Kripke-Joyal semantics

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### Context
#### Type theory
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[[!include type theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

**Kripke--Joyal semantics** is a higher order generalization of the [[semantics|semantic]] interpretation proposed initially by Beth, Grzegorczyk, and Kripke for [[intuitionistic logic|intuitionistic predicate logic]] (IPL). It provides a notion of _'local truth'_ or _'validity at a stage'_ in a [[topos]].

Since it is closely related to [[Paul Cohen|Paul Cohen's]] [[forcing]] technique in [[set theory]], a connection that was already observed by Grzegorczyk and Kripke, it is sometimes called _forcing_ semantics. Other terms in use for it are _external_ semantics, _cover_ semantics or _Beth--Kripke--Joyal_ semantics.

By giving a semantics to formulas written in the [[higher-order logic|higher-order]] [[type theory]] used to express [[ordinary mathematics]] in a topos, the Kripke--Joyal semantics serves as [[semantics|semantic]] interface between the [[internalisation|internal (syntactic) description]] of mathematical objects using the [[Mitchell–Bénabou language]] and the external description.

Kripke--Joyal semantics provides rules and prescriptions for semantic interpretation for general toposes but these prescriptions may simplify for special classes of toposes e.g. the rules resulting for [[presheaf toposes]] over posets (when restricted to first-order formulas) correspond to the original notion of model for IPL considered by [[Saul Kripke|Kripke]] et al.

Kripke--Joyal semantics for [[homotopy type theory]], as claimed [here](n-types+cover#InModels), can be found in  [Awodey, Gambino, and Hazratpour (2021)](#AGH21).

## Formulation

Let $\mathcal{E}$ be an [[elementary topos]]. We will now specify the Kripke--Joyal semantics for formulas $\varphi$ in the [[Mitchell–Bénabou language]] of $\mathcal{E}$ restricting ourselves mainly to formulas $\varphi (x)$ with one free variable $x$ of type $X$. The straightforward generalization to the case with (less or) more free variables can be found in Johnstone ([1977](JT77)) or Borceux ([1994](#Borceux94)).

Recall that such a formula $\varphi (x)$ is interpreted by a morphism $X\overset{\varphi(x)}{\to}\Omega$ and gets an extension $\{x|\varphi(x)\}$ assigned to by the pullback

$$
\array{
\{x|\varphi(x)\}&\to & 1& \\
\downarrow &  &\downarrow &\mathsf{true} \\
X &\underset{\varphi(x)}{\to} & \Omega&
}
$$

Now a global element $1\overset{x_0}{\to} X$ satisfies $\varphi(x)$ if it is contained in $\{x|\varphi(x)\}$ in the sense that the following commutes:

$$
\array{
& &\{x|\varphi(x)\}&\to & 1 &\\
 &\nearrow&\downarrow &  &\downarrow &\mathsf{true} \\
1&\underset{x_0}{\to}& X &\underset{\varphi(x)}{\to} & \Omega&
}
$$

Kripke--Joyal semantics results from the generalization of this satisfaction relation from global elements to _generalized_ elements $U\overset{\alpha}{\to} X$.

+-- {: .num_defn #forcing}
###### Definition
A generalized element $U\overset{\alpha}{\to} X$ is said to _satisfy the formula_ $\varphi(x)$ or more suggestively, $U$ _forces_ $\varphi$, denoted $U\models \varphi (\alpha)$, when $\alpha$ factors through $\{x|\varphi(x)\}$ i.e. there exists a map $m$ making the following commute:
$$
\array{
& &\{x|\varphi(x)\}&\to & 1& \\
 &\overset{m}\nearrow&\downarrow &  &\downarrow &\mathsf{true} \\
U&\overset{\alpha}{\to}& X &\underset{\varphi(x)}{\to} & \Omega&
}
$$
=--

This is the same as to say that $im\,\alpha\subseteq \{x|\varphi(x)\}$ as subobjects of $X$.

Whereas $\{x|\varphi(x)\}$ represents the set of elements satisfying $\varphi$ internally, the collection of global elements has in general no such representation thereby motivating the terminology 'external semantics'.

## Properties

One can now unwind the [[forcing]] relation $U\models \varphi (\alpha)$ recursively over the syntactic composition of $\varphi$. This results in a collection of semantic rules that is commonly referred to as the _Kripke--Joyal semantics_ and permits to make contact with the original rules proposed by Kripke for intuitionistic logic in 1965.

These rules are often useful for translating step by step an object defined by a formula of the Mitchell–Bénabou language into a concrete mathematical object in the topos.

First of all, the [[forcing]] relation is _monotone_ and _local_ :

+-- {: .num_prop #Monotone_Local}
###### Proposition
If $f:V\to U$ and $U\models \varphi (\alpha)$ then $V\models \varphi (\alpha\circ f)$. Conversely, if $f:V\to U$ is _epic_ and $V\models \varphi (\alpha\circ f)$ then $U\models \varphi (\alpha)$.
=--

The short proof can be found in MacLane--Moerdijk ([1994](#MM94), p.304).

Note that the proposition gives a first hint of the importance of epimorphisms or _covers_ in this semantics, testimony of its geometric underpinning.

+-- {: .num_prop #Kripke-Joyal}
###### Theorem
Let $\mathcal{E}$ be an elementary topos and $\alpha:U\to X\,$ a generalized element of $X\in\mathcal{E}$ . The forcing relation $\models$ satisfies

1. $U\models \varphi(\alpha)\wedge \psi(\alpha)$ iff $U\models \varphi(\alpha)$ and $U\models \psi(\alpha)\,$.

1. $U\models \varphi(\alpha)\vee \psi(\alpha)$ iff there exists maps $g:U_1\to U$ and $g_2:U_2\to U$ with $g_1+g_2:U_1+U_2\to U$ epic and such that $U_1\models \varphi(\alpha\circ g_1)$ and $U_2\models \psi(\alpha\circ g_2)\,$.

1. $U\models \varphi(\alpha)\Rightarrow \psi(\alpha)$ iff for any $g:V\to U\,$, $V\models \varphi(\alpha\circ g)$ implies $V\models\psi(\alpha\circ g)\,$.

1. $U\models\neg \varphi(\alpha)$ iff for all $g:V\to U\,$, $V\models \varphi(\alpha\circ g)$ implies that $V\simeq 0$.

Let $\varphi(x,y)$ be a formula with free variables x,y of type $X$ and $Y$ respectively. Then

1. $U\models \exists y\varphi(\alpha, y)$ iff there exists $g:V\to U$ epic and a generalized element $h:V\to Y$ such that $V\models \varphi(\alpha\circ g,h)\,$.

1. $U\models \forall y\varphi(\alpha, y)$ iff for every object $V$ and all pairs of generalized elements $g:V\to U$ and $h:V\to Y\,$, $V\models \varphi(\alpha\circ g,h)\,$.

=--

For a proof see MacLane--Moerdijk ([1994](#MM94), pp.305f).

## Lawvere on Kripke--Joyal semantics

> To workers in algebraic geometry and analysis, it may appear somewhat excessive to detour through an elaborate Mitchell–Bénabou language which in turn requires a Kripke--Joyal semantics in order to get back at the mathematical content of a specific topos. (That sometimes-recommended procedure is strictly analogous to defining a group to be the quotient of the free group generated by itself, which analogously is sometimes useful). The key clause in that semantics was presupposed in the title 'Quantifiers and Sheaves', but the linear case was a theorem in Godement 1958 and indeed just expresses in terms of 20th century concepts the content of Volterra's local existence theorem. Briefly,

> a) the rule of inference for existential quantification is just a symbolic expression of the universal property enjoyed by the geometric image of any map (not only in the category of sets where the axiom of choice holds, but) in any topos,

> b) a figure lying in such an image comes in fact only locally from figures in the domain of the map.

Lawvere ([2000](#Law00), pp.717f).

## Related concepts

* [[Mitchell–Bénabou language]]

* [[internal logic]]

* [[semantics]]

* [[stack semantics]]

* [[forcing]]

## References

For the origins in the semantics of intuitionistic logic consult

* E. W. Beth, _Semantic construction of intuitionistic logic_, Mededel. Kon. Ned. Akad. Wetensch. Afd. letterkunde N. S. **19** no.13 (1956) pp.357--388.

* A. Grzegorczyk, _A philosophically plausible formal interpretation of intuitionistic logic_, Indagationes Mathematicae **26** (1964) pp.596–601.

* [[Saul Kripke|S. A. Kripke]], _Semantical analysis of intuitionistic logic I_, pp.92--130 in Crossley, Dummett (eds.), _Formal Systems and Recursive Functions_, North-Holland Amsterdam 1965.

* H. C. M. de Swart, _An intuitionistically plausible interpretation of intuitionistic logic_, JSL **42** no. 4 (1977) pp.564--578.

* W. Veldman, _An intuitionistic completeness theorem for intuitionistic predicate calculus_, JSL **41** no. 1 (1976) pp.159--166.

A categorical-constructive take on these completeness results is in

* [[Erik Palmgren]], _Constructive Sheaf Semantics_, Math. Log. Quart. **43** (1997) pp.321-327.

A (non-categorical) textbook presentation of the original Kripke semantics can be found in

* [[John Lane Bell|J. Bell]], M. Machover, _A Course in Mathematical Logic_, North-Holland Amsterdam 1977. (section IX.6 pp.416--422)

A more recent overview is in

* D. van Dalen, _Intuitionistic Logic_, pp.224--257 in Goble (ed.), _The Blackwell Guide to Philosophical Logic_, Oxford 2001. (pp.237--240)

The topos-theoretic generalization is usually attributed to [[André Joyal]] who observed in the early 70s that this topos semantics subsumes various notions of forcing but his work was apparently not published. An early reference is

* [[Gerhard Osius|G. Osius]], _A note on Kripke--Joyal semantics for the internal language of topoi_, Springer LNM **445** (1975) pp.349--354.

The first section of the following sums up a 1973 talk and gives an interesting informal overview of the basics involved emphazising [[site|sites]]

* [[Anders Kock]], _Universal Projective Geometry via Topos Theory_, JPAA **9** (1976) pp.1-24.

The following texts stress the connection to Cohen and Kripke's work

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

* {#Lawvere76}[[F. William Lawvere]], _Variable quantities and variable structures in topoi_, pp.101--131 in Heller, Tierney (eds.), Algebra, Topology and Category Theory, Academic Press New York 1976.

