MAURICE KIANPI AND CELESTIN NKUIMI JUGNIA
Received 22 July 2005; Revised 21 June 2006; Accepted 5 July 2006
For an arbitrary-type functorF, the notion of split coalgebras, that is, coalgebras for which the canonical projections onto the simple factor split, generalizes the well-known notion of simple coalgebras. In caseFweakly preserves kernels, the passage from a coalge-bra to its simple factor is functorial. This is the simplification functor. It is left adjoint to the inclusion of the subcategory of simple coalgebras into the categorySetFofF-coalgebras, making it an epireflective one. If a product of split coalgebras exists, then this is split and preserved by the simplification functor. In particular, if a product of simple coalgebras exists, this is simple too.
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1. Introduction
Originally introduced by Aczel and Mendler [1] to model various types of transition sys-tems, coalgebras (or systems) offer a very rich field of mathematics. Basic tools and results in the theory of coalgebras can be found in [3,8].
Coalgebras of a givenSet-endofunctorF, also called the type functor with morphisms between them, form a category denoted bySetF. The notions of extensionality and sim-plicity are tightly connected to two of the main tools in the theory of coalgebras: bisimu-lations and congruence rebisimu-lations, respectively.
The coalgebraic definition of bisimulation as well as the notion of congruence relation made their first appearance in [1]. In the particular case of weak pullbacks preserving functors, Rutten in [8] has built on results and notions from them. In general, congru-ences need not be bisimulations, and even the largest bisimulation on a coalgebra need not be transitive in the general case. In [4], Gumm and Schr¨oder have shown that under a weaker condition, namely that of weak kernels preservation in which we are interested in this paper, every congruence relation is a bisimulation and in particular the largest bisimulation on a coalgebra is the largest congruence relation thereon.
This paper both gives a slight overview of some existing insights in the theory of coal-gebras in Sections2and3, and in addition, presents some new material such as the notion
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International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 56786, Pages1–9
of split coalgebra introduced inSection 4. This generalizes the notion of simple coalgebras as does [5] that of extensional coalgebra (seeSection 3). The fact that the quotient of a coalgebra with respect to a congruence relation [3] yields again a coalgebra helps to build many coalgebras from a given one. It turns out that amidst these, the quotient with re-spect to the largest congruence relation is its sole factor which is simple. We call it its simple factor (seeSection 3). This uniqueness draws our attention to a study inSection 5 of the passage from anF-coalgebra to its simple factor. Fortunately, as is shown here, in case the type functor weakly preserves kernels, this passage is functorial. We call it the simplication functor and show that it is left adjoint to the inclusion functor of the subcat-egorySimp(F) of simpleF-coalgebras making it an epireflective one. We also show that if a product of split coalgebras exists, then it is split and that the simplification functor is well-behaved with respect to split coalgebras, as far as the preservation of products is concerned. This implies that if a product of simple coalgebras exists, then it is simple too.
2. Preliminaries
2.1. Some categorical notions
2.1.1. Epireflective subcategory. If a functorI:C→Dis an inclusion, then a left adjoint S:D→Cis called a reflection. IfCis a full subcategory and there exists a reflectionSsuch that the unitηX:X→(I◦S)(X) is an epimorphism for all objectsXinD, the subcategory [9]Cis called epireflective. If there exists a reflectionSfor a subcategoryC, then (see [9]) there exists one which is the identity onC, that is, such thatS◦I=idC. If such a reflection
is chosen, then it is called an epireflector. 2.1.2. Weak limits preservation.
Definition 2.1. LetF :C→Cbe a functor andD:I→Ca diagram for whichD-limits exist inC. Then,
(i)Fweakly preservesD-limits, ifF transforms every limit cone overDinto a weak limit cone overF◦D, that is, for every limit (L, (vk)k∈κ) of the diagramD, (F(L),
F(vk)k∈κ) is a weak limit of the diagramF◦D;
(ii)Fpreserves weakD-limits if it transforms every weak limit cone overDinto a weak limit cone overF◦D.
It is quite clear that there is a fine linguistic difference between “Fpreserves weak D-limits” and “Fweakly preservesD-limits.” But fortunately, it has been proven [3,4] that the difference disappears in every category in whichD-limits exist.
