kinship

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### Context
#### Philosophy
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[[!include philosophy - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

**Kinship** is the collection of relations, rules and behaviors concerning alliance and fictive or real consanguinity that occur in the social organisation of the biological reproduction of individuals. Kinship relations play a significant role for individuals as well as for society as a whole.

In particular in pre-industrial societies, systems of marriage and kinship are often cast into rigid and intricate patterns that are a preferred playground for applications of mathematics to anthropology ever since [[André Weil]] contributed an algebraic appendix to [[Claude Lévi-Strauss|Claude Lévi-Strauss']] groundbreaking [_Les structures élémentaires de la parenté_](#LS67).

## Permutation models

It is generally assumed that all societies explicitly impose some restrictions on the types of possible marriages. The resulting kinship systems then differ whether these restrictions take on the form of prohibitions like e.g. in modern societies that merely ban certain marriages due to considerations of consanguinity or age, or the form of prescriptions that specify positively to marry into a determinate 'marriage class' or category of kins.

The latter prescriptive systems have been called _elementary structures_ by Lévi-Strauss and further divided into systems of _restricted exchange_ where two groups $A,B$ directly exchange spouses, or systems of _generalized exchange_ with longer cycles of exchange e.g. group $A$ gives spouses to $B$ that gives spouses to $C$ that gives spouses to $A$.

Since marriages become deterministic on the classes, elementary kinship systems are amenable to analysis via [[permutations]] on the set of marriage classes. This group-theoretic modeling of elementary structures has been pioneered by [[André Weil]] ([1949](#Weil49)). Here is Lévi-Strauss on Weil's contribution:

> Historiquement parlant, ces pages ont une grande importance. Toute la mathématique de la parenté qui s'est beaucoup développée depuis en est sortie. Et cela continue.[^LS]

[^LS]: Quoted from C. Lévi-Strauss, D. Eribon, _De près et de loin_ , Odile Jacob Paris (1990, p.78). See also the quotation at [[Claude Lévi-Strauss]] concerning his encounters with Hadamard and Weil during the preparation of his thesis in New York during WW2.

Weil's ideas have later been systematized by Philippe Courrège and others.

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

An _(elementary) kinship system_ is a triple $(S,\omega,\mu)$ where $S$ is a finite set, and $\omega,\mu$ are [[permutations]] of $S$.

A _morphism of kinship systems_ $(S_1,\omega_1,\mu_1)\to (S_2,\omega_2,\mu_2)$ is a function $f:S_1\to S_2$ that is equivariant with respect to $\omega$ and $\mu$ i.e. $f\circ\omega_1 =\omega_2\circ f$ and $f\circ\mu_1 =\mu_2\circ f$.

The resulting [[category]] $\mathcal{K}$ is called the _category of kinship (systems)_.
=--

+-- {: .num_remark}
###### Remark
The axiomatics was proposed in [Courrège (1965)](#Cour65) with the intended interpretation that $S$ represents the _set of (disjoint) marriage classes_, that the _conjugal_ function $\omega$ links the men in class $x$ to the class $\omega (x)$ that contains their wifes ('_men in $x$ marry women in $\omega (x)$_') and that the _matrilineal_ function $\mu$ links the mothers in class $x$ to the class $\mu (x)$ that contains their children ('_women in $x$ have children in $\mu(x)$_').

One can then define a _patrilineal_ function[^termin] $\pi:$$=\mu\circ\omega$ that links the fathers in $x$ to their children in $\mu(\omega (x))$ ('_men in $x$ have children in $\mu(\omega (x))$_').

Since a kinship structure $(S,\omega,\mu)$ contains an element $\omega$ of '_alliance_' and an element $\mu$ of '_descent_' one could view the axiomatics as a reconciliation of the Anglo-Saxon tradition in kinship studies that stressed descent and Lévi-Strauss' views that stressed alliance. But this 'reconciliation' is more one of appearances as one could equally well use $\pi$ and $\mu$ as undefined terms and define $\omega =\mu^{-1}\circ \pi$.

[^termin]: $\mu$ and $\pi$ are called the _maternal_ resp. _paternal_ function in Courrège (1965).
=--

+-- {: .num_defn #kinship}
###### Definition
Given a system of kinship $(S,\omega,\mu)$, the subgroup $G$ of $Aut(S)$ generated by $\omega$ and $\mu$ is called the _group of kinship terms_ of $(S,\omega,\mu)$. The system $(S,\omega,\mu)$ is called _irreducible_ (resp. _regular_) when the natural action of $G$ on $S$ is _transitive_ (resp. _free_) i.e. :

* For all $x,y\in S$ there is a $g\in G$ such that $g x=y$.

* (resp. For all $g\in G$: if $g\neq 1$ then $g x\neq x$ for all $x\in S$.)
=--

To equip a [[set]] $S$ with a kinship structure amounts to give a function $f:\{1,2\}\to |Aut(S)|$ that picks out $f(1)=\omega$ and $f(2)=\mu$ from the set |Aut(S)| of [[automorphism|automorphisms]] of $S$.
By the adjunction $F\dashv |\cdot|$ between the free-group functor $F:Set\to Grp$ and the underlying-set functor $|\cdot|: Grp\to Set$ this is the same as giving a group homorphism $\alpha: F(\{1,2\})\to Aut(S)$ from the [[free group]] on $\{1,2\}$, denoted $F_2$ in the following, to the group $Aut(S)$ of automorphisms of $S$, but this, in turn, is the same as giving an [[action]] of $F_2$ on the set $S$!

The group $G$ of kinship terms is none other than $im(\alpha)\subseteq Aut(S)$.

As this goes through regardless of whether $S$ is finite or not and makes sense with respect to the morphisms in $\mathcal{K}$ we have

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

The category $\mathcal{K}$ of kinship systems is equivalent to the [[topos]] of finite $F_2$-sets where $F_2$ is the [[free group]] on two generators. $\qed$
=--

Hence, the category $\mathcal{K}$ has a rich mathematical structure and naturally admits not only finite products $S\times T$ that play an important role already in [Courrège (1965)](#Cour65) but also more 'exotic' operations like [[exponential|exponentials]] $T^S$ and, in particular, power systems $\Omega^S$.

Here $\Omega$ is the truth-value object or [[subobject classifier]] which, in the case of a topos of group actions for a group $\tilde{G}$, is the set $\{\emptyset , \tilde{G}\}\cong\{0,1\}$ with trivial action $0\cdot \tilde{g}=0$ and
$1\cdot \tilde{g}=1$ for all $\tilde{g}\in \tilde{G}$. Whence in $\mathcal{K}$ the truth-value object $\Omega =(\{0,1\}, id, id)$ is a system with two classes with trivial structure. In fact, $\Omega = \mathbf{1}\times \mathbf{1}$ where $\mathbf{1}=(\{1\},id,id)$ is the [[terminal object]] of $\mathcal{K}$.

An $F_2$-[[torsor]] is a [[flat functor]] $T:F_2\to Set$ which by definition amounts to a functor $T$ such that the [[category of elements]] $\int _{F_2} T$ is [[filtered category|filtered]].

(...)

### Some terminology and properties

A system $\mathbf{S}=(S,\omega,\mu)\in \mathcal{K}$ is called _exogamic_ when $\omega\neq id$, and _endogamic_ otherwise. This simply says that spouses always stem from the same marriage class (resp. from different classes).

$\mathbf{S}$ is called _matrilineal_ (resp. _patrilineal_) when $\mu = id$ (resp. $\pi = id$). This says that the child is in the same class as the mother (resp. as the father).

$\mathbf{S}$ is called a _system of restricted exchange_ if $\omega^2=id$, a system of _generalized exchange_ otherwise.

A particular important category of kin are _cousins_ e.g. in many Western societies they are the closest type of kins eligible for marriage whereas in Arabian societies patrilateral parallel cousin marriage is often the preferred type of marriage.

This can be viewed as a consequence of the property that the cousin category is at the intersection of important poles of kinship relations as affinity and non-affinity, symmetry and asymmetry (in the case of cross-cousins e.g. the father'sister's daughter from male perspective and the mother's brother's son in reverse direction). This warrants

+-- {: .num_defn #cousin}
###### Definition
A system $\mathbf{S}=(S,\omega,\mu)\in \mathcal{K}$ is said to _admit marriage with the (female) matrilateral cross-cousin_ if

* $\omega\mu = \mu\omega$  (MCC)

$\mathbf{S}$ is said to _admit marriage with the (female) patrilateral cross-cousin_ if

* $\omega\mu = \mu\omega^{-1}$  (PCC)
=--

A direct calculation yields

+-- {: .num_prop}
###### Proposition
Let $\mathbf{S}=(S,\omega,\mu)\in \mathcal{K}$ satisfy MCC. $\mathbf{S}$ satifies PCC iff $\mathbf{S}$ is of restricted exchange. $\qed$
=--

Since $\omega$ and $\mu$ generate the group $G$ of kinship terms the following is immediate

+-- {: .num_prop #commutative}
###### Proposition
A system $\mathbf{S}=(S,\omega,\mu)\in \mathcal{K}$ satisfies MCC iff the group $G$ of kinship terms of $\mathbf{S}$ is commutative. $\qed$
=--

In fact, the fullsubcategory $\mathcal{M}\subset \mathcal{K}$ of kinships systems satisfying MCC is equivalent to the topos of $\mathbb{Z}\times\mathbb{Z}$-actions, where $\mathbb{Z}\times\mathbb{Z}$ is the free [[Abelian group]] on two generators because actions of $F_2$ such that $G$ is commutative correspond precisely to actions of $\mathbb{Z}\times\mathbb{Z}$.

A more general way to view this, is to consider the surjective homomorphism $\alpha:F_2\to \mathbb{Z}\times\mathbb{Z}$ viewed as a functor that sends the generators $1,2$ to the generators $(1,0),(0,1)$, respectively. By general results (cf. [MM94, p.359](#MM94); [RRZ04, p.226](#RRZ04)), this induces an [[essential geometric morphism]]

$$

\alpha_!\dashv \alpha^*\dashv\alpha_*:Set^{F_2{^{op}}}\to Set^{(\mathbb{Z}\times\mathbb{Z})^{op}}

$$

(...)

## Topos-theoretic perspectives

In our exposition above we have already taken advantage of the categorical point of view advanced in Lawvere and Schanuel ([1997](#LawSchan97), [1999](#Law99)) by adapting it to the classical algebraic approach to kinship. In this section we summarize their ideas in the original setting.

Lawvere and Schanuel proposed to study the concept of kinship starting from the notion of a (filiation structured) **society** modeled mathematically as a set $X$ (of _individuals_) together with two endofunctions $m,f$ that assign to an individual $x\in X$ his mother $m(x)\in X$ resp. his father $f(x)\in X$.

If one organizes these sets with filiation structure into a category the result is a [[topos]] namely the topos $Set^{M_2^{op}}$ of right actions of the free [[monoid]] $M_2$ on two symbols $\{m,f\}$.

Since the parental endofunctions $m,f$ are completely unconstrained[^Cour], the resulting 'kinship relations' are equally unconstrained, e.g. nothing prevents an individual's mother from coinciding with the same individual's father etc. It therefore becomes advantageous to equip the 'societies' with further structure. Lawvere and Schanuel propose to achieve this by introducing appropriate labeling objects $L$ and passage to the [[slice topos]] $Set^{M_2^{op}}/L$.

[^Cour]: so unconstrained indeed, that it might be better to think of the societies as being composed of 'sections' or 'marriage classes', or more generally of equivalence classes of _classificatory kinship_, instead of individuals and to think of $m,f$ as the assignment of the marriage classes  $m(x),f(x)$ of the respective parents of the individuals in the class $x$ in the spirit of [Courrège (1965)](#Cour65). This has the drawback that it admits (hardly plausible) societies composed of infinitely many classes, whereas societies composed of (countably) infinitely-many individuals can be conceived of as an ideal self-image of a sucessful society projecting itself to eternity.

A minimal model of a society that at least keeps the **gender** of the parents apart is the set $G=\{female, male\}$ consisting of a 'female' and a distinct 'male' individual[^group] together with parental functions $m,f$ complying with the usual gender rules e.g. $m(female)=female$, $f(female)=male$ etc. Then a consistent gender distinction on the parental structure of an arbitrary society corresponds to a structure preserving map to $G$ i.e. an object of the slice topos $Set^{M_2^{op}}/G$.

[^group]: Again it makes more sense to consider $G$ to consist of two gender groups instead of two individuals.

Similarly, **exogamy** can be modeled by a _clans object_ $C$ e.g. consisting of two clans $C=\{bear, wolf\}$ obeying matrilinearity in the sense that the mother-of function $m$ is constant but the father-of function $f$ switches the clans expressing thereby that the father of an individual comes from another clan than the individual and his mother.[^Fox]

By labeling with the product $G\times C$ one can achieve gender distinction and exogamy at the same time:

$$
G\times C \quad=\qquad\array{
he-wolf &\overset{f}{\underset{f}{\leftrightarrows}}& he-bear\\
_{m}\downarrow&_{f}{\nearrow}  {\nwarrow}{_{f}}& \downarrow{_{m}}\\
\underset{\overset{\circlearrowleft}{m}}{she-wolf}& &\underset{\overset{\circlearrowleft}{m}}{she-bear}
}\quad
$$

[^Fox]: The Fox tribe in Lawvere's home state Iowa indeed consists of two moieties called 'Bear' and 'Wolf' but they are patrilineal ([Liu 1986](#Liu86), p.21).

Occasionally, one can exploit the "biological unreasonability" inherent in the unconstrained parental mappings $m,f$: e.g. the Na society in Yunnan  (China) is a **fatherless society** lacking marriage or social acknowledgement of male contribution to procreation (cf. [Godelier, pp.494-509](#Godelier04)).

The society consists of matrilineal families consisting of a woman, her brothers and her children. Conception takes place via "nocturnal male visitors" from other houses. In first approximation we can model this absence of fatherhood by the triviality of the father-of-map $f=id$: every individual $x$ is his own father. This corresponds to the full subcategory of societies of form $(X,m:X\to X, id_X)$ which is equivalent to $Set^{M_1^{op}}$ where $M_1$ is the free monoid generated by $\{m\}$.

