From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/10571 Path: news.gmane.io!.POSTED.blaine.gmane.org!not-for-mail From: Phillip-Jan van Zyl Newsgroups: gmane.science.mathematics.categories Subject: Re: Parts and telling apart Date: Sun, 26 Sep 2021 08:59:42 +0000 Message-ID: Reply-To: Phillip-Jan van Zyl Mime-Version: 1.0 Content-Type: text/plain; charset="utf-8" Content-Transfer-Encoding: quoted-printable Injection-Info: ciao.gmane.io; posting-host="blaine.gmane.org:116.202.254.214"; logging-data="28632"; mail-complaints-to="usenet@ciao.gmane.io" Cc: categories To: Posina Venkata Rayudu Original-X-From: majordomo@rr.mta.ca Sun Sep 26 22:50:20 2021 Return-path: Envelope-to: gsmc-categories@m.gmane-mx.org Original-Received: from smtp2.mta.ca ([198.164.44.75]) by ciao.gmane.io with esmtps (TLS1.2:ECDHE_RSA_AES_256_GCM_SHA384:256) (Exim 4.92) (envelope-from ) id 1mUb6K-0007G0-Ke for gsmc-categories@m.gmane-mx.org; Sun, 26 Sep 2021 22:50:20 +0200 Original-Received: from rr.mta.ca ([198.164.44.159]:58498) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1mUb1t-0004D9-1D; Sun, 26 Sep 2021 17:45:45 -0300 Original-Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1mUb36-000624-O5 for categories-list@rr.mta.ca; Sun, 26 Sep 2021 17:47:00 -0300 Precedence: bulk Xref: news.gmane.io gmane.science.mathematics.categories:10571 Archived-At: Dear Posina I am certainly not able to answer all of your questions in a way to capture= s every subtlety therein, but I can offer the following insights: 1. Functions between sets are indeed prototypical in the creation of a cate= gory, but you can always regard a function simply as a relation. Then all t= he functions have inverse relations. If you want to argue about the "inside= " of such relations, you can do this by chasing subobjects. Chasing subobje= cts seems to be one fruitful way of getting around the fact that not all fu= nctions are invertible. When arguing about sets you can chase subsets, but = you can also chase partitions. 2. I am not sure what a coseparator is, but note that nothing stops you fro= m studying functions 2 -> X in a category. You can classify partitions in S= ets in this way, by selecting pairs of (possibly the same) elements in equi= valence classes. If you select distinct elements, this means that you selec= t instead elements in different equivalance classes. In the category of gro= ups a homomorphism 2 -> G is forced to give you a torsion element of order = 2. In other words, it selects such g in G that g * g =3D 1. 3. In so far as pointedness is a categorical concept, the category of sets = has certain behaviour that is pointed. If you take any function 1 -> X, the= n this induces a image, i.e. the singleton {f(x)}. But if you induce instea= d a partition, you always induce the discrete partition: the partition wher= e every point is in its own equivalence class. This is a way to study Set s= uch that it mimics the attribute in groups where f(1) =3D 1, where 1 is the= group identity, i.e. constant functions. You can instead study the categor= y of pointed sets, or the dual category of the category of sets. Partitions= are subobjects in the dual category of Sets. Note that I am not a regular (person) in category theory circles so my defi= nitions and arguments are not guaranteed to be standard. Best regards Phillip-Jan van Zyl =E2=80=90=E2=80=90=E2=80=90=E2=80=90=E2=80=90=E2=80=90=E2=80=90 Original Me= ssage =E2=80=90=E2=80=90=E2=80=90=E2=80=90=E2=80=90=E2=80=90=E2=80=90 On Saturday, September 25th, 2021 at 4:41 AM, Posina Venkata Rayudu wrote: > I just thought of adding that the conditions for the representability > > of a category C as a functor category B^A is given my Roos' theorem > > (kindly translated by Professor Lafforgue): > > Roos' Theorem, as stated on page 415 of SGA4 (volume 1), says that the > > three following conditions on a topos E are equivalent : > > i) The family of essential points of E is conservative. (NB : A point > > is called "essential" when its fiber functor not only has a right > > adjoint but also a left adjoint.) > > ii) The full sub-category of E consisting in objects which are > > connected - non empty and projective is generating. > > iii) E is a presheaf topos. > > At the end of the volume, there is a reference to three notes (in > > French) of Roos at "Comptes Rendus de l'Acad=C3=A9mie des Sciences", with > > the general title "Distributivity of colimits with respect to limits > > in toposes". This reference is to CR 259 (August and September 1964) : > > p. 969-972, 1605-1608 and 1801-1804. > > In the context of the mutual relationship between change/variation