirects Thompson's groups]]
[[!redirects Thompson's group F]]
[[!redirects Thompson's group T]]
[[!redirects Thompson's group V]]
[[!redirects Thompson's Groups]]
[[!redirects Thompson group F]]
[[!redirects Thompson group T]]
[[!redirects Thompson group V]]

##Idea

In the 1960s R. Thompson introduced three finitely presented [[group|groups]] with some remarkable properties that are today called **Thompson groups** and differentiated by prefixing the letters F, T, and V. The property of resurfacing in different guises in apparently unrelated mathematical fields earned them the nickname _Chameleon groups_.

##Related pages

* [[Jónsson-Tarski algebra]]

* [[Jónsson-Tarski topos]]

* [[Higman's embedding theorem]]

##References

* [Wikipedia entry](http://en.wikipedia.org/wiki/Thompson_groups): Thompson groups.

* [n-café blog](http://golem.ph.utexas.edu/category/2010/01/f_and_the_shibboleth.html): _T. Leinster - F and the Shibboleth_, January 2010.

The following has links to a lot of interesting material on the Thompson groups among them scans of handwritten notes by Thompson:

* M. Brin's [Reading list](http://www.math.binghamton.edu/matt/thompson/index.html) on the Thompson groups.

* M. Brin, _Higher dimensional Thompson groups_ , Geom. Ded. **108** no.1 (2004) pp.163-192. ([preprint](http://www.math.binghamton.edu/matt/thompson/hd3a.pdf))

* [[Marcelo Fiore]], [[Tom Leinster]], _An abstract characterization of Thompson's group F_ , arXiv.math/0508617 (2010). ([pdf](http://arxiv.org/pdf/math/0508617v2.pdf))

The following brings out connections between monoidal categories and the higher dimensional Thompson groups:

* J. A. Cohen, _Coherence for rewriting 2-theories_, Phd. Macquarie University 2008. ([arxiv:0904.0125v1](http://arxiv.org/pdf/0904.0125.pdf))


Thompson sporadic group

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## Idea

The _Thompson group_ is a [[sporadic finite simple group]] in the third generation of the [[Happy Family]]. It is of [[group order|order]] $2^{15}\cdot 3^{10}\cdot 5^3 \cdot 7^2 \cdot 13 \cdot 19 \cdot 31 = 90745943887872000$.

(Note that this group is different from the three [[Thompson groups]].)

##References

* Maryam Khaqan, _Elliptic curves and Thompson's sporadic simple group_, ([arXiv:2008.01607](https://arxiv.org/abs/2008.01607))


Thompson's group F

* [blog post](http://golem.ph.utexas.edu/category/2010/01/f_and_the_shibboleth.html)


Thomson's group

Richard Thompson's group, $T$ was the first known
finitely presented infinite simplegroup. This was followed by two related groups,usually denoted $F$ and $V$.

* [Notices AMS: 'What is' article](http://www.ams.org/notices/201108/rtx110801112p.pdf)


Thoralf Skolem

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

## related $n$Lab entries

* [[ZFC]]

* [[Löwenheim-Skolem theorem]]

* [[Skolem's paradox]]

* [[Some remarks on axiomatized set theory]]$\,\!$ [Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre], talk given in 1922

category: people


Thorsten Altenkirch

Thorsten Altenkirch is a professor of [[computer science]] at the University of Nottingham.

* [personal page](http://www.cs.nott.ac.uk/~txa/)

* [institute page](https://www.nottingham.ac.uk/computerscience/People/thorsten.altenkirch)

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

## Selected writings

On [[inductive types]] and [[W-types]]:

* [[Michael Abbott]], [[Thorsten Altenkirch]], [[Neil Ghani]]: *Representing Nested Inductive Types using W-types*, in: *Automata, Languages and Programming*, ICALP 2004, Lecture Notes in Computer Science, **3142**, Springer (2004) [[doi:10.1007/978-3-540-27836-8_8](https://doi.org/10.1007/978-3-540-27836-8_8), [pdf](http://www.cs.nott.ac.uk/~psztxa/publ/icalp04.pdf)]

  exposition: _Inductive Types for Free -- Representing nested inductive types using W-types_, talk at ICALP (2004) [[pdf](http://www.cs.nott.ac.uk/~txa/talks/icalp-slides.pdf)]

Introducing [[observational type theory]]:

* [[Thorsten Altenkirch]], [[Conor McBride]], _Towards observational type theory_, draft (2006) [[pdf](http://strictlypositive.org/ott.pdf), [pdf](https://www.cs.nott.ac.uk/~psztxa/publ/ott-conf.pdf), [[AltenkirchMcBrided-TowardsOTT.pdf:file]]]

* {#AltenkirchMcBrideSwierstra07} [[Thorsten Altenkirch]], [[Conor McBride]], [[Wouter Swierstra]], *Observational Equality, Now!*, PLPV '07: Proceedings of the 2007 workshop on Programming languages meets program verification (2007) 57-68 [ISBN:978-1-59593-677-6, [doi:10.1145/1292597.1292608](http://doi.org/10.1145/1292597.1292608), [pdf](https://www.cs.nott.ac.uk/~psztxa/publ/obseqnow.pdf)]

Introducing the [[quantum programming language]] [[QML]]:

* {#AltenkirchGrattage05} [[Thorsten Altenkirch]], [[Jonathan Grattage]], _A functional quantum programming language_, Logic in Computer Science, 2005. Proceedings. 20th Annual IEEE Symposium on, 26-29 June 2005 249 - 258 ([arXiv:quant-ph/0409065](https://arxiv.org/abs/quant-ph/0409065), [doi:10.1109/LICS.2005.1](https://doi.org/10.1109/LICS.2005.1), [pdf](http://www.cs.nott.ac.uk/~txa/publ/qml.pdf))

Introducing the [[quantum IO monad]]:

* [[Thorsten Altenkirch]], *The Quantum IO Monad*, Nottingham (2006) [[pdf](http://www.cs.nott.ac.uk/~txa/talks/qnet06.pdf), [[Altenkirch-QIOMonad.pdf:file]]]

* {#AltenkirchGreen10} [[Thorsten Altenkirch]], [[Alexander Green]], *The quantum IO monad*, Ch. 5 of: Simon Gay, Ian Mackie (eds.): *Semantic Techniques in Quantum Computation* (2010) 173-205 [[pdf](http://www.cs.nott.ac.uk/~txa/publ/qio-chapter.pdf), [doi:10.1017/CBO9781139193313.006](https://doi.org/10.1017/CBO9781139193313.006)]

Introducing [[relative monad|relative]] [[monads in computer science]]:

* [[Thorsten Altenkirch]], [[James Chapman]], [[Tarmo Uustalu]], *Monads need not be endofunctors*, Logical Methods in Computer Science **11** 1:3 (2015) 1–40 [[arXiv:1412.7148](https://arxiv.org/abs/1412.7148), [pdf](http://www.cs.nott.ac.uk/~txa/publ/jrelmon.pdf), doi:10.2168/LMCS-11(1:3)2015]

On [[weakly constant functions]] and [[propositional truncation]] in [[homotopy type theory]]:

* [[Nicolai Kraus]], [[Martín Escardó]], [[Thierry Coquand]], [[Thorsten Altenkirch]], *Notions of Anonymous Existence in Martin-Löf Type Theory*,  Logical Methods in Computer Science **13** 1 [doi:10.23638/LMCS-13(1:15)2017, [arXiv:1610.03346](https://arxiv.org/abs/1610.03346)]

On [[quotient inductive types]]:

* {#ADK2017} [[Thorsten Altenkirch]], [[Nils A. Danielsson]], [[Nicolai Kraus]], _Partiality, Revisited: The Partiality Monad as a Quotient Inductive-Inductive Type_, FoSSaCS 2017, Lecture Notes in Computer Science **10203** Springer (2017) [[arXiv:1610.09254](https://arxiv.org/abs/1610.09254)]

On [[free groups]] and [[infinity-groups|$\infty$-groups]] in [[homotopy type theory]]:

* [[Nicolai Kraus]], [[Thorsten Altenkirch]], *Free Higher Groups in Homotopy Type Theory*, LICS '18: Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, (2018) [[arXiv;1805.02069](https://arxiv.org/abs/1805.02069), [doi:10.1145/3209108.3209183](https://doi.org/10.1145/3209108.3209183)]

On [[quotient types]] and [[higher inductive types]]:

* [[Thorsten Altenkirch]], [[Paolo Capriotti]], [[Gabe Dijkstra]], [[Nicolai Kraus]], [[Fredrik Forsberg]], *Quotient inductive-inductive types*, Foundations of Software Science and Computation Structures (FoSSaCS 2018), Lecture Notes in Computer Science **10803** (2018) [[doi:10.1007/978-3-319-89366-2_16](https://doi.org/10.1007/978-3-319-89366-2_16), [arXiv:1612.02346](https://arxiv.org/abs/1612.02346)]

On the prehistory of [[homotopy type theory]] and the [[univalence axiom]] in the hands of [[Martin Hofmann]]:

* {#Altenkirch21} [[Thorsten Altenkirch]], *Martin Hofmann’s contributions to type theory: Groupoids and univalence*, Mathematical Structures in Computer Science **31** 9 (2021) 953-957 [[doi:10.1017/S0960129520000316](https://doi.org/10.1017/S0960129520000316)]

A fragment of [[higher observational type theory]]

* [[Thorsten Altenkirch]], [[Yorgo Chamoun]], [[Ambrus Kaposi]], [[Michael Shulman]], *Internal parametricity, without an interval* ([arXiv:2307.06448](https://arxiv.org/abs/2307.06448))

On naive/informal [[dependent type theory]]:

* [[Thorsten Altenkirch]], *Naive Type Theory*, in: *[[Reflections on the Foundations of Mathematics]]*, Synthese Library **407** Springer (2019) [[doi:10.1007/978-3-030-15655-8_5](https://doi.org/10.1007/978-3-030-15655-8_5), [pdf](https://www.cs.nott.ac.uk/~psztxa/publ/fomus19.pdf)]

## Related entries

* [[type theory]], [[homotopy type theory]], [[higher observational type theory]]

* [[verified software]]

* [[categorical model of dependent types]]

category: people


Thorsten B. Wahl

* [Institute page](https://www2.physics.ox.ac.uk/contacts/people/wahlt)

## Selected writings

On [[tensor networks]] in [[quantum many-body physics]]:

* [[Thorsten B. Wahl]], _Tensor network states for the description of quantum many-body systems_ ([arXiv:1509.05984](https://arxiv.org/abs/1509.05984))

category: people


Thorsten Palm

* [Mathematics Genealogy page](http://www.genealogy.math.ndsu.nodak.edu/id.php?id=97399)

* [archived web sites](http://www.zoominfo.com/p/Thorsten-Palm/697772458)

## Writings

* _A geometry for diagrammatic computads_ ([pdf](http://www.mat.uc.pt/~categ/ct2007/abstracts/Palm.pdf))

* _Categories with slicing_, Theory and Applications of Categories, Vol. 22, 2009, No. 5, pp 97-135. ([TAC](http://www.tac.mta.ca/tac/volumes/22/5/22-05abs.html))

## related $n$Lab entries

* [[opetopic omega-category]]

* [[opetopic type theory]]

category: people


thread

A **thread** is an element of a [[cofiltered limit]] of [[topological spaces]] (usually studied in the generality of [[projective spectrum|projective spectra]]) or (in a more rarely used terminology) of a cofiltered limit of sets.

