# nLab William Lawvere

Contents

foundations

category theory

topos theory

## Theorems

#### Synthetic differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Overview

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

• Interview by Felice Cardone, March 2013 (link)

for an interview on his contributions to categorical logic.

For a (somewhat random) list of further links see also at “conceptual mathematics”.

## Topics

### Main contributions

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.

### Mathematics relating to Physics

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 Wikipedia 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.

### Mathematics relating to Philosophy

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 92) 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.

## Talks, Lecture notes and Publications

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.)

category: people

Last revised on October 24, 2022 at 07:34:11. See the history of this page for a list of all contributions to it.