Proof Theory of Partially Normal Skew Monoidal Categories
DDavid I. Spivak and Jamie Vicary (Eds.):Applied Category Theory 2020 (ACT2020)EPTCS 333, 2021, pp. 230–246, doi:10.4204/EPTCS.333.16
Proof Theory of Partially Normal Skew Monoidal Categories
Tarmo Uustalu
Reykjavik University, Reykjavik, IcelandTallinn University of Technology, Tallinn, Estonia [email protected]
Niccol`o Veltri
Tallinn University of Technology, Tallinn, Estonia [email protected]
Noam Zeilberger ´Ecole Polytechnique, Palaiseau, France [email protected]
The skew monoidal categories of Szlach´anyi are a weakening of monoidal categories where the threestructural laws of left and right unitality and associativity are not required to be isomorphisms butmerely transformations in a particular direction. In previous work, we showed that the free skewmonoidal category on a set of generating objects can be concretely presented as a sequent calculus.This calculus enjoys cut elimination and admits focusing, i.e. a subsystem of canonical derivations,which solves the coherence problem for skew monoidal categories.In this paper, we develop sequent calculi for partially normal skew monoidal categories, whichare skew monoidal categories with one or more structural laws invertible. Each normality conditionleads to additional inference rules and equations on them. We prove cut elimination and we showthat the calculi admit focusing. The result is a family of sequent calculi between those of skewmonoidal categories and (fully normal) monoidal categories. On the level of derivability, these define8 weakenings of the I , ⊗ fragment of intuitionistic non-commutative linear logic. Substructural logics are logical systems in which one or more structural rules are not allowed. Structuralrules typically include exchange, weakening and contraction. More generally, in a sequent calculuswith sequents of the form Γ −→ C , with the antecedent Γ some type of a collection of formulae, a ruleis structural if it manipulates the antecedent and does not mention any connectives. Affine logics aresubstructural wrt. intuitionistic logics since contraction is disallowed. Linear logics are substructuralwrt. affine logics since weakening is also disallowed. By dropping the exchange rule as well, we obtainordered variants of (intuitionistic) linear logics [1], which include logical systems such as Lambek’ssyntactic calculus [16]. One can even identify more rudimentary structural rules, such as associativity,which is dropped in some variants of Lambek’s calculus [17].Given a sequent of the form Γ −→ C in a certain logical system, it is natural to think of formulae in Γ as types of resources at our disposal, while the formula C is a task that needs to be fulfilled with theresources at hand. Under this interpretation, the structural rules tell us how resources can be manipulatedand consumed. In intuitionistic logics, resources can be permuted, deleted and copied. In linear logics,they can be neither deleted nor copied, but they can be permuted. In non-commutative linear logicsresources cannot be permuted, so they must be consumed in the order they occur in the antecedent.In previous work [24], we started investigating the proof theory of (left-)skew monoidal categories ,a weakening of monoidal categories introduced by Szlach´anyi [22] where the unitors and associatorare not required to be isomorphisms but merely transformations in a particular direction. ExtendingZeilberger’s [27] analysis of the Tamari order (which is the free (left-)skew semigroup category) as a . Uustalu, N. Veltri & N. Zeilberger At of generating objects. This sequent calculus weakens the I , ⊗ -fragment ofintuitionistic non-commutative linear logic [1] by replacing unitality and associativity with semi-unitality and semi-associativity . This means that it is possible to derive sequents corresponding to the two unitors λ and ρ and the associator α of skew monoidal categories, but not sequents corresponding to the inversesof these structural laws. Sequents have the form S | Γ −→ C , where the antecedent is split into an optionalformula S , called the stoup , and a list of formulae Γ , the context . The left rules apply only to the formulain the special stoup position. The tensor right rule forces the formula in the stoup of the conclusion to bethe formula in the stoup of the first premise. Under the resource-as-formulae interpretation, resources arerequired to be consumed in the order they appear in the antecedent, with the formula in the stoup beingthe first. At any moment, only the formula then occupying the stoup can be decomposed on the left.This sequent calculus enjoys cut elimination and a focused subsystem, defining a root-first proofsearch strategy attempting to build a derivation of a sequent. The focused calculus finds exactly onerepresentative of each equivalence class of derivations and is thus a concrete presentation of the freeskew monoidal category, as such solving the coherence problem for skew monoidal categories.Skew monoidal categories differ from normal (i.e., ordinary) monoidal categories in that the twounitors and the associator are not invertible. Requiring one or more of the structural laws to be invertible,we obtain a less skew structure more like a normal monoidal category. In this paper, we perform a proof-theoretic analysis of each of the three normality conditions. The result is a family of sequent calculibetween those describing the free skew monoidal category and the free monoidal category, definingaltogether 8 weakenings of the I , ⊗ fragment of non-commutative intuitionistic linear logic.For each of these sequent calculi, we prove cut elimination and identify a focused subsystem ofcanonical derivations as a concrete presentation of the free skew monoidal category of the correspondingdegree of normality. We