Thus let $F: D^{op}\to Set$ be a functor where $D$ is a [[small category|small]] [[filtered category]] (for example, a [[directed set]]). Then $lim F = lim_{d\in D} F(d)$ consists of families $(s_d)_{d\in D}$ where $s_d \in F(d)$ and for every morphism $\delta:d\to e$ in $D$, $F(\delta)(s_e) = s_d$. Such families are called _threads_.

If $F: D^{op}\to Top$ is a functor where $D$ is a small filtered category then $lim F$ has the same underlying set (of threads) as the composition $U\circ F$ where $U:Top\to Set$ is the forgetful functor; the topology of $lim F$ is the subspace topology on $lim (U\circ F)$ understood as a subset of the Cartesian product $\prod_d F(d)$ equipped with the (product)[[Tihonov's topology]].

[[!redirects thread]]
[[!redirects threads]]


Three Roles of Quantum Field Theory

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In May 2011 [[Graeme Segal]] gave a series of lectures on the topic of [[quantum field theory]] at the Hausdorff Center for Mathematics in Bonn.

* [[Graeme Segal]], _Three Roles of Quantum Field Theory_ , Felix Klein Lectures 2011, Bonn ([website](http://www.mpim-bonn.mpg.de/node/3372/))

  > **Abstract.** [[quantum field theory|Quantum field theory]] has many roles, and the lectures will be about three of them. The primary role is to provide a description of all of fundamental [[physics]] when [[gravity]] is firmly excluded. A second, at first surprising, role has emerged from [[string theory]], which is a theory of gravitation: it turns out that a two-dimensional field theory can be regarded as a generalized manifold, and in particular can be a model for [[space-time]]. Thirdly, quite apart from physics, the concept of a field theory has taken on a new life as an organizing principle in other areas of mathematics - not only in [[geometry]] and [[representation theory]], but even in connection with [[quantum computing]].

The three roles can be seen together as aspects of [[noncommutative geometry]], and that will be a central theme of the lectures. The talks will aim to show how powerful the field theory idea is by jumping between a variety of contexts, beginning from the origins of the structure in particle physics, with a little about so-called "[[Wick rotation]]", and then moving towards more pure-mathematical applications to analysis, algebra, and the structure of [[manifold]]s.

* part I ([mp4](http://www.mpim-bonn.mpg.de/sites/default/files/videos/download/segal_qft_2011_part1.mp4))

  Basics of [[classical mechanics]] from an [[AQFT]]-point of view: [[symplectic manifold]], [[Hamiltonian]], [[deformation quantization]], [[C-star algebra]], [[state]], [[Lagrangian]], [[action functional]], [[covariant phase space]], [[correlator]], locality of the would-be [[path integral]] [[measure]]

* part II ([mp4](http://www.mpim-bonn.mpg.de/sites/default/files/videos/download/segal_qft_2011_part2.mp4))

* part III ([mp4](http://www.mpim-bonn.mpg.de/sites/default/files/videos/download/segal_qft_2011_part3.mp4))

* part IV ([mp4](http://www.mpim-bonn.mpg.de/sites/default/files/videos/download/segal_qft_2011_part4.mp4))

* part V ([mp4](http://www.mpim-bonn.mpg.de/sites/default/files/videos/download/segal_qft_2011_part5.mp4))

* part VI ([mp4](http://www.mpim-bonn.mpg.de/sites/default/files/videos/download/segal_qft_2011_part6.mp4))

category: reference


three-sorted set theory

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

# Three-sorted set theory
* table of contents
{: toc}

## Overview

Three-sorted set theories are [[set theories]] with three different sorts, representing [[sets]], [[elements]], and some third sort, usually [[functions]] or [[relations]]. Three-sorted set theories are usually [[structural set theories]]. In common with other [[simply sorted set theory]], the membership $\in$ is a [[relation]], in contrast to [[dependently sorted set theory]], where membership is a typing judgment. In contrast to [[unsorted set theory]] and [[two-sorted set theory]], sets and elements are different, and sets and elements are different from whatever relates sets and elements to each other ([[functions]] in [[categorical set theory]], [[relations]] in [[allegorical set theory]]).

## Examples

* [[three-sorted SEAR]]

* [[three-sorted structural ZFC]]

* Similarly, there should be three-sorted presentations of [[ETCS with elements]], although it has not been written out yet.

## See also

* [[unsorted set theory]]
* [[two-sorted set theory]]
* [[simply sorted set theory]]
* [[dependently sorted set theory]]


threshold correction

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[quantum field theory]] a _threshold correction_ to an [[effective field theory]] is an adjustment to its [[renormalization group flow]] due to effects in the full theory  that are being ignored in the effective field theory; for instance the presence of further [[particles]] of high [[mass]] in the full theory which do not preserve some [[symmetry]] seen in the effective field theory.

## References

* {#Wiczek98} [[Frank Wilczek]], p. 29-31 of 39 in: _Beyond The Standard Model: An Answer and Twenty Questions_ ([arXiv:hep-ph/9802400](https://arxiv.org/abs/hep-ph/9802400))

* Alexander Voigt, below (4.32) in: _Calculation of Threshold Corrections to Renormalization Group Running in the Exceptional Supersymmetric Standard Model_, 2010 ([pdf](https://iktp.tu-dresden.de/IKTP/pub/10/Diplomarbeit_Voigt.pdf))

[[!redirects threshold corrections]]


thunk-force category

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

* table of contents
{:toc}

## Idea

A **thunk-force category** is a category that models [[call-by-value]] programming languages with effects.
More commonly, terms in a call-by-value language are modelled as morphisms in the [[Kleisli category]] of a [[strong monad]].
A thunk-force category axiomatizes the Kleisli category _directly_ and is also called an **abstract Kleisli category**[^terminology].
The original category can be recovered from this category if it satisfies certain properties.

The name derives from programming terminology. A **thunk** means a value that represents an unevaluated computation, which can later be **forced** to be evaluated and perform its [[side effects]]. This is represented by the object $L A$ which could be read as a "thunk of $A$" or a "lazy $A$". The thunking arrow $\theta_A : A \to L A$ embeds a value as a trivial thunk, and the force $\epsilon_A : L A \to A$ forces the evaluation of its thunked input.

The type constructor $L$ and associated thunk and force don't normally appear as a standalone feature in a call-by-value language, but if the language supports first-class functions, they can be implemented in the usual way as a function of unit input.
On the model side, this corresponds to the fact that if a thunk-force category is [[premonoidal category|premonoidal closed]], then $L A$ is equivalent to $I \rightharpoonup A$ and $\theta, \epsilon$ can be defined using this structure.

The [[opposite category|opposite]] of a thunk-force category is sometimes called a category equipped with a **runnable monad**.  (Thus, a thunk-force structure could be called a "(co)runnable comonad", while a runnable monad could be called a "co-thunk-force category".)  Just as a thunk-force category models call-by-value programming, a runnable monad models [[call-by-name]] programming.

## Definition

A thunk-force category [Führmann 99](#F99) consists of

* a category $K$
* a functor $L : K \to K$
* a [[unnatural transformation|possibly unnatural transformation]] $\theta : Id_K \to L$ pronounced **thunk**
* a [[natural transformation]] $\epsilon : L \to Id_K$ pronounced **force**

such that

* $(L\theta)\theta = (\theta L)\theta$
* $\theta L$ is a natural transformation
* $\epsilon\theta = \id$
* $(L\epsilon)(\theta L) = \id$

These conditions on the data $(L,\epsilon,\theta)$ can also be rephrased as

* $(L,\epsilon,\theta L)$ is a [[comonad]].
* Each $\theta_A : A \to L A$ equips $A$ with the structure of an $L$-[[coalgebra over a comonad]].

The second of these axioms states that $\epsilon\theta = \id$ and that $(L\theta)\theta = (\theta L)\theta$, while the first states that $\theta L$ is natural and $(L\epsilon)(\theta L) = \id$ (plus two other properties that follow from the second axiom).

The definition can be extended to monoidal closed thunk-force category to model a context and strong monad [Führmann 99](#F99).

## Thunkable Morphisms

A morphism $f : A \to B$ represents an effectful program. The presence of the thunk $\theta$ allows us to make the distinction between the "pure"/"trivially effectful" programs and those that have non-trivial effects.
A morphism $f : A \to B$ in a thunk-force category is **thunkable** if $( L f )\theta = \theta f$, i.e., $\theta$ is "natural with respect to $f$".  Note that the condition $(L\theta)\theta = (\theta L)\theta$ in the definition says precisely that **$\theta$ is thunkable**.

