# General Concepts and Reality (Prof. F. William Lawvere)

Subject: Re: [cm] particulars vs. generals, Date: 25 Oct 2011, From: F. William Lawvere ,To: “posina”

Dear Posina

[...] In my 2003 Florence lectures and elsewhere I elaborated the relation between general concepts and reality in terms of two contrasts:

particulars vs general concepts

Within the general, abstract vs concrete.

A system (or “doctrine”) of general concepts involves a fibered category in which the objects in the base are abstract generals (often called “theories”) and the fibers are the concrete generals (categories of models). The fibration structure (Cat-valued contravariant functor # with domain the base) is precisely “functorial semantics”, because a morphism in the base (“interpretation of one theory in another”) induces contravariantly a functor from one fiber to the other. In case there is an initial theory, every fiber has a canonical “underlying” functor to the thus-distinguished fiber K (which is typically taken to be the category of abstract sets).

A major methodological advance exploited over the last 50 years (starting in linear algebra) came from the observation that a FURTHER FUNCTORIZATION of functorial semantics is often possible, because not only the concrete generals (which are “obviously” categories) but also the abstract generals can often be construed as categories, and the objects of the fibers “more concretely” as functors.

A particular by contrast is a category P equipped with a functor to K regarded as a measurement. This category itself may come from reality as a collection of experiments which are behaviourly comparable; that is it need not be construed conceptually as consisting of structures and structure-preserving maps. However such a conceptual theory P* may be derived (depending on the doctrine of generals) by considering the diagrams of natural transformations on K and close relatives (like AK where A is for example an exponential functor on K, or “arity”). That abstract general P* has over it the fiber P*# which of course is a concrete general. Typically there is a functor P->P*# into this double dual giving concrete models for the objects of the particular. However this functor is surely not surjective and typically not faithful. Thus there is no natural place in this account for any “concrete particular”; that would be a “category mistake”.

While in mathematical practice the part the particulars P may also be given as structures (just of a different kind or different doctrine), the possibility is deliberately included that P is derived from planned perception of reality. By contrast both aspects abstract and concrete of a general concept are purely mental (i.e., belong to collective thinking). To cite a notorious example the category of featherless bipeds as a class is equally mental with the theory of fbps. The contrast between the abstract and concrete aspects of a general concept is an important historically developed tool for developing thinking. It is all part of a method for understanding and dealing with reality, but to confuse reality with the concrete developed to model it is the philosophical mistake of objective idealism.

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From: F W Lawvere , Date: 3 Dec 2001

ONE MORE HISTORICAL CITATION

The Pumplun paper cited by Wyler as well as the Auderset paper cited by Mme Ehresmann illustrate that the study of generic structures in 2-categories has been going on for some time. My own paper ORDINAL SUMS AND EQUATIONAL DOCTRINES, SLNM 80 (1969) 141-155

shows that the augmented simplicial category Delta serves as the generic monad, but moreover goes on to actually apply this to show that the Kleisli construction is a tensor product left-adjoint to the Eilenberg-Moore construction which is an enriched Hom. The Hom/tensor formalism

appropriate to the case of strict monoid objects is all that is required here, as I will explain below.

AN EXTENSION AND A RESTRICTION

The important special case of FROBENIUS monads is explicitly characterized in three ways in my paper.

Concerning the IDEMPOTENT case discussed a few days ago by Grandis and Johnstone, note that the publication of Schanuel and Street proves among other things that the monoid Delta in Cat has very few quotients (see below for significance of the monoid structure).

THE GENERAL HOM/TENSOR FORMALISM AND A VERY PARTICULAR MONOID

In any cartesian-closed category with finite limits and co-limits, a non-linear version of the Cartan-Eilenberg Hom/tensor formalism applies to actions and biactions of monoid objects. In Cat, Delta is a (strict) monoid and its actions are precisely monads on arbitrary categories. A

crucial part of the formalism is that categories of actions are automatically enriched in the basic cartesian-closed category, which in this case is Cat. There is a particular biaction of Delta, which I called Delta plus, with the property that the enriched Hom of it into an arbitrary Delta-action is exactly the Eilenberg-Moore category of “algebras”, automatically equipped with its structure as a Delta^op action (co-monad). The left-adjoint tensor assigns to any category equipped with

a co-monad its Kleisli category, as a category with monad. Not only are the calculations in this particular case quite explicit, but the enriched Hom tensor formalism has a lot of content which is still under-exploited.