conclude by presenting a single parameterized focused sequent calculus thatcan handle any combination of the three normality aspects.We fully formalized the new results presented in Section 3 in the dependently typed programminglanguage Agda. The formalization uses Agda version 2.6.0. and it is available at https://github.com/niccoloveltri/skewmoncats-normal . A category C is said to be (left-)skew monoidal [22] if it comes together with a distinguished object I , afunctor ⊗ : C × C → C and three natural transformations λ , ρ , α typed λ A : I ⊗ A → A ρ A : A → A ⊗ I α A , B , C : ( A ⊗ B ) ⊗ C → A ⊗ ( B ⊗ C ) satisfying the equations ( m1 ) I ⊗ I λ I (cid:24) (cid:24) I ρ I (cid:70) (cid:70) I ( m2 ) ( A ⊗ I ) ⊗ B α A , I , B (cid:47) (cid:47) A ⊗ ( I ⊗ B ) A ⊗ λ B (cid:15) (cid:15) A ⊗ B ρ A ⊗ B (cid:79) (cid:79) A ⊗ B ( m3 ) ( I ⊗ A ) ⊗ B α I , A , B (cid:47) (cid:47) (cid:34) (cid:34) λ A ⊗ B I ⊗ ( A ⊗ B ) λ A ⊗ B (cid:124) (cid:124) A ⊗ B ( m4 ) ( A ⊗ B ) ⊗ I α A , B , I (cid:47) (cid:47) A ⊗ ( B ⊗ I ) A ⊗ B ρ A ⊗ B (cid:98) (cid:98) (cid:60) (cid:60) A ⊗ ρ B Proof Theory of Partially Normal Skew Monoidal Categories ( m5 ) ( A ⊗ ( B ⊗ C )) ⊗ D α A , B ⊗ C , D (cid:47) (cid:47) A ⊗ (( B ⊗ C ) ⊗ D ) A ⊗ α B , C , D (cid:15) (cid:15) (( A ⊗ B ) ⊗ C ) ⊗ D α A , B , C ⊗ D (cid:79) (cid:79) α A ⊗ B , C , D (cid:47) (cid:47) ( A ⊗ B ) ⊗ ( C ⊗ D ) α A , B , C ⊗ D (cid:47) (cid:47) A ⊗ ( B ⊗ ( C ⊗ D )) If λ , ρ or α is a natural isomorphism, we say that ( C , I , ⊗ ) is left-normal , right-normal resp. associative-normal (or Hopf ) [14]. A monoidal category [4] is a fully normal skew monoidal category.Equations (m1)–(m5) are directed versions of the original Mac Lane axioms [19] for monoidal cate-gories. Kelly [13] observed that, in the monoidal case, equations (m1), (m3), and (m4) follow from (m2)and (m5). In the skew situation, this is not the case.Skew monoidal categories arise naturally in many settings, for example in the study of relative mon-ads [2] and of quantum categories [14], and have been thoroughly investigated by Street, Lack andcolleagues [15, 8, 5, 6]. They, as well as Uustalu [23], present many examples. Here we show just oneexample where normalities also play a role.Consider two categories J and C and a functor J : J → C such that the left Kan extension along J exists for every functor J → C . The functor category [ J , C ] has a skew monoidal structure givenby I = J , F ⊗ G = Lan J F · G . The unitors and associator are the canonical natural transformations λ F : Lan J J · F → F , ρ F : F → Lan J F · J , α F , G , H : Lan J ( Lan J F · G ) · H → Lan J F · Lan J G · H . Thiscategory is right-normal if J is fully-faithful. It is left-normal if J is dense, which is to say that the nerveof J is fully-faithful. Finally, it is associative-normal if this nerve preserves left Kan extensions along J . This example is from the work of Altenkirch et al. [2] on relative monads. Relative monads on J are(skew) monoids in the skew monoidal category [ J , C ] . The free skew monoidal category
Fsk ( At ) over a set At (of atoms) can be viewed as a deductive system,which we refer to as the skew monoidal categorical calculus .Objects of Fsk ( At ) are called formulae , and are inductively generated as follows: a formula is eitheran element X of At (an atom ); I ; or A ⊗ B where A , B are formulae. We write Fma for the set of formulae.Maps between two formulae A and C are derivations of (singleton-antecedent, singleton-succedent)sequents A = ⇒ C , constructed using the following inference rules: A = ⇒ A id A = ⇒ B B = ⇒ CA = ⇒ C comp A = ⇒ C B = ⇒ DA ⊗ B = ⇒ C ⊗ D ⊗ I ⊗ A = ⇒ A λ A = ⇒ A ⊗ I ρ ( A ⊗ B ) ⊗ C = ⇒ A ⊗ ( B ⊗ C ) α (1) and identified up to the congruence . = induced by the equations:(category laws) id ◦ f . = f f . = f ◦ id ( f ◦ g ) ◦ h . = f ◦ ( g ◦ h ) ( ⊗ functorial) id ⊗ id . = id ( h ◦ f ) ⊗ ( k ◦ g ) . = h ⊗ k ◦ f ⊗ g λ ◦ id ⊗ f . = f ◦ λ ( λ , ρ , α nat. trans.) ρ ◦ f . = f ⊗ id ◦ ρα ◦ ( f ⊗ g ) ⊗ h . = f ⊗ ( g ⊗ h ) ◦ αλ ◦ ρ . = id id . = id ⊗ λ ◦ α ◦ ρ ⊗ id ( m1-m5 ) λ ◦ α . = λ ⊗ id α ◦ ρ . = id ⊗ ρα ◦ α . = id ⊗ α ◦ α ◦ α ⊗ id (2) . Uustalu, N. Veltri & N. Zeilberger g ◦ f for comp f g to agree with the standard categoricalnotation.Mac Lane’s coherence theorem [19] says that, in the free monoidal category, there is exactly one map A = ⇒ B if the formulae A and B have the same frontier of atoms and no such map otherwise.For the free skew monoidal category, this does not hold. We have pairs of formulae that have thesame frontier of atoms but no maps between them or multiple maps. There are no maps X = ⇒ I ⊗ X ,no maps X ⊗ I = ⇒ X and no maps X ⊗ ( Y ⊗ Z ) = ⇒ ( X ⊗ Y ) ⊗ Z . At the same time, we have two maps id (cid:54) . = α ◦ ρ ⊗ λ : X ⊗ ( I ⊗ Y ) = ⇒ X ⊗ ( I ⊗ Y ) and two maps id (cid:54) . = ρ ⊗ λ ◦ α : ( X ⊗ I ) ⊗ Y = ⇒ ( X ⊗ I ) ⊗ Y . In [24], we showed that the free skew monoidal category
Fsk ( At ) admits an equivalent presentation asa sequent calculus. In the latter, sequents are triples S | Γ −→ C . The antecedent is a pair of a stoup S together with a context Γ , while the succedent C is a single formula. A stoup is an optional formula,meaning that it can either be empty (written − ) or contain a single formula. A context is a list offormulae. For the empty list, we usually just leave a space, but where necessary for readability, we write () . Derivations in the sequent calculus are inductively generated by these inference rules: A | Γ −→ C − | A , Γ −→ C pass − | Γ −→ C I | Γ −→ C IL A | B , Γ −→ CA ⊗ B | Γ −→ C ⊗ L A | −→ A ax − | −→ I IR S | Γ −→ A − | ∆ −→ BS | Γ , ∆ −→ A ⊗ B ⊗ R (3) ( pass for ‘passivate’, L , R for introduction on the left (in the stoup) resp. right) and identified up to thecongruence (cid:36) induced by the equations: ( η -conversions) ax I (cid:36) IL IR ax A ⊗ B (cid:36) ⊗ L ( ⊗ R ( ax A , pass ax B )) (commutative conversions) ⊗ R ( pass f , g ) (cid:36) pass ( ⊗ R ( f , g )) ( for f : A (cid:48) | Γ −→ A , g : − | ∆ −→ B ) ⊗ R ( IL f , g ) (cid:36) IL ( ⊗ R ( f , g )) ( for f : − | Γ −→ A , g : − | ∆ −→ B ) ⊗ R ( ⊗ L f , g ) (cid:36) ⊗ L ( ⊗ R ( f , g )) ( for f : A (cid:48) | B (cid:48) , Γ −→ A , g : − | ∆ −→ B ) (4) Although these rules look very similar to the rules of the I , ⊗ fragment of intuitionistic non-commuta-tive linear logic [1] (the sequent calculus describing the free monoidal category)—in particular, there isno left exchange rule, weakening or contraction—, there are two crucial differences.