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Dialectica Categories... and Lax Topological Spaces? Valeria de Paiva Rearden Commerce University of Birmingham March, 2012 Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 1 / 38

Dialectica Categories... and Lax Topological Spaces?

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(Joint work with Apostolos Syropoulos) Talk at Dept Matematica, Universidade Federal da Bahia, Salvador, Bahia, March 2012

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Page 1: Dialectica Categories... and Lax Topological Spaces?

Dialectica Categories...and Lax Topological Spaces?

Valeria de Paiva

Rearden CommerceUniversity of Birmingham

March, 2012

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 1 / 38

Page 2: Dialectica Categories... and Lax Topological Spaces?

LLI nasslli2012.com

June 18–22, 2012

Johan van Benthem Logical Dynamics of Information and Interaction

Chris Po�s Stanford UniversityExtracting Social meaning and Sentiment

Craige Roberts Ohio State UniversityQuestions in Discourse

Mark Steedman University of EdinburghCombinatory Categorial Grammar: �eory and Practice

Noah Goodman Stanford UniversityStochastic Lambda Calculus

and its Applications in Semantics and Cognitive Science

University of Amsterdam / Stanford University

�e Fi�h North American Summer School of Logic, Language, and Information

Jonathan Ginzburg - University of ParisRobin Cooper - Göteborg UniversityType theory with records for natural language semantics

Jeroen Groenendijk - University of AmsterdamFloris Roelofsen - University of AmsterdamInquisitive semantics

Shalom Lappin - King’s College LondonAlternative Paradigms for Computational Semantics

Tandy Warnow - University of TexasEstimating phylogenetic trees in linguistics and biology

Hans Kamp - University of TexasMark Sainsbury - University of TexasVagueness and context

Steve Wechsler - University of Texas Eric McCready - Osaka UniversityWorkshop on Meaning as Use: Indexicality and Expressives (Speakers: Eric McCready, Steve Wechsler, Hans Kamp, Chris Po�s, Pranav Anand, and Sarah Murray)

Catherine Legg - University of WaikatoPossible Worlds: A Course in Metaphysics(for Computer Scientists and Linguists)

Adam Lopez - Johns Hopkins UniversityStatistical Machine Translation

Eric Pacuit - Stanford UniversitySocial Choice �eory for Logicians

Valeria de Paiva - Rearden CommerceUlrik Buchholtz - Stanford UniversityIntroduction to Category �eory

Adam Pease - Rearden CommerceOntology Development and Application with SUMO

Ede Zimmermann - University of FrankfurtIntensionality

�omas Icard - Stanford UniversitySurface Reasoning

Nina Gierasimczuk - University of GroningenBelief Revision Meets Formal Learning �eory

at the University of Texas at Austin

Registration: $175 (academic rate) / $400 (professional rate)Student scholarships available, see website for application instructions

Accommodation provided for $70 / night (single) or $35 / night (double)

June 16–17: Bootcamp Session June 23–24: Texas Linguistic Society Conference Special sessions on American Sign Language, Semantics, and Computational Linguistics Details regarding Call for Papers at nasslli2012.com June 23: Turing Centennial Symposium

Additional Courses: Special Events:

Poster by Derya Kadipasaoglu (dkadipas.weebly.com) and Christopher BrownNASSLLI is sponsored by the NSF (BCS 1019206), the UT College of Liberal Arts, and the UT Departments of Linguistics, Philosophy, and Psychology.

Twenty interdisciplinary graduate-level courses, 90

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Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 2 / 38

Page 3: Dialectica Categories... and Lax Topological Spaces?

Goals

Describe two kinds of dialectica categoriesShow they are models of ILL and CLL, respectivelyCompare and contrast dialectica categories with Chu spacesDiscuss how we might be able to use these ideas for topological spaces

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 3 / 38

Page 4: Dialectica Categories... and Lax Topological Spaces?

Functional Interpretations and Gödel’s Dialectica

Starting with Gödel’s Dialectica interpretation(1958) a series of“translation functions" between theoriesAvigad and Feferman on the Handbook of Proof Theory:

This approach usually follows Gödel’s original example: first,one reduces a classical theory C to a variant I based onintuitionistic logic; then one reduces the theory I to aquantifier-free functional theory F.

