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W. F. Newns | https://ncatlab.org/nlab/source/W.+F.+Newns | W. F. Newns was a professor at the University of Liverpool.
## Selected writings
On [[vector fields as derivations]]:
* [[W. F. Newns]], [[A. G. Walker]], _Tangent Planes To a Differentiable Manifold_. Journal of the London Mathematical Society s1-31:4 (1956), 400–407 ([doi:10.1112/jlms/s1-31.4.400](https://doi.org/10.1112/jlms/s1-31.4.400))
category: people |
W. Forrest Stinespring | https://ncatlab.org/nlab/source/W.+Forrest+Stinespring |
* [Wikipedia entry](https://en.wikipedia.org/wiki/W._Forrest_Stinespring)
* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=6468)
## Selected writings
Introducing what later came to be known as the [[Kraus decomposition]] of [[quantum operations]]:
* {#Stinespring55} [[W. Forrest Stinespring]], *Positive functions on $C^\ast$-algebras*, Proc. Amer. Math. Soc. **6** 2 (1955) 211-216 [[doi:2032342](https://www.jstor.org/stable/2032342), [doi:10.2307/2032342](https://doi.org/10.2307/2032342)]
category: people |
W. Hugh Woodin | https://ncatlab.org/nlab/source/W.+Hugh+Woodin | William Hugh Woodin is a Professor of Mathematics at Harvard University
and a Professor Emeritus of Mathematics at the University of California, Berkeley.
[Website](https://www.math.harvard.edu/people/woodin-hugh/)
[Wikipedia page](https://en.wikipedia.org/wiki/W._Hugh_Woodin)
## Related concepts
* [[Woodin cardinal]]
* [[ultimate-L]]
* [[Omega conjecture]]
* [[axiom of determinacy]]
[[!redirects William Hugh Woodin]]
[[!redirects Hugh Woodin]]
[[!redirects Woodin]] |
W. Stephen Wilson | https://ncatlab.org/nlab/source/W.+Stephen+Wilson | W. Stephen Wilson is an American [[homotopy theory|homotopy theorist]] based at John Hopkins University. He specializes 'in the use and development of [[MU|complex (co)bordism]], [[BP|Brown-Peterson (co)homology]] and [[Morava K-theory]]'.
* [Mathematical Home Page](http://www.math.jhu.edu/~wsw/)
## Selected writings
Introducing the [[Johnson-Wilson spectrum]] (see at _[[Morava K-theory]]_):
* [[David Copeland Johnson]], [[W. Stephen Wilson]], _BP operations and Morava's extraordinary K-theories_, Math. Z. 144 (1): 55−75 (1975) ([pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/jw-morava.pdf))
On unstable [[cohomology operations]] (hence: in [[non-abelian cohomology]]):
* [[John Michael Boardman]], [[David Copeland Johnson]], [[W. Stephen Wilson]], _Unstable Operations in Generalized Cohomology_ ([pdf](https://hopf.math.purdue.edu/Boardman-Johnson-Wilson/bjw.pdf)), in: [[Ioan Mackenzie James]] (ed.) _[[Handbook of Algebraic Topology]]_ Oxford 1995 ([doi:10.1016/B978-0-444-81779-2.X5000-7](https://doi.org/10.1016/B978-0-444-81779-2.X5000-7))
On [[Hopf rings]] in [[algebraic topology]]:
* [[Douglas Ravenel]], [[W. Stephen Wilson]], _The Hopf ring for complex cobordism_, Bull. Amer. Math. Soc. 80 (6) 1185 - 1189, November 1974 (<a href="https://doi.org/10.1016/0022-4049(77)90070-6">doi:10.1016/0022-4049(77)90070-6</a>, [euclid](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-80/issue-6/The-Hopf-ring-for-complex-cobordism/bams/1183536024.full?tab=ArticleLink), [pdf](https://people.math.rochester.edu/faculty/doug/mypapers/hopfring.pdf))
> (on [[MU]])
* [[W. Stephen Wilson]], _Hopf rings in algebraic topology_, Expositiones Mathematicae, 18:369–388, 2000 ([pdf](https://math.jhu.edu/~wsw/papers2/math/39-hopf-rings-expo-2000.pdf))
category: people
[[!redirects W.S. Wilson]]
[[!redirects W. S. Wilson]] |
w4-orientation of EO(2)-theory | https://ncatlab.org/nlab/source/w4-orientation+of+EO%282%29-theory |
#Contents#
* table of contents
{:toc}
## Idea
The real form $EO(2)$ of [[ER-theory]] (i.e. [[real-oriented cohomology theory|real oriented]] [[Morava E-theory]]) is [[orientation in generalized cohomology|oriented]] by [[Spin structure]] combined with [[w4-structure]], hence by trivialization of the [[Stiefel-Whitney classes]] $w_1$, $w_2$ and $w_4$. ([Kriz-Sati 04, section 5.2, esp. p. 29](#KrizSati04)).
## Related concepts
[[!include genera and partition functions - table]]
## References
* {#KrizSati04} [[Igor Kriz]], [[Hisham Sati]], _M Theory, Type IIA Superstrings, and Elliptic Cohomology_, Adv.Theor.Math.Phys. 8 (2004) 345-395 ([arXiv:hep-th/0404013](http://arxiv.org/abs/hep-th/0404013))
|
w4-structure | https://ncatlab.org/nlab/source/w4-structure |
## Idea
[[c-structure]] for $c$ the 4th [[Stiefel-Whitney class]] $w_4$
I.e. trivialization of $w_4$
## Related concepts
* [[p1-structure]]
* [[w4-orientation of EO(2)-theory]]
|
Wagner-Preston theorem | https://ncatlab.org/nlab/source/Wagner-Preston+theorem |
#Contents#
* table of contents
{:toc}
## Idea
The Wagner-Preston theorem is a generalization of [[Cayley's theorem]] to [[inverse semigroups]]. The same idea, when sets have additional local structure, is responsible for the appearance of [[pseudogroup]]s of symmetries in geometry.
## Statement
Every [[inverse semigroup]] has an injective homomorphism into the inverse semigroup of partial bijections of some set.
## Literature
* V. V. Wagner, _Generalised groups_. Proceedings of the USSR Academy of Sciences (in Russian) 84 (1952), 1119–1122.
* V. V. Wagner, _The theory of generalised heaps and generalised groups_, Matematicheskii Sbornik. Novaya Seriya (in Russian). 32 (74) (1953): 545–632.
* G. B. Preston, _Representation of inverse semi-groups_, J. London Math. Soc. 29 (1954) 411-419
* [planetmath](https://www.planetmath.org/WagnerPrestonRepresentationTheorem)
* wikipedia: [inverse semigroup](https://en.wikipedia.org/wiki/Inverse_semigroup)
category: algebra |
Waldhausen (infinity,1)-category | https://ncatlab.org/nlab/source/Waldhausen+%28infinity%2C1%29-category |
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###Context###
#### $(\infty,1)$-Category theory
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[[!include quasi-category theory contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The analogue of [[Waldhausen category]] for [[(infinity,1)-categories]].
## Examples
* Any ordinary [[Waldhausen category]] is a [[Waldhausen (infinity,1)-category]]. See ([Barwick 12](#Barwick12), Example 2.12).
* Any [[stable (infinity,1)-category]] has a canonical Waldhausen structure. See ([Barwick 12](#Barwick12), Example 2.11).
## Related entries
* [[algebraic K-theory]]
## References
* {#Barwick12} [[Clark Barwick]], _On the algebraic K-theory of higher categories_, [arXiv:1204.3607](http://arxiv.org/abs/1204.3607)
[[!redirects Waldhausen (infinity,1)-categories]] |
Waldhausen category | https://ncatlab.org/nlab/source/Waldhausen+category |
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###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A _Waldhausen category_ $C'$ is a [[homotopical category]] equipped with a bit of extra structure that allows us to consider it as a presentation (via [[simplicial localization]]) of an [[(infinity,1)-category]] $C$ such that the extra structure allows us to conveniently compute the [[K-theory]] [[Grothendieck group]] $\mathbf{K}(C)$ of $C$.
Notably a Waldhausen category provides the notion of co[[fibration sequence]]s, which are crucial structures controlling $\mathbf{K}(C)$. Dual to the discussion at [[homotopy limit]] and [[homotopy pullback]], ordinary [[pushout]]s in Waldhausen categories of the form
$$
\array{
A &\hookrightarrow& B
\\
\downarrow && \downarrow
\\
0 &\to& B//A
}
$$
with $A \hookrightarrow B$ a special morphism called a Waldhausen cofibration compute _homotopy pushout_s and hence coexact sequences in the corresponding [[stable (infinity,1)-category]].
Using this, the [[Waldhausen S-construction]] on $C'$ is an algorithm for computing the [[K-theory]] spectrum of $C$.
## Definition
Waldhausen in his work in [[K-theory]] introduced the notion of a category with cofibrations and weak equivalences, nowadays known as _Waldhausen category_. As the original name suggests, this is a category $C$ with zero object $0$, equipped with a choice of two classes of maps $\mathrm{cof}$ of the cofibrations and $w.e.$ of weak equivalences such that
* (C1) all isomorphisms are cofibrations
* (C2) there is a zero object $0$ and for any object $a$ the unique morphism $0\to a$ is a cofibration
* (C3) if $a\hookrightarrow b$ is a cofibration and $a\to c$ any morphism then the pushout $c\to b\cup_a c$ is a cofibration
* (W1) all isomorphisms are weak equivalences
* (W2) weak equivalences are closed under composition (make a subcategory)
* (W3) "glueing for weak equivalences": Given any
commutative diagram of the form
$$\array{
D &\leftarrow& A &\hookrightarrow &B\\
\downarrow^\sim&& \downarrow^\sim &&\downarrow^\sim\\
D' &\leftarrow &A' &\hookrightarrow &B'
}$$
in which the vertical arrows are weak equivalences and right horizontal maps cofibrations, the induced map
$B\cup_A D\hookrightarrow B'\cup_{A'} D'$
is a weak equivalence.
The axioms imply that for any cofibration $A\hookrightarrow B$ there is a cofibration sequence $A\hookrightarrow B\to B/A$ where $B/A$ is the choice of the cokernel $B\cup_A 0$.
Given a Waldhausen category $C$ whose weak equivalence classes from a set, one defines $K_0(C)$ as an abelian group whose elements are the weak equivalence classes modulo the relation $[A]+[B/A]=[B]$ for any cofibration sequence $A\hookrightarrow B\to B/A$.
Waldhausen then devises the so called S-construction $C\mapsto S_\bullet C$ from Waldhausen categories to simplicial categories with cofibrations and weak equivalences (hence one can iterate the construction producing multisimplicial categories).
The [[K-theory space]] of a Waldhausen construction is
given by formula $\Omega\mathrm{hocolim}_{\Delta^{\mathrm{op}}}([n]\mapsto N_\bullet(w.e.(S_n C)))$, where $\Omega$ is the loop space functor, $N$ is the simplicial [[nerve]], w.e. takes the (simplicial) subcategory of weak equivalence and $[n]\in\Delta$. This construction can be improved (using iterated [[Waldhausen S-construction]]) to the [[K-theory]] $\Omega$-[[spectrum]] of $C$; the K-theory space will be just the one-space of the K-theory spectrum.
Then the K-groups are the [[homotopy group]]s of the K-theory space.
## Remarks
* The axioms of a Waldhausen category $C$ are very similar to the axioms of a [[category of fibrant objects]] on the [[opposite category]] $C^{op}$ in which the initial object is also terminal. One difference is that the weak equivalences in a Waldhausen category are not required to satisfy [[2-out-of-3]]. For example, Waldhausen gives an example of a Waldhausen category where the weak equivalences are [[simple homotopy equivalences]]. Another difference is in axiom W3, whose analog in a [[category of fibrant objects]] is the axioms that every object has a [[path object]]. It still follows that one has fibration sequences in a [[category of fibrant objects]].
## Examples
### Waldhausen category of a small abelian category
For $C$ a [[small category|small]] [[abelian category]] the [[category of chain complexes|category of bounded chain complexes]] $Ch^b(C)$ becomes a Waldhausen category by taking
* a [[weak equivalence]] is a [[quasi-isomorphism]] of chain complexes;
* a cofibration $f : A_\bullet \to X_\bullet$ is a chain morphism that is a [[monomorphism]] in $C$ in each degree $f_n : A_n \to X_n$.
### Waldhausen category of a small exact category
For $C$ just a [[Quillen exact category]] with ambient [[abelian category]] $\hat C$ there is an analogous, slightly more sophisticated construction of a Waldhausen category structure on $Ch^b(C)$:
* weak equivalences are the morphisms that are [[quasi-isomorphism]]s when regarded as morphisms in $\hat C$;
* the cofibrations are the degreewise _admissible morphisms_, i.e. those morphisms $A \to X$ such that the pushout $A \to X \to A/X$ computed in the ambient [[abelian category]] $\hat C$ is in $C$.
## Related concepts
* [[category with weak equivalences]]
* [[category of fibrant objects]], [[category of cofibrant objects]]
* [[model category]]
## References
Waldhausen categories are discussed with an eye towards their application in the computation of [[Grothendieck group]]s in [chapter 2](http://www.math.rutgers.edu/~weibel/Kbook/Kbook.II.pdf) of
* [[Charles Weibel]], _The K-book: An introduction to algebraic K-theory_ ([web](http://www.math.rutgers.edu/~weibel/Kbook.html))
Section 1 of
* [[R. W. Thomason]], Thomas Trobaugh, _Higher algebraic K-theory of schemes and of derived categories_, _The Grothendieck Festschrift_, 1990, 247-435.([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/tt.pdf))
[[!redirects Waldhausen categories]] |
Waldhausen K-theory of a dg-category | https://ncatlab.org/nlab/source/Waldhausen+K-theory+of+a+dg-category | +-- {: .rightHandSide}
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#### Homological algebra
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[[!include homological algebra - contents]]
=--
#### Stable Homotopy theory
+-- {: .hide}
[[!include stable homotopy theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The **Waldhausen K-theory** of a [[dg-category]] is defined as the [[Waldhausen K-theory]] of a [[Waldhausen category]] associated to it. It is an _additive invariant_ in the sense of [[noncommutative motives]]. There is a [[non-connective algebraic K-theory|nonconnective]] variant studied by [[Marco Schlichting]], which is a _localizing invariant_ (again in the sense of [[noncommutative motives]]).
On the other hand, one could define the K-theory of a [[pretriangulated dg-category]] as the [[algebraic K-theory]] of its [[dg-nerve]], which is a [[stable infinity-category]]. This should coincide with the Waldhausen K-theory (presumably).
## Definition
Let $A$ be a [[dg-category]]. Consider the [[triangulated category]] $D(A)$ of [[dg-presheaves]] on $A$, and let $perf(A) \subset D(A)$ denote its [[full subcategory]] of [[compact objects]]. There is the structure of a [[Waldhausen category]] on $perf(A)$ where the [[weak equivalences]] are objectwise [[quasi-isomorphisms]] and the [[cofibrations]] are degreewise [[split monomorphisms]]. The **Waldhausen K-theory** of $A$ is the [[Waldhausen K-theory]] of $perf(A)$ with this Waldhausen structure.
## References
* [[Bernhard Keller]], _On differential graded categories_, International Congress of Mathematicians. Vol. II, 151--190, Eur. Math. Soc., Zürich, 2006, [pdf](https://www.imj-prg.fr/~bernhard.keller/publ/dgcatX.pdf).
* [[Marco Schlichting]], _Negative K-theory of derived categories_, Math. Z. 253 (2006), no. 1, 97 - 134, [pdf](http://homepages.warwick.ac.uk/~masiap/research/frob5.pdf).
* [[Goncalo Tabuada]], _Higher K-theory via universal invariants_, Duke Mathematical Journal, 145 (2008), no. 1, 121-206, [arXiv](http://arxiv.org/abs/0706.2420).
|
Waldhausen S-construction | https://ncatlab.org/nlab/source/Waldhausen+S-construction |
#Contents#
* table of contents
{:toc}
## Idea ##
The Waldhausen $S_\bullet$-construction is a procedure that produces [[algebraic K-theory]] (as an [[infinite loop space]] or [[connective spectrum]]) from a category or [[(infinity,1)-category]] equipped with a [[Waldhausen structure]].
The construction can take as input any of the following:
* [[Waldhausen category]]
* [[Waldhausen (infinity,1)-category]]
* [[stable (infinity,1)-category]]
* [[stable derivator]]
* ...
## Definition ##
### For Waldhausen categories
Recall from the definition at [[K-theory]] that the K-theory spectrum $K(C)$ of the [[(∞,1)-category]] $C$ is the diagonal simplicial set on the [[bisimplicial set]] $Core(Func(\Delta^n,C))$ of sequences of morphisms in $C$ and equivalences between these (the [[core]] of the [[Segal space]] induced by $C$).
The Waldhausen S-construction mimics precisely this: for $C'$ a [[Waldhausen category]] for every integer $n$ define a simplicial set $S_n C'$ to be the [[nerve]] of the category whose
* objects are sequences $0 \hookrightarrow A_{0,1} \hookrightarrow \cdots \hookrightarrow A_{0,n}$ of Waldhausen cofibrations;
* together with choices of quotients $A_{i j} = A_{0, j}/ A_{0,i}$, i.e. cofibration sequences $A_{0,i} \to A_{0,j} \to A_{i j}$
* morphisms are collections of morphisms $\{A_{i,j} \to B_{i,j}\}$ that commute with all diagrams in sight.
Then one finds that the [[nerve and realization|realization]] of the bisimplicial set $S_\bullet C'$ with respect to one variable is itself naturally a topological [[Waldhausen category]]. Therefore the above construction can be repeated to yield a sequence of topological categories $S^{(n)}_\bullet C'$. The corresponding sequence of thick [[geometric realization|topological realization]]s is a [[spectrum]]
$$
\mathbf{K}(C)_n = |S^{(n)}_\bullet C'|
$$
this is the **S-construction** of the **Waldhausen [[K-theory]] spectrum** of $C'$.
(... roughly at least, need to polish this, see link below meanwhile...)
## Related
* [[algebraic K-theory]]
* [[Quillen Q-construction]]
* [[K-theory of a stable (infinity,1)-category]]
* [[K-theory of a Waldhausen (infinity,1)-category]]
* [[categorified Dold-Kan correspondence]]
## References ##
### For Waldhausen categories
* {#Waldhausen85} [[F. Waldhausen]], _Algebraic K-theory of spaces_, Alg. and Geo. Top., Springer Lect. Notes Math. 1126 (1985), 318-419, [pdf](http://www.maths.ed.ac.uk/~aar/surgery/rutgers/wald.pdf).
The Waldhausen S-construction is recalled for instance in section 1 of
* {#ThomasonTrobaugh90} [[R. W. Thomason]], Thomas Trobaugh, _Higher algebraic K-theory of schemes and of derived categories_, _The Grothendieck Festschrift_, 1990, 247-435.
or in section 1 of
* Paul D. Mitchener, _Symmetric Waldhausen K-theory spectra of topological categories_ ([pdf](http://bib.mathematics.dk/unzip.php?filename=DMF-2001-05-001-v1.pdf.gz))
### For Waldhausen (infinity,1)-categories and stable (infinity,1)-categories
* [[C. Barwick]], _On the algebraic K-theory of higher categories_, [arXiv:1204.3607](http://arxiv.org/abs/1204.3607).
* [[Andrew J. Blumberg]], [[David Gepner]], [[Goncalo Tabuada]], _A universal characterization of higher algebraic K-theory_, [arXiv:1001.2282](http://arxiv.org/abs/1001.2282v4).
### Other
A combinatorial construction of symmetries due to Nadler has a relation to the S-construction in a special case:
* [[David Nadler]], _Cyclic symmetries of $A_n$-quiver representations_, [arxiv/1306.0070](http://arxiv.org/abs/1306.0070)
> This short note contains a combinatorial construction of symmetries arising in symplectic geometry (partially wrapped or infinitesimal Fukaya categories), algebraic geometry (derived categories of singularities), and K-theory (Waldhausen's S-construction). Our specific motivation (in the spirit of expectations of Kontsevich, and to be taken up in general elsewhere) is a combinatorial construction of quantizations of Lagrangian skeleta (equivalent to microlocal sheaves in their many guises). We explain here the one dimensional case of ribbon graphs where the main result of this paper gives an immediate solution.
Interpretation in terms of a [[categorified Dold-Kan correspondence]] is discussed in
* {#Dyckerhoff17} [[Tobias Dyckerhoff]], _A categorified Dold-Kan correspondence_ ([arXiv:1710.08356](https://arxiv.org/abs/1710.08356)) |
Walecka model | https://ncatlab.org/nlab/source/Walecka+model |
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### Context
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _Walecka model_ (also: _QHD-I model_) is an [[effective field theory]] related to [[chiral perturbation theory]], that describes the residual [[strong nuclear force]] between [[baryons]] and specifically between [[nucleons]], via exchange of [[sigma-mesons]] and [[omega-mesons]], with the [[nucleons]] appearing as explicit [[effective field theory|effective]] [[field (physics)|fields]] (as opposed to emergent [[Skyrmion]] fields), as is more generally the case in [[baryon chiral perturbation theory]]. Some authors use the term _[[quantum hadrodynamics]]_ specifically for the Walecka model of [[nuclear physics]].
The inclusion also of [[pions]] and of [[rho-mesons]] into the Walecka model came to be known as _[[quantum hadrodynamics]]_, specifically _QHD-II_. See there for more.
## Related concepts
[[!include effective field theories of nuclear physics -- contents]]
## References
[[!include quantum hadrodynamics -- references]]
[[!include baryon chiral perturbation theory -- references]]
[[!redirects baryon chiral perturbation theory]]
|
Walker coordinates | https://ncatlab.org/nlab/source/Walker+coordinates |
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###Context###
#### Riemannian geometry
+--{: .hide}
[[!include Riemannian geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Adapted [[coordinates]] for [[smooth manifolds]] with [[special holonomy]] in which the [[Riemannian metric]] takes a nice form.
## Related concepts
* [[Riemann normal coordinates]]
## References
The original article is
* A. G. Walker, _Canonical form for a Riemannian space with a parallel field of null planes_, Quart. J. Math., Oxford Ser. (2) 1 (1950), 69–79. MR MR0035085(11,688d)
Review includes
* [[Helga Baum]], _Holonomy groups of Lorentzian manifolds -- A status report_, 2011 ([pdf](http://www.mathematik.hu-berlin.de/~baum/publikationen-fr/Publikationen-ps-dvi/Baum-Holonomy-report-final.pdf))
[[!redirects Walker coordinate chart]]
[[!redirects Walker coordinate charts]] |
Walker-Wang model | https://ncatlab.org/nlab/source/Walker-Wang+model |
+-- {: .rightHandSide}
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### Context
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Topological physics
+--{: .hide}
[[!include topological physics -- contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _Walker-Wang model_ is a [[model (in theoretical physics)]] which is a [[4d TQFT]] analog of the [[Levin-Wen model]] [[3d TQFT]]. It is the [[Hamiltonian]] formulation of the TQFT whose [[partition function]] formulation is the [[Crane-Yetter model]]. It is speculated to have some application to [[topological insulators]] in [[solid state physics]].
## References
* [[Kevin Walker]], [[Zhenghan Wang]], _(3+1)-TQFTs and Topological Insulators_, Frontiers of Physics volume 7, pages 150–159 (2012) ([arXiv:1104.2632](http://arxiv.org/abs/1104.2632), [doi:10.1007/s11467-011-0194-z](https://doi.org/10.1007/s11467-011-0194-z))
Discussion of [[defect QFT|line defects]] and the [[loop braid group]] statistics, with application to [[topological phases of matter]] using [[higher gauge theory]]/[[higher parallel transport]]:
* C. W. von Keyserlingk, F. J. Burnell, Steven H. Simon, *Three-dimensional topological lattice models with surface anyons*, Phys. Rev. B 87, 045107 ([arXiv:1208.5128] (http://arxiv.org/abs/1208.5128), [doi:10.1103/PhysRevB.87.045107](https://doi.org/10.1103/PhysRevB.87.045107))
* Alex Bullivant, Marcos Calcada, Zoltán Kádár, [[João Faria Martins]], Paul Martin, *Higher lattices, discrete two-dimensional holonomy and topological phases in (3+1) D with higher gauge symmetry*, Reviews in Mathematical Physics, Vol. 32, No. 04, 2050011 (2020) ([arXi:1702.00868](https://arxiv.org/abs/1702.00868))
* AtMa P.O. Chan, Peng Ye, Shinsei Ryu, *Braiding with Borromean Rings in (3+1)-Dimensional Spacetime*, Phys. Rev. Lett. 121, 061601 (2018) [[arXiv:1703.01926](https://arxiv.org/abs/1703.01926)]
* Tian Lan, [[Liang Kong]], [[Xiao-Gang Wen]], *A classification of 3+1D bosonic topological orders (I): the case when point-like excitations are all bosons*, Phys. Rev. X 8, 021074 (2018) ([arXiv:1704.04221](https://arxiv.org/abs/1704.04221))
* [[Clement Delcamp]], *Excitation basis for (3+1)d topological phases*, Journal of High Energy Physics **2017** (2017) 128 ([arXiv:1709.04924](https://arxiv.org/abs/1709.04924) <a href="https://doi.org/10.1007/JHEP12(2017)128">doi:10.1007/JHEP12(2017)128</a>)
* Tian Lan, [[Xiao-Gang Wen]], *A classification of 3+1D bosonic topological orders (II): the case when some point-like excitations are fermions*, Phys. Rev. X 9, 021005 (2019) ([arXiv:1801.08530](https://arxiv.org/abs/1801.08530))
* Alex Bullivant, [[João Faria Martins]], Paul Martin, *Representations of the Loop Braid Group and Aharonov-Bohm like effects in discrete (3+1)-dimensional higher gauge theory*, Advances in Theoretical and Mathematical Physics Volume 23 (2019) Number 7 ([arXiv:1807.09551](https://arxiv.org/abs/1807.09551))
* Qing-Rui Wang, Meng Cheng, Chenjie Wang, Zheng-Cheng Gu, *Topological Quantum Field Theory for Abelian Topological Phases and Loop Braiding Statistics in (3+1)-Dimensions*, Phys. Rev. B 99, 235137 (2019) ([arXiv:1810.13428](https://arxiv.org/abs/1810.13428))
* [[Clement Delcamp]], Apoorv Tiwari, *On 2-form gauge models of topological phases*, JHEP 05 (2019) 064 ([arXiv:1901.02249](https://arxiv.org/abs/1901.02249))
With application to [[quantum computation]]:
* Sam Roberts, Stephen D. Bartlett, *Symmetry-protected self-correcting quantum memories*, Phys. Rev. X 10, 031041 (2020) ([arXiv:1805.01474](https://arxiv.org/abs/1805.01474))
* Sam Roberts and Stephen D. Bartlett, *Symmetry-Protected Self-Correcting Quantum Memories*, Phys. Rev. X 10, 031041 2020 ([doi:10.1103/PhysRevX.10.031041](https://doi.org/10.1103/PhysRevX.10.031041))
* Sam Roberts, Dominic J. Williamson, *3-Fermion topological quantum computation* ([arXiv:2011.04693](https://arxiv.org/abs/2011.04693))
* Charles Stahl, Rahul Nandkishore, *Symmetry protected self correcting quantum memory in three space dimensions* ([arXiv:2103.08622](https://arxiv.org/abs/2103.08622))
[[!redirects Walker-Wang models]] |
walking 2-isomorphism with trivial boundary | https://ncatlab.org/nlab/source/walking+2-isomorphism+with+trivial+boundary | \tableofcontents
\section{Idea}
The _walking 2-isomorphism with trivial boundary_ is, roughly speaking, the minimal [[2-category]] which contains a [[2-isomorphism]] between identity [[1-morphism|1-arrows]]. It is in fact a [[2-groupoid]], and a model of the [[homotopy type]] of the 2-truncation of [[2-sphere]].
It is an example of a [[walking structure]], and can be compared for example with the [[walking 2-isomorphism]].
\section{Definition and elementary observations}
\begin{defn} Let $F$ be the free strict 2-category on the 2-truncated [[reflexive globular set]] with exactly one object $\bullet$, no non-identity 1-arrows, a 2-arrow $\iota: id(\bullet) \rightarrow id(\bullet)$, and a 2-arrow $\iota^{-1}: id(\bullet) \rightarrow id(\bullet)$. The _walking 2-isomorphism with trivial boundary_ is the strict 2-category obtained as the quotient of $F$ by the equivalence relation on 2-arrows generated by forcing the equations $\iota \circ \iota^{-1} = id$ and $\iota^{-1} \circ \iota = id$ to hold. \end{defn}
\begin{rmk} \label{RemarkCharacterisationOf2Arrows} There are exactly $\mathbb{Z}$ 2-arrows $\id\left( \bullet \right) \rightarrow id\left( \bullet \right)$, namely one for each possible string of compositions of $\iota$ and $\iota^{-1}$, taking into account (strict) associativity. Here $\mathbb{Z}$ is of course the [[integer|integers]]. This amounts to a computation of $\pi_{2}\left(S^{2}\right)$, the second homotopy group of the [[2-sphere]]. \end{rmk}
\begin{rmk} Let $\cdot$ denote horizontal composition. By the interchange law, we have that
$$
\begin{aligned}
\left( \iota^{-1} \cdot \iota \right) \circ \iota^{-1} &= \left( \iota^{-1} \cdot \iota \right) \circ \left( id \cdot \iota^{-1} \right) \\
&= \left(\iota^{-1} \circ id \right) \cdot \left( \iota \circ \iota^{-1} \right) \\
&= \iota^{-1} \circ id \\
&= \iota^{-1}.
\end{aligned}
$$
The only possibility, given Remark \ref{RemarkCharacterisationOf2Arrows}, is then that $\iota^{-1} \cdot \iota = id$.
An entirely analogous argument demonstrates that $\iota \circ \iota^{-1} = id$. Thus horizontal composition in the walking 2-isomorphism with trivial boundary is trivial.
\end{rmk}
\section{Representing of 2-isomorphisms with trivial boundary}
\begin{prpn} Let $\mathcal{I}$ denote the walking 2-isomorphism with trivial boundary. Let $\mathcal{A}$ be a [[2-category]], and let $\phi$ be a 2-isomorphism $id(a) \rightarrow id(a)$ in $\mathcal{A}$, for some object $a$ of $\mathcal{A}$. Then there is a unique [[2-functor|functor]] $\mathcal{I} \rightarrow \mathcal{A}$ such that $\iota$ maps to $\phi$. \end{prpn}
\begin{proof} Immediate from the definitions. \end{proof} |
walking adjoint equivalence | https://ncatlab.org/nlab/source/walking+adjoint+equivalence | \tableofcontents
\section{Idea}
The _walking adjoint equivalence_ (as in "[[walking structure]]") or _free-standing adjoint equivalence_ is the [[2-category]] (in fact a [[(2,1)-category]]) which 'represents' [[adjoint equivalence|adjoint equivalences]] in a 2-category. It is a [[categorification]] of the [[free-standing adjoint isomorphism]].
\section{Definition}
\begin{defn} The _free-standing adjoint equivalence_ is the unique (up to [[isomorphism]]) [[2-category]] $\mathcal{AE}$ with exactly two [[objects]] $0$ and $1$, [[1-morphisms]] freely [[generators and relations|generated]] by an arrow $i: 0 \rightarrow 1$, and an arrow $j: 1 \rightarrow 0$, and [[2-morphisms]] generated by [[2-isomorphisms]] $\iota_{0}: j \circ i \cong id(0)$ and $\iota_{1}: i \circ j \cong id(1)$, subject to the equalities
$$id(i) = (\iota_1 i)\circ (i \iota_0^{-1}), id(j) =(\iota_0^{-1} j) \circ (j \iota_1)$$
\end{defn}
\begin{rmk} The [[1-morphism|arrow]] $i: 0 \rightarrow 1$ is an [[adjoint equivalence]].\end{rmk}
\begin{rmk} The free-standing adjoint equivalence is a [[(2,1)-category]]. \end{rmk}
\section{Representing of adjoint equivalences}
$\mathcal{AE}$ is the model for all adjoint equivalences in all 2-categories. In other words, any adjoint equivalence in a 2-category $\mathcal{A}$ is just a [[2-functor]] from $\mathcal{AE}$:
\begin{prpn} Let $\mathcal{C}$ be a 2-category (weak or strict). Let $f$ be a 1-arrow of $\mathcal{C}$ which is part of an [[adjoint equivalence]]. Then there is a [[2-functor]] $F: \mathcal{AE} \rightarrow \mathcal{C}$ such that the arrow $i:0 \rightarrow 1$ of $\mathcal{I}$ maps under $F$ to $f$. \end{prpn}
\begin{proof} Immediate from the definitions. \end{proof}
\section{Related entries}
* [[Adj]]
* [[walking equivalence]]
[[!redirects walking adjoint equivalences]]
[[!redirects free-standing adjoint equivalence]]
[[!redirects free-standing adjoint equivalences]] |
walking equivalence | https://ncatlab.org/nlab/source/walking+equivalence | \tableofcontents
\section{Idea}
The _walking equivalence_ (as in "[[walking structure]]") or _free-standing equivalence_ is the [[2-category]] (in fact a [[(2,1)-category]]) which 'represents' [[equivalence|equivalences]] in a 2-category. It is a [[categorification]] of the [[free-standing isomorphism]], though not the only one: the [[walking adjoint equivalence]] is another.
Roughly speaking, it is the minimal 2-category which contains a 1-arrow $f$ with an inverse-up-to-isomorphism $g$, that is to say, which contains in addition to $g$ a [[2-isomorphism]] between $g f$ and the identity, and a 2-isomorphism between $f g$ and the identity.
The _walking semi-strict equivalence_ is the same except that either $f g$ or $g f$ is required to be equal on the nose to the identity.
\section{Definitions}
\begin{defn} \label{DefinitionFreeStandingEquivalence} Let $F_{\leq 1}$ be the [[free category]] on the [[directed graph]] with exactly two objects $0$ and $1$, an arrow $i: 0 \rightarrow 1$, and an arrow $i^{-1}: 1 \rightarrow 0$. Let $F$ be the free strict 2-category on the 2-truncated [[reflexive globular set]] whose 1-truncation is the underlying reflexive directed graph of $F_{\leq 1}$, and which in addition has a 2-arrow $\iota_{0}: i^{-1} \circ i \rightarrow id(0)$, a 2-arrow $\iota_{0}^{-1}: id(0) \rightarrow i^{-1} \circ i$, a 2-arrow $\iota_{1}: i \circ i^{-1} \rightarrow id(1)$, and a 2-arrow $\iota_{1}^{-1}: i \circ i^{-1} \rightarrow id(1)$.
The _free-standing equivalence_ is the quotient $\mathcal{E}$ of $F$ by the relation on 2-arrows generated by forcing the equations $\iota_{0}^{-1} \circ \iota_{0} = id$, $\iota_{0} \circ \iota_{0}^{-1} = id$, $\iota_{1}^{-1} \circ \iota_{1} = id$, and $\iota_{1} \circ \iota_{1}^{-1} = id$ to hold. \end{defn}
\begin{rmk} The [[1-morphism|arrow]] $i: 0 \rightarrow 1$ is an [[equivalence]], whose inverse-up-to-isomorphism is the arrow $i^{-1}:1 \rightarrow 0$. \end{rmk}
\begin{rmk} The [[2-morphism|2-arrows]] $\iota_{0}$ and $\iota_{1}$ are [[2-isomorphism|2-isomorphisms]]. \end{rmk}
\begin{rmk} The free-standing equivalence is a [[(2,1)-category]], that is, all its 2-morphisms are invertible. \end{rmk}
\begin{defn} Let $F_{\leq 1}$ be as in Definition \ref{DefinitionFreeStandingEquivalence}. Let $F$ be the free strict 2-category on the 2-truncated [[reflexive globular set]] whose 1-truncation is the underlying reflexive directed graph of $F_{\leq 1}$, and which in addition has a 2-arrow $\iota_{1}: i \circ i^{-1} \rightarrow id(1)$, and a 2-arrow $\iota_{1}^{-1}: i \circ i^{-1} \rightarrow id(1)$.
The _free-standing semi-strict equivalence_ is the quotient $\mathcal{E}_{semi}$ of $F$ by the relation on 1-arrows which forces that $i^{-1} \circ i = id(0)$, and by the relation on 2-arrows which forces that $\iota_{1}^{-1} \circ \iota_{1} = id$, and $\iota_{1} \circ \iota_{1}^{-1} = id$.
\end{defn}
\begin{rmk} The free-standing semi-strict equivalence is the quotient of the free-standing equivalence by the relation on 1-arrows which identifies $i^{-1} \circ i$ and $id(0)$, and which identifies $\iota_{0}$ and $\iota^{-1}$ with $id\left(id(0)\right)$. \end{rmk}
\section{Representing of equivalences}
The 2-category $\mathcal{E}$ is the model for all equivalences in all 2-categories. In other words, any equivalence in a 2-category $\mathcal{A}$ is just a [[2-functor]] from $\mathcal{E}$:
\begin{prpn} Let $\mathcal{A}$ be a 2-category (weak or strict). Let $\mathcal{E}$ denote the free-standing equivalence. Let $f$ be a 1-arrow of $\mathcal{A}$ which is an [[equivalence]], the equivalence being exhibited by a 1-arrow $f^{-1}$, a 2-isomorphism $\phi_{0}: f^{-1} \circ f \rightarrow id$, and a 2-isomorphism $\phi_{1}: f \circ f^{-1} \rightarrow id$. Then there is a unique [[2-functor]] $F: \mathcal{E} \rightarrow \mathcal{A}$ such that the arrow $i:0 \rightarrow 1$ of $\mathcal{I}$ maps under $F$ to $f$, such that the arrow $i^{-1}: 1 \rightarrow 0$ of $\mathcal{I}$ maps under $F$ to $f^{-1}$, such that $\iota_{0}$ maps under $F$ to $\phi_{0}$, and such that $\iota_{1}$ maps under $F$ to $\phi_{1}$. \end{prpn}
\begin{proof} Immediate from the definitions. \end{proof}
\begin{prpn} Let $\mathcal{A}$ be a 2-category (weak or strict). Let $\mathcal{E}_{semi}$ denote the free-standing semi-strict equivalence. Let $f$ be a 1-arrow of $\mathcal{A}$ which is a [[semi-strict equivalence]], the equivalence being exhibited by a 1-arrow $f^{-1}$ and a 2-isomorphism $\phi_{1}: f \circ f^{-1} \rightarrow id$. Then there is a unique [[2-functor]] $F: \mathcal{E}_{semi} \rightarrow \mathcal{A}$ such that the arrow $i:0 \rightarrow 1$ of $\mathcal{I}$ maps under $F$ to $f$, and such that $\iota_{1}$ maps under $F$ to $\phi$. \end{prpn}
\begin{proof} Immediate from the definitions. \end{proof}
\section{Structured interval}
Let $\mathcal{E}_{semi}$ denote the free-standing semi-strict equivalence. We shall view it as an [[interval object]] equipped with all the structures required for Corollary XV.6 and Corollary XV.7 of [Williamson2011](#Williamson2011).
Throughout, we shall denote the category of strict 2-categories by [[2Cat]], and denote the final object of [[2Cat]] by $1$.
\begin{notn} We denote by $i_{0}$ (resp. $i_{1}$) the [[2-functor|functor]] $1 \rightarrow \mathcal{E}_{semi}$ which picks out the object $0$ (resp. $1$) of $\mathcal{E}_{semi}$. \end{notn}
\begin{notn} \label{NotationContractionStructure} We denote by $p$ the canonical functor $\mathcal{E}_{semi} \rightarrow 1$. It is immediate that it defines a _contraction structure_ on $\left(\mathcal{E}_{semi}, i_{0}, i_{1} \right)$ in the sense of VI.6 of [Williamson2011](#Williamson2011). \end{notn}
\begin{notn} We denote by $v$ the functor $\mathcal{E}_{semi} \rightarrow \mathcal{E}_{semi}$ determined by $i \mapsto i^{-1}$, $i^{-1} \mapsto i$, $\iota_{1}, \iota_{1}^{-1} \mapsto id\left(id(0)\right)$. It defines an _involution structure_ on $\left(\mathcal{E}_{semi}, i_{0}, i_{1} \right)$ in the sense of VI.10 of [Williamson2011](#Williamson2011).
Since $1$ is a final object of [[2Cat]], it is immediate that $v$ is compatible with the contraction structure $p$ of Notation \ref{NotationContractionStructure} in the sense of VI.12 of [Williamson2011](Williamson2011).
\end{notn}
\begin{notn} Let
\begin{tikzcd}
1 \ar[r, "i_{0}"] \ar[d, swap, "i_{1}"] & \mathcal{E}_{semi} \ar[d, "r_{0}"] \\
\mathcal{E}_{semi} \ar[r, swap, "r_{1}"] & S
\end{tikzcd}
be a [[pushout|co-cartesian square]] in [[2Cat]].
Explicitly, let $F_{\leq 1}$ be the [[free category]] on the [[directed graph]] with exactly three objects $0$, $1$, and $2$, and with non-identity arrows $r_1: 0 \rightarrow 1$, $r_0: 1 \rightarrow 2$, and $r_1^{-1}: 1 \rightarrow 0$, $r_0^{-1}: 2 \rightarrow 1$. Let $F$ be the free strict 2-category on the 2-truncated [[reflexive globular set]] whose 1-truncation is the underlying reflexive directed graph of $F_{\leq 1}$, and which in addition has 2-arrows $\iota_{1}, \iota_{1}^{-1}: r_1 \circ r_1^{-1} \rightarrow id(1)$ and $\kappa_{2}, \kappa_{2}^{-1}: r_0 \circ r_0^{-1} \rightarrow id(2)$.Then $S$ can be taken to be the quotient of $F$ by the relation on 2-arrows which forces $\iota_{0}$ to be a [[2-isomorphism]] with inverse $\iota_{0}^{-1}$, and similarly for $\kappa_{2}$. The functors $r_0$ and $r_1$ can be taken to be functors picking out the equivalences in $S$ of the same name.
There is a functor $s: \mathcal{E}_{semi} \rightarrow S$ which picks out the semi-strict equivalence in $S$ given by $r_{0} \circ r_{1}$. It is immediately checked that $\left( S, r_0, r_1, s \right)$ defines a _subdivision structure_ with respect to $\left( \mathcal{E}_{semi}, i_0, i_1 \right)$ in the sense of VI.14 of [Williamson2011](Williamson2011). Moreover, since $1$ is a final object, it is immediate that this subdivision structure is compatible with the contraction structure of Notation \ref{NotationContractionStructure} in the sense of VI.18 of [Williamson2011](Williamson2011).
The functor $q_{l}$ of VI.34 in [Williamson2011](Williamson2011) is in this case the functor $S \rightarrow \mathcal{E}_{semi}$ which is determined by $r_{0} \mapsto i$, $r_0^{-1} \mapsto i^{-1}$, $r_1 \mapsto id(0)$, $r^{-1} \mapsto id(0)$, $\iota_1 \mapsto id$, and $\kappa_2 \mapsto \iota_1$. We see then that $\left( \mathcal{E}_{semi}, i_0, i_1, p, S, r_0, r_1, s \right)$ has strictness of left identities in the sense of VI.34 of [Williamson2011](Williamson2011).
It is similarly the case that $\left( \mathcal{E}_{semi}, i_0, i_1, p, S, r_0, r_1, s \right)$ has strictness of right identities in the sense of VI.34 of [Williamson2011](Williamson2011).
\end{notn}
\begin{rmk} Explicitly, $\mathcal{E}_{semi} \times \mathcal{E}_{semi}$ can be described as follows. Let $F_{\leq 1}$ be the [[free category]] on the [[directed graph]] with objects $(0,0)$, $(0,1)$, $(1,0)$, $(1,1)$, and with arrows $(0, i)$, $(0, i^{-1})$, $(i, 0)$, $(i^{-1}, 0)$, $(1, i)$, $(1, i^{-1})$, $(i, 1)$, and $(i^{-1}, 1)$. Let $F$ be the free strict 2-category on the 2-truncated [[reflexive globular set]] whose 1-truncation is the underlying reflexive directed graph of $F_{\leq 1}$, and which in addition has 2-arrows $(0, \iota_{1}): (0, i \circ i^{-1}) \rightarrow id(0, 1)$, $(1, \iota_{1}): (1, i \circ i^{-1}) \rightarrow id(1, 1)$, $(\iota_{1}, 0): (i \circ i^{-1}, 0) \rightarrow id(1, 0)$, and $(\iota_{1}, 1): (i \circ i^{-1}, 1) \rightarrow id(1, 1)$.
