Types of quantum field thories
F. William Lawvere
is an influential pure category theorist.
for a survey of his academic path and work.
Lawvere invented categorical logic and introduced the eponymous Lawvere theories as a category-theoretic way to describe finitary algebraic theories. He generalised Grothendieck toposes to elementary toposes, revolutionising the foundations of mathematics; in this vein, he developed the foundation in structural set theory called ETCS. He also introduced and worked on synthetic differential geometry as a foundation for differential geometry and equations of motion in continuum physics. Later he introdcued the notion of cohesive topos as a more general foundation of geometry.
Lawvere’s work tends to adobt terminology from Georg Hegel’s Science of Logic, for instance in the term “categories of being and becoming” for cohesive toposes in Some Thoughts on the Future of Category Theory and in the emphasis of the duality between intensive and extensive in Categories in Continuum Physics.
Indeed, Lawvere essentially proposed that Hegel’s “unity of opposites” is usefully formalized by the notion of adjoint triples inducing adjoint pairs of idempotent (co-)monads. For more on this see at Science of Logic – Formalization of unity of opposites
A central motivation for Lawvere’s work is the search for a good mathematical foundations of physics, specifically of (classical) continuum mechanics (or at least some kinematical aspects thereof, Lawvere does not seem to mention Hamiltonians, Lagrangians or action functionals).
In (interview, p. 8) he recalls:
I had been a student at Indiana University from 1955 to January 1960. I liked experimental physics but did not appreciate the imprecise reasoning in some theoretical courses. So I decided to study mathematics first. Truesdell was at the Mathematics Department but he had a great knowledge in Engineering Physics. He took charge of my education there. in 1955 (and subsequently) had advised me on pursuing the study of continuum mechanics and kinetic theory.
In Summer 1958 I studied Topological Dynamics with George Whaples, with the agenda of understanding as much as possible in categorical terms. Categories would clearly be important for simplifying the foundations of continuum physics. I concluded that I would make category theory a central line of my study.
Then in (interview, p. 11) about the early 1960s:
The title of the early text Toposes of laws of motion, which is often cited as the text introducing synthetic differential geometry, clearly witnesses the origin and motivation of these ideas in classical mechanics.
On this, in (interview, p. 15):
Q: As an assistant professor in Chicago, in 1967, you taught with Mac Lane a course on Mechanics, where you “started to think about the justication of older intuitive methods in geometry”7 You called it “synthetic differential geometry”. How did you arrive at the program of Categorical Dynamics and Synthetic Differential Geometry?
A: From January 1967 to August 1967 I was Assistant Professor at the University of Chicago. Mac Lane and I soon organized to teach a joint course based on Mackey’s book “Mathematical Foundations of Quantum Mechanics”.
Q: So, Mackey, a functional analyst from Harvard mainly concerned with the relationship between quantum mechanics and representation theory, had some relation to category theory.
Then (interview, p. 16):
Q: Back to the origins of Synthetic Dierential Geometry, where did the idea of organizing such a joint course on Mechanics originate ? Apparently, Chandra had suggested that Saunders give some courses relevant to physics, and our joint course was the rst of a sequence. Eventually Mac Lane gave a talk about the Hamilton-Jacobi equation at the Naval Academy in summer 1970 that was published in the American Mathematical Monthly
A: I began to apply the Grothendieck topos theory that I had learned from Gabriel to the problem of simplied foundations of continuum mechanics as it had been inspired by Truesdell’s teachings, Noll’s axiomatizations, and by my 1958 efforts to render categorical the subject of topological dynamics.
A review of this with more comments on more relations to physics is in the introduction to the book collection Categories in Continuum Physics, which is the proceedings of a meeting organized by Lawvere in 1982.
In the talk Toposes of laws of motion in 1997, Lawvere starts with the following remark
I read somewhere recently that the basic program of infinitesimal calculus, continuum mechanics, and differential geometry is that all the world can be reconstructed from the infinitely small. One may think this is not possible, but nonetheless it’s certainly a program that has been very fruitful over the last 300 years. I think we are now finally in a position to actually make more explicit what that program amounts to.
I think that on the basis of these developments we can focus on this question of making very explicit how continuum physics etc. can be built up mathematically from very simple ingredients.
