Appendix
An Experiment in Formalizing the Structures of Commodity Exchange

by Guillaume Couffignal

The purpose of this text is to translate structural elements of the forms of commodity valuation we have identified into mathematical language. Our expectation is that the translation will make it possible both to assess the coherence of these elements and to compare the structures of commodity exchange with those of other types of exchange, ranging from kinship structures in anthropology to economic theories. One advantage of mathematical language is its degree of abstraction, which helps to establish connections among structures drawn from very different fields. In addition, the many discussions we had with Guillaume Couffignal in the course of our work served as a useful constraint, leading us to focus on the structural relations among things and on the maintenance of those relations when transformations occur. We see category theory as a potentially useful tool for developing a substantively less rigid form of structuralism that is capable of following the lineaments of action – a form that we call “pragmatic structuralism.”

Luc Boltanski and Arnaud Esquerre

The goal in using mathematical language is to clarify the processes of the commercial circulation of things, as Luc Boltanski and Arnaud Esquerre have described and analyzed them. Let us recall that, according to the authors, the forms of valuation they have identified do not exhaust either “every perspective that can be applied to anything at all” or all possible forms of circulation. For social significations lend themselves to an infinite number of formal organizations, no one of which can be definitive. Recourse to the language of category theory seems appropriate, then, for several reasons. First, the language of categories has an abstract, generalizing character that does not close off interpretation prematurely, while an overly rigid formal language, though it might appear more precise, tends to impose limits on meaning. Second, category theory is a natural formal framework for the problems that arise when objects and mathematical theories are compared.

Moreover, category theory itself has been formulated in many different ways, and no particular presentation can be privileged in any absolute sense. The multitude of presentations in fact offers a wide range of viewpoints and practices. Thus various concepts that may be considered central can also be expressed as particular cases of other concepts. While in this text I have chosen to privilege Kan extensions in order to introduce the notion of (co) limit as a particular case, it would have been entirely possible to make the inverse choice by presenting Kan extensions as particular cases of (co) limits. The same point could be made regarding the very important notion of adjunction, about which I shall say very little here for want of space.

However, this observation has implications that go well beyond the simple affirmation that category theory is self-reflexive. By “speaking about itself,” category theory enriches itself, in a sense (it would be more accurate to say “in more than one sense”) that can be formalized.

Finally, to synthesize the foregoing points (even if this brief text cannot really give their full flavor), category theory is a highly developed and highly ramified theory that is being further ramified every day. It is thus equipped with a considerable arsenal of mathematical tools with applications to physics and biology, information technology and engineering, and many other fields.1

Some basic elements of the language of category theory

The two sections that follow apply especially to the forms of commodity valuation; to make these sections comprehensible, I must present some basic elements of the language of category theory. A more thorough treatment of category theory would fill hundreds of pages; thus it is out of the question to offer anything but a rough outline of its formalization here. For a much more complete and detailed discussion, I can recommend Saunders Mac Lane’s Categories for the Working Mathematician, and the more recent Categories and Sheaves, by Masaki Kashiwara and Pierre Schapira.

Definition 1

A Category ² consists of

– a set Ob() whose elements are the objects of ,
– for each pair x, y of objects from of a set Hom (x, y) (or simply Hom(x, y)) whose elements are called arrows (or morphisms) with source x and target y and which are denoted by x → y. This set (like the preceding one) may be empty.
– for each object x in a morphism with source and target x, which is denoted by 1_x. This is called the identity morphism or identity on x.
– morphisms such that the target of the first matches the source of the second can be composed, and these compositions are associative and possess identity morphisms as neutral elements. The composition of two morphisms f : x ⟼ y and g : y → z is denoted by g ο f : x → z.

For example, let us assume that the following diagram is a category in which all morphisms (and objects) are represented. Thus the objects of this category are the “points” a, b, c, and d, and its morphisms are f, g, h as well as the composition g ο f and the identity morphisms corresponding to all four points.

Since we have assumed that this diagram represents a category, it follows by definition 1 that:

– the morphism g ο f is the composition of the morphism f followed by the morphism g
– The morphisms g and h can be composed, that is, the compositions h ο g : a → a and g ο h : b → b exist. Since according to the diagram there is only one single morphism with source a and target a (analogous for b), we have necessarily

h ο g = 1_a and g ο h = 1_b.

We say that in this situation g (and analogously h) is an isomorphism, i.e. a morphism g : a → b is an isomorphism if there exists a morphism h : b → a, verifying the two previous relations.

– In the same way the morphisms g ο f and h can be composed, and it follows from the diagram (with the same arguments as before) that

h ο (g ο f ) = f.

Since the composition of morphisms is associative, it follows that h ο (g ο f ) = (h ο g) ο f. After the previous point we know that h ο g = 1_a, so we rediscover by associativity (and the fact that the identity morphisms are neutral for composition) the previous result:

h ο (g ο f ) = (h ο g) ο f = 1_a ο f = f.

A category is thus an oriented graph (whose edges, here called morphisms, are oriented) that is equipped with an associative composition between edges/morphisms as well as with identity morphisms, which play the role of neutral elements for compositions.

Conversely, starting from an oriented graph one can produce a category generated by the oriented graph by formally adjoining the compositions of morphisms and the identity morphisms modulo the relations of associativity and neutrality for the compositions with identity morphisms.

