basic constructions:
strong axioms
further
synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality $\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality $\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
physics, mathematical physics, philosophy of physics
theory (physics), model (physics)
experiment, measurement, computable physics
Axiomatizations
Tools
Structural phenomena
Types of quantum field thories
Talking with Bill, I often feel like a fly buzzing around a cow. (It seems to me I can liken Bill to a cow, if I’m just a fly myself.) On any easy question, I’ll probably see the answer first. But his thoughts seem to move on a level where I don’t function, I can barely see down there. (From an interview with John Isbell.)
F. William Lawvere
is an influential category theorist.
See
for a survey of his academic path and work. See also
for an interview on his contributions to categorical logic.
For a (somewhat random) list of further links see also at “conceptual mathematics”.
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 introduced the notion of cohesive topos as a more general foundation of geometry.
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:
I felt a strong need to learn more set theory and logic from experts in that field, still of course with the aim of clarifying the foundations of category theory and of physics.
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 justification of older intuitive methods in geometry” 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 Differential 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 first 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 $T$, it says:
The basic spaces which are needed for functional analysis and theories of physical fields are thus in some sense available in any topos with a suitable object $T$.
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 $[25]$ 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.
Lawvere has proposed formalizations in category theory, categorical logic and topos theory of concepts which are motivated from philosophy, notably in Georg Hegel’s Science of Logic (see there for more). This includes for instance definitions of concepts found there such as:
(see the references in these entries for pointers).
In (Lawvere 02) it says:
It is my belief that in the next decade and in the next century the technical advances forged by category theorists will be of value to dialectical philosophy, lending precise form with disputable mathematical models to ancient philosophical distinctions such as general vs. particular, objective vs. subjective, being vs. becoming, space vs. quantity, equality vs. difference, quantitative vs. qualitative etc. In turn the explicit attention by mathematicians to such philosophical questions is necessary to achieve the goal of making mathematics (and hence other sciences) more widely learnable and useable. Of course this will require that philosophers learn mathematics and that mathematicians learn philosophy.
A precursor to this undertaking is Hermann Grassmann with his Ausdehnungslehre (Lawvere 95), see there for more.
The following is a list of texts by Lawvere, equipped with brief comments and hyperlinks to further material on the $n$Lab. See also the
and also this
(Some of Lawvere’s writings don’t exist as published articles, but circulate in some form or other. Notably the “Archive for Mathematical Sciences & Philosophy” run by Michael Wright has a lot of recordings or lectures by Lawvere, though presently few or none of the files in the archive are available online.)
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)
(on algebraic theories, now also called Lawvere theories)
The Category of Categories as a Foundation for Mathematics , pp.1-20 in Eilenberg, Harrison, MacLane, Röhrl (eds.), Proceedings of the Conference on Categorical Algebra - La Jolla 1965, Springer Heidelberg 1966.
(Axiomatizing the category of categories; a summary is at ETCC)
Categorical dynamics, 1967 Chicago lectures (pdf)
Categorical Dynamics Revisited, talk at Sets Within Geometry, Nancy, France 26-29 July 2011 (recording)
(on synthetic differential geometry as a context for axiomatizing ordinary differential equations as they appear in classical physics)
Adjointness in Foundations, (TAC), Dialectica 23 (1969), 281-296
(on the interpretation of quantifiers in categorical logic as base change adjoints)
Equality in hyperdoctrines and comprehension schema as an adjoint functor. In A. Heller, ed., Proc. New York Symp. on Applications of Categorical Algebra, pp. 1–14. AMS, 1970. (pdf)
(on comprehension in hyperdoctrines)
Quantifiers and sheaves, Actes, Congrès intern, math., 1970. Tome 1, p. 329 à 334 (pdf)
on geometric modalities/Lawvere-Tierney topology and universal/existential quantifiers related to dependent product/dependent sum further developed in:
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
(on basic category theory, the category of sets and elementary toposes, and an approach to indexed categories)
Metric spaces, generalized logic and closed categories Reprints in Theory and Applications of Categories, No. 1 (2002) pp 1-37 (tac)
(A classic text from 1973 on Cauchy complete categories and enriched category theory)
Variable sets etendu and variable structure in topoi , Lecture notes University of Chicago 1975.
(a review of topos theory: on structural sets, role of forcing and non-standard analysis and étendue toposes)
Variable quantities and variable structures in topoi , pp.101-131 in Heller, Tierney (eds.), Algebra, Topology and Category Theory, Academic Press New York 1976.
(an important text for the Eilenberg-festschrift, reviews toposes in algebraic geometry, first attention to gros topos)
Topos-theoretic methods in geometry, Aarhus 1997
Extensive and intensive quantities Workshop on Categorical Methods in Geometry, Aarhus 1983
(a proposal for a general axiomatization of homotopy/homology-like “extensive quantities” and cohomology-like “intensive quantities”)
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)
Toward the description in a smooth topos of the dynamically possible motions and deformations of a continuous body, Cah.top.géo.diff.cat, 21 nr.4, 1980, pp.377-392 (Numdam)
(contains a section with some remarks on Lagrangian mechanics)
with Stephen Schanuel, Categories in Continuum Physics, Lectures given at a Workshop held at SUNY, Buffalo 1982, Lecture Notes in Mathematics 1174, 1986
(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.)
