A Topological Perspective on Interacting Algebraic Theories

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by  Amar Hadzihasanovic

A Topological Perspective on
Interacting Algebraic Theories
AmarHadzihasanovic
UniversityofOxford
[email protected]
Techniquesfromhighercategoriesandhigher-dimensionalrewritingarebecomingincreasinglyim-
portantforunderstandingthefiner,computationalpropertiesofhigheralgebraictheoriesthatarise,
among other fields, in quantum computation. These theories have often the property of contain-
ingsimplersub-theories,whoseinteractionisregulatedinalimitednumberofways,whichrevealsa
topologicalsubstratewhenpicturedbystringdiagrams.Byexploringthedoublenatureofcomputads
aspresentationsofhigheralgebraictheories,andcombinatorialdescriptionsof“directedspaces”,we
developabasiclanguageofdirectedtopologyforthecompositionalstudyofalgebraictheories. We
present constructions of computads, all with clear analogues in standard topology, that capture in
greatgeneralitysuch notionsas homomorphismsand actions, and the interactionsof monoidsand
comonoidsthatleadtothetheoryofFrobeniusalgebrasandofbialgebras. Afteranumberofexam-
ples,wedescribehowafragmentoftheZXcalculuscanbereconstructedinthisframework.
1 Introduction
Atraditional presentation ofanalgebraic theory consists ofanumberofgenerating operations, together
with a number of equations that they satisfy. If we are concerned with computational aspects of the
presentation—lookingfornormalisationprocedures,forinstance—itiscommonplacetoreplaceequa-
tions with directed rewrite rules. Then, in the analysis of critical pairs and confluences of a rewrite
system, we are led to consider relations between different sequences of rewrites, which can in turn be
relaxedto“rewritesofrewrites”,andsoon,leadingintohigher-dimensional rewritingtheory[35]. From
thisperspective, thedichotomy betweengenerators andrelations inapresentation isresolved: theyboth
becomegenerators ofahigher-dimensional algebraic theory, onlydifferingindimension.
The natural setting for higher-dimensional rewriting is higher category theory, where, besides the
objects (0-cells) and morphisms (1-cells) of basic category theory, there can also be n-cells between
(n−1)-cells, for any n>0. The use of terminology borrowed from topology is not coincidental: there
is a sense in which the “directed n-cells” of higher categories behave like topological n-cells. This is
exemplified by the successful application of methods from homology theory in the study of rewriting
systems, based on this analogy [28, 29]; however, it is perhaps best pictured through the use of string
diagrams[38,26](or,morerecently, surface diagrams[18])forreasoning abouthighercategories.
Inoneespeciallyrelevantapplication,stringdiagramshaveemergedasastrongcontenderforahigh-
level,nativesyntaxforquantumprogramming[44,15],whosehighlysymmetricalsemantics—inpure,
finite-dimensional quantum theory, all processes are reversible, and inputs can be turned into outputs
and vice versa [37] — require a quite unusual amount of interplay between algebraic and coalgebraic
structures.
Betterunderstanding thecomputational properties oftheoriessuchastheZXcalculus[14,2]andits
refinement, the ZW calculus [22], is pivotal in making them viable for the efficient design of quantum
(cid:13)c A.Hadzihasanovic
R.DuncanandC.Heunen(Eds.):QuantumPhysicsandLogic(QPL)2016
Thisworkislicensedunderthe
EPTCS236,2017,pp.70–86,doi:10.4204/EPTCS.236.5
CreativeCommonsAttributionLicense.
A.Hadzihasanovic 71
algorithms and protocols. Although these theories include, as a whole, a relatively large number of
axioms, they contain a number of simpler sub-theories, whose interactions are regulated by the axioms
in fairly predictable ways: something is ahomomorphism of something else, something is an action on
something else...
Beyond thesemotivating examples, such afactorisation seemstobeaproperty ofmanytheories: as
asimplecase, thinkof∗-monoids, whichcanbeseenasaninteraction ofthetheory ofmonoids andthe
theoryofinvolutions. Soweaskedourselvesthequestion:
• Isthere awaytostudyalgebraic theories compositionally, sothatonecanderiveproperties ofthe
largertheoryfromitscomponents, andthefewwaysinwhichtheyareallowedtointeract?