• The left rules IL and ⊗ L are restricted to apply only to the formula within the stoup (in theconclusion-first reading of these rules). This restriction was also present in Zeilberger’s sequentcalculus for the Tamari order [27]. In this calculus, it is possible to derive sequents corresponding34 Proof Theory of Partially Normal Skew Monoidal Categories to the right unitor ρ : A = ⇒ A ⊗ I and the associator α : ( A ⊗ B ) ⊗ C = ⇒ A ⊗ ( B ⊗ C ) : A | −→ A ax − | −→ I IR A | −→ A ⊗ I ⊗ R A | −→ A ax B | −→ B ax C | −→ C ax − | C −→ C pass B | C −→ B ⊗ C ⊗ R − | B , C −→ B ⊗ C pass A | B , C −→ A ⊗ ( B ⊗ C ) ⊗ R A ⊗ B | C −→ A ⊗ ( B ⊗ C ) ⊗ L ( A ⊗ B ) ⊗ C | −→ A ⊗ ( B ⊗ C ) ⊗ L On the other hand, since IL and ⊗ L only act on the formula in the stoup, there is no way of derivingsequents corresponding to inverses of ρ and α for atomic A resp. A , B , C .• The stoup is allowed to be empty, permitting a distinction between antecedents of the form A | Γ (with A inside the stoup) and antecedents of the form − | A , Γ (with A outside the stoup). Thisdistinction plays an important role in the right rule ⊗ R , which sends the formula in the stoup, whenit is present, to the first premise. In this calculus, it is possible to derive a sequent correspondingto the left unitor λ : I ⊗ A = ⇒ A : A | −→ A ax − | A −→ A passI | A −→ A ILI ⊗ A | −→ A ⊗ L On the other hand, since ⊗ R always send the formula in the stoup to the first premise, it is notpossible to derive a sequent corresponding to the inverse of λ for atomic A .There are no primitive cut rules in this sequent calculus, but two forms of cut are admissible: S | Γ −→ A A | ∆ −→ CS | Γ , ∆ −→ C scut − | Γ −→ A S | ∆ , A , ∆ −→ CS | ∆ , Γ , ∆ −→ C ccut (5) Sequent calculus derivations can be turned into categorical calculus derivations by means of a func-tion sound : S | Γ −→ C → (cid:74) S | Γ (cid:75) = ⇒ C , where the interpretation of an antecedent as a formula (cid:74) S | Γ (cid:75) is defined as (cid:74) S | Γ (cid:75) = (cid:74) S (cid:104)(cid:104) (cid:104)(cid:104) Γ (cid:75) with (cid:74) −(cid:104)(cid:104) = I (cid:74) A (cid:104)(cid:104) = A A (cid:104)(cid:104) (cid:75) = A A (cid:104)(cid:104) B , Γ (cid:75) = ( A ⊗ B ) (cid:104)(cid:104) Γ (cid:75) which means that A (cid:104)(cid:104) A , A . . . , A n (cid:75) = ( . . . ( A ⊗ A ) ⊗ A ) . . . ) ⊗ A n . The interpretation of antecedents isfunctorial, i.e., we have the following inference rule: A = ⇒ B (cid:74) A | Γ (cid:75) = ⇒ (cid:74) B | Γ (cid:75) (cid:74) | Γ (cid:75) (6) The function sound is well-defined on (cid:36) -equivalence classes: given related derivations f (cid:36) g , then sound f . = sound g .Categorical calculus derivations can be interpreted as sequent calculus derivations via a function cmplt : (cid:74) S | Γ (cid:75) = ⇒ C → S | Γ −→ C , well-defined on . = -equivalence classes: given related derivations f . = g , then cmplt f (cid:36) cmplt g . The cut rule scut is fundamental for modelling the composition operation comp of the categorical calculus.The functions sound and cmplt establish a bijection between derivations in the categorical calculusand the sequent calculus: sound ( cmplt f ) . = f and cmplt ( sound g ) (cid:36) g , for all f : (cid:74) S | Γ (cid:75) = ⇒ C and g : S | Γ −→ C . . Uustalu, N. Veltri & N. Zeilberger The congruence relation (cid:36) can be considered as a term rewrite system, by directing every equation fromleft to right. The resulting rewrite system is weakly confluent and strongly normalizing, hence confluentwith unique normal forms.Normal-form derivations in our sequent calculus can be described as derivations in a focused sub-system. In the style of Andreoli [3], we present the focused subsystem as a sequent calculus with anadditional phase annotation on sequents. In phase L , sequents are of the form S | Γ −→ L C , where S is ageneral stoup. In phase R , sequents take the form T | Γ −→ R C , where T is an irreducible stoup, that isan optional atom: either empty or an atomic formula. Derivations in the focused calculus are inductivelygenerated by the following inference rules: A | Γ −→ L C − | A , Γ −→ L C pass − | Γ −→ L C I | Γ −→ L C IL A | B , Γ −→ L CA ⊗ B | Γ −→ L C ⊗ L T | Γ −→ R CT | Γ −→ L C switch X | −→ R X ax − | −→ R I IR T | Γ −→ R A − | ∆ −→ L BT | Γ , ∆ −→ R A ⊗ B ⊗ R (7) The focused rules define a sound and complete proof search strategy. The focused calculus is clearlysound: by erasing phase annotations, all of the rules are either rules of the original calculus or else (inthe case of switch ) have the conclusion equal to the premise. So focused calculus derivations can beembedded into sequent calculus derivations via a function emb P : S | Γ −→ P C → S | Γ −→ C , where P ∈ { L , R } .The focused calculus is also complete: we can define a normalization procedure focus : S | Γ −→ C → S | Γ −→ L C sending each derivation in the sequent calculus to a canonical representative of its (cid:36) -equivalence class in the focused calculus. This means in particular that focus maps (cid:36) -related derivationsto equal focused derivations.The functions focus and emb L establish a bijection between derivations in the full sequent calculus(up to (cid:36) ) and its focused subsystem: emb L ( focus f ) (cid:36) f and focus ( emb L g ) = g , for all f : S | Γ −→ C and g : S | Γ −→ L C .Putting this together with the results discussed in Section 2.2, it follows that the focused calculusis a concrete presentation of the free skew monoidal category Fsk ( At ) . As such, the focused calculussolves the problem of characterizing the homsets of the free skew monoidal category, a.k.a. the coherenceproblem . Moreover, it solves two related algorithmic problems effectively:• Duplicate-free enumeration of all maps A = ⇒ C in the form of representatives of . = -equivalenceclasses of categorical calculus derivations: For this, find all focused derivations of A | −→ L C ,which is solvable by exhaustive proof search, which terminates, and translate them to the categor-ical calculus derivations.