On the ﬁnite presentation of operads

Abstract: Operads were introduced to describe compositional structures arising in algebraic topology.

Recently, some researches were interested in using operads in applied mathematics, to model composition

of structures in logic, databases, and dynamical systems. In, we focus on ﬁnite presentation of an operad

and its associated algebra. More precisely, we prove the general result stating that if an operad Ohas

a ﬁnite presentation, then the associate O-algebra has also a corresponding one. Some application in

physics, especially in wiring diagrams will be discussed.

Key-Words: operad, operad algebra, wiring diagrams, ﬁnite presentation, compositionality.

1 Introduction

Operad theory is a ﬁeld of algebra (more precisely

abstract algebra) used to describe algebraic struc-

ture that models some algebraic properties such

as commutativity, associativity, Lie brackets, . . . .

An operad (or colored operad) is a structure

that consists of a bunch of elements that are

viewed as abstract operations, each one having

a multiple inputs, where inputs are ﬁnite ordered

list (possibly zero ones) of elements called colors

in a ﬁxed non-empty ﬁnite class S, these oper-

ations are equipped with a speciﬁcation of how

to compose them in one output (element in S),

and subject to associativity and unity axioms, [1],

these operations are represented by trees which

can be grafted onto each other to represent the

composition. Just like a monoid can be viewed as

a single object category, likewise an operad can

be seen as a single object multicategorie, hence

multicategories are also named as operads, or col-

ored operads, and in case of ambiguity, it can be

identiﬁed the class S, then it will be called an

S-colored operad.

An operad-algebras (algebra over on operad)

is a generalization of the notion of a module over

a ring. One can deﬁne an operad-algebra as a

concrete realization of the abstract operations of

the operad, in other words, it can be deﬁned as a

set combined with concrete operations on this set

whereby their behaviour is analogous to the ab-

stract operations in the operad : an object com-

bined with operations as deﬁned by the operad,

subject to the composition condition as deﬁned

by the operad, [1]. Operad-algebra is to its as-

sociate operad as group representation is to its

group. They form a category analog of that of

universal algebras. Operads were ﬁrstly intro-

duced in algebraic topology in the early of 1970s

by [2], notably to model iterated loop spaces,

and the original deﬁnition of operad is due to

J.Peter in his book ”The Geometry of Iterated

Loop Spaces”, and he was the ﬁrst one to coin

the term operad. Operads are basic in homo-

topy theory, and they have many applications in

many branches of mathematics, [3], [4], such as

string topology, category theory, combinatories

of trees, algebraic deformation theory, homotopi-

cal algebras and vertex operator algebras. Fur-

thermore, operads are essential in mathematical

physics, computer science, biology, and others.

For more details on operads, we refer the inter-

ested reader to, [5], [6].

Here, we focus on two speciﬁc operads, that

of wiring diagrams (resp. undirected wiring dia-

grams) denoted through this paper by WD (resp.

UWD). The original deﬁnition of wiring diagram

is given by Rupel and Spivak in [7] who observed

that the set of wiring diagrams form an operad

called the operad of wiring diagrams. Wiring di-

agrams are a simpliﬁed representation of electri-

cal systems or circuits (graphical language) com-

posed of such operations, each one having a mul-

tiple inputs and multiple outputs, each element

of which is allowed to carry a such value, and

describes how these operations are connected be-

tween them to form a larger one operation more

complicated. Contrary to a wiring diagram, an

undirected wiring diagram is version of wiring di-

agrams that each operation can be seen as a ﬁ-

nite set, which each element is allowed to carry a

value. Mathematically speaking, let Sbe a class,

Received: September 23, 2022. Revised: May 8, 2023. Accepted: May 25, 2023. Published: June 19, 2023.

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ETTAKI AYOUB1, ELOMARY MOHAMED ABDOU1, MAMOUNI MY ISMAIL2,

1IMII laboratory. Department of Mathematics and Computer Science. FST Settat, MOROCCO

2M@DA research team, Department of Mathematics, CRMEF Rabat, MOROCCO

aS-wiring diagram (we can drop S, and call it a

wiring diagram if Sis clear from the context) in

[1] is given by ψ= (X, Y, DN, v, s) where Yis the

output box of ψand X= (X1, ...., Xn) is a BoxS-

proﬁle with Xithe i-th input box of ψ, (DN, v) is

an S-ﬁnite set, and sis the supplier assignment

for ψ. Similarly a S-undirected wiring diagram

(we can drop S, and call it an undirected wiring

diagram if Sis clear from the context) in [1] is

given by ψ= (X, Y, C, f, g) where Yis the output

box of ψand X= (X1, ...., Xn) is a F inS-proﬁle

with Xithe i-th input box of ψ,C∈F inSthe set

of cables of ψ, and f, g are maps in the cospan di-

agram. More details about the explicit deﬁnition

of wiring diagram and undirected wiring diagram

can be found in [1].

In [1], D. Yau established that for each class

S, the collection of S-wiring diagrams is a BoxS-

colored operad denoted WD that has 8 generat-

ing wiring diagrams that they generate the op-

erad WD and 28 elementary generating relations

that they generate together with the associativ-

ity and the unity axioms deﬁned by the operad

WD all the relations in WD , and he also es-

tablished that for each class S, the collection of

S-undirected wiring diagrams is a F inS-colored

operad denoted UWD that has 6 generating undi-

rected wiring diagrams that they generate the op-

erad UWD and 17 elementary generating rela-

tions that they generate together with the asso-

ciativity and the unity axioms deﬁned by the op-

erad UWD all the relations in UWD. Then every

wiring diagram (respectively undirected wiring

diagram) has a presentation in terms of ﬁnitely

many generating wiring diagrams (respectively

undirected wiring diagrams) as a ﬁnite iterated

operadic composition, diagrams in both WD and

UWD can be built as an operadic composition

of generating diagrams in diﬀerent ways, so they

can have many diﬀerent presentations expressed

as a ﬁnite iterated operadic composition. Then

the concept of a simplex was crucial to develop

the necessary language to check if two any pre-

sentations of the same diagram are equivalent,

meaning connected by a ﬁnite sequence of ele-

mentary equivalences. According to [1] the op-

erad WD (resp UWD ) has a ﬁnite presentation

if and only if WD (resp UWD ) satisﬁes the two

following assertions : the ﬁrst is that every wiring

diagram (resp. undirected wiring diagram) can

be generated by the generating wiring diagrams

(resp. undirected wiring diagram) as a ﬁnite it-

erated operadic composition, and the second is

that if a wiring diagram (resp. undirected wiring

diagram) can be expressed as an operadic com-

position of the generating wiring diagrams (resp.

undirected wiring diagram) in two diﬀerent ways,

then it can be found a ﬁnite sequence of elemen-

tary equivalences from the ﬁrst operadic compo-

sition to the other one, that’s what it call a ﬁnite

presentation theorem for WD (resp. UWD).

In [1], D. Yau used these ﬁnite presentations

to describe the WD-algebra and UWD-algebra

in terms of ﬁnitely many generating structure

maps and generating axioms corresponding to

the generating wiring diagrams and elementary

relations in their associted operads. His proof

was based on an equivalence between two diﬀer-

ent deﬁnitions of O-algebra (where O=WD or

O=UWD). In fact, the WD-algebra (resp. the

UWD-algebra) has 8 generating structure maps

corresponding to the 8 generating wiring dia-

grams, and 28 generating axioms corresponding

to the 28 elementary relations in WD (resp. 6

generating structure maps corresponding to the

6 generating undirected wiring diagrams, and

17 generating axioms corresponding to the 17

elementary relations in UWD). Finally D.Yau

proved that the operad algebra WD-algebra has

a ﬁnite presentation corresponding to the one of

WD.

