
The RedPRL Proof Assistant (Invited Paper)
RedPRL is an experimental proof assistant based on Cartesian cubical com...
read it

Partial Functions and Recursion in Univalent Type Theory
We investigate partial functions and computability theory from within a ...
read it

Models of Type Theory Based on Moore Paths
This paper introduces a new family of models of intensional MartinLöf t...
read it

Callbyname Gradual Type Theory
We present gradual type theory, a logic and type theory for callbyname...
read it

Connecting Constructive Notions of Ordinals in Homotopy Type Theory
In classical set theory, there are many equivalent ways to introduce ord...
read it

Uniform Elgot Iteration in Foundations
Category theory is famous for its innovative way of thinking of concepts...
read it

Representing All Stable Matchings by Walking a Maximal Chain
The seminal book of Gusfield and Irving [GI89] provides a compact and al...
read it
Effective Kan fibrations in simplicial sets
We introduce the notion of an effective Kan fibration, a new mathematical structure that can be used to study simplicial homotopy theory. Our main motivation is to make simplicial homotopy theory suitable for homotopy type theory. Effective Kan fibrations are maps of simplicial sets equipped with a structured collection of chosen lifts that satisfy certain nontrivial properties. This contrasts with the ordinary, unstructured notion of a Kan fibration. We show that fundamental properties of Kan fibrations can be extended to explicit constructions on effective Kan fibrations. In particular, we give a constructive (explicit) proof showing that effective Kan fibrations are stable under push forward, or fibred exponentials. This is known to be impossible for ordinary Kan fibrations. We further show that effective Kan fibrations are local, or completely determined by their fibres above representables. We also give an (ineffective) proof saying that the maps which can be equipped with the structure of an effective Kan fibration are precisely the ordinary Kan fibrations. Hence implicitly, both notions still describe the same homotopy theory. By showing that the effective Kan fibrations combine all these properties, we solve an open problem in homotopy type theory. In this way our work provides a first step in giving a constructive account of Voevodsky's model of univalent type theory in simplicial sets.
READ FULL TEXT
Comments
There are no comments yet.