• Finding whether two given maps of type A = ⇒ C , presented as categorical calculus derivations, areequal, i.e., . = -related as derivations: For this, translate them to focused derivations of A | −→ L C and check whether they are equal, which is decidable.Different approaches to the coherence problem of skew monoidal categories have been consideredbefore. Uustalu [23] identified a class of normal forms of objects in Fsk ( At ) and showed that there existsat most one map between an object and an object in normal form, and exactly one map between an objectand that object’s normal form. In another direction, Lack and Street [15] addressed the problem of de-termining equality of maps by proving that there is a faithful, structure-preserving functor Fsk ( ) → ∆ ⊥ Proof Theory of Partially Normal Skew Monoidal Categories from the free skew monoidal category on one generating object to the category of finite non-empty or-dinals and first-element-and-order-preserving functions (which is an associative-normal skew-monoidalcategory under the ordinal sum with unit 1). This approach was further elaborated by Bourke and Lack[5] with a more explicit description of the homsets of
Fsk ( ) .Let us use the focused calculus to analyze where multiple maps A = ⇒ C can come from. There aretwo sources of non-determinism in root-first proof search in the focused calculus: (i) in phase L , whenthe stoup is empty, whether to apply pass or switch , and (ii) in phase R , when the succedent formula is atensor and the rule ⊗ R is to be applied, how to split the context into Γ and ∆ . In the latter situation, onlythose choices where | T | , | Γ | = | A | and | ∆ | = | B | can possibly lead to a complete derivation; we write | | for the frontier of atoms in a formula, an optional formula or a list of formulae. But there can be multiplesuch choices if in the middle of the context there are closed formulae (i.e., formulae made of I and ⊗ only): those can be freely split between Γ and ∆ .The two maps id (cid:54) . = α ◦ ρ ⊗ λ : X ⊗ ( I ⊗ Y ) = ⇒ X ⊗ ( I ⊗ Y ) translate to two different focused deriva-tions of the sequent X ⊗ ( I ⊗ Y ) | −→ L X ⊗ ( I ⊗ Y ) because of the non-determinism of type (i). Noticethe choice between pass and switch . X | −→ R X ax − | −→ R I IR Y | −→ R Y ax Y | −→ L Y switch − | Y −→ L Y pass − | Y −→ R I ⊗ Y ⊗ R − | Y −→ L I ⊗ Y switchI | Y −→ L I ⊗ Y ILI ⊗ Y | −→ L I ⊗ Y ⊗ L − | I ⊗ Y −→ L I ⊗ Y pass X | I ⊗ Y −→ R X ⊗ ( I ⊗ Y ) ⊗ R X | I ⊗ Y −→ L X ⊗ ( I ⊗ Y ) switch X ⊗ ( I ⊗ Y ) | −→ L X ⊗ ( I ⊗ Y ) ⊗ L X | −→ R X ax − | −→ R I IR Y | −→ R Y ax Y | −→ L Y switch − | Y −→ L Y passI | Y −→ L Y ILI ⊗ Y | −→ L Y ⊗ L − | I ⊗ Y −→ L Y pass − | I ⊗ Y −→ R I ⊗ Y ⊗ R − | I ⊗ Y −→ L I ⊗ Y switch X | I ⊗ Y −→ R X ⊗ ( I ⊗ Y ) ⊗ R X | I ⊗ Y −→ L X ⊗ ( I ⊗ Y ) switch X ⊗ ( I ⊗ Y ) | −→ L X ⊗ ( I ⊗ Y ) ⊗ L (8) The two maps id (cid:54) . = ρ ⊗ λ ◦ α : ( X ⊗ I ) ⊗ Y = ⇒ ( X ⊗ I ) ⊗ Y translate to distinct focused derivationsof the sequent ( X ⊗ I ) ⊗ Y | −→ L ( X ⊗ I ) ⊗ Y due to type-(ii) non-determinism. Here the first (from theendsequent) application of the tensor right rule ⊗ R splits the context in two different ways: in the firstderivation the unit in the context is sent to the first premise, while in the second derivation it is sent tothe second premise. X | −→ R X ax − | −→ R I IR − | −→ L I switchI | −→ L I IL − | I −→ L I pass X | I −→ R X ⊗ I ⊗ R Y | −→ R Y ax Y | −→ L Y switch − | Y −→ L Y pass X | I , Y −→ R ( X ⊗ I ) ⊗ Y ⊗ R X | I , Y −→ L ( X ⊗ I ) ⊗ Y switch X ⊗ I | Y −→ L ( X ⊗ I ) ⊗ Y ⊗ L ( X ⊗ I ) ⊗ Y | −→ L ( X ⊗ I ) ⊗ Y ⊗ L X | −→ R X ax − | −→ R I IR − | −→ L I switch X | −→ R X ⊗ I ⊗ R Y | −→ R Y ax Y | −→ L Y switch − | Y −→ L Y passI | Y −→ L Y IL − | I , Y −→ L Y pass X | I , Y −→ R ( X ⊗ I ) ⊗ Y ⊗ R X | I , Y −→ L ( X ⊗ I ) ⊗ Y switch X ⊗ I | Y −→ L ( X ⊗ I ) ⊗ Y ⊗ L ( X ⊗ I ) ⊗ Y | −→ L ( X ⊗ I ) ⊗ Y ⊗ L (9) . Uustalu, N. Veltri & N. Zeilberger The free left-normal skew monoidal category
Fsk LN ( At ) on a set At is obtained by extending the gram-mar of derivations in the fully skew categorical calculus (1) with a new inference rule: A = ⇒ I ⊗ A λ − and extending the equivalence of derivations (2) with two new equations: λ ◦ λ − . = id and λ − ◦ λ . = id .An equivalent sequent calculus presentation of Fsk LN ( At ) is obtained by adding another right rulefor the tensor ⊗ to the fully skew sequent calculus (3), which allows one to send the formula in thestoup to the second premise, provided that all of the context is also sent to the second premise (so theantecedent of the first premise is left completely empty). − | −→ A A (cid:48) | ∆ −→ BA (cid:48) | ∆ −→ A ⊗ B ⊗ R The introduction of ⊗ R makes it possible to derive a sequent corresponding to λ − : A = ⇒ I ⊗ A : − | −→ I IR A | −→ A ax A | −→ I ⊗ A ⊗ R In particular this allows us to interpret categorical derivations into sequent calculus derivations for theleft-normal case, extending the definition of the function cmplt introduced in Section 