Cofibrant complexes are free

0.21 MB·English

by  Francois Metayer

Cofibrant complexes are free
Franc¸ois M´etayer ∗
7
0 February 2, 2008
0
2

n
a
J Abstract

5
Wedefineanotionofcofibrationamong∞-categoriesandshowthatthecofibrantobjects
2
are exactly the free ones, that is those generated by polygraphs.

]
T
C 1 Introduction
.
h
Polygraphs[Bur91,Bur93], alsoknownas computads [Str76, Pow91] are structuredsystems
t
a of generators for ∞-categories, extending the familiar notion of presentation by generators
m
and relations beyond monoids or groups, and have recently proved extremely well-adapted
[ to higher-dimensional rewriting [Gui06a, Gui06b].

They also lead to a simple definition of a homology for ∞-categories [M´et03, LM06],
1
v based on the following construction: a polygraphic resolution of an ∞-category C is a pair
6 (S,p) where
4
• S is a polygraph, generating a free ∞-category S∗;
7
1 • the morphism p:S∗ →C is a trivial fibration (see 3.1).
0
7 S gives rise to an abelian complex ZS, whose homology only depends on C, so that we may
0 define a polygraphic homology by
/
h
t Hp∗ol(C)=def H∗(ZS).
a
m
Here the mainpropertyoffree∞-categoriesis thatthey arecofibrant. Inotherwords,given
: a polygraph S and a trivial fibration p : D → C, any morphism f : S∗ → C lifts to a
v
i morphism g :S∗ →C (figure 1).
X
r
a
D
}}g}}}}}}>> (cid:15)(cid:15)p
∗
S // C
f
Figure 1: lifting
The purpose of the present work is to prove the converse, namely that all cofibrant
∞-categoriesarefreelygeneratedbypolygraphs,thusestablishingasimple,abstractcharac-
terization of the free objects, otherwise defined by a rather complex inductive construction.
We firstgiveabriefreviewofthe basiccategoriesinplay(section2): Glob,Compland
Pol stand respectively for the category of globular sets, ∞-categories (or “complexes”) and
polygraphs. Then we investigate trivial fibrations and cofibrations (section 3). In section 4,
we reduce our theorem to the fact that the full subcategory of Compl whose objects are
∗E´quipe PPS Universit´e Paris 7 - CNRS
1
free is Cauchy-complete, in other words that all its idempotents split. This is proved in
appendix B.
LetussketchtheCauchy-completenessargumentinthesimplercaseofmonoids: thus,let
Mondenotethecategoryofmonoids,andFmonthefullsubcategoryofMonwhoseobjects
are the free monoids. It is well-known that a submonoid of a free monoid is not necessarily
free itself. However, if M = S∗ is the free monoid on the alphabet S and h : M → M is
an idempotent endomorphism of M, then the submonoid Fix(h) = {m ∈ M | h(m) = m}
of fixpoints of h is free, which easily leads to a splitting of h in Fmon, hence to the fact
that Fmon is Cauchy-complete. Here the keypoint is to find a set of generators of Fix(h)
without non-trivial relations in M. A simple way to build such a set is by considering the
subset S ⊂ S of those s ∈ S such that h(s) = usv where h(u) = h(v) = 1. Then we
1
define T = {h(s) | s ∈ S }. It turns out that the obvious inclusion T∗ → M sends T∗
1
isomorphically to Fix(h), as shown by the existence of a a retraction M →T∗.
Now the same ideas carry into higher dimensions, with ∞-categories instead of monoids
andpolygraphsinsteadofgeneratingsets. Thegeneralcaseinvolvesadditionaltechnicalities,
due to the presence of higher dimensional compositions. Of particular importance is the
notion of context, defined and explored in appendix A.
ManythankstoAlbertBurroni,YvesLafontandKrzysztofWorytkiewicz,whohavebeen
a great help in the preparationof the present work.