Examples of functional interpretations:Kleene’s realizabilityKreisel’s modified realizabilityKreisel’s No-CounterExample interpretationDialectica interpretationDiller-Nahm interpretation, etc...

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 4 / 38

Page 5: Dialectica Categories... and Lax Topological Spaces?

Gödel’s Dialectica Interpretation

Gödel’s Aim: liberalized version of Hilbert’s programme – to justifyclassical systems in terms of notions as intuitively clear as possible.

Gödel’s result: an interpretation of intuitionistic arithmetic HA in aquantifier-free theory of functionals of finite type Tidea: translate every formula A of HA to AD = ∃u∀x.AD where AD isquantifier-free.use: If HA proves A then T proves AD(t, y) where y is a string ofvariables for functionals of finite type, t a suitable sequence of terms notcontaining y.

Method of ‘functional interpretation’ extended and adapted to several othertheories systems, cf. Feferman.

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 5 / 38

Page 6: Dialectica Categories... and Lax Topological Spaces?

Gödel’s Dialectica Interpretation

For each formula A of HA we associate a formula of the formAD = ∃u∀xAD(u, x) (where AD is a quantifier-free formula of Gödel’ssystem T) inductively as follows: when Aat is an atomic formula, then itsinterpretation is itself.Assume we have already defined AD = ∃u∀x.AD(u, x) andBD = ∃v∀y.BD(v, y).We then define:

(A ∧B)D = ∃u, v∀x, y.(AD ∧BD)(A→ B)D = ∃f : U → V, F : U ×X → Y, ∀u, y.( AD(u, F (u, y))→ BD(fu; y))(∀zA)D(z) = ∃f : Z → U∀z, x.AD(z, f(z), x)(∃zA)D(z) = ∃z, u∀x.AD(z, u, x)

The intuition here is that if u realizes ∃u∀x.AD(u, x) then f(u) realizes∃v∀y.BD(v, y) and at the same time, if y is a counterexample to∃v∀y.BD(v, y), then F (u, y) is a counterexample to ∀x.AD(u, x).

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 6 / 38

Page 7: Dialectica Categories... and Lax Topological Spaces?

Gödel’s Dialectica Interpretation: logical implication

Most interesting aspect: If AD = ∃u∀x.AD and BD = ∃v∀y.BD

(A⇒ B)D = ∃V (u)∃X(u, y)∀u∀y[AD(u,X(u, y))⇒ BD(V (u), y)]

This can be explained (Spector’62) by the following equivalences:

(A⇒ B)D = [∃u∀xAD ⇒ ∃v∀yBD]D (1)= [∀u(∀xAD ⇒ ∃v∀yBD)D] (2)= [∀u∃v(∀xAD ⇒ ∀yBD)]D (3)= [∀u∃v∀y(∀xAD ⇒ BD)]D (4)= [∀u∃v∀y∃x(AD ⇒ BD)]D (5)= ∃V ∃X∀u, y(AD(u,X(u, y)⇒ BD(V (u), y)] (6)

All steps classically valid. But:Step (2) (moving ∃v out) is IP (not intuitionistic),Step (4) uses gener. of Markov’s principle (not intuitionistic)for (6) need to skolemize twice, using AC (not intuitionistic).

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 7 / 38

Page 8: Dialectica Categories... and Lax Topological Spaces?

Basics of Categorical Semantics

Model theory using categories, instead of sets or posets.Two main uses “categorical semantics”:categorical proof theory, and categorical model theoryCategorical proof theory models derivations (proofs) not simplywhether theorems are true or not

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 8 / 38

Page 9: Dialectica Categories... and Lax Topological Spaces?

Categorical Semantics: what?

Based on Curry-Howard Isomorphism:

Natural deduction (ND) proofs ⇔ λ-termsNormalization of ND proof ⇔ reduction in λ-calculus.

