arXiv subjects

Amar Hadzihasanovic

Publications and source records attributed to Amar Hadzihasanovic.

At least 19 recordsLinked to original sources

arXiv2026

Polarised noncrossing partititions and the coherent self-dual $ω$-equivalence

We construct an acyclic augmented chain complex of abelian groups whose entry in degree $n > 0$ is free on the set of noncrossing partitions of degree $n-1$ equipped with a $\{0, 1\}$-labelling of their gaps. The definition of the differential in this complex is related, via a restricted Leibniz rule, to the gap-insertion operad of Ebrahimi-Fard, Foissy, Kock, and Patras. We conjecture that this augmented chain complex is the linearisation of a polygraph presenting a self-dual model of the coherent walking $ω$-equivalence constructed by the author, Loubaton, Ozornova, and Rovelli, and provide evidence for this conjecture.

math.CT↗

arXiv2026

Proceedings Eighth International Conference on Applied Category Theory

The Eighth International Conference on Applied Category Theory took place at the University of Florida on June 2-6 2025. The conference consisted of 2 plenary invited talks, 28 contributed talks, an online community meeting, a general community meeting, and 4 talks by junior researchers who attended the Adjoint School to present the results of their research at the school. Information regarding the conference may be found at https://gataslab.org/act2025/act2025.html. Submission to ACT2025 had three tracks: extended abstracts, software demonstrations, and proceedings. Accepted proceedings track submissions are included in this volume. The contributions to ACT2025 ranged from pure to applied and included contributions in a wide range of disciplines. ACT2025 included talks related to computer science, probability theory, chemistry, string diagrams, game semantics, quantum computation, and more.

math.CT↗

arXiv2025

Equivalences in diagrammatic sets

We show that diagrammatic sets, a topologically sound alternative to polygraphs and strict $ω$-categories, admit an internal notion of equivalence in the sense of coinductive weak invertibility. We prove that equivalences have the expected properties: they include all degenerate cells, are closed under 2-out-of-3, and satisfy an appropriate version of the "division lemma", which ensures that enwrapping a diagram with equivalences at all sides is an invertible operation up to higher equivalence. On the way to this result, we develop methods, such as an algebraic calculus of natural equivalences, for handling the weak units and unitors which set this framework apart from strict $ω$-categories.

math.CT↗

arXiv2025

Semi-strictification of $(\infty, n)$-categories

We prove the first equivalence between a weak non-algebraic model and a semi-strict algebraic model of $(\infty, n)$-categories. This takes the form of a natural semi-strictification, whereby a weak $(\infty, n)$-category is embedded into a semi-strict one through an acyclic cofibration, in such a way that weak functors lift to semi-strict functors; this constitutes the derived unit of a Quillen equivalence between weak model categories whose fibrant objects are, respectively, the weak $(\infty, n)$-categories and (up to an acyclic fibration) the semi-strict ones. The semi-strict model has algebraic units and composition of round pasting diagrams, satisfying a strict form of associativity and interchange as in Henry's regular version of Simpson's weak units conjecture; semi-strict functors strictly preserve round composition, but only weakly preserve units. Globular composition operations are obtained from a combination of units and round composition. Since the models satisfy the homotopy hypothesis in the case $n = 0$, this result also exhibits the first semi-strict model of the classical homotopy types that has algebraic units and composition. The constructions are based on the combinatorics of regular directed complexes and are entirely explicit and combinatorial, in the spirit of Mac Lane's strictification of bicategories.

