Contents

philosophy

# Contents

## 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' groundbreaking Les structures élémentaires de la parenté.

## 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). 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.1

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

###### 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).

###### Remark

The axiomatics was proposed in Courrège (1965) 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 function2 $\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 $\omega =\mu^{-1}\circ \pi$.

###### 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 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

###### 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) but also more ‘exotic’ operations like 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.

(…)

### 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

###### 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

###### 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

###### 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; RRZ04, p.226), 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, 1999) 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 unconstrained3, 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 $Set^{M_2^{op}}/L$.

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’ individual4 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 $Set^{M_2^{op}}/G$.

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.5

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

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

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).

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, 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 essential subtoposes since by an result of Kelly-Lawvere (1989) 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) 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 $\int _W L$ !

• Wikipedia entry: kinship (link)

• Mathematical Anthropology and Cultural Theory (online journal)

## 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

• Robert Parkin, Linda Stone (eds.), Kinship and Family , Blackwell Oxford 2004.

A conspicuously clear and concise introduction can be found in

• 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.

Classical anthropological texts are

• Robin Fox, Kinship and Marriage , 2nd ed. Cambridge UP 1983.

• 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)

More recent texts include

• German Dziebel, The Genius of Kinship , Cambria Youngstown 2007.

• Maurice Godelier, Métamorphoses de la parenté , Fayard Paris 2004.

• Maurice Godelier, Thomas R. Trautmann, Franklin E. Tjon Sie Fat (eds.), Transformations of Kinship , Smithsonian Institute Washington 1998.

The permutation model originates with

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)

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

• 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:

• Philippe Courrège, Un modèle mathématique des structures élémentaires de parenté , L’Homme 5 no.3-4 (1965) pp.248-290. (link; 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:

• Pin-Hsiung Liu, Foundations of Kinship Mathematics , Academica Sinica Nankang 1986.

The category-theoretic point of view is sketched in

The following pursues an abstract general approach to kinship:

• 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.)

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.

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

• Mauro W. Barbosa De Almeida, On the Structure of Dravidian Relationship Systems , MACT 3 no.1 (2010). (pdf)

A connection between Dravidian kinship terminology and supersymmetry is proposed in

• Ruth M. Vaz, Relatives, Molecules and Particles , MACT 7 no. 1 (2014). (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)

• Alain Gottcheiner, On some Classes of Kinship Systems II: Nonabelian Systems , MACT 2 no. 5 (2008). (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)

• Frank Harary, Douglas R. White, P-Systems: A Structural Model for Kinship Studies , Connections 24 no.2 (2001) pp.35-46. (pdf)

• Paul Jorion, Gisèle De Meur, Trudeke Vuyk, Le mariage Pende , L’Homme 22 no.1 (1982) pp.53-73. (link)

• Claude Lévi-Strauss, Georges Guilbaud, Système parental et matrimonial au Nord Ambrym , J. Société des Océanistes 26 no. 26 (1970) pp.9-32. (link)

• 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)

• 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)

• Nina Otter, Mason J. Porter, A unified framework for equivalences in social networks , arXiv:2006.10733 (2020). (abstract)

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

• M. La Palme Reyes, G. E. Reyes, H. Zolfaghari, Generic Figures and their Glueings , Polimetrica Milano 2004.

to the standard textbook

The result concerning essential localizations is in

• G. M. Kelly, F. W. Lawvere, On the Complete Lattice of Essential Localizations, Bull. Soc. Math. de Belgique XLI (1989) pp.289-319.

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.

1. 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.

2. $\mu$ and $\pi$ are called the maternal resp. paternal function in Courrège (1965).

3. 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). 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.

4. Again it makes more sense to consider $G$ to consist of two gender groups instead of two individuals.

5. The Fox tribe in Lawvere’s home state Iowa indeed consists of two moieties called ‘Bear’ and ‘Wolf’ but they are patrilineal (Liu 1986, p.21).

Last revised on June 19, 2020 at 05:49:29. See the history of this page for a list of all contributions to it.