A natural extension, is to investigate the anal-

ogous of D. Yau’s claim for any Ooperad, then

our approach is based on the operad WD (resp.

UWD) which initially invented in Spivak, [7], [8].

This paper deals with an operadic approach to

formulating and proving a more general result

that consider both of these ﬁnite presentation

theorems as special cases, our main result stating

that if an operad Ohas a ﬁnite presentation, then

the associate O-algebra has also a corresponding

one, a ﬁnite presentation of O(resp. O-algebra)

means that the operad O(resp. O-algebra) has a

ﬁnite generating set and any two equivalent sim-

plices in O(resp. O-algebra) are connected by

a ﬁnite sequence of elementary equivalences in O

(resp. O-algebra). Our results state the following

Theorem 1.1. If the operad Ohas a ﬁnite gen-

erating set T, then its associated O-algebra, A,

has a corresponding ﬁnite generating set Tµ.

That will be suﬃcient to prove the ﬁrst part

of our main theorem, to establish the second part

of our theorem, we will deﬁne the concept of a

simplex and an elementary equivalence, we will

see that every elementary equivalence is induced

by an elementray relation or an operad associa-

tivity unity equivariance axiom, then we develop

the necessary language that allows us to deﬁne

what an elementary relation means, and prove

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the following result :

Theorem 1.2. Let Tbe a ﬁnite generating set

for O, and take ζand ξtwo simplices in O. If

|ζ|=|ξ|is an elementary relation in O, then its

corresponding elementary relation in Ais µ|ζ|=

µ|ξ|.

Afterwards, we will show that the operad as-

sociativity unity equivariance axiom in Ohas

a corresponding associativity unity equivariance

axiom in O-algebra, and this together with the

above results permits us to announce the next

result :

Theorem 1.3. If Wthe set of all elementary

equivalences in Tis a strong generating set of O

in T, then its corresponding set Wµof all ele-

mentary equivalences in Tµis a strong generating

set of Ain Tµ.

This last theorem yields the second part of our

main theorem. and that will be suﬃcient to prove

the following theorem :

Theorem 1.4. If Ohas a ﬁnite presentation,

then its associated O-algebra, Ahas a correspond-

ing ﬁnite presentation one.

The rest of the paper is broken down as fol-

lows : in section 2 we will summarize the nec-

essary background to prove our results, that will

be proved in section 3. In section 4, we apply see

how our result ﬁts in the cases of both directed

and undirected wiring diagrams operads.

2 Materials

Let Sbe a class, (n, m)∈N2, and P rof(S) the

class of ﬁnite ordered sequences of elements in S.

Elements a= (a1, . . . , an) of P rof (S) of length n

are also called S-proﬁles.

Deﬁnition 2.1. AS-colored operad (O,1,◦)is

deﬁned as follows:

To any b∈ S and any two S-proﬁles, a=

(a1, . . . , an)and c= (c1, . . . , cm),(O,1,◦)is

equipped with

•a class Ob

a=Ob

a1, . . . , anwhich ele-

ments are called multimaps;

•a bijection

Ob

aσ

−→ O b

aσ

where σ∈Sn, and aσ = (aσ(1), . . . , aσ(n))

•a speciﬁc element called the b-colored unit

1b∈ O b

b

•a map called the operadic composition ◦iis

deﬁned by

Ob

a× O ai

c◦i

−→ O b

a◦ic

where a◦ic=

(a1, . . . , ai−1, c1, . . . , cm, ai+1, . . . , an).

This is enough to satisfy the associativity, the

unity, and the equivariance axioms.

The best general refernce here is, [1].

Deﬁnition 2.2. Let Cbe a collection of mul-

timaps of O, and ψa multimap in O, we say that

ψadmits a presentation in Cif ψcan be expressed

as an iterated operadic composition of multimaps

in C.

For example, consider the collection C=

{ψ1, ψ2, ψ3}, where ψ1, ψ2, ψ3are multimaps in

Suppose ψ∈ O such that ψ= (ψ1◦3ψ2)◦4ψ3,

then ψhas a presentation in C.

Note that the iterated operadic composition ψ1◦3

(ψ2◦4ψ3) is not necessarily a presentation of ψ

in C, since the equality ψ= (ψ1◦3ψ2)◦4ψ3=

ψ1◦3(ψ2◦4ψ3) is not assured, for this reason, we

need to deﬁne the concept of a simplex later.

Deﬁnition 2.3. Let Obe an operad.

1. A set (or collection) Tof multimaps of O

is called a generating set for Oif every

multimap in Ohas a such presentation in

T, meaning that for all ψin Othere ex-

ist ψ1, . . . , ψrin Tsuch that ψcan be de-

composed as an iterated operadic composi-

tion (possibly inﬁnite) of ψ1, . . . , ψr,for some

r≥1.

2. A set (or collection) Tof multimaps of Ois

called a ﬁnite generating set for O, when T

is a ﬁnite set (or collection) of mutimaps of

O, and every multimap in Ohas a such pre-

sentation in T, meaning that for all ψin O

there exist ψ1, . . . , ψrin Tsuch that ψcan be

decomposed as a ﬁnite iterated operadic com-

position of ψ1, . . . , ψr,for some r≥1.

The elements of Tare called generating mul-

timaps.

Given an operad O, an algebra over an operad,

or just O-algebra for simplicity, roughly, is a left

module over Owith multiplications parametrized

by O. Formally meaning

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Deﬁnition 2.4. Let Obe an operad. An operad-

algebra (shortly O-operad), is a pair (A, µ)

equipped with a class , denoted Ab, for any b∈ S,

called the b-colored entry of A, and the multipli-

cation µis deﬁned to be

Aa=

i=1

Aai

µψ

−→ Ab,

called the structure map, where b∈ S, a =

(a1, . . . , an)∈P rof(S)and ψ∈ O b

a, such

that the associativity, unity, and equivariance ax-

ioms are respected.

Each structure map has one entry of O. The

best general refernce here is, [1].

Simplices:

Let Obe an operad with a ﬁnite generating set

T, let n∈N∗, we deﬁne inductively a n-simplex

ψand its composition |ψ|in Oas follows :

•A 1-simplex is a generating multimap, and

its composition |ψ|is deﬁned to be ψitself,

i.e |ψ|=ψ;

•An-simplex, for n≥2 is deﬁned to be a tuple

ψ= (φ, i, ϕ) consists of :

(i∈N∗

p−simplex φwith p≥1

q−simplex ϕwith q≥1

such that p+q=n, and the operadic com-

position |ψ|=|φ| ◦i|ϕ|.

Here, the k-simplices for all 1 ≤k≤n−1

and their compositions are supposed to be

well deﬁned in O.

For simplicity of notation, we also denote a

n-simplex by ψ= (ψ1, . . . , ψn), where the gener-

ating multimaps ψ1, . . . , ψnare ordered in which

they appear in the composition and we write

|ψ|=ψ1◦i1· · ·◦in−1ψn. Unless otherwise stated a

simplex in Ois a m-simplex in Ofor some m≥1.

Moreover we say that a simplex ψis a presenta-

tion of |ψ|, and the set of all presentations of ψ

in O, denoted by

ψ={Ψsimplex in O/|Ψ|=ψ}.