Originally: constructive prop. logic ⇔ simply typed λ-calculus.Applies to other (constructive) logics and λ-calculi, too.Cat Proof Theory:λ-calculus types ⇔ objects in (appropriate) categoryλ-calculus terms ⇔1−1 morphisms in categoryCat. structure models logical connectives (type operators).Isomorphism: transfers results/tools from one side to the otherλ-calculus basis of functional programming: logical view ofprogramming

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 9 / 38

Page 10: Dialectica Categories... and Lax Topological Spaces?

Where is my category theory?

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 10 / 38

Page 11: Dialectica Categories... and Lax Topological Spaces?

Categorical Logic?

Well known that we can do categorical models as well as settheoretical ones.Why would you?Because sometimes this is the only way to obtain models.E.g. Scott’sλ-calculus

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 11 / 38

Page 12: Dialectica Categories... and Lax Topological Spaces?

Categorical Logic?

Well known that we can do categorical models as well as settheoretical ones.

Why would you?Because sometimes this is the only way to obtain models.E.g. Scott’sλ-calculus

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 11 / 38

Page 13: Dialectica Categories... and Lax Topological Spaces?

Categorical Logic?

Well known that we can do categorical models as well as settheoretical ones.Why would you?

Because sometimes this is the only way to obtain models.E.g. Scott’sλ-calculus

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 11 / 38

Page 14: Dialectica Categories... and Lax Topological Spaces?

Categorical Logic?

Well known that we can do categorical models as well as settheoretical ones.Why would you?Because sometimes this is the only way to obtain models.

E.g. Scott’sλ-calculus

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 11 / 38

Page 15: Dialectica Categories... and Lax Topological Spaces?

Categorical Logic?

Well known that we can do categorical models as well as settheoretical ones.Why would you?Because sometimes this is the only way to obtain models.E.g. Scott’sλ-calculus

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 11 / 38

Page 16: Dialectica Categories... and Lax Topological Spaces?

Categorical Semantics: some gains..

Functional Programming: optimizations that do not compromisesemantic foundations (e.g. referential transparency)Logic: new applications of old theorems (e.g. normalization)Category Theory: new concepts not from mathsPhilosophy: new criteria for identities of proofs

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 12 / 38

Page 17: Dialectica Categories... and Lax Topological Spaces?

Basics of Linear Logic

A new(ish..) logic taking resource sensitivity into accountDeveloped mainly from proof theoretic perspective:— must keep track of where premises / assumptions are usedBut ability to ignore resource sensitivity whenever you want:(a) Modal ! allowing duplication and erasing of resources(b) A⇒ B = !A−◦B

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 13 / 38

Page 18: Dialectica Categories... and Lax Topological Spaces?

Where is my category theory?

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 14 / 38

Page 19: Dialectica Categories... and Lax Topological Spaces?

Linear categories

Model multiplicative-additive fragment is easymultiplicative fragment: need a a symmetric monoidal closedcategory (SMCC)additive fragment: need (weak) categorical products andcoproductslinear negation: need an involution

modalities (exponentials): the problem...several solutions, depending on linear λ-calculus chosen...

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 15 / 38

Page 20: Dialectica Categories... and Lax Topological Spaces?

Symmetric Monoidal Closed Categories (SMCC)

A category L is a symmetric monoidal closed category ifit has a tensor product A⊗B for all object A,B of C.for each object B of C, the tensor-product functor _⊗B has aright-adjoint, written B−◦_(ie have isomorphism between (A⊗B)→ C and A→ (B−◦C))

Recap:Categorical product A×B Tensor product A⊗B

1 Projections onto A and B Not necessarily2 For any object C with maps Not necessarily

to A and B, there is aunique map from C to A×B

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 16 / 38

Page 21: Dialectica Categories... and Lax Topological Spaces?

Modelling the Modality !

Objects A in a CCC have duplication A→ A×A and erasing A→ 1(satisfying equations of a comonoid).Objects A in a SMCC have no duplication A→ A⊗A nor erasingA→ I, in general. Neither have they projections A⊗B → A,B.But objects of the form !A should have duplication and erasing.Moreover they must satisfy S4-Box introduction and elimination rules.S4-axioms give you proofs !A→ A and !A→!!A, uniformly for any Aobject A in C. Hence want a functor (unary operator) on category,! : C → C such that there are such natural transformations. Such astructure already in category theory, a comonad.

