Recent talks.

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2020-07-05. Diagrammatic sets and rewriting in weak higher categories.
At First Workshop on Geometric and Categorical Structures in Computation and Deduction.
Slides -
In 1991, Kapranov and Voevodsky published a purported proof of the homotopy hypothesis for strict omega-categories; 7 years later, Carlos Simpson proved that the result is false even under a generous interpretation. Since then, the theory of polygraphs has re-invented strict omega-categories as a versatile framework for higher-dimensional rewriting, but that barrier remains, preventing us from doing sound homotopical rewriting with polygraphs.
As an intermediate step, Kapranov and Voevodsky's non-proof passed through a notion of “diagrammatic set”, a space modelled on a certain category of combinatorial pasting diagrams. I will revisit this idea with 3 decades of hindsight, and show how, with some retouching, it leads to
- a framework for rewriting with a combinatorial feel similar to polygraphs, that can be adopted by practitioners of diagrammatic methods with minimal adaptation, and
- a model of weak or semistrict higher categories that does satisfy the homotopy hypothesis.
In this talk, I will give an overview of the definition, its background and its motivation.
2020-06-11. Diagrammatic sets: weak higher categories for rewriting.
At TallCat Seminar, TalTech, Tallinn, Estonia.
Slides -
Diagrammatic sets are my attempt to do several things at once, including:
1. rescue some good ideas in a famous wrong paper by Kapranov and Voevodsky;
2. build a framework for higher-dimensional rewriting and algebra that is more flexible than polygraphs;
3. develop a model of higher categories that is actually enjoyable for those who love diagrams, without driving away the rest;
4. ultimately, try a “tinkerer's” approach to figuring out what “directed homotopy theory” should be all about.
In this talk, I will give an overview of the definition, its background and its motivation.
2019-12-17. Representable diagrammatic sets as a model of weak higher categories.
At Sixth Symposium on Compositional Structures, Leicester, United Kingdom.

Slides -
In a model of higher categories, the scope of diagrammatic reasoning is set by a “pasting theorem”, defining what diagrams are admitted and assigning them a unique interpretation. Mainstream models from categorical homotopy theory are mostly unconcerned with pasting, since the focus is less on presented higher categories than in ACT. On the other hand, models with stronger pasting theorems may not be “general enough”: it is false or unproven that they model all homotopy types in the sense of Grothendieck's homotopy hypothesis.
In this talk, I will present a model of weak omega-categories which has a rich combinatorics of pasting and also satisfies a version of the homotopy hypothesis. I will also try to convince you that one is led naturally to it by taking seriously an analogy between higher-dimensional rewrite systems and CW complexes. This talk is based on arXiv preprint 1909.07639.
2019-10-31. Diagrammatic sets between rewriting and topology.
Invited. At RIMS Computer Science Seminar, Kyoto, Japan.

Slides -
Higher-dimensional rewriting theory generalises abstract rewriting, string rewriting, and term rewriting. This field has some intriguing connections with combinatorial topology, which are not fully realised when rewrite systems are modelled, as commonly done, by strict higher categories.
I will show how following these connections can lead to an alternative framework: diagrammatic sets. Then, I will discuss how one can build a model of weak higher categories on top of diagrammatic sets, to function as a semantics of higher-dimensional rewrite systems.
2018-09-24. ZW calculi: diagrammatic languages for quantum computing.
Invited. At 7th Conference Logic and Applications, IUC, Dubrovnik, Croatia.

Slides -
This is an introductory talk on ZW calculi, string-diagrammatic axiomatisations of monoidal categories that are relevant to quantum computing. After a quick introduction to PROs in categorical universal algebra, and the way that string diagrams can turn algebraic structures into topological objects, I will present the two main variants of ZW calculi: the qubit ZW calculus, axiomatising the category of linear maps of qubits, and the fermionic ZW calculus (joint with G. de Felice and K. F. Ng), axiomatising the category of local fermionic modes and parity-preserving and parity-reversing maps.
2018-07-07. Merge-bicategories: towards semi-strictification of higher categories.
At 4th Workshop on Higher-Dimensional Rewriting and Algebra, Oxford, United Kingdom.