The presence of non-thunkable morphisms makes $\theta$ fail to be natural overall; naturality of $\theta$ is exactly the same as saying all morphisms are thunkable. This is the case for the Kleisli category of the identity monad. On the other hand, most monads produce many non-thunkable morphisms. For example in the Kleisli category of the [[maybe monad]], which is equivalent to the category of sets and partial functions, the thunkable morphisms are the total functions. As discussed below, most monads $T$ on a category $C$ used in [[denotational semantics]] satisfy an equalizing requirement which means the thunkable morphisms are in one-to-one correspondence to the morphisms of the original category $C$.

## Relation to Monads

Let $(T,\eta,\mu)$ be a [[monad]] on a category $C$. The [[Kleisli category]] of $T$ is a thunk-force category. Let $F \dashv G$ denote the adjunction between $C$ and the Kleisli category $C_T$. Define $L = FG$ and $\epsilon$ the counit of the comonad. To define $\theta$, take $F\eta : F \to FT$ and, using the fact that $F$ is an [[identity-on-objects functor]] (or at least bijective on objects), treat it as an [[unnatural transformation]] from $Id_{C_T}\to L$. Then it follows immediately by naturality that every morphism in the image of $F$ is thunkable.

More explicitly, using the "Kleisli arrow" definition of the Kleisli category, $L$
is just $T$, $\epsilon_A : T A \to T A$ is the identity, and $\theta_A = \eta_{T A} \eta_{A} : A \to T^2 A$.

In the opposite direction, starting with a thunk-force category $K$, we can define the (non-full) subcategory of thunkable morphisms $G_\theta K$. Since $\theta L$ is natural, the image of $L$ lands in this subcategory, and is right adjoint to the inclusion. Then there is a monad $L_\theta$ induced on $G_{\theta}K$ by this adjunction.
Furthermore, the thunk-force category generated by the Kleisli category of the monad is equivalent to $K$.

These constructions exhibit the category of thunk-force categories as a [[reflective subcategory]] of the category of monads. The subcategory can be characterized as the monads satisfying the "equalizing requirement", that the unit $\eta$ of the monad is an equalizer in the following diagram:

$$
1_{C}\underset{\quad \eta \quad}{\to}T\underoverset{\quad \eta T \quad}{T \eta}{\rightrightarrows}T^2
$$

Which is in practice often satisfied for monads used in [[denotational semantics]].
The definition of the category of morphism of thunk-force categories and proof of this theorem are in [Führmann 99 section 5](#F99).

## Applications

Since the thunk $L$ is a [[comonad]], the Kleisli category of the comonad is a model of a [[call-by-name]] language. This is a semantic counterpart of the "thunking translation" of call-by-name into call-by-value (described for example in [Hatcliff-Danvy 97](#HD97).

## Related Concepts

* The dual concept for [[call-by-name]] is a [[runnable monad]].
* A [[duploid]] is an analogous structure for effectful languages with both [[positive type|positive types]] and [[negative type|negative types]].

## References

Thunk-force categories were introduced (under the name "abstract Kleisli categories" in

* {#F99} Carsten Führmann, "Direct Models of the Computational Lambda-calculus" Electronic Notes in Theoretical Computer Science 1999.  [https://doi.org/10.1016/S1571-0661(04)80078-1](https://doi.org/10.1016/S1571-0661(04)80078-1)

See also:

* {#HD97} John Hatcliff and Olivier Danvy, "Thunks and the $\lambda$-Calculus", Journal of Functional Programming 1997.  [https://doi.org/10.1017/S0956796897002748](https://doi.org/10.1017/S0956796897002748)

[^terminology]: We prefer the term thunk-force category since it is ambiguous whether [[Kleisli category]] refers to the Kleisli category of a [[monad]] or a [[comonad]].

[[!redirects thunk-force categories]]
[[!redirects abstract Kleisli category]]
[[!redirects abstract Kleisli categories]]

[[!redirects runnable monad]]
[[!redirects runnable monads]]


Théo Winterhalter

* [personal page](https://theowinterhalter.github.io/)

## Selected writings

On [[dependent type theory]] and [[proof assistants]] ([[Coq]]):

* [[Théo Winterhalter]], *Formalisation and Meta-Theory of Type Theory*, Nantes (2020) [[pdf](https://github.com/TheoWinterhalter/phd-thesis/releases/download/v1.2.1/TheoWinterhalter-PhD-v1.2.1.pdf), [github](https://github.com/TheoWinterhalter/phd-thesis)]

## Talks

* [[Théo Winterhalter]], *A conservative and constructive translation from extensional type theory to weak type theory*, Strength of Weak Type Theory, [[DutchCATS]], 11 May 2023. ([slides](https://dutchcats.github.io/Weak_type_theories/slides_winterhalter.pdf))

## Related entries

* [[type telescope]]

category: people

[[!redirects Theo Winterhalter]]


Théophile de Donder

* [Wikipedia entry](http://en.wikipedia.org/wiki/Th%C3%A9ophile_de_Donder)

## Related entries

* [[de Donder-Weyl formalism]]

category: people

[[!redirects Theophile de Donder]]
[[!redirects de Donder]]
[[!redirects De Donder]]


Tian Lan

* [institute page](http://www.phy.cuhk.edu.hk/people/teaching/tlan.html)

## Selected writings

Discussion of [[anyon|anyonic]] [[topological order]] in terms of [[braided monoidal categories|braided]] [[fusion categories]]/[[modular tensor categories]]:

* [[Tian Lan]], *A Classification of (2+1)D Topological Phases with Symmetries* $[$[arXiv:1801.01210](https://arxiv.org/abs/1801.01210)$]$

On generalized [[Laughlin wavefunctions]] for non-abelian [[anyons]]:

* [[Tian Lan]], *Matrix formulation for non-Abelian families*, Phys. Rev. B **100** 241102 (2019) $[$[arXiv:1908.02599](https://arxiv.org/abs/1908.02599), [doi:10.1103/PhysRevB.100.241102](https://doi.org/10.1103/PhysRevB.100.241102)$]$

category: people


Tian Yang

* [institute page](http://mathematics.stanford.edu/people/name/tian-yang/)

## Selected writings

On the [[volume conjecture]]:

* {#ChenYang15} [[Qingtao Chen]], [[Tian Yang]], _A volume conjecture for a family of Turaev-Viro type invariants of 3-manifolds with boundary_ ([arXiv:1503.02547](http://arxiv.org/abs/1503.02547))

## Related $n$Lab entries

* [[volume conjecture]]

category: people


Tibor Beke

* [website](http://faculty.uml.edu/tbeke)

## Selected writings

* _Categorification, term [[rewriting]] and the Knuth-Bendix
procedure_, [pdf](http://faculty.uml.edu/tbeke/knuth.pdf)

* _Higher Čech theory_, K-Theory __32__ (2004), no.4, pp. 293--322

* _Simplicial torsors_, Theory and Applications of Categories __9__ (2001), no.3, pp. 43--60

On [[toposes]] and [[combinatorial model categories]]:

* [[Tibor Beke]] *Sheafifiable homotopy model categories*,  Math. Proc. Cambridge Philos. Soc. __129__ 3 (2000) 447-475 [[arXiv:math/0102087](http://arxiv.org/abs/math/0102087), [doi:10.1017/S0305004100004722](https://doi.org/10.1017/S0305004100004722)]

* _Sheafifiable homotopy model categories. II_, J. Pure Appl. Algebra  __164__  (2001), no. 3, 307--324; ([preprint version](http://faculty.uml.edu/tbeke/right.dvi))

category: people


Tibor Radó

* [Wikipedia entry](https://en.wikipedia.org/wiki/Tibor_Rad%C3%B3)

## Selected writings

The [[triangulation theorem]] for [[Riemann surfaces]]:

* {#Rado25} [[Tibor Radó]], *Über den Begriff der Riemannschen Fläche*, Acta Litt. Sci. Szeged, 2 (101-121), 10 ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/rado.pdf), [pdf](http://acta.bibl.u-szeged.hu/13312/1/math_002_101-121.pdf))

category: people


Tietze extension theorem

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

#Contents#
* table of contents
{:toc}

## Summary

The _Tietze extension theorem_ says that [[continuous functions]] [[extension|extend]] from [[closed subsets]] of a [[normal topological space]] $X$ to the whole space $X$.

This is a close cousin of [[Urysohn's lemma]] with many applications.

One implication is that [[topological vector bundles]] over a topological space $X$ that trivialize over a [[closed subspace]] $A$ are equivalent to vector bundles on the [[quotient space]] $X/A$ (see [there](topological+vector+bundle#OverClosedSubspaces)). This in turn is what implies the [[long exact sequence in cohomology]] for [[topological K-theory]] (see [there](topological+K-theory#ExactSequences)).

## Statement

### For continuous functions
 {#ForContinuousFunctions}

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

For $X$ a [[normal topological space]] and $A \subset X$ a [[closed subspace]], there is for every [[continuous function]] $f \colon A \to \mathbb{R}$ to the [[real line]] (with its [[Euclidean space|Euclidean]] [[metric topology]]) a continuous function $\hat f \colon X \to \mathbb{R}$ [[extension|extending]] it, i.e. such that $\hat f|_A = f$:

$$
  \array{
     A &\hookrightarrow & X
     \\
     {}^{\mathllap{f}}\downarrow & \swarrow_{\mathrlap{\exists \hat f} }
     \\
     \mathbb{R}
  }
$$

Therefore one also says that _$\mathbb{R}$ is an [[absolute extensor]]_ in [[topology]].

=--

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

We produce a [[sequence]] of approximations to the desired extension by [[induction]]. Then we will observe that the sequence is a [[Cauchy sequence]] and conclude by observing that this implies that its [[limit of a sequence|limit]] is an extension of $f$ as desired.

