String diagrams. a functorial semantics of proofs and programs. Paul-André Melliès. CNRS, Université Paris Denis Diderot.

Transcription

1 Sring diagram a uncorial emanic o proo and program Paul-ndré Melliè CNRS, Univerié Pari Deni Didero Hop-in-Lux Luxembourg 17 July

2 Connecing 2-dimenional cobordim and logic 2

3 Par 1 Caegorie a monad 3

4 Saring poin: caegorie a monad Depending on he bicaegory, he poin o view ederae: Enriched caegorie Inernal caegorie lgebra a bicaegory o V-marice a bicaegory o pan a bicaegory o comodule The la connecion appear in Marcelo guiar PhD hei 4

5 The bicaegory Ca() he 0-cell are he monad (, ) o he bicaegory he 1-cell (, ) : (, ) (, ) are he pair coniing o a 1-cell and o a 2-cell : : : 5

6 The bicaegory Ca() Diagrammaically: aiying he equaliie below: 6

7 The bicaegory Ca() id η = id id η µ = µ 7

8 n alernaive ormulaion in ring diagram = = 8

9 The bicaegory Ca() Same deiniion, excep ha he direcion o he 2-cell change: kind o ibraion wihou i uncor 9

10 The 2-cell o he bicaegory Ca() he 2-cell are he 2-cell θ : g : θ : g : uch ha he diagram o 2-cell commue: θ g θ µ g g g µ g Reduced orm 10

11 n alernaive ormulaion in ring diagram g g = Reduced orm 11

12 The 2-cell o he bicaegory Ca() he 2-cell are he 2-cell θ : g : θ : g : making he diagram o 2-cell commue: θ g θ µ g g g µ g Non reduced orm 12

13 lernaive ormulaion in ring diagram g g = Non reduced orm 13

14 From he reduced orm o he non reduced orm (and converely) g g g g = = 14

15 Fir equaion g g g g = = = 15

16 Second equaion g g g = = 16

17 Propery o he non reduced orm g g g = = 17

18 Illuraion: inernal caegorie Given a caegory C wih inie limi... he bicaegory Ca(Span) ha he ame 0-cell a he caegory C, he 1-cell are he pan, he 2-cell are morphim beween pan. Fac: a monad in Span i an inernal caegory in C 18

19 Par 2 Module beween caegorie caegorie een a ring wih everal objec 19

20 Repreenaion principle Every monad (in he bicaegorical ene) : induce a monad (in he caegorical ene) (X, ) : (X, ) (X, ) deined by po-compoiion X X or every 0-cell X o he bicaegory. 20

21 Repreenaion principle (dual) Dually, every monad (in he bicaegorical ene) : induce a monad (in he caegorical ene) (, X) : (, X) (, X) deined hi ime by pre-compoiion: X X or every 0-cell X o he bicaegory. 21

22 Repreenaion principle (on boh ide) Every pair o monad (in he bicaegorical ene) : : induce a monad (in he caegorical ene) (, ) : (, ) (, ) deined by pre- and po- compoiion: or every 0-cell X o he bicaegory. 22

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24 The bicaegory Module() he 0-cell are he monad (, ) o he bicaegory he 1-cell (, ) (, ) are he algebra o he monad (, ) : (, ) (, ). So, hey are pair (, φ) coniing o a 1-cell 23

25 and a 2-cell φ in he bicaegory, aiying he coherence diagram:

26 The bicaegory Module() µ φ µ = φ φ 24

27 id η φ η id = id id id

28 The bicaegory Module() he 2-cell (,φ) (, ) θ (, ) (g,ψ) are he morphim o (, )-algèbre in he caegory (, ). 25

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30 The bicaegory Module() In oher word, a 2-cell (,φ) (, ) θ (, ) i a 2-cell θ (g,ψ) g 26

31 aiying he equaliy: θ φ = g ψ θ g g

32 The bicaegory Module() The compoie o he wo 1-cell (, ) (,φ) (, ) (g,ψ) (C, u) i deined a he co-equalizer o he 2-cell decribed by he wo dieren way o compoe: g φ g C g g ψ g C 27

33 We make he hypohei (1) ha he caegory (, ) ha coequalizer, or every,, (2) he horizonal compoiion in preerve hee co-equalizer.

34 Par 3 Game cobordim 28

35 Dialogue caegorie ymmeric monoidal caegory C equipped wih a uncor and a naural bijecion : C op C ϕ,,c : C(, C) C(, ( C ) ) C C 29

36 Cobordim 30

37 Frobeniu objec Frobeniu objec F i a monoid and a comonoid aiying d m = d m = m d an alernaive ormulaion o cobordim 31

38 Frobeniu objec Equivalenly, a Frobeniu objec F i a monoid wih an iomorphim o i dual objec F uch ha S : F F S S = F F F F F F 32

39 Lax dualiie in a 3-dimenional caegory L L L η ε L R ε η R R R 33

40 aiying wo coherence properie (a) η R L R η L i he ideniy 3-cell on he uni η o he 2-adjuncion L R, 34

41 aiying wo coherence properie (b) L ε L R ε R i he ideniy 3-cell on he couni ε o he 2-adjuncion L R. 35

42 Peudo Frobeniu objec peudo Frobeniu objec in he bicaegory o module d m d m m d i he ame hing a a -auonomou caegory... when he wo module m and e are uncor. n obervaion by rian Day and Ro Sree (2003) 36

43 Lax Frobeniu objec Relax he el-dualiy equivalence C C op ino an adjuncion C L R C op hi connec game emanic and quanum group 37

44 Game emanic in ribbon diagram C (x, y) C (x, y) S = C op C op C op C op C op C op Idea: replace he elemenary paricle by he game board 38

45 Game emanic in ribbon diagram C (x, y) C (x, y) = C op C op C op C op C op C op Idea: replace he elemenary paricle by he game board 39

46 Game emanic in ribbon diagram C (x, y) C (x, y) = C op C op C op C op C op C op Idea: replace he elemenary paricle by he game board 40

47 Concluion Logic = Daa Srucure + Dualiy Thi poin o view i acceible hank o 2-dimenional algebra 41