This subcategory is the image under the full and faithful $q^*$ arising from the adjunction

$$ q_!\dashv q^*\dashv q_*: Set^{M_2^{op}}\to Set^{M_1^{op}}$$

induced by the surjective morphism of monoids $q:M_2\to M_1$ with $q(f)=1$ and $q(m)=m$.

Even societies that entertain an elaborate pattern of kinship relations and terminologies tend to blur distinctions with increasing genealogical depth e.g. whereas parallel and cross kin differ terminologically in the parental generation $G^{+1}$, they are named with the same term in the generation $G^{+2}$ of the grandparents. This lumping together of generations $G^{\leq i}$ to a cluster of 'ancestors' is achieved by quotienting $M_2$ with relations expressing the equivalence $G^{+j}\cong G^{+(j+1)}$ for $i\leq j$ e.g. for $i=2$ this amounts to equate all expressions of depth $3$ with an appropriate expression of depth $2$: $f^3=fmf=f^2$ etc. The resulting epimorphism of monoids $\alpha$:$M_2\to M_\approx$ induces functors $\alpha_!\dashv \alpha^*\dashv\alpha_*:Set^{M_2{^{op}}}\to Set^{{M_\approx}^{op}}$ like above.

At the end of his [paper](#Law99), Lawvere makes  the interesting suggestion to use as [[abstract general]] not the category corresponding to $M_2$ but the [[category]] $\mathcal{C}$ based on the diagram of three parallel arrows

$$
\bullet{\underset{\rightarrow}{\overset{\rightarrow}{\rightarrow}}}\bullet
$$

The result still contains the topos of $M_2$-actions as a quotient topos because inverting one of the arrows turns the resulting diagram up to equivalence into the diagram of one point with two endomaps. Hence the objects in topos $Set^{\mathcal{C}^{op}}$ permit a more finely grained analysis than the objects in the topos of $M_2$-actions.

### Some further remarks

* The topos $Set^{M_2^{op}}$ lacks non-trivial [[level|essential subtoposes]] since by an result of Kelly-Lawvere ([1989](#KL89)) they correspond to two-sided (=left+right) ideals $I$ of $M_2$ that are _idempotent_ i.e. $I=I\cdot I=\{xy|x,y\in I\}$. But a non-empty ideal $I$ has a minimal length $#I$ such that there is a word $w\in I$ of length $#I$ but no word $v\in I$ of strictly smaller length. Since concatenation with non-empty words strictly increases the length of a word, one has $#(I\cdot I)=2\cdot #I$ hence $#I\neq #(I\cdot I)$ iff $\epsilon\notin I$ and, accordingly, $I\neq I\cdot I$ iff $\epsilon\notin I$.

* The [[Jónsson-Tarski topos]] $\mathcal{J}_T$ is a (non-essential) [[subtopos]] of $Set^{M_2^{op}}$.

* The construction $X\mathsf{mod}A$ that [Lawvere (1999)](#Law99) discusses p.417, is defined as the [[pushout]] of the inclusion $A\hookrightarrow X$ along the map $A\to\Pi_0(A)$ that maps $a\in A$ to its [[connected component]] $[a]\in\Pi_0(A)$ (where the set $\Pi_0(A)$ of connected components of $A$ is equipped with the trivial functions $\omega=id$ and $\mu=id$):
$$
\begin{matrix}
A &\hookrightarrow & X \\
\downarrow &          &\downarrow\\
\Pi_0(A) &\hookrightarrow  & X\mathsf{mod} A
\end{matrix}
$$

* The categories of the form $W/G$ occurring on p.420 and p.424 denote the [[category of elements]] $\int _{W} G$ of the [[presheaf]] $G:W^{op}\to Set$. The background here is that a [[slice topos]] $Set^{\mathcal{C}^{op}}/X$ is equivalent to $Set^{(\int _{\mathcal{C}}X)^{op}} $ (see at [[category of presheaves]] for more details). Whence labeling with $L$ can equivalently be achieved by taking presheaves over $\int _W L$ !

* The construction of Grothendieck that permits internalising a relative age functor mentioned on p.416 is given by the [[category of elements#ColimitPreserving2|right adjoint to the category of elements]] construction (cf. [[Pursuing stacks]], §28).

## Related entries

* [[Claude Lévi-Strauss]]

* [[canonical formula of myth]]

* [[structuralism]]

* [[quasicategory]]

## Links

* Wikipedia entry: kinship ([link](https://en.wikipedia.org/wiki/Kinship))

* Mathematical Anthropology and Cultural Theory ([online journal](http://mathematicalanthropology.org))

## References

Early pioneering attempts include

* Alexander Macfarlane, _Analysis of relationships of
consanguinity and affinity_ , J. Royal Anthr. Inst. **12** (1882) pp.46-63.

* John H. Greenberg, _The logical analysis of kinship_ , Philosophy of Science **15** (1949) pp.58-64.

* Rudolf Carnap, _Introduction to Symbolic Logic and its Applications_ , Dover New York 1958. (pp.220-225, 230)

An obstinate 70s' rumour has it that 'kinship' boils down to a Western folk myth. Already better informed:

* Ernest Gellner, _The concept of kinship_ , Philosophy of Science **27** (1960) pp.187–204.

A comprehensive and highly recommendable anthology is

* {#PS04}Robert Parkin, Linda Stone (eds.), _Kinship and Family_ , Blackwell Oxford 2004.

A conspicuously clear and concise introduction can be found in

* {#Op09}Michael Oppitz, _Notwendige Beziehungen - Abriß der strukturalen Anthropologie_ , Suhrkamp Frankfurt am Main 2009².

Current textbooks include

* Robert Parkin, _Kinship: An Introduction to the Basic Concepts_ , Blackwell Oxford 1997.

* Linda Stone, _Kinship and Gender: An Introduction_ , Westview 2013.

* Murray J. Leaf, Dwight Read, _An Introduction to the Science of Kinship_, Lexington Lanham 2021.

Classical anthropological texts are

* {#Fox83}Robin Fox, _Kinship and Marriage_ , 2nd ed. Cambridge UP 1983.

* {#LS67}[[Claude Lévi-Strauss]], _Les structures élémentaires de la parenté_ , 2nd ed. Mouton The Hague 1967.

* [[Claude Lévi-Strauss]], _Réflexions sur l'atome de parenté_ , L'Homme **13** no.3 (1973) pp.5-30. ([link](http://www.persee.fr/doc/hom_0439-4216_1973_num_13_3_367355))

More recent texts include

* German Dziebel, _The Genius of Kinship_ , Cambria Youngstown 2007.

* {#Godelier04}Maurice Godelier, _Métamorphoses de la parenté_ , Fayard Paris 2004.

* {#GTT98}Maurice Godelier, Thomas R. Trautmann, Franklin E. Tjon Sie Fat (eds.), _Transformations of Kinship_ , Smithsonian Institute Washington 1998.

The permutation model originates with

* {#Weil49}[[André Weil]], _Sur l'étude algébrique de certains types de lois de mariage (Système Murngin)_ , pp.257-263 in [Lévi-Strauss (1949/1967)](#LS67).

A review of Weil's approach as well as an overview of mathematical approaches to kinship is provided by

* Pin-Hsiung Liu, _Theory of groups of permutations, matrices and kinship: a critique of mathematical approaches to prescriptive marriage systems_ , Bulletin of the Institute of Ethnology Academia Sinica **26** (1968) pp.29–38. ([pdf](http://c.ianthro.tw/sites/c.ianthro.tw/files/da/df/407/407386_0001.pdf))

An overview of structuralist kinship theory and a criticism of permutation models is

* {#Sperber68} Dan Sperber, _Le structuralisme en anthropologie_ , pp.167-238 in Ducrot et al. , _Qu'est-ce que le structuralisme?_ , Seuil Paris 1968.

Courrège's seminal paper is easily accessible and eminently readable:

* {#Cour65}Philippe Courrège, _Un modèle mathématique des structures élémentaires de parenté_ , L'Homme **5** no.3-4 (1965) pp.248-290. ([link](http://www.persee.fr/doc/hom_0439-4216_1965_num_5_3_366751); Engl. translation in Ballonoff (ed.), _Genetics and Social Structure_ , Hutchinson&Ross 1974)

A useful overview and synthesis of the algebraic approach is

* Franklin E. Tjon Sie Fat, _Representing Kinship: Simple Models of Elementary Structures_ , PhD. Leiden University 1990.

A visit to the library is richly rewarded by the clear and comprehensive exposition in the following outstanding monograph:

* {#Liu86}Pin-Hsiung Liu, _Foundations of Kinship Mathematics_ , Academica Sinica Nankang 1986.

with companion volume by his collaborator

* S. H. Gould, _A New System for the Formal Analysis of Kinship_, University of America Press Lanham (2000).

The category-theoretic point of view is sketched in

* {#Law99}[[William Lawvere|F. William Lawvere]], _Kinship and Mathematical Categories_ , pp.411-425 in Jackendoff, Bloom, Wynn (eds.), _Language, Logic, and Concepts_ , MIT Press Cambridge 1999.

* {#LawSchan97}[[William Lawvere|F. William Lawvere]], [[Stephen Schanuel|Stephen H. Schanuel]], _Conceptual Mathematics_ , Cambridge UP 1997. (pp.162-163)

The following pursues an abstract general approach to kinship:

* {#Allen86}N. J. Allen, _Tetradic Theory: An Approach to Kinship_ , J. Anthr. Soc. Oxford **17** (1986) pp.87-109. (Revised reprint pp.221-235 in [Parkin-Stone 2004](#PS04).)

The general role of group theory in culture is dicussed in

* P. Lucich, _Beyond Formalism: Group Theory in the Symmetries of Culture_ , J. Math. Sociology **16** (1991) pp.221-264.

* P. Lucich, _Genealogical Symmetries: Rational Foundations of Australian Kinship_, Light Stone Publications Armidale (1987).

A group-theoretic approach to Dravidian kinship and universal classification is proposed in

* {#Alm10}Mauro W. Barbosa De Almeida, _On the Structure of Dravidian Relationship Systems_ , MACT **3** no.1 (2010). ([pdf](http://mathematicalanthropology.org/pdf/BarbosadeAlmeida.0810.pdf))

A connection between Dravidian kinship terminology and [[supersymmetry]] is proposed in

* {#Vaz14}Ruth M. Vaz, _Relatives, Molecules and Particles_ , MACT **7** no. 1 (2014). ([pdf](http://mathematicalanthropology.org/pdf/RMVaz_MACT%200614.pdf))

Kinship systems are studied from a mathematical perspective in

* John P. Boyd, _The Algebra of Group Kinship_ , J. Math. Psychology **6** (1969) pp.139-167.

* John P. Boyd, John H. Haehl, Lee D. Sailer, _Kinship and Inverse Semigroups_ , J. Math. Sociology **2** no.1 (1972) pp.37-61.

* Aimé Fuchs, _Les structures de parenté: traitement mathématique_ , Revue Européenne des Sciences Sociales et Cahiers Vilfredo Pareto **19** (1981) pp.161-182.

* Alain Gottcheiner, _On some Classes of Kinship Systems I: Abelian Systems_ , MACT **2** no. 4 (2008). ([pdf](http://mathematicalanthropology.org/Pdf/GottscheinerI_0908.pdf))

* Alain Gottcheiner, _On some Classes of Kinship Systems II: Nonabelian Systems_ , MACT **2** no. 5 (2008). ([pdf](http://mathematicalanthropology.org/Pdf/GottscheinerII_0908.pdf))

* Labib Haddad, Yves Sureau, _Les groupes, les hypergroupes et l'énigme des Murngin_ , JPAA **87** (1993) pp.221-235.

* Per Hage, Frank Harary, _The Logical Structure of Asymmetric Marriage_ , L'Homme **36** no.139 (1996) pp.109-124. ([link](http://www.persee.fr/doc/hom_0439-4216_1996_num_36_139_370120))

* Frank Harary, Douglas R. White, _P-Systems: A Structural Model for Kinship Studies_ , Connections **24** no.2 (2001) pp.35-46. ([pdf](http://www.insna.org/PDF/Connections/v24/2001_I-2-6.pdf))

* Paul Jorion, Gisèle De Meur, Trudeke Vuyk, _Le mariage Pende_  , L'Homme **22** no.1 (1982) pp.53-73. ([link](http://www.persee.fr/doc/hom_0439-4216_1982_num_22_1_368256))

* [[Claude Lévi-Strauss]], Georges Guilbaud, _Système parental et matrimonial au Nord Ambrym_ , J. Société des Océanistes **26** no. 26 (1970) pp.9-32. ([link](http://www.persee.fr/doc/jso_0300-953x_1970_num_26_26_2281))

* Gisèle De Meur (ed.), _New Trends in Mathematical Anthropology_ , Routledge London 1986.

* Gisèle De Meur, Alain Gottcheiner, _Prescriptive Kinship Systems, Permutations, Groups, and Graphs_ , MACT **1** no. 1 (2000). ([pdf](http://mathematicalanthropology.org/Pdf/MACTdeMEUR1100.pdf))

* Harrison C. White, _An Anatomy of Kinship_ , Prentice-Hall Englewood Cliffs 1963.

For related ideas see also

* Wyllis Bandler, _Some Esomathematical Uses of Category Theory_ , pp.243-255 in Glir (ed.), _Applied General Systems Research_ , Springer Heidelberg 1978.

* John P. Boyd, _The Universal Semigroup of Relations_ , Social Networks **2** no.2 (1979) pp.91-117.

* John P. Boyd, _Social Semigroups_ , George Mason UP 1991.

* François Lorrain, _Réseaux sociaux et classifications sociales: Essai sur l 'algèbre et la géométrie des structures sociales_ , Hermann Paris 1975.

* François Lorrain, Harrison C. White, _Structural Equivalence of Individuals in Social Networks_ , J. Math. Sociology **1** (1971) pp.49-80.