For the induction step, let

$$
  \hat f_n \;\colon\; X \longrightarrow \mathbb{R}
$$

be a [[continuous function]] on $X$ such that the difference of its restriction to $A$ with $f$ is a [[bounded function]], for a bound $c_n \in (0,\infty) \subset \mathbb{R}$:

$$
  \underset{a \in A}{\forall}
  \left(
    { \left\Vert f(a) - \hat f_n (a) \right\Vert } \leq c_n
  \right)
  \,.
$$

Consider then the [[pre-image]] subsets

$$
  S_- \coloneqq \left( f - \hat f_n\vert_A \right)^{-1}\big( [-c_n, -c_n/3] \big)
  \phantom{AAAA}
  S_+ \coloneqq \left( f - \hat f_n\vert_A \right)^{-1}\big( [c_n/3, c_n] \big)
  \,.
$$

Since the [[closed intervals]] $[-c_n,-c_n/3], [c_n/3, c_n] \subset \mathbb{R}$ are [[closed subsets]], and since $f - \hat f_n\vert_A$ is a [[continuous function]], these are [[closed subsets]] of $A$. Moreover, since [[subsets are closed in a closed subspace precisely if they are closed in the ambient space]], these are also closed subsets of $X$.

Therefore, since $X$ is [[normal topological space|normal]] by assumption,  it follows with [[Urysohn's lemma]] that there is a continuous function

$$
  \phi \;\colon\; X \longrightarrow \mathbb{R}
$$

with

$$
  \underset{x \in X}{\forall} \left( 0 \leq \phi(x) \leq 1 \right)
$$

and

$$
  \phi\vert_{S_+} = 1
  \phantom{AAAA}
  \phi\vert_{S_-} = 0
  \,.
$$

Consider then the continuous function

$$
  g_{n+1}
    \;\coloneqq\;
  \tfrac{2 c_n}{3} \phi
    -
  \tfrac{c_n}{3}
$$

This now satisfies

$$
  g_{n+1}\vert_{S_+} = \frac{c_n}{3}
  \phantom{AAAA}
  g_{n+1}\vert_{S_-} = -\frac{c_n}{3}
  \,.
$$

with

$$
  \underset{x \in X}{\forall} \left(
    \left \Vert g_{n+1} (x) \right\Vert
     \leq
    \tfrac{c_n}{3}
  \right)
  \,.
$$

Moreover, observe that this function satisfies

$$
 \underset{a \in A}{\forall}
 \left(
    \left\Vert f - \hat f_n(a) - g_{n+1}(a)  \right\Vert
    \leq \tfrac{2 c_n}{3}
 \right)
 \,.
$$

To wit, this is because

1. for $a \in S_+$ we have $g_{n+1}(a) = \tfrac{c_n}{3}$ and $f(a) - \hat f_{n}(a) \in [c_n/3,c_n]$;

1. for $a \in S_-$ we have $g_{n+1}(a) = -\tfrac{c_n}{3}$ and $f(a) - \hat f_{n}(a) \in [-c_n/3,-c_n]$;

1. for $a \in Y \setminus \{S_+ \cup S_-\}$ we have $g(a) \in [-c_n/3,c_n/3]$ as well as $f(a) - \hat f_{n}(a) \in [-c_n/3, c_n/3]$.

It follows that if we set

$$
  \hat f_{n+1} \coloneqq \hat f_n + g_{n+1}
$$

then

$$
  \underset{a \in A}{\forall}
  \left(
     { \left\Vert f(a) - \hat f_{n+1}(a) \right\Vert }
     \leq
     \tfrac{2 c_n}{3}
  \right)
  \,.
$$

This gives the induction step.

To start the induction, first assume that $f$ is bounded by a constant $c_0$. Then we may set

$$
  \hat f_0 \coloneqq const_0
  \,.
$$

Hence [[induction]] now gives a [[sequence]] of continuous functions

$$
  (\hat f_n)_{n \in \mathbb{N}}
$$

with the property that

$$
  \underset{a \in A}{\forall}
  \left(
    \left\Vert f(a) -\hat f_n(a) \right\Vert
    \leq \left( \tfrac{2}{3}\right)^n c_0
  \right)
  \,.
$$

Moreover, for $n_1, n_2 \in \mathbb{N}$ with $n_2 \geq n_1$ and $x \in X$ we have

$$
  \begin{aligned}
    {\Vert \hat f_{n_2}(x) - \hat f_{n_1}(x) \Vert}
    & =
    {\Vert g_{n_1 + 1}(x) + g_{n_1 + 2}(c) + \cdots + g_{n_2}(x) \Vert}
    \\
    & \leq
    \underoverset{k = n_1+1}{n_2}{\sum}
    \tfrac{1}{3^{k}} c_0
    \\
    & \leq
    \underoverset{k = n_1+1}{\infty}{\sum}
    \tfrac{1}{3^{k}} c_0
  \end{aligned}
$$

That the [[geometric series]] $\sum_{k = 0}^\infty 1/3^k$ [[convergence of a sequence|converges]]

$$
  \underoverset{n}{k = 0}{\sum} 1/3 k
   \overset{n \to \infty}{\longrightarrow}
  \frac{1}{1 - 1/3} = 3/2
$$

this becomes arbitrarily small for large $n_1$.

This means that the sequence $(\hat f_{n+1})_{n\in \mathbb{N}}$ is a [[Cauchy sequence]] in the [[supremum norm]] for real-valued functions.

Since uniform Cauchy sequences of continuous functions with values in a [[complete space|complete]] [[metric space]] [[uniform convergence|converge uniformly]] to a [[continuous function]] ([this prop.](uniform+convergence#FunctionsUniformCauchySequence)) this implies that the sequence [[uniform convergence|converges uniformly]] to a [[continuous function]]. By construction, this is an extension as required.

Finally consider the case that $f$ is not a [[bounded function]]. In this case consider any [[homeomorphism]] $\phi \colon \mathbb{R}^1 \overset{\simeq}{\to} (-c_0,c_0) \subset \mathbb{R}^1$ between the [[real line]] and an [[open interval]] Then $\phi \circ f$ is a continous function bounded by $c_0$ and hence the above argument gives an extension $\widehat {\phi \circ f}$. Then $\phi^{-1} \circ \widehat{ \phi \circ f }$ is an extension of $f$.

=--

### For smooth functions
 {#Manifolds}

See _[[Whitney extension theorem]]_, also _[[Steenrod-Wockel approximation theorem]]_.

### For smooth loci

Let $\mathbb{L} = (C^\infty Ring^{fin})^{op}$ be the category of [[smooth loci]],
the [[opposite category]] of finitely generated [[generalized smooth algebra]]s.
By the theorem discussed there, there is a [[full and faithful functor]]
[[Diff]] $\hookrightarrow \mathbb{L}$.

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

For $A = C^\infty(\mathbb{R}^n)/J$ and $B = C^\infty(\mathbb{R}^n)/I$
with $I \subset J$ and $B \to A$ the projection of [[generalized smooth algebra]]s
the corresponding [[monomorphism]] $\ell A \to \ell B$ in $\mathbb{L}$
exhibits $\ell A$ as a **closed smooth sublocus** of $\ell B$.

=--

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

Let $X$ be a [[smooth manifold]] and let $\{g_i \in C^\infty(X)\}_{i = 1}^n$ be [[smooth function]]s that are independent in the sense that at each common zero point $x\in X$, $\forall i : g_i(x)= 0$ we have the [[derivative]] $(d g_i) : T_x X \to \mathbb{R}^n$ is a surjection, then the ideal $(g_1, \cdots, g_n)$ coincides with the ideal of functions that vanish on the zero-set of the $g_i$.

=--

This is lemma 2.1 in Chapter I of ([MoerdijkReyes](#MoerdijkReyes)).

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

If $\ell A \hookrightarrow \ell B$  is a closed
sublocus of $\ell B$ then every morphism $\ell A \to R$
extends to a morphism $\ell B \to R$

=--

This is prop. 1.6 in Chapter II of ([MoerdijkReyes](#MoerdijkReyes)).

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

Since we have $R = \ell C^\infty(\mathbb{R})$ and $C^\infty(\mathbb{R})$
is the free [[generalized smooth algebra]] on a single generator,
a morphism $\ell A \to R$ is precisely an element of $C^\infty(\mathbb{R}^n)/J$. This is represented by an element in $C^\infty(\mathbb{R}^n)$ which in particular defines an element in $C^\infty(\mathbb{R}^n)/I$.

=--

## Related theorems

[[!include extension theorems -- table]]

* [[Urysohn's lemma]]

* [[Hadamard lemma]]

* [[Borel's theorem]]

* [[Steenrod-Wockel approximation theorem]]

* [[Whitney extension theorem]]

* [[Taimanov theorem]]

* [[embedding of smooth manifolds into formal duals of R-algebras]]

* [[smooth Serre-Swan theorem]]

* [[derivations of smooth functions are vector fields]]

## References
 {#References}

Named after:

* [[Heinrich Tietze]], *Über Funktionen, die auf einer abgeschlossenen Menge stetig sind*, Journal für die reine und angewandte Mathematik, **145** (1915) [[doi:10.1515/crll.1915.145.9](https://doi.org/10.1515/crll.1915.145.9)]

Leture notes:

* Adam Boocher, _A proof of the Tietze Extension Theorem Using Urysohn's Lemma_, 2005 ([pdf](http://www.maths.ed.ac.uk/~aboocher/math/tietze.pdf))

Generalization
to maps into any [[locally convex topological vector space]]:

* {#Dugundji51} [[James Dugundji]], *An extension of Tietze's theorem*, Pacific Journal of Mathematics **1** 3 (1951) [[doi:10.2140/pjm.1951.1.353](http://dx.doi.org/10.2140/pjm.1951.1.353), [pdf](https://msp.org/pjm/1951/1-3/pjm-v1-n3-p04-p.pdf)]

Discussion via [[algebraic topology]]:

* [[Norman Steenrod]], _Cohomology operations, and obstructions to extending continuous functions_, Advances in Mathematics Volume 8, Issue 3, June 1972, Pages 371-416 (doi:10.1016/0001-8708(72)90004-7, [pdf](http://www.maths.ed.ac.uk/%7Eaar/papers/steen6.pdf))

Discussion of the smooth version includes

* {#MoerdijkReyes} [[Ieke Moerdijk]], [[Gonzalo Reyes]], chapters I and II of: _[[Models for Smooth Infinitesimal Analysis]]_

See also

* Wikipedia, _[Tietze extension theorem](https://en.wikipedia.org/wiki/Tietze_extension_theorem)_

* [[Planet Math]], _[Proof of the Tietze extension theorem](http://planetmath.org/proofoftietzeextensiontheorem)_

* [[Bruce Blackadar]], *Extending continuous functions* [[arXiv:1207.6147](https://arxiv.org/abs/1207.6147)]


Tietze transformation

# Tietze transformations
* table of contents
{: toc}

## Idea

Tietze transformations are a formalisation of the informal substitution methods that are natural when working with [[group presentations]].