There has already been, in fact, an attempt to develop a compositional algebra, through Lack’s “com-
posing PROPs” framework [27]. In this setting, a presentation of a fragment of the ZX calculus — the
theory of interacting bialgebras — was successfully constructed from the theories of monoids and of
comonoids [7]. Thereare,nevertheless, twodownsides tothisapproach, relativetoourobjectives.
Firstly, composition relies on the choice of a “distributive law”, which conceptually amounts to
stating what the normal form for operations of the resulting theory should be. Thus, we can already
tellwhattheresulting theory willglobally looklike,whichsubtracts something fromitsheuristic value,
especially whenweonly haveanalgebraic presentation athand. Infact, “composing PROPs”ismostly
usefultoderiveaxiomswhenconcretemodelsofthecomponenttheoriesareavailable,alsosuggestinga
“concrete” wayofcomposing them.
Secondly, it is a flat composition, in that it works in a strictly 2-categorical framework, and fails to
account for any of the topological properties of the interactions. For instance, two specular distributive
laws for monoids and comonoids lead, respectively, to the theory of special Frobenius algebras, and to
thetheoryofbialgebras. Boththetheoryofmonoidsandthetheoryofcomonoids areplanar —noneof
the axioms require the swapping of inputs or outputs of operations — and so isthe theory ofFrobenius
algebras; hence, the interaction leading to Frobenius algebras is not supposed to change the dimension
ofgenerators.
On the other hand, the bialgebra law — apart of the theory of bialgebras — isnot planar, and is in
factbestrepresented byastringdiagram in3dimensions, wherethemonoidpartandthecomonoidpart
belongtodifferent, orthogonal planes,asinthefollowingpicture.
=
Sotheinteraction leading tobialgebras should beofadifferent, dimension-increasing sort.
In this paper, we try to lay the groundwork for an alternative approach, and make a case for the
followingassertions:
1. thatthereexistsawayofstudyingalgebraictheoriescompositionally, withasmallnumberofbasic
constructions corresponding tothemostfrequentinteractions;
2. thatthelanguage forcompositional algebra isakindofcombinatorial directedtopology, allinter-
actionshavingclearanalogues instandard topology.
In Section 2, we briefly present our technical framework of choice, the theory of computads or
polygraphs, andbuildabasicvocabulary ofdirected topology inthiscontext. InSection3,weusethese
72 ATopologicalPerspectiveonInteractingAlgebraicTheories
toolstoconstructpresentationsofsomebasictheories,suchasthetheoryofmonoids,fromevensimpler
ones. In Section 4, we show how two kinds of interaction capture the notion of homomorphism and of
(co)action. Finally, in Section 5, we introduce the dimension-changing operation which enables us to
obtainthetheoryofbialgebras(andofcommutativemonoids)fromthetheoryofmonoids. Weconclude
bydescribing apartialreconstruction “fromscratch” ofthetheory ofinteracting bialgebras, anddiscuss
someofthemanypossiblefurtherdirections ofthisproject.
2 Computads and directed spaces
A computad is, informally, a presentation of a higher category “by generators and no relations”. This
notion, introduced byStreet [40]in the2-dimensional case, was rediscovered and expanded by Burroni
[13] in the context of higher-dimensional rewriting theory, where it is known under the name of poly-
graph. Wewillonlyusetheoriginalsettingofcomputadsforstrictw -categories,althoughweakvariants
havealsobeenconsidered [4].
Acomputad describes howtobuild ahigher category byprogressively adjoining generating cells of
higher dimension, whose border is a pasting of lower-dimensional cells, and letting composition be, at
eachstage,thefreepastingofoldandnewcells.
