CAUTION: The Sender of this email is not from within Dalhousie. Dear Professor Phillip-Jan van Zyl, Thanks so much for your detailed insights! Please allow me to write to you again after studying your clarifications. Thank you, posina P.S. I'd also like to take this opportunity to invite you all to my Conceptual Mathematics seminar series (Tuesdays, 3:00-4:00 PM, Bengaluru time). Please send me your email and I'll send you the Google Meet link. Tomorrow I will discuss the calculation of subobject classifiers of various categories in terms of inverse images of parts of the basic shapes along structural maps. The overall objective of the seminar series is to study Professor F. William Lawvere's functorial semantics of algebraic theories (http://tac.mta.ca/tac/reprints/articles/5/tr5.pdf), i.e. the mathematics of representation: given a category of particulars, how we conceptualize the category in terms of measured properties (of its objects) and their mutual determinations (how we abstract the essence--basic shapes and their incidences--of the category in which every object of the category partakes), and how we model/represent objects of the category, based on the thus abstracted essence/theory, as structures in a background category of structureless sets (http://www.math.union.edu/~niefiels/13conference/Web/). Given the representational nature of our conscious experiences, the mathematics of representation is particularly relevant for the maturation of consciousness studies. I'll also introduce Bastiani and Ehresmann's sketch theory (http://www.numdam.org/item/CTGDC_1972__13_2_104_0.pdf), which has already been applied to the study of brain, mind, consciousness, and the self (cf. Memory Evolutive Systems). Grothendieck's descent theory is also related (p. 15 in http://www.mat.uc.pt/~picado/lawvere/interview.pdf), but by the time I get to descent, it's very likely that I'll be with Grothendieck (given my pace ;-) From a mathematical perspective, I'll discuss the adjointness between product and exponentiation, which, in brief, is the 1-1 correspondence between two-variable functions and one-variable functions with functions as values. I'll do exercises on natural transformations (e.g. Exercise 7.22(a), Sets for Mathematics, p. 135), unity-and-identity of adjoint opposites (discrete --| points --| codiscrete), left- and right-actions of various monoids, and figure geometry vis-a-vis function algebra (Conceptual Mathematics, pp. 370-371). On Mon, Sep 27, 2021 at 2:19 AM Phillip-Jan van Zyl <mikorym@protonmail.com> wrote:
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
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