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An explicit formula for the free
exponential modality of linear logic
Paul-André Melliès
Nicolas Tabareau
Christine Tasson
Laboratoire Preuves Programmes Systèmes
CNRS & Université Paris 7 - Denis Diderot
The exponential modality of linear logic associates a commutative
comonoid !A to every formula A, in order to duplicate it. Here, we explain how
to compute the free commutative comonoid !A in various models of linear logic,
using a sequential limit of equalizers. The recipe is simple and elegant, and enables to unify for the rst time the miscellaneous constructions of the exponential
modality appearing in the literature. It also sheds light on the duplication policy
of linear logic. We illustrate its relevance by applying it to two familiar models
of linear logic based on coherence spaces, Conway games and we show its limits
in niteness spaces.
Abstract.
1
Introduction
Linear logic is based on the principle that every hypothesis
Ai
should appear exactly
once in a proof of the sequent
A1 , . . . , A n
`
B.
(1)
This logical restriction enables to represent the logic in monoidal categories, along
the idea that every formula denotes an object of the category, and every proof of the
sequent (1) denotes a morphism
A1 ⊗ · · · ⊗ An
−→
B
where the tensor product is thus seen as a linear kind of conjunction. Note that, for
clarity's sake, we use the same notation for a formula
A
and for its interpretation (or
denotation) in the monoidal category.
This linearity policy on proofs seems far too restrictive in order to integrate traditional forms of reasoning, where it is accepted to repeat or to discard an hypothesis in
the course of a logical argument. This diculty is nicely resolved by providing linear
A into
!A which may be repeated or discarded. From a semantic point of view, the
formula !A is most naturally interpreted as a comonoid of the monoidal category. Recall
that a comonoid (C, d, u) in a monoidal category L is dened as an object C equipped
logic with an exponential modality, whose task is to strengthen every formula
a formula
with two morphisms
d
where
1
:
C
−→
C ⊗C
u
:
C
denotes the monoidal unit of the category. The morphism
tively called the
multiplication
and the
unit
−→
1
d
u
and
are respec-
of the comonoid. The two morphisms
d
and
u
are supposed to satisfy
associativity
and
unitality
properties, neatly formulated
by requiring that the two diagrams
C
d
C ⊗C
C ⊗C
*
C⊗d
d
C
d
C ⊗C
id
C ⊗C
d
C ⊗C ⊗C
t
d⊗C
,Cr
u⊗C
C⊗u
commute. Note that we draw our diagrams as if the category were
although the usual models of linear logic are only
The comonoidal structure of the formula
and the
weakening rule
weakly
strictly
monoidal,
monoidal.
!A enables to interpret the contraction rule
of linear logic
π
π
.
.
.
.
.
.
Γ, !A, !A, ∆ ` B
Γ, !A, ∆ ` B
Γ, ∆ ` B
Γ, !A, ∆ ` B
Contraction
by pre-composing the interpretation of the proof
π
Weakening
with the multiplication
d in the case
of contraction
⊗ !A
Γ
and with the unit
⊗
u
∆
d
−→
⊗ !A ⊗ !A
Γ
⊗
π
−→
∆
B
in the case of weakening
⊗ !A
Γ
⊗
u
−→
∆
Besides, linear logic is generally interpreted in a
requires that the comonoid
⊗
Γ
∆
symmetric
π
−→
B.
monoidal category, and one
!A is commutative, this meaning that the following equality
holds:
/ A⊗A
d
A
/ A⊗A
symmetry
=
/ A⊗A
d
A
When linear logic was introduced by Jean-Yves Girard, twenty years ago, it was soon
realized by Robert Seely and others that the multiplicative fragment of the logic should
be interpreted in a
category
L
∗-autonomous
category, or at least, a symmetric monoidal closed
; and that the category should have nite products in order to interpret the
additive fragment of the logic, see [9]. A more dicult question was to understand what
categorical properties of the exponential modality
“!”
were exactly required, in order
to dene a model of propositional linear logic that is, including the multiplicative,
additive and exponential components of the logic. Nonetheless, Yves Lafont found in
his PhD thesis [6] a simple way to dene a model of linear logic. Recall that a comonoid
morphism between two comonoids
f : C1 →
− C2
(C1 , d1 , u1 ) and (C2 , d2 , u2 ) is dened as a morphism
such that the two diagrams
C1
f
d1
C1 ⊗ C1
/ C2
C1
f
/ C2
d2
f ⊗f
/ C2 ⊗ C2
u1
-1q
u2
!A is freely generated by an object A when there
commute. The commutative comonoid
exists a morphism
ε
:
!A
→
−
A
f
:
C
→
−
A
such that for every morphism
C
from a commutative comonoid
A,
to the object
there exists a unique comonoid
morphism
f†
:
C
→
− !A
such that the diagram
. !A
f†
C
(2)
ε
0A
f
commutes. Lafont noticed that the existence of a free commutative comonoid
every object
A
of a symmetric monoidal closed category
model of propositional linear logic.
L
!A
for
induces automatically a
But this is not the only way to construct a model
of linear logic. A folklore example is the coherence space model, which admits two
alternative interpretations of the exponential modality: the original one, formulated by
Girard [3] where the coherence space
construction, where
!A
!A
is dened as a space of
is dened as a space of
of the original coherence space
multicliques
cliques,
and the free
(cliques with multiplicity)
A.
