Category Theory Seminar

The category theory seminar is intended for anyone who has something to do with category theory. It takes place every Wednesday from 14:00 to 15:00. If you would like to receive announcements of the talks, please send an e-mail to coumans at math.ru.nl.

2012

01-02-2012, (HG00.108) Frank Roumen
Coalgebraic Trace Semantics for the Ball Monad
Abstract: Various dynamical systems in mathematics and computer science, for instance automata and Markov chains, can be modeled using coalgebras. The coalgebraic framework allows us to study different kinds of systems in a uniform way. After giving a general introduction to the theory of coalgebras, I will show how it can be used to describe the trace semantics of non-deterministic systems. So far, this has been done systematically for coalgebras on Kleisli categories that are enriched over complete partial orders. I will describe a metric analogue of this approach, which works for coalgebras for the so-called ball monad.

08-02-2012, (HG00.108) Moritz Groth
Simplicial sets and their various roles: homotopy types, categories, infinity-categories
Abstract: The aim of this talk is to give a short overview of some aspects of the theory of simplicial sets. We will try to advertise the fact that simplicial sets can play -at a first sight- quite different, though strongly related roles. Depending on the perspective (or, more precisely, depending on which extension properties are imposed) a simplicial set can be thought of as a homotopy type, as a category, or as an infinity-category. Corresponding to each of these 'interpretations' there is a nice adjunction between the category of simplicial sets and a further category. If time permits these adjunctions will be mentioned/constructed. Some of (all of/more than) this will be treated in more detail in a reading seminar on simplicial methods which starts in the week after this talk.

15-02-2012, (HG00.108) Joost Nuiten
Topos Theory in AQFT
Abstract: Two of the previous talks considered the topos approach to quantum mechanics by Heunen, Landsman and Spitters. I will shortly summarize these results, emphasizing the presentation of a quantum phase space as a ringed topos. I will then show how this perspective naturally extends from quantum mechanics to algebraic quantum field theory (AQFT): from a net of observables one can construct a presheaf of quantum phase spaces. This provides a more geometric description of AQFT, in which the causal locality of the net of observables is equivalently expressed as a gluing condition on the corresponding presheaf of ringed toposes.

07-03-2012, (HG00.086) Bart Jacobs
Scalars, Monads and Categories
Abstract: This talk will give an overview of relations between certain algebraic structures, certain monads, and certain categories. These relations take the form of "triangles of adjunctions" between for instances categories of monoids, of monads (on Sets), and of Lawvere theories. The basic relationships (for monoids and semirings) have been elaborated in joint work with Dion Coumans. Restrictions of these adjunctions to rings and effect monoids exist, and also to fields. The latter is joint work with Robert Furber.

14-03-2012, (HG00.086) Dmitry Roytenberg
What is Superspace? On Categorical Foundations of Supergeometry
Abstract: The notion of superspace first arose in physics as part of a unified framework for treating bosons and fermions on an equal footing, which came to prominence after the discovery of symmetries that mix the two kinds of particles -- so-called supersymemtries. To a physicist, a superspace is an extension of space by anti-commuting (or "fermionic") degrees of freedom. However, defining superspace rigorously in a way closest to a physicist's intuition requires category theory. This approach to supergeometry was pioneered in the early 1980s by A.S. Schwarz and V. Molotkov and, in modern terms, essentially amounts to doing differential geometry internally in a certain ringed topos. Most of the talk will be devoted to explaining this. If time allows, we shall also mention more recent applications of superspace to problems involving higher categorical structures in differential geometry.

21-03-2012, (HG00.086) Robert Furber
Entangled Quantum States and Algebraic Group Actions
Abstract: Multipartite quantum states can be considered as elements of a tensor product of vector spaces. Automorphisms of the individual vector spaces define a subgroup of the automorphisms of the tensor product, and this can be used to define an equivalence relation on quantum states. The physical interpretation of this equivalence relation is that two states are equivalent if they can be transformed into each other using only local operations. It is natural to try to mod out this equivalence relation and get a classification of multipartite states. I will explain why the naive way of doing this doesn't go right and how the fact that the automorphism group involved is an algebraic group is helpful.

28-03-2012, (HG00.108) Blaz Jelenc
Homotopy groups of topological groupoids
Abstract: In will present the notion of homotopy groups of a topological groupoid and explain its relation to the homotopy groups of the associated classifying space. Then I will define a class of maps between topological groupoids, called Serre fibrations, that enable us to calculate these homotopy groups with the help of long exact sequences.

04-04-2012, (HG00.108) Matija Basic
Dendroidal sets as a generalization of simplicial sets
Abstract: Dendroidal sets have been introduced as models for homotopy operads. Their theory is in many aspects analogous to the theory of simplicial sets. We will briefly recall the notion of simplicial sets and describe their relation to categories and topological spaces, emphasizing the role of Kan complexes. The main part of the talk will be dedicated to the notions of (coloured) operads and dendroidal sets. We will describe the adjunction between the categories of dendroidal sets and operads and also discuss in which sense dendroidal sets generalize simplicial sets. We will also mention some recent results concerning the geometric realization of dendroidal sets.

11-04-2012 No seminar

18-04-2012, (HG01.028) Michael Mueger
Braided and modular categories
Abstract: I will give a short introduction to braided tensor categories and ways of obtaining non-trivial examples. I will then focus on modular categories and Drinfeld's center construction. Some results in the way of structure theory will be mentioned, as well as recent developments concerning classification.