2.2.Equivalence of derivations in the sequent calculus is the least congruence (cid:36) induced by the equationsin (4) together with the following equations: ⊗ R ( f , pass g ) (cid:36) pass ( ⊗ R ( f , g )) ( for f : − | −→ A , g : A (cid:48) | ∆ −→ B ) ⊗ R ( f , IL g ) (cid:36) IL ( ⊗ R ( f , g )) ( for f : − | −→ A , g : − | ∆ −→ B ) ⊗ R ( f , ⊗ L g ) (cid:36) ⊗ L ( ⊗ R ( f , g )) ( for f : − | −→ A , g : A (cid:48) | B (cid:48) , ∆ −→ B ) The two cut rules in (5) are also admissible in the left-normal sequent calculus. The definition of ccut uses the following rule act (for ‘activate’), admissible in this sequent calculus (but not in the fullyskew one) and inverting pass up to the congruence (cid:36) : − | A , Γ −→ L CA | Γ −→ L C act The transformation sound introduced in Section 2.2, interpreting the fully skew sequent calculusderivations as categorical calculus derivations, extends to the left-normal case. Given f : − | −→ A and g : A (cid:48) | ∆ −→ B , define sound ( ⊗ R ( f , g )) as: (cid:74) A (cid:48) | ∆ (cid:75) sound g = ⇒ B B = ⇒ I ⊗ B λ − I ⊗ B sound f ⊗ id = ⇒ A ⊗ BB = ⇒ A ⊗ B comp (cid:74) A (cid:48) | ∆ (cid:75) = ⇒ A ⊗ B comp Again, the congruence relation (cid:36) read as a term rewrite system is weakly confluent and stronglynormalizing, and normal-form derivations in the sequent calculus can be described as derivations in a38
Proof Theory of Partially Normal Skew Monoidal Categories focused subsystem. Sequents in the left-normal focused calculus are annotated with two possible phaseannotations, as in the fully skew focused calculus (7). The condition for switching phase is different: theformula in the stoup is still required to be irreducible, but if the stoup is empty we are allowed to switchphase only when the context is empty as well. In phase R , we also include the new tensor right rule ⊗ R ,in which both premises are required to be R -phase derivations. Again T is an irreducible stoup: eitherempty or an atomic formula. A | Γ −→ L C − | A , Γ −→ L C pass − | Γ −→ L C I | Γ −→ L C IL A | B , Γ −→ L CA ⊗ B | Γ −→ L C ⊗ L T | Γ −→ R C T = − → Γ = () T | Γ −→ L C switch X | −→ R X ax − | −→ R I IR T | Γ −→ R A − | ∆ −→ L BT | Γ , ∆ −→ R A ⊗ B ⊗ R − | −→ R A X | ∆ −→ R BX | ∆ −→ R A ⊗ B ⊗ R All R -phase sequents in a derivation of a sequent S | Γ −→ L C have the context empty if the stoup isempty.The functions emb L and emb R embedding fully skew focused calculus derivations in the unfocusedsequent calculus, discussed in Section 2.3, can clearly be adapted to the left-normal case. The sameholds for the focus function, sending a sequent calculus derivation to its (cid:36) -normal form in the focusedcalculus. In particular, this means that focus maps two (cid:36) -equivalent derivations to the same focusedderivation. Similarly to the fully skew case, focus and emb L establish a bijection between maps in theleft-normal sequent calculus (up to (cid:36) ) and its focused subsystem.The left-normal focused calculus has less non-determinism than the fully skew focused calculus.The non-determinism of type (i) is not there: In phase L , switch cannot be applied unless the context isempty while pass only applies if it is non-empty. The non-determinism of type (ii) remains much like inthe fully skew case except that there are two tensor right-rules, ⊗ R and ⊗ R . In a choice that can leadto a completed derivation, we must be able to take | T | , | Γ | = | A | and | ∆ | = | B | in ⊗ R or () = | A | and X , | ∆ | = | B | in ⊗ R . There may be multiple such choices if in the middle of the context we have closedformulae.In the free left-normal skew monoidal category, we have id . = α ◦ ρ ⊗ λ : X ⊗ ( I ⊗ Y ) = ⇒ X ⊗ ( I ⊗ Y ) : id . = id ⊗ λ − ◦ id ⊗ λ . = id ⊗ λ − ◦ ( id ⊗ λ ◦ α ◦ ρ ⊗ id ) ◦ id ⊗ λ . = α ◦ ρ ⊗ λ This collapse is reflected by there being exactly one focused derivation of the sequent X ⊗ ( I ⊗ Y ) | −→ L X ⊗ ( I ⊗ Y ) . Compare this with the two distinct derivations of the sequent in the fully skew focusedcalculus, displayed in (8). In the left-normal focused calculus, we are forced to use pass , we cannotapply switch since the stoup is empty but the context is not. X | −→ R X ax − | −→ R I IR Y | −→ R Y ax Y | −→ R I ⊗ Y ⊗ R Y | −→ L I ⊗ Y switch − | Y −→ L I ⊗ Y passI | Y −→ L I ⊗ Y ILI ⊗ Y | −→ L I ⊗ Y ⊗ L − | I ⊗ Y −→ L I ⊗ Y pass X | I ⊗ Y −→ R X ⊗ ( I ⊗ Y ) ⊗ R X | I ⊗ Y −→ L X ⊗ ( I ⊗ Y ) switch X ⊗ ( I ⊗ Y ) | −→ L X ⊗ ( I ⊗ Y ) ⊗ L . Uustalu, N. Veltri & N. Zeilberger pass is invertible. Here we only show the focused subsystem of the stoup-free variant. In phase L , sequentsare of the form Γ −→ L C , where Γ is a general list of formulae. In phase R , sequents take the form Λ −→ R C , where Λ is an irreducible list of formulae, meaning that it is either empty or the formula in itshead is an atom. Γ −→ L C I , Γ −→ L C IL A , B , Γ −→ L CA ⊗ B , Γ −→ L C ⊗ L Λ −→ R C Λ −→ L C switch X −→ R X ax −→ R I IR Λ −→ R A ∆ −→ L B Λ , ∆ −→ R A ⊗ B ⊗ R Derivations of a sequent S | Γ −→ L C are in a bijection with derivations of (cid:74) S (cid:104)(cid:104) , Γ −→ L C where (cid:74) S (cid:104)(cid:104) isthe interpretation of a stoup as a formula introduced in Section 2.2. The free right-normal skew monoidal category
