EV Paton

¹Faculté de Mathématiques et d'Informatique, 33 rue Saint-Leu, 80039 Amiens, France

This paper assesses the role of metaphors in mathematical research, and more particularly in the case of Category Theory. It is shown how different metaphors for categories have had different consequences for their later development. The metaphor of category "as an" oriented graph is analysed more thoroughly, by stressing the relations between categories and graphs; it leads to a more geometric approach and is at the basis of the diagrammatic method. Metaphors are also constructive for the applications of Category Theory in Science. As an example, the last sections show how metaphors, and in particular the metaphor "colimit as a gluing", have helped develop a model for natural complex systems (e.g. biological, social or neural systems) based on Category Theory.

This paper seeks to assess the role of metaphors in mathematical research, and especially in the case of Category Theory. We shall take a working definition of metaphor to be the language (whether verbal, visual, or diagrammatic) used to talk about one thing in terms of something else (Harré, 1986; Paton, 2001; Soskice, 1985). At this level the idea is very general, and so analogy and simile become special cases of this inclusive idea. Category Theory was introduced in the middle of the 20^th century, provides a unified treatment of mathematical structures that allows domains as far apart as Algebra and Topology to be related. Its development has relied heavily on the use of metaphors and similes, such as the relations between category - collection of sets, category - group, category - oriented graph. More recently, it has led to applications in other Sciences.

There has been considerable discussion and debate regarding the nature and status of Category Theory in Mathematics and the Sciences. Category Theory provides conceptual tools to mobilise and manage ideas regarding mathematical structures. There have been a number of philosophical analyses, discussions and reflections on the role Category Theory could play in Mathematics (e.g., Bell, 1981; Landry, 1998). For some, following Lawvere (1966), it provides an alternative foundation to Set Theory; others have argued that it is more a complement to Set Theory (e.g., Marquis, 1995), possibly imposing some supplementary conditions on it (e.g. Muller, 2002). One of the founders of the subject (Saunders Mac Lane) ascribes it a major organising role within Mathematics (e.g., Mac Lane, 1986) because, as he explicitly argues (Mac Lane, 1992), Mathematics is not so much about things (objects) as about form (patterns or structures). Charles Ehresmann, who made many essential contributions to Category Theory, stresses its unifying role in a 1967 paper entitled "Trends toward unity in Mathematics". He explains that, after the development of a large variety of new mathematical structures in the 19th century and the beginning of the 20th,

"the necessity of unification was deeply felt: without a unifying theory following a period of rapid expansion, the mathematicians would fatally tend to use divergent, incompatible languages, like the builders of the tower of Babel. … this theory of categories seems to be the most characteristic unifying trend in present day Mathematics". (Ehresmann 1967, p. 762-3)

He proposed Category Theory as the current unifying tool. We shall begin by examining the role of metaphors and similes in Mathematics and, in particular, in Category Theory.

As noted previously, we use metaphor as an inclusive idea, for the language used to talk about one thing in terms of something else. For those who suggest that Mathematics is a language, it would seem reasonable to expect that in a similar way to other languages we should find mathematical tropes. There are a number of linguistic types of figure of speech such as analogy, metaphor, metonymy, simile, synecdoche. For the purposes of the present discussion, we shall focus on analogy and simile as special cases of metaphor as it is described above. We now examine this with regard to metaphor and mathematical practice and in terms of historical developments.

Consider a few examples from mathematicians who have reflected on the subject of mathematical tropes. In a chapter entitled “Relations to other disciplines” Kac and Ulam (1971) note how Minkowski had been impressed by the conceptual similarity between ideas of Einstein in physics and of Felix Klein in Geometry. In Polya’s (1954) analysis, analogy is a sort of similarity. It is similarity on a more definite or conceptual level. The essential difference between analogy and other kinds of similarity lies in the intentions of the thinker. Similar objects agree with each other in some aspect. Two systems are analogous if they agree in clearly definable relations of their respected parts. The relation of a triangle to a plane is the same as the relation of a tetrahedron to a space. He further proposes that analogy seems to have had a share in all (mathematical) discoveries and in some it was the major share. Mac Lane (1986) lists a variety of processes that may generate new ideas and new notions such as conundrums, completion, invariance, common structure (analogy), intrinsic structure, generalisation, abstraction and axiomatisation. He talks about analogies in terms of distances and angles, Boolean operations on sets and logical operations on propositions, and linear combinations, for example vectors in plane and complex numbers. Davies and Hersh (1990) compare and contrast analogue and analytical sides of Mathematics. In analytical mathematics symbolic material predominates. They relate this to the conscious side. The analogue, holistic and experimental, is related to the unconscious side. In a related way, Hesse (1966) contrasts two types of approach to Science: “Duhemist” and “Campbellian”. The former relies on a formal and analytical approach and the latter more on analogy; it is less precise.

