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¨ oteborg, 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. is isomorphic to . Prop. 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; open (and closed) sets are more important. A formal topology is given by a set of 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