* Oystein Ore, _Sex in Graphs_ , Proc. AMS **11** (1960) pp.533-539. ([pdf](http://www.ams.org/journals/proc/1960-011-04/S0002-9939-1960-0117174-7/S0002-9939-1960-0117174-7.pdf))

* Nina Otter, Mason J. Porter, _A unified framework for equivalences in social networks_ , arXiv:2006.10733 (2020). ([abstract](https://arxiv.org/abs/2006.10733))

First steps in [[topos theory]] can be undertaken with [Lawvere-Schanuel (1997)](#LawSchan97) whereas the level of sophistication then increases steadily from

* {#RRZ04} M. La Palme Reyes, [[Gonzalo E. Reyes|G. E. Reyes]], H. Zolfaghari, _Generic Figures and their Glueings_ , Polimetrica Milano 2004.

to the standard textbook

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

The result concerning essential localizations is in

* {#KL89}[[G. M. Kelly]], [[F. W. Lawvere]], *On the Complete Lattice of Essential Localizations*, Bull. Soc. Math. de Belgique **XLI** (1989) 289-319 [[[Kelly-Lawvere_EssentialLocalizations.pdf:file]]]

The topos $Set^{M_2}$ is studied under the name _category of autographs_ in

* R. Guitart, _Autocategories: I. A common setting for knots and 2-categories_ , Cah. Top. Géom. Diff. Cat. **LV** (2014) pp.66-82.

* R. Guitart, _Autocategories: II. Autographic algebras_ , Cah. Top. Géom. Diff. Cat. **LV** (2014) pp.151-160.

* R. Guitart, _Autocategories: III. Representations and expansions of previous examples , Cah. Top. Géom. Diff. Cat. **LVIII** (2017) pp.67-81.

[[!redirects kinship system]]
[[!redirects mathematical anthropology]]


Kip Thorne

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

## Selected writings

On the [[classical field theory]] of [[Einstein gravity]] ("[[general relativity]]"):

* [[Charles Misner]], [[Kip Thorne]], [[John Wheeler]]: *[[Gravitation]]* W. H. Freeman, San Francisco (1973) [ISBN:9780716703440, 9780691177793, [inspire:95654](https://inspirehep.net/literature/95654), Wikipedia entry]

## Related entries

* _[[Gravitation]]_

category: people


Kiran Kedlaya

* [webpage](http://www-math.mit.edu/~kedlaya/)

category: people


Kiran Luecke

* [institute page](https://math.berkeley.edu/people/grad/kiran-luecke)

## Selected writings

On [[equivariant cohomology|equivariant]] [[Tate K-theory]] (a corner of [[equivariant elliptic cohomology]]) for [[compact Lie groups]]:

* {#Luecke19}  [[Kiran Luecke]], *Completed K-theory and Equivariant Elliptic Cohomology*, Advances in Mathematics **410** B (2022) 108754 [[arXiv:1904.00085](https://arxiv.org/abs/1904.00085), [doi:10.1016/j.aim.2022.108754](https://doi.org/10.1016/j.aim.2022.108754)]

category: people


Kirby calculus

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
#### Knot theory
+--{: .hide}
[[!include knot theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

What is called _Kirby calculus_ is a way of identifying/classifying [[3-manifolds]] in terms of [[Dehn surgery]] along [[links]] in the [[3-sphere]].

## References

The original article is

* {#Kirby78} [[Rob Kirby]], _A Calculus for Framed Links in $S^3$_. Inventiones Mathematicae, vol. 45 (1978), pp. 35–56 ([dml:142535](https://eudml.org/doc/142535))

See also

* Wikipedia, _[Kirby calculus](https://en.wikipedia.org/wiki/Kirby_calculus)_


Kirchhoff's laws

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

# Contents
* table of contents
{: toc}

## Idea

In the context of _[[electromagnetism]]_, Kirchhoff's laws are a kind of coarse-grained form of [[Maxwell's equations]]. Where the latter deal with [[infinitesimal object|infinitesimal]] quantities, Kirchhoff's law involve macroscopic [[electrical current|current]] and [[voltage]] in electrical [[electrical circuit|circuits]].

One speaks of two laws,

1. the _Kirchhoff voltage law_ (abbreviated KVL)

1. the _Kirchhoff current law_ (abbreviated KCL)

The derivation of these two laws from [[Maxwell's equations]] is spelled out for instance in lecture 7 ([here](#Lecture)).

Nevertheless, Kirchhoff's laws preserve the [[cohomology|cohomological]] nature of Maxwell's equations: where the latter involves [[de Rham cohomology]] and hence "infinitesimal cochains", Kirchhoff's laws can neatly be formulated in terms of [[cochain on a simplicial set|cochains on finite cell complexes]]. (See for instance appendix B of ([Frankel](#Frankel)) or the section "Basic concepts" in ([Baez](#Baez))).

## Related concepts

* [[Maxwell's equations]]

## References

Reviews include

lecture 7 in

* ([pdf](http://web.archive.org/web/20100609120845/http://ieeeb.okstate.edu/lecturenotes/EET-4654%20Microwaves/Lecture%207%20and%208.pdf))
 {#Lecture}

appendix B in

* [[Theodore Frankel]], _[[The Geometry of Physics - An Introduction]]_ .
 {#Frankel}

section _Basic concepts_ in

* [[John Baez]], _[[johnbaez:Circuit theory]]_
 {#Baez}

The cohomological nature of Kirchoff's laws was maybe first made explicit in

* [[Hermann Weyl]], _Repartición de corriente en una red conductora, Rev. Mat. Hisp. Amer. 5 (1923), 153-164.

A textbook on electromagnetism amplifying this point of view is

* P. W. Gross and P. R. Kotiuga, _Electromagnetic Theory and Computation: A Topological Approach_, Cambridge University Press, (2004)

[[!redirects Kirchhoff's law]]
[[!redirects Kirchhoff's laws]]
[[!redirects Kirchhoff's law]]
[[!redirects Kirchhoff's laws]]
[[!redirects Kirchhoff law]]
[[!redirects Kirchhoff laws]]


Kirill Gubarev

* [GoogleScholar page](https://scholar.google.co.za/citations?user=nb_dagwAAAAJ&hl=en)

## Selected writings

On possible structures in [[M2-brane]] dynamics and [[M2-M5-brane bound states]] which could be [[M-theory]]-lifts of the familiar [[integrable system|integrability]] of the [[Green-Schwarz superstring]] on [[AdS spacetime|$AdS_5$]] $\times$ [[5-sphere|$S^5$]]:

* [[Kirill Gubarev]], [[Edvard Musaev]], *Integrability structures in string theory* [[arXiv:2301.06486](https://arxiv.org/abs/2301.06486)]

category: people


Kirill Krasnov

* [webpage](https://www.maths.nottingham.ac.uk/plp/pmzkk)

## Selected writings

Discussion of [[Euclidean field theory|Euclidean]] [[BTZ black holes]]/[[thermal AdS3]] (the [[hyperbolic solid torus]]) with an eye towards [[AdS3/CFT2]]:

* [[Kirill Krasnov]], _Holography and Riemann Surfaces_, Adv. Theor. Math. Phys. 4 (2000) 929-979 ([arXiv:hep-th/0005106](https://arxiv.org/abs/hep-th/0005106))

* [[Kirill Krasnov]], _Analytic Continuation for Asymptotically AdS 3D Gravity_, Class. Quant. Grav. 19 (2002) 2399-2424 ([arXiv:gr-qc/0111049](https://arxiv.org/abs/gr-qc/0111049))

* [[Kirill Krasnov]], _Black Hole Thermodynamics and Riemann Surfaces_, Class. Quant. Grav. 20 (2003) 2235-2250 ([arXiv:gr-qc/0302073](https://arxiv.org/abs/gr-qc/0302073))

On the exact [[gauge group]] of the [[standard model of particle physics]] (see [there](standard+model+of+particle+physics#GaugeGroup)) as [[isomorphism|isomorphic]] to the [[subgroup]] of [[Spin(9)]]  which respects a splitting $\mathbb{H} \oplus \mathbb{H} \simeq_{\mathbb{R}} \mathbb{C} \oplus \mathbb{C}^3$:

* {#Krasnov19} [[Kirill Krasnov]], _$SO(9)$ characterisation of the Standard Model gauge group_ ([arXiv:1912.11282](https://arxiv.org/abs/1912.11282))

New perspective on the [[symmetry breaking]] of [[Spin(10)]]-[[GUT]] to the [[standard model of particle physics|standard model]] [[gauge group]]:

* [[Kirill Krasnov]], *Geometry of $Spin(10)$ Symmetry Breaking* [[arXiv:2209.05088](https://arxiv.org/abs/2209.05088)]

On complex and Lorentzian analogs of [[Cayley forms]]:

* [[Kirill Krasnov]], *Lorentzian Cayley Form* [[arXiv:2304.01118](https://arxiv.org/abs/2304.01118)]

category: people


Kirill Mackenzie

Kirill Mackenzie was a Reader in the School of Mathematics and Statistics and had worked at the University since August 1989. His death was announced 2 May 2020.

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=97270)

* [Institute page](https://maths.dept.shef.ac.uk/maths/staff_info_11.html)

## Selected writings

On [[Lie groupoids]]and [[Lie algebroids]]:

* [[Kirill Mackenzie]], _Lie groupoids and Lie algebroids in differential geometry_, London Mathematical Society Lecture Note Series, 124. Cambridge University Press, Cambridge, 1987. xvi+327 pp ([doi:10.1017/CBO9780511661839](https://doi.org/10.1017/CBO9780511661839), [MR:896907](http://www.ams.org/mathscinet-getitem?mr=896907))

  (> Ch. II on [[topological groupoids]])

* [[Kirill Mackenzie]], _General Theory of Lie Groupoids and Lie Algebroids,_ Cambridge University Press, 2005 ([doi:10.1017/CBO9781107325883](https://doi.org/10.1017/CBO9781107325883))

category: people

[[!redirects Kirill C. H. Mackenzie]]


Kirill Shtengel

* [GoogleScholar page](https://scholar.google.com/citations?user=n9xywcAAAAAJ&hl=en)

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=102443)

## Selected writings

On potential [[quantum measurement]] of non-abelian [[anyons]]:

* {#BondersonShtengelSlingerland07} [[Parsa Bonderson]], [[Kirill Shtengel]], [[Joost Slingerland]], *Decoherence of Anyonic Charge in Interferometry Measurements*, Phys. Rev. Lett. **98** (2007) 070401 [[doi:10.1103/PhysRevLett.98.070401](https://doi.org/10.1103/PhysRevLett.98.070401), [arXiv:quant-ph/0608119](https://arxiv.org/abs/quant-ph/0608119)]

* [[Parsa Bonderson]], [[Kirill Shtengel]], [[Joost Slingerland]], *Interferometry of non-Abelian Anyons*, Annals Phys. **323** (2008) 2709-2755 [[doi:10.1016/j.aop.2008.01.012](https://doi.org/10.1016/j.aop.2008.01.012), [arXiv:0707.4206](https://arxiv.org/abs/0707.4206)]

category: people


Kirillov correspondence

## References

[web](http://www.ams.org/journals/proc/2007-135-08/S0002-9939-07-08781-3/)


Kirk Sturtz

* [DBLP page](https://dblp.org/pid/22/2992.html)

## Selected writings

On the [[Giry monad]] ([[category theory|category theoretic]] [[probability theory]]):

* {#Sturtz22} [[Kirk Sturtz]], _Giry algebras via the support of a probability measure_ $[$[arXiv:2202:10819](https://arxiv.org/abs/2202.10819)$]$

which has evolved from the preliminary work:

* [[Kirk Sturtz]], _Categorical Probability Theory_, $[$[arXiv:1406.6030](http://arxiv.org/abs/1406.6030)$]$

category: people


Kirsten Wickelgren

* [web](http://people.math.gatech.edu/~kwickelgren3/)

[[anabelian geometry]]


Kirti Joshi

* [personal page](https://www.math.arizona.edu/~kirti/)

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=170515)

## Selected writings

On [[p-adic Teichmüller theory]]:

* [[Kirti Joshi]], *Construction of Arithmetic Teichmüller Spaces and some applications* ([arXiv:2106.11452](https://arxiv.org/abs/2106.11452))

* [[Kirti Joshi]], *Untilts of fundamental groups: construction of labeled isomorphs of fundamental groups -- Arithmetic Holomorphic Structures* ([arXiv:2210.11635](https://arxiv.org/abs/2210.11635))

* [[Kirti Joshi]], *Construction of Arithmetic Teichmüller spaces II: Proof of a local prototype of Mochizuki's Corollary 3.12* ([arXiv:2303.01662](https://arxiv.org/abs/2303.01662))

* [[Kirti Joshi]], *Construction of Arithmetic Teichmüller spaces II($\frac{1}{2}$): Deformations of Number Fields* ([arXiv:2305.10398](https://arxiv.org/abs/2305.10398))

* [[Kirti Joshi]], *Comments on Arithmetic Teichmüller Spaces* ([arXiv:2111.06771](https://arxiv.org/abs/2111.06771))

* [[Kirti Joshi]], *Mochizuki's Corollary 3.12 and my quest for its proof* [pdf]

* [[Kirti Joshi]], *(Older-Version) Construction of Arithmetic Teichmuller spaces II: Towards Diophantine Estimates* ([arXiv:2111.04890](https://arxiv.org/abs/2111.04890))

* [[Kirti Joshi]], *(Older-Version) Untilts of fundamental groups: construction of labeled isomorphs of fundamental groups* ([arXiv:2010.05748](https://arxiv.org/abs/2010.05748))

Claim of proof of statements similar to [[Mochizuki's corollary 3.12]]:

* [[Kirti Joshi]], *Appendix to Construction of Arithmetic Teichmüller spaces II($\frac{1}{2}$): Deformations of Number Fields* ([arXiv:2305.10398](https://arxiv.org/abs/2305.10398))

* [[Kirti Joshi]], *Construction of Arithmetic Teichmuller Spaces III: A Rosetta Stone and a proof of Mochizuki's Corollary 3.12* ([arXiv:2401.13508](https://arxiv.org/abs/2401.13508))

On [[anabelian geometry]]:

* [[Kirti Joshi]], *Mochizuki's anabelian variation of ring structures and formal groups* ([arxiv:1906.06840](https://arxiv.org/abs/1906.06840))

* [[Kirti Joshi]], *On Mochizuki's idea of Anabelomorphy and its applications* ([arxiv:2003.01890](https://arxiv.org/abs/2003.01890))

On the Hitchin-Mochizuki morphism and Frobenius destablized vector bundles on curves:

* [[Kirti Joshi]], Christian Pauly, *Hitchin–Mochizuki morphism, opers and Frobenius-destabilized vector bundles over curves* ([Advances in Mathematics,
Volume 274, 9 April 2015, Pages 39-75] (https://doi.org/10.1016/j.aim.2015.01.004))

* [[Kirti Joshi]], *The Degree of the Dormant Operatic Locus* ([International Mathematics Research Notices, Volume 2017, Issue 9, May 2017, Pages 2599–2613] (https://doi.org/10.1093/imrn/rnw066))

## Related entries

* [[inter-universal Teichmüller theory|Inter-universal Teichmüller theory]]

category: people


Kiyonori Gomi

* [personal page](http://www.math.titech.ac.jp/~kgomi/)

* [Institute page](http://t2r2.star.titech.ac.jp/cgi-bin/researcherinfo.cgi?lv=en&q_researcher_content_number=CTT100804216)

## Selected writings

On [[fiber integration in differential cohomology]] (fiberwise [[Stokes theorem]] lifted to [[Deligne cohomology]]):

* {#GomiTerashima00} [[Kiyonori Gomi]], [[Yuji Terashima]], _A Fiber Integration Formula for the Smooth Deligne Cohomology_, International Mathematics Research Notices 2000, No. 13 ([pdf](http://imrn.oxfordjournals.org/content/2000/13/699.full.pdf), [pdf](http://numr.wdfiles.com/local--files/differential-cohomology/gomi-terashima.pdf), [doi:10.1155/S1073792800000386](https://doi.org/10.1155/S1073792800000386))

On [[higher parallel transport]] for [[circle n-bundles with connection]]:

* [[Kiyonori Gomi]], [[Yuji Terashima]], _Higher dimensional parallel transport_ Mathematical Research Letters 8, 25–33 (2001) ([pdf](http://mrlonline.org/mrl/2001-008-001/2001-008-001-004.pdf))

On [[equivariant ordinary differential cohomology]]:

* [[Kiyonori Gomi]], _Equivariant smooth Deligne cohomology_, Osaka J. Math. Volume 42, Number 2 (2005), 309-337 ([euclid:ojm/1153494380](https://projecteuclid.org/euclid.ojm/1153494380))

On [[twisted K-theory]] via [[vectorial bundles]]:

* {#Gomi} [[Kiyonori Gomi]], _Twisted K-theory and finite-dimensional approximation_ ([arXiv:0803.2327](http://arxiv.org/abs/0803.2327))

* {#GomiTerashima} [[Kiyonori Gomi]], [[Yuji Terashima]], _Chern-Weil Construction for Twisted K-Theory_, Communication ins Mathematical Physics, Volume 299, Number 1, 225-254 ([doi:10.1007/s00220-010-1080-1](https://doi.org/10.1007/s00220-010-1080-1))

On [[discrete torsion]]:

* [[Kiyonori Gomi]], [[Yuji Terashima]], _Discrete Torsion Phases as Topological Actions_, Commun. Math. Phys. (2009) 287: 889 ([doi:10.1007/s00220-009-0736-1](https://doi.org/10.1007/s00220-009-0736-1))

On combined [[twisted K-theory|twisted]] [[equivariant K-theory|equivariant]] [[KR-theory]] of [[orbifold|orbi-]] [[orientifolds]]:

* {#Gomi17} [[Kiyonori Gomi]], _Freed-Moore K-theory_ ([arXiv:1705.09134](https://arxiv.org/abs/1705.09134), [spire:1601772](http://inspirehep.net/record/1601772))

On the [[K-theory classification of topological phases of matter]] for the case of [[topological crystalline insulators]] classified by [[equivariant K-theory]]:

* [[Ken Shiozaki]], [[Masatoshi Sato]], [[Kiyonori Gomi]], *Topological Crystalline Materials - General Formulation, Module Structure, and Wallpaper Groups*, Phys. Rev. B **95** (2017) 235425 [[arXiv:1701.08725](https://arxiv.org/abs/1701.08725), [doi:10.1103/PhysRevB.95.235425](https://doi.org/10.1103/PhysRevB.95.235425)]

and using the [[Atiyah-Hirzebruch spectral sequence]]:

* [[Ken Shiozaki]], [[Masatoshi Sato]], [[Kiyonori Gomi]], *Atiyah-Hirzebruch Spectral Sequence in Band Topology: General Formalism and Topological Invariants for 230 Space Groups*, Phys. Rev. B **106** (2022) 165103 [[arXiv:1802.06694](https://arxiv.org/abs/1802.06694), [doi:10.1103/PhysRevB.106.165103](https://doi.org/10.1103/PhysRevB.106.165103)]

On [[T-duality]] in the [[K-theory classification of topological phases of matter]]:

* [[Kiyonori Gomi]], [[Guo Chuan Thiang]], *Crystallographic T-duality*. J.Geom.Phys **139** (2019) 50-77 [[arXiv:1806.11385](https://arxiv.org/abs/1806.11385), [doi:10.1016/j.geomphys.2019.01.002](https://doi.org/10.1016/j.geomphys.2019.01.002)]

On [[differential K-theory|differential]] [[KO-theory]] via [[mass terms]]:

* [[Kiyonori Gomi]], [[Mayuko Yamashita]], *Differential KO-theory via gradations and mass terms* [[arXiv:2111.01377](https://arxiv.org/abs/2111.01377)]

category: people


Kiyoshi Igusa

* [website](http://people.brandeis.edu/~igusa/)

###Seleted papers

* Kiyoshi Igusa, _Iterated integrals of superconnections_ ([arXiv](http://arxiv.org/abs/0912.0249))

* Kiyoshi Igusa, _[Twisting cochains and higher torsion](https://tcms.org.ge/Journals/JHRS/xvolumes/2011/n2a3/v6n2a3.pdf)_, J. Homotopy Relat. Struct, 6(2):213–238, 2011.

category: people


Kiyoshi Oka

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

## Selected writings

* [[Kiyoshi Oka]], _Sur les fonctions des plusieurs variables. III: Deuxième problème de Cousin_, J. Sc. Hiroshima Univ. __9__, 7–19 (1939) ([doi:10.32917/hmj/1558490525](https://projecteuclid.org/journals/hiroshima-mathematical-journal/volume-9/issue-none/Sur-les-fonctions-analytiques-de-plusieurs-variables-IIIDeuxi%c3%a8me-probl%c3%a8me-de/10.32917/hmj/1558490525.full))

## Related $n$Lab entries

* [[Oka principle]]

* [[Oka manifold]]

category: people


KK principle

* table of contents
{:toc}

## Idea

In [[epistemic modal logic]], the **KK principle** or **KK thesis** states that for any proposition $p$, if one knows that $p$, then one knows that one knows it. While admitted by one of the early workers in the field, [[Jaakko Hintikka]], it is the subject of considerable philosophical debate (see [Hemp](#Hemp)).

## References

* {#Hemp} David Hemp, _The KK (Knowing that One Knows) Principle_, Internet Encyclopedia of Philosophy ([entry](https://iep.utm.edu/kk-princ/))

[[!redirects KK thesis]]


KK-bootstrap category

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

#Contents#
* table of contents
{:toc}

## Idea

The [[thick subcategory]]

$$
  KK_{boot} \hookrightarrow KK
$$

of the [[triangulated category]] $KK$ of [[KK-theory]] generated by the [[tensor unit]] is called the _bootstrap category_ ([Rosenberg-Schochet 87](#RosenbergSchochet87)), because on this subcategory [[KK-theory]] has a good [[axiom|axiomatic]] description (see [Blackadar 22.3](#Blackadar)).

The canonical [[functor]] from [[KK-theory|KK]] to (the [[homotopy category]] of) [[KU]]-[[module spectra]] restricts to a [[full and faithful functor]] on the bootstrap category (prop. \ref{StrongMonoidalFullInclusionIntoKUMod} below), and so the KK-bootstrap category may also be thought of as a [[full subcategory]] of $KU Mod$

$$
  KK_{boot} \hookrightarrow Ho(KU Mod)
  \,.
$$

Many [[C*-algebras]] that "appear in practice" are in the bootstrap category, notably the [[groupoid convolution algebras]] of [[amenable Lie groupoids]] (prop. \ref{ConvolutionOfAmenableLieGroupoid} below). Generally, a [[C*-algebra]] is in the bootstrap category precisely if it is KK-[[equivalence|equivalent]] to a [[commutative C*-algebra]].

## Definition

See ([Blackadar, def. 22.3.4](#Blackadar)).

## Examples of objects in the bootstrap category

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

The [[groupoid convolution algebra]] of an [[amenable topological groupoid]] is in the bootstrap category.

=--

This is due to ([Tu 99, prop. 10.7](#Tu99)), recalled for instance as ([Uuye 11, example 3.6](#Uuye11)).

## Properties

### Künneth theorem

In the bootstrap category a [[Künneth theorem]] for [[operator K-theory]] is true. ([Rosenberg-Schochet 87](RosenbergSchochet87)).

### Embedding into $KU$-module spectra

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

On the bootstrap category the operator K-theory spectrum functor $KK \to Ho(KU Mod)$ is

1. a [[fully faithful functor]],

1. a [[strong monoidal functor]]

=--

The first statement is ([DEKM 11, section 3](DEKM11)), the second is ([DEKM 11, prop. 4.2](#DEKM11)).

See at _[[KK-theory]]_ the section _[Triangulated and spectrum-enriched structure](KK-theory#TriangulatedAndSpectrumEnrichedStructure)_ for more details.

## References

The notion originates in

* [[Jonathan Rosenberg]], [[Claude Schochet]], _The Künneth theorem and the universal coefficient theorem for Kasparov's generalized K-functor_, Duke Math. J. 55 (1987), no. 2, 431–474.
 {#RosenbergSchochet87}

A texbook account is around def. 22.3.4 of

* [[Bruce Blackadar]], _[[K-Theory for Operator Algebras]]_
 {#Blackadar}

The corresponding [[localizing subcategories]] are discussed in

* [[Ivo Dell'Ambrogio]], _Localizing subcategories in the Bootstrap category of separable C*-algebras_, Journal of K-Theory 8 (2011), 493-50 ([arXiv:1003.0183](http://arxiv.org/abs/1003.0183))

The full embedding into the homotopy category of KU-modules is discussed in section 3 of

* [[Ivo Dell'Ambrogio]], [[Heath Emerson]], [[Tamaz Kandelaki]], [[Ralf Meyer]], _A functorial equivariant K-theory spectrum and an equivariant Lefschetz formula_ ([arXiv:1104.3441](http://arxiv.org/abs/1104.3441))
 {#DEKM11}

The inclusion of [[groupoid convolution algebras]] of [[amenable topological groupoids]] into the bootstrap category is discussed in

* [[Jean-Louis Tu]], _La conjecture de Baum-Connes pour les feuilletages moyennables_, K-Theory 17 (1999), no. 3, 215–264. MR 1703305 (2000g:19004)
 {#Tu99}

* [[Otgonbayar Uuye]], _A note on the Künneth theorem for nonnuclear $C^\ast$-algebras_ ([arXiv:1111.7228](http://arxiv.org/abs/1111.7228))
 {#Uuye11}

Discussion of the bootstrap category in [[equivariant K-theory]] is in

* [[Heath Emerson]], _Localization techniques in circle-equivariant KK-theory_ ([arXiv:1004.2970](http://arxiv.org/abs/1004.2970))

This show that there are algebras in the plain bootstrap category with [[circle group]]-equivariant structure which are not $U(1)$-equivariant KK-equivalent to a commutative $U(1)$-$C^\ast$-alegbra.

[[!redirects KK-bootstrap categories]]

[[!redirects bootstrap category]]
[[!redirects bootstrap categories]]


KK-compactifications of M-theory -- table

**[[KK-compactification]] of [[M-theory]]**

* [[M-theory on a circle]]

* [[M-theory on 8-manifolds]]/[[M-theory on Spin(7)-manifolds|on Spin(7)-manifolds]]

* [[M-theory on G2-manifolds]]

  * [[G2-MSSM]]

* [[M-theory on Sp(1)-manifolds]]

* [[M-theory on hyperbolic manifolds]]

* [[heterotic M-theory on ADE-orbifolds]]

* [[M-theory on Calabi-Yau manifolds]]

  * [[F/M-theory on elliptically fibered Calabi-Yau 2-folds|on CY2-folds]]

  * [[F/M-theory on elliptically fibered Calabi-Yau 3-folds|on CY3-folds]]

  * [[F/M-theory on elliptically fibered Calabi-Yau 4-folds|on CY4-folds]]

  * [[F/M-theory on elliptically fibered Calabi-Yau 5-folds|on CY5-folds]]


KK-monopole geometries -- table

| geometry transverse to [[KK-monopoles]]  | [[Riemannian metric]] |  remarks |
|---|---|---|
| **[[Taub-NUT space]]**:
 geometry transverse to
 $N+1$ distinct [[KK-monopoles]]
 at $\vec r_i \in \mathbb{R}^3 \;\; i \in \{1, \cdots, N+1\}$ | $\array{d s^2_{TaubNUT} \coloneqq U^{-1}(d x^4 + \vec \omega \cdot d \vec r)^2 + U (d \vec r)^2 \,, \\ \vec r \in \mathbb{R}^3,\, x^4 \in \mathbb{R}/(2 \pi R\mathbb{Z}) \\ U \coloneqq 1 + \underoverset{i = 1}{N+1}{\sum} U_i\,, \phantom{AA} \vec \omega \coloneqq \underoverset{i = 1}{N+1}{\sum} \vec \omega_i \\ U_i \coloneqq \frac{R/2}{ {\vert \vec r - \vec r_i\vert} }\,, \phantom{AA} \vec \nabla \times \vec \omega= \vec \nabla U_i}$ | (e.g. [Sen 97b, Sect. 2](Kaluza-Klein+monopole#Sen97b))  |
| **[[ALE space]]**
 Taub-NUT close to $N$ close-by [[KK-monopoles]]
 e.g. close to $\vec r = 0$: $\frac{{\vert \vec r_i\vert}}{R/2}, \frac{{\vert \vec r\vert}}{R/2} \ll 1$
  | $\array{d s^2_{ALE} \coloneqq U'^{-1}(d x^4 + \vec \omega \cdot d \vec r)^2 + U' (d \vec r)^2 \,, \\ \vec r \in \mathbb{R}^3,\, x^4 \in \mathbb{R}/(2 \pi R\mathbb{Z}) \\ U' \coloneqq \underoverset{i = 1}{N+1}{\sum} U'_i\,, \phantom{AA} \vec \omega \coloneqq \underoverset{i = 1}{N+1}{\sum} \vec \omega_i \\ U'_i \coloneqq \frac{R/2}{ {\vert \vec r - \vec r_i\vert} }\,, \phantom{AA} \vec \nabla \times \vec \omega= \vec \nabla U_i}$ | e.g. via [[Euler angles]]: $\vec \omega = (N+1)R/2(\cos(\theta)-1) d\psi$
 (e.g. [Asano 00, Sect. 2](Kaluza-Klein+monopole#Asano00)) |
| **$A_N$-type [[ADE singularity]]**:
 [[ALE space]] in the [[limit of a sequence|limit]]
 where all $N+1$ [[KK-monopoles]] coincide at $vec r_i = 0$ | $\array{d s^2_{A_N Sing} \coloneqq \frac{\vert\vec r\vert }{(N+1)R/2}(d x^4 + \vec \omega \cdot d \vec r)^2 + \frac{ (N+1)R/2}{\vert \vec r\vert} (d \vec r)^2 \,, \\ \vec r \in \mathbb{R}^3,\, x^4 \in \mathbb{R}/(2 \pi R\mathbb{Z}) } $ | (e.g. [Asano 00, Sect. 3](Kaluza-Klein+monopole#Asano00))  |


