(eBook)(Mit Press)(Philosophy) Jackendoff-Etal - Language, Logic, And Concepts - 16

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Language, Logic, and Concepts.Ray S. Jackendoff, Paul Bloom and Karen Wynn, editors. 2002 The MIT Press.

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Chapter 16

Those concepts hat are historically stable tend to be those that in someway reflect reality, and in turn tend to be those that are teachable" Mathematical" should mean in particular " teachable" and the modernmathematical theory of flexible categoriescan indeed be used to sharpenthe teachability of basic concepts A concept that has enjoyed some his-torical stability for 40,000 years is the one involving that structure of asociety that arises from the biological process of reproduction and itsreflection n the collective consciousness s deasof genealogyand kinship.Sophisticatedmethods for teaching this concept were devised ong ago,making possible regulation of the process tself. A more accurate modelof kinship than heretofore possible can be sharpenedwith the help ofthe modern theory of flexible mathematical categories at the same ime,established cultural acquaintance with the concept helps to illuminateseveralaspectsof the general heory, which I will therefore try to explainconcurrently.Abstracting the genealogicalaspect of a given society yields a mathe-matical structure within which aunts, cousins and so on can be preciselydefined. Other objects, in the samecategory of such structures that seemvery different from actual societies are nonethelessshown to be impor-tant tools in a societys conceptualizing about itself, so that for examplegender and moiety become labeling morphisms within that category.Topological operations, such as contracting a connected subspace o apoint, are shown to permit rationally neglecting he remote past. But suchoperations also lead to the qualitative transformation of a topos of pureparticular Becoming into a topos of pure general Being; the latter twokinds of mathematical toposesare distinguished rom each other by pre-cise conditions. The mythology of a primal couple is thus shown to bea naturally arising didactic tool. The logic of genealogy s not at all 2-

Kinship and Mathematical F. William Lawvere...Categories


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valued or Boolean, because he truth -value space naturally associatedwith the ancestorconcept has a rich lattice structure.As John Macnamara emphasized an explicit mathematical frameworkis required for progress n the scienceof cognition, much as multidimen-

sional differential calculus has been equired for 300 years as a frameworkfor progress n such sciencesas thermomechanicsand electromagneticsI hope that the exampIes consideredhere will con ri bu e to constructingthe sort of general ramework John had in mind.

Although a framework for thinking about thinking would be called" logic" by someancient definitions, in this century logic has come o havea much more restrictive connotation that emphasizes tatementsas such,rather than the objects to which the statements efer; this restrictive con-notation treats systemsof statementsalmost exclusively n terms of pre-sentations via primitives and axioms) of such systems thus obscuring heobjective aspectsof abstract generals hat are invariant under change ofpresentation About 100years ago the necessary bjective support for thissubjective ogic began o be relegated o a rigidified set theory that shareswith mereology the false presupposition that given any two sets it ismeaningful to ask.whether one is included in the other or not. Althoughthe combination of narrow logic and rigidified set theory is often said tobe the framework for a " foundation of mathematics" it has in fact neverservedas a foundation in practice; for example the inclusion question sposed n practice only betweensubsetsof a given set, such as the set ofreal numbers or the set of real functions on a given domain. There aremany different explicit transformations betweendifferent domains, but thepresumption that there is a preferred one leads o fruitless complications.

The development of multidimensional differential calculus led in par-ticular to functional analysis algebraic geometry, and algebraic opology,that is, to subjectswhose nterrelationships and internal qualitative leapsare difficult to account for by the narrow and rigidified " foundations."This forced the development of a newer, more adequate framework(which of course takes full account of the positive results of previousattempts) 50 years ago, and 20 years later the resulting notions of cate-gory, functor, natural transformation, adjoint functor, and so on hadbecome he standard explicit framework for algebraic topology and alge-braic geometry, a framework that is even indispensable or the commu-nication of many concepts Then, inspired by developments n mechanicsand algebraic geometry, it was shown how a broader logic and a less igidnotion of set are naturally incorporated in the categorical ramework.

