David Corfield homotopification

Idea

Part of the revolution in mathematics. Backed in via higher category theory, and intrinsically by HoTT.

Quotations

• Anel The homotopy quotient:

The price to pay to access this structure is to enhance sets into ∞-groupoids. Bye bye sets…

This is a big revolution, a shift of paradigm.

• Toën, B. 2014. ‘Derived Algebraic Geometry,’ arXiv:1401.1044.

the expression homotopical mathematics reflects a shift of paradigm in which the relation of equality relation is weakened to that of homotopy. (Toën 2014, p. 3)

At the same time, he points the reader to the HoTT program as the new foundational language for this homotopical mathematics.

• Also Preface of Denis-Charles Cisinski’s Higher Categories and Homotopical Algebra. (‘A couple of perspectives and a tribute’)

The theory of $\infty$-categories is not only a new approach to the foundations of mathematics: it appears in many spectacular advances, such as the proof of Weil’s conjecture on Tamagawa numbers over function fields by Lurie and Gaitsgory, or the modern approach to $p$-adic Hodge Theory by Bhatt, Morrow and Scholze, for instance. (p. vii)

some traditions seem to put classical Category Theory and classical Homotopy Theory apart. The story that we want to tell here is that the theory of $\infty$-categories involves a reunion… (p. viii)

In some sense, this is the natural outcome of a historical process. Indeed Category Theory was born as a convenient language to express the constructions of Algebraic Topology, and the fact that these two fields were separated is a kind of historical accident whose effects only started to fade in the late 1990’s, with the rise of $\infty$-categories as we know them today (p. viii)

The title of this book is less about putting Higher Category Theory and Homotopy Theory side by side, than observing that Higher Category Theory and Homotopical Algebra are essentially the same thing. (pp. viii-ix)

In the interview with Urs:

the unfortunate schism in mathematicians’ awareness between category theory and homotopy theory was bridged a few years back, finally ending a huge delay in intellectual progress of a whole community.

What’s possible in homotopical algebra

MO question