https://doi.org/10.31449/inf.v42i4.1282 Informatica 42 (2018) 563–576 563
A Category-theoretic Approach to Organization-based Modeling of Multi
Agent Systems on the Basis of Collective Phenomena and Organizations in
Human Societies
Abderrahim Siam
ICOSI Lab University, Abbes Laghrour khenchela BP 1252 El Houria 40004 Khenchela, Algeria
E-mail: siamabderrahim@gmail.com
Ramdane Maamri
Lab Lire University Mahri Abdelhamid Contantina, Algeria
E-mail: rmaamri @yahoo.fr
Keywords: multi-agent systems, organization, category theory
Received: June 30, 2016
This paper presents an idea of using category theory for developing organizational multi-agent systems
by taking inspiration from collective phenomena and organizations in human societies. Category theory
is used for studying and formalizing organizations and collective phenomena in human societies with the
aim of capturing their logics into categorical models. Afterward, the captured models are mapped categor-
ically to categorical MAS organizational models. This way of thinking allows studying properties of result
MAS organizational models as well as properties of organizations in human societies such as stability and
adaptation before taking them as landmarks for developing MAS organizational models.
Povzetek: Predstavljeno je modeliranje multiagentnih sistemov na osnovi kolektivnega delovanja
ˇ cloveških združb.
1 Introduction
Several characteristics have emerged as essential in nowa-
days computer applications seeing that the widespread of
software use in the various fields and the pervasiveness of
information processing tools in all equipments around us
and which are embedded with more and more means of
communication. Applications must be increasingly dis-
tributed, open, adaptable and robust. Moreover, a great
complexity characterizes software and their development
processes.
Multi-agent systems often abbreviated MASs and orga-
nizational MASs approaches, in which MASs are analyzed
and designed as computational organizations using social
concepts, present privileged solutions to develop applica-
tions outlined above. This is due to their interesting fea-
tures such as the proposed abstractions for structuring the
software as combinations of entities in interactions; the in-
troducing of concepts with very high levels of abstraction
as agents, plans, roles and organizations; and the flexi-
ble coupling that MASs offer through indirect interaction
modes and the late binding between agents in such a way
that the determination of the action to execute and the en-
tity responsible for its execution can be postponed as late
as possible.
The most interesting features of agent-based approaches
include the possibility to combine agents and MASs with
other development paradigms and technologies in order to
strengthen agents and MASs with other interesting fea-
tures. In several works, the agent based development
paradigm is combined with the component based devel-
opment [1]. Although agents and MASs are real techno-
logical advances if compared to software components, the
component based development has reached a stage of ma-
turity and preserves some assets such as good structuring
of applications with variable granularities, reuse of com-
ponents as well as possibilities of dynamic adaptation of
component based applications by adding, removing, substi-
tuting components or reassembling and reconfiguring com-
ponent assemblies. Agents and components are combined
in various forms in several works. In [2],[3],[4], and [5]
as well as in numerous other works, agents via their ca-
pacities of negotiation are used to assist component based
development. For example, this assistance takes forms of
classification, research and rapid selection of reusable com-
ponents, matching between components and assisting ef-
fective assemblies of components. In another alternative,
the maturity of components in structuring and deploying
software as well as possibilities of automatic assemblies of
components encouraged the use of components as building
units to construct agents. This form of combination ap-
pears is several works such as [6],[7] and [8]. Other forms
of combination of agents and components may be found
in the literature. Among the works where the components
are used to build agents, we are particularly interested in
the works of Siam et al [8] where software components are

564 Informatica 42 (2018) 563–576 S. Abderrahim et al.
used to build self-adaptable agents that form an organiza-
tion in which agents are grouped into coalition groups.
The consideration of a social or organizational stand-
point as framework for analysis and design presents an im-
portant key to develop MASs, notably open and adaptive
ones. The organization is concerned with defining, manag-
ing and changing the relationships between agents. It cor-
responds to the concept linking the description of a society
of agents and that of control and coordination activities.
The concept of organization was the subject of numerous
works giving birth to an even greater number of MAS or-
ganizational models. These works are often inspired from
sociology, social psychology, economics, ethology or cog-
nitive science. However, inspirations processes from these
disciplines were not driven with formal approaches that al-
low for reasoning about inspired solutions as well as inspi-
ration processes.
In this paper, we are interested in looking for solu-
tions for the organization of MASs by taking inspiration
from collective phenomena and organizations in human so-
cieties. First, we use the category theory CT concepts
[9][10] to capture the logics of some collective phenomena
and organizations in human societies. After that, the cap-
tured logics serve as the basis for modeling categorically
adaptive organizations of MASs in which agents are built
based on software components. Category theory presents
a sophisticated mathematical toolbox. It provides instru-
ments that facilitate the modeling of complex situations and
which involves structured objects. Concepts in category
theory are typically formalized in terms of the relationships
that each object exhibits to the other objects in the universe
of discourse. Particularly, agents based methods as well as
component based development methods; typically model
the universe as a society of interacting elements. Also, col-
lective phenomena in human societies may be described via
interactions between individuals. Category theory proves a
great power of expression. This power is the result of the
composition of very natural and comprehensible construc-
tions. Although category theory is not based on complex
concepts, it is extremely powerful and it derives its power
from the idea of composition. Category definition itself
contains composition. In addition, it is possible to build
categories of categories. Categories are to be composed in
order to generate more abstract structures. With the ideas
of compositional thinking and the possibilities of diagram-
matical representation, categories can be very useful in ad-
dressing a large range of problems and consequently that
addressed in this paper.