412 La wv ere


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Toposesare categoriesof sets hat have specified nternal cohesion andvariation and whose transformations are continuous or equivariant. Theinfinite variety of toposesnow studied arisesnot in order to justify someconstructivist or other philosophical prejudice, but in order to make use-fully explicit the modes of cohesion and variation that are operative inmultidimensional differential calculus. The internal logic of to poses urnedout to be that which had beenpresentedearlier by Heyting. Heyting logic(and co-Heyting logic) describesnclusions and transfoffi1ationsof subsetsof such sets n a way that takes account of the internal cohesion andvariation, refining the fragmented and static picture that the still earlierapproximation by Boolean logic provides.I will explore here two examplesshowing how the categorical insightcan be used o make more explicitly calculable, not only the more overtlyspatial and quantitative notions, but also other recurring generalconceptsthat human consciousness as perfected over millennia for aiding in theaccurate reflection of the world and in the planning of action. The twoexamplesare kinship, in particular, and the becoming of parts versus hebecoming of wholes, in general Since the narrowing of Logic 100 yearsago, it has been customary to describe kin ships in teffi1s of abstract" relations" ; however, if we make the theory slightly less abstract, thecategory of examplesgains considerably n concreteness nd in power torepresentconcepts hat naturally arise. Becoming has long been studiedby resolving it into two aspects time and states with a map of timeinto the states but when we consider the becoming of a whole " body"together with that of its parts (e.g., the development of collective con-sciousness r the motion of the solar system, there are several " conflict-ing" aspects hat we need o manage the collective state " is" nothing butthe ensembleof the statesof the parts, yet the constitutive law deteffi1ining what the state (even of a part) " becomes may depend on the state ofall; the time of the whole may be taken as a prescribedsynchronization ofthe times of each part, yet the news of the collective becoming may reachdifferent parts with differing delays. A general philosophical idea is thatthe cohesiveness f being is both the basis n which these conflicts takeplace and also partly the result of the becoming (Hegel: " Wesen istgewesen- that is, the essenceof what there is now is the product ofthe process t has gone through). I will try to show more precisely howthe being of kinship in a given historical epoch relates o the becoming ofreproduction and of intermarriage betweenclans, even when this kinshipand reproduction are consideredpurely abstractly, neglecting heir naturaland societal setting.

Kinship and Mathematical Categories 413


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(T1) Each individual has exactly one mother and exactly one father,who are also individuals.As a first approximation, we can explore the ramifications of this idea,taken in itself, as an abstract theory T 1. Though we may have thoughtoriginally of " individual" as meaning a member of a particular tribe,there is in fact in a "theory" no information at all about what individualsare; the mathematical concretecorresponding o the stated abstract heoryTl is the category whose objects are sets consisting of abstract elementsbut structured by two specifiedself-mappings m and f Such a structuredset can be considered as a " society." We can immediately define suchconcepts as "a and b are siblings" by the equations am == bm and af ==bf , or " b's maternal grandfather is also c's paternal grandfather" by theequation bmf == cff . ( In the Danish language morfar' means mother'sfather, ' farfar' means father's father, similarly 'farmor' and 'mormor';and young children commonly address their grandparents using thosefour compound words.) There is a great variety of such systems most ofwhich seemat first glance o be ineligible, even mathematically, for morethan this verbal relation to the kinship concepts for example I cannotbe my own maternal grandfather. However, if we take the theory and thecategory seriously we will find that some of these strange objects areactually useful for the understandingof kinship.

As is usual with the concrete realizations of any given abstract theory,these orm a category n the mathematical sense hat there is a notion ofstructure-preservingmorphism X - t Y betweenany two examples in thiscase he requirement on ljJ s that jJxm) == jJx)m and ljJxf ) == jJx)f forall x in X. (It is convenient to write morphisms on the left, but structuralmaps on the right of the elements in the two equations required, theoccurrencesof m and f on the right sides of the equality refer to thestructure in the codomain Y of the morphism.) There are typically manymorphisms from given domain X to given codomain Y. If X - t Y ! z

414 Lawvere16.1 A ConcreteCategory Abstracting the Notion of Reproduction

I want to consider general notions of kinship system as suggestedbyknown particular examples The following fundamental notion was ela-borated in collaboration with Steve Schanuel (Lawvere and Schanuel1997.