Most textbooks on topos theory have a section on Kripke--Joyal semantics. Particularly thorough are

* {#Borceux94}[[Francis Borceux]], _Handbook of Categorical Algebra vol.3_, Cambridge UP 1994. (section 6.6)

* [[Saunders MacLane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_, Springer Heidelberg 1994. (section VI.6--7)

More concise are

* [[Peter Johnstone]], _Topos Theory_, Academic Press New York 1977. (Reprinted by Dover Mineola 2014; pp.157--161)

* [[Colin McLarty]], _Elementary Categories, Elementary Toposes_, Oxford UP 1992. (chapter 18)

Freely available online are

* {#Goldblatt84} R. Goldblatt, _Topoi -- The Categorical Analysis of Logic_, 2nd ed. North-Holland Amsterdam 1984. (Dover reprint New York 2006; [project euclid](http://projecteuclid.org/euclid.bia/1403013939), section XIV.6)

* [[Ieke Moerdijk]],  [[Jaap van Oosten]], _Topos theory_, lecture notes (2007). ([pdf](http://www.staff.science.uu.nl/~ooste110/syllabi/toposmoeder.pdf), page 58ff)

The above Lawvere quote stems from

* {#Law00}F. W. Lawvere, _Comments on the development of topos theory_, pp.715--734 in Pier (ed.), _Development of Mathematics 1950--2000_, Birkhäuser Basel 2000. Reprinted as TAC reprint **24** (2014) pp.1--22.  ([TAC](http://www.tac.mta.ca/tac/reprints/articles/24/tr24abs.html))

For a philosophical assessment and comparison to ordinary Tarski semantics see

* Jean Petitot, _La neige est blanche ssi… Prédication et perception
_, Mathématiques et Sciences humaines, Tome **140** (1997) pp.35--50. ([pdf](http://archive.numdam.org/article/MSH_1997__140__35_0.pdf))

A generalization to [[quantales]] is in

* R. Goldblatt, _A Kripke--Joyal Semantics for Noncommutative Logic in Quantales_, pp.209--225 in Governatori, Hodkinson, Venema (eds.), _Advances in Model Logic vol. 6_, College Publications London 2006. ([draft](http://www.mcs.vuw.ac.nz/~rob/papers/aiml11printed.pdf))

Semantics via generalized elements in categories with pullbacks is studied in

* [[Anders Kock]], _Elementwise semantics in categories with pull-backs_, arXiv:2004.14731 (2020). ([abstract](http://arxiv.org/abs/2004.14731))

Kripke-Joyal semantics for [[Homotopy type theory]] are developed in

* {#AGH21} [[Steve Awodey]], [[Nicola Gambino]], [[Sina Hazratpour]], *Kripke-Joyal forcing for type theory and uniform fibrations* (2021) [[arXiv:2110.14576](https://arxiv.org/abs/2110.14576)]

[[!redirects Kripke-Joyal semantics]]
[[!redirects Kripke–Joyal semantics]]
[[!redirects Kripke--Joyal semantics]]

[[!redirects Beth-Kripke-Joyal semantics]]
[[!redirects Beth–Kripke–Joyal semantics]]
[[!redirects Beth--Kripke--Joyal semantics]]

[[!redirects Kripke model]]
[[!redirects Kripke models]]

[[!redirects external semantics]]


Kristen Schemmerhorn

* [faculty website](https://www.cuchicago.edu/faculty/mathematics-computer-science/kristen-schemmerhorn/)

## Selected writings

On [[model categories]] and [[simplicial homotopy theory]] ([[simplicial model categories]]):

* [[Paul Goerss]], [[Kirsten Schemmerhorn]], _Model categories and simplicial methods_, Notes from lectures given at the University of Chicago in August 2004, published as: _Interactions between Homotopy Theory and Algebra_, Contemporary Mathematics **436** AMS (2007) [[arXiv:math.AT/0609537](http://arxiv.org/abs/math.AT/0609537), [doi:10.1090/conm/436](http://dx.doi.org/10.1090/conm/436)]

## Related entries

* [[stack]]

* [[model structure on simplicial presheaves]]

* [[model structure on dg-algebras]]

category: people

[[!redirects Kirsten Schemmerhorn]]


Kristina Sojakova

* [webpage](http://www.informatik.uni-trier.de/~ley/db/indices/a-tree/s/Sojakova:Kristina.html)

## Selected writings

On [[inductive types]] in [[homotopy type theory]] via homotopy-[[initial algebras over an endofunctor]]:

* {#AwodeyGambinoSojakova12} [[Steve Awodey]], [[Nicola Gambino]], [[Kristina Sojakova]], *Inductive types in homotopy type theory*, LICS'12: (2012) 95–104 [[arXiv:1201.3898](http://arxiv.org/abs/1201.3898), [doi:10.1109/LICS.2012.21](https://doi.org/10.1109/LICS.2012.21), [Coq code](https://github.com/HoTT/HoTT/tree/master/Coq/IT)]

  \linebreak

  Exposition:

  [[Steve Awodey]], *Inductive types in HoTT* (Jan 2012) [[blog post](http://homotopytypetheory.org/2012/01/19/inductive-types-in-hott/)]

and analogously on [[higher inductive types]]:

* [[Kristina Sojakova]], *Higher Inductive Types as Homotopy-Initial Algebras*, ACM SIGPLAN Notices **50** 1 (2015) 31–42 [[arXiv:1402.0761](http://arxiv.org/abs/1402.0761), [doi:10.1145/2775051.2676983](https://doi.org/10.1145/2775051.2676983)]

* [[Steve Awodey]], [[Nicola Gambino]], [[Kristina Sojakova]], *Homotopy-initial algebras in type theory*, Journal of the ACM **63** 6 (2017) 1–45 [[arXiv:1504.05531](http://arxiv.org/abs/1504.05531), [doi:10.1145/3006383](https://doi.org/10.1145/3006383)]

On the [[Eckmann-Hilton argument]] in [[homotopy type theory]]:

* [[Kristina Sojakova]], *Syllepsis in Homotopy Type Theory*, 2021 ([[Sojakova_Syllepsis_20210728.pdf:file]])

## Related $n$Lab entries

* [[higher inductive type]]

* [[initial algebra over an endofunctor]]

* [[W-type]]

category: people


Kristine Bauer

* [webpage](https://contacts.ucalgary.ca/info/math/profiles/101-152919?thickbox=1_target%25253D)

## Selected writings

On [[directional derivatives]] in [[Goodwillie calculus]]:

* [[Kristine Bauer]], Brenda Johnson, Christina Osborne, [[Emily Riehl]], Amelia Tebbe, _Directional derivatives and higher order chain rules for abelian functor calculus_ ([arxiv/1610.01930](https://arxiv.org/abs/1610.01930))

Expressing [[Goodwillie calculus]] in terms of [[tangent (∞,1)-categories]]:

* {#BauerBurkeChing21} [[Kristine Bauer]], [[Matthew Burke]], [[Michael Ching]], _Tangent $\infty$-categories and Goodwillie calculus_ ([arXiv:2101.07819](https://arxiv.org/abs/2101.07819))

category: people


Kronecker delta

# Contents
* table of contents
{: toc}

## Definition

For $I$ a [[set]], the _Kronecker delta-function_ is the [[function]] $I \times I \to \{0,1\}$ which takes the value 0 everywhere except on the [[diagonal]], where it takes the value 1.

Often one writes for [[elements]] $i,j \in I$

$$
  \delta^{i}_j \coloneqq \delta(i,j)
  \,.
$$

Then

$$
  \delta^i_j
  =
  \left\{
   \array{
     1 & if i = j
     \\
     0 & otherwise
   }
  \right.
$$

In [[constructive mathematics]], it is necessary that $I$ have [[decidable equality]]; alternatively, one could let the Kronecker delta take values in the [[lower reals]].

## Generalizations

* The Kronecker delta is the [[characteristic map]] of the [[diagonal function]] into $I \times I$. More generally, we may call the [[characteristic morphism]] of a [[diagonal morphism]] in any category with a [[subobject classifier]] a "Kronecker delta".

* In the context of [[profunctors]]/[[graph of a functor|graphs of functors]], we can view the Kronecker delta as the decategorification of the [[Hom]] profunctor.

## Related concepts

* [[Einstein summation convention]]

* [[Levi-Civita symbol]]

* [[Dirac delta distribution]]

* [[graph of a functor]]

## References

Named after [[Leopold Kronecker]].

See also

* Wikipedia, _[Kronecker delta](https://en.wikipedia.org/wiki/Kronecker_delta)_

[[!redirects Kronecker delta]]
[[!redirects Kronecker delta function]]


Kronecker pairing

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###Context###
#### Algebraic topology
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[[!include algebraic topology - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[ordinary homology]]/[[ordinary cohomology]] represented as [[singular homology]]/[[singular cohomology]], _Kronecker pairing_ refers to the defining pairing of a [[chain]] with a [[cochain]].

More generally, in [[generalized (Eilenberg-Steenrod) cohomology]]/[[generalized homology]] represented by a [[ring spectrum]] $E$, then the _Kronecker pairing_
is a canonical pairing of the $E$-[[generalized cohomology]] [[cohomology group|groups]] $E^\bullet(X)$ with the $E$-[[generalized homology]] groups $E_\bullet(X)$ of suitable spaces ([[homotopy types]]/[[spectra]]) $X$

$$
  \langle-,-\rangle_X
    \;\colon\;
   E^{\bullet_1}(X) \otimes E_{\bullet_2}(X) \longrightarrow \pi_{\bullet_2-\bullet_1}(E)
  \,.
$$

The combination of the Kronecker pairing with a [[diagonal]] map yields the [[cap product]] pairing in generalized (co-)homology.

If $E_\bullet(X)$ is a [[projective module|projective]] [[graded module]] over the [[graded ring]] $\pi_\bullet(E)$ then the [[adjunct]]

$$
  E^0(X) \longrightarrow Hom_{\pi_\bullet(E)}(E_\bullet(X), \pi_\bullet(E))
$$

of the Kronecker pairing is an [[isomorphism]] and hence exhibits $E$-[[generalized cohomology]] as the $\pi_\bullet(E)$-[[linear dual]] of the $E$-[[generalized homology]] of $X$; an instance of a [[universal coefficient theorem]] for generalized (co-)homology (prop. \ref{KroneckerPairingAdjunctIsIsomorphism} below).

On [[CW-complexes]] $X$ of [[finite number|finite]] [[dimension]], the Kronecker pairing induces a pairing of the corresponding [[Atiyah-Hirzebruch spectral sequences]] (prop. \ref{KroneckerPairingOnAHSS} below).

## Definition

Let $E$ be a [[ring spectrum]] with product denoted $\mu \colon E \wedge E \longrightarrow E$. Let $X,Y$ be any [[spectra]].

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

Given $[f] \in E^k(X)$ with representative $f \colon X \longrightarrow \Sigma^k E$ and given $[w] \in E_{n+k}(X \wedge Y)$ with representative $w \colon S^{n+k} \longrightarrow E \wedge X \wedge Y$, then their _Kronecker pairing_ is the element

$$
  \langle f,w\rangle \in E_n(Y)
$$

represented by the composite

$$
  S^{k+n}
    \stackrel{w}{\longrightarrow}
  E\wedge X \wedge Y
    \stackrel{id_E \wedge f \wedge id_Y}{\longrightarrow}
  E \wedge \Sigma^k E \wedge Y
   \stackrel{\Sigma^k \mu \wedge id_Y}{\longrightarrow}
  \Sigma^k E \wedge Y
  \,.
$$

This yields a homomorphism of [[graded abelian groups]]

$$
  \langle-,-\rangle
   \;\colon\;
  E^{\bullet_1}(X)
   \otimes
  E_{\bullet_2}(X \wedge Y)
   \longrightarrow
  E_{\bullet_2-\bullet_1}(Y)
  \,.
$$

(and similarly for $Y$ on the other side...)

For $Y = \mathbb{S}$ this is

$$
  \langle-,-\rangle
   \;\colon\;
  E^{\bullet_1}(X)
   \otimes
  E_{\bullet_2}(X)
   \longrightarrow
  \pi_{\bullet_2-\bullet_1}(E)
  \,.
$$

=--

(e.g. [Kochman 96, (4.2.1)](#Kochman96), [Schwede 12, construction 6.13](#Schwede12))

## Properties

### Universal coefficient theorem
 {#UniversalCoefficientTheorem}

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

If $E_\bullet(X)$ is a [[projective module|projective]] [[graded module]] over the [[graded ring]] $\pi_\bullet(E)$ then the [[adjunct]]

$$
  \pi_0[X, E \wedge Y]
  \longrightarrow
  Hom_{\pi_\bullet(E)}( E_\bullet(X), E_\bullet(Y) )
$$
$$
  f \mapsto \langle f,-\rangle
$$

of the Kronecker pairing, def. \ref{KroneckerPairing}, is an [[isomorphism]].

=--

(e.g. [Schwede 12, chapter II, prop.6.20](#Schwede12))

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

By the formula for [[adjuncts]], the morphism factors through the [[free-forgetful adjunction]] for $E$-[[module spectra]]

$$
  \pi_0[X, E \wedge Y]
    \stackrel{\simeq}{\longrightarrow}
  \pi_0[E\wedge X, E \wedge Y]_{E Mod}
    \stackrel{\pi_\bullet}{\longrightarrow}
  Hom_{\pi_\bullet}( E_\bullet(X), E_\bullet(Y) )
  \,.
$$

Hence one is reduced to showing that under the given conditions the second morphis is an iso. (...)

=--

This may be regarded as a [[universal coefficient theorem]] ([Adams 74, part III, around prop. 13.5](#Adams74)).

