towards a mathematical theory of substitutionmpf23/talks/ct2007.pdf · substitution aspects i...
TRANSCRIPT
Towards a
Mathematical Theory
of Substitution
Marcelo FioreComputer Laboratory
University of Cambridge
Invited talk at CT200720.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.w0X1w1 . . . Xnwn Xi 7→ Wi
_
��w0W1w1 . . . Wnwn
/2
I Proof theory.
P1 · · · Pn
P
OP1
· · · OPn
_
��
OP1 · · ·
OPn
P
I Structural combinatorics.
•
• •
•
••
•
•
• ◦
◦◦
◦
•
•
◦
••
•
�
��???
????
??
•
• ••
•
••
•
•
•
••
•
•
/3
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
Plan
ANALYSE substitution from a foundationalstandpoint in a variety of scenarios andSYNTHESISE a mathematical theory.
/4
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
Plan
ANALYSE substitution from a foundationalstandpoint in a variety of scenarios andSYNTHESISE a mathematical theory.
/4-a
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
/5
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
/5-a
Cartesian mono-sortedsubstitution
monoid structure for the substitutiontensor product on Set
F
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]
/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
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
/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
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
/7-a
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
/8
The substitution tensor product ...
free cartesian categoryon one generator
↑1 � � //
Y
,,
∼=
F◦ � � //
Y(−)
CCCC
C
!!CCC
CCC a
SetF
− •Y
||yyyyyyyyyyyyy
SetF
Lan∼=
〈Y,−〉
<<yyyyyyyyyyyyy
〈Y,Z〉(n) = [Yn,Z]
... is closed
/9
Algebraic theories in SetF
syntax with variable binding
Example: Σλ = { app : 2, abs : V } NB: V = y(1)
Then,
SetF
Σλ(X) = X2+XV
oo
and
(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
F◦
(−)+1
��
� � //Set
F
a(−)×V��
F◦ � � //Set
F
(−)V = ((−)+1)∗
OO
/10
Algebraic theories in SetF
syntax with variable binding
Example: Σλ = { app : 2, abs : V } NB: V = y(1)
Then,
SetF
Σλ(X) = X2+XV
oo
and
(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
F◦
(−)+1
��
� � //Set
F
a(−)×V��
F◦ � � //Set
F
(−)V = ((−)+1)∗
OO
/10-a
Algebraic theories in SetF
syntax with variable binding
Example: Σλ = { app : 2, abs : V } NB: V = y(1)
Then,
SetF
Σλ(X) = X2+XV
oo
and
(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
F◦
(−)+1
��
� � //Set
F
a(−)×V��
F◦ � � //Set
F
(−)V = ((−)+1)∗
OO
/10-b
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
/11
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
/11-a
General theorySETTING:
I/C
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.
ΣX
ξ
��an algebra
X ⊗ X m// X Ie
oo a monoidsuch that
Σ(X) ⊗ X
ξ⊗X
��
σX,e // Σ(X ⊗ X)Σm // ΣX
ξ
��X ⊗ X m
// X
/12
General theorySETTING:
I/C
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.
ΣX
ξ
��an algebra
X ⊗ X m// X Ie
oo a monoidsuch that
Σ(X) ⊗ X
ξ⊗X
��
σX,e // Σ(X ⊗ X)Σm // ΣX
ξ
��X ⊗ X m
// X
/12-a
Thefree Σ-algebra and free monoid
constructionsΣ-alg(C )
//> Coo // Mon(C )>
oo
µX.C + ΣX C�oo � // µX. I + C ⊗ X
unify to(Σ, σ)-Mon(C )
��a
C
OO
S
XX
whereS(C) = µX. I + C ⊗ X + ΣX
NB: The initial (Σ, σ)-monoid has underlyingobject S0 = µX. I + ΣX = Σ?I.
see [26, 30]
/13
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]
/14
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〉
OO
〈D,D〉V // 〈D,D〉
∼=HH
HH
$$HHH
H
〈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]
/15
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]
/16
Second-order syntaxwith variable binding
and substitution
Example:
Σλ-Mon(SetF)
��a
SetF
OO
wwoooooooooooSλ
SS
SetN
g(−)77ooooooooooo a
For M ∈ SetN,
Sλ(M) ∼= V + M • Sλ(M) + Σλ(SλM)
see [22]
/17
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]
/18
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]
/19
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:
S
X
������
����
�
X •Y
S� � //
Y++
∼=
F[S]◦ � � //
Y#
DDD
""DDD
DD
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]
/20
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]
/21
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)
)
/22
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)
)
/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
Γ -alg
��
(†)
77ooooooooooooo
77oooooooooooooΣ-Mon/CC ⊥� �
eq// Σ-Mon
oo
%%KKKKKKKKKK
a
(Set
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.]