25-04-2012, (HG01.139) Jelle Herold
Category theory and computer science
Abstract: Abstract: Programmers, especially in the industry, need to build large systems, usually using impoverished tools. Dealing with this, computer scientists early on realized that abstractions are essential to scaling software. Tools like functional programming languages ('type theory') are particularly suited to implement such abstractions. Similar principles can be seen in category theory and for the mathematically oriented programmer this then provides a much needed bridge between mathematics and computer science. It might even be said that category theory is the programming language of mathematics. We will elaborate a bit on the above and other commonalities between our fields and then illustrate the point (hopefully) by showing the Haskell programming language and exploring how monads and monad transformers are used in it.
(I will try to keep the talk interesting to both mathematicians and computer scientists.)

16-05-2012, (HG00.108) Bert Lindenhovius
Categorical Quantum Mechanics
Abstract: The standard axiomatic presentation of quantum mechanics is given in terms of Hilbert spaces. The aim of this talk is to discuss Abramsky and Coecke's paper "A categorical semantics of quantum protocols" in which the authors develop an alternative mathematical framework for quantum mechanics in terms of strongly compact closed categories with biproducts, which they claim to be more suitable for describing quantum protocols like quantum teleportation.

23-05-2012, (HG00.108) Benno van den Berg
The Herbrand topos
Abstract: In this talk I will introduce a new realizability topos, called the Herbrand topos, and investigate some of its properties. The internal logic of the Herbrand topos is given by "Herbrand realizability", a realizability interpretation defined in earlier joint work with Eyvind Briseid and Pavol Safarik.

2011

18-10-2011, (HG00.068) Ieke Moerdijk
The Classifying Space of a Category

25-10-2011 No seminar!!

01-11-2011, (HG00.086) Sander Wolters
Topos Theory and Foundations of Quantum Physics
Abstract: In this talk, which assumes familiarity with neither operator algebras nor topos theory, I want to give an idea of how the topos approach of Heunen, Landsman and Spitters to quantum theory works. In this approach a quantum system is described by a C*-algebra. By considering all the commutative subalgebras (which are thought of as different classical contexts relative to which we can study the quantum system) one arrives at a nontrivial topos (a category that has so much structure that we reason with its objects and arrows as if these were sets and functions). One of the appealing features of this approach (if not one of its guiding principles) is that typical constructions in algebraic quantum theory (such as self-adjoint operators representing physical quantities and positive normalized linear functionals representing physical states) considered in the topos of sets, look a lot more like constructions from classical physics when viewed from within nontrivial topos of the approach (observables are then continuous functions on some "state space" and the functionals that represent states now correspond to probability valuations on the state space). Subsequently I will discuss the process of "daseinisation" which, among other things, gives the best approximation of a physical quantity in any (classical) context.

08-11-2011, (HG01.028) Bas Spitters
Topos Theory and Foundations of Physics, the Interplay Between Internal and External Logic
Abstract: I will continue on the topic of Sander's lecture, emphasizing the interplay between the internal logic of a topos and its external interpretation.

15-11-2011 (HG01.028) Jorik Mandemaker
The Expectation Monad
Abstract: This talk will be about the expectation monad. This monad originated in computer science where it is used for programming semantics and security proofs. However, the expectation monad is usually defined in an ad-hoc manner in this line of work. In this talk I will give a succinct definition of the expectation monad in terms of a dual adjuction between convex sets and effect modules. This will lead to probabilistic versions of Manes and Gelfands theorems. We will show that certain well behaved algebras of the expectation monad are equivalent to convex compact Hausdorff spaces and dually equivalent to Banach effect modules.

22-11-2011 (HG01.028) Dion Coumans
Generalizing Canonical Extension to the Categorical Setting
Abstract In the 1950s Jonsson and Tarksi introduced the notion of canonical extension of a Boolean algebra with operators. In this setting, canonical extension provides an algebraic description of Stones topological duality. By now, the theory of canonical extensions has been developed further and it has proven be a powerful tool in the algebraic study of propositional logics. After a brief introduction in this theory, I'll define a notion of canonical extension for coherent categories, the categorical analogue of distributive lattices. This construction opens the door to applications of the theory of canonical extension in the study of first order logics.

29-11-2011 No seminar!!

06-12-2011 (HG00.310) Urs Schreiber
Cohesive Toposes
Abstract: Two simple axioms on a topos (locality and strong connectedness) ensure that it behaves like a category of "geometrical spaces", for instance like a category of smooth spaces. When the same axioms are formulated in the logic of intensional type theory, hence "up to homotopy", they enode a category of "geometrical groupoids", for instance Lie groupoids.
I give an introduction to some of the basic ideas and facts about such "cohesive toposes".

13-12-2011 (HG00.310) Thomas Nikolaus
Equivariant Dijkgraaf-Witten Theory
Abstract: For a finite group G there is a well known Quantum field theory called Dijkgraaf-Witten theory. From this theory one can extract an interesting tensor category. This category can also be described as the representation category of a Quantum group (ribbon Hopf algebra) D(G) called the Drinfel'd double of G. We present an equivariant extension of Dijkgraaf-Witten theory. This leads us to equivariant generalizations of D(G) and its represenation categories. They have a rich, but still accessible structure. We furthermore discuss the issue of modularity and the orbifold theory.