Generalised Species of Structures: Cartesian Closed and Differential Structure (Preliminary Notes)
Marcelo Fiore∗ Computer Laboratory University of Cambridge
May and December 2003
Abstract Species of structures. Joyal species of structures [9] are an abstract categorical formalisation of a type of We generalise Joyal’s notion of species of struc- unlabelled combinatorial structures. tures and develop their combinatorial calculus. In In its basic form, species are defined as functors particular, we provide operations for their composi- B / Set where B is the category of finite sets and tion, addition, multiplication, pairing and projection, bijections, and Set is the category of sets and func- abstraction and evaluation, and differentiation; devel- tions. Such a species P is to be thought of as pro- oping both the cartesian closed and linear structures viding, for each finite set of tokens T, the set P(T) of of species. P-structures (i.e., structures of type P) built form to-
kens in T together with, for each bijective renaming
¢¡ £
σ : T / T ¤ between sets of tokens, a bijective cor-
¡
£ ¤ Contents respondence ( ) ¥ σ : P(T ) / P(T ) between the · respective sets of P-structures subject to the laws
1. Categorical background 2
¥ ¦ ( ) id = id ¥¨§©¦ : P(T) / P(T) · 2. The calculus of generalised species 3 and
2.1. The bicategory of species ...... 3
¥ ¤ ¥ ¤ 2.2. Addition and multiplication ...... 4 ( ) ¥ σ σ = ( ) (σ σ ) : P(T ) / P(T )
2.3. Linear structure ...... 5 · · · ◦
2.4. Differential structure ...... 5 ¤ ¡
for all T and T ¡ / T / T in . £ B £ B 2.5. Cartesian closed structure ...... 6 ∈ An important aspect of the theory of species is that 2.6. Graph models of the lambda calculus . 7 it provides a calculus for the structural manipulation 2.7. Higher-order differential structure . . . 7 of combinatorial structures. Indeed, Joyal provided a 2.8. Operators on generalised Fock space . 8 calculus of species with operations such as composi- tion, addition, multiplication, derivation, etc. subject to the usual algebraic laws. Furthermore, he showed that the calculus is a powerful tool for extracting com- binatorial information by associating generating series to species in such a way that the combinatorial opera-
∗Research supported by an EPSRC Advanced Research Fel- tions on species correspond to the usual respective op- lowship. erations on power series. (See [3] for a full treatment.)
1 Outline. We generalise the notion of species and in- when these isomorphisms are equalities. A monoidal vestigate the resulting calculus. functor F : (C, , I, a, l, r) / (C , , I , a , l , r ) ⊗ 0 ⊗0 0 0 0 0 The basis of the generalisation is the observation is a functor F : C / C0 equipped with a morphism that the category of finite sets and bijections B is equiv- I / F(I) and a natural transformation with compo- alent to the free symmetric strict monoidal category nents F(A) F(B) / F(A B) subject to coherence ⊗ 0 ⊗ on one generator !1 (i.e., the category of finite car- axioms [7, 11]. We have a strict monoidal functor if dinals and bijections). In this view, a species is a these morphisms are identities. functor !1 / Set and we are naturally led to con- A symmetric (strict) monoidal category is a (strict)