math.CT↗

arXiv2024

Model structures for diagrammatic $(\infty, n)$-categories

Diagrammatic sets admit a notion of internal equivalence in the sense of coinductive weak invertibility, with similar properties to its analogue in strict $ω$-categories. We construct a model structure whose fibrant objects are diagrammatic sets in which every round pasting diagram is equivalent to a single cell -- its weak composite -- and propose them as a model of $(\infty, \infty)$-categories. For each $n < \infty$, we then construct a model structure whose fibrant objects are those $(\infty, \infty)$-categories whose cells in dimension $> n$ are all weakly invertible. We show that weak equivalences between fibrant objects are precisely morphisms that are essentially surjective on cells of all dimensions. On the way to this result, we also construct model structures for $(\infty, n)$-categories on marked diagrammatic sets, which split into a coinductive and an inductive case when $n = \infty$, and prove that they are Quillen equivalent to the unmarked model structures when $n < \infty$ and in the coinductive case of $n = \infty$. Finally, we prove that the $(\infty, 0)$-model structure is Quillen equivalent to the classical model structure on simplicial sets. This establishes the first proof of the homotopy hypothesis for a model of $\infty$-groupoids defined as $(\infty, \infty)$-categories whose cells in dimension $> 0$ are all weakly invertible.

math.AT↗

arXiv2024

Combinatorics of higher-categorical diagrams

This is a book on higher-categorical diagrams, including pasting diagrams. It aims to provide a thorough and modern reference on the subject, collecting, revisiting and expanding results scattered across the literature, informed by recent advances and practical experience with higher-dimensional diagram rewriting. We approach the subject as a kind of directed combinatorial topology: a diagram is a map from a "directed cell complex", encoded combinatorially as a face poset together with orientation data. Unlike previous expositions, we adopt from the beginning a functorial viewpoint, focussing on morphisms and categorical constructions. We do not tie ourselves to a specific model of higher categories, and instead treat diagrams as independent combinatorial structures that admit functorial interpretations in various contexts. Topics covered include the theory of layerings of diagrams; acyclicity properties and their consequences; constructions including Gray products, suspensions, and joins; special shapes such as globes, oriented simplices, cubes, and positive opetopes; the interpretation of diagrams in strict omega-categories and their geometric realisation as simplicial and CW complexes; and Steiner's theory of directed chain complexes.

math.CT↗

arXiv2024

Acyclicity conditions on pasting diagrams

We study various acyclicity conditions on higher-categorical pasting diagrams in the combinatorial framework of regular directed complexes. We present an apparently weakest acyclicity condition under which the $ω$-category presented by a diagram shape is freely generated in the sense of polygraphs. We then consider stronger conditions under which this $ω$-category is equivalent to one obtained from an augmented directed chain complex in the sense of Steiner, or consists only of subsets of cells in the diagram. Finally, we study the stability of these conditions under the operations of pasting, suspensions, Gray products, joins and duals.

math.CT↗

arXiv2024

Diagrammatic sets as a model of homotopy types

Diagrammatic sets are presheaves on a rich category of shapes, whose definition is motivated by combinatorial topology and higher-dimensional diagram rewriting. These shapes include representatives of oriented simplices, cubes, and positive opetopes, and are stable under operations including Gray products, joins, suspensions, and duals. We exhibit a cofibrantly generated model structure on diagrammatic sets, as well as two separate Quillen equivalences with the classical model structure on simplicial sets. We construct explicit sets of generating cofibrations and acyclic cofibrations, and prove that the model structure is monoidal with the Gray product of diagrammatic sets.

math.AT↗

arXiv2024

A model for the coherent walking $ω$-equivalence

We prove that a certain $ω$-category, which was constructed in previous work by the third and fourth author, is a model for the fully coherent walking $ω$-equivalence. Further, appropriate truncations of it give models for the fully coherent walking $n$-equivalence for each $n\geq1$.

math.CT↗

arXiv2023

Obstructions to Compositionality

Compositionality is at the heart of computer science and several other areas of applied category theory such as computational linguistics, categorical quantum mechanics, interpretable AI, dynamical systems, compositional game theory, and Petri nets. However, the meaning of the term seems to vary across the many different applications. This work contributes to understanding, and in particular qualifying, different kinds of compositionality. Formally, we introduce invariants of categories that we call zeroth and first homotopy posets, generalising in a precise sense the pi0 and pi1 of a groupoid. These posets can be used to obtain a qualitative description of how far an object is from being terminal and a morphism is from being iso. In the context of applied category theory, this formal machinery gives us a way to qualitatively describe the "failures of compositionality", seen as failures of certain (op)lax functors to be strong, by classifying obstructions to the (op)laxators being isomorphisms. Failure of compositionality, for example for the interpretation of a categorical syntax in a semantic universe, can both be a bad thing and a good thing, which we illustrate by respective examples in graph theory and quantum theory.

