Categorical models of DiLL pol

pN ,˛q pP,˛q

L

R

%

with natural transformationsI ÝÝÑ^uN N,N ÝÝÑ^ON N˛N, P ÝⁿÝÑ^P J andP˛P ÝÝÑ^∆P P such thatN, u,Oqis a commutative monoid andpP, n,∆qis a commutative comonoid.

Then the additive structures on the setsN pˆP, Nqinterpreting proofs of$N, P is defined as:

f`g:ˆP //ˆpP˛Pq » pˆPq ˛ pˆPqÝÝÑ^f˛g N˛N //N.

2.5.2.3 Categorical models ofDiLLpol

Proposition 2.5.9. Consider a model ofLL_polwith a polarized biproduct:

• A negative chiralitypP,b,1qandpN ,`,Kqwith a strong monoidal left closurep´q^KL :P //N^op % p´q^KR :N ^op //P,with a polarized closure´:N //P%ˆ:P //N suchˆ˝´ “ Id_N,

• A polarized biproductp˛, IqonN andp˛, JqonP,

• A co-cartesian categorypP⁸,‘₈,0qand a cartesian categorypN ^8,op,ˆ₈,Jqwith a strong monoidal left closure

p´q^KL,8:P⁸ //N ^8,op% p´q^KR,8:N ^8,op //P⁸.

• A strong monoidal right closure

? :pP^8,op,‘,0q //pN ^op,`,1q %U :pN <sup>op</sup>,`,1q //pP^8,K,‘,Jq.

• A natural transformation:

d¯: ?U //Id_N which is thus defined between endofunctors ofN.

Then these four adjunctions, together with the monoidal and biproduct structures, define a denotational model of DiLLpol.

Outlook3.We would like to interpret dereliction and co-dereliction as shifts between the categoriesP⁸andN, and thus axiomatizing a categorical model ofDiLLthe data of two chiralities: one between positives inP and negatives inN, and the other betweenP⁸ andN. However, as the codereliction does not lead to a functor fromP⁸ to N because of non-compositionality (see Section2.5.2.1), this is still work in progress. It should incorporate the biproduct presentation, from which the interpretation of sums in hom-sets follows.

Chapter 3

Topological vector spaces

"Nej, jeg lever måske tusinder af dine dage, og min dag er hele årstider! Det er noget så langt, du slet ikke kan udregne det!"

"Nej, for jeg forstår dig ikke! Du har tusinder af mine dage, men jeg har tusinder af øjeblikke til at være glad og lykkelig i! Holder al denne verdens dejlighed op, når du dør?"

"Nej," sagde træet, "den bliver vist ved længere, uendeligt længere, end jeg kan tænke det!"

"Men så har vi jo lige meget, kun at vi regner forskelligt!"

Andersen, Det gamle egetræs sidste drøm, 1858.

Traditionally, one constructs a model of Linear Logic by defining the interpretation of formulas, then the one of linear proofs, and last one studies the possible interpretation for the exponential and non-linear proofs. We will inverse this order here: as we are looking for a nice and smooth interpretation of non-linear proofs, the necessary interpretation of the exponential connective will give us constraints on the topologies on our spaces.

This chapter tackles in its first section the basic definitions of the theory of topological vector spaces.

In Section 3.2we give examples of topological vector spaces made of functions (whether they are sequences or continuous measurable or smooth, functions). In Section3.3we introduce linear maps and the notions of dual pairs and weak topologies. Section3.4.3exposes notion of bounded subset, of bornologies, and explains how these bornologies characterize the different topologies on spaces of linear functions. Once that spaces of linear functions are well-understood, one can look for spaces on which the linear negation is involutive: the different possibilities for reflexivity are detailed in Section 3.5. At last, Section3.6explains the theory of topological tensor products. Nothing exposed in this chapter is new and every result is referred to the literature. The book by Jarchow [44] is the major reference.

As a foretaste for this chapter, we review the notion of orthogonalities and closures which are used throughout chapters5, chapter6and2.

Contents

Closure and orthogonalities . . . 55 Filters and topologies . . . 55 3.1 First definitions. . . 56 3.1.1 Topologies on vector spaces. . . 56 3.1.2 Metrics and semi-norms. . . 57 3.1.3 Compact and precompact sets. . . 58 3.1.4 Projective and inductive topolgies. . . 58 3.1.5 Completeness . . . 59 3.2 Examples: sequences and measures . . . 60 3.2.1 Spaces of continuous functions . . . 60 3.2.2 Spaces of smooth functions . . . 60 3.2.3 Sequence spaces. . . 61 3.3 Linear functions and their topologies . . . 62 3.3.1 Linear continuous maps . . . 62 3.3.2 Weak properties and dual pairs . . . 63 3.3.3 Dual pairs . . . 65 3.4 Bornologies and uniform convergence . . . 66 3.4.1 Polars and equicontinuous sets . . . 66 3.4.2 Boundedness for sets and functions . . . 67 3.4.3 Topologies on spaces of linear functions . . . 68 3.4.4 Barrels. . . 70 3.5 Reflexivity. . . 70 3.5.1 Weak, Mackey and Arens reflexivities . . . 71 3.5.2 Strong reflexivity . . . 72 3.5.3 The duality of linear continuous functions . . . 73 3.6 Topological tensor products and bilinear maps. . . 74 3.6.1 The projective, inductive andBtensor products . . . 74 3.6.2 The injective tensor product. . . 76 3.6.3 Theεtensor product . . . 78

Closure and orthogonalities

Let us describe a the notions of orthogonality and closure operators [74]: they generalize for example the operations of completion, bornologification, of taking the Mackey or the Weak topology on a space.