An analysis of kinship can begin with the following biologicalobservation:


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Kinship and Mathematical Categories 415aremorphisms there s a well-determined ompositemorphismX ~ Z,and for eachobjectX, there s always he identity morphism rom X toitself. Two objectsXl and X2 are isomorphic n their category f thereexistsa pair of morphisms l --t X2 --t Xl with both composites qual otherespectivedentitymorphisms Most morphisms renot somorphismsand they indeeddo not even satisfy he requirement f " injectivity"which ust means hat the cancellationaw, ljJXl== fjJx2mpliesXl == X2holds or fjJ If a morphism~ is in fact injective it is alsocalled"a subobjectof its codomain which aken n itself s the domain" (Contrary osome igidified versionsof set theory a given object may occur as thedomainof manydifferentsubobjects f anothergivenobject A subobjectmay be thought of as a specifiedway of " including its domain nto itscodomain) An element is said o "belong o" a subobject (notationy G~) if there is x for which y == ~x; this x will be unambiguous becauseof the injectivity requirement on ~. Here by an element of X is oftenmeant any morphism T - t X ; such elementsare also referred to as figuresin X of shape T, but often we also restrict the word 'element to mean'figure of a sufficient few preferred shapes. Sometimeswe have betweentwo subobjects an actual inclusion map ~ S ~2 as subsetsof Y (i.e.,Xl ~ X2 for which ~l == ~2Ct. It is these nclusions that the propositionallogic of Y is about.16.2 GenealogicalTruth, SeenTopos-TheoreticallyThe category sketchedabove, that is, the mathematically concrete corre-spondent of the abstract theory TI of m and/, is in fact a " topos." Thatmeans among other properties, that there s a " truth-value" object Q that"classifies" via characteristic morphisms, all the subobjectsof any givenobject. In the simpler topos of abstract (unstructured) sets the set 2,whose elementsare " true" and " false," plays that role; for if X ~ Y isany injective mapping of abstract sets there is a unique mapping Y ~ 2such that for any element y of Y, rP == true if and only if y belongs tothe subset~; and if two maps rPI rP2 rom Y to 2 satisfy rPIentails rP2nthe Boolean algebra of 2, then the corresponding subsetsenjoy a uniqueinclusion map ~l ~ ~2 as subsetsof Y (i.e., Xl ~ X2 for which ~l == ~2a).However, in our topos, 2 must be replaced by a bigger object of truth-values n order to have those features or all systems Y and for all sub-systemsX, ~, as explained below.In our tapas of all Tl -societies there s a particular object I , which as anabstract set consistsof all finite strings of the two symbols m and f, with


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416 La wverethe obvious right -action that increases length by 1, giving its cohesivenessand variation ; in particular , the empty string 1 is the generic individual ,all of whose ancestors are distinct . A concrete individual in a society X isany morphism I ---+ X ; in particular , the infinitely many endomorphismsI ~ I are easily seen to be in one-to -one correspondence with the stringsin I . Composing with the two endomorphisms m and f , which correspondto the two strings of length 1, completely describes, in the context of thewhole category , everything we need to know about the particular internalstructure of X : the actions xm , x f are realized as special cases of thecomposition f morphisms The condition hat everymorphismX ! Yin the topos must satisfy is then seen to be just a special case of the asso-ciative law of composition , namely , the special case in which the first ofthe three morphisms being composed is an endomorphism of I . The factsstated in this paragraph constitute the special case of the Cayley -Y onedalemma , applied to the small category W that has one object , whoseendomorphisms are the strings in two letters, composed by juxtaposition ;a category with one object only is often referred to as a monoid .