## The four transformations

Let $G= \langle X: R\rangle$ be a [[group presentation]], where the 'specified isomorphism to $G$' is unspecified!

The following transformations do not change the group $G$:

T1:  Adding a superfluous relation

$\langle X: R\rangle$ becomes $\langle X: R^'\rangle$, where $R^' = R \cup \{r\}$ and $r\in N(R)$ the normal closure of the relations in the free group on $X$, i.e., $r$ is a consequence of $R$;

T2: Removing a superfluous relation

$\langle X: R\rangle$ becomes $\langle X: R^'\rangle$ where $R^' = R - \{r\}$, and $r$ is a consequence of $R^'$;

T3: Adding a superfluous generator

$\langle X: R\rangle$ becomes $\langle X^': R^'\rangle$, where $X^' = X\cup \{ g\}$, $g$ being a new symbol not in $X$, and $R^' = R\cup\{wg^{-1}\}$, where $w$ is a word in the other generators, that is $w$ is in the image of the inclusion of $F(X)$ into $F(X^')$;

T4: Removing a superfluous generator

$\langle X: R\rangle$ becomes $\langle X^': R^'\rangle$, where $X^' = X - \{ g\}$, and $R^' = R-\{wg^{-1}\}$ with $w\in F(X^')$ and $wg^{-1}\in R$ and no other members of $R\prime$ involve $g$.

## Tietze's theorem

+-- {: .un_theo}
###### Theorem
Given two finite presentations of the same group, one can be obtained from the other by a finite sequence of Tietze transformations.
=--

## References

Tietze's original paper is

* H. Tietze, _Über die topologischen Invarianten mehrdimensionaler Mannigfaltigkeiten_, Monatsschr. Math. Phys., 19 (1908) 1 --118.

See also

* W. Magnus and B. Chandler, _The history of combinatorial group theory_, Springer (1982).

category : group theory

[[!redirects Tietze transformation]]
[[!redirects Tietze transformations]]
[[!redirects Tietze's transformation]]
[[!redirects Tietze's transformations]]


tight and loose morphisms

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--

*This page is about double categories. For the use of 'tight' and 'loose' in the context of [[F-categories]], see there.*

#Contents#
* table of contents
{:toc}

## Idea

In the context of [[double category|double category theory]], *vertical/horizontal* or *tight/loose* are terminologies used to distinguish the two kinds of 1-cells comprising a [[double category]].

Since a double category $\mathbb{D}$ is a [[internal category|category internal to]] $\mathbf{Cat}$, it comes with two kinds of 1-cells: the ones between objects of $\mathbb{D}_0$ (called *tight*) and the ones which are the objects of $\mathbb{D}_1$ (called *loose*).
As explained in [[double category]], *loose* cells are the protagonists in the specification of the internal category structure, since such structure tells us (1) what are their boundaries, (2) which one are the identities and (3) how they compose.

Historically, double categories were defined to be *strict* internal categories hence you could freely swap tight and loose directions without mathematical consequences (see (§3.2.2, §3.2.5, [Grandis 2019](#Grandis19)). Recently, however, double categories are more commonly defined to be [[internal pseudocategories]], meaning the category structure on the loose arrows is associative and unital only up to coherent isomorphism. As a consequence, loose cells do not organize in a category anymore, instead yielding a bicategory. Thus loose and tight cells are nowadays qualitatively different kinds of 1-cells and cannot be swapped (loose morphisms couldn't form an object of $\mathbf{Cat}$).

The interplay between the weakness of loose cells and the strictness of tight cells is a major principle of double category theory (extending to [[multiple categories|multiple category theory]] too). Since the lack of strictness in the loose direction is described in terms of tight cells instead of other loose cells, double categories are tamer 2-dimensional structures than [[bicategories]]. For instance [[monoidal double categories]] (and other algebraic structures) are generally easier to define than [[monoidal bicategories]] (see refs [[monoidal double categories|here]]), or the fact lax [[double functors]] form a double category in a more natural way than lax [[2-functors]] form a 2-category (see at [[icons]]).

## Notation and terminology
Since 2-cells in double categories are naturally denoted by squares, their boundary 1-cells are predominantly referred to of 'horizontal' and 'vertical'.
This terminology is the most common in the published literature, albeit used inconsistently (i.e. sometimes vertical corresponds to loose, chiefly in Grandis and Parè's papers, sometimes to tight). As discussed above, the modern usage of double categories doesn't allow one to swap loose and tight cells freely, hence confusing vertical for horizontal is not just a nuisance but could make the difference between a true and a false theorem.

The terminology 'tight' and 'loose' comes from the world of [[F-categories]], in particular from ([Lack and Shulman, 2012](#LackShulman12)). In that setting tight arrows are a special kind of loose arrows, but in the broader setting of double category theory that doesn't have to be the case.

Relatedly, in a [[framed bicategory]] loose 1-cells are called 'proarrows' and tight 1-cells are called arrows. That's because framed bicategories present [[proarrow equipments]].

|**1-cells in $\mathbb{D}_0$**|**0-cells in $\mathbb{D}_1$**| |
|--|--|--|
|longitudinal|lateral|([Ehresmann 1963](#Ehresmann63))|
|horizontal|vertical|Dawson and Paré; Grandis and Paré|
|vertical|horizontal|([Leinster 2004](#Lein04)), ([Cruttwell and Shulman, 2010](#CS10))|
|tight|loose|([Lack and Shulman 2012](#LackShulman12))|
|dynamic|static|([Grandis 2019](#Grandis19))|
|transversal|1-objects|([Grandis 2019](#Grandis19))|

## Examples

1. In the double category [[Rel|$\mathbf{\mathbb{R}el}$]] of sets, functions, relations and inclusions, functions are the tight cells and relations are the loose ones
2. In the double category [[profunctors|$\mathbf{\mathbb{P}rof}$]] of small categories, functors, profunctors and natural transformations, functors are the tight cells and profunctors are the loose ones

## Related concepts

* [[double category]]
* [[framed bicategory]]

## References

(Strict) double categories were introduced in

* [[Charles Ehresmann]], *Catégories double et catégories structurées*, C.R. Acad. Paris 256 (1963) 1198-1201 [[[Ehresmann-CategoriesDoubles.pdf:file]], [gallica](https://gallica.bnf.fr/ark:/12148/bpt6k3208j/f1246)]

where the two kinds of 1-cells were called longitudinal and lateral.

The less notationally-bound terminology tight/loose was introduced by

* {#LackShulman12} [[Steve Lack]], [[Mike Shulman]], _Enhanced 2-categories and limits for lax morphisms_, Advances in Mathematics, vol. 229, 2012, ([arxiv:1104.2111](https://arxiv.org/abs/1104.2111), [doi:j.aim.2011.08.014](https://doi.org/10.1016/j.aim.2011.08.014))

Other references:

* {#Grandis19} [[Marco Grandis]], _Higher dimensional categories: from double to multiple categories_, World Scientific, 2019. ([doi:10.1142/11406](https://doi.org/10.1142/11406))

* {#CS10} [[Geoffrey Cruttwell]], [[Mike Shulman]], _A unified framework for generalized multicategories_, TAC, vol. 24, 21, 2010, ([arxiv:0907.2460](https://arxiv.org/abs/0907.2460)), ([tac](http://www.tac.mta.ca/tac/volumes/24/21/24-21abs.html))

* {#Lein04} [[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)]

[[!redirects horizontal morphism]]
[[!redirects horizontal morphisms]]

[[!redirects vertical morphism]]
[[!redirects vertical morphisms]]

[[!redirects tight morphism]]
[[!redirects tight morphisms]]

[[!redirects loose morphism]]
[[!redirects loose morphisms]]

[[!redirects loose map]]
[[!redirects loose maps]]

[[!redirects tight map]]
[[!redirects tight maps]]

[[!redirects tight]]
[[!redirects loose]]


tight relation

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

\tableofcontents

## Definition

A [[binary relation]] $R$ on a set $A$ is __tight__ if for all elements $a \in A$ and $b \in A$, $\neg R(a, b)$ implies that $a = b$.

Every tight relation is a [[connected relation]]. Every connected [[symmetric relation]] is a tight relation.

## Examples

* A [[tight apartness relation]] is an [[apartness relation]] which is tight.

## Weakly tight relations

Sometimes in [[constructive mathematics]], what matters for a relation is not that the negation implies equality, but the negation implies the [[double negation]] of equality. Such a relation is called **weakly tight**.

A [[binary relation]] $R$ on a set $A$ is __weakly tight__ if for all elements $a \in A$ and $b \in A$, $\neg R(a, b)$ implies that $\neg \neg (a = b)$. By intuitionistic [[contraposition]], this implies that for all elements $a \in A$ and $b \in A$, $\neg (a = b)$ implies that $\neg \neg R(a, b)$.

Important examples of weakly tight relations include [[denial inequality]].

## In dependent type theory

An [[irreflexive relation]] $\sim$ with terms $a:A \vdash \mathrm{irr}(a):(a \sim a) \to \emptyset$ is tight if the canonical function
$$\mathrm{idtonotrel}(a, b):(a =_A b) \to ((a \sim b) \to \emptyset)$$
inductively defined by
$$\mathrm{idtonotrel}(a, a)(\mathrm{refl}_A(a)) \equiv \mathrm{irr}(a):(a \sim a) \to \emptyset$$
is an [[equivalence of types]]
$$\mathrm{conn}(a, b):\mathrm{isEquiv}(\mathrm{idtonotrel}(a, b))$$

## See also

* [[connected relation]]
* [[tight inequality relation]]
* [[tight apartness relation]]

[[!redirects tight]]
[[!redirects tight relation]]
[[!redirects tight relations]]

[[!redirects weakly tight]]
[[!redirects weakly tight relation]]
[[!redirects weakly tight relations]]


tileorder

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

## Idea

A **tileorder** is a *double order*, i.e. a set $T$ equipped with two [[partial orders]] $\le$ and $\sqsubseteq$, which can be realized by a dissection of a rectangle into finitely many subrectangles, the subrectangles being the elements of $T$, in such a way that $a\sqsubseteq b$ iff $a$ "lies below" $b$, while $a\le b$ iff $a$ "lies to the left of" $b$.  Interestingly and importantly, such orders can also be characterized in purely combinatorial, and effectively verifiable, terms.