Then $\mathcal{E}_{semi} \times \mathcal{E}_{semi}$ is the quotient of $F$ by the relation on 1-arrows which forces $(0, i^{-1} \circ i)$ to be equal to $(0,0)$, and similarly for $(i^{-1} \circ i, 0)$, $(1, i^{-1} \circ i)$, and $(i^{-1} \circ i, 1)$; and on 2-arrows which forces $(0, \iota_{1})$, $(\iota_{1}, 0)$, $(1, \iota_1)$ and $(\iota_1, 1)$ to be 2-isomorphisms.
\end{rmk}
\begin{notn} Let $\Gamma_{ul}$ be the functor $\mathcal{E}_{semi} \times \mathcal{E}_{semi} \rightarrow \mathcal{E}_{semi}$ determined by
$$
\begin{aligned}
(0, i), (i,0) &\mapsto i, \\
(0, i^{-1}), (i^{-1}, 0) &\mapsto i^{-1}, \\
(1, i), (1, i^{-1}), (i, 1), (i^{-1}, 1) &\mapsto id(1), \\
(0, \iota_{1}), (\iota_{1}, 0) &\mapsto \iota_{1}, \\
(1, \iota_{1}), (\iota_{1}, 1) &\mapsto id\left(id(1)\right).
\end{aligned}
$$
Then $\Gamma_{ul}$ defines an _upper left connection structure_ with respect to $\left( \mathcal{E}_{semi}, i_0, i_1, p \right)$ in the sense of VI.22 of [Williamson2011](Williamson2011). \end{notn}
\begin{notn} Let $\Gamma_{lr}$ be the functor $\mathcal{E}_{semi} \times \mathcal{E}_{semi} \rightarrow \mathcal{E}_{semi}$ determined by
$$
\begin{aligned}
(0, i), (i,0), (0, i^{-1}), (i^{-1}, 0) &\mapsto id(0), \\
(1, i), (i, 1) &\mapsto i, \\
(1, i^{-1}), (i^{-1}, 1) &\mapsto i^{-1}, \\
(0, \iota_{1}), (\iota_{1}, 0) &\mapsto id\left(id(0)\right), \\
(1, \iota_{1}), (\iota_{1}, 1) &\mapsto \iota_{1}.
\end{aligned}
$$
Then $\Gamma_{lr}$ defines an _lower right connection structure_ with respect to $\left( \mathcal{E}_{semi}, i_0, i_1, p \right)$ in the sense of VI.24 of [Williamson2011](Williamson2011).
Since $1$ is a final object of [[2Cat]], it is immediate that $\Gamma_{lr}$ is compatible with $p$ in the sense of VI.26 of [Williamson2011](Williamson2011).
\end{notn}
\begin{notn} Let $\Gamma_{ur}$ be the functor $\mathcal{E}_{semi} \times \mathcal{E}_{semi} \rightarrow \mathcal{E}_{semi}$ determined by
$$
\begin{aligned}
(i,0) &\mapsto i, \\
(i^{-1}, 0) &\mapsto i^{-1}, \\
(0, i), (0, i^{-1}) &\mapsto id(0), \\
(1, i) &\mapsto i^{-1}, \\
(1, i^{-1}) &\mapsto i, \\
(i, 1), (i^{-1}, 1) &\mapsto id(0), \\
(\iota_{1}, 0) &\mapsto \iota_{1}, \\
(0, \iota_{1}) &\mapsto id\left(id(0)\right), \\
(1, \iota_{1}) &\mapsto id\left(id(0)\right), \\
(\iota, 1) &\mapsto id\left(id(0)\right).
\end{aligned}
$$
Then $\Gamma_{ur}$ defines an _upper right connection structure_ with respect to $\left( \mathcal{E}_{semi}, i_0, i_1, p, v \right)$ in the sense of VI.29 of [Williamson2011](Williamson2011).
The functor $x \circ \left( \mathcal{E}_{semi} \times s \right) : \mathcal{E}_{semi} \times \mathcal{E}_{semi} \rightarrow \mathcal{E}_{semi}$ of VI.32 in [Williamson2011](Williamson2011) is determined by
$$
\begin{aligned}
(i, 0), (i,1) &\mapsto i, \\
i^{-1}, 0), (i^{-1}, 1) &\mapsto i^{-1}, \\
(\iota_{1}, 0) &\mapsto \iota_{1}, \\
(\iota_{1}, 1) &\mapsto \iota_{1}, \\
(0, i), (0,i^{-1}) &\mapsto id(0), \\
(1, i), (1, i^{-1}) &\mapsto id(1), \\
(0, \iota_{1}) &\mapsto id\left(id(0)\right), \\
(1, \iota_{1}) &\mapsto id\left(id(1)\right).
\end{aligned}
$$
The key observation here is that $(1, i), (1, i^{-1}) \mapsto id(1)$, which relies on the fact that $i^{-1} \circ i = id(0)$ in $\mathcal{E}_{semi}$.
We deduce that $x \circ \left( \mathcal{E}_{semi} \times s \right) = \mathcal{E}_{semi} \times p$, and thus that $\Gamma_{lr}$ and $\Gamma_{ur}$ are compatible with $\left(S, r_0, r_1, s \right)$ in the sense of VI.32 of [Williamson2011](Williamson2011).
\end{notn}
\begin{rmk} It follows from the above, Corollary XV.6, and Corollary XV.7 of [Williamson2011](Williamson2011) that there is a model structure on [[2Cat]], the category of strict 2-categories, whose fibrations and cofibrations are 'Hurewicz' fibrations and cofibrations respectively with respect to the [[interval object]] $\left( \mathcal{E}_{semi}, i_0, i_1 \right)$. See [[canonical model structure on 2-categories]] for more. \end{rmk}
\begin{rmk} All of the structures in this section have an analogue for $\mathcal{E}$, the free-standing equivalence, as well. All the required compatibilities hold except for one: $\Gamma_{lr}$ and $\Gamma_{ur}$ are not compatible with $\left(S, r_0, r_1, s \right)$. This is exactly where the semi-strictness of $\mathcal{E}_{semi}$ is needed. \end{rmk}
\section{References}
* {#Williamson2011} [[Richard Williamson]], _Cylindrical model structures_, DPhil thesis, University of Oxford, 2011. [author's webpage](http://rwilliamson-mathematics.info/cylindrical_model_structures/cylindrical_model_structures.html) [arXiv:1304.0867](https://arxiv.org/abs/1304.0867)
[[!redirects walking equivalences]]
[[!redirects walking semi-strict equivalence]]
[[!redirects walking semi-strict equivalences]]
[[!redirects free-standing equivalence]]
[[!redirects free-standing equivalences]]
[[!redirects free-standing semi-strict equivalence]]
[[!redirects free-standing semi-strict equivalences]] |
walking isomorphism | https://ncatlab.org/nlab/source/walking+isomorphism | \tableofcontents
\section{Idea}
The _[[interval object]] in [[groupoids]]_ or _free-standing isomorphism_ or, if you insist, the "[[walking structure|walking]] isomorphism", is the [[groupoid]]:
\[
I_\sim
\;\coloneqq\;
\big\{
a \overset{\;\; \sim \;\;}{\leftrightarrows} b
\big\}
\]
with precisely two [[objects]] and (besides their [[identity morphisms]]) one [[isomorphism]] and its [[inverse morphism]] connecting them.
This is such that for $\mathcal{C}$ any [[category]], a [[functor]] of the form
$$
I_\sim \longrightarrow \mathcal{C}
$$
is precisely a choice of [[isomorphism]] in $\mathcal{C}$.
The interval groupoid can be [[categorified]] in a number of ways: to the [[walking equivalence]], to the [[walking adjoint equivalence]], or to the [[walking 2-isomorphism]]. Related, though not quite a categorification in one of the usual senses, is the [[walking 2-isomorphism with trivial boundary]].
\section{Definition}
\begin{defn}
The _free-standing isomorphism_ is the category generated by two objects $0$ and $1$, one arrow $0 \rightarrow 1$, one arrow $1 \rightarrow 0$, and the equations:
\[
0 \to 1 \to 0 = \mathrm{id}_0, \quad 1 \to 0 \to 1 = \mathrm{id}_1,
\]
which make $0 \to 1$ an [[isomorphism]].
\end{defn}
\begin{rmk} The arrow $0 \rightarrow 1$ is an isomorphism, whose inverse is the arrow $1 \rightarrow 0$. \end{rmk}
\begin{rmk} The free-standing isomorphism is a groupoid. \end{rmk}
\begin{rmk} The free-standing isomorphism can also be described as the [[free groupoid]] on the [[interval category]], that is to say, the walking arrow. Because it is an [[interval object]] for [[Cat]] and [[Grpd]], it is also known as the _interval groupoid_.\end{rmk}
\section{Representing of isomorphisms}
\begin{prpn}
Let $\mathcal{A}$ be a category. Let $\mathcal{I}$ denote the free-standing isomorphism.
Evaluation at the arrow $0\to 1$ establishes a natural bijective
correspondence between functors $\mathcal{I}\to\mathcal{A}$
and isomorphisms in $\mathcal{A}$.
Thus, for any isomorphism $f$ of $\mathcal{A}$ there is a unique [[functor]] $F: \mathcal{I} \rightarrow \mathcal{A}$ such that the arrow $0 \rightarrow 1$ of $\mathcal{I}$ maps under $F$ to $f$.
\end{prpn}
\begin{proof} Immediate from the definitions. \end{proof}
\section{Properties}
\begin{prpn} $\{ 1 \} \subseteq \mathcal{I}$ is the full subcategory classifier of the 1-category $Cat$. That is, for any small category $\mathcal{A}$, there is a bijective correspondence between full subcategories of $\mathcal{A}$ and functors $\mathcal{A} \to \mathcal{I}$, with the reverse direction given by taking the pullback of the inclusion $\{ 1 \} \subseteq \mathcal{I}$.
\end{prpn}
\begin{proof} Functors into $\mathcal{I}$ are uniquely determined by functions on the sets of objects. $\{ 1 \}$ is a full subcategory of $\mathcal{I}$, and any pullback of a full subcategory can be given as a full subcategory.
\end{proof}
In fact, this generalizes. If $X$ is a simplicial set, say that a _full subspace_ of $X$ is a subsimplicial set $S \subseteq X$ with the property there is some subset $S_0 \subseteq X_0$ such that $S$ contains exactly the simplices whose vertices are all contained in $S_0$.
\begin{prpn} The nerve $N(\mathcal{I})$ is the full subspace classifier for $sSet$,
and thus $N(\mathcal{I})$ represents the subpresheaf $FullSub \subseteq Sub$ of full subspaces. \end{prpn}
\begin{proof} This can be determined from the explicit description of $N_n(\mathcal{I}) \cong \{ 0, 1 \}^{n+1}$ given by listing the vertices of a path. However, it's more informative to observe that $N(\mathcal{I}) = indisc(\{ 0, 1 \})$, where $indisc$ is the _indiscrete space_ functor, which is the direct image part of the geometric embedding $Set \subseteq sSet$ whose inverse image is $X \mapsto X_0$.
\end{proof}
\begin{rmk} This also implies $N(\mathcal{I})$ is the full subcategory classifier of $qCat$, the 1-category of [[quasi-category|quasi-categories]], since those are given by full subspaces of simplicial sets. \end{rmk}
## References
A generalization of the notion of the interval groupoid to [[simplicial groupoids]] is considered in
* {#DwyerKan84} [[William Dwyer]], [[Daniel Kan]], §2.8 of: *Homotopy theory and simplicial groupoids*, Indagationes Mathematicae (Proceedings) **87** 4 (1984) 379-385 [<a href="https://doi.org/10.1016/1385-7258(84)90038-6">doi:10.1016/1385-7258(84)90038-6</a>]
and plays a key role in the discussion of the [[model structure on simplicial groupoids]], see [there](model+structure+on+simplicial+groupoids#SimplicialIntervalGroupoid).
[[!redirects free-standing isomorphism]]
[[!redirects interval groupoid]] |
walking structure | https://ncatlab.org/nlab/source/walking+structure |
# Walking structures
* tables of contents
{: toc}
## Idea
Around the nLab and elsewhere, one occasionally sees an expression "the walking _____" where the blank is some mathematical concept. This is a colloquial way of referring to an archetypal model of the concept or type, and usually refers to a *[[free construction|free]]* or *[[initial object|initial]]* form of such a kind of structure.
Pronunciation is just as in 'John is a walking almanac' or 'Eugene Levy is a walking pair of eyebrows'. The term is [believed](http://golem.ph.utexas.edu/category/2010/01/f_and_the_shibboleth.html#c031066) to have been introduced by [[James Dolan]].
Sometimes, "the free-living _____" or "the free-standing _____" is used instead; this terminology is probably much older.
## Definition
The idea is probably easier to apprehend through examples rather than through a formal definition, but for the record:
If $X$ is a type of [[mathematical structure]] that can be defined in a [[category]], [[higher category]], or category with some [[extra structure]], then the *walking X* refers to the *[[free construction|free]] category (resp. higher category, category with suitable structure) containing an $X$.
More precisely, if $StructCat$ denotes some (higher) category of categories with an appropriate type of structure, then the **walking X** is an object $[X] \in StructCat$ together with a [[natural transformation|natural]] [[equivalence]]
$$
StructCat([X],C)
\;\simeq\;
\big\{Xs \; in \; C\}
$$
between the [[hom-set]]/[[hom-category|category]]/[[hom-space|space]] from $[X]$ to $C$, for any $C\in StructCat$, and the set/category/space of all Xs in $C$.
+-- {: .num_remark}
###### Remark
In other words, the structured category $[X]$ equipped with its canonical type $X$ is [[initial object|initial]] among such structured categories that come equipped with such types $X$. A fancier expression is that $[X]$ 'coclassifies' such types: this is analogous to how a [[classifying space]] $B G$ for a [[topological group]] $G$ classifies $G$-[[bundles]], in that every $G$-bundle $p: E \to X$ over a suitable space $X$ has a classifying map $\chi_p: X \to B G$ (unique up to homotopy) such that pulling back the canonical $G$-bundle type $\pi: E G \to B G$ along $\chi_p$ reproduces the type $p$. Only here we say 'coclassifies' (as for example in this [comment](http://nforum.mathforge.org/discussion/1529/walking-structures/#Item_5)), since here we instead "push forward" the canonical type $X$ of $[X]$ along a structured-category morphism $[X] \to C$ to obtain a given type of $C$.
=--
## Examples
* The [[interval category]] is the *walking morphism*.
* The [[interval groupoid]] is the [[walking isomorphism]].
* [[equivalence|Equivalences]] in a [[2-category]] are represented by the [[walking equivalence]].
* The [[2-category]] [[Adj]] is the [[walking adjunction]].
* The [[walking 2-isomorphism with trivial boundary]] is a 2-groupoid model for the 2-truncation of the [[2-sphere]].
* The augmented/algebraist's [[simplex category]] is "the *walking [[monoid]]*" (in a [[monoidal category]]). That is to say: the simplex category is initial (in a 2-categorical sense) among monoidal categories equipped with a monoid object. Intuitively, it is [[the]] monoidal category that results if all one need be told about it is that it has a monoid object -- all the morphisms of the category are obtainable from the monoid structure by applying the operations of a monoidal category, and they are subject to no further relations beyond those implied by the monoid axioms.
* Similarly, the [[Lawvere theory]] of groups can be described as "the *walking [[group object|group]]*" (qua [[cartesian monoidal categories]]). This gives a good intuitive description: this Lawvere theory can be understood as the category (with finite products) that results if all one need be told about it is that it has a [[group object]]; the rest of the structure of this category can be deduced from this one fact.
The last two examples indicate the need for a little care: the [[doctrine]] or type of structured category in which the '$X$' of "the walking $X$" lives should either be specified or clear from context. For example, if one simply says "the
walking monoid", this means the simplex category if the surrounding context is the doctrine of monoidal categories -- but means something else (the Lawvere theory of monoids) if the ambient context is the doctrine of categories with finite products, and it means the category opposite to that of finitely presentable monoids if we are in the doctrine of [[finitely complete categories]].
If the doctrine is not specified, then a reasonable default is a 'minimal' doctrine in which the concept makes sense; for example, to make sense of monoids, one doesn't need more than monoidal categories. See also [[microcosm principle]].
Thus, more generally,
* The [[syntactic category]] of a [[theory]] $T$ in some [[doctrine]] $D$ is the "walking $T$-model" (in a $D$-category). In particular, the [[classifying topos]] of a [[geometric theory]] $T$ is "the *walking $T$-model*" *qua* [[Grothendieck toposes]] (where the morphisms are the left-adjoint parts of [[geometric morphisms]]).
## Relation to initial objects
The walking X is, of course, not the same as the [[initial object|initial]] X.
Consider for example the case when X is a pointed monoid (a monoid equipped with an element). The initial pointed monoid (in $Set$) is the [[natural numbers]] equipped with $1\in \mathbb{N}$. Whereas the walking pointed monoid (qua category with finite products, say) is a category $C_M$ with finite products containing a [[monoid object]] $M\in C_M$ and an "[[generalized element|element]]" $e:1\to M$. They have different types and different universal properties: $\mathbb{N}$ has a universal property mapping into other pointed monoids in $Set$, while $C_M$ has a universal property mapping into other categories with products equipped with pointed monoids.
Nevertheless, the first sits inside the second! Specifically, for any category $C$ with products, the "underlying set" functor $C(1,-):C\to Set$ is product-preserving and hence carries monoid objects to monoid objects, and in the case of the walking pointed monoid we have $C_M(1,M) \cong \mathbb{N}$.
This is true rather generally: *the initial X is the underlying X of the walking X*. One general theorem of this sort is the following:
+--{: .un_theorem}
###### Theorem
Let $K$ be a [[2-category]] containing an object $S$, and suppose that:
1. The domain projection $K\sslash S \to K$ from the [[lax slice 2-category]] has a section. Explicitly, for every object $X$ we have a map $s_X : X\to S$ and for every morphism $f:X\to Y$ we have a 2-cell $\sigma_f : s_X \to s_Y f$, such that for every 2-cell $\alpha :f\to g$ we have $\sigma_g . s_Y\alpha = \sigma_f$, and these vary functorially.
2. We have $s_S \cong 1_S$, and for any $X$ the composite $s_X \xrightarrow{\sigma_{s_X}} s_S s_X \cong s_X$ is the identity.
Then for any $X$, the morphism $s_X:X\to S$ is the initial object of the hom-category $K(X,S)$.
=--
+--{: .proof}
###### Proof
We will use the characterization of initial objects via [cones over the identity](/nlab/show/initial+object#ConesOverTheIdentity). Thus, we must construct a natural transformation from the constant functor $\Delta_{s_X} : K(X,S) \to K(X,S)$ to the identity, which is the identity at $s_X$. However, given any $f:X\to S$, we have the 2-cell $s_X \xrightarrow{\sigma_f} s_S f \cong f$, and the assumption $\sigma_g . s_Y\alpha = \sigma_f$ makes this a natural transformation; and the final assumption says exactly that this is the identity at $s_X$.
=--
To see how this theorem implies that the initial X is the underlying X of the walking X, consider again the case of pointed monoids. Let $K$ be the 2-category of categories with products and product-preserving functors, let $S=Set$, and let $s_C : C \to Set$ be $C(1,-)$. The hypotheses are easy to verify; thus the theorem tells us that $C(1,-)$ is the initial functor $C\to Set$ for any $C$.
Now take $C$ to be the walking pointed monoid $C_M$ above. Then its universal property tells us that functors $F:C_M\to Set$ are equivalent to pointed monoids $F(M)$ in $Set$; so we see that $C(1,M)$ is the initial pointed monoid in $Set$, i.e. $\mathbb{N}$.
A similar argument applies whenever we have a "$Set$-like" object of a 2-category with "underlying set" morphisms to it. For instance, in place of categories with finite products we could consider categories with finite limits. However, in other cases, such as monoidal categories, there is a disconnect: the natural "underlying set" functor for a monoidal category $C$ is $C(I,-)$, where $I$ is the unit object; but in general this is only *lax* monoidal, whereas the universal property of a "walking X qua monoidal category" is relative to *strong* monoidal functors.
It does often happen that this disconnect can be bridged. For instance, suppose X is something that can be defined in any [[multicategory]] (like a pointed monoid). Then we can apply the above argument to the 2-category of multicategories, as the walking X qua multicategory has a universal property relative to functors of multicategories, even though such functors correspond to *lax* functors between monoidal categories. It follows that the underlying set functor $C_M(;-) : C_M\to Set$ of the walking X qua multicategory corresponds to the initial X in $Set$.
Moreover, we can deduce from this that the underlying pointed monoid of the walking X qua monoidal category is also the initial X in $Set$. This is because the walking X qua monoidal category is the monoidal category freely generated by the walking X qua multicategory, and a multicategory embeds fully-faithfully in the monoidal category that it freely generates. In terms of the [[type theory]] that generates walking Xs, the latter fact can be seen as a sort of [[canonicity]] for the tensor-product constructor.
Similar arguments apply in other cases. For instance, in the 2-category of [[double categories]] we can take $S$ to be the double category [[Span]]; this has the same problem as that of monoidal categories, but we can solve it similarly by considering [[virtual double categories]] instead.
On the other hand, there are also cases where this argument does not apply. For instance, X could be something that can be defined in a monoidal category but *not* in a multicategory, such as a [[Frobenius monoid]]. In this case the claim doesn't even make sense: the functor $C(I,-)$, being only lax monoidal in general, need not preserve Frobenius monoids. It seems that in most cases where an "underlying set" functor $C(I,-)$ preserves Xs, there is a kind of [[generalized multicategory]] in which Xs can be defined and this argument carried through, but I do not know of a general theory suggesting this.
## References
* A [Café post](http://golem.ph.utexas.edu/category/2010/01/f_and_the_shibboleth.html) essentially about walking objects (among other things), including a [comment](http://golem.ph.utexas.edu/category/2010/01/f_and_the_shibboleth.html#c031066) that explains the terminology.
The terminology "free-living" appears in:
* {#FoltzLairKelly80} [[François Foltz]], [[Christian Lair]], [[G. M. Kelly]], *Algebraic categories with few monoidal biclosed structures or none*, Journal of Pure and Applied Algebra **17** 2 (1980) 171-177 [[pdf](https://core.ac.uk/download/pdf/82322397.pdf), <a href="https://doi.org/10.1016/0022-4049(80)90082-1">doi:10.1016/0022-4049(80)90082-1</a>]
[[!redirects walking]]
[[!redirects walking structure]]
[[!redirects walking structures]]
[[!redirects walking object]]
[[!redirects walking objects]]
[[!redirects free-living]]
|
wall crossing | https://ncatlab.org/nlab/source/wall+crossing |
> This entry is about discontinuities in parameter dependence of (often asymptotic) solutions of [[differential equations]] and similar phenomena (notably in the context of [[BPS states]] formalized via [[Bridgeland stability conditions]]) with stability parameters (and their stability slopes) in [[algebraic geometry]] which are often interpreted as crossing the walls of marginal stability in [[physics]]. For the different notions of the same name in [[Morse theory]] see at _[[Cerf wall crossing]]_ and for the (Weyl chamber wall) crossing functors in representation theory see [[wall crossing functor]].
# Contents
* table of contents
{: toc}
## Overview
In the study of [[solitons]], one may try a [[WKB approximation|WKB-style approximation]] to a nonlinear [[wave equation]] (see also _[[eikonal equation]]_, _[[Maslov index]]_, etc.). Stokes has observed that when trying to connect the local solutions, one has discontinuities along certain lines, now called **Stokes lines**. This is called the [[Stokes phenomenon]]. Similar issues appear in study of [[isomonodromic deformation]]s of nonlinear [[ODEs]] in the [[complex plane]], which is also relevant in [[soliton]] theory, and [[integrable systems]], and special functions like [[Painlevé transcendent]]s. This has especially been studied by the Kyoto school (Jimbo, Miwa, Sato, [[Masaki Kashiwara|Kashiwara]] etc.), including the use of [[D-modules]] and [[microlocal analysis]]. The Kyoto school found a connection of isomonodromic theory to what is called [[holonomic quantum field]]s.
The solutions of meromorphic differential equations can be expressed in terms of [[meromorphic connection]]s. Then the slopes related to the solutions can be viewed as features of particular objects in a category of $D$-[[D-module|modules]]. More generally, slope filtrations are structures which appear in many other additive categories, e.g. in [[Hodge theory]], theory of Dieudonné modules and so on. Many of those are related to the stability of the objects, which is important in the construction of [[moduli spaces]].
In [[algebraic geometry]], [[Grothendieck]] has shown how to correctly define and construct some fundamental [[moduli spaces]], like Hilbert schemes and Quot schemes for [[coherent sheaves]]. The work has been continued by [[David Mumford]] who geometrized classical invariant theory into [[geometric invariant theory]]. To keep moduli under control, one needs to impose stability conditions on objects and also look at classes with some fixed data: those involve slopes or equivalently phase factors. This is thus similar to the phases of eikonal in the case of Stokes phenomenon.
Cf. also Harder-Narasimhan filtration, [[Castelnuovo-Mumford regularity]] (cf. [wikipedia](http://en.wikipedia.org/wiki/Castelnuovo%E2%80%93Mumford_regularity)) etc.
### In supersymmetric field theory
In [[super Yang-Mills theory]] the number of [[BPS states]] is locally constant as a function of the parameters of the theory, but it may jump at certain "walls" in the [[moduli spaces]] of parameters. The precise behaviour of the BPS states as one crosses these walls is studied as "wall crossing phenomena".
Another example are the [[moduli spaces]] of [[Higgs bundles]], studied by [[Carlos Simpson]] and others, which have special cases with interpretations both in geometry and in the gauge theory (instantons). It appears that sometimes they can be linked to the geometric picture. [[Riemann-Hilbert correspondence]], spectral transform and similar correspondences again play a major role.
Surely, one often works at the derived level. An adaptation of the notion of stability into the setup of [[triangulated categories]] has been introduced by Bridgeland. Bridgeland stability for the derived categories of (boundary conditions of) D-branes (B-model) are relevant for string theory.
## Related concepts
* [[BPS state]], [[D-module]], [[cluster algebra]], [[quiver]], [[representation theory]], [[Donaldson-Thomas invariant]].
[[!include field theory with boundaries and defects - table]]
## References
### Introductions and lectures
* [[Sergio Cecotti]], _Trieste lectures on wall-crossing invariants_ (2010) [[pdf](http://people.sissa.it/~cecotti/ictptext.pdf)&rbrack
* [[Greg Moore]], _PiTP Lectures on BPS states and wall-crossing in $d = 4$, $\mathcal{N} = 2$ theories_ ([pdf](http://www.physics.rutgers.edu/~gmoore/PiTP_July26_2010.pdf))
* {#Dimofte10} [[Tudor Dimofte]], _Refined wall crossing_ ([pdf](http://thesis.library.caltech.edu/5808/4/TD_part1.pdf)), part I of _Refined BPS invariants, Chern-Simons theory, and the quantum dilogarithm_, 2010 ([pdf](http://thesis.library.caltech.edu/5808/1/DimofteTDofficial.pdf), [web](http://thesis.library.caltech.edu/5808/))
### Original articles
#### General
* [[Maxim Kontsevich]], [[Yan Soibelman]], _Stability structures, motivic Donaldson-Thomas invariants and cluster transformations_, [arXiv:0811.2435](http://arxiv.org/abs/0811.2435); _Motivic Donaldson-Thomas invariants: summary of results_, [0910.4315](http://arxiv.org/abs/0910.4315); _Wall-crossing structures in Donaldson-Thomas invariants, integrable systems and Mirror Symmetry_, [arxiv/1303.3253](http://arxiv.org/abs/1303.3253)
* Arend Bayer, [[Yuri Manin|Yuri I. Manin]], _Stability conditions, wall-crossing and weighted Gromov-Witten invariants_, [math.AG/0607580](http://arxiv.org/abs/math.AG/0607580), Mosc. Math. J. __9__ (1), 2009.
* [[Mina Aganagic]], Hirosi Ooguri, [[Cumrun Vafa]], Masahito Yamazaki, _Wall crossing and M-theory_, [arxiv/0908.1194](http://arxiv.org/abs/0908.1194)
* [[Mina Aganagic]], _Wall crossing, quivers and crystals_, [arxiv/1006.2113](http://arxiv.org/abs/1006.2113)
* [[Gregory Moore]], _PiTP lectures on BPS states and wall-crossing in d = 4, N = 2 theories_ [pdf](http://www.physics.rutgers.edu/grad/695/PiTP-LectureNotes.pdf)
* S. Cecotti, [[Cumrun Vafa]], _BPS wall crossing and topological strings_,
[arXiv/0910.2615](http://arxiv.org/abs/0910.2615)
* [[Davide Gaiotto]], [[Greg Moore]], [[Andrew Neitzke]], _Wall-crossing, Hitchin systems, and the WKB approximation, Advances in Mathematics __234__ (2013) 239--403 ([arXiv:0907.3987](https://arxiv.org/abs/0907.3987) [doi](https://doi.org/10.1016/j.aim.2012.09.027))
* E. Diaconescu, [[Greg Moore]], _Crossing the wall: branes vs. bundles_, [arXiv/0706.3193](http://arxiv.org/abs/0706.3193)
* E. Andriyash, [[Frederik Denef]], [[Daniel Jafferis]], [[Greg Moore|G. W. Moore]], _Wall-crossing from supersymmetric galaxies_, [arxiv/1008.0030](http://arxiv.org/abs/1008.0030)
* Tom Bridgeland, [[Valerio Toledano-Laredo]], _Stability conditions and Stokes factors_, [arxiv/0801.3974](http://arxiv.org/abs/0801.3974)
* M. C. N. Cheng, E. P. Verlinde, _Wall crossing, discrete attractor flow and Borcherds algebra_, SIGMA 4 (2008), 068, 33 pages, [pdf](http://www.emis.de/journals/SIGMA/2008/068/sigma08-068.pdf)
* Masahito Yamazaki, _Crystal melting and wall crossing phenomena_, Ph.D. thesis, [arxiv/1002.1709](http://arxiv.org/abs/1002.1709)
* Michele Cirafici, Annamaria Sinkovics, [[Richard Szabo]], _Instanton counting and wall-crossing for orbifold quivers_, [arxiv/1108.3922](http://arxiv.org/abs/1108.3922)
* H.-Y. Chen, N. Dorey, K. Petunin, _Moduli space and wall-crossing formulae in higher-rank gauge theories_, JHEP 11 (2011) 020, <a href="http://dx.doi.org/10.1007/JHEP11(2011)020">doi</a>; _Wall crossing and instantons in compactified gauge theory_, JHEP 06 (2010) 024 [arXiv:1004.0703](ttp://arxiv.org/abs/1004.0703)
### In supergravity
* {#Denef00} [[Frederik Denef]], _Supergravity flows and D-brane stability_, JHEP 0008:050,2000 ([arXiv:hep-th/0005049](http://arxiv.org/abs/hep-th/0005049))
* [[Frederik Denef]], _Quantum Quivers and Hall/Hole Halos_, JHEP 0210:023,2002 ([arXiv:hep-th/0206072](http://arxiv.org/abs/hep-th/0206072))
* [[Daniel Jafferis]], [[Gregory Moore]], _Wall crossing in local Calabi Yau manifolds_ ([arXiv:0810.4909](http://arxiv.org/abs/0810.4909))
### Conferences and seminars
* (past) [[Kontsevich]] in Aarhus, August 2010, [master class on wall crossing](http://qgm.au.dk/events/show/artikel/masterclass-aug-2010/); we will keep a [[wall crossing in Aarhus|nlab page]] on it
* (past) [Focus Week on New Invariants and Wall Crossing](http://member.ipmu.jp/domenico.orlando/FocusInvariants.html), May 18-22, 2009, Kashiwa Campus of the University of Tokyo
* (past) [Wall-crossing in Mathematics and Physics](http://www.math.uiuc.edu/wallcrossing), May 24-28, 2010,
Department of Mathematics, University of Illinois at Urbana-Champaign
* Description of seminar on stability conditions and Stokes factors in Bonn, [pdf](http://www.math.uni-bonn.de/people/compgeo/Hall.pdf)
Also ([Gaiotto-Moore-Witten 15](#GaiottoMooreWitten15)).
### Categorification
A [[categorification]] of wall crossing formulas to an [[(infinity,2)-category]] of sorts is discussed in
* {#GaiottoMooreWitten15} [[Davide Gaiotto]], [[Gregory Moore]], [[Edward Witten]], _An introduction to the web-based formalism_ ([arXiv.1506.04086](http://arxiv.org/abs/1506.04086))
[[!redirects wall crossing]]
[[!redirects wall-crossing]] |
wall crossing functor | https://ncatlab.org/nlab/source/wall+crossing+functor | Various objects in Lie theory have the combinatorics of their representations described in terms of roots and weights; the structure of Weyl chamber walls is crucial for many criteria in their study. Certain important functors in this representation theory cross the chamber walls. We talk about the wall crossing functors in representation theory.
Symmetries related to Weyl groups, Weyl chambers and chamber walls are involved (what is sometimes also in the other notion of [[wall crossing]] in the BPS setup). A priori wall crossing functors in representation theory (introduced in 1970s by the Moscow school, Gelfand, Bernstein etc.) are about certain functors which in take as input an infinite-dimensional representation, tensor it with finite-dimensional and look for certain pieces in their decomposition. In this study, the position with respect to the chamber walls is crucial.
* A. [[Beilinson]], [[Victor Ginzburg|V. Ginzburg]], _Wall-crossing functors and $D$-modules_, Representation Theory __3__ (electronic), 1--31 (1999) [pdf](http://www.ams.org/ert/1999-003-01/S1088-4165-99-00063-1/S1088-4165-99-00063-1.pdf).
> The composition of the translation functor that sends the category at a regular maximal ideal to the category at a non-regular maximal ideal with the translation functor acting in the opposite direction is called a wall-crossing functor.
* J. Bernstein, S. Gelfand, _Tensor products of finite and infinite dimensional representations of semisimple Lie algebras_, Comp. Math. 41 (1980), 245–285. MR 82c:17003
[[!redirects wall crossing functor]]
[[!redirects wall crossing in representation theory]] |
wall crossing in Aarhus 2010 | https://ncatlab.org/nlab/source/wall+crossing+in+Aarhus+2010 | [[!redirects wall crossing in Aarhus]]
[[Maxim Kontsevich]] taught a master class on [[wall crossing]] in Aarhus, August 16-20, 2010
* ([master class webpage](http://qgm.au.dk/events/show/artikel/masterclass-aug-2010/), [video recording](http://qgm.au.dk/video/mc/wall-crossing/))
Kontsevich spoke about two a priori unrelated theories which lead to the same class of wall-crossing formulae.
> The first one is the theory of (motivic, or refined) [[Donaldson-Thomas invariants]]. It applies to
> 1) [[quivers]] (maybe with potentials), including cluster categories and matrix models,
> 2) moduli of [[coherent sheaves]] on algebraic surfaces, moduli of representations of an arbitrary finitely presented algebra,
> 3) 3-dimensional [[Calabi-Yau categories]].
> One can define in each of these cases appropriate cohomology groups of moduli spaces, carrying a generalization of mixed Hodge structure. The generating series for Serre polynomials splits into an infinite product of quantum dilogarithms, depending on stability structure. Exponents in this expansion are refined Donaldson-Thomas invariants, and they jump in a universal way when we cross a wall in the space of stability structures.
> The second theory is about the behavior of collapsing hyperkahler metrics, and of WKB asymptotics for complex integrable systems. Here one counts holomorphic discs with boundaries on Lagrangian submanifolds.
> Physics partially explains the relation between two theories, via BPS counting and geometry of hypermultiplet moduli spaces.
category: reference
|
Wall's finiteness obstruction | https://ncatlab.org/nlab/source/Wall%27s+finiteness+obstruction |
#Contents#
* table of contents
{:toc}
## Idea
The [[compact object in an (infinity,1)-category|compact objects]] in [[∞Grpd]] are the [[retracts]] of [[finite homotopy types]] ([[finite CW-complexes]]). Not every such retract is itself a finite homotopy type; the vanishing of _Wall's finiteness [[obstruction]] is a necessary and sufficient condition for this to happen.
## References
* [[C. T. C. Wall]], _Finiteness conditions for CW-complexes I_ [pdf](http://math.uchicago.edu/~shmuel/tom-readings/wall%20finiteness%201.pdf)
* [[C. T. C. Wall]], _Finiteness conditions for CW-complexes II_ [pdf](http://www.maths.ed.ac.uk/~aar/papers/findom2.pdf)
[[!redirects Wall finiteness obstruction]]
[[!redirects Wall finiteness]]
[[!redirects Wall finiteness condition]]
[[!redirects Wall's finiteness condition]]
|
Wallman compactification | https://ncatlab.org/nlab/source/Wallman+compactification | __Wallman compactification__ is a particular compactification of $T_1$-topological spaces introduced in
* Henry Wallman, _Lattices and topological spaces_, Annals of Math. __39__:1 (Jan., 1938), pp. 112-126 [jstor](http://www.jstor.org/stable/1968717)
There are recent applications related to topoi and noncommutative geometry
* [[Olivia Caramello]], _Gelfand spectra and Wallman compactifications_, [arxiv/1204.3244](http://arxiv.org/abs/1204.3244); _Dualité de Gelfand et bases de Wallman_, [seminar talk videos](https://sites.google.com/site/logiquecategorique/autres-seminaires/20130404-caramello-gelfand-wallman)
A __Wallman base__ $B$ for a topological space $X$ is a sublattice of of the [[frame]] $Open(X)$ of open sets of $X$ which is a [[base of a topology|base for the topology]] and satisfies the property that for any $U\in B$ and
$x\in U$ there exists $V\in B$ such that $U\cup V = X$ and $x\notin V$.
Standard references are
* P. T. Johnstone, _[[Stone Spaces]]_, Cambridge Studies in Advanced Math. __3__ (1982).
* [eom]: [Wallman compactification](http://www.encyclopediaofmath.org/index.php?title=Wallman_compactification)
* wikipedia: [Wallman compactification](http://en.wikipedia.org/wiki/Wallman_compactification)
category: topology
[[!redirects Wallman base]]
[[!redirects Wallman basis]]
[[!redirects Wallman bases]] |
wallpaper group | https://ncatlab.org/nlab/source/wallpaper+group |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
A _wallpaper group_ is a [[crystallographic group]] ([[space group]]) in [[dimension]] 2, hence a [[subgroup]] $G \subset Iso(\mathbb{R}^2)$ of the [[isometry group]] of the [[Euclidean plane]] such that
1. the part of $G$ inside the [[translation group]] is generated by two [[linearly independent subset|linearly independent]] [[vectors]];
1. the [[point group]] is [[finite group|finite]].
{#Tilley2020Fig310} [Fig. 3.10 in Tilley 2020](#Tilley2020):
<center>
<img src="https://ncatlab.org/nlab/files/SymmetryElementsOfWallpaperGroups.jpg" width="600">
</center>
## Related concepts
* [[flat orbifold]]
* [[finite rotation group]]
* [[tesselation]]
## References
e. g.
* {#Tilley2020} Richard Tilley, Sec. 3.5 in: *Crystals and Crystal Structure*, Wiley (2020) [[ISBN:978-1-119-54838-6](https://www.wiley.com/en-us/Crystals+and+Crystal+Structures,+2nd+Edition-p-9781119548386)]
* [[Patrick Morandi]], _The Classification of Wallpaper Patterns: From Group Cohomology to Escherís Tessellations_ ([pdf](http://sierra.nmsu.edu/morandi/notes/Wallpaper.pdf))
See also
* Wikipedia, _[Wallpaper group](https://en.wikipedia.org/wiki/Wallpaper_group)_
Discussion of [[meromorphic functions]] on the [[complex plane]] invariant under wallpaper groups:
* Richard Chapling, _Invariant Meromorphic Functions on the Wallpaper Groups_ ([arXiv:1608.05677](https://arxiv.org/abs/1608.05677))
[[!redirects wallpaper groups]]
|
Walter D. Neumann | https://ncatlab.org/nlab/source/Walter+D.+Neumann |
* [Wikipedia entry](https://en.wikipedia.org/wiki/Walter_Neumann)
## Selected writings
Introducing the notion of [[cutting and pasting of manifolds]]:
* U. Karras, [[Matthias Kreck]], [[Walter D. Neumann]], E. Ossa, *Cutting and Pasting of Manifolds; SK-Groups*, Publish or Perish (1973) [[pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/skbook.pdf), [[Karras-Kreck-Neumann-Ossa_CutNPaste.pdf:file]]]
* [[Walter D. Neumann]], *Manifold cutting and pasting groups*, Topology **14** 3 (1975) 237-244 [<a href="https://doi.org/10.1016/0040-9383(75)90004-X">doi:10.1016/0040-9383(75)90004-X</a>]
category: people
[[!redirects Walter Neumann]]
[[!redirects Walter David Neumann]]
|
Walter Feit | https://ncatlab.org/nlab/source/Walter+Feit |
* [Wikipedia entry](https://en.wikipedia.org/wiki/Walter_Feit)
## related $n$Lab entries
* [[Feit-Thompson theorem]]
* [[permutation representation]]
category: people |
Walter Felscher | https://ncatlab.org/nlab/source/Walter+Felscher | * [German Wikipedia entry](https://de.wikipedia.org/wiki/Walter_Felscher)
## Related entries
* [[game semantics]]
category: people |
Walter Gordon | https://ncatlab.org/nlab/source/Walter+Gordon | German theoretical physicist, 1893-1939.
* [Wikipedia](http://en.wikipedia.org/wiki/Walter_Gordon_%28physicist%29)
## related $n$Lab entries
* [[Klein-Gordon equation]]
category: people |
Walter Greiner | https://ncatlab.org/nlab/source/Walter+Greiner |
* [Wikipedia entry](https://en.wikipedia.org/wiki/Walter_Greiner) ([de](https://de.wikipedia.org/wiki/Walter_Greiner))
## Selected writings
On [[electrodynamics]]:
* [[Walter Greiner]], *Classical Electrodynamics*, Springer (1998) [[doi:10.1007/978-1-4612-0587-6](https://doi.org/10.1007/978-1-4612-0587-6)]
category: people
|
Walter Gubler | https://ncatlab.org/nlab/source/Walter+Gubler |
* [webpage](http://www.mathematik.uni-regensburg.de/gubler/)
category: people
|
Walter Lewis Baily | https://ncatlab.org/nlab/source/Walter+Lewis+Baily |
* [Wikipedia entry](https://en.wikipedia.org/wiki/Walter_Lewis_Baily,_Jr.)
