# Conflict Resolution in Mechatronic Collaborative Design Using Category Theory
## 3 State-of-the-Art
### Category Theory (CT)
#### 3.3.2 Category Theory in Collaborative Design  
In recent years, CT has found wide applications in several domains such as artificial intelligence [34], computer science [35], Systems Engineering (SE) [8] and collaborative design [36]. In the context of collaborative design, [PERSON] et al. [32] presented an approach using category theory in order to construct a unified multi-agent systems model. Through this model it is possible to represent the communication between the different agents by means of CT. This proposition intends to check the system properties by the construction of categories and functors. Furthermore, another research work based on CT was proposed by [PERSON] and [PERSON] [8]. The researchers applied CT to verify and validate the modelled system design. The authors introduced CT as a formal foundation  
Figure 1: Fundamental concepts of Category theory: (**a**) A mapping from category C to D through a functor “F”, (**b**) Comma category and (**c**) a pullback structure, adopted from [33].  
for model-based systems engineering (MBSE). In this work, the global system is considered as a category and its components as objects. The interrelations between these objects are presented by means of arrows and the different alternatives of the system are introduced as categories. The solution category is related to the system category by functors where each alternative is a result of a particular functor. Additionally, [PERSON] and [PERSON] [37] presented a categorical framework to formally design and implement complex systems as well as to verify the consistency of communication between design and implementation. The presented framework is based on the use of functors in order to verify design categorical models against implementation categorical models. Moreover, [PERSON] et al. [38] proposed a formalization of the verifiable design process (VDP) through CT. The system models are presented by categories and their constituent parts are defined through objects and morphisms. Categorical structures such as pullback and pushout are used to analyze the different representations of the system and functors are applied to define the abstraction layer of the VDP.  
In summary, a multitude of methodologies based on the mathematical formalism of CT can be found in the literature. Although their diversity, a common point between these approaches is presented, that is, the use of CT as a formal tool to support the heterogeneous representations of the system under design. Similar to these approaches cited beforehand, CT will be applied in this research paper as a common formalism to support conflict detection and handling. However, this theory will be used in order to manipulate fine granularity data (i.e., in the form of parameters and constraints) in contrast with previous works where the CT is used to manipulate either system components or system agents without taking into account the parameters and constraints of the system. Categorical structures, such as objects, morphisms, functors, and comma category, will be used in our formalism, as detailed in the next section.