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 17 / 38

Page 22: Dialectica Categories... and Lax Topological Spaces?

Modelling the Modality !

An object !A must have four maps:

ε : !A→ A, δ : !A→!!A, er : !A→ I, dupl : !A→!A⊗!A

To make identity of proofs sensible, must have lots of equations. Theones relating to duplicating and erasing say !A is a comonoid withrespect to tensor; the ones relating to S4-axioms say ! is a comonad.First idea (Seely’87): Logical equivalence !(A&B) ≡ !A⊗!BIntroduce a comonad (!, δ, ε) defining ! such that

!(A×B) ∼= !A⊗!B

Say this comonad takes additive comonoid structure(A, A→ A×A, A→ 1) (if it exists) to desired comonoid(!A, dupl : !A→!A⊗!A, er : !A→ I).

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 18 / 38

Page 23: Dialectica Categories... and Lax Topological Spaces?

Categorical Semantics of LL: problem

Modelling uses isomorphism !(A&B) ∼=!A⊗!BBut can’t model multiplicatives & modalities without additives.So, better idea:If we don’t have additive conjunction, ask for morphisms

mA,B : !(A⊗B)→ !A⊗!B

mI : !I → I

i.e. monoidal comonad to deal with contexts

Must also ask for (coalgebra) conditions relating comonad structure tocomonoid structure.Hence:

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 19 / 38

Page 24: Dialectica Categories... and Lax Topological Spaces?

Categorical Semantics of LL: solution

A linear category comprises— an SMCC C, (with products and coproducts),— a symmetric monoidal comonad (!, ε, δ,mI ,mA,B)(a) For every free co-algebra (!A, δ) there are nat. transfns.erA : !A→ I and duplA : !A→!A⊗!A, forming a commutative comonoid,which are coalgebra maps.(b) Every map of free coalgebras is also a map of comonoids.Bierman 1995 (based on Benton et al’93)

long definition, lots of diagrams to check..Prove A⇒ B ∼=!A−◦B by constructing CCC out of SMCC plus comonad!. Extended Curry-Howard works!!

Classical linear category = linear category with involution,()⊥ : Cop → C, to model negation, same as ∗-autonomous category, with(weak) products and coproducts plus a symm monoidal comonad satisfying(a) and (b) above.

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 20 / 38

Page 25: Dialectica Categories... and Lax Topological Spaces?

Alternative Categorical Semantics

(Benton 1995) A model of LNL (Linear/NonLinear Logic) consists of aSMC category L together with a cartesian closed category C and amonoidal adjunction between these categories.

Modelling two logics, intuitionistic and linear at the same time.Logics on (more) equal footing..

First definition builds the cartesian closed category out of the monoidalcomonad, this def asks for the adjunction

Easier to remember, easier to say, same diagrams to check..

Extended Curry-Howard works (Barber’s thesis)(reduction calculus (+explicit subst) needed for implementation, Wollic’98)

classical version: *-autonomous cat, plus monoidal adjunction, plusproducts

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 21 / 38

Page 26: Dialectica Categories... and Lax Topological Spaces?

Categorical Semantics: Back to Dialectica

Have a definition of what a categorical model of ILL ought to be, canwe find some concrete ones?Dialectica categories (Boulder’87) one such model..Assume C is cartesian closed categoryDC objects are relations between U and X in C,monics AαU ×X written as (UαX).DC maps are pairs of maps of Cf : U → V , F : U × Y → Xmaking a certain pullback condition hold.Condition in Sets: uαF (u, y)⇒ f(u)βy.

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 22 / 38

Page 27: Dialectica Categories... and Lax Topological Spaces?

Reformulating DC

C is Sets: objects are maps A = U ×Xα2, and B = V × Y β2.morphisms are pairs of maps f : U → V , F : U × Y → X such thatα(u, F (u, y)) ≤ β(f(u), y):

U × Y〈π1, F 〉- U ×X

V × Y

f × idY

?

