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The Scientific World Journal

Volume 2013 (2013), Article ID 285071, 9 pages

http://dx.doi.org/10.1155/2013/285071

## Cubical Sets and Trace Monoid Actions

Faculty of Computer Technology, Komsomolsk-on-Amur State Technical University, Prospect Lenina 27, Komsomolsk-on-Amur 681013, Russia

Received 17 August 2013; Accepted 6 September 2013

Academic Editors: J. Hoff da Silva and C. Wang

Copyright © 2013 Ahmet A. Husainov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper is devoted to connections between trace monoids and cubical sets. We prove that the category of trace monoids is isomorphic to the category of generalized tori and it is a reflective subcategory of the category of cubical sets. Adjoint functors between the categories of cubical sets and trace monoid actions are constructed. These functors carry independence preserving morphisms in the independence preserving morphisms. This allows us to build adjoint functors between the category of weak asynchronous systems and the category of higher dimensional automata.

#### 1. Introduction

In this paper, it is established that the category of generalized tori is isomorpic to the category of trace monoids and basic homomorphisms. It is shown that the category of generalized tori is reflective subcategory of the category of cubical sets. Adjoint functors between the category of cubical sets and the category of trace monoids acting on sets are constructed. These functors carry independence preserving morphisms in the independence preserving morphisms. The results are used to compare asynchronous systems and higher dimensional automata for modelling of concurrent systems.

The problem of comparing the mathematical models of concurrent systems using adjoint functors has always been of great interest [1–5]. In [2], Goubault introduced automata with concurrency relations for generalization of asynchronous systems. The category of automata with concurrency relations is isomorphic to the category of 2-skeletion of a category of higher dimensional transition systems. The truncation gives the desired left adjoint to the inclusion functor . Moreover, in the thesis [3], Goubault proved that there are adjoint functors between the category of semiregular higher dimensional automata and the category of asynchronous transition systems satisfying the axiom of confluence.

But the asynchronous system (1) consisting of three states and two transitions with the independence relation does not satisfy the confluence condition and is not an automation with concurrency relation. Therefore, it remains an open problem of existence of a left adjoint functor from the category of higher dimensional automata in the category containing asynchronous systems which are not confluent.

This problem is solved in this paper.

We propose a category that includes all asynchronous systems and admits adjoint functors with the category of higher dimensional automata. By [6, 7], this category may be useful for studying the homology groups of asynchronous systems.

This work consists of two sections. In the first, we construct the adjoint functors between the categories of trace monoids and cubical sets. In the second, section we start with the construction of adjoint functors between the category of trace monoids acting on sets and the category of cubical sets. Then, we build adjoint functors between the categories of asynchronous systems and higher dimensional automata.

#### 2. Trace Monoids and Cubical Sets

Introduce a category of trace monoids and basic homomorphisms. Consider a category of cubical sets. Introduce generalized tori. Construct the adjoint functors between the category of trace monoids and cubical sets.

##### 2.1. Preliminaries

Throughout the paper, denotes the category of sets and maps.

For any category and objects , let denotes the set of morphisms . Let be the opposite category. A* diagram* (of objects) in is any functor from some small category to . Let denotes a diagram . will always denote the category of diagrams and natural trasformations between them.

Let be a functor between small categories and let be an arbitrary category. Consider the functor defined on objects as and on morphisms as . Left derived functor to is called a *left Kan extension along *.

Let be the Yoneda functor described in [8, III.2, page 62]. For each functor , we consider the functor defined on objects by its values with the obvious extension to morphisms.

Proposition 1 (see [9, Proposition II.1.3]). *Let be a cocomplete category and let be a small category. For every functor the functor has a left adjoint functor isomorphic to the left Kan extension :
*(2)

For any , we have where denotes the comma category of objects -over for the Yoneda functor and is the projection [8].

We can obtain by the theory of ends [8, Theorem X.4.1, page 240] the following.

Proposition 2 (see [8]). *For each , equals the colimit of a diagram defined on a directed bipartite graph consisting of morphisms
*(3)*
where runs all morphisms of the category .*

##### 2.2. Trace Monoids

Let be a set. A binary relation is *irreflexive* if . One is *symmetric* if . An *independence relation* on is an arbitrary irreflexive symmetric binary relation . In this case, elements are *independent* if .

Let be the set of all *words* composed of *letters * for all . Then, is the monoid with the operation of *concatenation* by the following formula:

Identity element is the empty word.