With loss in precision we can say that x ' is an ancestor of x in case thereexists some endomorphism w of I for which x ' == xw . In the particularsociety X , the w may not be unique ; for example , xmff == xfmf mighthold. But the precision s retained n the category W X , discretely iberedover W, which is essentially the usual genealogical diagram of X . Thecorrelation of the reproductive process in X with past solar time could begiven by an additional age functor from this fibered category to a fixedordered set; a construction of Grothendieck (1983) would permit inter -nalizing such a chronology too in Tl 'S topos .

N ow we can explain the truth -value system. The subsets of I in thesense of our topos are essentially just sets A of strings that , however , arenot arbitrary but subject to the two conditions that if I ~ A is a memberof A , then also wm and w must be members of A . Thus , A can bethought of as all my ancestors before a certain stage, the antiquity of thatstage depending however on the branch of the family . These As constitutethe truth -values! To justify that claim , we must first define the action of mandfon As; this is what is often thought of as " division " :A : m == {w s W : mw sA } ,A : f == { w G W : fw GA} .It is easily verified that these two are again subobjects of I if A is. Notethat A : f consists of all my [ather 's ancestors who were in A . The truth -


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measurednswer

417Kinship and Mathematical Categories

tj>y) == {W8 W : yw 8 ~} ., to the question Precisely ow Irish is y?"his s the

value true is A := W, the whole set of strings, and union and intersectionA u B, A n B are the basic propositional operations or, and on truth -values. The operation not A of negation does not satisfy all the Booleanlaws, since t must again yield a subobject of I :not A == { W B W : for all v B W, wv does not belong to A}is the set of all my ancestorsnone of whose ancestorswere in A, so (Aor not A) is usually much smaller than true. Why does the object Q thusdefined serveas the unique notion of truth-value set for the whole topos?Consider any subset X ~ Y in the senseof the topos; then there is aunique morphism Y ~ Q such that for all individuals I ~ Y in Y,y G~ if and only if cjJy) == true.The value of on any elementy is forced to be

16.3 How to Rationally Neglect the RemotePastEvery X is a disjoint union of minimal components hat have no mutualinteraction; if we parameterize his set of componentsby a set IIo (X ) andgive the latter the trivial (identity) action of m, f, then there is an obviousmorphism X --+ IIo (X ). If llo (X ) := 1, a one-point set, we say X is con-nected that doesnot necessarilymean that any two individuals in X havesome common ancestor becausemore generally connecting can be veri-fied through cousinsof cousins of cousins and so on.

An important construction borrowed from algebraic topology is thefollowing:Given a subobject A ~ X , we can form the "pushout" X mod A thatfits into a commutative rectangular diagram involving a collapsing mor-phism X --t XmodA and an inclusion llo (A) ~ XmodA as well as thecanonical A --t llo (A) and satisfies he universal property that, if we aregiven any morphism X - t Y whose restriction to A depends only onno (A), then there is a unique morphism XmodA ~ Y for which t/I is thecomposite, t/I0 following X -+ X mod A, and which also agreeson no (A)with that restriction.This X mod A is a very natural thing to consider becausepracticalgenealogicalcalculations cannot cope with an infinite past. For examplethe Habsburg X are mainly active over the past 1,000 years and the Orsini


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418 Lawvereovrer the past 2 , 000 years . But this collapsing of the remote ancestors A toIIo ( A ) introduces an interesting idealization ; for simplicity we considerthe case where A is connected , IIoA == 1 . Then the construction has givenrise to a morphism 1 - * X mod A that is the residuum of A . But what is a" point " ( == figure of shape 1) in a topos l ike ours ? Since m and factas identity in 1 , a point ( e .g . , the point of X mod A that has arisen ) is aspecial sort of individual x that is its own mother and father :

xm == x == xi .

This idea of a superindividual is of course merely a convenience in genea -logical bookkeeping .

