**Higher theories and monads**

Abstract:

We extend Bourke and Garner's idempotent adjunction between monads and pretheories to the framework of $\infty$-categories, and exploit this to prove many classical theorems about monads in the $\infty$-categorical setting. Amongst other things, we prove that the category of algebras for an accessible monad on a locally presentable $\infty$ category is locally presentable. We also apply the result to construct examples of $\infty$-categorical monads from pretheories.

**The nerve of a relative monad**

Abstract:

In this talk we will consider an embedding of monads into double categories, which sends a monad P to its (concrete) double category of P-embeddings (also called P split monos), as well as generalizations of this embedding the setting of (well-behaved) relative monads. By considering a more general notion of monad morphism (motivated by the relative setting), we will understand this embedding as a fully faithful nerve which exhibits the terminal monad as 'dense' with respect to these more general morphisms.

We will then give two applications of this construction. Firstly, we will give another approach by which one arrives at the “decagon type” axioms of last time, and secondly we will give a simpler proof of the reduced form of pseudo-distributive laws involving lax-idempotent pseudomonads. Note that some parts are a work in progress.

Extreme hike! The 32nd mathematical hike is planned on the

We go by train (leaves at 06:12),

All information and photos can be found at https://conference.math.muni.cz/vylety/. (in CZ)

Have a nice summer, we look forward to you joining us.

Jana Bartoňová and Jonatan Kolegar, organizers

Jan Slovák, Director of the Department of Mathematics and Statistics

**Vopěnka's principle in infinity-categories**

Abstract:

Vopěnka's principle has arisen as a model theoretical statement, provably independent of ZFC set theory. However, there are a number of categorical ways of formulating it, preventing the existence of proper classes of objects with some conditions in presentable categories, and these are what our attention will be focused on. In particular, we will look at analogous statements in the context of oo-categories and we will ask how these new statements interact with the older ones. Moreover, some of the consequences of Vopěnka's principle on classes of subcategories of presentable categories are investigated and to some extent generalized to oo-categories. A parallel discussion is undertaken about the similar but weaker statement known as weak Vopěnka's principle.

This year, the first regular Department Assembly will take place on Wednesday, June 16, at 4pm in the so called "hybrid" form. Thus, the real meeting in M1 will be streamed in real time, too, see the ZOOM meeting invitation below.

The online meeting will be chaired by doc. Jan Kolacek and the programme will involve my report on the department budgeting and development foresight, discussion and the item "other". After the Assembly, there will be a light refreshment served outside the building at the desks in front of the library (if the weather allows).

**Lax factorisation systems and categories of partial maps**

Abstract:

Lax factorisation systems and categories of partial maps

Factorisation systems describe morphisms in a category by factorising them into pairs of composable morphisms. Their definition depends on a kind of orthogonality relation between morphisms, which entails the existence of some diagonal morphisms for certain squares. In this seminar we present the new notion of lax weak orthogonality between morphisms, which involves lax squares and the factorisation systems it generates. Then we will introduce lax versions of functorial and algebraic weak factorisation systems and some of their properties. These lax factorisation systems are discussed, keeping the theory of ordinary factorisation systems as a blueprint and providing useful properties.

An overview of the examples of such lax factorisation systems is presented in the context of partial maps. We conclude with a discussion of general constructions of these examples and their description in the particular case of sets with partial maps.

The purpose of the series of IWFOS is to highlight the major trends in different areas of functional statistics through the exchange of ideas and the promotion of collaboration between researchers from different countries. It aims at contributing to future developments of such areas. The workshop will be a platform for communication, exchange of ideas and interaction for researchers in statistics for infinite-dimensional and high-dimensional problems.

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**Skew monoidal categories and the proof-theoretic anatomy of associativity (and unitality)**

Abstract:

Based on joint work with Tarmo Uustalu and Niccolò Veltri.

The talk will survey a recent line of work, which takes a proof-theoretic approach to solving the coherence problem(s) for skew monoidal categories and related structures. I will begin by discussing the so-called Tamari order on fully-bracketed words induced by a semi-associative law (AB)C <= A(BC), and explain how a simple sequent calculus may account for some of its fascinating properties, such as the fact that the set of fully-bracketed words on n+1 letters forms a lattice Y_n under this order, as well as a remarkable formula counting the number of intervals in Y_n.

Then I will recall the definition of skew monoidal categories, and explain how a more refined sequent calculus may be used to solve two related coherence problems: deciding equality of maps and enumerating homsets in free skew monoidal categories. Closely related to recent work by Bourke and Lack, this sequent calculus may be considered as a canonical construction of the free left representable skew multicategory over a set of atoms.

Finally, I will briefly discuss variations of the sequent calculus capturing "partially skew" monoidal categories with different normality conditions.

References:

[1] https://arxiv.org/abs/1803.10080

[2] https://arxiv.org/abs/2003.05213

[3] https://arxiv.org/abs/2101.10487

**June 7**, **10am**, **online on MS Teams and the seminar room on the second floor**

Join via this **LINK.**

**Some remarks on variational nature of Monge-Ampère equations in dimension four**

Abstract:

I will present a necessary condition for the local solvability of the strong inverse variational problem in the context of Monge-Ampère partial differential equations and first-order Lagrangians. In contrast with the previous talk by Marcus Dafinger, this condition is given by comparing differential forms on the first jet bundle and is valid only for the aforementioned PDEs. To illustrate how this approach can be applied, we will examine the linear Klein-Gordon equation, first and second heavenly equations of Plebanski, Grant equation, and Husain equation.

I will also speak about the drawbacks of the method when trying to generalize it to a system of equations.

**May 31**, **10am**, **online on MS Teams and the seminar room on the second floor**

Join via this **LINK.**

**On formal calculus of variations, Cartan formula and Noether's theorem**

Abstract:

We give an introductory talk on the formal calculus of variations. Concepts from differential geometry, like Lie-derivative, exterior derivative and Cartan formula will be related to first variation, Helmholtz map and a kind of related Cartan formula. With the help of these operations we then formulate Noether's theorem and present some results on the so-called Takens' problem.

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