Let be an independence relation on . Define an equivalence relation on assuming if the word can be obtained from by a finite sequence permutations of adjacent independent elements. For any , its equivalence class is called a *trace*. It is easy to see that the operation transforms the set of equivalence classes in a monoid. This monoid is called a *trace monoid *.

Our definition of trace monoid is different from that given in [10]. We suppose that can be infinite.

In some cases, we omit square brackets and write instead of . If , then is the *free monoid *. If , then is the *free commutative* monoid .

*Definition 3. *A homomorphism is basic if .

If for some , then is the *length* of trace .

Proposition 4. *Every diagram has the colimit in .*

*Proof. *Let for and let be the coproduct in the category of monoids . Then, the colimit is equal to the quotient monoid where is the smallest congruence relation satisfying for all , , , . A more detailed proof can be found in the preprint [11].

*Definition 5. *A basic homomorphism is *independence preserving* if for all the following implication holds:

This is equivalent to the following condition:

It follows that the composition of independence preserving homomorphisms is independence preserving. Let be a subcategory consisting of all trace monoids and basic independence preserving homomorphisms.

##### 2.3. Cubical Sets and Trace Monoids

A *cubical set * is a sequence of sets , with two family of maps(i), , , (face operators)(ii), , (degeneracies)

satisfying the following equations:

A *morphism ** of cubical sets* is a family of maps commuting with the face operators and degeneracies. Let be a category of cubical sets and morphisms.

Let be the category consisting of the partially ordered sets and maps admitting decompositions by the following maps:

By setting , , , we can consider every cubical set as functor . A morphism of cubical sets can be considered as natural transformations. We will identify the category with .

Introduce generalized tori.

*Definition 6. *Let be a trace monoid. *Generalized torus * is a cubical set consisting of sets and
for all . The maps , are defined by

The map extends to a functor which assigns to each basic homomorphism a morphism of cubical sets given by a family of maps defined as And conversely, since every morphism of cubical sets commutes with face operators, any morphism can be given by the maps defined by some . So, it holds the following.

Proposition 7. *The functor is full and faithful.*

We will prove the following.

Theorem 8. *The functor has a left adjoint . For every cubical set , the trace monoid can be given by the generator set by a smallest equivalence relation on identifying for all and , with the following relations for the equivalence classes:
*

*Proof. *For the construction of left adjoint to , we use Propositions 1 and 2. With this aim, define a functor by setting on objects of . Let for all and . Set by
Here, are elements of the free monoid generated by one element . Let be the tuple in which the generator is located on the th place. In particular, .

Basic homomorphisms are given by values at , . Therefore, . It follows that the functor is isomorphic to acting by . Hence, the functor has a left adjoint .

By Proposition 2, the object is isomorphic to quotient monoid of by the smallest congruence relation identificating the pairs:
Element equals for some nonnegative integers .

Substituting by and and using the commutativity of monoids , we conclude that monoid can be given by the generators with the relations
The relation (15) follows from commutativity of . The relations (16) are obtained by the identifications:

The relations (18) can be obtained from
by the substitution .

Identity (neutral) elements in are equal. It follows that for all and .

Hence, the relations (17) can be removed leaving the .

It follows from that each generator equals for some . Moreover for , for every , the similar sequence of relations leads to
Consequently, it is enough to leave the generators corresponding to -cubes. All relations can be obtained by the relations between those generators corresponding to -cubes.

It is easy to see that, among the relations (18), can be left only the following identifications (for and ):

So, can be given by generators , . The class of equals . Since and , every 2-cube (23) gives , ,
(23)

We have the following relations between the generators,
Since for all pairs with , we can take the generators instead of generators . Thus, we obtain the desired generators and relations. And the class of equals .

For example, the cubical set shown in (23) has the monoid isomorphic to the free commutative monoid generated by two elements.

The category of generalized tori is the image of the functor . By Proposition 7, that is, a full subcategory of we have shown that has the left adjoint . By [8, Theorem IV.3.1], reflectivity is equivalent to that the counit of adjunction is an isomorphism. Therefore, we have the following.

Corollary 9. *The subcategory of generalized tori is reflective in the category of cubical sets. In particular, counit of the adjunction is an isomorphism.*

##### 2.4. Independence Preserving Morphisms

We introduce independence preserving morphisms of cubical sets. We prove that the category of cubical sets and independence preserving morphisms are linked with by adjoint functors.

*Definition 10. *A morphism of cubical sets are *independence preserving* if

Here, denotes the congruence class of considered in Theorem 8.

Let denotes the category of cubical sets and independence preserving morphisms. We construct the adjoint functors between and . Definition 10 and Corollary 9 follow.

