Thursday, May 6, 2021

Research Timeline (starting from January 2021)

May-June 2021

  • The paper on linear exactness properties, joint work with Pierre-Alain Jacqmin, was accepted for publication in Journal of Algebra.
  • Started supervision of PhD studies of Brandon Laing on the SOFiA project.
  • To be starting supervision of PhD studies of Ineke van der Berg.
  • Current plan: to finalize the papers on binary matrix properties, noetherian form of sets and matches of digraphs.

March-April 2021

  • Started research on matches of digraphs: pioneering joint work with Francois van Niekerk and Jade Viljoen (research grew out from her honors project).
  • Started research on binary matrix properties: joint work with Michael Hoefnagel, Pierre-Alain Jacqmin and Emil van der Walt (undergraduate student) on the structure of the poset of matrix properties. The project grew out from Emil solving problems that Michael and I suggested to him in the fall of 2021, which naturally evolved from the joint work with Michael and Pierre-Alain. My co-authors finalized a paper based on Emil's results and I have rejoined them as a co-author.
  • Revisited research SOFiA: joint work with Louise Beyers, Gregor Feierabend and Brandon Laing. First python implementation of the SOFiA proof system produced. As a result, its deduction rules were refined.
  • Started supervision of MSc studies of Daniella Mooore on categorical aspects of near-vector spaces (cosupervised by Karin Howell)

January-February 2021

  • The paper on stability of exactness properties under pro-completion, a 7 year old joint work with Pierre-Alain Jacqmin, was published in Advances in Mathematics.
  • Revisited research on the noetherian form of sets: together with Francois van Niekerk, we are elaborating the proof of our recent theorem that the category of sets provides a model for the self-dual axiomatic setup for homomorphism theorems proposed in my publication no. 35. The paper is almost complete.

