Towards a Mathematical Theory of Substitution

Towards a Mathematical Theory of Substitution Marcelo Fiore Computer Laboratory University of Cambridge Invited talk at CT2007 20.VI.2007 /1 Subst...
Author: Tracy York
7 downloads 0 Views 369KB Size
Towards a Mathematical Theory of Substitution Marcelo Fiore Computer Laboratory University of Cambridge

Invited talk at CT2007 20.VI.2007

/1

Substitution Examples: I Logic/algebra/rewriting.

t [ u/x ]

t [ u1/x1 , . . . ,un /xn ]

I Type theory.

T [ t/x ] I Formal languages.

w0 X 1 w1 . . . X n wn _

Xi 7→ Wi



w0 W 1 w1 . . . W n wn

/2

I Proof theory.

O ··· O P P

P1 · · · P n

n

1

P _ 

O O P ··· P 1

n

P I Structural combinatorics. • •







• • •

◦ •





◦ ◦

• •





◦ ??? ?? ?? ??  • •



• •





• • •









/3

Substitution Aspects I syntactic vs. semantic models I homogeneous vs. heterogeneous I typed vs. untyped I variables vs. occurrences I single vs. simultaneous I binding I higher order I algorithms

/4

Substitution Aspects I syntactic vs. semantic models I homogeneous vs. heterogeneous I typed vs. untyped I variables vs. occurrences I single vs. simultaneous I binding I higher order I algorithms

Plan ANALYSE

substitution from a foundational

standpoint in a variety of scenarios and SYNTHESISE

a mathematical theory.

/4-a

Algebraic theories Clone of operations ≡ ≡ ≡

{ Cn ×(Cm )n → Cm | ··· }

Lawvere theories Finitary monads Monoids for the substitution tensor product

/5

Algebraic theories Clone of operations ≡ ≡ ≡

{ Cn ×(Cm )n → Cm | ··· }

Lawvere theories Finitary monads Monoids for the substitution tensor product

Substitution tensor product F on Set finite sets and functions ↑

Endofin (Set) ' SetF    V(n) = n

Id, ◦ ↔ V, •

  (X • Y)(n) = Rk∈F X(k) × (Yn)k /5-a

Cartesian mono-sorted substitution monoid structure for the substitution tensor product on SetF

Examples: I Finitary algebraic syntax.

Σ = signature of operators with arities in 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 property of Σ? by structural recursion (; correct substitution algorithm). see e.g. [31]

/6

I Lambda-calculus syntax.

Λ(n) = { λ-terms with free variables in n }

                

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

x∈n

t1 , t2 ∈ Λ(n)

x ∈ Λ(n)

t1 (t2 ) ∈ Λ(n)

t ∈ Λ n ] {x}



(†)

λx. t ∈ Λ(n)

(†) SUBTLETY: α-equivalence

/7

I Lambda-calculus syntax.

Λ(n) = { λ-terms with free variables in n }

                

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

x∈n

t1 , t2 ∈ Λ(n)

x ∈ 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) simultaneous substitution

/7-a

I Clone of maps.

The clone of maps hC, Ci on an object C in a cartesian category is given by hC, Ci(n) = [Cn , C] SUBSTITUTION STRUCTURE :

• n

/ [Cn , C] : i 7→ πi

• [Cn , C] ×L [Cm , C]n LLL

∼ LL = L

LL%

/ [Cm , C] 9 r r r r r rr◦ r r r

[Cn , C] × [Cm , Cn ]

/8

The substitution tensor product ...

