Polish Seminar on Category Theory and its Applications

Ogólnopolskie Seminarium z Teorii Kategorii i jej Zastosowań

2011-2012

See what happened on past meetings: 8th meeting, 7th meeting, 6th meeting, 5th meeting, 4th meeting, 3rd meeting, 2nd meeting, 1st meeting.
In case of questions please contact Marek Zawadowski (zawado 'at' mimuw.edu.pl).

9th meeting
The meeting will take place on
Saturday, June 2nd, 2012
at the Faculty of Mathematics, Informatics, and Mechanics of the University of Warsaw, room 4050.

Speaker

Eric Finster (EPFL, Lausanne), Opetopic Type Theory.


Program

10:30--12:00 Eric Finster Dependent Type Theory
12:20--13:50 Eric Finster Towards an Opetopic Type Theory

Abstracts

Dependent Type Theory
Eric Finster (EPFL, Lausanne)
Abstract: In the past few years some very interesting connections have been discovered between a class of formal systems know as dependent type theories, well studied in computer science and logic for many years, and concepts from homotopy theory and higher category theory. In short, these languages exhibit a kind of internal “decoherence” which forces the objects they describe to act more like homotopy types than sets. This observation has become the motivating priciple for Voevodsky’s Univalent Foundations Project, which seeks a new foundational system in which homotopy plays a central role. In my first talk, I will give a brief overview of some concepts from type theory, focusing on how one can use them to encode homotopy theoretic statements.

Towards an Opetopic Type Theory
Eric Finster (EPFL, Lausanne)
Abstract: Given the success of dependent type theories in describing homotopy types, one is naturally led to look for connections with other concepts from higher category theory. One intriguing possibility is the Opetopes: a collection of shapes recursively defined in each dimension by considering all possible “many-in-one-out” pasting diagrams in the previous definition. Michael Makkai, in setting out his definition of weak higher category using these shapes and his concept of a FOLDS signature, explicitly mentions the influence of type theory in his thinking. In my second talk, I will give a type-theoretic prespective on the definition of the Opetopes, relating them to the computer scientist’s “inductive family”, and describe some work in progress for developing a type theory which incorporates them into its structure.




Past meetings

8th meeting
The meeting took place on
Saturday, December 10th, 2011
at the Faculty of Mathematics, Informatics, and Mechanics of the University of Warsaw, room 4050.

Speaker

Urs Schreiber (Department of Mathematics, University of Nijmegen), Differential cohomology and action functionals in a cohesive infinity-topos.


Photo gallery.


Program

10:30--12:00 Urs Schreiber Differential cohomology and action functionals in a cohesive infinity-topos
12:20--13:50 Urs Schreiber Differential cohomology and action functionals in a cohesive infinity-topos (part 2)

Abstract

Differential cohomology and action functionals in a cohesive infinity-topos
Urs Schreiber (University of Nijmegen)
Abstract: I explain how in higher topos theory there is a natural formulation of differential cohomology, which is the theory of higher gauge fields / higher connections. Then I indicate the natural construction of higher Chern-Simons action functionals on moduli stacks of such higher gauge fields. I'll assume a knowledge of some basic category theory and some basic differential geometry, but otherwise will be self-contained. It will be helpful, though, to have heard of Lie groupoids/differentiable stacks. Pointers to the literature and further background material is here.


7th meeting
The meeting took place on
Saturday, November 26th, 2011
at the Faculty of Mathematics, Informatics, and Mechanics of the University of Warsaw, room 4050.

Speaker

Mihaly Makkai (Department of Mathematics, McGill University), Canonical versus non-canonical constructions in category theory. A selective survey of concepts in higher category theory, with examples and problems.


Photo gallery.


Program

10:30--12:00 Mihaly Makkai Canonical versus non-canonical constructions in category theory
12:20--13:50 Mihaly Makkai Canonical versus non-canonical constructions in category theory (part 2)

Abstract

Canonical versus non-canonical constructions in category theory. A selective survey of concepts in higher category theory, with examples and problems
Mihaly Makkai (McGill University)
Abstract: The title is descriptive enough, I think, so that you will not need an abstract, I hope.


6th meeting
The meeting took place on
Saturday, October 22nd, 2011
at the Faculty of Mathematics, Informatics, and Mechanics of the University of Warsaw, room 4050.

Speaker

Bartek Klin (Department of Computer Science, University of Warsaw), Coalgebras and bialgebras.


Photo gallery.


Program

10:30--12:00 Bartek Klin Coalgebras and bialgebras
12:20--13:50 Bartek Klin Coalgebras and bialgebras (part 2)

5th meeting
The meeting took place on
Saturday, April 16th, 2011
at the Faculty of Mathematics, Informatics, and Mechanics of the University of Warsaw, room 4050.

Speakers

Bartek Klin (Department of Computer Science, University of Warsaw), Coalgebras and coinduction

Jerzy Król (University of Silesia), Breaking a general tovariance...

Program

10:30--12:00 Bartek Klin Coalgebras and coinduction
12:20--13:50 Jerzy Król Breaking a general tovariance...

4th meeting
Joint meeting of the Seminar on Quantum Field Theories & the Polish Seminar on Category Theory took place on
Saturday, March 12th, 2011
at the Faculty of Mathematics, Informatics, and Mechanics of the University of Warsaw, room 4070.