In this paper, we explain how to construct the free commutative comonoid in the
symmetric monoidal categories
L
typically encountered in the semantics of linear logic.
Our starting point is the well-known formula dening the
SA
M
=
symmetric algebra
A⊗n / ∼n
(3)
n∈N
generated by a vector space
A.
monoid associated to the object
The formula computes indeed the free commutative
A
in the category of vector spaces over a given eld
Σn of permutations on {1, . . . , n} acts on
⊗n
the vector space A
/ ∼n of equivalence classes (or orbits)
dened as the coequalizer of the n! symmetries
Here, the group
A⊗n
symmetry
···
symmetry
/
/ A⊗n
coequalizer
the vector space
Lop ,
/ A⊗n / ∼n
category
L.
L is the same thing as
it is tempting to apply the
in order to dene the free commutative comonoid
!A
k.
and
modulo the group action is
in the category of vector spaces. Since a comonoid in the category
a monoid in the opposite category
A⊗n ,
dual
formula to (3)
generated by an object
A
in the
Although the idea is extremely naive, it is surprisingly close to the truth...
Indeed, one signicant aspect of our work is to establish that the equalizer
n!
An
of the
symmetries
/
/ A⊗n
symmetry
···
symmetry
/ A⊗n
equalizer
An
(4)
n-th layer of the
A. As we will see in Sections 2
n
and 3, this principle is nicely illustrated by the equalizer A in the category of coherence
spaces, which contains the multicliques of cardinality n in the coherence space A ; and
n
by the equalizer A in the category of Conway games, which denes the game where
Opponent may open up to n copies of the game A, one after the other, in a sequential
exists in many familiar models of linear logic, and provides there the
free commutative comonoid
!A
generated by the object
order.
Of course, the construction of the free exponential modality does not stop here: one
still needs to combine the layers
An
together in order to dene
solution is to apply the dual of formula (3) and to dene
¯
product
!A
=
!A
!A properly. One obvious
as the innite cartesian
An .
(5)
n∈N
This formula works perfectly well for symmetric monoidal categories
L where the tensor
product distributes over the innite product, in the sense that the canonical morphism
X
⊗
¯
An
n∈N
→
−
¯
( X ⊗ An )
(6)
n∈N
is an isomorphism. This algebraic miracle is not so uncommon: it often happens in
models of linear logic enriched over commutative monoids where morphisms (and
thus proofs) may be added. A typical illustration is provided by the relational model of
linear logic, where the free exponential
each
An
!A
is dened as the set of nite multisets of
describing the set of multisets of cardinality
A,
n.
On the other hand, the formula (5) is far too optimistic, and does not work in
the typical models of linear logic, like coherence spaces, or game semantics. It is quite
instructive to apply it to the category of Conway games: the formula denes in that case
!A where the rst move by Opponent selects a component An , and thus decides
the number n of copies of the game A played subsequently. This departs from the free
commutative comonoid !A which we shall describe in Section 3, where Opponent is
allowed to open a new copy of the game A at any point of the interaction. So, there
n
remains to understand how the various layers A should be combined together, in order
to ensure that !A performs this particular copy policy. The temptation is to ask that
n
n+1
every layer A is glued inside the next layer A
in order to permit the computation
a game
to transit from one layer to the next in the course of interaction.
The most natural way to perform this glueing is to introduce the notion of pointed
L, one means a pair (A, u)
A and a morphism u : A →
− 1 to the monoidal unit. So, a pointed
(or ane) object. By pointed object in a monoidal category
consisting of an object
object is the same thing as a comonoid, without a comultiplication.
It is folklore that
the category of pointed objects and pointed morphisms (dened in the expected way)
is symmetric monoidal, and
ane
in the sense that its monoidal unit
1
is terminal.
Once this notion of pointed object introduced, the construction of the free commutative
comonoid
!A
is excessively simple and elegant, and proceeds in three elementary steps.
First step.
The object
A
is transported to the free pointed object
when this object exists in the monoidal category
pointed object
A•
L.
(A• , u)
it generates,
Intuitively, the purpose of the
is to describe one copy of the object
A,
or none... It is usually quite
A• = A & 1 is obtained by
A ; in the case of Conway games, the game A• is the game
easy to dene: in the case of coherence spaces, the space
adding a point to the web of
A
itself, at least when the category is restricted to the Opponent-starting games.
Second step.
The object
A≤n
is dened as the equalizer
of the diagram
/ ⊗n
/ A•
symmetry
···
symmetry
/ A⊗n
•
equalizer
A≤n
(A• )n
(7)
L. The purpose of A≤n is to describe all the layers Ak at the same time,
≤n
for k ≤ n. Typically, the object A
computed in the category of coherence spaces is
the space of all multicliques in A of cardinality less than n.
in the category
Third step.
It appears that there exists a canonical morphism
A≤n o
induced by the unit
generated by
1o
A
u
A≤n+1
of the pointed object
A• .
The free commutative comonoid
!A
is then dened as the sequential limit of the sequence
A≤1 o
A≤2 o
··· o
A≤n o
A≤n+1 o
···
The 2-dimensional description of algebraic theories and PROPs recently performed
by Melliès and Tabareau [8] ensures then that this recipe in three steps denes the
free commutative comonoid
!A
generated by the object
A...
as long as the following
fundamental property is satised by the symmetric monoidal category
L:
its tensor
product should distribute over
1. the equalizer computing the object
A≤n ,
2. the sequential limit computing the object
!A.