math.CT↗

arXiv2023

Data Structures for Topologically Sound Higher-Dimensional Diagram Rewriting

We present a computational implementation of diagrammatic sets, a model of higher-dimensional diagram rewriting that is "topologically sound": diagrams admit a functorial interpretation as homotopies in cell complexes. This has potential applications both in the formalisation of higher algebra and category theory and in computational algebraic topology. We describe data structures for well-formed shapes of diagrams of arbitrary dimensions and provide a solution to their isomorphism problem in time O(n^3 log n). On top of this, we define a type theory for rewriting in diagrammatic sets and provide a semantic characterisation of its syntactic category. All data structures and algorithms are implemented in the Python library rewalt, which also supports various visualisations of diagrams.

math.CT↗

arXiv2023

Higher-dimensional subdiagram matching

Higher-dimensional rewriting is founded on a duality of rewrite systems and cell complexes, connecting computational mathematics to higher categories and homotopy theory: the two sides of a rewrite rule are two halves of the boundary of an (n+1)-cell, which are diagrams of n-cells. We study higher-dimensional diagram rewriting as a mechanism of computation, focussing on the matching problem for rewritable subdiagrams within the combinatorial framework of diagrammatic sets. We provide an algorithm for subdiagram matching in arbitrary dimensions, based on new results on layerings of diagrams, and derive upper bounds on its time complexity. We show that these superpolynomial bounds can be improved to polynomial bounds under certain acyclicity conditions, and that these conditions hold in general for diagrams up to dimension 3. We discuss the challenges that arise in dimension 4.

math.CT↗

arXiv2021

The smash product of monoidal theories

The tensor product of props was defined by Hackney and Robertson as an extension of the Boardman-Vogt product of operads to more general monoidal theories. Theories that factor as tensor products include the theory of commutative monoids and the theory of bialgebras. We give a topological interpretation (and vast generalisation) of this construction as a low-dimensional projection of a "smash product of pointed directed spaces". Here directed spaces are embodied by combinatorial structures called diagrammatic sets, while Gray products replace cartesian products. The correspondence is mediated by a web of adjunctions relating diagrammatic sets, pros, probs, props, and Gray-categories. The smash product applies to presentations of higher-dimensional theories and systematically produces higher-dimensional coherence cells.

math.CT↗

arXiv2020

Diagrammatic sets and rewriting in weak higher categories

We revisit Kapranov and Voevodsky's idea of spaces modelled on combinatorial pasting diagrams, now as a framework for higher-dimensional rewriting and the basis of a model of weak omega-categories. In the first part, we elaborate on Steiner's theory of directed complexes as a combinatorial foundation. We individuate convenient classes of directed complexes and develop the theory of diagrammatic sets relative to one such class. We study a notion of equivalence internal to a diagrammatic set, and single out as models of weak omega-categories those diagrammatic sets whose every composable diagram is connected by an equivalence to a single cell. We then define a semistrict model providing algebraic composites and study the embedding of strict omega-categories into this model. Finally, we prove a version of the homotopy hypothesis for the infinity-groupoids in the weak model, and exhibit a specific mistake in a proof by Kapranov and Voevodsky that had previously been refuted indirectly.

