### Workshop on Algebra, Coalgebra and Topology (part of the Wessex Theory Seminar)

### University of Bath, Friday 1 March 2013

The shorter url go.bath.ac.uk/wessex20 also points to this page.

## Announcement

The Technische Universität Dresden is one of the world's leading centres in clone theory.

An abstract clone, with nullary operations, is a mild rephrasing of what category theorists call a Lawvere theory. They are also equivalent to finitary monads, but the equivalence with the latter is far more than mild rephrasing.

The notions of abstract clone and Lawvere theory have been developed for several decades largely independently of each other. Much of the development of Lawvere theories has been done in the UK, in recent years primarily in association with the University of Cambridge. Duality has been studied for both, with notions such as coclone, comodel and coalgebra prominent. Both have interacted with computer science, and both have involved relationships with topology.

Several of the key researchers in the two fields will meet for a workshop at the University of Bath on Friday 1 March. Before lunch, Mike Behrisch, Sebastian Kerkhoff and Martin Schneider from Dresden will speak on clone theory, explaining the background and current research, with a view towards categories. After lunch, Jiri Adamek and Martin Hyland will speak from a category theoretic perspective; and late in the afternoon, we will have talks more broadly about the topic from Nicolai Vorobjov, Uday Reddy, James Davenport and Sam Staton.

We would like to invite others to participate too. If you are interested in coming, please contact Cai Wingfield (C.A.J.Wingfield@bath.ac.uk).

## Abstracts

**"A gentle introduction to clones, their Galois theory, and applications" Mike Behrisch (TU Dresden)**

We present the notion of clone of operations (within the category of sets) via three prototypical examples: clones arising within concrete catego- ries over Set having finite powers, clones of term operations of algebras, operations being compatible with a given set of finitary relations. Fur- thermore, we generally discuss the classical Galois theoretical approach (Pol-Inv) towards clone theory, based on the concept of preservation (compatibility), thereby linking clones to certain relational structures, called relational clones. In particular, we deal with the differences be- tween clones on finite and infinite carrier sets. Finally, we consider a few classical results of clone theory, as well as recent applications.

**"(Concrete) Clones and Pol-Inv vs. (Models of) Lawvere Theories and ???" Sebastian Kerkhoff (TU Dresden)**

Building on the previous talk, we discuss the relationship between (mod- els of) abstract clones and (models of) Lawvere theories. Although the two notions are easily seen to be equivalent, the talk aims at suggesting how they might benefit from each other. More precisely, we will out- line how the concept of duality might motivate universal algebraists to treat clones as (models of) Lawvere theories, and we will discuss how the Galois connection Pol-Inv (introduced in the previous talk) might be generalized into a category-theoretic setting. In particular, we will address the question why the latter might also be useful for those that mainly care about classical clone theory and hence already have such a Galois connection at their hands.

**"Clones and invariant relations in categories" Martin Schneider (TU Dresden)**

Continuing the two talks by Mike Behrisch and Sebastian Kerkhoff, we give some details on a possible category-theoretic generalization of the Galois connection Pol-Inv and related operators. A particular focus will be laid on the situation in the category of compact Hausdorff spaces.

**"Nondeterministic Closure Automata" Jiri Adamek (TU Braunschweig)**

(Joint work with R. Myers, S. Milius and H. Urbat.) Based on the coalgebraic automata theory the concept of a nondeterministic closure automaton is introduced as an automaton whose states form a closure space. The minimal one is constructed for every regular language. If its closure is a topology, we obtain a minimal nondeterministic automaton in the classical sense.

**"Algebraic theories: a general bicategorical approach" Martin Hyland (University of Cambridge)**

TBA

**"Monotone functions and maps." Nicolai Vorobjov (University of Bath)**

A subset X of R^n is called a (topologically) regular k-cell if the pair (\bar X,X) is homeomorphic to the pair ([0,1]^k, (0,1)^k). It is usually not easy to prove that a given X, defined by a formula or a construction, is a regular cell. Sometimes this can be done by proving a stronger but more explicit property, that X is a graph of a monotone map. The latter is a multi-dimensional generalization of a usual univariate continuous monotone function, while the closure of the graph of a monotone map is a generalization of a compact convex set. Graps of monotone maps are regular cells. The described above methodology of proving regularity can be demonstrated on the example of toric cubes - images of the standard cube under monomial maps. We prove the conjecture of Sturmfels et al. that toric cubes are regular cells.

**"Automata theory for program semantics" Uday Reddy (University of Birmingham)**

TBA

**"What can algebra tell us about pedagogical correctness?" James Davenport (University of Bath)**

**"Abstract clones and computational effects" Sam Staton (University of Cambridge)**

We can build computational monads in a principled way by considering equational theories of computation and using a free algebra construction. I will discuss this idea from the perspective of abstract clones. I will present an enriched version of abstract clones, focusing particularly on presheaf enrichment. I will revisit earlier work on Freyd categories and provide a clone perspective on my syntactic "parameterized algebraic theories".