Taking categories seriously, Reprints in Theory and Applications of Categories, No. 8, 2005, pp. 1–24. (pdf)
(reprint of a 1986 paper on the relevance of fundamental concepts in category theory, such as Isbell duality)
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)
F. W. Lawvere, Qualitative Distinctions between some Toposes of Generalized Graphs , Cont. Math. 92 (1989) pp.261-299.
(Uses toposes of graphs to introduce his ideas on petit/gros toposes)
Display of graphics and their applications, as exemplified by 2-categories and the Hegelian “taco”, Proceedings of the first international Conference on algebraic methodology and software technology University of Iowa, May 22-24 1989, Iowa City pp. 51-74
(on Aufhebung, graphic categories, the Hegelian taco; the most explicit text with Lawvere’s ideas on Hegel)
More on graphic toposes, Cah. Top. Géom. Diff. Cat. XXXII no. 1 (1991) pp.5-10. (Numdam)
(on graphic categories and their presheaf categories)
Some Thoughts on the Future of Category Theory in A. Carboni, M. Pedicchio, G. Rosolini (eds.), Category Theory , Proceedings of the International Conference held in Como, LNM 1488 Springer Heidelberg 1991.
(on – implicitly – cohesive toposes);
with Stephen Schanuel, Conceptual Mathematics - A first introduction to categories , Cambridge UP 1997.
(first steps into category theory; the Spanish translation is available here)
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 and on formalizing concepts from philosophy as in the Science of Logic in terms of category theory and categorical logic)
Cohesive Toposes and Cantor's "lauter Einsen" Philosophia Mathematica (3) Vol. 2 (1994), pp. 5-15. (pdf)
(an early version of the notion of cohesive toposes)
Tools for the advancement of objective logic: closed categories and toposes, in J. Macnamara and Gonzalo Reyes (Eds.), The Logical Foundations of Cognition, Oxford University Press 1993 (Proceedings of the Febr. 1991 Vancouver Conference “Logic and Cognition”), pages 43-56, 1994.
(on formalizing philosophy as in the Science of Logic in terms of category theory and categorical logic.)
William Lawvere, A new branch of mathematics, “The Ausdehnungslehre of 1844,” and other works. Open Court (1995), Translated by Lloyd C. Kannenberg, with foreword by Albert C. Lewis, Historia Mathematica Volume 32, Issue 1, February 2005, Pages 99–106 (publisher)
(on Hermann Grassmann’s Ausdehnungslehre and hence also on extensive and intensive quantity)
Grassmann’s Dialectics and Category Theory, in Hermann Günther Graßmann (1809–1877): Visionary Mathematician, Scientist and Neohumanist Scholar, Boston Studies in the Philosophy of Science Volume 187, 1996, pp 255-264 (publisher)
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)
Toposes of laws of motion , transcript of a talk in Montreal, Sept. 1997 (pdf)
(on the description of differential equations in terms of synthetic differential geometry and the notion of toposes of laws of motion)
Outline of synthetic differential geometry , seminar notes (1998) (pdf)
(the origin of the concept of synthetic differential geometry)
F. William Lawvere, Kinship and Mathematical Categories , pp.411-425 in Jackendoff, Bloom, Wynn (eds.), Language, Logic, and Concepts , MIT Press Cambridge 1999.
(Lawvere’s venture into anthropology; a summary is at kinship)
Comments on the development of topos theory , pp.715-734 in Jean-Paul Pier (ed.), Development of mathematics 1950-2000, Birkhäuser Basel 2000.
(review of the history of topos theory)
Linearization of graphic toposes via Coxeter groups, JPAA 168 (2002) pp. 425-436. (pdf)
Categorical algebra for continuum micro physics, JPAA 175 (2002) pp. 267-287. (pdf)
Foundations and Applications: Axiomatization and Education, Bulletin of Symbolic Logic 9 no.2 (2003) pp.213-224. (ps-preprint)
(a review of (not only set-theoretic) foundations in mathematics)
William Lawvere, Robert Rosebrugh, Sets for Mathematics , Cambridge UP 2003. (web)
(on set theory from a practical theoretic point of view (informal ETCS))
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)
The Dialectic of the Continuous and the Discrete in the history of the struggle for a usable guide to mathematical thought, talk at Sets Within Geometry, Nancy, France 26-29 July 2011 (recording)
What are Foundations of Geometry and Algebra, Keynote lecture at the Fifty Years of Functorial Semantics conference, Union College, October 2013 (recording)
Alexander Grothendieck and the Concept of Space, address CT15 Aveiro 2016. (link)
(a talk given at CT15 reviewing 50 years of developments since the 1965 La Jolla conference)