2.2. Some basic facts about coalgebras. LetFbe a type functorC→C. AnF-coalgebra is just a pair (A,αA:A→F(A)) whereAis an object called the carrier andαAa morphism called the structure of the coalgebra. This will often be denoted by (A,αA) or by the single letterᏭ. A homomorphism from a coalgebraᏭ=(A,αA) to a coalgebraᏮ=(B,αB) is a map f :A→Bsuch thatF(f)◦αA=αB◦f.
in [2] in the setting of categorySet which serves as the basic category from now on. As for limits, the forgetful functor [7] creates those which are preserved by the functorF. Epis inSetFare just surjective homomorphisms and every injective homomorphism is a mono [8].
2.2.1. Generation of homomorphisms. The following theorem can be used to prove that a map is a homomorphism.
Theorem 2.2 [7]. LetᏭ,Ꮾ, andᏯbe coalgebras,ϕ:Ꮽ→Ꮿa homomorphism,f :A→B andg:B→Cmaps withϕ=g◦f.
(1) If f is a surjective homomorphism, thengis a homomorphism. (2) Ifgis an injective homomorphism, then f is a homomorphism.
2.2.2. Bisimulations. The following definition of bisimulation was introduced by Aczel and Mendler [1]: a bisimulation between two coalgebrasᏭandᏮis a relationR⊆A×B for which there exists a structureαR:R→F(R) such that the projectionsπ1:R→Aand π2:R→Bare homomorphisms of coalgebras. In particular,∅is always a bisimulation betweenᏭandᏮ. On the other hand, bisimulations are closed under arbitrary unions so that there is always a largest bisimulation∼Ꮽ,Ꮾbetween two coalgebrasᏭandᏮ. When Ꮽ=Ꮾ, one talks of bisimulation onᏭand∼Ꮽdenotes the largest bisimulation onᏭ. For every coalgebraᏭ, the diagonalΔAis [3,7,8] a bisimulation onᏭ.
We have the following facts about bisimulations.
Theorem 2.3 [7]. LetᏭandᏮbe coalgebras.
(1) Given two homomorphismsϕ:ᏼ→Ꮽandψ:ᏼ→Ꮾ, then (ϕ,ψ)(P) := {(ϕ(p), ψ(p))|p∈P}is a bisimulation betweenᏭandᏮ.
(2) Letϕ:Ꮽ→Ꮾbe a homomorphism. For all bisimulationRonᏭ,ϕ(R) := {(ϕ(a), ϕ(a))|(a,a)∈R}is a bisimulation onᏮ.
2.2.3. Congruence relations. A congruence relation on a coalgebraᏭis a relation onA which is the kernel (inSet) of a homomorphism with domainᏭ. For any congruence relationθonᏭthere is [3] a unique coalgebra structure onA/θfor which the canonical projectionπθ:A→A/θis a homomorphism. The corresponding coalgebra is denoted by Ꮽ/θand called a factor coalgebra forᏭ. In [1], it has been shown that there is a largest congruence relation contained in every reflexive relationRdenoted by Con[R]. In partic-ular,∇Ꮽ:=Con[A×A] denotes the largest congruence relation onᏭ. This is in general a proper subset ofA×A.
Lemma 2.4 [4]. IfFweakly preserves kernels, then the largest bisimulation∼Ꮽon a coalge-braᏭis transitive, in fact it is the largest congruence relation∇ᏭonᏭ.
3. Extensionality and simplicity
3.1. Definitions and some basic results
Definition 3.1 [5]. LetᏭbe a coalgebra for an arbitrary-type functorF.
The following theorem is a characterization of extensional coalgebras. This can be found in [3], and, in [8] (in caseF preserves weak pullbacks), as a characterization of “simple” coalgebras.
Theorem 3.2. For a coalgebraᏭ, the following are equivalent.
(1)Ꮽis extensional.
(2) Every homomorphism with domainᏭis a monomorphism.
(3) For every coalgebraᏮ, there is at most one homomorphismψ:Ꮾ→Ꮽ. The following result shows that extensionality generalizes simplicity.
Corollary 3.3 [5]. Every simple coalgebra is extensional.
For functors weakly preserving kernels, both notions agree. The following gives an easy description of simple coalgebras when the terminal coalgebra exists.
Lemma 3.4 [3]. If the terminal coalgebra exists, then simple coalgebras are precisely isomor-phic copies of its subcoalgebras.
For everySet-endofunctorF, we denote byExt(F) (Simp(F), resp.) the fully faithful subcategory ofSetF whose objects are extensional coalgebras (simple coalgebras, resp.). Despite the fact thatSetF is cocomplete [2] for any type functor F, the subcategories Ext(F) andSimp(F) are not cocomplete in general. More precisely, we have the following important observation.