Metaphor in Science and Mathematics fulfils catechretic, ontological and didactic roles, like it does in everyday language as well as in other disciplines. These roles may overlap. For example, Miller (1996) notes how metaphor provides a means of explaining a poorly understood entity in terms the reader or listener understands better. Metaphors are an essential part of creativity not least because they provide a means for seeking literal descriptions of the world. For example, a vector is like a scalar in that you can perform certain types of operation on each and also represent aspects of each with some notion of space (or dimension). However, they are not the same in other respects. Number can be represented as a scalar (e.g., real or integer) or as a vector (e.g., complex) and by setting i to zero reals can be represented on the complex plane or described in terms of vectors. So in this case we can think in terms of a simile (i.e., a vector is like a scalar) but not in terms of a model (i.e., a vector is not a model of a scalar). Consequently generalisations can be made.

In mathematical research, metaphors, analogies and similes give important insights in constructing new mathematical tools well adapted to specific problems of Pure or Applied Mathematics; they allow the furtherance of mathematical knowledge by relating distant domains, and the development of applications in Science. A particular concept may give rise to various similes, each one bearing on a different aspect of the concept, and thus by analogy with this aspect leading to intuitions of a specific nature; the confrontation of these different intuitions often helps one cross the frontiers between distant domains. For instance in Sections 3 to 6, we'll see how various similes for the concept "category" led to different developments of Category Theory. And in Sections 7 and 8 we will see the role of metaphors in the development of a mathematical model for complex natural systems.

The metaphorical nature of Mathematics has been subject to recent debate not least with the approach developed by Lakoff and Nunez (2000); itself based on Lakoff’s considerable work on conceptual metaphor. These metaphors preserve inference structures between source and target domains, and in terms of Mathematics provide a description of how the abstract is comprehended in terms of the concrete. It is not the purpose of the current discussion to review or debate the views of Lakoff and Nunez although considerable analysis is needed to deal with their proposal. Lakoff and Nunez distinguish two kinds of basic metaphorical ideas in Mathematics, grounding metaphors and linking metaphors. The former provides directly grounded ideas such as a set as a container. Note that the container metaphor is a well-established notion that affords more than just membership issues. Linking metaphors can generate ideas that are more abstract, for example, the notion of real numbers as points on a line. Typically, one might view Category Theory as a linking metaphor in that it has originated in the highly abstract mathematics associated with objects and morphisms. However, one could also argue that it is or could be a grounding metaphor. For example, in dialogues where process is at least as important as object, the grounding is the former and the linking becomes the latter. Category Theory provides this ontological switch in the sense that the arrows or morphisms or processes can be given the priority over objects or things.

Eilenberg and Mac Lane first introduced categories in the early forties (Eilenberg & Mac Lane, 1945) as a tool for studying problems connecting Topology and Algebra, and to make calculation possible in Topology, in particular to compute the homology and cohomology of a topological space. Their initial definition of a category (which is the one most people use) is a collection of objects (say A, B,…) and of sets Hom(A,B) of morphisms between them satisfying some rules. This definition is given in a theory of sets and classes, so that the 'collection' of objects can be a proper class, thus allowing one to speak of 'large' categories such as the category of 'all' sets (and maps between them) Set, the category of abelian groups Ab, the category of topological spaces Top,… But it suppresses the 'geometrical insight'.