Lemma 11. *Let be the functor given in Definition 6 and let be the left adjoint to . *(i)*A morphism of cubical sets is independence preserving if and only if one is carried by the functor to an independence preserving basic homomorphism .*(ii)*Basic homomorphism is carried by the functor to independence preserving morpism if and only if it is independence preserving.*(iii)*All components of the counit are independence preserving.*(iv)*All components of the unit are independence preserving.*

*Proof. *The property (i) follows from Definition 10. Let us prove the last assertion. By Definition 10, it is equivalent to the statement that is independence preserving. Since is left adjoint to , the composition is equal to the identity morphism. The morphism is isomorphism by Corollary 9. Consequently, is an isomorphism. Thus, is independence preserving by (i).

We obtain by Theorem 8 and by Lemma 11 the following.

Theorem 12. *Restrictions of the functors and on the independence preserving morphisms give the adjoint functors
*(26)*
where is left adjoint to .*

#### 3. Category of Trace Monoid Actions

We construct adjoint functors between a category of monoids acting on sets and the category of cubical sets.

##### 3.1. Trace Monoid Actions

*Definition 13. *A trace monoid action consists of a trace monoid with a set and a map called a right action, , satysfying to the following conditions:(i),(ii).

For example, for any trace monoid , we have the trace monoid action defined as for all . It is called the action *by right translations*.

*Definition 14. *A *morphism * of trace monoid actions consists of a homomorphism and a map such those

For example, for any homomorphism , the pair is the morphism as satifying .

A morphism of trace monoid actions is *basic* if is a basic homomorphism. It is called *independence preserving* if is independence preserving.

Let be a category of trace monoid actions and basic morphisms.

Lemma 15. *The category is cocomplete.*

*Proof. *Monoid is a category with unique object. A monoid action can be considered as a functor with the value on the unique object of the category . assigns to morphisms the maps such those . The following natural transformations corresponds to morphisms :
(28)

Consider an arbitrary diagram in as the diagram of functors . There exists in the category . Let be the colimit cone. Take left Kan extensions belonging to the category of functors from to . It is easy to see that desired colimit equals .

We study colimits in the category for diagrams of trace monoid actions by right translations. For an arbitrary small category, , denotes the set of connected components of . For a set , denotes a trace monoid with the action *by right translations * for all , .

Let be a diagram of trace monoids and basic homomorphisms defined for all and . Then, we have the diagram of trace monoid actions by right translations, consisting of morphisms .

Proposition 16. * is isomorphic to the trace monoid action by right translations.*

*Proof. * will denote . Let be the colimit cone. We prove the universality of the cone of morphisms defined as
for all . Here, is the connected components of containing .

Consider an arbitrary cone and construct a morphism making the following commutative diagram in :
(30)
The existence and uniqueness of the homomorphism follow from the universal property of in .

Construct . Since is cone, for all with same connected components. It follows that, for every , the values must be determined by an arbitrary object belonging to connected component . These values give the unique extension to the set . It follows the uniqueness of . The equality of values at gives coincidence of compositions for all and with it is the commutative triangles (30).

Further, let , , denote the functors involved in Theorem 8. Let be the functor assigning to every trace monoid action the cubical set

Recall that denotes the comma category of objects -over (Proposition 1).

Theorem 17. *The functor has a left adjoint such that are the trace monoid actions by right translation for all cubical sets .*

*Proof. *Let be a functor defined as (with actions by right translations) at objects and for morphisms . The category is cocomplete. Hence, this functor fits into the diagram
(32)
where is left adjoint to taking values
Elements are determined by values where is identity element of the monoid and by the values of on generators of the monoid . It follows that the functors and are isomorphic. For , is the colimit of a diagram consisting of trace monoids with actions by right translations . The colimit in the first component equals by Theorem 8. We obtain by Proposition 16.

##### 3.2. Independent Morphisms of Trace Monoid Actions

By Lemma 11, a morphism of cubical set is independence preserving if and only if the homomorphism of monoids is independence preserving.

*Definition 18. *A morphism of the category is independence preserving if is independence preserving homomorphism.

Since , it follows from this definition that is independence preserving if and only if is independence preserving. So, the functor takes independence preserving morphisms into independence preserving.

The same is true for .

Lemma 19. *The functor takes independence preserving morphisms into independence preserving.*

*Proof. *By definition, a morphism is independence preserving if and only if the homomorphism of monoids is independence preserving. The monoid is generated by classes such that if . We have also . Therefore, . It follows that if and only if . We need to prove the following implication:
By the equivalences in and in , the premise of implication (34) is transformed into formula
Since the homomorphism is independence preserving, we obtain the formula leading to the conclusion of (34).

