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Showing posts from October, 2021

Matrix Properties

Matrix properties are a particular type of exactness properties that can be seen as category-theoretic analogues of linear Mal'tsev conditions in Universal Algebra. See this list for relevant papers in this research area. The study of matrix properties led to the theory of "approximate operations" developed jointly with Dominique Bourn, and a general theory of exactness properties developed jointly with Pierre-Alain Jacqmin. Work in progress on matrix properties: Open problem on finding an algorithm for implication of basic matrix properties solved - see the working version of the  preprint .  Even for binary matrices, the preorder of implications is quite complex. Some new results on this appear in this work in progress. Python implementation of the algorithm for deducing implication of (basic) matrix properties can be found here . The program needs to be improved in some future.

2021 Academic Activities

November-December 2021 Finalized marks for Foundations of Abstract Mathematics I, II and an honors module in Logic. Resumed research on a  noetherian form of sets . The binary matrix properties paper submitted for publication (corresponding author: M. Hoefnagel) - see the submitted version of the paper here . The revised version of the paper on  matrix taxonomy re-submitted for publication (corresponding author: M. Hoefnagel) - see the new version here . Talk given at SAMS Congress on the matrix project - see the write-up of the talk here . At the SAMS AGM held during the congress, I was appointed to serve on the SAMS Council as the President of the South African Mathematical Society for 2022-2023. Plenary talk given at the 4th BRICS Mathematics Conference  on noetherian forms   ( slides , recording ). This talk was given jointly with Amartya Goswami. Funding awarded for the NITheCS research programme in Mathematics, entitled " Space-like mathematical structures and related

Séminaire Itinérant de Catégories October 2021

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Forms vs monoidal categories

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Below is a summary of the talk given at the Séminaire Itinérant de Catégories  (8 October 2021), prepared before the talk.  The talk is mainly based on Zurab Janelidze's joint work in progress with Francois van Niekerk, as well as his earlier work on forms with former collaborators. The talk assumes that the listener is familiar with basic ideas and concepts of category theory found in Categories for the Working Mathematician by Saunders Mac Lane (in particular, Chapters I, VII and VIII), as well as with the notions of factorization system and Grothendieck fibration. 1. Biproducts, products, sums and monoidal categories The goal of this talk is to explain the following diagram: The notion of an abelian category brings together various important categories of abstract mathematics, such as the categories of modules, which includes the category of vector spaces as well as the category of abelian groups. In an abelian category, the monoidal structure of product and the monoidal stru