On the other hand, Charles Ehresmann (cf. "Charles Ehresmann: Œuvres complètes et commentées") came to categories in the late fourties in a different context, namely to study problems in Differential Geometry. He knew of a generalization of groups given by Brandt (1926), that is, the notion of a groupoid. And he remarked that the isomorphisms between fibres in a fibre bundle form such a groupoid. Now a groupoid is just a category in which each morphism is invertible. Thus he first thought of 'small' categories seen 'as group'. This group metaphor relies on the definition of a category (also given by Eilenberg and Mac Lane, 1945), as a set equipped with a partially defined composition law satisfying two axioms:

1. Identity: there exist 'identity' elements such that each element f can be composed on the right and on the left with an identity called, respectively, its source and target; and the composite gf is defined if and only if the source of g is equal to the target of f;

And the foundational basis adopted by Charles Ehresmann was a Theory of Sets with at least one inaccessible ordinal. A category is 'small' if its set of morphisms has a cardinality strictly less than the first inaccessible, 'large' otherwise. Then Set corresponds to the category of small sets. When necessary, the existence of more inaccessibles is assumed (Grothendieck universes)[1].

The second definition is based on the unique data of the composition law, and so seems essentially algebraic. However it is natural to think of a morphism f as an arrow from its source to its target, and Charles Ehresmann whose mind was geometrically oriented, soon rephrased the definition by making apparent the graph so obtained: A category is a graph equipped with a rule for composing two successive arrows, satisfying the Identity and Associativity axioms. Here graph will always mean oriented multi-graph, that is the data of a set of vertices and of arrows between them, with possibly several arrows from a vertex to another one and closed arrows.

Thus there are 3 main metaphors for categories analogous to the relation number/vector introduced above each one suggesting new results: collection of Hom sets/category, group/category; graph/category. A metaphor such as group/category leads to a generalisation, not to a model, as it is often the case in Mathematics.

It might seem that the metaphor at the basis of the definition of a category is not so important since in the end we get the same notion. But it is not true, and here the role of metaphors in Mathematics is particularly clear. When Category Theory became developed in the early 1960s, it evolved very differently depending on the metaphor one employed. For instance the manner of introducing supplementary structures was strongly influenced by the metaphor used:

§ Categoricians who thought of a category as a (large) collection of sets Hom(A,B), introduced 'enriched categories' (in particular abelian categories), in which a supplementary structure (e.g. an abelian group) is introduced in a coherent way on the sets Hom(A,B). Grothendieck introduced the topos of presheaves for problems in Algebraic Geometry (Grothendieck & Verdier, 1963-64), and Lawvere and Tierney (Lawvere, 1972) 'abstracted' its properties in the general concept of an 'elementary topos', which can be thought of as a generalization of the category of sets allowing for an intuitionistic logic, into which a translation of most mathematical concepts is possible (Natural Number Object, Synthetic Differential Geometry,…).

§ The thought of 'small' categories generalizing groups led Ehresmann in 1963 to define 'structured categories' (later named 'internal categories') in which a supplementary structure (e.g. a topology) is given on the whole category, in such a way that the composition law become compatible with this structure. He developed their general theory and gave various applications, e.g. topological and differentiable categories in Differential Geometry, ordered categories, double categories,… (there are numerous papers on this subject in his "Œuvres", Ehresmann, 1980-82). It is only in the early seventies that the importance of this notion was recognised by the other tendency.

· The metaphorical context of category/graph leads to a more diagrammatic approach, when the morphisms are represented as arrows in a graph. It is at the basis of the theory of sketches that Charles Ehresmann introduced in 1967 as a general theory of structures, and developed in a series of papers (reprinted in Ehresmann, 1980-82, Part IV),. Fifteen years later, it was recognised as an important tool in Computer Science (e.g., Barr & Wells, 1984).

We may ask whether the diagrammatic method in Category Theory is a helpful tool or a device for novices and ‘working mathematicians’ though perhaps not needed in a fully formal setting. Naturally the diagrammatic method can be avoided in a formal setting; but then the main benefit of categories as a suggestive and flexible notion seems to disappear. One of the advantages of Category Theory is that it allows us to reason in a diagrammatic and geometric way, though at the same time it can also be approached in a logical setting.

A related question is whether these diagrammatic/analogical arguments relate to the cognitive styles of mathematicians. We note that mathematicians' styles are extensively various. Some can do complicated computations and even creative reasoning on pure formulas. This was however not the case with Charles Ehesmann who had a mainly iconic imagination and translated everything into geometrical images. In his courses he would often draw a diagram and say: "look, there is nothing else to prove". This is effective because proofs of categorical theorems (e.g., the 'Snake Lemma' for abelian categories) often successively exploit the commutativity of certain diagrams, for instance by computing the diagonal of a square in the two different possible ways. Naturally it is possible to do this by writing the corresponding formulas, but it is more intuitive simply to 'look' at the figure; it is generally in this manner that the result has been initially obtained. Moreover a diagrammatic representation also allows one to imagine 'motions' on it. For instance paths in a graph are thought of as a way of 'jumping' from an object to another one; and the composition rule of a category becomes a way of measuring when two paths represent the same global motion. How to work with diagrams is well explained in the following citation of Guitart (2000):

