With deep sorrow we announce that prof. RNDr. Ivan Kolář, DrSc., passed away on December 15, 2020.

Ivan Kolář was born on May 22, 1936 in Brno and most of his academic career was linked to Brno, too. Already during his study at Masaryk University, he joined the research group following the geometric footsteps of Eduard Čech. Influenced by the abstract and coordinate free approach to infinitesimals which had been recently introduced by Charles Ehresmann, Ivan Kolář worked out an abstract general version of the Cartan moving frame method and employed the newly emerged theory of jets and iterated differentials in numerous areas of differential geometry, variational calculus, and mathematical physics. Soon, these interests naturally moved his focus towards the abstract theory of natural objects and operations in differential geometry.

After a decade spent at the Brno military academy, in 1969 Ivan Kolář accepted Borůvka’s invitation to join the Brno branch of the Mathematical Institute of the Academy of Sciences of the Czech Republic. He later led this institution with grace through the difficult period of normalization until 1989. During this time, he also pushed forward the research culminating in the celebrated Springer monograph Natural Operations in Differential Geometry (published 1993, co-authored by Peter Michor and Jan Slovák, the most highly cited item ever in the 58A MathSci category, and perhaps also the initial impulse towards the 58A32 subcategory Natural Bundles in MSC2000).

In 1980, Ivan Kolář organized the very first meeting of the (still running) series of international conferences Differential Geometry and its Applications, and from 1985 he cochaired the (also still running) Central European Seminar with Peter Michor, perhaps the only regular seminar happening across the “iron curtain” before 1990. From 1991 onwards, Ivan Kolář was working at the Faculty of Science of Masaryk University, where he contributed in an essential way to several further research directions, including the covariant approach to the so-called Weil bundles, the abstract axiomatic geometric approach to infinitesimals.

Ivan Kolář was the main driving force in the differential geometry group in Brno for at least forty years, and there is a large group of grateful students and collaborators pushing his views and hopes further.

Announcement in PDF

We will continue online on Thursday, December 17th, at 1pm on ZOOM platform (for information how to acces seminar and next programme visit this page) by the talk:

Gabriele Lobbia (University of Leeds)

Distributive Laws for Relative Monads

Abstract:
Monads are useful tools both in mathematics (especially in universal algebra) and in computer science. An important notion is that of a distributive law between two monads, which goes back to fundamental work of Jon Beck in the late '60s. This notion describes how two monads can interact with each other, an analogue of the ring distributivity of product over sum.

In recent years, a generalisation of monads has been studied, relative monads, where we drop the endofunctor requirement. This definition relies on an extension operator instead of a multiplication. We will start by reviewing the notion of distributive law. Then we will introduce relative monads and see what the right counterpart of distributive laws is when we consider a monad and a relative monad.

We will continue online on Thursday, December 10th, at 1pm on ZOOM platform (for information how to acces seminar and next programme visit this page) by the talk:

John Bourke (Masaryk University)

Accessible Infinity-Cosmoi

Abstract:
Riehl and Verity introduced infinity-cosmoi - certain simplicially enriched categories - as a framework in which to give a model-independent approach to infinity categories.  For instance, there is an infinity cosmos of infinity-categories with finite limits or colimits, or of cartesian fibrations.  In this talk, I will introduce the notion of an accessible infinity-cosmos and explain that most, if not all, infinity-cosmoi arising in practise are accessible.  Applying results of earlier work, it follows that accessible infinity-cosmoi have homotopy weighted colimits and admit a broadly applicable homotopical adjoint functor theorem.  This is a report on joint work with Steve Lack, and builds on recent work with Lack and Lukáš Vokřínek.

We will continue online on Thursday, November 26th, at 1pm on ZOOM platform (for information how to acces seminar and next programme visit this page) by the talk:

Martin Bidlingmaier (Aarhus University)

Model categories of lcc categories and the gros model of dependent type theory

Abstract:
In this talk we discuss various model categories of locally cartesian closed (lcc) categories and their relevance to coherence problems, in particular the coherence problem of categorical semantics of dependent type theory. We begin with Lcc, the model category of locally cartesian closed (lcc) sketches. Its fibrant objects are precisely the lcc categories, though without assigned choices of universal objects. We then obtain a Quillen equivalent model category sLcc of strict lcc categories as the category of algebraically fibrant objects of Lcc. Strict lcc categories are categories with assigned choice of lcc structure, and their morphisms preserve these choices on the nose. Conjecturally, sLcc is precisely Lack’s model category of algebras for a 2-monad T, where T is instantiated with the free lcc category functor on Cat. We then discuss the category of algebraically cofibrant objects of sLcc and show how it can serve as a “gros” model of dependent type theory.