The n-Category Café

The following paper is interesting for many reasons:

• Bill Lawvere, The category of probabilistic mappings with applications to stochastic processes, statistics, and pattern recognition, Spring 1962, featuring Lawvere’s abstract and author commentary from 2020, reformatted for Lawvere Archive Posthumous Publications by Tobias Fritz, July 14, 2025.

The mathematics is interesting. But Lawvere also describes his classified work on arms control in the 1960s, back when he was writing his thesis! Applied category theorists who worrry about military applications of their work will see their concerns are not new. Lawvere’s employer goes unnamed, but its description fits the RAND Corporation.

My acceptance of the job offered by the “Think Tank” in Southern California depended on an agreement that the main topic treated would be Kennedy’s Arms Control and Disarmament Agency. The preliminary interview in the Pentagon was requested by that Agency. Somewhat more precisely, the aim of the study would be: planning for the technical support of an Arms Control Treaty between the Superpowers, for example, of a reliable verification protocol to be agreed upon.

It was envisaged that a protocol would involve three tiers of verification: Space, Stratosphere, and On-site. The passage from one tier to the next would follow probabilistically from continuing observations. What would be the mathematical framework under which this whole fantasy would function? Someone described it as a “network of probabilistic mappings”. “What would that mean?” I asked myself: “It must involve diagrams in a category extending the monoid of Markov processes”, and then I produced the present document, which served as an Appendix to an Appendix of a large SECRET document.

The proposal was to study a projected system of verification and inspection for a possible Arms Control Agreement between the Superpowers. The system would be organized into the three levels: satellite surveillance, which could trigger the request for over-flight inspection, that in turn could trigger an on-site inspection. Of course, the trigger thresholds would be a matter of diplomacy, but the system as a whole would involve an elaborate network of “probabilistic mappings”.

The whole thing had to be scrutinized by the Pentagon before the Arms Control Agency could do anything. Probably, passing through so many hands increased its exposure to espionage. The leader of the group within the Think Tank stated that an important calculation to be done by the study would be the determination of the probability of the discovery of missiles concealed on the ocean floor as part of a planned circumvention of any treaty. That was also the year of the Cuban missile crisis

A few years later I came across a Russian document containing several of the results of my unpublished thesis, including the mistakes, (as well as the missing two lines that we later discovered had been missed by the typist). But there was no attribution. And in Moscow the lectures were beginning on a very similar category, called the Markov category (not without justification, of course, although I don’t believe Markov himself used categories).

I was surprised a couple of years later by being offered a job with French military intelligence. The one who transmitted that offer was a collaborator of M. Giry, which may explain why she knew about the “secret” developments in the US.

Apparently regarding the contact with the Arms Control Agency as dormant, the leaders of the Think Tank had a further proposal, disregarding their initial agreement with me: First, I should study books by Mao Tse-tung and Che Guevara as a preparation for evaluating a large system designed to eliminate the guerilla threat in Vietnam. My last paychecks were for studying that proposal. Of course, I advised against it, after having verified mathematically that the proposed system was unfeasible. The last time I saw the director of the “Vietnam Proposal” was at the old Waltham Watch Factory, which had been taken over as a subsidiary of the California Think Tank. Naturally, my report met with utter disapproval. I took a bus from Waltham to NYC in order to defend my thesis at Hamilton Hall, in front of Eilenberg, Kadison, and Morgenbesser. Now I could complete my application for a teaching job at Reed College.

A few years later the New York Times reported on the failure of a large system that differed only in detail from the one I had analyzed. The supporters of the proposal had taken the plan and defected to another Think Tank.

Near the end of the manuscript there’s a bit from a 2016 letter from Lawvere to Rosebrugh:

I realize that there is a group of younger researchers who would like to know more about this topic (so do I). Some are claiming that it will become a key ingredient in DARPA’s thrust toward “genuine” artificial intelligence.