This point will play an important role in the formalization proposed here. We shall assume that commodity structures form categories whose objects are “commodities” and whose morphisms are relations of valuation. For example, a morphism f : x → y in this category expresses the fact that f is a point of view that assigns a higher metaprice to commodity x than to commodity y. The identity morphism 1_x : x → x is a way to express the fact that the commodity x has one and the same metaprice, in a “tautological” way (translating the fact that identity morphisms are neutral elements for the composition of morphisms).

Thus, in practice, commodities are most often considered in the form of an oriented graph: x has a higher metaprice than y, which itself has a higher metaprice than z, and so on. This oriented graph can be completed and become a category. For example, the situation just described can be represented graphically as follows:

This generates the following category (the identity morphisms are usually not represented but are present nevertheless):

The morphism g ο f is thus the point of view composed from f and g which expresses the fact that x has a higher metaprice than z.

Categories are omnipresent in mathematics. Here are just a few examples.

Example. A category that appears trivial but plays an important “structural” role is the³ terminal category, which we denote by ∗. It consists of exactly one object and one morphism (the identity morphism).

Example. The category Ens, whose objects are sets and whose morphisms are set-theoretical mappings.

Example. Each ordered set can be regarded as a category. Here we denote by ℝ₊ the positive real numbers, which we regard as a category as follows:

– the objects of ℝ₊ are the positive real numbers;
– morphisms between positive real numbers are given by the order relation ≥, i.e. for two positive real numbers x and y:

x ≥ y means that there is a morphism x → y.

We usually use the category which denotes positive real numbers to which we “adjoin infinity,” that is, we insert a new object, which we denote by ∞ for infinity, thus extending the morphisms of the order relation to infinity.

Without going into detail, we should note that numerous mathematical structures give rise to categories (for example, groups form the category Grp, topological spaces the category Top, etc.⁴).

It is also possible, starting from any category, to create a new category by formally “reversing” the direction of the morphisms. To illustrate this with an example, we can put the following sentence into the passive mode:

The cat eats the mouse The mouse has been eaten by the cat.

If we display the underlying relations with arrows, we get the following:

A similar “form” of operation (setting into passive) also exists in category theory, where it is called the opposite category of a category. In the case of a category representing commodities under certain relations of value, we shall interpret the concept of the opposite category as a reversal of the temporal orientation of the relations of valuation. This point will be useful for the description of different forms of commodity valuation.

Definition 2

Let be a category. The opposite category of , which we denote by ^opp, is the category with the same objects as , but in which any morphism x → y is induced by a morphism y → x in , so that the composition f ο g in ^opp corresponds to the composition g ο f in .

As soon as a categorical concept for is defined, with the help of this construction, there exists a “dual” or opposite version of the concept that results from applying the foregoing definition to the opposite category ^opp.

We have now introduced an object/notion of category, and in order not to leave the notion “floating on its own in the (mathematical) air,” we need the means to “compare” categories with each other. This leads us to the concept of a functor.

Definition 3

Let and ℬ be two categories. A functor F from the category to the category ℬ, which we denote by F : → ℬ, is given by

– a function that we also denote by F by abuse of notation, from the set of objects from to the set of objects from ℬ:

– for each pair of x, y of objects of A a function that we also denote by F by abuse of notation,

– such that these functions respect the compositions between morphisms and the identity morphisms in , i.e.

where ο also denotes composition in the category ℬ.

A functor F : → ℬ is the datum of a diagram of objects and morphisms in ℬ, indexed by objects and morphisms in the category . We shall often use the terms functor and diagram synonymously. A possible view of the functor concept is that it forms a point of view that allows us to “shift,” i.e. the structure of the category is perceived from the category ℬ. In another logical perspective, a functor F : → ℬ can be understood as the provision of a model of a theory in a theory ℬ. In this view, the theory ℬ assumes the role of semantics.

Examples

For any category there are always at least two functors:

– the identity functor id_c : → to , which maps every object and morphism to itself;
– the “terminal” functor T_c : → ∗, which maps every object from to the only object in ∗ and every morphism in to the only morphism in ∗.

Pointing out that these two “trivial” functors exist may sound like an “innocent” remark, but behind this “innocence” lies the fact that the trivial existence of these two functors is of great theoretical importance. In particular, the language provided here is not “empty” and has a certain well-defined content (these functors are obviously defined).

An important property of functors is the fact that they can be composed. If F : → ℬ and G: ℬ → are two functors, we define the composition functor G ο F : → as the functor we get by first applying F and then G. This leads us to the following example of a category.

Example. The categories (as objects) and the functors (as morphisms) form a category called C at, the category of categories.⁵

Now that we have introduced the concept of the functor, we must find a means to compare functors.

Definition 4

Let F : → ℬ and G : ℬ → be two functors. A natural transformation α of the functor F to the functor G, which we denote by α : F → G, is the datum of a family of morphisms in the category ℬ, indexed in the following way by the objects of the category :

so that for each morphism f : x → y in the following diagram is commutative⁶:

Natural transformations can also be composed. With the above notation, let α : F → G and β : G → H be two natural transformations. Then we define the composition β ο α : F → H by setting for each object x in :

( β ο α)_x = β_x ο α_x : F (x) → H (x).

In this way we get a new kind of category whose objects are the functors.