Like Lawvere theories, PROs, and operads [32], computads can be used to describe algebraic the-
ories, to be internalised in arbitrary higher categories through appropriate “semantical” functors. The
equational lawsbywhichsuchtheories arenormallypresented needtohavecorresponding highercells,
and are, therefore, directed by default: this makes computads a natural choice for studying computa-
tional aspects of presentations (normalisation, confluence...), and has led to a particular interest among
rewritingtheorists [21,28].
The information contained in a computad is conceptually analogous to the description of a topo-
logical space as a CW complex, with directed cells replacing the undirected, topological cells. The
formalism of higher categories can be seen as an auxiliary, combinatorial tool for specifying how cells
are glued together, corresponding to the gluing maps of point-set topology; in other words, acomputad
is a combinatorial description of a directed space, in the spirit of Grandis [20]. There may be ways to
makeanexplicitconnection; fornow,wetakethisasnomorethanaguiding heuristics.
Thesimplestandmoredirectcombinatorics ofpasting areprovidedbyglobular w -categories. Since
technicaldetailsarenotparticularlyimportantintheremainder,weleavethepresentationquiteinformal,
andrefer, forinstance, to[34]formoredetails. PreliminarydefinitionsareinAppendixA.
Definition 2.1. A computad X is apair ofan w -category X and asubset |X|⊆X, such that, if X is
n
then-skeleton ofX,and|X| :={s ∈|X||d(s )=n},forn≥0:
n
• X =|X| ;
0 0
• X isobtained fromX byfreelyadjoining thecellsof|X| .
n n−1 n
A map f :X →Y of computads is a function of sets |f|:|X|→|Y| that induces a functor of free
w -categories f :X →Y . Computadsandtheirmapsformacategory Cpt.
Thus, a computad is described by giving, for each n, a set X of generating n-cells, and specifying
n
their border as aformalcomposition ofthelower-dimensional generating cells. Amapof computads is
amappingofthegenerating cellsofX ontogenerating cellsofY preserving thecomposition ofborders.
Definition2.2. ThedisjointunionX⊕Y oftwocomputadsX andY isthecomputadwithgenerating set
|X⊕Y|:=|X|+|Y|,and
¶ a s , s ∈|X|,
¶ a s = X,n
n (¶ Ya,ns , s ∈|Y|.
A.Hadzihasanovic 73
Asubcomputad A⊆X isamapA→X ofcomputads whoseunderlying set-function isaninclusion
of sets |A|⊆|X|. Given a set B of cells of X, we also denote by B the smallest subcomputad of X that
contains them.
An equivalence relation E on a computad X is an equivalence relation on the set |X| of generating
cells, whose extension to composite cells commutes with all border operators. We denote by X/E the
quotient ofX bytheequivalencerelationE,withgeneratingset|X|/E. GivenasubcomputadA⊆X,we
alsowriteX/Aforthequotient ofX bytheequivalence relation |A|×|A|.
Theseoperationsareanalogoustothecorrespondingoperationsontopologicalspaces,somethingwe
try tohighlight through notation and terminology: webuild the disjoint union of X andY byseparately
attaching thegenerating cellsofX andofY,andobtainaquotient spacebyidentifying cellscompatibly
with their borders. The disjoint union is a categorical coproduct in Cpt [31]; however, the product
of topological spaces does not correspond to the categorical product of computads, but to a different
monoidalstructure —thecomputadic versionoftheCrans-Graytensorproduct ofw -categories [16].
Definition 2.3. The tensor product X⊗Y of two computads X andY is the computad with generating
set|X⊗Y|:=|X|×|Y|,where
n
(cid:229)
|X⊗Y| = |X| ×|Y| ,
n k n−k
k=0
and, for all generating n-cells s ⊗t , s ∈|X| , t ∈|Y| , attaching is characterised by the following
k n−k
condition. Let
+ ifniseven,
e (n)=
(− ifnisodd.