**Participants**

**Technische Universität Dresden**- Sebastian Kerkhoff
- Martin Schneider
- Mike Behrisch
- Cynthia Glodeanu

**University of Bath**- Guy McCusker
- Jim Laird

- John Power
- Cai Wingfield
- Nicolai Vorobjov
- Russell Bradford
- Alessio Guglielmi
- Anupam Das
- Paola Bruscoli
- James Davenport
- Teo Floor
- Willem Heijltjes
- David Wilson
- Matthew England
- G. K. Sankaran
- Benjamin Pring
- Florian Gellert
- Daniel Joseph Finnegan
- Fatemeh Jahedpari
- Leon Watts
- Alan Hayes
- Alex Collins
- Marina De Vos
- James Rooney

**University of Cambridge**- Martin Hyland
- Ben Price
- Sam Staton
- Jonas Frey

**Queen Mary, University of London**- Edmund Robinson

**University of Birmingham**- Uday Reddy
- Chuangjie Xu
- Paul Levy
- Liang-Ting Chen

**Technische Universität Braunschweig**- Jiri Adamek

**University of Oxford**- Kwok-Ho Cheung
- Yoshihiro Maruyama

**University of Swansea**- Anton Setzer
- Ulrich Berger
- Oliver Kullmann

**Imperial College London**- Faris Abou-Saleh

**Ludwig-Maximilians-Universität München**- Catrin Campbell-Moore

**Università degli Studi di Genova**(via video)- Giuseppe Rosolini
- Eugenio Moggi
- Ruggero Pagnan
- Fabio Pasquali
- Francesco Aloe
- Enrico Ghiorzi

**University of Leicester**- Roy Crole

**Université Paris 13**- Etienne Duchesne

**Università di Bologna**- Maurizio Gabbrielli

## Programme

- 10:45–11:15 Mike Behrisch (TU Dresden)
*"A gentle introduction to clones, their Galois theory, and applications"* - 11:15–11:45 Sebastian Kerkhoff (TU Dresden)
*"(Concrete) Clones and Pol-Inv vs. (Models of) Lawvere Theories and ???"* - 11:45–12:15 Martin Schneider (TU Dresden)
*"Clones and invariant relations in categories"* - 12:15–13:30 Lunch
- 13:30–14:15 Jiri Adamek (TU Braunschweig)
*"Nondeterministic Closure Automata"* - 14:15–15:00 Martin Hyland (University of Cambridge)
*"Algebraic theories: a general bicategorical approach"* - 15:00–15:30 Break
- 15:30–16:00 Nicolai Vorobjov (University of Bath)
*"Monotone functions and maps"* - 16:00–16:30 Uday Reddy (University of Birmingham)
*"Automata theory for program semantics"* - 16:30–17:00 James Davenport (University of Bath)
*"What can algebra tell us about pedagogical correctness?"* - 17:00–17:30 Sam Staton (University of Cambridge)
*"Abstract clones enriched in presheaf categories"*

## Venue

The meeting will be held in the (very pleasant) department lounge of the Department of Computer Science, East Building, University of Bath.

- Campus map, showing East Building [pdf]
- Campus map with accessibility information [pdf]
- Getting to the University of Bath

By lucky coincidence, the Bath Literature Festival and Bristol Jazz and Blues Festival will both be happening on the 1st and the days following.

## Registration Fee

There is no registration fee, but the meeting will be held on a pay-your-own-everything basis. We will show you suitable places for morning and afternoon teas and for lunch, either sit-down or take-away.

## Travel Funding

The workshop will be part of the Wessex Theory Seminar, so there is a (very) small amount of travel funding available to those people working at Wessex sites. If you would like to apply for it, please email Guy McCusker (G.A.McCusker@bath.ac.uk).

## Accomodation

Few people are likely to need accommodation for a one-day meeting, but if you do and need help, please contact us, although you can probably find it as easily on the web as we can. You are strongly advised to book accommodation soon, as the Bath Literature Festival starts that day and much of the accommodation we normally recommend is already sold out.

## Proceedings

We plan to publish an ENTCS post-proceedings of the workshop if we receive a reasonable number of high quality submissions of relevant articles. We can discuss details at the time. The one caveat is that Elsevier now charges $50/paper for publishing in ENTCS, which we may need to pass on to successful authors.

## Live Video Stream

We will attempt to broadcast a live video feed of the talks throughout the day using the service Livestream. The live feed will be available here (free account required to view). Remote viewers can ask questions via Skype (text only) by contacting user "caiwingfield" during the event. *Both of these options will be subject to our internet connection (and other technology) working as planned.*