For example, we can consider for some

i, j, k, l ∈Nthe case of a 5-simplex in Owhich is

an iterated operadic composition in the operad

Oof the form:

(((ψ1◦iψ2)◦jψ3)◦kψ4)◦lψ5,shortly

((((ψ1, i, ψ2), j, ψ3), k, ψ4), l, ψ5)

((ψ1◦i(ψ2◦jψ3)) ◦kψ4)◦lψ5,shortly

(((ψ1, i, (ψ2, j, ψ3)), k, ψ4), l, ψ5)

(ψ1◦i((ψ2◦jψ3)◦kψ4)) ◦lψ5,shortly

((ψ1, i, ((ψ2, j, ψ3), k, ψ4)), l, ψ5)

(ψ1◦i(ψ2◦j(ψ3◦kψ4))) ◦lψ5,shortly

((((ψ1, i, ψ2), j, ψ3), k, ψ4), l, ψ5)

((ψ1◦iψ2)◦j(ψ3◦kψ4)) ◦lψ5,shortly

(((ψ1, i, ψ2), j, (ψ3, k, ψ4)), l, ψ5)

ψ1◦i(ψ2◦j(ψ3◦k(ψ4◦lψ5))),shortly

(ψ1, i, (ψ2, j, (ψ3, k, (ψ4, l, ψ5))))

ψ1◦i(ψ2◦j((ψ3◦kψ4)◦lψ5)),shortly

(ψ1, i, (ψ2, j, ((ψ3, k, ψ4), l, ψ5)))

ψ1◦i((ψ2◦j(ψ3◦kψ4)) ◦lψ5),shortly

(ψ1, i, ((ψ2, j, (ψ3, k, ψ4)), l, ψ5))

ψ1◦i(((ψ2◦jψ3)◦kψ4)◦lψ5),shortly

(ψ1, i, (((ψ2, j, ψ3), k, ψ4), l, ψ5))

ψ1◦i((ψ2◦jψ3)◦k(ψ4◦lψ5)),shortly

(ψ1, i, ((ψ2, j, ψ3), k, (ψ4, l, ψ5)))

(ψ1◦iψ2)◦j((ψ3◦kψ4)◦lψ5),shortly

((ψ1, i, ψ2), j, ((ψ3, k, ψ4), l, ψ5))

(ψ1◦iψ2)◦j(ψ3◦k(ψ4◦lψ5)),shortly

((ψ1, i, ψ2), j, (ψ3, k, (ψ4, l, ψ5)))

((ψ1◦iψ2)◦jψ3)◦k(ψ4◦lψ5),shortly

(((ψ1, i, ψ2), j, ψ3), k, (ψ4, l, ψ5))

(ψ1◦i(ψ2◦jψ3)) ◦k(ψ4◦lψ5),shortly

((ψ1, i, (ψ2, j, ψ3)), k, (ψ4, l, ψ5)).

In our example, for ψ= (((ψ1◦iψ2)◦j

ψ3)◦kψ4)◦lψ5,there exists a 5-simplex Ψ =

((((ψ1, i, ψ2), j, ψ3), k, ψ4), l, ψ5) (shortly Ψ =

(ψ1, ψ2, ψ3, ψ4, ψ5)) whose composition is ψ=

|Ψ|= (((ψ1◦iψ2)◦jψ3)◦kψ4)◦lψ5,then

the 5-simplex Ψ is a presentation of ψ=|Ψ|,

but neither ((((ψ1, j, ψ2), i, ψ3), k, ψ4), l, ψ5) nor

(((ψ1, i, (ψ2, j, ψ3)), k, ψ4), l, ψ5) is not necessarily

a presentation of ψ.

Deﬁnition 2.5. Two simplices ψand ϕare said

to be equivalent in O, whenever their composi-

tions |ψ|and |ϕ|are equal in O.

A sub-simplex b

ψof ψis deﬁned to be itself

if ψis a 1-simplex, since otherwise ψ= (φ, i, ϕ)

is deﬁned as above, hence a sub-simplex of ψis

deﬁned to be either a sub-simplex of φ, or of ϕ,

or ψitself, and we write b

ψ⊆ψ.

Once again in our example, for ψ=

(ψ1◦i(ψ2◦jψ3))◦k(ψ4◦lψ5),both ψ1◦i(ψ2◦jψ3) =

(ψ1, i, (ψ2, j, ψ3)), ψ2◦jψ3= (ψ2, j, ψ3),or

ψ4◦lψ5= (ψ4, l, ψ5) are sub-simplices of

ψ, however, both ψ1◦iψ2= (ψ1, i, ψ2) and

ψ3◦k(ψ4◦lψ5)=(ψ3, k, (ψ4, l, ψ5)) are not.

In what follows, we will see how two such sim-

plices that are presentations of the same mul-

timap can be related, more precisely we develop

the precise concept that will allow us to be able to

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replace (or substitue) a sub-simplex of a simplex

by another one.

Deﬁnition 2.6. Given a n-simplex ψin Owith

ψ= (ψ1, ...., ψn)and k < n. A relaxed k-moves

of ψis the given of some 1≤k′, k”≤k; 1 ≤

s≤n−k′and a (n−k′+k”)-simplex ϕsuch that

1. ψj=ϕjfor all 1≤j≤s−1,

2. ψj=ϕj−k′+k”for all s+k′≤j≤n,

3. |ζ|=|ξ|, for ζ= (ψs, ...., ψs+k′−1)a sub-

simplex of ψ, and ξ= (ϕs, ...., ϕs+k”−1)a

sub-simplex of ϕsuch that

Remarks.

•Firstly it is worth to point out the compat-

ibility with the operadic composition ◦iof

ψs+k′=ϕs+k”,

•a relaxed k-moves of ψ, is ﬁnally a l-simplex

ϕ, where l=n−k′+k” such that n−k+1 ≤

l≤n+k−1. Since k < n then 2 ≤l≤2n−2,

•for s= 1, we obtain ζ= (ψ1, ...., ψk′), ξ=

(ϕs, ...., ϕk”) and ψj=ϕj−k′+k”for all 1+k′≤

j≤n,

•if k= 1, then ϕis a n-simplex, such that

ψj=ϕjfor all 1 ≤j≤n,

•one may consider k=n, but in this case we

should have k′< k and k”< k,

•ak-relaxed moves is a substitution of a k′-

simplex (k′≤k) with k”-simplex (k”≤k),

but keeping ﬁxed the operadic composition

into which they are substituted.

Deﬁnition 2.7. Under the same hypotheses of

the previous deﬁnition, we say that ψand ϕare

equivalent by a relaxed k-moves.

Vocabulary.

Let Obe an operad with a ﬁnite generating set

T, then :

•every multimap in Ohas a decomposition (or

presentation) in T, then there exists a sim-

plex ψin Twhich is a presenation of this

multimap. The above equality relation

|ζ|=|ξ|

is either an operad associativity or unity or

an equivariance axiom, otherwise is called an

elementary relation,

•an elementary sub-simplex ˆ

ψof ψis a sub-

simplex of one of two following forms:

–ˆ

ψis one side (either left or right) of a

speciﬁed elementary relation,

–ˆ

ψis one side (either left or right) of a

speciﬁed operad associativity or unity

or equivariance axiom involving only the

generating multimaps.

