### The HoTT Approach to Physics

#### Posted by David Corfield

Summer saw the foundations of mathematics rocked by the publication of The HoTT Book. Here we are a few months later and the same has happened to physics with the appearance on the ArXiv of Urs’s Differential cohomology in a cohesive infinity-topos.

Physics clearly needs more than the bare homotopy types of HoTT. Field configurations may be groupoids (1-types) under gauge equivalence, or indeed $\infty-groupoids$ (homotopy types) under gauge-of-gauge-of-… equivalence, but they also possess differentiable structure. The question then is how to cater for all of those principal bundles, connections, curvature forms, and in more recent times 2-bundles, orbifolds, Lie $\infty$-algebroids,…, while building on HoTT, or, in terms of the environments in which HoTT (or Univalent Foundations) functions, while adding structure to $(\infty, 1)$-toposes.

We’ve become rather used to caring little about how elaborate are the constructions we would need to place on our foundational language were we to formalize the mathematics of current theoretical physics. What does it matter if it would be a nightmare to go ahead and fully express the ingredients of, say, the Green-Schwarz mechanism in heterotic supergravity in the language of ZFC set theory?

But perhaps we should never have let ourselves fall into this state of indifference. One person who has fought against this trend is Bill Lawvere. His simple notion of cohesion, how bare points are held together in spaces, was just what Urs needed to add to the axioms for $(\infty, 1)$-toposes. Now all the differential cohomology you ever wanted to do can be done in the setting of cohesive $(\infty, 1)$-toposes, or, if you prefer a syntactic approach, through cohesive homotopy type theory.

I’m looking forward to cohesion cropping up in a wide range of places within mathematics over coming years. Perhaps we’ll start to understand why number theorists and physicists have so much to talk about together.

## Re: The HoTT Approach to Physics

Your link on

cohesive homotopy type theoryhas an invisible Left-to-Right Mark at its end which keeps it from working. Did you copy and paste from near some Hebrew or Arabic text?