For $Y = \mathbb{S}$ prop. \ref{KroneckerPairingAdjunctIsIsomorphism} gives:

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

If $E_\bullet(X)$ is a [[projective module|projective]] [[graded module]] over the [[graded ring]] $\pi_\bullet(E)$ then the [[adjunct]]

$$
  E^0(X)
  \longrightarrow
  Hom_{\pi_\bullet(E)}( E_\bullet(X), \pi_\bullet(E) )
$$
$$
  f \mapsto \langle f,-\rangle
$$

of the Kronecker pairing, def. \ref{KroneckerPairing}, is an [[isomorphism]].

=--

### Pairing on Atiyah-Hirzebruch spectral sequences

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

For $E$ a [[ring spectrum]] and $X$ a [[CW complex]] of [[finite number|finite]] [[dimension]], then the Kronecker pairing $\langle -,-\rangle \colon E^\bullet(X)\otimes E_\bullet(X)\to \pi_\bullet(E)$ of def. \ref{KroneckerPairing} passes to a page-wise pairing of the corresponding [[Atiyah-Hirzebruch spectral sequences]] for $E$-cohomology/homology

$$
  \langle-,-\rangle_r
  \;\colon\;
  \mathcal{E}_r^{n,-s}
  \otimes
  \mathcal{E}^r_{n,t}
  \longrightarrow
  \pi_{s+t}(E)
$$

such that

1. on the $\mathcal{E}_2$-page this restricts to the Kronecker pairing for [[ordinary cohomology]]/[[ordinary homology]] with [[coefficients]] in $\pi_\bullet(E)$;

1. the [[differentials]] act as [[derivations]]

   $$
     \langle d_r(-),-\rangle = \langle -, d^r(-)\rangle
     \,,
   $$

1. The pairing on the $\mathcal{E}_\infty$-page is compatible with the Kronecker pairing.

=--

([Kochman 96, prop. 4.2.10](#Kochman96))

## References

Presumeably named after [[Leopold Kronecker]].

* {#Adams74} [[Frank Adams]], Part III, section 13 of _[[Stable homotopy and generalised homology]]_, 1974

* {#Kochman96} [[Stanley Kochman]], (4.2.1) and prop. 4.2.10 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* {#Schwede12} [[Stefan Schwede]], chapter II, section 6 of _[[Symmetric spectra]]_, 2012 ([pdf](http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf))


Kronecker-Weber theorem

#Contents#
* table of contents
{:toc}

## Idea

__Kronecker-Weber theorem.__ Every abelian [[field extension]] of the field $\mathbf{Q}$ of [[rational numbers]] is contained in some [[cyclotomic field]] (extension of $\mathbf{Q}$ by roots of unity).

## Related concepts

* [[Artin reciprocity law]]


Krull dimension

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###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

The __Krull dimension__ of a [[commutative ring]] $R$ is the [[supremum]] of lengths of chains
$$
  P_1\subset P_2\subset\ldots \subset P_r
$$
of distinct [[prime ideals]] in $R$.

If $R$ is a possibly noncommutative ring $R$ and $M$ a left $R$-module, then the __Krull dimension__ of $M$ is by definition a deviation of the poset of subobjects of $M$.

A deviation of a poset is defined recursively.

* the deviation of a trivial poset is $-\infty$
* the deviation of a poset satisfying a [[descending chain condition]] is $0$
* a poset has a deviation less than $\alpha$ for an ordinal $\alpha$ if for every descending chain
$$
a_1\geq a_2\geq \ldots\geq a_n\geq\ldots
$$
the subposets of all elements
between $a_n$ and $a_{n+1}$
do not have deviation of less than $\alpha$
for at most finitely many $n$.

## Literature

* wikipedia: [deviation of a poset](https://en.wikipedia.org/wiki/Deviation_of_a_poset)


Kruna S. Ratkovic

* [ResearchGate page](https://www.researchgate.net/profile/Kruna-Ratkovic)

## Selected writings

On [[Morita equivalence|Morita theory]] for/via [[strong monads]]:

* [[Kruna S. Ratkovic]], *Morita theory in enriched context* (2012) [[arXiv:1302.2774](https://arxiv.org/abs/1302.2774), [hal:tel-00785301](https://theses.hal.science/tel-00785301/document)]

* {#BergerRatkovic19} [[Clemens Berger]], [[Kruna Ratkovic]], *Gabriel-Morita theory for excisive model categories*, Appl Categor Struct **27** (2019) 23–54 [[arXiv:1705.03863](https://arxiv.org/abs/1705.03863), [doi;10.1007/s10485-018-9539-x](https://doi.org/10.1007/s10485-018-9539-x)]

## Related entries

* [[strong functor]]

* [[enriched monad]]

category: people

[[!redirects Kruna Ratkovic]]


Kruskal–Szekeres coordinates

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## Idea

What are called _Kruskal–Szekeres coordinates_ is a choice of [[coordinate system]] for maximaly extended [[Schwarzschild spacetime]].

## References

* Wikipedia, _[Kruskal–Szekeres coordinates](https://en.wikipedia.org/wiki/Kruskal%E2%80%93Szekeres_coordinates)_

[[!redirects Kruskal–Szekeres coordinate system]]
[[!redirects Kruskal–Szekeres coordinate systems]]


Krzysztof A. Meissner

* [personal page](https://www.fuw.edu.pl/~meissner/home.html)

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

## Selected writings

On super-heavy [[gravitinos]] as [[dark matter]] candidates, by embedding [[D=4 N=8 supergravity]] into [[E10]]-[[U-duality]]-invariant [[M-theory]]:

* {#MeissnerNicolai18a} [[Krzysztof A. Meissner]], [[Hermann Nicolai]], _Standard Model Fermions and Infinite-Dimensional R-Symmetries_, Phys. Rev. Lett. 121, 091601 (2018) ([arXiv:1804.09606](https://arxiv.org/abs/1804.09606))

* {#MeissnerNicolai18b} [[Krzysztof A. Meissner]], [[Hermann Nicolai]], _Planck Mass Charged Gravitino Dark Matter_, Phys. Rev. D 100, 035001 (2019) ([arXiv:1809.01441](https://arxiv.org/abs/1809.01441))

* [[Krzysztof A. Meissner]], [[Hermann Nicolai]], _Supermassive gravitinos and giant primordial black holes_ ([arXiv:2007.11889](https://arxiv.org/abs/2007.11889))

* [[Krzysztof A. Meissner]], [[Hermann Nicolai]], *Evidence for a stable supermassive gravitino with charge $2/3$?* [[arXiv:2303.09131](https://arxiv.org/abs/2303.09131)]

On relation of the quantum [[M2-brane]] to the [[string theory]]-[[dilaton]] under [[double dimensional reduction]]:

* [[Krzysztof A. Meissner]], [[Hermann Nicolai]], *Fundamental Membranes and the String Dilaton* [[arXiv:2208.05822](https://arxiv.org/abs/2208.05822)]

category: people

[[!redirects Krzysztof Meissner]]


Krzysztof Bar

* [website](https://www.cs.ox.ac.uk/people/krzysztof.bar/)

## Selected entries

On the online proof assistant *[[Globular]]* for higher dimensional [[rewriting]] via [[semistrict]] [[globular set|globular]] [[higher categories]]:

* [[Krzysztof Bar]], [[Aleks Kissinger]], [[Jamie Vicary]], *Globular: an online proof assistant for higher-dimensional rewriting*,  Logical Methods in Computer Science **14** 1 (2018) [[doi:10.23638/LMCS-14(1:8)2018](https://doi.org/10.23638/LMCS-14(1:8)2018), [arXiv:1612.01093](https://arxiv.org/abs/1612.01093)]

category: people


Krzysztof Galicki

* [webpage](http://www.galicki.com/math/homepage.html)

* [obituary](http://www.galicki.com/)

## Selected writings

On [[Sasakian geometry]]

* {#BoyerGalicki07} [[Charles Boyer]], [[Krzysztof Galicki]], _Sasakian Geometry_, Oxford Mathematical Monographs, Oxford University Press, 2007 ([doi:10.1093/acprof:oso/9780198564959.001.0001](https://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780198564959.001.0001/acprof-9780198564959))

See also:

*   [[Charles Boyer]], _Sasakian geometry: The recent work of Krzysztof Galicki_ ([arXiv:0806.0373](https://arxiv.org/abs/0806.0373))

category: people


Krzysztof Gawedzki

**Krzysztof Gawędzki** is a mathematical physicist. Much of his works is related to [[conformal field theory]], especially geometric aspects of [[WZNW models]]. More recently he studies turbulence. He has helped identidy the [[Kalb-Ramond field]]/[[B-field]] with a [[cocycle]] in [[ordinary differential cohomology]]  (in [[Deligne cohomology]]/a [[bundle gerbe]] with connection).

Pronounciation note: the Polish 'ę' (cf. [wikipedia](http://en.wikipedia.org/wiki/Polish_alphabet)) has a nazalization.

## Selected writings

On [[2d conformal field theory]] and introducing [[Deligne cohomology]] in degree 3 ([[bundle gerbes]] [[connection on a bundle gerbe|with connection]]) as global models for the [[background field|background]] [[B-field]], such as in the [[WZW model]]:

* [[Krzysztof Gawędzki]], *Topological Actions in two-dimensional Quantum Field Theories*, in: _Nonperturbative quantum field theory_, Nato Science Series B **185**, Springer (1988) [[spire:257658](http://inspirehep.net/record/257658), [doi:10.1007/978-1-4613-0729-7_5](https://doi.org/10.1007/978-1-4613-0729-7_5), [[Gawedzki-TopologicalActions.pdf:file]]]

* _Lectures on CFT_ (from Park City, published in QFT and strings for mathematicians, Dijkgraaf et al editors, [site](http://www.math.ias.edu/qft)) [source](http://www.math.ias.edu/QFT/fall/NewGaw.tex), [dvi](http://www.math.ias.edu/QFT/fall/NewGaw.dvi), [ps](http://www.math.ias.edu/QFT/fall/NewGaw.ps)

and with emphases on the [[WZW model]] and its relation to [[Chern-Simons theory]] via the [[CS-WZW correspondence]]:

* [[Krzysztof Gawędzki]], *Conformal field theory: a case study*, in Y. Nutku, C. Saclioglu, T. Turgut (eds.) *Conformal Field Theory -- New Non-perturbative Methods In String And Field Theory*, CRC Press (2000) [[arXiv:hep-th/9904145](https://arxiv.org/abs/hep-th/9904145), [doi:10.1201/9780429502873](https://doi.org/10.1201/9780429502873)]

On the [[basic bundle gerbe]] over a [[compact Lie group|compact]] [[simple Lie group]] in relation to the [[WZW model]]:

* [[Krzysztof Gawędzki]], [[Nuno Reis]], *WZW branes and gerbes*, Rev. Math. Phys. **14** (2002) 1281-1334 [[arXiv:hep-th/0205233](https://arxiv.org/abs/hep-th/0205233), [doi:10.1142/S0129055X02001557](https://doi.org/10.1142/S0129055X02001557)]

* [[Krzysztof Gawędzki]], [[Nuno Reis]], _Basic gerbe over non-simply connected compact groups_, Journal of Geometry and Physics **50** 1-4 (2004) 28–55 [[arXiv:0307010](https://arxiv.org/abs/math/0307010), [doi:10.1016/j.geomphys.2003.11.004](http://dx.doi.org/10.1016/j.geomphys.2003.11.004)]

## Related entries

* [[2d CFT]]

* [[WZW model]]

* [[quantization of Chern-Simons theory]]

category: people

[[!redirects Krzysztof Gawedzki]]
[[!redirects Krzysztof Gawędzki]]
[[!redirects Krzysztof Gawȩdzki]]

[[!redirects K- Gawedzki]]
[[!redirects Gawedzki]]


Krzysztof Pilch

* [webpage](https://physics.usc.edu/Faculty/Pilch/)

## Selected writings

On the [[D'Auria-Fré formulation of supergravity]]:

* {#CFGPN83} [[Leonardo Castellani]], [[Pietro Fré]], F. Giani, [[Krzysztof Pilch]], [[Peter van Nieuwenhuizen]], *Gauging of $d = 11$ supergravity?*, Annals of Physics **146** 1 (1983) 35-77 [[spire:11998](https://inspirehep.net/literature/11998), doi:10.1016/0003-4916(83)90052-0]

On [[gauged supergravity|gauged]] [[D=7 supergravity]]:

* M. Pernici, [[Krzysztof Pilch]], [[Peter van Nieuwenhuizen]], _Gauged Maximally Extended Supergravity in Seven Dimensions_ , Phys. Lett. 143B (1984) 103 ([spire:15064](https://inspirehep.net/literature/15064), doi:10.1016/0370-2693(84)90813-X)

* M. Pernici, [[Krzysztof Pilch]], [[Peter van Nieuwenhuizen]], [[Nicholas Warner]], _Noncompact Gauging and Critical Points of Maximal Supergravity in Seven Dimensions_, Nucl. Phys. B249 (1985) 381 (doi:10.1016/0550-3213(85)90046-X>doi:10.1016/0550-3213(85)90046-X)

On [[W-algebras]]:

* [[Peter Bouwknegt]], Jim McCarthy, [[Krzysztof Pilch]], _The $W_3$ algebra: modules, semi-infinite cohomology, and BV algebras_, Lec. Notes in Phys. __42__, Springer 1996 ([doi:10.1007/978-3-540-68719-1](https://link.springer.com/book/10.1007/978-3-540-68719-1))

On [[Freund-Rubin compactifications]] of [[D=11 supergravity]]:

* [[Hermann Nicolai]], [[Krzysztof Pilch]], _Consistent truncation of $d = 11$ supergravity on $AdS_4 \times S^7$_, JHEP 03 (2012) 099 ([arXiv:1112.6131](http://arxiv.org/abs/1112.6131))

On the [[Myers effect]] in [[M-theory]] for [[M2-branes]] polarizing into [[M5-branes]] of [[fuzzy 3-sphere|fuzzy]] [[3-sphere]]-shape, with emphasis on the role of the [[Page charge]]/[[Hopf WZ term]]:

* [[Krzysztof Pilch]], Alexander Tyukov, [[Nicholas Warner]], _Flowing to Higher Dimensions: A New Strongly-Coupled Phase on M2 Branes_, JHEP11 (2015) 170 ([arXiv:1506.01045](https://arxiv.org/abs/1506.01045))

On (in-)stability of non-[[supersymmetry|supersymmetric]] [[AdS spacetime|AdS]] [[string theory vacua|vacua in string theory]]:

* [[Iosif Bena]], [[Krzysztof Pilch]], [[Nicholas Warner]], _Brane-Jet Instabilities_,  J. High Energ. Phys. 2020, 91 (2020) ([arXiv:2003.02851](https://arxiv.org/abs/2003.02851))

category: people


Krzysztof Worytkiewicz

* [website](http://lama.univ-savoie.fr/~worytkiewicz/main.html)

* [Papers](http://lama.univ-savoie.fr/~worytkiewicz/papers.html)

[[!redirects K. Worytkiewicz]]
category: people


KSC-theory

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## Idea

What is called _self-conjugate K-theory_ of spaces $X$ ([Anderson 64](#Anderson64)) is  [[KR-theory]] on [[real spaces]] of the form $X \times S^{0,2}$, where the second factor denotes the [[circle]] equipped with the antipodal $\mathbb{Z}_2$-[[action]] (see at _[[real space]]_ for the notation).

## Applications

In the context of [[type II string theory]] on [[orientifolds]] $KSC$-theory is the [[cohomology theory]] in which the [[RR-fields]] of the $\tilde I$-variant of [[type I superstring theory]] are [[cocycles]] ([Witten 98](Witten98), [DMR 13, section 3.3.](#DMR13))

## Related concepts

[[!include string theory and cohomology theory -- table]]

## References

The definition of KSc theory is due to

* {#Anderson64} D. W. Anderson, _The real K-theory of classifying spaces_ Proc. Nat. Acad. Sci. U. S. A., 51(4):634–636, 1964.

Discussion of applications to [[superstring theory]] on [[orientifolds]] is in

* {#Witten98} [[Edward Witten]], section 5 of _D-branes and K-theory_, J. High Energy Phys., 1998(12):019, 1998 ([arXiv:hep-th/9810188](http://arxiv.org/abs/hep-th/9810188))

* {#DMR13} Charles Doran, Stefan Mendez-Diez, [[Jonathan Rosenberg]], _T-duality For Orientifolds and Twisted KR-theory_ ([arXiv:1306.1779](http://arxiv.org/abs/1306.1779))


ku

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## Idea

The [[ring spectrum]] commonly denoted $ku$ is the [[connective cover]] of the spectrum [[KU]], which [[Brown representability theorem|represents]] [[complex vector bundle|complex]] [[topological K-theory]].

## Related concepts

* [[iterated algebraic K-theory]]


KU-local stable homotopy theory

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## Idea

The [[stable homotopy theory]] of [[local spectra]], local with respect to the [[topological K-theory]]-spectrum [[KU]].

## Related concepts

* [[Bousfield localization of spectra]]

* [[K(n)-local stable homotopy theory]]

## References

* [[Jeroen van der Meer]], _A Stack-Theoretic Perspective on $\mathrm{KU}$-Local Stable Homotopy Theory_. 2019 ([pdf](https://www.universiteitleiden.nl/binaries/content/assets/science/mi/scripties/master/2018-2019/jvd-meer-thesisroot.pdf))


Kuiper's theorem

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## Statement

Throughout, let $\mathcal{H}$ be a [[separable Hilbert space|separable]] infinite-dimensional [[complex numbers|complex]] [[Hilbert space]].

\begin{theorem}\label{KuiperTheorem}
**(Kuiper's theorem)**
\linebreak
The [[topological group|topological]] [[unitary group]] [[U(ℋ)|$\mathrm{U}(\mathcal{H})$]] in either the

* [norm topology](UH#NormTopology)

or the equivalent

* [strong operator topology](UH#StrongOperatorTopology)

* [weak operator topology](UH#WeakOperatorTopology)

* [compact-open topology](UH#CompactOpenTopology)

is [[contractible]] in that there is a [[left homotopy]] between the [[identity]] $id \;\colon\; U(\mathcal{H}) \to U(\mathcal{H})$  and the [[constant function]] $const_{\mathrm{e}} \;\colon\; U(\mathcal{H}) \to U(\mathcal{H})$.

Similarly, since the [[general linear group]] $GL(\mathcal{H})$ of [[bounded operators]] is homotopy-equivalent to [[U(ℋ)|$\mathrm{U}(\mathcal{H})$]]  by the [[Gram-Schmidt process]], it, too, is [[contractible]].
\end{theorem}

See the commented list of references [below](#References).

## Related statements

* [[the infinite-dimensional sphere is weakly contractible]]

* [[U(ℋ)|$U(\mathcal{H})$]]

## References
 {#References}

Proof for the [norm topology](UH#NormTopology) on [[U(ℋ)]]:

* {#Kuiper64} [[Nicolaas Kuiper]], *Contractibility of the unitary group in Hilbert space*, Topology, 3, 19-30 (1964)

* [[Luc Illusie]], *Contractibilité du groupe linéaire des espaces de Hilbert de dimension infinie*, Séminaire Bourbaki: années 1964/65 1965/66, exposés 277-312, Séminaire Bourbaki, no. 9 (1966), Exposé no. 284, 9 p. ([numdam:SB_1964-1966__9__105_0](http://www.numdam.org/item/SB_1964-1966__9__105_0))

Proof for the [strong operator topology](UH#StrongOperatorTopology) on [[U(ℋ)]]:

* {#DixmierDOuady63} [[Jacques Dixmier]], [[Adrien Douady]], Lem. 3 on p. 251 of: *Champs continus d’espaces hilbertiens et de $C^\ast$-algèbres*, Bulletin de la Société Mathématique de France, Tome 91 (1963), pp. 227-284 ([numdam:BSMF_1963__91__227_0](http://archive.numdam.org/item/BSMF_1963__91__227_0))

Direct proof for the [compact-open topology](UH#CompactOpenTopology) on [[U(ℋ)]]:

* {#AtiyahSegal04} [[Michael Atiyah]], [[Graeme Segal]], Prop. A2.1 in: _Twisted K-theory_, Ukrainian Math. Bull. **1**, 3 (2004) ([arXiv:math/0407054](http://arxiv.org/abs/math/0407054), [journal page](http://iamm.su/en/journals/j879/?VID=10), [published pdf](http://iamm.su/upload/iblock/45e/t1-n3-287-330.pdf))

Proof that the [compact-open topology](UH#CompactOpenTopology) agrees with the [strong operator topology](UH#StrongOperatorTopolgy) on [[U(ℋ)]], by which contractibility of the former follows from that of the latter:

* {#Schottenloher} [[Martin Schottenloher]], _The Unitary Group In Its Strong Topology_ ([arXiv:1309.5891](https://arxiv.org/abs/1309.5891)), Advances in Pure Mathematics **08** 05 (2018) ([doi:10.4236/apm.2018.85029](https://www.scirp.org/html/4-5301487_84967.htm))

* {#EspinozaUribe} [[Jesus Espinoza]], [[Bernardo Uribe]], _Topological properties of the unitary group_, JP Journal of Geometry and Topology **16** (2014) Issue 1, pp 45-55 ([arXiv:1407.1869](https://arxiv.org/abs/1407.1869), [journal](http://www.pphmj.com/abstract/8730.htm))

See also:

* Wikipedia, _[Kuiper's theorem](http://en.wikipedia.org/wiki/Kuiper%27s_theorem)_

[[!redirects Kuiper theorem]]


Kumar S. Gupta

__Kumar S. Gupta__ is a theoretical physicist from Kolkatta, India. His main interests are [[black hole]]s (especially the study of [[quasinormal mode]]s), field theory on [[noncommutative space]]-times, [[self-adjoint extension]]s in quantum mechanics, [[graphene]].

* [web](https://www.saha.ac.in/theory/kumars.gupta)

## Selected writings

On [[renormalization]] in [[quantum mechanics]]:

* [[Kumar S. Gupta]], [[Sarada G. Rajeev]], _Renormalization in quantum mechanics_, Phys. Rev. D **48** (1993) 5940 [[doi:10.1103/PhysRevD.48.5940](https://doi.org/10.1103/PhysRevD.48.5940)]

On [[quasinormal modes]] of [[Schwarzschild black holes]]

* Sayan K. Chakrabarti, [[Kumar S. Gupta]], _Asymptotic quasinormal modes of d-dimensional Schwarzschild black hole with Gauss-Bonnet correction_, Int. J. Mod. Phys. A **21** (2006) 3565-3574 [[hep-th/0506133](https://arxiv.org/abs/hep-th/0506133), [doi:10.1142/S0217751X06031612](https://doi.org/10.1142/S0217751X06031612)]

On [[quantum anomalies]] and [[renormalization]] in [[solid state physics]]:

* [[Kumar S. Gupta]], Amilcar Queiroz, _Anomalies and renormalization of impure states in quantum theories_, Modern Physics Letters __A29__ 13 (2014) [[arxiv/1306.5570](http://arxiv.org/abs/1306.5570) [doi:10.1142/S0217732314500643](http://dx.doi.org/10.1142/S0217732314500643)]

category: people


Kumar S. Narain

* [PhysicsTree page](https://academictree.org/physics/peopleinfo.php?pid=496836)

## Selected writings

Relating [[parafermions]] to the [[affine Lie algebra|affine]] [[su(2)]]-[[current algebra]]/[[WZW model]] ([[su(2)-anyons]]):

* T. Jayaraman, [[Kumar S. Narain]], M. H. Sarmadi, *$SU(2)_k$ WZW and parafermion models on the torus*, Nuclear Physics B **343** 2 (1990) 418-449 $[$doi:10.1016/0550-3213(90)90477-U$]$

category: people

[[!redirects Kumar Narain]]


Kummer sequence

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## Idea

For $(-)^n \colon \mathbb{G}_m \longrightarrow \mathbb{G}_m$ the [[endomorphism]] of powering by $n$ on the [[multiplicative group]] over an [[étale site]], then if $n$ is invertible over the site then there is a [[short exact sequence]]

$$
  0 \to \mu_n \to \mathbb{G}_m \stackrel{(-)^n}{\to} \mathbb{G}_m \to 0
  \,,
$$

where $\mu_n$ is the [[group of units]] of order $n$, the group of $n$th [[roots of unity]].