Suppose that we have a n-simplex ψ(for

some integer n≥2) in Owhich is a

presentation of the multimap |ψ|in O,

and ζis a sub-simplex of ψsuch that

|ζ|=|ξ|, where ξis a simplex in O, then

the relation |ζ|=|ξ|is either an elemen-

tary relation or an operad associativity

or unity or equivariance axiom involving

only the generating multimaps, hence,

one can obtain a relaxed k-moves of the

simplex ψby substituting the elemen-

tary sub-simplex ζby the other one ξ,

•two simplicies ψand ϕare called to be el-

ementarily equivalent in O, if ψand ϕare

equivalent by a relaxed k-moves (for some in-

teger k≥2), then we write ψk

∼ϕ(if there is

no confusion, we can drop kand write ψ∼ϕ)

and call this an elementary equivalence in O,

in other words an eleelementary equivalence

is a subsitution of a elementary sub-simplex

of one side by the other one,

•two simplicies ψand ϕare said to be con-

nected by a ﬁnite sequence of elementary

equivalences in Oif and only if there exist

some simplices ψ1, ...., ψrin Osuch that ψ1

∼

. . . kr−1

∼ψr, and ψk

∼ψ1

∼. . . kr−1

∼ψr

∼ϕfor

some integers k, k1, . . . , kr≥2.

Denotations. Let Obe an operad with a ﬁnite

generating set T. Let ψbe an n-simplex in Ofor

some integer n≥2.

•The set of all relaxed k-moves of ψin Tis

denoted by Wk

ψ,

•The set of all relaxed k-moves for all k∈

{2, . . . , n −1}is

ψ:=

n−1

[

k=2

ψ,

and this is the set of all relaxed n-moves of

ψin T. In other words, Wn

ψis the set of all

elementary equvalences of ψ.

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•The set of all relaxed n-moves in Tis denoted

by W, where

W:= [

ψ∈O

In other words, Wis the set of all elementary

equivalences in T.

Deﬁnition 2.8. Let Obe an operad with a ﬁnite

generating set T, and Wthe set of all its ele-

mentary equivalences. We say that Wis a strong

generating set for Oin Tif any two equivalent

simplices in Tare connected by a ﬁnite sequence

of elementary equivalences in W. Roughly speak-

ing, Wis the set that generate all relations in

Finally, one may understand that a multimap

can be built by using the generating multimaps in

two ways diﬀerent thanks to a ﬁnite sequence of

steps connecting them such that each step is re-

lated to the next one by replacing an elementary

sub-simplex of one side (either left or right) of a

speciﬁed elementary relation or a speciﬁed operad

associativity/unity/equivariance axiom involving

only the generating multimaps with the elemen-

tary sub-simplex of the other side.

3 Results and proofs

In this section we will present and prove our re-

sults related the ﬁnite presentations of an operad

and its associate algebra. Firstly let us recall

what that means. In all the remainder of this

paper, let (O,1,◦) be an S-colored operad and

(A, µ) the associated O-algebra.

Deﬁnition 3.1. We say that O(respectively O-

algebra A) has a ﬁnite presentation if O(respec-

tively O-algebra A) satisﬁes the two following as-

sertions :

1. O(respectively O-algebra A) has a ﬁnite gen-

erating set,

2. any two simplices in O(respectively O-

algebra A) that are presentation of the same

multimap (respectively structure map) are

connected by a ﬁnite sequence of elementary

equivalences.

Our ﬁrst result states the following :

Theorem 3.1. If the operad Ohas a ﬁnite gen-

erating set, then the associated O-algebra A, has

a corresponding ﬁnite generating set.

Proof. Let T={ψ1, . . . , ψd}be a ﬁnite generat-

ing set for Owhere d∈N∗.Then every multimap

ψin Ohas a presentation in Tof the form

ψ=ψl1◦i1· · · ◦ik−1ψlk,

where lr, k ∈N∗for all integers 1 ≤r≤k,

and ψl1, . . . , ψlkin T(note that some of the ψlr

may be repeated). It follows that there exists

al-simplex (ψl1, . . . , ψlk) in O(especially in T)

whose composition is ψ. Then the structure map

µψin O-algebra Aassociated to the multimap

ψin Ois µψ=µψl1◦i1···◦ik−1ψlk, applying the

associativity axiom of the O-algebra A, we get

µψ=µψl1◦i1· · · ◦ik−1µψlk.Then µψhas a pre-

sentation in the set Tµ={µψ1, . . . , µψd}. Since ψ

is an arbitrary element in O, then this is true for

all ψin O, so each structure map µψin Ahas a

presentation in Tµ, then Tµis a ﬁnite generating

set for A.

Deﬁnition 3.2. The structure maps of Tµare

called generating structure maps.

Lemma 3.1. Let n≥2, and ψ1, . . . , ψnbe some

multimaps in O. If (ψ1, . . . , ψn)is a n-simplex

in O, then its corresponding n-simplex in Ais

(µψ1, . . . , µψn).

Proof. We will lead an induction proof. For

n= 1, a 1-simplex ψin Ois a generating mul-

timap, and its composition is itself |ψ|=ψ, since

ψis a generating multimap in O, then by the

previous theorem, its associated structure maps

µψin Ais a generating structure map, hence

the corresponding 1-simplex of ψin Ais the 1-

simplex µψ, and its composition is deﬁned as itself

|µψ|=µ|ψ|=µψ, in other words, one can deﬁne

the 1-simplices in Aas the generating structure

maps.

For n= 2, a 2-simplex is an iterated operadic

composition in Oof the form ψ= (ψ1, i, ψ2)

for some integer i≥1, and its composition is

|ψ|=ψ1◦iψ2, consider ψ1∈ O b

a1, . . . , an, and

ψ2∈ O ai

c1, . . . , cm, are two generating mul-

timaps in O, then by the operadic composition

◦ideﬁned by O, we can assert that ψ1◦iψ2∈

Ob

a◦ic, then its associated structure map is

deﬁned by

µψ=µψ1◦iψ2:=

i−1

k=1

Aai×

j=1

Acj×

k=i+1

Aai

µψ

−→ Ab

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Follows, [1], the associativity axiom in Astates

that

µψ=µψ1◦iψ2=µψ1◦iµψ2

where µψ1and µψ2are generating structures maps

in A(1-simplices) corresponding to the generat-

ing multimaps (1-simplices) ψ1and ψ2respec-

tively in O. Hence, the corresponding 2-simplex

of ψin Ais µψ= (µψ1, i, µψ2) for some integer

i≥1, and its composition is deﬁned in Aas

|µψ|=µ|ψ|=µψ1◦iψ2=µψ1◦iµψ2

Then the statement holds for n= 2.

Assume that the statement holds for any given

integer p, q ≤n, and let ψ= (ψ1, . . . , ψn+1) be a

(n+ 1)-simplex, then ψis an iterated operadic

composition in Oof the form ψ= (ϕ, i, φ) for

some integer i≥1, where ϕis a p-simplex for

some integer p≥1, and φis a q-simplex for some

integer q≥1 such that p+q=n+ 1, with-

out loss of generality we can suppose that ϕ=

(ψ1, . . . , ψr) and φ= (ψr+1, . . . , ψn+1) then by in-

duction on n, the corresponding p-simplex of ϕin

Ais µϕ= (µψ1, . . . , µψr), its composition |µϕ|=

µ|ϕ|=µψ1◦i1· · · ◦ir−1µψr, and the corresponding

q-simplex of φin Ais µφ= (µψr+1 , . . . , µψn+1 ), its

composition |µφ|=µ|φ|=µψr+1 ◦ir+1 · · ·◦inµψn+1 ,

by deﬁnition of the (n+1)-simplex ψ, the next op-

eradic composition |ψ|=|ϕ| ◦i|φ|is well deﬁned

in O, then µ|ψ|=µ|ϕ|◦i|φ|, and by the associativ-

ity axiom in A, we get

|µψ|=µ|ψ|=µ|ϕ|◦i|φ|=µ|ϕ|◦i1µ|φ|

Hence the n+ 1-simplex in Acorresponding the

n+ 1-simplex ψ= (ψ1, . . . , ψn+1) in Ois µψ=

(µψ1, . . . , µψn+1 ).