We present an approach for modeling organizations of
MASs and verifying their proprieties. In this approach, we
propose constructing models of organizations of MASs us-
ing category theory on the basis of categorical models of
social processes and organizations in human societies. The
constructed models permit checking the organization pro-
prieties such as stability. The stability of an organization of
a multi agent system includes three important aspects. The
first one concerns the degree to which social structures, es-
tablished by the organization, resist disturbance. The sec-
ond aspect concerns the convergence of a MAS to a valid
organization when organizing, i.e., when a MAS decides to
change the current organization for different reasons, does
the process of organizing converges always to a valid orga-
nization? The third aspect of organization stability depends
on the frequency of changing organization; if the changes
are less frequent, the organization is more stable.
The paper is structured as follows: the second section
presents important concepts related to multi-agent systems
and organizations; the third section is devoted to present
collective phenomena and organizations in human soci-
eties; the fourth one provides a view on the category theory
and its important concepts; in the fifth section, we show
how category theory can assist in the articulation and the
modeling of collective phenomena and organizations in hu-
man societies primarily and how the logics of collective
phenomena and organizations in human societies captured
categorically serve as basis for modeling organizations of
MAS; in the last section, we conclude the paper and define
directions for future work.
2 Multi-agent systems and
organization
Multi-agent systems present an important and very use-
ful development paradigm through an armada of tools and
methodologies for constructing complex, open, distributed
systems that can autonomously adapt to their environ-
ments. The goal of research in multi-agent systems is to fig-
ure out how a society of autonomous entities called agents
can organize themselves in order to solve problems and
produce global phenomena that each agent cannot solve
or/and produce individually. MASs use social metaphor
of the so-called social insects when agents are reactive or
of human organizations when agents are cognitive.
The organization of MASs is concerned with defining,
managing and the change of relationship between agents.
It presents a powerful technique to constrain autonomous
agents to behave in such a way to meet overall goals. It
corresponds to a concept linking the description of a soci-
ety of agents and that of the control and coordination ac-
tivities. The definition of the organization of MASs is not
a subject of consensus. However, two significations may
present extremities of the interval of organization defini-
tions. In [11], an organization is defined as a collective
entity with its own identity which is represented by a group
of agents exhibiting highly formalized social structures. In
[12], an organization is a scheme or a stable structure of
joint activity that can constrain or affect the actions and
interactions of agents for a specific goal. The MAS de-
sign guided by the organization makes later implementa-
tion questions and introduces concepts of very high level
of abstraction such as roles and groups. MASs designed
according to an organizational view may be equipped with
mechanisms of reorganization or self-organization allow-

A Category-theoretic Approach to Organization-based Modeling of. . . Informatica 42 (2018) 563–576 565
ing them to change their organizations by passing from one
organization to another one in order to respond to environ-
ments’ dynamics. When the organization is treated accord-
ing to a self-organizational point of view, the organization
is seen as a process where the degree of order increases
with time, from less to more organized.
Several organizational models of MAS have been devel-
oped. Many of these models are inspired from organiza-
tions in human societies. A survey of such models can be
found in [13].
The organization of multi-agent systems may be seen
from an angle of emergent phenomena in complex systems
[14]. Indeed, the design of such systems focuses mainly on
agents that compose the system. Accordingly, agents are
the engines of the organization; organizations exist only as
emergent and observable phenomena. According to this
agent-centered view [15], emphasis is put on the agents by
specifying and designing local behaviors and peer interac-
tions. The global functionality of the system is the result
of complex and dynamic interactions in the agent’s society.
However, such design approaches often introduce unpre-
dictable phenomena which cannot be checked in advance.
The overall behavior is more than a simple juxtaposition of
agents’ behaviors. Several studies on emergent phenomena
in multi agent systems and multi agent simulations such as
[16],[17] and [18] may be found in literature.
In the opposite direction of the agent-centered point
of view, organizations exist as explicit entities of system.
Thus, the observer of the system can obtain a description of
the organization. A pattern of cooperation is determined by
the designer to define or constrain the behavior of agents.
Works that are included into this direction may be catego-
rized into two classes: approaches where the organization
is specified and formalized but agents do not know about
it and cannot reason on this subject. Examples of such ap-
proaches are agent-oriented methods focusing on organiza-
tional models as in INGENIAS [19] and MaSE [20]. The
second class includes approaches where organizations are
manipulated by both designer and agents. Agents have a
representation of the organization and can perform organi-
zational acts and potentially change the organization. AGR
[21], ISLENDER [22] and the model proposed in [8] are
examples of this class.
It is remarkable that most of existing organizations of
MAS are focused on the concept of role. A role presents
an important organizational concept. It permits an organi-
zational structuring of MASs. The concept of role has vari-
ous meanings depending on the MAS organizational model
and the context of its use. We will see later that the con-
cept of role is an important concept also for organizations
in human societies.
3 Human societies and
organizations, landmarks and
inspirations
Collective phenomena and organizations in human soci-
eties present a great source of inspiration for artificial orga-
nizations and especially for MASs. Human societies have
always produced collective behaviors patterns, allowing
them to overcome the most of encountered problems and
difficulties. Capturing logics of collective phenomena and
organizations in human societies for the sake of reproduc-
ing them requires thorough observation and in-depth stud-
ies of human societies. Nevertheless, these observations
and studies are the core of sociology. Sociological studies
and the studies produced by psychologists, economists and
AI searchers offer tremendous beneficial potential for ex-
ploration and exploitation to bring out landmarks suscep-
tible to serve as basis for developing organization based
multi-agent systems. We can distinguish two main ele-
ments in human societies, whose study would allow the
discovery of several models that could be operationalized
and reproduced for developing more effective and efficient
MASs. The first element is social processes, and the sec-
ond, organizations.