So, one main purpose of the paper is to establish that this pair of distributivity properties holds for the category of coherence spaces (in Section 2) and for the category of
Conway games (in Section 3). In this way, we demonstrate the extraordinary fact that
despite their dierence in style, the free exponential modalities of coherence spaces and
Conway games are based on
exactly
the same limiting process.
In contrast, in the setting of topological vector spaces, both the formula (5) and the free
pointed recipe meet their limits. We take the opportunity (in Section 4) to explain the
reason why these methods fail in the niteness space model of dierential linear logic
recently introduced by Thomas Ehrhard [2].
2
Coherence spaces
In this section, we compute the free exponential modality in the category of coherence
spaces dened by Jean-Yves Girard [3]. A
coherence space E = (|E|, _
^)
consists of a
set
|E|
clique
called its
of
E
web,
is a set
X
and of a binary reexive and symmetric relation
_
^
over
E.
A
of pairwise coherent elements of the web:
e1 _
^ e2 .
∀e1 , e2 ∈ X,
We do not recall here the denition of the category
Coh
of coherence spaces (however,
the reader will nd a brief description of the category in Appendix 1). Just remember
that a morphism
particular,
R
R : E → E0
in
Coh
is a clique of the coherence space
0
E ( E0,
so in
|E| × |E |.
is a relation on the web
It is easy to see that the tensor product does not distribute over cartesian products:
simply observe that the canonical morphism
A ⊗ (1 & 1)
→
−
(A ⊗ 1) & (A ⊗ 1)
is not an isomorphism. This explains why formula (5) does not work, and why the
construction of the free exponential modality requires a sequential limit, along the line
described in the introduction.
First step: compute the free ane object. Computing the free pointed (or ane)
object on a coherence space
E
is easy, because the category
Coh has cartesian products:
it is simply given by formula
E• = E & 1.
It is useful to think of
E & 1 is the space of multicliques of E
with
at most
one element:
the very rst layer of the construction of the free exponential modality. Recall that a
multiclique of
E
is just a multiset on
|E|
whose underlying set is a clique of
Second step: compute the symmetric tensor power E ≤n .
see that the equalizer
E ≤n
E.
It is not dicult to
of the symmetries
(E & 1)⊗n
is given by the set of multicliques of
E
symmetry
···
symmetry
/
⊗n
/ (E & 1)
with at most
n
elements, two multicliques being
coherent if their union is still a multiclique. As explained in the introduction, one also
needs to check that the tensor product distributes over those equalizers. Consider a
cone
Y
R
y
X ⊗ (E & 1)⊗n
R0
X⊗symmetry
···
X⊗symmetry
%
⊗n
/ X ⊗ (E & 1)
/
We can choose the identity among the symmetries. This ensures already that
Next, we show that the morphism
X ⊗ E ≤n
R
factors uniquely through the morphism
X⊗ equalizer
/ X ⊗ (E & 1)⊗n
To that purpose, one denes the relation
R≤n : Y −→ X ⊗ E ≤n
by
y R≤n (x, µ)
i
y R (x, u)
(8)
R = R0 .
µ is a multiset of |E| of cardinal less than n, and u is any word of length n whose
|E & 1| = |E| ] {∗} dene the multiset µ. We let the reader
≤n
≤n
check that the denition is correct, that it denes a clique R
of Y ( (X ⊗ E
),
and that it is the unique way to factor R through (8).
where
letters with multiplicity in
Third step: compute the sequential limit
E ≤1 = (E & 1) o
E ≤0 = 1 o
whose arrows are (dualized) inclusions from
that the limit
!E
E ≤2 o
E ≤n
into
E ≤3 · · ·
E ≤n+1 .
Again, it is a basic fact
of the diagram is given by the set of all nite multicliques, two
multicliques being coherent if their union is a multiclique. One also needs to check that
the tensor product distributes over the sequential limit. So, consider a cone
Y
R0
v
X ⊗ (E & 1) o
t
X ⊗1 o
R3
R2
R1
X ⊗ E ≤2 o
&
X ⊗ E ≤3 · · ·
and dene the relation
R∞ : Y −→ X⊗!E
y R∞ (x, µ)
by
∃n,
i
y Rn (x, u)
µ is a multiset of elements of |E| and the element u of the web of E ≤n is any word
n whose letters with multiplicity in |E & 1| = |E| ] {∗} dene the multiset µ.
We let the reader check that R∞ is a clique of Y ( (X⊗!E) and denes the unique
way to factor the cone. This concludes the proof that the sequential limit !E denes the
free commutative comonoid generated by E in the category Coh of coherence spaces.
where
of length
3
Conway games
In this section, we compute the free exponential modality in the category of Conway
games introduced by André Joyal in [4]. One unifying aspect of our approach is that
the construction works in exactly the same way as for coherence spaces.
Conway games. A Conway game A is an oriented rooted graph (VA , EA , λA ) consisting
of (1) a set VA of vertices called the positions of the game; (2) a set EA ⊂ VA × VA
of edges called the moves of the game; (3) a function λA : EA → {−1, +1} indicating
whether a move is played by Opponent (−1) or by Proponent (+1). We write
the root of the underlying graph. A Conway game is called
negative
?A
for
when all the moves
starting from its root are played by Opponent.