1



free cartesian category on one generator ↑ / F◦   CC CC C ∼

= Y

CC Y (−)

Lan



CC = CC ! ,

/ F Set y yy y< y − •Y yy yy yya yyy y yy yyyhY,−i y |y yy y y F

Set

hY,Zi(n) = [Y n ,Z]

... is closed

/9

Algebraic theories in SetF syntax with variable binding Example: Σλ = { app : 2, abs : V }

NB: V = y(1)

/10

Algebraic theories in SetF syntax with variable binding Example: Σλ = { app : 2, abs : V }

NB: V = y(1)

Then,

and

SetF o

Σλ (X) = X2 +XV

∼ Λ (Σλ )? V = µX. V + X2 + XV =

see [16, 31]

/10-a

Algebraic theories in SetF syntax with variable binding Example: Σλ = { app : 2, abs : V }

NB: V = y(1)

Then, SetF o

and

Σλ (X) = X2 +XV

∼ Λ (Σλ )? V = µX. V + X2 + XV =

see [16, 31]

NB: X2 → X ≡ { (Xn)2 → Xn | · · · }

XV → X ≡ { X(n + 1) → Xn | · ·O · } O

α-equivalence

F





SetO F

(−)×V a (−)V = ((−)+1)



(−)+1

F

/



  ◦ /



SetF /10-b

Λ is (universally characterised as) the free Σλ -algebra on V , and its substitution structure is derived by parameterised structural recursion as follows: Σλ (Λ) • Λ

/ Σλ (Λ • Λ)



s

Λ •O Λ V•Λ

Σλ (s)

/ Σλ (Λ) 

3/ Λ g g g g gg g g g g ∼ ggg

gg= g g g g gg g g g g g

; correct (capture-avoiding) simultaneous substitution algorithm

see [16, 31]

/11

Λ is (universally characterised as) the free Σλ -algebra on V(†) , and its substitution structure is derived by parameterised structural recursion as follows: Σλ (Λ) • Λ 

/ Σλ (Λ • Λ)

s

Σλ (s)

/ Σλ (Λ) 

3/ Λ g g g g gg g g g g ∼ ggg

Λ •O Λ V•Λ

(‡)

gg= g g g g gg g g g g g

; correct (capture-avoiding) simultaneous substitution algorithm

see [16, 31]

(†)

yields an induction principle

(‡)

SUBTLETY :

see [19, 31]

pointed strength !

capture avoidance

/11-a

SETTING :

General theory I/C U Σ



7C 

a monoidal closed category 

an endofunctor with a U-strength: Σ(X) ⊗ Y

σX,(I→Y)

/ Σ(X ⊗ Y)

/12

SETTING :

General theory I/C U Σ



7C 

a monoidal closed category 

an endofunctor with a U-strength: Σ(X) ⊗ Y MODELS :

σX,(I→Y)

/ Σ(X ⊗ Y)

(Σ, σ)-monoids. ΣX

an algebra

ξ

X⊗X

m

 /Xo

e

a monoid

I

such that Σ(X) ⊗ X

σX,e

/ Σ(X ⊗ X)

Σm

/ ΣX ξ

ξ⊗X



X⊗X



m

/X /12-a

The free Σ-algebra and free monoid constructions Σ-alg(C ) o µX. C + ΣX o

>

/

o



C C



>

/ Mon(C ) / µX. I + C ⊗ X

unify to (Σ, σ)-Mon(C ) O a

CX



S

where S(C) = µX. I + C ⊗ X + ΣX NB: The initial (Σ, σ)-monoid has underlying object S0 = µX. I + ΣX = Σ? I. see [26, 30]

/13

Initial-algebra semantics with substitution The unique (Σ, σ)-monoid homomorphism from the initial (Σ, σ)-monoid provides an initial-algebra semantics that is both compositional and respects substitution. see [16, 24]

/14

Example: Lambda calculus. For D C DD in a cartesian closed category, the clone of maps hD, Di has a canonical Σλ -algebra structure hD, Di × hD, Di 

hD, Di × hD, DD i V hD, Di H

HHH

∼H = H

HH $

/ hD, Di O ∼ =

/ hD, D × DD i

/ hD, Di v: v vv v vv v v

hD, DD i

making it into a Σλ -monoid for the canonical pointed strength. The induced initial-algebra semantics amounts to the standard interpretation of the λ-calculus. see [16, 19]