3.1 Social processes
A social process may be defined as the observable and
repetitive patterns of social interaction that have a consis-
tent direction or quality [23]. It may be seen as a way in
which individuals and groups interact and establish rela-
tionships and models of behaviors. These relationships and
models are permanently adjusted and modified through so-
cial interactions. This concept refers to some of the general
and persistent forms that social interaction may assume,
i.e., forms of social interactions or mutual activities that
occur repeatedly [23].
We use a large-scale process, drawn from classical so-
cial theory to illustrate how category theory can assist, at
first, in the articulation and the capturing of logics of com-
plex social phenomena. In a second time, captured log-
ics formalized in terms of categorical concepts serve as
metaphors and starting points for categorical MASs mod-
eling. The used process which is presented below is the
circulation of elites [24]. In sociological literature, several
social processes may be found along with various works
aimed at using formal models to describe them. For ex-
amples, in [25] and [26] as well as in several other works,
investigations have explored the possible applicability of
mathematical chaos theory [27] to the social sciences. In
[28], Attractor models have been used in the formulations
of some social processes.
In the circulation of elite process, sociologist and
economist Vilfredo Pareto divided social classes into mass
class and elite class [24]. According to him, the elite dis-
tinguishes itself by its eminent qualities, its natural and

566 Informatica 42 (2018) 563–576 S. Abderrahim et al.
psychological superiority. It is made up of all those who
exhibit exceptional qualities and eminent aptitudes within
their sphere of activity. It contains both governmental and
non-governmental elites. Elite individuals perpetually rise
from the mass; the ruling elite have the choice to fight
emerging non ruling elite or integrate it, until the ruling
one is finally defeated or replaced. Pareto observes elite re-
newal and replacement, or, on the contrary, its closure. The
formation of counter-elite at the moment of the appearance
of degeneration symptoms of the ruling class appeared to
him as the dynamic constituent of history.
This social process may be abstracted as the interaction
of three elements. The first one is the gradual degradation
of the ruling elite; the second consists of the inherent will-
ingness of the most talented members of the non-elite to up-
grade; the third element is the degree of facility for rising
to a higher social position permitted by prevailing social
structure imposed by the ruling elite. Facility for rising to
a higher social or economic position is known in sociology
as "upward mobility". Depending on whether this mobility
is relatively open or closed, the elite is continually reconsti-
tuted by incorporating new talented members and eliminat-
ing unworthy ones; or, talented and aggressive non-ruling
elites accumulate within the non-ruling elite causing insta-
bility.
3.2 Organization
An organization is a social entity in which the members
thereof perform predefined functions. In human societies,
the organization allows the coordination of complex behav-
iors. The changes in human societies, as the emergence of
large organizations such as the army and the administra-
tion instead of artisan corporations, large families and other
forms of organizations, demonstrates that human societies
change its trends of thinking about social life. The replace-
ment of the behaviors based on common values or social
norms by behaviors directed by a more systematic search
for efficiency is an instrumental logic which subordinates
any action to objectives, purposes. Organizational theory
[29] tries to query this instrumental rationality and its con-
crete expression in the organization in several directions
such as: analysis of rationality and decision making, anal-
ysis of relationships of dependence and power and analysis
of the dynamics of change.
The study of organizations in the human societies en-
ables for the identification of several different organizations
which can be classified and cataloged according to various
points of view and criteria. For examples, the following
kinds of organization may be identified in human societies:
the single person organization which is the simplest orga-
nization, bureaucracy structure organizations [29], matrix
structure organizations [29] and team structure organiza-
tions [29]. The organization refers at the same time to the
process of organizing and the result of this process. The
study of an organization (existing or a new one to define)
consists in: analyzing how it works, identifying its main
features, accentuating the pertinent characteristics of the
members, pointing out the roles adopted by members and
the member relationships, identifying norms and rules that
oversee the organization as well as the chain of command,
specifying how tasks are subdivided into independent and
formalized subtasks, identifying which of subtasks have
dependencies, defining how and by what means tasks may
be grouped and pointing out where/when decisions have to
be made.
We can notice that the issues discussed above present the
same concerns to be addressed by a designer in the anal-
ysis and the design of an organization based multi-agent
system added to the defining of environment proprieties to
be considered such as openness and dynamicity as well as
constraints related to the operationalization of organization
computing.
In this paper, we present team and group structure as ex-
amples of organizations for the purpose of illustrating how
category theory can help in capturing of functioning logics
of these organizations and which are subsequently mapped
to organization based multi-MASs. Group and team are
not the same. We use the term crew to cover both groups
and teams. A crew structure is an organic organization.
Contrary to mechanical organizations where tasks are pre-
cisely defined and broken down into separated and special-
ized parts, in organic organizations, tasks are adjusted and
redefined by means of collaborative work in crews. In a
crew based organization, an organization consists of a set
of crews. Crews represent a system with several actors that
share the common goal of accomplishing the crew’s global
task. The global task of a crew is fractionalized into sub-
tasks which are assigned to members of the crew vis-à-vis
members’ degrees of qualifications.
3.2.1 Group structure organizations
In human societies, a group may be defined as two or
more individuals, interacting and interdependent, who have
come together to achieve particular objectives [29]. Groups
may be formal or informal. A formal group is defined by
the organization’s structure, with designated work assign-
ments establishing tasks. In formal groups, the behaviors
group members should engage in are stipulated by and di-
rected toward organizational goals [29]. An informal group
is a group that is neither formally structured nor organiza-
tionally determined; such a group appears in response to
the need for social contact [29].
