Rectification : Exercises 1 and 2 : Build a functor of (D) in (C).( For reasons of homogeneity of the notations with other papers.)
Le jeu. 7 mai 2020 à 12:05, Robert Marty <robertmarty...@gmail.com> a écrit : > Jon Alan, Helmut, Edwina, List > > JAS > "Unfortunately I am not adept enough with mathematical category > theory to make heads or tails of Robert's exposition below. It still seems > to me that "category" means something quite different in that context than > it does for Peirce when he is writing about 1ns, 2ns, and 3ns. Am I > wrong? If so, I would appreciate some further explanation of how they > relate to each other." > > > > RM > I'd love to. The term "category" can be used without problem in a > field other than the one you are used to. In the category theory it refers > to a mathematical object in the usual sense chosen by Peirce as" "a > construction independent of its real existence". An industrial property law > allows the same name to be used for products that are in very remote areas > of the economy such as Corona for a beer or for a virus (which is not > without danger, hence Covid-19!). You see the comparison ...😉 > > Nb: I noted that you are a professional engineer; your training should > allow you to understand the following where it is only definitions. No > theorem, no specific technique; definitions, I stress that point. > > > > 1 - https://en.wikipedia.org/wiki/Category_(mathematics) > > *Definition of category* > > There are many equivalent definitions of a category.[2] > <https://en.wikipedia.org/wiki/Category_(mathematics)#cite_note-2> One > commonly used definition is as follows. A *category* *C* consists of > > · a class <https://en.wikipedia.org/wiki/Class_(set_theory)> ob( > *C*) of *objects* > > · a class hom(*C*) of *morphisms > <https://en.wikipedia.org/wiki/Morphism>*, or *arrows*, or *maps*, > between the objects. Each morphism *f* has a *source object a* and a *target > object b* where *a* and *b* are in ob(*C*). We write *f*: *a* → *b*, and > we say "*f* is a morphism from *a* to *b*". We write hom(*a*, *b*) (or hom > *C*(*a*, *b*) when there may be confusion about to which category hom(*a*, > *b*) refers) to denote the *hom-class* of all morphisms from *a* to *b*. > (Some authors write Mor(*a*, *b*) or simply *C*(*a*, *b*) instead.) > > · for every three objects *a*, *b* and *c*, a binary operation > hom(*a*, *b*) × hom(*b*, *c*) → hom(*a*, *c*) called *composition of > morphisms*; the composition of *f* : *a* → *b* and *g* : *b* → *c* is > written as *g* ∘ *f* or *gf*. (Some authors use "diagrammatic order", > writing *f;g* or *fg*.) > > such that the following axioms hold: > > · (associativity <https://en.wikipedia.org/wiki/Associativity>) > if *f* : *a* → *b*, *g* : *b* → *c* and *h* : *c* → *d* then *h* ∘ (*g* ∘ > *f*) = (*h* ∘ *g*) ∘ *f*, and > > · (identity <https://en.wikipedia.org/wiki/Identity_(mathematics)>) > for every object *x*, there exists a morphism 1*x* : *x* → *x* (some > authors write *idx*) called the *identity morphism for x*, such that for > every morphism *f* : *a* → *x* and every morphism *g* : *x* → *b*, we > have 1*x* ∘ *f* = *f* and *g* ∘ 1*x* = *g*. > > From these axioms, one can prove that there is exactly one identity > morphism for every object. Some authors use a slight variation of the > definition in which each object is identified with the corresponding > identity morphism. > > > > *( ! **) By extraordinary the example at the top right is interpretable > with* > > > > A = 3ns ; B = 2ns ; C = 1ns ; f = involvesβ ; g = involvesα ; g o f = β o > α ; let's name C this category > > But also > > > > A = O ; B = S ; C = I ; f = det1 ; g = det2 ; g o f = det2 o det1 ; > let's name *D* this category > > > > *Remember these two interpretations, * *they will* *serve ....