/15

Single-variable and simultaneous substitution The theory of monoids in SetF for the substitution tensor product is enriched algebraic for the cartesian closed structure. MonV,• (SetF ) is (equivalent to) the category of algebras X with operations

subject to

XV+1 → X

and

1 → XV

. . . 4 axioms . . .

see [16]

/16

Second-order syntax with variable binding and substitution Example: F Σλ -Mon(Set ) O a



7 SetF S ooa oooo o o Sλ oo oooo wooo g ooo (−) oo

SetN For M ∈ SetN ,

∼ V+M f = f • Sλ (M) f + Σλ (Sλ M) f Sλ (M)

see [22]

/17

f can be syntactically presented as follows: Sλ (M) x∈n

f var(x) ∈ Sλ (M)(n) f t1 , t2 ∈ Sλ (M)(n) f app(t1 , t2 ) ∈ Sλ (M)(n)

 f t ∈ Sλ (M) n ] { x }  f abs (x)t ∈ Sλ (M)(n)

f t1 , . . . , tk ∈ Sλ (M)(n) f T [t1 , . . . , tk ] ∈ Sλ (M)(n)

(up to α-equivalence)

T ∈ M(k)



and the monoid multiplication structure f • Sλ (M) f → Sλ (M) f Sλ (M)

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

/18

the action of Sλ enriches over SetF with respect to its cartesian closed structure, and we obtain a further substitution structure MOREOVER ,

f M f e e Sλ (M) × (Sλ N) → Sλ (N)

that amounts to SECOND - ORDER SUBSTITUTION for metavariables. As usual, this arises by universal properties and is given by parameterised structural recursion; yielding a correct substitution algorithm. see [31]

/19

Many-sorted contexts For S a set of sorts, let F[S] be the free cocartesian category on S. The substitution tensor product on Set given as follows:

S

/ F[S]◦   DD D ∼



=

Y

/

 F[S] S

  X      

is

S

SetF[S]

xx x Lan − •Y xx Y #DD ∼ xx DD = x D" x x | + SetF[S] q

def X •Y

That is, (X • Y)σ (Γ ) =

Z ∆∈F[S]

Xσ (∆) ×

Y

Yτ (Γ )

τ∈∆

see [19, 23, 24, 26]

/20

Simply typed lambda calculus, algebraically Let T be a set of base types, and let T be its closure under 1, *, =>. Consider Σ

? SetF[T ]

T

induced by (†) app(σ,τ) : (σ=>τ, σ) → τ  (σ,τ) (‡) abs : (σ)τ → σ=>τ proj1(σ,τ) : (σ, τ) → σ , pair(σ,τ) : (σ, τ) → σ*τ

proj2(σ,τ) : (σ, τ) → τ

ter : () → 1

(†) Xσ=>τ × Xσ → Xτ

(‡) Xτ Vσ → Xσ=>τ

see [19, 23]

/21

Furthermore, let CC be the following equational theory 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

/22

Furthermore, let CC be the following equational theory for Σ-monoids: (β)

F : [σ]τ, T : [ ]σ



app abs (x : σ)F[var(x)] , T [ ]

` =

F [T [ ]] : τ



informally: (λx. F)T = F [ T/x ]

(η)

F : [ ](σ=>τ) abs (x : σ)app F[ ], var(x)

` =



F[ ] : σ=>τ λx. Fx = F  x 6∈ FV(F)

/22-a

(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

/23

Then

o

Σ-Mon/CC  

⊥ eq

o7 Γ -alg o o oo o7 o o o o

o o o o o

o o

oo ooo

o o

o o

K (†)

/ Σ-Mon KKK eKK KKK



KK KKa KKK%

KK 

KK 

Set

F [T ] T

and the Lawvere theory associated to the initial Σ-Mon/CC is the free cartesian closed category on T . (†) induced by CC