One of the key concepts behind the concept of group
in human societies is what is called social identity. It is
perspective that considers when and why individuals con-
sider themselves members of groups [29]. It helps peo-
ple reduce uncertainty about who they are and what they
should do [29]. Several characteristics make a social iden-
tity important to a person. Mainly, similarity and distinc-
tiveness; people who have the same values or character-
istics as other members of their organization have higher
levels of group identification; people are more likely to no-

A Category-theoretic Approach to Organization-based Modeling of. . . Informatica 42 (2018) 563–576 567
tice identities that show how they are different from other
groups [29]. According to [29], Groups generally pass
through a sequence of five stages in their evolution. The
first stage (forming stage) is characterized by uncertainty
about the group’s purpose, structure, and leadership. Mem-
bers try to determine what types of behaviors are accept-
able. This stage is completed when members have begun
to think of themselves as part of a group. In the second
stage called storming stage , members accept the existence
of the group but resist the constraints it imposes on individ-
uality. There is conflict over who will control the group. In
the third stage, the group demonstrates cohesiveness with
a strong sense of group identity. This stage is called norm-
ing stage, it is complete when the group structure solidifies
and the group has assimilated a common set of expectations
of what defines correct member behavior. For temporary
groups, the stage four (performing stage) is for wrapping
up activities and preparing to disband. Some group mem-
bers are upbeat, basking in the group’s accomplishments.
In the last stage called adjourning stage, the structure is
fully functional. Group energy has moved from getting to
know and understand each other to performing the task at
hand. For permanent work groups, this stage is the last
one in development. Groups have properties that format
members’ behavior and allow the explaining of individual
behavior within the group besides the performance of the
group. These proprieties include [29] roles, norms, status,
size, cohesiveness, and diversity.
According to [29], a role may be defined as a set of par-
ticular behavior patterns attributed to an actor occupying a
given position in a social unit. Norms are acceptable stan-
dards of behavior shared by members of a group. Norms
express what members ought and ought not to do under cer-
tain circumstances. When become a member of a group7,
its norms influence members’ behavior with a minimum
of external controls. Members are susceptible to conform-
ing to the group’s norms. This conformity means the ad-
justment of one’s behavior to align with the norms of the
group. Different groups, communities, and societies have
different norms, but they all have them. Status is a socially
defined position given to groups or group members by oth-
ers. Status can help to differentiate members in group and
may be an important motivator when individuals perceive a
disparity between what they believe their status is and what
others perceive it to be. Status tends to derive from one of
three sources: the power a person wields over others; the
person’s ability to contribute to a group’s goals and the in-
dividual’s personal characteristics. Each group is charac-
terized by a size. It has influence on its overall behavior in
the sense that individuals perform better in smaller groups
which are faster at completing tasks than larger ones. Con-
versely, large groups consistently get better results than
their smaller ones. It seems that individuals have tendency
to expend less effort when working collectively than alone.
The Cohesiveness of a group expresses the degree to which
members are attracted to each other and motivated to stay
in this one. Cohesiveness has influence on group’s produc-
tivity. Finally, the diversity in a group expresses the degree
to which its members are similar to, or different from, one
another. This propriety appears to increase group conflict.
One of the most important points when studying a group
is the group decision making. Group decisions compared
to those made by an individual alone have advantages in
the sense that groups generate more complete information
and knowledge; groups take more input and heterogeneity
into the decision process as well as they offer a wider range
of points of view. This allows obtaining more approaches
and alternatives. However, Group decisions are time con-
suming; discussions in the group can be dominated by a
few members; and group decisions suffer from ambiguous
responsibility.
3.2.2 Team structure organizations
Work teams are different from work groups. In groups,
members interact primarily to share information and make
decisions to help each member perform within his area of
responsibility [29]. The group performance is simply the
sum of contributions of the group members. The group is
not able to reach the advanced level of collaboration and
cooperation that would create an overall level of perfor-
mance superior than the sum of the inputs. Work teams are
characterized by positive synergy that means cooperation
of two or more substances, or to produce a combined ef-
fect greater than the sum of their separate effects. When an
organization is oriented work teams rather than groups, it
creates the potential for to generate more outputs without
increasing inputs. In teams, skills are complementary, un-
like groups where skills are random and varied. Teams are
crews as well as groups; according to [29], effective teams
have characteristics related to their key components orga-
nized into three general categories summarized in Table 1.
The most common types of teams in an organization
are: problem-solving teams, self managed work teams,
cross-functional teams, and virtual teams [29]. In problem-
solving teams, members discuss ways of improving quality,
efficiency, and the work environment. Generally, problem-
solving teams have not authority to implement any of their
suggestions, i.e., problem-solving teams make recommen-
dations only. Self managed work teams make recommen-
dations, propose solutions and implement them. Members
of such teams perform: planning and scheduling work, as-
signing tasks to members, making operating decisions, tak-
ing action on problems, and working with suppliers and
customers. Cross-functional teams are constituted of em-
ployees from the same hierarchical level but different work
areas, who are regrouped together to accomplish a task.
Virtual teams are teams that use computer technology to
link together physically detached members in considera-
tion of achieving a common goal.

568 Informatica 42 (2018) 563–576 S. Abderrahim et al.
Context. Composition. Process.
Adequate resources Abilities of members Common purpose
Leadership Personality Specific goals
Structure Allocating roles Team efficacy
Climate of trust Diversity Conflict levels
Performance Size of teams Social loafing
Evaluation Member flexibility
reward systems Member preferences
Table 1: Effective teams characteristics.
4 Category theory, key concepts,
applicability and applications
The effective way of dealing with complexity in order to
represent and reasoning about required information to build
computer systems is via formal methods. Among the most
effective methods and which is adequately capable of ad-
dressing structures is the category theory [9],[10].