Byinduction: (X ⊗Y ) isjustX ×Y . Foralln>0,in(X ⊗Y ) ,extendthe−⊗−operation
0 0 0 n−1
tocompositecellsbywriting(r # r ′)⊗p forthepastingofr ⊗p andr ′⊗p alongtheirsharedborder
m
¶ + r ⊗p =¶ − r ′⊗p ,andsimilarlyforr ⊗(p # p ′).
m−1 m−1 m
Then,¶ a (s ⊗t )isobtained bypastingthe(n−1)-cells
n−1
¶ a s ⊗t and s ⊗¶ e(k)a t
k−1 n−k−1
alongtheirsharedborder¶ a s ⊗¶ e(k)a t .
k−1 n−k−1
The tensor product determines a non-symmetric monoidal structure on Cpt, with 1, the terminal
computad ofasingle0-cell,asunit.
TheexplicitcombinatoricsfortheborderoperatorsinX⊗Y arequitecomplicated;wewill,however,
onlyconsiderlow-dimensionalcases,ofwhichsomeexpressionsaregiveninAppendixB.Theapproach
tow -categories bycubical setswithconnections [9,10]leads toamuchsleeker definition oftheCrans-
Gray tensor product [1], but it is unclear whether a cubical description of computads would lead to an
overallsimplification, duetotheadditional complications relatedtothehandling ofthincells.
Remark 2.4. That this is a valid definition can be seen as following from the results of [39]. Our con-
ditions determine the tensor product for the category of loop-free augmented directed complexes with
unitalbases,which,moduloanadjustmentofterminology,isequivalenttoasubcategoryofw -categories
presented by computads that are loop-free inasuitable sense. These include, inparticular, the n-globes
Gn,generated byonen-cell⊤andonecellka forallk<n,a ∈{+,−},suchthat¶ a (⊤)=ka ,andthe
k
“walkingm-compositions” Gn# Gn (calledG[n]andG[n;m]inthereferenced paper).
m
74 ATopologicalPerspectiveonInteractingAlgebraicTheories
Thistensorproductdetermines,inturn,theCrans-Graytensorproductonthewholeofw Cat. LetX,
Y becomputads, andX ⊗Y bethe tensor product of thew -categories theygenerate. Foralls ∈|X| ,
k
t ∈|Y| ,s ⊗t corresponds toafunctor(s ⊗t )∗:Gk⊗Gn−k→X ⊗Y sending⊤⊗⊤tos ⊗t .
n−k
In particular, the border of s ⊗t is the image ofa cell of Gk⊗Gn−k. Butsince Gk⊗Gn−k is freely
generated by the ia ⊗ jb , and (s ⊗t )∗(ia ⊗ jb )=¶ a s ⊗¶ b t , the border of s ⊗t is generated by the
i j
¶ a s ⊗¶ b t . If¶ a s ,¶ b t aregenerators, thereisnothingtodo.
i j i j
Otherwise, suppose the firstone is a composite cell; by the interchange law, it can be written in the
form r # r ′ for somem>0, sothatnocomposition ofdimension d >mappears inr ,r ′. Byasimilar
m
argumentasbefore,usingthefactthat(r # r ′)⊗p correspondstoafunctor(Gi# Gi)⊗Gj→X ⊗Y ,
m m
we can rewrite the tensor as a composition of r ⊗p , r ′⊗p , and lower-dimensional cells. Continuing
likethisuntil allthehighest-dimensional compositions areeliminated, since generators canappear only
in finite number in the cells of X and Y , we finally obtain an expression of the border of s ⊗t as
a composition of tensors of lower-dimensional generators; so X⊗Y, as defined, is a presentation of
X ⊗Y .
All these constructions have analogues for topological spaces; the following, however, is purely
directed.
Definition2.5. LetX beacomputad, S⊆N. Then,Xop(S) isthecomputadwiththesamegenerating set
ofX,butwiththedirectionofn-cellsandn-compositionreversedforalln∈S;thatis,lettings op(S):=s
forall0-cellsofX,defineinductively
(¶ −a s )op(S), n∈S,
¶ a (s op(S)):= n−1
n−1 ((¶ na−1s )op(S), n6∈S,
t op(S)# s op(S), n∈S,
(s # t )op(S):= n
n (s op(S)#nt op(S), n6∈S,
for all numbers n, cells s , t of X , and a ∈{+,−}. Then, Xop(S) is generated by the s op(S), for all
n
s ∈|X|. WewriteXop:=Xop(N).
