On the transport of finiteness structures

Finiteness spaces were introduced by Ehrhard as a model of linear logic, which relied on a finitess property of the standard relational interpretation and allowed to reformulate Girard’s quantitative semantics in a simple, linear algebraic setting. We review recent results obtained in a joint work with Christine Tasson, providing a very simple and generic construction of finiteness spaces: basically, one can transport a finiteness structure along any relation mapping finite sets to finite sets. Moreover, this construction is functorial under mild hypotheses, satisfied by the interpretations of all the positive connectives of linear logic. Recalling that the definition of finiteness spaces follows a standard orthogonality technique, fitting in the categorical framework established by Hyland and Schalk, the question of the possible generalization of transport to a wider setting is quite natural. We argue that the features of transport do not stand on the same level as the orthogonality category construction; rather, they provide a simpler and more direct characterization of the obtained structure, in a webbed setting. 1 Finiteness spaces and finitary relations Sets and relations. We write P (A) for the powerset of A, Pf (A) for the set of all finite subsets of A and !A for the set of all finite multisets of elements of A. Let A and B be sets and f be a relation from A to B: f ⊆ A × B. We then write f for the transpose relation {(β, α) ∈ B ×A; (α, β) ∈ f}. For all subset a ⊆ A, we write f · a for the direct image of a by f : f · a = {β ∈ B; ∃α ∈ a, (α, β) ∈ f}. If α ∈ A, we will also write f · α for f · {α}. We say that a relation f is quasi-functional if f · α is finite for all α. If b ⊆ B, we define the division of b by f as f \ b = {α ∈ A; f · α ⊆ b}. Notice that in general f · (f \ b) may be a strict subset of b, and f \ (f · a) may be a strict superset of a. We write Rel for the category of sets and relations. It is a very simple model of linear logic: multiplicatives are given by the compact closed structure associated with cartesian products of sets (linear negation is then the transposition of relations, which is also a dagger); additives are modelled by disjoint union of sets, which gives a biproduct; the exponential modality is that of finite multisets. Let T and U be two endofunctors of Rel, and let f be the data of a relation fA (which we may also write f) from TA to UA for all set A: we say f is a lax natural transformation from T to U if, for all relation g from A to B, fB ◦ (Tg) ⊆ (Ug) ◦ fA. As an example, consider the finite multiset functor !, and for all A, let σA be the only relation from !A to A such that for all α ∈ !A, σA ·α is the support set of α. This defines a quasi-functional lax natural transformation from ! to the identity functor: notice that in that case, the inclusion σ ◦ !g ⊆ g ◦σ may be strict. Finiteness spaces. We briefly recall the basic definition of finiteness spaces as given by Ehrhard [Ehr05]. Let A and B be sets, we write A ⊥f B if A ∩ B is finite. If A ⊆ P (A), we define the predual of A on A as A⊥ = {a′ ⊆ A; ∀a ∈ A, a ⊥f a′}. A finiteness structure on A is a set A of subsets of A such that A⊥⊥ = A. A finiteness space is then a pair A = (|A| ,F (A)) where |A| is the underlying set, called the web of A, and F (A) is a finiteness structure on |A|. We write A⊥ for the dual finiteness space: ∣∣A⊥∣∣ = |A| and F (A⊥) = F (A)⊥. The elements of F (A) are called the finitary subsets of A. Standard arguments on closure operators defined by orthogonality apply and in particular A⊥ = A⊥⊥⊥, for all A ⊆ P (A); hence finiteness structures are exactly preduals. More specific to the orthogonality ⊥f , for all finiteness structure A on A, we obtain: (1) A is downwards closed for inclusion, i.e. a ⊆ a′ ∈ A implies a ∈ A; (2) Pf (A) ⊆ A and A is closed under finite unions, i.e. a, a′ ∈ A implies a ∪ a′ ∈ A. The first property is similar to the one for coherence spaces. The second one is distinctive of finiteness spaces, and is a non-uniformity property: union of finitary subsets models some form of computational non-determinism, which is crucial to interpret the differential λ-calculus [ER03]. Finitary relations. Let A and B be two finiteness spaces: we say a relation f from |A| to |B| is finitary from A to B if: for all a ∈ F (A), f ·a ∈ F (B), and for all b′ ∈ F ( B⊥ ) , f · b′ ∈ F ( A⊥ ) . The identity relation is finitary from A to itself, and finitary relations compose: this defines the category Fin whose objects are finiteness spaces and morphisms are finitary relations. Finitary relations form a finiteness structure: remark that f ⊆ |A| × |B| is finitary iff f ∈ { a× b′; a ∈ F (A) and b′ ∈ F ( B⊥ )}⊥. This reflects the ∗-autonomous structure of Fin, with tensor product given by |A ⊗ B| = |A| × |B| and F (A⊗ B) = {a× b; a ∈ F (A) and b ∈ F (B)}⊥⊥, and ∗-functor given by duality on finiteness spaces and transposition on finitary relations: f ∈ Fin(A,B) 7→ f ∈ Fin(B⊥,A⊥).