Category theory is a sophisticated mathematical tool-
box. It provides instruments that make easier the model-
ing of complex situations and that involve structured ob-
jects. Concepts in category theory are typically formalized
in terms of the relationships that each object exhibits to
the other objects in the universe of discourse. This way of
thinking is totally different from set theory in which con-
cepts are formalized extensionally in the sense that a set
is defined by its elements. The focus on social aspect of
objects lives is the reason of applying CT to software en-
gineering as well as for synthesizing social theory. Partic-
ularly, agents based methods as well as component based
development methods; typically model the universe as a
society of interacting elements. Likewise, human societies
are subject to the same vision.
Our objective is to develop organization based MASs
by taking inspiration from social processes and organiza-
tions that have proven their success and effectiveness in
human societies. For the reasons outlined in the intro-
duction, agents that constitute such MASs are self adap-
tive component-based agents in accordance with the agent
model presented in [8]. MASs development methods, com-
ponent based development methods and the social theory
present three different areas. In these areas, and partic-
ularly in the social theory, the perspective of represent-
ing collective phenomena and social processes at multi-
ple scales and with any level of details is highly probable
to produce complex and interlaced models. The lack of
mathematical formalization can make the results of apply-
ing these models hard to validate and render them difficult
to reproduce. One solution to managing the conceptual and
computational complexity of social models is to move them
toward a higher level of abstraction.
The CT may be involved to formalize these models and
exploring their potential interaction. In one hand, category
theory is able to integrate diverse areas and provides a com-
mon language which can be applied to deal with the di-
versity of social theories in the synthesis of social theory,
as well as dealing with agents, MAS organizational con-
cepts and software components with the same terms. Cat-
egory theory not only allows these areas to be expressed
in the same categorical terms, it also provides a basis for
their possible integration. In the other hand, category the-
ory supplements firm equality with weakened definitions of
equivalence relations and/or classes. This propriety makes
important contribution in studying areas in which concepts
involve proximity assessments which is the case of social
analysis as well as organizations in MASs. The potential is
to address qualitative concepts with greater precision. Cat-
egory theory can assist in the articulation of complex social
phenomena and organizations in human societies; it helps
to capture logics of validated phenomena from the domain
of human societies and move them to the domain of organi-
zations of MASs, categorically. The coherence of such cat-
egorical transformations makes a direct and important con-
tribution to the process of validating MAS organizational
models.
4.1 Some category theory concepts and
notations
A categoryC consists of objects and morphisms with com-
position, identity, associativity and unit. Objects a;b;c;
. . . etc are denoted ob(C). Morphisms f;g;h; . . . etc de-
notedhom(C)where, for each morphismg there are given
objects: dom(g) and cod(g) called the domain and the
codomain of g. The notation g : a! b indicates that
a = dom(g) and b = cod(g). Given the objects a;b;c
andd ofC and morphismsf : a! b andg : b! c,
withcod(f) = dom(g), there is a given morphism: gof :
a! c, called the composite offandg. For each objecta
there is a given morphimida :a! a, called the identity
morphism ofa.
A categoryC is characterized by the associativity which
means that for the morphisms f : a! b;g : b! c
andh : c! d;ho(gof) = (hog)of. The unit propriety
means that foida = f = idbof, for all f : a ! b.
A discrete category is a category where the morphisms
are only identity morphisms. An initial object in a cate-
gory C is an object for which every other object of C is
a codomain of a unique morphism with the initial object

A Category-theoretic Approach to Organization-based Modeling of. . . Informatica 42 (2018) 563–576 569
as a domain. A terminal object has every object of C as
the domain of a unique morphism where the terminal ob-
ject is a codomain. Formally, in a categoryC an object is
called initial objecto if for any other objecta inC, there
is a unique morphismo! a. An object is called termi-
nal objectt if for any other objecta inC, there is unique
morphism a! t. In the case where objects of a cate-
gory are categories, morphisms are called functor. A func-
torF :C! D between categoriesC andD is a structure
preserving mapping of objects to objects and morphisms to
morphisms with: F (f : A! B) = F (f) : F (A)!
F (B);F (gof) =F (g)oF (f); andF (idA) =idF (A).
This section offers a very brief overview of some cate-
gory theory concepts. To obtain a broad or complete view
on the concepts of this theory, works [9] or [10] can serve
as references.
5 Categorical modeling of
organizations of MASs on the
basis of capturing logics of some
collective phenomena in human
societies
We use two examples to explain the ideas of looking for
organizational solutions by studying collective phenom-
ena and organizations in human societies when modeling
a multi agent system for a given problem. For the first
example, we assume that it is a question of proposing an
organization for a multi-agent system comprising a num-
ber of heterogeneous agents. Each agent has abilities and
skills that differ from other agents. The best agents in terms
of abilities and skills decide the overall strategy of the sys-
tem. For example, in the case of a multi-agent system for
information retrieval [30], the agents best equipped with
planning and researcher capabilities decide how or which
information retrieval algorithm will be applied. Given that
agents are self-adaptive and can acquire skills and quali-
ties as the system progresses in its functioning, the set of
agents that decide changes over time. The second example
presents a case of MASs where agents must be structured
into crews in which agents plays roles while respecting cer-
tain norms. For example, a distributed auction system in
which three roles are played. The role Auctioneer played
by agents that want to sell items; the role Seller played by
agents that sell items on behalf of Auctioneers; and the role
Bidder played by agents that make a bid on an item being
auctioned [8].