* > > > > 2 - https://en.wikipedia.org/wiki/Functor > > *Definition** of functor* > > Let *C* and *D* be categories > <https://en.wikipedia.org/wiki/Category_(mathematics)>. > A *functor* *F* from *C* to *D* is a mapping that > <https://en.wikipedia.org/wiki/Functor#cite_note-FOOTNOTEJacobson2009p._19,_def._1.2-3> > associates to each object X {\displaystyle X} X in *C* an object F ( X ) > {\displaystyle F(X)} f(X) in *D*, > > · associates to each morphism f : X → Y {\displaystyle f\colon > X\to Y} f : XàY in *C* a morphism F ( f ) : F ( X ) → F ( Y ) > {\displaystyle F(f)\colon F(X)\to F(Y)} f(X)àY in *D* such that the > following two conditions hold: F ( i d X ) = i d F ( X ) {\displaystyle > F(\mathrm {id} _{X})=\mathrm {id} _{F(X)}\,\!} > > · F(idX) = idF(X) for every object X {\displaystyle X} in *C*,F ( > g ∘ f ) = F ( g ) ∘ F ( f ) {\displaystyle F(g\circ f)=F(g)\circ F(f)} > > > > · F(g o f) = F(g) o F(f) for all morphisms f : X → Y > {\displaystyle f\colon X\to Y\,\!} f ; XàY and g : Y → Z {\displaystyle > g\colon Y\to Z} g : YàZ in *C*. > > That is, functors must preserve identity morphisms > <https://en.wikipedia.org/wiki/Morphism#Definition> and composition > <https://en.wikipedia.org/wiki/Function_composition> of morphisms. > > __________________________ > > *Exercise 1* 😉 > > Build a functor of *(C)* in (*D)*). > *Graphic Hint:* for this it is necessary to connect the elements of C to > those of D by 3 arrows avoiding any intersection > > *Exercice 2* > > Build all the functors of *(C)* in (*D*) > > *Answer: *there are exactly 10 funtors (C*C*) in (*D*) > > > > 3 – https://en.wikipedia.org/wiki/Natural_transformation > > *Definition of natural transformation of functors* > > If F {\displaystyle F} F and G {\displaystyle G} G are functors > <https://en.wikipedia.org/wiki/Functor> between the categories *C > {\displaystyle C} C* and *D* D {\displaystyle D} , then a *natural > transformation* η {\displaystyle \eta } µ from F {\displaystyle F} F to G > {\displaystyle G} G is a family of morphisms that satisfies two > requirements. > *3.1* - The natural transformation must associate, to every object X > {\displaystyle X} X in *C {\displaystyle C} C* , a morphism > <https://en.wikipedia.org/wiki/Morphism> > > η X : F ( X ) → G ( X ) {\displaystyle \eta _{X}:F(X)\to G(X)} µX : F(X) > à G(X) between objects of *D {\displaystyle D} D*. The morphism η X : F > ( X ) → G ( X ) {\displaystyle \eta _{X}:F(X)\to G(X)} µX η X > {\displaystyle \eta _{X}} is called the *component* of η {\displaystyle > \eta } µ at X {\displaystyle X} X. > > *3.2-* Components must be such that for every morphism f : X → Y > {\displaystyle f:X\to Y} f : XàY in *C {\displaystyle C} C *we have: > > µY o F(f) = G(f) o µXη Y ∘ F ( f ) = G ( f ) ∘ η X {\displaystyle \eta > _{Y}\circ F(f)=G(f)\circ \eta _{X}} > > The last equation can conveniently be expressed by the commutative diagram > <https://en.wikipedia.org/wiki/Commutative_diagram> > > X F(X) ---------µX--------à G(X) > | | | > f | F(f) | | G(f) > v v v > Y F(Y)---------- µY --------à G(Y) > > *Nb*: *I am obliged to rewrite all the diagrams and even the letters that > are images in Wikipedia and I put **µ** in place of "eta".* > > If µ η {\displaystyle \eta } is a natural transformation from F > {\displaystyle F} to G {\displaystyle G} F to G, we also write µ : F àG η > : F → G {\displaystyle \eta :F\to G} η : F ⟹ G {\displaystyle \eta > :F\implies G} . > > *Exercise* 3: Choose from the 10 functors found in Exercise 2 two > functors for which there is a natural transformation. > > *Graphic Hint*: To do this, you have to link the elements of F to the > elements of G by 3 arrows, avoiding any intersection. > > *Exercice 4* : find all possible natural transformations and make sure > they get the lattice of the sign classes! > > ! > > That all ! 😊 > > Best regards, > > Robert > > > > Le jeu. 7 mai 2020 à 03:32, Jon Alan Schmidt <jonalanschm...