towards a mathematical theory of substitutionmpf23/talks/ct2007.pdf · substitution aspects i...

Towards a

Mathematical Theory

of Substitution

Marcelo FioreComputer Laboratory

University of Cambridge

Invited talk at CT200720.VI.2007

Substitution

Examples:

I Logic/algebra/rewriting.t [ u/x ] t [ u1/x1

, . . . ,un/xn]

I Type theory.T [ t/x ]

I Formal languages.w0X1w1 . . . Xnwn Xi 7→ Wi

��w0W1w1 . . . Wnwn

I Proof theory.

P1 · · · Pn

· · · OPn

��

OP1 · · ·

I Structural combinatorics.

• •

••

• ◦

◦◦

••

��???

• ••

••

Substitution

Aspects

I syntactic vs. semantic modelsI homogeneous vs. heterogeneousI typed vs. untypedI variables vs. occurrences

I single vs. simultaneousI bindingI higher order

I algorithms

ANALYSE substitution from a foundationalstandpoint in a variety of scenarios andSYNTHESISE a mathematical theory.

Substitution

Aspects

I syntactic vs. semantic modelsI homogeneous vs. heterogeneousI typed vs. untypedI variables vs. occurrences

I single vs. simultaneousI bindingI higher order

I algorithms

ANALYSE substitution from a foundationalstandpoint in a variety of scenarios andSYNTHESISE a mathematical theory.

Algebraic theories

Clone of operations { Cn×(Cm)n → Cm | ··· }

≡Lawvere theories

≡Finitary monads

≡Monoids for the substitution tensor product

Substitution tensor product

on SetF

finite sets and functions↑

Endofin(Set) ' SetF

Id, ◦ ↔ V, •

V(n) = n

(X • Y)(n) =∫k∈F

X(k) × (Yn)k

Algebraic theories

Clone of operations { Cn×(Cm)n → Cm | ··· }

≡Lawvere theories

≡Finitary monads

≡Monoids for the substitution tensor product

Substitution tensor product

on SetF

finite sets and functions↑

Endofin(Set) ' SetF

Id, ◦ ↔ V, •

V(n) = n

(X • Y)(n) =∫k∈F

X(k) × (Yn)k

Cartesian mono-sortedsubstitution

monoid structure for the substitutiontensor product on Set

Examples:I Finitary algebraic syntax.

Σ = signature of operators with aritiesin N

Σ? = free monad on Σ(X) =∐

o∈Σ X|o|

SUBSTITUTION STRUCTURE:• n → Σ?(n)

• Σ?(n) × (Σ?m)n → Σ?(m)

NB: Arises from the universal propertyof Σ? by structural recursion (; correctsubstitution algorithm).

see e.g. [31]

I Lambda-calculus syntax.Λ(n) = { λ-terms with free variables in n }

with functorial action given by(capture-avoiding) variable renaming

x ∈ n

x ∈ Λ(n)

t1, t2 ∈ Λ(n)

t1(t2) ∈ Λ(n)

t ∈ Λ(n ] { x }

λx. t ∈ Λ(n)(†)

(†) SUBTLETY: α-equivalence

SUBSTITUTION STRUCTURE:• n → Λ(n)

• Λ(n) × (Λm)n → Λ(m)

t, (i 7→ ti)i∈n 7→ t [ ti/i ]i∈n

b� (capture-avoiding)

simultaneoussubstitution

I Lambda-calculus syntax.Λ(n) = { λ-terms with free variables in n }

with functorial action given by(capture-avoiding) variable renaming

x ∈ n

x ∈ Λ(n)

t1, t2 ∈ Λ(n)

t1(t2) ∈ Λ(n)

t ∈ Λ(n ] { x }

λx. t ∈ Λ(n)(†)

(†) SUBTLETY: α-equivalenceSUBSTITUTION STRUCTURE:• n → Λ(n)

• Λ(n) × (Λm)n → Λ(m)

t, (i 7→ ti)i∈n 7→ t [ ti/i ]i∈n

b� (capture-avoiding)

simultaneoussubstitution

I Clone of maps.The clone of maps 〈C,C〉 on an object C

in a cartesian category is given by

〈C,C〉(n) = [Cn, C]

SUBSTITUTION STRUCTURE:

• n // [Cn, C] : i 7→ πi

• [Cn, C] × [Cm, C]n // [Cm, C]

∼=LLLL

%%LLLLL

[Cn, C] × [Cm, Cn]