"Un diagramme est un réseau de segments orientés entre des points (compris comme lieux abstraits) dont la "vérité" se découvre en y circulant, en réglant cette circulation, et en insérant ce réseau dans d'autres réseaux. Ce qui se fait en ecrivant des diagrammes dans le diagramme, en écrivant le diagramme en situation comme point abstrait dans un autre diagramme."[2]

Borrowing from Harré’s classification of theories (Harré, 1986), what he called Type 2 (iconic theories – theory families) are described as a “statement-picture” complex. In what sense does the category/graph metaphor place an emphasis on the complex of a picture (the graph) and a statement (the composition law)? To answer this question, in this section we develop the intricate relations between categories and (oriented multi-) graphs.

As said above, we can always speak of a category as a graph (diagram) equipped with a composition law (rule) which associates an arrow to 2 successive arrows, and satisfies the axioms of Identity and Associativity. In this sense, the notion of a category is more restricted than that of a graph since it is not always possible to define such an internal composition on a given graph. For instance in the graph in Figure 1, G is not (the graph underlying) a category since there is no way of defining an internal composition law on it.

But a graph always generates a category, in the following sense: there exists at least one category of which the given graph is a sub-graph and such that each morphism in the category is the composite of one or more arrows of this sub-graph. Naturally the same graph can generate several categories. For example, in the graph in Figure 1, G generates a 'largest' category, namely the category P of its paths, in which each 'diagonal' arrow is the composite of a unique 2-path, and a 'smallest' one, the category C with only one 'diagonal' which becomes the composite of the two 2-paths (cf. Figure 2). Notice that there also exists a category H which has the same underlying graph as P but in which the composition law is different: the lowest 'diagonal' is the composite of each of the 2-paths (so that C is a sub-category of H), while the highest one is not the composite of any 2-path (hence H is not generated by G). This example shows that the drawing of the graph underlying a category is not sufficient to determine the composition law; and to obtain this one must be given an additional rule.

More formally, let us summarize the relations between category and (oriented multi-)graph:

· Given a graph G, it always generates a 'largest' category, namely the category P(G) of its paths; its objects are the vertices of the graph, and the morphisms from A to B are the paths in the graph from A to B. In this way we define a 1-1 correspondence between graphs and between categories of paths (i.e., 'free categories' with respect to the forgetful functor from the category Cat of categories to the category Graph of graphs.) Thus the 2 following notions appear as being 'equivalent': Graph ßà Free category. A graph G also generates a 'smallest' category obtained as the quotient of P(G) by the relation identifying two paths with the same source and the same target.

· Conversely, any category K can be defined as follows: considering K as a graph, take the category P(K) of the paths of K. Then K is the quotient category of P(K) by the equivalence relation R generated by:

(g,f) R gf whenever (g,f) is a path of K and gf is its composite in the category K.

In other terms, each category is the quotient of a free category by an equivalence relation defining the composition. Or still (in analogy with the construction of a group by generators and relations): any category can be defined by generators (arrows of the underlying graph) and the relations indicated above.

It follows that any construction or result established for graphs can be applied to categories, by 'forgetting' the composition law. Conversely, the results about categories can be extended to graphs by replacing a graph by the category of its paths. But free categories (i.e., categories of paths of a graph) are very special, so that many important categorical constructions (such as the gluing process, cf. Section 6) become trivial for them.

An essential feature of Category Theory is its capacity to give a unified treatment of various mathematical structures and of their interrelations, since each structure gives rise to a category (e.g. the structure of group leads to the category of groups and homomorphisms of groups, that of topology to the category of topological spaces and continuous maps,…). Maybe it could also be argued that using the idea of a structure in Mathematics (e.g., groups, rings, fields, lattices etc) requires a notion of similarity (and so simile). The notion of structure, a powerful mathematical construct, also has a number of metaphorical dimensions – related to pattern, relation, space and organisation.