Remark 3.5. For every type functor different from the empty constant functor, neither the class of simple coalgebras nor that of extensional ones is closed under sums. However, the former is closed under homomorphic images and domains of injective homomorphisms and the latter under domains of monomorphisms but not under homomorphic images.
3.2. Some examples
3.2.1. Nondeterministic labelled transition systems with output. LetΣbe a set with|Σ| ≥2 andaandbtwo different elements inΣ.
Example 3.6. Let ᏼbe the power set functor defined byᏼ(X) := {A|A⊆X}, and for all map f :X→Y,ᏼ(f) :ᏼ(X)→ᏼ(Y) whereᏼ(f)(A) := f(A),Σ×Ᏽd the functor which associates every setXwithΣ×Xand every map f :X→Ywith idΣ×f :Σ×X→
Σ×Y defined by (idΣ×f)(e,x)=(e,f(x)). For the arrow representation of coalgebras
and the characterizations of homomorphisms and bisimulations between coalgebras in Setᏼ◦(Σ×Ᏽd), see [8].
The ᏼ◦(Σ×Ᏽd)-coalgebras :=(S,αS) defined by αS(0)= {(a, 1)} and αS(1)= {(b, 1)}and᐀:=(T,αT) defined by T= {0, 1, 2}withαT(0)= {(a, 1), (b, 2)},αT(1)=
{(b, 1)}andαT(2)= {(a, 2)}represented below are extensional.
It is well known [6] thatᏼpreserves weak pullbacks. Similarly does the functorΣ×
Ᏽd. Thus the type functorᏼ◦(Σ×Ᏽd) weakly preserves pullbacks. Consequently, it does weakly preserve kernels so that the coalgebrasand᐀are also simple.
3.2.2. (−)32-coalgebras. The functor (−)32given on sets as (X)32= {(x1,x2,x3)∈X3|x1= x2orx1=x3orx2=x3} and on maps f :X →Y as (f)32(x1,x2,x3)=(f(x1),f(x2), f(x3)) constitutes a great source of counterexamples in the theory of coalgebras. It has been introduced in [1] and is an example of functor which does not weakly preserve ar-bitrary pullbacks [3].
Example 3.7. Consider the (−)3
2-coalgebrasᏭ:=(A,αA), whereA= {a,b}andαA:A→ (A)32is defined byαA(a)=(a,a,b) andαA(b)=(a,b,a) and 11 :=(1,α1) with 1= {∗}. 11
is the terminal coalgebra and hence the only nonempty simple coalgebra (up to isomor-phism) andᏭis extensional.
3.3. Simple factor coalgebras. Every coalgebra “can be made” simple by taking the quo-tient with respect to its largest congruence relation. It is well known [3] that when ordered by the set inclusion, congruences on a coalgebra form a complete lattice. The following which extends a result from Rutten [8] for weak pullbacks preserving functors is imme-diately checked using the fact that [5] the congruence lattice ofᏭ/θis isomorphic to the interval aboveθof the congruence lattice ofᏭ.
Proposition 3.8. For any functorF, every coalgebraᏭ, and every congruence relationθ onᏭ, the quotientᏭ/θis simple if and only ifθ= ∇Ꮽ.
We callᏭ/∇Ꮽthe simple factor of the coalgebraᏭ.
4. Split coalgebras
In this section, we introduce the concept of split coalgebra. As the concept of extensional coalgebra, this generalizes that of simple coalgebra.
4.1. Definitions and some basic results
Definition 4.1. LetFbe any type functor. A coalgebraᏭis called split if the canonical projectionπ∇ᏭofᏭontoᏭ/∇Ꮽsplits.
We have the following characterization of simple coalgebras.
Proposition 4.2. For every type functorF, a coalgebra is simple if and only if it is split and extensional.
From Propositions3.8and4.2, it follows that for any coalgebra if there exists a ho-momorphismχ:Ꮽ/∇Ꮽ→Ꮽ, then automaticallyπ∇Ꮽ◦χ=idᏭ/∇Ꮽ, that is,Ꮽis a split
coalgebra.
This constitutes a key factor in tackling the proof of the following result which permits obtaining split coalgebras when the terminal coalgebra exists.
4.2. Some examples
4.2.1. (−)32-coalgebras (continued).