16 . 4 Deepening the Theory to Make Gender Explicit

A reasonable gender structure is not definable in the abstract theory T 1above , since there is no requirement that the values of the operator m bedisjoint from the values of the operatorfand also no way of determiningthe gender of inferti le individuals . So as usual we refine our theory to aricher one :

( T 2 ) In addition to T 1 , every individual has a definite gender , everyindividual ' s mother is female , and every individual ' s father is male .

The corresponding category of all concrete applications of T2 will also bea topos and in fact have a direct relation to the topos of TI ' Na ~ ely , inthe first topos there is a particular object G with only two individualscalled male andfemale and where the two structural operations act as theconstant maps . Our refined topos has objects that may be described aspairs X , y where X ~ G is a morphism in the first tapas and has as mor -phisms all triangular diagrams [X ~ G , X ~ Y , Y ~ G ] in the first tapasfor which Jtj ; == y ; the codomain of this triangle is defined to be the pair Y ,J . The conditions

y ( xm ) == ( yx ) mand

y ( xf ) == ( yx ) f

that morphisms must satisfy state in this case that y is a compatiblegender - labeling , and the further condi tion on a tr iangular diagram meansthat morphisms in our refined tapas moreover preserve gender . Thatapplies in particular to the individuals in X that happen to be ( in our


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A still further refinemen makes precise the idea of a society equippedwith a strategy or incest avoidance The simplest dea T3 of such s that of(matrilineal) moiety labeling, which can be explained n terms of anothertwo-individual application C of T 1. The two " individuals" are called Bearand Wolf in Lawvere and Schanuel1997 The idea is that the moiety ofany individual x is the sameas that ofx 's mother and distinct from that ofx's father. Thus, on the two individuals in C, m acts as the identity,whereas 2 == d but f # id . Any T -respecting morphism X -+ C is alabeling with the expected properties, and commuting triangles over Cconstitute the morphisms of a further topos that should be investigatedHowever, I will denote by T3 the richer abstract theory whose applica-tions are all the T -applications X equipped with a labeling morphismX -+ G x C. Thus, in T3s topos there are four generic ndividuals I (onefor each pair in G x C), morphisms from which are concrete ndividualsof either gender and of either Bear or Wolf moiety.16.6 Congealing ecomingnto Being

419

16.5 Moieties

Knship and Mathematical Categories

purely reproductive sense ) spinsters or bachelors . This tapas has two

generic individuals , because label ng maps I - + G can map 11 either to m

or to ! The terminal object of this new tapas is actualy IG so that a point

in the new sense is realy the " locus of a point moving virtualy between m

and f " ; more exactly , a morphism IG - + Y in this tapas amounts to an

Eve / Adam pair for which

e f = = a f = = a & em = = am = = e .

If we construct the pushout along A - + IIoA of a subobject A < : - + X in the

sense of this topos , we get in particular l oA C - ? - X / A , which specifies a

di fferent Eve / Adam pair for each of the mutualy oblvious subsocieties

among the specified ancestors .

According to the criteria proposed in La wvere 1991 the three toposesthus far introduced representparticular forms of pure becoming, becausethey satisfy the condition that every object X receivesa surjective mor-phism from another one E that has the specialpropertyZl W== Z2W implies ZI == Z2


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420 Lawveref or any g iv en w i n W and any two i ndi v dua ls Z l , Z2 i n E . Thi s " s eparab le

covering " E of X is achieved through refining each individual ( if neces -

s ar y ) in to se ve ral in div id ual s w th f onn al y d ist inc t g ene alo gi es ( e . g . ,

xmf # yf in E even if y = = xm in X ) . However , by restricting to certain

subcategories conssting only of objects that contain something roughly

l ke the Eve / Adam pair , we obtain toposes that have instead the qual ta -

tive character of pure being ; for example , the Habsburg era had a certain

qual ty of being ( and was t he basis for a lot of motion ) .