## Selected writings
On [[orbifolds]] in [[complex geometry]]:
* [[Walter Lewis Baily]], _On the quotient of an analytic manifold by a group of analytic homeomorphisms_, PNAS 40 (9) 804-808 (1954) ([doi:10.1073/pnas.40.9.804](https://doi.org/10.1073/pnas.40.9.804))
* [[Walter Lewis Baily]], _The Decomposition Theorem for V-Manifolds_, American Journal of Mathematics Vol. 78, No. 4 (Oct., 1956), pp. 862-888 ([jstor:2372472](https://www.jstor.org/stable/2372472))
category: people
|
Walter Michaelis | https://ncatlab.org/nlab/source/Walter+Michaelis |
* [Mathematics Genealogy page](http://www.genealogy.math.ndsu.nodak.edu/id.php?id=18941)
## related $n$Lab entries
* [[coalgebra]]
* [[cofree coalgebra]]
category: people |
Walter Neumann | https://ncatlab.org/nlab/source/Walter+Neumann |
* [webpage](http://www.math.columbia.edu/~neumann/)
## related $n$Lab entries
* [[hyperbolic manifold]]
category: people |
Walter Noll | https://ncatlab.org/nlab/source/Walter+Noll |
* [Wikipedia entry](http://en.wikipedia.org/wiki/Walter_Noll)
* [website](http://www.math.cmu.edu/~wn0g/)
category: people
|
Walter Rudin | https://ncatlab.org/nlab/source/Walter+Rudin |
* [Wikipedia entry](https://en.wikipedia.org/wiki/Walter_Rudin)
## Related $n$Lab entries
* [[analysis]]
category: people |
Walter Thirring | https://ncatlab.org/nlab/source/Walter+Thirring |
* [Wikipedia entry](https://en.wikipedia.org/wiki/Walter_Thirring) ([de](https://de.wikipedia.org/wiki/Walter_Thirring))
## Selected writings
On [[mathematical physics]]:
* [[Walter Thirring]], *A Course in Mathematical Physics* -- 1 *Classical Dynamical Systems* and 2 *Classical Field Theory*, Springer (1988, 1992) [[doi:10.1007/978-1-4684-0517-0](https://doi.org/10.1007/978-1-4684-0517-0)]
category: people
|
Walter Tholen | https://ncatlab.org/nlab/source/Walter+Tholen |
* [website](https://tholen.mathstats.yorku.ca/)
## Selected writings
On ([[monadic descent|monadic]]) [[descent]]:
* {#JanelidzeTholen94} [[George Janelidze]], [[Walter Tholen]], *Facets of descent I*, Applied Categorical Structures **2** 3 (1994) 245-281 [[doi:10.1007/BF00878100](https://doi.org/10.1007/BF00878100)]
* {#JanelidzeTholen97} [[George Janelidze]], [[Walter Tholen]], *Facets of descent II*, Applied Categorical Structures **5** 3 (1997) 229-248 [[doi:10.1023/A:1008697013769](https://doi.org/10.1023/A:1008697013769)]
On [[free coproduct completions]]:
* {#HuTholen95} Hongde Hu, [[Walter Tholen]], *Limits in free coproduct completions*, Journal of Pure and Applied Algebra 105 (1995) 277-291 (<a href="https://doi.org/10.1016/0022-4049(94)00153-7">doi:10.1016/0022-4049(94)00153-7</a>, [pdf](https://core.ac.uk/download/pdf/82604415.pdf))
On a generalization of [[coproducts]] to [[colimits]] over certain [[diagrams]] of the shape of [[skeletal groupoids]]:
* [[Hongde Hu]], [[Walter Tholen]], *Quasi-coproducts and accessible categories with wide pullbacks*, Appl Categor Struct **4** (1996) 387–402 [[doi:10.1007/BF00122686](https://doi.org/10.1007/BF00122686)]
See also:
* [[Reinhard Börger]], [[Walter Tholen]], _Strong regular and dense generators_, [[Cahiers de Topologie et Géométrie Différentielle Catégoriques]] **32** 3 (1991) 257-276 [[numdam:CTGDC_1991__32_3_257_0](http://www.numdam.org/item?id=CTGDC_1991__32_3_257_0), [MR1158111](http://www.ams.org/mathscinet-getitem?mr=1158111)]
On (co-)[[reflective factorization systems]]:
* {#RosickyTholen08} [[Jiri Rosicky]], [[Walter Tholen]], _Factorization, Fibration and Torsion_, Journal of Homotopy and Related Structures, **2** 2 (2007) 295-314 [[arXiv:0801.0063](http://arxiv.org/abs/0801.0063), [publisher](http://www.emis.de/journals/JHRS/volumes/2007/n2a14/)]
Presenting a [[pretorsion theory]] on [[Cat]] whose torsion(-free) objects are the [[groupoids]] ([[skeletal categories]], respectively), hence whose "trivial objects" are the [[skeletal groupoids]]:
* [[Francis Borceux]], [[Federico Campanini]], [[Marino Gran]], [[Walter Tholen]], *Groupoids and skeletal categories form a pretorsion theory in $Cat$* [[arXiv:2207.08487](https://arxiv.org/abs/2207.08487)]
## Related entries
* [[duality]]
* [[Gelfand duality]]
category: people
[[!redirects W. Tholen]]
[[!redirects Tholen]] |
Wanda Szmielew | https://ncatlab.org/nlab/source/Wanda+Szmielew |
* [Wikipedia entry](https://de.wikipedia.org/wiki/Wanda_Szmielew)
## related $n$Lab entries
* [[Euclidean geometry]]
category: people |
Ward identity | https://ncatlab.org/nlab/source/Ward+identity |
> under construction
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In [[perturbative quantum field theory]] the _quantum master Ward identity_ (def. \ref{OnRegularPolynomialObservablesMasterWardIdentity} below) expresses the relation between the [[quantum field theory|quantum]] (measured by [[Planck's constant]] $\hbar$) [[interacting field theory|interacting]] (measured by the [[coupling constant]] $g$) [[equations of motion]] to the [[classical field theory|classical]] [[free field]] [[equations of motion]] at $\hbar, g\to 0$ (remark \ref{QuantumMasterEuqationRelatesQuantumInteractingELEquationsToClassicalFreeELEquations} below). As such it generalizes the [[Schwinger-Dyson equation]], to which it reduces for $g = 0$ (example \ref{QuantumMasterEuqationRelatesQuantumInteractingELEquationsToClassicalFreeELEquations} below) as well as the _classical master Ward identity_, which is the case for $\hbar = 0$ (example \ref{MasterWardIdentityClassical} below).
Applied to products of the [[equations of motion]] with any given [[observable]], the master Ward identity becomes a particular _Ward identity_.
This is of interest notably in view of [[Noether's theorem]], which says that every [[infinitesimal symmetry of the Lagrangian]] of, in particular, the given [[free field theory]], corresponds to a [[conserved current]], hence a [[horizontal differential form]] whose [[total spacetime derivative]] vanishes up to a term proportional to the [[equations of motion]]. Under [[transgression of variational differential forms|transgression]] to [[local observables]] this is a relation of the form
$$
div \mathbf{J} = 0 \phantom{AAA} \text{on-shell}
\,,
$$
where "on shell" means up to the ideal generated by the [[classical field theory|classical]] [[free field theory|free]] [[equations of motion]]. Hence for the case of [[local observables]] of the form $div \mathbf{J}$, the quantum Ward identity expresses the possible failure of the original [[conserved current]] to actually be conserved, due to both quantum effects ($\hbar$) and interactions ($g$). This is the form in which Ward identities are usually understood (example \ref{NoetherCurrentConservationQuantumCorrection} below).
In terms of [[BV-BRST formalism]], the master Ward identity is equivalent to the _[[quantum master equation]]_ on [[regular polynomial observables]] ([this prop.](quantum+master+equation#QuantumMasterEquation)).
Neither of these equations is guaranteed to hold for any choice of [[renormalization|("re"-)normalization]]. If a Ward identity is violated by the [[renormalization|("re"-)normalized]] [[perturbative QFT]], specifically if there is no possible choice of [[renormalization|("re"-)normalization]] that preserves it, the one speaks of a _[[quantum anomaly]]_. Specifically if the [[conserved current]] corresponding to a [[gauge symmetry]] is _anomalous_ in this way, one speaks of a _[[gauge anomaly]]_.
## Details
### Before renormalization
{#BeforeRenormalization}
+-- {: .num_defn #OnRegularPolynomialObservablesMasterWardIdentity}
###### Definition
Consider a [[free field theory|free]] [[gauge fixing|gauge fixed]] [[Lagrangian field theory]] $(E_{\text{BV-BRST}}, \mathbf{L}')$ ([this def.](A+first+idea+of+quantum+field+theory#GaugeFixingLagrangianDensity)) with global [[BV-differential]] on [[regular polynomial observables]]
$$
\{-S',(-)\}
\;\colon\;
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
\longrightarrow
PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
$$
([this def.](A+first+idea+of+quantum+field+theory#ComplexBVBRSTGlobal)).
Let moreover
$$
g S_{int}
\;\in\;
PolyObs(E_{\text{BV-BRST}})_{reg}[ [ \hbar , g ] ]
$$
be a [[regular polynomial observable]] (regarded as an [[adiabatic switching|adiabatically switched]] non-point-[[interaction]] [[action functional]]) such that the total action $S' + g S_{int}$ satisfies the [[quantum master equation]] ([this prop.](quantum+master+equation#QuantumMasterEquation)); and write
$$
\mathcal{R}^{-1}(-)
\;\coloneqq\;
\mathcal{S}(g S_{int})^{-1}
\star_H
(\mathcal{S}(g S_{int}) \star_F (-))
$$
for the corresponding [[quantum Møller operator]] ([this def.](quantum+master+equation#MollerOperatorOnRegularPolynomialObservables)).
Then by [this prop.](quantum+master+equation#QuantumMasterEquation) we have
$$
\label{OnRegularObservablesQuantumMasterWardIdentityViaTimeOrdered}
\{-S',(-)\} \circ \mathcal{R}^{-1}
\;=\;
\mathcal{R}^{-1}
\left(
\left\{ -(S' + g S_{int}) \,,\, (-) \right\}_{\mathcal{T}}
-
i \hbar \Delta_{BV}
\right)
$$
This is the _quantum master Ward identity_ on [[regular polynomial observables]], i.e. before [[renormalization]].
=--
([Rejzner 13, (37)](#Rejzner13))
+-- {: .num_remark #QuantumMasterEuqationRelatesQuantumInteractingELEquationsToClassicalFreeELEquations}
###### Remark
**([[quantum master Ward identity]] relates [[quantum field theory|quantum]] [[interacting field theory|interacting field]] [[equation of motion|EOMs]] to [[classical field theory|classical]] [[free field]] [[equation of motion|EOMs]])**
For $A \in PolyObs(E_{\text{BV-BRST}})_{reg}[ [ \hbar, g] ] $ the [[quantum master Ward identity]] on [[regular polynomial observables]] (eq:OnRegularObservablesQuantumMasterWardIdentityViaTimeOrdered) reads
$$
\label{RearrangedMasterQuantumWard}
\mathcal{R}^{-1}
\left(
\left\{ -(S' + g S_{int}) \,,\, A \right\}_{\mathcal{T}}
-
i \hbar \Delta_{BV}(A)
\right)
\;=\;
\{-S', \mathcal{R}^{-1}(A) \}
$$
The term on the right is manifestly in the [[image]] of the global [[BV-differential]] $\{-S',-\}$ of the [[free field theory]] ([this def.](A+first+idea+of+quantum+field+theory#ComplexBVBRSTGlobal)) and hence vanishes when passing to [[on-shell]] observables along the [[isomorphism]] ([this equation](A+first+idea+of+quantum+field+theory#eq:OnShellPolynomialObservablesAsBVCohomology))
$$
\underset{
\text{on-shell}
}{
\underbrace{
PolyObs(E_{\text{BV-BRST}}, \mathbf{L}')
}}
\;\simeq\;
\underset{
\text{off-shell}
}{
\underbrace{
PolyObs(E_{\text{BV-BRST}})_{def(af = 0)}
}}/im(\{-S',-\})
$$
(by [this example](A+first+idea+of+quantum+field+theory#BVDifferentialGlobal)).
Hence
$$
\mathcal{R}^{-1}
\left(
\left\{ -(S' + g S_{int}) \,,\, A \right\}_{\mathcal{T}}
-
i \hbar \Delta_{BV}(A)
\right)
\;=\;
0
\phantom{AAA}
\text{on-shell}
$$
In contrast, the left hand side is the [[interacting field observable]] (via [this def.](S-matrix#MollerOperatorOnRegularPolynomialObservables)) of the sum of the [[time-ordered product|time-ordered]] [[antibracket]] with the [[action functional]] of the [[interacting field theory]] and a quantum correction given by the [[BV-operator]]. If we use the definition of the [[BV-operator]] $\Delta_{BV}$ ([this def.](BV-operator#RearrangedMasterWardWithOnShell)) we may equivalently re-write this as
$$
\label{RearrangedMasterWardWithOnShell}
\mathcal{R}^{-1}
\left(
\left\{ -S' \,,\, A \right\}
+
\left\{ -g S_{int} \,,\, A \right\}_{\mathcal{T}}
\right)
\;=\;
0
\phantom{AAA}
\text{on-shell}
$$
Hence the [[quantum master Ward identity]] expresses a relation between the ideal spanned by the [[classical field theory|classical]] [[free field theory|free field]] [[equations of motion]] and the [[quantum field theory|quantum]] [[interacting field theory|interacting field]] equations of motion.
=--
+-- {: .num_example #SchwingerDysonReductionOfQuantumMasterWardIdentity}
###### Example
**([[free field]]-limit of [[master Ward identity]] is [[Schwinger-Dyson equation]])**
In the [[free field]]-limit $g \to 0$ (noticing that in this limit $\mathcal{R}^{-1} = id$) the [[quantum master Ward identity]] (eq:OnRegularObservablesQuantumMasterWardIdentityViaTimeOrdered) reduces to
$$
\left\{ -S' \,,\, A \right\}_{\mathcal{T}}
-
i \hbar \Delta_{BV}(A)
\;=\;
\{-S', A \}
$$
which is the defining equation for the [[BV-operator]] ([this equation](BV-operator#eq:BVOperatorDefiningRelation)), hence is isomorphic (under $\mathcal{T}$) to the [[Schwinger-Dyson equation]] ([this prop.](BV-operator#DysonSchwinger))
=--
+-- {: .num_example #MasterWardIdentityClassical}
###### Example
**([[classical limit]] of [[quantum master Ward identity]])**
In the [[classical limit]] $\hbar \to 0$ (noticing that the classical limit of $\{-,-\}_{\mathcal{T}}$ is $\{-,-\}$) the [[quantum master Ward identity]] (eq:OnRegularObservablesQuantumMasterWardIdentityViaTimeOrdered) reduces to
$$
\mathcal{R}^{1}
\left(
\left\{ -(S' + g S_{int}) \,,\, A \right\}
\right)
\;=\;
\{-S', \mathcal{R}^{-1}(A) \}
$$
This says that the [[interacting field observable]] corresponding to the global [[antibracket]] with the action functional of the [[interacting field theory]] vanishes on-shell, classically.
Applied to an observable which is [[linear map|linear]] in the [[antifields]]
$$
A
\;=\;
\underset{\Sigma}{\int}
A^a(x)
\mathbf{\Phi}^\ddagger_a(x)
\,
dvol_\Sigma(x)
$$
this yields
$$
\begin{aligned}
0
& =
\{-S', \mathcal{R}^{-1}(A)\}
+
\mathcal{R}^{-1}
\left(
\left\{ -(S' + S_{int}) \,,\, A \right\}_{\mathcal{T}}
\right)
\\
& =
\underset{\Sigma}{\int}
\frac{\delta S'}{\delta \mathbf{\Phi}^a(x)} \mathcal{R}^{-1}(A^a(x))
\, dvol_\Sigma(x)
+
\mathcal{R}^{-1}
\left(
\underset{\Sigma}{\int}
A^a(x) \frac{\delta (S' + S_{int})}{\delta \mathbf{\Phi}^a(x)}
\, dvol_\Sigma(x)
\right)
\end{aligned}
$$
This is the _classical master Ward identity_ according to ([Dütsch-Fredenhagen 02](#DuetschFredenhagen02), [Brennecke-Dütsch 07, (5.5)](#BrennecketDuetsch07)), following ([Dütsch-Boas 02](#DuetschBoas02)).
=--
+-- {: .num_example #NoetherCurrentConservationQuantumCorrection}
###### Example
**(quantum correction to [[Noether's theorem|Noether current]] [[conserved current|conservation]])**
Let $v \in \Gamma^{ev}_\Sigma(T_\Sigma(E_{\text{BRST}}))$ be an [[evolutionary vector field]], which is an [[infinitesimal symmetry of the Lagrangian]] $\mathbf{L}'$, and let $J_{\hat v} \in \Omega^{p,0}_\Sigma(E_{\text{BV-BRST}})$ the corresponding [[conserved current]], by [[Noether's theorem|Noether's theorem I]] ([this prop.](A+first+idea+of+quantum+field+theory#NoethersFirstTheorem)), so that
$$
\begin{aligned}
d J_{\hat v}
& =
\iota_{\hat v} \delta \mathbf{L}'
\\
& =
(v^a dvol_\Sigma) \frac{\delta_{EL} L'}{\delta \phi^a}
\phantom{AAA}
\in
\Omega^{p+1,0}_\Sigma(E_{\text{BV-BRST}})
\end{aligned}
$$
(by [this equation](A+first+idea+of+quantum+field+theory#eq:CurrentNoetherConservation)), where in the second line we just rewrote the expression in components (using [this equation](A+first+idea+of+quantum+field+theory#eq:EulerLagrangeEquationGeneral))
$$
v^a
\,,
\frac{\delta_{EL} L'}{\delta \phi^a}
\;\in \Omega^{0,0}_\Sigma(E_{\text{BV-BRST}})
$$
and re-arranged suggestively.
Then for $a_{sw} \in C^\infty_{cp}(\Sigma)$ any choice of [[bump function]], we obtain the [[local observables]]
$$
\begin{aligned}
A_{sw}
& \coloneqq
\underset{\Sigma}{\int}
\underset{
A^a(x)
}{
\underbrace{
a_{sw}(x)
v^a( \mathbf{\Phi}(x), D\mathbf{\Phi}(x), \cdots )
}
}
\mathbf{\Phi}^\ddagger_a(x)
\,
dvol_\Sigma(x)
\\
& \coloneqq
\tau_\Sigma( a_{sw} v^a \phi^{\ddagger}_a \, dvol_\Sigma)
\end{aligned}
$$
and
$$
\begin{aligned}
(div \mathbf{J})_{sw}
& \coloneqq
\underset{\Sigma}{\int}
\underset{
A^a(x)
}{
\underbrace{
a_{sw}(x)
v^a( \mathbf{\Phi}(x), D\mathbf{\Phi}(x), \cdots )
}
}
\frac{\delta S'}{\delta \mathbf{\Phi}^a(x)}
\,
dvol_\Sigma(x)
\\
& \coloneqq
\tau_\Sigma
\left(
a_{sw} v^a \frac{\delta_{EL} \mathbf{L}'}{\delta \phi^a}
\, dvol_\Sigma
\right)
\end{aligned}
$$
by [[transgression of variational differential forms]].
This is such that
$$
\left\{
-S' , A_{sw}
\right\}
=
(div \mathbf{J})_{sw}
\,.
$$
Hence applied to this choice of local observable $A$, the quantum master Ward identity (eq:RearrangedMasterWardWithOnShell) now says that
$$
\mathcal{R}^{-1}
\left(
{\, \atop \,}
(div \mathbf{J})_{sw}
\right)
\;=\;
\mathcal{R}^{-1}
\left(
\{g S_{int}, A_{sw} \}_{\mathcal{T}}
{\, \atop \,}
\right)
\phantom{AAA}
\text{on-shell}
$$
Hence the [[interacting field observable]]-version $\mathcal{R}^{-1}(div\mathbf{J})$ of $div \mathbf{J}$ need not vanish itself on-shell, instead there may be a correction as shown on the right.
=--
## Examples
* [[chiral anomaly]]
* [[conformal blocks]]
## References
Named after _[[John Clive Ward]]_.
Discussion of the [[master Ward identity]] in the rigorous context of [[relativistic field theory|relativistic]] [[perturbative quantum field theory]] formulated via [[causal perturbation theory]]/[[perturbative AQFT]] is in
* {#DuetschBoas02} [[Michael Dütsch]], F.-M. Boas, _The Master Ward Identity_, Rev. Math. Phys 14, (2002) 977-1049 ([pdf](http://cds.cern.ch/record/526377/files/0111101.pdf))
* {#DuetschFredenhagen02} [[Michael Dütsch]], [[Klaus Fredenhagen]], equation (90) in _The Master Ward Identity and Generalized Schwinger-Dyson Equation in Classical Field Theory_, Commun.Math.Phys. 243 (2003) 275-314 ([arXiv:hep-th/0211242](https://arxiv.org/abs/hep-th/0211242))
* {#BrennecketDuetsch07} Ferdinand Brennecke, [[Michael Dütsch]], equation (5.5) in _Removal of violations of the Master Ward Identity_, in perturbative QFT, Rev.Math.Phys. 20 (2008) 119-172 ([arXiv:https://arxiv.org/abs/0705.3160](https://arxiv.org/abs/0705.3160))
* {#Hollands07} [[Stefan Hollands]], around (322) and (333) and (345) of _Renormalized Quantum Yang-Mills Fields in Curved Spacetime_, Rev. Math. Phys.20:1033-1172, 2008 ([arXiv:0705.3340](https://arxiv.org/abs/0705.3340))
* {#Rejzner11} [[Katarzyna Rejzner]], section 5.3 of _Batalin-Vilkovisky formalism in locally covariant field theory_ ([arXiv:1111.5130](https://arxiv.org/abs/1111.5130))
* {#Rejzner13} [[Katarzyna Rejzner]], equation (37) of _Remarks on local symmetry invariance in perturbative algebraic quantum field theory_ ([arXiv:1301.7037](https://arxiv.org/abs/1301.7037))
* {#Duetsch18} [[Michael Dütsch]], equation (4.2) of _[[From classical field theory to perturbative quantum field theory]]_, 2018
See also
* Wikipedia, _[Ward-Takahashi identity](http://en.wikipedia.org/wiki/Ward%E2%80%93Takahashi_identity)_
[[!redirects Ward identities]]
[[!redirects Ward–Takahashi identity]]
[[!redirects Ward–Takahashi identities]]
[[!redirects classical master Ward identity]]
[[!redirects classical master Ward identities]]
[[!redirects quantum master Ward identity]]
[[!redirects quantum master Ward identities]]
[[!redirects master Ward identity]]
[[!redirects master Ward identities]]
[[!redirects quantum Ward identity]]
[[!redirects quantum Ward identities]]
|
Warren Dicks | https://ncatlab.org/nlab/source/Warren+Dicks |
* [personal page](https://mat.uab.cat/~dicks/)
* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=64122)
## Selected writings
On algebraic presentations of [[mapping class groups]] and [[braid groups]]:
* [[Warren Dicks]], Edward Formanek, *Algebraic Mapping-Class Groups of Orientable Surfaces with Boundaries*, in: *Infinite Groups: Geometric, Combinatorial and Dynamical Aspects*, Progress in Mathematics **248** Birkhäuser (2005) [[doi;10.1007/3-7643-7447-0_4](https://doi.org/10.1007/3-7643-7447-0_4)]
On [[group action|actions]] of [[braid groups]] (such as via [[automorphisms]] of [[free groups]]):
* [[Lluís Bacardit]], [[Warren Dicks]], *Actions of the braid group, and new algebraic proofs of results of Dehornoy and Larue*, Groups Complexity Cryptology **1** (2009) 77-129 [[arXiv:0705.0587](https://arxiv.org/abs/0705.0587), [doi;10.1515/GCC.2009.77](https://doi.org/10.1515/GCC.2009.77)]
category: people
|
Warren Siegel | https://ncatlab.org/nlab/source/Warren+Siegel |
* [webpage](http://insti.physics.sunysb.edu/~siegel/plan.html)
* [Wikipedia entry](http://en.wikipedia.org/wiki/Warren_Siegel)
## Selected writings
On [[supersymmetry]] in [[superspace]]:
* {#GatesGrisaruRocekSiegel83} [[Jim Gates]] Jr, [[Marcus Grisaru]], [[Martin Roček]], [[Warren Siegel]], _Superspace, or One thousand and one lessons in supersymmetry_, Front.Phys. 58 (1983) 1-548 (1983) ([arXiv:hep-th/0108200](https://arxiv.org/abs/hep-th/0108200), [spire:195126](inspirehep.net/record/195126))
On [[doubled supergeometry]]:
* Machiko Hatsuda, Kiyoshi Kamimura, [[Warren Siegel]], _Superspace with manifest T-duality from type II superstring_, J. High Energ. Phys. (2014) 2014: 39 ([arXiv:1403.3887](https://arxiv.org/abs/1403.3887))
category: people |
Warsaw circle | https://ncatlab.org/nlab/source/Warsaw+circle |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topology
+-- {: .hide}
[[!include topology - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
The **Warsaw circle** is a [[topological space]] that serves to illustrate some of the ideas of [[shape theory]].
A topological space may have very little separating it from '[[manifold|manifoldness]]', yet a 'singularity' can cause havoc! The simple example, here, is known as the __Warsaw Circle__ as it was studied extensively by K. Borsuk and his Polish collaborators, see the book ([Borsuk 75](#Borsuk75)).
## Definition
The **Warsaw circle** $S_W $ is the [[subset]] of the [[plane]], $\mathbb{R}^2$, specified by
$$\{(x,\sin\left(\frac{1}{x}\right)) \mid -\frac{1}{2\pi} \lt x \leq \frac{1}{2\pi}, x \neq 0\}\cup \{(0,y) | -1 \leq y \leq 1\} \cup C,$$
where $C$ is an arc in $\mathbb{R}^2$ joining $(\frac{1}{2\pi}, 0)$ and $(-\frac{1}{2\pi}, 0)$, disjoint from the other two subsets specified above except at its endpoints.
It looks something like [[warsaw.pdf|this:file]]:
<img src="https://ncatlab.org/nlab/files/warsaw.gif" alt="The Warsaw circle" title="A picture of the Warsaw circle">
**Note** There is a variant version $S_W'$ with no $(x,\sin(\frac{1}{x}))$-bit for the $x\lt 0$ and the curve $C$ joins $(0,0)$ to $(\frac{1}{2\pi}, 0)$. The discussion adapts very easily to that. For this version, there is a surjective continuous map $\mathbb{R} \to S_W'$. See eg [Wikipedia](https://en.wikipedia.org/wiki/Shape_theory_%28mathematics%29#/media/File:Warsaw_Circle.png) for a picture.
## Properties
The Warsaw circle is a [[compact space|compact]] [[metric space]], but is not [[locally connected space|locally connected]] along the line corresponding to $\{(0,y) \mid -1 \leq y \leq 1\} $, so is not a [[manifold]], nor for that matter a [[polyhedron]]. It is [[connected space|connected]], but not [[pathwise connected space|pathwise connected]] as no path can get out from the 'line'. (The variant version noted above _is_ pathwise connected.) We note
* $\pi_0(S_W)$ is two points;
* $\pi_1(S_W)$ is trivial at any base point.
There is a simple continuous map from $S^0$, the 0-circle, $\{-1,1\}$, to $S_W$ which is a [[weak homotopy equivalence]]. (For instance define $f(-1) = (0,0)$ and $f(1)$ to be any point in the outer $sin(1/x)$ part of the space, it does not matter which one.) This is not a [[homotopy equivalence]]. (In fact it is instructive to look at maps from $S_W$ to $S^0$! It does not take long.)
A striking thing about the picture is that it 'clearly' divides the plane into two components, an inside and an outside, and has a definite sense of being 'almost' a circle. It has a line of singularities, but otherwise ... .
If we consider, not just $S_W$ as a compact metric space, but as a subspace of the plane, then we can take small [[open neighbourhood|open neighbourhoods]] of $S_W$, to be definite take
$$N_{\frac{1}{n}}(S_W) = \{ \underline{x}\in \mathbb{R}^2 | d(\underline{x},S_W) \lt \frac{1}{n}\}.$$
This looks like an [[annulus]] with a thickenning at one small section. It has the [[homotopy type]] of a [[circle]]. If $N \gt n$, $N_{\frac{1}{N}}(S_W)\subset N_{\frac{1}{n}}(S_W)$, of course, (we will write $i^N_n$ for this map, and this is a homotopy equivalence. The Warsaw circle, $S_W$, is clearly the intersection of all these almost annular neighbourhoods. (Note, also clearly, that the complements of these neighbourhoods are gradually occupying more and more of the two components of $\mathbb{R}^2- S_W$.)
We have a inverse system ([[pro-object]]) of topological spaces all of which have the [[homotopy type]] of a [[polyhedron]], ... in fact always the same polyhedron, $S^1$. Note that by our use of a specific [[cofinal diagrams|cofinal]] family of neighbourhoods of $S_W$, indexed by the [[natural number|natural numbers]], we have an _inverse sequence_. That was a choice and we could have chosen differently or not at all. The ability to pick a sequence of neighbourhoods is related to the fact that we are considering a **compact metric space**.
Another point to note is that not only is each of the neighbourhoods [[homotopic]] to $S^1$, but the inclusion maps making up the 'bonds' of the inverse sequence, are homotopy equivalence. This is a particularity of $S_W$ and other examples, such as the [[solenoid|solenoids]] need not have this 'stability' property. The Warsaw circle is an example of what is called a [[stable space]].
### A (Borsuk) shape map $f\colon S^1 \to S_W$
There is a sequence of maps, $\{f_n : S^1 \to N_{\frac{1}{n}}(S_W)\mid n\in \mathbb{N}\}$, so that for each pair, $(n,N)$, with $N\gt n$, there is a homotopy $f_n \sim i^N_n f_N$. This makes a (Borsuk) [[shape map]] from the [[circle]] to the Warsaw circle. Each $f_n$ is in fact a [[homotopy equivalence]] and we can use a choice of homotopy inverses to get another shape map $g : S_W\to S^1$ and these make up a **shape equivalence**.
(A more detailed description of shape maps and shape equivalences in the Borsuk version of shape theory, is given in the entry [[Borsuk shape theory]]. The version given here skates over some points. It is, in fact, near the ANR-systems approach to shape.)
### From a Čech point of view
To get $\check{C}(S_W,-)$ Čech nerve complex of $S_W$, (see [[Čech methods]]), we can calculate $\check{C}(S_W,\alpha)$ for an arbitrary [[open cover]] $\alpha$ of $S_W$, but we need not do that (in fact that is a silly thing to do!). We first note that $S_W$ is [[compact space|compact]] so we need only consider finite open covers, as these form a [[cofinal subcategory]] of the category of all open covers. ('Cofinal subcategory' means that its inclusion into the bigger category is a [[cofinal functor]].)
Next we look at any finite open cover and note that it has a refinement in the form of open balls of radius $\frac{1}{n}$, in other words we can restrict to (well chosen) such covers, giving a countable family of open covers that have to be worked with.
For such open covers the nerve will look a bit like [[circle.pdf|this:file]].
There may be fine detail in the rectangle depending on the choice of cover, but that detail will disappear as one passes to finer and finer scales. (New holes may occur, but again going finer those disappear.) Cofinally it looks like a space obtained by adding in a thin rectangle transverse to a circle at one small segment. For different open coverings, the only difference will be where the region of attachment (marked $*$) will occur and the relative thinness of the rectangle. The line of singularities given by the interval $[-1,1]$ on the $y$-axis cannot be 'observed', of course. If one passes to finer and finer covers, most of the curve does not change appreciably. It just gets subdivided, but the part near $*$ will lengthen, 'spawning' a very large number of new vertices.
There are two important points to note:
* (essentially) each $\check{C}(S_W,\alpha)$ has the homotopy type of a circle
* the transition maps, $C(S_W,\alpha)\to C(S_W,\beta)$, will be (cofinally) homotopy equivalences.
(With a bit more care in the choice of the covers these can be made exact statements, not just 'essentially' or cofinally true.)
We note that there are obvious maps of pro-objects $\check{C}(S^1,-)\to \check{C}(S_W,-)$, and back again. These give an isomorphism in $pro-Ho(sSets)$. This is the Čech homotopy versions of the observations made for Borsuk's shape above.
## References
* {#Borsuk75} [[K. Borsuk]], _Theory of Shape_, Monografie Matematyczne Tom 59,Warszawa (1975)
[[!redirects Warsaw circle]]
[[!redirects Warsaw Circle]]
|
Washington Taylor | https://ncatlab.org/nlab/source/Washington+Taylor |
* [webpage](http://web.mit.edu/physics/people/faculty/taylor_washington.html)
## Selected writings
On [[2d QCD]]:
* [[David Gross]], [[Washington Taylor]], _Two Dimensional QCD is a String Theory_, Nucl. Phys. B400:181-210, 1993
On longitudinal [[M5-branes]] in the [[BFSS matrix model]] and introducing the [[fuzzy 4-sphere]]:
* Judith Castelino, Sangmin Lee, [[Washington Taylor]], _Longitudinal 5-branes as 4-spheres in Matrix theory_, Nucl. Phys. B526:334-350, 1998 ([arXiv:hep-th/9712105](https://arxiv.org/abs/hep-th/9712105))
(via [[D5-branes]])
On the [[non-abelian DBI action]] for [[intersecting branes]] with non-abelian [[gauge enhancement]] on their worldvolume:
* [[Akikazu Hashimoto]], [[Washington Taylor]], _Fluctuation Spectra of Tilted and Intersecting D-branes from the Born-Infeld Action_, Nucl. Phys. B503: 193-219, 1997 ([arXiv:hep-th/9703217](https://arxiv.org/abs/hep-th/9703217))
* [[Washington Taylor]], [[Mark Van Raamsdonk]], _Multiple D$p$-branes in Weak Background Fields_, Nucl. Phys. B573:703-734, 2000 ([arXiv:hep-th/9910052](https://arxiv.org/abs/hep-th/9910052))
On [[D0-D6 brane bound states]]:
* [[Washington Taylor]], _Adhering 0-branes to 6-branes and 8-branes_, Nucl. Phys. B508: 122-132, 1997 ([arXiv:hep-th/9705116](https://arxiv.org/abs/hep-th/9705116))
On [[F-theory]] [[string phenomenology]]:
* {#TaylorTurner19} [[Washington Taylor]], Andrew P. Turner, _Generic construction of the Standard Model gauge group and matter representations in F-theory_ ([arXiv:1906.11092](https://arxiv.org/abs/1906.11092))
Realization of [[E7|$E_7$]]-[[GUT]] models in [[F-theory]]:
* Shing Yan Li, [[Washington Taylor]], *Towards natural and realistic $E_7$ GUTs in F-theory* [[arXiv:2401.00040](https://arxiv.org/abs/2401.00040)]
category: people
|
Wasserstein metric | https://ncatlab.org/nlab/source/Wasserstein+metric |
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#### Measure and probability theory
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## Idea
A _Wasserstein metric_ is a certain [[metric]] over a space of [[probability measure]]s on a [[measurable space]] $X$.
By ([JKO](#JKO)) the [[heat flow]]/[[diffusion equation]] on $X$ is the [[gradient flow]] of the [[entropy|Boltzman-Shannon entropy]] functional with respect to the Wasserstein metric.
The Wasserstein metric does not seem to arise from a [[Riemannian metric|Riemann metric tensor]]. A detailed discussion of the relevant gradient flows in non-smooth [[metric space]]s is in ([AGS](#AGS)).
## References and links
* Related concepts [[information metric]], [[entropy]]
The characterization of heat flow as the gradient flow of Shannon-entropy is due to
* R. Jordan, D. Kinderlehrer, F. Otto, _The variational formulation of the [[Fokker-Planck equation]]_ , SIAM J. Math. Anal. 29 (1998), no. 1, 1-17.([pdf](http://www.imati.cnr.it/~savare/Ravello2010/JKO.pdf))
{#JKO}
The analog of this for finite probability spaces is discussed in
* Jan Maas, _Gradient flows of the entropy for finite Markov chains_ ([pdf](http://www.janmaas.org/papers/discrete.pdf))
A comprehensive discussion of the corresponding gradient flows is in
* l Ambrosio, N. Gigli, G. Savaré, _Gradient flows in metric spaces and in the space of probability measures_, Second edition. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2008. x+334 pp. ([pdf](http://www.imati.cnr.it/~savare/research/optimal_transportation/Introduction.pdf) of toc and introduction)
[MR2009h:49002](http://www.ams.org/mathscinet-getitem?mr=2401600) {#AGS}
* Luigi Ambrosio, Nicola Gigli, _Construction of the parallel transport in the Wasserstein space_, Methods Appl. Anal. 15 (2008), no. 1, 1–29, [MR2010c:49082](http://www.ams.org/mathscinet-getitem?mr=2482206)
[[!redirects earth mover distance]]
[[!redirects earth mover's distance]]
[[!redirects Wasserstein metrics]]
[[!redirects Wasserstein distance]]
[[!redirects Wasserstein distances]]
[[!redirects Kantorovich-Wasserstein metric]]
[[!redirects Kantorovich-Wasserstein distance]]
[[!redirects Kantorovich-Wasserstein norm]]
[[!redirects Kantorovich metric]]
[[!redirects Kantorovich distance]]
[[!redirects Kantorovich norm]]
[[!redirects Monge-Kantorovich metric]]
[[!redirects Monge-Kantorovich distance]]
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[[!redirects Monge-Kantorovich-Wasserstein metric]]
[[!redirects Monge-Kantorovich-Wasserstein distance]]
[[!redirects Monge-Kantorovich-Wasserstein norm]] |
wave | https://ncatlab.org/nlab/source/wave |
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## Idea
An [[oscillation]] is a *periodic motion of a physical system*, i.e. as a function of time, the position of the system of particles expressed as a coordinate in a configuration space depends as $t\mapsto x(t)$ where $x(t) = x(t+I)$ where $I$ is a constant called the period of the motion. The motion can pertain to a system of particles, to the coordinates determining the position of a rigid body or can express the magnitude of some physical quantity like pressure which evolves in time. In a mechanical system, the point in which the potential energy is minimal is called the equlibrium position; the relative position with respect to the equilibrium position is called the __elongation__. The maximal elongation is usually called the __amplitude of oscillation__.
A __wave__ is the (result of) *propagation of oscillations through an extended medium*.
Examples of extended media are a liquid, the vacuum (where physical fields can propagate, for example the electromagnetic field) or a discrete grid of interacting particles. Many partial differential equations for the mechanics of extended objects have wave solutions. Waves are typically viewed via quantity called __amplitude__ which describes the relative position of each particle in a continuum system from its equilibrium position (this conflicts a bit with the terminology on oscillations, where the amplitude would be the maximnal one). The amplitude depends both on the time and the position in space. Most typical wave propagation is of the form $A(x,t) = f(x-vt)$ where $x$ is the position, $v$ is the speed of propagation (which may depend on the direction of propagation, and for inhomogeneous systems on the coordinate) and $t$ is the time. Partial differential equations (PDE) which have __wave solutions__ are typically parabolic or hyperbolic PDEs. Here wave solutions are recognized by their form (variations of the formula $f(x-vt)$ from above). Wave solutions can add up together linearly or not; in both cases the phenomenon is called __superposition__. Finite superpositions have many typical features (e.g. Lissajous figures).
The superposition can stretch over infinite families of waves, then the sum passes into the integral and many periodic modes add up to something what is not periodic whatsoever, and may be even localized. One then talks about a __wave packet__. A wave packet can from far away look like a point particle. Wave packets can decay (damp) quickly as they propagate and in nonlinear situations can be unstable under collisions with other waves. In simple linear cases, the individual cases in the wave packet correspond to elements of a basis in a Hilbert space of functions, then the expression of a wave packet in terms of basic waves (say plane waves) amount to a case of [[Fourier transform|Fourier mode expansion]].
In some special circumstances, the waves can preserve their features in nonlinear situations even asymptotically after collisions, in that case we talk about [[soliton]]s. Solitons and multisolitons
are especially well-known in gauge theories (including the 4d case localized in time, so called instantons) and in many [[integrable systems]] like the [[sine-Gordon equation]], KdV and the nonlinear Schroedinger equation. The inverse scattering method in those cases can be interpreted as the existence of a certain nonlinear analogue of the Fourier transform (the spectral transform) for the nonlinear wave solutions.
The [[quantum mechanics]], when described mathematically via the solutions of a time-dependent Schroedinger equation, is a wave mechanics where the modulus square of the amplitude is interpreted as the density of probability to find the particle in the corresponding position. Linear superposition is thus one of the basic features of quantum mechanics.
## Related concepts
* [[plane wave]], [[wave vector]]
* [[harmonic analysis]], [[Fourier analysis]]
* [[oscillation]],
* [[semiclassical approximation]]
* [[geometrical optics]]
* [[soliton]]
* [[Stokes phenomenon]]
* [[Berry's phase]].
* [[wave function]]
## References
* [[Howard Georgi]], *The Physics of Waves*, Prentice Hall (1993) $[$[web](https://www.people.fas.harvard.edu/~hgeorgi/new.htm), [pdf](https://www.people.fas.harvard.edu/~hgeorgi/onenew.pdf)$]$
[[!redirects waves]]
[[!redirects wave mechanics]]
|
wave equation | https://ncatlab.org/nlab/source/wave+equation |
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## Idea
For $(X,g)$ a [[pseudo-Riemannian manifold]] and $\Box : C^\infty(X) \to C^\infty(X)$ its [[Laplace operator]], the **wave equation** on $X$ is the [[linear differential equation|linear]] [[differential equation]]
$$
\Box_g f = 0
\,.
$$
where $\Box_g$ denotes the _wave operator_ /[[Laplace operator]], a [[hyperbolic differential operator]].
For $m \in \mathbb{R}$ the inhomogenous equation
$$
\Box_g f = m^2 f
$$
is called the _[[Klein-Gordon equation]]_.
## Properties
### Fundamental solution
On a [[globally hyperbolic spacetime]] the wave equation/Klein-Gordon equation has unique advanced and retarded [[Green functions]].
Their difference is the [[Peierls bracket]] which gives the [[Poisson bracket]] on the [[covariant phase space]] of the [[free field|free]] [[scalar field]]. This in turn defines the [[Wick algebra]] of the free scalar field, which yields the [[quantization]] of the free scalar field to a [[quantum field theory]].
### Bicharacteristic flow and propagation of singularities
The [[bicharacteristic strips]] of the Klein-Gordon operator are [[cotangent vectors]] along [[lightlike]] [[geodesics]] ([this example](bicharacteristic+flow#BicharachteristicFlowOfKleinGordonOperator)).
## Related concepts
* [[wave]]
* [[plane wave]],
[[wave vector]], [[wavelength]], [[frequency]]
* [[superposition]]
* [[linear differential equation]]
* [[scalar field]]
* [[Green's function]]
* [[Feynman propagator]]
* [[spectral geometry]], [[hearing the shape of a drum]]
* [[heat equation]]
* [[Klein-Gordon equation]]
* [[sine-Gordon equation]]
* [[Schrödinger equation]]
* [[Dirac equation]]
## References
* F. Friedlander, _The Wave Equation on a Curved Space-Time_, Cambridge: Cambridge University Press, 1975
* [[Howard Georgi]], *The Physics of Waves*, Prentice Hall (1993) $[$[web](https://www.people.fas.harvard.edu/~hgeorgi/new.htm), [pdf](https://www.people.fas.harvard.edu/~hgeorgi/onenew.pdf)$]$
* {#BaerGinouxPfaeffle07} [[Christian Bär]], [[Nicolas Ginoux]], [[Frank Pfäffle]], _Wave Equations on Lorentzian Manifolds and Quantization_, ESI Lectures in Mathematics and Physics, European Mathematical Society Publishing House, ISBN 978-3-03719-037-1, March 2007, Softcover ([arXiv:0806.1036](https://arxiv.org/abs/0806.1036))
* {#Ginoux08} [[Nicolas Ginoux]], _Linear wave equations_, Ch. 3 in [[Christian Bär]], [[Klaus Fredenhagen]], _Quantum Field Theory on Curved Spacetimes: Concepts and Methods_, Lecture Notes in Physics, Vol. 786, Springer, 2009
* [[Sergiu Klainerman]], chapter 4, section 3 of _Lecture notes in analysis_, 2011 ([pdf](https://web.math.princeton.edu/~seri/homepage/courses/Analysis2011.pdf))
[[!redirects wave equations]]
[[!redirects wave operator]]
|
wave function collapse | https://ncatlab.org/nlab/source/wave+function+collapse |
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{: toc}
## Idea
In the context of [[quantum mechanics]], the **collapse of the wave function**, also known as the **reduction of the wave packet**, is said to occur after [[observation]] or [[measurement]], when a wave function expressed as the sum of [[eigenfunctions]] of the observable is projected randomly onto one of them. Different [[interpretations of quantum mechanics]] understand this process differently.
The perspective associated with the _[[Bayesian interpretation of quantum mechanics]]_ observes (see [below](#RelationToConditionalExpectationValues)) that the apparent collapse is just the mathematical reflection of the formula for [[conditional expectation values]] in [[quantum probability theory]].
