[2] He learned of category theory while teaching a course on functional analysis for Truesdell, specifically from a problem in John L. Kelley's textbook General Topology.

Anders Kock later found further simplifications so that a topos can be described as a category with products and equalizers in which the notions of map space and subobject are representable.

Lawvere had pointed out that a Grothendieck topology can be entirely described as an endomorphism of the subobject representor, and Tierney showed that the conditions it needs to satisfy are just idempotence and the preservation of finite intersections.

Clifford Truesdell participated in that meeting, as did several other researchers in the rational foundations of continuum physics and in the synthetic differential geometry that had evolved from the spatial part of Lawvere's categorical dynamics program.

Lawvere continued to work on his 50-year quest for a rigorous flexible base for physical ideas, free of unnecessary analytic complications.

He mentions that he and Mac Lane co-taught a course on mechanics, which led him to consider the justification of older intuitive methods in geometry, eventually coining the term "synthetic differential geometry" This course was based on Mackey's book Mathematical Foundations of Quantum Mechanics, indicating Mackey's influence on category theory.

This course was the first in a series, and Mac Lane later gave a talk on the Hamilton-Jacobi equation at the Naval Academy in 1970, which was published in The American Mathematical Monthly.

He explains that he began applying Grothendieck topos theory, learned from Gabriel, to simplify the foundations of continuum mechanics, inspired by Truesdell's teachings, Noll's axiomatizations, and his own efforts in 1958 to categorize topological dynamics.

[7] In his 1997 talk "Toposes of Laws of Motion", Lawvere remarks on the longstanding program of infinitesimal calculus, continuum mechanics, and differential geometry, which aims to reconstruct the world from the infinitely small.

He believes that recent developments have positioned mathematicians to make this program more explicit, focusing on how continuum physics can be mathematically constructed from "simple ingredients".

[6] In the same talk, Lawvere mentions that the essential spaces required for functional analysis and physical field theories can be found in any topos with an appropriate object (T).

[10] Lawvere was a committed Marxist-Leninist; for instance, in 1976, he gave a talk called "Applying Marxism-Leninism-Mao Tse-Thung Thought to Mathematics & Science".

According to Anders Kock's obituary, in 1971:[11][...] The [Dalhousie] university administration refused to renew the contract with [Lawvere], due to his political activities in protesting against the Vietnam war and against the War Measures Act proclaimed by Trudeau, suspending civil liberties under the pretext of danger of terrorism.As per the obituary on the Communist Party of Canada (Marxist–Leninist) site:[12]More than 1,000 students rallied in the lobby of the Dal Student Union Building to oppose the arbitrary dismissal of Professor Lawvere.He saw his political commitments as related to his mathematical work in sometimes surprising and unexpected ways: for instance, here's a passage from Quantifiers and Sheaves (1970):[13]When the main contradictions of a thing have been found, the scientific procedure is to summarize them in slogans which one then constantly uses as an ideological weapon for the further development and transformation of the thing.

Doing this for "set theory" requires taking account of the experience that the main pairs of opposing tendencies in mathematics take the form of adjoint functors, and frees us of the mathematically irrelevant traces (∈) left behind by the process of accumulating (∪) the power set (P) at each stage of a metaphysical "construction".In the earlier sections of the paper, he discusses the "unity of opposites" between logic and geometry.