Example. Let and B be two categories. We denote by C at( , ℬ) the category such that:

– the objects are the functors from to ℬ,
– the morphisms are the natural transformations between functors from to ℬ.
– the identity morphism 1_F is the natural transformation resulting from the identity morphisms 1_{F (a)} F (a) → F (a) for a ∈ Ob(),
– the composition of morphisms is defined as the composition of the associated natural transformations.

Example: Let be a category representing a relational commodity structure, and the category of positive real numbers with infinity ∞ adjoined. We denote by the category of the prices associated to the category of the functors C at (, ). From a logical point of view, this is the category of models of the structure embodied by in the semantics of positive real numbers.

Let V : → be a functor that represents a selling price for a seller of commodities in . A purchase of the same commodities in the same way also represents a functor A : → . We say that there is a (total) added value if a natural transformation α : A → V exists. The natural transformation α measures exactly the seller’s monetary profit at purchase. Conversely, if a natural transformation β : V → A exists, it measures the (total) loss.

Remark. The above example can be generalized. It assumes numbers (here positive real numbers) as “measures” for prices. But one can now also imagine categories other than (that is, other than numbers), which here implies the intervention of a currency with absolute liquidity, which is used like numbers and can play the role of a “metric” (in the sense of Boltanski and Esquerre) for prices and metaprices. After all, we have already seen that it is always possible to define a category C at(, ) of functors for two categories and . Nevertheless, we shall posit that not all categories are suitable and that a category must have certain structures like in order to play this role for metaprices. Some of these properties, because they translate linguistic operations, will be examined below.⁷

Definition 5

Let be a category. We call an object x from initial object, if for each object y in there exists a uniquely determined morphism of the form

x → y

It results from this definition that an initial object is not necessarily unique, but only unique up to unique isomorphism.

By applying the previous definition to the opposite category ^opp, we define a terminal object of a category . An object x of is called terminal if for every object y of a uniquely determined morphism of the form

y → x

exists.

Example: In the category the number 0 is the (only) terminal object and ∞ is the only initial object. For the category Ens the empty set øø is an initial object and the single-point sets are terminal objects (which are uniquely determined up to unique isomorphism).

We are now approaching the key mathematical tools we shall use in the discussion that follows. The essential concept is that of a Kan extension, which, if it exists, in a given situation (a given diagram) provides a “well defined” functor that satisfies certain given properties in the situation under consideration.

Definition 6

Given a diagram

where , ℬ and are categories and F and P are functors, we say that F has a left Kan extension along P if there is a pair (Lan_P F, α) consisting of a functor

Lan_P F : ℬ →

and a natural transformation α : F → Lan_PF ο P which satisfy the following universal property: For each pair (G, β) consisting of a functor G : ℬ → and a natural transformation β : F → G ο P there is a unique natural transformation η: Lan_P F → G, so that for each object a in the following diagram is commutative:

This situation can be presented in a diagram as follows:

We could reformulate this definition as follows: The set of pairs (G, β), where G : ℬ → is a functor and β : F → G ο P is a natural transformation, forms the objects of a category whose morphisms are given by natural transformations between such pairs. That a functor F has a left Kan extension along P means that this category has an initial object, the Kan extension in question.

In a dual way one defines the right Kan extension of a functor along another functor. If it exists, we denote the right Kan extension of the functor F along the functor P by Ran_P F.

We shall primarily use the following special cases of Kan extensions.

Definition 7

Let F : → ℬ be a functor. We say that F has colimits if there exists a left Kan extension of F along the terminal functor T_o:

In this case we denote the Kan extension of F along the terminal functor by colimF (instead of Lan_To F ).

In a dual way, using the previous definition, we say that F has limits if there is a right Kan extension of F along the terminal functor. This is denoted by limF.

A functor of the form ∗ → ℬ corresponds to the specification of an object of the category ℬ (which is the image of the unique object in ∗ under the functor to ℬ). So a colimit

if it exists, corresponds to an object in ℬ, which we also abusively denote by colimF and which satisfies the universal property described above.

Example. In the category , a functor F : → always admits

– a limit that is given by the supremum sup_a∈ F (a) of the F (a), and
– a colimit that is given by the infimum inf_a∈ F (a) of the F (a).

We call a category complete if it contains all limits, which means that all functors in that category have limits, and we call it cocomplete if it contains all colimits. The previous example shows that the category is complete and cocomplete.

Example. The category Ens is also complete and cocomplete. Let us now look at two special cases of limits and colimits, namely the fiber product and the amalgamated sum.

Given the following category, which we shall call I

which consists of three objects 0, 1 and 2 and two morphisms a : 0 → 2 and b : 1 → 2, apart from the identity morphisms (which, as usual, are not represented).

The datum of a functor F : I → Ens corresponds to the datum of a diagram:

where F(0), F(1) and F(2) are sets and F(a) and F(b) are functions. For greater clarity, we denote these sets and functions as follows:

F(0) := A, F(1) := B, F(2) := , F(a) := f, F(b) := g.

Since the category of sets is complete, this functor has a limit, which we call the fiber product of the above diagram, that is, there exists a set denoted by limF (or B ×_C A) and morphisms, which we show dotted below, so that the following diagram commutates:

These morphisms satisfy a universal property, which cannot be described in detail here for want of space. The interesting point is that the elements of the set limF can be fully described:

limF ≃ {(x, y) ∈ B × A | g(x) = f (y)},

i.e. the set that represents the limit is the set of elements from A and B, whose images under the functions f and g respectively agree in C.