sider A-species, for A a small category, as functors monoidal category equipped with a natural isomor- !A / Set. More generally, for small categories phism c, called the symmetry, with components c : A and B, we introduce (A, B)-species as profunctors A B / B A satisfying further coherence ax- ⊗ ⊗ !A / B (i.e., functors !A / B), where !A is the ioms [10]. A symmetric (strict) monoidal functor free symmetric strict monoidal completion of A (and between symmetric (strict) monoidal categories is a B is the functor category [Bo, Set]bof B-variable sets). (strict) monoidal functor that satisfies a further coher- Such a species P provides, for each !A-object of to- ence axiom associated to the symmetries [7, 11]. kbens A (given by a finite sequence of A-objects), the We write Cat for the category of small cate- B-variable set P(A) of P-structures over A. This no- gories and functors, and SMCat for the category tion of species encompasses many of the combinatorial of small symmetric strict monoidal categories and species considered in the literature, including permu- strict monoidal functors. The forgetful functor tationals [9, 2] and partitionals [16]; see also [15]. SMCat / Cat : (C, , I) / C has a left adjoint ⊗ From this standpoint, the calculus of generalised !( ) : Cat / SMCat that maps a small category into species is developed within the framework of gener- its symmetric strict monoidal completion. An explicit alised logic [11]. In particular, we provide operations description of !C is given by the category with objects
for the composition, addition, multiplication, pairing c n consisting of finite sequences [ ] ( N) of ob-
and projection, abstraction and evaluation, and differ- h i ∈ "
! c , d = k = `
! "# $ jects of C with C [ ] [ ] iff
entiation of species. All these operations are shown to h i h i ∅ 6
c c
and morphisms [ ] / [ 0] given by pairs
satisfy the expected algebraic laws. h i h i
σ, f σ S [ ] % consisting of a permutation
The treatment of differentiation necessarily calls for h i ∈
&
f : c c and a sequence of maps [ / 0 ] in C. the consideration of both linear and closed structure, as h i (Composition is essentially given pointwise modulo differentiation operators are maps permutation
hom [X, Y] / hom X, lin [X, Y]
&
σ , f σ, f = σ σ, f f
% 0 [ 0] % [ ] 0 [ 0 ] where hom and lin respectively represent the full and h i ◦ h i ◦ h ◦ i linear function spaces. This development in the con- and identities are given pointwise.) The symmetric text of generalised species is also included. strict monoidal structure of !C is given by concate- nation with unit the empty sequence and the obvious 1. Categorical background symmetry. We recall the basic notions needed throughout the The symmetric strict monoidal completion comes paper. equipped with canonical natural coherent equivalences as follows
Monoidal categories. A monoidal category is a tu- ' ple (C, , I, a, l, r) where C is a category, ⊗ ⊗ / ! is a bifunctor C C / C, I is an object of C, 1 0 × () / [ ]
and a, l, r are natural isomorphisms with components h i
¨ a : (A B) C / A (B C), l :
⊗ ⊗ ⊗ ⊗
( ( ⊕ I A / A, r : A I / A subject to coher- !C !C / !(C + C ) ×
⊗ ⊗ '
( ( ( ence axioms [10]. We have a strict monoidal category (C , C ) / ! (C ) ! (C ) q ⊗ q 2 Presheaves. For a small category C, we write C for Permutationals [9, 2] are CP-species for CP the • the functor category [Co, Set] of presheaves on C and groupoid of finite cyclic permutations. natural transformations between them, and let y ) b de- Partitionals [16] are B -species for B the groupoid : c [ , c] • ∗ ∗ note the Yoneda embedding C / C / C . of non-empty finite sets. For (C, , I) a (symmetric) monoidal category, the ⊗ presheaf category C acquires a (symmetric)b monoidal Further examples that fit into generalised species are
structure via Day’s tensor product construction [5, 8] coloured species and permutationals [14], and species + given, for X * , X bC, as on graphs and digraphs [13].
∈
,.-0/ ,21
)
+ * * + + ) * + X * X = ∈ bX (c ) X (c ) y (c c ) Basic general examples of species follow. ⊗ × × ⊗
R Presheaves on C are essentially species 0 5 / C, whereb is the coend construction whose definition and • basic properties can be found in [12, Chapter X]. The whilst presheaves on !C also correspond to species
R 5 unit for Day’s tensor product is y ) (I). C / 1.