Slides -
A 2-polygraph is regular if its 2-cells have non-degenerate, “interval-shaped” input and output boundaries. A merge-bicategory is a regular 2-polygraph with an algebraic composition of 2-cells, where the composable diagrams are the “disk-shaped” ones. A merge-bicategory is representable if it contains enough “divisible” 1-cells and 2-cells, satisfying certain universal properties.
I will show that representable merge-bicategories and morphisms that preserve divisible cells are equivalent to bicategories and pseudofunctors; through a natural monoidal biclosed structure on merge-bicategories, one can also recover higher morphisms. I will use this to develop a semi-strictification argument for bicategories, combining the explicit combinatorial aspects of string-diagram based coherence proofs, with the main features of Hermida's abstract proof based on representable multicategories.
All the constructions can be generalised to higher dimensions: I will sketch how this could lead to semi-strictification (excluding weak units) for higher categories, where it is still an open problem.
2018-07-05. Regular polygraphs and semi-strictification of higher categories.
Invited. At Séminaire Catégories supérieures, polygraphes et homotopie, IRIF, Paris, France.

Slides -
A regular polygraph is one whose cells have k-dimensional boundaries “shaped as k-dimensional balls”, via a suitable geometric realisation. I will describe a combinatorial approach, where regular polygraphs are defined as presheaves on a shape category, and subsequently proved to form a full subcategory of the category of polygraphs. I will also show how several operations and constructions on polygraphs – such as lax Gray products and joins – admit a sleek definition in this setting.
I will then give a non-algebraic, fully weak definition of higher category, as a regular polygraph satisfying a representability property, and sketch a complementary algebraic, semi-strict definition. Finally, I will sketch how the two are combined in a semi-strictification construction, where semi-strictness should be read in the sense of “Simpson's conjecture for regular compositions”, as in the earlier seminar entry by Simon Henry.
2018-03-06. Units without degeneracy, from polycategories to sequent calculi.
At Second Workshop on Mathematical Logic and its Applications, Kanazawa, Japan.

Slides -
Divisibility properties of cells in a polycategory or multicategory have a deep connection to the rules of sequent calculi for multiplicative linear logic, as first evidenced by Cockett and Seely: the left and right introduction rules for each connective correspond either to composition with a divisible cell, or the use of its divisibility property.
Hermida relied on the same notions to devise an abstract proof of strictifiability for monoidal categories, a classic theorem of category theory. The obvious generalisation of this theorem to higher-dimensional variants, such as braided monoidal categories, fails, and weaker versions are mostly still open problems.
An analysis of the failure of Hermida's proof to generalise indicates that the modelling of units with divisible cells that have a degenerate (0-ary) boundary may be responsible. We show that it is possible to recover units from the existence of certain divisible cells of lower dimension, even when degenerate boundaries are disallowed. This leads to a weaker semi-strictification theorem, which, however, we hope to extend to arbitrary dimensions.
While the polycategorical units so obtained are equivalent to the usual ones at the level of the induced monoidal category structure, they behave very differently at the level of sequent calculus: most notably, their polarity as connectives changes. We pose some questions on the logical and computational aspects of our notion of unit, in the light, also, of the increase in the complexity of proof equivalence brought by the usual notion.
2018-01-23. Bringing compositionality to rewriting theory via polygraphs.
Invited. At NII Shonan Meeting on Intensional and extensional aspects of computation, Japan.

Slides -
In recent years, the theory of polygraphs has emerged as a general framework for rephrasing classic results and developing connections between rewriting theory, universal algebra, and algebraic topology. These connections suggest that the compositional reasoning and methods that are essential in topology may be imported into rewriting theory. In this talk, with the help of string diagrams, I will describe tensor products of polygraphs, and show how they can be used to construct composite presentations — including higher-dimensional coherence rewrites — of familiar structures that involve two algebras, such as homomorphisms, bialgebras, and distributive laws of monads.
2017-09-08. A combinatorial-topological shape category for polygraphs.
At 3rd Workshop on Higher-Dimensional Rewriting and Algebra, Oxford, United Kingdom.

Slides -
Building on ideas of poset topology, I give a combinatorial description of a class of shapes of higher-categorical cells, with a mild constraint of topological nature (n-boundaries are homeomorphic to n-disks), and good technical properties, such as closure under tensor products. I also report on an associated notion of weak unit cell, obtained from a representability condition, which could bridge the gap with the needs of string diagram rewriting.