Research Feed
November 2024
consider the class of all one-one transformations of the space, or universe of discourse, or “world”, onto itself. What will be the science which deals with the notions invariant under this widest class of transformations? Here we will have very few notions, all of a very general character. I suggest that they are the logical notions, that we call a notion “logical” if it is invariant under all possible one-one transformations of the world onto itself (“What are logical notions?”, Alfred Tarski, 1986, p. 149).
“Finitely complete co-pretoposes are ideally exact.” James R. A. Gray
abstract. Semi-abelian categories were introduced in (JMT) to provide a categorical context in which to give a general treatment of the properties of the categories of groups, Lie algebras, and other related algebraic structures. Certain categories which have been of interest in classical algebra are not semi-abelian; these include rings with identity, Boolean algebras, and Heyting algebras (among others). Ideally exact categories were introduced in (J) as a non-pointed counterpart to semi-abelian categories and include the above-mentioned categories as examples.
A topos provides a categorical context in which one can “do mathematics.” Examples of toposes include the category of sets or, more generally, functor categories of sets, as well as the category of sheaves on a site. It turns out that every cotopos is ideally exact (see (B) for exactness and protomodularity). The category of compact Hausdorff spaces is not a topos but satisfies two important properties of a topos, namely, it is extensive and exact. It was shown in (BC) that the opposite of the category of pointed objects of the category of compact Hausdorff spaces is semi-abelian.
In this talk, we will briefly outline the known proof showing that co-toposes are ideally exact and give an outline of the proof that finitely complete co-pretoposes are ideally exact.
(B) D. Bourn, Protomodular aspect of the dual of a topos, Adv. Math. 187 (2004), no. 1, 240-255
(BC) F. Borceux, and M. M. Clementino, On toposes, algebraic theories, semi-abelian categories and compact Hausdorff spaces,
(JMT) G. Janelidze, L. M´arki, and W. Tholen, Semi-abelian categories, Category theory 1999 (Coimbra), J. Pure Appl. Algebra 168 (2002), no. 2-3, 367-386
(J) G. Janelidze, Ideally exact categories, Theory and Applications of Categories, Vol. 41, No. 11, 2024, pp. 414–425.
Category Theory Seminar Talk, November 6
October 2024
“On extensivity of morphisms.” Michael Hoefnagel
abstract. Extensivity of a category may be described as a property of coproducts in the category, namely, that they are disjoint and universal. An alternative viewpoint is that it is a property of morphisms in a category. This talk explores this point of view through a natural notion of extensive and coextensive morphism. There is an interesting interplay between the algebraic and the categorical. One the one hand, through these notions several topics in algebra related to (unique) factorisation and refinement of direct products, such as the strict refinement or Fraser-Horn properties, take categorical form, and thereby enjoy all the benefits of categorical generalisation. On the other hand, the algebraic theory surrounding these topics inspire categorical results. One such result is that a Barr-exact category is coextensive if and only if every split monomorphism in the category coextensive.
Category Theory Seminar Talk, October 30
Facets of descent I by George Janelidze & Walter Tholen.
Sheaves are presheaves with an equalizer diagram
Prof. George Janelidze. (University of Cape Town) Effective descent morphisms of filtered preorders. Strict monadic topology II: Descent for closure spaces. Strict monadic topology I: First separation axioms and reflections. Split extensions and semidirect products of unitary magmas. Picard-Vessiot and categorically normal extensions in differential-difference Galois theory. The monads of classical algebra are seldom weakly cartesian.
Prof. Marino Gran. (Université catholique de Louvain)
Prof. Zurab Janelidze. (Univeristy of Stellenbosch)
Dr. Michael Hoefnagel. (Univeristy of Stellenbosch)
Dr. Pierre-Alain Jacqmin. (Université catholique de Louvain) Embedding theorems in non-abelian categorical algebra.
Thomas Mbewu. (University of Cape Town) Coverings and descent theory of finite spaces.
Jacob Lund. (University of Cape Town) Finite covering spaces. (unpublished).
September, 2024
Attending weekly seminars held by Prof. G. Janelidze at UCT with Stellenbosch University. Seminars are composed of research talks by members of the category theory research group. August 2024 - present.
August, 2024
Concept design minechain.gg (previously coalonsolana). PoW mining of linked cryptocurrencies with crafting mechanics. Situated within the $ORE (ore.supply) ecosystem on Solana. [under continuous development].
Open source contribution. Prototype proof of work mining client for ore-hq-client First $ORE mining pool with delegated staking.
September, 2023
I was contacted by Yvonne Lundie from the resource centre unit of SAOTA to consult with the unit on archiving architectural materials with a view towards the right use of artificial intelligence.