KK-theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Index theory
+-- {: .hide}
[[!include index theory - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Operator algebra
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### Functional analysis
+--{: .hide}
[[!include functional analysis - contents]]
=--
#### Noncommutative geometry
+--{: .hide}
[[!include noncommutative geometry - contents]]
=--
#### Motivic cohomology
+--{: .hide}
[[!include motivic cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_KK-theory_ is a "[[bivariant cohomology theory|bivariant]]" joint generalization of [[operator K-theory]] and [[K-homology]]: for $A, B$ two [[C*-algebras]], the _KK-group_ $KK(A,B)$ is a natural [[homotopy]] [[equivalence class]] of $(A,B)$-[[Hilbert bimodules]] equipped with an additional left weak [[Fredholm module]] structure. These KK-groups $KK(A,B)$ behave in the first argument as [[K-homology]] of $A$ and in the second as [[K-cohomology]]/[[operator K-theory]] of $B$.

Abstractly, KK-theory is an [[additive category]] of [[C*-algebras]] which is the split-[[exact functor|exact]] and [[homotopy]]-invariant [[localization]] of [[C*Alg]] at the [[compact operators]]. Hence, abstractly KK-theory is a fundamental notion in [[noncommutative topology]], but its standard presentation by [[Fredholm module|Fredolm]]-[[Hilbert bimodules]] as above is rooted in [[functional analysis|functional]] [[analysis]]. A slight variant of this localization process is called _[[E-theory]]_.

Due to this joint root in [[functional analysis]] and ([[noncommutative topology|noncommutative]]) [[cohomology]]/[[homotopy theory]] ("[[noncommutative stable homotopy theory]]"), KK-theory is a natural home of [[index theory]], for [[elliptic operators]] on [[smooth manifolds]] as well as for their generalization to [[equivariant cohomology|equivariant]] situations, to [[foliations]] and generally to [[Lie groupoid]]-theory (via their [[groupoid convolution C*-algebras]]) and [[noncommutative geometry]].

As a special case of this, [[quantization]] in its incarnation as [[geometric quantization by push-forward]] has been argued to naturally proceed by [[index theory]] in KK-theory ([Landsman 03](#Landsman03), [Bos 07](#Bos07)). Also the coupling of [[D-branes]] and their [[Chan-Paton bundles]] in [[twisted K-theory]] with [[RR-charge]] in [[string theory]] is naturally captured by the coupling between [[K-homology]] and [[K-cohomology]] in KK-theory (e.g. [Szabo 08](#Szabo)).

## Definition

We state first the original and standard definition of $KK$-groups in terms of [[equivalence classes]] of [[Fredholm module|Fredholm]]-[[Hilbert C*-module|Hilbert C*]]-[[Hilbert bimodules|bimodules]] in

* [In terms of Fredholm Hilbert C-star-bimodules](#InTermsOfFredholmHilbertBimodules)

Then we state the abstract [[category theory|category-theoretic]] characterization by [[localization]] in

* _[Universal category-theoretic characterization](#UniversalCharacterization).

An equivalent and explicity [[homotopy theory|homotopy theoretic]] characterization akin to that of the standard [[homotopy category]] [[Ho(Top)]] is in

* [In terms of homotopy classes of star-homomorphisms](#RelationToHomotopyClassesOfStarHomomorphisms).

### In terms of Fredholm-Hilbert $C^\ast$-bimodules
 {#InTermsOfFredholmHilbertBimodules}

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

In all of the following, "$C^\ast$-algebra" means _[[separable topological space|separable]]_ [[C*-algebra]]. We write [[C*Alg]] for
for the [[category]] whose [[objects]] are separable $C^\ast$-algebras and whose [[morphisms]] are $\ast$-[[homomorphisms]] between these.

=--

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

We write

* $\mathcal{B} \coloneqq \mathcal{B}(H)$ for the $C^\ast$-algebra of [[bounded operators]] on a complex, infinite-dimensional separable [[Hilbert space]];

* $\mathcal{K} \coloneqq \mathcal{K}(H) \hookrightarrow \mathcal{B}(H)$ for the [[compact operators]].

=--

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

For $B \in $ [[C*Alg]], a [[Hilbert C*-module]] over $B$ is

1. a [[complex numbers|complex]] [[vector space]] $\mathcal{H}$;

1. equipped with a [[C*-representation]] of $B$ from the right;

1. equipped with a [[sesquilinear map]] (linear in the second argument)

   $$
     \langle -,-\rangle \colon \mathcal{H} \times \mathcal{H} \to B
   $$

   (the $B$-valued [[inner product]])

such that

1. $\langle -,-\rangle$ behaves indeed like a positive definitine inner product over $B$:

   1. $\langle x,y\rangle^\ast = \langle y,x\rangle$

   1. $\langle x,x\rangle \geq 0$ (in the sense of [[positive elements]] in $B$)

   1. $\langle x,x\rangle = 0$ precisely if $x = 0$;

   1. $\langle x,y \cdot b\rangle = \langle x,y \rangle \cdot b $

1. $H$ is [[complete space|complete]] with respect to the [[norm]]:

   ${\Vert x \Vert_H} \coloneqq {\Vert \langle x,x\rangle\Vert_B}$.

=--

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

For $A,B \in C^\ast Alg$ an $(A,B)$-[[Hilbert C*-bimodule]] is an $B$-[[Hilbert C*-module]], def. \ref{HilbertCStarModule} $(\mathcal{H}, \langle \rangle)$ equipped with a [[C-star representation]] of $A$ from the left such that all $a \in A$ are "adjointable" in the $B$-valued inner product, meaning that

$$
  \langle a^\ast \cdot x,y\rangle = \langle x, a y\rangle
  \,.
$$

=--

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

For $A, B \in $ [[C*Alg]], **Kasparov $(A,B)$-bimodule**
is a $\mathbb{Z}_2$-[[graded vector space|graded]] $(A,B)$-[[Hilbert bimodules]] $\mathcal{H}, \langle -,-\rangle$, def. \ref{HilbertBimodule}, equipped with an adjointable odd-graded [[bounded operator]] $F \in \mathcal{B}_B(\mathcal{H})$ such that

1. $(F^2 - 1)\pi(a) \in \mathcal{K}_B(\mathcal{H})$

1. $[F, \pi(a)] \in \mathcal{K}_B(\mathcal{H})$

1. $(F - F^\ast) \pi(a)\in \mathcal{K}_B(\mathcal{H})$

for all $a \in A$,

hence such that $F$ squares to the identity, commutes with multiplication operators and is self-adjoint _up to_ [[compact operators]].

=--

For instance ([Blackadar 99, p. 144](#Blackadar99)).

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

For $B = \mathbb{C}$ a Kasparov $(A,B)$-bimodule is equivalently an $A$-[[Fredholm module]] for an essentially self-adjoint [[Fredholm operator]]

=--

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

A [[homotopy]] between two Kasparov $(A,B)$-bimodules is an $(A, C([0,1],B))$-bimodule which interpolates between the two.

(...)

=--

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

Writes $KK(A,B)$ for the set of [[equivalence classes]] of Kasparov $(A,B)$-bimodules under homotopy, def. \ref{HomotopyOfKasparovBimodules}.

=--

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

$KK(A,B)$ is naturally an [[abelian group]] under [[direct sum]] of bimodules and operators.

=--

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

There is a composition operation

$$
  KK(A,B) \times KK(B,C) \to KK(A,C)
$$

such that (...). This is called the **Kasparov product**.

=--

A streamlined version of the definition of the Kasparov product is in ([Skandalis 84](#Skandalis84)).

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

From the point of view of [[E-theory]] the Kasparov product is equivalently just the composition of homotopy classes of completely positive [[asymptotic C*-homomorphisms]]. See at _[[E-theory]]_ for more on this.

=--

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

On the other hand, at least between $C^\ast$-algebras which are [[algebras of functions]] on [[smooth manifolds]]  $A_i = C(X_i)$ , KK-classes are presented by [[correspondences]] $X_1 \leftarrow Z \to X_2$ and the Kasparov product is given just by the [[fiber product]]-composition operation on correspondences ([Connes-Skandalis 84, theorem 3.2](#ConnesSkandalis84), [Block-Weinberger 99, section 3](#BlockWeinberger99)).

=--

### Universal category-theoretic characterization
 {#UniversalCharacterization}

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

The Kasparov product, def. \ref{KasparovProduct}, is [[associativity|associative]]. Thus under the Kasparov product

$$
  KK(-,-) \;\colon\; C^\ast Alg \times C^\ast Alg \to C^\ast Alg
$$

is the [[hom-functor]] of an [[additive category]].

=--

([Higson 87, theorem 4.1](#Higson))

The category $KK$ is a kind of [[localization]] of the category of [[C-star-algebras]]:

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

The canonical functor

$$
  Q \colon C^\ast Alg \to KK
$$

exhibits $KK$ as the [[universal property|universal category]] receiving a functor from [[C*-algebras]] such that

1. $KK$ is an [[additive category]];

1. $Q$ is homotopy-invariant;

1. $Q$ [[isomorphism|inverts]] the [[tensor product]] with the [[C*-algebra]] of [[compact operators]]

   (for all $C^\ast$-homomorphisms of the form $id \otimes e \langle e,- \rangle \;\colon A\; \to A \otimes \mathcal{K}$ the morphism $Q(id \otimes e \langle e)$ is an [[isomorphism]]).

1. $Q$ preserves [[split short exact sequences]].

=--

This is due to ([Higson 87, theorem 4.5](#Higson)). The generalization to the equivariant case is due to ([Thomsen 98](#Thomsen98)).

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

The localization conditions here are analogous to those that define the localization of [[stable (∞,1)-categories]] to [[noncommutative motives]] See there for more and see the references [below](#AsHomotopyOfStableInfinityCategoryReferences).

=--

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

The minimal [[tensor product of C-star-algebras]]

$$
  \otimes \colon C^\ast Alg \times C^\ast Alg \to C^\ast Alg
$$

extends uniquely to a [[tensor product]] $\otimes_{KK}$ on $KK$ such that there is a [[commuting diagram]] of [[functors]]

$$
  \array{
    C^\ast Alg \times C^\ast Alg &\stackrel{Q \times Q}{\to}& KK
    \\
    \downarrow^{\mathrlap{\otimes}} && \downarrow^{\mathrlap{\otimes_{KK}}}
    \\
    C^\ast Alg &\stackrel{Q}{\to}& KK
  }
  \,.
$$

=--

([Higson 87, theorem 4.8](#Higson))

For more discussion of more explicit presentations of this [[localization]] process for obtaining KK-theory see at _[[homotopical structure on C*-algebras]]_ and also at _[[model structure on operator algebras]]_.

### In terms of homotopy-classes of $\ast$-homomorphisms
 {#RelationToHomotopyClassesOfStarHomomorphisms}

Theorem (Cuntz)

If $A,B$ are [[C-star-algebras]] with $A$ separable and $B$ $\sigma$-unital, then

$$
  KK(A,B) \simeq [q A, B \otimes \mathcal{K}]
  \,,
$$

where

* $q A$ is the [[kernel]] of the [[codiagonal]] $A \star A \to A$,

* $\mathcal{K}$ is the $C^\ast$-algebra of [[compact operators]].

* $[-,-]$ is the set of [[homotopy]] [[equivalence classes]] of $\ast$-[[homomorphisms]].

(reviewed in ([Joachim-Johnson07](#JoachimJohnson07))).

### In terms of correspondences/spans of groupoids
 {#InTermsOfCorrespondences}

At least to some extent, KK-classes between [[C*-algebras]] of [[continuous functions]] on manifolds/spaces, and maybe more generally between [[groupoid convolution algebras]] can be represented by certain [[equivalence classes]] of [[spans]]/[[correspondences]]

$$
  X \leftarrow (Z,E) \to Y
$$

of such [[spaces]].

See the corresponding [references below](#ReferencesInTermsOfCorrespondences).

Such a description by abelianizations of [[correspondences]] is reminiscent of similar constructions of [[motivic cohomology]]. See [below](#AsAnAnalogOfMotives). For more on this see also the pointers at at _[[motivic quantization]]_.

(...)

* category of equivariant correspondences equipped with cocycle: $\hat F_{\mathcal{G}}^\ast$ (theorem 2.26);

* specifically for K-theory cocycles: $\widehat {KK}_{\mathcal{G}}^\ast$  (section 4, page 27)

* pull-push from correspondences to KK in proof of theorem 4.2, bottom of p. 27

(...)

### As an analog of motives in noncommutative topology
 {#AsAnAnalogOfMotives}

To some extent [[KK-theory]]/[[E-theory]] look like an analogue in [[noncommutative topology]] of what in [[algebraic geometry]] is the category of [[motives]]. ([Connes-Consani-Marcolli 05](#ConnesConsaniMarcolli05)). ([Meyer 06](#Meyer06)).