and > > unity/cohesion, I'd like to add: > > A monoid with a single constant (as a kind of variation) gives rise to > > a topos (of right actions of the monoid), i.e. idempotents (Ex. 5 in > > Conceptual Mathematics, p. 367; which is a quality type; Definition 1 > > in http://tac.mta.ca/tac/volumes/19/3/19-03.pdf), whose subobject > > classifier is not connected, and hence fails to satisfy one of the > > axioms of cohesion (Axiom 2 in > > http://tac.mta.ca/tac/reprints/articles/9/tr9.pdf). > > On the other hand, a monoid with two constants (another kind of > > variation) gives rise to a cohesive topos (reflexive graphs; > > Conceptual Mathematics, p. 367). > > With regard to UNIVERSAL MAPPING PROPERTY, it was also not discovered > > in its "simplest" (structureless) instantiation of initial set (as > > that which has exactly one function to any set); it required a more > > structured setting: > > Samuel, P. (1948) On universal mappings and free topological groups, > > Bull. A.M.S. 54: 591-598 > > (https://www.ams.org/journals/bull/1948-54-06/S0002-9904-1948-09052-8/S00= 02-9904-1948-09052-8.pdf). > > Is it related to the difficulty of seeing function as a > > structure-preserving map, given that the structure preserved is > > structurelessness? > > Lastly, is the > > coseparator ~ subobject classifier > > (isomorphism, after discounting that the 'pointed' in pointed object > > definition of subobject classifier) valid in the discrete/constant > > subcategory of any topos (category with subobject classifier)? > > I eagerly look forward to your corrections. > > Thank you, > > posina > > On Fri, Sep 24, 2021 at 3:54 AM Posina Venkata Rayudu > > posinavrayudu@gmail.com wrote: > >> Dear All, >> >> I hope and pray you and your families are all well. >> >> If I may, as I was preparing to give a lecture on SUBOBJECT >> >> CLASSIFIER, as part of the Conceptual Mathematics seminar series at >> >> Poornaprajna Institute of Scientific Research (https://ppisr.res.in/), >> >> I started recollecting how much I loved the formula 2^|X| for the >> >> number of subsets of a set X, when I first learned it in high school >> >> (it was simple :-). I remember listing out all the eight subsets in >> >> the case of X =3D {a, b, c}. (Grothendieck's profound insight of >> >> defining subsets as 1-1 functions was definitely not part of my >> >> understanding of SUBSET.) And about learning that the number of >> >> functions from a domain set X to a codomain set Y is |Y|^|X|, and with >> >> |Y| =3D 2 we could clearly see that: >> >> the number of subsets of a set X =3D the number of functions from the >> >> set X to a two-element set 2 =3D 2^|X| >> >> It's not out of the realm of possibilities that I might have listed >> >> all the functions from X to 2, which, along with all the subsets of X, >> >> would have brought the 1-1 correspondence: >> >> parts of a set X >> ---------------- >> >> functions from X to 2 >> >> into a clear view. >> >> Be that as it may, what occluded SUBOBJECT CLASSIFIER from set >> >> theorists and mathematical logicians; even Grothendieck missed it, but >> >> in his characteristic kindness called it Lawvere element, upon >> >> Professor F. William Lawvere's definition of subobject classifier as >> >> part of his axiomatization of topos (please see p. 7, >> >> http://www.mat.uc.pt/~picado/lawvere/interview.pdf). >> >> This is not an isolated incident in science; it appears to be a >> >> pattern--wherein far-reaching constructions are not initially >> >> conceptualized/recognized in their simplest instantiation, which is >> >> where the figural salience of the concepts is clearly visible for all >> >> to see and use--in scientific practice. A similar case can be made >> >> about universal mapping properties (e.g. terminal set 1 =3D {*}) and >> >> about category theory itself. That sets and functions form a >> >> mathematical category is not easy to ignore, but category theory took >> >> birth in an inaccessible realm rather remote for sets and functions >> >> (thanks to the then prevalent practice of identifying a function with >> >> its graph ((a, f(a)); cf. Conceptual Mathematics, pp. 293-294). >> >> Be that as it may, subset [and its representability by maps to >> >> subobject classifier] is also related to the telling apart the figures >> >> constituting the inside of an object of a category (with subobject >> >> classifier or topos; see Sets for Mathematics, pp. 18-21). In the >> >> case of sets, with coseparator 2 =3D {*, } as the property type, there >> >> are enough properties to tell apart any two different 1-shaped figures >> >> (points/elements) in any