The Category of Models

December 18, 2011 § Leave a comment

It is models that link us to the world.

We can not do (anything) without them. The model is one of the four transcendental (as well as “basic”) conditions of anything, as we are arguing in another chapter. We propose the concept of model as a proper operationalization of interpretation. At the same time, modeling comprises a practical dimension that exceeds that of the other conditions. Models, and modeling may be conceived as an eminently generative double-articulation, much in the sense as Deleuze and Guattari introduced the term in their book Mille Plateaux [1]. So, there is a good reason to say that models articulate us to the world.

Putting models into such a prominent place may raise certain objections. One could say, o.k., models are important, but to lift them out in this way results in the claim that epistemology is the main philosophical discipline. I think, such a characterization does not reflect the status of the generalized model in an appropriate manner. We will discuss this in much more detail in the chapter about the choreostemic constitution. For now, we just want to propose not to neglect that the operation of comparison may be considered a very fundamental operation. Actually, there are good reasons to think that it is the operation of comparison, including the implied concept of similarity, that saves us from being propositional logic, hence deterministic machines. Regardless what is happening in our brains and minds, comparison and hence modeling is taking place everywhere and all the time, even if it feels as if we would apply propositional logic.

Generally spoken, concrete models may be conceived as a particular class of transformations. A generalization of the notion of model could probably be taken also as a generalization of the notion of transformation. This should be checked, of course.

Taking an abstract perspective, we could ask about the (formal) properties of a particular class of transformations or its generalization and the difference to other classes. The purpose of such an endeavor of formalization serves, as it is the case of the formalization of model itself, just a single one: to get clear (1) about what could be said in principle about the object of the formalization, in this case: models, and what not, and (2) to describe the appropriate way of how to speak about models, that is, which structures could not be applied without falling back into inconsistencies. Insofar there are no singularized items also for the exemplars of the class we call “models,” we also may ask about the possibility to make any proposals about the relationships between models, that is, about the possibilities to combine them.

Looking for a formal theory where the notion of transformation is central in some way or another, we find topology and category theory. Since the latter is more general than the first, we may suggest to take a brief view into the mathematical category theory. Category theory is also said to be the abstract theory of functions, that is, mappings between sets. Models, on the other hand, also perform a mapping between two sets.

Category Theory

Before stepping deeper into the subject I have to admit that I am not a mathe-matician. Unfortunately, so, regarding the subject of this chapter. Thus, all we can do here is to justify some suggestions, mainly about the level where we can start to talk correctly about models. Of course, we feel that regarding the relationship between models and category theory there is much more about it than we can accomplish here.

Equally obvious, we can’t provide any thing that could be rated as an introduction, even the entries on Wikipedia are more complete. Our attempts here are based on the book by Steve Awodey [2].

Category theory has been proposed first in the beginning of the 1940ies. Today, it is an eminently important tool in mathematics, since it provides means to formalize (viz., to speak mathematically about ) the relation between mathematical objects.

A mathematical category is defined by the following two elements [2, p.4]:

  • Objects: A, B, C … , i.e. at least three basic objects;
  • Arrows: f, g, … relating the objects A, B, C in a directed manner;

The following conditions are needed to establish a category given A, B, C and f, and g .

  • – for each arrow f, there are implied objects that are called “domain” and “codomain” of f, dom(f) and cod(f) , such that we can write f : A → B in order to indicate that A = dom(f) and B = cod(f); the domain of an arrow is the object that serves as an argument to f (“input”), while the result of the transformation of f is called its codomain.
  • – given the arrows f : A → B and g : B → C, that is cod(f) = dom(g), there is an arrow g∘f : A → C, called the composite of f and g ;
  • – for each object A, there is given an arrow  IA : A → A , called the identity of A;

From these axiomatic conditions the following laws derive:

  • associativity
    h∘(g∘f) = (h∘g)∘f
    for all f : A → B , g : B → C ,  h : C → D
  • unit
    f∘IA = f = IB∘g
    for all  f : A → B

Anything satisfying these definitions is considered a category. The objects need not be sets and the arrows need not be functions. The arrows are also called “morphism” while the composition operator ∘ is a primitive.

“Anything” here means for example mathematical groups, graphs, number spaces, differentiable manifolds and, last but not least, categories itself. In a somewhat limited sense, one could say that a category is a generalized (mathematical) group. Arrows (morphisms) are not only mappings, but also comprise things like, as in the case of the category Rel, the set of all relations between any two objects. Actually, category theory builds a framework for propositions about morphisms, i.e. transformations between structures; it is a “theory about arrows,” as Awodey phrased it.