Fsk RN ( At ) on a set At is obtained by extending thegrammar of derivations in the fully skew categorical calculus (1) with a new inference rule: A ⊗ I = ⇒ A ρ − and extending the equivalence of derivations (2) with two new equations: ρ ◦ ρ − . = id and ρ − ◦ ρ . = id .An equivalent sequent calculus presentation of Fsk RN ( At ) is realized by adding additional left rulesfor I and ⊗ to the fully skew sequent calculus (3), relaxing the condition for deleting the unit and decom-posing tensors in the antecedent: S | Γ , Γ −→ CS | Γ , I , Γ −→ C IC Γ S | Γ , J , J (cid:48) , Γ −→ CS | Γ , J ⊗ J (cid:48) , Γ −→ C ⊗ C c Γ Here and later, J and J (cid:48) stand for closed formulae. The introduction of IC makes it possible to derive asequent corresponding to ρ − : A ⊗ I = ⇒ A for any A , including A = X : A | −→ A ax A | I −→ A IC () A ⊗ I | −→ A ⊗ L The rule ⊗ C c is needed since it is important to allow deletion in the context of any closed formula andnot just I : we need to be able to derive, e.g., the sequent X | I ⊗ I −→ X . Analogously, in the left-normalsequent calculus of Section 3.1, it was important to be able to derive the sequent X | −→ ( I ⊗ I ) ⊗ X ,which was possible since the first premise of the inference rule ⊗ R is a sequent − | −→ A , which isderivable precisely when A is closed.Equivalence of derivations in the sequent calculus is the least congruence (cid:36) induced by the equations40 Proof Theory of Partially Normal Skew Monoidal Categories in (4) together with the following equations: pass ( IL f ) (cid:36) IC () f ( for f : − | Γ −→ C ) IC Γ ( IC Γ , Γ f ) (cid:36) IC Γ , I , Γ ( IC Γ f ) ( for f : S | Γ , Γ , Γ −→ C ) ⊗ C c Γ ( IC Γ , J , J (cid:48) , Γ f ) (cid:36) IC Γ , J ⊗ J (cid:48) , Γ ( ⊗ C c Γ f ) ( for f : S | Γ , J , J (cid:48) , Γ , Γ −→ C ) pass ( IC Γ f ) (cid:36) IC A , Γ ( pass f ) ( for f : A | Γ , Γ −→ C ) IL ( IC Γ f ) (cid:36) IC Γ ( IL f ) ( for f : − | Γ , Γ −→ C ) ⊗ L ( IC B , Γ f ) (cid:36) IC Γ ( ⊗ L f ) ( for f : A | B , Γ , Γ −→ C ) ⊗ R ( IC Γ f , g ) (cid:36) IC Γ ( ⊗ R ( f , g )) ( for f : S | Γ , Γ −→ A , g : − | ∆ −→ B ) ⊗ R ( f , IC ∆ g ) (cid:36) IC Γ , ∆ ( ⊗ R ( f , g )) ( for f : S | Γ −→ A , g : − | ∆ , ∆ −→ B ) (10) plus the same number of similar equations for ⊗ C c . The only difference is in the 1st equation, with ⊗ L instead of IL in the left-hand side, in which an extra application of pass in the right-hand side is requiredfor the equation to be well-typed: pass ( ⊗ L f ) (cid:36) ⊗ C c () ( pass f ) ( for f : J | J (cid:48) , Γ −→ C ) (11) The two cut rules in (5) are admissible in the right-normal sequent calculus. In this case, they needto be defined by mutual induction with another cut rule A (cid:48) | Γ −→ A S | ∆ , A , ∆ −→ CS | ∆ , A (cid:48) , Γ , ∆ −→ C ccut Fma
In the fully skew sequent calculus of Section 2.2, the rule ccut
Fma is definable by first applying pass to the first premise and then using ccut . In the right-normal case, we have to define it simultaneouslywith scut and ccut because of the added cases for the added primitive rules. The definition of ccut
Fma relies on a lemma, stating that, if a sequent A | Γ −→ J is derivable, with J a closed formula, then both A and all the formulae in Γ are also closed.The interpretation sound of fully skew sequent calculus derivations as categorical calculus derivationsextends to the right-normal case. Given f : S | Γ , Γ −→ C , define sound ( IC Γ f ) as: (cid:74) S | Γ (cid:75) ⊗ I = ⇒ (cid:74) S | Γ (cid:75) ρ − (cid:74)(cid:74) S | Γ (cid:75) ⊗ I | Γ (cid:75) = ⇒ (cid:74)(cid:74) S | Γ (cid:75) | Γ (cid:75) (cid:74) | Γ (cid:75) (cid:74) S | Γ , Γ (cid:75) sound f = ⇒ C (cid:74)(cid:74) S | Γ (cid:75) | Γ (cid:75) = ⇒ C (cid:74)(cid:74) S | Γ (cid:75) ⊗ I | Γ (cid:75) = ⇒ C comp (cid:74) S | Γ , I , Γ (cid:75) = ⇒ C The double-line rules correspond to applications of the provable equality (cid:74) S | Γ , ∆ (cid:75) = (cid:74)(cid:74) S | Γ (cid:75) | ∆ (cid:75) , while (cid:74) | (cid:75) is the inference rule introduced in (6).Given f : S | Γ , J , J (cid:48) , Γ −→ C , define sound ( ⊗ C c Γ f ) as: (cid:74) S | Γ (cid:75) ⊗ ( J ⊗ J (cid:48) ) = ⇒ ( (cid:74) S | Γ (cid:75) ⊗ J ) ⊗ J (cid:48) α − (cid:74)(cid:74) S | Γ (cid:75) ⊗ ( J ⊗ J (cid:48) ) | Γ (cid:75) = ⇒ (cid:74) ( (cid:74) S | Γ (cid:75) ⊗ J ) ⊗ J (cid:48) | Γ (cid:75) (cid:74) | Γ (cid:75) (cid:74) S | Γ , J , J (cid:48) , Γ (cid:75) sound f = ⇒ C (cid:74) ( (cid:74) S | Γ (cid:75) ⊗ J ) ⊗ J (cid:48) | Γ (cid:75) = ⇒ C (cid:74)(cid:74) S | Γ (cid:75) ⊗ ( J ⊗ J (cid:48) ) | Γ (cid:75) = ⇒ C comp (cid:74) S | Γ , J ⊗ J (cid:48) , Γ (cid:75) = ⇒ C (12) . Uustalu, N. Veltri & N. Zeilberger α − : A ⊗ ( J ⊗ J (cid:48) ) = ⇒ ( A ⊗ J ) ⊗ J (cid:48) is the inverse of a restricted form of the associator α in which the second and third formula are closed. It is defined and shown to invert α by induction on J (cid:48) (distinguishing the two cases of J (cid:48) being I and the tensor of two closed formulae).Let us mention that, instead of IC and ⊗ C c , one could of course choose to work with one “big-step”inference rule S | Γ , Γ −→ CS | Γ , J , Γ −→ C I Γ but we prefer the “small-step” IC and ⊗ C c especially because, in the situation of simultaneous right- andassociative-normality, ⊗ C c is subsumed by the ⊗ C rule that we will introduce in the next section.Again, the congruence relation (cid:36) read as a term rewrite system is weakly confluent and stronglynormalizing, and normal-form derivations in the sequent calculus can be described as derivations in afocused subsystem. Sequents in the focused calculus are annotated by three possible phase annotations.Phases L and R are as in the fully skew focused calculus (7). In the new phase C , sequents are of the form S | Ω ... Γ −→ C C , where the context is split in two parts: an anteroom Ω and a passive context Γ . In phase C , each formula D in the anteroom is inspected, starting from the right end of the anteroom. In case D isthe unit, then it is removed from the anteroom. If D is a tensor J ⊗ J (cid:48) , with J and J (cid:48) closed formulae, thenit is decomposed and J (cid:48) is inspected next. Otherwise, D is moved to the left end of the passive context. S | Ω ... Γ −→ C CS | Ω , I ... Γ −→ C C IC S | Ω , J , J (cid:48) ... Γ −→ C CS | Ω , J ⊗ J (cid:48) ... Γ −→ C C ⊗ C c S | Ω ... D , Γ −→ C C D (cid:54) = JS | Ω , D ... Γ −→ C C move S | Γ −→ L CS | ... Γ −→ C C switch LC A | Γ −→ L C − | A , Γ −→ L C