## Relation to conditional expectation values
{#RelationToConditionalExpectationValues}
There is a close relation between wave function collapse and [[conditional expectation values]] in [[quantum probability]] (e.g. [Kuperberg 05, section 1.2](#Kuperberg05), [Yuan 12](#Yuan12)):
Let $(\mathcal{A},\langle -\rangle)$ be a [[quantum probability space]], hence a [[complex numbers|complex]] [[star algebra]] $\mathcal{A}$ of [[quantum observables]], and a [[state on a star-algebra]] $\langle -\rangle \;\colon\; \mathcal{A} \to \mathbb{C}$.
This means that for $A \in \mathcal{A}$ any [[observable]], its _[[expectation value]]_ in the given [[state on a star-algebra|state]] is
$$
\mathbb{E}(A) \;\coloneqq\; \langle A \rangle \in \mathbb{C}
\,.
$$
More generally, if $P \in \mathcal{A}$ is a [[real part|real]] [[idempotent]]/[[projector]]
$$
\label{RealIdempotent}
P^\ast = P
\,,
\phantom{AAA}
P P = P
$$
thought of as an event, then for any observable $A \in \mathcal{A}$ the [[conditional expectation value]] of $A$, conditioned on the observation of $P$, is (e.g. [Redei-Summers 06, section 7.3](#RedeiSummers06), see also [Fröhlich-Schubnel 15, (5.49)](#FroehlichSchubnel15), [Fröhlich 19 (45)](#Froehlich16))
$$
\label{ConditionalExpectation}
\mathbb{E}(A \vert P)
\;\coloneqq\;
\frac{
\left \langle P A P \right\rangle
}{
\left\langle P \right\rangle
}
\,.
$$
Now assume a [[star-representation]] $\rho \;\colon\; \mathcal{A} \to End(\mathcal{H})$ of the [[algebra of observables]] by [[linear operators]] on a [[Hilbert space]] $\mathcal{H}$ is given, and that the state $\langle -\rangle$ is a [[pure state]], hence given by a [[vector]] $\psi \in \mathcal{H}$ ("[[wave function]]") via the [[Hilbert space]] [[inner product]] $\langle (-), (-)\rangle \;\colon\; \mathcal{H} \otimes \mathcal{H} \to \mathbb{C}$ as
$$
\begin{aligned}
\langle A \rangle
& \coloneqq
\left\langle\psi
\vert A \vert \psi
\right\rangle
\\
& \coloneqq
\left\langle\psi,
A \psi
\right\rangle
\end{aligned}
\,.
$$
In this case the expression for the [[conditional expectation value]] (eq:ConditionalExpectation) of an observable $A$ conditioned on an idempotent observable $P$ becomes (notationally suppressing the [[representation]] $\rho$)
$$
\begin{aligned}
\mathbb{E}(A\vert P)
& =
\frac{
\left\langle
\psi \vert P A P\vert \psi
\right\rangle
}{
\left\langle
\psi \vert P \vert \psi
\right\rangle
}
\\
& =
\frac{
\left\langle
P \psi \vert A \vert P \psi
\right\rangle
}{
\left\langle
P \psi \vert P \psi
\right\rangle
}
\,,
\end{aligned}
$$
where in the last step we used (eq:ConditionalExpectation).
This says that _assuming_ that $P$ has been observed in the [[pure state]] $\vert \psi\rangle$, then the corresponding [[conditional expectation values]] are the same as actual [[expectation values]] but for the new pure state $\vert P \psi \rangle$.
This is the statement of "wave function collapse"
$$
\vert \psi \rangle \mapsto P \vert \psi \rangle
\,.
$$
The original [[wave function]] is $\psi \in \mathcal{H}$, and after observing $P$ it "collapses" to $P \psi \in \mathcal{H}$ (up to normalization).
## Related concepts
* [[interpretation of quantum mechanics]]
* [[Bayesian interpretation of quantum mechanics]]
* [[propositions as projections]]
* [[deferred measurement principle]]
[[!include states and observables -- content]]
## References
{#References}
Early explicit discussion of wavefunction collapse in [[quantum measurements]], including discussion relating to historical experiments:
* {#vonNeumann32} [[John von Neumann]], §III.3 and §VI of:
*Mathematische Grundlagen der Quantenmechanik*, Springer (1932, 1971) [[doi:10.1007/978-3-642-96048-2](https://link.springer.com/book/10.1007/978-3-642-96048-2)]
*Mathematical Foundations of Quantum Mechanics* Princeton University Press (1955) [[doi:10.1515/9781400889921](https://doi.org/10.1515/9781400889921), [Wikipedia entry](https://en.wikipedia.org/wiki/Mathematical_Foundations_of_Quantum_Mechanics)]
But the formulation in [von Neumann 1932](#vonNeumann32) postulated that a [[pure state]] would collapse to a [[mixed state]] when the [[quantum observable]]'s [[eigenvalues]] are degenerate.
This problem was pointed out and the modern formulation of the collapse postulate was formulated by:
* {#Lüders51} [[Gerhart Lüders]]:
*Über die Zustandsänderung durch den Meßprozeß*, Ann. Phys. **8** (1951) 322–328 [[doi:10.1002/andp.19504430510](https://doi.org/10.1002/andp.19504430510)]
*Concerning the state-change due to the measurement process*, Ann. Phys. **15** 9 (2006) 663-670 [[pdf](http://myweb.rz.uni-augsburg.de/~eckern/adp/history/historic-papers/2006_518_663-670.pdf), [[Lueders-StateChange.pdf:file]]]
Early review:
* {#Scheibe73} [[Erhard Scheibe]], pp 137 in: _*The logical analysis of quantum mechanics&, Pergamon Press Oxford (1973)
Textbook accounts in [[quantum information theory]]:
* [[Michael A. Nielsen]], [[Isaac L. Chuang]], §2.2.3 of: *Quantum computation and quantum information*, Cambridge University Press (2000) [[doi:10.1017/CBO9780511976667](https://doi.org/10.1017/CBO9780511976667), [pdf](http://csis.pace.edu/~ctappert/cs837-19spring/QC-textbook.pdf), [[NielsenChuangQuantumComputation.pdf:file]]]
* [[Joseph M. Renes]], eq. (3.2) in: *Quantum Information Theory* (2015) [[pdf](https://edu.itp.phys.ethz.ch/hs15/QIT/renes_lecture_notes14.pdf)], (A.2) in: De Gruyter (2022) [[doi:10.1515/9783110570250](https://doi.org/10.1515/9783110570250)]
Discussion from the point of view of [[quantum probability]]:
* {#Kuperberg05} [[Greg Kuperberg]], _A concise introduction to quantum probability, quantum mechanics, and quantum computation_ (2005) [[pdf](http://www.math.ucdavis.edu/~greg/intro-2005.pdf), [[Kuperberg-ConciseQuantum.pdf:file]]]
* {#RedeiSummers06} [[Miklos Redei]], [[Stephen Summers]], _Quantum Probability Theory_ ([arXiv:quant-ph/0601158](https://arxiv.org/abs/quant-ph/0601158))
* {#Yuan12} [[Qiaochu Yuan]], _[Finite noncommutative probability, the Born rule, and wave function collapse](https://qchu.wordpress.com/2012/09/09/finite-noncommutative-probability-the-born-rule-and-wave-function-collapse/)_, 2012
* {#FroehlichSchubnel15} [[Jürg Fröhlich]], B. Schubnel, _Quantum Probability Theory and the Foundations of Quantum Mechanics_. In: Blanchard P., Fröhlich J. (eds.) _The Message of Quantum Science_. Lecture Notes in Physics, vol 899. Springer 2015 ([arXiv:1310.1484](https://arxiv.org/abs/1310.1484), [doi:10.1007/978-3-662-46422-9_7](https://doi.org/10.1007/978-3-662-46422-9_7))
* {#Froehlich16} [[Jürg Fröhlich]], _The structure of quantum theory_, Chapter 6 in _The quest for laws and structure_, EMS 2016 ([doi](https://www.researchgate.net/publication/308595814_The_Quest_for_Laws_and_Structure), [doi:10.4171/164-1/8](https://www.ems-ph.org/books/show_abstract.php?proj_nr=207&vol=1&rank=8))
See also
* Wikipedia, _[Wave function collapse](https://en.wikipedia.org/wiki/Wave_function_collapse)_
* Wikipedia, _[Born rule](https://en.wikipedia.org/wiki/Born_rule)_
More speculative discussion of state collapse as caused by stochastic terms added to the [[Schrödinger equation]]:
from the point of view of the [[symplectic formulation of quantum mechanics]]:
* {#Hughston96} [[Lane P. Hughston]], *Geometry of stochastic state vector reduction*, Proceedings of the Royal Society A **452** 1947 (1996) [[doi:10.1098/rspa.1996.0048](https://doi.org/10.1098/rspa.1996.0048), [jstor:52944](https://www.jstor.org/stable/52944)]
[[!redirects wave function collapse]]
[[!redirects wavefunction collapse]]
[[!redirects collapse of the wave function]]
[[!redirects collapse of the wavefunction]]
[[!redirects collapse of the wave-function]]
[[!redirects quantum state collapse]]
[[!redirects Born's rule]]
[[!redirects projection postulate]]
|
wave packet | https://ncatlab.org/nlab/source/wave+packet |
## Related concepts
* [[plane wave]]
## References
See also
* Wikipedia, *[Wave packet](https://en.wikipedia.org/wiki/Wave_packet)*
[[!redirects wave packets]]
[[!redirects wave-packet]]
[[!redirects wave-packets]]
|
wave polarization | https://ncatlab.org/nlab/source/wave+polarization |
> for other concepts of a similar name see at _[[polarization]]_
***
#Contents#
* table of contents
{:toc}
## Idea
A _[[plane wave]]_ with more than one component, i.e. being a [[section]] of a [[trivial vector bundle]], is characterized not just by its [[wave vector]] $k$, but also by that component vector $e$. Broadly speaking this $e$ is the _polarization_ of the wave.
Specifically a [[plane wave]] with [[coefficients]] in the [[cotangent bundle]] of a [[Minkowski spacetime]] $\Sigma$, such as an [[electromagnetic field|electromagnetic]] [[field history]] "[[vector potential]]" $A \in \Gamma_\Sigma(T^\ast \Sigma)$ is given by
$$
A_\mu(x) = e_\mu e^{i k_\mu x^\mu}
\,.
$$
In the case of [[free field theory|free]] [[electromagnetic waves]] the [[wave vector]] $k$ is [[light-like]], $k_\mu k^\mu = 0$, and in [[Gaussian-averaged Lorenz gauge]] the polarization vector $e$ has to satisfy $e^\mu k_\mu = 0$ and polarization vectors proportional to the wave vector $e_\mu \propto k_\mu$ are [[gauge equivalence|gauge equivalent]] to [[zero]]. Therefore in this case the space of physically distinguishable polarizations for given [[wave vector]] $k$ is the [[quotient space]]
$$
\frac{
\left\{ e \,\vert\, e^\mu k_\mu = 0 \right\}
}{
\left\{ e \,\vert\, e_\mu \propto k_\mu \right\}
}
\,.
$$
This is also called the space of _transversal polarizations_.
If one chooses [[coordinates]] such that $k = (\kappa, 0, \cdots, 0, \kappa)$ then this may be identified with the space of vectors of the form $e = (0, *, \cdots, *, 0)$.
## References
* Wikipedia, _<a href="http://en.wikipedia.org/wiki/Polarization_(waves)">Polarization (waves)</a>_
* {#Dermisek09} [[Radovan Dermisek]], _Quantum Electrodynamics (QED)_ ([pdf](http://www.physics.indiana.edu/~dermisek/QFT_09/qft-II-5-4p.pdf), [[DermisekQED.pdf:file]])
[[!redirects wave polarizations]]
[[!redirects transversal polarization]]
[[!redirects transversal polarizations]]
|
wave vector | https://ncatlab.org/nlab/source/wave+vector |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Harmonic analysis
+-- {: .hide}
[[!include harmonic analysis - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A _wave vector_ is a [[vector]] that encodes [[wavelength]] and [[direction of a vector|direction]] of a _[[plane wave]]_.
## Definition
Let $n \in \mathbb{N}$ and write $\mathbb{R}^n$ the [[Cartesian space]] of [[dimension]] $n$. Thinking of $\mathbb{R}^n$ as a [[vector space]], then each point in it is a [[vector]] $\vec x \in \mathbb{R}^n$ and hence a [[smooth function]] $f \colon \mathbb{R}^n \to \mathbb{C}$ may be thought of as a function of these "position vectors".
If $f$ is a [[function with rapidly decreasing partial derivatives]], then its [[Fourier transform]] $\hat f \;\colon\; \mathbb{R}^n \to \mathbb{C}$ exists. By the [[Fourier inversion theorem]], this function is such that it expresses $f$ as a [[superposition]] of "[[plane wave]]" functions $\vec x \mapsto e^{2\pi i \vec x \cdot \vec k}$ as
$$
f(\vec x)
\;=\;
\underset{\vec k \in \mathbb{R}^n}{\int}
\hat f(k) \, e^{2 \pi i \vec k \cdot \vec x}
\, d \vec k
\,.
$$
Here the [[vector]] $\vec k \in \mathbb{R}^n$ determines
1. the [[wavelength]] $\lambda \coloneqq 1/{\vert \vec k\vert}$ (the inverse of the [[norm]] of $\vec k$);
1. the direction $\frac{\vec k}{{\vert \vec k\vert }} \in S(\mathbb{R}^n)$ (the corresponding unit vector in the [[unit sphere]])
of the "[[plane wave]]" $\vec x \mapsto e^{2 \pi i \vec x \cdot \vec k}$.
The product $2 \pi {\vert k \vert}$ is also called the _wave number_ and $2 \pi k$ then the _wave number vector_. Beware that elsewhere the wave number vector is denoted "$k$", which makes the "wave vector" become $k / 2 \pi$. (See e.g. [Wikipedia](#Wikipedia), "Physics definition" as opposed to "Crystallography definition".)
If here $\mathbb{R}^n \simeq \mathbb{R}^{p,1}$ is identified with [[Minkowski spacetime]] with canonical coordinates denoted $(x^0, x^1, \cdots, x^p)$, then the 0-component of the wave vector
$$
\nu \coloneqq k_0
$$
is called the _[[frequency]]_ of the corresponding [[plane wave]] (in the chosen [[coordinate system]]); this $\omega = 2 \pi \vu$ is the _[[angular frequency]]_.
[[!include plane waves -- table]]
## Related concepts
* [[plane wave]]
* [[Fourier transform]]
* [[wave]]
* [[frequency]], [[amplitude]]
* [[wave polarization]]
## References
See also
* {#Wikipedia} Wikipedia, _[Wave vector](https://en.wikipedia.org/wiki/Wave_vector)_
[[!redirects wave vectors]]
[[!redirects wavevector]]
[[!redirects wavevectors]]
|
waveform | https://ncatlab.org/nlab/source/waveform | \tableofcontents
\section{Idea}
In the context of [[signal processing]], a _waveform_ is a [[periodic wave]] [[signal]] viewed as a [[function]] from [[time]] to [[amplitude]], both viewed as [[real number|real-valued]] variables. One speaks of this as viewing the signal 'in the time domain'.
This is contrast to describing the wave as a function from [[frequency]] to amplitude, where we typically have only a [[finite set]] of frequencies. One speaks in this case of viewing the signal 'in the frequency domain'.
A [[discrete Fourier transform]] can be used to pass from the time domain to the frequency domain; an inverse discrete Fourier transform can be used to pass from the frequency domain to the time domain.
\section{See also}
* [[frequency]]
* [[fundamental frequency]]
[[!redirects waveforms]] |
waveform modulation | https://ncatlab.org/nlab/source/waveform+modulation | \tableofcontents
\section{Idea}
In [[signal processing]], _modulation_ refers to modification of a standard [[waveform]] (for example a [[sine wave]]) according to a fixed algorithm in order to create a [[signal]] conveying the input to that algorithm. To create a [[digital signal]], for example, the input to such an algorithm would typically be a sequence $S$ of [[bit|bits]], and the output of the algorithm would be a [[waveform]] from whose properties $S$ can be recovered.
\section{Digital modulation}
The following are forms of digital modulation.
* [[On-off keying]] (OOK) |
wavefront set | https://ncatlab.org/nlab/source/wavefront+set |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Functional analysis
+-- {: .hide}
[[!include functional analysis - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In [[microlocal analysis]], the _wave front set_ ([Hörmander 70](#Hoermander70)) of a [[generalized function]] such as a [[distribution]] or a [[hyperfunction]] is a characterization of the singularity structure of the generalized function, hence of how it deviates from being an ordinary smooth function.
The wave front set is the sub-bundle of the [[cotangent bundle]] that consists of all those [[direction of a vector|directions]] (non-zero [[covectors]]) such that the local [[Fourier transform]] of the distribution is not rapidly decaying in this [[direction of a vector|direction]] ([Hörmander 90, section 8.1](#Hoermander90)). Such covectors are stable under multiplication by positive scalars, so the wave front set can also be considered as a [[sub-bundle]] of the [[unit sphere bundle]] of the [[cotangent bundle]].
The [[projection]] of the wave front set down to the base space is the [[singular support of a distribution|singular support]] of the distribution. The additional information in the "wave front" [[covectors]] over this singular support may be understood as providing the directions of _propagation of these singularities_. This is made precise by the _[[propagation of singularities theorem]]_
A notorious issue with [[distributions]] is that, when thought of as generalized functions, generally neither their [[composition of distributions]] nor their pointwise [[product of distributions]] is defined. However, closer inspection shows that the [[obstruction]] to these operations being defined for any given pair of distributions is exactly characterized by the wave front set:
For instance the [[product of distributions]] is well defined precisely if the sum of their wave front sets does not intersect the zero-section ([[Hörmander's criterion]], [Hörmander 90, theorem 8.2.10](#Hoermander90)).
## Definition
### Motivation
The definition of wavefront sets is motivated by a version of a [[Paley-Wiener theorem]] that characterizes smooth compactly supported functions ($\mathbb{R}^n \to \mathbb{R}$) by a growth condition on their [[Fourier transform]] $\mathcal{F}$:
+-- {: .num_theorem}
###### Theorem
**([[Paley-Wiener-Schwartz theorem]])**
The vector space $C_0^{\infty}(\mathbb{R}^n)$ of [[smooth function|smooth]] [[compact support|compactly supported]] functions ([[bump functions]]) is (algebraically and topologically) [[isomorphism|isomorphic]], via the [[Fourier transform]], to the space of [[entire functions]] $F$ which satisfy the following estimate: there is a positive constant $B$ such that for every [[integer]] $m \gt 0$ there is a constant $C_m$ such that:
$$
F(z) \le C_m (1 + |z|)^{-m} \exp{ (B \; |\operatorname{Im}(z)|)}
$$
=--
### Smoothness
We call a smooth compactly supported function that is identically $1$ in a neighbourhood of a point $x_0$ a **cutoff** function at $x_0$.
Let $U \subset \mathbb{R}^n$ be open, we identify the [[cotangent bundle]] of $U$ with $U \times \mathbb{R}^n$. A subset of $U \times \mathbb{R}^n$ is said to be **conic** if it is stable under the transformation
$$
(x, \zeta) \mapsto (x, \rho \zeta) \quad \text{with} \; \rho \gt 0
$$
Note that a [[conical set|conic]] subset is uniquely determined by its intersection with the [[unit sphere]] bundle $U\times S^{n-1}$.
+-- {: .num_defn}
###### Definition
Let $f$ be a distribution and $(x_0, \zeta_0)$ with $\zeta_0 \neq 0$ be a point of the cotangent bundle of $U$. $f$ is **smooth** in $(x_0, \zeta_0)$ if there is a cutoff function $\chi$ in $x_0$ and an open cone $\Gamma_0$ in $\mathbb{R}^n$ containing $\zeta_0$ such that for every $m \gt 0$ there is a nonnegative constant $C_m$ such that for all $\zeta \in \Gamma_0$:
$$
\| \mathcal{F}(\chi f) (\zeta) \| \le C_m (1 + \| \zeta \|)^{-m}
$$
where $\mathcal{F}(\chi f)$ is the [[Fourier transform]] (of the variable $\zeta$) of the function $\chi f$ (of the variable $x$).
=--
+-- {: .num_defn}
###### Definition
A distribution $f$ is smooth in a [[conical set|conic]] subset $\Gamma$ of the cotangent bundle of $U$ if $f$ is smooth in a neighbourhood of every point in $\Gamma$.
=--
### Wavefront set
Let $U \subseteq \mathbb{R}^n$ be an open subset, $T^* U$ its cotangent bundle and $f$ be a distribution on $U$. The complement of the union of all [[conical set|conic]] subsets of $T^* U$ where $f$ is smooth is the **wavefront set $WF(f)$**. Since the wavefront set is therefore itself [[conical set|conic]], it is equivalently determined by a subset of the unit sphere bundle of $T^* U$.
([Hörmander 70 (2.4.1)](#Hoermander70), [Hörmander 90, section 8.1](#Hoermander90))
This definition turns out to make invariant sense ([Hörmander 90, p. 256](#Hoermander90)).
## Examples
+-- {: .num_example #WaveFrontOfDeltaDistribution}
###### Example
**(wave front set of [[delta distribution]])**
For $n \in \mathbb{N}$, consider the [[delta distribution]]
$$
\delta(0) \in \mathcal{D}'(\mathbb{R}^n)
$$
on $n$-dimensional [[Cartesian space]], given by [[evaluation]] at the origin. Its wave front set is
$$
WF(\delta(0))
=
\left\{
(0,k)
\;\vert\;
k \in \mathbb{R}^n \setminus \{0\}
\right\}
\subset
\mathbb{R}^n \times \mathbb{R}^n
\simeq
T^\ast \mathbb{R}^n
\,.
$$
=--
+-- {: .proof}
###### Proof
First of all the [[singular support of a distribution|singular support]] of $\delta(0)$ is clearly $singsupp(\delta(0)) = \{0\}$, hence the wave front set vanishes over $\mathbb{R}^n \setminus \{0\}$.
At the origin, any bump function $b$ supported around the origin with $b(0) = 1$ satisfies $b \cdot \delta(0) = \delta(0)$ and hence the wave front set over the origin is the set of covectors along which the [[Fourier transform of distributions|Fourier transform]] $\hat \delta(0)$ does not suitably decay. But this Fourier transform is in fact a [[constant function]] and hence does not decay in any direction.
=--
+-- {: .num_example #WaveFrontSetOfHeavisideDistribution}
###### Example
**(wave front set of [[Heaviside distribution]])**
Let $H \in \mathcal{D}'(\mathbb{R}^1)$ be the [[Heaviside distribution]] given by
$$
\langle H, b\rangle \coloneqq \int_0^\infty b(x)\, d x
\,.
$$
Its wave front set is
$$
WF(H) = \{(0,k) \vert k \neq 0\}
\,.
$$
=--
+-- {: .num_example}
###### Example
For $(X,e)$ a [[globally hyperbolic spacetime]] and $P$ a [[hyperbolic differential operator]] such as the [[wave operator]]/[[Klein-Gordon operator]], then the [[propagation of singularities theorem]] says that the wave front set of any solution $f$ to $P f = 0$ is a union of [[lightlike]] [[geodesics]] and their [[cotangent vectors]].
Specifically for the [[Klein-Gordon operator]] such ditributional solutions include the [[causal propagator]] and the [[Feynman propagator]].
=--
+-- {: .num_example #WaveFrontOfTensorProductDistribution}
###### Example
**(wave front set of [[tensor product distribution]])**
Let $u \in \mathcal{D}'(X)$ and $v \in \mathcal{D}'(Y)$ be two distributions. then the wave front set of their [[tensor product distribution]] $u \otimes v \in \mathcal{D}'(X \times Y)$ satisfies
$$
WF(u \otimes v)
\;\subset\;
\left(
WF(u) \times WF(v)
\right)
\cup
\left(
\left( supp(u) \times \{0\} \right)
\times WF(v)
\right)
\cup
\left(
WF(u)
\times
\left(
supp(v) \times \{0\}
\right)
\right)
\,,
$$
where $supp(-)$ denotes the [[support of a distribution]].
=--
([Hörmander 90, theorem 8.2.9](#Hoermander90))
## Properties
+-- {: .num_prop #EmptyWaveFrontSetCorrespondsToOrdinaryFunction}
###### Proposition
**(empty wave front set corresponds to ordinary functions)**
The wave front set of a [[compactly supported distribution]] is empty precisely if the distribution comes from an ordinary [[smooth function]] (hence a [[bump function]]).
=--
e.g. ([Hörmander 90, below (8.1.1)](#Hoermander90))
+-- {: .num_prop #DerivativeOfDistributionRetainsOrShrinksWaveFrontSet}
###### Proposition
**([[derivative of distributions]] retains or shrinks wave front set)**
Taking [[derivatives of distributions]] retains or shrinks the wave front set:
For $u \in \mathcal{D}'(\mathbb{R}^n)$ a distribution and $\alpha \in \mathbb{N}^n$ a multi-index with $D^\alpha$ denoting the corresponding [[partial derivative|partial]] [[derivative of distributions]], then
$$
WF(D^\alpha u) \subset WF(u)
\,.
$$
Hence if $P$ is any [[differential operator]] with [[smooth function]] [[coefficients]], then
$$
WF(P u) \subset WF(u)
\,.
$$
=--
([Hörmander 90, (8.1.10) (8.1.11), p. 256](#Hoermander90))
+-- {: .num_prop #WaveFrontSetOfCompactlySupportedDistributions}
###### Proposition
**([[wave front set]] of [[convolution of distributions|convolution of]] [[compactly supported distributions]])**
Let $u,v \in \mathcal{E}'(\mathbb{R}^n)$ be two [[compactly supported distributions]]. Then the [[wave front set]] of their [[convolution of distributions]] is
$$
WF(u \star v)
\;=\;
\left\{
(x + y, k)
\;\vert\;
(x,k) \in WF(u) \,\text{and}\, (y,k) \in WF(u)
\right\}
\,.
$$
=--
([Bengel 77, prop. 3.1](convolution+of+distributions#Bengel77))
## Related concepts
* [[ultraviolet divergence]]
## References
### General
The concept of wave front set is due to
* {#Hoermander70} [[Lars Hörmander]], _Linear differential operators_, Actes Congr. Int. Math. Nice 1970, 1, 121-133 ([pdf](http://www.mathunion.org/ICM/ICM1970.1/Main/icm1970.1.0121.0134.ocr.pdf))
A textbook account for distributions on open subsets of [[Euclidean space]] is in
* {#Hoermander90} [[Lars Hörmander]], section 8.1 of _The analysis of linear partial differential operators_, vol. I, Springer 1983, 1990
and for distributions more generally on smooth manifolds is in
* {#Hoermander94} [[Lars Hörmander]], _The analysis of linear partial differential operators_, vol. III, Springer 1994
A history of the concept of wave front sets with extensive pointers to the
literature is given in [Hörmander 90, p. 322-324](#Hoermander90).
See also
* Wikipedia: [wavefront set] (http://en.wikipedia.org/wiki/Wavefront_set)
### In quantum field theory
The application of [[microlocal analysis]] via wave front sets to the discussion of [[n-point functions]] in [[quantum field theory]] and especially [[quantum field theory on curved spacetimes]] originates with the results of
* {#DuistermaatHoermander72} [[Johann Duistermaat]], [[Lars Hörmander]], sections 6.5, 6.6 of _Fourier integral operators II_, Acta Mathematica 128, 183-269, 1972 ([Euclid](https://projecteuclid.org/euclid.acta/1485889724))
which were first picked up in
* C. Moreno, _Spaces of positive and negative frequency solutions of field equations in curved space- times. I. The Klein-Gordon equation in stationary space-times, II. The massive vector field equations in static space-times_, J. Math. Phys. 18, 2153-61 (1977), J. Math. Phys. 19, 92-99 (1978)
* [[Jonathan Dimock]], _Scalar quantum field in an external gravitational background_, J. Math. Phys. 20, 2549-2555 (1979)
and brought into context with the [[Hadamard distributions]] needed for the [[construction]] of [[Wick algebras]] in
* {#Radzikowski96} [[Marek Radzikowski]], *Micro-local approach to the Hadamard condition in quantum field theory on curved space-time*, Commun. Math. Phys. **179** (1996) 529-553 [[doi:10.1007/BF02100096](https://doi.org/10.1007/BF02100096), [euclid:cmp/1104287114](http://projecteuclid.org/euclid.cmp/1104287114)]
* {#BrunettiFredenhagen00} [[Romeo Brunetti]], [[Klaus Fredenhagen]], *Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds*, Commun. Math. Phys. **208** (2000) 623-661 [[math-ph/9903028](https://arxiv.org/abs/math-ph/9903028), [doi:10.1007/s002200050004](https://doi.org/10.1007/s002200050004)]
A textbook account amplifying this usage (on [[Minkowski spacetime]]) in the mathematically rigorous construction of [[perturbative quantum field theory]] via [[causal perturbation theory]] is in
* {#Scharf95} [[Günter Scharf]], _[[Finite Quantum Electrodynamics -- The Causal Approach]]_, Berlin: Springer-Verlag, 1995, 2nd edition
* {#Scharf01} [[Günter Scharf]], _[[Quantum Gauge Theories -- A True Ghost Story]]_, Wiley 2001
For more see the references at _[[locally covariant perturbative quantum field theory]]_.
### In differential cohomology
Wave-front sets of [[currents (distribution theory)|currents]] play a role in the construction of "geometric cycles" for [[differential cobordism cohomology]] by actual [[cobordism]]-classes equipped with [[differential geometry|differential geometric]] data:
* [[Ulrich Bunke]], [[Thomas Schick]], Ingo Schroeder, Moritz Wiethaup, §4.2.6 in: *Landweber exact formal group laws and smooth cohomology theories*, Algebr. Geom. Topol. **9** (2009) 1751-1790 [[arXiv:0711.1134](https://arxiv.org/abs/0711.1134), [doi:10.2140/agt.2009.9.1751](https://doi.org/10.2140/agt.2009.9.1751)]
and in the [[Hodge-filtered differential cohomology|Hodge-filtered]] version:
* {#Haus22} [[Knut Bjarte Haus]], §2.6 in: *Geometric Hodge filtered complex cobordism*, PhD thesis (2022) [[ntnuopen:3017489](https://ntnuopen.ntnu.no/ntnu-xmlui/handle/11250/3017489)]
* {#HausQuick22} [[Knut Bjarte Haus]], [[Gereon Quick]], §2.10 of: *Geometric Hodge filtered complex cobordism* [[arXiv:2210.13259](https://arxiv.org/abs/2210.13259)]
[[!redirects wavefront sets]]
[[!redirects wave front set]]
[[!redirects wave front sets]]
[[!redirects wave-front set]]
[[!redirects wave-front sets]]
[[!redirects wavefront]]
[[!redirects wavefronts]]
|
wavelength | https://ncatlab.org/nlab/source/wavelength |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Harmonic analysis
+-- {: .hide}
[[!include harmonic analysis - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The [[distance]] between two [[crests]] of a [[wave]].
## Definition
For $n \in \mathbb{N}$ and
$$
\vec x \mapsto e^{2 \pi i \vec k \cdot \vec x}
$$
a [[plane wave]] with [[wave vector]] $\vec k \in \mathbb{R}^n$, then its _wavelength_ is the inverse of the [[norm]] of $k$:
$$
\lambda \coloneqq 1/{\vert k\vert}
\,.
$$
The quotient $k/{\vert k \vert} \in S(\mathbb{R}^n)$ is the _[[direction of a vector|direction]]_ of the [[plane wave]].
The product $2\pi/\lambda$ is also called the _wave number_.
[[!include plane waves -- table]]
## Examples
* [[Compton wavelength]]
## Related concepts
* [[wave vector]]
* [[frequency]]
* [[Fourier analysis]]
* [[momentum]], [[energy]]
[[!redirects wavelengths]]
[[!redirects wave-length]]
[[!redirects wave length]]
[[!redirects wave-lengths]]
[[!redirects wave lengths]]
[[!redirects wave number]]
[[!redirects wave numbers]]
[[!redirects wavenumber]]
[[!redirects wavenumbers]]
|
weak (star)-autonomous category | https://ncatlab.org/nlab/source/weak+%28star%29-autonomous+category | ## Idea
Some [[symmetric monoidal categories]] have a notion of [[duality]] without being a [[*-autonomous category]]. However, they can be a weak $*$-autonomous category.
## Definition
\begin{definition}
A _weak $*$-autonomous category_ is a [[symmetric monoidal
category|symmetric]] [[closed monoidal category]] $\langle
C,\otimes, I,\multimap\rangle$ with an object
$\bot$ such that the canonical morphism
$$ d_A: A \to (A \multimap \bot) \multimap \bot ,$$
which is the transpose of the [[evaluation map]]
$$ ev_{A,\bot}: (A \multimap \bot) \otimes A \to \bot ,$$
is a [[monomorphism]] for all $A$. (Here, $\multimap$ denotes the [[internal hom]].)
\end{definition}
A $*$-autonomous category is a weak $*$-autonomous category such that for every object $A$, $d_{A}: A \to (A \multimap \bot) \multimap \bot$ is an isomorphism.
A strictly weak $*$-autonomous category is a weak $*$-autonomous category such that for at least one object $A$, $d_{A}$ is not an isomorphism. Equivalently, a strictly weak $*$-autonomous category is a weak $*$-autonomous category which is not a $*$-autonomous category.
## Example
* For every [[field]] $\mathbb{K}$, the category $Vec_{\mathbb{K}}$ is a strict weak $*$-autonomous category.
* For every [[commutative ring]] $R$, the category $Mod_{R}$ is a strict weak $*$-autonomous category.
## Related concepts
[[*-autonomous category]]
[[closed monoidal category]]
|
weak adjoint | https://ncatlab.org/nlab/source/weak+adjoint | # Weak adjoints
* table of contents
{: toc}
## Idea
A **weak adjoint** is like an [[adjoint functor]] but without the uniqueness of factorizations.
Importantly, this means that unlike adjoint functors, weak adjoints are not unique, and so a weak adjoint of a given functor is a [[structure]] whereas a functor having an adjoint is a [[property]].
## Definition
Let $G:D\to C$ be a [[functor]]. We say $G$ has a **left weak adjoint** if for each $x\in C$ there exists a morphism $\eta_x:x\to G z$ such that for any morphism $f:x\to G y$ there exists a (not necessarily unique) morphism $g:y\to z$ such that $f = G g \circ\eta_x$.
## Examples
* A [[weak limit]] is a weak right adjoint to a constant-diagram functor.
## Remarks
Of course, if the factorizations $g$ are always unique, this is precisely an ordinary [[left adjoint]]. A weak adjoint can also be regarded as a stronger form of the [[solution set condition]] in which the solution sets are required to be singletons. The "dual" property, in which the solution sets need not be singletons but the factorizations are unique, is called a [[multi-adjoint]].
## Related Concepts
* [[weak representation of a functor]]
* [[weakly initial object]]
## References
{#References}
Weak adjoint functors along with [[weak colimits]] were defined in:
* [[Paul Kainen]], *Weak adjoint functors*, Mathematische Zeitschrift **122** 1 (1971) 1-9 [[dml:171575](https://eudml.org/doc/171575)]
[[!redirects weak adjoints]]
[[!redirects weak adjoint functor]]
[[!redirects weak adjoint functors]]
[[!redirects weak adjunction]]
[[!redirects weak left adjoint]]
[[!redirects weak left adjoint]]
[[!redirects weak right adjoint]]
[[!redirects weak right adjoint]]
[[!redirects left weak adjoint]]
[[!redirects left weak adjoints]]
[[!redirects right weak adjoint]]
[[!redirects right weak adjoints]]
|
weak bialgebra | https://ncatlab.org/nlab/source/weak+bialgebra |
#Contents#
* table of contents
{:toc}
## Idea
The notion of _weak bialgebra_ is a generalization of that of [[bialgebra]] in which the comultiplication $\Delta$ is weak in the sense that $\Delta(1)\neq 1\otimes 1$ in general;
similarly the compatibility of counit with the multiplication
map is weakened (counit might fail to be a morphism of algebras). (Still a special case of [[sesquialgebra]].)
Correspondingly [[weak Hopf algebras]] generalize [[Hopf algebras]] accordingly. Every weak Hopf algebra defines a
[[Hopf algebroid]].
### Physical motivation
This kind of structures naturally comes in [[CFT]] models relation to quantum groups a root of unity: the full symmetry algebra is not quite a quantum group at root of unity, because if it were one would have to include the nonphysical quantum dimension zero finite-dimensional quantum group representations into the (pre)Hilbert space; those are the zero norm states which do not contribute to physics (like ghosts). If one quotients by these states then the true unit of a quantum group becomes an idempotent (projector), hence one deals with weak Hopf algebras instead as a price of dealing with true, physical, Hilbert space.
## Definitions
A __weak bialgebra__ is a tuple $(A,\mu,\eta,\Delta,\epsilon)$
such that $(A,\mu,\eta)$ is an associative unital algebra,
$(A,\Delta,\epsilon)$ is a coassociative counital coalgebra
and the following compatibilities, (i),(ii) and (iii), hold:
(i) the coproduct $\Delta$ is multiplicative $\Delta(x)\Delta(y)= \Delta(x y)$. If only (i) is satisfied, following Böhm, Caenapeel and Janssen 2011, we may speak of a __prebialgebra__.
(ii) the counit $\epsilon$ satisfies weak multiplicativity
$$
\epsilon(x y z) = \epsilon(x y_{(1)})\epsilon(y_{(2)} z),
$$
$$
\epsilon(x y z) = \epsilon(x y_{(2)})\epsilon(y_{(1)} z).
$$
A prebialgebra satisfying the first (the second) of the above properties is said to be left (right) monoidal.
(iii) Weak comultiplicativity of the unit:
$$
\Delta^{(2)} (1) = (\Delta(1) \otimes 1)(1\otimes \Delta(1))
$$
$$
\Delta^{(2)} (1) = (1 \otimes\Delta(1))(\Delta(1) \otimes 1)
$$
A prebialgebra satisfying the first (the second) of the above properties is said to be left (right) comonoidal.
As usually in the context of coassociative coalgebras, we denoted $\Delta^{(2)} := (id\otimes\Delta)\Delta =
(\Delta\otimes id)\Delta$.
A weak $k$-bialgebra $A$ is a __weak Hopf algebra__ if it has a $k$-linear map $S:A\to A$ (which is then called an antipode)
such that for all $x\in A$
$$
x_{(1)} S(x_{(2)}) = \epsilon(1_{(1)} x)1_{(2)},
$$
$$
S(x_{(1)})x_{(2)} = 1_{(1)} \epsilon(x 1_{(2)}),
$$
$$
S(x_{(1)})x_{(2)} S(x_{(3)}) = S(x)
$$
It follows that the antipode is antimultiplicative, $S(x y)=S(y)S(x)$, and
anticomultiplicative, $\Delta(S(x)) = S(x)_{(1)}\otimes S(x)_{(2)} =
S(x_{(2)})\otimes S(x_{(1)})$.
## Properties
### Idempotents ("projections")
For every weak bialgebra there are $k$-linear maps
$\Pi^L,\Pi^R:A\to A$ defined by
$$
\Pi^L(x) := \epsilon(1_{(1)} x) 1_{(2)},\,\,\,\,
\Pi^R(x) := 1_{(1)}\epsilon(x 1_{(2)}).
$$
Expressions for $\Pi^L(x),\Pi^R(x)$ are already met above as the right hand sides in two of the axioms for the antipode. Maps $\Pi^L,\Pi^R$
are idempotents, $\Pi^R\Pi^R = \Pi^R$ and $\Pi^L\Pi^L = \Pi^L$:
$$\array{
\Pi^L(\Pi^L(x)) &=& \epsilon\left(1_{(1')}\epsilon(1_{(1)}x) 1_{(2)}\right)1_{(2')}
= \epsilon(1_{(1)}x)\epsilon(1_{(1')}1_{(2)}) 1_{(2')}
\\ &=&\epsilon(1_{(1)}x)\epsilon(1_{(2)}) 1_{(3)}
= \epsilon(1_{(1)}x)1_{(2)} = \Pi^L(x).
}$$
Notice $\epsilon(x z) = \epsilon(x 1 z) = \epsilon(x 1_{(2)})\epsilon(1_{(1)}z)) = \epsilon(x \epsilon(1_{(1)}z))1_{(2)} = \epsilon(x\Pi^L(z)) = \epsilon(\Pi^R(x)z)$.
The images of the idempotents $A^R = \Pi^R(A)$ and $A^L = \Pi^L(R)$ are dual as $k$-linear spaces: there
is a canonical nondegenerate pairing $A^L\otimes A^R\to k$
given by $(x,y) \mapsto \epsilon(y x)$.
Also $\Pi^L(x\Pi^L(y)) = \Pi^L(x y)$
and $\Pi^R(\Pi^R(x)y) = \Pi^R(x y)$,
dually $\Delta(A^L)\subset A\otimes A^L$
and $\Delta(A^R)\subset A^R\otimes A$,
and in particular $\Delta(1)\in A^R\otimes A^L$.
Sometimes it is also useful to consider the idempotents $\bar\Pi^L,\bar\Pi^R:A\to A$ defined by
$$
\bar\Pi^L(x) := \epsilon(1_{(2)} x) 1_{(1)},\,\,\,\,
\bar\Pi^R(x) := 1_{(2)}\epsilon(x 1_{(1)}).
$$
$$\array{
\bar\Pi^L(\bar\Pi^L(x))&=&\epsilon(1_{(2')}\epsilon(1_{(2)}x)1_{(1)})1_{(1')} = \epsilon(1_{(2)}x)\epsilon(1_{(2')}1_{(1)})1_{(1')} \\
&=& \epsilon(1_{(3)}x)\epsilon(1_{(2)})1_{(1)}=
\epsilon(1_{(2)}x)1_{(1)} = \bar\Pi^L(x).
}$$
### Relation to fusion categories
Under [[Tannaka duality]] (semisimple) weak Hopf algebras correspond to (multi-)[[fusion categories]] ([Ostrik](#Ostrik)).
### Relation to Frobenius algebras
As explained in [[Hopf algebra]], any finite-dimensional Hopf algebra can be given the structure of a [[Frobenius algebra]]. There is a similar result for weak Hopf algebras.
+-- {: .num_prop #WeakHopfAlgebraIsQuasiFrobeniusAlgebra}
###### Proposition
Any finite-dimensional weak Hopf algebra can be given the structure of a [[quasi-Frobenius algebra]].
=--
This is due to [Bohm, Nill, and Szlachanyi (1999)](#BNS99). While [Vecsernyés (2003)](Vec03) seems to show that finite-dimensional weak Hopf algebras can be turned into Frobenius algebras, it is observed in [Iovanov & Kadison (2008)](IK08) that the proof only implies they are *quasi-*Frobenius algebras.
## Literature
Weak comultiplications were introduced in
* G. Mack, [[Volker Schomerus]], _Quasi Hopf quantum symmetry in quantum theory_, Nucl. Phys. B370(1992) 185.
where also weak quasi-bialgebras are considered and physical motivation is discussed in detail. Further work in this vain is in
* [[G. Böhm]], [[K. Szlachányi]], _A coassociative $C^\ast$-quantum group with non-integral dimensions_, Lett. Math. Phys. __35__ (1996) 437--456, [arXiv:q-alg/9509008](http://arxiv. rg/abs/q-alg/9509008); _Weak $C*$-Hopf algebras: the coassociative symmetry of non-integral dimensions_, in: Quantum groups and quantum spaces (Warsaw, 1995), 9-19, Banach Center Publ. __40__, Polish Acad. Sci., Warszawa 1997.