Then the statement holds for n+ 1.

Corollary 3.1. Let b

ψbe a sub-simplex of the

n-simplex ψin O, then the corresponding sub-

simplex of b

ψin Ais µb

ψ.

Proof. Given a n-simplex ψin Ofor an integer

n≥1, then its corresponding n-simplex in Ais

µψ. Once again we will lead an induction proof ;

for n= 1, ψis a 1-simplex, then ψis a generating

multimap, and a sub-simplex of ψis deﬁned to be

itself b

ψ=ψ, then the corresponding sub-simplex

of b

ψ=ψin Ais µb

ψ=µψthat is a generat-

ing structure map. Then the statement holds for

n= 1.

Suppose that the statement holds for any integers

p, q ≥1, and ψis a n+ 1-simplex for an integer

n≥1, then according to the deﬁnition of a sim-

plex ψcan be written as the form ψ= (ϕ, i, φ)

for some integer i≥1, with ϕis p-simplex and

φis q-simplex such that p+q=n+ 1, and the

operadic composition in O,|ψ|=|ϕ|◦i|φ|, then a

sub-simplex b

ψof ψis deﬁned to be a sub-simplex

of either ϕ, or of φ, or ψitself, hence, if b

ψis

a sub-simple of either ϕ, or of φ, by induction

on n, the corresponding sub-simplex of b

ψin A

is µb

ψwhich is a sub-simplex of either µϕ, or of

µφ, otherwise, if b

ψis deﬁned to be ψitself, then

the corresponding sub-simplex of b

ψ=ψin A,

is the corresponding simplex of ψin Awhich is

µb

ψ=µψ.

Then the statement holds for n+ 1.

Corollary 3.2. Let ψand ϕbe two simplices in

O, then ψand ϕare equivalent in Oif and only

if their associated simplices µψand µϕare equiv-

alent in A. More presicely

|ψ|=|ϕ|⇔|µψ|=|µϕ|

Proof. Suppose ψ= (ψ1, . . . , ψr) and ϕ=

(ϕ1, . . . , ϕs) are two simplices in Ofor some inte-

gers r, s ≥1 such that |ψ|=|ϕ|.

Then ther exist i1,...ir−1, j1, . . . , js−1∈N∗such

that |ψ|=ψ1◦i1· · · ◦ir−1ψr, and |ϕ|=ϕ1◦j1

· · · ◦js−1ϕs, since the corresponding simplex of

ψin Ais µψ= (µψ1, . . . , µψr) with composition

|µψ|=µ|ψ|=µψ1◦i1· · · ◦ir−1µψr, and the corre-

sponding simplex of ϕin Ais µϕ= (µϕ1, . . . , µϕs)

with composition |µϕ|=µ|ϕ|=µϕ1◦j1· · ·◦js−1µϕs,

then the relation |ψ|=|ϕ|implies

|µψ|=µ|ψ|=µψ1◦i1···◦ir−1µψr=µϕ1◦j1···◦js−1µϕs=µ|ϕ|=|µϕ|

Conversely, we know that the construction of a

simplex in Arequires the existence of a simplex

in O. Let ψand ϕbe two simplices in Oand

their associated simplices respectively µψand µϕ

are equivalent in A, i.e |µψ|=µ|ψ|=µ|ϕ|=

|µϕ|, since each structure map has one entry in

O, let it be for example Oa

b1, . . . , bmwith

(a, b)∈ S × P rof (S) for some m∈N∗, then

|ψ|,|ϕ| ∈ O a

b1, . . . , bm, therefore they have the

same structure map, hence |ψ|=|ϕ|, so ψand ϕ

are two simplices equivalent in O. This ﬁnishes

the proof.

Our second lemma corresponding to the k-

moves states that

Lemma 3.2. Given a n-simplex ψin Owith ψ=

(ψ1, . . . , ψn), for any k < n, and any ϕ∈ Wk

ψ,

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one have

µϕ∈ Wk

µψ

in other words, if ψand ϕare equivalent by a

relaxed k-moves in O, then µψand µϕare equiv-

alent by a relaxed k-moves in O-Algebra A.

In fact

ϕ∈ Wk

ψ⇒µϕ∈ Wk

µψ

Proof. Let ϕ∈ Wk

ψ, then there exist 1 ≤k′, k”≤

k , 1≤s≤n−k′, and (n−k′+k”)-simplex ϕ

with

1. ψj=ϕjfor all 1 ≤j≤s−1,

2. ψj=ϕj−k′+k”for all s+k′≤j≤n,

3. ζ= (ψs, . . . , ψs+k′−1) is a sub-simplex of ψ,

and ξ= (ϕs, . . . , ϕs+k”−1) is a sub-simplex

of ϕsuch that |ζ|=|ξ|, which is compatible

with the operadic composition ◦iof ψs+k′=

ϕs+k”.

We conclude by the Lemma 3.1 and the Corollary

3.2

1. µψj=µϕjfor all 1 ≤j≤s−1,

2. µψj=µϕj−k′+k”for all s+k′≤j≤n,

3. µζ= (µψs, . . . , µψs+k′−1) is a sub-simplex of

µψ, and µξ=µϕs, . . . , µϕs+k”−1is a sub-

simplex of µϕsuch that µ|ζ|=µ|ξ|, which is

compatible with the operadic composition ◦i

of µψs+k′=µϕs+k”.

Then µψand µϕare equivalent by a relaxed k-

moves in O-Algebra A.

As a direct consequence, now we can announce

the following :

Theorem 3.2. Under the same hypotheses of

Deﬁnition 2.6, consider ζand ξtwo simplices in

Osuch that |ζ|=|ξ|is an elementary relation in

O, then its corresponding elementary relation in

Ais µ|ζ|=µ|ξ|.

Proof. Let |ζ|=|ξ|be an elementary relation

in O, then ζand ξare two simplices that are

equivalent in O, by the Corollary 3.2, their cor-

responding simplices in Aare equivalent, hence

|µζ|=µ|ζ|=µ|ξ|=|µξ|, and the previous Lemma

3.2 provides that this relation is an elementary

relation in Awhich is the corresponding of the

elementary relation |ζ|=|ξ|in O.

Remarks

•For the operad WD (resp. UWD) , we

know from the ﬁnite presentation theorem

in Chapter 5, in [1] (resp. in Chapter

10, in [1]) that one may substitute a sub-

simplex within a simplex presentation by an-

other, only by allowing substitution of an el-

ementary sub-simplex of one side of an el-

ementary relation or an operad associativ-

ity/unity/equivariance axiom involving only

generating wiring diagrams by the elemen-

tary sub-simplex of the other side,

•in theorems 3.2, we saw that each elementary

relation in Ohas a corresponding elementary

relation in A. Now we will prove the cor-

responding associativity/unity/equivariance

axiom for generating multimaps in A.

Proposition 3.1. The associativity axiom holds

in A(for the deﬁnition see 2.11 and 2.12 in [1]).

Proof. For some integers n≥2, m, l ≥1, and

1≤i<j≤nwhere |c|=n,|b|=m, and |a|=l.