⊗
5 ) The Yoneda embedding y 5 is a C / !C species. For small categories A and B, an (A, B)-profunctor, •
b
5 ) indicated as A / B, is a functor A / B. Small cat- The species ) : !C / C is defined as (C) = • egories, profunctors, and natural transformations be- !!C [ [ ] ] , C .
tween them form a bicategory [1]. Theb profunctor h h i i ) Thespecies S : C 5 / C is defined as composition V U : A / C of U : A / B and • ◦
V : B / C is given by )
S ) (C) = y (c) (2)
3
4 ,
(V U)(a)(c) = ∈ V(b)(c) U(a)(b) (1) 9 ◦ × X∈ R
with identities y ) : C / C.
/ /
: 5 : 4 The species E 4 : A / B is defined by E (A) = For more on the bicategory of profunctors see [1, • 1. 11, 4].
2. The calculus of generalised species 2.1. The bicategory of species
For small categories A and B, an (A, B)-species of We introduce the bicategory (Especes` de Struc- structures is a profunctor !A / B. In particular, ES tures) of generalised species of structures. (A, 1)-species are referred to as A-species. The no- tation P : A 5 / B is used to indicate that P is an
Composition. For species P : 5 / and
(A, B)-species. A B 5 Q : B 5 / C, the composition Q P : A / C is de- Structures in P(A)(b), for a species P : A 5 / B, ◦
are pictorially represented as follows fined as
; 4 A (Q P)(A)(c) = ∈ 5 Q(B)(c) P#(A)(B) ◦ × R P where b
P#(A)(B)
3
<>= ;BA
:@?
5
3 3
3 3 ; Concrete examples of combinatorial species = ∈ ∈ ; P(A )(b) !A A , A × abound in the literature. ∈ ∈
R Q N
6 7 8
Joyal’s k-sorted species [9, 3] are * 1 - One can visualise the structures in (Q P)(A)(c) as • ◦ species. follows P
3 A 2.2. Addition and multiplication
A C ...... Each hom-category [A, B] acquires a commuta- ES P tive rig (= ring without negatives) structure given by the addition and multiplication of species. b · · · · · ·
Addition. For P, Q : A D / B, the addition P + Q :
Q A D / B is defined by Q P c ◦ (P + Q)(A)(b) = P(A)(b) + Q(A)(b)
A A We give explicit descriptions of sample pre- and
post-compositions with a species P : A D / B.
E
C
H
G H y F
h i D For b B, the composite species A D / B / 1
• ∈ Q D is isomorphic to the species P C : A / 1 defined as P P + Q P + Q
P C (A)() = P(A)(b) (3) b b
I
E
N N D D For X A, the composite 0 D / A / B, is as More generally, for X B and P : A / B (i I),
• ∈ ∈ ∈
N N D the linear combination N X P : / is defined follows O A B ∈
b by b
J K ∼ D P (P X) [ ] (b) = ∈ P(A)(b) A S K A, X [ ]
◦ h i × h i
N N N N N
N X P (A)(b) = X (b) P (A)(b) O R ∈ O ∈ × b where S K : !A / A is as in (2). P P
Addition together with the species 0 : A D / B de- D Identities. The identity species I L : / is defined b C C fined as as 0(A) = 0
I L (C)(c) = !C [c] , C h i satisfy commutative monoid laws: C (P + Q) + R =∼ P + (Q + R) ∼ ∼ [ ] P + 0 = P P + Q = Q + P
h•i D I for P, Q, R : A D / B. Further, for P, Q : A / B and
c R : C D / A, we have