* Florian Nill, _Axioms for weak bialgebras_, [math.QA/9805104](http://arxiv.org/abs/math/9805104)
* [[G. Böhm]], F. Nill, [[K. Szlachányi]], _Weak Hopf algebras. I. Integral theory and $C^\ast$-structure, J. Algebra __221__ (1999), no. 2, 385-438, [math.QA/9805116](https://arxiv.org/abs/math/9805116) #{BohmNillSzlachanyi}
A book exposition is in [chapter](https://link.springer.com/chapter/10.1007/978-3-319-98137-6_6) weak (Hopf) bialgebras in
* [[Gabriella Böhm]], _Hopf algebras and their generalizations from a category theoretical point of view_, Lecture Notes in Math. __2226__, Springer 2018, [doi](https://link.springer.com/book/10.1007/978-3-319-98137-6)
Now these works are understood categorically from the point of view of weak monad theory:
* [[Gabriella Böhm]], Stefaan Caenepeel, Kris Janssen, _Weak bialgebras and monoidal categories_, Comm. Algebra __39__ (2011), no. 12 (special volume dedicated to Mia Cohen), 4584-4607. [arXiv:1103.226](http://arxiv.org/abs/1103.2261)
* [[Gabriella Böhm]], [[Stephen Lack]], [[Ross Street]], _Weak bimonads and weak Hopf monads_, J. Algebra 328 (2011), 1-30, [arXiv:1002.4493](http://arxiv.org/abs/1002.4493)
* [[Gabriella Böhm]], José Gómez-Torrecillas, _On the double crossed product of weak Hopf algebras_, [arXiv:1205.2163](arXiv/1205.2163)
The relation to [[fusion categories]] is discussed in
* Takahiro Hayashi, _A canonical Tannaka duality for finite semisimple tensor categories_ ([arXiv:math/9904073](http://arxiv.org/abs/math/9904073))
* [[Victor Ostrik]], _Module categories, weak Hopf algebras and modular invariants_ ([arXiv:math/0111139](http://arxiv.org/abs/math/0111139))
{#Ostrik}
On the relation to [[Frobenius algebra|Frobenius algebras]]
* {#BNS99} [[Gabriella Bohm]], [[Florian Nill]], [[Kornel Szlachanyi]]. *Weak Hopf Algebras I: Integral Theory and* $C^*$*-structure*. (1999). ([arXiv:math/9805116](https://arxiv.org/abs/math/9805116))
* {#Vec03} [[Peter Vecsernyés]]. *Larson–Sweedler theorem and the role of grouplike elements in weak Hopf algebras*. Journal of Algebra. Volume 270, Issue 2, 15 December 2003, Pages 471-520. ([doi](https://doi.org/10.1016/j.jalgebra.2003.02.001))
* {#IK08} [[Miodrag Iovanov]], [[Lars Kadison]]. *When weak Hopf algebras are Frobenius*. (2008). ([arXiv:0810.4777](https://arxiv.org/abs/0810.4777))
category: algebra
[[!redirects weak bialgebras]]
[[!redirects weak Hopf algebra]]
[[!redirects weak Hopf algebras]]
|
weak bimonad | https://ncatlab.org/nlab/source/weak+bimonad | ## Idea
[[weak bialgebra|Weak bialgebra]] is like bialgebra, except that the requirement that the counit map is an algebra map and the unit is a coalgebra map are appropriately relaxed. Like bialgebras, this notion can also be properly generalized to a monoidal/bicategorical context leading to weak bimonads.
Distributive laws among monads are monads in appropriate bicategory/2-category of monads. Similarly, one can understand [[weak distributive law]]s.
## Literature
* [[Gabriella Böhm]], [[Stephen Lack]], [[Ross Street]], _Weak bimonads and weak Hopf monads_, J. Algebra 328 (2011), 1-30, [arXiv:1002.4493](http://arxiv.org/abs/1002.4493)
* [[Gabriella Böhm]], [[Stephen Lack]], [[Ross Street]], _On the 2-categories of weak distributive laws_, Comm. Alg. _39_:12 (2011) 4567--4583 [doi](https://doi.org/10.1080/00927872.2011.616436)
* Yuanyuan Chen, [[Gabriella Böhm]], _Weak bimonoids in duoidal categories_, Journal of Pure and Applied Algebra __218__:12 (2014) 2240--2273 [doi](https://doi.org/10.1016/j.jpaa.2014.04.001)
|
weak Cayley table | https://ncatlab.org/nlab/source/weak+Cayley+table | For $G$ a group, its weak Cayley table is the map which sends elements in $G\times G$ to their [[higher group character|biconjugacy class]].
#References#
* Kenneth W. Johnson, Sandro Mattarei and Surinder K. Sehgal, _Weak Cayley tables_, [Journal of the London Mathematical Society 2000 61(2):395-411](http://jlms.oxfordjournals.org/cgi/content/abstract/61/2/395) ([pdf](http://journals.cambridge.org/download.php?file=%2FJLM%2FJLM61_02%2FS0024610799008571a.pdf&code=e119cd0ef145cd79d298f403f4d38dc7))
|
weak colimit | https://ncatlab.org/nlab/source/weak+colimit |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A **weak colimit** for a [[diagram]] in a [[category]] is a [[cocone]] over that diagram which satisfies the existence property of a [[colimit]] but not necessarily the uniqueness. The dual concept is a [[weak limit]], see there for more.
For example, a **[[weakly initial object]]** in a category, $C$, is such that there is at least one arrow from it to any object in $C$.
## Related concepts
* [[weak limit]]
* [[weak multilimit]]
* [[WISC]]
* [[weakly initial set]]
* [[weakly initial object]]
## References
[[weak adjoint|Weak adjoint functors]] along with [[weak colimits]] were defined in:
* [[Paul Kainen]], *Weak adjoint functors*, Mathematische Zeitschrift **122** 1 (1971) 1-9 [[dml:171575](https://eudml.org/doc/171575)]
[[!redirects weak colimits]]
[[!redirects weak multicolimit]]
[[!redirects weakly initial object]]
|
weak complicial set | https://ncatlab.org/nlab/source/weak+complicial+set |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Weak complicial sets are [[simplicial sets]] with [[stuff, structure, property|extra structure]] that are closely related to the [[∞-nerve]]s of weak [[∞-categories]].
The goal of characterizing such nerves, without an *a priori* definition of "weak $\omega$-category" to start from, is called [[simplicial weak ∞-category]] theory. It is expected that the (nerves of) weak $\omega$-categories will be weak complicial sets satisfying an extra "saturation" condition ensuring that "every [[equivalence]] is [[thin element|thin]]." General weak complicial sets can be regarded as "presentations" of weak $\omega$-categories.
Weak complicial sets are a joint generalization of
* [[strict omega-category|strict ∞-categories]];
* [[Kan complexes]];
* [[quasi-category|quasi-categories]].
## Definition
+-- {: .num_defn}
###### Definition
Let
* $\Delta^k[n]$ be the [[stratified simplicial set]] whose underlying simplicial set is the $n$-[[simplex]] $\Delta[n]$, and whose marked cells are precisely those simplices $[r] \to [n]$ that contain $\{k-1, k, k+1\} \cap [n]$;
* $\Lambda^k[n]$ be the stratified simplicial set whose underlying simplicial set is the $k$-[[horn]] of $\Delta[n]$, with marked cells those that are marked in $\Delta^k[n]$;
* $\Delta^k[n]'$ be obtained from $\Delta^k[n]$ by making the $(k-1)$st $(n-1)$-face and the $(k+1)$st $(n-1)$ face thin;
* $\Delta^k[n]''$ be obtained from $\Delta^k[n]$ by making all $(n-1)$-faces thin.
An **elementary anodyne extension** in $Strat$, the category [[stratified simplicial sets]] is
* a **complicial horn extension** $\Lambda^k[n] \stackrel{\subset_r}{\hookrightarrow} \Delta^k[n]$
or
* a **complicial thinness extension** $\Delta^k[n]' \stackrel{\subset_e}{\hookrightarrow} \Delta^k[n]''$
for $n = 1,2, \cdots$ and $k \in [n]$.
=--
+-- {: .num_defn}
###### Definition
A [[stratified simplicial set]] is a **weak complicial set** if it has the [[right lifting property]] with respect to all
$\Lambda^k[n] \stackrel{\subset_r}{\hookrightarrow} \Delta^k[n]$ and
$\Delta^k[n]' \stackrel{\subset_e}{\hookrightarrow} \Delta^k[n]''$
A [[complicial set]] is a weak complicial set in which such liftings are unique.
=--
## Model structure
There is a [[model category]] structure that presents the [[(infinity,1)-category]] of weak complicial sets, hence that of weak $\omega$-categories. See
* [[model structure for weak complicial sets]].
## Examples
* For $C$ a [[strict ∞-category]] and $N(C)$ its [[oriental|∞-nerve]], the _Roberts stratification_ which regards each identity morphism as a thin cell makes $N(C)$ a strict [[complicial set]], hence a weak complicial set. This example is not "saturated."
* There is also the [[stratified simplicial set|stratification]] of $N(C)$ which regards each $\omega$-equivalence morphism as a thin cell. $N(C)$ with this stratification is a weak complicial set (example 17 of [Ver06](http://arxiv.org/abs/math/0604414)). This should be the "saturation" of the previous example, and exhibits the inclusion of strict $\omega$-categories into weak ones.
* A simplicial set is a weak complicial set when equipped with its maximal [[stratified simplicial set|stratification]] (every simplex of dimension $\gt 0$ is thin) if and only if it is a [[Kan complex]]. This example is, of course, saturated, and is viewed as embedding $\omega$-groupoids into $\omega$-categories.
* A simplicial set is a [[quasi-category]] if and only if it is a weak complicial set when equipped with the stratification in which every simplex of dimension $\gt 1$ is thin, and only degenerate 1-simplices are thin. This example is not saturated; in its saturation the thin 1-simplices are the internal equivalences in a quasi-category (equivalently, those that become isomorphisms in its [[homotopy category]]). It presents the embedding of $(\infty,1)$-[[(infinity,1)-category|categories]] into weak $\omega$-categories.
Note that 1-simplex equivalences in a quasi-category are automatically preserved by simplicial maps between quasi-categories; this is why $QCat$ can "correctly" be regarded as a full subcategory of $sSet$. This is not true at higher levels; for instance not every simplicial map between nerves of strict $\omega$-categories necessarily preserves $\omega$-equivalence morphisms.
## References
The definition of weak complicial sets is definition 14, page 9 of
* [[Dominic Verity]], _Weak complicial sets Part I: Basic homotopy theory_ ([arXiv](http://arxiv.org/abs/math/0604414))
Further developments are in
* Dominic Verity, _Weak complicial sets Part II: Nerves of complicial Gray-categories_ ([arXiv](http://arxiv.org/abs/math/0604416))
A [[model category]] structure on [[stratified simplicial sets]] modelling [[(infinity,n)-categories|$(\infty,n)$-categories]] in the guise of [[n-complicial sets|$n$-complicial sets]]:
* [[Viktoriya Ozornova]], [[Martina Rovelli]], *Model structures for (∞,n)-categories on (pre)stratified simplicial sets and prestratified simplicial spaces*, Algebr. Geom. Topol. **20** (2020) 1543-1600 [[arxiv:1809.10621](https://arxiv.org/abs/1809.10621), [doi:10.2140/agt.2020.20.1543](https://doi.org/10.2140/agt.2020.20.1543)]
A [[Quillen adjunction]] relating [[n-complicial sets|$n$-complicial sets]] to [[n-fold complete Segal spaces|$n$-fold complete Segal spaces]]:
* [[Viktoriya Ozornova]], [[Martina Rovelli]], *A Quillen adjunction between globular and complicial approaches to $(\infty,n)$-categories*, Advances in Mathematics
**421** (2023) 108980 [[doi:10.1016/j.aim.2023.108980](https://doi.org/10.1016/j.aim.2023.108980), [arXiv:2206.02689](https://arxiv.org/abs/2206.02689)]
Review:
* {#Rovelli2023} [[Martina Rovelli]], *$n$-Complicial sets as a model for $(\infty,n)$-categories*, [talk at](Center+for+Quantum+and+Topological+Systems#RovelliMay2023) *[[CQTS]]* (2023) [[web](Center+for+Quantum+and+Topological+Systems#RovelliMay2023), video: [YT](https://www.youtube.com/watch?v=T9Bg1AdaKv8)]
[[!redirects weak complicial sets]]
[[!redirects weak compicial set]]
[[!redirects n-complicial set]]
[[!redirects n-complicial sets]]
|
weak counterexample | https://ncatlab.org/nlab/source/weak+counterexample |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Mathematics
+-- {: .hide}
[[!include mathematicscontents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Recall that a _[[counterexample]]_ to a [[conjecture]] is a way to satisfy the hypotheses of the conjecture such that the conclusion is a contradiction.
In [[constructive mathematics]] and the [[foundations of mathematics]], a __weak counterexample__ to a theorem from (typically) [[classical mathematics]] is a way to satisfy the hypotheses of the theorem such that the conclusion is unacceptable in some way, without being an outright contradiction.
Weak counterexamples were first (at least under that name) used informally by [[Jan Brouwer]] in [[intuitionistic mathematics]]. They can be made into bona fide counterexamples by specifying a formal system and checking one of the following:
* [[proof theory|Proof-theoretically]], if the unacceptable conclusion is known to be unprovable in the system, then we have a counterexample to the conjecture that every instance of the classical theorem can be proved in the system.
* [[model theory|Model-theoretically]], if a model is known in which the unacceptable conclusion fails, then we have a counterexample to the conjecture that the classical theorem is true in every model of the system.
Either way, we now know that the classical theorem cannot be proved in the formal system. One typically uses a weak counterexample when the classical theorem cannot be outright refuted.
## Related concepts
* [[theory]]
* [[axiom]]
* [[definition]]
* [[lemma]]
* [[proposition]]
* [[theorem]]
* [[proof]]
* [[example]]
* [[counterexample]]
* [[conjecture]]
* [[folklore]]
* [[analogy]]
* [[paradox]]
## See also
* [[taboo]]
[[!redirects weak counterexamples]]
|
weak distributive law | https://ncatlab.org/nlab/source/weak+distributive+law |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
* table of contents
{:toc}
## Idea
The notion of a **weak distributive law** between two [[monads]] is a generalisation of that of a [[distributive law]], in which forming the [[composition|composite]] monad requires [[split idempotent|splitting]] an [[idempotent]] on the [[underlying]] [[composition|composite]] [[endofunctor]].
## Related concepts
* [[weak wreath product]]
## References
See also [[weak bimonad]].
Weak distributive laws among monads:
* [[Gabriella Böhm]], _The weak theory of monads_, Adv. in Math. __225__:1 (2010) 1--32 [doi](https://doi.org/10.1016/j.aim.2010.02.015)
For the weak _mixed_ distributive law (monad and comonad) version see
* [[Ross Street]], *Weak distributive laws*, Theory and Appl. of Categ. **22** (2009) 313--320 [[tac:22-12](http://www.tac.mta.ca/tac/volumes/22/12/22-12abs.html)]
2-categorical context in the sense of formal theory of monads is also exposed in
* [[Gabriella Böhm]], [[Stephen Lack]], [[Ross Street]], _On the 2-categories of weak distributive laws_, Comm. Alg. _39_:12 (2011) 4567--4583 [doi](https://doi.org/10.1080/00927872.2011.616436)
* [[Daniela Petrisan]], Ralph Sarkis. _Semialgebras and weak distributive laws_, Proceedings 37th Conference on
Mathematical Foundations of Programming Semantics, EPTCS 351 (2021) 218--241. ([doi:10.4204/EPTCS.351.14](https://doi.org/10.4204/EPTCS.351.14))
An application of weak distributive laws to explain [[weak wreath product]]s (comparable to the treatment of [[wreath]]s in bicategories) and also related bilinear factorization structures
* [[Gabriella Böhm]], _On the iteration of weak wreath products_, Theory and Appl. of Categories __26__:2 (2012) 30--59 [arXiv:1110.0652](https://arxiv.org/abs/1110.0652)
* [[Gabriella Böhm]], [[José Gómez-Torrecillas]], _Bilinear factorization of algebras_, Bull. Belg. Math. Soc. Simon Stevin 20(2): 221-.244 (may 2013) [doi](https://doi.org/10.36045/bbms/1369316541)
[[!redirects weak distributive laws]]
|
weak enrichment | https://ncatlab.org/nlab/source/weak+enrichment |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### Enriched category theory
+--{: .hide}
[[!include enriched category theory contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
An analog of [[enriched category]] in [[higher category theory]]. Enrichment in a monoidal category $(V,\otimes,I)$ requires composition morphisms
$$Hom (A,B) \otimes Hom(B,C) \to Hom(A,C)$$
and identity morphisms
$$I \to Hom(A,A)$$
in $V$ satisfying the axioms of associativity and unitality. To weakly enrich, we wish to have the axioms of associativity and unitality hold up to coherent 2-morphisms. To accomplish this we can replace $V$ with a [[monoidal bicategory]] which has sufficient structure to accommodate this. This describes one approach to weakly enriching in the 2-dimensional case. For higher values of n we need to replace $V$ with a weak n-category.
## Definition
A category weakly enriched in a [[monoidal bicategory]] $W$ is called a $W$-bicategory. The data of a $W$-bicategory $C$ includes the data of an ordinary enriched category in the [[decategorification]] of $W$. In addition, $C$ must contain associator and unitor 2-morphisms in $W$ which fill the appropriate diagrams of composition and unit 1-morphisms. These 2-cells are required to satsify higher dimensional coherence axioms.
Note that this should not be confused with a [[category enriched in a bicategory]] which allows for multiple bases of enrichment. A $Cat$-bicategory is an ordinary bicategory. Many examples come from weakly enriching in the [[monoidal 2-category]] $V$-$Cat$ of categories enriched in $V$.
In the context of [[(∞,1)-category theory]] see at
_[[enriched (∞,1)-category]]_.
In the context of [[(∞,1)-operad]] theory see ([Lurie, def. 4.2.1.12](#Lurie)):
Write $\mathcal{LM}^\otimes$ the [[operad for modules over an algebra]] regarded as an [[(∞,1)-operad]], regarded as the [[(∞,1)-category of operators]]. Similarly write $\mathcal{Ass}^\otimes$ for the [[(∞,1)-category of operators]] of the [[associative operad]].
+-- {: .num_defn }
###### Definition
For $\mathcal{V}^\otimes \to \mathcal{Ass}^\otimes$ exhibiting a [[planar (∞,1)-operad]], a **weak enrichment** of an [[(∞,1)-category]] $\mathcal{C}$ over $\mathcal{C}^\otimes$ is a [[fibration of (∞,1)-operads]]
$$
q \colon \mathcal{O}^\otimes \to \mathcal{LM}^{\otimes}
$$
which exhibits $\mathcal{C}$ as a module over $\mathcal{V}^\otimes$ in that it is equipped with [[equivalence in an (∞,1)-category|equivalences]]
$$
\mathcal{V}^\otimes \simeq \mathcal{O}^\otimes_{\mathfrak{a}}
$$
and
$$
\mathcal{C} \simeq \mathcal{O}^\otimes_{\mathfrak{n}}
\,.
$$
=--
([Lurie, def. 4.2.1.12](#Lurie))
> maybe better: weak _[[tensoring]]_?
## Related concepts
* [[homotopical enrichment]];
* [[enriched (∞,1)-category]]
## References
$W$-bicategories are briefly explained in Section 4 of
* [[Richard Garner]], [[Nick Gurski]], The low-dimensional structures formed by tricategories, [arXiv:0711.1761](https://arxiv.org/abs/0711.1761)
Section 4.2.1 of
* [[Jacob Lurie]], _[[Higher Algebra]]_
{#Lurie}
* [[David Gepner]], [[Rune Haugseng]], _Enriched ∞-categories via non-symmetric ∞-operads_, [arXiv:1312.3178](https://arxiv.org/abs/1312.3178)
## Discussion
This was originally at [[bicategory]]:
+-- {: .query}
_Sebastian_: Is there a formal meaning of _weak enrichment_?
[[John Baez]]: Yes there is; indeed Clark Barwick is writing a huge book on this.
_Sebastian_: If not, is there at least a method how to get the definition of a weak $n$-category if I know the definition of a (strict) $n$-category?
[[John Baez]]: that sounds harder! That's like pushing a rock uphill. It's easier to go down from weak to strict.
_Sebastian_: Of course, I have recognised that there are actually different definitions of what a weak n-category should be... so to give my question a bit more precision: How do I get a definition of a weak $n$-category that is as close as possible to the definition of a strict $n$-category? The weak $n$-category should be what you call "globular", I think. (Are there different definitions of globular (weak or strict) n-categories?)
[[John Baez]]: globular strict n-categories have been understood since time immemorial, or at least around 1963, and there is just one reasonable definition. Globular weak n-categories were defined in the 1990s by Michael Batanin, and his theory relates them quite nicely to the globular strict ones. But there is also a _different_ definition of globular weak n-categories due to Penon. It had a mistake in it which has now been fixed.
From the preface to [[johnbaez:Towards Higher Categories|Towards Higher Categories]]:
There is a quite different and more extensively developed operadic approach to globular [[nlab:weak omega-category|weak infinity-categories]] due to Batanin (Bat1, Str2), with a variant due to Leinster (Lein3). Penon (Penon) gave a related, very compact definition of infinity-category; this definition was later corrected and improved by Batanin (Bat2) and Cheng and Makkai (ChMakkai).
You can get the references there.
I think this discussion should be moved over to some page on n-categories, since it's not really about bicategories.
=--
[[!redirects weak tensoring]]
|
weak equivalence | https://ncatlab.org/nlab/source/weak+equivalence |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Equality and Equivalence
+--{: .hide}
[[!include equality and equivalence - contents]]
=--
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A _weak equivalence_ is a [[morphism]] in a [[category]] $C$ which is supposed to be a true [[equivalence]] in a [[higher category theory|higher categorical]] refinement of $C$.
The bare minimum of axioms to be satisfied by a weak equivalence are encoded in the concepts of [[category with weak equivalences]] and [[homotopical category]]. For such categories one can consider
* the corresponding [[homotopy category]], which is the universal solution to turning all weak equivalences into [[isomorphisms]];
* the corresponding $(\infty,1)$-[[(infinity,1)-category|category]], which is, roughly, the universal solution to turning all weak equivalences into higher categorical [[equivalences]]. There are various versions of this construction depending on what model for $(\infty,1)$-categories is chosen.
* The [[Dwyer-Kan localization]] uses [[simplicially enriched category|simplicially enriched categories]] to model $(\infty,1)$-categories.
* If we use [[complete Segal spaces]] or [[quasicategories]] to model $(\infty,1)$-categories, then the construction is a version of _fibrant replacement_.
Often, categories having weak equivalences also have extra structure that makes them easier to work with. A very powerful, and commonly occurring, level of such structure is called a [[model category|model structure]]. There are also various weaker levels of structure, such as a [[category of fibrant objects]].
## Examples
* A [[weak homotopy equivalence]] is a weak equivalence in the [[classical model structure on topological spaces]].
* A [[simplicial weak equivalence]] is a weak equivalence in the [[classical model structure on simplicial sets]].
* An [[equivariant weak homotopy equivalence]] is a weak equivalence in the [[fine model structure on topological G-spaces]].
* A weak equivalence in the [[folk model structure on Cat]] is a functor that is [[full and faithful functor|fully faithful]] and [[essentially surjective functor|essentially surjective]].
* {#EquivalencesOfCategoriesInHomotopyTypeTheory} In [[homotopy type theory]], a [[functor]] $F: A \to B$ between [[internal categories in HoTT]] $A$ and $B$ is called a "weak equivalence" if it is [[fully faithful]] and [[essentially surjective]].
For [[univalent categories]] there is no difference between weak equivalences and [[equivalences]].
## Related concepts
* [[equality]]
* [[isomorphism]]
* [[equivalence]]
* **weak equivalence**
* [[homotopy equivalence]], [[weak homotopy equivalence]]
* [[equivalence in an (∞,1)-category]]
* [[equivalence of (∞,1)-categories]]
[[!redirects weak equivalences]] |
weak equivalence of internal categories | https://ncatlab.org/nlab/source/weak+equivalence+of+internal+categories |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Internal categories
+--{: .hide}
[[!include internal infinity-categories contents]]
=--
#### Equality and Equivalence
+--{: .hide}
[[!include equality and equivalence - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The naive [[2-category]] $Cat(S)$ of [[internal categories]] [[internalization|in]] an ambient [[category]] $S$ does in general not have enough [[equivalences of categories]], due to the failure of the [[axiom of choice]] in $S$. Those [[internal functors]] which *should* but may not have [[inverses]] up to internal [[natural isomorphism]], namely those which are suitably [[fully faithful functor|fully faithful]] and [[essentially surjective functor|essentially surjective]], may be regarded as [[weak equivalences]] of internal categories ([Bunge & Paré 1979](#BungePare79)). The [[bicategory of fractions|2-category theoretic localisation]] of $Cat(S)$ at this class of [[1-morphisms]] then serves as a more natural [[2-category]] of [[internal categories]].
## Definition
Let $f:X\to Y$ be a functor between categories internal to some category $S$. $f$ is **fully faithful** if the following diagram is a pullback
$$
\begin{matrix}
X_1& \stackrel{f_1}{\to} & Y_1 \\
\downarrow&& \downarrow \\
X_0\times X_0 &\underset{f_0\times f_0}{\to} & Y_0\times Y_0
\end{matrix}
$$
To discuss the analogue of essential surjectivity, we need a notion of 'surjectivity', as this does not generalise cleanly from $Set$. If we are working in a topos, a natural choice is to take epimorphisms, but weaker ambient categories are sometimes needed. A natural choice is to work in a unary [[site]], where the covers are taken as the 'surjective' maps.
Given a functor $f:X\to Y$ internal to a unary site $(S,J)$, $f$ is **essentially $J$-surjective** if the map $t\circ pr_2:X_0 \times_{f_0,Y_0,s}Y_1 \to Y_0$ is a $J$-cover.
We then define an internal functor to be a **$J$-equivalence** if it is fully faithful and essentially $J$-surjective.
## Related concepts
* [[canonical model structure on Cat]]
* [[anafunctor]]
* [[equivalence of categories]], [[weak equivalence]]
[[!redirects weak equivalencs of internal categories]]
## References
* {#BungePare79} [[Marta Bunge]], [[Robert Paré]], _Stacks and equivalence of indexed categories_, [[Cahiers|Cahiers de Topologie et Géométrie Différentielle Catégoriques]], 20 no. 4 (1979), p. 373-399 ([numdam:CTGDC_1979__20_4_373_0](http://www.numdam.org/item?id=CTGDC_1979__20_4_373_0))
* [[Tomas Everaert]], [[R.W.Kieboom]] and [[Tim Van der Linden]], _Model structures for homotopy of internal categories_, Theory and Applications of Categories 15 (2005), no. 3, 66-94. ([journal](http://www.tac.mta.ca/tac/volumes/15/3/15-03abs.html))
* [[David Roberts]], _Internal categories, anafunctors and localisation_, Theory and Applications of Categories, 26 (2012) No. 29, pp 788-829. ([journal](http://www.tac.mta.ca/tac/volumes/26/29/26-29abs.html))
[[!redirects weak equivalences of internal categories]]
|
weak excluded middle | https://ncatlab.org/nlab/source/weak+excluded+middle | # Weak excluded middle
* table of contents
{: toc}
## Definition
In [[logic]], the principle of **weak excluded middle** says that for any [[proposition]] $P$ we have $\neg P \vee \neg\neg P$.
This follows from the full principle of [[excluded middle]], which says that for any $Q$ we have $Q \vee \neg Q$ (take $Q = \neg P$). Thus, in [[classical mathematics]] weak excluded middle is just true. But in [[constructive mathematics]] (i.e. [[intuitionistic logic]]), it is a weaker assumption than full excluded middle.
## Equivalent forms
### De Morgan's Law
In intuitionistic logic, **de Morgan's law** often refers to the one of [[De Morgan duality|de Morgan's four laws]] that is *not* an intuitionistic tautology, namely $\neg (P \wedge Q) \to (\neg P \vee \neg Q)$ for any $P,Q$.
+-- {: .un_theorem}
###### Theorem
De Morgan's law is equivalent to weak excluded middle.
=--
+-- {: .proof}
###### Proof
If de Morgan's law holds, then since $\neg (P \wedge \neg P)$, we have $\neg P \vee \neg\neg P$, as desired. Conversely, if weak excluded middle holds and we have $\neg (P\wedge Q)$, then from weak excluded middle we get $\neg P \vee \neg\neg P$ and $\neg Q \vee \neg\neg Q$ which give four cases. In three of those cases $\neg P \vee \neg Q$ holds, while in the fourth we have $\neg\neg P$ and $\neg\neg Q$, which together imply $\neg\neg(P\wedge Q)$ (see the first lemma [here](https://ncatlab.org/nlab/show/Heyting+algebra#ToBooleanAlgebras) and its proof), contradicting the assumption of $\neg (P\wedge Q)$; so the fourth case is impossible.
=--
## See also
* [[De Morgan's law]]
* [[De Morgan Heyting algebra]]
* [[De Morgan Heyting category]]
[[!redirects principle of weak excluded middle]]
[[!redirects law of weak excluded middle]]
[[!redirects weak principle of excluded middle]]
[[!redirects weak law of excluded middle]]
[[!redirects weak PEM]]
[[!redirects weak LEM]]
|
weak factorization system | https://ncatlab.org/nlab/source/weak+factorization+system |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Factorization systems
+--{: .hide}
[[!include factorization systems - contents]]
=--
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A _weak factorization system_ on a [[category]] is a [[pair]] $(\mathcal{L},\mathcal{R})$ of [[classes]] of [[morphisms]] ("[[projective morphisms]]" and "[[injective morphisms]]") such that 1) every morphism of the category factors as the composite of one in $\mathcal{L}$ followed by one in $\mathcal{R}$, and 2) $\mathcal{L}$ and $\mathcal{R}$ are closed under having the [[lifting property]] against each other.
If the liftings here are unique, then one speaks of an _orthogonal factorization system_. A classical example of an orthogonal factorization system is the [[(epi,mono)-factorization system]] on the category [[Set]] or in fact on any [[topos]].
Non-orthogonal weak factorization systems are the key ingredient in [[model categories]], which by definition carry a weak factorization system called ($\mathcal{L} = $ [[cofibrations]],$\mathcal{R} = $ [[acyclic fibrations]]) and another one called ($\mathcal{L} =$ [[acyclic cofibrations]], $\mathcal{R} =$ [[fibrations]]). Indeed most examples of non-orthogonal weak factorization systems arise in the context of model category theory. A key tool for constructing these, or verifying their existence, is the _[[small object argument]]_.
There are other [[properties]] which one may find or impose on a weak factorization system, for instance [[functorial factorization]]. There is also [[extra structure]] which one may find or impose, such as for [[algebraic weak factorization systems]]. For more variants see at _[[factorization system]]_.
## Definition
### Weak factorization systems
+-- {: .num_defn #WeakFactorizationSystem}
###### Definition
A **weak factorization system** (WFS) on a [[category]] $\mathcal{C}$ is a [[pair]] $(\mathcal{L},\mathcal{R})$ of [[classes]] of [[morphisms]] of $\mathcal{C}$ such that
1. Every [[morphism]] $f \colon X\to Y$ of $\mathcal{C}$ may be factored as the [[composition]] of a morphism in $\mathcal{L}$ followed by one in $\mathcal{R}$
$$
f\;\colon\; X \overset{\in \mathcal{L}}{\longrightarrow} Z \overset{\in \mathcal{R}}{\longrightarrow} Y
\,.
$$
1. The classes are closed under having the [[lifting property]] against each other:
1. $\mathcal{L}$ is precisely the class of morphisms having the [[left lifting property]] against every morphism in $\mathcal{R}$;
1. $\mathcal{R}$ is precisely the class of morphisms having the [[right lifting property]] against every morphism in $\mathcal{L}$.
=--
+-- {: .num_defn #FunctorialFactorization}
###### Definition
For $\mathcal{C}$ a [[category]], a **[[functorial factorization]]** of the morphisms in $\mathcal{C}$ is a [[functor]]
$$
fact \;\colon\; \mathcal{C}^{\Delta[1]} \longrightarrow \mathcal{C}^{\Delta[2]}
$$
which is a [[section]] of the [[composition]] functor $d_1 \;\colon \;\mathcal{C}^{\Delta[2]}\to \mathcal{C}^{\Delta[1]}$.
=--
+-- {: .num_remark}
###### Remark
In def. \ref{FunctorialFactorization} we are using the following notation, see at _[[simplex category]]_ and at _[[nerve of a category]]_:
Write $\Delta[1] = \{0 \to 1\}$ and $\Delta[2] = \{0 \to 1 \to 2\}$ for the [[ordinal numbers]], regarded as [[posets]] and hence as [[categories]]. The [[arrow category]] $Arr(\mathcal{C})$ is equivalently the [[functor category]] $\mathcal{C}^{\Delta[1]} \coloneqq Funct(\Delta[1], \mathcal{C})$, while $\mathcal{C}^{\Delta[2]}\coloneqq Funct(\Delta[2], \mathcal{C})$ has as objects pairs of composable morphisms in $\mathcal{C}$. There are three injective functors $\delta_i \colon [1] \rightarrow [2]$, where $\delta_i$ omits the index $i$ in its image.
By precomposition, this induces [[functors]] $d_i \colon \mathcal{C}^{\Delta[2]} \longrightarrow \mathcal{C}^{\Delta[1]}$. Here
* $d_1$ sends a pair of composable morphisms to their [[composition]];
* $d_2$ sends a pair of composable morphisms to the first morphism;
* $d_0$ sends a pair of composable morphisms to the second morphism.
=--
+-- {: .num_defn #FunctorialWeakFactorizationSystem}
###### Definition
A weak factorization system, def. \ref{WeakFactorizationSystem}, is called a **functorial weak factorization system** if the factorization of morphisms may be chosen to be a [[functorial factorization]] $fact$, def. \ref{FunctorialFactorization}, i.e. such that $d_2 \circ fact$ lands in $\mathcal{L}$ and $d_0\circ fact$ in $\mathcal{R}$.
=--
+-- {: .num_remark}
###### Remark
Not all weak factorization systems are functorial, although most (including those produced by the [[small object argument]], with due care) are. But all [[orthogonal factorization systems]], def. \ref{OrthogonalFactorizationSystem}, automatically are functorial. An example of a weak factorization that is not functorial can be found in [Isaksen 2001](https://arxiv.org/abs/math/0106152).
=--
### Orthogonal factorization systems
+-- {: .num_defn #OrthogonalFactorizationSystem}
###### Definition
An **[[orthogonal factorization system]]** (OFS) is a weak factorization system $(\mathcal{L},\mathcal{R})$, def. \ref{WeakFactorizationSystem} such that the lifts of elements in $\mathcal{L}$ against elements in $\mathcal{R}$ are _unique_.
=--
+-- {: .num_remark }
###### Remark
While every OFS (def. \ref{OrthogonalFactorizationSystem}) is a WFS (def. \ref{WeakFactorizationSystem}), the primary examples of each are different:
A "basic example" of an OFS is [[(epi,mono)-factorization]] in [[Set]] (meaning $L$ is the collection of [[epimorphisms]] and $R$ that of [[monomorphisms]]), while a "basic example" of a WFS is (mono, epi) in $Set$.
The superficial similarity of these two examples masks the fact that they generalize in very different ways.
The OFS (epi, mono) generalizes to any [[topos]] or [[pretopos]], and in fact to any [[regular category]] if we replace "epi" with [[regular epimorphism|regular epi]]. Likewise it generalizes to any [[quasitopos]] if we instead replace "mono" with [[regular monomorphism|regular mono]].
On the other hand, saying that (mono,epi) is a WFS in [[Set]] is equivalent to the [[axiom of choice]]. A less loaded statement is that $(L,R)$ is a WFS, where $L$ is the class of inclusions $A\hookrightarrow A\sqcup B$ into a binary [[coproduct]] and $R$ is the class of [[split epimorphism|split epis]]. In this form the statement generalizes to any [[extensive category]]; see also [[weak factorization system on Set]].
=--
### Algebraic weak factorization systems
An [[algebraic weak factorization system]] enhances the *properties* of lifting and factorization to algebraic [[stuff, structure, property|structure]].
### Accessible weak factorization systems
An [[accessible weak factorization system]] is a wfs on a [[locally presentable category]] whose factorization is given by an [[accessible functor]].
## Properties
### Closure properties
{#ClosureProperties}
+-- {: .num_prop #ClosuredPropertiesOfWeakFactorizationSystem}
###### Proposition
Let $(\mathcal{L},\mathcal{R})$ be a weak factorization system, def. \ref{WeakFactorizationSystem} on some [[category]] $\mathcal{C}$. Then
1. Both classes contain the class of [[isomorphism]] of $\mathcal{C}$.
1. Both classes are closed under [[composition]] in $\mathcal{C}$.
$\mathcal{L}$ is also closed under [[transfinite composition]].
1. Both classes are closed under forming [[retracts]] in the [[arrow category]] $\mathcal{C}^{\Delta[1]}$ (see remark \ref{RetractsOfMorphisms}).
1. $\mathcal{L}$ is closed under forming [[pushouts]] of morphisms in $\mathcal{C}$ ("[[cobase change]]").
$\mathcal{R}$ is closed under forming [[pullback]] of morphisms in $\mathcal{C}$ ("[[base change]]").
1. $\mathcal{L}$ is closed under forming [[coproducts]] in $\mathcal{C}^{\Delta[1]}$.
$\mathcal{R}$ is closed under forming [[products]] in $\mathcal{C}^{\Delta[1]}$.
=--
+-- {: .proof}
###### Proof
We go through each item in turn.
**containing isomorphisms**
Given a [[commuting square]]
$$
\array{
A &\overset{f}{\rightarrow}& X
\\
{}_{\mathllap{\in Iso}}^{\mathllap{i}}\downarrow && \downarrow^{\mathrlap{p}}
\\
B &\underset{g}{\longrightarrow}& Y
}
$$
with the left morphism an isomorphism, the a [[lift]] is given by using the [[inverse]] of this isomorphism ${}^{{f \circ i^{-1}}}\nearrow$. Hence in particular there is a lift when $p \in \mathcal{R}$ and so $i \in \mathcal{L}$. The other case is [[formal dual|formally dual]].
**closure under composition**
Given a [[commuting square]] of the form
$$
\array{
A &\longrightarrow& X
\\
\downarrow && \downarrow^{\mathrlap{p_1}}_{\mathrlap{\in \mathcal{R}}}
\\
{}^{\mathllap{i}}_{\mathllap{\in \mathcal{L}}}\downarrow && \downarrow^{\mathrlap{p_2}}_{\mathrlap{\in \mathcal{R}}}
\\
B &\longrightarrow& Y
}
$$
consider its [[pasting]] decomposition as
$$
\array{
A &\longrightarrow& X
\\
\downarrow
&\searrow& \downarrow^{\mathrlap{p_1}}_{\mathrlap{\in \mathcal{R}}}
\\
{}^{\mathllap{i}}_{\mathllap{\in \mathcal{L}}}\downarrow && \downarrow^{\mathrlap{p_2}}_{\mathrlap{\in \mathcal{R}}}
\\
B &\longrightarrow& Y
}
\,.
$$
Now the bottom commuting square has a lift, by assumption. This yields another [[pasting]] decomposition
$$
\array{
A &\longrightarrow& X
\\
{}^{\mathllap{i}}_{\mathllap{\in \mathcal{L}}}\downarrow
&& \downarrow^{\mathrlap{p_1}}_{\mathrlap{\in \mathcal{R}}}
\\
\downarrow &\nearrow& \downarrow^{\mathrlap{p_2}}_{\mathrlap{\in \mathcal{R}}}
\\
B &\longrightarrow& Y
}
$$
and now the top commuting square has a lift by assumption. This is now equivalently a lift in the total diagram, showing that $p_1\circ p_1$ has the right lifting property against $\mathcal{L}$ and is hence in $\mathcal{R}$. The case of composing two morphisms in $\mathcal{L}$ is [[formal dual|formally dual]]. From this the closure of $\mathcal{L}$ under [[transfinite composition]] follows since the latter is given by [[colimits]] of sequential composition and successive lifts against the underlying sequence as above constitutes a [[cocone]], whence the extension of the lift to the colimit follows by its [[universal property]].
**closure under retracts**
Let $j$ be the [[retract]] of an $i \in \mathcal{L}$, i.e. let there be a [[commuting diagram]] of the form.
$$
\array{
id_A \colon & A &\longrightarrow& C &\longrightarrow& A
\\
& \downarrow^{\mathrlap{j}} && \downarrow^{\mathrlap{i}}_{\mathrlap{\in \mathcal{L}}} && \downarrow^{\mathrlap{j}}
\\
id_B \colon & B &\longrightarrow& D &\longrightarrow& B
}
\,.
$$
Then for
$$
\array{
A &\longrightarrow& X
\\
{}^{\mathllap{j}}\downarrow && \downarrow^{\mathrlap{f}}_{\mathrlap{\in \mathcal{R}}}
\\
B &\longrightarrow& Y
}
$$
a [[commuting square]], it is equivalent to its [[pasting]] composite with that retract diagram
$$
\array{
A &\longrightarrow& C &\longrightarrow& A &\longrightarrow& X
\\
\downarrow^{\mathrlap{j}} && \downarrow^{\mathrlap{i}}_{\mathrlap{\in \mathcal{L}}} && \downarrow^{\mathrlap{j}} && \downarrow^{\mathrlap{f}}_{\mathrlap{\in \mathcal{R}}}
\\
B &\longrightarrow& D &\longrightarrow& B &\longrightarrow & Y
}
\,.
$$
Now the pasting composite of the two squares on the right has a lift, by assumption,
$$
\array{
A &\longrightarrow& C &\longrightarrow& A &\longrightarrow& X
\\
\downarrow^{\mathrlap{j}}
&&
\downarrow^{\mathrlap{i}}_{\mathrlap{\in \mathcal{L}}}
&&
\nearrow
&&
\downarrow^{\mathrlap{f}}_{\mathrlap{\in \mathcal{R}}}
\\
B &\longrightarrow& D &\longrightarrow& B &\longrightarrow & Y
}
\,.
$$
By composition, this is also a lift in the total outer rectangle, hence in the original square. Hence $j$ has the left lifting property against all $p \in \mathcal{R}$ and hence is in $\mathcal{L}$. The other case is [[formal duality|formally dual]].
**closure under pushout and pullback**
Let $p \in \mathcal{R}$ and and let
$$
\array{
Z \times_f X &\longrightarrow& X
\\
{}^{\mathllap{{f^* p}}}\downarrow && \downarrow^{\mathrlap{p}}
\\
Z &\stackrel{f}{\longrightarrow} & Y
}
$$
be a [[pullback]] diagram in $\mathcal{C}$. We need to show that $f^* p$ has the [[right lifting property]] with respect to all $i \in \mathcal{L}$. So let
$$
\array{
A &\longrightarrow& Z \times_f X
\\
{}^{\mathllap{i}}_{\mathllap{\in \mathcal{L}}}\downarrow && \downarrow^{\mathrlap{\mathrlap{f^* p}}}
\\
B &\stackrel{g}{\longrightarrow}& Z
}
$$
be a [[commuting square]]. We need to construct a diagonal lift of that square. To that end, first consider the [[pasting]] composite with the pullback square from above to obtain the commuting diagram
$$
\array{
A &\longrightarrow& Z \times_f X &\longrightarrow& X
\\
{}^{\mathllap{i}}\downarrow && \downarrow^{\mathrlap{f^* p}}
&& \downarrow^{\mathrlap{p}}
\\
B &\stackrel{g}{\longrightarrow}& Z &\stackrel{f}{\longrightarrow}& Y
}
\,.
$$
By the right lifting property of $p$, there is a diagonal lift of the total outer diagram
$$
\array{
A &\longrightarrow& X
\\
\downarrow^{\mathrlap{i}} &{}^{\hat {(f g)}}\nearrow& \downarrow^{\mathrlap{p}}
\\
B &\stackrel{f g}{\longrightarrow}& Y
}
\,.
$$
By the [[universal property]] of the [[pullback]] this gives rise to the lift $\hat g$ in
$$
\array{
&& Z \times_f X &\longrightarrow& X
\\
&{}^{\hat g} \nearrow& \downarrow^{\mathrlap{f^* p}}
&& \downarrow^{\mathrlap{p}}
\\
B &\stackrel{g}{\longrightarrow}& Z &\stackrel{f}{\longrightarrow}& Y
}
\,.
$$
In order for $\hat g$ to qualify as the intended lift of the total diagram, it remains to show that
$$
\array{
A &\longrightarrow& Z \times_f X
\\
\downarrow^{\mathrlap{i}} & {}^{\hat g}\nearrow
\\
B
}
$$
commutes. To do so we notice that we obtain two [[cones]] with tip $A$:
* one is given by the morphisms
1. $A \to Z \times_f X \to X$
2. $A \stackrel{i}{\to} B \stackrel{g}{\to} Z$
with universal morphism into the pullback being
* $A \to Z \times_f X$
* the other by
1. $A \stackrel{i}{\to} B \stackrel{\hat g}{\to} Z \times_f X \to X$
2. $A \stackrel{i}{\to} B \stackrel{g}{\to} Z$.
with universal morphism into the pullback being
* $A \stackrel{i}{\to} B \stackrel{\hat g}{\to} Z \times_f X$.