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

/24

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

`

For N[X] = n∈N X , let M ∈ Set defined from the context of (β) as n

Mτ (σ) = { F }

,

 N[ T ] T

be

Mσ () = { T }

and empty otherwise. The terms of (β) correspond to global elements f 1 −→ S(M)

that induce functors

Σ-Mon ' S-alg −→ (−)M -alg  F[ T ] T over Set as follows f

f

XM 

S(X) 

X

7→

f

S(X)S(M) 

S(X)1 

X

see [30]

/25

Dependent sorts For a small category C, let FC ' (Set )fp be the free finite colimit completion of C. C◦

The substitution tensor product on (Set is given as follows:

F C C◦

)

C◦

C





/ (FC)◦   EE E ∼ =

Y

/

}} } X } }} } }~ }

SetF C

yy y Lan − •Y yy Y #EE ∼ yy EE = y E" y y | , SetF C q

def X •Y

That is, (X • Y)C (Γ ) =

Z ∆∈F C

XC (∆) × lim Yp∆ D (Γ ) D∈El(∆)

/26

PROGRAMME

The various developments of the previous slides carry over to this more general setting. Following Makkai [13] , after Lawvere [9] and Otto [11] , the syntactic theory is considered for simple categories (= skeletal and one-way, with finite fan-out). This amounts to extending the theory to incorporate DEPENDENT SORTS. NB: The limit in the substitution tensor product accounts for the heavy dependency required in the substitution operation. see [26]

/27

General theory : Idea For T a 2-monad on CAT, consider C I

 / TC  ??? ?

C

∼ =

Y

)

Lan

∼ Y #?? =

??  ,

− •Y

/ c TC

  X     

def



X •Y

Tc C q !

equipped with a T -algebra structure Examples: I T = identity I T = free cartesian completion I T = free finite limit completion

/28

I T = free monoidal completion I T = free symmetric monoidal

completion

/29

   I T = free monoidal completion   I T = free symmetric monoidal

completion

    x

the T -algebra structure on Tc C is given by Day’s tensor product [2, 7]

&

(I, •)-monoids =

planar

  

 symmetric 

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

QUESTION :

Are there applications of the previous theory to the theory of operads!?

/29-a

More generally : From substitution to composition For (T, η, µ) a 2-monad on CAT, consider A IB

B

ηB

F

/ TB   AA AA

∼ = G

yB

)



/ c TB ~ ~ ~ def Lan − •G ~ G#AA = ∼ ~ ~ AA ~ A ~~ , Tc C q

F •G

with respect to T -algebra structures  c τC : T T C → Tc C such that yC : (T C, µC ) → (Tc C, τC ) and LanyX (h) : (Tc C, τC ) → (Tc D, τD ) for all h : (T C, µC ) → (Tc D, τD )

NB: The above can be axiomatised further.

[Hyland, Gambino, Fiore] (see also [24]) /30

Kleisli bicategory T C _ / B

T B _ / A

T C _ / A I T = identity

; profunctors

I T = free symmetric monoidal completion

; J OYAL SPECIES OF

STRUCTURES [6, 8]

as the endomorphisms of 1

arise

; GENERALISED SPECIES OF STRUCTURES

see [23, 25]

Coherence (idea): #

Lany (Lany H ) G  # # ∼ = Lany (Lany H ) G ∼ (Lany H# ) (Lany G# ) = #

/31

Kleisli bicategory T C _ / B

T B _ / A

T C _ / A I T = identity

; profunctors

I T = free symmetric monoidal completion

; J OYAL SPECIES OF

STRUCTURES [6, 8]

as the endomorphisms of 1

arise

; 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 of coherence. The coherence laws can be then established elementwise. PROGRAMME :

Extend generalized logic to a type theory within which the coherence laws may be also established formally. /31-a

Linear models Substitution operations on linear species[6] finite linear orders and monotone bijections ↑

For X, Y ∈ SetL with Y(∅) = ∅: X Y 1. (X • Y)(L) = X(P) × Y(`) P∈LinPart(L)