Let f∈ O d

c,g∈ O ci

a, and h∈ O cj

b, be

some generating multimaps in O, then the hori-

zontal associativity in Ostates that

(f◦jh)◦ig= (f◦ig)◦j−1+lh

the corresponding equality in Aof this equality

is given by

µ(f◦jh)◦ig=µ(f◦ig)◦j−1+lh

by the associativity axiom in A, we obtain

(µf◦jµh)◦iµg= (µf◦iµg)◦j−1+lµh

Since f, g and hare generating multimaps in O,

then µf,µg, and µhare generating structure

maps in Acorresponding respectively to f, g and

h. Hence the last equality is the corresponding

associativity of the horizontal associativity in A.

Suppose now n, m ≥1, 1 ≤i≤n, and 1 ≤

j≤m.

Let f∈ O d

c,g∈ O ci

b, and h∈ O bj

a,

are some generating multimaps in O, then the

vertical associativity in Ostates that

f◦i(g◦jh)=(f◦ig)◦i−1+jh

the corresponding equality in Aof this equality

is given by

µf◦i(g◦jh)=µ(f◦ig)◦i−1+jh

by the associativity axiom in A, we have

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µf◦i(µg◦jµh)=(µf◦iµg)◦i−1+jµh

where µf,µg, and µhare generating structure

maps in A, corresponding respectively to the gen-

erating multimaps f, g and hin O. Then the last

equality is the corresponding associativity of the

vertical associativity in A.

Remark: By the operad associativity in O, we

mean both the horizontal and the vertical asso-

ciativity, and by the O-operad associativity in A,

we mean their corresponding in A.

Proposition 3.2. The unity axiom holds in A

(for the deﬁnition see 2.13 and 2.14 in [1]).

Proof. Let f∈ O d

cbe a generating multimap,

1d∈ O d

dand 1dthe d-colored unit in O, and

1ci∈ O ci

cithe ci-colored unit in O.

The left unity in Ostates 1d◦1f=f. Then,

the corresponding equality in Aof this equality

is given by

µ1d◦1f=µf

by associtivity axiom in A, we get

µ1d◦1µf=µf

On the other hand, the right unity in Ostates

that

f◦i1ci=f.

the corresponding equality in Aof this equality

is given by

µf◦i1ci=µf

Once again, the associtivity axiom in A, assert

µf◦iµ1ci=µf

where µfis a generating structure map corre-

sponding to the generating multimap fin O, and

µ1ci,µ1dare the identity maps in Acorrespond-

ing respectively to the ci-colored and d-colored

unit in O. Then the corresponding unity axiom

in Aholds.

Proposition 3.3. The equivariance axiom holds

in A(for the deﬁnition see 2.15 in [1]).

Proof. For some integers n, m ≥1 and 1 ≤i≤n

where |c|=n,|b|=m,σ∈Sn, and τ∈Sm.

Let f∈ O d

c, and g∈ O cσ(i)

b, be generating

multimaps in O, then the equivariance axiom in

Ostates

fσ◦igτ= (f◦σ(i)g)σ◦iτ

the corresponding equality in Aof this equality

is given by

µfσ◦igτ=µ(f◦σ(i)g)σ◦iτ

by the associativity axiom in A, we obtain

µfσ◦iµgτ=µ(f◦σ(i)g)σ◦iτ

where fσ∈ O d

cσ,gτ∈ O cσ(i)

bτ and (f◦σ(i)

g)σ◦iτ∈ O d

(c◦σ(i)b)(σ◦iτ).

Since f, g are generating multimap in O, then fσ,

gτare too, hence µfσ,µgτare generating struc-

ture maps in Acorresponding respectively to the

generating multimaps fσ,gτin O.

Then the last equality is the equivariance axiom

in Acorresponding to the equivariance axiom in

Vocabulary.

Let Obe an operad equipped with a ﬁnite gener-

ating set T, and Athe O-algebra equipped with

the corresponding ﬁnite generating set Tµ.

•every structure map in Ahas a presentation

in Tµ, then there exists a simplex in Tµwhich

is presentation of this structure map. The

equality relation deﬁned in the Theorem 3.1

|µζ|=µ|ζ|=µ|ξ|=|µξ|

is either an O-operad associativity or unity

or an equivariance axiom, or an elementary

relation in A,

•an elementary sub-simplex µb

ψof µψis a sub-

simplex of one of two following forms:

–µb

ψis one side (either left or right) of a

speciﬁed elementary relation in A,

–µˆ

ψis one side (either left or right) of a

speciﬁed O-operad associativity or unity

or equivariance axiom involving only the

generating sturture maps.

Suppose that we have a n-simplex µψ

(for some integer n≥2) in Awhich is

a presentation of the structure map |µψ|

in A, and µζis a sub-simplex of µψsuch

that |µζ|=|µξ|, where µξis a simplex

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in A, then the relation |µζ|=|µξ|is ei-

ther an elementary relation in Aor an O-

operad associativity or unity or equivari-

ance axiom involving only the generat-

ing structur maps, hence, one can obtain

a relaxed k-moves of the simplex µψby

substituting the elementary sub-simplex

µζby the other one µξ,

•two simplicies µψand µϕare called to be el-

ementarily equivalent in A, if µψand µϕare

equivalent by a relaxed k-moves (for some

integer k≥2), then we write µψ

∼µϕ(if

there is no confusion, we can drop kand write

µψ∼µϕ) and call this an elementary equiv-

alence in A, in other words an elementary

equivalence is a subsitution of a elementary

sub-simplex of one side by the other one.

•two simplicies µψand µϕare said to be

connected by a ﬁnite sequence of elementary

equivalences in Aif and only if there exist

some simplicies µψ1, . . . , µψrin Asuch that

µψ1

∼. . . kr−1

∼µψr, and µψ

∼µψ1

∼. . . kr−1

∼

µψr

∼µϕfor some integers k, k1, . . . , kr≥2.

Denotations. Let Obe an operad with a ﬁnite

generating set Tand Aits associated O-algebra

with the corresponding ﬁnite generating set Tµ.

Let ψbe an n-simplex in Oand µψits corre-

sponding n-simplex in A.

•The set of all relaxed k-moves of µψin Tµis

denoted by Wk

µψ,

•The set of all relaxed k-moves for all k∈

{2, . . . , n −1}is

µψ:=

n−1

[

k=2

µψ,

and this is the set of all relaxed n-moves of

µψin Tµ. In other words, Wn

µψis the set of

all elementary equvalences of µψin A.

•The set of all relaxed n-moves in Tµis de-

noted by Wµ, where

Wµ:= [

µψ∈A

µψ

In other words, Wµis the set of all elementary

equivalences in Tµ.

Lemma 3.3. Let ψand ϕbe two simplices ele-

mentarily equivalent in O, then their correspond-

ing simplices µψand µϕare elementarily equiva-

lent in A. In other words

ψ∼ϕ⇒µψ∼µϕ

Proof. Let ψand ϕbe two simplices elementarily

equivalent in O, then ψand ϕare equivalent by

a relaxed k-moves, then ϕ∈ Wk

ψ, by the Lemma

3.2, we get µϕ∈ Wk

µψ, where µψand µϕare the

corresponding simplices of ψand ϕrespectively in

A, in other words, µψand µϕare equivalent by a

relaxed k-moves in A, then they are elementarily

equivalent in A.

Theorem 3.3. Let ψand ϕbe two simplices con-

nected by a ﬁnite sequence of elementary equiva-

lences in O, then their corresponding simplices

µψand µϕare connected by a ﬁnite sequence

of elementary equivalences in A. More precisely

if ψ1, . . . , ψris a ﬁnite sequence of elementary

equivalences in O, for an integer r≥1such that

ψ∼ψ1∼ · · · ∼ ψr∼ϕ

Then

µψ∼µψ1∼ · · · ∼ µψr∼µϕ

where µψ1, . . . , µψris the corresponding ﬁnite se-

quence of elementary equivalences of ψ1, . . . , ψr

in A.

