David Corfield higher category theory as revolution

Higher category theory, and its radically new point of view, are bringing about great changes in many areas.

• Allowing a new geometry
• Rethinking the nature of space
• It’s tying in with physics

There are a number indications that today we are in a period where the fundamental mathematical nature of quantum field theory (QFT) and of the worldvolume aspects of string theory is being identified. It is not unlikely that future generations will think of the turn of the millennium and the beginning of the 21st century as the time when it was fully established that QFT in general and worldvolume theories in particular are precisely the representations of higher categories of cobordisms with structure or, dually, encoded by copresheaves of local algebras of observables, vertex operator algebras, factorization algebras and their siblings. (Hati and Schreiber, Preface, Mathematical Foundations of Quantum Field and Perturbative String Theory)

• It’s tying in with logic in particular via the associated type theory. Note the transformation from predicate logic + set theoretic axioms to type theory, a shift in the constitutive language of mathematics, with an ability to write the mathematical principle of equivalence right into the foundations.

It marks a change in foundations, in the most appropriate sense of the word:

In my own education I was fortunate to have two teachers who used the term “foundations” in a common-sense way (rather than in the speculative way of the Bolzano-Frege-Peano-Russell tradition). This way is exemplified by their work in Foundations of Algebraic Topology, published in 1952 by Eilenberg (with Steenrod), and The Mechanical Foundations of Elasticity and Fluid Mechanics, published in the same year by Truesdell. The orientation of these works seemed to be “concentrate the essence of practice and in turn use the result to guide practice”. (Lawvere 2003: 213)

‘Foundations and applications: axiomatization and education’, Bulletin of Symbolic Logic 9 (2003), 213-224. Also available as Postscript file.

Rather than as mere periodic episodes of rapid change, we can point to longer chains of transformations. Just as, in a sense, the Einsteinian revolution is a chapter in the story of the mathematisation of nature, one can see higher category theory as part of the realisation of ideas from late nineteenth understandings of mathematics, especially those of Klein and Dedekind.

• Set theory

• Every entity is a set, so can ask about membership everywhere.

• Isomorphic structures behave the same

• Category Theory

• Universal properties, adjunctions, “All concepts are Kan extensions” Better to work with nice categories than nice objects. Need to work with operations defined up to homotopy.

• Homotopy Type Theory/Higher category theory

• Entities are the same if equivalent at right level.

You can understand why set theory was so successful from later systems, but you wouldn’t be able to reconstruct it in all its details. Set as well-pointed boolean topos, free co-complete on one object, chain of 5 adjunctions.

Physics

Dynamics encoded by cohesion, describing transformations along paths.

Cohomology plays a fundamental role in modern physics. (Zeidler, Quantum Field Theory, Volume 1, p. 14).

Fundamental physics is all controled by cohomology. (Schreiber)

See articles in Mathematical Foundations of Quantum Field and Perturbative String Theory

Logic

Whilst it is possible to encode all of mathematics into Zermelo-Fraenkel set theory, the manner in which this is done is frequently ugly; worse, when one does so, there remain many statements of ZF which are mathematically meaningless. (Voevodsky, Oberwolfach report)

Bourbaki

Bourbaki “Introduction to the Theory of Sets” (http://sites.mathdoc.fr/archives-bourbaki/feuilleter.php?chap=2_REDAC_E1) “… nowadays it is known to be possible, logically speaking, to derive practically the whole of known mathematics from a single source, the Theory of Sets. By so doing we do not claim to legislate for all time. It may happen at some future date that mathematicians will agree to use modes of reasoning which cannot be formalized in the language described here; according to some, the recent evolution of axiomatic homology theory would be a sign that this date is not so far. It would then be necessary, if not to change the language completely, at least to enlarge its rules of syntax. But this is for the future to decide.”