Without going into detail, a similar consideration yields the amalgamated sum (given by the fiber product in the opposite category), which we represent by the following commutative diagram:

The set colimF, which we also call B ⊔_C A, is itself given by

colimF ≃ B ⊔ A/ ∼,

where B ⊔ A denotes the disjoint union of the sets B and A and where we identify in the quotient the points from B ⊔ A related by ∼:

x ∼ y, if there is a z ∈ C with g(z) = x and f(z) = y,

which means, in short, that the colimit is the set of all elements in B and A that are not in the image of the functions g and f, to which the above “quotient” is adjoined.

We can reformulate the previous examples as follows in the form of a “scheme” (using this term here in an illustrative sense rather than as a precise mathematical term), concerning the limits and colimits:

a limit “embodies” what appears in a diagram as a common and/or comparable feature and “forgets” the differences in a way determined by the functor (universal property).
a colimit “embodies” what appears in the diagram as different and “deletes” what is common and/or comparable in a way determined by the functor (universal property).

We shall now formulate a theorem about a condition of existence of Kan extensions without proving it or providing details. It suffices to state that this theorem allows us to produce, under certain conditions, a functor that fulfills certain properties and that will enable us to compare categories of commodities in a well-defined way (universal property).

Theorem Consider the following diagram of categories and functors:

where is a (small) category and is cocomplete.
Then there exists a left Kan extension:

given for every b in ℬ by the colimit:

Lan_P F (b) ≃ colim_{(P (a)→b)∈(P ↓b)} F (a).

This notation under the colim sign means that we regard the colimit of the (composite) functor:

where U is a “forgetful functor” (“projection”) and (P ↓ b) denotes the category whose objects are the morphisms in ℬ of the form P(a) → b for objects a of A. In this category the morphisms g → g ′ are given by commutative diagrams of the form:

where f : a → a′ is a morphism in the category .⁸

In a dual way, we have an equivalent result for complete categories. Hence there exists a right Kan extension.

Let us look at an example of the use of Kan extensions in price-setting.

A price is the datum of a functor F : → . The category is complete and cocomplete. So by the previous results there exists for each functor P : → ℬ a left Kan extension

and a right Kan extension

What “sense” can these constructions have? If one understands F as a representative of the metaprice of a seller and the functor P as a shift (actual or narrative) of commodities seen from the viewpoint of toward commodities seen from the viewpoint of ℬ (for example, if P represents the “inclusion” of commodities “structured” according to into a more comprehensive structure ℬ, or if P provides the model for a change in the “discourse” about the commodities or an exchange, etc.), the following interpretations result⁹:

– the left Kan extension (Lan_P F, α) stands, so to speak, for the maximum profit that is “coherent” with the price F under the shift P, where this new price is given by the functor Lan_P F and the quantification of profit by the natural transformation α (the universal property specifying the “maximum”). The theorem stated above even gives a formula for its calculation.
– in a dual way, the right Kan extension (Ran_PF, α) stands, so to speak, for the maximum loss “coherent” with the price F under the shift P.

This brief attempt at a categorical formulation of the investigation of commodity structures has to suffice for now. Category theory and its ramifications have produced an abundance of concepts that could prove useful for an understanding of these structures, and for what Boltanski and Esquerre call “pragmatic structuralism”.¹⁰ The limited scope of this text only allows the presentation of some basic elements of the language of such an endeavor. After this general introduction, let us now turn our attention to the formalization of the forms of valuation that the authors of Enrichment have identified.

Forms of commodity valuations

In this section we shall look at the initial elements in the formalization of the four “pure” forms of commodity valuation: the standard form, the trend form, the collection form, and the asset form. We use the term “pure” because commodities can of course be considered from different viewpoints. For example, depending on the context of its valuation, the same watch may be considered to fall under the standard form or the collection form. We shall examine these possible types of cases later, especially if they can be understood as “shifts” from one form to another.

Let us now summarize the basic assumptions that have guided our undertaking thus far:

A set of commodities under consideration and the relations of “valuation” among these commodities form a category (in the formal sense). For example, the commodities offered in a shop and the valuation relations that exist between them and that justify their price are modeled in a category.
The four forms of commodity valuations are modeled in four different types of categories. Each form of valuation is based on its own type of structure, which is formally reflected in the fact that the modeling categories have different structures. Thus, in the above example, if the relations among the commodities offered in a shop are structured by the standard form, the category that models this situation would have a structure specific to the category associated with the standard form.
In modeling the different forms, we shall limit ourselves to the concepts of category theory that we have introduced above. Some structural elements of the individual forms, whose modeling would be worthwhile, will therefore not be considered in the following.
We have so far limited ourselves to the concepts of limits and colimits. The interest shown in this term is justified by the following scheme: What is structurally determined within a category is given in the form of a limit or colimit of a functor. Conversely, the limit (or colimit) of a functor, if it exists, is a “determination procedure.” This concept of determination seems to us to be indispensable for the understanding of commodity structures. For example, assume that a style is given if one uses the example of the trend form: A style must be sufficiently determined for it to be a style, i.e. for it to be regarded as such and imitated by the various actors.
Furthermore, we have assumed that it is necessary for the price formation and more generally still for shifts between the different forms that things be sufficiently determined in their quality as commodities, as perceived by the various actors involved. A “naked” watch, so to speak, that has neither analytic nor narrative characteristics, has no definable price or metaprice. It is only when a watch is associated with a brand, a style and technical characteristics, and only when the brands, styles and technical characteristics have been sufficiently determined (i.e. recognized as such) by and for the actors, that it is possible to determine its value and associate it with a metaprice.
In view of these various observations, our aim in this section is to raise the question of the existence (or non-existence) of limits and colimits for the different types of (formal) categories associated with the four forms of commodity valuation.
Finally, this questioning has been guided by the (dual) scheme of limits and colimits, presented earlier, in which a functor is understood as a diagram in a category (the target category):
1. A limit “embodies” what appears in a diagram as a common and/ or comparable element and “forgets” the differences in a way that is determined by the functor (universal property).
2. A colimit “embodies” what appears different in a diagram and “deletes” what is common and/or comparable in a way determined by the functor (universal property).

