Monthly Archives: April 2011
The idea of higher inductive types, as described here and here, purports (among other things) to give us objects in type theory which represent familiar homotopy types from topology. Perhaps the simplest nontrivial such type is the circle, , which … Continue reading
The proof assistant Coq is based on a formal system called the “Predicative Calculus of (Co)Inductive Constructions” (pCiC). But before pCiC, there was the “Calculus of Constructions” (CoC), in which inductive types were not a basic object, but could be … Continue reading
(This was written in inadvertent parallel with Mike’s latest post at the Café, so there’s a little overlap — Mike’s discusses the homotopy-theoretic aspects more, this post the type-theoretic nitty-gritty.) Higher inductive types have been mentioned now in several other … Continue reading
Higher-dimensional inductive types are an idea that many people have been kicking around lately; for example, in An Interval Type Implies Function Extensionality. The basic idea is that you define a type by specifying both what it’s elements are (as … Continue reading
From a homotopy theorist’s point of view, identity types and their connection to homotopy theory are perfectly natural: they are “path objects” in the category of types. However, from a type theorist’s point of view, they are somewhat more mysterious. … Continue reading
Richard Garner has now completed and posted the report from the Oberwolfach meeting.
I have adapted Dan Licata’s Agda proof that the higher homotopy groups are abelian to Coq, and I have added a link to the code on the code page of this blog.
Several current proof assistants, such as Agda and Epigram, provide uniqueness of identity proofs (UIP): any two proofs of the same propositional equality are themselves propositionally equal. Homotopy type theory generalizes this picture to account for higher-dimensional types, where UIP … Continue reading