pass − | Γ −→ L C I | Γ −→ L C IL A | B ... Γ −→ C CA ⊗ B | Γ −→ L C ⊗ L T | Γ −→ R CT | Γ −→ L C switch RL X | −→ R X ax − | −→ R I IR T | Γ −→ R A − | ∆ −→ L BT | Γ , ∆ −→ R A ⊗ B ⊗ R (13) All C -phase, L -phase and R -phase sequents in a derivation of a sequent S | Ω ... −→ C C have the(passive) context free of closed formulae.By dropping the phase annotations (also turning ... into a comma), we can define three functions emb C , emb L and emb R embedding right-normal focused calculus derivations into the unfocused calculus.We can also define a normalization function focus : S | Ω −→ C → S | Ω ... −→ C C , which identifies (cid:36) -related derivations. The functions focus and emb C (restricted to sequents whose passive context is empty)establish a bijection between the right-normal sequent calculus and its focused subsystem.The central design element of this focused calculus, the anteroom, together with the associatingorganization of the phases, is due to Chaudhuri and Pfenning [9].In the right-normal focused calculus, the type (i) non-determinism in phase L between the pass and switch rules is present. But the type (ii) non-determinism in phase R concerning the split of the contextat ⊗ R is inessential. Since the context cannot contain any closed formulae, at most one of the splits ofthe context into Γ , ∆ can lead to a complete derivation.In the free right-normal skew monoidal category, we have id . = ρ ⊗ λ ◦ α : ( X ⊗ I ) ⊗ Y = ⇒ ( X ⊗ I ) ⊗ Y : id . = ρ ⊗ id ◦ ρ − ⊗ id . = ρ ⊗ id ◦ ( id ⊗ λ ◦ α ◦ ρ ⊗ id ) ◦ ρ − ⊗ id . = ( ρ ⊗ λ ) ◦ α There is accordingly a single focused derivation of the sequent ( X ⊗ I ) ⊗ Y | ... −→ C ( X ⊗ I ) ⊗ Y . Comparethis with the two distinct derivations of the sequent in the fully skew focused calculus, displayed in (9).In the right-normal focused calculus, the unit is removed from the antecedent (more precisely, from the42 Proof Theory of Partially Normal Skew Monoidal Categories anteroom) with an application of the IC rule, so the ⊗ R does not have to choose in which premise to sendit. X | −→ R X ax − | −→ R I IR − | −→ L I switch RL X | −→ R X ⊗ I ⊗ R Y | −→ R Y ax Y | −→ L Y switch RL − | Y −→ L Y pass X | Y −→ R ( X ⊗ I ) ⊗ Y ⊗ R X | Y −→ L ( X ⊗ I ) ⊗ Y switch RL X | ... Y −→ C ( X ⊗ I ) ⊗ Y switch LC X | I ... Y −→ C ( X ⊗ I ) ⊗ Y IC X ⊗ I | Y −→ L ( X ⊗ I ) ⊗ Y ⊗ L X ⊗ I | ... Y −→ C ( X ⊗ I ) ⊗ Y switch LC X ⊗ I | Y ... −→ C ( X ⊗ I ) ⊗ Y move ( X ⊗ I ) ⊗ Y | −→ L ( X ⊗ I ) ⊗ Y ⊗ L ( X ⊗ I ) ⊗ Y | ... −→ C ( X ⊗ I ) ⊗ Y switch LC The free associative-normal skew monoidal category
Fsk AN ( At ) on a set At is obtained by extending thegrammar of derivations in the fully skew categorical calculus (1) with a new inference rule: A ⊗ ( B ⊗ C ) = ⇒ ( A ⊗ B ) ⊗ C α − and extending the equivalence of derivations (2) with two new equations: α ◦ α − . = id and α − ◦ α . = id .The associative-normal sequent calculus is obtained by adding to (3) a new logical left rule for thetensor, relaxing the condition for decomposing a formula A ⊗ B in the antecedent: S | Γ , A , B , Γ −→ CS | Γ , A ⊗ B , Γ −→ C ⊗ C Γ Including the rule ⊗ C in the calculus makes it possible to derive the sequent corresponding to α − : A ⊗ ( B ⊗ C ) = ⇒ ( A ⊗ B ) ⊗ C : A | −→ A ax B | −→ B ax − | B −→ B pass A | B −→ A ⊗ B ⊗ R C | −→ C ax − | C −→ C pass A | B , C −→ ( A ⊗ B ) ⊗ C ⊗ R A | B ⊗ C −→ ( A ⊗ B ) ⊗ C ⊗ C () A ⊗ ( B ⊗ C ) | −→ ( A ⊗ B ) ⊗ C ⊗ L We do not include here all the new equations that need to be added to the ones in (4) as generatorsof the least congruence (cid:36) . They are obtained from the equalities in (10) (except for the 1st and 2rd) byreplacing IC and ⊗ C c with ⊗ C and the equation (11) by replacing ⊗ C c with ⊗ C .The cut rules in (5) are also admissible in the associative-normal sequent calculus. In this case, theyneed to be defined by mutual induction with a third cut rule: S (cid:48) | Γ −→ A S | ∆ , A , ∆ −→ CS | ∆ , (cid:74) S (cid:48) (cid:104)(cid:104) , Γ , ∆ −→ C ccut Stp . Uustalu, N. Veltri & N. Zeilberger ccut
Stp relies on an additional admissible rule that is a restricted form of the IC rulein the right-normal sequent calculus: a unit in the context can be removed, provided that the part of thecontext on its right is non-empty. S | Γ , B , Γ −→ CS | Γ , I , B , Γ −→ C IC c Γ The case of the function sound for the new inference rule ⊗ C is defined as for the rule ⊗ C c in theright-normal sequent calculus, given in (12), with the closed formulae J and J (cid:48) replaced by arbitraryformulae, and the application α − replaced by α − .We need not prove soundness for the rule IC c since it is not primitive. But it is motivated by thepresence in the fully skew categorical calculus already of a derivation ( id ⊗ λ ) ◦ α : ( A ⊗ I ) ⊗ B = ⇒ A ⊗ B that, by (m2), post-inverts ρ ⊗ id .An equivalent focused subsystem is obtained similarly to the right-normal case. It is in fact the sameas the focused calculus in (13) with the rule IC removed, the rule ⊗ C c replaced by ⊗ C and the rule move modified accordingly. If the rightmost formula D in the anteroom is a tensor A ⊗ B , then it is decomposedand B is inspected next. Otherwise, D is moved to the left end of the passive context. S | Ω , A , B ... Γ −→ C CS | Ω , A ⊗ B ... Γ −→ C C ⊗ C S | Ω ... D , Γ −→ C C D (cid:54) = A ⊗ BS | Ω , D ... Γ −→ C C move All C -phase, L -phase and R -phase sequents in a derivation of a sequent S | Ω ... −→ C C have the(passive) context free of formulae A ⊗ B .Similarly to the right-normal case discussed in Section 3.2, the associative-normal sequent calculuscan be proved equivalent to its focused subsystem by means of two functions emb C and focus .Lack and Street [15] observed that the free associative-normal skew monoidal category on one gener-ator Fsk AN ( ) is isomorphic to ∆ ⊥ , the category of finite non-empty ordinals