In the same talk, a few lines later after discussion of infinitesimally thickened points , it says:
In 2000 in the article_Comments on the development of topos theory_ Lawvere writes in the closing section 7 titled “From and to continuum physics”:
What was the impetus which demanded the simplification and generalization of Grothendieck’s concept of topos, if indeed the application to logic and set theory were not decisive. My own motivation came from my earlier study of physics. The foundation of the continuum physics of general materials, in the spirit of Truesdell, Noll and others, involves powerful and clear physical ideas, which unfortunately have been submerged under a mathematical apparatus . But, as Fichera lamented, all this apparatus gives often a very uncertain fit to the phenomena. The apparatus may well be helpful in the solution of certain problems, but can the problems themselves and the needed axioms be stated in a direct and clear manner? And might this not lead to a simpler, equally rigorous account? These were the questions to which I began to apply the topos method in my 1967 Chicago lectures Categorical dynamics . It was clear that work on the notion of topos itself would be needed to achieve the goal. I had spent 1961-62 with the Berkeley logicians, believing that listening to experts on foundations might be the road to clarifying foundational questions.
Several books treating the simplified topos theory (MacLane-Moerdijk being the most recent and readable text), together with the three excellent books on synthetic differential geometry provide a solid basis on which further treatment of functional analysis and the needed development of continuum physics can be based.
The Wikipdia entry concludes:
Lawvere continues to work on his 50-year quest for a rigorous flexible base for physical ideas, free of unnecessary analytic complications.
See also at higher category theory and physics for more on this.
(Many of Lawvere’s writing don’t exist as published articles, but circulate in some form or other. )
Functorial Semantics of algebraic theories Originally published as: Ph.D. thesis, Columbia University, 1963 and in Reports of the Midwest Category Seminar II, 1968, 41-61, Republished in: Reprints in Theory and Applications of Categories, No. 5 (2004) pp 1-121 (tac)
Adjointness in Foundations, (TAC), Dialectica 23 (1969), 281-296
Quantifiers and sheaves, Actes, Congrès intern, math., 1970. Tome 1, p. 329 à 334 (pdf)
Equality in hyperdoctrines and comprehension schema as an adjoint functor, Proceedings of the AMS Symposium on Pure Mathematics XVII (1970), 1-14.
Teoria delle categorie sopra un topos di base, lecture notes from Perugia (1972–73) – available online here
Topos-theoretic methods in geometry, Aarhus 1997
Extensive and intensive quantities Workshop on Categorical Methods in Geometry, Aaerhus (1983)
State Categories, Closed Categories, and the Existence Semi-Continuous Entropy Functions , IMA reprint 86, 1984 (pdf)
Functional Remarks on the General Concept of Chaos , IMA reprint 87, 1984 (pdf)
on continuum mechanics, variational calculus and laws of motion in synthetic differential geometry and in diffeological spaces and Frölicher spaces, and on intensive and extensive properties in terms of ring objects and modules in a topos.
Some Thoughts on the Future of Category Theory in A. Carboni, M. Pedicchio, G. Rosolini, Category Theory , Proceedings of the International Conference held in Como, Lecture Notes in Mathematics 1488, Springer (1991)
(on – implicitly – cohesive toposes);
Categories of space and quantity in: J. Echeverria et al (eds.), The Space of mathematics , de Gruyter, Berlin, New York (1992)
(on space and quantity)
(an early version of the notion of cohesive toposes)
Volterra’s functionals and covariant cohesion of space Perugia Studies in Mathematics (Proceedings of the May 1997 Meeting in Perugia) (pdf)
(with first remarks on cohesion)
Outline of synthetic differential geometry , seminar notes (1998) (pdf)
(the origin of the concept of synthetic differential geometry) j s
Comments on the development of topos theory in Jean-Paul Pier (ed.) Development of mathematics 1950-2000, Birkhäuser, 2000
Metric spaces, generalized logic and closed categories Reprints in Theory and Applications of Categories, No. 1 (2002) pp 1-37 (tac)
Taking categories seriously, Reprints in Theory and Applications of Categories, No. 8, 2005, pp. 1–24. (pdf)
Categories of spaces may not be generalized spaces, as exemplified by directed graphs , preprint, State University of New York at Buffalo, (1986) Reprints in Theory and Applications of Categories, No. 9, 2005, pp. 1–7 (pdf)
(on the notion of gros toposes)
Axiomatic cohesion Theory and Applications of Categories, Vol. 19, No. 3, 2007, pp. 41–49. (pdf)
(a more formal definition of cohesive toposes)
Cohesive Toposes -- Combinatorial and Infinitesimal Cases , lecture in Como (2008)
(on cohesive toposes)
Open problems in topos theory, April 2009 (pdf)
(problem 7 presented there “The algebra of time”, concerns the characterization of Toposes of laws of motion)