Thursday, April 8, 2021

Publications

  1. Z. Janelidze, Characterization of pointed varieties of universal algebras with normal projections, Theory and Applications of Categories 11, 2003, 212-214
  2. Z. Janelidze, Varieties of universal algebras with normal local projections, Georgian Mathematical Journal 11, 2004, 93-98
  3. Z. Janelidze, Subtractive categories, Applied Categorical Structures 13, 2005, 343-350
  4. Z. Janelidze, Closedness properties of internal relations I: A unified approach to Mal’tsev, unital and subtractive categories, Theory and Applications of Categories 16, 2006, 236-261
  5. Z. Janelidze, Closedness properties of internal relations II: Bourn localization, Theory and Applications of Categories 16, 2006, 262-282
  6. Z. Janelidze, Closedness properties of internal relations IV: Expressing additivity of a category via subtractivity, Journal of Homotopy and Related Structures 1, 2006, 219-227
  7. Z. Janelidze, Closedness properties of internal relations III: Pointed protomodular categories, Applied Categorical Structures 15, No. 3, 2007, 325-338
  8. Z. Janelidze, Closedness properties of internal relations V: Linear Mal’tsev conditions, Algebra Universalis 58, 2008, 105-117
  9. D. Bourn and Z. Janelidze, Approximate Mal’tsev operations, Theory and Applications of Categories 21, 2008, 152-171
  10. Z. Janelidze, Cover relations on categories, Applied Categorical Structures 17, 2009, 351-371
  11. D. Bourn and Z. Janelidze, Subtractive categories and extended subtractions, Applied Categorical Structures 17, 2009, 317-343
  12. D. Bourn and Z. Janelidze, Pointed protomodularity via natural imaginary subtractions, Journal of Pure and Applied Algebra 213, 2009, 1835-1851
  13. Z. Janelidze, Closedness properties of internal relations VI: Approximate operations, Cahiers de Topologie et G&eacuteom&eacutetrie Diff&eacuterentielle Cat&eacutegoriques 50, 2009, 298-319
  14. Z. Janelidze, The pointed subobject functor, 3x3 lemmas, and subtractivity of spans, Theory and Applications of Categories 23, 2010, 221-242
  15. Z. Janelidze and A. Ursini, Split short five lemma for clots and subtractive categories, Applied Categorical Structures 19, 2011, 233-255
  16. D. Bourn and Z. Janelidze, Categorical (binary) difference terms and protomodularity, Algebra Universalis 66, 2011, 277-316
  17. Z. Janelidze and N. Martins-Ferreira, Weakly Mal’tsev categories and strong relations, Theory and Applications of Categories 27, 2012, 65-79
  18. M. Gran, Z. Janelidze, D. Rodelo, and A. Ursini, Symmetry of regular diamonds, the Goursat property, and subtractivity, Theory and Applications of Categories 27, 2012, 80-96
  19. M. Gran, Z. Janelidze, and A. Ursini, A good theory of ideals in regular multi-pointed categories, Journal of Pure and Applied Algebra 216, 2012, 1905-1919
  20. M. Gran, Z. Janelidze, and D. Rodelo, 3x3 lemma for star-exact sequences, Homology, Homotopy and Applications 14(2), 2012, 1-22
  21. Z. Janelidze, An axiomatic survey of diagram lemmas for non-abelian group-like structures, Journal of Algebra 370, 2012, 387-401
  22. D. Rodelo, Z. Janelidze, and T. Van der Linden, Hagemann’s theorem for regular categories, Journal of Homotopy and Related Structures 9, 2014, 55-66
  23. Z. Janelidze, On the Form of Subobjects in Semi-Abelian and Regular Protomodular Categories, Applied Categorical Structures 22, 2014, 755-766
  24. M. Gran and Z. Janelidze, Star-regularity and regular completions, Journal of Pure and Applied Algebra 218, 2014, 1771-1782
  25. Z. Janelidze and T. Weighill, Duality in non-abelian algebra I. From cover relations to grandis ex2-categories, Theory and Applications of Categories 29, 2014, 315-331
  26. D. Bourn and Z. Janelidze, A note on the abelianization functor, Communications in Algebra 44, 2016, 2009-2033
  27. Z. Janelidze and T. Weighill, Duality in non-abelian algebra II. From Isbell bicategories to Grandis exact categories, Journal of Homotopy and Related Structures 11, 2016, 553-570
  28. Z. Janelidze, Duality for diagram chasing a la Mac Lane in non-abelian categories, Homology, Homotopy and Applications 18, 2016, 85-106
  29. Z. Janelidze and T. Weighill, Duality in non-abelian algebra III. Normal categories and 0-regular varieties, Algebra Universalis 77, 2017, 1-28
  30. Z. Janelidze and N. Martins-Ferreira, Involution-rigidness – a new exactness property, and its weak version, Journal of Algebra and Its Applications 16, 2017, 1750074
  31. Z. Janelidze and E. Vitale, Snail lemma in a pointed regular category, Journal of Pure and Applied Algebra 221, 2017, 135-143
  32. M. Duckerts-Antoine, M. Gran and Z. Janelidze, Epireflective subcategories via epi-closure operators, Theory and Applications of Categories 32, 2017, 526-546
  33. A. Goswami and Z. Janelidze, On the structure of zero morphisms in a quasi-pointed category, Applied Categorical Structures 25, 2017, 1037-1043
  34. Z. Janelidze and A. M. Abdalla, An order-theoretic perspective on categorial closure operators, Quaestiones Mathematicae 41, 2018, 529-539
  35. A. Goswami and Z. Janelidze, Duality in non-abelian algebra IV. Duality for groups and a universal isomorphism theorem, Advances in Mathematics 349, 2019, 781-812
  36. M. Hoefnagel, Z. Janelidze, and D. Rodelo, On difunctionality of class relations, Algebra Universalis 81, Article 19, 2020 (15 pages)
  37. P.-A. Jacqmin and Z. Janelidze, On stability of exactness properties under the pro-completion, Advances of Mathematics 377, 2021, 107484 (55 pages)
  38. Z. Janelidze, H. Prodinger, and F. van Niekerk, Combinatorics arising from lax colimits of posets, submitted for publication in 2020 (29 pages)
  39. Z. Janelidze, I. van der Berg, A Dedekind-style axiomatization and the corresponding universal property of an ordinal number system, submitted for publication in 2020 (22 pages)
  40. M. Hoefnagel, P.-A. Jacqmin, and Z. Janelidze, The matrix taxonomy of finitely complete categories, submitted for publication in 2020 (38 pages)
  41. P.-A. Jacqmin and Z. Janelidze, On linear exactness properties, submitted for publication in 2020 (43 pages)
  42. Z. Janelidze and F. K. van Niekerk, An introduction to noetherian forms admitting exact join decomposition, in preparation since 2019
  43. Z. Janelidze, F. K. van Niekerk and J. Viljoen, On the endomatch index of a digraph, in preparation since 2020
  44. M. Hoefnagel, P.-A. Jacqmin, Z. Janelidze, and E. van der Walt, On binary matrix properties, in preparation since 2020

Saturday, April 3, 2021

Lecture Notes

Thursday, March 25, 2021

Research Topic: 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 realised 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 Maclarty's article.

See this list for relevant papers in this research area.

Research Topic: 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 Piere-Alain Jacqmin.