Mo re pr ec se ly , le t u s co ns de r v ar iou s th eor ies co rr es po ndi ng t o

monoid homomorphisms

W ~ M

that are epimorphic . Epimorphic homomorphisms may be surjective

( induced by a congruence on W ) , or may adjoin at most new operator

s ym bo ls t ha t a re t wo - s d ed i nv er se s o f o pe ra to r s ym bo ls c om in g f ro m W ,

or may involve a combination of these two kinds of " simpl fications " of

W . Such a homomorphism gives rise to a wel - defined subcategory of Tl ' S

topos , namely , the one cons sting of al those systems X satisfying the

co ngr uen ce or i nv er ti bi t y co ndi tio ns t ha t be co me t ru e i n M . ( F or a nal -

o go us s ub ca te go ri es o f t he o th er t wo t op os es , w e w ou ld n ee d to c on si de r

epimorphic functors

W / G ~ D

and

W / Gx C ~ E

to smal categories D or E of two ( respectively four ) objects . ) The result -

ing subcategories , although toposes , are not subtoposes but " quotient

toposes . " Here a quotient morphism of toposes , compressing a bigger

si tua tio n i nto a sma le r s it uat ion , i nv ol ve s a f ul a nd fa ith fu l in ve rse

functor p * ( the obv ous inclusion in our example ) and two forward func -

tors p ! and p * that are respectively left and right adjoint to p * in the sense

that there are natural one - to - one correspondences

PIX - * Y Z - * p * X

X - * p * Y p * Z - * X

between morphisms , for al sets X defined over the big situation and al

Y , Z defined over the smaler one . Note that the notion of morphism is

the same in both situations ; that is what it means to be ful a nd faithful .


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These adjointnesses are the refined objective version of relations whosesubjective reflections are the rules of inference for existential quantifica -tion p !, resp. universal quantification p * of predicates relative to a sub-stitution operation p * (indeed, p * preserves coproducts and products ofobjects, as is subjectively reflected in the fact that a substitution preservesdisjunction and conjunction of predicates; it may fail to preserve theanalogous negations however , for example in the case of continuouspredicates in topology ). Specifically , p*X extracts those individuals fromX whose genealogy happens to conform to the requirements specified inM , whereas PIX applies to all individuals in X, the minimal forcing re-quired to merge them into a new object that conforms to M .

These facts will be used below in investigating elementary examplesof quotient toposes, in particular , some of those that seem to congealbecoming into being.

To distinguish a topos of general pure being , I proposed in La wvere1986 and 1991 the two criteria that llo (X x Y) == llo (X ) x llo (Y) andthat every object is the domain of a subobject of some connected object .The first of these means that any pair consisting of a component of X anda component of Y comes from a unique single component of the Cartesianproduct X x Y; in particular , the product of connected objects should beagain connected (since 1 x 1 == 1), which is rarely true in a tapas of par -ticular becoming . That condition will be important for the qualitative ,homotopical classification of the objects, but I will not discuss it furtherhere .

The second criterion mentioned above, that every object can beembedded as a subobject of some connected object , has the flavor " Weare all related ," but is much stronger than merely requiring that a partic -ular object be connected; for example , the disjoint sum of two connectedobjects, which is always disconnected, should be embeddable in a con-nected object .

It follows from a remark of Grothendieck (1983) that both proposedcriteria will be satisfied by a topos of M -actions if the generic individual I(== M acting on itself ) has at least two distinct points 1 ~ I . The distinct -ness must be in a strong sense that will be automatic in our examplewhere 1 has only two subobjects; it is also crucial that the generic indi -vidual I itself be connected. Part of the reason why this special conditionof Grothendieck implies the two general criteria can be understood in thefollowing terms: it is not that becoming is absent in a topos of generalbeing; rather , it is expressed in a different way . An object X may be a