/24
(†)Example: The parallel pair induced by (β).
For N[X] =∐
n∈NXn, let M ∈
(Set
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
S(X)
��X
7→
XfM
��
S(X)S(fM)
��S(X)1
��X
see [30]
/25
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:
C◦
X
~~}}}}
}}}}
}
X •Y
C◦ � � //
Y,,
∼=
(FC)◦ � � //
Y#
EEE
""EEE
EE
SetF C
def− •Y
||yyyyyyyyyy
SetF C
Lan∼=
That is,
(X • Y)C(Γ) =
∫∆∈F C
XC(∆) × limD∈El(∆)
Yp∆D(Γ)
/26
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]
/27
General theory : Idea
For T a 2-monad on CAT, consider
C
X
������
����
�
X •Y
C //
Y,,
I
))
∼=
TC� � //
Y#
????
��???
?
TC
def− •Y
������
����
�
TC
Lan∼=
!!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 monoidalcompletion
xx
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!?
/29
I T = free monoidal completion
I T = free symmetric monoidalcompletion
xx
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!?
/29-a
More generally :
From substitution to composition
For (T, η, µ) a 2-monad on CAT, considerA
F
������
����
�
F •G
BηB //
G,,
IB
))
∼=
TB� � yB //
G#
AAAA
AAA
AA
TB
def− •G
��~~~~
~~~~
~
TC
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])
/30
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.
/31
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.
/31-a
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) ×∏
`∈P
Y(`)
; composition of ordinary generating series [Joyal]see [6]
2. (X • Y)(L) =∑
P∈Part(L)
X(P) ×∏
`∈P
Y(`)
; composition of exponential generating series [Foata]see [1, 14]
/32
Substitution tensor productson linear species
For X, Y ∈ SetL:
1. (X • Y)(`)
=
∫P∈LX(P) ×
∫ `p(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)∏
p∈P
Y(`p) × Monbij( ∨
p∈P
`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
/33
Substitution tensor productson linear species
For X, Y ∈ SetL:
1. (X • Y)(`)
=
∫P∈LX(P) ×
∫ `p(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)∏
p∈P
Y(`p) × Monbij( ∨
p∈P
`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
/33-a
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
/34
MATHEMATICAL MODEL
The category of (mono-sorted) mixed contexts Mis the free symmetric monoidal category over thefollowing symmetric monoidal theory:
0
������
�
L // I I ⊕ Ioo
a commutativemonoid
•
weakening& contractionlinear variables canbecome intuitionistictype theoretically:
Γ ; x,∆ ` t
Γ, x ; ∆ ` t
/35
CONTEXT INDEXING
In fact
1
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
/36
CONTEXT INDEXING
In fact
1
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
/36-a
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
M(C)
⊕C��
M
I Monoids = Mixed operadsgeneralise and combine Lawveretheories and (symmetric) operads
I A combinatorial model of DILL. . . and more
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A unifying framework
C
D +3
zzuuuuuuuuuuuu
C
��
S //Cat
◦
C(−)
wwCat ;C����C dom
// Cat
contexts shapes
CΓ
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]
/38
Further generalisations
D-sorted(
Cat ;C����D
)◦
C
D +3
zzuuuuuuuuuuuu
C
��
S //Cat
◦
C(−)
uuCat ;C����C
′ dom// Cat
CΓ
ΣΓ ��C
′
C� � // C ′
<<zzzzzzzzzzzzzz ��
SΓσΓ��
D
I Substitution tensor product ?For X, Y ∈
(Set
C◦)D:
(X • Y)τ(C)
=
∫ Γ∈C
Xτ(Γ) ×
∫∆∈CΓ (limi∈S∆
Yσ∆(i)(DΓ∆)i
)× [ΣΓ∆,C]
/39
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.
/40
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
/41
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])
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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]
/43
References
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[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.
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[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.
[8] A. Joyal. Foncteurs analytiques et especes destructures, Combinatoire enumerative, Lecture Notesin Mathematics, vol. 1234, pages 126–159, 1986.