A
play s = m1 · m2 · . . . · mk−1 · mk
from the root
?A
m
of a Conway game
m
mk−1
A
is a path
s : ? A xk
starting
m
1
2
k
s : ?A −−→
x1 −−→
. . . −−−→ xk−1 −−→
xk
Two paths are parallel when they have the same initial and nal positions. A play is
alternating
when
∀i ∈ {1, . . . , k − 1},
λA (mi+1 ) = −λA (mi ).
We note PlayA the set of plays of a game
A.
Dual. Every Conway game A induces a dual
game
A∗
obtained simply by reversing the
polarity of moves.
Tensor product. The tensor product A⊗B of two Conway games A and B is essentially
the asynchronous product of the two underlying graphs. More formally, it is dened as:
VA⊗B = VA × VB ,
its moves are of two kinds :
x⊗y →
z⊗y
x⊗z
if
if
x→z
y→z
the polarity of moves is inherited from games
The unique Conway game
1
in the game
in the game
A
and
with a unique position
A
B,
B.
?
and no move is the neutral
element of the tensor product. As usual in game semantics, every play
A ⊗ B can
game B .
s|A
be seen as the interleaving of a play
of the game
A
s
of the game
and a play
s|B
of the
Strategies. A strategy σ of a Conway game A is dened as a non empty set of alternating plays of even length such that (1) every non empty play starts with an Opponent
move; (2) σ is closed by even length prex; (3) σ is deterministic, i.e. for all plays s,
and for all moves
m, n, n0 ,
s · m · n ∈ σ ∧ s · m · n0 ∈ σ ⇒ n = n0 .
The category of Conway games.
objects, and strategies
σ
of
A∗ ⊗ B
Conway has Conway games as
σ : A → B . The composition is based
The category
as morphisms
on the usual parallel composition plus hiding technique and the identity is dened
by a copycat strategy. The resulting category
Conway
is compact-closed in the sense
of [5].
The category
Conway
does not have nite or innite products. For that reason,
we compute the free exponential modality in the full subcategory
Conway
of negative
Conway games, which has products. We explain in a later stage how the free construction
on the subcategory
Conway
induces a free construction on the whole category.
First step: compute the free ane object. The monoidal unit 1 is terminal in the
category
Conway
. In other words, every negative Conway game may be seen as an
ane object in a unique way, by equipping it with the empty strategy
particular, the free ane object
A•
is simply
A
Second step: compute the symmetric tensor power An
n!
symmetries
A⊗n
symmetry
···
symmetry
A simple argument shows that the equalizer
tA : A → 1.
In
itself.
/
as the equalizer of the
/ A⊗n .
An = A≤n
is the following Conway game:
An
w = x1 · · · xn of length n, whose
xi+1 = ?A is the root of A
whenever xi = ?A is the root of A, for every 1 ≤ i < n. The intuition is that the
n
letter xk in the position w = x1 · · · xn of the game A describes the position of the
k -th copy of A, and that the i + 1-th copy of A cannot be opened by Opponent
unless all the i-th copy of A has been already opened.
its root is the word ?An = ?A · · · ?A where the n the positions xk are at the root ?A
of the game A,
a move w → w0 is a move played in one copy:
the positions of the game
letters are positions
xi
are the nite words
of the game
A,
and such that
w1 x w2 → w1 y w2
where
x→y
is a move of the game
implies that when a new copy of
w1
A
A.
Note that the condition on the positions
is opened (that is, when
is at the root, and all the positions in
w2
the polarities of moves are inherited from the game
Note that
An
may be also seen as the subgame of
always opened after the
i-th
copy of
x = ?A )
no position in
are at the root.
A⊗n
A
in the obvious way.
where the
i + 1-th
copy of
A
is
A.
Third step: compute the sequential limit
A1 = A o
A0 = 1 o
A2 o
whose morphisms are the partial copycat strategies
An ← An+1
···
identifying
An
as the
n copies of A are played. The limit of this diagram
∞
≤n
in the category Conway is the game A
dened in the same way as A
except that
its positions w = x1 · x2 · · · are innite sequences of positions of A, all of them at the
∞
root except for a nite prex x1 · · · xk . We establish in Appendix 2 that A
is indeed
subgame of
A
n+1
A3 o
where only the rst
the limit of this diagram, and that the tensor product distributes with this limit. From
this, we deduce that the sequential limit
in the category
Conway
A∞
describes the free commutative comonoid
.
It is nice to observe that the free construction extends to the whole category
Conway
of Conway games. A careful study shows that every commutative comonoid in the
category of Conway games is in fact a negative game. Moreover, the inclusion functor
Conway has a right adjoint, which associates to every Conway game
A obtained by removing all the Proponent moves from
∞
the root ?A . By combining these two observations, we obtain that (A )
is the free
commutative comonoid generated by A in the category Conway of Conway games.
from
A,
4
Conway
to
the negative Conway game
An interesting counter-example: Finiteness spaces
In Sections 2 and 3 we have seen how formula (5) can be rened into formula (
??)
which suits the models of Coherent spaces and Conway games. To conclude this paper,
we want to give the limits of this approach with the niteness spaces counter-example.