The commutativity of the diagrams that we have established so far shows that the first and second morphisms here equal each other, respectively. By the fact that the universal morphism into a pullback diagram is _unique_ this implies the required identity of morphisms.
The other case is [[formal dual|formally dual]].
**closure under (co-)products**
Let $\{(A_s \overset{i_s}{\to} B_s) \in \mathcal{L}\}_{s \in S}$ be a set of elements of $\mathcal{L}$. Since [[colimits]] in the [[presheaf category]] $\mathcal{C}^{\Delta[1]}$ are computed componentwise, their [[coproduct]] in this [[arrow category]] is the universal morphism out of the coproduct of objects $\underset{s \in S}{\coprod} A_s$ induced via its [[universal property]] by the set of morphisms $i_s$:
$$
\underset{s \in S}{\sqcup} A_s
\overset{(i_s)_{s\in S}}{\longrightarrow}
\underset{s \in S}{\sqcup} B_s
\,.
$$
Now let
$$
\array{
\underset{s \in S}{\sqcup} A_s &\longrightarrow& X
\\
{}^{\mathllap{(i_s)_{s\in S}}}\downarrow && \downarrow^{\mathrlap{f}}_{\mathrlap{\in \mathcal{R}}}
\\
\underset{s \in S}{\sqcup} B_s
&\longrightarrow&
Y
}
$$
be a [[commuting square]]. This is in particular a [[cocone]] under the [[coproduct]] of objects, hence by the [[universal property]] of the coproduct, this is equivalent to a set of commuting diagrams
$$
\left\{
\;\;\;\;
\array{
A_s &\longrightarrow& X
\\
{}^{\mathllap{i_s}}_{\mathllap{\in \mathcal{L}}}\downarrow
&&
\downarrow^{\mathrlap{f}}_{\mathrlap{\in \mathcal{R}}}
\\
B_s
&\longrightarrow&
Y
}
\;\;\;\;
\right\}_{s\in S}
\,.
$$
By assumption, each of these has a lift $\ell_s$. The collection of these lifts
$$
\left\{
\;\;\;\;
\array{
A_s &\longrightarrow& X
\\
{}^{\mathllap{i_s}}_{\mathllap{\in \mathcal{L}}}\downarrow
&{}^{\ell_s}\nearrow& \downarrow^{\mathrlap{f}}_{\mathrlap{\in \mathcal{R}}}
\\
B_s
&\longrightarrow&
Y
}
\;\;\;\;
\right\}_{s\in S}
$$
is now itself a compatible [[cocone]], and so once more by the [[universal property]] of the coproduct, this is equivalent to a lift $(\ell_s)_{s\in S}$ in the original square
$$
\array{
\underset{s \in S}{\sqcup} A_s &\longrightarrow& X
\\
{}^{\mathllap{(i_s)_{s\in S}}}\downarrow
&{}^{(\ell_s)_{s\in S}}\nearrow&
\downarrow^{\mathrlap{f}}_{\mathrlap{\in \mathcal{R}}}
\\
\underset{s \in S}{\sqcup} B_s
&\longrightarrow&
Y
}
\,.
$$
This shows that the coproduct of the $i_s$ has the left lifting property against all $f\in \mathcal{R}$ and is hence in $\mathcal{L}$. The other case is [[formal dual|formally dual]].
=--
+-- {: .num_remark}
###### Remark
Beware, in the situation of prop. \ref{ClosuredPropertiesOfWeakFactorizationSystem},
that $\mathcal{L}$ is not in general closed under all [[colimits]] in $\mathcal{C}^{\Delta[1]}$, and similarly $\mathcal{R}$ is not in general closed under all [[limits]] in $\mathcal{C}^{\Delta[1]}$. Also $\mathcal{L}$ is not in general closed under forming [[coequalizers]] in $\mathcal{C}$, and $\mathcal{R}$ is not in general closed under forming [[equalizers]] in $\mathcal{C}$. However, if $(\mathcal{L},\mathcal{R})$ is an [[orthogonal factorization system]], def. \ref{OrthogonalFactorizationSystem}, then $\mathcal{L}$ is closed under all colimits and $\mathcal{R}$ is closed under all limits.
=--
+-- {: .num_remark #RetractsOfMorphisms}
###### Remark
Here by a _retract_ of a [[morphism]] $X \stackrel{f}{\longrightarrow} Y$ in some [[category]] $\mathcal{C}$ is meant a [[retract]] of $f$ as an object in the [[arrow category]] $\mathcal{C}^{\Delta[1]}$, hence a morphism $A \stackrel{g}{\longrightarrow} B$ such that in $\mathcal{C}^{\Delta[1]}$ there is a factorization of the identity on $g$ through $f$
$$
id_g \;\colon\;
g \longrightarrow f \longrightarrow g
\,.
$$
This means equivalently that in $\mathcal{C}$ there is a [[commuting diagram]] of the form
$$
\array{
id_A \colon & A &\longrightarrow& X &\longrightarrow& A
\\
& \downarrow^{\mathrlap{g}} && \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{g}}
\\
id_B \colon & B &\longrightarrow& Y &\longrightarrow& B
}
\,.
$$
=--
### Retract argument
+-- {: .num_lemma #RetractArgument}
###### Lemma
**([[retract argument]])**
Consider a [[composition|composite]] [[morphism]]
$$
f \;\colon\; X\stackrel{i}{\longrightarrow} A \stackrel{p}{\longrightarrow} Y
\,.
$$
Then:
1. If $f$ has the [[left lifting property]] against $p$, then $f$ is a [[retract]] of $i$.
1. If $f$ has the [[right lifting property]] against $i$, then $f$ is a [[retract]] of $p$.
=--
+-- {: .proof}
###### Proof
We discuss the first statement, the second is [[formal duality|formally dual]].
Write the factorization of $f$ as a [[commuting square]] of the form
$$
\array{
X &\stackrel{i}{\longrightarrow}& A
\\
{}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{p}}
\\
Y &= & Y
}
\,.
$$
By the assumed [[lifting property]] of $f$ against $p$ there exists a diagonal filler $g$ making a [[commuting diagram]] of the form
$$
\array{
X &\stackrel{i}{\longrightarrow}& A
\\
{}^{\mathllap{f}}\downarrow &{}^{\mathllap{g}}\nearrow& \downarrow^{\mathrlap{p}}
\\
Y &= & Y
}
\,.
$$
By rearranging this diagram a little, it is equivalent to
$$
\array{
& X &=& X
\\
& {}^{\mathllap{f}}\downarrow
&&
{}^{\mathllap{i}}\downarrow
\\
id_Y \colon & Y &\underset{g}{\longrightarrow}& A &\underset{p}{\longrightarrow}&
Y
}
\,.
$$
Completing this to the right, this yields a diagram exhibiting the required retract according to remark \ref{RetractsOfMorphisms}:
$$
\array{
id_X \colon & X &=& X &=& X
\\
& {}^{\mathllap{f}}\downarrow
&&
{}^{\mathllap{i}}\downarrow
&&
{}^{\mathllap{f}}\downarrow
\\
id_Y \colon & Y &\underset{g}{\longrightarrow}& A &\underset{p}{\longrightarrow}&
Y
}
\,.
$$
=--
## Examples
* [[model category|Model categories]] provide many examples of weak factorization systems. In fact, most applications of WFS involve model categories or model-categorical ideas.
* The existence of certain [[weak factorization system on Set|WFS on Set]] is related to the [[axiom of choice]].
* See the [[joyalscatlab:Weak factorisation systems|Catlab]] for more examples.
## Related concepts
* [[retract argument]]
* [[injective and projective morphisms]]
* [[algebraic weak factorization system]]
[[!include algebraic model structures - table]]
## References
* [[joyalscatlab:HomePage|Joyal's CatLab]], _[[joyalscatlab:Weak factorisation systems]]_
* {#Hirschorn} Philip S. Hirschhorn, _Model Categories and Their Localizations_ ([AMS](http://www.ams.org/bookstore?fn=20&arg1=whatsnew&item=SURV-99), [pdf toc](http://www.gbv.de/dms/goettingen/360115845.pdf), [pdf](http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf))
* Jiri Rosicky, Walter Tholen , *Factorization, Fibration and Torsion*, [arxiv](https://arxiv.org/abs/0801.0063) (2007)
Introductory texts:
* [[Introduction to Homotopy Theory]]
* {#Riehl2008} [[Emily Riehl]], [_Factorization Systems_](https://math.jhu.edu/~eriehl/factorization.pdf), 2008
* [[Emily Riehl]], Chapter 11 in: *[[Categorical Homotopy Theory]]*, Cambridge University Press (2014) [[doi:10.1017/CBO9781107261457](https://doi.org/10.1017/CBO9781107261457), [pdf](http://www.math.jhu.edu/~eriehl/cathtpy.pdf)]
The following dissertation section is entirely written after learning of [Riehl (2008)](#Riehl2008) above, but has complementary examples and may dive deeper into some proofs:
* {#Nuyts2020} [[Andreas Nuyts]], _Contributions to Multimode and Presheaf Type Theory, section 2.4: Factorization Systems_, [PhD thesis](https://lirias.kuleuven.be/retrieve/581985), KU Leuven, Belgium, 2020
[[!redirects weak factorization systems]]
|
weak factorization system on Set | https://ncatlab.org/nlab/source/weak+factorization+system+on+Set | The "prototypical" example of a [[weak factorization system]] is ([[injection|mono]], [[surjection|epi]]) on [[Set]]. That this is a WFS is equivalent to the [[axiom of choice]], but various weaker things can be said in more generality.
* (complemented mono, split epi) is a WFS in any [[extensive category]], without any need for choice and even in [[intuitionistic logic]]. A factorization of $X\to Y$ is $X\to X+Y \to Y$. In this WFS, every object is "cofibrant" (that is, $0\to X$ is in the left class) while an object is "fibrant" (i.e. $X\to 1$ is in the right class) if and only if it admits a [[global element]].
* Thus, [[classical logic]] (which says that any subset is complemented) is equivalent to saying that (mono, split epi) is a WFS on Set.
* Since the [[axiom of choice]] is equivalent to saying that any epic in Set is split, (mono, epi) is a WFS if and only if AC holds.
We might also ask for WFS on Set (or more general categories) whose left class or right class consists of monics or epics, respectively, with the other class being whatever it must be.
* On any [[topos]], there is a WFS (mono, relative injective), where a map $f\colon X\to Y$ in a category $C$ is *relatively injective* if it is an [[injective object]] in the [[slice category]] $C/Y$. Factorizations are provided by the fact that any topos has enough injectives (i.e. any object can be embedded in an injective), and any slice category of a topos is again a topos. See also [this answer](http://mathoverflow.net/questions/10246/model-category-structure-on-set-without-axiom-of-choice/10368#10368) by [[Denis-Charles Cisinski]] on MO. In this model structure every object is "cofibrant," while the "fibrant" objects are the injectives; thus we might call this the *injective model structure*.
* On any extensive category, there is a WFS whose right class is the epimorphisms if and only if there exist enough [[projective objects]] (i.e. every object admits an epimorphism from a projective one). For if such a WFS exists, then clearly the "cofibrant" objects are the projectives, so cofibrant replacement provides a projective cover of any object. Conversely, if there are enough projectives, then we take the left class to be the complemented monics with projective complement; a factorization of $X\to Y$ is provided by $X\to X+Y' \to Y$, where $Y'\to Y$ is a projective cover. In particular, there is a WFS on Set whose right class is the epimorphisms if and only if [[COSHEP]] holds. As remarked above, the "cofibrant" objects here are the projective ones, while the "fibrant" objects are the "well-supported ones;" thus we might call this the *projective model structure*.
Looking at the "projective" and "injective" model structures above, by analogy with the [[model structures on chain complexes]] we might also be tempted to call (complemented mono, split epi) the *Hurewicz WFS*.
[[!redirects weak factorization systems on Set]]
[[!redirects WFS on Set]]
[[!redirects wfs on Set]]
|
weak function extensionality | https://ncatlab.org/nlab/source/weak+function+extensionality | \tableofcontents
## Definition
The principle of **weak function extensionality** states that the [[dependent product type]] of a family of contractible types is itself contractible.
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B \; \mathrm{type} \quad \Gamma, x:A \vdash p(x):\mathrm{isContr}(B(x))}{\Gamma \vdash \mathrm{weakfunext}(\lambda x.p(x)):\mathrm{isContr}\left(\prod_{x:A} B(x)\right)}$$
This is equivalent to the condition that the dependent product type of a family of [[h-propositions]] itself is an [[h-proposition]]:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B \; \mathrm{type} \quad \Gamma, x:A \vdash p(x):\mathrm{isProp}(B(x))}{\Gamma \vdash \mathrm{weakfunext}(\lambda x.p(x)):\mathrm{isProp}\left(\prod_{x:A} B(x)\right)}$$
Weak function extensionality is not equivalent to the [[principle of unique choice]]. The principle of unique choice states that the dependent product type of a family of contractible types is pointed
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B \; \mathrm{type} \quad \Gamma, x:A \vdash p(x):\mathrm{isContr}(B(x))}{\Gamma\vdash \mathrm{uniquechoice}(\lambda x.p(x)):\prod_{x:A} B(x)}$$
## Properties
* Weak function extensionality is equivalent to [[function extensionality]].
## See also
* [[function extensionality]]
* [[principle of unique choice]]
## References
Weak function extensionality appears in section 13.1 of
* [[Egbert Rijke]], *[[Introduction to Homotopy Type Theory]]*, Cambridge Studies in Advanced Mathematics, Cambridge University Press ([pdf](https://raw.githubusercontent.com/martinescardo/HoTTEST-Summer-School/main/HoTT/hott-intro.pdf)) (478 pages) |
weak gravity conjecture | https://ncatlab.org/nlab/source/weak+gravity+conjecture |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _weak gravity conjecture_ is a [[conjecture]] ([Arkani-Hamed, Motl, Nicolis, Vafa 06](#ArkaniHamedMotlNicolisVafa06)) often summarized as saying that for [[gravity]] or [[quantum gravity]] to be consistent, the [[force]] of the field of gravity must be weaker than that of any of the [[gauge field]] forces, in suitable units.
## Loose statement
More precisely, what is argued in ([AH-M-N-V 06, section 2](#ArkaniHamedMotlNicolisVafa06)) is that there alwayst must be "[[elementary particles]]" for which the ratio $m/{\vert q\vert}$ of their [[mass]] over their [[gauge field]] [[charge]] (e.g. [[electric charge]], [[magnetic charge]]) is smaller than one: $m/{\vert q\vert} \lt 1$ (in natural [[units]]).
The argument for this in ([AH-M-N-V 06, section 2.2](#ArkaniHamedMotlNicolisVafa06)) is the following. The [[extremal black hole|extremal]] ([[BPS state]]) bound for [[charged black holes]] is that their mass $M$ _exceeds or equals_ their charge ${\vert Q\vert}$, hence $M/{\vert Q\vert} \geq 1$. Now via [[black hole radiation]] any black hole is supposed to eventually decay by [[radiation|radiating]] away elementary particle quanta. If there were no elementary particle with $m/{\vert q\vert} \leq 1$, so the argument goes, then by radiating away particles with $m/{\vert q\vert} \gt 1$ the black hole would eventually lose all its mass, while still having some non-vanishing charge left. But such "black hole remnants" have been argued to lead to inconsistency ([Susskind 96](#Susskind96)).
## In string theory
The weak gravity conjecture was originally motivated from [[string theory]], which is argued to validate the "weak gravity" assumption.
In ([AH-M-N-V 06, section 4](#ArkaniHamedMotlNicolisVafa06)) it is argued that the closed [[superstring]] excitation states in [[heterotic string theory]] [[KK-compactification|KK-compactified]] on to 4d satisfy the condition $m/{\vert q\vert} \lt 1$:
<img src="https://ncatlab.org/nlab/files/WeakGravityConjectureHetStates.jpg" width="600">
## Relation to cosmic censorship
In 2015 [[Cumrun Vafa]] has argued that the weak gravity conjecture implies the [[cosmic censorship hypothesis]]: The latter turns out to be generally false, even in in 4 dimensions ([Crisford-Santos 17](CrisfordSantos17)), but there are arguments that configurations violating it also violate the assumption of "weak gravity".
> Subsequent calculations by Santos and Crisford supported Vafa's hunch; the simulations they're running now could verify that naked singularities become cloaked in black holes right at the point where gravity becomes the weakest force. ([Wolchover, June 20 2017](https://www.quantamagazine.org/where-gravity-is-weak-and-naked-singularities-are-verboten-20170620/))
For more on this see [Horowitz-Santos 19](#HorowitzSantos19)
## Related concepts
* [[cosmic censorship hypothesis]]
* [[landscape of string theory vacua]]
## References
The statement is due to
* {#ArkaniHamedMotlNicolisVafa06} [[Nima Arkani-Hamed]], Lubos Motl, Alberto Nicolis, [[Cumrun Vafa]], _The String Landscape, Black Holes and Gravity as the Weakest Force_, JHEP 0706:060,2007 ([arXiv:hep-th/0601001](https://arxiv.org/abs/hep-th/0601001))
with reference to arguments in
* {#Susskind96} [[Leonard Susskind]], _Trouble For Remnants_ ([arXiv:hep-th/9501106](https://arxiv.org/abs/hep-th/9501106))
Review is in
* [[Eran Palti]], _The Swampland: Introduction and Review_, lecture notes ([arXiv:1903.06239](https://arxiv.org/abs/1903.06239))
Further discussion includes
{#FurtherDiscussion}
* Clifford Cheung, Grant N. Remmen, _Naturalness and the Weak Gravity Conjecture_, Phys. Rev. Lett. 113, 051601 (2014) ([arXiv:1402.2287](https://arxiv.org/abs/1402.2287))
* Yu Nakayama, Yasunori Nomura, _Weak Gravity Conjecture in AdS/CFT_, Phys. Rev. D 92, 126006 (2015) ([arXiv:1509.01647](https://arxiv.org/abs/1509.01647))
* [[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))
* Clifford Cheung, Junyu Liu, Grant N. Remmen, _Proof of the Weak Gravity Conjecture from Black Hole Entropy_ ([arXiv:1801.08546](https://arxiv.org/abs/1801.08546))
* Miguel Montero, _A Holographic Derivation of the Weak Gravity Conjecture_, JHEP 03 (2019) 157 ([arXiv:1812.03978](https://arxiv.org/abs/1812.03978))
* [[Ben Heidenreich]], [[Matthew Reece]], [[Tom Rudelius]], _Repulsive Forces and the Weak Gravity Conjecture_ ([arXiv:1906.02206](https://arxiv.org/abs/1906.02206))
Relation to [[cosmic censorship hypothesis]]
* {#CrisfordSantos17} Toby Crisford, [[Jorge Santos]], _Violating weak cosmic censorship in AdS$_4$_, Phys. Rev. Lett. 118, 181101 (2017) ([arXiv:1702.05490](https://arxiv.org/abs/1702.05490))
* {#HorowitzSantos19} [[Gary Horowitz]], [[Jorge Santos]], _Further evidence for the weak gravity - cosmic censorship connection_ ([arXiv:1901.11096](https://arxiv.org/abs/1901.11096))
Relation, via [[AdS/CFT]] and the [[conformal bootstrap]], to the solution of the [[sphere packing problem]] in [[dimensions]] 8 and 24:
* {#HMR19} [[Thomas Hartman]], [[Dalimil Mazáč]], [[Leonardo Rastelli]], _Sphere Packing and Quantum Gravity_ ([arXiv:1905.01319](https://arxiv.org/abs/1905.01319))
See also:
* Lars Aalsma, [[Alex Cole]], Gregory J. Loges, [[Gary Shiu]],
_A New Spin on the Weak Gravity Conjecture_ ([arXiv:2011.05337](https://arxiv.org/abs/2011.05337))
Proof of the weak gravity conjecture in [[bosonic string theory]] is claimed in:
* [[Ben Heidenreich]], Matteo Lotito, *Proving the Weak Gravity Conjecture in Perturbative String Theory, Part I: The Bosonic String* [[arXiv:2401.14449](https://arxiv.org/abs/2401.14449)]
|
weak homotopy equivalence | https://ncatlab.org/nlab/source/weak+homotopy+equivalence |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Homotopy theory
+-- {: .hide}
[[!include homotopy - contents]]
=--
#### Equality and Equivalence
+--{: .hide}
[[!include equality and equivalence - contents]]
=--
=--
=--
# Weak homotopy equivalences
* table of contents
{: toc}
## Idea
A _[[weak homotopy equivalence]]_ is a map between [[topological spaces]] or [[simplicial sets]] or similar which induces [[isomorphisms]] on all [[homotopy groups]]. (The analogous concept in [[homological algebra]] is called a _[[quasi-isomorphism]]_.)
The [[localization]] or [[simplicial localization]] of the categories [[Top]] and [[sSet]] at the weak homotopy equivalences used as [[weak equivalences]] yields the standard [[homotopy category]] [[Ho(Top)]] and [[Ho(sSet)]] or the [[(∞,1)-category]] of [[∞-groupoids]]/[[homotopy types]], respectively.
Weak homotopy equivalences are named after _[[homotopy equivalences]]_. They can be identified with homotopy equivalences after one allows to replace the [[domains]] by a [[resolution]]. The corresponding notions in [[homological algebra]] are [[quasi-isomorphisms]] and [[chain homotopy]]-equivalences.
From another perspective, the notion of _weak homotopy equivalence_ is 'observational', in that a map is a weak homotopy equivalence if when we look at it through the observations that we can make of it using [[homotopy groups]] or even the [[fundamental infinity-groupoid]], it looks like an equivalence. In contrast, _[[homotopy equivalence]]_ is more 'constructive'; in that $f$ is a homotopy equivalence if there exists an inverse for it (up to homotopy, of course). Note that both of these notions are weaker than mere [[isomorphism]] of topological spaces (homeomorphism) and so can be considered examples of [[weak equivalence]]s.
There are actually two related concepts here: whether two spaces are weakly homotopy equivalent and whether a map between spaces is a weak homotopy equivalence. The former is usually defined in terms of the latter.
## Definition (for topological spaces and simplicial sets)
+-- {: .num_defn #WeakHomotopyEquivalence}
###### Definition
For $X, Y \in $ [[Top]] or $\in $ [[sSet]] two [[topological spaces]] or [[simplicial sets]], a [[continuous function]] or [[simplicial set|simplicial map]] $f : X \to Y$ between them is called a **weak homotopy equivalence** if
1. $f$ induces an [[isomorphism]] of [[connected components]] (path components in the case of topological spaces)
$$
\Pi_0(f) \colon \Pi_0(X) \stackrel{\simeq}{\to} \Pi_0(Y)
$$
in [[Set]];
1. for all [[points]] $x \in X$ and for all $(1 \leq n) \in \mathbb{N}$ $f$ induces an isomorphism on [[homotopy groups]]
$$
\pi_n(f,x) \colon \pi_n(X,x) \stackrel{\simeq}{\to} \pi_n(Y,f(x))
$$
in [[Grp]].
=--
+-- {: .num_remark }
###### Remark
If $X$ and $Y$ are [[connected space|path-connected]], then (1) is trivial, and it suffices to require (2) for a single (arbitrary) $x$, but in general one must require it for at least one $x$ in each path [[connected component]].
=--
\begin{remark}
It is not enough to require that each pair of homotopy groups are isomorphic. For example, let $Y = S^1 \vee S^2$, and let $X$ be its double cover, which glues two spheres at the north and south poles of a circle. Then there are isomorphisms on the fundamental group $\pi_1(X) \simeq \mathbb{Z} \simeq \pi_1(Y)$, although the map $\pi_1(X) \to \pi_1(Y)$ induced by $f$ is not an isomorphism. And by the properties of covering spaces, all higher homotopy groups are isomorphic. But these two are not weakly homotopy equivalent.
\end{remark}
\begin{remark}
There are many alternative definitions of weak homotopy equivalences.
A simplicial map $f$ is a weak equivalence of simplicial sets
if and only if $Ex^\infty f$ is a simplicial homotopy equivalence
if and only if $Hom(f,A)$ is a simplicial homotopy equivalence for any [[Kan complex]] $A$
if and only if $f$ has a right relative-homotopy-lifting property with respect
to the maps $\partial\Delta^n\to\Delta^n$
if and only if $f$ is a composition of an acyclic cofibration
(i.e., a map with a left lifting property with respect to all maps with a right lifting
property with respect to horn inclusions)
and an acyclic fibration (i.e., a map
with a right lifting property with respect to inclusions $\partial\Delta^n\to\Delta^n)$.
A continuous map $f$ is a weak equivalence of topological spaces
if and only if $|Sing(f)|$ is a homotopy equivalence of topological spaces
if and only if $Hom(A,f)$ is a homotopy equivalence for any [[CW-complex]] $A$
if and only if $f$ has a right relative-homotopy-lifting property with respect
to the maps $S^{n-1}\to D^n$
if and only if $f$ is a composition of an acyclic Serre cofibration
(a retract of a relative CW-complex) and an acyclic [[Serre fibration]].
Both functors $|-|$ and $Sing$ preserve and reflect weak equivalences,
so any of the two classes defines the other.
\end{remark}
+-- {: .num_defn }
###### Definition
The [[homotopy category]] of [[Top]] with respect to weak homotopy equivalences is [[Ho(Top)]]${}_{whe}$.
=--
+-- {: .num_remark }
###### Remark
Accordingly, weak homotopy equivalences are the [[weak equivalences]] in the standard [[Quillen model structure on topological spaces]] and the [[Quillen model structure on simplicial sets]], and also in the [[mixed model structure]].
=--
## Properties
### Equivalent characterizations
{#EquivalentCharacterizations}
+-- {: .num_prop }
###### Proposition
A continuous map $f : X \to Y$ is a weak homotopy equivalence
precisely if for all $n \in \mathbb{N}$ and for all [[commuting diagrams]] of continuous maps of the form
$$
\array{
S^{n-1} &\to& X
\\
\downarrow && \downarrow^{\mathrlap{f}}
\\
D^n &\to& Y
}
\,,
$$
where the left morphism is the inclusion of the $(n-1)$-[[sphere]] as the [[boundary]] of the $n$-[[ball]],
there exists a continuous map $\sigma : D^n \to X$
that makes the resulting upper triangle commute and such that the lower triangle commutes up to a [[homotopy]]
$$
\array{
S^{n-1} &&\to&& X
\\
\\
\downarrow && \nearrow & \swArrow& \downarrow^{\mathrlap{f}}
\\\\
D^n &&\to&& Y
}
$$
which is constant along $S^{n-1} \hookrightarrow D^n$.
=--
In this form the statement and its proof appears in
([Jardine](#Jardine)) (where it is also generalized to weak equivalences in a [[model structure on simplicial presheaves]]).
See also around ([Lurie, prop. 6.5.2.1](#Lurie)). The relevant arguments are spelled out in ([May, section 9.6](#May)). A variant is called the _HELP lemma_ in ([Vogt](#Vogt)).
### Relation to homotopy equivalences
{#RelationToHomotopyEquivalences}
+-- {: .num_prop }
###### Proposition
Every [[homotopy equivalence]] is a weak homotopy equivalence.
=--
+-- {: .proof}
###### Proof
It requires a little bit of thought to prove this, because $f$ and its homotopy inverse $g$ need not preserve any chosen basepoint. But for any $x\in X$ and any $n\ge 1$, we have a [[commuting diagram]]
$$
\array{
\pi_n(X,x) & & \to & & \pi_n(X,g(f(x)))
\\
&
\searrow
&&
\nearrow
&&
\searrow
\\
&&
\pi_n(Y,f(x))
&&
\longrightarrow
&&
\pi_n\big(
Y,f(g(f(x)))
\big)
}
$$
in which the two horizontal morphisms are [[isomorphisms]], because $g f$ and $f g$ are [[homotopy|homotopic]] to [[identity morphism|identities]]. Hence, by the [[two-out-of-six property]] for isomorphisms, the diagonal morphhisms are also all isomorphisms.
=--
+-- {: .num_prop }
###### Proposition
Conversely, any weak homotopy equivalence between [[m-cofibrant spaces]] (spaces that are homotopy equivalent to [[CW complexes]]) is a [[homotopy equivalence]].
=--
### Relation to homotopy types
{#RelationToHomotopyTypes}
We discuss the [[equivalence relation]] generated by weak homotopy equivalence, called _(weak) [[homotopy type]]_. For the "abelianized" analog of this situation see at [[quasi-isomorphism]] the section _[Relation to homology type](quasi-isomorphism#RelationToChainHomologyType)_.
+-- {: .num_prop #ReflexiveAndTransitiveButNotSymmetric}
###### Proposition
The existence of a weak homotopy equivalence from $X$ to $Y$ is a [[reflexive relation|reflexive]] and [[transitive relation]] on [[Top]], but it is not a [[symmetric relation]].
=--
+-- {: .proof}
###### Proof
Reflexivity and transitivity are trivially checked. A counterexample to symmetry
is example \ref{CircleAndPseudoCircle} below.
=--
But we can consider the genuine equivalence relation _generated_ by weak homotopy equivalence:
+-- {: .num_defn }
###### Definition
We say two spaces $X$ and $Y$ have the same **(weak) [[homotopy type]]** if they are equivalent under the [[equivalence relation]] _generated_ by weak homotopy equivalence.
=--
+-- {: .num_remark }
###### Remark
Equivalently this means that $X$ and $Y$ have the same (weak) homotopy type
if there exists a [[zigzag]] of weak homotopy equivalences
$$
X \leftarrow \to\leftarrow \dots \to Y
\,.
$$
This in turn is equivalent to saying that $X$ and $Y$ become [[isomorphism|isomorphic]] in the [[homotopy category]] [[Ho(Top)]]/[[Ho(sSet)]] with the weak homotopy equivalences [[localization|inverted]].
=--
+-- {: .num_remark }
###### Remark
Two spaces $X$ and $Y$ may have isomorphic homotopy groups without being weak homotopy equivalence: for this all the isomorphisms must be induced by an actual map $f : X \to Y$, as in the above definition.
However, if, roughly, one remembers, how all the homotopy groups [[nLab:action|act]] on each other, then this is enough information to exhibit the full homotopy type. This collection of data is called the _[[Postnikov tower]]_ decomposition of a homotopy type.
=--
### Relation to free homotopy sets
{#RelationToFreeHomotopySets}
For $K, X \,\in\, TopSp$, write $Map(K, X)$ for their [[mapping space]], i.e. not considering or respecting any [[basepoint]]. For example, $Map(S^1, Y)$ is the [[free loop space]] of $Y$, in contrast to the [[based loop space]]
$$
\Omega_x X
\xhookrightarrow{\; fib_x(ev_\ast) \;}
Map(S^1, X)
\overset{\; ev_x \;}{\twoheadrightarrow}
X
$$
for any base point $x \,\in\, X$.
Moreover, when $K$ is a [[CW complex]], write
\[
\label{FreeHomotopySets}
{[K,X]} \,\coloneqq\, \tau_0 \,Map(K,X)\,
\]
for the *free homotopy set* of maps from $K$ to $X$, hence the set of [[homotopy classes]] of map $K \to X$, hence the set of [[connected components]] of the mapping space.
\begin{proposition}
**(weak homotopy equivalences detected on free homotopy sets)**
\label{WeakHomotopyEquivalencesDetectedOnTermsOfFreeHomotopySet}
\linebreak
For $f \,\colon\, X \to Y$, a [[continuous function]] between [[connected topological spaces]], the following are equivalent:
1. $f$ is a [[weak homotopy equivalence]]
$$
\underset{n \in \mathbb{N}}{\forall}
\;
\;\;
p_n(X)
\underoverset
{\simeq}
{f_\ast}
{\longrightarrow}
\pi_n(Y)
\,;
$$
1. $f$ induces an [[isomorphism]] on all free homotopy sets (eq:FreeHomotopySets) out of [[CW-complexes]]:
$$
\underset{ K \in CWCplx}{\forall}
\;\;
[K, X]
\underoverset
{\simeq}
{f_\ast}
{\longrightarrow}
[K, Y]
\,;
$$
* $f$ induces
1. an [[isomorphism]] on all free homotopy sets (eq:FreeHomotopySets) out of $K = $any [[n-sphere]] of [[positive number|positive]] [[dimension]],
$$
\underset{
n \in \mathbb{N}_+
}{\forall}
\;\;
[S^n, X]
\underoverset
{\simeq}
{f_\ast}
{\longrightarrow}
[S^n, Y]
\,.
$$
1. a [[surjection]] on the free homotopy set out of the [[wedge sum]] of [[circles]] [[index set|indexed]] by (the set [[underlying]]) the [[fundamental group]] of $Y$:
$$
\big[
\underset{\pi_1(Y)}{\vee} S^1
,\,
X
\big]
\overset{\; f_\ast \;}{\twoheadrightarrow}
\big[
\underset{\pi_1(Y)}{\vee} S^1
,\,
Y
\big]
$$
\end{proposition}
([Matumoto, Minami and Sugawara 1984, Thm. 2](#MatumotoMinamiSugawara84))
\begin{proof}
The implication $(1) \Rightarrow (2)$ is a standard/classical conclusion in homotopy theory, for example it is a small special case of the fact that $Map(K,-)$ out of any cell complex $K$ preserves weak homotopy equivalences ([this Prop.](classical+model+structure+on+topological+spaces#HomProductAdjunctionForCofibrantObjectInTopCGIsQuillen)).
The implication $(2) \Rightarrow (3)$ is trivial, as the conditions in (3) are just a special case of the condition in (2).
So the point of the statement is that (3) is already sufficient to recover (1). This is the content of [Matumoto, Minami and Sugawara 1984, Thm. 1 & Lem. 1.3](#MatumotoMinamiSugawara84).
\end{proof}
## For other kinds of spaces
A map of [[simplicial sets]] is called a weak homotopy equivalence equivalently if its [[geometric realization]] is a weak homotopy equivalence of topological spaces, as above. (Since the geometric realization of any simpicial set is a
[[CW complex]], in this case its geometric realization is actually a [[homotopy equivalence]].)
Likewise, a [[functor]] between [[small category|small]] categories is sometimes said to be a weak homotopy equivalence if its [[nerve]] is a weak homotopy equivalence of simplicial sets, hence of topological spaces after [[geometric realization of categories]]. These are the weak equivalences in the
[[Thomason model structure]] on categories (not the [[canonical model structure]]). The statement of [[Quillen's theorem A]] and [[Quillen's theorem B]] in in this contex.
Similarly, one can define weak homotopy equivalences between any sort of object that has a [[geometric realization]], such as a [[cubical set]], a [[globular set]], an [[n-category]], an [[n-fold category]], and so on.
Note that in some of these cases, such as as simplicial sets, symmetric sets, and probably cubical sets, there is also a notion of "homotopy equivalence" from which this notion needs to be distinguished. A simplicial homotopy equivalence, for instance, is a simplicial map $f:X\to Y$ with an inverse $g:Y\to X$ and simplicial homotopies $X\times \Delta^1 \to X$ and $Y\times \Delta^1 \to Y$ relating $f g$ and $g f$ to identities.
A different direction of generalization is the notion of a [[homotopy equivalence of toposes]].
## Examples
### Of non-reversible weak homotopy equivalences
{#ExamplesOfNonReversibleWHEs}
We discuss examples of weak homotopy equivalences that have no weak homotopy equivalence going the other way, according to prop. \ref{ReflexiveAndTransitiveButNotSymmetric} above.
+-- {: .num_example #CircleAndPseudoCircle}
###### Example
Let $S^1 \in $ [[Top]] denote the ordinary [[circle]] and $\mathbb{S}$ the
[[pseudocircle]].
There is a [[continuous function]] $S^1 \to \mathbb{S}$ which is a weak homotopy equivalence, hence in particular $\pi_1(\mathbb{S}) \simeq \mathbb{Z}$. But every continuous map the other way round has to induce the trivial map on $\pi_1$.
=--
This is the simplest in a class of counter-examples discussed in ([McCord](#McCord)).
## Related concepts
* [[equality]]
* [[isomorphism]]
* [[equivalence]]
* [[weak equivalence]]
* [[homotopy equivalence]], **weak homotopy equivalence**
* [[rational homotopy equivalence]]
* [[n-equivalence]]
* [[equivariant weak homotopy equivalence]]
* [[stable weak homotopy equivalence]]
* [[Bousfield equivalence]]
* [[homotopy equivalence of toposes]]
* [[equivalence in an (∞,1)-category]]
* [[equivalence of (∞,1)-categories]]
## References
A general account is for instance in section 9.6 of
* {#May} [[Peter May]], _[[A Concise Course in Algebraic Topology]]_
The characterization of weak homotopy equivalences by lifts up to homotopy seems is in
* {#Jardine} [[J. F. Jardine]], _Simplicial Presheaves_, Journal of Pure and Applied Algebra 47, 1987, no.1, 35-87.
* {#Vogt} [[Rainer Vogt]], _The HELP-Lemma and its converse in Quillen model categories_ ([arXiv:1004.5249](http://arxiv.org/abs/1004.5249))
For related and general discussion see also section 6.5 of
* {#Lurie} [[Jacob Lurie]], _[[Higher Topos Theory]]_
Examples for the non-symmetry of the weak homotopy equivalence relation are in
* {#McCord} [[Michael McCord]], _Singular homology groups and homotopy groups of finite topological spaces_, Duke Math. J. Volume 33, Number 3 (1966), 465-474. ([euclid.dmj/1077376525](http://projecteuclid.org/euclid.dmj/1077376525))
See also:
* Topospaces-Wiki, _[Weak homotopy equivalence of topological spaces](http://topospaces.subwiki.org/wiki/Weak_homotopy_equivalence_of_topological_spaces)_
Detection in terms of free homotopy sets:
* {#MatumotoMinamiSugawara84} [[Takao Matumoto]], [[Norihiko Minami]], [[Masahiro Sugawara]], *On the set of free homotopy classes and Brown's construction*, Hiroshima Math. J. 14(2): 359-369 (1984) ([doi:10.32917/hmj/1206133043](https://projecteuclid.org/journals/hiroshima-mathematical-journal/volume-14/issue-2/On-the-set-of-free-homotopy-classes-and-Browns-construction/10.32917/hmj/1206133043.full))
[[!redirects weak homotopy equivalences]] |
weak inverse | https://ncatlab.org/nlab/source/weak+inverse |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A weak inverse or quasi-inverse is like an inverse, but weakened to work in situations where being an inverse on the nose would violate the [[principle of equivalence]].
## Definitions
Given a [[functor]] $F: C \to D$, a __weak inverse__ of $F$ is a functor $G: D \to C$ with [[natural isomorphism]]s
$$ \iota: id_C \to G \circ F,\; \epsilon: F \circ G \to id_D .$$
If it exists, a weak inverse is unique up to natural isomorphism, and furthermore can be improved to form an [[adjoint equivalence]], where $\iota$ and $\epsilon$ sastisfy the [[triangle identities]].
More generally, given a $2$-[[2-category|category]] $\mathcal{B}$ and a morphisms $F: C \to D$ in $\mathcal{B}$, a __weak inverse__ of $F$ is a morphism $G: D \to C$ with $2$-isomorphisms
$$ \iota: id_C \to G \circ F,\; \epsilon: F \circ G \to id_D .$$
Weak inverses give the proper notion of [[equivalence of categories]] and [[equivalence]] in a $2$-category. Note that you must use [[anafunctor]]s to get the weak notion of equivalence of categories here without using the [[axiom of choice]].
## Links with homotopy theory
Given the [[geometric realization of categories]] functor $ \vert -\vert: Cat \to Top$, weak inverses are sent to [[homotopy]] inverses. This is because the product with the [[interval groupoid]] is sent to the product with the topological interval $[0,1]$. In fact, less is needed for this to be true, because the classifying space of the [[interval category]] is also the topological interval. If we define a _lax inverse_ to be given by the same data as a weak inverse, but with $\iota$ and $\epsilon$ replaced by [[natural transformation|natural transformations]], then the classifying space functor sends lax inverses to homotopy inverses. An example of a lax inverse is an [[adjunction]], but not all lax inverses arise this way, as we do not require the triangle identities to hold.
([[David Roberts]]: I'm just throwing this up here quickly, it probably needs better layout or even its own page.)
## Related concepts
* [[inverse]]
[[!redirects weak inverses]]
|
weak Kan complex | https://ncatlab.org/nlab/source/weak+Kan+complex |
#Contents#
* table of contents
{:toc}
## Idea
A **weak Kan complex** [[Boardman & Vogt (1973)](#BoardmanVogt73)] is a [[simplicial set]] for which all _inner_ [[horns]] have a filler.
The notion was later popularized under the name *[[quasi-category]]*. See there for more.
## References
* {#BoardmanVogt73} [[Michael Boardman]], [[Rainer Vogt]], *Homotopy invariant algebraic structures on topological spaces*, Lecture Notes in Mathematics **347** Springer (1973) [[doi:10.1007/BFb0068547](https://doi.org/10.1007/BFb0068547)]
[[!redirects weak Kan complexes]]
|
weak limit | https://ncatlab.org/nlab/source/weak+limit |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A **weak limit** for a [[diagram]] in a [[category]] is a [[cone]] over that diagram which satisfies the existence property of a [[limit]] but not necessarily the uniqueness. The dual concept is a [[weak colimit]].
Beware that "weak" here does **not** correspond to that in "[[weak n-category]]", in particular it does **not** refer to [[homotopy limits]]. Nevertheless, there is a relation, see [below](#RelationToHomotopyLimits). It is due to this relation that weak limits in [[homotopy categories]] play a key role in the [[Brown representability theorem]].
## Weak pullbacks
A **weak pullback** of a [[cospan]]
$$
A\overset{f}{\to} C \overset{g}{\leftarrow} B
$$
(in some [[category]]) is a [[commutative diagram|commutative square]]
$$
\array{
P & \overset{p}{\to} & A
\\
{}^{\mathllap{q}} \downarrow && \downarrow^{\mathrlap{f}}
\\
B & \overset{g}{\to} & C
}
$$
such that for every commuting square
\[\array{ X & \overset{x}{\to} & A\\
{}^{\mathllap{y}} \downarrow && \downarrow^{\mathrlap{f}}\\
B & \overset{g}{\to} & C}\]
there exists a morphism $h: \colon X\to P$, not necessarily unique, such that $x = h p$ and $y = h q$;
If the actual [[pullback]] $A \underset{C}{\times}B$ exists, then this condition means equivalently that the universal morphism
$$
P \longrightarrow A \underset{C}{\times}B
$$
is a [[split epimorphism]].
## Weak terminal objects
Every [[inhabited set]] is a weak [[terminal object]] in [[Set]], since there always exists a [[function]] from any [[set]] to any inhabited set. But only a [[singleton]] is a terminal object.
## Projective objects and exact completion
In any category with finite limits and [[projective object|enough projectives]], the full [[subcategory]] of [[projective object]]s has weak finite limits. For example, given a cospan $A\overset{f}{\to} C \overset{g}{\leftarrow} B$ of projective objects, let $P\to A\times_C B$ be a projective cover of the actual pullback; then any square
\[\array{ X & \overset{x}{\to} & A\\
^y \downarrow && \downarrow ^f\\
B & \overset{g}{\to} & C}\]
with $X$ projective induces a morphism $X\to A\times_C B$, which lifts to a morphism $X\to P$ since $X$ is projective.