SKETCHES VERSUS PLATONISM

The often repeated slander that mathematicians think “as if” they were “platonists” needs to be combatted rather than swallowed. What mathematicians and other scientists use is the objectively developed human instrument of general concepts. (The plan to misleadingly use that fact as a support for philosophical idealism may have been an honest mistake by Plato, or it may have been part of his job as disinformation officer for the Athenian CIA organization; it probably would not have survived until now had it not been for the special efforts of Cosimo de’ Medici.)

It seems that a general concept has two related aspects, as I began to realize more explicitly in connection with my paper Adjointness in foundations, Dialectica vol. 23, 1969 281-296; I later learned that some philosophers refer to these two aspects as “abstract general vs. concrete general”. For example, there is the algebraic theory of rings vs. the category of all rings, or

a particular abstract group vs. the category of all permutation representations of the group. While it is “obvious” that, at least in mathematics, a concrete general should have the structure of a category, because all the instances embody the same abstract general and hence any two instances can be compared in preferred ways, by contrast it was not until the late fifties that one realized that an abstract general can also be construed as a category in its own right. That realization essentially made explicit the fact that substitution is a logical operation and indeed is the most fundamental logical operation.

Thus an abstract general is essentially a special algebraic structure indeed a category with additional structure such as finite limits or still richer doctrines. As with other algebraic structures there are again two aspects, the structures themselves and their presentations which are closely related, yet quite distinct; for example, more than one presentation may be needed for efficient calculations determining features of the same algebraic structure. What is meant by a presentation depends on the doctrine: for example Delta as a mere category has an infinite

presentation used in topology, but as a strict monoidal category it has a finite presentation.

The notion of SKETCH is the most efficient scheme yet devised for the general construction of PRESENTATIONS OF ABSTRACT GENERALS. The fact that particular abstract generals and the idea of sketches exist within the historically developed objective science does not mean that they somehow always existed; to call them “platonic” seems to detract from the honor of their actual discoverers.

Bill Lawvere

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F. William Lawvere

Mathematics Department, State University of New York

HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere

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http://facultypages.ecc.edu/alsani/ct01(9-12)/msg00110.html

From: “F. William Lawvere” , Date: 05 Dec 2001

In my 1972 Perugia Notes I had made an attempt to characterize the relation between these sorts of mathematical considerations and philosophy by saying that while platonism is wrong on the relation between Thinking and Being, something analogous is correct WITHIN the realm of Thinking. The relevant dialectic there is between abstract general and concrete general.

Not concrete particular (“concrete” here does not mean “real”).There is another crucial dialectic making particulars (neither abstract nor concrete) give rise to an abstract general; since experiments do not mechanically give rise to theory, it is harder to give a purely mathematical outline of how that dialectic works, though it certainly does work. A mathematical model of it can be

based on the hypothesis that a given set of particulars is somehow itself a category (or graph), i.e., that the appropriate ways of comparing the particulars are given but that their essence is not. Then their “natural structure” (analogous to cohomology operations) is an abstract general and the corresponding concrete general receives a Fourier-Gelfand-Dirac functor from the original particulars. That functor is usually not full because the real particulars are infinitely deep and the natural structure is computed with respect to some limited doctrine; the doctrine can be varied, or “screwed up or down” as James Clerk Maxwell put it, in order to see various phenomena.

http://facultypages.ecc.edu/alsani/ct01(9-12)/msg00116.html

From: Andree Ehresmann , Date: 01 Dec 2001

I don’t know the Pumplun’s paper cited by Wyler. But there is another reference at about the same time; indeed, the “walking adjunction” has been explicitly constructed and studied in the paper of Auderset:

“Adjonction et monade au niveau des 2-cat=E9gories”

published in “Cahiers de Top. et Geom. Diff.” XV-1 (1974), 3-20.

More formally it could also be called “the 2-sketch of an adjunction” in the terminology in my paper with Charles Ehresmann:

“Categories of sketched structures”, in the “Cahiers” XIII-2 (1972),

reprinted in

“Charles Ehresmann: Oeuvres completes et commentees” Part IV-2.

