The exponential

Monomials

Definition 5.2.1. LⁿpE, Fqis the space of symmetricn-linear separately continuous functions fromEⁿtoF, We writeLnpE, Fqfor the space of all symmetricn-linear maps fromEtoF.

Ann-monomial fromEtoF is a functionf :E ÑF such that there isfˆPLⁿpE, Fqverifying that for all xPE fpxq “fˆpx, ..., xq. It is symmetric when for every permutationσ PS_n, for everyx₁, ..x_n P Ewe have fpx_σp1q, ..., x_σpnqq “fpx₁, ..., x_nq.

Proposition 5.2.2(The Polarization formula [53, 7.13]). Considerf an-monomial fromEtoF. Then we have fpxq “fˆpx, ..., xqwherefˆis a symmetricn-linear function fromEtoFdefined by:

For everyx1, ...xnPE,fˆpx1, ..., xnq “_n!¹ ř1

d1,...,dn“0p´1q^n´řkdkfpř

kdkxkq.

Thus the sum in the polarization formula is indexed by the subsets ofr1, ns. Another way to write it would be the following:

For everyx₁, ...x_n PE,fˆpx₁, ..., x_nq “ _n!¹ ř1 IĂJ1,nK

p´1q^n´cardIfpř

kPIx_kq.

Proof. Let us write for the multinomial coefficient:

ˆ n

1It appears that the weakest completeness condition necessary to model quantitative linear logic should be Mackey completeness [49].

2A very weak completeness condition, studied in sections2.4.3and6, which was shown to be enough for power series inC-vector spaces in [49].

Let us show thatř1

d1,...,dn“0p´1q^n´řjdjd^k₁¹...d^k_nⁿis non-zero if and only ifk1“ ¨ ¨ ¨ “ kn “1. Indeed, if there is anisuch thatkią1, then there isjsuch thatkj“0, ask1`...`kn“n. Let us supposek1“0. Then

1

ÿ

d₁,...,d_n“0

p´1q^pn´řjdjqd^k₁¹...d^k_nⁿ“

1

ÿ

d₂,...,d_n“0

p´1q^{pn´1´d2´...´dnq}d^k₂²...d^k_nⁿ

`

1

ÿ

d2,...,dn“0

p´1q^{n´d2´...´dn}d^k₂²...d^k_nⁿ

“0

Thus _n!¹ ř1

d₁,...,d_n“0p´1q^pn´řjdjqfpř

jdjxjq “fˆpx1, ..., xnq.

Definition 5.2.3. Let us writeHⁿpE, Fqfor the space of n-monomials overE endowed with the topology of simple convergence on points ofE. For every lcsEandF,HⁿpE, Fqis a lcs.

As a consequence of the polarization formula5.2.2, we know that there is is a unique symmetricn-linear map fˆassociated to an-monomialf.

Corollary 5.2.4. There is a bijection betweenHⁿpE, FqandLⁿpE, Fq.

As we will endowHⁿpE, Fqwith its weak topology, we need to get a better understanding of its dual. To do so, we retrieve information from the dual ofLⁿ_σpE, Fq.

Proposition 5.2.5. For every lcsEandF, for everynPN, we have HⁿpE, Fq »Lⁿ_σpE, Fq.

Proof. The algebraic isomorphism between the two vector spaces follows from the previous corollary, as the function mapping a n-linear symmetric mapping to the correspondingn-monomial is clearly linear. As they are both endowed with the topology of simple convergence of points ofE (respEˆ...ˆE), this mapping is bicontinuous.

Notation 5.2.6. We writeE_s^bn for the symmetrizedn^th-tensor product ofEwith itself³. We denote byLⁿ_spE, Fq the vector space of alln-linear symmetric functions fromEtoF. Thus

Lⁿ_spE, Fqw»LσpE^b_sⁿ, Fq.

As also we haveHⁿpEw, Fwq »HⁿpE, Fwq »Lⁿ_σpE, Fwqby Proposition5.2.5, the dual ofHⁿpE, Fq¹is the dual ofLσpE_s^bn, Fq. Proposition5.1.4thus gives us a way to compute it:

Proposition 5.2.7. For every lcsEandF,HⁿpE,Kq¹“E_s^bnbiF¹. That is, every continuous linear formθon HⁿpE, Fqcan be written as a finite sum of functions of the typel˝evx₁b...b_ix_nwithlPF¹andx1, ...xn PE.