99rrrrrrrrrr

The substitution tensor product ...

free cartesian categoryon one generator

↑1 � � //

F◦ � � //

Y(−)

− •Y

||yyyyyyyyyyyyy

Lan∼=

〈Y,−〉

<<yyyyyyyyyyyyy

〈Y,Z〉(n) = [Yn,Z]

... is closed

Algebraic theories in SetF

syntax with variable binding

Example: Σλ = { app : 2, abs : V } NB: V = y(1)

Σλ(X) = X2+XV

(Σλ)?V = µX. V + X2 + XV ∼= Λ

see [16, 31]

NB:X2 → X ≡ { (Xn)2 → Xn | · · · }

XV → X ≡ {X(n + 1) → Xn | · · · }

��

�O�O

α-equivalence

(−)+1

��

� � //Set

a(−)×V��

F◦ � � //Set

(−)V = ((−)+1)∗

Σλ(X) = X2+XV

(Σλ)?V = µX. V + X2 + XV ∼= Λ

see [16, 31]

NB:X2 → X ≡ { (Xn)2 → Xn | · · · }

XV → X ≡ {X(n + 1) → Xn | · · · }

��

�O�O

α-equivalence

(−)+1

��

� � //Set

a(−)×V��

F◦ � � //Set

(−)V = ((−)+1)∗

Σλ(X) = X2+XV

(Σλ)?V = µX. V + X2 + XV ∼= Λ

see [16, 31]

NB:X2 → X ≡ { (Xn)2 → Xn | · · · }

XV → X ≡ {X(n + 1) → Xn | · · · }

��

�O�O

α-equivalence

(−)+1

��

� � //Set

a(−)×V��

F◦ � � //Set

(−)V = ((−)+1)∗

Λ is (universally characterised as) the freeΣλ-algebra on V

, and its substitution structureis derived by parameterised structural recursionas follows:

Σλ(Λ) • Λ

��

// Σλ(Λ • Λ)Σλ(s) // Σλ(Λ)

��Λ • Λ

s // Λ

V • Λ

OO∼=ggggggggggggg

33ggggggggggggg

; correct (capture-avoiding) simultaneoussubstitution algorithm

see [16, 31]

(†) yields an induction principle see [19, 31]

(‡) SUBTLETY: pointed strength

!!capture avoidance

Λ is (universally characterised as) the freeΣλ-algebra on V(†), and its substitution structureis derived by parameterised structural recursionas follows:

Σλ(Λ) • Λ

��

(‡) // Σλ(Λ • Λ)Σλ(s) // Σλ(Λ)

��Λ • Λ

s // Λ

V • Λ

OO∼=ggggggggggggg

33ggggggggggggg

; correct (capture-avoiding) simultaneoussubstitution algorithm

see [16, 31]

(†) yields an induction principle see [19, 31]

(‡) SUBTLETY: pointed strength

!!capture avoidance

General theorySETTING:

U��

CΣ 77

��

��a monoidal closed category

an endofunctor with a U-strength:

Σ(X) ⊗ YσX,(I→Y) // Σ(X ⊗ Y)

MODELS: (Σ, σ)-monoids.

��an algebra

X ⊗ X m// X Ie

oo a monoidsuch that

Σ(X) ⊗ X

ξ⊗X

��

σX,e // Σ(X ⊗ X)Σm // ΣX

��X ⊗ X m

General theorySETTING:

U��

CΣ 77

��

��a monoidal closed category

an endofunctor with a U-strength:

Σ(X) ⊗ YσX,(I→Y) // Σ(X ⊗ Y)

MODELS: (Σ, σ)-monoids.

��an algebra

X ⊗ X m// X Ie

oo a monoidsuch that

Σ(X) ⊗ X

ξ⊗X

��

σX,e // Σ(X ⊗ X)Σm // ΣX

��X ⊗ X m

Thefree Σ-algebra and free monoid

constructionsΣ-alg(C )

//> Coo // Mon(C )>

µX.C + ΣX C�oo � // µX. I + C ⊗ X

unify to(Σ, σ)-Mon(C )

��a

whereS(C) = µX. I + C ⊗ X + ΣX

NB: The initial (Σ, σ)-monoid has underlyingobject S0 = µX. I + ΣX = Σ?I.

see [26, 30]

Initial-algebra semanticswith substitution

The unique (Σ, σ)-monoid homomorphismfrom the initial (Σ, σ)-monoid provides aninitial-algebra semantics that is bothcompositional and respects substitution.