The Standard Form

We call St a category of commodities considered according to the standard form, i.e. according to a form whose relations of valuation, the morphisms of the category, are given by a structure specific to the standard form.

Consider a diagram in this category, i.e. a functor F : → St, where is a category.

If we apply the scheme to the limits and colimits of the standard form, it appears that

– the limit of F, if it exists, as a common “core” of the objects in the diagram whose differences are excluded, corresponds to the common properties of the commodities and the relations considered by the functor F. But since a limit, if it exists, is embodied by an object (unique to unique isomorphism) in the category St, which we call limF, it seems reasonable to assume that the kind of thing represented by limF embodies the prototype common to the objects and relations of the diagram under consideration.
– According to the previous point, not all limits necessarily exist in a category of the standard type. Not all commodity diagrams have a common prototype. A prototype of prototypes is not imaginable, whereas one could imagine the possibility of imagining a collection of collections.
– The colimit of F, if it exists, stands for the “grouping” of the differences among commodities by identifying what they have in common. In the standard form, the differences among commodities are taken into account only if they are comparable and compared, which according to our hypothesis is not compatible with the existence of a colimit. It would be possible for colimits to exist in a category of the standard type, but they would not have a systematic character; rather, they would be contingent for the category under consideration.

Therefore, a commodities category St considered according to the standard form is characterized by the existence of some limits embodying the prototypes.

A price associated with such a category is the specification of a functor St → . According to point 5 above, the prototypes are indispensable for the specification of such a functor. The following facts can be derived from this:

The prototypes function as upper price and metaprice markers insofar as they do not determine the prices and metaprices of the commodities of which they are the prototypes, but insofar as the price or metaprice of any such commodity cannot be higher than the price or metaprice of the prototype. To put it bluntly, the price or metaprice of such commodities cannot be higher than the price or metaprice of the prototype. In this way, in the case of a seller, the price of a prototype linked to a commodity diagram acts as an upper “limit” in the sense of an upper limit on the profit that can be expected from the sale of a given commodity.
Conversely, if the prices or metaprices of the commodities falling under the same prototype (formally represented by a commodity diagram having a limit) are determined, these prices or metaprices constitute the lower limit of the price or metaprice of the prototype.

We shall now use the same approach to analyze the trend form, the collection form, and the asset form. Let us sketch the elements of analysis that allowed us to settle on the proposed solutions.

The Trend Form

We denote by Td a category of commodities considered according to the trend form, i.e. according to a form whose relations of valuation, the morphisms of the category, are given by a structure specific to the trend form.

Let us consider a functor F : → Td.

If we apply the scheme as above to the limits and colimits of the trend form, we get:

– The limit of F, if it exists, embodies the common “core” of the objects in the diagram whose differences are excluded. For the “most pertinent dimension of the trend form” is “the fact of anticipating and controlling the life cycle not only of products as material entities but also and especially of the differences supported by those products.” While in the standard form there are no colimits, in the trend form the concept of limits is not relevant (while limits may conceivably exist in a category of the trend form type, such limits have no systematic character and are contingent on the specific category considered).
– The colimit of F, on the other hand, can be useful for the trend form. Since it represents the “grouping” of the differences of the commodities by identifying what they have in common, the colimit of F, if it exists, embodies the concept of the style under which the commodities are considered. The style is regarded as something that can be recognized by the various actors from a trend perspective, which justifies its modeling as colimit. Since the “style” has no concrete embodiment, it forms an “ideal” and therefore potentially volatile object (after the model of the ideal collections).
– It follows from the previous point that not all colimits necessarily exist in a category of the trend type.

Not every diagram of commodities considered according to the trend form necessarily has a style. Therefore, a category Td of commodities considered according to the trend form is characterized by the existence of certain colimits that embody the respective styles.

A price associated with such a category is given by a functor P: Td → . For the specification of such a functor thus the styles are formative. The following facts about styles can be derived from their property as colimits:

The styles function as lower price or metaprice markers to the extent that they determine an amount that is below that of the commodities considered. This reflects the fact that the commodities purchased within the framework of the trend form are acquired primarily on the basis of their style or in their capacity as signs. The prices or metaprices of such commodities cannot be lower than the prices and metaprices of the style to which they belong. In this way, the price or metaprice of a style associated with a commodity diagram acts, in the case of a seller, as a lower “limit” in the sense of a lower limit on the price that can be expected in the sale of a commodity.
Conversely, if the price or metaprice of a commodity falling under the same style is determined, that price or metaprice constitutes an upper limit for the price or metaprice associated with the style.

The Collection Form

We denote by Col a category of commodities considered according to the collection form, i.e. according to a form whose relations of valuation are given by a structure specific to the collection form.