From this, we deduce thatHⁿpEw, Fwqis a weak space: it is already endowed with its weak topology.

Corollary 5.2.8. For every lcsEandF, we have thatHⁿpE_w, F_wqw»HⁿpE, F_wq »HⁿpE_w, F_wq.

Proof. The topology onHⁿpE_w, F_wqis the topology of simple convergence onE^b_symⁿ , with weak convergence on F. This is exactly the topology induced by its dualE_sym^bn biF¹.

3That is, the vector spaceEbi¨ ¨ ¨ biE, quotiented by the equivalence relationx1bi...bixn”xσp1qbi¨ ¨ ¨ bixσpnqfor allσPSn.

The exponential The exponential! : WEAK ÑWEAKis defined as a functor on the category of linear maps.

In Equation 2.7, we detailed how the exponential in a model with smooth functions should be interpreted by a space of distributions. The same reasonning applies here. Indeed, suppose we want non-linear proofsE ñ F to be interpreted in some space of functionsFpE, Fq. As the category of weak spaces and these functions is the co-Kleisli category WEAK!, we have:

p!Eqw» pp!Eqwq²

»L_σp!E,Kq¹

»FpE,Kq¹

As we want our non-linear proofs to be interpreted by sequences of monomials, the definition of!Eis straight-forward.

Definition 5.2.9. Let us define!Eas the lcsÀ

nPNHⁿpE,Kq¹. As usual, we need to endow!Ewith its weak topology.

Proposition 5.2.10. We havep!Eq¹“ś

nHⁿpE,Kq, and thusp!Eq_w» pś

nHⁿpE,Kqq¹. Proof. According to Proposition5.1.19, we have that

p!Eq¹ “ź

n

HⁿpE,Kq²“ź

n

HⁿpE,Kq.

Thus,p!Eq¹ » pś

nHⁿpE,Kqqw, as both spaces in this equality are endowed by the topology of simple con-vergence on !E. Then p!Eq¹ » ś

nHⁿpE,Kqw » ś

nHⁿpE,Kq. Taking the dual of these spaces, we get

!E_w» pś

nHⁿpE,Kqq¹.

As in spaces of linear functions, see Proposition5.1.6, we have always thatHⁿpE, F_wq »HⁿpE_w, F_wq. Thus

!pE_wq »À

nPNHⁿpE_w,Kq¹»À

nPNHⁿpE,Kq¹»!E.

Notation 5.2.11. We will write without any ambiguity!Efor!pE_wqand!E_wforp!Eq_w. Definition 5.2.12. Forf PLσpEw, Fwqwe define

!f :

$

&

%

!EwÑ!Fw

φÞÑ ppgnq Pź

n

HⁿpF,Kq ÞÑφppgn˝fqnq Proposition 5.2.13. This makes!a covariant functor onWEAK.

Proof. One has immediatly that for any lcsE,!IdE“IdocE. Now consider three lcsE,F andG, and two linear continuous mapsf PLσpEw, Fwqandgf PLσpFw, Gwq. Then by definition, forφP!Eone has :

!pg˝fqpφq “ phnq Pź

n

HⁿpG,Kq ÞÑφpphn˝g˝fqnq.

On the other hand, one has immediatly:

!g˝!fpφq “ phnq Pź

n

HⁿpG,Kq ÞÑ p!fpφqqpphn˝gqnq ph_nq ÞÑφpph_n˝g˝fqn.

Arithmetic of the composition˝! We now would like to endow!with its co-monadic structure whose structure is based on a good notion of composition in the co-Kleisli category WEAK!. Forf P ś

mH^mpE, Fqandg P Proposition 5.2.14. The operation˝! :ś

mH^mpE, Fq ˆś

nHⁿpE, Gq // ś

pH^ppE, Gqis indeed a commu-tative and associative operation.