` ∈P

; composition of ordinary generating series [Joyal] 2. (X • Y)(L) =

X

P∈Part(L)

X(P) ×

Y

see [6]

Y(`)

` ∈P

; composition of exponential generating series [Foata]

see [1, 14]

/32

Substitution tensor products on linear species For X, Y ∈ SetL : 1. (X • Y)(`) =

Z P∈L

X(P) ×

Z `p (p∈P) Y

Y(`p ) × L(⊕p∈P `p , `)

p∈P

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

/33

Substitution tensor products on linear species For X, Y ∈ SetL : 1. (X • Y)(`) =

Z P∈L

X(P) ×

Z `p (p∈P) Y

Y(`p ) × L(⊕p∈P `p , `)

p∈P

arises from the general theory for T the free monoidal completion, noting that L is (equivalent to) the free monoidal category on one object. 2. (X • Y)(`) =

Z P∈L

X(P) ×

Z `p (p∈P)

Y

p∈P

W

Y(`p ) × Monbij

_

`p , `



p∈P

U

where p∈P `p has underlying set p∈P `p ordered by (p, x) ≤ (p 0 , x 0 ) iff either p = p 0 and x ≤ x 0 , or p < p 0 and x = min(`p ) and x 0 = min(`p 0 ). ;

GENERALISED LINEAR SPECIES OF STRUCTURES /33-a

Mixed models Example: DILL = Dual Intuitionistic Linear Logic

see [12]

Γ; ∆ ` t : τ intuitionistic p (cartesian) context

linear (symmetric monoidal) context /

CUT RULE :

x1 : σ 1 , . . . , x m : σ m ; y 1 : τ 1 , . . . , y n : τ n ` t : α Γ ; − ` ui : σi

(1 ≤ i ≤ m)

Γ ; ∆ j ` v j : 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

/34

MATHEMATICAL MODEL

The category of (mono-sorted) mixed contexts M is the free symmetric monoidal category over the following symmetric monoidal theory: a commutative monoid 0

L

   • /Io

linear variables can become intuitionistic type theoretically:

I⊕I

weakening & contraction

Γ ; x, ∆ ` t Γ, x ; ∆ ` t

/35

CONTEXT INDEXING

Mo

?_ F

?1    

/F×F

in Cat

induces M◦ ··· ⊕ L ⊕ ··· ⊕ I ⊕ ···



M

/ Cat

/

... × M × ... × F × ...

/36

CONTEXT INDEXING

In fact ?1      _ /F×F MMo M ? F=  ;C MMM ==  3+ MMM == + MMM==  M&   _ ? F Mo

in Cat ;C  M

induces M◦ ··· ⊕ L ⊕ ··· ⊕ I ⊕ ···

M

/ Cat ;C 



M

/

... × M × ... × F × ... ↓ M

/36-a

Mixed substitution tensor product For X, Y ∈ SetM : (X • Y)(D) Z C∈M Z ∆∈M(C) Y  = X(C) × Y(∆i ) × M ⊕C (∆), D i∈|C|

NEW FEATURE

M(C) ⊕C



M I Monoids = Mixed operads

generalise and combine Lawvere theories and (symmetric) operads I A combinatorial model of DILL

. . . and more

/37

A unifying framework shapes

contexts uC

uu u uu u C uu u u uz  w CatC; / Cat  dom

S

/ Cat◦

D +3 C(−)

C

DΓ : CΓ → CSΓ

CΓ ΣΓ



C I Substitution tensor product ? For X, Y ∈ Set : C◦

(X • Y)(C) Z Γ ∈C Z ∆∈CΓ   lim Y(DΓ ∆)i × [ΣΓ ∆, C] = X(Γ ) × i∈S∆

/38

Further generalisations SΓ 

σΓ

D uC

Cat;C  0 C C



D-sorted  ◦ C ; Cat  z< z z D zz

z z zz z zz z z

uu u uu u C uu u u uz  u / Cat



S

/ Cat◦

D +3 C(−)

dom

/ C0

CΓ ΣΓ



C0 I Substitution tensor product ?  C◦ D : For X, Y ∈ Set (X • Y)τ (C) Z Γ ∈C Z ∆∈CΓ   lim Yσ∆(i) (DΓ ∆)i × [ΣΓ ∆, C] Xτ (Γ ) × = i∈S∆