β- 2

α

?

Usually write maps asU α X

⇓ ∀u ∈ U,∀y ∈ Y α(u, F (u, y)) ≤ β(f(u), y)

V

f

?β Y

F

6

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 23 / 38

Page 28: Dialectica Categories... and Lax Topological Spaces?

Relationship to Dialectica

Objects of DC represent essentially AD.Maps some kind of normalisation class of proofs.abstractly: maps are realisation of the formula AD → BD.Expected: cat DC is ccc, as Gödel int. as constructive as possible.Surprise: DC is a model of ILL, not IL.But (very neat result!)Diller-Nahm variant gives CCC.Note: reformulation makes basic predicates AD decidable, as in Gödel’s work, but first version is more general.

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 24 / 38

Page 29: Dialectica Categories... and Lax Topological Spaces?

Structure of DC

DC is a SMCC with categorical products:

A�B = (U × V α� βX × Y )

A−◦B = (V U ×XU×Y α−◦βU × Y )

A&B = (U × V αβX + Y )

only weak coproducts:

A+B = (U + V α+ βXU × Y V )

Morever!A = (Uα∗X∗)

is cofree comonoid on A.

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 25 / 38

Page 30: Dialectica Categories... and Lax Topological Spaces?

Conjunction: neat detail

A 6` A�A as need a DC map (∆, δ) such that

U α X

U × U

?α� αX ×X.

δ

6

∀u ∈ U,∀x, x′ ∈ X ×X α(u, δ(u, x, x′))⇒ (α� α)(∆(u), (x, x′)).But (α� α)(∆(u), (x, x′)) = α(u, x) ∧ α(u, x′)hence unless α decidable can’t do anything...If α decidable define:δ(u, x, x′) = x′ if α(u, x)δ(u, x, x′) = x otherwise. Same logical argument..

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 26 / 38

Page 31: Dialectica Categories... and Lax Topological Spaces?

Dialectica categories DC: summary

DC pluses:structure needed to model ILL,including non-trivial modality !related to Diller-Nahm variantconjunction explainedindependent confirmation of LL’s worth;Dialectica is interesting (and mysterious).DC minuses:accounting of recursion?mathematics involved;terse style – AMS vol.92

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 27 / 38

Page 32: Dialectica Categories... and Lax Topological Spaces?

Dialectica model GC

Girard’s suggestion ’87C cartesian closed category, say Sets.GC objects are relations between U and X, in C as before.GC maps are pairs of maps of C, f : U → V , F : Y → Xsuch that uαF (y)⇒ f(u)βy.As before, can ‘reformulate’ and say A is map U ×X → 2

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 28 / 38

Page 33: Dialectica Categories... and Lax Topological Spaces?

Structure of GC

GC is ∗-autonomous with products and coproducts. Involution is given byimplication into ⊥, unit for extra monoidal structure .....................................................

.......................................... .

A−◦B = (V U ×XY α−◦βU × Y )A⊗B = (U × V α⊗ βXV × Y U )A

...............................................................................................B = (V X × UY α

............................................................................................... βX × Y )

A&B = (U × V α.βX + Y )A⊕B = (U + V α.βX × Y )

Moreover, GC has Exponentials

!A = (U !α(X∗)U )

?A = ((U∗)X?αX)

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 29 / 38

Page 34: Dialectica Categories... and Lax Topological Spaces?

Dialectica-like categories GC

GC pluses:the best model of LL ?..no collapsing of unitiesmaths easier for modality-free fragmentmodalities adaptableshould think of Sets!GC minuses:MIX ruleC must be ccc

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 30 / 38

Page 35: Dialectica Categories... and Lax Topological Spaces?

Chu construction ChuKC

Suppose C is cartesian closed category, (but could be smcc with pullbacks)and K any object of C.ChuKC construction in Barr’s book, 1972.ChuKC objects are (generalised) relations between U and X or maps ofthe form U ×XαK.ChuKC maps are pairs of maps of C f : U → V , F : Y → X such that

uαF (y) = f(u)βy

To make comparison with GC transparent read equality as

uαF (y)⇔ f(u)βy)

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 31 / 38

Page 36: Dialectica Categories... and Lax Topological Spaces?