There are two layers of niteness spaces. Relational niteness spaces constitute a renement of the relational model, built through
bi-orthogonality. Linear niteness spaces
are linearly topologized spaces [7] built on the relational layer. We explain the failure
at both levels. For an introduction to niteness spaces, we refer the reader to [2] or to
Appendix 4.
Relational niteness spaces. Two subsets u, u0
E are said orthogu ∩ u0 is nite. The orthogonal of
G ⊆ P(E) is then dened by G ⊥ = {u0 ⊆ E | ∀u ∈ G, u ⊥ u0 }.
A relational niteness space E = (|E|, F(E)) is given by its web (the countable
⊥⊥
set |E|) and by a set F(E) ⊆ P(|E|) orthogonally closed, i.e. F(E)
= F(E). The
⊥
elements of F(E) (resp. F(E) ) are said nitary (resp. antinitary ). A nitary relation
R between two niteness spaces E1 and E2 is a subset of |E1 | × |E2 | such that
∀u ∈ F(E1 ), R · u := b ∈ |E2 | ∃a ∈ u, (a, b) ∈ R ∈ F(E2 ),
∀v 0 ∈ F(E2 )⊥ , t R · v 0 := a ∈ |E1 | ∃b ∈ v 0 , (a, b) ∈ R ∈ F(E1 )⊥ .
onal, denoted by
The
u ⊥ u0 ,
RelFin
category
∗-autonomous.
of
relational
niteness
spaces
and
nitary
relations
is
Thus, it provides a model of linear logic.
The exponential modality of a niteness space
by its web
of a countable set
whenever their intersection
|!E| = Mfin (|E|)
E
made of nite multisets
is the niteness space
µ : |E| → N
!E
dened
and by its niteness
structure
F(!E) = {M ∈ Mfin (|E|) | πE (M ) ∈ F(E)},
where for every
M ∈ Mfin (|E|), πE (M ) := {x ∈ |E| | ∃µ ∈ M, µ(x) 6= 0}.
E n of the n! symmetries
The equalizer
exists in
RelFin
and provides the
of the multisets of cardinality
n
n-th
/
/ E ⊗n
symmetry
···
symmetry
/ E ⊗n
equalizer
En
layer of
!E .
Its web
|E n | = Mnfin (|E|)
is made
and its niteness structure is
F(E n ) = {Mn ⊆ Mnfin (|E|) | πE (Mn ) ∈ F(E)}.
However, the innite cartesian product
E ∞ of the layers E n , i.e. E ∞ = &n∈N E n ,
RelFin. Indeed, although the webs are
does not compute the exponential modality of
both equal to
|E ∞ | = Mfin (|E|),
their niteness structures are dierent. As shown by
F(!E) ( F (E ∞ ) since
Mn = M ∩ Mnfin (|E|),
∞
F (E ) = M ∈ Mfin (|E|) ∀n ∈ N,
.
πE (Mn ) ∈ F(E).
the counter-example in Appendix 3, we have in general
(9)
Besides, the tensor product does not distribute over innite product, i.e.
X
⊗
¯
n∈N
En
→
−
¯
( X ⊗ En )
n∈N
is not an isomorphism since in general the niteness structures are dierent in general
F
F(X ⊗ E ∞ ) = {M ⊆ |X| × Mfin (|E|) | πX (M ) ∈ F(X) ; πE (M ) ∈ F(E)} ,
! ¯
Mn = M ∩ |X| × Mnfin (|E|),
n
.
(10)
(X ⊗ E ) = M ∀n ∈ N,
πX (Mn ) ∈ F(X), πE (Mn ) ∈ F(E).
n∈N
Otherwise, in contrast with coherence spaces, the rened formula (
??)
does not pro-
!E . Roughly speaking, the tensor product ⊗ commutes with
∞
the nite cartesian product & in RelFin. Therefore the limit E
coincides with the
≤n
sequential limit of pointed objects &n∈N E
.
vide more information on
To better understand this result, we move to the linear niteness spaces layer. In
this setting, the lack of uniformity observed in (9) and (10) can explained in terms of
uniform convergence.
Linear niteness spaces. In the sequel, k is an innite eld endowed with the discrete
topology i.e. every subset of
vector space, the
k is open. Every relational niteness space E
linear niteness space
khEi =
x ∈
where for any sequence
parts of
E,
k|E| ,
x ∈ k|E| |x| ∈ F(E) ,
|x| = {a ∈ |E| | xa 6= 0}.
Using the antinitary
this space is endowed by the topology generated by the
neighborhood
of
generates a
fundamental linear
0
VJ 0 = x ∈ khEi |x| ∩ J 0 = ∅ , ∀J 0 ∈ F(E)⊥ .
U of khEi is open if
x + VJx0 ⊆ U . Endowed
More precisely, a subset
and only if for each
Jx0 ∈ F(E)⊥
with this topology,
such that
x ∈ U
khEi
there is
is a linearly
topologized space [7].
The category
LinFin,
with linear niteness spaces as objects and linear continuous
functions as morphisms, is
∗-autonomous and provides a model of linear logic.
khEi is entirely determined by its underlying relational
Though a linear niteness space
space
E,
the constructions of linear logic can be described from an algebraic and topo-
logical viewpoint
independent
from
E.
