CAUTION: The Sender of this email is not from within Dalhousie. 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/S0002-9...).
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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