# Category theory and a new approach to transport and logistics

There is a strategic need for generalizing models, theories and concepts that should bring the transport modes in an equivalent logistic space where the inter- and multimodal systems are comparable and relational. The structural and relational patterns valid in any transport mode must be identified. Modal and intermodal cooperation models transcending the transport modes may be achieved based on them. As a matter of fact, the transport of the loading units from the producer to the consumer, irrespective of the transport mode includes the same logistic stages such as their loading-unloading into and from the vehicles, the transhipment, storage, consolidation and dismantling of the loads, the shipment-delivery formalities and customs operations etc.

The category theory is such a theoretical framework that may be used for the construction of some generalizing models, providing a way of identifying the invariant configurations and processes based on which the unitary treatment of the systems belonging to different transport modes should be possible. By the category theory, the systems and subsystems in transports, their components, the respective models and procedures, the specific institutions and organizations become objects among which there are certain relations referred to as morphisms.

The category theory is based on the idea of system of functions between certain objects. The category is an algebraic structure being different from the notion of group by the fact that the composition law is not compulsorily defined everywhere.

In the category theory, an object A belonging to Ob(**C**) is is determined by its relations with other objects B belonging to Ob(**C**). The analogy with logistics, for instance, with the logistic chains in a set of logistic systems is self-required.

There are given:

- The class of objects A, B,C, … , noted Ob (K);
- The set of morphisms from A to B:

HomK (A,B), for any pair (A,B) belonging to Ob(K) x Ob (K);

- The law of composing the morphisms:

the application from HomK (A,B) x HomK (B,C) to HomK(A,C), for any triplet (A,B,C) belonging to Ob (K) x Ob(K); (u,v) to v o u, meaning v o u : from A to C, or vu belonging to HomK (A,C).

We say that these data define a category K, if there are checked some axioms, like associativity, identity (self identical morphism) etc. The enclosed diagram commutes.

If K_{1} and K_{2} are two categories, we say that it has been defined a covariant (contravariant) functor F from K_{1} to K_{2}, F : K_{1} to K_{2}, if:

- it has been defined an application Ob(K
_{1}) to Ob(K_{2}), which associates to any object A from K_{1}an object F(A) from K_{2}; - for any pair (A,B) of objects from K
_{1}it has been defined the operation

F(A,B) : from Hom_{K1}(A,B) to Hom_{K2}(F(A),F(B))

(respective F(A,B) : from Hom_{K1}(A,B) to Hom_{K2}(F(B),F(A))) such as, if instead of F(A,B)(u) we write F(u), we have:

- F (1
_{A}) = 1_{F(A)}for any A belonging to Ob (K_{1}), - F(vou) = F (v) o F (u) (respective F(vou) = F (u) o F (v)), for any morphisms u and v of the K
_{1}, for which, the compound v◦u has a meaning.

The composition vou is a kind of product of the functions u and v. A category is an algebra formed of the objects A,B,C… and morphisms u, v, f, g, h,… among the objects. The category theory has been created as a modality of studying and characterizing different types of structures in terms of their admissible transformations, which preserves the structures.

The transport modes are objects forming a separate topological space. Each transport mode is a topologic space. The current division into modes is a convention from the past, but on the future the transport modes could be unified or restructured in other topological spaces. The logistic objects may be aggregated by the relationships between them into more and more complex objects corresponding to the more and more complex requirements of the users. By the relationships between objects becoming morphisms from a class of objects to another one, the logistic systems may be created, modeled and managed. The more or less complex notions become objects and morphisms defining the logistic systems. Logistic patterns may appear from this, the systems’ heredity can be studied and a genetic helix of each logistic system could be defined.

The dynamics of the logistic objects consists of the change of the connections between objects, of the connection diagrams, of the object structure, of the connection intensity. As the sets are a basic elements of mathematics, the sets of various logistic objects are the basic elements of logistics.

Why category theory? Because:

- It puts the current transport and logistic concepts into a new perspective;
- It points out the unity of the logistic and transport concepts.
- It simplifies and standardizes the way of modeling the logistic and transport concepts.
- The results proven in the theory of categories generate automatically results regarding the categories in logistics and transport (mutatis mutandis).
- For each category there is a dual obtained by reversing the morphisms, which consequently suggests new logistic patterns.
- The difficult problems in logistics and transport may be translated and solved into other fields by using the functors that move the processes from a category to another.
- It specifies some notions that were quite vague: universality, accessibility, inter- and multi-modality, co-modality, inter-operability etc.
- By the theory of categories, it is carried out the systemic treatment, there are modeled the relationships between a part and the whole, the analysis / synthesis ratio.
- The concepts of transport and logistics can be generalized by category of categories.

The sequence in the diagram below may suggest the category of transport modes (objects) being in intermodal relations (morphisms). It may be considered together with its dual. If the transport modes form an ordered set A where a, b belong to A, and a precedes b shows that the transport modes a and b are ordered according to the preference criteria, then A is a small category.

In a traffic network, there may be considered that the traffic flows (connections) represent a relationship over the group of nodes. But if it is considered that, for instance., the iron ore that arrives in B, coming from A, can be found in the laminates flow that departs from B to C, then we are referring to the morphisms A to B, B to C, … between the objects of the category of nodes A, B, C, … If a node is isolated, it will regress because it is not included in relationships or morphisms. That is why, the county issues can be decided considering the strategies at the national level, the national issues at the European level and the European issues at the global level, i.e. at the upper level of the set or category in which it is integrated by relationships or morphisms.

Trade, as well as other activities can be similarly modeled.

The proposed theory shows that the models in transports can be translated into the other fields of activities, for instance in biology, following the same behavioural patterns. This means the proximity to the original, universally valid, matrices of the human activities.