arXiv subjects

Ioannis Markakis

Publications and source records attributed to Ioannis Markakis.

10 recordsLinked to original sources

arXiv2026

Computads with invertible generators for weak {\omega}-categories

We extend the notion of computads for weak $\omega$-categories to allow marking certain generators as invertible, and describe inductively the free $\omega$-categories they generate. This gives a simple, finite description of the walking equivalences, the $\omega$-categories classifying invertible cells. We then construct a coreflection from generalised to ordinary computads, preserving the generated $\omega$-categories, and conclude that $\omega$-categories generated by generalised computads are cofibrant. Finally, we study the subcategory of generalised computads and generator-preserving morphisms, and show that it is a presheaf topos, similarly to the case of ordinary computads.

math.CT↗

arXiv2026

A type theory for invertibility in weak $\omega$-categories

We present a conservative extension ICaTT of the dependent type theory CaTT for weak $\omega$-categories with a type witnessing coinductive invertibility of cells. This extension allows for a concise description of the "walking equivalence" as a context, and of a set of maps characterising $\omega$-equifibrations as substitutions. We provide an implementation of our theory, which we use to formalise basic properties of invertible cells. These properties allow us to give semantics of ICaTT in marked weak $\omega$-categories, building a fibrant marked $\omega$-category out of every model of ICaTT.

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arXiv2025

Beyond Eckmann-Hilton: Commutativity in Higher Categories

We show that in a weak globular $\omega$-category, all composition operations are equivalent and commutative for cells with sufficiently degenerate boundary, which can be considered a higher-dimensional generalisation of the Eckmann-Hilton argument. Our results are formulated constructively in a type-theoretic presentation of $\omega$-categories. The heart of our construction is a family of padding and repadding techniques, which gives an equivalence relation between cells which are not necessarily parallel. Our work has been implemented, allowing us to explicitly compute suitable witnesses, which grow rapidly in complexity as the dimension increases. These witnesses can be exported as inhabitants of identity types in Homotopy Type Theory, and hence are of relevance in synthetic homotopy theory.

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arXiv2025

Naturality for higher-dimensional path types

We define a naturality construction for the operations of weak omega-categories, as a meta-operation in a dependent type theory. Our construction has a geometrical motivation as a local tensor product with a directed interval, and behaves logically as a globular analogue of Reynolds parametricity. Our construction operates as a ``power tool'' to support construction of terms with geometrical structure, and we use it to define composition operations for cylinders and cones in omega-categories. The machinery can generate terms of high complexity, and we have implemented our construction in a proof assistant, which verifies that the generated terms have the correct type. All our results can be exported to homotopy type theory, allowing the explicit computation of complex path type inhabitants.

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arXiv2024

Invertible cells in $\omega$-categories

We study coinductive invertibility of cells in weak $\omega$-categories. We use the inductive presentation of weak $\omega$-categories via an adjunction with the category of computads, and show that invertible cells are closed under all operations of $\omega$-categories. Moreover, we give a simple criterion for invertibility in computads, together with an algorithm computing the data witnessing the invertibility, including the inverse, and the cancellation data.

math.CT↗

arXiv2024

CaTT contexts are finite computads

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.

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arXiv2024

Hom $\omega$-categories of a computad are free

We provide a new description of the hom functor on weak $\omega$-categories, and we show that it admits a left adjoint that we call the suspension functor. We then show that the hom functor preserves the property of being free on a computad, in contrast to the hom functor for strict $\omega$-categories. Using the same technique, we define the opposite of an $\omega$-category with respect to a set of dimensions, and we show that this construction also preserves the property of being free on a computad. Finally, we show that the constructions of opposites and homs commute.

math.CT↗

arXiv2023

Computads for generalised signatures

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.

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arXiv2022

Computads for weak $\omega$-categories as an inductive type

We give a new description of computads for weak globular $\omega$-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 $\omega$-category is equivalent to a free one, and that the category of computads with generator-preserving 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.

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arXiv2017

Computing minimal generating systems for some special toric ideals

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.

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