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Ioannis Markakis

PhD Candidate
University of Cambridge
ioannis.markakis@cl.cam.ac.uk


About Me

I am a Ph.D. candidate at the Computer Lab of the University of Cambridge, and member of Fitzwilliam College. My research interests lie in the intersection of mathematics and theoretical computer science. More specificallly, I am interested in higher category theory, and type-theoretic methods in it. I have studied and generalised the notion of computads to theories that are not necessarily globular, and I am currently thinking about comparisons of globular and simplicial notions of higher categories. My studies are supported by a scholarship of the Onassis Foundation.

Papers

  1. Thibaut Benjamin, Ioannis Markakis, Chiara Sarti
    40th Mathematical Foundations of Programming Semantics (2024)
    Abstract

    Two novel descriptions of weak \(\omega\)-categories have been recently proposed, using type-theoretic ideas. The first one is the dependent type theory CaTT whose models are \(\omega\)-categories. The second is a recursive description of a category of computads together with an adjunction to globular sets, such that the algebras for the induced monad are again \(\omega\)-categories. We compare the two descriptions by showing that there exits a fully faithful morphism of categories with families from the syntactic category of CaTT to the opposite of the category of computads, which gives an equivalence on the subcategory of finite computads. We derive a more direct connection between the category of models of CaTT and the category of algebras for the monad on globular sets, induced by the adjunction with computads.

  2. Thibaut Benjamin, Ioannis Markakis
    Preprint (2024)
    Abstract

    We define inductively the opposites of a weak globular \(\omega\)-category with respect to a set of dimensions, and we show that the properties of being free on a globular set or a computad are preserved under forming opposites. We then provide a new description of hom \(\omega\)-categories, and show that the opposites of a hom \(\omega\)-category are hom \(\omega\)-categories of opposites of the original \(\omega\)-category.

  3. Ioannis Markakis
    Journal of Pure and Applied Algebra (2024)
    Abstract

    We introduce a notion of signature whose sorts form a direct category, and study computads for such signatures. Algebras for such a signature are presheaves with an interpretation of every function symbol of the signature, and we describe how computads give rise to signatures. Generalising work of Batanin, we show that computads with certain generator-preserving morphisms form a presheaf category, and describe a forgetful functor from algebras to computads. Algebras free on a computad turn out to be the cofibrant objects for certain cofibrantly generated factorisation system, and the adjunction above induces the universal cofibrant replacement, in the sense of Garner, for this factorisation system. Finally, we conclude by explaining how many-sorted structures, weak \(\omega\)-categories, and algebraic semi-simplicial Kan complexes are algebras of such signatures, and we propose a notion of weak multiple category.

  4. Christopher J. Dean, Eric Finster, Ioannis Markakis, David Reutter, Jamie Vicary
    Advances in Mathematics (2024)
    Abstract

    We give a new description of computads for weak globular ω-categories by giving an explicit inductive definition of the free words. This yields a new understanding of computads, and allows a new definition of \(\omega\)-category that avoids the technology of globular operads. Our framework permits direct proofs of important results via structural induction, and we use this to give new proofs that every ω-category is equivalent to a free one, and that the category of computads with variable-to-variable maps is a presheaf topos, giving a direct description of the index category. We prove that our resulting definition of \(\omega\)-category agrees with that of Batanin and Leinster and that the induced notion of cofibrant replacement for \(\omega\)-categories coincides with that of Garner.

  5. Dimitrios Dais, Ioannis Markakis
    Preprint (2017)
    Abstract

    Let \(X_P\) be the projective toric surface associated to a lattice polytope \(P\). If the number of lattice points lying on the boundary of \(P\) is at least \(4\), it is known that \(X_P\) is embeddable into a suitable projective space as zero set of finitely many quadrics. In this case, the determination of a minimal generating system of the toric ideal defining \(X_P\) is reduced to a simple Gaussian elimination.

Talks

  1. Masaryk University Algebra Seminar
    Masaryk University (Apr 03, 2024)
  2. Peripatetic Seminar on Sheaves and Logic 107
    National Technical University of Athens (Apr 01, 2023)
  3. Yorkshire and Midlands Category Theory Seminar 30
    University of Birmingham (Mar 15, 2023)
  4. Topology, Algebra and Categories in Logic 2022
    University of Coimbra (Jun 23, 2022)

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