Definition 3.0.1. Consider a setX. An orthogonality relationK ĂXˆX is a symmetric relation. We define the dual of X:X^K :“ ty PX | @xPX,px, yq P Ku. Then one has alwaysX ĂX^KK, andX^K “X^KKK. A set is said to beK-reflexive whenX “X^KK.

Examples of orthogonality relations include polars of subsets (Section3.4.1) and duality in sequence spaces (Sec-tion3.2). Orthogonalities are defined in general in the context of posets, but we give here a definition in the context of a category. A relevant example is the category of lcs and continuous linear injective maps between them.

Definition 3.0.2. ConsiderCa category. A closure operator¯¨is an idempotent endofunctor onC, with a natural transformationw:Id //¯¨such thatwĎ_A“idA¯“wA¯.

Note that according to this definition a closure operator is in particular an idempotent monad, where the multipli-cation is the identity.

Another way of looking to closure operators are thus adjunction between a forgetful functor and a completion:

C K C

U

¯¨

such that¯¨ ˝U “IdC.

Definition 3.0.3. A full subcategoryCof a categoryLis said to be reflective when the inclusion functorU :C //L has a left adjoint¯¨. ThenU˝¯¨is a closure operator.

Closure operators and reflective subcategories are associated with many topological constructions: the topological closure¯¨(making a subset closed), the absolutely convex closed closure¯¨^abs, the completion˜¨consists of particular of closure operators. In Chapter2we will in particular define a polarized version of this notion of closure.

Filters and topologies

The reader can be used to speak about topologies in terms of convergence of sequences: the set of open sets on a metric spaceF is entirely determined by the convergence of sequencespxnqnin this space. Otherwise said, if for two norms on a vector space the convergent sequences are exactly the same, and have the same limits, then the norms are equivalent. This is true only when the topology is determined by ametric(Definition3.1.2): then there exists a countable basis of open sets.

In general, one cannot describe a topology only by its convergent sequences. A more general, non-countable, notion is needed:

Definition 3.0.4. ConsiderEa set. Then a filterFonEis a family of subsets ofE, such that:

• H RF,

• for allA, BPF,AXBPF,

• IfAPFandBĂEis such thatAĂB, thenBPF.

If pxnqn P E^N, then the family tSn “ txk|k ě Nuu is a filter onpxnqn. A basisof a filter F is a family F¹ ĂFsuch that every element ofFcontains an element ofF¹: a basis describes the behaviour ofFin terms of convergence. The convergence is itself described in terms of filters.

Definition 3.0.5. A topological space is a setE, such that for everyxwe have a filterFpxqsuch that:

• every element ofFpxqcontainsx,

• for everyU PFpxqthere existsV PFpxqsuch that for allyPV,U PFpyq.

ThenFpxqis called the filter of neighborhoods ofx.

For example, a sub basis for the filter of neighborhood of a pointxP F in a metric spaceF is the family of all open balls of centerxand of radius _n¹, for allnPN^˚

Definition 3.0.6. ConsiderFa filter on a topological vector spaceE.Fis said to be convergent towardsxPEif Fpxq ĂF.

In the example of a filter corresponding to a sequencepxkqk inF, it means exactly that for every n ě1, there existsknsuch that for everykěkn,kxk´xkă _n¹.

We have an equivalent definition of topological space, via a topology:

Definition 3.0.7. A topological space is a setEendowed with a collectionT of subsets ofE, called itstopology, such that:

• H RT andEPT,

• T is stable by arbitrary union,

• T is stable by finite intersection.

When comparing different topologies on a same vector spaceE, we will say that the topologyT is coarser than the topologyT¹, and thatT¹is finer thanT, if any open set ofT is in particular an open set ofT¹. We will denote this preorder on the topologies ofEbyď:

Definition 3.0.8. IfT andT¹are two topologies on a spaceE, we say thatT ďT¹if and only ifT ĂT¹. Convergence in topological vector spaces can also be characterized in terms of convergence of nets:

Definition 3.0.9. A net in a topological spaceEis a familypxaqaPAof elements ofEindexed by an ordered set A. A net is said to be converging towardsxPEis for every neighborhoodUofx, there existsbPA, such that for allaěb,xaPU. A sub-net ofpxaqaPAis a familypxbqbPBwithB ĂA.