Conversely, from any category with weak finite limits one can construct an [[exact category|exact completion]] in which the original category sits as the projective objects, and the exact categories constructible in this way are precisely those having enough projectives.
## Relation to homotopy limits
{#RelationToHomotopyLimits}
Unlike usages of 'weak' in terms like [[weak n-category]], a weak limit is not be like a [[homotopy limit]] or a [[2-limit]], which satisfy uniqueness (as well as existence) albeit only up to higher [[homotopy|homotopies]] or [[weak equivalence|equivalences]].
However, some homotopy limits induce the corresponding type of weak limit in the corresponding [[homotopy category]]. For example, suppose that
$$\array{ P & \overset{p}{\to} & A\\
^q \downarrow && \downarrow ^f\\
B & \overset{g}{\to} & C}
$$
is a homotopy pullback in some category $M$ having a notion of [[homotopy]], such as a [[model category]]. In particular, this square commutes up to homotopy, and thus it commutes in the homotopy category $Ho(M)$. Then any square
$$\array{ X & \overset{x}{\to} & A\\
^y \downarrow && \downarrow ^f\\
B & \overset{g}{\to} & C}$$
that commutes in $Ho(M)$ commutes up to homotopy in $M$, and thus (by the ("local") universal property of homotopy pullbacks), there is a map $h:X\to P$ and homotopies $p h \simeq x$ and $q h\simeq y$; thus the given square is a weak pullback in $Ho(M)$. While the universal property of a homotopy pullback means that $h$ is unique up to homotopy, this is only true for a given _choice_ of homotopy $f x \simeq g y$, and different such homotopies can induce inequivalent $h$'s. Thus in $Ho(M)$, which remembers only the _existence_ of homotopies, we have only a weak pullback.
Note, though, that not _all_ homotopy limits produce weak limits in the homotopy category, because in general it will not be possible to lift a cone that commutes in $Ho(M)$ to a cone that commutes up to _coherent_ homotopy in $M$. However, in "simple" cases such as pullbacks, products, equalizers, sequential inverse limits, and so on, this is always true (and it will be true whenever the diagram category is a [[quiver]]). On the other hand, homotopy products in $M$ give actual (not weak) products in $Ho(M)$, since there are no homotopies necessary.
## Related concepts
* [[weak adjoint]]
* [[weak multilimit]]
## References
* [[Peter Freyd]], _Representations in abelian categories_ 1966 Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965) pp. 95–120 Springer, New York
[[!redirects weak limits]]
[[!redirects weak pullback]]
[[!redirects weak pullbacks]]
[[!redirects weak pushout]]
[[!redirects weak pushouts]]
[[!redirects weak finite limit]]
[[!redirects weak finite limits]]
[[!redirects weak terminal object]]
[[!redirects weak terminal objects]]
[[!redirects weakly terminal]]
|
weak local ring | https://ncatlab.org/nlab/source/weak+local+ring |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
\tableofcontents
## Idea
A weak notion of [[local ring]] in the context of [[constructive mathematics]].
## Definition
A **weak local ring** is a [[commutative ring]] such that
* $0 \neq 1$; and
* the sum of two non-invertible elements is non-invertible
These are the same as [[local rings]] in classical mathematics, but are a weaker (and thus more general) notion than local rings in [[constructive mathematics]].
## Properties
The non-invertible elements in a weak local ring form an [[ideal]]. Thus, the quotient of a weak local ring by its ideal of non-invertible elements form a [[weak Heyting field]] (cf. [Richman 2020](#Richman20)) or a [[Johnstone residue field]] (cf. [Johnstone 1977](#Johnstone77)).
Every weak local ring has an [[equivalence relation]] $\approx$, defined as $x \approx y$ if and only if $x - y$ is non-invertible. Then weak Heyting fields are precisely the weak local rings for which $\approx$ implies [[equality]].
## Examples
* Every [[weak Heyting field]] is an weak local ring where every non-invertible element is equal to zero.
* The dual algebra $\mathbb{R}[\epsilon]/\epsilon^2$ of the [[MacNeille real numbers]] $\mathbb{R}$ is a weak local ring where the [[nilpotent]] [[infinitesimal]] $\epsilon \in \mathbb{R}[\epsilon]/\epsilon^2$ is a non-zero non-invertible element.
* For any prime number $p$ and any positive natural number $n$, the [[prime power local ring]] $\mathbb{Z}/p^n\mathbb{Z}$ is an weak local ring, whose ideal of non-invertible elements is the ideal $p(\mathbb{Z}/p^n\mathbb{Z})$. The quotient of $\mathbb{Z}/p^n\mathbb{Z}$ by its ideal of non-invertible elements is the [[finite field]] $\mathbb{Z}/p\mathbb{Z}$.
* Every [[local ring]] is a weak local ring with an [[apartness relation]] $\#$ such that for all $a \in R$ and $b \in R$, $a \# b$ if and only if $a - b$ is invertible. The negation of $a \# b$ is an [[equivalence relation]] which holds if and only if $a - b$ is non-invertible, making every local ring a weak local ring.
## Weakly ordered local rings
A weakly ordered local ring is a weak local ring $R$ with a [[preorder]] $\leq$ such that
* for all $a \in R$ and $b \in R$, $a \approx b$ if and only if $a \leq b$ and $b \leq a$
* for all $a \in R$, $b \in R$, and $c \in R$, if $a \leq b$, then $a + c \leq b + c$
* for all $a \in R$ and $b \in R$, if $0 \leq a$ and $0 \leq b$, then $0 \leq a \cdot b$
If additionally, for all $a \in R$ and $b \in R$, $a \approx b$ implies that $a = b$, then a weakly ordered local ring becomes a weakly ordered Heyting field, and the preorder becomes a [[partial order]].
Every [[ordered local ring]] and thus every [[ordered field]] is a weakly ordered local ring.
## See also
* [[local ring]]
* [[ordered local ring]]
## References
* {#Johnstone77} [[Peter Johnstone]], *Rings, Fields, and Spectra*, Journal of Algebra **49** (1977) pp 238-260. doi:[10.1016/0021-8693(77)90284-8](https://doi.org/10.1016/0021-8693%2877%2990284-8)
* {#Richman20} [[Fred Richman]], *Laurent series over $\mathbb{R}$*. Communications in Algebra, Volume 48, Issue 5, 11 Jan 2020 Pages 1982-1984 [[doi:10.1080/00927872.2019.1710166](https://doi.org/10.1080/00927872.2019.1710166)]
[[!redirects weak local ring]]
[[!redirects weak local rings]] |
weak model category | https://ncatlab.org/nlab/source/weak+model+category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The concept of _weak model categories_ is a relaxation of that of [[model categories]], even weaker than the concept of [[semimodel categories]], but such that it still allows for a rich theory largely analogous to that of actual [[model categories]]:
The weak analogue of the construction of the [[homotopy category of a model category]] still exists, as do notions of [[Quillen adjunction]] and [[Quillen equivalence]].
Also, for example, an analogue of left or right _[[Bousfield localization of model categories]]_ still makes sense for weak model categories; and, as a bonus in contrast to the usual case, it does not require the assumption of left or right [[proper model category|properness]].
## Definition
A __weak model category__ is a [[premodel category]] that satisfies the following two [[axioms]]:
1. __Cylinder axiom__: Every [[cofibration]] $A\to X$ from a [[cofibrant object]] to a [[fibrant object]] admits a __relative strong cylinder object__
$$X\sqcup_A X\to I_A X\to X,$$
where the left map is a [[cofibration]] and its first component $X\to I_A X$
is an [[acyclic cofibration]].
1. __Path object axiom__: Every [[fibration]] $A\to X$ from a [[cofibrant object]] to a [[fibrant object]] admits a __relative strong path object__
$$A\to P_X A\to A\times_X A,$$
where the right map is a [[fibration]] and its first component $P_X A\to A$ is an [[acyclic fibration]].
## Properties
### Relation to premodel categories
A [[premodel category]] can be upgraded to a weak model category
as follows.
\begin{theorem}
If a [[premodel category]] admits a weak Quillen cylinder,
then it is a weak model category.
\end{theorem}
\begin{definition}
A __weak Quillen cylinder__ on a [[premodel category]] $C$
is a pair of [[left adjoint functors]] $I,D\colon C\to C$
together with the following commutative square of
[[natural transformations]] of [[functors]] $C\to C$:
$$\begin{matrix}
id_C\sqcup \id_C&\mathop{\longrightarrow}\limits^i&I\cr
\downarrow\nabla&&\downarrow e\cr
id_C&\mathop{\longrightarrow}\limits_j&D,\cr
\end{matrix}$$
where $\nabla$ is the [[codiagonal]],
$i$ is a [[cofibration]],
$j$ is a [[trivial cofibration]],
and the first component of $i$ is a [[trivial cofibration]].
\end{definition}
Here a [[natural transformation]] $\lambda\colon F\to G$
of [[functors]] $C\to C$ is a cofibration
if for any (trivial) cofibration $X\to Y$
the map $F(Y)\sqcup_{F(X)}G(X)\to G(Y)$ is a (trivial) cofibration.
Likewise, $\lambda$ is a trivial cofibration
if for any cofibration $X\to Y$ the above map is a trivial cofibration.
Reference: [Henry 20, Section 6](#Henry20).
This is essentially a reformulation of [[Cisinski-Olschok theory]].
### Relation to model categories
[[model category|Model categories]] can be singled out from weak model categories
by adding the following properties:
1. Every [[fibrant object]]
admits a strong path object and every [[cofibrant object]] admits a strong [[cylinder object]].
1. All [[acyclic cofibrations]] are [[trivial cofibrations]]
and all [[acyclic fibrations]] are [[trivial fibrations]].
(Trivial maps are given as data for a [[premodel category]],
whereas acyclic (co)fibrations are defined as (co)fibrations
that satisfy a right (left) lifting property with respect to the class of cofibrations with cofibrant source (respectively fibrations with fibrant target.)
1. The two classes of [[weak equivalence]] corresponding
to the left and right induced [[semimodel structures]] coincide.
### Relation to combinatorial model categories
Every [[combinatorial]] weak model category can
be connected to a [[combinatorial model category]]
by a zigzag of [[Quillen equivalences]].
## Examples
\begin{example}\label{WeakModelStructureOnSemiSimplicialSets}
**([[weak model structure on semi-simplicial sets]])** \linebreak
There is a weak model structure on [[semi-simplicial sets]] which is [[Quillen equivalence|Quillen equivalen]] to [[classical model structure on simplicial sets|that on simplicial sets]]. ([Henry 18, Thm. 5.5.6](#Henry18)).
\end{example}
## Related concepts
* [[model category]]
* [[premodel category]]
## References
* {#Henry18} [[Simon Henry]], _Weak model categories in classical and constructive mathematics_, Theory and Applications of Categories, Vol. 35, 2020, No. 24, pp 875-958. ([arXiv:1807.02650](https://arxiv.org/abs/1807.02650), [tac:35-24](http://www.tac.mta.ca/tac/volumes/35/24/35-24abs.html))
* {#Henry20} [[Simon Henry]], _Combinatorial and accessible weak model categories_ ([arXiv:2005.02360](https://arxiv.org/abs/2005.02360))
[[!redirects weak model categories]]
[[!redirects weak model structure]]
[[!redirects weak model structures]]
|
weak multilimit | https://ncatlab.org/nlab/source/weak+multilimit | # Weak multilimits
* table of contents
{: toc}
## Idea
A *weak multilimit* is a common generalization of [[multilimits]] and [[weak limits]].
## Definition
If $F\colon D\to C$ is a diagram in a [[category]] $C$, then a **weak multilimit** of $F$ is a (small) set $L$ of [[cones]] over $F$ such that any other cone over $F$ factors (not necessarily uniquely) through some (not necessarily unique) element of $L$. If the factorization, and the cone factored through, are unique, then $L$ is a multilimit, whereas if $L$ is a singleton, then it is a *weak limit*.
The existence of weak multilimits is a "pure size condition" on $C$, in the sense that if $C$ is a [[small category]], then every small diagram in $C$ (that is, every functor $F\colon D\to C$ where $D$ is also small) has a weak multilimit, namely the set of *all* cones over $F$.
Of course, weak multilimits in $C^{op}$ are called **weak multicolimits** in $C$.
## Examples
* A weak multilimit of the empty diagram is a *weak multi-terminal-object*, also called a **weakly terminal set**: a small set $T$ of objects such that every object admits a morphism to some object in $T$. The dual concept is a **weakly initial set**. These notions play a role in some statements of the [[adjoint functor theorem]].
[[!redirects weak multilimit]]
[[!redirects weak multicolimit]]
[[!redirects weakly initial set]]
[[!redirects weakly terminal set]]
|
weak omega-category | https://ncatlab.org/nlab/source/weak+omega-category |
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### Context
#### Higher category theory
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[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In [[higher category theory]] the term _weak $\omega$-categories_ is essentially synonymous with _[[infinity-category]]_ in the fully general sense of _[[(infinity,infinity)-category]]_. The terms "$\omega$-category" and "$\infty$-category" originate in different schools and their choice of use is mostly a matter of the preference of individual authors.
One slight difference is that "$\infty$-category" usually implies a "weak" (fully general) notion, while in addition to weak $\omega$-categories there are also [[strict omega-category|strict ones]]. Another difference is that definitions of weak $\omega$-categories tend to be [[algebraic definitions of higher categories|algebraic]] instead of [[geometric definition of higher categories|geometric]] (accordingly typically the central open question is whether a definition really satisfies the [[homotopy hypothesis]]), though some definitions of weak $\omega$-categories are geometry (for instance some flavors of definition of [[opetopic omega-category]]).
## Examples
The following are examples for proposals of definitions of weak $\omega$-categories.
* [[Trimble n-category|Trimble ∞-category]]
* [[simplicial weak omega-category|simplicial weak ∞-category]]
* [[Batanin ∞-category]]
* [[opetopic omega-category]]
## References
> Fore more see general references at *[[higher category theory]]*, such as:
* [[Andre Joyal]], [[Tim Porter]], [[Peter May]], _Weak categories_ ([pdf](https://web.archive.org/web/20150326110254/http://www.ima.umn.edu/talks/workshops/SP6.7-18.04/may/PorterMay.pdf))
Discussion of weak $\omega$-categories via [[computads]] construed as [[inductive types]]:
* [[Christopher J. Dean]], [[Eric Finster]], [[Ioannis Markakis]], [[David Reutter]], [[Jamie Vicary]], *Computads for weak $\omega$-categories as an inductive type* [[arXiv:2208.08719](https://arxiv.org/abs/2208.08719)]
[[!redirects weak omega-categories]]
[[!redirects weak ∞-category]]
[[!redirects weak ∞-categories]] |
weak order | https://ncatlab.org/nlab/source/weak+order |
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### Context
#### Relations
+-- {: .hide}
[[!include relations - contents]]
=--
=--
=--
## Disambiguation
A **weak order** could refer to either
* a [[total preorder]]
* a [[strict weak order]]
## References
* Wikipedia, [Weak order](https://en.wikipedia.org/wiki/Weak_order)
category: disambiguation
[[!redirects weak order]]
[[!redirects weak orders]]
[[!redirects weak ordering]]
[[!redirects weak orderings]]
[[!redirects weakly ordered]]
[[!redirects weakly ordered set]]
[[!redirects weakly ordered sets]] |
weak quotient > history | https://ncatlab.org/nlab/source/weak+quotient+%3E+history | < [[weak quotient]]
[[!redirects weak quotient -- history]] |
weak representation of a functor | https://ncatlab.org/nlab/source/weak+representation+of+a+functor | # Weak representation of a functor
* table of contents
{: toc}
## Idea
A _weak representation of a functor_ $P : C^o \to Set$ ([[presheaf]]) is like a [[representable functor]] but only satisfying the representability condition up to [[retract]] rather than [[isomorphism]]. Unlike a representable, this structure is not unique up to isomorphism, and so we speak of _representations_ as [[structure]] rather than _representability_ as a [[property]].
In [[type theory|type theoretic]] terms, this is an object that only satisfies the $\beta$ equation of a universal property and not the $\eta$ equation.
## Definition
A weak representation of a presheaf $P : C^o \to Set$ consists of
1. An object $X$ in $C$
2. [[natural transformation|natural transformations]] $s : P(\Gamma) \to C(\Gamma, X)$ and $r : C(\Gamma, X) \to P(\Gamma)$ such that $r$ is a [[retract]] of $s$.
Equivalently, by the [[Yoneda lemma]], a weak representation is
1. An object $X$
2. A "weak universal morphism" $\epsilon : P(X)$
3. A natural transformation $I : P(\Gamma) \to C(\Gamma, X)$ that is a [[section]] of $P(-)(\epsilon) : C(\Gamma, X) \to P(\Gamma)$
## Relation to Representable Functors
A weak representation $(X, \epsilon, I)$ of $P$ induces an [[idempotent]] $e$ on the representable sets $C(-, X)$ such that $P$ is the [[idempotent splitting]] of the representable $C(-,X)$ in the presheaf category. By the Yoneda lemma, this induces an idempotent $e$ on $X$ in $C$, and an object $Y$ represents $P$ if and only if $Y$ is the idempotent splitting of $e$. As a consequence, a representable $Y$ has a canonical section to any weak representation $X$.
In type theoretic terms, the idempotent performs [[eta expansion|$\eta$ expansion]], and so this says that a representable can be constructed from a weak representable as the [[fixed points]] of $\eta$ expansion, or dually as a quotient equating the $\eta$ expansion with the identity.
## Related Concepts
* [[weak adjoint]]
* [[weak limit]]
* [[semifunctor]]
* [[lax idempotent 2-adjunction]]
* [[weak 2-representation of a functor]] |
weak topology | https://ncatlab.org/nlab/source/weak+topology |
> For the strong topology in functional analysis, see at [[strong operator topology]].
***
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### Context
#### Topology
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[[!include topology - contents]]
=--
=--
=--
# Induced topologies
* table of contents
{:toc}
## Definitions
See also at *[[Top]]* the section *[Universal constructions](Top#UniversalConstructions)*.
### Weak/coarse/initial topology
{#WeakTopologyDefinition}
Suppose
1. $S$ is a [[set]],
1. $\{ (X_i, T_i) \}_{i \in I}$ is a [[family]] of [[topological spaces]]
1. $\big\{ f_i \colon S \to X_i \big\}_{i \in I}$ an [[indexed set]] of [[functions]] from $S$ to the family $\{ X_i \}_{i \in I}$.
Let $\Gamma$ denote the set of all topologies $\tau$ on $S$ such that $f_i$ is a [[continuous map]] for every $i \in I$. Then the [[intersection]] $\bigcap_{\tau \in \Gamma} \tau$ is again a topology and also belongs to $\Gamma$. Clearly, it is the coarsest/weakest topology $\tau_0$ on $X$ such that each function $f_i\colon S \to X_i$ is a [[continuous map]].
We call $\tau_0$ the **weak/coarse/initial topology induced** on $S$ by the family of mappings $\{ f_i \}_{i \in I}$. Note that all terms 'weak topology', 'initial topology', and 'induced topology' are used. The [[subspace topology]] is a special case, where $I$ is a [[singleton]] and the unique function $f_i$ is an [[injection]].
### Strong/fine/final topology
{#StrongTopologyDefinition}
[[formal duality|Dually]], suppose
1. $S$ is a [[set]],
1. $\{ (X_i, T_i) \}_{i \in I}$ a [[family]] of [[topological spaces]]
1. $\big\{ f_i \colon X_i \to S \big\}_{i \in I}$ a family of [[functions]] to $S$ from the family $\{ X_i \}_{i \in I}$.
Let $\Gamma$ denote the set of all topologies $\tau$ on $S$ such that $f_i$ is a [[continuous map]] for every $i \in I$. Then the [[union]] $\bigcup_{\tau \in \Gamma} \tau$ is again a topology and also belongs to $\Gamma$. Clearly, it is the finest/strongest topology $\tau_0$ on $S$ such that each function $f_i\colon X_i \to S$ is a [[continuous map]].
We call $\tau_0$ the **strong/fine/final topology induced** on $S$ by the family of mappings $\{ f_i \}_{i \in I}$. Note that all terms 'strong topology', 'final topology', and 'induced topology' are used. The [[quotient topology]] is a special case, where $I$ is a [[singleton]] and the unique function $f_i$ is a [[surjection]].
## Generalisations
We can perform the first construction in any [[topological concrete category]], where it is a special case of an [[initial structure]] for a [[sink|source or cosink]].
We can also perform the second construction in any [[topological concrete category]], where it is a special case of an [[final structure]] for a [[sink]].
## In functional analysis
In [[functional analysis]], the term 'weak topology' is used in a special way. If $V$ is a [[topological vector space]] over the [[ground field]] $K$, then we may consider the [[continuous linear functional]]s on $V$, that is the [[continuous map|continuous]] [[linear maps]] from $V$ to $K$. Taking $V$ to be the set $X$ in the general definition above, taking each $T_i$ to be $K$, and taking the continuous linear functionals on $V$ to comprise the family of functions, then we get the __weak topology__ on $V$.
The _weak-star topology_ on the dual space $V^*$ of continuous linear functionals on $V$ is precisely the weak topology induced by the dual (evaluation) functionals on $V^*$
$$
\left\{V^* \overset{\operatorname{ev}_v}{\to} K, \text{ by } f \mapsto f(v)\right\}_{v \in V}.
$$
For the strong topology in functional analysis, see the [[strong operator topology]].
## Related entries
* [Category of topological spaces -- Universal constructions](Top#UniversalConstructions)
## References
The original version of this article was posted by [[Vishal Lama]] at [[nlabmeta:induced topology]].
See also
* Wikipedia, _[Initial topology](https://en.wikipedia.org/wiki/Initial_topology)_, _[Final topology](https://en.wikipedia.org/wiki/Final_topology)_,
[[!redirects weak topology]]
[[!redirects weak topologies]]
[[!redirects coarse topology]]
[[!redirects coarse topologies]]
[[!redirects coarsest topology]]
[[!redirects coarsest topologies]]
[[!redirects initial topology]]
[[!redirects initial topologies]]
[[!redirects strong topology]]
[[!redirects strong topologies]]
[[!redirects fine topology]]
[[!redirects fine topologies]]
[[!redirects final topology]]
[[!redirects final topologies]]
[[!redirects weak-star topology]]
[[!redirects induced topology]]
[[!redirects induced topologies]]
|
weak type theory | https://ncatlab.org/nlab/source/weak+type+theory |
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### Context
#### Type theory
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[[!include type theory - contents]]
=--
#### Deduction and Induction
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[[!include deduction and induction - contents]]
=--
#### Constructivism, Realizability, Computability
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[[!include constructivism - contents]]
=--
#### Foundations
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[[!include foundations - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
**Weak type theory** or **propositional type theory** is a [[dependent type theory]] without judgmental conversion; all the [[computation rules]]/[[beta-conversion|$\beta$-conversion rules]] and [[uniqueness rules]]/[[eta-conversion|$\eta$-conversion rules]] for types use [[identity types]] instead of [[judgmental equality]]. As a result, the results in weak type theory are more general than in models which use judgmental equality for computational and uniqueness rules, since judgmental equality implies typal equality, while the converse isn't necessarily the case.
Formally, weak type theories come into two general versions:
* Dependent type theories which do not have judgmental equality and its [[structural rules]]. Instead, [[definitions]] of [[types]] and [[terms]] are made using [[identifications]] or [[equivalences of types]]. See [[objective type theory]] for more details on this.
* Dependent type theories which do have judgmental equality and its [[structural rules]], and thus definitions of [[types]] and [[terms]] are still made using [[judgmental equality]]. However, in addition to the [[congruence rules]] for [[judgmental equality]] for the [[formation rules]], [[introduction rules]], and [[elimination rules]] of each type former, there are additional [[congruence rules]] for the computation and uniqueness rules, since the conversion rules are represented by the structure of an identification rather than judgmental equality, and thus this structure has to be preserved across judgmental equality.
The latter includes weak versions of [[Martin-Löf type theory]], [[cubical type theory]], and [[observational type theory]], as well as extensions thereof such as [[type theory with shapes]] and [[simplicial type theory]]. Hypothetically, the latter would also be seen in [[proof assistants]], where the base [[programming language]] used to implement the weak type theory, such as [[Coq]] or [[Agda]], already has a judgmental equality.
A hybrid of weak and non-weak type theories can occur in [[two-level type theory]] and variants thereof like [[Homotopy Type System]], where one level has weak types and the other one has strict types.
## Open problems
There are plenty of questions which are currently unresolved in weak type theory. Van der Berg and Besten listed the following problems in the context of [[objective type theory]], but equally this applies to any weak type theory:
* Does [[univalence]] imply [[function extensionality]] for types in the universe in [[weak type theory]]?
* Is (the [[homotopy type theory]] based upon) [[Martin-Löf type theory]] conservative over (the [[homotopy type theory]] based upon) [[weak type theory|weak]] Martin-Löf type theory?
* How much of the [[HoTT book]] could be done in [[weak type theory]]?
* Does [[weak type theory]] have [[homotopy canonicity]] and [[normalization]]?
Other problems include
* What is the [[categorical semantics]] of the [[homotopy type theory]] based upon [[weak type theory]], with all [[higher inductive types]] and [[weakly Tarski]] [[univalent universes]]?
* Is [[weak function extensionality]] equivalent to [[function extensionality]] in [[weak type theory]]?
* Does [[product extensionality]] hold in [[weak type theory]]? Namely, given types $A$ and $B$ and elements $a:A \times B$ and $b:A \times B$, is the canonical function $\mathrm{idtoprojectionids}:(a =_{A \times B} b) \to (\pi_1(a) \simeq \pi_1(b)) \times (\pi_2(a) \simeq \pi_2(b))$ an [[equivalence of types]]?
* Is [[function extensionality]] still provable in weak [[cubical type theory]]?
See also [[open problems in homotopy type theory]]
## See also
* [[dependent type theory]]
## References
The original paper on weak type theory, in the context of [[objective type theory]]:
* {#BB21} [[Benno van den Berg]], [[Martijn den Besten]], *Quadratic type checking for objective type theory* ([arXiv:2102.00905](https://arxiv.org/abs/2102.00905))
Talks at *Strength of Weak Type Theory*, hosted by *[[DutchCATS]]*:
* [[Daniël Otten]], *Models for Propositional Type Theory* (11 May 2023) [[slides pdf](https://dutchcats.github.io/Weak_type_theories/slides_otten.pdf)]
* [[Théo Winterhalter]], *A conservative and constructive translation from extensional type theory to weak type theory*, 11 May 2023. ([slides](https://dutchcats.github.io/Weak_type_theories/slides_winterhalter.pdf))
* [[Sam Speight]], *Higher-Dimensional Realizability for Intensional Type Theory*, 11 May 2023. ([slides](https://dutchcats.github.io/Weak_type_theories/slides_speight.pdf))
* [[Matteo Spadetto]], *Weak type theories: a conservativity result for homotopy elementary types* (12 May 2023) [[slides pdf](https://dutchcats.github.io/Weak_type_theories/slides_spadetto.pdf)]
* [[Benno van den Berg]], *Towards homotopy canonicity for propositional type theory*, 12 May 2023. ([slides](https://dutchcats.github.io/Weak_type_theories/slides_van_den_berg.pdf))
* [[Rafaël Bocquet]], *Towards coherence theorems for equational extensions of type theories*, 12 May 2023. ([slides](https://dutchcats.github.io/Weak_type_theories/slides_bocquet.pdf))
Project to convert [[extensional type theory]] to [[weak type theory]]:
* Github, [ett-to-wtt](https://github.com/TheoWinterhalter/ett-to-wtt)
[[!redirects propositional type theory]]
[[!redirects propositional type theories]]
[[!redirects weak type theory]]
[[!redirects weak type theories]] |
weak wreath | https://ncatlab.org/nlab/source/weak+wreath | [[!redirects weak wreath product]]
## Idea
A **weak wreath** is to a [[weak distributive law]] what a [[wreath]] is to a [[distributive law]].
## Related pages
* [[weak distributive law]]
## References
* Stefaan Caenepeel and Erwin De Groot. _Modules over weak entwining structures_, Contemporary Mathematics 267 (2000): 31-54.
* [[Ross Street]], *Weak distributive laws*, Theory and Appl. of Categ. **22** (2009) 313--320 [[tac:22-12](http://www.tac.mta.ca/tac/volumes/22/12/22-12abs.html)]
* [[Gabriella Böhm]], _On the iteration of weak wreath products_, Theory and Appl. of Categories __26__:2 (2012) 30--59 [arXiv:1110.0652](https://arxiv.org/abs/1110.0652)
* [[Gabriella Böhm]], [[José Gómez-Torrecillas]], _Bilinear factorization of algebras_, Bull. Belg. Math. Soc. Simon Stevin 20(2): 221-.244 (may 2013) [doi](https://doi.org/10.36045/bbms/1369316541)
* [[Gabriella Böhm]], [[Steve Lack]], [[Ross Street]], _Idempotent splittings, colimit completion, and weak aspects of the theory of monads_, Journal of Pure and Applied Algebra __216__:2 (2012) 385--403 [doi](https://doi.org/10.1016/j.jpaa.2011.07.003)
[[!redirects weak wreath]]
[[!redirects weak wreaths]]
[[!redirects weak wreath products]] |
weakening rule | https://ncatlab.org/nlab/source/weakening+rule |
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### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
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# The weakening rule
* table of contents
{: toc}
## Idea
In [[formal logic]] and [[type theory]], by "weakening rules" one means [[inference rules]] for [[context extension]],
which state that any [[premises]] may be added to the [[hypotheses]] ([[antecedents]]) of a valid [[judgement]].
Along with the [[variable rule]], the [[contraction rule]] and the [[exchange rule]], the weakening rule is one of the most commonly adopted [[structural rules]]. But the weakening rule is not used in all [[logical frameworks]], for instance in [[linear logic]] it is discarded.
Concretely, in [[intuitionistic type theory|intuitionistic]] [[dependent type theory]] the weakening rule is the [[inference rule]]
$$
W
\frac{
\Gamma
,\,
\Delta
\;
\vdash
\;
j \colon J
\;\;\;\;\;\;\;\;
\Gamma
\;
\vdash
\;
A \colon Type
}{
\Gamma
,\;
A
,\,
\Delta
\;
\vdash
\;
j \colon J
}
$$
As usual in [[dependent type theory]], the meaning of this rule is a little less trivial than it may superficially seem, due to the generic [[dependent type|type dependency]] involved: In [[context extension|extending the contents]] in the [[antecedent]] we are implicitly making the [[succedent]] [[dependent type|depend]] on this extended context, albeit trivially.
Accordingly, in the [[categorical semantics]] [[categorical semantics of dependent types|of dependent types]] the weakening rule corresponds to [[pullback]] of [[bundles]]/[[display maps]] to the [[fiber product]] which interprets the extended context:
<center>
<img src="/nlab/files/StructuralTypeInferenceRules-230326b.jpg" width="650">
</center>
## Semantics
Weakening rules correspond to having [[semicartesian monoidal category|projections for the monoidal structure]] that corresponds to the logical binary operator at hand.
## Related concepts
* [[inference rule]]
* [[structural rules]]
* [[variable rule]]
* [[substitution rule]]
## References
The notion of weakening as a [[structural inference rule]] originates (under the German name *Verdünnung*) with:
* {#Gentzen35} [[Gerhard Gentzen]], §1.2.1 in: _Untersuchungen über das logische Schließen I_ _Mathematische Zeitschrift_ **39** 1 (1935) [[doi:10.1007/BF01201353](http://dx.doi.org/10.1007/BF01201353)]
* {#Gentzen69} [[Gerhard Gentzen]], §1.21 in: *Investigations into Logical Deduction*, in M. E. Szabo (ed.), *The Collected Papers of Gerhard Gentzen*, Studies in Logic and the Foundations of Mathematics **55**, Springer (1969) 68-131 [[ISBN:978-0-444-53419-4](https://www.sciencedirect.com/bookseries/studies-in-logic-and-the-foundations-of-mathematics/vol/55), [pdf](https://logic-teaching.github.io/prop/texts/Gentzen%201969%20-%20Investigations%20into%20Logical%20Deduction.pdf)]
On the [[categorical semantics]]
* [[Bart Jacobs]], *Semantics of weakening and contraction*, Annals of Pure and Applied Logic **69** 1 (1994) 73-106 [<a href="https://doi.org/10.1016/0168-0072(94)90020-5">doi:10.1016/0168-0072(94)90020-5</a>]
Discussion in [[intuitionistic type theory|intuitionistic]] [[dependent type theory]]:
* {#Jacobs98} [[Bart Jacobs]], p. 122, 585 in: *Categorical Logic and Type Theory*, Studies in Logic and the Foundations of Mathematics **141**, Elsevier (1998) [[ISBN:978-0-444-50170-7](https://www.sciencedirect.com/bookseries/studies-in-logic-and-the-foundations-of-mathematics/vol/141), [pdf](https://people.mpi-sws.org/~dreyer/courses/catlogic/jacobs.pdf)]
> (emphasis on the [[categorical model of dependent types]])
* {#UFP13} [[Univalent Foundations Project]], §A of *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]* (2013) [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)]
> (in the context of [[homotopy type theory]])
* {#Rijke18} [[Egbert Rijke]], *Dependent type theory* [[pdf](https://www.andrew.cmu.edu/user/erijke/hott/dtt.pdf)], Lecture 1 in: *[[Introduction to Homotopy Type Theory]]*, lecture notes, CMU (2018) [[pdf](http://www.andrew.cmu.edu/user/erijke/hott/hott_intro.pdf), [[Rijke-IntroductionHoTT-2018.pdf:file]], [webpage](https://www.andrew.cmu.edu/user/erijke/hott/)]
> (in the context of [[homotopy type theory]])
[[!redirects weakening rule]]
[[!redirects weakening rules]]
|
weakly constant function | https://ncatlab.org/nlab/source/weakly+constant+function |
# Weakly constant function
* table of contents
{: toc}
## Idea
By a *weakly constant function* one means a [[function]] which is equipped with a certain sort of "witness of [[constant function|constancy]]".
However, in [[higher category theory]] and [[homotopy theory]], it is debatable whether or not this witness really exhibits "constancy", hence the use of a different word. (The term "steady function" was suggested by [[Andrej Bauer]] but did not catch on.)
## Definition
In [[homotopy type theory]], a [[function type|function]]
$$
f\colon A\to B
$$
is *weakly constant* if we have a [[term]] of [[type]] of the [[dependent product]] over the [[identification types]] of all the values of $f$ on all [[pairs]] of arguments:
$$
\prod_{(x,y:A)}
(f x = f y)
\,.
$$
By regarding [[homotopy type theory]] as the [[internal logic]] of an [[(∞,1)-topos]], we obtain a definition that makes sense in any [[(infinity,1)-category|$\infty$-category]] with [[binary products]]: a morphism $f:A\to B$ is *weakly constant* if the two composites $A\times A \rightrightarrows A \xrightarrow{f} B$ are [[equivalence in an (infinity,1)-category|equivalent]].
## Relationship to constancy
If $f$ is [[constant function|constant]] in the sense that it factors through the [[terminal object]] (i.e. we have $f = \lambda x. b$ for some $b:B$), then $f$ is obviously weakly constant. The converse holds if we know that the [[domain]] $A$ is [[inhabited type|inhabited]], for if $a_0:A$, then $f a = f a_0$ for all $a:A$. However, the [[identity function]] of the [[empty type]] is weakly constant, yet not equal to $\lambda x.b$ for any $b:\varnothing$ (since no such $b$ exists).
More generally, if $f$ factors through the [[propositional truncation]] ${\|A\|}$, then it is weakly constant, since any two [[elements]] of ${\|A\|}$ are equal (i.e. it is an [[h-proposition]]). In fact, this is true if $f$ factors through *any* [[h-proposition]] (in which case it in fact also factors through ${\|A\|}$, by the [[universal property]] of the latter).
The converse to this last implication does hold for some specific $f:A\to B$, such as:
* If $B$ is an [[h-set]]. For then $f$ factors through the [[0-truncation]] ${\|A\|_0}$, and the set-[[coequalizer]] of the two [[projections]] ${\|A\|_0} \times {\|A\|_0} \to {\|A\|_0}$ is the [[propositional truncation]].
* If $A$ has [[split support]]. For then we have a composite ${\|A\|} \to A \xrightarrow{f} B$, whose restriction to $A$ is equal to $f$ by weak constancy.
* If $A=P+Q$, with $P$ and $Q$ [[h-propositions]]. For then ${\|A\|}$ is the [[join]] $P*Q$ of $P$ and $Q$, i.e. the [[pushout]] of the two [[projections]] $P \leftarrow P\times Q \to Q$. The [[universal property]] of this pushout says exactly that any weakly constant map $P+Q\to B$ factors through $P*Q = \Vert P+Q\Vert$.
* If $A=B$ (see below).
However, it can fail in general, even when $A$ is [[mere proposition|merely]] inhabited (i.e. ${\|A\|}= 1$). For instance, let $A=P+Q+R$ for h-propositions $P$, $Q$, and $R$, and let $B$ be the triple pushout of $P$, $Q$, and $R$ under $P\times Q$, $P\times R$, and $Q\times R$. Then there is a steady map $f:A\to B$, but there exist models in which ${\|A\|} = 1$ but $B$ has no [[global element]]. The most straightforward such model is [[presheaves]] on the [[poset]] of proper [[subsets]] of $\{a,b,c\}$, with $P=\{a,b\}$, $Q=\{b,c\}$, and $R=\{a,c\}$. In this model, we have $B(S) = 1$ for all nonempty proper subsets $S$, while $B(\emptyset) = S^1$, and $B$ has no global sections.
See [this discussion](https://groups.google.com/d/msg/homotopytypetheory/FeBAScTgwzg/Dx6E3-ezdxIJ).
In general, being weakly constant may be regarded as an "incoherent approximation" to constancy in the sense of factoring through an [[h-proposition]]. Indeed for a set $A$, its propositional truncation is the set-[[coequalizer]] of $A\times A\rightrightarrows A$. However, in general such a construction requires the realization of a whole [[simplicial diagram]] (the simplicial [[kernel]] of the map $A\to 1$).
## Weakly constant endomaps
While an arbitrary weakly constant function is not very [[coherence law|coherent]], a weakly constant [[endofunction]] $f:A\to A$ has some extra degree of "coherence", as witnessed by the following results of [(KECA)](#KECA).
+-- {: .un_lemma}
###### Lemma
If $f:A\to A$ is weakly constant, then the type $Fix(f) \coloneqq \sum_{x:A} (f x = x)$ is an [[h-proposition]], and equivalent to ${\|A\|}$.
=--
+-- {: .proof}
###### Proof
Suppose $H: \prod_{(x,y:A)} (f x = f y)$, and let $(a,p),(b,q):Fix(f)$; we want to show $(a,p)=(b,q)$. Let $r:a=b$ be the concatenated path
$$ a \xrightarrow{p^{-1}} f a \xrightarrow{H_{a,a}^{-1}} f a \xrightarrow{H_{a,b}} f b \xrightarrow{q} b. $$
It will suffice to show that $p \bullet r = ap_f(r) \bullet q$, where $ap_f$ denotes the action of $f$ on paths. However, the dependent action of $H$ on paths implies that $H_{x,y} \bullet ap_f(s) = H_{x,y'}$ for any $x:A$ and any $s:y=y'$, and in particular $ap_f(r) = H_{a,a}^{-1} \bullet H_{a,b}$. From this $p \bullet r = ap_f(r) \bullet q$ is immediate. Thus, $Fix(f)$ is an h-proposition.
Now we have a map $g:A\to Fix(f)$ defined by $g(x) \coloneqq (f x, H_{f x,x})$, so by the universal property of ${\|A\|}$, we have ${\|A\|} \to Fix(f)$. On the other hand, we have the first projection $pr_1:Fix(f) \to A$, and hence $Fix(f) \to {\|A\|}$. Thus, these two h-propositions are equal.
=--
+-- {: .un_theorem}
###### Theorem
A type $A$ has [[split support]] if and only if it admits a weakly constant endomap $f:A\to A$.
=--
+-- {: .proof}
###### Proof
Given ${\|A\|} \to A$, the composite $A\to {\|A\|} \to \A$ is weakly constant. Conversely, if $f$ is steady, $Fix(f) = {\|A\|}$ by the lemma, so $pr_1:Fix(f) \to A$ splits the support of $A$.
=--
Note that $pr_1 \circ g = f$, so that if we start with a weakly constant endomap of $A$, deduce a splitting of the support of $A$, and then reconstruct a weakly constant endomap, we obtain the same map. However, the proof of steadiness is generally different from the one we began with, so this "logical equivalence" is not an equivalence of types.
+-- {: .un_cor}
###### Corollary
A type $A$ is an [[h-set]] if and only if every [[identity type]] $x=_A y$ admits a weakly constant endomap.
=--
+-- {: .proof}
###### Proof
We know that $A$ is an h-set just when all $x=_A y$ have split support.
=--
## References
* {#KECA} [[Nicolai Kraus]] and [[Martin Escardo]] and [[Thierry Coquand]] and [[Thorsten Altenkirch]], "Generalizations of Hedberg's theorem", M. Hasegawa (Ed.): TLCA 2013, LNCS 7941, pp. 173-188. Springer, Heidelberg 2013. [PDF](http://www.cs.bham.ac.uk/~mhe/papers/hedberg.pdf). In this paper, steady functions are called "constant".
* [[Nicolai Kraus]], [[Martín Escardó]], [[Thierry Coquand]], [[Thorsten Altenkirch]], *Notions of Anonymous Existence in Martin-Löf Type Theory*, Logical Methods in Computer Science **13** 1 [<a href="https://doi.org/10.23638/LMCS-13(1:15)2017">doi:10.23638/LMCS-13(1:15)2017</a>, [arXiv:1610.03346](https://arxiv.org/abs/1610.03346)]
[[!redirects weakly constant function]]
[[!redirects weakly constant functions]]
[[!redirects steady function]]
[[!redirects steady functions]]
|
weakly distributive category > history | https://ncatlab.org/nlab/source/weakly+distributive+category+%3E+history |
see _[[linearly distributive category]]_ |
weakly globular n-fold category | https://ncatlab.org/nlab/source/weakly+globular+n-fold+category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Classically, [[n-category|models of weak $n$-categories]] comprise sets of cells in dimension 0 up to $n$. This is also called the globularity condition. Geometrically, it corresponds to cells having a globular shape
\begin{imagefromfile}
"file_name": "Glob_Shape_N.jpg",
"width": 300
\end{imagefromfile}
On the other hand, non-globular structures exist in higher category theory: for instance, [[n-fold category|$n$-fold categories]], defined by iterated [[internalization|internalization]]
\[
Cat^{0}=Set,\qquad\qquad Cat^{n}=Cat(Cat^{n-1})\;.
\]
The category $n\text{-}Cat$ of [[strict n-category|strict $n$-categories]] is defined by iterated [[enriched category|enrichment]]:
$$
0\text{-}Cat=Set,\qquad\qquad n\text{-}Cat=((n-1)\text{-}Cat,\times)\text{-}Cat\;.
$$
There is an embedding
$$
n\text{-}Cat\hookrightarrow Cat^{n}
$$
such that a strict $n$-category is an $n$-fold category in which certain substructures are discrete (that is, sets).
This discreteness condition is precisely the globularity condition and the sets underlying these discrete substructures are the sets of cells in the strict $n$-category.
For instance, in the case $n=2$, strict 2-categories are [[double category|double categories]] in which the category of objects and vertical arrows is discrete. In pictures:
\begin{imagefromfile}
"file_name": "2-Cat.jpg",
"width": 800
\end{imagefromfile}
We see that the picture on the right becomes the one on the left when all vertical morphisms are identities.
In the weakly globular approach to higher categories, the cells in each dimension instead of forming a set, have a higher categorical structure which is homotopically discrete, that is only equivalent (in a higher categorical sense) to a set. This condition, called weak globularity condition, is a new paradigm to weaken higher categorical structures and allows to use rigid structures, namely $n$-fold categories, to model weak $n$-categories. There are strict embeddings
$$
n\text{-} Cat \hookrightarrow Cat_{wg}^{n} \hookrightarrow Cat^{n}
$$
so that the category $Cat_{wg}^{n}$ of weakly globular $n$-fold categories is intermediate between strict $n$-categories and $n$-fold categories.
