nLab
Bill Lawvere

F. William Lawvere

• (web site)

is an influential pure category theorist.

Topics

Lawvere introduced 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 ETCS. He also introduced and worked on synthetic differential geometry. Building on the he recently introduced the notion of cohesive topos.

His motivation for all of this, believe it or not, is to better understand classical physics. See higher category theory and physics for more on this.

Writings and publications

(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)

  (on algebraic theories, now also called Lawvere theories)

• Extensive and intensive quantities Workshop on Categorical Methods in Geometry, Aaerhus (1983)

  (a proposal for a general axiomatization of homotopy/homology-like “extensive quantities” and cohomology-like “intensive quantities”)

• 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)

• State Categories, Closed Categories, and the Existence Semi-Continuous Entropy Functions , IMA reprint 86 (pdf)

• Functional Remarks on the General Concept of Chaos , IMA reprint 87 (pdf)

• 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)

• 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)

• Outline of synthetic differential geometry , seminar notes (1998) (pdf)

  (the origin of the concept of synthetic differential geometry)

• Metric spaces, generalized logic and closed categories Reprints in Theory and Applications of Categories, No. 1 (2002) pp 1-37 (tac)

  (on Cauchy complete categories)

• 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)

• Taking categories seriously, Reprints in Theory and Applications of Categories, No. 8, 2005, pp. 1–24. (pdf)

  (on the relevance of fundamental concepts in category theory, such as Isbell duality)

• 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)

category: people