Thus, forgetting the underlying relational layer,
linear niteness spaces will be denoted by
We will now build
itive. In
LinFin,
X∞
X, Y,. . .
using its dual since its functional denition is more intu-
the dual space
X ⊥ = (X (
k)
is the linearly topologized space of
continuous linear forms. This space is endowed with the topology of uniform conver-
linearly compact subspaces,
i.e. subspaces K ⊆ X that are closed and have a
|K| := ∪x∈K |x|. This linearly compact open topology is generated by
W(K) = {f | f (K) = 0} where K ranges over linearly compact subspaces 1 .
⊗n
Similarly, X
( k is the space of n-linear forms φ : X ×n → k which are hypocon2
tinuous , i.e. ∀K ⊆ X linearly compact, ∃V open s.t. ∀1 ≤ i ≤ n, φ(K ×(i−1) × V ×
K ×(n−i) ) = 0. This space is endowed with the linearly compact open topology gener×n
) = 0} where K ranges over linearly compact subspaces.
ated by W(K) = {φ | φ(K
n
n
The dual X ( k of the equalizer X of the n! symmetries is then the space of symmetric n-linear forms which are hypocontinuous. It is endowed with the linearly compact
open topology. It can also be described as the space of homogeneous polynomials P of
degree n over X , i.e. P : X → k is associated to an hypocontinuous symmetric n-linear
×n
form φ : X
→ k such that P (x) = φ(x, . . . , x).
gence on
nitary support
1
2
Linear compactness can be dened adapting the intersection property to the linearly topologized setting [7]. We prefer their nitary characterisation which is here more useful.
Hypocontinuity is a notion of continuity in between continuity and separated continuity
Finally, we combine the dierent layers by taking the innite cartesian product. The
dual
(&n∈N X n ) (
k
is the space of
polynomials
(nite linear combinations of ho-
mogeneous polynomials). In this linear niteness space, a vector subspace
n ∈ N,
{P ∈ X n ( k | P (Kn ) = 0} ⊆ V .
if and only if for every
there is
However, as shown in [1], the dual
Kn ⊆ X
!X (
k,
is the
V
is open
linearly compact such that
completion
of the space of
polynomials endowed with the linearly compact open topology. In this topology, a vector
subspace V is open if and only if there is a linearly compact subspace K ⊆ X such that
∀n, {P ∈ X n ( k | P (K) = 0} ⊆ V . In other words, the topology is uniform over the
dierent layers. Thanks to the Taylor formula shown in [2], the functions in !X ( k
are analytic, i.e. they coincide with the sums of converging series whose n-th term is an
homogeneous polynomial of degree n.
Finally, using either combinatorial computations or topological viewpoint, we have
!E is dierent from &n E n . Indeed, this latter is related to the local information
level n though the exponential modality is related to a global information. The
seen that
at each
second formula (
??) proposed in this article was a step towards understanding how the
dierent layers combine to construct the exponential modality. Although it is sucient
in coherence spaces and Conway games, it does not in the niteness spaces where
innitely many layers are needed.
References
1. Thomas Ehrhard. On niteness spaces and extensional presheaves over the lawvere theory
of polynomials. Journal of Pure and Applied Algebra, to appear.
2. Thomas Ehrhard. Finiteness spaces. Mathematical Structures in Computer Science, 15(4),
2005.
3. Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50:1102, 1987.
4. André Joyal. Remarques sur la théorie des jeux à deux personnes. Gazette des Sciences
Mathématiques du Québec, 1(4):4652, 1977. English version by Robin Houston.
5. Max Kelly and Miguel Laplaza. Coherence for compact closed categories. Journal of Pure
and Applied Algebra, 19:193213, 1980.
6. Yves Lafont. Logique, catégories et machines. Thèse de doctorat, Université de Paris 7,
Denis Diderot, 1988.
7. Solomon Lefschetz et al. Algebraic topology. American Mathematical Society, 1942.
8. Paul-André Melliès and Nicolas Tabareau. Free models of T-algebraic theories computed
as Kan extensions. Submitted to Journal of Pure and Applied Algebra.
9. Robert Seely. Linear logic, ∗-autonomous categories and cofree coalgebras. In Applications
of categories in logic and computer science, number 92. Contemporary Mathematics, 1989.
Appendix 1: Coherence spaces
The coherence relation induces an
e1 ˚ e2
incoherence relation ˚ dened by
¬(e1 _
^ e2 )
⇐⇒
e1 = e2 .
or
Finite product.
The product E1 & E2 of two coherence spaces E1 and E2 is dened
|E1 & E2 | = |E1 | ] |E2 | and two elements (e, i) and (e0 , j) of the web are coherent
0
when i 6= j or when i = j and e _
^e.
by
Tensor product.
dened by
The tensor product
|E1 ⊗ E2 | = |E1 | × |E2 |
E1 ⊗ E2
E1
of two coherence spaces
and two elements
(e1 , e2 )
and
(e01 , e02 )
and
E2
is
are coherent
when
0
e1 _
^ e1
0
e2 _
^ e2 .
and
Linear implication. The linear implication E1 ( E2
E2
is dened by
|E1 ( E2 | = |E1 | × |E2 |
E1 and
(e01 , e02 ) of the
of two coherence spaces
and two elements
(e1 , e2 )
and
web are incoherent when
0
e1 _
^ e1
e2 ˚ e02 .
and
The category of coherence spaces. The category Coh of coherence spaces has coherence spaces as objects and cliques of
web of
|E2 |,
E1 ( E2
is
|E1 | × |E2 |,
E1 ( E2
as morphisms from
E1
to
E2 . As the
|E1 | and
a morphism can be seen as a relation between
satisfying additional consistency properties. In particular, identity and composi-
tion are dened in the same way as identity and composition in the category of sets
and relations. This category is
∗-autonomous
and provides a model the multiplicative
fragment of linear logic.