and first-element-and-order-preserving functions, and their main coherence theorem states that the canonical functor Fsk ( ) → ∆ ⊥ isfaithful. This seems to imply that the embedding of the fully skew sequent calculus into the associative-normal sequent calculus is faithful, but we leave a direct proof of this fact to future work. The additional inference rules and equations for the three normality aspects can be enabled simulta-neously, to yield, e.g., a sequent calculus for the free simultaneous left-normal and right-normal skewmonoidal category.The following rules define a single focused sequent calculus that can handle any combination of thethree normality aspects. It is parameterized in three flags ln , rn and an for left-, right- resp. associativenormality. (Recall that J , J (cid:48) are metavariables for closed formulae.) S | Ω ... Γ −→ C C rnS | Ω , I ... Γ −→ C C IC S | Ω , A , B ... Γ −→ C C ( rn ∧ A ⊗ B = J ⊗ J (cid:48) ) ∨ anS | Ω , A ⊗ B ... Γ −→ C C ⊗ C S | Ω ... D , Γ −→ C C rn → D (cid:54) = J an → D (cid:54) = A ⊗ BS | Ω , D ... Γ −→ C C move S | Γ −→ L CS | ... Γ −→ C C switch LC A | Γ −→ L C − | A , Γ −→ L C pass − | Γ −→ L C I | Γ −→ L C IL A | B ... Γ −→ C CA ⊗ B | Γ −→ L C ⊗ L T | Γ −→ R C ln ∧ T = − → Γ = () T | Γ −→ L C switch RL X | −→ R X ax − | −→ R I IR T | Γ −→ R A − | ∆ −→ L BT | Γ , ∆ −→ R A ⊗ B ⊗ R − | −→ R A X | ∆ −→ R B lnX | ∆ −→ R A ⊗ B ⊗ R Proof Theory of Partially Normal Skew Monoidal Categories
In the case of simultaneous left- and right-normality (i.e., ln ∧ rn ), the non-determinism of type (i) inphase L is not present and the non-determinism of type (ii) in phase R is inessential as there cannot beany closed formulae in the context. Consequently, any sequent can have at most one focused derivation:the free simultaneously left- and right-normal skew-monoidal category is thin.Laplaza [18] proved that the free skew semigroup category is thin (and this was reproved by Zeil-berger [27]). We have now seen that this remains true also when a both left-normal and right-normal unitis freely added. We showed that, similarly to the free skew monoidal category and the free monoidal category, the freeskew monoidal categories of different degrees of partial normality can be described as sequent calculi.These sequent calculi define logics weaker than the multiplicative fragment of the intuitionistic non-commutative linear logic. They enjoy cut elimination and they also admit focusing, a deductive descrip-tion of a root-first proof search strategy that finds exactly one representative from each equivalence classof derivations.We intend to continue this study by broadening its scope to fully skew and partially normal closedand monoidal closed categories and also prounital-closed categories (where the unit is present in a non-represented way). Skew closed categories (the skew variant of closed categories of Eilenberg and Kelly[10]) were first considered by Street [21], prounital-closed categories by Shulman [20, Rev. 49]. Zeil-berger [26] used a thin variant of skew closed categories, which he called imploids, in his study of therelation between typing of linear lambda terms and flows on 3-valent graphs. In the recent work [25],we dissected the Eilenberg-Kelly theorem about adjoint monoidal and closed structures on a category,revisited by Street [21] for the skew situation, to establish it in the general partially normal case. Somesurprising phenomena occur around skew closed categories, e.g., the free skew closed category on a setis left-normal, but this is lost when the tensor is added. We also want to study the proof theory of skewbraided monoidal categories, as recently introduced by Bourke and Lack [7].We have in this paper explained how the free skew monoidal category of each possible degree ofpartial normality can be described as a sequent calculus (a “logic”). This correspondence extends to non-free partially normal monoidal categories. But in this case, rather than inductively defining the maps andtheir equality, the inference rules and equations of the sequent calculus merely impose closure conditionson some homset predicate and some equality relation given upfront. One could compare a non-freecategory to a “theory” (a set of judgments closed under some inference rules) as opposed to a “logic”(the least set of judgments closed under them, i.e., the set of derivable judgments). Cut elimination (inthe sense that a set of judgments closed under the inference rules adopted minus cut would necessarily beclosed also under cut) cannot be expected, neither can focusing. Bourke and Lack [6] showed that skewmonoidal categories are equivalent to representable skew multicategories, a weakening of representablemulticategories [11]. In our previous work [24], we showed that the map constructors and equations ofa (nullary-binary) representable skew multicategory are very close to and mutually definable with thoseof the sequent calculus for the corresponding skew monoidal category (viewed as a deductive calculus,the representable skew multicategory uses exactly the same sequent forms, but has the basic inferencerules and equations chosen differently). We expect that partially normal skew monoidal categories canbe analyzed in similar terms. Specifically, we hope that the correct variations of representable skewmulticategories can be systematically derived in the framework of (op)fibrations of multicategories [12],adapted for skew multicategories. . Uustalu, N. Veltri & N. Zeilberger
Acknowledgments
T.U. was supported by the Icelandic Research Fund grant no. 196323-052 and the Estonian Ministryof Education and Research institutional research grant no. IUT33-13. N.V. was supported by the ESFfunded Estonian IT Academy research measure (project 2014-2020.4.05.19-0001).
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