Speakers

Aleks Kissinger (Computing Laboratory, Oxford University), Frobenius States and a Graphical Language for Multipartite Entanglement

Marek Zawadowski (Department of Mathematics, University of Warsaw), Monoidal categories, an introduction

Program

10:00 AM Welcome coffee
10:30 AM Marek Zawadowski Monoidal categories, an introduction
12:00 AM Lunch
12:30 PM Aleks Kissinger Quantum mechanics and quantum computation
1:30 PM Coffee break
2:00 PM Aleks Kissinger Categorical quantum mechanics and the graphical calculus
3:00 PM Coffee break
3:30 PM Aleks Kissinger Graphs of many-body systems: the algebraic structure of entanglement
4:30 PM End of the meeting!
6:00 PM Dinner at "U Szwejka"!

Abstracts

Frobenius States and a Graphical Language for Multipartite Entanglement
Aleks Kissinger (Oxford University)
Abstract: While multipartite quantum states constitute a key resource for quantum computations and protocols, obtaining a high-level, structural understanding of entanglement involving arbitrarily many qubits is a long-standing open problem in quantum computer science. We approach this problem by identifying the close connection between a special class of highly entangled tripartite states, which we call Frobenius states, and the standard notion of a commutative Frobenius algebra. We use these states (and their induced algebras) as a primitive in a graphical language that is both universal for quantum computation and capable of highlighting the behavioural differences in types of Frobenius states. This graphical language then suggests methods for state preparation, classification, and efficient classical modeling of certain kinds of quantum systems.


Monoidal categories, an introduction
Marek Zawadowski (MIM UW)
Abstract: This talk is meant to be an introduction to the theory of monoidal categories and some of its extensions. After reviewing the basic notions concerning monoidal categories I will discuss the coherence, the graphical language, and some 2-dimensional aspect of the theory. Then I will present some extension of the notion of a monoidal category: braided, autonomous, and traced categories among others. These notions will be illustrated with some typical examples.


3rd meeting
The meeting took place on
Saturday, January 15th, 2011
at the Faculty of Mathematics, Informatics, and Mechanics of the University of Warsaw, room 4050.

Speakers

Paweł Waszkiewicz (Jagellonian University), V-enriched categories---an introduction

Jerzy Król (University of Silesia), Sheaves, categories, and physics

Program

10:30--12:00 Paweł Waszkiewicz V-enriched categories---an introduction
12:20--13:50 Jerzy Król Sheaves, categories, and physics

2nd meeting
The meeting took place on
Saturday, December 11th, 2010
at the Faculty of Mathematics, Informatics, and Mechanics of the University of Warsaw, room 4050.

Speakers

Marek Zawadowski (Depertment of Mathematics, University of Warsaw) An introduction to Higher-dimensional Categories

Stanisław Szawiel (Department of Mathematics, University of Warsaw), A Constructuion of Opetopic Sets

Program

10:30--12:00 Marek Zawadowski An introduction to Higher-dimensional Categories
12:20--13:50 Stanisław Szawiel A Constructuion of Opetopic Sets

1st meeting
The meeting took place on
Saturday, November 6th, 2010
at the Faculty of Mathematics, Informatics, and Mechanics of the University of Warsaw, room 4070.

Speaker

Thomas Streicher (Universität Darmstadt) Types and Kan Complexes

Program

10:15--11:15 Thomas Streicher Type Theory and its Categorical Models
11:30--12:30 Thomas Streicher Basics of Simplicial Homotopy Theory
12:45--13:45 Thomas Streicher Interpreting Type Theory in SSet and Voevodsky's Equivalence Axiom

Abstract

Types and Kan Complexes
Thomas Streicher (Universität Darmstadt)
Abstract: Constructive type theory was introduced by Martin-Loef as a foundation for constructive mathematics. Its impredicative version is implemented within the Coq system which allows to construct proofs interactively as elements of a given type representing a proposition. In this sense it provides a synthesis of (constructive) logic and functional programming.
In order to make type checking decidable one has to distinguish between "judgemental" and "propositional" equality. The former corresponds to "conversion" and is decidable whereas the latter is (closer to) mathematical equality which is highly non-decidable.
This distinction between two notions of equality makes life a bit cumbersome. But it allows one to interpret propositional equality as being isomorphic as in the Groupoid Model of Hofmann and Streicher from 1993. Already there it was suggested that actually types should not be interpretad as groupoids but as weak higher-dimensional groupoids". The most accessible notion of weak higher dimensional groupoid is provided by Kan complexes within the topos SSet of simplicial sets as suggested independently by Streicher and Voevodsky in 2006.
Recently Voevodsky has shown that this model validates his "Equivalence Axiom" stating that types are equal iff there is a "weak equivalence" between them. Here "weakly equivalent" means that there is a map between the types which is surjective when this condition is expressed in terms of propositional equality. The intention is that extending Coq this way allows one to consider isomorphic types as equal as it is common in category theory.
Our lectures are organized into 3 parts
1. Type Theory and its Categorical Models
2. Basics of Simplicial Homotopy Theory
3. Interpreting Type Theory in SSet and Voevodsky's Equivalence Axiom

This page was last updated on October 19th, 2011 by Aleksander Horawa.