A general theory of structures was initiated by Bourbaki in the late 1930s and early 1940s; it was influenced by discussions with thinkers and philosophers (e.g. Lautman, Cavailles, Queneau, Levi-Strauss,…) who were friends of some of the first Bourbakists. Its development is parallel to the development of structuralism in France. But this theory was not really formalized. It is Category Theory that provided an adequate setting for a theory of structures. This link with structures explains why Charles Ehresmann called his book "Catégories et Structures" (1966). Already in his 1960 paper on "Foncteurs types" (Ehresmann, 1960), he tried to translate the Bourbaki's definitions of species of structures into categorical terms. It was this paper which initially suggested the concept of a double category, which later, matched up to with the topological or differentiable categories Charles had met in Differential Geometry, led to the general concept of internal categories. Lawvere defined the notion of an algebraic structure in categorical terms in his thesis (Lawvere, 1963). Ehresmann's theory of sketches, introduced in 1967 and thoroughly developed by his school in the seventies and eighties, encompassed more general structures, such as categories themselves and topologies.

If we consider a category itself as a structure, the natural question is to define the 'morphisms' between such structures. These are the functors between categories. Indeed, it was to get a 'good' notion of functors that categories were initially introduced.

A functor from a category C to a category C' is a rule associating to each object of C an object of C' and to each morphism of C a morphism of C', in a way compatible with their graph structures and their composition laws. Thus the construction of a functor is related to constructing in a geometrical sense, through the selection of which object of C' will be associated to each object of C.

Often metaphors rely on the replacement of a category (of some kind) by an equivalent one, or more generally by the construction of a functor from a category to another one. Indeed, the functors allow to compare different domains of Mathematics and to carry intuitions between them in a metaphorical way. In particular, in Algebraic Topology functors such as the cohomology functors from the category of complexes to that of abelian groups allow to picture geometric situations by algebraic relations. If two domains of knowledge seem to have such interrelations, then one can be used to hypothesise the existence of (as yet unknown) properties in the other.

Another problem related to the construction of a functor is the case where a category C is given and the problem is to construct a category C' and a functor from C to C' satisfying some constraints in the 'best' way (such is the meaning of 'universal problems', leading, e.g., to the technical construction of adjoint functors, Kan, 1958). This kind of construction is in close similarity to design and architecture.

Categorical thinking allows us to reason as easily with complicated diagrams as with linear sequences; thus a kind of 'parallel' computation is done at once instead of only a serial one as in language.

Consider an illustration from literature. To get meaning from a text one may seek to highlight thematic strands and connect them into a graph. The graph that is produced helps provide a local semantics, a semi-local ‘glue’ that ties various parts of the text together in an internal cohesion. The graph conveys summary information that would be very long-winded to convey in any other way. In many ways categories have a lot in common with the collective nouns of everyday and scientific usage (Paton, 2002). Maybe one way to convey the nature of categories is by way of thinking about the nature and internal composition of texts?

This leads to a metaphor category/text: a category defines a language in which the nouns are its objects, the verbs its morphisms, and the composition law gives grammatical rules. In particular, at a more formal level, we can consider the language so associated to the (large) category Cat whose objects are the (small) categories and the morphisms the functors between them. In this case, the categories correspond to nouns, while the functors play the role of verbs. At a still more formal level, a functor can itself be taken as an object (hence a noun) in a new category in which the morphisms (hence the verbs) between two functors are the natural transformations. This construction leads to the 2-category of categories, functors and natural transformations.

If a hypertext 'glues' parts of a text a similar construction can be done in categories, by forming a new category in which some diagrams of a former category are 'glued'. To give a precise meaning to this, we describe the categorical process which corresponds metaphorically to 'glue', namely the notion of the colimit of a diagram (or of a pattern). (This notion can be used analogically in biosystems to apply ideas related to semi-local glues (Paton, 2001).)

Let C be a category. A pattern P in C is a family of objects N_i of C and some distinguished arrows between them representing a diagram in the category. A collected link from P to an object N' is a family of morphisms f_i from N_i to N' commuting with the distinguished arrows of P; diagramatically it can be represented as a cone with basis P and vertex N' (cf. Figure 3). (Formally, P defines a functor to the category, and a collective link is a natural transformation from this functor to the constant functor on N'.) There may exist several such collective links (or none).