Inthedefinition ofthetensor product ofcomputads, theborder ofthetensor oftwocellsisreversed
when the border of the twocells, separately, is: itfollows that, for allcomputads X andY, (X⊗Y)op ≃
Xop⊗Yop.
Inwhatfollows,ourconstructionswilloftenresultinlaxversionsofalgebraictheories,whereequal-
ities are replaced by cells pointing inaspecific direction. Thesemay notalways bethe best choices for
computational purposes;wewillleaveitimplicitthatonecanalwaysreversecellsofagivendimension,
if needed. We will also tend to not distinguish between strict and lax versions of a theory: the theories
that weconsider are usually interpreted in low-dimensional categories, so we can leave it to semantical
functors tostrictify asnecessary, turning directedcellsintoequalities orisomorphisms.
3 Basic examples: Yang-Baxter, associativity, units
In this section, we look at some fundamental examples of interacting computads, with a focus on their
algebraicinterpretation. Thelatterisbetterunderstoodwhencellsarevisualisedasstringdiagramsinthe
styleof[26];forthefirstexamples,however,wewillalsoprovidethemoretraditional, dualpresentation
bypastingdiagrams.
A.Hadzihasanovic 75
The simplest 1-dimensional computad, and the basic ingredient of many later constructions, is the
directed intervalI,whichisthesameasthe1-globe G1.
a
0 1
Definition3.1. Thedirectedn-cubeisthecomputad In obtainedbytensoring ncopiesofI.
Thedirected2-cubemodelsan“interaction” operation withnoadditional relations. Wewritest for
s ⊗t inthepasting diagrams.
10
a0 1a
00 aa 11
0a a1
01
Further on, the directed 3-cube can be seen as the presentation of a coloured, directed version of the
Yang-Baxterequation.
100 1a0 110 100 1a0 110
a00 a10 11a a00 10a 11a
aa0 1aa
aaa
000 010 a1a 111 000 a0a 101 111
0a0 1a1
0aa aa1
00a 01a a11 00a a01 a11
001 0a1 011 001 0a1 011
(1)
Notingthatallcellsofthesamedimension havethesameshapeinI3,wecaneliminateallcolouring by
quotienting bytheequivalence relationE :={(s ,s ′)|d(s )=d(s ′)}. ModelsofI3/E,thatis,functors
from the 3-category it generates to a monoidal category C (as a 2-category with a single 1-cell), are
R-matricesinC [36].
Thereismuchmoretoquotientsofcubes: infact,theycoverallsortsofassociativity-like equations.
Weintroduce acouplegeneralconstructions forlateruse.
Definition 3.2. Let X be a computad. The cylinder of X is the computad I⊗X. The future cone of X
is the quotient computad C+(X):= (I⊗X)/({1}⊗X). The past cone of X is the quotient computad
C−(X):=(I⊗X)/({0}⊗X).
76 ATopologicalPerspectiveonInteractingAlgebraicTheories
One can obtain different variants of these constructions by reversing the directions of cells of any
dimension. All directed variants collapse for the corresponding constructions of CW complexes, the
usualcylinderandconeofatopological space.
We can see the n-cube as the n-th iteration of the cylinder construction on the terminal computad.
The corresponding iterations of the future cone produce Street’s oriented simplexes, or orientals [41].
Thisisprovenindetailin[12].
Definition3.3. Then-oriental isthecomputadC+.n..C+(1).
The2-oriental isjustabinaryoperation.
a0
00 1
aa
0a a1
01
The3-oriental isacoloured versionofanassociator.
a00 a00
000 1 000 1
0a0 a1a aaa a0a a01
00a 0aa a1 00a aa1 a1
001 0a1 01 001 0a1 01
The 4-oriental corresponds to the pentagonator, a directed version of MacLane’s pentagon [30]and
apartofthetheoryofpseudomonoids; weonlydrawitsstring-diagrammatic version.