The relation between A, B and C as established by f, g, establishes a category. Remarkably, a category is not just a relation between two objects, which is important for any possibility to build “daisy chains” of overlapping categories; a basic category can be visualized in a diagram like the following:

Note that the category theoretic notion of “diagram” is a very special one, to which we return later.
If we take A, B and C as a classical set, then the arrows simply turn into (ordinary) functions.

There are also deep relations to (quasi-) logics. Category theory hence may be considered as a theory of abstract mappings, and, as far as this gets symbolized itself, an algebra of abstract mappings, where abstract mappings include any kind of transformation.

Different categories can be related to each other. If such a relation exists as a structure preserving mapping, that relation is called a functor. In [2, p.155] we find: “For fixed categories 𝒞 and 𝒟 (not just “simple” objects as above), we can regard the functors 𝒞𝒟 as the objects of a new category, and the arrows between these objects (actually: functors) are what we are going to call natural transformation. They are to be thought of as different ways of “relating” functors to each other, …”.  Likewise, natural equivalency expresses the isomorphy of the category of functors.

In contrast to the theory of (analytic) functions, category theory allows to express morphisms (“transitions”, “transformations”) between structures of different “kinds.” Interestingly, such cross-structural morphisms are well defined in category theory; if now the describe the relation between, say from a group to a set of sets, where this set is no equivalent to all sets, we have a “forgetful” functor.

The concept of functor developed into one of the most important concepts in (structural) mathematics. In [3] we find: “The Yoneda lemma is an abstract result on functors of the type morphisms into a fixed object. […] It allows the embedding of any category into a category of functors […] defined on that category. It also clarifies how the embedded category, of representable functors and their natural transformations, relates to the other objects in the larger functor category. It is an important tool that underlies several modern developments in algebraic geometry and representation theory. It is named after Nobuo Yoneda.

Such, the Yoneda lemma allows for a reduction of complexity: instead of studying the particular category C, one could instead study the respective functors of sets (as a category). In the opposite direction, it allows to transfer (generalize) concepts from the category of sets to arbitrary categories, even if they can not be conceived as “sets”, i.e. enumerable items.

This is very close to a self-referential situation, which is called a 2-category in category theory. A 2-category is a category with “morphisms between morphisms,” that is, in layman terms, a transformation between transformations. Even higher abstractions are possible: Higher category theory is the part of category theory at a higher order, which—quite remarkably—means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities [4].

It is indeed a remarkable step, since this ultimately allows to start with an “empty structure,” which in some way is equivalent to the transcendental difference as Deleuze has conceived it in “Difference and Repetition.” It also allows to get rid of the apriori of logics, astonishingly by means of mathematics itself.

Models and Categories

Models are generative transformations. In some way, they start either in the undefined, or in another model, i.e. in the same category. Such, they resemble to the (compressed) form of the Yoneda-lemma: X ↦ Hom(—,X). Models thus must not be conceptualized as “initial” or “primitive” objects. As such they would be items of sets, which are reduced, “primitive” categories. Yet, this would be rather inappropriate, since it would require an axiom stating/claiming its existence; in other words, formal concepts and ontological claims are (inappropriately) equated, leading to considerable mess. Additionally, as we have already seen in the chapter about the generalized model, models can’t be generalized into (mathematical) groups.

Now note that the inverse element is not part of a category, in contrast to the definition of a group, and also remember that the inverse element was precisely the element that could not be satisfied by our notion of model!

There are indeed good arguments that models could be formalized as (category theoretic) functors, or better as a particular category of functors. Replacing set theory (including groups) by category theory and its notions allows to study the structure of equivalence, instead of introducing it axiomatically. One stops to talk about equivalence, instead one is interested in isomorphism. Indeed, any equivalence should always be understood as , i.e. equivalent (just) by definition. There is no equivalence in the empirical world, or in any aspect of our relation to the world. Ultimately, we may set things not as equivalent if they match in “all” their criteria, but only if we can’t measure them, if they, in other words, are indiscernible because they are outside of any available scale for comparison.

Isomorphism, in contrast, requires some criteria that have first to be set or selected. This process may be repeated (recursively), as the rules for modeling may be considered as results of models. Again, we have an equivalent structure in our theory, which we called “ortho-regulation.” In both cases we are not threatened by the infinite regress, since in both case the progressive abstraction comes to an end quite naturally.

Yet, in this way, category theory clearly allows for a constructive attitude of equivalence. This situation is very similar to our notion of assignates in modeling. Funny enough, 2-category gives rise to statements like “The objects of the 2-category are called theories, the 1-morphisms  f : A → B are called models of the A in B, and the 2-morphisms are called morphisms between models.” [5] We will return to this in our discussion about the theory of theory. From the perspective of category theory, theories are closely related to “diagrams”:  they are the categorical analogue of an indexed family in set theory.  An indexed family of sets is a collection of sets, indexed by a fixed set; a diagram is a collection of objects and morphisms, indexed by a fixed category, or, equivalently, a functor from a fixed index category to some category.

