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

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