Appendix 2: Conway games
Proposition 1. The game A∞ is the free exponential of the negative Conway game A.
Proof.
Instead of showing in two steps that
A∞
A and that
X ⊗ A∞ is the
is the limit of the diagram
the tensor product distributes with this limit, we will directly show that
limit of the diagram
X⊗symmetry
X ⊗A : X o
tA
o
X ⊗A o
o
X ⊗ A⊗2 o
X⊗A⊗tA
X⊗tA ⊗A
Let us dene a cone on the diagram
X ⊗A
s·m
of
A∞ ,
A
∞
to plays
we dene
hs · mi = hsi · m
X ⊗ A⊗3 · · ·
X⊗tA ⊗A⊗2
X ⊗ A∞ . We
n
of A , and then
whose origin is
dening a interleaving function from plays of
a copycat strategy. Given a play
X⊗symmetry
X⊗A⊗2 ⊗tA
···
proceed by
by dening
where
m is the underlying move of m in A. We then dene the strategy εn : A∞ → A⊗n
by its set of plays
εn
The cone of
def
{s ∈ Playeven
| ∀t ≺even s , t|A1 ∞ = ht|A⊗n i}.
A1 ∞ (A⊗n
=
2
2
X⊗A
∞
X⊗A
on
is then given by the strategies
X ⊗ εn .
Note that
the scheduling of the opening of moves is enforced by the presence of symmetry in
A∞ is not just the innite tensor product of A. Let
(B, α : B → A) be a cone on X ⊗ A. We have to dene a strategy from B to X ⊗ A∞ .
⊗n
Let us introduced the strategy in : X ⊗ A
→ X ⊗ A∞ which mimics Opponent on X
and on the n rst copies of A, and which does not answer when Opponent opens the
n + 1th copy of A∞ . The strategy α†(n) is dened for all n by the commutative diagram
the diagram. This explains why
/ A⊗n
u
u
u
uu
uu in
u
uz
αn
B
α†(n)
A∞
Consider now the diagram
αn
B
α†(n+1)
αn+1
/ A⊗n+1
in+1
A∞
'
A
⊗n
/ A⊗n
⊗tA
A∞
in
A∞
It commutes on all faces except for the right down one which satises
in+1 .
The clockwise external path is equal
α
†(n)
in ◦ (An ⊗ tB ) ⊆
, so we deduce that
α†(n) ⊆ α†(n+1) .
The comonoidal lifting
α†
is then dened by the monotone limit of the
α†
def
[
=
α†(n) 's:
α†(n)
n
This strategy is a cone morphism because
εn ◦ in = An ,
which implies
εn ◦ α†(n) = εn ◦ in ◦ αn = αn .
It remains to show that this strategy is unique as a cone morphism. Let
cone morphism from
B
to
A∞ .
β
be another
Let us dene
β (n) = in ◦ εn ◦ β
and remark the two following things
β (n) = α†(n)
and
β=
[
β (n) .
n
We deduce that
†
α = τ,
which concludes the proof.
We clearly have the symmetric property for the diagram
for
X = 1,
we obtain that
A∞
is the limit of the diagram
A.
A ⊗ X.
By using this fact
Appendix 3: Finiteness spaces
Constructions of linear logic in RelFin.
Orthogonal:
Multiplicatives:
|1| = |⊥| = {∗}
|E ⊥ | = |E|
F (1) = F(⊥) = {∅, {∗}}
F(E ⊥ ) = F(E)⊥
|E1 ` E2 | = |E1 ⊗ E2 | = |E1 | × |E2 |
Products and coproducts:
|0| = |>| = ∅
F(0) = F(>) = {∅}
|&i Ei | = |⊕i Ei | =
i |Ei |

]j∈J uj s.t. J nite
∀j ∈ J, uj ∈ F(Ej )

]i ui s.t.
∀i ∈ I, ui ∈ F (Ei )
F(⊕i Ei ) =
F(&i Ei ) =
F
ff
ff
8
9
< R ⊆ |E1 | × |E2 | s.t.
=
F(E1 ` E2 ) = ∀u ∈ F (E1 )⊥ , R · u ∈ F (E2 )
:
;
∀v ∈ F (E2 )⊥ , t R · v ∈ F (E1 )

F(E1 ⊗ E2 ) =
w ⊆ |E1 | × |E2 | s.t. π1 (w) ∈ F(E1 )
π2 (w) ∈ F(E2 )
ff
where π1 (w) := {x1 ∈ |E1 | | ∃x2 ∈ |E2 |, (x1 , x2 ) ∈ w}
π2 (w) := {x2 ∈ |E2 | | ∃x1 ∈ |E1 |, (x1 , x2 ) ∈ w}
Exponentials:
|!E| = |?E| = Mfin (|E|) = {µ : |E| → N | µ(a) > 0 for nitely many a ∈ |E|}
F(!E) = {M ⊆ Mfin (|E|) | ∪{|µ|, µ ∈ M } ∈ F(E)}
n
o
F(?E) = M ⊆ Mfin (|E|) | ∀u ∈ F (E)⊥ , Mfin (u) ∩ M nite
Example 1.