math.CT↗

arXiv2019

Weak units, universal cells, and coherence via universality for bicategories

Poly-bicategories generalise planar polycategories in the same way as bicategories generalise monoidal categories. In a poly-bicategory, the existence of enough 2-cells satisfying certain universal properties (representability) induces coherent algebraic structure on the 2-graph of single-input, single-output 2-cells. A special case of this theory was used by Hermida to produce a proof of strictification for bicategories. No full strictification is possible for higher-dimensional categories, seemingly due to problems with 2-cells that have degenerate boundaries; it was conjectured by C. Simpson that semi-strictification excluding units may be possible. We study poly-bicategories where 2-cells with degenerate boundaries are barred, and show that we can recover the structure of a bicategory through a different construction of weak units. We prove that the existence of these units is equivalent to the existence of 1-cells satisfying lower-dimensional universal properties, and study the relation between preservation of units and universal cells. Then, we introduce merge-bicategories, a variant of poly-bicategories with more composition operations, which admits a natural monoidal closed structure, giving access to higher morphisms. We derive equivalences between morphisms, transformations, and modifications of representable merge-bicategories and the corresponding notions for bicategories. Finally, we prove a semi-strictification theorem for representable merge-bicategories with a choice of composites and units.

math.CT↗

arXiv2019

A combinatorial-topological shape category for polygraphs

We introduce constructible directed complexes, a combinatorial presentation of higher categories inspired by constructible complexes in poset topology. Constructible directed complexes with a greatest element, called atoms, encompass common classes of higher-categorical cell shapes, including globes, cubes, oriented simplices, and a large sub-class of opetopes, and are closed under lax Gray products and joins. We define constructible polygraphs to be presheaves on a category of atoms and inclusions, and extend the monoidal structures. We show that constructible directed complexes are a well-behaved subclass of Steiner's directed complexes, which we use to define a realisation functor from constructible polygraphs to omega-categories. We prove that the realisation of a constructible polygraph is a polygraph in restricted cases, and in all cases conditionally to a conjecture. Finally, we define the geometric realisation of a constructible polygraph, and prove that it is a CW complex with one cell for each of its elements.

math.CT↗

arXiv2019

Representable diagrammatic sets as a model of weak higher categories

Developing an idea of Kapranov and Voevodsky, we introduce a model of weak omega-categories based on directed complexes, combinatorial presentations of pasting diagrams. We propose this as a convenient framework for higher-dimensional rewriting. We define diagrammatic sets to be presheaves on a category of directed complexes presenting pasting diagrams with spherical boundaries. Diagrammatic sets have structural face and degeneracy operations, but no structural composition. We define a notion of equivalence cell in a diagrammatic set, and say that a diagrammatic set is representable if all pasting diagrams with spherical boundaries are connected to individual cells -- their weak composites -- by a higher-dimensional equivalence cell. We develop the basic theory of representable diagrammatic sets (RDSs), and prove that equivalence cells satisfy the expected properties in an RDS. We study nerves of strict omega-categories as RDSs and prove that 2-truncated RDSs are equivalent to bicategories. Finally, we connect the combinatorics of diagrammatic sets and simplicial sets, and prove a version of the homotopy hypothesis for "groupoidal" RDSs: there exists a geometric realisation functor for diagrammatic sets, inducing isomorphisms between combinatorial and classical homotopy groups, which has a homotopical right inverse.

math.CT↗

arXiv2019

A diagrammatic calculus of fermionic quantum circuits

We introduce the fermionic ZW calculus, a string-diagrammatic language for fermionic quantum computing (FQC). After defining a fermionic circuit model, we present the basic components of the calculus, together with their interpretation, and show how the main physical gates of interest in FQC can be represented in our language. We then list our axioms, and derive some additional equations. We prove that the axioms provide a complete equational axiomatisation of the monoidal category whose objects are systems of finitely many local fermionic modes (LFMs), with maps that preserve or reverse the parity of states, and the tensor product as monoidal product. We achieve this through a procedure that rewrites any diagram in a normal form. As an example, we show how the statistics of a fermionic Mach-Zehnder interferometer can be calculated in the diagrammatic language. We conclude by giving a diagrammatic treatment of the dual-rail encoding, a standard method in optical quantum computing used to perform universal quantum computation.

cs.LO↗