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Online algebra seminar - December 17th, 1pm PDF Print

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.

Last Updated on Wednesday, 16 December 2020 11:16
 
Online algebra seminar - December 10th, 1pm PDF Print

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.

Last Updated on Wednesday, 09 December 2020 10:26
 
Online algebra seminar - November 26th, 1pm PDF Print

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.

Last Updated on Wednesday, 25 November 2020 10:35
 
Online algebra seminar - November 19th, 1pm PDF Print

We will continue online on Thursday, November 19th, at 1pm on ZOOM platform (for information how to acces seminar and next programme visit this page) by the talk:

Maru Sarazola (Cornell University)

The stable homotopy hypothesis

Abstract:
The homotopy hypothesis is a well-known bridge between topology and category theory. Its most general formulation, due to Grothendieck, asserts that topological spaces should be "the same" as infinity-groupoids. In the stable version of the homotopy hypothesis, topological spaces are replaced with spectra.

In this talk we will review the classical homotopy hypothesis, and then focus on the stable version. After discussing what the stable homotopy hypothesis should look like on the categorical side, we will use the Tamsamani model of higher categories to provide a proof. This is based on joint work with Moser, Ozornova, Paoli and Verdugo.

Last Updated on Wednesday, 18 November 2020 11:31
 
Online algebra seminar - November 12th, 1pm PDF Print

We will continue online on Thursday, November 12th, at 1pm on ZOOM platform (for information how to acces seminar and next programme visit this page) by the talk:

Marcos Mazari-Armida (Carnegie Mellon University)

Model-theoretic stability and superstability in classes of modules

Abstract:
Dividing lines in complete first-order theories were introduced by Shelah in the early seventies. A dividing line is a property such that the classes satisfying such a property have some nice behaviour while those not satisfying it have a bad one. Two of the best understood dividing lines are those of stability and superstability.
In this talk, I will study the notion of stability and superstability in abstract elementary classes of modules with respect to pure embeddings, i.e., classes of the form (K,≤p) where K is a class of R-modules for a fixed ring R and ≤p is the pure submodule relation. In particular, using that the class of p-groups with pure embed- dings is a stable AEC, I will present a solution to Problem 5.1 in page 181 of Abelian Groups by Laszlo Fuchs. Moreover, I will show how the notion of superstability can be used to give new characterizations of noetherian rings, pure-semisimple rings, and perfect rings.

Last Updated on Tuesday, 10 November 2020 15:14
 
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