Author Archives: Mike Shulman

Modules for Modalities

As defined in chapter 7 of the book, a modality is an operation on types that behaves somewhat like the n-truncation. Specifically, it consists of a collection of types, called the modal ones, together with a way to turn any … Continue reading

Posted in Code, Programming | 19 Comments

Not every weakly constant function is conditionally constant

As discussed at length on the mailing list some time ago, there are several different things that one might mean by saying that a function is “constant”. Here is my preferred terminology: is constant if we have such that for … Continue reading

Posted in Univalence | 10 Comments

The HoTT Book does not define HoTT

The intent of this post is to address certain misconceptions that I’ve noticed regarding the HoTT Book and its role in relation to HoTT. (At least, I consider them misconceptions.) Overall, I think the HoTT Book project has been successful … Continue reading

Posted in Uncategorized | 6 Comments

Splitting Idempotents, II

I ended my last post about splitting idempotents with several open questions: If we have a map , a witness of idempotency , and a coherence datum , and we use them to split as in the previous post, do … Continue reading

Posted in Code, Homotopy Theory | Leave a comment

Splitting Idempotents

A few days ago Martin Escardo asked me “Do idempotents split in HoTT”? This post is an answer to that question.

Posted in Code, Homotopy Theory, Univalence | 13 Comments

Universal properties without function extensionality

A universal property, in the sense of category theory, generally expresses that a map involving hom-sets, induced by composition with some canonical map(s), is an isomorphism. In type theory we express this using equivalences of hom-types. For instance, the universal … Continue reading

Posted in Code, Foundations, Higher Inductive Types | 16 Comments

Fibrations with fiber an Eilenberg-MacLane space

One of the fundamental constructions of classical homotopy theory is the Postnikov tower of a space X. In homotopy type theory, this is just its tower of truncations: One thing that’s special about this tower is that each map has … Continue reading

Posted in Homotopy Theory | 2 Comments