Such orders are of interest in [[double category]] theory, since these are the arrangements of 2-cells in a double category which could potentially be composed (although in a general double category, not all of them can actually be composed).

## Definition

Suppose given a dissection of a rectangle into finitely many subrectangles.  Define $T$ to be the set of subrectangles, and for $a,b\in T$ write $a\le b$ if there exists a finite list $a= x_0, x_1, \dots, x_n = b$ such that for each $i$, the right edge of $x_i$ intersects the left edge of $x_{i+1}$ in more than one point.  Define $a\sqsubseteq b$ similarly using top and bottom edges instead of left and right.  Clearly $\le$ and $\sqsubseteq$ are [[partial orders]].

A set $T$ equipped with two partial orders $\le$ and $\sqsubseteq$ (a priori, unrelated) is called a **double order**.  A double order which arises in this way from a rectangle dissection is called a **tileorder**.

+--{.query}
[[Todd Trimble|Todd]]: Maybe I'm missing what you mean by "dissection", but off the bat it looks like you are allowing the "pinwheel" as a possible dissection (I hope it is clear what the pinwheel is; Dawson discusses it in one of his papers), but this is the kind of configuration not interpretable in double categories.

[[Mike Shulman]]: Yes, we are allowing the pinwheel, even though it is not always composable in a double category.  That's what I meant to imply above by

> arrangements of 2-cells in a double category which could potentially be composed (although in a general double category, not all of them can actually be composed)

but maybe that isn't sufficiently clear.  The notion of tileorder is purely geometric/combinatorial, and you then have to ask which tileorders are composable in a double category (essentially, all that don't contain a pinwheel).  (BTW, a picture of the pinwheel can be found [here](http://ncatlab.org/nlab/show/double+category#unbiased_composition_and_associativity_9).)
=--

## Characterizations

As proven by Dawson and Paré, tileorders can be characterized in purely combinatorial terms in a couple of interesting ways.

### Using maximal chain properties

A double order $T$ is said to have the **$\sqsubset$-parallel maximal chain property** if whenever $K$ and $L$ are maximal $\sqsubseteq$-chains in $T$, and we have $k_1,k_2\in K$ and $l_1,l_2\in L$ with $k_1 \sqsubset k_2$, $k_1 \le l_1$, and $l_2 \le k_2$, then there exists an $e\in K\cap L$ such that $k_1\sqsubset e$, $l_1\sqsubset e$, $e\sqsubset k_2$, and $e\sqsubset l_2$.  In other words, two maximal $\sqsubseteq$-chains cannot "swap places" in the $\le$-order without intersecting.

Dually, we define the **$\lt$-parallel maximal chain property**.

A double order $T$ is said to have the **orthogonal maximal chain property** if every maximal $\le$-chain intersects every maximal $\sqsubseteq$-chain exactly once.

+-- {: .un_theorem}
###### Theorem
A double order is a tileorder iff it has both parallel maximal chain properties and the orthogonal maximal chain property.
=--

### Using local configuration structure

A double order is **strongly antisymmetric** if two unequal elements are never related (in either direction) by both $\lt$ and $\sqsubset$.  That is, if $a\neq b$, then at most one of $a\lt b$, $b\lt a$, $a\sqsubset b$, and $b\sqsubset a$ holds.

A double order is **rectangular** if $\exists c.(a\sqsubseteq c \le b)$ iff $\exists d.(a\le d \sqsubseteq b)$, and similarly $\exists c.(a\sqsubseteq c \ge b)$ iff $\exists d.(a\ge d \sqsubseteq b)$.

A double order is **total** if for any $a,b$, there exists a $c$ such that either $a\sqsubseteq c \le b$, or $a\sqsubseteq c \ge b$, or $b \le c \sqsubseteq a$, or $b\ge c \sqsubseteq a$.

A double order has the **first $\sqsubset$-orthogonal butterfly factorization property** if given $a,b,c,d$ with $c\sqsubseteq a$, $c\sqsubseteq b$, $d\sqsubseteq b$, and $d\le a$, there exists an $e$ with $c\sqsubseteq e \sqsubseteq b$ and $d\le e \le a$.  The **second** such property is defined by replacing $\le$ by $\ge$, but not changing $\sqsubseteq$.  The **$\lt$-orthogonal** such properties are defined by switching the roles of $\sqsubseteq$ and $\le$.

A double order has the **$\sqsubset$-parallel butterfly factorization property** if given $a,b,c,d$ with $c\sqsubseteq a$,  $b\le c$, $d\sqsubseteq b$, and $d\le a$, there exists an $e$ with $b\le e \le c$ and $d\le e \le a$.  The **$\lt$-parallel** such property is defined by switching the roles of $\sqsubseteq$ and $\le$.

+-- {: .un_theorem}
###### Theorem
A double order is a tileorder iff it is strongly antisymmetric, rectangular, total, and has all the orthogonal and parallel butterfly factorization properties.
=--

## References

* Dawson and Paré, "Characterizing tileorders"

[[!redirects tileorders]]
[[!redirects double order]]
[[!redirects double orders]]


Tilman Bauer

* [webpage](https://people.kth.se/~tilmanb/)

## Selected writings

On [[Bousfield localization of spectra]] and [[fracture squares]]:

* {#Bauer11} [[Tilman Bauer]], _Bousfield localization and the Hasse square_ (2011) ([pdf](https://math.mit.edu/events/talbot/2007/tmfproc/Chapter09/Bousfield.pdf), [[Bauer_BousfieldLocalization.pdf:file]]), chapter 6 in:  [[Christopher Douglas]], [[John Francis]], [[André Henriques]], [[Michael Hill]] (eds.), _Topological Modular Forms_, Mathematical Surveys and Monographs Volume 201, AMS 2014  ([ISBN:978-1-4704-1884-7](https://bookstore.ams.org/surv-201))

On [[cohomology operations]]:

* [[Tilman Bauer]], _Formal plethories_, Advances in Mathematics Volume 254, 20 March 2014, Pages 497-569 ([arXiv:1107.5745](https://arxiv.org/abs/1107.5745), [doi:10.1016/j.aim.2013.12.023](https://doi.org/10.1016/j.aim.2013.12.023))

### Related pages

* [[Dwyer-Wilkerson H-space]]
* [[localization of a space]]

category: people


Tilman Sauer

* [webpage](http://www.hss.caltech.edu/~tilman/)

## Selected writings

On [[David Hilbert]]'s lectures on [[general relativity]], [[quantum physics]] and [[epistemology]]:

* {#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)]

## Related entries

* [[general relativity]]

* [[Einstein-Hilbert action]]

category: people


Tim > history

< [[Tim]]
[[!redirects Tim -- history]]


Tim Adamo

* [webpage](http://www.damtp.cam.ac.uk/people/ta358/)

## related $n$Lab entries

* [[twistor space]]

* [[super Yang-Mills theory]]

category: people


Tim Berberich

#Contents#
* table of contents
{:toc}

##Contact
Mail at <berxue@outlook.com> or <Tim.Benedict.Berberich@studium.uni-hamburg.de>
##Master Thesis


Tim Berberich/master_thesis

#Contents#
* table of contents
{:toc}

##Status

## Tasks/ Quetions to answer
* Understand local BV-BRST complex
  * See that in $(E_{BV-BRST}, -L+L_{BRST})$ there are no nontrivial gauge symmetries.

##Notes

###Cauchy surfaces
There are two seemingly very different definitions for cauchy surfaces

+-- {: .num_defn #CauchySurfaceGeometric }
###### Definition
Given a Spacetime $\mathcal{M}$ a closed subset $\Sigma\in\mathcal{M}$ is called \textit{cauchy surface} if:

1. (achronal) Time like curves in $\mathcal{M}$ intersect $\Sigma$ at most once.
2. () $\forall x\in\mathcal{M}$ all future- or past-inextensible causal curve passing through $x$ intersect $\Sigma$.
=--

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

Given a Lagrangian field theory $(E,L)$

=--


Tim Byrnes

* [Institute page](https://shanghai.nyu.edu/academics/faculty/directory/tim-byrnes)

* [[CQTS]]

## Selected writings

Proposal to use [[Bose-Einstein condensates]] for [[quantum computation]]:

* [[Tim Byrnes]], Kai Wen, Yoshihisa Yamamoto, *Macroscopic quantum computation using Bose-Einstein condensates*, Phys. Rev. A **85** (2012)  040306(R) $[$[doi:10.1103/PhysRevA.85.040306](https://doi.org/10.1103/PhysRevA.85.040306)$]$

On [[quantum coherence]]:

* [[Radhakrishnan Chandrashekar|R. Chandrashekar]], P. Manikandan, J. Segar, [[Tim Byrnes]], *Distribution of quantum coherence in multipartite systems*, Phys. Rev. Lett. **116** 150504 (2016) [[arXiv:1602.00286](https://arxiv.org/abs/1602.00286), [doi:10.1103/PhysRevLett.116.150504](https://doi.org/10.1103/PhysRevLett.116.150504)]

On [[quantum optics]] in [[quantum information theory]]:

* [[Tim Byrnes]], [[Ebubechukwu Ilo-Okeke]], *Quantum Atom Optics: Theory and Applications to Quantum Technology*, Cambridge University Press 2021 ([arXiv:2007.14601](https://arxiv.org/abs/2007.14601), [CUP](https://www.cambridge.org/core/books/quantum-atom-optics/2D867888B5C666D3A936F1C942C99568))

On [[topological lasers]]:

* Natsuko Ishida*, Yasutomo Ota*, Wenbo Lin, [[Tim Byrnes]], Yasuhiko Arakawa and Satoshi Iwamoto, *A large-scale single-mode array laser based on a topological edge mode*, Nanophotonics **11** 9 (2022) 2169–2181 [[doi:10.1515/nanoph-2021-0608](https://doi.org/10.1515/nanoph-2021-0608)]

## Related nLab entries

* [[Bose-Einstein condensate]]

category: people


Tim Campion

Tim Campion is a mathematician at Johns Hopkins University.

