CAUTION: The Sender of this email is not from within Dalhousie. 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'Académie 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/S0002-9...). 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 = {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| = 2 we could clearly see that:
the number of subsets of a set X = the number of functions from the set X to a two-element set 2 = 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 = {*}) 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 = {*, *} 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(*) = 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
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