@gmail.com> a > écrit : > >> Gary R., Robert, List: >> >> I take no exception to anything in Gary R.'s reply, and in light of his >> and Robert's comments along with Jon A.'s remark in the other thread, I am >> now persuaded to embrace the formulation that 3ns involves 2ns and 1ns, >> while 2ns involves 1ns. However, I would welcome some further discussion >> of whether involution (or presupposition) in this context is really a >> *genuine >> *triadic relation in the case of 3ns, rather than a *degenerate* triadic >> relation that can be reduced to transitive dyadic relations (3ns involves >> 2ns, which involves 1ns). >> >> Unfortunately I am not adept enough with mathematical category theory to >> make heads or tails of Robert's exposition below. It still seems to me >> that "category" means something quite different in that context than it >> does for Peirce when he is writing about 1ns, 2ns, and 3ns. Am I wrong? >> If so, I would appreciate some further explanation of how they relate to >> each other. >> >> I will also point out once again my disagreement regarding the logical >> order of the three interpretants in the hexad. I believe that it should be >> If→Id→Ii, and it seems to me that this is *more *consistent with "the >> immutable suite of the three 3ns→2ns→1ns" as 3→2/3→1/2/3, just as the two >> objects are Od→Oi as 2→1/2. Am I overlooking something in the underlying >> logic that *requires* the sequence of the interpretants to be Ii→Id→If? >> >> Regards, >> >> Jon Alan Schmidt - Olathe, Kansas, USA >> Professional Engineer, Amateur Philosopher, Lutheran Layman >> www.LinkedIn.com/in/JonAlanSchmidt - twitter.com/JonAlanSchmidt >> >> On Wed, May 6, 2020 at 6:10 AM robert marty <robert.mart...@gmail.com> >> wrote: >> >>> Gary, Jon Alan, Jon Awbrey, List >>> >>> *1 *-First I note that the formulation "3ns involves 2ns, which >>> involves 1ns" is very dangerous car it forgets that 2ns has its autonomy >>> and 1ns too. If you look at the podium on remains in the inner cylinder. It >>> seems to me that Peirce's reproach to Hegel is: >>> >>> "*He has usually overlooked external Secondness, altogether. In other >>> words, he has committed the trifling oversight of forgetting that there is >>> a real world with real actions and reactions. **Rather a serious >>> oversight that".* >>> >>> It is therefore important to prefer"3ns involves 2ns and 1ns, while >>> 2ns involves 1ns" which preserves the autonomy of the Peircian >>> categories so as not to encourage the idea of a possible peircean >>> hegelianism. " >>> >>> 2 – On the specific question *"about **the conceptual relationship >>> between Peirce's trichotomic category theory and contemporary mathematical >>> category theory if any"*, I will mainly have a limited response to the >>> field at hand, that is, the classification of signs. Then I can give >>> some personal reflections on the general scope of mathematical category >>> theory in the humanities. >>> >>> I was just preparing a text on the comparison of the ways in which >>> classes of signs are generated by different authors and I chose the most >>> interesting and successful in my eyes, i.e. Gary's trikônics, the triangles >>> of Priscila Farias and Joao Queiroz and the signtree of Priscila Borges. >>> It can be said at first glance that they are equivalent since they >>> generate the same classes of signs that can be characterized by sequences >>> of numbers of length n = 3, 6 or 10 taken in the set {1,2,3} and verifying >>> that each number must be less or equal to the previous number. But the >>> question is, since these are classes based on different graphic metaphors, >>> what is the common formal structure - if there is one - of which they are >>> the graphic inscriptions. For this it is necessary to go in the field of >>> posets and more precisely totally ordered sets the simplest that are the >>> chains: >>> >>> " A