In the case $n=2$, a weakly globular double category $X$ is a double category satisfying two conditions:
the first is the weak globularity condition, stating that the category $X_0$ of objects and vertical arrows is equivalent to a discrete category, that is it is an equivalence relation. The set $pX_0$ of connected components of $X_0$ plays the role of 'set of objects' in the weak structure. We therefore need to define a composition of horizontal arrows whose source and target are in the same vertical connected component
\begin{imagefromfile}
"file_name": "Horiz-Comp.jpg",
"width": 700
\end{imagefromfile}
For this purpose, we impose a second condition: for each 'staircase path'
\begin{imagefromfile}
"file_name": "Staircase.jpg",
"width": 600
\end{imagefromfile}
A more concise way to express this second condition is in term of the so called induced Segal maps condition.
Weakly globular double categories have been shown by Paoli and Pronk to be suitably equivalent to [[bicategory|bicategories]].
In dimension $n\geq2$, a weakly globular $n$-fold category $X$ is an $n$-fold category which first of all needs to satisfy the weak globularity condition; namely those substructures that are discrete in the image of the embedding $n\text{-} Cat\hookrightarrow Cat^{n}$, are now instead only equivalent to sets. The precise notion we use for these substructures is the one of homotopically discrete $n$-fold categories. The latter are defined inductively, starting with equivalence relations in the case of $n=1$. When $n\geq1$, the idea of a homotopically discrete $n$-fold category is that it is an $n$-fold category suitably equivalent to a discrete one both 'globally' and in each simplicial dimension.
The additional conditions in the definition of $Cat_{wg}^{n}$ are given inductively in terms of induced Segal maps conditions. They guarantee the existence of weakly associative and weakly unital compositions but, like in the Tamsamani-Simpson model, their [[coherence law|coherences]] are not given explicitly but they are automatically encoded in the [[multisimplicial set|multi-simplicial]] combinatorics.
Weakly globular $n$-fold categories have been shown to satisfy the [[homotopy hypothesis|homotopy hypothesis]] and to be suitably equivalent to the [[Homotopy Theory of Higher Categories|Tamsamani-Simpson model]].
## Homotopically discrete $n$-fold categories
The definition of weakly globular $n$-fold category requires a preliminary notion, the one of homotopically discrete $n$-fold category.
\begin{definition}\label{def01}
Let $Cat_{hd}^{0}=Set$. Suppose, inductively, we have defined the subcategory $Cat_{hd}^{n-1}\subset Cat^{n-1}$ of homotopically discrete $(n-1)$-fold categories. We say that the $n$-fold category $X\in Cat^{n}$ is homotopically discrete if:
a) $X$ is a levelwise equivalence relation, that is for each $(k_1,\ldots,k_{n-1})\in\Delta^{{n-1}^{op}}$, $X_{k_1,\ldots,k_{n-1}}\in Cat$ is an [[equivalence relation|equivalence relation]] (that is, a category equivalent to a discrete one).
b) $p^{(n-1)}X\in Cat_{hd}^{n-1}$ where $(p^{(n-1)}X)_{k_1,\ldots,k_{n-1}}=p X_{k_1,\ldots,k_{n-1}}$ with $p: Cat\rightarrow Set$ the [[isomorphism class|isomorphism classes of objects]] functor.
When $n=1$ we denote by $Cat_{hd}^{1}=Cat_{hd}^{}$ the subcategory of $ Cat$ consisting of equivalence relations.
\end{definition}
\begin{definition}\label{def02}
Let $X\in Cat_{hd}^{n}$. Denote by $\gamma^{(n-1)}_X:X\rightarrow p^{(n-1)}X$ the morphism given by
$$
(\gamma^{(n-1)}_X)_{s_1...s_{n-1}} :X_{s_1...s_{n-1}} \rightarrow p X_{s_1...s_{n-1}}\;.
$$
Denote by
$$
X^d =p p^{(1)}...p^{(n-1)}X
$$
and by $\gamma_{n}$ the composite
$$
X\xrightarrow{\gamma^{(n-1)}}p^{(n-1)}X \xrightarrow{\gamma^{(n-2)}} p^{(n-2)}p^{(n-1)}X \rightarrow \cdots \xrightarrow{\gamma^{(0)}} X^d\;.
$$
We call $\gamma_{n}$ the discretization map.
\end{definition}
\begin{definition}\label{def03}
Given $X\in Cat_{hd}^{n}$, for each $a,b\in X_0^d$ denote by $X(a,b)$ the fiber at $(a,b)$ of the map
$$
X_1 \xrightarrow{(d_0,d_1)} X_0\times X_0 \xrightarrow{\gamma_{n}\times\gamma_{n}} X_0^d\times X_0^d\;.
$$
$X(a,b)\in Cat_{hd}^{n-1}$ should be thought of as a hom-$(n-1)$-category.
\end{definition}
\begin{definition}\label{def04}
Define inductively $n$-equivalences in $Cat_{hd}^{n}$. For $n=1$, a 1-equivalence is an [[equivalence of categories|equivalence of categories]]. Suppose we have defined $(n-1)$-equivalences in $Cat_{hd}^{n-1}$. Then a map $f:X\rightarrow Y$ in $Cat_{hd}^{n}$ is an $n$-equivalence if
a) For all $a,b \in X_0^d$,
$$
f(a,b):X(a,b) \rightarrow Y(f a,f b)
$$
is an $(n-1)$-equivalence.
b) $p^{(n-1)}f$ is an $(n-1)$-equivalence.
\end{definition}
The next proposition means that homotopically discrete $n$-fold categories are a higher categorical 'fattening' of sets.
\begin{proposition}\label{pro01}
Let $X\in Cat_{hd}^{n}$. Then the maps $\gamma^{(n-1)}:X\rightarrow p^{(n-1)}X$ and $\gamma_{n}:X\rightarrow X^d$ are $n$-equivalences.
\end{proposition}
## The three Segal-type models
There are three Segal-type models of weak $n$-categories
\begin{tikzcd}
& Ta_{wg}^{n} \\
Ta^{n} && Cat_{wg}^{n}
\arrow[hook, from=2-1, to=1-2]
\arrow[hook', from=2-3, to=1-2]
\end{tikzcd}
Here $Ta^{n}$ is the Tamsamani-Simpson model and $Ta_{wg}^{n}$ (called weakly globular Tamsamani $n$-categories) is a generalization of it using weak globularity; the latter contains weakly globular $n$-fold categories as special case.
### Weakly globular Tamsamani ${n}$-categories
The definition of $Ta_{wg}^{n}$ is by induction on $n$, starting with $Ta_{wg}^{1}= Cat$ and 1-equivalences being equivalences of categories. Suppose, inductively, that we defined $Ta_{wg}^{n-1}$ and $(n-1)$-equivalences. Then we define $Ta_{wg}^{n}$ through the following conditions:
a) There is an embedding of $Ta_{wg}^{n}$ into [[functor category|functor categories]]
\[
\label{eq3}
Ta_{wg}^{n}\hookrightarrow [\Delta^{{}^{op}},{Ta_{wg}^{n-1}}]\hookrightarrow [\Delta^{{n-1}^{op}}, Cat]
\]
b) There is a truncation functor
\[
\label{eq4}
p^{(n-1)}:Ta_{wg}^{n}\rightarrow Ta_{wg}^{n-1}
\]
$$
(p^{(n-1)}X)_{k_1,\ldots, k_{n-1}}=p X_{k_1,\ldots, k_{n-1}}
$$
for each $(k_1,\ldots, k_{n-1})\in\Delta^{{n-1}^{op}}$ and $X\in Ta_{wg}^{n}$, where $p: Cat\rightarrow Set$ is the isomorphism classes of objects functor.
c) $X_0$ is a homotopically discrete $(n-1)$-fold category. This comes with a $(n-1)$-equivalence $\gamma:X_0\rightarrow X_0^d$ where $X_0^d$ is a discrete $(n-1)$-fold category.
d) For each $k\geq 2$ the induced Segal maps
\[
\label{eq5}
\hat{\mu}_k:X_k\rightarrow {X_1\times_{X_0^d}\overset{k}{\cdots}\times_{X_0^d}X_1}
\]
are $(n-1)$-equivalences in $Ta_{wg}^{n-1}$. The maps $\hat{\mu}_k$ arise from the commutativity of the diagram
\begin{tikzcd}[column sep=scriptsize]
&&&& {X_k} \\
& {X_1} && {X_1} && \cdots && {X_1} \\
{X_0^d} && {X_0^d} && {X_0^d} & \cdots & {X_0^d} && {X_0^d}
\arrow["{\nu_1}"', from=1-5, to=2-2]
\arrow["{\nu_k}", from=1-5, to=2-8]
\arrow["{\nu_2}"', from=1-5, to=2-4]
\arrow["{\gamma d_1}"', from=2-2, to=3-1]
\arrow["{\gamma d_0}", from=2-2, to=3-3]
\arrow["{\gamma d_0}", from=2-8, to=3-9]
\arrow["{\gamma d_0}", from=2-4, to=3-5]
\arrow["{\gamma d_1}"', from=2-4, to=3-3]
\arrow["{\gamma d_1}"', from=2-8, to=3-7]
\end{tikzcd}
e) To complete the inductive step, we define $n$-equivalences in $Ta_{wg}^{n}$. For this, given $X\in Ta_{wg}^{n}$ and $(a,b)\in X_0^d\times X_0^d$, we let $X(a,b)\subset X_1$ be the fiber at $(a,b)$ of the map
$$
X_1\xrightarrow{(d_0,d_1)} X_0\times X_0 \xrightarrow{\gamma\times\gamma} X^d_0\times X^d_0\;.
$$
We define a map $f:X\rightarrow Y$ in $Ta_{wg}^{n}$ to be an $n$-equivalence if the following conditions hold:
i) For all $a,b\in X_0^d$
$$
f(a,b): X(a,b)\rightarrow Y(f a,f b)
$$
is a $(n-1)$-equivalence.
ii) $p^{(n-1)}f$ is a $(n-1)$-equivalence.
### Tamsamani ${n}$-categories
The category $Ta^{n}$ of Tamsamani $n$-categories is the full subcategory of $Ta_{wg}^{n}$ whose objects $X$ are such that $X_0$ and $X_{k_1\ldots k_r 0}$ are discrete for all $(k_1\ldots k_r)\in \Delta^{{r}^{op}}$, $1\leq r\leq n-2$.
The embedding \eqref{eq3} restricts to an embedding
$$
Ta_{n}\hookrightarrow [\Delta^{{}^{op}},{Ta_{n-1}}]
$$
and the functor $p^{(n-1)}$ in \eqref{eq4} restricts to
$$
p^{(n-1)}:Ta^{n}\rightarrow Ta^{n-1}
$$
The induced Segal maps $\hat{\mu}_k$ coincide with the [[Segal map|Segal maps]]
$$
X_k\rightarrow {X_1\times_{X_0}\overset{k}{\cdots}\times_{X_0}X_1}
$$
and are $(n-1)$-equivalences in $Ta^{n-1}$.
### Weakly globular $n$-fold categories
Let $Cat_{wg}^{1}= Cat$. Having defined the full subcategory $Cat_{wg}^{n-1}\subset Ta_{wg}^{n-1}$, the category $Cat_{wg}^{n}$ of weakly globular $n$-fold categories is the full subcategory of $Ta_{wg}^{n}$ whose objects $X$ are such that $X\in Cat^{n}$, $X_k\in Cat_{wg}^{n-1}$ for all $k\geq 0$ and $p^{(n-1)}X\in Cat_{wg}^{n-1}$.
## Main results
\begin{theorem}\label{th1}
a) There is a functor 'rigidification'
$$
Q_n: Ta_{wg}^{n}\rightarrow Cat_{wg}^{n}
$$
and for each $X\in Ta_{wg}^{n}$ an $n$-equivalence in $Ta_{wg}^{n}$ natural in $X$
$$
s_n(X):Q_n X\rightarrow X.
$$
b) There is a functor 'discretization'
$$
Disc_{n}:Cat_{wg}^{n}\rightarrow Ta^{n}
$$
and, for each $X\in Cat_{wg}^{n}$, a zig-zag of $n$-equivalences in $Ta_{wg}^{n}$ between $X$ and $Disc_{n} X$.
\end{theorem}
The functors discretization and rigidification are used in the comparison result between Tamsamani $n$-categories and weakly globular $n$-fold categories as follows:
\begin{theorem}\label{th2}
The functors
$$
Q_n:Ta^{n}\leftrightarrows Cat_{wg}^{n}:Disc_{n}
$$
induce an equivalence of categories after [[localization|localization]] with respect to the $n$-equivalences
$$
Ta^{n}/\!\!\sim^n\;\simeq \; Cat_{wg}^{n}/\!\!\sim^n\;.
$$
\end{theorem}
## The groupoidal case
There is a subcategory $GCat_{wg}^{n}\subset Cat_{wg}^{n}$ of groupoidal weakly globular $n$-fold categories which is an algebraic model of [[homotopy n-type|homotopy $n$-types]]. This means that weakly globular $n$-fold categories satisfy the homotopy hypothesis.
To define $GCat_{wg}^{n}$ we first consider the groupoidal version of the largest of the three Segal-type models:
\begin{definition}\label{def1}
The full subcategory $GTa_{wg}^{n}\subset Ta_{wg}^{n}$ of groupoidal weakly globular Tamsamani $n$-categories is defined inductively as follows.
For $n=1$, $GTa_{wg}^{1}=Gpd$ is the category of [[groupoid|groupoids]]. Note that $Cat_{hd}^{}\subset GTa_{wg}^{1}$. Suppose inductively we have defined $GTa_{wg}^{n-1}\subset Ta_{wg}^{n-1}$. We define $X\in GTa_{wg}^{n}\subset Ta_{wg}^{n}$ such that
i) $X_k\in GTa_{wg}^{n-1}$ for all $k\geq 0$.
ii) $p^{(n-1)}X\in GTa_{wg}^{n-1}$.
\end{definition}
From this definition it is immediate that homotopically discrete $n$-fold categories are a full subcategory of $GTa_{wg}^{n}$.
\begin{definition}\label{def2}
The category $GCat_{wg}^{n}\subset Cat_{wg}^{n}$ of groupoidal weakly globular $n$-fold categories is the full subcategory of $Cat_{wg}^{n}$ whose objects $X$ are in $GTa_{wg}^{n}$.
The category $GTa^{n}\subset Ta^{n}$ of groupoidal Tamsamani $n$-categories is the full subcategory of $Ta^{n}$ whose objects $X$ are in $GTa_{wg}^{n}$.
\end{definition}
\begin{proposition}\label{pro1}
The functors
$$
Q_n:GTa^{n}\leftrightarrows GCat_{wg}^{n}:Disc_{n}
$$
induce an equivalence of categories after localization with respect to the $n$-equivalences
$$
GTa^{n}/\!\!\sim^n \;\simeq\; GCat_{wg}^{n}/\!\!\sim^n\;.
$$
\end{proposition}
Since Tamsamani $n$-groupoids are a model of $n$-types, as a consequence of the previous proposition we obtain that weakly globular $n$-fold categories are an algebraic model of $n$-types.
There is an alternative more explicit way to obtain a fundamental functor
$$
\mathcal{H}_{n}:n\text{-types}\rightarrow GCat_{wg}^{n}
$$
based on the work of Blanc and Paoli. This work shows in particular how the notion of weak globularity arises naturally in topology.
The functor $\mathcal{H}_n$ is given by the composite
$$
\mathcal{H}_n: n\text{-types}\xrightarrow{\mathcal{S}} [\Delta^{{}^{op}},Set] \xrightarrow{Or_{n}} [\Delta^{{n}^{op}},Set]\xrightarrow{\mathcal{P}_n} Gpd^{n}\;.
$$
Here $\mathcal{S}$ is the [[singular simplicial complex|singular functor]] and $Or_{n}$ is the functor induced by [[ordinal sum|ordinal sum]] $or_{n}: \Delta^{{n}^{op}} \rightarrow \Delta$, that is
$$
(Or_{n} X)_{p_1\ldots p_n}=X_{n-1+p_1+\ldots+ p_n}
$$
The functor $\mathcal{P}_n$ is left adjoint to the $n$-fold [nerve](https://ncatlab.org/nlab/show/nerve#internal_nerve) functor $N_n:Gpd^{n}\rightarrow
[\Delta^{{n}^{op}},Set]$. While the computation of $\mathcal{P}_n$ is in general very difficult, this becomes easy on those multi-simplicial sets in the essential image of the functor $Or_{n}\mathcal{S}$ and one obtain
$$
\mathcal{H}_n X=\hat{\pi}^{1} \hat{\pi}^{2} \cdots \hat{\pi}^{n} Or_{n} \mathcal{S} X\;.
$$
where $\hat{\pi}^{(i)}$ denotes the [[fundamental groupoid|fundamental groupoid]] functor in the $i^{th}$ direction. Using this expression of $\mathcal{H}_n X$ one can check that $\mathcal{H}_n X\in GCat_{wg}^{n}$. In conclusion
\begin{theorem}
The functors
$$
B\circ Disc_{n}:GCat_{wg}^{n}\rightleftarrows n\text{-types}:\mathcal{H}_n
$$
induce an equivalence of categories
$$
GCat_{wg}^{n}/\!\!\sim^{n}\,\simeq \,\mathcal{H}\!o(n\text{-types})\;.
$$
where $\mathcal{H}\!o(n\text{-types})$ is the [[homotopy category|homotopy category]] of $n\text{-types}$.
\end{theorem}
## References
The main reference for the theory of weakly globular $n$-fold categories is the following [research monograph](https://doi.org/10.1007/978-3-030-05674-2), which contains an account of all three Segal-type models
* [[Simona Paoli|S. Paoli]]. _Simplicial Methods for Higher Categories: Segal-type models of weak n-categories_, volume 26 of Algebra and Applications. Springer, 2019 [toc pdf](https://link.springer.com/content/pdf/bfm%3A978-3-030-05674-2%2F1.pdf).
The case $n=2$ was originally introduced in the following paper
* S. Paoli, [[Dorette Pronk|D. Pronk]], A double categorical model of weak 2-categories, _Theory and Application of categories_, 28, (2013), 933-980.
The theory or Tamsamani weak $n$-categories was originally developed in
* Z. Tamsamani. Sur des notions de $n$-categorie et $n$-groupoide non-strictes via des ensembles multisimpliciaux. _K-theory_, 16:51–99, 1999.
Tamsamani weak $n$-categories were further studied (from a model category theoretic perspective) in
* [[Carlos Simpson|C. Simpson]]. _Homotopy theory of higher categories_, volume 19 of New Math. Monographs. Cambridge University Press, 2012.
The groupoidal case of weakly globular $n$-fold structures started in
* [[David Blanc|D. Blanc]] and S. Paoli. Segal-type algebraic models of $n$-types. _Algebr. Geom. Topol._, 14:3419–3491, 2014.
An even earlier precursor in the groupoidal case, restricted to modelling path connected $n$-types and related to the [[cat-n-group|$cat_n$-groups]] model, can be found in
* S. Paoli. Weakly globular $cat_n$-groups and Tamsamani’s model. _Adv. in Math._, 222:621–727, 2009.
[[!redirects weakly globular n-fold categories]]
|
weakly Hausdorff topological space | https://ncatlab.org/nlab/source/weakly+Hausdorff+topological+space |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--
# Weakly Hausdorff spaces
* table of contents
{: toc}
## Definition
A [[topological space]] $X$ is **weakly Hausdorff** (or weak Hausdorff) if for any [[compact Hausdorff space]] $K$ and every [[continuous map]] $f\colon K\to X$, the [[image]] $f(K)$ is [[closed subspace|closed]]. Every weakly Hausdorff space is [[T1-space|$T_1$]] (that is every point is [[closed point|closed]]), and every [[Hausdorff space]] is weakly Hausdorff. For the most common purposes for which Hausdorff spaces are used, the assumption of being weakly Hausdorff suffices. See also [[compactly generated space]].
The notion also also makes sense as stated for [[locales]]. Where their theory overlaps (in [[sober spaces]] and [[topological locales]]), the notions of weak Hausdorffness agree, given the [[ultrafilter theorem]] (which implies that all compact Hausdorff spaces/locales are sober/topological).
## Properties
### Weak Hausdorffification
(this is a [[left adjoint]] ...) See for now ([Strickland 2009](#Strickland), Proposition 2.22), for the weak Hausdorffification of a compactly-generated topological space. Note in particular that the construction of the weak Hausdorffification of a compactly-generated space is a quotient by a closed equivalence relation given by a one-step construction (the intersection of all closed equivalence relations on the space), as opposed to the equivalence relation generated by a possibly transfinite procedure as in the Hausdorffification of an arbitrary topological space.
### Pushouts
Write CGWH for the category of [[compactly generated topological spaces|compactly generated]] weakly Hausdorff topological spaces, and $CGH$ for compactly generated [[Hausdorff topological spaces]]. Both are [[convenient categories of topological spaces]] that both admit a [[homotopy hypothesis]]-comparison to [[simplicial sets]], but CGWH has a key further property:
>The construction of [[pushouts]] is better behaved in CGWH than in CGH. Specifically, CHWH is closed under pushouts, one leg of which is the inclusion of a closed subspace. CGH does not have such nice behavior, and pushouts like that are used all over _The Geometry of Iterated Loop Spaces_, specifically in the construction of a monad from an operad and in the use of geometric realizations of simplicial spaces.
([[Peter May]], [MO answer, April 2015](http://mathoverflow.net/a/204221/))
## Related concepts
* [[model structure on compactly generated weak Hausdorff spaces]]
## References
The category of [[compactly generated topological space|compactly generated]] weakly Hausdorff topological spaces was introduced in
* {#McCord69} [[Michael C. McCord]], Section 2 of: *Classifying Spaces and Infinite Symmetric Products*, Transactions of the American Mathematical Society, Vol. 146 (Dec., 1969), pp. 273-298 ([jstor:1995173](https://www.jstor.org/stable/1995173), [pdf](https://www.ams.org/journals/tran/1969-146-00/S0002-9947-1969-0251719-4/S0002-9947-1969-0251719-4.pdf))
as a more convenient setting than Steenrod's compactly generated Hausdorff spaces, given that the latter is not closed under many colimits (for instance quotients) as computed in $Top$.
* [[André Henriques]] et al, _Why the “W” in CGWH (compactly generated weakly Hausdorff spaces)?_, [MO discussion](http://mathoverflow.net/q/47702/), 2010.
A survey is in
* {#Strickland} Neil Strickland, _The category of CGWH spaces_ (2009) (<a class='existingWikiWord' href='/nlab/files/StricklandCGHWSpaces.pdf' title='The category of CGWH spaces'>pdf</a>)
[[!redirects weak Hausdorff space]]
[[!redirects weak Hausdorff spaces]]
[[!redirects weakly Hausdorff space]]
[[!redirects weakly Hausdorff spaces]]
[[!redirects weakly Hausdorff topological spaces]] |
weakly initial object | https://ncatlab.org/nlab/source/weakly+initial+object |
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### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Definition
{#Definition}
An [[object]] in a [[category]] is **weakly initial** if there exists a [[morphism]] from it to every other object in the category.
(So a weakly initial object is an actual [[initial object]] if this morphism is unique.)
Weak initiality is an instance of a [[weak colimit]]. It is also an instance of a [[weakly initial set]] that happens to be a [[singleton]] set.
## Related concepts
* [[strict initial object]]
## References
[[weak adjoint|Weak adjoint functors]] along with [[weak colimits]], with weakly initial objects as a special case, were defined in:
* [[Paul Kainen]], *Weak adjoint functors*, Mathematische Zeitschrift **122** 1 (1971) 1-9 [[dml:171575](https://eudml.org/doc/171575)]
See also:
* [[Philip Wadler]], _[Recursive types for free!](https://homepages.inf.ed.ac.uk/wadler/papers/free-rectypes/free-rectypes.txt)_
[[!redirects weak initial object]]
[[!redirects weakly initial objects]]
[[!redirects weak initial objects]]
[[!redirects weak initial]]
[[!redirects weakly initial]]
[[!redirects weakly initial algebra]]
[[!redirects weakly initial algebras]]
[[!redirects weak initial algebra]]
[[!redirects weak initial algebras]]
[[!redirects weakly terminal object]]
[[!redirects weakly terminal objects]]
|
weakly interacting massive particle | https://ncatlab.org/nlab/source/weakly+interacting+massive+particle | [[!redirects weakly interacting massivle particle]]
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###Context###
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
What are called _weakly interacting massive particles_ or "WIMP"s, for short, are hypothetical [[fundamental particles]]/[[fields (physics)|field quanta]] which have [[mass]] but which, besides via [[gravity]], [[interaction|interact]] only via the [[weak nuclear force]], or via some yet weaker force, but in particular not via [[electromagnetism]].
WIMPs used to be thought of as likely candidates for [[dark matter]], a conclusion suggested by analysis of relic abundancies (see [below](#MotivationViaRelicAbundancy)), but more recently a series of direct detection [[null results]] puts that assumption increasingly into question -- unless one assumes that the WIMPs interact strictly _only_ via [[gravity]], in which case they would have to be very massive ("WIMPzillas", [CKR 98](#CKR98), [CCKR 01](#CCKR01)).
## Motivation via relic abundancy
{#MotivationViaRelicAbundancy}
In detail, the argument for WIMP dark matter proceeds as follows (recalled e.g. in [CCKR 01](#CCKR01)):
> The case for [[dark matter|dark]], non[[baryon|baryonic]] [[matter]] in the [[observable universe|universe]] is today stronger than ever [1]. The observed [[cosmic structure formation|large-scale structure]] suggests that [[dark matter]] (DM) accounts for at least 30% of the critical mass density of the universe $\rho_C = 3 H_0^2 M_{Pl}^2 / 8 \pi = 1.88 \times 1-^{-29} g cm^{-3}$, where $H_0 = 100 h km sec^{-1} Mpc^{-1}$ is the present [[Hubble constant]] and $M_{Pl}$ is the [[Planck mass]].
> The most familiar assumption is that dark matter is a thermal relic, i.e., it was initially in chemical equilibrium in the early universe. A particle species, $X$, tracks its equilibrium abundance as long as reactions which keep the species in chemical equilibrium can proceed on a timescale more rapid than the expansion rate of the universe, $H$. When the reaction rate becomes smaller than the expansion rate, the particle species can no longer track its equilibrium value. When this occurs the particle species is said to be "frozen out.”" The more strongly interacting the particle, the longer it stays in local thermal equilibrium and the smaller its eventual freeze-out abundance. Conversely, the more weakly interactingthe particle, the larger its present abundance. If freeze out occurs when the particles $X$ areno nrelativistic, the freeze-out value of the particle number per comoving volume $Y$ is related to the mass of the particle and its annihilation cross section (here characterized by $\sigma_0$) by $Y \propto (1/M_X M_{Pl} \sigma_0)$ where $M_X$ is the mass of the particle $X$. Since the contribution to $\Omega_X = \rho_X/\rho_C$ is proportional to $M_X Y$, the present contribution to $\Omega_X$ from a thermal relic roughly is independentof its mass and depends only upon the annihilation cross section. The cross section that results in $\Omega_X h^2\sim 1$ is of order $10^{-37 } cm^2$, which is of the order the [[weak nuclear force|weak]] scale.
## References
* Jonathan L. Feng, *The WIMP Paradigm: Theme and Variations* [[arXiv:2212.02479](https://arxiv.org/abs/2212.02479)]
See also
* Wikipedia _[Weakly interacting massive particle](https://en.wikipedia.org/wiki/Weakly_interacting_massive_particles)_
On super-heavy WIMPs
* {#CKR98} Daniel J. H. Chung, Edward W. Kolb, Antonio Riotto, _Superheavy dark matter_, Phys. Rev. D 59, 023501 (1998) ([arXiv:hep-ph/9802238](https://arxiv.org/abs/hep-ph/9802238))
* {#CCKR01} Daniel J. H. Chung, Patrick Crotty, Edward W. Kolb, Antonio Riotto, _On the gravitational production of superheavy dark matter_, Phys. Rev. D64:043503, 2001 ([arXiv:hep-ph/0104100](https://arxiv.org/abs/hep-ph/0104100))
[[!redirects weakly interacting massive particles]]
[[!redirects WIMP]]
[[!redirects WIMPs]]
|
weakly Lindelöf spaces with countably locally finite base are second countable | https://ncatlab.org/nlab/source/weakly+Lindel%C3%B6f+spaces+with+countably+locally+finite+base+are+second+countable | ## Statement
Recall that a [[topological space]] is [[weakly Lindelöf topological space|weakly Lindelöf]] if every [[open cover]] has a [[countable set|countable]] subcollection the union of which is dense.
\begin{theorem}
Every [[weakly Lindelöf topological space|weakly Lindelöf]] space with [[countably locally finite set of subsets|$\sigma$-locally finite]] [[topological base|base]] is [[second countable space|second countable]].
\end{theorem}
\begin{proof}
Let $\mathcal{V}$ be a countably locally finite base. For each $x \in X$, there is a neighborhood $N_x$ meeting countably many members of $\mathcal{V}$. If $X$ is weakly Lindelöf, there is a countable $\{N_n\}_n$ which covers a dense subset of $X$. Then $ \mathcal{U} = \{V\in \mathcal{V} \mid N_n \cap V \neq \emptyset\}$ is a countable basis for X.
\end{proof}
## Related statements and properties
[[!include topology - global countability axioms]] |
weakly Lindelöf topological space | https://ncatlab.org/nlab/source/weakly+Lindel%C3%B6f+topological+space | +-- {: .rightHandSide}
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### Context
#### Topology
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[[!include topology - contents]]
=--
=--
=--
Named after [[Ernst Leonard Lindelöf]].
#Contents#
* table of contents
{:toc}
## Definition
A [[topological space]] is a _Lindelöf space_ if every [[open cover]] has a [[countable set|countable]] subcover.
## Related properties and theorems
[[!include topology - countability axioms]]
## References
* [Weakly Lindelöf](https://topology.jdabbs.com/properties/P000062), $\pi$-Base.
[[!redirects weakly Lindelöf topological spaces]]
[[!redirects weakly Lindelöf space]]
[[!redirects weakly Lindelöf spaces]]
[[!redirects weakly Lindeloef topological space]]
[[!redirects weakly Lindeloef topological spaces]]
[[!redirects weakly Lindeloef space]]
[[!redirects weakly Lindeloef spaces]]
[[!redirects weakly Lindelof topological space]]
[[!redirects weakly Lindelof topological spaces]]
[[!redirects weakly Lindelof space]]
[[!redirects weakly Lindelof spaces]]
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weakly open subtopos | https://ncatlab.org/nlab/source/weakly+open+subtopos |
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###Context###
#### Topos Theory
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[[!include topos theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The concept of **weakly open subtopos** is weakening of the concept of a [[dense subtopos]] in topos theory.
## Definition
Let $\mathcal{E}$ be a [[topos]] and $Sh_j(\mathcal{E})$ a subtopos of $\mathcal{E}$ with $j$ the corresponding [[Lawvere-Tierney topology]], $a_j$ the corresponding [[associated sheaf functor]] and $\neg$ and $\neg^j$ the pseudo-complementation operators on the subobject lattices in $\mathcal{E}$ and $Sh_j(\mathcal{E})$, respectively. The subtopos $Sh_j(\mathcal{E})$ is called _weakly open_ if for all subobjects $A\rightarrowtail E$ in $\mathcal{E}$, $a_j(\neg A)\cong \neg^j a_j(A)$.
## Remark
In other words, the sheafification $a_j$ commutes with pseudo-complementation.
If $i:Sh_j(\mathcal{E})\hookrightarrow\mathcal{E}$ is weakly open then the topology $j$ and the inclusion $i$ are called _weakly open_ as well.
Since the associated sheaf functor $a_j$ is just the [[inverse image]] of the inclusion $i: Sh_j(\mathcal{E})\hookrightarrow\mathcal{E}$ the definition can be generalized to general [[geometric morphisms]].
## Properties
As already the name suggests, the concept of a weakly open subtopos can equally be viewed as a weakening of the concept of an [[open subtopos]]. Indeed, since the associated sheaf functor of an open inclusion is [[logical functor|logical]] we have
+-- {: .num_prop}
###### Proposition
An open subtopos $i:Sh_j(\mathcal{E})\hookrightarrow\mathcal{E}$ is weakly open. $\qed$
=--
Less straightforward is the following
+-- {: .num_prop}
###### Proposition
A dense subtopos $i:Sh_j(\mathcal{E})\hookrightarrow\mathcal{E}$ is weakly open.
=--
This occurs as prop.6.3 in [Caramello (2012a)](#Caramello12a). See also [Johnstone (1982)](#Johnstone82).
## Remark
The concept and terminology goes back to [Johnstone (1982)](#Johnstone).
The equivalent concept for [[frames]] is called '_nearly open_' in [Banaschewski-Pultr (1996)](#Banaschewski_Pultr96). The maps they call 'weakly open' are called '[[skeletal geometric morphism|skeletal]]' by Johnstone.
The concept of map in point-set topology that corresponds to weakly open is that of a continuous map $f$ such that $f(U)$ is dense in some open set, for $U$ open (cf. [Pták (1958)](#Ptak58)).
## Related entries
* [[dense subtopos]]
* [[open subtopos]]
* [[closed subtopos]]
* [[locally closed subtopos]]
* [[skeletal geometric morphism]]
* [[double negation]]
* [[De Morganization]]
## References
* {#Banaschewski_Pultr94} [[B. Banaschewski]], A. Pultr, *Variants of openness*, Appl. Cat. Struc. **2** (1994) 1-21 [[doi:10.1007/BF00873038](https://doi.org/10.1007/BF00873038)]
* {#Banaschewski_Pultr96} [[B. Banaschewski]], A. Pultr, *Booleanization*, Cah. Top. Géom. Diff. Cat. **XXXVII** **1** (1996) 41-60 [[numdam:CTGDC_1996__37_1_41_0](http://numdam.mathdoc.fr/numdam-bin/fitem?id=CTGDC_1996__37_1_41_0)]
* {#Caramello12a}[[Olivia Caramello]], _Universal models and definability_ , Math. Proc. Cam. Phil. Soc. (2012) pp.279-302. ([arXiv:0906.3061](http://arxiv.org/abs/0906.3061))
* {#Caramello12}[[Olivia Caramello]], _Topologies for intermediate logics_ , arXiv:1205.2547 (2012). ([abstract](http://arxiv.org/abs/1205.2547))
* {#Johnstone82}[[Peter Johnstone]], _Factorization theorems for geometric morphisms II_ , pp.216-233 in LNM **915** Springer Heidelberg 1982.
* {#Johnstone02} [[Peter Johnstone]], _[[Sketches of an Elephant]] I_, Oxford UP 2002. (pp.209f)
* {#Ptak58}V. Pták, _Completeness and the open mapping theorem_ , Bull. Soc. Math. France **86** (1958) pp.41-74. ([pdf](http://archive.numdam.org/article/BSMF_1958__86__41_0.pdf))
[[!redirects weakly open subtoposes]]
[[!redirects weakly open topology]]
[[!redirects weakly open topologies]]
[[!redirects implicationally open subtopos]]
[[!redirects implicationally open topology]]
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weakly periodic cohomology theory | https://ncatlab.org/nlab/source/weakly+periodic+cohomology+theory |
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###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
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#### Elliptic cohomology
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[[!include elliptic cohomology -- contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
A [[multiplicative cohomology theory|multiplicative]] [[cohomology theory]] $A$ is **weakly periodic** if the natural map
$$
A^2({*}) \otimes_{A^0({*})} A^n({*}) \stackrel{\simeq}{\to}
A^{n+2}({*})
$$
is an [[isomorphism]] for all $n \in \mathbb{Z}$.
e.g. [Lurie, remark 1.4](#Lurie)
Compare with the notion of a _[[periodic cohomology theory]]_.
## Properties
### Relation to formal groups
One reason why weakly periodic cohomology theories are of interest is that their [[cohomology ring]] over the space $\mathbb{C}P^\infty$ defines a [[formal group]].
To get a [[formal group]] from a [[weakly periodic cohomology theory|weakly periodic]], [[even cohomology theory|even]] [[multiplicative cohomology theory|multiplicative]] [[cohomology theory]] $A^\bullet$, we look at the induced map on $A^\bullet$ from a morphism
$$
i_0 : {*} \to \mathbb{C}P^\infty
$$
and take the kernel
$$
J := ker(i_0^* : A^0(\mathbb{C}P^\infty) \to A^0({*}))
$$
to be the [[ideal]] that we complete along to define the [[formal scheme]] $Spf A^0(\mathbb{C}P^\infty)$ (see there for details).
Notice that the map from the point is unique only up to [[homotopy]], so accordingly there are lots of chocies here, which however all lead to the same result.
The fact that $A$ is weakly periodic allows to reconstruct the [[cohomology theory]] essentially from this [[formal scheme]].
To get a [[formal group law]] from this we proceed as follows: if the [[Lie algebra]] $Lie(Spf A^0(\mathbb{C}P^\infty))$ of the [[formal group]]
$$
Lie(Spf A^0(\mathbb{C}P^\infty))
\simeq
ker(i_0^*)/ker(i_0^*)^2
$$
is a free $A^0({*})$-module, we can pick a generator $t$ and this gives an [[isomorphism]]
$$
Spf(A^0(\mathbb{C}P^\infty)) \simeq Spf(A^0({*})[[t]])
$$
if $A^0(\mathbb{C}P^\infty) A^0({*})[ [t] ]$ then $i_0^*$ "forgets the $t$-coordinate".
## Examples
* [[complex K-theory]]
* [[elliptic cohomology]]
## Related entries
* [[periodic cohomology theory]]
* [[periodic ring spectrum]]
* [[A Survey of Elliptic Cohomology - formal groups and cohomology]]
## References
* {#Lurie} [[Jacob Lurie]], _[[A Survey of Elliptic Cohomology]]_
[[!redirects weakly periodic cohomology theories]]
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weakly reductive semigroup | https://ncatlab.org/nlab/source/weakly+reductive+semigroup |
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###Context###
#### Algebra
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[[!include higher algebra - contents]]
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=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A [[semigroup]] is called _left/right weakly reductive_ if it coincides with the semigroup of its left/right translations.
## Definition
We only define left weakly reductive semigroups, right weakly reductive semigroups are defined similarly.
Let $S_l$ be the set of left translations of $S$. That is, this is the set of maps $x_l:S\to S$ defined by $x_l(y) := x\cdot y$. The semigroup $(S_l, \circ)$, where $\circ$ denotes composition of maps, is called the semigroup of left translations of $S$.
The map $f:x\mapsto x_l$ is then a morphism in the category of semigroups. We call $(S, \cdot)$ _left weakly reductive_, if $f$ is an isomorphism.
Explicitly, and this is where the name comes from, if $a, b\in S$, and $x\cdot a = x\cdot b$ for all $x\in S$, then $a = b$.
A _weakly reductive semigroup_ is a semigroup that is both right and left weakly reductive.
## In terms of varieties
A left weakly reductive semigroup can be thought of as a class of structures $(S, \cdot , w, r)$, where $\cdot, w$ are binary operations, and $r$ is a ternary operation, satisfying the following axioms for all $x, y, z$: $ x\cdot (y\cdot z) = (x\cdot y)\cdot z, r(x, y, w(x, y)\cdot x) = x, r(x, y, w(x, y)\cdot y) = y $.
## Examples
Any _left monoid_, a semigroup with a left identity element, is a left weakly reductive semigroup. In particular, any [[monoid]] is weakly reductive.
Any left weakly reductive commutative semigroup is weakly reductive.
A monogenic semigroup that isn't a [[group]] is not weakly reductive.
There exists unique smallest left weakly reductive semigroup which isn't a left monoid. It can be defined as the idempotent semigroup $(\{x, y, z\}, \cdot)$ such that $a\cdot b = z$ for $a\neq b$.
## References
* A. H. Preston and G. B. Clifford, _The algebraic theory of semigroups: Volume I_, American Mathematical Society (1961)
[[!redirects weakly reductive semigroups]]
|
weakly separated space | https://ncatlab.org/nlab/source/weakly+separated+space | weakly separated space |
weakly étale morphism of schemes | https://ncatlab.org/nlab/source/weakly+%C3%A9tale+morphism+of+schemes | [[!redirects weakly étale morphism]]
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###Context###
#### Geometry
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[[!include higher geometry - contents]]
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#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A variant of [[étale morphism of schemes]] where the finiteness conditions on [[étale morphisms]] are relaxed.
Used in the definition of _[[pro-étale site]]_ and _[[pro-étale cohomology]]_.
## Definition
+-- {: .num_defn #WeaklyEtale}
###### Definition
A [[morphism]] $f \colon X \longrightarrow Y$ of [[schemes]]
is called _weakly étale_ if
1. $f$ is a [[flat morphism of schemes]];
1. its [[diagonal]] $X \longrightarrow X \times_Y X$ is also flat.
=--
([Bhatt-Scholze 13, def. 4.1.1](#BhattScholze13))
## Properties
+-- {: .num_prop}
###### Proposition
Every [[weakly étale morphism]] is a [[formally étale morphism]].
=--
([Gabber-Ramero 03, theorem 2.5.36, prop. 3.2.16](#GabberRamero03) [Bhatt-Scholze 13, prop. 2.3.3. (2)](#BhattScholze13))
+-- {: .num_remark}
###### Remark
As discussed there, an [[étale morphism]] is a [[formally étale morphism]] which is [[locally of finite presentation]].
=--
+-- {: .num_cor}
###### Corollary
A [[weakly étale morphism]] which is [[locally of finite presentation]] is an [[étale morphism of schemes|étale morphism]].
[[étale morphism of schemes|étale morphism]] $\Rightarrow$ [[weakly étale morphism of schemes|weakly étale morphism]] $\Rightarrow$ [[formally étale morphism of schemes|formally étale morphism]]
=--
In fact a weakly étale morphism is equivalently a [[formally étale morphism]] which is "locally [[pro-object|pro-finitely]] presentable" (dually locally of [[ind-object|ind]]-finite rank) in the following sense
+-- {: .num_defn #IndEtale}
###### Definition
For $A \to B$ a [[homomorphism]] of [[rings]], say that it is an **[[ind-étale morphism]]** if that $A$-[[associative algebra|algebra]] $B$ is a [[filtered colimit]] of $A$-[[étale algebras]].
=--
+-- {: .num_prop}
###### Proposition
Let $f \;\colon\; A \longrightarrow B$ be a [[homomorphism]] of [[rings]].
* If $f$ is ind-étale, def. \ref{IndEtale}, then it is weakly étale, def. \ref{WeaklyEtale}.
Almost conversely
* If $f$ is weakly étale, then there is a [[faithfully flat morphism]] $g \colon B \to C$ which is ind-étale such that the [[composition|composite]] $g\circ f$ is ind-étale.
=--
([Bhatt-Scholze 13, theorem 1.3](#BhattScholze13))
+-- {: .num_cor}
###### Corollary
The [[sheaf toposes]] over the [[sites]] of weak étale morphisms and of [[pro-étale morphisms of schemes]] into some base [[scheme]] are [[equivalence of categories|equivalent]], both define the _[[pro-étale topos]]_ over the _[[pro-étale site]]_.
=--
## Related concepts
[[étale morphism of schemes|étale morphism]] $\Rightarrow$ [[pro-étale morphism of schemes|pro-étale morphism]] $\Rightarrow$ [[weakly étale morphism of schemes|weakly étale morphism]] $\Rightarrow$ [[formally étale morphism of schemes|formally étale morphism]]
## References
* {#GabberRamero03} [[Ofer Gabber]] and Lorenzo Ramero, _Almost ring theory_, volume 1800 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2003. ([arXiv:math/0201175](http://arxiv.org/abs/math/0201175))
* {#BhattScholze13} [[Bhargav Bhatt]], [[Peter Scholze]], _The pro-étale topology for schemes_ ([arXiv:1309.1198](http://arxiv.org/abs/1309.1198))
[[!redirects weakly étale morphisms]]
[[!redirects weakly etale morphism]]
[[!redirects weakly etale morphisms]] |