June, 2022
In the algebraic topology course at UCT we used the textbook A first course in algebraic topology by Czes Kosniowski and in the differential geometry course we used Geometry, Topology, and Physics by M. Nakahara.
January, 2022
In preperation for a course in category theory at UCT, I’ve started reading Categories for the Working Mathematician by Saunders Mac Lane.
I was given 99 Variations on a Proof by Philip Ording and have been periodically going through the proofs. I find the way the book illuminates the importance of style in mathematical writing interesting and it has solidifed certain ideas about my own style in mathematics. The introduction mentions how the author took inspiration from the Oulipo, specifically Raymond Queneau’s Exercises in Style. I had come across the Oulipo a number of times before this and was interested in their work in relation to their use of mathematics in literature as well as their focus on structural constraints on literature. To fill this gap in my knowledge, I started reading the Oulipo Compendium and All That Is Evident Is Suspect: Readings from the Oulipo: 1963 - 2018.
I recently finished The Shape of Space by Jeffrey R. Weeks. The book stretched my imagination and I had to expand my capability of visualisation to imagine the weird spaces mentioned in the book. I used this book as a visual complement to the differential geometry course I took as well as using the new textbook Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts by Tristan Needham. I found his exposition, based very much on visual intuition, to be a good complement to the much more rigorous and algebraic course notes I received. I came to this textbook after reading Needham’s Visual Complex Analysis which took a very different approach to my complex analysis course however I found the visual intuiton of the book quite beautiful.
In order to gain more exposure to general relativity and find out more about Albert Einstein’s work I started reading Subtle is the Lord: The Science and the Life of Albert Einstein by Abraham Pais. This has got to be one of the greatest biographies of Einstein, as Roger Penrose wrote, “Here, surely, is the biography that Einstein would have wanted.” Much of the book is an exposition of Einstein’s work and how he came to his essential ideas. I’ve also been reading the new biography Journey to the Edge of Reason: The Life of Kurt Gödel by Stephen Budiansky which not only covers Gödel’s life and work but also sets the tone by describing early 1900s Vienna as well as the inception of the Vienna Circle.
As a result of my interest in modern continental philosophy, I finished reading Alain Badiou’s In Praise of Mathematics which explores, through a dialogue, mathematics’ crucial influence on philosophy as well as Badiou’s fascination with the subject. Although, he doesn’t mention much contemporary mathematics.
February, 2021
Probably my favorite piece of “fiction”, When we cease to understand the world by Benjamín Labatut, is a masterpiece of magical realism. I put fiction in inverted commas since the novel is entirely based on true events, but as you move through the work, more artistic liberty is taken by the author. It follows the lives and discoveries of some of the greatest mathematical minds of the 20th century. I particularly loved Labatut’s magical rendering of Grothendieck’s life.
Unfortunately, I’m about to finish Borge’s Labyrinths. I’ve loved reading every short story, and in all of them, Borge’s overwhelming intellect and knowledge is present. I am now moving on to his Selected Non Fictions.
I’m working through a few textbooks. Conceptual Mathematics: A first introduction to categories - Lawvere & Schanuel Abel’s Theorem in Problems and Solutions - V.B. Alekseev (based on the lectures by Professor V.I. Arnold)
Although our prescribed reading for topology is Munkres, I’m also reading the following as supplements. Topology and Groupoids - Ronald Brown Counterexamples in Topology - Steen & Seebach
For my group theory course I decided to follow Algebra - Birkhoff & MacLane for its use of categories.
Mathematics and Computation: A Theory Revolutionizing Technology - Avi Wigderson. such a beautiful book.
supplement. finite languages and automata course. Introduction to Theory of Computation. Mathematics and Computation.
Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning - Coecke & Kissinger.
After Finitude: An Essay on the Necessity of Contingency - Quentin Meillassoux. Meillassoux’s work. Quentin Meillassoux: Philosophy in the Making - Graham Harman.
philosophy of contemporary mathematics. Synthetic Philosophy of Contemporary Mathematics - Fernando Zalamea. Update: I’ve finished reading most of the book, mainly jumping between sections. A lot of the content went over my head, however I found the sections on Albert Lautman and Michael Atiyah particularly illuminating, especially being introduced to Lautman’s work on structuralism which I had not been exposed to before. I found the Bibligoraphic Survey and biographical sketches most accessible and thoroughly enjoyed those aspects of the book. I’ll definitely be coming back to this book often.
Reading quite a bit about Mark Fisher. Capitalist Realism. contact with the work of CCRU. Cyberpositive o(rphan) d(rift>)
Deleuze and Guattari are mentioned a lot in Mark Fisher’s work. Critical Lives: Gilles Deleuze - Frida Beckman.
I’m not reading enough African philosophers. Necropolitics - Achille Mbembe.