2 Basic categories
2.1 Globular sets
Let O be the small category defined as follows:
• the objects of O are integers 0,1,...;
• the arrows are generated by composition of s ,t : n → n+1, n ∈ N subject to the
n n
following equations
s ◦s = t ◦s ,
n+1 n n+1 n
s ◦t = t ◦t .
n+1 n n+1 n
As a consequence, O(m,n) has exactly twoelements if m<n, namely s =s ◦···◦s
m,n n−1 m
and t = t ◦···◦t . O(m,n) = ∅ if m > n, and contains the unique element id if
m,n n−1 m m
m=n.
Definition 1 A globular set is a presheaf on O.
In other words, a globular set is a functor from Oop to Sets. Globular sets and natural
transformations form a category Glob. The Yoneda embedding
O→Glob
takes each integer n to the standard globe O[n]. We still denote by s ,t :O[n] →O[n+1]
n n
the morphisms of globular sets representing the corresponding arrows from n to n+1.
Let X be a globular set and p an integer, the set X(p) will be denoted by X , and its
p
elements called cells of dimension p or p-cells. Hence O[n] has exactly two p-cells for p<n,
exactly one n-cell, and no p-cells for p > n. Let ∂O[n] be the globular set with the same
cells as O[n] except for (∂O[n]) =∅, and
n
i :∂O[n]→O[n]
n
the canonical injection: ∂O[n] has two p-cells for p<n and no other cells.
Let us point out a few facts about i :
n
• s ◦i =t ◦i ;
n n n n
• there are unique maps s and t such that s =i ◦s and t =i ◦t ;
n n n n+1 n n n+1 n
b b b b
2
• the following diagram is a pushout:
∂O[n] in // O[n]
in sn
c
(cid:15)(cid:15) (cid:15)(cid:15)
O[n] //∂O[n+1]
ctn
Now let X be a globular set, Yoneda’s lemma yields a natural equivalence
X ∼=Glob(O[n],X) (1)
n
hence in particular s , t give rise to a double sequence of maps
n n
σ ,τ :X ⇐X
n n n n+1
satisfying the boundary conditions:
σ ◦σ = σ ◦τ ,
n n+1 n n+1
τ ◦σ = τ ◦τ .
n n+1 n n+1
Whenever m<n, we set σ =σ ◦···◦σ and τ =τ ◦···◦τ . Let 0≤i<n,
m,n m n−1 m,n m n−1
we say that the n-cells x,y ∈ X are i-composable if τ x = σ y, a relation we denote by
n i,n i,n
x⊲ y.
i
If u ∈ X and σ (u) = x, τ (u) = y, x and y are respectively the source and the
n n−1 n−1
target of u, which we simply denote by u : x → y. Likewise, if σ u = x and τ u = y, we
i,n i,n
shall write u:x→ y. In case u:x→y and v :x→y, we say that u, v are parallel, which
i
we denote by u k v (see figure 2). Any two 0-cells are also considered to be parallel. Let
u
$$
• •
::
v
Figure 2: parallel cells
P (X) denote the set of orderedpairs of paralleln-cells in X. We get a natural equivalence
n
P (X)∼=Glob(∂O[n+1],X) (2)
n
similarto(1). Theequivalences(1)and(2)assertthat,foreachn,thefunctorsX 7→X and
n
X 7→P (X)fromGlobtoSetsarerepresentable,therepresentingobjectsbeingrespectively
n
O[n] and ∂O[n+1].
2.2 Complexes
Recall that an ∞-category is a globular set C endowed with
• a product u∗ v :x→z defined for all u:x→y and v :y →z in C ;
n−1 n
• a product u∗ v : x∗ y → z ∗ t defined for all u : x → z and v : y → t in C with
i i i n
i<n−1 and u⊲ v;
i
• a unit 1 (x):x→x defined for all x∈C .
n+1 n
These operations satisfy the conditions of associativity, left and right unit, and exchange:
• (x∗ y)∗ z =x∗ (y∗ z) for all x⊲ y ⊲ z in C with i<n;
i i i i i i n
• 1 (x) ∗ u = u = u∗ 1 (y) for all u : x → y in C with i < n, where 1 =
n,i i i n,i i n n,i
1 ◦1 ◦···◦1 ;
n n−2 i+1
3
• (x∗ y)∗ (z∗ t) = (x∗ z)∗ (y∗ t) for all x,y,z,t∈ C with i < j < n and x ⊲ y,
i j i j i j n i
x⊲ z, y ⊲ t.
j j
Throughout this work, complex means ∞-category. Let C, D be complexes. A morphism
f : C → D is a morphism of the underlying globular sets preserving units and products.
