A category of games for topology Pierre Hyvernat (joint work with - - PowerPoint PPT Presentation

A category of games for topology

Pierre Hyvernat (joint work with Peter Hancock)

hyvernat@iml.univ-mrs.fr

Chalmers, Computing science (G¨

• teborg, Sweden)

Institut math´ ematique de Luminy, (Marseille, France)

APPSEM-II, Friday March 28th 2003 – p.1

Cultural bit...

Alexander Grothendieck 28 of March in Berlin (Germany). – Bourbaki (with Weil, Cartan and Dieudonné) – topology, algebraic geometry; – Fields medal in 1966.

APPSEM-II, Friday March 28th 2003 – p.2

The games

Emphasis on states rather than on moves:

• Def. an interaction structure is given by:

– a set of states:

– for each state, a set of moves:

– after each move, a set of counter-moves:

– after each counter-move, a new state:

• APPSEM-II, Friday March 28th 2003 – p.3

Strategies, Angel’s side

For

• (final states) and

• ,

is the set of

• winning strategies.

for the Angel

It is an inductive definition:

if

then

;

if

and

, then

.

APPSEM-II, Friday March 28th 2003 – p.4

Strategies, Demon’s side

For and

• ,

is the set of

• restrictive strategies.

for the Demon

It is an co-inductive definition:

• ✎

if

• ✟

APPSEM-II, Friday March 28th 2003 – p.5

Simulations

Generalization of usual simulation on automata. Def.: if

and

are 2 interaction structures, and

a relation on

• ✁

• ✄

,

is a simulation if:

with

,

,

and

.

APPSEM-II, Friday March 28th 2003 – p.6

Refinement

Def.: a refinement from

• ✝

to

• ✟

... ...is a simulation from

• ✝

to “

• ✁

”.

where “_

” is a reflexive / transitive closure.

Prop.

• ✝
• ✁

is isomorphic to

• ✁

• ✁

.

APPSEM-II, Friday March 28th 2003 – p.7

Saturation (difficult!)

For a refinement , we write for the relation:

• ✁

Prop. is still a refinement. and have the same strength if

(We write

.)

APPSEM-II, Friday March 28th 2003 – p.8

Space, covering relation

Idea:

points are too complex;

• pen (and closed) sets are more important.

A formal topology is given by a set

• f basic open sets.

(a base)

There is a covering relation:

• .

If

• and

• ,

• ✁

means that “

is covered by

”

(“

is smaller than the

”...)

This is enough to start doing topology!

APPSEM-II, Friday March 28th 2003 – p.9

And so...

• Th. The category of “non-distributive” formal topologies “is”

the category of interaction structures...

topological spaces are interaction structures;

continuous relations are (inverses) of refinements;

equality is

• .

APPSEM-II, Friday March 28th 2003 – p.10

If you (really) want more...

Details for the above – P . Hancock, P . Hyvernat: “Interaction, computer science and topology”. Basic topologies Giovanni Sambin’s “basic picture” articles, especially: – G. Sambin and S. Gebellato: “Pointfree continuity and convergence” Inductive generation of formal topologies – T. Coquand, G. Sambin, J. Smith, S. Valentini: “Inductively generated topologies.”

APPSEM-II, Friday March 28th 2003 – p.11