set with a partial order is called a *partially ordered set* (also >>> called a *poset*). The term *ordered set* is sometimes also used, as >>> long as it is clear from the context that no other kind of order is meant. >>> In particular, totally ordered sets >>> <https://en.m.wikipedia.org/wiki/Total_order> can also be referred to >>> as "ordered sets", especially in areas where these structures are more >>> common than posets. >>> >>> For *a, b*, elements of a partially ordered set *P*, if *a* ≤ *b* or *b* >>> ≤ *a*, then *a* and *b* are *comparable >>> <https://en.m.wikipedia.org/wiki/Comparability>*. A partial order under >>> which every pair of elements is comparable is called a *total order >>> <https://en.m.wikipedia.org/wiki/Totally_ordered_set>* or *linear order*; >>> a totally ordered set is also called a *chain* (e.g., the natural >>> numbers with their standard order)". >>> >>> ( https://en.wikipedia.org/wiki/Partially_ordered_set) >>> >>> >>> >>> Clearly the abstract diagram *3ns**à**2ns**à**1ns* (let's call the P) >>> is a chain which is common to all three approaches. >>> >>> >>> >>> We have also the maps between partially ordered sets >>> >>> >>> >>> " Definition 6: A function f : P → Q between partially ordered sets is >>> order-preserving if x ≤P y ⇒ f(x) ≤Q f(y). >>> >>> >>> >>> Definition 7: Two partially ordered sets P and Q are isomorphic if there >>> exists a bijective, order-preserving map between them whose inverse is also >>> order-preserving" >>> >>> (http://www-math.mit.edu/~levine/18.312/alg-comb-lecture-7.pdf ) >>> >>> >>> >>> To use this notion of the preservation of order, it is necessary to >>> identify in each graphic metaphor a Q chain . >>> >>> >>> >>> *I claim that these Q chains are materialized at the moment when >>> everyone chooses the convention that consists of locating the sign, the >>> object of the sign and its interpretant on the graphic icon he has chosen.* >>> >>> >>> >>> *As for Gary*: starting from the object at the lower corner of his >>> trikône he goes up to the sign at the top corner following the "vertical" >>> side and then from there he should go to interpretant it from the top side >>> but if his graph indicates a direct relationship between the object and the >>> interpretant, which is the same because it is the concatenation of the >>> first two paths. What is thus traced is an OàS àI chain. >>> >>> >>> >>> *As for **Priscila Farias et Joao Queiroz :* ( >>> https://www.researchgate.net/publication/249933979_On_diagrams_for_Peirces_10_28_and_66_classes_of_signs >>> ) >>> >>> it is the same ; the way is taken from Peirce for which the categories >>> are assigned thereby : the object to the upper left corner to go to the >>> sign at the bottom corner and from there to interpretant it in the upper >>> right corner. They create what they call "triangular coordinates" but it's >>> the same OàS àI chain and it's do the same it when they process >>> graphically n = 6 and n = 10 which has the effect of multiplying the >>> triangles. >>> >>> >>> >>> *Priscila Borges* uses the graphic metaphor of the developing tree: >>> >>> >>> https://www.researchgate.net/publication/263463845_THE_SIGNTREE_FROM_SIGN_STRUCTURE_TO_PEIRCE'S_PHILOSOPHY_THROUGH_READING_A_VISUAL_MODEL_OF_THE_66_CLASSES_OF_SIGNS >>> >>> >>> >>> "So, the diagram construction begins by the idea of tree rings. They >>> are used in dendrocronology to count the age of trees. As years go by >>> rings grow in trees, but they are also affected by climate factors. More >>> than sign of time, tree rings show interaction between systems. All >>> these concepts are welcome in semiotic process. Each ring corresponds >>> to one trichotomy: the first trichotomy comes in the centre, the second >>> trichotomy in the second ring and so on. " >>> >>> >>> >>> In this text she does it for n =10: >>> >>> >>> >>> "Consequently, since