After having explored the collective phenomena or the
organizations which have proved their effectiveness and
which seem to have a certain similarity with the problem to
be solved, it is necessary to target a phenomenon or an or-
ganization to be the basis for the organization of the multi-
agent system to be modeled. Once a phenomenon or an
organization is targeted, we proceed to its categorical mod-
eling. Such a modeling allows the study of the properties
considered as important such as the stability presented in
the introduction. A categorical modeling is to construct
categories for which it is possible to find functors allowing
the construction of categories that model a MAS. These
functors are schematized in the figure 1 by the different
connections between the concepts of the category theory
and social concepts on the one hand, and the concepts of
the domains agent, MAS and the component based devel-
opment on the other. This path of reflection can be summa-
rized as follows:
– Analysis of the problem for which we are seeking an
organizational solution;
– Explore collective phenomena or the organizations
which have proved their effectiveness and which seem
to have a certain similarity with the problem;
– Modeling with categories the selected phenomenon or
organization;
– The study of the properties of the categorical model (If
the properties of the chosen model are not satisfactory,
it is necessary to return to step 2);
– Use the correspondences schematized in the figure 1
between categorical and social concepts and agent and
MASs concepts to obtain a categorical organizational
model of MAS;
– The study of the properties of the categorical MAS
model (If the properties of the chosen model are not
satisfactory, it is necessary to return to step 5 or 2);
– Refine the categorical model of the organization of the
MAS by introducing more details where objects of a
category are modeled as new categories. For exam-
ple an agent that is an object in a MAS category can
be modeled as a category whose objects are software
components;
– Make implementation choices.
When modeling a social process or a collective phe-
nomenon in human societies categorically, it is possible to
take social object as objects of categories, then, According
to the behavior to be modeled, adapted sorts of morphisms
can be identified. Social objects include several kinds of
objects. Among these objects, social actors and Social ag-
gregates are identified [31]. A Social actor may be defined
as an active social entity that is considered at any scale,
such as, individuals, Alliances, groups, teams... etc. Social
actors make decisions and try actively to arouse coordina-
tion with internal and external actors. Social aggregates
are passive collections of social actors such as populations,
generation and aggregations. A social aggregate is deter-
mined by exogenous structures, events and process. It is
changed through proximate adaptation and evolution.
After modeling a social process or a collective phe-
nomenon, we use the same set of social mappings to

570 Informatica 42 (2018) 563–576 S. Abderrahim et al.
metaphorize the process of organizing multi agent systems.
This categorical way of thinking is immensely important
for how we understand organizations both in human soci-
eties and multi agent systems. Thereby, a MAS can be seen
as a category which objects are agents and morphisms are
interactions between agents. We can also build a category
of MASs which objects are MASs and morphisms present
reorganization processes. An agent can be seen as a cat-
egory also. Its objects are different components that con-
stitute the agent and morphisms are interactions between
components or compositions. A functor between two cat-
egories agent A and agent B may present an adaptation
of the agent A to become similar to B, as well as it may
present all sorts of interactions between the agents A and B.
These same principles are valid for components develop-
ment concepts as depicted in figure 1 that illustrates map-
pings from Categories to social concepts, MAS and agents’
concepts and components development concepts. For ex-
ample, component based architecture can be seen as a cat-
egory in which components present objects and interac-
tions present morphisms. Concerning architectural styles,
the choice of a specific category can be seen to reflect the
choice of a specific architectural style.
It is clear that there is a sort of universality in the way
that social theory, MAS, organizations, and component
based development can be approached through categories.
This universality supports the integration of different ap-
proaches.
5.1 Organizing a MAS on the basis of the
circulation of elite process
Assuming that we want to propose a multi-agent
organization-oriented solution for the problem presented
earlier as a first example. In this class of problems, a sub-
set of agents that show good capacities decide about the
global strategy of the system. For example, in the case of
a multi-agent system for information retrieval [30], the de-
cision to choose the information retrieval strategy is taken
by the most qualified agents. An organizational solution
for this class of problems can be inspired from the process
of circulation of elites. According to the process of circula-
tion of elites presented above, a society may be represented
by the set of all individuals noted  . At the timei, the so-
ciety is given as the triplet (A
i
;B
i
;C
i
) which is a partition
of the set  where: A
i
is the set of the mass individuals at
the timei; B
i
is the set of the non ruling elite individuals
at the timei;C
i
the set of the ruling elite individuals at the
timei. Thus,  = A
i
[B
i
[C
i
.
At times i(i = 1; 2;:::;n), a so-
ciety may be described with the stats
M
1
(A
1
;B
1
;C
1
);M
2
(A
2
;B
2
;C
2
);:::;M
n
(A
n
;B
n
;C
n
).
The transition of the society from a stateM
j
to another one
M
k
means that: (i) an element from the non ruling elite
is integrated in the ruling elite; (ii) an element form the
ruling elite is excluded from this one; or (iii) an element
from the mass is integrated in the elite.
If we consider two successive statesM
i
andM
i+1
, D
i
is the transformation passing the society fromM
i
toM
i+1
.
For two non-successive stats M
j
and M
k
, there is a se-
quence of transformations D : M
j
! M
k
. If His a
transformation formM
k
toM
k+1
, the compositionH D :
M
j
! M
k+1
, changes the society from the stat M
j
to
M
k+1
. We are in front of a category of social transforma-
tions.
Each stat M
i
may be described categori-
cally. Let be the set E defined as: E =
f(x;X
i
)=x2X
i
;X
i
=A
i
_B
i
_C
i
g.
Let be the discrete categoryCatE which objects are the
elements ofE and morphisms are identity morphisms. We
define categories CatE
m
by recurrence. The objects of
a category CatE
n
are the elements of the setE
n
defined
below. Morphisms ofCatE
n
are identities.
E
n
=f(x;X
n
i
)=x2X
n
i
;X
n
i
=A
n
_B
n
_C
n
g.