This is called the _Kummer sequence_.

The analog for the [[additive group]] is the _[[Artin-Schreier sequence]]_. Both are unified in the [[Kummer-Artin-Schreier-Witt exact sequence]].

## Related concepts

* [[Kummer theory]]

* [[exponential exact sequence]], [[Artin-Schreier sequence]]

* [[Kummer-Artin-Schreier-Witt exact sequence]]

* [[Tate twist]]

## References

Named after [[Ernst Kummer]].

* [[Günter Tamme]], section II 4.4 of _[[Introduction to Étale Cohomology]]_

* Jordan Watts, _The Kummer sequence_ ([pdf](http://www.math.toronto.edu/jwatts/papers/galois.pdf))

* {#Lieblich04} [[Max Lieblich]], around prop. 2.3.4.4 _Moduli of twisted sheaves_ ([arXiv:math/0411337](http://arxiv.org/abs/math/0411337))

[[!redirects Kummer sequences]]

[[!redirects Kummer exact sequence]]
[[!redirects Kummer exact sequences]]


Kummer theory

## Related entries

* [[Kummer sequence]]

## References

Named after [[Ernst Kummer]].

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


#Contents#
* table of contents
{:toc}

## Idea

A joint generalization of the concept of [[Kummer exact sequence]] and of _[[Artin-Schreier exact sequence]]_. A review is in [Mezard-Romagny-Tossici 11, section 1](#MezardRomagnyTossici11).

The case $n =1$ (Kummer-Artin-Schreier sequence) is due to ([Waterhouse 87](#Waterhouse87)) and independently ([Sekiguchi-Oort-Suwa 89](#SekiguchiOortSuwa89)). The general case is due to ([Sekiguchi-Suwa 94](#SekiguchiSuwa94), [Sekiguchi-Suwa 99](#SekiguchiSuwa99), [Sekiguchi-Suwa 01](#SS01)) and accordingly also referred to as _Sekiguchi-Suwa theory_ ([Mezard-Romagny-Tossici 11](#MezardRomagnyTossici11)).

## Related concepts

* [[ring of Witt vectors]]

* [[exponential exact sequence]]

## References

* {#Waterhouse87} William Waterhouse, _A unified Kummer-Artin-Schreier sequence_, Mathematische Annalen 1987, Volume 277, Issue 3, pp 447-451 ([publisher](http://link.springer.com/article/10.1007/BF01458325))

* {#SekiguchiOortSuwa89} T. Sekiguchi, F. Oort and N. Suwa, On the deformation of Artin-Schreier to Kummer, Ann. Scient. Éc. Norm. Sup., 4e série 22 (1989), 345–375

* {#SekiguchiSuwa94} T. Sekiguchi, N. Suwa, Théories de Kummer-Artin-Schreier-Witt, C. R. Acad. Sci. Paris, 319 Série I,
(1994), 105–110.

* {#SekiguchiSuwa99} T. Sekiguchi, N. Suwa, On the Unified Kummer-Artin-Schreier-Witt theory, Prépublications du laboratoire de Mathématiques Pures de Bordeaux, Prépublication n◦ 111 (1999).

* {#SekiguchiSuwaS01} T. Sekiguchi, N. Suwa, A note on extensions of algebraic and formal groups IV, Kummer-Artin- Schreier-Witt theory of degree p2, Tohoku Math. J. 53 (2001), 203–240.

* Kazuyoshi Tsuchiya, _On the descriptions of $\mathbb{Z}/p^2\mathbb{Z}$-Torsors by the Kummer-Artin-Schreier-Witt theory_, Tokyo J. of Math. Volume 26, Number 1 (2003), 147-177. ([ProjectEuclid](https://projecteuclid.org/euclid.tjm/1244208687))

* {#MezardRomagnyTossici11} Ariane Mézard, Matthieu Romagny, Dajano Tossici, _Sekiguchi-Suwa theory revisited_ ([arXiv:1104.2222](http://arxiv.org/abs/1104.2222))

[[!redirects Kummer-Artin-Schreier-Witt exact sequences]]

[[!redirects Kummer-Artin-Schreier-Witt sequence]]
[[!redirects Kummer-Artin-Schreier-Witt sequences]]

[[!redirects Kummer-Artin-Schreier-Witt sequence]]
[[!redirects Kummer-Artin-Schreier-Witt sequences]]

[[!redirects Kummer-Artin-Schreier sequence]]
[[!redirects Kummer-Artin-Schreier sequences]]

[[!redirects Sekiguchi-Suwa theory]]

[[!redirects Sekiguchi-Suwa sequence]]
[[!redirects Sekiguchi-Suwa sequences]]

[[!redirects Sekiguchi-Suwa exact sequence]]
[[!redirects Sekiguchi-Suwa exact sequences]]


Kunihiko Kodaira

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

## Related entries

* [[Kodaira vanishing theorem]]

* [[Kodaira-Spencer theory]]

* [[elliptic fibration]]

category: people


Kunihito Uzawa

* [Institute page](https://nrid.nii.ac.jp/nrid/1000050378931/)

## Selected writings

* {#Uzawa18} [[Kunihito Uzawa]], _No-Go theorems for ekpyrosis from ten-dimensional supergravity_, J. High Energ. Phys. (2018) 2018: 41 ([arXiv:1803.11084](https://arxiv.org/abs/1803.11084), doi:10.1007/JHEP06(2018)041)

  reviewed in:

  Kunihito Uzawa, _Ekpyrotic scenario in string theory_ ([pdf](http://www2.yukawa.kyoto-u.ac.jp/~gc2018/slides/4th/Mar2/Uzawa.pdf))

  (on [[no-go theorems]] for embedding [[ekpyrotic cosmology]] into [[type IIA supergravity]])

## Related $n$Lab entries

* [[ekpyrotic cosmology]]

* [[de Sitter spacetime]]

category: people


Kuno Lorenz

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

## Related entries

* [[game semantics]]

category: people


Kuo Tsai Chen

* [biography](http://www-history.mcs.st-andrews.ac.uk/Biographies/Chen.html)

* [MathSciNet publication list](http://www.ams.org/mathscinet/search/publications.html?pg1=IID&s1=47865)

* [[Richard Hain]], [[Philippe Tondeur]], _The life and work of Kuo-Tsai Chen_,  Illinois J. Math. Volume 34, Issue 2 (1990), 175-190. ([euclid:ijm/1255988263](http://projecteuclid.org/euclid.ijm/1255988263))

## Selected writings

Introducing [[Chen spaces]] ([[diffeological spaces]]):

* {#Chen73} [[Kuo Tsai Chen]], _Iterated integrals of differential forms and loop space homology_, Ann. Math. 97 (1973), 217–246 ([jstor:1970846](https://www.jstor.org/stable/1970846))

* {#Chen75} [[Kuo Tsai Chen]], _Iterated integrals, fundamental groups and covering spaces_, Trans. Amer. Math. Soc. 206 (1975), 83–98 ([jstor:1997148](https://www.jstor.org/stable/1997148))

* {#Chen77} [[Kuo Tsai Chen]], _Iterated path integrals_, Bull. Amer. Math. Soc. 83, (1977), 831–879 ([euclid:1183539443](https://projecteuclid.org/euclid.bams/1183539443))

## Related entries

* [[iterated integral]]

* [[diffeological space]], [[Chen space]]

* [[Chen connection]]

[[!redirects Kuo Chen]]

category: people

[[!redirects Kuo-Tsai Chen]]
[[!redirects K.-T. Chen]]


Kuranishi deformation theory

#Contents#
* table of contents
{:toc}

## Idea

[[deformation theory]] of [[compact topological space|compact]] [[complex manifolds]].

## References

* K.Kodaira, _Complex manifolds and deformation of complex structures_ Translated from the 1981 Japanese original by Kazuo Akao. Reprint of the 1986 English edition. Classics in Mathematics. Springer-Verlag, Berlin, 2005. ISBN 3-540-22614-1 (very detailed, exposition unfortunately rather disorientating)

* [[Masatake Kuranishi]], _Deformations of compact complex manifolds_ Séminaire de Mathématiques Supérieures, No. 39 (Été 1969). Les Presses de l'Université de Montral, Montreal, Que., 1971.

[[!redirects complex deformation theory]]


Kuratowski pairs

# Contents
* table of contents
{: toc}

## Idea

The concept of _Kuratowski pair_ is one possible way of encoding the concept of an [[ordered pair]] in [[material set theory]] (say in the construction of [[Cartesian products]]): A pair of the form $(a,b)$ is represented by the [[set]] of the form $\{\{a\}, \{a,b\}\}$.

The main point is to cook up a definition whereby $(a, b) = (a', b')$ if and only if $a = a'$ and $b = b'$, where set equality may be verified via the [[axiom of extensionality]]. Obviously many such definitions are possible, but Kuratowski's is a simple one.

## Related concepts

* [[tuple]]

* [[Cartesian product]]

## References

* Wikipedia, _[Ordered pair - Kuratowski definition](https://en.wikipedia.org/wiki/Ordered_pair#Kuratowski_definition)_

[[!redirects Kuratowski pair]]
[[!redirects Kuratowski pairs]]


Kurt Gödel

# Kurt Gödel
* table of contents
{: toc}

## Introduction

Kurt Gödel is one of the major figures of [[logic]] and [[foundations]] from the early 20th century.

## References

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

## Related $n$Lab entries

* [[completeness theorem]]

* [[incompleteness theorem]]

* [[continuum hypothesis]]

* [[NBG]] set theory

* [[Dialectica interpretation]]

category: people

[[!redirects Kurt Gödel]]
[[!redirects Kurt Goedel]]
[[!redirects Kurt Godel]]
[[!redirects K. Gödel]]
[[!redirects K. Goedel]]
[[!redirects K. Godel]]
[[!redirects K Gödel]]
[[!redirects K Goedel]]
[[!redirects K Godel]]
[[!redirects Gödel]]
[[!redirects Goedel]]
[[!redirects Godel]]


Kurt Hensel

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

## Related entries

* [[p-adic numbers]]

category: people


Kurt Lechner

* [cv pdf](http://persone.csia.unipd.it/persone/curriculum/75E8A2ADC5E3DD4BF7D8D80816D2568E.pdf)

## Selected writings

On [[10d supergravity]] and [[10d super Yang-Mills theory]]:

* {#BonoraBregolaLechnerPastiTonin87} [[Loriano Bonora]], M. Bregola, [[Kurt Lechner]], [[Paolo Pasti]], [[Mario Tonin]], _Anomaly-free supergravity and super-Yang-Mills theories in ten dimensions_, Nuclear Physics B
Volume 296, Issue 4, 25 January 1988 (doi:10.1016/0550-3213(88)90402-6)

* {#BonoraBregolaLechnerPastiTonin90} [[Loriano Bonora]], M. Bregola; [[Kurt Lechner]], [[Paolo Pasti]], [[Mario Tonin]], _A discussion of the constraints in $N=1$ SUGRA-SYM in 10-D_,  International Journal of Modern Physics A **05** 03 (1990) 461-477 [doi:10.1142/S0217751X90000222]

* {#BBDFLPPRRTZ} L. Bonora, M. Bregola, [[Riccardo D'Auria]], [[Pietro Fre]], [[Kurt Lechner]], [[Paolo Pasti]], I. Pesando, M. Raciti, F. Riva, [[Mario Tonin]] and D. Zanon, _Some remarks on the supersymmetrization of the Lorentz Chern-Simons form in $D = 10$ $N= 1$ supergravity theories_, Physics Letters B **277** (1992) [[[BonoraSuperGS.pdf:file]]]