ComplexesandmorphismsbuildacategoryCompl,andthereisanobviousforgetfulfunctor
Compl → Glob. Its left adjoint Glob → Compl associates to each globular set the free
complex generated by it. Note that Glob is a topos of presheaves and that the forgetful
functor Compl → Glob is finitary monadic over Glob. Hence Compl is complete and
cocomplete, and we shall take limits and colimits in Compl without further explanations
(see also [Bat98, Str00]).
By restricting a complex C to its cells of dimension ≤n, we get an n-category
C| :C ⇐C ⇐···⇐C .
n 0 1 n
Thisn-categorycanbeextendedtoacomplexC(n)byadjoiningunitstoC| inalldimensions
n
>n:
C(n) :C ⇐···⇐C ⇐C ⇐··· .
0 n n
Let us call C(n) the n-skeleton of C. It will be convenient to define C(−1) as the initial
complex 0 with no cells. There is a canonical inclusion
j :C(n) →C(n+1).
n
Here again j denotes the unique morphism 0→C(0).
−1
The following result is an easy consequence of the definitions:
Lemma 1 Any complex C is the colimit of its n-skeleta:
C(−.1) j−1 //C(0) j0 // ··· jn−1//C(n) jn //···
2.3 Polygraphs
Let us describe a process of attaching n+1-cells to an n-category C ⇐ C ⇐ ··· ⇐ C .
0 1 n
Let S be a set, and σ ,τ : C ⇐ S a graph where σ , τ satisfy the boundary
n+1 n n n n+1 n n
conditions σ ◦σ =σ ◦τ andτ ◦σ =τ ◦τ . We build the free n+1-category
n−1 n n−1 n n−1 n n−1 n
C ⇐ C ⇐ ··· ⇐ C ⇐ S∗ , where S∗ consists of formal compositions of elements of
0 1 n n+1 n+1
S , including identities on cells of C , and subject to the equations of units, associativity
n+1 n
and exchange. We refer to [Bur93] or [M´et03] for formal definitions.
Nown-polygraphsandfreegeneratedn-categoriesaredefinedbysimultaneousinduction
on n:
• a 0-polygraph is a set S , generating the 0-category (i.e. set) S∗ =S ;
0 0 0
• givenann-polygraphS∗ ⇐S ,...,S∗ ⇐S withthefreen-categoryS∗ ⇐...⇐S∗
0 1 n−1 n 0 n
itgenerates,ann+1-polygraphisdeterminedbyagraphσ ,τ :S∗ ⇐S satisfying
n n n n+1
the boundary conditions, and the free n+1-category generated by it is S∗ ⇐ S∗ ⇐
0 1
···S∗ ⇐S∗ .
n n+1
Inparticular,a1-polygraphissimplyagraph,andthenotionoffree1-categorygenerated
by it coincides with the usual notion of a free category generated by a graph.
Definition 2 A polygraph is an infinite sequence
S :S∗ ⇐S ,S∗ ⇐S ,...,S∗ ⇐S ,...
0 1 1 2 n n+1
whose first n items define an n-polygraph for each n.
A free complex is a complex generated by a polygraph, that is of the form
S∗ :S∗ ⇐S∗ ⇐···S∗ ⇐S∗ ··· .