With
A
0
=A;B
0
=B;C
0
=C
and
A
n
=A
(n 1)
 D
(n 1)
;
B
n
= (B
(n 1)
 F
(n 1)
)[G
(n 1)
;
C
n
=C
(n 1)
 G
(n 1)
Where setsD
i
,F
i
andG
i
are defined as follows:
D
i
=fx2A
i
^T 1(x) =Cte
1
g
F
i
=fx2B
i
^T 2(x) =Cte
2
g
G
i
=fx2C
i
^T 3(x) =Cte
3
g
T 1, T 2 and T 3 are functions expressing different situ-
ations allowing an individual x from  to pass from one
class to another at the timei as depicted in figure 2. The
arrows to the symbol  express the disappearance of an
individual.
We define now a category CAT of the categories
CatE
m
which objects are categories CatE
1
, CatE
2
,. . . ,
CatE
m
and morphisms (functors) are inclusion maps
(canonical injections)= defined as:
= : (x;X
i
)! (x;X
j
);X
i
 X
j
The formalization presented above stipulates that the tran-
sition of the society to the state n from the state n 1
means that: some elements from the mass class have be-
come members of the class of non-governing elite (A
n
=
A
(
n 1) D
(
n 1)) ; some elements from the non ruling
elite class have become members of the class of the gov-
erning elite and vice-versa (B
n
= (B
(
n 1) F
(
n 1))[
G
(
n 1)) ; or some elements from the ruling elite class
have become members of the class of the non ruling elite
(C
n
=C
(
n 1) G
(
n 1)).

A Category-theoretic Approach to Organization-based Modeling of. . . Informatica 42 (2018) 563–576 571
Figure 1: Correspondences between categorical concepts and social, MAS and component based development concepts.

572 Informatica 42 (2018) 563–576 S. Abderrahim et al.
Figure 2: A schematic of passages of individuals between
classes according to the circulation of elite process.
Once the categorical model is established, the study of
all properties becomes possible. For example, the study
of the stability of the circulation of elites process can be
achieved through monitoring cardinalities of the setsA
i
,B
i
and C
i
. With the progressive changes in a society, if the
cardinality of one of the three sets becomes lower com-
pared to a minimum limit or upper compared to a max-
imum limit, the instability is the result. At the time i,
the number of elements of the population  is N
i
. Ele-
ments of  are divided into the three subsetsA
i
,B
i
andC
i
which cardinalities are respectivelyni
1
,ni
2
andni
3
with
N
i
= ni
1
+ni
2
+ni
3
. For example, One of the factors
of system stability is the comply with the condition formu-
lated as:
 j
 nij
Ni
  j
;8j = (1::3)
The functionsT 1,T 2 andT 3 and the valuesCte1,Cte2
andCte3 express conditions allowing an individualx from
 to pass from one class to another. T 1, T 2 and T 3 ex-
pressions may be determined in several ways. For exam-
ple, for the defining of the functions T 1, T 2 and T 3 that
produce the probability that a member of a class moved to
another class or disappear at the timei, it is possible to take
inspiration from the Maxwell-Boltzmann law used in sta-
tistical physics to determine the distribution of particles be-
tween different energy levels [32]. At the basis of Maxwell-
Boltzmann law and according to the case of one of its vari-
ants relative to bosons (Bose-Einstein) or fermions (Fermi-
Dirac) taking into account the parameters giving the char-
acteristic of elements. It is possible to correspond the level
0 to the elements that disappear; to the elements that remain
in their class the level 1 and those who pass to another class
the level 2. The number of elements in a given level takes
a similar form to that presented in the previous paragraph
by considering as parameters some characteristics of the
elements and the characteristic of each pre-defined level.
The circulation of elite process categorically modeled
can be the basis of modeling multi-agent systems. This so-
cial process may be reproduced in the modeling of a MAS
for information retrieval as well as for a great class of sim-
ilar problems. Agents of such system are partitioned into
steering agents and executor ones. Steering agents produce
plans and decide about strategies of information retrieval
that executor agents execute. The Steering agents are the
best agents in terms of planning capacities. The same cate-
gorical modeling may be used in such a way steering agents
are the ruling elite, executor agents form the mass class.
The agents are self adaptive and can acquire new skills.
As the system progresses in its execution, agents acquire
new skills. The best executor agents in terms of capaci-
ties, progress to join the steering agents passing by an in-
termediary status of a non-ruling elite (non ruling steering
agents). The stability of such organization of MAS may be
studied in the same way in which the stability of societies is
treated i.e. via cardinalities of the groups of ruling steering
agents, non ruling elite agents and executor agents.
For such system we use the model of agent presented
in [8]. In this model, agents are composed of software
components assembled automatically and dynamically by
an assembly engine. The different capabilities of an agent
are implemented based on software components. The ar-
chitecture of the agent includes a control module. Among
the tasks for which the control module is responsible is the
decision to make changes in the internal composition of the
agent. These changes consist in perform assemblies or re-
assemblies of components. By achieving changes in the
composition of an agent, it acquires new skills. Agents are
to present categorically in order to study their capacities.
Each agent may be modeled as a category which objects
are software components and morphisms are interactions
between components. Agent’s capacities may be studied
as proprieties determined using the capacities fuzzy mea-
sure proposed by Siam et al [8]. The figure 3 offers a gen-
eral view of such modeling in which the different status of
the agents society constitutes a category in which objects
are status and morphisms are the transitions of the society
from one status to another one. A transition expresses the
fact that an agent has changed class. A status which is an
object of the category described above is also a category. Its
objects are agents and morphims are interactions between
agents. Agents in their turns are categories where objects
are software components and morphisms are interactions
and compositions of components.