see [16, 24]

Example: Lambda calculus.For D C DD in a cartesian closed category,the clone of maps 〈D,D〉 has a canonicalΣλ-algebra structure

〈D,D〉 × 〈D,D〉 // 〈D,D〉

��〈D,D〉 × 〈D,DD〉 ∼= // 〈D,D × DD〉

〈D,D〉V // 〈D,D〉

∼=HH

〈D,DD〉

::vvvvvvvvv

making it into a Σλ-monoid for the canonicalpointed strength.The induced initial-algebra semantics amountsto the standard interpretation of the λ-calculus.

see [16, 19]

Single-variable andsimultaneous substitution

The theory of monoids in SetF for the

substitution tensor product is enrichedalgebraic for the cartesian closedstructure.

MonV,•(SetF) is (equivalent to) the category of

algebras X with operations

XV+1 → X and 1 → XV

subject to

. . . 4 axioms . . .see [16]

Second-order syntaxwith variable binding

and substitution

Example:

Σλ-Mon(SetF)

��a

wwoooooooooooSλ

g(−)77ooooooooooo a

For M ∈ SetN,

Sλ(M) ∼= V + M • Sλ(M) + Σλ(SλM)

see [22]

Sλ(M) can be syntactically presented as follows:

x ∈ n

var(x) ∈ Sλ(M)(n)

t1, t2 ∈ Sλ(M)(n)

app(t1, t2) ∈ Sλ(M)(n)

t ∈ Sλ(M)(n ] { x }

abs((x)t

)∈ Sλ(M)(n)

(up to α-equivalence)

t1, . . . , tk ∈ Sλ(M)(n)

T [t1, . . . , tk] ∈ Sλ(M)(n)

(T ∈ M(k)

and the monoid multiplication structure

Sλ(M) • Sλ(M) → Sλ(M)

amounts to (capture-avoiding) simultaneoussubstitution. see [31]

MOREOVER, the action of Sλ enriches over SetF

with respect to its cartesian closed structure,and we obtain a further substitution structure

Sλ(M) × (SλN)fM → Sλ(N)

that amounts to SECOND-ORDER SUBSTITUTIONfor metavariables.

As usual, this arises by universal properties andis given by parameterised structural recursion;yielding a correct substitution algorithm.

see [31]

Many-sorted contexts

For S a set of sorts, let F[S] be the freecocartesian category on S.

The substitution tensor product on(Set

F[S])S is

given as follows:

��

X •Y

S� � //

F[S]◦ � � //

SetF[S]

def− •Y

||xxxxxxxxxx

SetF[S]

Lan∼=

That is,

(X • Y)σ(Γ) =

∫∆∈F[S]

Xσ(∆) ×∏

τ∈∆

Yτ(Γ)

see [19, 23, 24, 26]

Simply typed lambda calculus,

algebraically

Let T be a set of base types, and let T be itsclosure under 1, *, =>.Consider

Σ ??(Set

F[T ])T

induced by

(†) app(σ,τ) : (σ=>τ, σ) → τ

(‡) abs(σ,τ) :((σ)τ

)→ σ=>τ

proj1(σ,τ) : (σ, τ) → σ , proj2(σ,τ) : (σ, τ) → τ

pair(σ,τ) : (σ, τ) → σ*τ

ter : () → 1

(†) Xσ=>τ × Xσ → Xτ

(‡) XτVσ → Xσ=>τ

see [19, 23]

Furthermore, let CC be the following equationaltheory for Σ-monoids:

(β) F : [σ]τ, T : [ ]σ

` app(abs

((x : σ)F[var(x)]

), T [ ]

F [T [ ]] : τ

(η) F : [ ](σ=>τ)

` abs((x : σ)app

(F[ ], var(x)

F[ ] : σ=>τ

informally:(λx. F)T = F [ T/x ]

λx. Fx = F(x 6∈ FV(F)

Furthermore, let CC be the following equationaltheory for Σ-monoids:

(β) F : [σ]τ, T : [ ]σ

` app(abs

((x : σ)F[var(x)]

), T [ ]

F [T [ ]] : τ

(η) F : [ ](σ=>τ)

` abs((x : σ)app

(F[ ], var(x)

F[ ] : σ=>τ

informally:(λx. F)T = F [ T/x ]

λx. Fx = F(x 6∈ FV(F)

(proj) M : [ ]σ,N : [ ]τ

` proj1(M[ ],N[ ]) = M[ ] : σ

` proj2(M[ ],N[ ]) = N[ ] : τ

(pair) T : [ ](σ*τ)

` pair(proj1(T [ ]), proj2(T [ ])

T [ ] : σ*τ

(ter) T : [ ]1 ` T [ ] = ter : 1

Γ -alg

��

77ooooooooooooo

77oooooooooooooΣ-Mon/CC ⊥� �

eq// Σ-Mon

%%KKKKKKKKKK

F[T ])T

eeKKKKKKKKKK

and the Lawvere theory associated to the initialΣ-Mon/CC is the free cartesian closed categoryon T .