/39

PROGRAMME

Obtain a substitution tensor product from (cartesian (compare [5, 10]) ) monad structure on e S(X) = S;C X induced by structure on C.

e ∼ C NB: S(1) =

/40

Developments I Mathematical theory of substitution 

typed vs. untyped



homogeneous vs. heterogeneoussee [18]



single variable vs. simultaneous substitution



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

/41

I Equational and inequational theories 

free constructions



modularity



rewriting

I Structural combinatorics 

Generalised species  − cartesian closed  − differential



see [23]



structure

Groupoids and generalised analytic functorssee [27]

I Profunctors 



Groupoids and strong (= †) compact closuresee [28] Annihilation/creation operators

see [28, 32] (and also [17, 29])

/42

Programme I Categories of contexts as free monoidal

theories

I Comparison with/extension to Kelly’s

clubs[5, 10]

I Generalized logic type-theoretically and

coherence

I Extraction of syntactic theory from model

theory

I Applications 

Theory of operads



Combinatorics



Domain Theorysee [27]

/43

References ` [1] P. Cartier and D. Foata. Problemes Combinatoires ´ de Commutation et de Rearrangements. Lecture Notes in Mathematics, vol. 85, 1969. [2] B.J. Day. On closed categories of functors. Lecture Notes 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 of Categories, No. 13, pages 1-13, 2005.) [4] F.W. Lawvere. Metric spaces, generalized logic and closed categories. Rendiconti del Seminario Matematico e Fisico di Milano, vol. XLIII, pages 135–166, 1973. (Reprints in Theory and Applications of Categories, No. 1, pages 1-37, 2002.) [5] G.M. Kelly. On clubs and doctrines. In Category Seminar, Lecture Notes in Mathematics, vol. 420, pages 181–256, 1974. ´ [6] A. Joyal. Une theorie combinatoire des series formelles. Advances in Mathematics, vol. 42, pages 1–82, 1981. /44

[7] G.B. Im and G.M. Kelly. A universal property of the convolution monoidal structure. Journal of Pure and Applied Algebra, vol. 43, pages 75–88, 1986. ` de [8] A. Joyal. Foncteurs analytiques et especes ´ ´ structures, Combinatoire enum erative, Lecture Notes in Mathematics, vol. 1234, pages 126–159, 1986. [9] F.W. Lawvere. More on graphic toposes. Cah. de Top. et Geom. Diff, vol. 32, pages 5–10, 1991. [10] G.M. Kelly. On clubs and data-type constructors. In Applications of Categories in Computer Science, London Mathematical Society Lecture Note Series, vol. 177, 1992. [11] J. Otto. Complexity doctrines. Ph.D. thesis, Department of Mathematics and Statistics, McGill University, Montreal, 1995. [12] A. Barber. Dual Intuitionistic Linear Logic. Technical report ECS-LFCS-96-347, University of Edinburgh, School of Informatics, 1996. [13] M. Makkai. First order logic with dependent sorts, with applications to category theory. Manuscript, 1997. (Available from hhttp://www.math.mcgill.ca/∼makkai/i.) /45