Comparing GC and ChuKC

ChuKC pluses: smcc instead of ccctopological spaces, plus other representabilities (see Pratt)ChuKC minuses:No K can make I and ⊥ different!!modalities Barr(1990) harder, also more pullbacks.Lafont and Streicher (LICS’91), ChuKSets is GAMEK , but with modalitybased on GCStructure:

A−◦ChuB = (L1α−◦CβU × Y )

where L1 = pullback ⊂ V U ×XY

similarly for par and tensor..

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 32 / 38

Page 37: Dialectica Categories... and Lax Topological Spaces?

A mild generalisation of GC

Categories DialLC, where C is smcc with products and L a closed poset (orlineale).Objects of DialLC:generalised relations between U and X or maps of the form

U ⊗XαL

DialLC maps are pairs of maps of C f : U → V , F : Y → Xsuch that uαF (y)⇒ f(u)βy(Now implication inherited from L).

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 33 / 38

Page 38: Dialectica Categories... and Lax Topological Spaces?

Vicker’s Topological Systems

A triple (U,α,X), where X is a frame, U is a set, and α : U ×X → 2 abinary relation, is called by Vicker’s a topological system if

when S is a finite subset of X, then

α(u,∧S) ≤ α(u, x) for all x ∈ S.

when S is any subset of X, then

α(u,∨S) ≤ α(u, x) for some x ∈ S.

α(u,>) = 1 and α(u,⊥) = 0 for all u ∈ U .

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 34 / 38

Page 39: Dialectica Categories... and Lax Topological Spaces?

A Frame, did you say?

A partially ordered set (X,≤) is a frame if and only ifEvery subset of X has a join;

if S is a finite subset subset of X it has a meet;

Binary meets distribute over joins:x ∧

∨Y =

∨{x ∧ y : y ∈ Y }

Frames are sometimes called locales. They confuse me a lot.

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 35 / 38

Page 40: Dialectica Categories... and Lax Topological Spaces?

Maps of Topological Systems

If (U,α,X) and (V, β, Y ) are topological systems, a map consists off : U → V and F → X such that uαFy = fuβy.As in Chu spaces this condition can be broken down into two halves:uαFy implies fuβy and fuβy implies uαFy.If we only have one half, we are back into a dialectica-like situation.

Call this the category of lax topological systems, TopSysL.What can we prove about TopSysL?We know it has an SMCC structure.Anything else? Would topologists be interested?

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 36 / 38

Page 41: Dialectica Categories... and Lax Topological Spaces?

Further Work?

Presented 2 kinds of dialectica categories and four theorems aboutthem, explaining where they came fromHinted at some others...Not enough work done on these dialectica categories all togetherIteration,(co-)recursion not touched, spent three days thinking aboutit...We’re thinking about instances of the construction appropriate forcardinal invariantsAlso about doing primitive recursion categoricallyWork on logical extensions, model theory should be attempted (vanBenthem little paper)

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 37 / 38

Page 42: Dialectica Categories... and Lax Topological Spaces?

Some References

The Dialectica Categories, V.C. V de Paiva, Categories in ComputerScience, Contemporary Maths, AMS vol 92, 1989.Dialectica-like Categories, V.C. V de Paiva, Proc. of CTCS’89, LNCSvol 389.The Dialectica categories, Ph D thesis, available as Technical Report213, Computer Lab, University of Cambridge, UK, 1991.Categorical Proof Theory and Linear Logic –preliminary notes forSummer SchoolLineales, Hyland, de Paiva, O que nos faz pensar? 1991A Dialectica Model of the Lambek Calculus, Proc. 8th AmsterdamColloquium’91.A linear specification language for Petri nets, Brown, Gurr & de Paiva,TR ’91Full Intuitionistic Linear Logic, Hyland & de Paiva APAL’93 273–291Abstract Games for Linear Logic, Extended Abstract, Hyland & SchalkCTCS’99.

Valeria de Paiva (Rearden Commerce University of Birmingham ) March, 2012 38 / 38