The possibility of self-referential definitions also opens a fresh view to the concept of data. Data are no longer “givens,” as the etymological roots suggest. Instead, in category theoretic terms, data are the domain and codomain of particular functors, which we usually call “models.” Since data are compounds, it is natural to conceive of them as categories, too.

We already said that models “are” functors, i.e. models may be conceived most appropriately as functors; more precisely, they are functors between two categories C, D, where those categories are related bodies of data. These data C and D, respectively, have a different factual structure, but their abstract structural value is the same. Only for this reason we can “transform” them. If any two categories are “related” through functors we say that those categories are adjuncted.

𝒞 𝒟 U (m1) F (m2)

In our case U on model m1 and F on model m2, as well as 𝒞 and 𝒟, respectively, even belong to the same family of structures.   

Now, Awodey describes [2, p.253] that

… every adjunction describes, in a “syntax invariant” way, a notion of an”algebra” for an abstract “equational theory.”

[quotation marks by Awodey]

Awodey then emphasizes [p.255] that concepts that are defined by adjoints can be defined itself without referring to more “complicated,” say derived or semantic, concepts like limits, quantifiers, homomorphism-sets, infinite conditions, etc.

As it can be seen from the formal definition, the category is an arrangement that fulfills the law of associativity for the arrows:  h∘(g∘f) = (h∘g)∘f.

The important question for us here is, whether this relation also holds for models, which we have set to be equivalent to arrows. Why is this important? First,we think that it is important to be able to conceptualize the relationships between models in a formal manner. Second, we saw that group theory does not provide the means to talk generally about the relationships between models. Third, we see that category theory provides concepts that seem to characterize the structure of models quite well. So, the intention is to keep models aligned to category theory. Put it into different words we also could ask, in which way we should talk about models and their relationships and what should we avoid in order to keep this alignment alive?

In our previous investigation about the relationships between models and groups we already mentioned that a combination of models will fulfill the law of associativity only if the models are completely disjoint. Here, “disjoint” means that the respective arrows f, g, and h result in differently structured solution spaces. Usually, one can regard solution spaces as secondary data spaces, which are disjoint in the case if a transition between them would require a folding of one or of both of them. We can’t proof it here, but we guess that neither a difference in methods nor a difference in the used variables (assignates) represent a “necessary” condition for disjointness.

If we accept that it is meaningful at all to take the perspective of category theory in dealing with the investigation of generalized models, then the previous statement has an important consequence regarding the practice of modeling. In order to be able to conceive of models as a category we have to preserve associativity. In common parlance, however, models are perceived as different if (1) the use different constitutive variables, or if (2) they have been built using different methods. Yet, according to our results, in many cases we simply have different parts of the same, still incomplete model. Perfect disjoint models need to be able to indicate to which data they are applicable and they need to indicate different data as applicable.

The important consequence then is that a sound modeling process should be built as an iterative process, which establishes both a sorting, rating and selection of the input data, and which builds disjoint models on disjoint (primary) input data. Otherwise we would not achieve an instance of the category of models, we just would build a single mapping, a 3-valued relation, a single arrow. Yet and again, we would be trapped in set theory of which we know that it is inappropriate. To put it short we may say that

Category theory directly implies comparative theory.

There is still another correlate to this result. Modeling, correctly understood (where “correctly” means: in a way that avoids formal inconsistencies), has to include a self-focusing mechanism. From other contexts we know this property as “idealistic” style of thinking. For us it implies that we have to implement such an “idealizing” mechanism, which works for any “subject” being “contained” in a SOM.

Categories, Conditions and Machines

These result now seem to be quite important for our perspective onto modeling as part of of machine-based epistemology.

First, we may take it as a further confirmation of the generality of the concept of model, at least if taken in the general form as we’ve proposed it here. To achieve at a model we need not to start with a formal theory, contrary to what is believed by logical constructivism. Of course, we still need theories for modeling, but for quite different reasons. There is no generalization necessary and possible, which would exceed the general notion of a model.

Second, the formal properties of models revealed by category theory emphasize their character of being a double-articulating entity between matter and symbols. Since the formal properties of categories comprise a “cartesic closedness” we are formally allowed to conceive modeling as a purely mechanic activity—within the choreostemic conditions. Hence, we also conclude (1) that modeling can be automated, (2) that modeling could even take place in “purely” material arrangements, and (3) that any model comprises an associative part (here, “associative” does not refer to the “law of association” as in group theory).

Last, but not least, we will have to investigate two further issues: the generalization of the notion of the condition, and the role of the entity we usually call “theory.”

This article was first published 18/12/2011, last revision is from 25/12/2011


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