* {#LechnerTonin08} [[Kurt Lechner]], [[Mario Tonin]], _Superspace formulations of ten-dimensional supergravity_, JHEP 0806:021 (2008) [[arXiv:0802.3869](https://arxiv.org/abs/0802.3869)]

On [[anyons]] described by [[Chern-Simons theory]]:

* [[Roberto Iengo]], [[Kurt Lechner]], *Anyon quantum mechanics and Chern-Simons theory*, Physics Reports **213** 4 (1992) 179-269 [doi:10.1016/0370-1573(92)90039-3]

On [[pregeometric electromagnetism|duality-symmetric]] [[supergravity]]:

* Antonio Candiello, [[Kurt Lechner]], *Duality in supergravity theories*, Nuclear Physics B **412** 3 (1994) 479-501 \[doi:10.1016/0550-3213(94)90389-1\]

On [[dual heterotic string theory]] and the heterotic [[NS5-brane]] [[Green-Schwarz sigma model]]:

* [[Kurt Lechner]], [[Mario Tonin]], _Worldvolume and target space anomalies in the D=10 super--fivebrane sigma--model_ ([arXiv:hep-th/9603094](http://arxiv.org/abs/hep-th/9603094))

* {#Lechner10} [[Kurt Lechner]], _Quantum properties of the heterotic five-brane_, Phys.Lett.B693:323-329, 2010 ([arxiv:1005.5719](https://arxiv.org/abs/1005.5719))

On [[torsion constraints in supergravity]]:

* {#CandielloLechner93} A. Candiello, [[Kurt Lechner]], _Duality in Supergravity Theories_, Nucl.Phys. B412 (1994) 479-501 ([arXiv:hep-th/9309143](http://arxiv.org/abs/hep-th/9309143))

On the [[Green-Schwarz sigma-model]] for the [[M5-brane]]:

* {#BLNPST97} [[Igor Bandos]], [[Kurt Lechner]], [[Alexei Nurmagambetov]], [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], _Covariant Action for the Super-Five-Brane of M-Theory_, Phys. Rev. Lett. 78 (1997) 4332-4334 ([arXiv:hep-th/9701149](http://arxiv.org/abs/hep-th/9701149))

* [[Igor Bandos]], [[Kurt Lechner]], [[Alexei Nurmagambetov]], [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], _On the equivalence of different formulations of the M Theory five--brane_, Phys. Lett. B408 (1997) 135-141 ([arXiv:hep-th/9703127](http://arxiv.org/abs/hep-th/9703127))

On [[intersecting brane|intersecting]] [[M2-M5 brane bound states]]:

* [[Kurt Lechner]], _Intersecting M2- and M5-branes_, Phys. Lett. B589 (2004) 147-154 ([arXiv:hep-th/0402078](https://arxiv.org/abs/hep-th/0402078), [doi:10.1016/j.physletb.2004.03.056](https://doi.org/10.1016/j.physletb.2004.03.056))

## Related $n$Lab entries

* [[M5-brane]]

category: people


Kurt Otto Friedrichs

* wikipedia, [Kurt Otto Friedrichs](http://en.wikipedia.org/wiki/Kurt_Otto_Friedrichs)
* [Friedrichs' biography written by by C. Reid](http://www.friedrichs.us/history-KOF-life-by-C-Reid.html)
* [[Friedrichs' system]] (a PDE problem, designed for mixed type)
* [[Friedrichs extension]]
* K. O. Friedrichs, _Asymptotic phenomena in mathematical physics_, (short survey) Bull. Amer. Math. Soc. 61 (1955), 485-504 [doi](http://dx.doi.org/10.1090/S0002-9904-1955-09976-2)

category: people

[[!redirects K. O. Friedrichs]]


Kurt Reidemeister

* [Wikipedia (German)](http://de.wikipedia.org/wiki/Kurt_Reidemeister)

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

## Selected writings

On [[twisted ordinary homology]]:

* [[Kurt Reidemeister]], _Überdeckungen von Komplexen_, Crelle **173** (1935), 164-173 [[doi:10.1515/crll.1935.173.164](http://dx.doi.org/10.1515/crll.1935.173.164)]

## Related entries

* [[Reidemeister move]]

* [[Reidemeister torsion]]

[[!redirects Reidemeister]]
[[!redirects K. Reidemeister]]

category:people


Kurt Strebel

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=57470)

## Selected writings

On [[quadratic differentials]]:

* {#Strebel84} [[Kurt Strebel]], _Quadratic differentials_, Springer Verlag, Berlin, 1984 ([doi:10.1007/978-3-662-02414-0](https://link.springer.com/book/10.1007/978-3-662-02414-0))

category: people


Kurt Symanzik

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

## Selected writings

Introducing the [[LSZ reduction formula]]:

* {#LehmannSymanzikZimmermann55} [[Harry Lehmann]], [[Kurt Symanzik]], [[Wolfhart Zimmermann]], _Zur Formulierung quantisierter Feldtheorien_, Nuovo Cimento 1(1), 205 (1955) ([doi:10.1007/BF02731765](https://doi.org/10.1007/BF02731765))

Introducing [[Symanzik effective field theory]] for [[lattice QCD]]:

* {#Symanzik82} [[Kurt Symanzik]], Section 3 of: _Some topics in quantum field theory_ In: Schrader R., Seiler R., Uhlenbrock D.A. (eds.) _Mathematical Problems in Theoretical Physics_ Lecture Notes in Physics, vol 153, Springer 1982 ([doi:10.1007/3-540-11192-1_11](https://doi.org/10.1007/3-540-11192-1_11))

* {#Symanzik83} [[Kurt Symanzik]], _Continuum limit and improved action in lattice theories: (I). Principles and $\phi^4$ theory_, Nuclear Physics B Volume 226, Issue 1, 26 September 1983, Pages 187-204 Nuclear Physics B (doi:10.1016/0550-3213(83)90468-6)

## Related $n$Lab entries

* [[LSZ reduction formula]]

* [[Symanzik effective field theory]]

category: people


Kurt TeKolste

I studied category theory briefly the University of Rochester long ago.

The editing process needs a preview function...

+-- {: .query}
Welcome!  You may want to read [[About]], [[HowTo]], and [[FAQ]].

The preview function is as follows:

*  Submit your entry.
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*  Go back and forth between the entry and your edit window as you fix any errors.
*  If you don\'t wish to modify it further, cancel your edit so that others can edit the page.
*  But if you submit again within half an hour (using the same signature) and nobody else overrides your lock on the entry, then it will all count as one edit.

—[[Toby Bartels|Toby]]
=--

tekolste@fastmail.net

testing...

$\left\{n_i:Y_i\to X\right\}$

category: people


Kurusch Ebrahimi-Fard

__Kurusch Ebrahimi-Fard__ is a German mathematician, currently working in Norway. His main works are mainly concerned with combinatorial and algebraic structures appearing in [[renormalization]] in mathematical physics, in solving differential equations, in Lie theory and so on.
Related entries include: [[dendriform algebra]], [[Rota-Baxter algebra]], [[Birkhoff decomposition]].

## Selected writings

On [[Rota-Baxter algebra]]:

* K. Ebrahimi-Fard, L. Guo, _Rota-Baxter algebras and dendriform dialgebras_, Jour. Pure Appl. Algebra __212__ (2008), 320-339, [arXiv:math/0503647](https://arxiv.org/abs/math/0503647)

* [[Kurusch Ebrahimi-Fard]], [[Li Guo]],  _Rota--Baxter algebras in renormalization of perturbative quantum field theory_,  In  Universality and Renormalization, I. Binder and D. Kreimer, editors, Fields Institute Communicatins, v. 50, AMS 2007, 47-105, [arXiv:hep-th/0604116](https://arxiv.org/abs/hep-th/0604116).

* K. Ebrahimi-Fard, L. Guo, D. Manchon, _Birkhoff type decompositions and the Baker--Campbell--Hausdorff recursion_, Comm. Math. Physics __267__ (2006) 821-845, [arXiv:math-ph/0602004](https://arxiv.org/abs/math-ph/0602004)

On [[Dyson-Schwinger equations]] via [[chord diagrams]]:

* Paul-Hermann Balduf, Amelia Cantwell, [[Kurusch Ebrahimi-Fard]], Lukas Nabergall, Nicholas Olson-Harris, [[Karen Yeats]], *Tubings, chord diagrams, and Dyson-Schwinger equations* \[arXiv:2302.02019\]

category: people


KW format test

blah $f(a) = a'$ more blah blah $\hat\Sigma_\infty$ and $\mathbb{C}P^\infty$


Kyle Miller

* [personal page](https://math.berkeley.edu/~kmill/)

## Selected writings

On [[formal proof|formalized proof]] in the *[[Lean]]* [[proof assistant]]:

* [[Kyle Miller]], [pp. 22](https://math.berkeley.edu/~kmill/talks/2020-06-26-lean-seminar.pdf#page=22) in: *Doing math the Lean way*, Berkeley Lean Seminar (2020) [[pdf](https://math.berkeley.edu/~kmill/talks/2020-06-26-lean-seminar.pdf#page=22), [[Miller-MathTheLeanWay.pdf:file]]]

category: people


Kyle Wamer

* [personal page](https://sites.google.com/view/kyle-wamer/home)

* [GoogleScholar page](https://scholar.google.ca/citations?user=9G1GH8wAAAAJ&hl=en)

## Selected writings

On [[flag manifold sigma-models]]:

* [[Ian Affleck]], [[Dmitri Bykov]], [[Kyle Wamer]], *Flag manifold sigma models: spin chains and integrable theories*,  Phys. Rept. **953** (2022) 1-93 [[arXiv:2101.11638](https://arxiv.org/abs/2101.11638), [doi:10.1016/j.physrep.2021.09.004](https://doi.org/10.1016/j.physrep.2021.09.004)]

category: people


Kähler differential

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

## Idea and definition

The notion of _Kähler differential_ is a very general way to encode a notion of [[differential form]]: something that is dual to a [[derivation]] or [[vector field]].

Conceptually, in dual language of algebras, a symmetry of a commutative algebra $A$ is an [[automorphism]] $g\colon A\to A$, i.e., $g(a b)=g(a)g(b)$. The 'infinitesimal' symmetries are the [[derivations]] $X\colon A\to A$, with $X(a b)=X(a)b+X(b)a$. The [[module]] of Kähler differentials $\Omega^1_K(A)$  parametrizes derivations, in the sense that every derivation $X$ corresponds uniquely to a morphism of $A$-modules $\mu_X: \Omega_K^1 (A)\to A$.

### The ordinary definition and its insufficiency

Kähler differentials are traditionally conceived in terms of an algebraic construction of a certain [[module]] $\Omega_K^1(Spec R)$ on a given ordinary [[ring]] $R$. On [[space]]s modeled (in the sense described at [[space]]) on the [[site]] [[CRing]]$^{op}$, such as [[varieties]], [[scheme]]s, [[algebraic space]]s, [[Deligne-Mumford stack]]s, this produces the correct notion of [[differential form]] in this context. This is the case discussed in the section

* [Over ordinary rings](#OrdinaryRings)

below.

The definition, concrete as it is, applies of course also to function rings on spaces not modeled on $CRing^{op}$, such as rings $C^\infty(X)$ of smooth functions on a [[smooth manifold]]. One might expect that the module of Kähler differentials $\Omega_K(Spec C^\infty(X))$ of $C^\infty(X)$, regarded as an ordinary [[ring]], does reproduce the familiar notion of smooth [[differential form]]s on a [[manifold]]. But it does not. This is discussed in the section

* [Over smooth rings regarded as ordinary rings](#SmoothOrPlain).