5.2 Organization of a MAS based on
organizations in human societies
It is possible to describe with category theory concepts,
organizations in human societies and their organizational
concepts. All concepts presented in the section 3 such as
organizations, groups, roles, teams, norms...etc may be rep-
resented using categorical concepts. The figure 1 illustrates
correspondences between categorical concepts and organi-
zational ones.
To propose an organizational modeling of a MAS allow-
ing the implementation of the distributed auction system

A Category-theoretic Approach to Organization-based Modeling of. . . Informatica 42 (2018) 563–576 573
Figure 3: A categorical modeling on the basis of the circu-
lation of elite process.
described above, all organizations in human societies in
which a society is structured into groups or teams of mem-
bers who play roles can serve as a source of inspiration.
For example, it is possible to define a category ROL of
roles. The objects of this category are triplets (R
i
;S
i
;P
i
)
whereR
i
is a set of roles;S
i
are protocols that connect the
rolesR
i
,P
i
denote pair of roles connected by each proto-
col. Morphisms of ROL are relations  between triplets
(R
i
;S
i
;P
i
). A relation between two triplets (R
1
;S
1
;P
1
)
and (R
2
;S
2
;P
2
) may be defined as a functionf = (; )
with  : R
1
! R
2
and  : S1 ! S2such that
8s2S1;P (s) = (r1;r2) =) (s) = ( (r1); (r2)).
We can verify that the objects (R
i
;S
i
;P
i
) with the re-
lations  are a category by checking the different propri-
eties of a category. Objects of the category are (R
i
;S
i
;P
i
)
and its morphisms are relations  . The identity morphism
is id
 = (id
R
;id
S
) such that id
R
and id
S
are identity
functions;id
R
maps each role to itself andid
S
maps each
protocol to itself. To verify the composition propriety, Let
M
1
;M
2
;M
3
be three objects and f
1
= ( 1
; 1
);f
2
=
( 2
; 2
) be two relations such that f
1
: M
1
! M
2
and f
2
: M
2
! M
3
. The composition is defined as:
f
2
 f
1
= ( 2
  1; 2  1). Regarding the associativ-
ity propriety, the relations between triplet objects consist
of functions between sets. Thence, the assiociativity is de-
rived from the associativity of functions between sets.
In the same way, all organizational and social concepts
such as norms, groups . . . etc, may be modeled categori-
cally. After modeling all concepts of an organization with
categories C
i
, the organization categorical model may be
obtained by building a new category from the categories
C
i
. Several ways are available to build a new category
from existing ones. For examples, by compositions, by
adding structures, using sub categories, using product of
categories. . . etc.
After modeling an organization with categories, it be-
comes possible to study formally its properties such as
stability and adaptation by using advanced categorical
concepts and notions such as natural transformations,
Pushout...etc.
By using correspondences between categorical concepts
and organizational ones and correspondences between cat-
egorical concepts and multi-agent organizational ones il-
lustrated in figure1, the categorical model of an organi-
zation in a human society may be moved in a categorical
model of an organization of MAS. The resulting model
may be extended categorically in the same way with that
presented in the first example. Once the organizational cat-
egorical model of MAS produced, studying organization
proprieties becomes possible. Finally, an implementation
of the MAS is to realize. Several ways to implement MASs
are available; the use of MAS platforms [33] is one of the
most effective.

574 Informatica 42 (2018) 563–576 S. Abderrahim et al.
6 Conclusion
The important property of category theory is that it allows
studying structures and their proprieties in one domain and
moving them to another one together with the capacities to
deal with all concepts of all domains with the same cate-
gorical terms. In this paper, we presented an idea of study-
ing and formalizing organizations and collective phenom-
ena in human societies with category theory with the aim
of capturing their logics into categorical models. After-
ward, the captured models are mapped categorically to cat-
egorical MAS organizational models. This way of think-
ing allows taking inspiration from efficient organizations
in human societies in order to develop efficient organiza-
tional models for MAS. Categories are compositional; in
a categorical MAS organizational model, agents are often
presented as the objects of certain categories. Such model
may be refined and extended by considering each agent as a
category which objects are software components and mor-
phisms are interactions between components. So, agent’s
proprieties may also be studied. With this way of thinking,
it becomes possible to study the properties of organizations
in human societies such as stability and adaptation before
taking them as landmarks for developing MAS organiza-
tional models. The inspirations from human societies are
conducted with formal concepts. So, it is possible to rea-
son about the inspiration processes themselves regardless
of the source or inspired models.
The present work presents a nucleus around which sev-
eral research projects can be initiated. These projects are to
be managed according to three main axes. In the first one,
it is necessary to study the maximum possible of collective
phenomena and organizations in human societies and sub-
sequently to model them with category theory concepts. A
classification of the produced categorical models must be
carried out. Each class of models responds to the specific
needs of one or more types of problems. This classification
requires a parameterization operation to output a signature
that characterizes and allows classifying each model in one
or more classes. Once reached at an advanced stage in the
study, categorical modeling and classifying collective phe-
nomena and organizations, a library of models is the result.
To look for an organizational solution for a multi-agent sys-
tem designed for a given problem, we have to do the param-
eterization of the problem. The result signature directs the
search for a solution in the model library. The search for a
solution in the model library may be done in an automatic
way. Thus, the reuse of solutions is pushed very far. In
addition, it is possible to evaluate the appropriateness of
each chosen solution with the type of problem for which
the solution is chosen. The traces of the different evalu-
ations allow better directing the search for organizational
solutions for each type of problems. This approach is sim-
ilar to component-oriented development approaches where
off-the-shelf components are ready to use. In the second
axis, works must be driven to study on the basis of the cat-
egorical models the emergent phenomena in both MASs
and human societies. In the third axis, sociology benefits
from the possibilities of making simulations to study the
impact of the variations that societies can undergo. .
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