Specifically the characterization in terms of spans/correspondences [above](#InTermsOfCorrespondences) is reminiscent to the definition of [[pure motives]], see the rferences below: _[In terms of correspondences](#ReferencesInTermsOfCorrespondences). A relation between bivariant [[algebraic K-theory]] and [[motivic cohomology]] is discussed in ([Garkusha-Panin 11](#GarkushaPanin11)).

A universal [[functor]] from KK-theory to [[noncommutative motives]]

$$
  KK \longrightarrow NCC_{dg}
$$

was given in ([Mahanta 13](#Mahanta13)). This sends a [[C*-algebra]] to the [[dg-category]] of [[perfect complexes]] over (the [[unitalization]] of) its underlying [[associative algebra]].

### Equivariant KK-theory
 {#EquivariantKKTheory}

Pretty much all of KK-theory has a generalization to [[equivariant cohomology]] where all algebras and modules are equipped with [[actions]] of a given [[topological group]] or more generally [[topological groupoid]] $\mathcal{G}$, and all operators are suitably [[invariant|invariant]]/[[equivariance|equivariant]] under this action. See at _[[equivariant KK-theory]]_ for more.

The [[Baum-Connes conjecture]] and the [[Green-Julg theorem]] assert that under some conditions $\mathcal{G}$-equivariant KK-theory is equivalent to the plain KK-theory of the [[groupoid convolution algebras]] of the corresponding [[action groupoids]]. See at _[[Green-Julg theorem]]_ for details.

## Examples
 {#Examples}

### Basic examples

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

For $f \colon A \to B$ a [[homomorphism]] of $\mathbb{Z}_2$ [[graded C*-algebras]], take $B$ as a right [[Hilbert module]] over itself and equip it with the left [[action]] of $A$ induced by $f$. This makes it a [[Hilbert bimodule]]. Together with the 0-[[Fredholm operator]], this represents an element

$$
  (B, f, 0) \in KK(A,B)
  \,.
$$

=--

For instance ([Blackadar 99, example 17.1.2 a)](#Blackadar99)).

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

For

$$
  (H_i, F_i) \in KK(A,B)
$$

a [[Fredholm module|Fredholm]] $(A_i,B)$-[[Hilbert bimodule]] for $i \in \{1,2\}$, the [[direct sum]] is

$$
  (H_1 \oplus H_2, F_1 \oplus F_2) \in KK(A_1\oplus A_2, B)
  \,.
$$

=--

For instance ([Blackadar 99, example 17.1.2 c)](#Blackadar99)).

### The archetypical examples

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

Let $(X,g)$ be a [[closed manifold|closed]] [[smooth manifold|smooth]] [[Riemannian manifold]], and let $V_0, V_1$ be two smooth [[vector bundles]] over $X$ with Hermitian strucure ([[associated bundle|associated]] to a chosen [[unitary group]]-[[principal bundle]]).

Then given an [[elliptic operator|elliptic]] [[pseudodifferential operator]]

$$
  P \colon \Gamma(V_0) \to \Gamma(V_1)
$$

on smooth [[sections]] it extends to an essentially [[unitary operator|unitary]] [[Fredholm operator]] on [[square integrable function|square integrable]] sections $L^2(V_i)$.

Consider then the $\mathbb{Z}_2$-graded [[Hilbert space]]

$$
  H \coloneqq L^2(V_0) \oplus L^2(V_1)
$$

equipped with the evident action of $C(X)$ (by "[[multiplication operators]]"). Then with $P$ a [[parametrix]] for $Q$, the
operator

$$
  F \coloneqq
  \left[
    \array{
       0 & Q
       \\
       P & 0
    }
  \right]
$$

is a [[Fredholm operator]] on $H$, so that

$$
  \left(
    L^2(V_1) \oplus L^2(V_2),
  \left[
    \array{
       0 & Q
       \\
       P & 0
    }
  \right]
  \right)
  \in
  KK(C(X),\mathbb{C})
  \,.
$$

=--

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

Let $(X,g)$ be an [[almost complex manifold]]
and let $D \colon \overline{\partial} + \overline{\partial}^\ast$ be the [[Dolbeault-Dirac operator]]. This extends to an operator on

$$
  H \coloneqq L^2(\Omega^{0,\bullet})
$$

and

$$
  F \coloneqq \frac{D}{\sqrt{1 + D^2}}
$$

(defined by [[functional calculus]]) is then a [[Fredholm operator]] on that.  Then

$$
  \left(
    L^2(\Omega^{0,\bullet}),
    \frac{\overline{\partial} + \overline{\partial}^\ast}{\sqrt{1+ (\overline{\partial} + \overline{\partial}^\ast)^2}}
  \right)
  \in
  KK(C(X), \mathbb{C})
  \,.
$$

=--

## Properties

### Relation to operator K-cohomology, K-homology, twisted K-theory
 {#RelationToKCohomologyAndTwistedKTheory}

KK-theory is a joint generalization of [[operator K-theory]], hence also of [[topological K-theory]], as well as of [[K-homology]] and of [[twisted K-theory]].

For $A \in $ [[C*Alg]] we have that

* $KK(\mathbb{C}, A) \simeq K_0(A)$

is the [[operator K-theory]] group of $A$ in degree 0 and

* $KK(C(\mathbb{R}^1),A) \simeq K_1(A)$

is the [[operator K-theory]] group of $A$ in degree 1. (e.g. ([Introduction, p. 20](#Introduction)). If here $A = C(X)$ is the [[C*-algebra]] of functions on a suitable [[topological space]] $X$, then this is the [[topological K-theory]] of that space

* $KK(\mathbb{C}, C(X)) \simeq K^0(X)$

* $KK(C(\mathbb{R}), C(X)) \simeq K^1(X)$.

More generally, if $A = C_r(\mathcal{G}_\bullet)$ is the reduced [[groupoid convolution algebra]] of a [[Lie groupoid]], then

* $KK(\mathbb{C}, C_r(\mathcal{G}_\bullet)) \simeq K^0(\mathcal{G})$

is the K-theory of the corresponding [[differentiable stack]]. If moreover $c \colon \mathcal{G} \to \mathbf{B}^2 U(1)$ is a [[circle 2-group]]-[[principal 2-bundle]] ($U(1)$-[[bundle gerbe]]) over $\mathcal{X}$ and if $A = C(\mathcal{X}_\bullet, c)$ is the  [[twisted groupoid convolution algebra]] of the corresponding [[centrally extended Lie groupoid]], then

* $KK(\mathbb{C}, C_r(\mathcal{X}_\bullet,x)) = K^0(\mathcal{X}, c)$

is the corresponding [[twisted K-theory]] ([Tu, Xu, Laurent-Gengoux 03](#TXLG)).

On the other hand, with $A$ in the first argument and the complex numbers in the second we have that

* $K(A,\mathbb{C}) \simeq K^0(A)$

ar equivalence classes of $A$-[[Fredholm modules]] and hence the [[K-homology]] of $A$.

(...)

### Relation to extensions

There is an [[isomorphism]]

$KK(A,B) \simeq Ext^1(A,B)$

to a suitable group of suitable [[extensions]] of $A$ by $B$. ([Kasparov 80](#Kasparov80), reviewed in [Inassaridze](#Inassaridze)).

### Triangulated structure and $KU$-module structure
 {#TriangulatedAndSpectrumEnrichedStructure}

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

$KK$ is naturally a stable [[triangulated category]].

=--

([Meyer 07](#Meyer), [Uuye 10, theorem 2.29](#Uuye10)).

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

There is a [[functor]]

$$
  \mathbb{K}(-) \;\colon\; C^\ast Alg \to Ho(Spectra)
$$

to the [[stable homotopy category]] such that

1. $\pi_n(\mathbb{K})(A) \simeq K_n(A)$, for all $A \in C^\ast Alg$, (hence the spectrum is a cohomology spectrum for the [[operator K-theory]] of $A$);

1. $\mathbb{K}(\mathbb{C})$ is naturally a [[ring spectrum]];

1. $\mathbb{K}(A)$ is naturally a [[symmetric spectrum|symmetric]] $\mathbb{K}(\mathbb{C})$-[[module spectrum]]

1. $\mathbb{K}$ lifts to a [[lax monoidal functor]]

   $$
     \mathbb{K} \;\colon\; C^\ast Alg \to Ho(\mathbb{K}(\mathbb{C}) Mod)
   $$

   to the [[homotopy category]] of [[module spectra]], and this in turn extends to a [[lax monoidal functor]] on the KK-category

   $$
     \mathbb{K} \;\colon\; KK \to Ho(\mathbb{K}(\mathbb{C}) Mod)
     \,.
   $$

1. $\mathbb{K}$ restricts to a [[fully faithful functor]] on the [[thick subcategory]] of the [[triangulated category]] $KK$ generated by the [[tensor unit]] (the "[[bootstrap category]]").

=--

This is the main result of ([DEKM 11, section 3](#DEKM11)).

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

Since $\mathbb{K}$ is a [[lax monoidal functor]] in particular it preserves [[dual objects]] and [[dual morphisms]], hence [[Poincaré duality algebras]] and their [[Umkehr maps]].

=--

### Künneth theorem

The [[thick subcategory]] of the [[triangulated category]] $KK$ generated from the [[tensor unit]] is called the _[[bootstrap category]]_ $Boot \hookrightarrow KK$. For $A \in Boot \hookrightarrow KK$ one has that $KK(A,B)$ satisfies a [[Künneth theorem]]. See at _[[bootstrap category]]_ for more.

### Excision and relation to E-theory

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

Given a [[short exact sequence]] of [[C*-algebras]] one says that $KK$ satisfies **[[excision]]** or that it is **excisive** for this sequence if it preserves its [[exact sequence|exactness]] in the middle.

=--

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

By theorem \ref{KKIsAdditiveSplitExactLocalizationAtCompacts}, $KK$ is excisive over [[split exact sequences]].

=--

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

$KK$ is excisive for [[nuclear C*-algebras]] in the first argument.

=--

This is discussed ([Kasparov 80, section 7](#Kasparov80)), ([Cuntz-Skandalis 86](#CuntzSkandalis86)).

More generally:

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

$KK$ is excisive for [[K-nuclear C*-algebras]] in the first argument.

=--

([Skandalis 88](#Skandalis88))

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

It is not expected that excision is satisfied fully generally by $KK$. Instead, the [[universal property|universal]] improvement of $KK$-theory under excision can be constructed. This is called _[[E-theory]]_. See there for more.

=--

### Poincaré duality and Thom isomorphism
 {#PoincareDualityAndThomIsomorphism}

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

A [[C*-algebra]] is a [[Poincaré duality algebra]]
if it is a [[dualizable object]] in the  [[symmetric monoidal category]] $KK$ with dual its [[opposite algebra]].

=--

([Brodzki-Mathai-Rosenberg-Szabo 07, def. 2.1](#BrodzkiMathaiRosenbergSzabo07))

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

Let $X$ be a [[smooth manifold]] which is [[compact topological space|compact]]. Then the [[C*-algebra]] $C(X) \otimes C_0(T^\ast X)$
(the tensor product of the algebra of functions of [[compact support]] on $X$ and on its [[cotangent bundle]]) is isomorphic, in $KK$, to $\mathbb{C}$:

$$
  d \colon C(X) \otimes C_0(T^\ast X) \stackrel{\simeq}{\to}
  \mathbb{C}
  \,.
$$

=--

([Kasparov 80](#Kasparov80))

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

For $X$ a [[compact topological space|compact]] [[smooth manifold]], there is a [[natural isomorphism]] ([[Thom isomorphism]])

$$
  K_0( C_0(T^\ast X))
    \simeq
  KK(\mathbb{C}, C_0(T^\ast X))
    \stackrel{KK(C,(-)\otimes C(X))}{\to}
  KK(C(X), C(X) \otimes C_0(T^\ast X) )
    \underoverset{\simeq}{KK(C(X), d)}{\to}
  KK(C(X), \mathbb{C} )
  \,.
$$

=--

For more discussion see at _[[Poincaré duality algebra]]_.

### Push-forward in KK-theory
 {#UmkehrMap}

[[Umkehr map]] in KK-theory ([Brodzki-Mathai-Rosenberg-Szabo 07, section 3.3](#BrodzkiMathaiRosenbergSzabo07))

If $A$, $B$ are [[Poincaré duality algebras]],
def. \ref{PoincareDualityAlgebra}, then for $f \colon A \to B$ a morphism, the corresponding [[Umkehr map]] is (postcomposition) with the [[dual morphism]] of its [[opposite algebra]] version:

$$
  f! \coloneqq (f^op)^\ast
  \,.
$$

([Brodzki-Mathai-Rosenberg-Szabo 07, p. 14](#BrodzkiMathaiRosenbergSzabo07))

For more and a discussion of [[twisted Umkehr maps]] see at _[[Poincaré duality algebra]]_ and at _[[Freed-Witten-Kapustin anomaly cancellation]]_.

## Further Theorems

* The [[Baum-Connes conjecture]] is naturally formulated within KK-theory.

* The [[Novikov conjecture]] has been verified in many cases using KK-theory. (see for instance [Rosenberg 80](#Rosenberg80)).

* The [[Atiyah-Singer index theorem]] is naturally formutaled in KK-theory/[[E-theory]]. (See ([Higson-Roe](#HigsonRoe))).

## Related concepts

* [[KK-bootstrap category]]

* [[equivariant KK-theory]]

* [[index theory]]

* [[bivariant cohomology theory]]

* [[bivariant algebraic K-theory]], [[K-motive]]

* [[operator K-theory]]

* [[E-theory]]

* [[homotopical structure on C*-algebras]], [[model structure on operator algebras]]

* [[assembly map]], [[Baum-Connes conjecture]]

[[!include noncommutative motives - table]]

## References

### General

KK-theory was introduced by [[Gennady Kasparov]] in

* {#Kasparov80} [[Gennady Kasparov]], *The operator $K$-functor and extensions of $C^{\ast}$-algebras*, Izv. Akad. Nauk SSSR Ser. Mat. __44__  (1980), no. 3, 571-636, 719, [MR81m:58075](http://www.ams.org/mathscinet-getitem?mr=582160), [Zbl](http://www.zentralblatt-math.org/zmath/search/?an=Zbl%200464.46054%7CZbl%200448.46051), [abstract](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=im&paperid=1739&option_lang=eng), [english doi](http://dx.doi.org/10.1070%2FIM1981v016n03ABEH001320), free Russian original: [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=im&paperid=1739&volume=44&year=1980&issue=3&fpage=571&what=fullt&option_lang=eng)

prompted by the advances in [[Brown-Douglas-Fillmore theory]], especially in the last 1977 article.