Example 4.4. Nonempty split coalgebras are just coalgebrasᏭfor which there exists a homomorphismϕ: 11→Ꮽ, that is, there exists an elementa0∈A such thatαA(a0)= (a0,a0,a0).
4.2.2.ᏼ◦(Σ×Ᏽd)-coalgebras (continued).
Example 4.5. The coalgebraᏻ= a o b is obviously split andᏹ=ᏻ+ᏻis split but not simple.
5. The simplification functor
It follows fromProposition 3.8that for every type functorFand all coalgebrasᏭ,Ꮽ/∇Ꮽ is the only simple factor coalgebra forᏭ. Can the passage fromᏭtoᏭ/∇Ꮽbe made into a functor fromSetFtoSimp(F)?
5.1. The simplification functor. The following theorem provides a positive answer to the aforementioned question and shows the adjunction of the thus obtained functor with the inclusion functor I :Simp(F)→SetF.
Theorem 5.1. Assume that the type functorFweakly preserves kernels. Consider S :SetF→
Simp(F) defined by S(Ꮽ)=Ꮽ/∇Ꮽand for all homomorphismϕ:Ꮽ→Ꮾand allainA, S(ϕ) :Ꮽ/∇Ꮽ→Ꮾ/∇Ꮾwith S(ϕ)(a/∇Ꮽ) :=ϕ(a)/∇Ꮾ. Then the following holds.
(1) S is a covariant functor.
(2) S is left adjoint to the inclusion functor I with unit η: idSetF →I◦S and counit
ε: S◦I→idSimp(F)defined byηᏭ:=π∇Ꮽ, the canonical projection ofᏭontoᏭ/∇Ꮽ for allᏭinSetFandεᏭ:Ꮽ/∇Ꮽ→ᏭwithεᏭ(a/∇Ꮽ) :=afor allᏭinSimp(F) and allainA.
Proof. (1) S is well defined on objects sinceᏭ/∇Ꮽis, as seen inProposition 3.8, always a simple coalgebra. Let nowϕ:Ꮽ→Ꮾbe a homomorphism. We want to show that S(ϕ) is a homomorphism too. Letaandabe elements ofAwith (a,a)∈ ∇Ꮽ. Then (a,a)∈∼Ꮽ for∇Ꮽ=∼ᏭbyLemma 2.4. Nowϕ(∼Ꮽ) is a bisimulation onᏮbyTheorem 2.3. There-fore (ϕ(a),ϕ(a))∈∼Ꮾand consequently (ϕ(a),ϕ(a))∈ ∇Ꮾfor∇Ꮾ=∼Ꮾ, that is, S(ϕ) is a map. We now show that it is even a homomorphism. By the definition of S(ϕ), we have S(ϕ)◦π∇Ꮽ=π∇Ꮾ◦f, whereπ∇Ꮽ andπ∇Ꮾare the canonical projections ofᏭandᏮonto
Ꮽ/∇ᏭandᏮ/∇Ꮾ, respectively. Nowπ∇Ꮾ◦ϕis a homomorphism andπ∇Ꮽis a surjective
homomorphism. Therefore byTheorem 2.2, S(ϕ) is a homomorphism. One can easily check that S preserves composition of homomorphisms and identity homomorphisms.
(2) We want to show that we have the following identities:
We first show that I∗ε◦η∗I=idI. LetᏭbe a coalgebra inSimp(F). We have
(I∗ε◦η∗I)Ꮽ=IεᏭ◦ηI(Ꮽ)
=εᏭ◦ηᏭ. (5.2)
NowᏭis simple, thus it is extensional byCorollary 3.3. Therefore, it follows from Theo-rem3.2thatεᏭ◦ηᏭ=idᏭ.
For the second identity, letᏭbe an arbitrary coalgebra inSetF. We have
(ε∗S◦S∗η)Ꮽ=(ε∗S)Ꮽ◦(S∗η)Ꮽ
=εS(Ꮽ)◦S
ηᏭ. (5.3)
Now S(ηᏭ) : S(Ꮽ)→S(S(Ꮽ)) andεS(Ꮽ): S(S(Ꮽ))→S(Ꮽ) and S(Ꮽ) is simple and hence extensional. Thus, automatically, we haveεS(Ꮽ)◦S(ηᏭ)=idS(Ꮽ).
The functor S will be called the simplification functor for the functorF. FromTheorem 5.1, we immediately deduce the following.
Corollary 5.2. If the type functorFweakly preserves kernels, thenSimp(F) is an epire-flective subcategory ofSetFand the simplification functor S is an epireflector (up to isomor-phism).