Proof. Let ψand ϕbe two simplices connected

by a ﬁnite sequence of elementary equivalences in

O, then there exist simplices ψ1, . . . , ψrsuch that

ψ∼ψ1∼ · · · ∼ ψr∼ϕ, since ψ∼ψ1, then

the Lemma 3.3 assert that µψ∼µψ1, similraly

µψ1∼µψ2, . . . , µψr∼µϕ.

Hence

µψ∼µψ1∼ · · · ∼ µψr∼µϕ

Lemma 3.4. Let Obe an operad and Aits asso-

ciated algebra, let Tbe a ﬁnite generating set for

O, and Tµits corresponding ﬁnite generating set

of A, if Wthe set of all elementary equivalences

in Tis a strong generating set of Oin T, then

Wµthe set of all elementary equivalences in Tµis

a strong generating set of Ain Tµ.

Proof. Let µψ= (µψ1, . . . , µψr) and µϕ=

(µϕ1, . . . , µϕs) be two simplices in Tµfor some

integers r, s ≥1 that are equivalent, then their

composition are equal in A, i.e |µψ|=|µϕ|,

the Corollary 3.2 assert that the two simplices

ψ= (ψ1, . . . , ψr) and ϕ= (ϕ1, . . . , ϕs) are equiva-

lent in O, i.e |ψ|=|ϕ|, since Wis a strong gener-

ating set of Oin T, then ψand ϕare connected

by a ﬁnite sequence of elementary equivalences in

W, then there exist φ1, . . . , φlfor an integer l≥1

such that ψ∼φ1∼ · · · ∼ φl∼ϕ, by the previous

Lemma 3.3, we get µψ∼µφ1∼ · · · ∼ µφl∼µϕ,

where µφ1, . . . , µφlis a ﬁnite sequence of ele-

mentary equivalences in Wµ. This ﬁnishes the

proof.

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As we have collected all necessary tools, we can

now formulate and prove our following main the-

orem :

Theorem 3.4. Suppose that Ohas a ﬁnite pre-

sentation, then Ahas a corresponding ﬁnite pre-

sentation one.

Proof. Firstly, we have to prove that every struc-

ture map can be expressed as a ﬁnite iter-

ated operadic composition in terms of generating

structue maps in A.

Indeed, let µψbe a simplex in Afor a ﬁxed

simplex ψin Owhose composition is µ|ψ|, let

T={ψ1, . . . , ψd}be a ﬁnite generating set for

Owhere d∈N∗, then there exist ψ1, . . . , ψl∈ T

for l∈N∗sucht that ψ= (ψ1, . . . , ψl) and

|ψ|=ψ1◦i1· · ·◦il−1ψlfo some integers ı1, . . . , il−1≥1

By Theorem 3.1, the corresponding ﬁnite gen-

erating set of Tin Ais Tµ={µψ1, . . . , µψd},

and by Lemma 3.1 the corresponding l-simplex

of (ψ1, . . . , ψl) in Ais (µψ1, . . . , µψl) whose com-

position is given by

|µψ|=µ|ψ|=µψ1◦i1···◦il−1ψl=µψ1◦i1· · · ◦il−1µψl

Since this is true for every structure map in A,

then every structure map can be expressed as a

ﬁnite iterated operadic composition in terms of

generating structue maps in Tµwhich is a ﬁnite

generating set of A.

Secondly, it remains to prove that if a struc-

ture map can be operadically generated by the

generating structure maps in two diﬀerent ways,

then there exists a ﬁnite sequence of elementary

equivalences in Wµfrom the ﬁrst iterated op-

eradic composition to the other one.

Let µ|φ|be a structure map in A, for a ﬁxed |φ|

in Owhich is generated by the generating struc-

ture maps in two diﬀerent ways, then there exist

two simplices in O,ψ= (ψ1, . . . , ψr) in which the

composition is |ψ|and ϕ= (ϕ1, . . . , ϕs) in which

the composition is |ϕ|for some integers r, s ≥1

such that

µ|φ|=µ|ψ|=µ|ϕ|

By the corollary 3.2, we get |ψ|=|ϕ|, then ψ

and ϕare equivalent in T, since Wis a strong

generating set for Oin T, then there exist a ﬁnite

sequence of elementary equivalences φ1, . . . , φm

in W, such that ψ∼φ1∼ · · · ∼ φm∼ϕfor

some integer m≥1,the Theorem 3.3 assert that

µψand µϕare connected by a ﬁnite sequence of

elementary equivalences in Wµ

µψ∼µφ1∼ · · · ∼ µφm∼µϕ

By the Lemma 3.4, Wµis a strong generating set

of Ain Tµ, hence µψand µϕare connected by

ﬁnite sequence of elementary equivalences in Wµ.

This ﬁneshes the proof

Our approach’s advantage that it allow us

to get a ﬁnite presentation of the O-algebra

Adirectly out of the ﬁnite presentation of the

operad O.

Unfortunately, we wanted to prove that any

multimap (resp. structure map) in the Ooperad

with a ﬁnite generating set has a stratiﬁed pre-

sentation which is a simplex presentation where

the same generating multimaps within the sim-

plex must appear in a consecutive serie, for ex-

ample, suppose Oan operad with the ﬁnite gen-

erating set T= (ψ1, ψ2, ψ3), consider the follow-

ing 9-simplex (ψ1, ψ2, ψ3, ψ1, ψ3, ψ2, ψ3, ψ1, ψ3)

then the stratiﬁed 9-simplex is given by

(ψ1, ψ1, ψ1, ψ2, ψ2, ψ3, ψ3, ψ3, ψ3). D.Yau had

proved that every wiring diagram in WD (resp.

undirected wiring diagram in UWD) has a strat-

iﬁed presentation (see Theorem 5.11 and 10.12 in

[1]), but we couldn’t prove that for a such simplex

in O.

4 Applications

As an application of our main theorem, we will

consider the operad of wiring diagrams, denoted

WD, and that of undirected wiring diagrams, de-

noted UWD. We suppose here that the reader is

familiar with the basic tools, vocabulary of the

operad of wiring diagrams (directed or not) and

its structure. If not, we recommend, [1]. Let

us recall that a multimap in WD (respectively

in UWD) is called a wiring diagram (respec-

tively undirected wiring diagram), and a gener-

ating multimap in WD (respectively in UWD) is

called a generating wiring diagram (respectively

a generating undirected wiring diagram). In fact,

D. Yau has especially proved that both WD and

UWD have a ﬁnite presentation (Theorem 5.22 in

[1], for WD), (Theorem 10.19 in [1], for UWD).

In the chapter 3 in [1] D. Yau described the

set

TWD ={ϵ, δ, τ, θ, λ, σ∗, σ∗, ω}

of 8 wiring diagrams which is a ﬁnite generating

set for the operad WD. In the chapter 8 in [1] he

had described the set

TUWD =nϵ, ω∗, τf, θ(X,Y ), λ(X,x±), σ(X,x1,x2)o

of 6 undirected wiring diagrams which is a ﬁnite

generating set for the operad UWD. Then ev-

ery wiring diagram (resp. undirected wiring dia-

gram) has a presentation in TWD (resp. TUWD)

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as a ﬁnite iterated operadic composition. That is

the ﬁrst assumption of our theorem.