(†) induced by CC

[NB: This generalises to free cartesian closedcategories on small categories.]

(†)Example: The parallel pair induced by (β).

For N[X] =∐

n∈NXn, let M ∈

N[T ])T bedefined from the context of (β) as

Mτ(σ) = { F } , Mσ() = { T }

and empty otherwise.The terms of (β) correspond to global elements

1 −→ S(M)

that induce functorsΣ-Mon ' S-alg −→ (−)

fM-alg

over(Set

F[T ])T as follows

��X

��

S(X)S(fM)

��S(X)1

��X

see [30]

Dependent sorts

For a small category C, let FC ' (SetC◦

)fp bethe free finite colimit completion of C.

The substitution tensor product on (SetF C)C◦

is given as follows:

~~}}}}

X •Y

C◦ � � //

(FC)◦ � � //

SetF C

def− •Y

||yyyyyyyyyy

SetF C

Lan∼=

That is,

(X • Y)C(Γ) =

∫∆∈F C

XC(∆) × limD∈El(∆)

Yp∆D(Γ)

PROGRAMME

The various developments of the previousslides carry over to this more general setting.

Following Makkai [13], after Lawvere [9] andOtto [11], the syntactic theory is consideredfor simple categories (= skeletal and one-way,with finite fan-out).

This amounts to extending the theory toincorporate DEPENDENT SORTS.

NB: The limit in the substitution tensor productaccounts for the heavy dependency required inthe substitution operation.

see [26]

General theory : Idea

For T a 2-monad on CAT, consider

��

X •Y

TC� � //

��???

def− •Y

��

Lan∼=

!!equipped with a T -algebra structure

Examples:

I T = identity

I T = free cartesian completion

I T = free finite limit completion

I T = free monoidal completion

I T = free symmetric monoidalcompletion

the T -algebra structure on TC

is given by Day’s tensor product [2, 7]

planar(I, •)-monoids =

symmetric

operads

see e.g. [3, 15, 20]

QUESTION: Are there applications of theprevious theory to the theory of operads!?

I T = free monoidal completion

I T = free symmetric monoidalcompletion

the T -algebra structure on TC

is given by Day’s tensor product [2, 7]

planar(I, •)-monoids =

symmetric

operads

see e.g. [3, 15, 20]

QUESTION: Are there applications of theprevious theory to the theory of operads!?

More generally :

From substitution to composition

For (T, η, µ) a 2-monad on CAT, considerA

��

F •G

BηB //

TB� � yB //

def− •G

��~~~~

Lan∼=

with respect to T -algebra structures

τC : T(TC

)→ TC

such thatyC : (TC, µC) → (TC, τC)

andLanyX

(h) : (TC, τC) → (TD, τD)

for all h : (TC, µC) → (TD, τD)

NB: The above can be axiomatised further.[Hyland, Gambino, Fiore]

(see also [24])

Kleisli bicategory

TC�_ // B TB

�_ // A

TC�_ // A

I T = identity; profunctors

I T = free symmetric monoidal completion; JOYAL SPECIES OF STRUCTURES[6, 8] arise

as the endomorphisms of 1; GENERALISED SPECIES OF STRUCTURES

see [23, 25]

Coherence (idea):

Lany((Lany H#)G

∼= Lany((Lany H#)G#

∼= (Lany H#) (Lany G#)

(†) The coherence isomorphisms can also be given formally (compare [3])

in a theory of Lawvere’s generalized logic [4], providing a logical view ofcoherence. The coherence laws can be then established elementwise.

PROGRAMME: Extend generalized logic to a type theory within whichthe coherence laws may be also established formally.