Taking past instead of future cones gives the coalgebraic duals of these constructions: respectively, a
co-multiplication, co-associator, andco-pentagonator.
Clearly,thesecomputadscanalsobeobtaineddirectlyasquotientsofthen-cubes. Now,letE instead
betheequivalence relationonI3 with
0⊗s ⊗t ∼0⊗s ′⊗t ,
1⊗s ⊗t ∼1⊗s ⊗t ′,
A.Hadzihasanovic 77
for all s ,s ′,t ,t ′∈|I|. Then, the quotient I3/E — which roughly corresponds to taking apast cone on
one copyofI,and afuture cone onanother, “fibrewise” oncellsoftheother copy —presents aversion
of the Frobenius law (see [25, Chapter 4] for a review). This can be checked graphically by merging
regionsinDiagram(1).
(2)
Again, since both orientals and their duals have congruent shapes for cells of equal dimension, we
can quotient in order to obtain presentations of the theory of semigroups and co-semigroups, as lax —
thatis,withasmanynon-invertible highercells—aswewantthemtobe.
Now, we show how to obtain the theory of monoids by cones, in two steps; for comonoids, it will
sufficetodualiseeverything. Firstofall,weneedthe2-computadKpresentingthe“theoryofconstants”.
a h
Then, the future cone of K with reversed 1-cells, C+(K)op(1), is a 3-computad presenting an operation
witharightunit.
Now,takethefutureconeC+(C+(K)op(1));thiscanbepicturedasfollows.
ThisisadirectedversionofMacLane’striangleequality. Asusual,wecanidentifyallcellsofcongruent
shape; since we have a right unitor 3-cell for the “white-yellow” operation, and a new left unitor 3-cell
forthe“yellow-magenta” operation, uponidentification, theconstant becomesatwo-sidedunit.
Thus, byquotientingC+(C+(K)op(1))(orfurther iterations offuturecones), weobtain presentations
ofarbitrarily laxtheoriesofmonoids.
4 Homomorphisms and actions
Now that we have a list of basic examples, let us reconsider the constructions of Section 3 more in
general. GivenacomputadX,thecylinderofX hasthefollowingstructure:
78 ATopologicalPerspectiveonInteractingAlgebraicTheories
1. itcontains twocopiesofX,{0}⊗X and{1}⊗X;
2. for all generating n-cells s of X, it contains an (n+1)-cell a⊗s , which has 0⊗s in its input
border, and1⊗s initsoutputborder.
The category w Cat with the Crans-Gray tensor product actually admits a biclosed structure, so that a
functor h:I ⊗X →C corresponds to a functor h˜ :I →(X ⊸C), where X ⊸C is an internal
hom-object. ButI is just the “walking arrow” category, so h˜ is a morphism in X ⊸C — which can
be seen as ahigher-dimensional version ofa category of X-algebras in C; hence, h˜ is ahomomorphism
ofX-algebras.
This suggests that I⊗X captures the theory of homomorphisms for the theory X: a functor h :
I ⊗X → C is an internal homomorphism in C between the two models h : {0}⊗X → C and
0
h :{1}⊗X → C. Observe that the undirected analogue is just a homotopy of maps of topological
1
spaces.
Example4.1. LetMbeacomputadpresentingatheoryof(possiblylax)monoids,withasingle“object”
1-cell a, a multiplication 2-cell m , and a unit 2-cell h . In I⊗M, the 3-cells a⊗m and a⊗h can be
pictured inthefollowingway.
,
(3)
These cells embody the action of “sliding” the multiplication and unit past the mediating 1-cell a⊗
∗, where ∗ is the unique 0-cell of M. The use of sliding for reasoning about naturality is discussed
extensively in[26];thetensorproductofcomputadsprovides itwithacompositional semantics.
Models of I⊗M in the 2-category Cat are pairs of monads related by a Kleisli law, an asymmetric
version of a distributive law [5]. The dual notion of Eilenberg-Moore law is obtained as a model of
(I⊗M)op(1).