[14] F. Bergeron, G. Labelle, and P. Leroux. Combinatorial Species and Tree-Like Structures. Encyclopedia of Mathematics, vol. 67. Cambridge University Press, 1998. [15] J. Baez and J. Dolan. Higher-dimensional algebra III: n-categories and the algebra of opetopes. Advances in Mathematics, vol. 135, pages 145–206, 1998. [16] M. Fiore, G. Plotkin and D. Turi. Abstract syntax and variable binding. In Proceedings of the 14th Annual IEEE Symposium on Logic in Computer Science (LICS’99), pages 193–202, 1999. [17] J. Baez and J. Dolan. From finite sets to Feynman diagrams. Mathematics unlimited - 2001 and beyond, 2001. [18] M. Fiore and D. Turi. Semantics of name and value passing. In Proceedings of the 16th Annual IEEE Symposium on Logic in Computer Science (LICS’01), pages 93–104, 2001. [19] M. Fiore. Semantic analysis of normalisation by evaluation for typed lambda calculus. In Proceedings of the 4th International Conference on Principles and Practice of Declarative Programming (PPDP 2002), pages 26–37, 2002. /46

[20] B.J. Day and R. Street. Lax monoids, pseudo-operads, and convolution. In Diagrammatic Morphisms and Applications, Contemporary Mathematics, vol. 318, pages 75–96, 2003. [21] G.L. Cattani and G. Winskel. Profunctors, open maps and bisimulation. Mathematical Structures in Computer Science, Vol. 15, Issue 03, pages 553–614, 2005. [22] M. Hamana. Free Σ-monoids: A higher-order syntax with metavariables. In Proceedings of the 2nd Asian Symposium on Programming Languages and Systems (APLAS 2004), Lecture Notes in Computer Science, vol. 3202, pages 348–363, 2005. [23] M. Fiore. Mathematical models of computational and combinatorial structures. Invited address for Foundations of Software Science and Computation Structures (FOSSACS 2005) at the European Joint Conferences on Theory and Practice of Software (ETAPS), Lecture Notes in Computer Science, vol. 3441, pages 25-46, 2005. [24] J. Power and M. Tanaka. A unified category-theoretic formulation of typed binding signatures. In Proceedings of the 3rd ACM SIGPLAN workshop on /47

Mechanized reasoning about languages with variable binding, pages 13–24, 2005. [25] M. Fiore, N. Gambino, M. Hyland and G. Winskel. The cartesian closed bicategory of generalised species of structures. To appear in the Journal of the London Mathematical Society. (Available from hhttp://www.cl.cam.ac.uk/∼mpf23/i.) [26] M. Fiore. On the structure of substitution. Invited address for the 22nd Mathematical Foundations of Programming Semantics Conference (MFPS XXII), DISI, University of Genova, 2006. (Available from hhttp://www.cl.cam.ac.uk/∼mpf23/i.) [27] M. Fiore. Analytic functors and domain theory. Invited talk at the Symposium for Gordon Plotkin, LFCS, University of Edinburgh, 2006. (Available from hhttp://www.cl.cam.ac.uk/∼mpf23/i.) [28] M. Fiore. Adjoints and Fock space in the context of profunctors. Talk given at the Cats, Kets and Cloisters Workshop (CKC in OXFORD), Computing Laboratory, Oxford University, 2006. (Available from hhttp://www.cl.cam.ac.uk/∼mpf23/i.) [29] J. Morton. Categorified algebra and quantum mechanics. Theory and Applications of Categories, /48

Vol. 16, No. 29, pages 785–854, 2006. [30] M. Fiore and C.-K. Hur. Equational systems and free constructions. In International Colloquium on Automata, Language and Programming (ICALP 2007), Lecture Notes in Computer Science, vol. 4596, pages 607-619, 2007. [31] M. Fiore. A mathematical theory of substitution and its applications to syntax and semantics. Invited tutorial for the Workshop on Mathematical Theories of Abstraction, Substitution and Naming in Computer Science, International Centre for Mathematical Sciences (ICMS), Edinburgh, 2007. [32] M. Fiore. An axiomatics and a combinatorial model of creation/annihilation operators and differential structure. Invited talk at the Categorical Quantum Logic Workshop (CQL), Computing Laboratory, Oxford University, 2007. (Available from hhttp://www.cl.cam.ac.uk/∼mpf23/i.)

/49