**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, (HG00.086) *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.

**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.