D D For P : A D / B, Q : B / C, and R : C / D, (P + Q) R =∼ (P R) + (Q R) we have canonical natural coherent isomorphisms as ◦ ◦ ◦ follow Multiplication. For P, Q : A D / B, the multiplication
(R Q) P =∼ R (Q P) P Q : A D / B is defined by ·
◦ ◦ ∼ ∼ ◦ ◦ M
P I K = P = I P ◦ ◦ (P Q)(A)(b)
·
JP.Q JSR K D (4)
establishing the associativity of composition and the ∈
U T U = P(A T )(b) Q(A )(b) !A [A A , A] unit laws of identities. × × ⊗
4 R
Y o
e _ and V : K _ / B C is the matrix V U : K / × •
Ao C defined by Y]\ Y[Z ×
V e U (K)(a, c)
•
Z \
^
fgih g h j c
∈ c ∈ k
= V(K )(b, c) U(K k )(a, b) !K [K K , K] W V × × ⊗
(Compare with the composition of profunctors (1) and
V W the multiplication of species (4).) Using (3), we obtain X · R
the familiar formula for matrix multiplication
l l#m
l#m
^
f
e ^
V U ∈ V U ^ g n.o o
= g n.o That is, using (3), • g ·
for a A and c C. R
^ ^ (P Q) ^ = P Q ∈ ∈
· ⊗ The associativity of matrix multiplication and the _ unit laws with respect to the identity matrix ∆ p : C / for b B. b ∈ o defined as _ A A Multiplication together with the species 1 : A / B × defined by ∆ p (C)(a0, a) = !C [ ] , C A [a, a0] h i × 1(A)(b) = !A [ ] , A hold
h i ∼
q e q W e (V U) = (W V) U satisfy commutative monoid and distrib utive laws: • • • •
∼ ∼
q e p
U ∆ = U = ∆f U • •
(P Q) R =∼ P (Q R) o o _ where U : _ / , V : / , and W : · · · · K A B a K B C
∼ ∼ o × ×o _ P 1 = P P Q = Q P K _ / C D. Further, for U : K / A B (i I),
· · · × o × ∈ _ P 0 =∼ 0 P (Q + R) =∼ (P Q) + (P R) and V r : K / B C (j J), we have
· · · · × ∈
l
a a a
a ∼
r e r e r
r V U = V U
s t g o s t _ for P, Q, R : _ / . Further, for P, Q : / and A B A B ∈ • ∈ ∈ × • R : C _ / A, we have P P P 2.4. Differential structure (P Q) R =∼ (P R) (Q R) · ◦ ◦ · ◦ We introduce differentiation in the context of gen- eralised species and establish its basic properties. 2.3. Linear structure Higher-order differential operators are further consid- o ered in Subsection 2.7. We refer to a C _ / A B species as a × Differentiation. For P : _ / and a , the (C-parameterised) A B-matrix. The transpose of an m A B A
× ∈ _ o o o u P
A B-matrix U : C _ / A B is the B A -matrix partial derivative : A / B is defined as u t× o o o × × m U : _ / ( ) defined as C B A u × P (A)(b) = P(A [a] )(b) u ⊗ h i
Ut(C)(b, a) = U(C)(a, b) Y
a
` a b c v More generally, for a species P : _ / we
C a A
`
ab c _ define the transposition P d : C / A accord- Qd ing to the permutation σ S ` by
∈ Q
V
c c
` `
P d (C)(a , . . . , a ) = P(C)(a , . . . , a )
d
d
w
V w0x
Matrix multiplication. The matrix multiplication (or X o linear composition) of the matrices U : K _ / A B ×
5 For all P, Q : A y / B and X B, we have the C ∈
following basic properties
z z z
z b
z|{ z|{ z|{ z|{ P =∼ P
0 0
z z z
z|{ z|{
z|{ (P + Q) =∼ ( P) + ( Q)
z
z P P z|{
z|{ (XP) = X P
c c z
z|{ (E) = E
( ) ( ) q • q • P , P h i
and the Leibniz’s rule c
z z z
z|{ z|{ z|{ (P Q) =∼ P Q + P Q For i I, the projection species π : y / · · · C C ∈ ∈
is defined as y
P : y Q :