Kleisli bicategory

TC�_ // B TB

�_ // A

TC�_ // A

I T = identity; profunctors

I T = free symmetric monoidal completion; JOYAL SPECIES OF STRUCTURES[6, 8] arise

as the endomorphisms of 1; GENERALISED SPECIES OF STRUCTURES(†)

see [23, 25]

Coherence (idea):

Lany((Lany H#)G

∼= Lany((Lany H#)G#

∼= (Lany H#) (Lany G#)

(†) The coherence isomorphisms can also be given formally (compare [3])

in a theory of Lawvere’s generalized logic [4], providing a logical view ofcoherence. The coherence laws can be then established elementwise.

PROGRAMME: Extend generalized logic to a type theory within whichthe coherence laws may be also established formally.

Linear modelsSubstitution operations on

linear species[6]

finite linear orders and monotone bijections↑

For X, Y ∈ SetL with Y(∅) = ∅:

1. (X • Y)(L) =∑

P∈LinPart(L)

X(P) ×∏

; composition of ordinary generating series [Joyal]see [6]

2. (X • Y)(L) =∑

P∈Part(L)

X(P) ×∏

; composition of exponential generating series [Foata]see [1, 14]

Substitution tensor productson linear species

For X, Y ∈ SetL:

1. (X • Y)(`)

∫P∈LX(P) ×

∫ `p(p∈P) ∏

Y(`p) × L(⊕p∈P `p, `)

arises from the general theory for T thefree monoidal completion, noting that L is(equivalent to) the free monoidal categoryon one object.

2. (X • Y)(`)

∫ P∈L

X(P) ×

∫ `p(p∈P)∏

Y(`p) × Monbij( ∨

`p, `)

where∨

p∈P `p has underlying set⊎

p∈P `p

ordered by (p, x) ≤ (p ′, x ′) iff either p = p ′

and x ≤ x ′, or p < p ′ and x = min(`p) andx ′ = min(`p ′).

; GENERALISED LINEAR SPECIES OF STRUCTURES

Substitution tensor productson linear species

For X, Y ∈ SetL:

1. (X • Y)(`)

∫P∈LX(P) ×

∫ `p(p∈P) ∏

Y(`p) × L(⊕p∈P `p, `)

arises from the general theory for T thefree monoidal completion, noting that L is(equivalent to) the free monoidal categoryon one object.

2. (X • Y)(`)

∫ P∈L

X(P) ×

∫ `p(p∈P)∏

Y(`p) × Monbij( ∨

`p, `)

where∨

p∈P `p has underlying set⊎

p∈P `p

ordered by (p, x) ≤ (p ′, x ′) iff either p = p ′

and x ≤ x ′, or p < p ′ and x = min(`p) andx ′ = min(`p ′).

; GENERALISED LINEAR SPECIES OF STRUCTURES

Mixed models

Example: DILL = Dual Intuitionistic Linear Logicsee [12]

Γ ;∆ ` t : τ

ppintuitionistic(cartesian)

context

// linear(symmetric monoidal)

context

CUT RULE:

x1 : σ1, . . . , xm : σm ; y1 : τ1, . . . , yn : τn ` t : α

Γ ; −−− ` ui : σi (1 ≤ i ≤ m)

Γ ;∆j ` vj : τj (1 ≤ j ≤ n)

Γ ;∆1, . . . , ∆n ` t [ ui/xi,vj/yj

]1≤i≤m,1≤j≤n : α

I NEW FEATURE absent in mathematical examples

MATHEMATICAL MODEL

The category of (mono-sorted) mixed contexts Mis the free symmetric monoidal category over thefollowing symmetric monoidal theory:

��

L // I I ⊕ Ioo

a commutativemonoid

weakening& contractionlinear variables canbecome intuitionistictype theoretically:

Γ ; x,∆ ` t

Γ, x ; ∆ ` t

CONTEXT INDEXING

In fact

M F? _oo

??��// F × F in Cat

;C��M

&&MMMMMMMMMMMM

��===

==== +3

��M F? _oo��

;C��

induces

M ◦ M //Cat

;C��M

· · · ⊕ L ⊕ · · · ⊕ I ⊕ · · · � // . . . × M × . . . × F × . . .

