Category Archives: Constructive math

Topics related to constructive mathematics.

The HoTT book

The HoTT book is finished!

Since spring, and even before that, I have participated in a great collaborative effort on writing a book on Homotopy Type Theory. It is finally finished and ready for public consumption. You can get the book freely at Mike Shulman has written about the contents of the book, so I am not going to repeat that here. Instead, I would like to comment on the socio-technological aspects of making the book, and in particular about what we learned from open-source community about collaborative research.

Continue reading The HoTT book

The topology of the set of all types

It is well known that, both in constructive mathematics and in programming languages, types are secretly topological spaces and functions are secretly continuous. I have previously exploited this in the posts Seemingly impossible functional programs and A Haskell monad for infinite search in finite time, using the language Haskell. In languages based on Martin-Löf type theory such as Agda, there is a set of all types. This can be used to define functions $\mathbb{N} \to \mathrm{Set}$ that map numbers to types, functions $\mathrm{Set} \to \mathrm{Set}$ that map types to types, and so on.

Because $\mathrm{Set}$ itself is a type, a large type of small types, it must have a secret topology. What is it? There are a number of ways of approaching topology. The most popular one is via open sets. For some spaces, one can instead use convergent sequences, and this approach is more convenient in our situation. It turns out that the topology of the universe $\mathrm{Set}$ is indiscrete: every sequence of types converges to any type! I apply this to deduce that $\mathrm{Set}$ satisfies the conclusion of Rice’s Theorem: it has no non-trivial, extensional, decidable property.

To see how this works, check:

The Agda pages can be navigated be clicking at any (defined) symbol or word, in particular by clicking at the imported module names.

On the Bourbaki-Witt Principle in Toposes

With Peter LeFanu Lumsdaine.

Abstract: The Bourbaki-Witt principle states that any progressive map on a chain-complete poset has a fixed point above every point. It is provable classically, but not intuitionistically. We study this and related principles in an intuitionistic setting. Among other things, we show that Bourbaki-Witt fails exactly when the trichotomous ordinals form a set, but does not imply that fixed points can always be found by transfinite iteration. Meanwhile, on the side of models, we see that the principle fails in realisability toposes, and does not hold in the free topos, but does hold in all cocomplete toposes.

Download paper: bw.pdf
ArXiv version: arXiv:1201.0340v1 [math.CT]

This paper is an extension of my previous paper on the Bourbaki-Witt and Knaster-Tarski fixed-point theorems in the effective topos (arXiv:0911.0068v1).

Constructive gem: an injection from Baire space to natural numbers

I am not sure whether to call this one a constructive gem or stone. I suppose it is a matter of personal taste. I think it is a gem, albeit a very unusual one: there is a topos in which $\mathbb{N}^\mathbb{N}$ can be embedded into $\mathbb{N}$. Continue reading Constructive gem: an injection from Baire space to natural numbers