0 1 n n+1
4
Let S, T be polygraphs. A morphism f : S → T amounts to a sequence of maps
f :S →T such that, for all ξ :x→y in S , f (ξ):f∗ (x) →f∗ (y), where f∗ is the
n n n n n n−1 n−1 n
unique extension of f which is compatible with products and units.
n
We denote by Pol the category of polygraphs and morphisms. The functor
Pol → Compl,
S 7→ S∗,
is left-adjoint to a forgetful functor
Compl → Pol,
C 7→ |C|.
A detailed description of C 7→|C| is given in [M´et03], where this functor is called P.
Remark that any globular set X can be viewed as a particular polygraph and that this
identification makes Glob a full subcategory of Pol. Moreover the free complex generated
by a globularsetis the same as the free complex generatedby the correspondingpolygraph.
Howevermost free complexes generatedby polygraphscannotbe generatedby globular sets
alone.
Forinstancethe globularsetsO[n]and∂O[n]canbe viewedaspolygraphs,andgenerate
complexesO[n]∗ and∂O[n]∗. Remarkthatinthiscase,the freeconstructiondoesnotcreate
new non-trivial cells. Therefore, from now on, we drop the “∗” in the notation of these
complexes. Likewise, i will denote a morphism of globular sets, polygraphs, or complexes
n
according to the context. Note also that the natural equivalences (1) and (2) extend to
Compl:
C ∼= Compl(O[n],C) (3)
n
P (C) ∼= Compl(∂O[n+1],C) (4)
n
Let S be a polygraph, S∗ the free complex it generates, and n an integer. By ∂O[n]
PSn
(resp. O[n]), we mean the direct sum of copies of ∂O[n] (resp. O[n]) indexed by the
PSn
elements of S . As a consequence of (4), the source and target maps S∗ ⇐S determine
n n−1 n
a morphism
ρ:X∂O[n]→(S∗)(n−1).
Sn
Then the following result is merely a reformulation of the definition of polygraphs:
Lemma 2 The diagram
ρ
PSn∂O[n] //(S∗)(n−1)
PSnin jn
(cid:15)(cid:15) (cid:15)(cid:15)
PSnO[n] //(S∗)(n)
is a pushout in Compl.
3 Two classes of morphisms
Let C be a category, and f : A → B, g : C → D morphisms. f is left-orthogonal to g (or,
equivalently, g is right-orthogonal to f)if, for eachpair of morphisms u:A→C, v :B →D
suchthatg◦u=v◦f,thereexistsanh:B →C makingthefollowingdiagramcommutative:
u
A //C
~>>
~
~
f h g
~
~
(cid:15)(cid:15) ~ (cid:15)(cid:15)
B //D
v
For any class M of morphisms in C, ⊥M (resp. M⊥) denotes the class of morphisms in
C which are left-(resp. right-) othogonal to all morphisms in M.
5
3.1 Trivial fibrations
Let I be the set {i |n∈N} as morphisms in Compl.
n
Definition 3 A morphism of complexes is a trivial fibration if and only it belongs to I⊥.
In other words,p:C →D is a trivialfibrationif for all n, f :∂O[n]→C, and g :O[n]→D
suchthatp◦f =g◦i ,thereisanh:O[n]→C makingthefollowingdiagramcommutative:
n
f
∂O[n] //C
zzzz==
in (cid:15)(cid:15) zzzzh (cid:15)(cid:15)p
O[n] //D
g
Definition 4 Let C be a complex. A polygraphicresolution of C is a pair (S,p) where S is
a polygraph and p:S∗ →C is a trivial fibration.
It was shown in [M´et03] that, for each complex C, the counit of the adjunction (.)∗ ⊣|.|,
∗
ǫ :|C| →C,
C
isatrivialfibration. Hence(|C|,ǫ )isapolygraphicresolutionofC,andwegetthefollowing
C
result, which will play an essential part in section 4 below :
Proposition 1 Each complex C has a polygraphic resolution.
3.2 Cofibrations
Definition 5 A morphism of complexes is a cofibration if and only if it is left-orthogonal
to all trivial fibrations.