If X is PRO-like, in the sense that it has only one 0-cell ∗ and one cell a of lowest, non-zero di-
mension k (so that k is seen as the dimension of “objects” in the theory), one is usually interested in
homomorphisms between X-algebras in a higher category C that are localised at the same 0-cell, so
thatthelowest-dimensional componentisa(k+1)-morphism betweenthetwounderlyingobjectsofthe
algebras.
In order to achieve this, we quotient I⊗X to obtain the reduced cylinder of X, (I⊗X)/(I⊗{∗}).
Again, thisconstruction has awell-established undirected analogue: mapsfrom the reduced cylinder of
atopological spaceintoanotherspacearepointedhomotopies.
Example4.2. ForthecomputadMofthepreviousexample,the3-cellsa⊗m anda⊗h takethefollowing
forminthereducedcylinder ofM.
,
IntheconeofacomputadX,oneofthecopiesofX inacylinderistrivialisedbyaquotient. Afterthis
analysis, thefuture cone ofX can beseen asthe theory ofhomomorphisms from anarbitrary X-algebra
tothetrivialX-algebra. WhenX is,forinstance,thetheoryofmonoids,thishappenstocaptureprecisely
A.Hadzihasanovic 79
thenotionofleftaction ofamonoid, ascanmosteasilybeseenbymerging theregions beyonda⊗∗in
Diagram(3).
,
Dually, the past cone captures a notion of right co-action, and reversing cells gives all the usual mirror
variants. Generalising how weobtained (2),wecanalsoconstruct theories ofobjects withacompatible
left(co-)action ofX andright(co-)action ofY,asaquotient ofI⊗X⊗Y.
Remark 4.3. It may be worth observing that, in the compositional style of building algebraic theories,
the“simplest”theoriesaretheoneswiththeleastartificialidentificationsofcells—hence,theoneswith
the most cells, or most “colours”. Thus, for instance, the theory of semigroups arises more naturally as
a quotient of the theory of actions of non-associative operations; or the theory of Frobenius algebras as
a quotient of the theory of objects with a compatible left action of a monoid and right co-action of a
comonoid. This is in contrast with direct, symbolic presentations, where one is led to consider theories
withasmallsignature assimpler.
5 Smash products: commutativity, bialgebras
When X andY are two PRO-like computads, hence presentations of algebraic theories in the standard,
narrower sense, we may want to be able to compose them in order to obtain another. Starting from the
tensor product, ifa ,a arethebasicobject cellsofX⊗Y,theobvious candidate forthenewbasiccell
X Y
is a ⊗a ; thus, we need to quotient out the a ⊗∗ and ∗⊗a . This comes naturally if we work with
X Y X Y
pointed spaces.
Definition5.1. Apointedcomputad(X,∗ )isacomputadX withadistinguished0-cell∗ ,itsbasepoint.
X X
Amapofpointed computads f :(X,∗ )→(Y,∗ )isamapofcomputads with f(∗ )=∗ .
X Y X Y
Given two pointed computads (X,∗ ), (Y,∗ ), their wedge sum is the pointed computad (X∨Y,∗),
X Y
whereX∨Y :=(X⊕Y)/({∗ }⊕{∗ }),and∗istheidentification of∗ and∗ .
X Y X Y
There is an inclusion of computads X∨Y ⊆X⊗Y, given by X 7→X⊗{∗ }, Y 7→{∗ }⊗Y. The
Y X
smash product of X andY is the pointed computad X∧Y :=(X⊗Y)/(X∨Y), with the image of X∨Y
through thequotientasbasepoint.
Example 5.2. The reduced cylinder of (X,∗ ), mentioned in the previous section, can be described as
X
thesmashproduct (X,∗ )∧(I+1,∗ ).
X 1
Muchliketheiranaloguesintopology,thewedgesumandsmashproductdefinemonoidalstructures
on the category Cpt of pointed computads, with (1,∗) and (1+1,∗), respectively, as units. We will
∗
sometimesleavethebasepoint implicit.
LetS1 betheoriented 1-sphere, thatis,thecomputad
a
withitsunique0-cellasbasepoint.