set. Does subobject classifier (in every >> >> category with it/topos) always serve as the property type coadequate >> >> to tell apart figures in any object of the category/topos? What >> >> difference does the fact that subobject classifier is defined as a >> >> pointed object (map from the terminal object of the category to the >> >> subobject classifier, i.e., t: 1 --> 2, where t() =3D t; see Exercise 8 >> >> in Conceptual Mathematics, p. 337), which is not the case with >> >> coseparator, i.e. it is just a constant/discrete/abstract 2-element >> >> set. It seems a little odd that representing parts of an object >> >> amounts to telling apart all different figures constituting the >> >> object, simply going by the fact that PART (monomorphism) is a special >> >> type of FIGURE (a morphism A --> B is an A-shaped figure in B; >> >> Conceptual Mathematics, pp. 81-85; for additional context, among other >> >> mathematical clarifications, see >> >> https://cgasa.sbu.ac.ir/article_12425_b4ce2ab0ae3a843f00ff011b054f918b.= pdf). >> >> Telling apart figures in a domain object also appears to be the job of >> >> epimorphisms (with subobject classifier of the category as codomain >> >> object). >> >> Is >> >> coseparator ~ subobject classifier >> >> (isomorphism; after discounting that the 'pointed' in pointed object >> >> definition of subobject classifier) specific to localic (as opposed to >> >> cohesive) toposes (cf. Sets for Mathematics, pp. 93-94)? >> >> Also from a pedagogical perspective, given that our everyday >> >> experience is that of categories of objects, all of which partake in >> >> the essence/theory T characterizing a category C (Sets for >> >> Mathematics, pp. 154-155), and since all objects of any category >> >> partake in the essence T of the category, the transformations between >> >> objects of the category necessarily preserve the essence, and hence >> >> are structure-respecting maps (see Conceptual Mathematics, pp. >> >> 149-151), which, in turn, are representable as natural transformations >> >> (ibid, p. 378). Equally accessible is the idea that mathematical >> >> objects, which are about everyday objects, are not unlike everyday >> >> objects in that they are also made up of figures of various basic >> >> shapes and their incidences (my attempt to introduce basic shapes and >> >> their incidence relations to designers didn't get far; >> >> https://zenodo.org/record/3924760#.YUzpptJBzZ4). And, then, in >> >> addition to cohesion/essence/theory characterizing a category (in a >> >> sense, we are limiting ourselves to presheaves or those categories C >> >> that can be represented as contravariant functors M: T^op --> S, i.e. >> >> as diagrams in the category S of sets), we can see that the natural >> >> transformation (not unlike the transformations that we encounter in >> >> our everyday life such as a water flowing downhill) respects the >> >> essence of the object of a category that is being transformed (which >> >> is T in the natural transformation of M: T^op --> S to N: T^op --> S). >> >> Equally importantly, we can also begin with a mode of variation/change >> >> and arrive at a category of objects (topos of right actions of a >> >> monoid [objectifying change/variation]; Conceptual Mathematics, pp. >> >> 360-361). This direction, i.e. objectification of change is >> >> particularly important, given that we are given change (the basic >> >> building block of our conscious experiences is contrast) and those >> >> changes (natural transformations), in respecting/preserving the way >> >> figures of various basic shapes stick together, make it possible to >> >> reconstruct objects (as functors) based on the given change/variation >> >> (but for natural transformations preserving the unity of objects >> >> transformed, by the time I reach Malabar Cafe for ginger tea, there is >> >> nothing stopping my leg being up in the clear skies of Bengaluru ;-) >> >> Is it inappropriate to claim that a universe of everyday experience is >> >> accounted for by the notion of: >> >> categories of objects >> >> along with the mutual determination of: >> >> change/variation <--> cohesion/unity >> >> and does so in a manner accessible to total beginners (here's an >> >> everyday objectification of change; https://youtu.be/r0kLC-pridI). >> >> Your time permitting, please correct any mistakes I might have made in >> >> my characterization of categories of objects. >> >> I eagerly look forward to your corrections. >> >> Thanking you, >> >> posina > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] [For admin and other information see: http://www.mta.ca/~cat-dist/ ]