We say that an object N is a colimit (or inductive limit, Kan, 1958) of P if there exists a collective link (c_i) from P to N such that any other collective link (f_i) from P to any N' factors uniquely through (c_i), in the sense that there is a unique f: N -> N' such that c_i f = fi for each i. Metaphorically, the colimit N corresponds to the gluing of the pattern, and the factor f to the gluing of the collective link (f_i).

If a pattern admits a colimit (which is not always the case), this colimit is unique (up to an isomorphism). For instance if the category is associated with a poset, a family of objects has a colimit N if and only if it has a least upper bound N in the poset. If a pattern does not have a colimit in the category, we can construct a 'larger' category in which it becomes glued (into a colimit). This is the essence of the complexification process used (in Ehresmann & Vanbremeersch, 1987) to model the state transitions of a system: given a category C and some patterns in C which have no colimit, the problem is to construct a ('universal') category C' and a functor F from C to C' such that the image by F of each of the given patterns acquires a gluing in C', and conversely some given objects or gluings in C are suppressed.

We have seen that Category Theory seems to be a major metaphorical conduit for transferring or displacing mathematical ideas and constructs between different branches of the subject. More generally, Category Theory can be seen as a common language for talking about complex systems. Not however in the sense that the Science is reduced to the Mathematics (or vice versa) but rather as serving catechrestic roles. The search for integrative and integrating approaches in the Biosciences is very important at the present time because of the rapid increases in amounts of data, knowledge and information. In particular the displacement of ideas concerned with ‘gluing’ metaphors is a challenging research issue (see e.g., Paton, 1997).

Already graphs have been extensively used in the theory of systems. Indeed, the well-known definition of a system as 'objects with relations between them' (Bertalanffy, 1973) corresponds to the definition of a graph (oriented if the relations are directed), whence the metaphor 'system as graph'. It becomes more 'computable' if the 'relations' are quantified, so that the graph is labelled, whence the metaphor 'system as labelled graph'. In dynamical models the labels vary in time, e.g. become solutions of (partial) differential equations.

We have seen that a graph 'is' a free category (by taking its paths), so the 'system as graph' metaphor leads to a 'system as category' metaphor. But here the category can be a general (non-free) category. The difference is that in the free category of paths of a graph there is only one path with a given composite, while in a general category several paths may have the same composite (for instance, the two 2-paths in the category C of the example Section 3). This is important in categories modeling natural systems, where the data of the composition law identifies which paths of the (underlying) graph are 'functionally equivalent' (have the same composite).

In particular a labelled graph completely determines a non-free category by identifying two parallel paths when some specified function (e.g., the sum or the product) of the labels of their factors takes the same value on both. For instance, this construction is used to model a category of neurons (Ehresmann & Vanbremeersch , 1987).

The identification of paths imposes constraints which play an essential role in a number of properties of categories, in particular those related to the existence of colimits. Indeed, a pattern in a free category cannot have a colimit, unless it has a very, very special form, that makes free categories (hence 'simple' graphs) not well adapted to the study of gluing. This theorem has been proved in (Ehresmann, 1996) in answer to Mac Lane who had asked why graphs were not sufficient for the study of neural systems.

Several authors have used categories as models of natural complex systems. For instance Rosen proposed their use in Biology in the 1950s (Rosen, 1958). Lawvere and other authors (1980) emphasised their role in Physics.

If we want to study problems related to the complexity of the objects (e.g., define gluings of objects and recognise a hierarchy of objects in the system) or of the system (how it can be complexified by adding or deleting gluings), then we need to represent the system as a category with an 'effective' composition, not as a free category. Various models for hierarchical natural systems have been developed (e.g., Goguen, 1970; Baas, 1992). In Ehresmann and Vanbremeersch (1987), the metaphor for hierarchies has led to the notion of a hierarchical category (Ehresmann & Vanbremeersch, 1987): it is a category in which the objects are divided into several 'complexity' levels, with an object A of level n+1 being the gluing (colimit) of a pattern of linked objects of level n. Then A is also an iterated colimit (gluing of gluings of…) of a ramification down to each lower level. Roughly such a ramification can be thought of as a fractal structure (though we are somewhat weary of terms such as 'fractal' which have been most often used 'metaphorically', without a real match with a precise concept. When biological or other physical entities are described as fractals there may be a number of ways of mismatching between an abstract mathematical form with a real physical entity. So assigning fractal to a non mathematical entity is a simile.)

(a) One graph gives the static structure [there could be many static structures – for example, associated with varieties of systemic metaphors, or with different levels of complexity or various observers].