5.2. Some categorical properties. It is well known [2] thatSetFis cocomplete. On the other hand, every left adjoint [9] preserves colimits but does not necessarily preserve lim-its. The following shows amongst other things that the simplification functor preserves some types of limits, as far as they exist.
Proposition 5.3. Assume thatFis a type functor which weakly preserves kernels. The sim-plification functor S preserves all colimits, the terminal coalgebra (as far as it exists) trans-forms every homomorphism into an injective homomorphism, and, in particular, epimor-phisms into isomorepimor-phisms.
The following result shows that the simplification functor preserves some products whenever they exist.
Theorem 5.4. IfFweakly preserves kernels and the productΠi∈IᏭiof a family (Ꮽi)i∈Iof split coalgebras exists, then this is split and (Πi∈IᏭi)/∇Πi∈IᏭi∼=Πi∈I(Ꮽi/∇Ꮽi).
Proof. Let (Ꮽi)i∈I be a family of split coalgebras such that the product Πi∈IᏭi exists in SetF. We want to show that it is a split coalgebra too. Thus we need to find a ho-momorphismχ: (Πi∈IᏭi)/∇Πi∈IᏭi→Πi∈IᏭisuch thatπ∇Πi∈IᏭi◦χ=id(Πi∈IᏭi)/∇Πi∈IᏭi. Let
pi:Πi∈IᏭi→Ꮽibe the canonical projection ofΠi∈IᏭitoᏭi,i∈I. SinceᏭiis split, there exists a homomorphismχi:Ꮽi/∇Ꮽi→Ꮽisuch thatπ∇Ꮽi◦χi=idᏭi/∇Ꮽi for alli∈I.
But then we haveχi◦S(pi) : (Πi∈IᏭi)/∇Πi∈IᏭi→Ꮽifor eachi∈I. Thus there exists a
unique homomorphismχ: (Πi∈IᏭi)/∇Πi∈IᏭi→Πi∈IᏭisuch that, for alli∈I,pi◦χ=χi◦
Thus byTheorem 3.2,π∇Πi∈IᏭi◦χ=id(Πi∈IᏭi)/∇Πi∈IᏭi.
Πi∈IᏭi
π∇Πi∈
IᏭi
pi
Ꮽi
π∇Ꮽ
i
(Πi∈IᏭi)/∇Πi∈IᏭi S(p i)
χ
Ꮽi/∇Ꮽi
χi (5.4)
To end the proof, we show that (Πi∈IᏭi)/∇Πi∈IᏭi∼=Πi∈I(Ꮽi/∇Ꮽi). Letqi:Ꮿ→Ai/∇Ai
be homomorphisms inSetF. We haveui:=χi◦qi:Ꮿ→ᏭiinSetF. Thus, there is a unique u:Ꮿ→Πi∈IᏭisuch that pi◦u=uifor alli∈I. Now (Πi∈IᏭi)/∇Πi∈IᏭi is a simple
coal-gebra. Thus it is extensional and consequentlyπ∇Πi∈IᏭi◦uis the unique homomorphism
fromᏯto (Πi∈IᏭi)/∇Πi∈IᏭi. On the other hand, the coalgebrasᏭi/∇Ꮽi’s are simple. Thus
they are extensional so that, automatically,qi=S(pi)◦π∇Πi∈IᏭi◦ufor eachi.
If (Ꮽi)i∈Iis a family of simple coalgebras then byProposition 4.2, for eachi,Ꮽiis split withᏭi/∇Ꮽi∼=Ꮽi. Thus, if their product exists, then byTheorem 5.4, (Πi∈IᏭi)/∇Πi∈IᏭi∼=
Πi∈IᏭi. Hence we have the following.
Corollary 5.5. IfFweakly preserves kernels and a product of simple coalgebras exists, then this is simple.
Acknowledgment
This paper is a final form and no version of it will be submitted to publication elsewhere. The authors gratefully acknowledge helpful and substantial suggestions of the referees for the improvement of this paper.
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Maurice Kianpi: Laboratory of Algebra, Department of Mathematics, Faculty of Science, University of Yaounde 1, P.O. Box 812, Yaounde, Cameroon
E-mail address:[email protected]
Celestin Nkuimi Jugnia: Laboratory of Algebra, Department of Mathematics, Faculty of Science, University of Yaounde 1, P.O. Box 812, Yaounde, Cameroon