* [old personal page](http://sites.nd.edu/tcampion/) (at Notre Dame)

## Selected writings

On the [[homotopy types]] of [[classifying spaces]] of [[abstract elementary classes]]:

* [[Tim Campion]], Jinhe Ye, *Homotopy Types of Abstract Elementary Classes*, Journal of Pure and Applied Algebra
**225** 5 (2021) 106461 [[arXiv:1909.07965](https://arxiv.org/abs/1909.07965), [doi:10.1016/j.jpaa.2020.106461](https://doi.org/10.1016/j.jpaa.2020.106461)]

On a [[universal property]] characterizing [[equivariant stable homotopy theory]]:

* [[Tim Campion]], _Free duals and a new universal property for stable equivariant homotopy theory_ [[arXiv:2302.04207](https://arxiv.org/abs/2302.04207)]

category: people


Tim Gowers

* [website](http://www.dpmms.cam.ac.uk/~wtg10/)

category: people

[[!redirects Timothy Gowers]]


Tim Hosgood

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

## Selected writings

On the [[field with one element]]:

* [[Tim Hosgood]], _Under $\mathop{Spec}\mathbb{Z}$: a reader's companion_, master thesis, 2016 ([pdf](https://thosgood.com/assets/files/under-spec-z.pdf)).

On [[simplicial Chern--Weil theory]]:

* [[Tim Hosgood]], _Simplicial Chern--Weil theory for coherent analytic sheaves, part I_, (2020) ([arXiv:2003.10023](https://arxiv.org/abs/2003.10023)).

* [[Tim Hosgood]], _Simplicial Chern--Weil theory for coherent analytic sheaves, part II_, (2020) ([arXiv:2003.10591](https://arxiv.org/abs/2003.10591)).

* [[Tim Hosgood]], _Chern classes of coherent analytic sheaves: a simplicial approach_, PhD thesis (2020) ([tel-02882140](https://tel.archives-ouvertes.fr/tel-02882140)).

Translations:

* English translation of [[Grothendieck]]'s *[[Fondements de la Géométrie Algébrique]]* ([[FGA]])

  [webpage](https://thosgood.com/fga/).

* Various English mathematical translations

  [webpage](https://thosgood.com/translations/).

category: people


Tim Maudlin

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

## related $n$Lab entries

* [[philosophy of physics]]

category: people


Tim Porrter > history

< [[Tim Porrter]]
[[!redirects Tim Porrter -- history]]


Tim Porter

I was until 2006 a professor of mathematics at the University of Wales Bangor (aka University of Bangor). The University closed down the Mathematics degree and I was 'retired'! Since then I have been on various interesting visiting positions, whilst being Emeritus, until recently retaining an office in the Computer Science section of the university and continuing to work at least as hard as before (so I claim). I am also an emeritus member of [WIMCS](http://www.wimcs.ac.uk/), the Wales Institute of Mathematical and Computational Sciences and a Fellow of the [Learned Society of Wales](https://www.learnedsociety.wales/), a recently formed  society that seeks to act as an Academy of learning of the kind that has long existed elsewhere in the United Kingdom and worldwide, but not in Wales.

+--{.query}
Not offering a maths degree is one thing, but it seems that they don\'t even have a maths *department*! Who teaches basic maths courses to the CS (and chemistry, biology, electronics, ...) majors? —[[Toby Bartels|Toby]]

In the UK, there are several universities that do not have mathematics departments and in which engineers, computer scientists, physicists and chemists do not get any theoretically based mathematics to speak of. Biologists have traditionally received no mathematics training after the age of 16. (Sometimes it shows and is the bane of the statisticians detailed to teach biology students statistics, and don't mention psychology students! Note also that quite a few of the so called 'financial  experts' in the UK have little or no maths. Some do, and I am not saying that they were any better in the causes of the recent crisis!)

In Bangor, it is even worse. Not only is there no mathematics department. There are very few mathematicians _per se_, employed as full time members of staff and they are not employed as mathematicians as such.  There are some mathematically competent people in some departments who do teach their undergraduates some mathematics.  I, of course, do not think that is as good.

-[[Tim Porter|Tim]]

At least a couple of the top ten CS departments in the USA teach all of their own Maths classes.  The actual Math department at most schools is too focused on calculus and analysis (at the undergraduate level).  I think it has to be this way -- it would be obscene for the CS department to dictate curriculum to the Math department.

Don Knuth proposed that every professor should be required to have both a "major" and "minor" department, just like undergraduates must at most universities.  That might be one step towards improving the situation -- a way to influence the curriculum without compromising autonomy.

=--

##What you can find here and on my personal pages##

I have given a summary in my [private $n$Lab area](http://ncatlab.org/timporter/show/HomePage) of some of the research area that I have worked in with references for further 'enlightenment'. Some of these are in areas that continue to interest me, and I have various 'projects' and lists of 'work in progress' there as well, some of which are mentioned below. Some more recent ones have been added (August 2018).

I am writing a series of notes on the theory and application of crossed gadgetry in algebra and topology, and some parts of these notes (approximately first 14 chapters) have been made available on the web at various times. I am currently trying to write new sections on the links between homotopy quantum field theory and non-Abelian cohomology.

I am also very interested in directed homotopy theory and the application of ideas from the general area of the infinity category/homotopy toolkit in topological data analysis, artificial intelligence, and computer science. This relates both to the use of [[stratified spaces]] and possibly to [[defects]] in TQFTs. Some material can be found on those personal pages or should be added .

See also my [private $n$Lab area](http://ncatlab.org/timporter/show/HomePage), where I have put a link to a recent version of the first 14 chapters of the Menagerie.

Other goodies there include parts of a draft monograph on [profinite algebraic homotopy](https://ncatlab.org/timporter/show/profinite+algebraic+homotopy), and a (slightly reformatted) version of a research proposal from 2002, that did not get funded, but may be useful as it does have some ideas in it (or related to it) that are worth pursuing especially since the recent progress on the cobordism and TQFT problems. There are links to notes for the Lisbon school and workshop on Higher Gauge Theory etc.,  (Feb. 2011). There are also some lecture notes from lectures in Hagen and La Laguna, and a survey article that grew out of talks in 1991 in Italy.

I have also put the start of a brief discussion of the relationships between the Cech and Vietoris methods and the newly developed methods of Topological Data Analysis.

###List of Goodies!!###
Please go to [private $n$Lab area](http://ncatlab.org/timporter/show/HomePage) for more downloads and fuller details.

* _The [[Menagerie]]_: a set of notes on crossed modules, higher dimensional analogues, simplicial techniques, homotopy coherence, and non-Abelian cohomology of various sorts. This version consists of 10 chapter. It will be updated periodically.

Several 'cut down' versions of these notes have been prepared for various workshops and as they are less long may be of use.  For instance in February, 2011, there was a Workshop and School on Higher Gauge Theory, TQFT and Quantum Gravity in the IST, Lisbon, (7-13 February, 2011).  A set of notes was prepared using the Menagerie as a 'mine' from which to extract the sections relevant to the theme of the workshop, which were then edited slightly. The result was entitled: **Homotopy Quantum Field Theories meets the Crossed Menagerie: an introduction to HQFTs and their relationship with things simplicial and with lots of crossed gadgetry.** A copy can be found [[HQFT-XMenagerie.pdf|here:file]], whilst a table of contents and some more introductory material can be found via the other pages.

*  The 'Luminy notes': A cut down version of the Menagerie has been prepared for the LI2012 week on Algebra and computation (27 February – 2 March, 2012). A copy can be obtained by contacting me by e-mail or [[nLab:Rewriting-and-homotopy.pdf|here:file]].

* _Abstract Homotopy Theory: The Interaction of Category Theory and Homotopy
theory_ This article is an expanded version of notes for a series of lectures given at the _Corso estivo Categorie e Topologia_ in 1991. They appeared in more or less this form in Cubo, 5 (2003) 115-165, 2003.

* _$\mathcal{S}$-categories, $\mathcal{S}$-groupoids, Segal categories and quasicategories_ The notes were prepared for a series of talks that I gave in Hagen in late June and early July 2003, and, with some changes, in the University of La Laguna, the Canary Islands, in September, 2003.

*  _Profinite Algebraic Homotopy_.   There is a link to  eight chapters of a draft monograph.  Hopefully, this is likely to be published, so this is an incomplete version. (At present count the full version will probably have more than 1000 pages.) I have added new sections and even chapters to the earlier part, so the first 8 chapters of the current version do not correspond to these here.

* Grothendieck's letters. Thirty years ago I was involved in a brief exchange of letters with Alexander Grothendieck. I will be putting up some excerpts. It may happen that commented versions of these will be made available by the _Documents mathématiques_ project.

* [[Spaces as infinity-groupoids.pdf|Spaces as Infinity Groupoids:file]]: this is a draft chapter of a (hopefully) forthcoming book, _New Spaces in Mathematics and Physics_.

##More Information

You may also find me on: [MathOverflow](http://mathoverflow.net/users/3502/tim-porter) | [ORCiD](http://orcid.org/0000-0001-7962-8471)

and on Linkedin. (I have a gmail account and you can also contact me via that.  The e-mail is initial\dot familyname\dot maths and then the normal ending for a gmail account.)

category: people

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Tim Portr > history

< [[Tim Portr]]
[[!redirects Tim Portr -- history]]


Tim van Beek

Hi,
after spending some time at the universities of Göttingen and Heidelberg, I graduated in physics and started to work as a software developer, currently living in Munich, Germany.

But I still like to think about mathematical physics and pure mathematics, which is why I visit the café in my spare time -- and since one learns best by trying to explain things, I like to add a little bit to the nLab iff I have anything to say (doesn't happen too often).

