Showing posts from October, 2021

Noetherian Forms

Link to a talk on noetherian forms at the PALS semnar: written summary , recording of the talk . Noetherian forms are mathematical structures defined by self-dual axioms, that include all lattices, Janelidze-Marki-Tholen semi-abelian categories and Grandis exact categories. They can be seen as a realization of Saunders Mac Lane's hypothesis from his 1950 paper on Duality for Groups that self-dual axioms can be found to treat isomorphism theorems for non-abelian groups, as this is realized for abelian groups with the notion of an abelian category. Abelian categories are actually given by the overlap of semi-abelian and exact categories. The term "noetherian" refers to the fact that these forms can seen as a fulfilment of Emmy Noether's program to "disregard the elements and operations in algebraic structures in favor of selected subsets, linked to homomorphisms between structures by the homomorphism and isomorphism theorems" - quote from Colin Macl

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

September-October 2021 Paper published:  linear exactness properties  in Journal of Algebra . See the paper on the journal's website here . Some progress made on the SOFiA project , including implementation of an intuitive command-line proof building interface for . Used this tool in the delivery of the Foundations of Abstract Mathematics I seminar.   The binary matrix properties paper finalized from my side. Worked on the revision of the matrix taxonomy paper. Revision in progress can be found here .  Together with Amartya Goswami, gave a NITheCS mini-school (October 2021)  on Elementary Introduction to Category Theory . See this blog for the write-up and links to video-recordings of lectures.  Talk given at SIC on forms vs monoidal categories . See the write-up and the recording of the talk here . See the recordings of all talks here . Examiners nominated for the MSc Thesis of Paul Hugo, to be finalized by the end of the year. July-August 2021 Further progress m

Séminaire Itinérant de Catégories October 2021


Forms vs monoidal categories

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