Some streamlining of the definitions appeared in

* {#Skandalis84} [[Georges Skandalis]], *Some remarks on Kasparov theory*, J. Funct. Anal. 59 (1984) 337-347.

A textbook account is in

* {#Blackadar99} [[Bruce Blackadar]], *[[K-Theory for Operator Algebras]]*, 2nd ed. Cambridge University Press, Cambridge (1999)

Introductions and surveys include

* [[Gennady Kasparov]], *Operator K-theory and its applications: elliptic operators, group representations, higher signatures $C^\ast$-extensions*, Proceedings ICM 1983 Warszawa, PWN-Elsevier (1984) 987-1000.

* [[Nigel Higson]], _A primer on KK-theory_. Proc. Sympos. Pure Math. 51, Part 1, 239–283. (1990) ([pdf](http://www.math.psu.edu/higson/math/Papers_files/Higson%20-%201990%20-%20A%20primer%20on%20KK-theory.pdf))

* [[Georges Skandalis]], _Kasparov's bivariant K-theory and applications_ Exposition. Math. 9, 193–250 (1991) ([pdf slides](http://www.math.vanderbilt.edu/~bisch/ncgoa08/talks/skandalis2.pdf))

* {#Introduction} *Introduction to KK-theory and E-theory*, Lecture  notes (Lisbon 2009) ([pdf slides](http://oaa.ist.utl.pt/files/cursos/courseD_Lecture4_KK_and_E1.pdf))

* [[Heath Emerson]], [[R. Meyer]] (notes taken by S. Hong), *KK-theory and Baum-Connes conjecture*, Lectures at _Summer school on operator algebras and noncommutative geometry_ (June 2010) ([pdf](http://www.pims.math.ca/files/EM_%20KK-theory_BC_Conjecture.pdf))

* {#Meyer06} [[R. Meyer]], *How analysis and topology interact in bivariant K-theory*, 2006 ([pdf](http://mate.dm.uba.ar/~gcorti/article.pdf))

### Excision

Excision for KK-theory is further studied in

* {#CuntzSkandalis86} [[Joachim Cuntz]], [[Georges Skandalis]], *Mapping cones and exact sequences in KK-theory*, J. Operator Theory 15 (1986) 163-180.

* {#Skandalis88} [[Georges Skandalis]], _Une notion de nuclearité en K-theorie_, K-Theory 1 (1988) 549-574.

### In Category theory and Homotopy theory

KK-theory is naturally understood in terms of [[universal properties]] in [[category theory]] and in [[homotopy theory]].

That $KK(A,B)$ is naturally thought of as a collection of "generalized [[homomorphisms]]" of $C^\ast$-algebras was amplified in

* [[Joachim Cuntz]], *Generalized Homomorphisms Between $C^\ast$-algebras and KK-theory*, Springer Lecture Notes in Mathematics, 1031 (1983), 31-45. doi:[10.1007/BFb0072109](https://doi.org/10.1007/BFb0072109)

* [[Joachim Cuntz]], _K-theory and C*-algebras,_ Springer Lecture Notes in Mathematics, 1046 (1984), 55-79.

That under the Kasparov product these are indeed the [[hom-objects]] in a [[category]] was first observed in

* [[Nigel Higson]], _A characterization of KK-theory_,  Pacific J. Math. Volume 126, Number 2 (1987), 253-276. ([EUCLID](http://projecteuclid.org/euclid.pjm/1102699804))

where moreover this category is realized as the [[universal property|universal]] additive and split exact "[[localization]]" of $C^\ast Alg$ at the $C^\ast$-algebra of [[compact operators]].

The generalization of this statement to [[equivariant cohomology|equivariant]] KK-theory is in

* {#Thomsen98} Klaus Thomsen, _The universal property of equivariant KK-theory_ ([K-theory preprint](http://www.math.uiuc.edu/K-theory/0249/))

Characterization of KK-theory as the [[satellites]] of a functor is in

* {#Inassaridze} [[Hvedri Inassaridze]], _Universal property of Kasparov bivariant K-theory_ ([[InassaridzeOnKK.pdf:file]], [ps](http://www.math.uni-bielefeld.de/~bak/kasp.ps))

A [[triangulated category]] structure for KK-theory is discussed in

* {#Meyer} [[Ralf Meyer]], _Categorical aspects of bivariant K-theory_, ([arXiv:math/0702145](http://arxiv.org/abs/math/0702145))

* [[Ralf Meyer]], [[Ryszard Nest]], _Homological algebra in bivariant K-theory and other triangulated categories_ ([arXiv:math/0702146](http://arxiv.org/abs/math/0702146))

* {#Meyer} [[Ralf Meyer]], _KK-theory as a triangulated category_, Lecture notes (2009) ([pdf](http://wwwmath.uni-muenster.de/u/echters/Focused-Semester/lecturenotes/Meyer_--_KK-theory_as_a_triangulated_category.pdf))

A [[model category]] realization of KK-theory is discussed in

* {#JoachimJohnson07} [[Michael Joachim]], [[Mark Johnson]], _Realizing Kasparov's KK-theory groups as the homotopy classes of maps of a Quillen model category_ ([arXiv:0705.1971](http://arxiv.org/abs/0705.1971))

A [[category of fibrant objects]]-structure on [[C*Alg]] which unifies the above homotopical pictures is discussed in

* {#Uuye10} [[Otgonbayar Uuye]], _Homotopy theory for $C^\ast$-algebras_ ([arXiv:1011.2926](http://arxiv.org/abs/1011.2926))

More on this is at _[[homotopical structure on C*-algebras]]_.

Further discussion in the context of [[stable homotopy theory]] and [[E-theory]] is in

* Martin Grensing, _Noncommutative stable homotopy theory_ ([arXiv:1302.4751](http://arxiv.org/abs/1302.4751))

* {#Mahanta11} [[Snigdhayan Mahanta]], _Higher nonunital Quillen $K'$-theory, KK-dualities and applications to topological $\mathbb{T}$-duality_, Journal of Geometry and Physics, Volume 61, Issue 5 2011, p. 875-889.   ([pdf](http://wwwmath.uni-muenster.de/u/snigdhayan.mahanta/papers/KQ.pdf))

Refinement of [[operator K-theory]] to cohomology [[spectra]] is discussed in

* {#BunkeJoachimStolz03} [[Ulrich Bunke]], [[Michael Joachim]], [[Stephan Stolz]], _Classifying spaces and spectra representing the K-theory of a graded $C^\ast$-algebra_, High-dimensional manifold topology, World Sci. Publ., River Edge, NJ, 2003, pp. 80–102

This construction is functorial (only) for _essential_ $\ast$-homomorphisms of [[C*-algebras]].

### As the homotopy category of a stable $\infty$-category
 {#AsHomotopyOfStableInfinityCategoryReferences}

A refinement of the KK-category to a [[spectrum]]-[[enriched category]] ($\sim$ [[stable (∞,1)-category]]) is claimed in

* {#JoachimStolz09} [[Michael Joachim]], [[Stephan Stolz]], _An enrichment of $KK$-theory over the category of symmetric spectra_ Münster J. of Math. 2 (2009), 143–182 ([pdf](http://www3.nd.edu/~stolz/KKenrich.pdf))

and the generalization of this to [[equivariant K-theory]] over geometrically discrete groupoids is discussed in

* {#Mitchener07} [[Paul Mitchener]], _$KK$-theory spectra for $C^\ast$-categories and discrete groupoid $C^\ast$-algebras_ ([arXiv:0711.2152](http://arxiv.org/abs/0711.2152))

but this construction is stated to be mistaken on p. 3 of

* {#DEKM11} [[Ivo Dell'Ambrogio]], [[Heath Emerson]], [[Tamaz Kandelaki]], [[Ralf Meyer]], _A functorial equivariant K-theory spectrum and an equivariant Lefschetz formula_ ([arXiv:1104.3441](http://arxiv.org/abs/1104.3441))

This article in turn considers a variant of the construction in ([Bunke-Joachim-Stolz 03](#BunkeJoachimStolz03)) which gives operator K-theory spectra that are functorial for general $\ast$-homomorphisms.

Observations relating to a genuine [[stable (∞,1)-category]] structure maybe at least of [[E-theory]] are in

* {#Mahanta12} [[Snigdhayan Mahanta]], _Noncommutative stable homotopy and semigroup $C^*$-algebras_,  ESI preprint 2394 ([arXiv:1211.6576](http://arxiv.org/abs/1211.6576))

A complete realization of ([[equivariant K-theory|equivariant]], even) KK-theory as the [[homotopy category of an (infinity,1)-category|homotopy category]] of a [[stable (infinity,1)-category|stable $(\infty,1)$-category]]:

* {#BunkeEngelLand21} [[Ulrich Bunke]], [[Alexander Engel]], [[Markus Land]], *A stable $\infty$-category for equivariant KK-theory* $[$[arXiv:2102.13372](https://arxiv.org/abs/2102.13372)$]$

### In the context of the Novikov conjecture

* {#Rosenberg80} [[Jonathan Rosenberg]], _Group C*-algebras and Topological Invariants_ , Proc. Conf. in Neptun, Romania, 1980, Pitman (London, 1985)

### In the context of the Atiyah-Singer index theorem

The classical [[Atiyah-Singer index theorem]] is reviewed in [[operator K-theory]] (with some hints towards KK-theory) in

* {#HigsonRoe} [[Nigel Higson]], [[John Roe]], _Lectures on operator K-theory and the Atiyah-Singer Index Theorem_ (2004) ([pdf](http://folk.uio.no/rognes/higson/Book.pdf))

Generalization to the relative case in [[KK-theory]], hence for indices of fiberwise [[elliptic operators]] on [[Hilbert C*-module]]-[[fiber bundles]] is in

* Jody Trout, _Asymptotic Morphisms and Elliptic Operators over $C^\ast$-algebras_, K-theory, 18 (1999) 277-315 ([arXiv:math/9906098](http://arxiv.org/abs/math/9906098))

### For convolution algebras and In geometric quantization

Discussion of KK-theory with an eye towards [[representation of a C-star algebra|C-star representations]] of [[groupoid convolution

algebras]] in the context of [[geometric quantization]] [[geometric quantization by push-forward|by push-forward]] is in

* [[Klaas Landsman]], _Quantization as a functor_ ([arXiv:math-ph/0107023](http://arxiv.org/abs/math-ph/0107023))

* {#Landsman03} [[Klaas Landsman]], _Functorial quantization and the Guillemin-Sternberg conjecture_ ,  Proc. Bialowieza 2002 ([arXiv:math-ph/0307059](http://arxiv.org/abs/math-ph/0307059))

* {#Bos07} [[Rogier Bos]], _Groupoids in geometric quantization_ PhD Thesis (2007) ([pdf](http://www.math.ist.utl.pt/~rbos/ProefschriftA4.pdf))

with a summary/exposition in

* {#Landsman10} [[Klaas Landsman]], _Functoriality of quantization: a KK-theoretic approach_, talk at ECOAS, Dartmouth College, October 2010 ([web](http://www.academia.edu/1992202/Functoriality_of_quantization_a_KK-theoretic_approach))

See also the related references at _[[Guillemin-Sternberg geometric quantization conjecture]]_.

The KK-theory of twisted convolution algebras and its relation to [[twisted K-theory]] of [[differentiable stacks]] is discussed in

* {#TXLG} [[Jean-Louis Tu]], [[Ping Xu]], [[Camille Laurent-Gengoux]], _Twisted K-theory of differentiable stacks_ ([arXiv:math/0306138](http://arxiv.org/abs/math/0306138))

Discussion of groupoid 1-[[cocycles]] and their effect on the [[groupoid algebra]] KK-theory is discussed in

* Bram Mesland, _Groupoid cocycles and K-theory_ ([arXiv:1005.3677](http://arxiv.org/abs/1005.3677))

### In terms of correspondences/spans
 {#ReferencesInTermsOfCorrespondences}

#### For plain KK-theory

KK-classes between [[algebras of functions]] on [[smooth manifolds]] are described in terms of [[equivalence classes]] of [[correspondence]] manifolds carrying a [[vector bundle]] in section 3 of

* {#ConnesSkandalis84} [[Alain Connes]], [[Georges Skandalis]], *The longitudinal index theorem for foliations*, Publ. Res. Inst. Math. Sci. **20** 6, 1139-1183 (1984) ([pdf](http://www.alainconnes.org/docs/longitudinal.pdf))

This generalizes the [[Baum-Douglas geometric cycles]] from [[K-homology]] to KK-theory.

A further generalization of this, where one algebra $C(Y)$ is generalized to
$C(Y) \otimes A$ for $A$ a unital separable $C^\ast$-algebra, is in section 3 of:

* {#BlockWeinberger99} [[Jonathan Block]], [[Shmuel Weinberger]], *Arithmetic manifolds of positive scalar curvature*, J. Diff. Geom. 52, no. 2, 375-406 (1999). ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.30.297))

In section 5 of

* {#BrodzkiMathaiRosenbergSzabo07} [[Jacek Brodzki]], [[Varghese Mathai]], [[Jonathan Rosenberg]], [[Richard Szabo]], _Noncommutative correspondences, duality and D-branes in bivariant K-theory_, Adv. Theor. Math. Phys.13:497-552,2009 ([arXiv:0708.2648](http://arxiv.org/abs/0708.2648))

this is reviewed and then a characterization in terms of [[co-spans]] of [[C*-algebras]] is given. This version is effectively a restatement of [the characterization by Cuntz](#RelationToHomotopyClassesOfStarHomomorphisms) as reproduced in ([Blackadar 99, corollary 17.8.4](#Blackadar99)).

For similar structures see also at _[[motive]]_ in the section _[Relation to bivariant K-theory](motive#RelationToKKTheory)_.