Let
Nat
be the relational niteness space whose web is the set of inte-
F(Nat) is the nite powerset Pn (N). It is the
RelFin.
∞
The niteness spaces !Nat and Nat
have the same web Mfin (N) but their nitiness
gers
N
and whose niteness structure
interpretation of
(!1)⊥
in
structures are dierent:
F(!Nat) = M ⊆ Mfin (N) πNat (M ) nite
= M ⊆ Mfin (N) ∃N ∈ N; M ⊆ Mfin (0, . . . , N ) ,
F(Nat∞ ) = M ⊆ Mfin (N) ∀n ∈ N, πNat (M ∩ Mnfin (N)) nite
µn the multiset made of n copies of n and M = {µn | n ∈ N}.
M ∈ F(Nat∞ ) but M ∈
/ F(!Nat).
∞
n
Similarly, Nat ⊗ Nat
and &n∈N (Nat ⊗ Nat ) have the same web N × Mfin (N) but
For instance, let us denote
We have
dierent niteness structures:
F(Nat ⊗ Nat∞ ) = {M ⊆ N × Mfin (N) | ∃N ; M ⊆ {0, . . . , N } × Mfin (0, . . . , N )} ,
! ¯
Mn = M ∩ (N × Mnfin (N)),
n
F
(Nat ⊗ Nat ) = M ∀n ∈ N, ∃Nn
.
Mn ⊆ {0, . . . , Nn } × Mfin (0, . . . , Nn ).
n∈N
For instance, let us denote
0
n
M 0 = {(n, µn ) | n ∈ N}.
M ∈
/ F(&n∈N (Nat ⊗ Nat ).
We have
M 0 ∈ F(Nat ⊗ Nat∞ )
but
Constructions of linear logic in LinFin.
Product and Coproduct. The coproduct X ⊕ Y
Y
is made of linear combinations of elements of
of of linear niteness spaces
X
and
Y
X
and
and is endowed with the
product topology. Finite product coincide with nite coproduct. However, the innite
⊕i Xi
&i Xi .
coproduct
product
of the collection of niteness space
Xi
Linear implication. The linear implication X ( Y
and
Y
is a strict subspace of the innite
of two linear niteness spaces
X
is the linearly topologized space of continuous linear functions endowed with the
topology of uniform convergence on closed spaces with nitary support. This
compact open
linearly
topology is generated by
W(K, V ) = {f | f (K) ⊆ V }
K ranges over linearly compact subspaces of khXi, i.e. K is closed and |K| =
∪x∈K |x| is nitary, and V ranges over fundamental neighbourhoods of 0.
⊥
Let ⊥ = k. The topological dual X
= X ( ⊥ of X is endowed with the compact
0
⊥
0
open topology generated by W(K) = {x ∈ X | ∀x ∈ K, hx , xi = 0} where K ranges
over linearly compact subspaces of X .
where
Inductive tensor product. An n-linear form φ : (Xi )i → k over linear niteness
(Xi )i≤n is hypocontinuous if for any (Ki ) collection of linearly compact subspaces
spaces
Xi s (respectively), for any i0 there exists a fundamental linear neighborhood Ui0 such
φ(×Xi ) = 0 where Xi = Ki if i 6= i0 and Xi0 = Ui0 . The inductive tensor product3
X ` Y of two linear niteness spaces X and Y is the space of hypocontinuous bilinear
⊥
⊥
forms over X × Y , endowed with the linearly compact open topology generated by
0
0
0
0
W(KX
, KY0 ) = {φ | φ(KX
, KY0 ) = 0} where KX
(resp. KY ) range over linearly compact
0
0
subspaces of X (resp. Y ).
of
that
Tensor product.
and
Y
The tensor product
X ⊥ ` Y ⊥ . It is
khXi ⊗ khY i endowed with
is the dual of
product
e khY i
khXi ⊗
of two linear niteness spaces
X
the topological completion of the algebraic tensor
the topology induced by
(X ⊥ ` Y ⊥ )⊥ .
Exponential modality. The linear niteness space kh!E ⊥ i coincides with the completion of the space of polynomial functions over
khEi endowed with the linearly compact
open topology.
Example 2.
The linear niteness space associated with
quences, denoted
khNati = k(ω) .
Nat
is the space of nite se-
Its topology is discrete since
N ∈ F(Nat)⊥ .
infer that the linearly compact subspaces are the nite dimensional subspaces
We can
k(ω) .
khNatn ( ki is the space of every symmetric n-linear forms or equivalently of
(ω)
→ k of degree n.
any homogeneous polynomial P : k
∞
The space khNat
( ki is made of every polynomials over k(ω) . Its topology is gener-
Hence,
Vsuch that ∀n ∈ Nat, there
exists
Wn (Kn ) = P ∈ khNatn i0 P (Kn ) ⊆ V .
ated by the fundamental system made of subspaces
a nite dimensional
3
Kn ⊆ k(ω)
such that
X ` Y is an adaptation of the inductive tensor product ⊗ε to linearly topologized space.
The space
kh!Nat ( ki is made of every formal sums ofhomogeneous polynomial
(pos
W(K) = P ∈ kh!Ei0 P (K) where K
sibly innite). Its topology is generated by
ranges over niteness subspaces.