Re: Parts and telling apart

Re: Parts and telling apart

[Posina Venkata Rayudu]

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/S0002-9904-1948-09052-8.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 = {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


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Parts and telling apart

[Phillip-Jan van Zyl]

Dear Posina

I am certainly not able to answer all of your questions in a way to captures every subtlety therein, but I can offer the following insights:

1. Functions between sets are indeed prototypical in the creation of a category, but you can always regard a function simply as a relation. Then all the functions have inverse relations. If you want to argue about the "inside" of such relations, you can do this by chasing subobjects. Chasing subobjects seems to be one fruitful way of getting around the fact that not all functions 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 from studying functions 2 -> X in a category. You can classify partitions in Sets in this way, by selecting pairs of (possibly the same) elements in equivalence classes. If you select distinct elements, this means that you select instead elements in different equivalance classes. In the category of groups 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 = 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, then this induces a image, i.e. the singleton {f(x)}. But if you induce instead a partition, you always induce the discrete partition: the partition where every point is in its own equivalence class. This is a way to study Set such that it mimics the attribute in groups where f(1) = 1, where 1 is the  group identity, i.e. constant functions. You can instead study the category 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 definitions and arguments are not guaranteed to be standard.

Best regards
Phillip-Jan van Zyl

‐‐‐‐‐‐‐ Original Message ‐‐‐‐‐‐‐

On Saturday, September 25th, 2021 at 4:41 AM, Posina Venkata Rayudu <posinavrayudu@gmail.com> 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'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/S0002-9904-1948-09052-8.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 = {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
>
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]