[9] F.W. Lawvere. More on graphic toposes. Cah. deTop. et Geom. Diff, vol. 32, pages 5–10, 1991.
[10] G.M. Kelly. On clubs and data-type constructors. InApplications 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, McGillUniversity, Montreal, 1995.
[12] A. Barber. Dual Intuitionistic Linear Logic. Technicalreport 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〈http://www.math.mcgill.ca/∼makkai/〉.)
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[14] F. Bergeron, G. Labelle, and P. Leroux.Combinatorial Species and Tree-Like Structures.Encyclopedia of Mathematics, vol. 67. CambridgeUniversity Press, 1998.
[15] J. Baez and J. Dolan. Higher-dimensional algebra III:n-categories and the algebra of opetopes. Advancesin Mathematics, vol. 135, pages 145–206, 1998.
[16] M. Fiore, G. Plotkin and D. Turi. Abstract syntax andvariable binding. In Proceedings of the 14th AnnualIEEE Symposium on Logic in ComputerScience (LICS’99), pages 193–202, 1999.
[17] J. Baez and J. Dolan. From finite sets to Feynmandiagrams. Mathematics unlimited - 2001 andbeyond, 2001.
[18] M. Fiore and D. Turi. Semantics of name and valuepassing. In Proceedings of the 16th Annual IEEESymposium on Logic in ComputerScience (LICS’01), pages 93–104, 2001.
[19] M. Fiore. Semantic analysis of normalisation byevaluation for typed lambda calculus. In Proceedingsof the 4th International Conference on Principles andPractice of Declarative Programming (PPDP 2002),pages 26–37, 2002.
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[20] B.J. Day and R. Street. Lax monoids,pseudo-operads, and convolution. In DiagrammaticMorphisms and Applications, ContemporaryMathematics, vol. 318, pages 75–96, 2003.
[21] G.L. Cattani and G. Winskel. Profunctors, openmaps and bisimulation. Mathematical Structuresin Computer Science, Vol. 15, Issue 03,pages 553–614, 2005.
[22] M. Hamana. Free Σ-monoids: A higher-order syntaxwith metavariables. In Proceedings of the 2nd AsianSymposium on Programming Languages andSystems (APLAS 2004), Lecture Notes in ComputerScience, vol. 3202, pages 348–363, 2005.
[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.
[24] J. Power and M. Tanaka. A unified category-theoreticformulation of typed binding signatures. InProceedings of the 3rd ACM SIGPLAN workshop on
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Mechanized reasoning about languages withvariable binding, pages 13–24, 2005.
[25] M. Fiore, N. Gambino, M. Hyland and G. Winskel.The cartesian closed bicategory of generalisedspecies of structures. To appear in the Journal ofthe London Mathematical Society. (Available from〈http://www.cl.cam.ac.uk/∼mpf23/〉.)
[26] M. Fiore. On the structure of substitution. Invitedaddress for the 22nd Mathematical Foundations ofProgramming Semantics Conference (MFPS XXII),DISI, University of Genova, 2006. (Available from〈http://www.cl.cam.ac.uk/∼mpf23/〉.)
[27] M. Fiore. Analytic functors and domain theory.Invited talk at the Symposium for Gordon Plotkin,LFCS, University of Edinburgh, 2006. (Availablefrom 〈http://www.cl.cam.ac.uk/∼mpf23/〉.)
[28] M. Fiore. Adjoints and Fock space in the contextof profunctors. Talk given at the Cats, Kets andCloisters Workshop (CKC in OXFORD), ComputingLaboratory, Oxford University, 2006. (Availablefrom 〈http://www.cl.cam.ac.uk/∼mpf23/〉.)
[29] J. Morton. Categorified algebra and quantummechanics. Theory and Applications of Categories,
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Vol. 16, No. 29, pages 785–854, 2006.
[30] M. Fiore and C.-K. Hur. Equational systems andfree constructions. In International Colloquium onAutomata, Language and Programming (ICALP2007), Lecture Notes in Computer Science,vol. 4596, pages 607-619, 2007.
[31] M. Fiore. A mathematical theory of substitution andits applications to syntax and semantics. Invitedtutorial for the Workshop on Mathematical Theoriesof Abstraction, Substitution and Naming in ComputerScience, International Centre for MathematicalSciences (ICMS), Edinburgh, 2007.
[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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