Proof. Commutativity is immediate. For formal power seriesf,g, andhone has : ppf ˝!gq ˝!hqp“ÿ

Remark5.2.15. At this point we must pay attention to the arithmetic employed here. So as to avoid infinite sums and a diverging term forpg˝fq04, we allow for only0to divide0. Thuspg˝fq0“g₀˝f₀.

Beware that even in the case of finite sums, this composition does not behave as the traditional composition between functions fromRtoR. If we considerf :xÞÑx`x<sup>2</sup>andg :yÞÑy<sup>2</sup>, we haveg˝!f :z ÞÑx<sup>2</sup>`x⁴, while as functions ofRone hasg˝f “x²`2x<sup>3</sup>`x⁴.

Remark5.2.16. Another composition of Formal power series, which coincide with the composition of real func-tions for converging power series, is given by the Faa di Bruno formula and detailed in Section5.2.5.

The co-monadic structure

Theorem 5.2.17. The functor! :LinÑLinis a co-monad. Its co-unitd: !Ñ1is defined by d_E

#!EwÑEw

φÞÑφ₁PE²»E

The co-unit is the operator extracting from φ P !E its part operating on linear maps. The co-multiplication µ: !Ñ!!is defined as:

4The problem of the possible divergence of the nonzero term can be found also in the theory of formal power series [38, IV.4], where composition is only allowed for series with no constant component.

µE want to have on our co-Kleisli category a composition such thatpg˝fqp“ř

k|pg_k˝f^p

k.The co-multiplication µ: !Ñ!!can be seen as a continuation-passing style transformation of this operation. Indeed, considerφP!E.

We constructµpφqas a function inpś

nHⁿp!E,Kqq¹mapping a sequencepgnqntoφapplied to the sequences of

This co-multiplication corresponds indeed to the composition˝!between power series: iff Pś

nHⁿpE, Fq andgPś

nHⁿpF, Gq, then:

g˝!f “g˝!f˝µ (5.1)

So as to show that !is in fact a co-monad, we need to understand better the elements of!E. The space!E is defined as‘nHⁿpE,Kq¹, soφ P !E can be described as a finite sum φ “ řN

n“1φn withφn P HⁿpE,Kq¹. The proofs presented below are based more on the idea of non-linear continuations than on a combinatoric point of view. The space!Ew “ pś

pH^ppE,Kqq¹can be thought of as a space of quantitative-linear continuations,K being the space of the result of a computation.

Proof. We have to check the two equations of a co-monad, that is:

1. d_!µ“ p!dqµ“Id_! 2. µ_!µ“ p!µqµ

‚ Let us detail the computations of the first equation. Remember that we write ev^HnpF,Kq for ev : F ÞÑ HⁿpF,Kq¹. For everyφ“ř element in!E, we have without using the isomorphism!E²»!E:

d!EµEpφq “g1P!E¹ÞÑÿ

d!EµEpφq “g1P!E¹ ÞÑÿ The equation!dµ“Idis proved likewise: considerφ“ř

φpP!E. Then

‚The computations of the second equations follow immediatly from the functoriality of!(proposition5.2.13) and the associativity of the composition˝! (proposition5.2.14). Indeed considerEwa weak lcs, andIdE as an element ofLpE, Eq, and thus as an element ofś

nHⁿpE, Eq. One has by associativity:

pId!!!E˝!Id!!Eq ˝!Id!E“Id!!!E˝!pId!!E˝!Id!Eq Id!!!E˝!Id!!E˝µ!E˝!Id!E˝µE“Id!!!E˝!!pId!!E˝!Id!Eq ˝µE

Id!!!E˝!Id!!E˝µ!E˝!Id!E˝!µE“Id!!!E˝!Id!!E˝!Id!E˝!µE˝µE

As!Id_E“Id_!Ewe have thusµ_!Ecircµ_E “!µ_E˝µ_E. This co-monad is in fact strong monoidal by proposition5.2.24.

Definition 5.2.18. The?connective of linear logic is interpreted as the dual of!, that is

?E» p!Eq¹»ź

n

HⁿpE¹,Kq.

We will write WEAK! for the co-Kleisli category of WEAK with!. We first show that morphisms of this category are easy to understand, as they are just sequences ofn-monomials.

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