A 'generators and relations' description of KK-theory in terms of spans is given in

* Bernhard Burgstaller, _The generators and relations picture of KK-theory_, (2016) arXiv:[1602.03034](http://arxiv.org/abs/1602.03034)

#### For equivariant KK-theory
 {#ReferencesCorrespondencesEquivariant}

Generalization of such [[correspondence]]-presentation to [[equivariant KK-theory]] (and hence, by the [[Green-Julg theorem]] essentially to KK-theory of [[groupoid algebras]] of [[action groupoids]] of [[compact topological groups]]) -- was introduced in

* [[Heath Emerson]], [[Ralf Meyer]], _Bivariant K-theory via correspondences_, Adv. Math. 225 (2010), 2883-2919 ([arXiv:0812.4949](http://arxiv.org/abs/0812.4949))

based on

* {#EmersonMeyer09} [[Heath Emerson]], [[Ralf Meyer]] _Equivariant embedding theorems and topological index maps_, Adv. Math. 225 (2010), 2840-2882 ([arXiv:0908.1465](http://arxiv.org/abs/0908.1465))

* [[Heath Emerson]], [[Ralf Meyer]], _Dualities in equivariant Kasparov theory_ ([arXiv:0711.0025](http://arxiv.org/abs/0711.0025))

based on technical aspects of the construction of pushforward along and comoposition of equivariant correspondences in

* [[Paul Baum]], [[Jonathan Block]], _Equivariant bicycles on singular spaces_. C.R. Acad. Sci. Paris, t. 311 Serie I, 1990 ([pdf](http://www.math.upenn.edu/~blockj/papers/BlockEquivariantbi.pdf))

* [[Heath Emerson]], [[Ralf Meyer]], *Equivariant embedding theorems and topological index maps*, Adv. Math. 225 (2010), 2840-2882 ([arXiv:0908.1465](http://arxiv.org/abs/0908.1465))

Further developments of this are in

* [[Heath Emerson]], *Duality, correspondences and the Lefschetz map in equivariant KK-theory: a survey* ([arXiv:0904.4744](http://arxiv.org/abs/0904.4744))

* [[Heath Emerson]], Robert Yuncken, _Equivariant correspondences and the Borel-Bott-Weil theorem_ ([arXiv:0905.1153](http://arxiv.org/abs/0905.1153))

### Relation to motives and algebraic KK-theory
 {#RelationToMotives}

The general analogy between KK-cocycles and [[motives]] is noted explicitly in

* {#ConnesConsaniMarcolli05} [[Alain Connes]], Caterina Consani, [[Matilde Marcolli]], *Noncommutative geometry and motives: the thermodynamics of endomotives* ([arXiv:math/0512138](http://arxiv.org/abs/math/0512138))

* [[Alain Connes]], [[Matilde Marcolli]], *[[Noncommutative Geometry, Quantum Fields and Motives]]*

and also very briefly in ([Meyer 06](#Meyer06)).

A relation between [[motivic cohomology]] and bivariant [[algebraic K-theory]] is discussed in

* [[Guillermo Cortiñas]], [[Andreas Thom]], *Bivariant algebraic K-theory*, J. Reine Angew. Math. 510 (2007), 71–124. ([arXiv:math/0603531](http://arxiv.org/abs/math/0603531))

* {#Mahanta09} [[Snigdahayan Mahanta]], *Noncommutative correspondence categories, simplicial sets and pro $C^\ast$-algebras* ([arXiv:0906.5400](http://arxiv.org/abs/0906.5400))

* {#Mahanta11} [[Snigdahayan Mahanta]], *Higher nonunital Quillen $K'$-theory, KK-dualities and applications to topological $\mathbb{T}$-duality*, Journal of Geometry and Physics, Volume 61, Issue 5 2011, p. 875-889.   ([pdf](http://wwwmath.uni-muenster.de/u/snigdhayan.mahanta/papers/KQ.pdf))

* {#GarkushaPanin11} [[Grigory Garkusha]], Ivan Panin, _K-motives of algebraic varieties_ ([arXiv:1108.0375](http://arxiv.org/abs/1108.0375))

* [[Grigory Garkusha]], _Algebraic Kasparov K-theory. II_ ([arXiv:1206.0178](http://arxiv.org/abs/1206.0178))

For a collection of literature see also paragraph 1.5 in

* [[Andrew Blumberg]], [[David Gepner]], [[Gonçalo Tabuada]], _A universal characterization of higher algebraic K-theory_ ([arXiv:1001.2282](http://arxiv.org/abs/1001.2282))

(in the context of [[noncommutative motives]]).

In

* {#Mahanta13} [[Snigdhayan Mahanta]], _Higher nonunital Quillen $K'$-theory, KK-dualities, and applications to topological T-duality_,  J. Geom. Phys., 61 (5), 875-889, 2011 ([pdf](http://wwwmath.uni-muenster.de/u/snigdhayan.mahanta/papers/KQ.pdf), [talk notes](http://wwwmath.uni-muenster.de/u/snigdhayan.mahanta/papers/KKTD.pdf))

it is shown that there is a universal functor $KK \longrightarrow NCC_{dg}$ from [[KK-theory]] to the category of [[noncommutative motives]], which is the category of [[dg-categories]] and dg-[[profunctors]] up to homotopy between them. This is given by sending a [[C*-algebra]] to the [[dg-category]] of [[perfect complexes]] of (the unitalization of) its underlying [[associative algebra]].

See also at _[[motivic quantization]]_ and _[[motives in physics]]_.

### In D-brane theory

KK-theory also describes [[RR-field]] [[charges]] and sources in [[D-brane]] theory.

A review is in

* {#Szabo} [[Richard Szabo]], _D-branes and bivariant K-theory_, Noncommutative Geometry and Physics 3 1 (2013): 131. ([arXiv:0809.3029](http://arxiv.org/abs/0809.3029))

based on

* [[Jacek Brodzki]], [[Varghese Mathai]], [[Jonathan Rosenberg]], [[Richard Szabo]],

  *D-Branes, RR-Fields and Duality on Noncommutative Manifolds*, Commun. Math. Phys. **277** (2008) 643-706 [[arXiv:hep-th/0607020](http://arxiv.org/abs/hep-th/0607020), [doi:10.1007/s00220-007-0396-y](https://doi.org/10.1007/s00220-007-0396-y)]

  _Noncommutative correspondences, duality and D-branes in bivariant K-theory_, Adv. Theor. Math. Phys.13:497-552,2009 ([arXiv:0708.2648](http://arxiv.org/abs/0708.2648))

  _D-branes, KK-theory and duality on noncommutative spaces_, J. Phys. Conf. Ser. 103:012004,2008 ([arXiv:0709.2128](http://arxiv.org/abs/0709.2128))

### Smooth refinement and spectral triples

Discussion of KK-theory for [[spectral triples]] is discussed in

* [[Bram Mesland]], _Spectral triples and KK-theory: A survey_ ([arXiv:1304.3802](http://arxiv.org/abs/1304.3802))

[[!redirects Kasparov K-theory]]
[[!redirects bivariant K-theory]]

[[!redirects KK-group]]
[[!redirects KK-groups]]

[[!redirects Kasparov KK-group]]
[[!redirects Kasparov KK-groups]]

[[!redirects Kasparov KK-groups]]


Klaus Dietz

* [InSpire page](https://inspirehep.net/authors/1049993)

## Selected writings

On the [[Nicolai map]]:

* [[Klaus Dietz]], [[Olaf Lechtenfeld]], *Nicolai maps and stochastic observables from a coupling constant flow*, Nuclear Physics B **255** (1985) 149-173 [doi:10.1016/0550-3213(85)90132-4]

* [[Klaus Dietz]], [[Olaf Lechtenfeld]], *Ghost-free quantisation of non-abelian gauge theories via the Nicolai transformation of their supersymmetric extensions*, Nuclear Physics B **259** 2–3 (1985) 397-411 [doi:10.1016/0550-3213(85)90642-X]

category: people


Klaus Fredenhagen

* [webpage](http://unith.desy.de/people/allmembers/fredenha/)

## Selected writings

On [[anyons]]:

* [[Klaus Fredenhagen]], Karl-Henning Rehren, [[Bert Schroer]], _Superselection sectors with braid group statistics and exchange algebras -- I: General theory_,   Comm. Math. Phys. Volume 125, Number 2 (1989), 201-226.  ([euclid:cmp/1104179464](http://projecteuclid.org/euclid.cmp/1104179464))

* [[Klaus Fredenhagen]], Karl-Henning Rehren, [[Bert Schroer]], _Superselection sectors with braid group statistics and exchange algebras -- II: Geometric aspects and conformal covariance_,  Reviews in Mathematical PhysicsVol. 04, No. spec01, pp. 113-157 (1992)  ([doi:10.1142/S0129055X92000170](https://doi.org/10.1142/S0129055X92000170) [pdf](ftp://ftp.theorie.physik.uni-goettingen.de/pub/papers/rehren/92/superselection_sectors_II_RMP.pdf))

*  [[Klaus Fredenhagen]], [[Matthias Gaberdiel]] and Stefan M. Rüger, *Scattering states of plektons (particles with braid group statistics) in 2+1 dimensional quantum field theory*, Communications in Mathematical Physics **175** (1996) 319–335 $[$[doi:10.1007/BF02102411](https://doi.org/10.1007/BF02102411)$]$

Examples of [[non-perturbative quantum field theory|non-perturbative]] [[interacting quantum field theory|interacting]] [[scalar field theory]] in _any_ [[spacetime]] [[dimension]] (in particular in $d \geq 4$):

* [[Detlev Buchholz]], [[Klaus Fredenhagen]], _A $C^\ast$-algebraic approach to interacting quantum field theories_,  Commun. Math. Phys. **377** (2020) 947–969  [[arXiv:1902.06062](https://arxiv.org/abs/1902.06062), [doi:10.1007/s00220-020-03700-9](https://doi.org/10.1007/s00220-020-03700-9)]

On the [[perturbative algebraic quantum field theory|locally covariant pAQFT approach]] to effective [[quantum gravity]] and applications to experiment:

* [[Romeo Brunetti]], [[Klaus Fredenhagen]], [[Kasia Rejzner]], *Locally covariant approach to effective quantum gravity*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2212.07800](https://arxiv.org/abs/2212.07800)]

## Related pages

* [[AQFT]], [[local net of observables]]

* [[AQFT on curved spacetimes]]

* [[locally covariant perturbative quantum field theory]]

category: people


Klaus Heiner Kamps

Klaus Heiner Kamps was a German mathematician who worked in Abstract Homotopy Theory.

He was the joint author, with [[T. Porter]], of the book [[Abstract Homotopy and Simple Homotopy Theory]].

[Maths Genealogy:](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=35747)

## Selected writings

### Books

On [[homotopy theory]]:

* {#Homotopietheorie} [[Tammo tom Dieck]], [[Klaus Heiner Kamps]], [[Dieter Puppe]], *Homotopietheorie*, Lecture Notes in Mathematics **157** Springer 1970 ([doi:10.1007/BFb0059721](https://link.springer.com/book/10.1007/BFb0059721))

On [[simplicial homotopy theory]]:

* K. H. Kamps and [[T. Porter]], _Abstract Homotopy and Simple Homotopy Theory_, World Scientific 1997 ([doi:10.1142/2215](https://doi.org/10.1142/2215), [GoogleBooks](http://books.google.de/books?id=7JYKxInRMdAC&dq=Porter+Kamps&printsec=frontcover&source=bl&ots=uuyl_tIjs4&sig=Lt8I92xQBZ4DNKVXD0x76WkcxCE&hl=de&sa=X&oi=book_result&resnum=3&ct=result#PPP1,M1))

### Dissertation

* [[K. H. Kamps]] (1968), _Faserungen und Cofaserungen in Kategorien mit Homotopiesystem_, Dissertation, Saarbrücken

### Articles

On [[Kan complexes]] and [[homotopy theory]]:

* [[K. H. Kamps]], _Kan-Bedingungen und abstrakte Homotopietheorie_, Math. Z. 124,1972, 215 - 236.

* [[K. H. Kamps]], _Zur Homotopietheorie von Gruppoiden_, Arch. Math., 23 (1972) 610-618.

On the [[Toda bracket]] understood via [[homotopy coherent diagram|homotopy-coherent]] [[pasting diagrams]] in a [[pointed (infinity,1)-category|pointed]] [[homotopy 2-category]]:

* {#HardieKampsKieboom99} [[Keith Hardie]], [[Klaus Heiner Kamps]], [[Rudger  Kieboom]], _Higher homotopy groupoids and Toda brackets_, Homology Homotopy Appl. Volume 1, Number 1 (1999), 117-134 ([euclid:hha/1139840198](https://projecteuclid.org/euclid.hha/1139840198))

* {#HardieKampsMarcum02} [[Keith Hardie]], [[Klaus Heiner Kamps]], [[Howard Marcum]], _The Toda bracket in the homotopy category of a track bicategory_, Journal of Pure and Applied Algebra Volume 175, Issues 1–3, 8 November 2002, Pages 109-133 (doi:10.1016/S0022-4049(02)00131-7)

On the [[fundamental 2-groupoid]] of [[Hausdorff topological spaces]] as a [[strict 2-groupoid]] ([[Grpd-enriched category]]):

* [[Keith A. Hardie]], [[Klaus H. Kamps]], [[Rudger Kieboom]], *A homotopy 2-groupoid of a Hausdorff space*. Papers in honour of Bernhard Banaschewski (Cape Town, 1996). Appl. Categ. Structures **8** (2000) 209-234 [[doi:10.1023/A:1008758412196](https://doi.org/10.1023/A:1008758412196)]

and as a [[weak 2-groupoid]]:

* [[Keith A. Hardie]], [[Klaus H. Kamps]], [[Rudger Kieboom]], *A Homotopy Bigroupoid of a Topological Space*, Applied Categorical Structures **9** (2001) 311-327 [[doi:10.1023/A:1011270417127](https://doi.org/10.1023/A:1011270417127)]

category:people

[[!redirects K.H.Kamps]]
[[!redirects K.H. Kamps]]

[[!redirects Heiner Kamps]]
[[!redirects K. H. Kamps]]
[[!redirects Klaus H. Kamps]]