Let us consider a functor F: D → Col.

If we apply the scheme to the limits and colimits of the collection form, we find:

– The limit of F embodies a collection (ideal or non-ideal), for which the functor embodies the governing principle, to borrow the terms used by Boltanski and Esquerre.
– It follows from the previous point that, unlike the standard form (as far as limits are concerned) and the trend form (as far as colimits are concerned), all limits exist in a category of the collection type. Any diagram of commodities considered according to the collection form always provides a guideline for a collection.
– In a category of commodities considered according to the collection form, colimits do not necessarily exist (i.e., except in a contingent manner). Since commodities viewed through the prism of the collection form are characterized by the importance of more or less divided conventions supporting an (ideal or non-ideal) collection, the concept of colimits does not seem to have a systematic effect.

A category of commodities Col considered according to the collection form is characterized by the existence of all limits in this category that embody the collections.

A price associated with such a category is given by a functor P: Col → . The collections are decisive for the specification of such a functor. The following facts can be derived from the limits:

Collections act as upper price or metaprice markers to the extent that they supply an amount which is higher than that of the commodities considered. From this perspective, the price or metaprice of a commodity cannot be higher than the price or metaprice given by the (ideal or non-ideal) collection to which it belongs. In this way, the price or metaprice of a collection associated to a diagram of commodities acts, in the case of a seller, as an upper “limit” in the sense of an upper limit of the profit that can be expected upon the sale of a given commodity.
Conversely, when it comes to the determined price or metaprice of commodities belonging to the same collection, these prices or metaprices constitute a lower limit for the price and metaprice of the collection. In everyday language, this expresses the banal observation that a painting cannot cost more than the collection of which it is a part.

The Asset Form

We denote by Ast a category of commodities considered according to the asset form, i.e. according to a form whose relations of valuation, the morphisms of the category, are given by a structure specific to the asset form.

Consider a functor F : → Ast.

If we apply the scheme for the last time to the limits and colimits of the asset form, the following facts arise:

– Limits do not necessarily exist in a category considered according to the asset form. The reason for this is that the commodities considered through the prism of the asset form are identified by “the differences related to the degree to which objects can easily be converted into money.” Since it focuses on their common characteristics, the concept of the limit does not really seem to make sense.
– The colimit of F, on the other hand, embodies the liquidity displayed by commodities viewed from the perspective provided by the functor F. Liquidity, as it is widely used in the discourses of social actors when referring to commodities considered according to the asset form, presupposes ideal liquidity along the lines of ideal collections in the collection form. But the difference is that this ideal liquidity is common to all commodities.
– It does not seem exaggerated to assume that all colimits exist in a category of the asset type. The differences between commodities representing a commodity diagram seen in the asset form can always be bracketed, with their liquidity alone being taken into account.

Therefore, a category of commodities Ast considered according to the asset form is characterized by the existence of all colimits in that category that embody the various liquidities or liquidity markers.

A price associated with such a category is given by a functor P : Ast → . In this case, the liquidity markers are decisive for the specification of such a functor. The following properties can be derived from the properties of colimits:

The liquidity markers function as lower price or metaprice markers to the extent that they provide a lower numerical value than the commodities under consideration. From this perspective, the price or metaprice of a commodity cannot be lower than the price or metaprice given by the liquidity of which it is a part. Thus, in the case of a seller, the price or metaprice of the liquidity associated with a commodity diagram functions as lower “limit” in the sense of a lower limit on the profit that can be expected from the sale of a commodity.
Conversely, when the price or metaprice of a commodity with the same liquidity is referred to, that price or metaprice constitutes an upper limit on the price or metaprice of this liquidity.

The universal properties of limits and colimits, which we have not elaborated in detail, can also be used to model possible strategies for commodity valuation as well as structural “incoherences” in the exchange of commodities.

Some comments on this formalization

We summarize the above formalization in the following table:

This table invites several comments:

– The categories of the standard form and the trend form are “incomplete” in comparison with the collection form and the asset form insofar as they are not closed under the operations limits and colimits. If one assumes that the social practices inherent in commodities tend to be somewhat closed, the collection form and the asset form appear more stable in this respect. From a “linguistic” point of view, if limits and colimits are regarded as linguistic operations, this means that the structure of the standard form and the trend form is incomplete with respect to these operations. Thus, when one considers that social practices lead to a “transgression” of these operations, the collection form and the asset form constitute, or may constitute, a “refuge” for the expression of these operations.
– In addition, forms of “duality” occur. If we recall that the limits in a category become colimits in the opposite category (and vice versa), and if one interprets this latter term as a “reversal of the temporal orientation of the relations of valuation,” the following can be noted: Just like the collection form and the asset form, the standard and trend forms are in a reversed temporal relation to each other. The structure of the standard form is oriented toward duration, while the structure of the trend form is oriented toward brevity. The collection form gives preference to narratives turned toward the past, the asset form to narratives turned toward the future.

Taking into account the above remarks, let us now turn our attention to the shifts between the different forms.

Transitions between the forms

In the previous section, we have associated each of the forms with the existence (or non-existence) of limits and colimits. To the extent that “they are determined,” the (co-)limits form markers that play an important role in the attribution of metaprices and prices. Are there shifts (formally speaking: functors) from a category of commodities considered according to the standard form to a category of commodities considered according to the collection form that “respect” the limits? Or shifts from the trend form to the asset form which respect the colimits? etc. If a functor sends the (co-)limits of a category to the (co-)limits of the target category, we say that this functor commutates with (co-)limits.