My primary interest lies in mathematical aspects of quantum field theories.

My Email address is: <Tim.van.Beek@capgemini-sdm.com>

But primarily this  page exists to get rid of this odd question mark behind my name that appears after I have changed a page!

category: people

[[!redirects Tim Van Beek]]


Tim Van der Linden

Tim Van der Linden works on simplicial homotopy and homological algebra in [[semi-abelian categories]].

* [Home Page](http://perso.uclouvain.be/tim.vanderlinden/)

## Selected writings

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

[[!redirects Tim van der Linden]]
[[!redirects Tim van den Linden]]

[[!redirects T. van den Linden]]


Tim Weelinck

* [webpage](https://www.maths.ed.ac.uk/~tweelinck/)

## Selected writings

On [[factorization homology]] for [[orbifolds]]

* [[Tim Weelinck]], _Equivariant factorization homology of global quotient orbifolds_ ([arXiv:1810.12021](https://arxiv.org/abs/1810.12021), [talk pdf](https://www.maths.ed.ac.uk/~tweelinck/efhqsp.pdf))

category: people


Timaeus dialogue

[[!redirects Timaeus]]
[[!redirects Timaios]]

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The **Timaeus** is one of [[Plato's]] dialogues in which he gives an account of the construction of the universe by a rational craftsman, or Demiurge. The four "elements" (earth, air, fire, water) are associated with four of the five [[Platonic solids]].

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## The Platonic solids
 {#ThePlatonicSolids}

The dialogue discusses the five [[Platonic solids]] and argues that they are building blocks of nature, in some way. See for instance _[Platonic Solids and Plato's Theory of Everything](http://www.mathpages.com/home/kmath096/kmath096.htm)_

In modern mathematics the Platonic solids are still a source of rich structure, via the [[McKay correspondence]].

[[!include ADE -- table]]

By way of this correspondence, in the speculative [[physics]] [[theory (physics)|theory]] called _[[M-theory on G2-manifolds]]_, the Platonic solids do again serve to exhibit fundamental aspects of nature, even if in a more intricate way than Timaeus had imagined.

## Cosmogeny
 {#Cosmogeny}

> [35a] and in the fashion which I shall now describe. Midway between the Being which is indivisible and remains always the same and the Being which is transient and divisible in bodies, He blended a third form of Being compounded out of the twain, that is to say, out of the Same and the Other; and in like manner He compounded it midway between that one of them which is indivisible and that one which is divisible in bodies. And He took the three of them, and blent them all together into one form, by forcing the Other into union with the Same, in spite of its being naturally difficult to mix.

> [35b] And when with the aid of Being He had mixed them, and had made of them one out of three, straightway He began to distribute the whole thereof into so many portions as was meet; and each portion was a mixture of the Same, of the Other, and of Being. And He began making the division thus: First He took one portion from the whole; then He took a portion double of this; then a third portion, half as much again as the second portion, that is, three times as much as the first; he fourth portion He took was twice as much as the second; the fifth three times as much as the third;

In the _[[Science of Logic]]_ [[Hegel]] picks up this concept of  _[Something and an Other](Science+of+Logic#SomethingAndAnOther)_. His students paraphrased him as attributing this to Timaeus:

> [EL§92Zusatz](Science+of+Logic#EL92Zusatz) Gott hat die Welt aus der Natur des Einen und des Anderen (του ετεου) gemacht; diese hat er zusammengebracht und daraus ein Drittes gebildet, welches von der Natur des Einen und des Anderen ist. (vgl. Timaios, Steph. 34 f.)

## Related entries

* [[Parmenides dialogue]]

* [[Republic dialogue]]

## Reference

* Stanford Encyclopedia of Philosophy, _[Plato's Timaeus](http://plato.stanford.edu/entries/plato-timaeus/)_

* Wikipedia, _[Timaeus (dialogue)](https://en.wikipedia.org/wiki/Timaeus_(dialogue)_

category: reference


time

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## Idea

_Time_ is what passes along a [[time-like curve]]. The [[pseudo-Riemannian manifold|pseudo-Riemannian]] [[volume]] of a piece of such curve is the _proper time_.

## Philosophy of Time

Much debate here has focused on the rival pictures of time present by '$A$ series' and 'B series' views (see [SEP: Time](http://plato.stanford.edu/entries/time/)). The $A$ series view sees time as something that flows or passes, and privileges the particularity of the present moment. The $B$ series takes all moments to have equal ontological status. '$A$ properties' make reference to the present (20 years ago, next week), while '$B$ properties' only invoke relational notions of before and after (1987, the first Sunday after this year's Spring equinox).

$B$-theorists claim they have science on the side, in particular the [[theory of general relativity]] viewing the world as a four-dimensional manifold. $A$-theorists may maintain that the $B$-theory makes no sense of the lived dynamic experience of cause and change.

##Connes on the Emergence of Time

* [Temps et Aléa du Quantique](https://www.youtube.com/watch?v=ODAngTW8deg), IHES, April 2015. (English)
> When I was in preparatory school, my teacher asked me (...) "what is a variable?". I reflected and reflected, and after a while, I said "time". (...) The topic of my talk is that I believe we are all used, because of our constitution and so on, to attribute variability to the passing of time. The thesis which I will propose and try to back with mathematical results is the following: I believe that the true variability is [[quantum]], and that the true variability is the fact that when you take a quantum observable it doesn't have a single value, but it has many possible values which are given by the [[spectrum]] of the [[operator]], plus the fact that [[discrete]] variables cannot coexist with [[continuous function|continuous]] variables without the quantum formalism. I will explain how time emerges from these facts. I have never tried to explain this idea, I know it's difficult, and its difficult because in my mind it is backed up by an intuition which comes from many years of work, and this is the most difficult thing to transmit. (...) How to explain this? (...) The answer I believe comes from [[von Neumann]] (suitably implemented and very much ameliorated). (...) In the 40's and 50's [[von Neumann]] was asking what does it mean to have a subsystem? What does it mean that somehow, the [[Hilbert space]] in which you work is a Hilbert space in which you have partial knowledge of things because the system is a composite system and there is a part of the system which you know and a part which you ignore? What von Neumann was trying to understand was factorizations. (gives lecture on factorizations...) By the way, I should say that this is why I spent many years studying [[Noncommutative Geometry]]: the simplest geometric origin of von Neumann factorization is [[foliations]]. If you take the simplest [[foliation]] (well, I don't know if it's the simplest), the [???] [[foliation]] of the [[sphere bundle]] of a [[Riemann surface]], you get the most exotic factorization of [[von Neumann]] (type III1).

## Related concepts

* [[proper time]]

* [[comoving time]]

* [[frequency]]

* [[second]], [[year]]

* [[space]]

* [[spacetime]]

* [[mass]]

## References

See also

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

On [[Tomita-Takesaki modular flow]] as emergent [[time]] evolution in [[quantum physics]] ([[AQFT]]):

* {#Longo19} [[Roberto Longo]], _The emergence of time_ ([arxiv:1910.13926](https://arxiv.org/abs/1910.13926))

[[!redirects proper time]]


time series

Time series is a (finite) sequence of measurement data of the same quantity taken at equal intervals in time.

* wikipedia [time series](https://en.wikipedia.org/wiki/Time_series)
* related $n$Lab entries include [[statistics]], [[sampling theory]], [[signal processing]], [[information geometry]]
* N. H. Packard, J. P. Crutchfield, J. D. Farmer, R. S. Shaw, _Geometry from a time series_, Phys. Rev. Lett. 45, 712
* G. Schwartz, _Estimating the dimension of a model_, Ann. Stat. __6__, pp. 461-464,1978.

category: applications, probability


time slice axiom

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## Idea

In [[quantum field theory]] the _time slice axiom_ is an additional axiom often considered on top of the basic [[Haag-Kastler axioms]]: it encodes the property that the time evolution of the classical field theory underlying a [[QFT]] is governed by well-defined initial value problems.

## Definition

A [[local net of observables]] $A : Op(X) \to C^\ast Alg$ satisfies the **time slice axiom** if whenever a causal inclusion of [[Lorentzian space]]s $O_1 \hookrightarrow O_2$ is such that $O_1$ contains a [[Cauchy surface]] of $O_2$ then $A(O_1 \hookrightarrow O_2)$ is an [[isomorphism]].

## Related concepts

* [[split property]]

## References

### General

See references at *[[AQFT]]*.

Discussion for [[homotopical AQFT]]:

* [[Marco Benini]], [[Victor Carmona]], [[Alexander Schenkel]], *Strictification theorems for the homotopy time-slice axiom* [[arxiv:2208.04344](https://arxiv.org/abs/2208.04344)]

### Time tube theorem
 {#ReferencesTimeTubeTheorem}

* [[Hans-Jürgen Borchers]], *Über die Vollständigkeit lorentzinvarianter Felder in einer zeitartigen Röhre*, Il Nuovo Cimento (1955-1965) **19** (1961) 787–793 [[doi:10.1007/BF02733373](https://doi.org/10.1007/BF02733373)]

* [[Huzihiro Araki]], *A generalization of Borchers' theorem*, Helvetica Physica Acta **36** (1963) 132-139

Generalization to [[AQFT on curved spacetimes|AQFT on curved]] but [[real analytic space|real analytic]] [[spacetimes]]:

* [[Alexander Strohmaier]], *On the Local Structure of the Klein–Gordon Field on Curved Spacetimes*, Letters in Mathematical Physics **54** (2000) 249–261 [[doi:10.1023/A:1010927625112](https://doi.org/10.1023/A:1010927625112), [arXiv:math-ph/0008043](https://arxiv.org/abs/math-ph/0008043)]

* [[Alexander Strohmaier]], [[Edward Witten]], *Analytic states in quantum field theory on curved spacetimes* [[arXiv:2302.02709](https://arxiv.org/abs/2302.02709)]

* [[Alexander Strohmaier]], [[Edward Witten]], *The Timelike Tube Theorem in Curved Spacetime* [[arXiv:2303.16380](https://arxiv.org/abs/2303.16380)]

[[!redirects time-slice axiom]]

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