This shows that the concrete algebraic construction  of Kähler differential forms over plain rings, traditionally thought of as their very definition, does in fact not correctly capture their nature. There is another definition -- obtained from the [[nPOV]] -- which does capture the situation correctly:

### The correct definition of the notion of _module_ ...

In fact, already the definition of [[module]] has to be freed from it concrete realization in the context of ordinary rings, to exhibit its true nature. What this is has been established long ago in

* Dan Quillen, _Homotopical algebra_

and in Jon Beck's thesis, and is discussed in more detail in the entries _[[module]]_ and _[[Beck module]]_ : Beck and Quillen noticed that the [[category]] $R Mod$ of modules over an ordinary commutative ring $R$ is canonically [[equivalence of categories|equivalent]] to the category $Ab(CRing/R)$ of abelian [[group object]]s in the [[overcategory]] $CRing/R$ of all rings, over the given ring $R$:

$$
  R Mod \simeq Ab(CRing/R)
  \,.
$$

Under this equivalence an $R$-module $N$ is sent to the square-0-extension ring $R \oplus N$ that is canonically equipped with a ring homomorphism $R \oplus N \to R$ and with a unital and associative and commutative product operation

$$
  (R \oplus N) \times_R (R \oplus N) \to (R \oplus N)
$$

$$
  ((r, n_1), (r,n_2)) \mapsto (r, n_1 + n_2)
$$

that makes it first an object in the [[overcategory]] $CRing/R$ and in there an abelian group object, hence an object in $Ab(CRing/R)$.

Conversely, every module arises this way, up to [[isomorphism]]. So this gives an equivalent way of defining modules over rings.

And this is the right definition. Notably, this definition does not assume anything about the ring $R$. It does not even assume that $R$ is a ring at all! It could be anything.

Concretely: for $C$ _any_ category of test objects -- so that we may think of objects in the [[opposite category]] $C^{op}$ as function rings on the  test objects $C$ -- we may define the category of _module_ s over an object $R \in C^{op}$ by the above equation:

$$
  R Mod := Ab(C^{op}/R)
  \,.
$$

Notice that this is now a definition. And that $R$ could be anything, and the definition still makes sense.

The category of all modules over _all_ possible objects $R$ is then nothing but the [[codomain fibration]]

$$
  Mod_C := Ab([I,C^{op}]) \stackrel{p_C}{\to} C^{op}
$$

where $I$ is the [[interval category]] and fiberwise (over $C^{op}$) we form abelian group objects.

This turns out to be the correct [[category theory|category theoretic]] definition of [[module]] (as discussed there). In fact, this is is the special case of the [[higher category theory|higher categorical]] definition that works for $C$ any [[(∞,1)-category]]. In that case the construction $Ab(C^{op}/R)$ of abelian group objects in the [[overcategory]] is generalized (straightforwardly! and in fact even more elegantly) to the notion of [[tangent (∞,1)-category]].

### ... and the correct definition of _derivations_ and _Kähler modules_ {#AbstractDef}

With the above correct notion of [[module]] in hand, all the other concepts of [[deformation theory]], notably those of [[derivation]]s and of Kähler differentials follow straightforwardly

1. given an $R$-module $N$ regarded as an object $p_N : R \oplus N \to R$; the **[[derivation]]s** on $R$ with coefficients in $N$ are precisely the [[section]]s $\delta : R \to R \oplus N$ of $p_N$.

1. The assignment $Spec R \mapsto \Omega_K(Spec R)$ of modules of **Kähler differential**s is the assignment universal with respect to derivations, which means that

   $$
     \Omega_K : C^{op} \to Mod_C
   $$

   is the
   [[left adjoint]] to the above projection $p_C : Mod_C \to C^{op}$:

   this means that every [[derivation]] $\delta : R \to \mathcal{N}$
   (being a [[section]] in $C$ of the module which is the overcategory
   element $\mathcal{N} \to R$) is identified conversely with a
   morphism $\Omega_K(R) \to \mathcal{N}$ in the category of
   abelian [[group object]]s in the [[overcategory]]
   $C^{op}/R$:

   $$
      Hom_{Mod_C}(\Omega_K(R), \mathcal{N})
      \simeq
      Hom_{C^{op}}(R, \mathcal{N})
      \,.
   $$

Notice that in all of the above now, $C$ is still a
completely arbitrary [[category]].

### The fully general definition

By allowing $C$ -- the collection of test
spaces -- to be a general [[(∞,1)-category]], the above
story gives the following completely general [[nPOV]]
on the nature of Kähler differentials:

+-- {: .standout}

For $C$ any [[(∞,1)-category]] of test spaces, write
$p_{C^{op}} : Mod := T_{C^{op}} \to C^{op}$ for the
[[tangent (∞,1)-category]] of its [[opposite category]].

1. the fiber of $Mod \to C^{op}$ over $R \in C^{op}$ is
   the [[(∞,1)-category]] $R Mod$ of [[module]]s over $R$;

1. for $(p_{\mathcal{N}} : \mathcal{N} \to R) \in R Mod$,
   a [[derivation]] on $R$ with coefficients in $\mathcal{N}$
   is a [[section]] $\delta : R \to \mathcal{N}$ of $p_{\mathcal{N}}$.

1. The assignment of **modules of Kähler differentials** or
   **cotangent complexes** is the [[left adjoint]]

   $$
     \Omega_K : C^{op} \to Mod
   $$

   of the [[tangent (∞,1)-category]] projection $p_{C^{op}}$.

   Its value $\Omega_R(R)$ on an object $R \in C^{op}$ is the
   module of Kähler differentials on $Spec R \in C$.

=--

## Specific definitions

We spell out very concretely definitions of Kähler differentials
for special concrete choices of base category $C$
as special cases of the above general story. We start with the
familiar cases and then work our way up to more general or richer
cases.

### Over ordinary rings {#OrdinaryRings}

In terms of the above discussion, we now take $C = CRing^{op}$ to be the [[opposite category]] of the category of ordinary (commutative unital) [[ring]]. In fact without changing anything of the discusson we may assume that the ring $R$ in question is equipped with a ring homomorphism $k \to R$ from a [[ring]] or [[field]] $k$. This makes $R$ a $k$-[[algebra]], and we shall often speak of algebras in the following, where we could just as well speak of rings.

Suppose $A$ is a [[commutative algebra]] over a field $k$.   We may define Kähler differentials either by an explicit construction or by a universal property. In fact there are two explicit constructions.

The simplest construction, maybe, is as follows. The module of **Kähler differentials** $\Omega^1_K(A)$ over $A$ is generated by symbols $d a$ for all $a\in A$, subject to these relations:

* $d c = 0$ when $c$ is a 'constant', that is, an element of $k$ regarded as an element of $A$.

* $d(a b)=(d a)b+a(d b)$.

* $d(a+b)=d a+d b$.

* $(d a)b = b(d a)$.

In particular there are only finite sums in the module of Kähler differentials.

Another more sophisticated construction of $\Omega^1_K(A)$ is given below.  But turning to the universal property, note that we can define **derivations** from $A$ to any $A$-[[module]] $M$: they are $k$-linear maps $X : A \to M$ satisfying the product rule:

$$ X(a b) = X(a) b + a X(b) $$

Then $\Omega^1_K(A)$ may be defined as the [[universal property|universal]] $A$-module equipped with a derivation.  In other words, there is a derivation

$$d : A \to \Omega^1_K(A), $$

and if $X:A\to M$ is any [[derivation]] from $A$ to some $A$-module $M$, then there is a unique $A$-module morphism

$$\mu_X:\Omega_K^1(A)\to M$$

such that the following diagram commutes:

$$
\array{
A&\overset{X}\to     & M\\
 & \underset{d}\searr&\uparrow \mu_X\\
 &                   & \Omega_K^1{A}
}
$$

We say that $X$ **factors through** $d$.

#### Relative version

We can replace the commutative algebra $A$ more generally by  a [[morphism]] of [[commutative unital rings]] $f:R\to S$. Then the __module of Kähler differentials__ is the $S$-[[module]] $\Omega^1_K(S/R)$ corepresenting the functor

$$Der_R(S,f_*(-)) : S Mod\to Set : M\mapsto Der_R(S,f_* M)$$

that assigns to every $S$-[[module]] $M$ the [[set]] of [[derivations]] on $S$ with values in the (bi)module $f_* M$, where $f_*:S Mod\to R Mod$ is the restriction of scalars.

In other words,  $Der_R(S,f_*M)\cong Hom_S(\Omega^1_K(S/R),M)$. In a diagram: for every $R$-derivation $X\colon S \to M$ there is a unique morphism (of $S$-modules) $\mu\colon \Omega^1_K(S/R) \to M$ making the following diagram commute:

$$
\array{
S&\overset{X}\to     & M\\
 & \underset{d}\searr&\uparrow \mu\\
 &                   & \Omega^1_K(S/R)
}
$$

This framework also gives another construction of the module of Kähler differentials, instead of the generators and relations definition given above.

Let $I$ be the [[augmentation ideal]], i.e. the [[kernel]] of the multiplication map

$$ I := Ker(m:S\otimes_R S\to S)$$

Then $\Omega^1_K_{S/R}= I/I^2$ and there is a canonical induced map $d: S\to \Omega^1_{S/R}$
given by $d s = [1\otimes s - s\otimes 1]$.

#### Higher degree Kähler forms

Furthermore, if $R$ is in characteristic zero, one may introduce **Kähler $p$-forms** , which are elements of the $p$-th [[exterior power]] $\Omega^p_K_{S/R}:=\Lambda_R^p \Omega^1_K_{S/R}$. The [[module]] of Kähler differentials readily generalizes as a [[sheaf]] of Kähler differentials for a separated morphism $f:X\to Y$ of (commutative) [[schemes]], namely it is the [[pullback]] along the embedding of the ideal sheaf of the [[diagonal subobject|diagonal subscheme]] $X\hookrightarrow X\times_Y X$.

Compare the role of [[universal differential envelope]] and [[Amitsur complex]] for  analogous constructions in the noncommutative case. The appropriate extension of the module of relative Kähler differentials to the derived setting is the [[cotangent complex]] of Grothendieck--Illusie.

#### Relation to Hochschild homology

The module of Kähler differentials on $R$ is isomorphic to the first [[Hochschild homology]] of $R$

$$
  \Omega_K^1(R) \simeq HH_1(R,R)
  \,.
$$

Under mild conditions the analogous statement is true for higher Kähler differentials and higher Hochschild homology: this is the [[Hochschild-Kostant-Rosenberg theorem]].

### Over smooth rings regarded as ordinary rings {#SmoothOrPlain}

We have seen that we define Kähler differentials $\Omega^1_K(A)$ for any [[commutative algebra]] $A$.

The following special case deserves special attention:

The algebra $A = C^\infty(X)$ of smooth functions on some smooth space $X$ (a [[smooth manifold]] or a [[generalized smooth space]]) is in particular a commutative algebra. So one might think that its Kähler differentials form the ordinary [[differential form]]s on $X$ -- in analogy to the case when $A$ consists of the algebraic functions on an affine [[algebraic variety]] in which case Kähler differentials are often taken as a _definition_ of 1-forms.

#### The problem and its solution

However, when $A = C^\infty(X)$ consists of smooth functions on a manifold, the _ring theoretic_ Kähler differentials do _not_ agree with the ordinary smooth [[differential form|1-form]]s on this manifold! (Unless $X$ is, for instance, the [[point]], of course).
However, there is a canonical map from the Kähler differentials to the ordinary 1-forms.

But there is a solution to this, and an explanation for why something goes wrong:

Smooth spaces such as [[manifold]]s are _not_ modeled on the category $C =$ [[CRing]]${}^{op}$, as [[varieties]] are. Instead, they are modeled on the category $C = \mathbb{L}$ of [[smooth loci]], which is $= C^\infty Ring^{op}$ the opposite of the category of [[generalized smooth algebra|C-infinity rings]].

In particular, the algebra $A = C^\infty(X)$ of smooth functions on a [[manifold]] carries naturally the structure of such a $C^\infty$-ring. This does have "underlying" it the ordinary commutative ring of functions that forget the $C^\infty$-ring structure, but forgetting this structure is precisely what makes the definition of Kähler differentials fail to reproduce that of ordinary smooth 1-forms.