Hence the classof cofibrationsis exactly ⊥(I⊥). Immediate examples of cofibrationsarethe
i ’s themselves. The following lemma summarizes standard properties of maps defined by
n
left-orthogonality conditions.
Lemma 3 LetC beacategory, andMan arbitrary class of morphisms of C. LetL=⊥M.
Then
• L is stable by direct sums: if f : X → Y , i ∈ I is a family of maps of L with direct
i i i
sum f = f : X → Y , then f ∈L;
Pi∈I i Pi∈I i Pi∈I i
• L is stable by pushout: whenever f ∈L and
X //Z
f g
(cid:15)(cid:15) (cid:15)(cid:15)
Y // T
is a pushout square in C, then g ∈L;
• suppose
X l0 // ··· ln−1 // X ln //···
0 n
is a sequence of maps l ∈ L, with colimit (X,m : X → X). Then m : X → X
n n n 0 0
belongs to L.
Proof. We leavethe firsttwo claims as exercises. As for the third point, let f :Y →Z
beamorphisminM,andu:X →Y,v :X →Z suchthatthefollowingdiagramcommutes:
0
u
X0 //Y
m0 f
(cid:15)(cid:15) (cid:15)(cid:15)
X // Z
v
6
Let us define v =v◦m for each n≥0. Thus, for each n≥0, v ◦l =v◦m ◦l =
n n n+1 n n+1 n
v◦m = v , so that v : X → Z determines an inductive cone on the base (X ) to the
n n n n n
vertex Z. Let us define a family of maps u :X →Y satisfying the following equations:
n n
f ◦u = v , (5)
n n
u ◦l = u . (6)
n+1 n n
Let n = 0. Define u = u. We get f ◦u = f ◦u = v ◦m = v , and (5) holds. Thus
0 0 0 0
f ◦ u = v ◦ l , and because f ∈ M and l ∈ L, there is an u : X → Y such that
0 1 0 0 1 1
u ◦l = u , so that (6) holds. Suppose now that (5) and (6) hold for an n ≥ 0. By the
1 0 0
induction hypothesis, the following diagram commutes:
un
Xn //Y
ln f
(cid:15)(cid:15) (cid:15)(cid:15)
Xn+1vn+1 // Z
with f ∈ M and l ∈ L. Hence there is a u : X → Y such that f ◦u = v
n n+1 n+1 n+1 n+1
and u ◦l = u , and our equations hold for n+1. In particular, (6) means that (u )
n+1 n n n
determines an inductive cone on the base (X ) to the vertex Y. As X is the colimit of the
n
X ’s, there is a morphism h:X →Y such that, for each n≥0, h◦m =u . In particular,
n n n
h◦m = u = u. Also, for each n ≥ 0, f ◦h◦m = f ◦u = v = v◦m . Uniqueness of
0 0 n n n n
connecting morphisms show that f ◦h=v. Hence the following diagram is commutative
u
X0 }}}}//>>Y
m0 (cid:15)(cid:15) }}}}h (cid:15)(cid:15)f
X // Z
v
and we have shown that m ∈L, as required. ⊳
0
Definition 6 A complex C is cofibrant if 0→C is a cofibration.
Proposition 2 Free complexes are cofibrant.
Proof. Let S be a polygraph and C = S∗. By lemma 2, for each n ≥ −1, the canonical
inclusion j :C(n) →C(n+1) is a pushout of i . Now lemma 3 applies in the particular
n PSn n
case where L is the class of cofibrations: by the first point, i is a cofibration, and by
PSn n
the second point, so is j . By lemma 1, C is a colimit of the sequence
n
C(−1) j−1 //C(0) j0 // ··· jn−1//C(n) jn //···
hence the third point of lemma 3 applies, with X = C(n−1) and l = j , so that 0 →C
n n n−1
is a cofibration. In other words, C is cofibrant. ⊳
4 Main result
The main goal of this work is to establish the converse of proposition 2:
Theorem 1 Any cofibrant complex is isomorphic to a free one.