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422 Lawverespace of locations or states in the tradition of Aristotle ; that is , X is anarena in which becoming can take place. In particular , some such spacesT can measure time intervals and hence a particular motion in X during Tmay give rise to a morphism T ~ X describing the result of the becomingprocess; that is, if T is connected, if we can distinguish two instants I 4 Tcalled to , tl and if jltk == Xk for k == 0 , 1, then we can say that Xo becameXl during the process 11 this is the common practice in mathematicalengineering (at least for certain categories within which it has, in effect,become customary to work in the past 300 years). But then if we considertl as a subobject of T, its characteristic map from T to the truth -valuespace will be a process along which false becomes true ; this in turn impliesthat the truth -value space is itself connected since T is , and because of thepropositional structure of truth -values, it follows that indeed the hyper -space, whose points parameterize the subspaces of X , is connected too . Onthe other hand, in any topos any object X is embedded as a subobject ofits hyperspace via the " singleton " map. Taking T == I , it follows that inthe cases under discussion every object can be embedded in an object thatis not only connected, but moreover has no holes or other homotopicalirregularities . Note that now each individual I ~ X involves also a pro -cess in his or her society . There are many such situations , but what cansuch a process mean ?

As a simple example , suppose that we want a category of models ofsocieties in which genealogical records go back to grandparents . Thus , weconsider the homomorphism W ~ M onto a seven-element monoid inwhichm3 == mlm == m2andml2 == m2 == mlhold , together with similar equations withfand m interchanged . Drawingthe obvious genealogical diagram , we see that the generic individual / inthe topos of M actions has parents that represent the first intermarriagebetween two distinct lines, and that / 's grandparents are the " Eves andAdams " of those two distinct lines. Indeed, this object [ permits a uniquegender structuring and so we may pass instead into the topos whose theoryis M / G, finding there that each of the two generic ndividuals [ G has twodistinct points IG : 4 [ G. The " process" involved in an individual boy[ G ~ X in a society X in this tapos is that whereby his father 's line unitedwith his mother 's.


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Given any object L in a topos, one can form a new topos of objects fur-ther structured by a given labeling morphism to L . However, it is worthyof note that in our examples one L == G (leading to T2) and the other oneL == G x C (leading to T3), these labeling objects belong to a muchrestricted subcategory quotient topos) within the topos of all Tl -objects.Although the Tl topos, consisting of all right actions of the free monoidWon two symbols is too vast to hope for a complete survey of its objects(as is borne out by theorems of Vera Trnkova stating that any categorycan be embeddedas a full subcategoryof it ; seePultr and Tmkova 1980,by contrast thesesmaller toposesare more tractable.Consider first the case G of gender in itself. It belongs to the sub-category corresponding o W -+ W3, the three-element monoid in whichxl == I , xm == m for all three x == 1, I , m. If we restrict attention to theright actions of W3 only, we see hat the generic ndividual I in the senseof that topos has only one nontrivial subobject which is G itself. There-fore, the truth -value space Q3 or that topos (in contrast to the infinite .Qfor W) has only three elements also. The most general " society" X isa sum of a number of noninteracting nuclear families, some with twoparents and some with a single parent, and a society s determined up toisomorphism by the double coefficient array that counts families of allpossible sizesof these wo kinds. Cartesian products are easily computed.Second consider the simple moiety-labeler C in itself; it belongs o thequotient topos determinedby W -+ W2, the two-elementgroup generatedby I with 12 == 1, where we moreover interpret m as 1. Here C is itselfthe generic ndividual and has no nontrivial subobjects so that 02 is thetwo-element Boolean algebra. All right W2-actions are of the form X ==a + bC, where a is the number of individuals fixed, and b half the numbermoved, by ! Theseobjects are multiplied by the rule c2 == 2C.Third , we can include both G and C in a single subcategoryas follows:the infinite free monoid W maps surjectively to each of W3 and W2;therefore, it maps to the six-element product monoid W3 x W2, but notsurjectively since the image Ws is a five-element submonoid (the missedelement s the only nonidentity invertible element n the product monoid).The category of right Ws-setscan be probed with the help of its genericindividual I , which again (surprisingly) has only three subobjectsso thatQs has again three truth -value-individuals. In I (i.e., Ws) there are besidesme my four distinct grandparents the Tl -structure coming from the fact