To add a remark on the terminology: When Charles introduced the concept of a sketch (already in a Kansas report of1966, cf. “Oeuvres” Parts III-2 and IV-1), the aim was to define the ‘Platonist idea’ of a structure, not only of a purely algebraic one, but also of structures like categories

(partially defined operations), fields, or even topologies. He thought first of calling a sketch an idea, but then reserved the word “idea” for the smallest part which helps reconstruct the sketch; for instance for a category, the arrows which ‘represent’ the domain and codomain maps and the

composition law.

Sincerely

Andree C. Ehresmann

http://facultypages.ecc.edu/alsani/ct01(9-12)/msg00109.html

Union College Category Theory: Celebrating Bill Lawvere & 50 Years of Functorial Semantics

What are Foundations of Geometry and Algebra?

F.W. Lawvere

From observation of, and participation in, the ongoing actual practice of Mathematics, Decisive Abstract General Relations (DAGRs) can be extracted; when they are made explicit, these DAGRs become a guide to further rational practice of mathematics. The worry that these DAGRs may turn out to be as numerous as the specific mathematical facts themselves is overcome by viewing the ensemble of DAGRs as a ‘Foundation‘, expressed as a single algebraic system whose current description can be finitely-presented. The category of categories (as a cartesian closed category with an object of small discrete categories) aims to serve as such a Foundation. One basic DAGR is the contrast between space and quantity, and especially the relation between the two that is expressed by the role of spaces as domains of variation for intensively and extensively variable quantity; in that way, the foundational aspects of cohesive space and variable quantity inherently includes also the conceptual basis for analysis, both for functional analysis and for the transformation from continuous cohesion to combinatorial semi-discreteness via abstract homotopy theory. Function spaces embody a pervasive DAGR.

The year 1960 was a turning point. Kan, Isbell, Grothendieck and Yoneda had further developed the Eilenberg-Mac Lane Theory of Naturality. Their work implicitly pointed towards such a Foundation as a foreseeable goal. Although the work of those four great mathematicians was still unknown to me, I had independently traversed a sufficient fragment of a similar path to encourage me to become a student of Professor Eilenberg. As I slowly became aware of the importance of those earlier developments, I attempted to participate in the realization of a Foundation in the sense described above, first through concentration on the particular doctrine known as Universal Algebra, making explicit the fibered category whose base consists of abstract generals (called theories) and whose fibers are concrete generals (known as algebraic categories). The term âFunctorial Semanticsâ simply refers to the fact that in such a fibered category, any interpretation Tâ â T of theories induces a map in the opposite direction between the two categories of concrete meanings; this is a direct generalization of the previously observed cases of linear algebra, where the abstract generals are rings and the fibers consist of modules, and of group theory where the dialectic between abstract groups and their actions had long been fundamental in practice. This kind of fibration is special, because the objects T in the base are themselves categories, as I had noticed after first rediscovering the notion of clone, but then rejecting the latter on the basis of the principle that, to compare two things, one must first make sure that they are in the same category; when the two are (a) a theory and (b) a background category in which it is to be interpreted, comparisons being models, the category of categories with products serves. Left adjoints to the re-interpretation functors between fibers exist in this particular doctrine of general concepts, unifying a large number of classical and new constructions of algebra. Isbell conjugacy can provide a first approximation to the general space vs quantity pseudo-duality, because recent developments (KIGY) had shown that also spaces themselves are determined by categories (of figures and incidence relations inside them).

My 1963 thesis clearly explains that presentations (having a signature consisting of names for generators and another signature consisting of names for equational axioms) constitute one important source of theories. This syntactical left adjoint directly generalizes the presentations known from elimination theory in linear algebra and from word problems in group theory. No one would confuse rings and groups themselves with their various syntactical presentations, but previous foundations of algebra had under-emphasized the existence of another important method for constructing examples, namely the Algebraic Structure functor. Being a left adjoint, it can be calculated as a colimit over finite graphs. Fundamental examples, like cohomology operations as studied by the heroes of the 50′s, show that typically an abstract general (such as an isometry group) arises by naturality; to find a syntactical presentation for it may then be an important question. This extraction, by naturality from a particular family of cases, provides much finer invariants, and as a process bears a profound resemblance to the basic extraction of abstract generals from experience.