LetFcompldenotethe full subcategoryofComplwhoseobjectsarethe freecomplexes
S∗ generated by polygraphs. Then, theorem 1 reduces to the following statement:
Theorem 2 Fcompl is Cauchy-complete.
7
Recall that a category C is Cauchy-complete if all its idempotents split, that is, for each
object C, and each endomorphism h : C → C such that h◦h = h, there is an object D,
together with morphisms r :C →D, u:D →C, satisfying
u◦r = h,
r◦u = id.
Theorem 2 will be proved in annex B. Let us assume the result for the moment, and let
C be a cofibrantcomplex. By proposition1, C has a free resolutionp:S∗ →C, with S∗ an
objectofFcompl. BecauseC iscofibrant,andpisatrivialfibration,theidentitymorphism
id : C → C lifts through p, whence a morphism q : C → S∗ such that p◦q = id . Let
C C
h=q◦p,h◦h=q◦p◦q◦p=q◦id ◦p=q◦p=h,hencehisanidempotentendomorphism
C
of S∗. By using Cauchy completeness, we get a polygraph T, and morphisms r : S∗ → T∗,
u : T∗ → S∗ such that r ◦u = idT∗ and u◦r = h. Now, let f = p◦u : T∗ → C and
g =r◦q :C →T∗. We get
g◦f = r◦q◦p◦u
= r◦h◦u
= r◦u◦r◦u
= idT∗ ◦idT∗
= idT∗.
Likewise
f ◦g = p◦u◦r◦q
= p◦h◦q
= p◦q◦p◦q
= id ◦id
C C
= id .
C
Hence f :T∗ →C is an isomorphism with inverse g =f−1 so that C is isomorphic to a free
object, as required.
References
[Bat98] Michael Batanin. Computads for finitary monads on globular sets. Contemp.
Math., 230:37–57,1998.
[Bur91] Albert Burroni. Higher-dimensional word problem. In Category theory and com-
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Springer Verlag, 1991.
[Bur93] AlbertBurroni. Higher-dimensionalwordproblemswithapplicationstoequational
logic. Theoretical Computer Science, 115:43–62,1993.
[Gui06a] Yves Guiraud. The three dimensions of proofs. Annals of Pure and Applied Logic,
141(1–2):266–295,2006.
[Gui06b] YvesGuiraud. TwopolygraphicpresentationsofPetrinets. Theoretical Computer
Science, 360(1–3):124–146,2006.
[LM06] Yves Lafont and Franc¸ois M´etayer. Polygraphic resolutions and homology of
monoids. submitted, 2006.
[M´et03] Franc¸ois M´etayer. Resolutions by polygraphs. Theory and Applications of Cate-
gories, 11(7):148–184,2003. http://www.tac.mta.ca/tac/.
[Pow91] John Power. An n-categorical pasting theorem. In Category Theory, Proc. Int.
Conf., Como/Italy 1990, number 1488in Lect. Notes Math., pages 326–358,1991.
[Str76] Ross Street. Limits indexed by category-valued 2-functors. Journal of Pure and
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[Str00] RossStreet. Thepetittoposofglobularsets. Journalof Pureand Applied Algebra,
154(1–3):299–315,2000.
8
A Contexts
A.1 Indeterminates
Let C be a complex, andn≥1. An n-type is an orderedpair (x,y) of parallelcells in C ,
n−1
that is an element of P (C). By (4), n-types amount to morphisms θ : ∂O[n] → C. We
n−1
shall use the same notations for both sides of the natural equivalences (3) and (4).
Definition 7 The type of an n-cell x∈C is the pair (σ x,τ x).
n n−1 n−1
Hence the type of an n-cell is a particular n-type.
Given an n-type θ, we may adjoin to C an indeterminate n-cell of type θ by taking the
following pushout in Compl:
θ
∂O[n] //C
in jθ
(cid:15)(cid:15) (cid:15)(cid:15)
O[n] // C[x]
x
We let boldface variables x,y,... range over indeterminates.