CONTEXT INDEXING

In fact

M F? _oo

??��// F × F in Cat ;C

��M&&MMMMMMMMMMMM

��===

==== +3

��M F? _oo��

;C��

induces

M ◦ M //Cat ;C

��M

· · · ⊕ L ⊕ · · · ⊕ I ⊕ · · · � // . . . × M × . . . × F × . . .↓M

Mixed substitution tensor product

For X, Y ∈ SetM:

(X • Y)(D)

∫C∈MX(C) ×

∫∆∈M(C) ∏

i∈|C|

Y(∆i) × M(⊕C(∆),D

NEW FEATURE

⊕C��

I Monoids = Mixed operadsgeneralise and combine Lawveretheories and (symmetric) operads

I A combinatorial model of DILL. . . and more

A unifying framework

zzuuuuuuuuuuuu

��

S //Cat

C(−)

wwCat ;C��C dom

// Cat

contexts shapes

ΣΓ ��C

DΓ : CΓ → CSΓ

I Substitution tensor product ?

For X, Y ∈ SetC◦ :

(X • Y)(C)

∫ Γ∈C

X(Γ) ×

∫∆∈CΓ (limi∈S∆

Y(DΓ∆)i

)× [ΣΓ∆,C]

Further generalisations

D-sorted(

Cat ;C��D

zzuuuuuuuuuuuu

��

S //Cat

C(−)

uuCat ;C��C

′ dom// Cat

ΣΓ ��C

C� � // C ′

<<zzzzzzzzzzzzzz ��

SΓσΓ��

I Substitution tensor product ?For X, Y ∈

C◦)D:

(X • Y)τ(C)

∫ Γ∈C

Xτ(Γ) ×

∫∆∈CΓ (limi∈S∆

Yσ∆(i)(DΓ∆)i

)× [ΣΓ∆,C]

PROGRAMME

Obtain a substitution tensor product from(cartesian (compare [5, 10])) monad structure on

S(X) = S ;C��X

NB: eS(1) ∼= C

induced by structure on C.

Developments

I Mathematical theory of substitution� typed vs. untyped� homogeneous vs. heterogeneoussee [18]

� single variable vs. simultaneoussubstitution

� cartesian, linear, mixed, etc. substitution� specification and algorithms� syntax and semantics

I Reduction of type theory to algebra� admissibility of cut� second-order theories� dependent sorts

I Equational and inequational theories� free constructions� modularity� rewriting

I Structural combinatorics� Generalised species

− cartesian closed− differentialsee [23]

structure

� Groupoids and generalised analyticfunctorssee [27]

I Profunctors� Groupoids and strong (= †) compact

closuresee [28]

� Annihilation/creation operatorssee [28, 32] (and also [17, 29])

Programme

I Categories of contexts as free monoidaltheories

I Comparison with/extension to Kelly’sclubs[5, 10]

I Generalized logic type-theoretically andcoherence

I Extraction of syntactic theory from modeltheory

I Applications� Theory of operads� Combinatorics� Domain Theorysee [27]

References

[1] P. Cartier and D. Foata. Problemes Combinatoiresde Commutation et de Rearrangements. LectureNotes in Mathematics, vol. 85, 1969.

[2] B.J. Day. On closed categories of functors. LectureNotes in Mathematics, vol. 137, pages 1–38, 1970.

[3] G.M. Kelly. On the operads of J.P. May. Manuscript,1972. (Reprints in Theory and Applications ofCategories, No. 13, pages 1-13, 2005.)

[4] F.W. Lawvere. Metric spaces, generalized logic andclosed categories. Rendiconti del SeminarioMatematico e Fisico di Milano, vol. XLIII,pages 135–166, 1973. (Reprints in Theory andApplications of Categories, No. 1, pages 1-37, 2002.)

[5] G.M. Kelly. On clubs and doctrines. In CategorySeminar, Lecture Notes in Mathematics, vol. 420,pages 181–256, 1974.

[6] A. Joyal. Une theorie combinatoire des seriesformelles. Advances in Mathematics, vol. 42,pages 1–82, 1981.

[7] G.B. Im and G.M. Kelly. A universal property of theconvolution monoidal structure. Journal of Pure andApplied Algebra, vol. 43, pages 75–88, 1986.

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[23] M. Fiore. Mathematical models of computationaland combinatorial structures. Invited address forFoundations of Software Science and ComputationStructures (FOSSACS 2005) at the European JointConferences on Theory and Practice of Software(ETAPS), Lecture Notes in Computer Science,vol. 3441, pages 25-46, 2005.

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[32] M. Fiore. An axiomatics and a combinatorial modelof creation/annihilation operators and differentialstructure. Invited talk at the Categorical QuantumLogic Workshop (CQL), Computing Laboratory,Oxford University, 2007. (Available from〈http://www.cl.cam.ac.uk/∼mpf23/〉.)

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