If we do regard $C^\infty(X)$ as a [[generalized smooth algebra|C-infinity ring]], the its Kähler differentials do agree with ordinary 1-forms on $X$.

#### Detailed comparison

We discuss how Kähler differential forms relate to the ordinary notion of [[differential form]]s.

Since there are only finite sums in the module of Kähler differentials, the usual $d f=f' d t$ works only if $f$ is a finite polynomial in $t$, say, if $A$ is $C^\infty(\mathbb{R})$ ([[smooth map|smooth maps]]) or $\mathbb{R}[\![t]\!]$ ([[power series]]). For example, let $f = t^n$ then

$$\begin{aligned}
d f &=
d(t^n)  \\
&= t^{n-1} d t  + t(d t^{n-1}) \\
&= 2 t^{n-1} d t + t^2 d(t^{n-2}) \\
&= r t^{n-1} d t + t^r d(t^{n-r}) \\
&= n t^{n-1} d t \\
&= f' d t.
\end{aligned}$$

However, we have

$$d e^t \ne e^t d t$$

as Kähler differentials.  Intuitively, the reason is that $d$ cannot pass through the infinite sum

$$d e^t = d\left(\sum_{n=0}^{\infty} \frac{t^n}{n!}\right) \ne \sum_{n=0}^{\infty} \frac{d(t^n)}{n!} = e^t d t.$$

However, the only proof we know that $d e^t \ne e^t d t$ is quite tricky: in fact it uses the Axiom of Choice!

* David Speyer, Kähler differentials and ordinary differentials.  ([Math Overflow](http://mathoverflow.net/questions/6074/kahler-differentials-and-ordinary-differentials/9723#9723))

It would be  desirable to either find a proof that avoids the Axiom of Choice, or show that axioms beyond ZF are necessary for this result.

To avoid this annoying property of Kähler differentials we can proceed as follows.  Given a commutative algebra $A$, let $Der(A)$ be the $A$-bimodule of derivations.

Define $\Omega^1(A)$ to be the dual of $Der(A)$:

$$ \Omega^1(A) = Der(A)^* $$

in other words, the set of $A$-module maps $\omega : Der(A) \to A$, made into an $A$-module in the usual way.   There is a derivation

$$ d : A \to \Omega^1(A) $$

given by

$$ d f (X) = X(f)$$

for $X \in Der(A)$.

Now, suppose  $A=C^\infty(M)$ where $M$ is any smooth manifold.  Then elements of $\Omega^1(A)$ can be identified with ordinary smooth 1-forms on $M$:

from the [[Hadamard lemma]] it follows that $Der(C^\infty(M)) = \Gamma(T M)$ is precisely the $C^\infty(X)$-module of [[vector field]]s on $X$, and [[differential form|1-forms]] are the $C^\infty(X)$-linear duals of vector fields, by definition.

And in this case, one can show that any [[derivation]] $X: A \to M$ factors through $\Omega^1(A)$ when $M$ is free, and in particular if $M=A$.

We can expand on this remark as follows.  Quite generally, for any commutative algebra $A$ over a field $k$, we have

$$ Der(A) \cong \Omega^1_K(A)^* $$

by the universal property of Kähler differentials, which identifies derivations $A \to A$ with $A$-module morphisms $\Omega^1_K(A) \to A$.

Using the definition

$$\Omega^1(A) = Der(A)^* $$

(of ordinary smooth 1-forms in the case that $A = C^\infty(M)$) we have that these are the linear bidual of the Kähler differentials:

$$\Omega^1(A) \cong \Omega^1_K(A)^{**} $$

There is always a homomorphism from a module to its double dual, so we have a morphism

$$ j: \Omega^1_K(A) \to \Omega^1(A) $$

In the case when $A = C^\infty(M)$ this map is onto but typically not one-to-one, as witnessed by the fact that
$d e^t = e^t d t$ in the ordinary 1-forms $\Omega^1(A)$
but not in the Kähler differentials $\Omega^1_K(A)$.  However, one can show that when $M$ is a free $A$-module, any derivation $X: A \to M$ not only factors through $\Omega^1_K(A)$ (as guaranteed by the universal property of
Kähler differentials), but also $\Omega^1(A)$.
More generally this is true for torsionless $M$, i.e. $A$-modules that inject into their double duals, since $\Omega^1(A)$ is torsionless. So $\Omega^1_K(A)$ is the universal differential module for all modules, while $\Omega^1(A)$ is the universal differential module for torsionless $A$-modules.

+-- {: .query}
[[Martin Gisser]] Couldn't resist the above uncredentialed drive-by edit...
Is it torsionlessness
that ultimately discerns differential geometry from algebraic geometry?
(Need to fill in more dots between torsionless modules and "geometric modules"
in the sense of Jet Nestruev.)

------------------
[[Eric]]: Does this universal property mean that there is some diagram in some category for which the Kähler differentials can be thought of as a (co)limit?

[[John Baez]]: Yeah, take the category all $A$ modules $M$ equipped with a derivation $X : A \to M$, and take the diagram which consists of every object in this category and every morphism, and take the colimit of that, and you'll get $\Omega^1_K(A)$.

But this is just a cutesy way to say that $\Omega^1_K(A)$ is the [[initial object]] of this category.

And _this_, in turn, is just a cutesy way to say that there is a derivation

$$d : A \to \Omega^1_K(A), $$

such that if $X:A\to M$ is also a derivation, then there exists a unique $A$-module morphism

$$\mu_X:\Omega_K^1(A)\to M$$

such that the following diagram commutes:

$$
\array{
A&\overset{X}\to     & M\\
 & \underset{d}\searr&\uparrow \mu_X\\
 &                   & \Omega_K^1{A}
}
$$

All this is general abstract nonsense, nothing special to this example!  Any universal property involving maps out of an object says that object is initial in some category --- and that, in turn, is equivalent to saying that object is the colimit of the enormous diagram consisting of all objects of the same kind!  There's a lot less here than meets the eye.

[[Eric]]: Thank you! That actually makes a little sense to me. As trivial as it may seem, the fact that I was even able to ask this question represents tremendous progress :)

[[Herman Stel]]: Dear Eric and Prof. Baez. There is a mistake in the explanation by John Baez here. The two latter properties (both being that the derivation $d : A \to \Omega^1_K(A)$ is initial) are correct. The first one is not, though. If $\Omega^1_K(A)$ were the colimit of the huge diagram then for every derivation there would be a morphism from that derivation to the universal derivation, which is not true. Instead, note that an initial object is the vertex of a colimit of the empty diagram in any category (use that $\forall x\in X P(x)$ is true if $X$ is empty).

=--

### Over smooth rings {#CooCase}

A _$C^\infty$-ring_ (see [[generalized smooth algebra]]) is a ring that remembers that it carries extra smooth structure akin to the smooth structure carried by a ring of smooth functions on a smooth [[manifold]].

If we take a smooth function ring $C^\infty(X)$, regard it as a $C^\infty$-ring and then determine its module of Kähler differentials with respect to the category $C = \mathbb{L} = C^\infty Ring^{op}$, we do recover the ordinary notion of smooth [[differential form]]s.

For the moment, this case is described in detail
in the entry on [[Fermat theory]].

### Over general monoids

An ordinary (commutative) [[ring]] is precisely a comutative  [[monoid]] in the [[category]] $Ab$ of abelian [[group]]s.
The case of Kähler differentials over ordinary rings discussed [above](#OrdinaryRings) may therefore be thought of as the case where the category of test objects is taken to be

$$
  C = (CMon(Ab))^{op}
  \,.
$$

This has an evident generalization: we may replace here $Ab$ with _any_ category $K$ and consider

$$
  C = (CMon(K))^{op}
  \,.
$$

In practice $K$ is usually required to be an [[abelian category]], but our definitions so far are general enough not to be concerned about this:

for any such $K$ fixed we follow the [general definition](#AbstractDef), consider the [[functor]]

$$
  p : Mod := Ab([I,CMon(K)^{op}]) \to CMon(K)^{op}
$$

and define the assignment of **Kähler differentials**

$$
  \Omega_K : CMon(K)^{op} \to Ab([I,CMon(K)^{op}])
$$

to be the [[left adjoint]] of this functor.

For this to make good sense, everything here should be regarded as taking place in [[(∞,1)-categories]], typically [[model category|modeled]] by the [[simplicial ring|model structure on simplicial rings]].

Then one finds that for $C = sCRing^{op}$ the corresponding
notion of module [[(∞,1)-category]] reproduces the
[[derived category]] of ordinary ring modules.
This is Example 8.6. in

* [[Jacob Lurie]], _Stable $\infty$-categories_ .

#### Over simplicial rings

If in the above setup we choose $K = sAb = [\Delta^{op}, Ab]$ the category of abelian [[simplicial group]]s, then $Mon(K)$ is the category of [[simplicial ring]]s. The category $Mon(K)^{op}$, regarded as a higher category, is the site used in [[higher geometry]] in place of $CRing^{}$

## Related concepts

* [[canonical bundle]]

* [[cotangent complex]], [[André-Quillen homology]], [[topological André-Quillen homology]]

## References

### On smooth rings regarded as ordinary rings

For a proof that every derivation of $A = C^\infty(\mathbb{R})$ comes from a smooth vector field on the real line, and an extensive discussion of Kähler differentials versus ordinary 1-forms, see:

* This Week's Finds in Mathematical Physics ([Week 287](http://math.ucr.edu/home/baez/week287.html))

See also the discussion at the $n$-Café:

*  Blog discussion of Week 287.  ([Summary](http://golem.ph.utexas.edu/category/2009/12/this_weeks_finds_in_mathematic_48.html#c030626))

* [The module of Kähler differentials](http://ocw.mit.edu/NR/rdonlyres/Mathematics/18-726Spring-2009/8C4F62C5-7AE2-482B-9643-890EE76499F5/0/MIT18_726s09_lec13_differentials.pdf), MIT OpenCourseWare: 18.726 Algebraic Geometry, Spring 2009.

### On the fully general case

A detailed discussion of Kähler differentials and their generalization from [[algebra]] to [[higher algebra]] is in

* [[Jacob Lurie]], _[[Deformation Theory]]_

### For a categorical approach, ###

introducing a setting in which Kahler differentials live quite naturally (but not yet in as much generality as possibly one might hope), see

* [[R. Blute]], [[J.R.B. Cockett]], [[T. Porter]], [[R.A.G.Seely]],  _Kähler categories_,  Cahiers Top.
Géom. Diff. cat., 52 (2011) 253 – 268 ([pdf](http://www.math.mcgill.ca/rags/difftl/kahlercahiers.pdf))

####Abstract####
This paper establishes a relation between the recently introduced notion of differential category and the more classic theory of Kähler differentials in commutative algebra. A codifferential category is an additive symmetric monoidal category with a monad, which is furthermore an algebra modality. An algebra modality for a monad T is a natural assignment of an associative algebra structure to each object of the form T(M). In a (co)differential category, one should imagine the morphisms in the base category as being linear maps and the morphisms in the (co)Kleisli category as being infinitely differentiable. Finally, a differential category comes equipped with a differential combinator satisfying typical differentiation axioms, expressed coalgebraically.

The traditional notion of Kähler differentials defines the notion of a module of A-differential forms with respect to A, where A is a commutative k-algebra. This module is equipped with a universal A-derivation. With this in mind, a Kähler category is an additive monoidal category with an algebra modality and an object of differential forms associated to every object. This object of differential forms satisfies a universal property with respect to derivations. Surprisingly, we are able to show that, under some natural conditions, codifferential categories are Kähler.

[[!redirects Kahler differential]]
[[!redirects Kahler differentials]]
[[!redirects Kähler differentials]]

[[!redirects Kaehler differential]]
[[!redirects Kaehler differentials]]

[[!redirects Kahler differential form]]
[[!redirects Kahler differential forms]]
[[!redirects Kähler differential form]]
[[!redirects Kähler differential forms]]