(b) Another graph could give the state transitions of the static structure over time

One could argue that there is a need to be able to manage the various representations within a common framework. The more complex a system is, the more useful is the interweaving of several different ways to approach it. This need has been at the root of the development of the Memory Evolutive Systems (MES) of Ehresmann and Vanbremeersch (1987; cf. also their Internet site) which give a mathematical model for complex natural systems, e.g. biological, social, or neural systems, based on Category Theory. Each concept introduced in MES (evolutive system, hierarchical system, stability span, complexification, CR, interplay among strategies,…) has been suggested by metaphors from specific examples, such as a cell, an organism, a neural system or an enterprise.

First to capture its dynamics and evolution, the system is not represented by a unique category, but by an Evolutive System. It is a family of categories indexed by the time, modeling the successive states of the system; these state-categories are connected by partial functors modeling the state transitions. The functors are 'partial' in the sense that they may not be defined on the whole category, to allow for the loss or destruction of objects in time. Moreover the state-categories are hierarchical, and the state transitions correspond to a complexification process (cf. Section 5) to model the birth/death, scission/gluing processes.

But this leads to the question: how are these processes directed? Whence the introduction first of one CoRegulator (CR), then of a whole net of cooperative/competitive such CRs which direct the evolution of the system through the selection of strategies, and memorize the successive experiences in a central flexible Memory for a better adaptation.

In MES, the temporal aspects are most important. Indeed, a MES with its net of CRs 'is like' a parallel distributed processing system with a modular asynchronous organization, since each CR has its own timescale and operates in accordance with it. These 'local' times must be globally synchronized at the level of the whole system, through an equilibration process called the interplay among the strategies of the CRs. This process may lead to fractures for some CRs if their 'temporal structural constraints' cannot be respected. For instance an application has been given to the ageing process for an organism, which could result from a 'cascade of de/resynchronizations' of successive CRs which must modify their periods for their temporal constraints to be respected.

A MES modeling cognitive systems has to go further to study the development of a semantic memory (and possibly of 'consciousness', cf. Ehresmann & Vanbremeersch 2002).

The problem of semantics impacts on many issues in Biology. There is also the common mistake of thinking that ‘throwing’ more syntax at an issue will improve the semantics – it just adds syntactic complexity. The use of such ideas as CRs and colimits addresses issues of local semantics. There needs to be clarity on how much remains syntactic within this frame. In MES the construction of a semantic memory is described by a supplementary 3-step process:

· First a 'local' classification of objects with respect to a particular attribute (say, their color) is 'acted' by a lower CR, this CR treating in the same way the objects for which the value of this attribute is the same (all blue objects, whatever their shape,…).

· Then this classification is perceived by a higher CR, which memorizes an intransitivity class as a higher object (categorically represented by a limit), called a 'concept' (the concept 'blue').

Thus a 'web of meaning' will be formed by constructing concepts associated simultaneously to several attributes (blue circles), and links which describe their interconnections.

The notion of a MES was suggested by metaphors so as to obtain a sufficiently flexible model for natural systems. In the end it provides a model for cognitive systems which may explain which processes can support the formation of metaphors (Multiplicity principle allowing for balances between several ramifications of an object or instances of a concept). The whole circle is closed.

Taking the example of Category Theory, we have seen how metaphors can influence the development of a mathematical domain of knowledge. Different metaphors lead to different (though equivalent) definitions of a category, each one suggesting different lines of development. We have more thoroughly analysed the metaphor of category - oriented graph, which highlights a powerful diagrammatic method. A category is an oriented (mult)graph equipped with an internal composition of paths; it can also be constructed by the data of a 'generating' graph and an equivalence relation on its paths (that identifies 2 paths having the same composite). The diagrammatic approach is used in sketch theory to obtain a categorical notion of a mathematical structure, encompassing Bourbaki structures. In the last sections, we have briefly shown the role of metaphors, especially the metaphor colimit - gluing, in the development of a mathematical model for natural complex systems based on Category Theory.

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1. The recourse to inaccessibles has been criticized as "excessively comprehensive" (e.g., Muller, 2001). We agree with this, when the category-theoretician "chooses" to be a founding-theoretician. However it is very convenient for the "working categorician" who, in his everyday work, just wants a frame allowing to construct the categories he presently needs, leaving aside more complex foundational problems.