For the wirings diagrams, in the chapter 3 in

[1] D. Yau described 28 elementary relations in

TWD, and in the chapter 5 in [1] he had proved

that any two equivalent simplices in WD are ei-

ther equal or are connected by a ﬁnite sequence

of elementary equivalences where each elemen-

tary equivalence is induced by either one of the

28 elementary relations or an operad associativ-

ity/unity/equivariance axiom for the generating

wiring diagrams. Following our denotations, the

set WWD of all elementary equivalences in TWD

is a strong generating set of WD. That is the

second assumption of our theorem.

Hence, by our main theorem 3.4, the WD-algebra

has a corresponding ﬁnite presentation.

By using our approach in Theorem 3.1, the cor-

reseponding ﬁnite generating set for the WD-

algebra is given by

TWD

µ={µϵ, µδ, µτ, µθ, µλ, µσ∗, µσ∗, µω}.

which elements are called the generating struc-

ture maps corresponding to the 8 generating

wiring diagrams. Then every structure map can

be obtained from ﬁnitely many generating struc-

ture maps via some iterated operadic composi-

tions. These generating structure maps are ex-

actly those deﬁned by D. Yau (see deﬁnition 6.9

in [1]). For example the ﬁrst generating wiring

diagram in WD is the empty wiring diagram

ϵ∈ WD ∅(see Deﬁnition 3.1 in [1]), then

its corresponding generating structure map in the

WD-algebra is deﬁned to be µϵ:∗ −→ A∅(see

Deﬁnition 6.9 in [1]).

In other hand, by applying Theorem 3.2 to

the operad WD, we ﬁnd that every elementary

relation in WD has a corresponding elementary

relations in the WD-algebra. Then every one

of the 28 elementary relations in WD (see def-

inition 3.43 in [1]) has a corresponding elemen-

tary relation in the associated WD-algebra that

are exactly the generating axioms deﬁned by D.

Yau (see deﬁnition 6.9 in [1]). For example,

the ﬁrst elementary relation in WD (see propo-

sition 3.15 in [1]) says that if τb,a ∈ WD a

b

and τc,b ∈ WD b

care two consecutive name

changes, then they can be composed into one

name change τc,a ∈ WD a

c, i.e

τb,a ◦τc,b =τc,a

By using our Theorem 3.2, its corresponding ele-

mentary relation in WD is given by

µτb,a ◦µτc,b =µτc,a

where µτx,y := Ax−→ Ay.

This last equality is exactly the ﬁrst generating

axiom in WD-algebra deﬁned by D.Yau (see def-

inition 6.9 in [1]). The corresponding operad as-

sociativity, unity and equivariance axiom can be

obtained immediately from the propositions 3.1,

3.2 and 3.3.

In [1] D.Yau had proved that (Theorem 5.22)

any two equivalent simplices in WD are either

equal or are connected by a ﬁnite sequence of ele-

mentary equivalences in WD which each elemen-

tary equivalence is induced by either one of the

28 elementary equivalences or an operad associa-

tivity/unity/equivariance axiom for the generat-

ing wiring diagrams in WD, hence, by our The-

orem 3.3 any two equivalent simplices in WD-

algebra are either equal or are connected by a ﬁ-

nite sequence of elementary equivalences in WD-

algebra, which each elementary equivalence in

WD-algebra is induced by either one of the 28

generating axioms or an WD-algebra operad as-

sociativity/unity/equivariance axiom for the gen-

erating structures maps. Following our denota-

tions, the set WWD

µof all elementary equivalences

in TWD

µis a strong generating set of WD-algebra.

Then, WD-algebra has a ﬁnite presentation.

The same can be done for the operad UWD,

of undirected wiring diagrams by considering its

ﬁnite generating set

T=nϵ, ω∗, τf, θ(X,Y ), λ(X,x±), σ(X,x1,x2)o,

whose corresponding ﬁnite generating set for the

WD-algebra is

Tµ=nµϵ, µω∗, µτf, µθ(X,Y ), µλ(X,x±), µσ(X,x1,x2)o.

and the elementary relations in UWD is given by

D. Yau (see deﬁnition 8.26 in [1] ), and their cor-

responding elementary relations in UWD-algebra

that are called by D. Yau generating axioms (see

deﬁnition 11.1 in [1]).

For more application, the reader can see the op-

erad of normal wiring diagrams and strict wiring

diagrams and their algebras.

5 Future Work

As an extension of our study, we will try to in-

vestigate our result and discribe the operad with

two inputs and one output, a such operad is of

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Volume 22, 2023

the form OZ

X Y , where X, Y, Z ∈ S.

Let ψbe a multimap in OZ

X Y , by the ◦i-

composition in OZ

X Y , we can decompose ψ

as follows :

OZ

I× O I

I I× O I

X× O I

Y

(◦1;Id;Id)

−→ O Z

I I× O I

X× O I

Y

(◦1;Id)

−→ O Z

X I× O I

Y◦2

−→ O C

X Y 

Here, Iis a speciﬁc element in S.

For a speciﬁc elements X, Y and Zin S, we will

try to ﬁnd a ﬁnite generating set Tfor this op-

erad.

By putting

ψZ∈ O Z

I

ψI∈ O I

I I

ψX∈ O I

X

We obtain

ψ= ((ψZ◦1ψI)◦1ψX)◦2ψY

By the associativity axiom in O, we get also

ψ= ((ψZ◦1ψI)◦2ψY)◦1ψX

ψ=ψZ◦1((ψI◦1ψX)◦2ψY)

ψ=ψZ◦1((ψI◦2ψY)◦1ψX)

ψ= (ψZ◦1(ψI◦1ψX)) ◦2ψY

ψ= (ψZ◦1(ψI◦2ψY)) ◦1ψX

These equalities give all the presentation possible

of ψ, then there is 6 simplices, in fact, six

4-simplices that can be a presentation of ψ.

from these equalities, we check the following

elementary relations in OZ

X Y :

(ψZ◦1ψI)◦1ψX=ψZ◦1(ψI◦1ψX)

(ψZ◦1ψI)◦2ψY=ψZ◦1(ψI◦2ψY)

(ψI◦1ψX)◦2ψY= (ψI◦2ψY)◦1ψX

The six 4-simplices are :

ψ1= (((ψZ,1, ψI),1, ψX),2, ψY)

ψ2= (((ψZ,1, ψI),2, ψY),1, ψX)

ψ3= (ψZ,1,((ψI,1, ψX),2, ψY))

ψ4= (ψZ,1,((ψI,2, ψY),1, ψX))

ψ5= ((ψZ,1,(ψI,1, ψX)),2, ψY)

ψ6= ((ψZ,1,(ψI,2, ψY)),1, ψX)

References:

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[2] May, J.P. What Is ... an Operad. Vol. 271.

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AMS, 2002.

[4] Stasheﬀ J.D. .The geometry of iterated loop

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[5] Yau, D., Colored Operads, Graduate Studies

in Math. 170, Amer. Math. Soc., Providence,

RI, 2016.

[6] Loday, J. L., Vallette, B. . Algebraic Oper-

ads. Vol. 346. Grundlheren der Mathematis-

chen Wissenschaften, Berlin Springer, 2012.

[7] Rupel, D., Spivak, D.I., The operad of tem-

poral wiring diagrams: formalizing a graph-

ical language for discrete-time processes,

arXiv:1307.6894.

[8] Spivak, D.I. The operad of wiring dia-

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databases, recursion, and plug-and-play cir-

cuits, arXiv:1305.0297.

[9] Spivak, D.I., Category Theory for the Sci-

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[10] Yau, D., and Johnson, M.W., A Foundation

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