John Bourke

Associate Professor
Department of Mathematics and Statistics
Masaryk University
Brno 61137
Czech Republic
bourkej@math.muni.cz

News

  • The annual international conference in category theory CT2025 will take place at Masaryk University from July 13-19 2025!

    • Seminar

  • Algebra Seminar webpage, occasionally updated.

    • Editor

  • Journal of Pure and Applied Algebra.
  • Theory and Applications of Categories.

    • Preprints

    Enhanced 2-categorical structures, two-dimensional limit sketches and the symmetry of internalisation (with Nathanael Arkor and Joanna Ko), 2024. Submitted for publication.

    Publications

    On 2-categorical infinity-cosmoi (with Steve Lack), Journal of Pure and Applied Algebra, 228(9) 2024.
    A skew approach to enrichment for Gray-categories (with Gabriele Lobbia), Advances in Mathematics 434 (2023), 109327.
    An orthogonal approach to algebraic weak factorisation systems, Journal of Pure and Applied Algebra, 227(6), 2023, 107294.
    Accessible infinity-cosmoi (with Steve Lack), Journal of Pure and Applied Algebra 227 (2023) 107255.
    Adjoint functor theorems for homotopically enriched categories (with Steve Lack and Lukáš Vokřínek), Advances in Mathematics 412 (2023) 108812.
    Algebraically cofibrant and fibrant objects revisited (with Simon Henry), Homology, Homotopy and Applications 24(1), 2022, 271-298.
    Accessible aspects of 2-category theory, Journal of Pure and Applied Algebra 225(3), 2021.
    Iterated algebraic injectivity and the faithfulness conjecture, Higher Structures 4(2), 2020, 183–210.
    Braided skew monoidal categories (with Stephen Lack), Theory and Applications of Categories, Vol. 35, 2020, No. 2, pp 19-63.
    Equipping weak equivalences with algebraic structure, Mathematische Zeitschrift, 294(3), 2020, 995-1019
    Monads and theories (with Richard Garner), Advances in Mathematics 351 (2019), 1024-1071.
    Free skew monoidal categories (with Stephen Lack), Journal of Pure and Applied Algebra 222 (2018), no. 10, 3255-2381.
    Skew monoidal categories and skew multicategories (with Stephen Lack), Journal of Algebra 506 (2018), 237-266.
    Note on the construction of globular weak omega-groupoids from types, topological spaces etc , Cahiers de topologie et geometrie differentielle categoriques Vol. LVII (2016), 281-285.
    Skew structures in 2-category theory and homotopy theory , Journal of Homotopy and Related Structures 12 (2017), no. 1, 31-81.
    The Gray tensor product via factorisation (with Nick Gurski), Applied Categorical Structures 25 (2017), no.4, 603–624
    Algebraic weak factorisation systems II: categories of weak maps (with Richard Garner), Journal of Pure and Applied Algebra 220 (2016), no. 1, 148-174.
    Algebraic weak factorisation systems I: accessible AWFS (with Richard Garner), Journal of Pure and Applied Algebra 220 (2016), no. 1, 108-147.
    A cocategorical obstruction to tensor products of Gray-categories (with Nick Gurski), Theory and Applications of Categories 30 (2015), no. 11, 387-409.
    Two-dimensional regularity and exactness (With Richard Garner), Journal of Pure and Applied Algebra 218 (2014), no. 7, 1346-1371.
    Two-dimensional monadicity, Advances in Mathematics 252 (2014) 708-747.
    A colimit decomposition for homotopy algebras in Cat, Applied Categorical Structures 22 (2014), no. 1, 13-28.
    On semiflexible, flexible and pie algebras (with Richard Garner), Journal of Pure and Applied Algebra 217 (2013), no. 2, pages 293–321.

    Phd thesis

    Codescent objects in 2-dimensional universal algebra , Phd Thesis, University of Sydney, 2011.

    Past local conferences

  • The 106th Peripatetic Seminar on Sheaves and Logic was held in Brno on 14-15 May 2022. Here is the webpage.
  • The 95th Peripatetic Seminar on Sheaves and Logic was held in Brno in April 2014. Here is the webpage and here is the conference photo.

    • Phd students (past and present)

  • Miloslav Štěpán (graduated Feb 2025)
  • Jan Jurka (current)
  • Joanna Ko (current)

    • Some lecture notes for courses I taught

    Higher categories (Spring 2023): This was a course for graduate students. The first half of the course studied globular higher groupoids (Grothendieck infinity-groupoids) in detail, formulating the homotopy hypothesis and constructing the globular infinity-groupoid associated to identity types. Along the way, we look at generalised flavours of universal algebra, and the relationship between the small object argument and free constructions in universal algebra. The second half of the course looks briefly at different simplicial definitions of higher category. It starts with (infinity,1)-categories, before discussing infinity cosmoi, and then looking at several types of (infinity,2)-category, before ending with complicial sets.

    Representation theory (Spring 2019): These were notes for a first course on representation theory. The students all had some background in category theory, so the course is a bit categorical in places.

    If you read these notes and find any mistakes or have any comments, I'd be glad to hear from you!