423Kinship and Mathematical Categories16.7 Finer Analysis of the Coarser Theories


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424 La wverethat my parents are identical with my maternal grandparents in thistheory . The unique nontrivial subobject of I is precisely G x C, the labelobject for our central noncollapsed theory T 3; it is generated by any singleone of its four elements . I leave it to the reader to determine a structuraldescription of all right W s-actions , with or without G x C labelings .16 .8 Possible Further Elaborations

To the T2 topos (whose abstract general is the two -object category W / G)we can apply separately our concrete construction idealizing remoteancestors and also our refinement of the abstract general by moiety -labeling . However , it does not seem consistent to apply both of thesesimultaneously , owing to the simple group -theoretic nature of our C. Areasonable conjecture is that tribal elders in some part of the world havedevised a more subtle abstract general C' that relaxes the incest avoidancefor the idealized remote ancestors, thus permitting a smoothly functioninggenealogical system enjoying both kinds of advantages.

Some more general examples of simplifying abstractions include thatbased on a homomorphism W ~ M n that makes every string of lengthgreater than n (where, e.g., n == 17) congruent to a well -defined associatedstring of length n. An interesting construction to consider is the appli -cation of the pushout construction to an arbitrary Tl -object X , not withrespect to some arbitrarily chosen remoteness of ancestry A , but withrespect to the canonical morphism p * p * X --+ X .

The need for Eve and Adam arises from the central role of self -maps asstructure in the above theories and might be alleviated by consideringinstead a two -object category as the fundamental abstract general, althoughpossibly at the expense of too great a multiplicity of interpretations . Thiscategory simply consists of three parallel arrows , as described in Lawvere1989. A concrete application of this theory To involves two sets (ratherthan one) and three internal structural mapsX no~' - - t X up to nowcalled m, f , and 8, where s specifies, for each contemporary individual , theplace of his or her " self " in genealogical history . There is a natural notionof morphism between two such objects (involving two set-maps subject tothree equations ) yielding again a topos, whose truth -value object is finiteand illuminating to work out . To's topos has Tl 's topos as a quotient ,because forcing the structural map 8 to become invertible yields a small


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Lawvere, F. W., and Schanuel S.Cambridge University Press

category equivalent to the monoid W . I f s is not inverted , iteration ofm andfis a more complicated partial affair : in case xm has the propertythat there is a contemporary mother y for which ys = = xm , then yf is a

maternal grandfather of x ; but there may not be such a y , and on theother hand , mathematical experience counsels against excluding in gen -

eral the possibility of several such y for a given x . If we do invert s , we

make expl icit that which , in a larger sense , is the theme of this b ook , thecontinuing role of each past individual in our cognition at present .

References

Grothendieck , A . ( 1983 ) . Pursuing stacks . Unpublished manuscript , University ofMontpellier , France .

Lawvere , F . W . ( 1986 ) . Taking categories seriously . Revista Colombiana deMatematicas , 20 , 147 - 178 .

Lawvere , F . W . ( 1989 ) . Qualitative distinctions between some toposes of general -ized graphs . In J . Gray and A . Scedrov ( Eds . ) , Logic , categories , and computerscience ( Contemporary Mathematics 92 , pp . 261 - 299 ) . Providence , RI : AmericanMathematical Society .

Lawvere , F . W . ( 1991 ) . Some thoughts on the future of category theory . In A .Carboni , M . C . Pedicchio , and G . Rosolini ( Eds . ) , Category theory ( LectureNotes in Mathematics 1488 , pp . 1 - 13 ) . Heidelberg : Springer - Verlag .

( 1997 ) . Conceptual mathematics . Cambridge :

Macnamara, J., and Reyes G. E. (1994. The logical foundations of cognition.Oxford: Oxford University PressPultr, A ., and Tmkova, V. (1980. Combinatorial algebraic and topological rep-resentations f groups semigroups and categories Amsterdam: North -Holland .


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