Lemma 20. *The unit of the adjunction is independence preserving.*

* Proof. *Since is left adjoint to , the morphism is a coretraction. It follows that the homomorphism of monoids is a coretraction. It takes independent elements in the not equal. Hence, is dependence preserving.

Let be the counit of adjunction .

Lemma 21. *The counit of adjunction is independence preserving.*

* Proof. *The morphism consists of some pair . It is independence preserving if and only if the homomorphism is independence preserving. The monoid is generated by classes of -cubes . The homomorphism assigns to each class the element . If and are independed, then . Hence, is independence preserving.

It follows from obtained Lemmas.

Theorem 22. *The restrictions of functors and on the subcategories consisting of independence morphisms give adjoint functors
*(36)

##### 3.3. Asynchronous Systems and Higher Dimensional Automata

A *higher dimensional automation * consists of a cubical set and an *initial point *. A *morphism ** of higher dimensional *automata is a morphism of cubical sets such that . Let be the category of higher dimensional automata and let be the category of higher dimensional automata and morphisms for which are independence preserving.

Let be the cubical set such that consists of unique element for all . Face operators and degeneracies are the identity maps.

It is easy to see that .

Introduce a category of weak asynchronous systems. Consider partial maps of sets.

For any set , let . The set is called *pointed*. The element is the same for all pointed sets. A map is *pointed* if . Let be the category of pointed sets and pointed maps.

Any partial map will be considered as the pointed map defined as follows: This allows us to identify the category of sets and partial maps with the category .

A *(right) monoid action ** on a pointed set* consists of a monoid and a pointed set with an arbitrary map satisfying the following conditions:(i),(ii),(iii).

Let be a monoid action on a pointed set. Since the monoid is a category with an unique object, can be considered as a functor assigning to the unique object the pointed set and assigning to morphisms the pointed maps such those .

It was shown in [12] that each asynchronous system can be considered as a trace monoid action on a pointed set with an initial point.

*Definition 23. **A weak asynchronous system * is a trace monoid action on a pointed set with an *initial point *.

If we require and , then we get the definition of an asynchronous system in the sense of Bednarczyk [1].

A *polygonal morphism * of weak asynchronous systems is an independence preserving basic homomorphism with a pointed map satisfying the following conditions:(i),(ii).

The map is pointed from where . Let be the category of weak asynchronous systems and polygonal morphisms. Construct a functor assigning to every asynchronous system the cubical set consisting of a sequence of pointed sets and . Face operators are defined as follows:

Degeneracies are defined by . Initial point in equals .

Theorem 24. *The functor has a left adjoint .*

* Proof. *Let be the functor forgetting the point and let be the left adjoint to adding the point . Let be the functor assigning to every trace monoid action the composition and assigning to morphisms the morphisms . Similarly, we define the functor . Let be a unit and let be a counit of adjunction . Since the functor is left adjoint to , the compositions and are equal to identity natural transformations. Construct natural transformations and as follows: , . Components of natural transformations and on objects equal identity natural transformations. Therefore, is left adjoint to .

Consider the compositions(40)where and are the adjoint functors from Theorem 22. The functor is left adjoint to .

The monoid has the unique action on the set . This trace monoid action equals . We have the adjoint functors between comma categories(41)

It easy to see that the category is isomorphic to . The composition of this isomorphism with is equal to the functor . We obtain from this the functor left adjoint to .

#### 4. Conclusion

We have considered the category of trace monoids and basic homomorphisms and proved that this category has all colimits. This allowed us to show that the category of generalized tori is a reflective subcategory of the category of cubical sets. Then, we considered the category of trace monoid actions and proved that it is cocomplete. We built adjoint functors between the category of trace monoid actions and the category of cubical sets. We unexpectedly found that these adjoint functors translate the independence preserving morphisms in independence preserving. As a result, we have completely solved the problem of comparing the category of asynchronous systems with the category of higher dimensional automata. Earlier, the problem had been solved only for asynchronous systems satisfying the confluence condition.

#### Acknowledgments

The paper was partially financed by the Scientific-Educational Center of supercomputer technology in the Far East Federal Region. This work was also supported in 2012-2013 as a part of strategic development program of state educational institutions of higher education, Grant no. 2011-PR-054. The author is grateful to participants of the seminar “Applications of category theory in computer science” in Komsomolsk-on-Amur State Technical University for their attention to this work.

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