Further, for / and / , we have the P A B B C π (C)(c) = ! (c) , C chain rule C [ ]
∈ hq i
z z z
~
P
z z|{ z|{ ∼ C
(Q P) = ∈ ~ (Q) P P
} } ◦ ◦ · ~ R where a A and c C. [ ( )] ∈ ∈ hq • i The differential application (or Jacobian matrix)
o π y dP : A y / A B of P : A / B is defined as
× z c
(dP)(A)(a, b) = z|{ P (A)(b) The usual laws of pairing and projections are satisfied The basic properties of partial derivatives translate up to isomorphism:
in terms of differentials; in particular, the chain rule
amounts to the identity ∼
y π P = P : / (k I)
C C
◦ h i ∈ ∈ ∼ ∼ π P P y
= : / d(Q P) = d(Q) P dP C C
◦ ◦ • h ◦ i ∈ ∈ P For a species P : y / , one may introduce
Note that in the presence of cartesian structure the
A B
∈ o o
y
y j-differentials d P : / (j I) as P : / k I
differentials d ( )
A A B A A B P ∈ × ∈ ∈ × ∈ follows are derivable from the differential dP : y /
A
P z o P ∈ {
z , as (d P)(A)(a, b) = P (A)(b) A B ∈ × P q
P ∼ However, as we show below, these arederivable. d (P) = π d(P) (k I)
◦ ∈
2.5. Cartesian closed structure for all P : y / . A B ∈
AbstractionP and evaluation. For P : C + A y / B,
We describe the cartesian closed structure of o y the abstraction λ P : C / !A B is defined as species. ×
(λ P)(C)(A, b) = P(C A)(b)
Pairing and projections. There is exactly one species
⊕ y
y / 0. More generally, for P : / (i I), the
C C C C ∈
pairing P : y / is defined as C C
h i ∈ ∈ P (C)(c) P
h i ∈
}2
P
= ∈ P (C)(c ) [c, (c )] C
∈ × ∈ q λP
=∼ P(C)(Rc ) where c = (cP) A b q
6
o
and the evaluation ε : (!A B) + A / B by of reflections between finite ordinals, with morphisms ×
given by natural isomorphisms.
ε (M)(b)
o ∈ × ∈ (5) 2.7. Higher order differential structure = !(!Ao B)[ [(A, b)] , F ] × h i ! (!Ao B) + A [F A, M] We relate the linear and cartesian closed structures, × × ⊕ and introduce an operator which is shown to satisfy the R M basic properties of differentiation. Linear and cartesian closed structure. For a matrix
• ⊕ • o U : C / A B we define the species U : C + A / F A × B as e [( , )] U M b
h • • i ( )( ) ¢
ε ¤
§ ¥0¦
b ∈ ∈ ∈ e= U(C)(a, b) !A [a] , A !(C + A) [C A, M] × h i × ⊕
o For P : C / !A B, we write υ (P) for the
× This construction internalises as an embedding of ma-
composite ε P π , π : C + A / B. The usual ◦ h ◦ i trices into exponentials as laws of abstraction and evaluation are satisfied up to R
o o
isomorphism: ı ¨ = λ (I]o ) : A B / !A B × × × ∼ given explicitly by υ(λP) = P : C + A / B
∼ o
λ(υP) = P : C / !A B ı (U)(A, b)
× ¤ ∼ o = ∈ !(A B) [(a, b)] , U !A [a] , A We further note the following interesting commuta- × h i × h i o tion property between abstraction and linear composi- Indeed,Rfor all P : C /A B, wehave that × σ (12)(3)
tion: for the permutation , ∼ o
ı P = λ (P) : C / !A B ◦ ×
∼
(λ Q) P = λ Q (P π ) (6) • • ◦ Further, the embedding commutese with identities and composition; since, for
o o for all P : K / B C and Q : K + A / C D.
