Cubical Sets and Trace Monoid Actions

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.


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][2][3][4][5]. In [2], Goubault introduced automata with concurrency relations for generalization of asynchronous systems. The category ACR of automata with concurrency relations is isomorphic to the category of 2skeletion of a category HTS of higher dimensional transition systems. The truncation gives the desired left adjoint to the inclusion functor ACR → HTS. Moreover, in the thesis [3], Goubault proved that there are adjoint functors between the category of semiregular higher dimensional automata and the category ATS of asynchronous transition systems satisfying the axiom of confluence. But the asynchronous system 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. 2 The Scientific World Journal 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.

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

Preliminaries.
Throughout the paper, Set denotes the category of sets and maps.
For any category A and objects , ∈ A, let A( , ) denotes the set of morphisms : → . Let A op be the opposite category. A diagram (of objects) in A is any functor from some small category to A. Let { ( )} ∈ denotes a diagram : → A. A will always denote the category of diagrams → A and natural trasformations between them.
Let : C → D be a functor between small categories and let A be an arbitrary category. Consider the functor (−) ∘ : where : → runs all morphisms of the category D.

Trace Monoids. Let be a set. A binary relation
∈ . 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 1 2 ⋅ ⋅ ⋅ composed of letters 1 , 2 , . . . , ∈ for all ⩾ 0. Then, * is the monoid with the operation of concatenation by the following formula: Identity element 1 is the empty word.
Let be an independence relation on . Define an equivalence relation ≡ on * assuming 1 ≡ 2 if the word 2 can be obtained from 1 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 [ 1 ][ 2 ] = [ 1 2 ] 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.
It follows that the composition of independence preserving homomorphisms is independence preserving. Let FPCM ‖ ⊂ FPCM be a subcategory consisting of all trace monoids and basic independence preserving homomorphisms.

Cubical Sets and Trace Monoids.
A cubical set ( , ,] , ) is a sequence of sets , ∈ N with two family of maps satisfying the following equations: A morphism : → of cubical sets is a family of maps : → commuting with the face operators and degeneracies. Let Cube be a category of cubical sets and morphisms.
Let ◻ be the category consisting of the partially ordered sets I = {0, 1} and maps admitting decompositions by the following maps: By setting (I ) = , ( , ) = , , ( ) = , we can consider every cubical set as functor ◻ op → Set. A morphism of cubical sets can be considered as natural transformations. We will identify the category Cube with Set ◻ op . Introduce generalized tori.

Theorem 8. The functor
: For every cubical set , the trace monoid ( ) can be given by the generator set 1 / ≡ by a smallest equivalence relation on 1 identifying 2,0 2 ≡ 2,1 2 for all 2 ∈ 2 and ∈ {1, 2}, 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 : ◻ → FPCM by setting (I ) = N on objects of ◻. Let The Scientific World Journal ≅ ( , ). It follows that the functor is isomorphic to acting by ( ) = FPCM( (−), ) : ◻ op → Set. Hence, the functor has a left adjoint .
It follows from ( , −1 ( ), 1 ) that each generator equals ( , 1 ) for some ∈ 1 . Moreover for dim( ) = , for every 1 ⩽ ⩽ , the similar sequence of relations leads to Consequently, it is enough to leave the generators corresponding to 1-cubes. All relations can be obtained by the relations between those generators corresponding to 1-cubes.
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 Set ◻ op 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.

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 FPCM ‖ by adjoint functors.
Here, [ ] denotes the congruence class of ∈ 1 considered in Theorem 8. Let Cube ‖ denotes the category of cubical sets and independence preserving morphisms. We construct the adjoint functors between FPCM ‖ and Cube ‖ . Definition 10 and Corollary 9 follow.   ( ))) 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
where ‖ is left adjoint to ‖ .

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

Trace Monoid Actions
Recall that ◻ ↓ denotes the comma category ↓ of objects -over (Proposition 1).

Since
= ( , × 0 (◻ ↓ )), 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 20. The unit of the adjunction
: → is independence preserving.
Proof. Since is left adjoint to , the morphism It follows from obtained Lemmas.

Asynchronous Systems and Higher Dimensional Automata.
A higher dimensional automation ( , 0 ) consists of a cubical set and an initial point 0 ∈ 0 . A morphism : ( , 0 ) → ( , 0 ) of higher dimensional automata is a morphism of cubical sets : → such that 0 ( 0 ) = 0 . Let Cube be the category of higher dimensional automata and let Cube ‖ be the category of higher dimensional automata and morphisms : ( , 0 ) → ( , 0 ) for which : → are independence preserving. Let pt be the cubical set such that pt consists of unique element for all ⩾ 0. Face operators and degeneracies are the identity maps.
It is easy to see that Cube ‖ ≅ pt ↓ Cube ‖ . 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 : 1 * → 2 * is pointed if ( * ) = * . Let Set * be the category of pointed sets and pointed maps.
Any partial map : 1 ⇀ 2 will be considered as the pointed map * : 1 * → 2 * defined as follows: This allows us to identify the category of sets and partial maps with the category Set * . A (right) monoid action ( , * ) on a pointed set consists of a monoid and a pointed set * with an arbitrary map ⋅ : * × → * satisfying the following conditions: 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 op → Set * 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.
If we require ̸ = 0 and 0 ∈ , then we get the definition of an asynchronous system in the sense of Bednarczyk [1].

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.