The formalism we have introduced gives rise to twelve possible cases. The analysis of the different types of functors reveals that some valuation processes are not inherent in the forms, but result from shifts between different forms.

We analyze in list form the “meaning” (or meaninglessness) of each of these cases in the study of the cosmos of commodities:

A functor F : T d^op → St commuting with limits (if they exist). Such a functor transforms a style (colimit in the trend form) into a “prototype” (limit in the standard form). If one imagines it in the form of a shift, such a functor corresponds to the embodiment of a style in something we shall call the model.
A functor F : St^op → Td commuting with colimits (if they exist). This case is “dual” to the previous one. It transforms a “prototype” into a style. Such a process leads to the creation of a brand (such as Apple and the iPhone, the DS brand for the PSA group, etc.).
A functor F : St → Col commuting with limits. A functor of this type transforms a “prototype” into a (potentially ideal) collection.
A functor F : Col → St commuting with limits. This transforms a collection into a “prototype.” Within the framework of the collection form, however, such a shift may be prevented by the “prohibition of reproduction.” This kind of prohibition is not based on material impossibility, because there are copies of works that take the place of the originals in museums, which are kept under conditions that are suitable for their best possible preservation. But such a shift would lead to a decrease in the price of a piece and consequently a decrease in the metaprices of all items in the same collection.
A functor F : Col^op → Ast commuting with colimits. An example of such a functor, which transforms a collection into a liquidity marker, is any case in which a collectable object is used as a money substitute (stamps, coins, medals, etc.).
A functor F : Ast^op → Col commuting with limits. This functor type, on the other hand, embodies the situations in which stamp collectors or coin collectors find themselves when they ensure that things that were previously investments fall under the collection form.
A functor F : T d^op → Col commuting with limits. An example of a functor that shifts from a style to a collection is Hermès handbags, etc.
A functor F : Col^op → Td commuting with colimits. In this case, collections serve as resources for styles. For example, designers or fashion designers refer to collectables to create objects that are to become a new style.
A functor F : Td → Ast commuting with colimits. This case refers to things that were bought as equipment when they were not fashionable, but would possibly become so and could then be resold at a higher price.
A functor F : Ast → Td commuting with colimits. This functor type shifts a liquidity marker (the colimit in the asset form) to a style. Within this framework belong phenomena such as demonstrative or ostentatious wealth, etc.
A functor F : St^op → Ast commuting with colimits. In this case, “prototypes” become liquidity markers. One can think of patents that are not applied for in order to use them directly for production, but because of their future market potential.
A functor F : Ast^op → St commuting with limits. This case, in which liquidity markers are shifted to “prototypes,” is dual to the previous one.

Possible openings

In the previous section, commodities were considered insofar as they occur in “pure” form (corresponding to the standard, trend, collection and asset forms) and the various displacements between these “pure” forms. In our eyes, however, the social “reality” seems to be characterized for the most part by compositions between these different points of view. And it is also not necessary to assume that such a category ℳ is generated by only four types of morphisms: standard, trend, collection and asset. It would be interesting to see whether other types of morphisms, i.e. other forms of valuation, can occur. The difficulty lies in knowing how best to investigate these non-homogeneous structures, beyond pursuing the necessary development of the basic categorical theoretical concepts mentioned in the first section. I should like to point out two other approaches that I see as complementary:

The first is the approach of the theory of enriched categories as developed in Max Kelly’s book. In our view, such a formalism is motivated by the fact that a category of commodities can only “model” a “proper world” reasonably in its specificity if it is “sufficiently” extensive to be able to include in its own world elements that cross its path. Formally speaking, the category ℳ must have enough operations/structures (limits, colimits, monoidal structures, etc.) and internal possibilities of expression within the framework to which we limit ourselves here, for example (depending on the use/need),¹¹ as is the case when ℳ is a cosmos (but other structures are also possible). For example, if ℳ is a cosmos, we have the concept of an enriched category over ℳ, which is a “type” of category, for which the morphisms between objects no longer form a set, but are given by objects of ℳ, that is, for which the Hom(x, y) are not sets, but objects of ℳ. The category theory we are dealing with in our text already falls under this concept, namely the theory of categories enriched over Ens (the category of sets), and the concepts we have introduced can also be defined in the world of categories enriched over a cosmos. The relations between objects in this language are given by the “logic” of ℳ. In the case of categories enriched over , relations between two objects can be given by a positive real number (for example Hom(x, y) = 5)¹² or by commodities in the enriched over ℳ case, where ℳ is a cosmos of commodities. In the latter case, for example, if the objects of an ℳ -category are interpreted as persons, one can imagine an ℳ -category as a structure in which the relations between persons are embodied by commodities from ℳ and the relations among these commodities (phenomena of reification of social relations, etc.). A point of criticism that we can put forward against our formalization of the forms of valuation is that it says nothing or almost nothing about the relations among the commodities: they exist, as if out of the blue, given by “external” considerations and form sets. With the help of the formalization of enriched categories one can again lend more content to the relations/differences between the commodities/persons and above all establish more quantitative theories (the example of the categories enriched over , which are generalized metric spaces). Perhaps it is also possible in this context to present more precise formalizations of the forms of valuation?
The other possible research approach we would like to mention is given by the concept of localization and derived functors (for the formal aspect, we refer to the book by William G. Dwyer et al. and to the article by Bruno Kahn and Georges Maltsiniotis). Where, briefly put, the previous point refers to the step of creating terms that take into account the differentiations between objects, the second research approach that we propose is, in a sense, a complementary step: the ability to identify what already exists as “similar.” Let C be a category. Furthermore, let W be a class of morphisms in the category . In this case there exists a new category W ^–1 , which we call the localization of to W, which has the same objects as , but in which we have formally inverted the morphisms in W, that is, we have forced the morphisms in W to be isomorphisms. We recall that two objects in a category are isomorphic if there is an isomorphism “connecting” them, and that isomorphic objects are structurally (i.e. from the point of view of the category considered) identical/identifiable. The difficulty here is to “understand” the category W ^–1 (if one formally adds morphisms to a category to make it another category, the new compositions and the associativity of these compositions generally produce a large number of new relationships between the morphisms). Less formally, the morphisms in W provide criteria for identifying the objects they connect and the localization W ^–1 is the result of these identifications. In this way, the class of morphisms W provides/describes the objects that are identified, that is, the objects that one wants to see as “equal” (formally isomorphic), and at the same time the way these objects are made “equal” (given by the morphisms in W). There exists a functor called localization functor:

γ : → W ^–1 .

This sends every morphism from W to an isomorphism in W ^–1 and satisfies a universal property that states in a certain sense that this category is “the smallest” category with this property. One can apply the above to a category of commodities ℳ and, for example, take a class of morphisms for W that represent valuation relations that, for example, fall under the standard form. The category W ^–1 C stands for the structure produced when commodities have been “identified” by these morphisms. Conversely, one can ask oneself whether there is a class of relations of valuation W for a category of commodities ℳ, such that, if one identifies (according to the morphisms) those objects that it contains, one gets a category of the standard type, that is, if W ^–1 is a category of the standard type (that is, has certain limits). This formal question can be translated into the following form: If certain analytic features of the relations of valuation between given commodities are expanded, can one see these commodities and their relations as being of the standard form type? We might also turn our attention to the existence and/or the production of functors between categories and to the behavior of these functors under localization:

Here we enter the world of derived functors (which are special cases of Kan extensions). For example, if D = , F represents a price and the existence of derived functors makes it possible to raise the question of the existence of prices, after having “expanded,” among certain commodities, the “features” that remain coherent with price F. In my opinion, these considerations would make it possible to produce formal results with regard to a qualitative analysis of the structuring of commodities.

Translated from the French by Annette Werner

Notes

1. On this topic, the curious reader may wish to consult the articles by the physicist and mathematician John Baez on his website: math.ucr.edu/home/baez/ as well as on his blog: johncarlosbaez.wordpress.com/.
2. Translator’s note: Here the text deviates from the usual definition in category theory. In the standard definition of a category, the objects form a class, not necessarily a set. The special case considered here, in which the objects form a set, is usually referred to as a small category.
3. The use of the definite article “the” is intended. The terminal category is unique in a categorical sense, i.e. unique up to a uniquely determined isomorphism (of categories).
4. Translator’s note: This is only correct if one uses the usual category concept instead of the concept of small category designated in definition 1.
5. We exclude in this text cardinality problems and refer to the book of Saunders MacLane as well as to the books of Masaki Kashiwara and Pierre Schapira.
6. The commutativity of this diagram means the following equality of compositions of morphisms: α_y ο F (f ) = G(f ) ο α_x.
7. At this stage of our thinking, good candidates are those categories which Jean B´enabou calls cosmos, that is, complete and cocomplete closed symmetric monoidal categories, of which two examples are and Ens. A cosmos allows the development of a “good” theory of enriched categories (see for example the classic work of Max Kelly in the references). Even if one can define a concept of weaker enriched categories, the framework of a cosmos seems to provide a more expressive language to transfer a fine-grained critical analysis into the sphere of commodities.
8. The category (P ↓ b) is an example of a so-called comma category.
9. For these interpretations to be more “justified,” we would have to examine special features of the functor P (such as being fully faithful); however, we lack the space to do that here.
10. In particular, we have not addressed the very rich concepts of equivalence of categories and adjunction that can be easily introduced with the help of the material provided, nor the construction techniques for categories such as comma categories. The notion of comma category makes it possible to produce a category, i.e. in our case relations of valuation, based on functors. For example, for two given (price) functors F : → and G : ℬ → , the comma category (F ↓ G) embodies the generation of relations of valuation (and the coherence between these relations) derived from the prices F and G.
11. Here I am thinking in particular of the concept of internal homomorphisms, i.e. the concept of the existence of an object of the category that (internally) represents “morphisms.” For example, in the category of sets the morphisms are functions and the “set” of functions from the set X to the set Y is also a set, which we denote by Y^X.
12. The categories enriched over represent a generalization of metric spaces (see the groundbreaking article by F. William Lawvere). This case allows us to consider commodities as directly differentiated by a numeric value, a positive real number, which embodies the relative metaprice/price between commodities. In contrast to the framework of metric spaces, this framework allows a formulation of the fact that different commodities may have the same metaprice/price (i.e. relative metaprice/price zero). In a metric space in fact, two points with distance zero are equal. In this enriched framework this simply means that they are isomorphic (in terms of enriched categories over ), but not necessarily the same as the structure of a metric space would require.