× × ¢ o o o
B
© ` = π π : (B C) + (A B) / A C • × × × 2.6. Graph models of the lambda calculus we have that It does not take much to construct models of the ∼ ` P, Q = P © Q lambda calculus. Indeed, for a small category C, the ◦ h i •
o C o o free !( ) ( )-algebra C on C in at yields P : Q : × for all K / A B and K / B C, and
¡ × × ¢
∼ o
C C∼ C in ı ∆ = λ (I ) : 0 / !A A
e ES ◦ ×
¢ ¢
o ¢ ∼
B ª ¨ B Further, the final !( ) ( )-coalgebra U in Gpd yields ı ` = ı π , ı π m e × e ◦ h ◦ ◦ i ◦
an equivalence
¢ ¢
B ª ¨ for m = λ ε π , ε π the internal U o ◦ h o ◦ i o U U in composition (!B C) + (!A B) / !A C. ' ES × × × Interestingly, U has the following explicit description: Differentiation operator. We introduce the differenti-
o o o
the objects are given by the class of planar trees de- ation operator D : !A B / !A A B as × × × scribed by ω-chains follows
o o o o o
£ {0} O O (n N) D (F)(A, a, b) = !(!A B) [(A [a] , b)] , F ⊥ / ⊥ / · · · ⊥ / ⊥ / · · · ∈ × h ⊗ h i i 7
This operator is linear, as and
³ ¬
γ(U, V) = ∈ γ ³ (U, V) D =∼ δ For u, v A the followingR hold o o o ∈ for δ the (!A B) (!A eA B)-matrix given by
× × × × ∼
µ ¶ ¶ µ ¬ o ¯ α γ = γ α + A[v, u] ∆ δ U, A, a, b o A a , b , U • • ×
( ( )) = !A B ( [ ] ) ∼ ∼
¶ ¶ µ µ ¶ ¶ µ × ⊗ h i αµ α = α α γ γ = γ γ • • • • and internalises differential application since Further, for non-empty A, we also have that
∼ o
¬ ¬ d « P = υ (D λ P) : C + A / A B (7)
◦ × ∼ ¯ ¬ o α γ = γ α + ∆ • • ×
for all P : C + A / B.
³ ³ ³ Further, it is constant on linear maps, as Let A ³ = α , C = γ and A = α, C = γ.
o o
³ For A ³ , C : !A B / !A B, we have that × ×
∼ o o o f f e e
¬® ¯ ¬ ı λ (π ° ) : !
D = A B / A A B ∼ · ◦ × × × ³
A (F)(A, b) = ( ³ F)(A)(b) · It follows that and
∼ ³
∼ o ³ F A, b F χ A b
¬ d(I ¬ ) = ∆ : A / A A C ( )( ) = ( )( ) × ·
¯ ³ where F(A)(b) = ε ¬® (F A)(b) and χ (A)(b) = and we have from (6) and (7) above that, for σ the ⊕
¬ (A)(a) permutation (12)(3), I . Further, for u, v A the following hold
∈ ±
± ∼
² ¬ « ² °
D λ ¬ P U = λ d (P) (U π ) ∼
¶ ¶ µ ¬ ¯ µ o A C = C A + A[v, u] I ◦ • • ◦ ◦ ◦ ×
∼ ∼
¶ ¶ µ µ ¶ ¶ µ A µ A = A A C C = C C
o for all P : C + A / B and U : C / D A. This ◦ ◦ ◦ ◦ × identity corresponds to the β-rule of the differential and, for non-empty A, we also have that lambda calculus [6].
∼ ¯ ¬ o A C = C A + I 2.8. Operators on generalised Fock space ◦ ◦ ×
Annihilation and creation. For a A define the an- Acknowledgements. I am grateful to Nicola Gambino ∈ nihilation and creation operators as the (!Ao B) and Martin Hyland for discussions and joint work on
o × × ³ (!A B)-matrices α ³ and γ given by the cartesian closed structure of analytic functors be- × tween groupoids. I am also grateful to Prakash Panan-
α ³ (U, (A, b)) = δ(U, (A, a, b)) gaden and Jon Wolf for conversations. and References
γ ³ (U, (A, b)) ´
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