the object determines the sign, and not the sign >>> determines >>> the object, it was necessary to put the dynamical object in the central >>> ring, followed by the immediate object and the ground of sign. Given the >>> first three correlates, comes the first relation: between sign and >>> dynamical object. This relation determines the possible interpretants, >>> called immediate interpretants that when are existent become dynamical >>> interpretants. So, the elements that compose the second relation are >>> given: between sign and dynamical interpretant." >>> >>> >>> >>> Moving from a growth ring of its tree to the next it builds the de facto >>> chain for n = 6: >>> >>> >>> >>> Od àOiàSàIiàIdàIf >>> >>> >>> >>> And for n =1 0 she obtient very beautiful diagrams intelligently >>> colored. >>> >>> >>> >>> *My conclusion is that all these iconographic constructions are >>> isomorphic; they are produced in the same way using the applications of the >>> immutable suite of the three **3ns**à**2ns* *à**1ns **and the f >>> application in n-length chains similar to the protosigns I defined in the >>> article on the trichotomic machine. They all lead - we just saw - to sets >>> (in the sense of set theory). The results: classes of signs without >>> explicit relations between them. * >>> >>> >>> >>> *Now here's the jump in the category theory :* >>> >>> >>> >>> *"Every poset (and every **preordered set >>> <https://en.m.wikipedia.org/wiki/Preorder>**) may be considered as a >>> **category >>> <https://en.m.wikipedia.org/wiki/Category_(mathematics)>** where, for >>> objects x and y, there is at most one **morphism >>> <https://en.m.wikipedia.org/wiki/Morphism>** from x to y. More >>> explicitly, let hom(x, y) = {(x, y)} if x ≤ y(and otherwise the empty set) >>> and (y, z)**∘**(x, y) = (x, z). Such categories are sometimes called >>> **posetal >>> <https://en.m.wikipedia.org/wiki/Posetal_category>**."* >>> >>> >>> >>> "Posets are equivalent >>> <https://en.m.wikipedia.org/wiki/Equivalence_of_categories> to one >>> another if and only if they are isomorphic >>> <https://en.m.wikipedia.org/wiki/Isomorphism_of_categories>. In a >>> poset, the smallest element, if it exists, is an initial object >>> <https://en.m.wikipedia.org/wiki/Initial_object>, and the largest >>> element, if it exists, is a terminal object >>> <https://en.m.wikipedia.org/wiki/Terminal_object>. Also, every >>> preordered set is equivalent to a poset. Finally, every subcategory of a >>> poset is isomorphism-closed >>> <https://en.m.wikipedia.org/wiki/Isomorphism-closed>." ( >>> https://en.wikipedia.org/wiki/Partially_ordered_set#Mappings_between_partially_ordered_sets >>> ) >>> >>> >>> >>> so the same mathematical objects that are involved in the ensemblist >>> mathematical models that I have just listed can be looked at differently; >>> they are now algebraic categories. On can use all the conceptual apparatus >>> of the categories and first the functors and especially the bonus of >>> natural transformations of functors which brings us back to the trichotomic >>> machine. This machine naturally produces the same classes of signs of >>> course but with the order of a lattice revealed by the natural >>> transformations of functors that we will be able to exploit to increase our >>> knowledge of the signs and especially to create a methodology as an example >>> I did in the case of nicotine. >>> >>> >>> >>> The general idea that has guided me for a long time is that Peirce's >>> thought is "functorial" and that his universe of thought is above all >>> relational. This is the reason for the fact that I continued work started >>> in my book "The Algebra of Signs". I try to express all its semiotics by >>> starting with a formalization of the "percipuum" in the category of >>> relational structures. But that's another story... >>> >>> >>> >>> For now