Let x be an indeterminate n-cell of type θ and z :O[n]→C an n-cell in C. To say that
z is of type θ amounts to the commutativity of the following diagram:
θ
∂O[n] // C
in id
(cid:15)(cid:15) (cid:15)(cid:15)
O[n] //C
z
The pushout property gives a unique morphismsub :C[x]→C such that sub ◦x=z and
z z
sub ◦j =id. sub isnothingbuttheoperationofsubstitutingthecellz forx(seefigure3).
z θ z
Nown-cellsofC[x]areformalcompositesofelementsinC ∪{x}. Differentexpressionsmay
n
θ
∂O[n] //C
1
1
1
in jθ11
1
O[(cid:15)(cid:15)n]RRRxRRR//RCzRR[(cid:15)(cid:15)RxR]CRCsRCu1R1b1R1iCz1dRC1C1RC11!!))1(cid:24)(cid:24)
C
Figure 3: substitution
denote the same cell: howeverall those expressions containthe same number of occurrences
of x.
Definition 8 An n-context over x is an n-cell of C[x] having exactly one occurrence of x.
We denote n-contextsoverx by c[x],d[x],.... An n-cell z ofC is adapted to the contextc[x]
if it has the same type as x. Contexts are subject to the following operations:
• for each n-context c[x]:O[n]→C[x] and each adapted n-cell z, we denote by c[z] the
new n-cell of C obtained by substituting z for x, in other terms c[z]=sub ◦c[x];
z
9
• let u:C →D be a morphism of complexes and c[x]:O[n]→C[x] an n-context of C.
Define a new indeterminate y:O[n]→D[y] by the following pushout square
u◦θ
∂O[n] //D .
in ju◦θ
(cid:15)(cid:15) (cid:15)(cid:15)
O[n] // D[y]
y
This determines a unique morphism
uˆ:C[x]→D[y]
such that j ◦u = uˆ◦j and uˆ◦x = y (see figure 4). Thus uˆ◦c[x] is an n-context
u◦θ θ
θ
∂O[n] //C
in (cid:15)(cid:15) x (cid:15)(cid:15)jθFFFFFFuFFFF""
O[n] //C[x] D
SSSSSSSSySSSSFFSFSFuSˆFSFSFF)) "" (cid:15)(cid:15)jθ◦u
D[y]
Figure 4: context transformation
over y, denoted by cu[y].
Note that, if z is an n-cell adapted to c[x], then u(z) is adapted to cu[y] and
u(c[z])=cu[u(z)]. (7)
A.2 Thin contexts
Let us introduce a few additional terminology about cells and contexts. If S is a polygraph,
the elements of S∗ are the cells of dimension n, or n-cells. The generators of dimension n,
n
or n-generators are the elements of S . Each n-generator α determines an n-cell α∗. Such
n
cells are called atomic. All 0-cells are atomic, and if n>0, each n-cell may be expressed as
acompositionofatomiccellsandunits onn−1-cells. Foreachn-cellx,andgeneratorα, the
number of occurrences of α∗ in an expression of x only depends on x, not on the particular
expression. We call this number the weight of x at α, and denote it by w (x). The total
α
weight of x is
w(x)= X wα(x).
α∈Sn
The same definitions hold for contexts, where we take into account all generators but the
indeterminate. Thus, for instance. w(x)=0 for any indeterminate x.
Definition 9 An n-context c[x] is thin if its total weight is zero.
Now, if x ∈ S∗, either w(x) > 0 or there is a cell y ∈ S∗ such that x = 1 (y). More
n n−1 n
generally, if w(x)=0, there is a unique integer p<n with the following property:
• there is a p-cell z in S∗ such that w(z)>0 and x=1 (z)
n,p
Letuscallpthethickness of x,anddenoteitbyp=th(x). Ifw(x)6=0,wedefineth(x)=n.
The same definitions immediately apply to contexts. In particular, an n-context c[x] is thin
if and only if th(c[x])<n. We finally associate to each cell x an integer size(x) by:
• if w(x)6=0, size(x)=w(x);
10