# The Theoretical Cosynus Seminar

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**Louis Lemonnier** (LMF, University Paris-Saclay) – *Central Submonads and Notions of Computation*

07 June 2022 at 14:00 in Nicole Reine Lepaute

The notion of “centre” has been introduced for many algebraic structures in mathematics. A notable example is the centre of a monoid which always determines a commutative submonoid. Monads (in category theory) can be seen as generalisations of monoids and in this paper we show how the notion of centre may be extended to strong monads acting on symmetric monoidal categories. We show that the centre of a strong monad T, if it exists, determines a commutative submonad Z of T , such that the Kleisli category of Z is isomorphic to the premonoidal centre (in the sense of Power and Robinson) of the Kleisli category of T. We provide three equivalent conditions which characterise the existence of the centre of T. While not every strong monad admits a centre, we show that every strong monad on well-known naturally occurring categories does admit a centre, thereby showing that this new notion is ubiquitous. We also provide a computational interpretation of our ideas which consists in giving a refinement of Moggi’s monadic metalanguage. The added benefit is that this allows us to immediately establish a large class of contextually equivalent terms for monads that admit a non-trivial centre by simply looking at the richer syntactic structure provided by the refinement.

**Bryce Clarke** – *An introduction to delta lenses*

19 May 2022 at 14:00 in Nicole Reine Lepaute

Delta lenses are functors equipped with a suitable choice of lifts, and were first introduced in 2011 to model so-called bidirectional transformations between systems. In this talk, I will provide an introduction to several category-theoretic perspectives of delta lenses which were developed throughout my PhD research. Categories, functors, and delta lenses organise into a double category, and I will demonstrate how many interesting properties of delta lenses may be studied via this two-dimensional categorical structure.

**Cameron Calk** – *Abstract strategies and formal coherence*

21 April 2022 at 14:15 in Nicole Reine Lepaute

Kleene algebra have widespread use in mathematics and computer science, from formal language theory to program correctness. Following formalisations of abstract rewriting results in modal Kleene algebra, globular 2-Kleene algebras were introduced, providing a formal setting for reasoning about coherence proofs in abstract rewriting systems. On the other hand, normalisation strategies give a categorical interpretation of the notion of contracting homotopies, constructed via confluent and terminating rewriting. This approach relates standardisation to coherence results in the context of higher dimensional rewriting systems. In this work, we formalise the notion of normalisation strategy in the setting of globular 2-Kleene algebras. In such structures, normalisation strategies allow us to prove a formal coherence theorem via convergent abstract rewriting.

**Aly-Bora Ulusoy** – *A Really Short Hike Through Model Theoretical Galois Theory*

29 March 2022 at 14:00 in Nicole Reine Lepaute

Short talk about the Galois theory applied to models of first-order theories. Using symmetry arguments to express non-computability.

**Roman Kniazev** – *Why should study directed loop spaces?*

22 March 2022 at 14:00 in Nicole Reine Lepaute

In the first part of the talk, we will discuss classical links of loop spaces with homotopy and (co)homology, together with recognition principle relating loop spaces and operads. In the second part, we will tentatively introduce analogous constructions and suggest possible developments in the context of directed spaces.