I am sorry to find that I submitted my nicotine analysis to the >>> criticism on May 3rd and that I did not get any reaction. I believe that I >>> show and demonstrate how a positive image of semiotics is formed and how it >>> gains in "semioticity" until it becomes able to compete with the negative >>> image of nicotine installed in a Dicent Symbol, at the top of the lattice. >>> >>> See >>> https://www.academia.edu/42930701/Nicotine_a_semiotic_confrontation_between_life_and_death >>> >>> >>> >>> Best regards, >>> >>> Robert Marty >>> >>> Le mer. 6 mai 2020 à 06:47, Gary Richmond <gary.richm...@gmail.com> a >>> écrit : >>> >>>> Jon, Robert, List, >>>> >>>> JAS: Overall, we seem to be more or less on the same page. >>>> GR: I think that's so. >>>> >>>> JAS: I understand the impetus for using "presupposition" rather than >>>> "involution," since the former term is more familiar to modern >>>> mathematicians and logicians than the latter. >>>> GR: I too understand the impetus for Robert's using "presupposition" as >>>> being more familiar to modern mathematicians than "involution." But how >>>> many of them are familiar with Peirce's three category theory at all? I >>>> continue to believe that in consideration of Peirce's semeiotic (and all >>>> that follows from it) that "involution" is the more accurate and evocative >>>> term. >>>> >>>> JAS: I have no objection to saying that 3ns involves 2ns and 1ns, while >>>> 2ns involves 1ns. I just find it more succinct and equally accurate to say >>>> that 3ns involves 2ns, which involves 1ns; this already entails that 3ns >>>> also involves 1ns. >>>> GR: Logically, of course, you are correct and your more succinct >>>> version is equivalent. But saying that "3ns involves 2ns and 1ns" >>>> brings the fundamental trichotomy into high relief immediately. But it is a >>>> minor point, perhaps one merely of emphasis. >>>> >>>> JAS: I am not wedded to Peirce's adaptation of Aristotelian terminology >>>> (1ns/2ns/3ns = form/matter/entelechy), which is most prevalent in his >>>> writings around 1904--e.g., in "New Elements" (EP 2:303-305) and "Sketch of >>>> Dichotomic Mathematics" (NEM 4:292-300)--but I find it helpful in certain >>>> contexts. >>>> GR: I suppose it is helpful in certain contexts to employ Peirce's >>>> tricategorial adaptation of Aristotelian terminology. But is it possible >>>> that the movement from one equivalent terminology to another -- especially, >>>> but not only, within a single analysis -- has impeded the more general >>>> acceptance of some core Peircean ideas? I don't think there's an easy >>>> solution to this or any of the terminological questions we've taken up on >>>> the list over the last few years, but I think that there may be a >>>> communicational problematic here worth considering. >>>> >>>> JAS: I share Gary R.'s interest in learning more about "the conceptual >>>> relationship between Peirce's trichotomic category theory and contemporary >>>> mathematical category theory if any." I am more familiar with Fernando >>>> Zalamea's opinion (which I share) that Peirce's mathematical conception of >>>> continuity is more consistent with category theory (synthetic/top-down) >>>> than set theory (analytic/bottom-up). >>>> GR: Zalamea is, in my estimation, one of, if not the leading >>>> contemporary expert writing on mathematical continuity today. Again, I >>>> would be most interested in your thoughts, Robert, about "the >>>> conceptual relationship between Peirce's trichotomic category theory and >>>> contemporary mathematical category theory if any." >>>> >>>> Best. >>>> >>>> Gary R >>>> >>>> "Time is not a renewable resource." gnox >>>> *Gary Richmond* >>>> *Philosophy and Critical Thinking* >>>> *Communication Studies* >>>> *LaGuardia College of the City University of New York* >>>> >>>>>
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