21 research outputs found
Weak omega-categories from intensional type theory
We show that for any type in Martin-L\"of Intensional Type Theory, the terms
of that type and its higher identity types form a weak omega-category in the
sense of Leinster. Precisely, we construct a contractible globular operad of
definable composition laws, and give an action of this operad on the terms of
any type and its identity types
Functions out of Higher Truncations
In homotopy type theory, the truncation operator ||-||n (for a number n > -2)
is often useful if one does not care about the higher structure of a type and
wants to avoid coherence problems. However, its elimination principle only
allows to eliminate into n-types, which makes it hard to construct functions
||A||n -> B if B is not an n-type. This makes it desirable to derive more
powerful elimination theorems. We show a first general result: If B is an
(n+1)-type, then functions ||A||n -> B correspond exactly to functions A -> B
which are constant on all (n+1)-st loop spaces. We give one "elementary" proof
and one proof that uses a higher inductive type, both of which require some
effort. As a sample application of our result, we show that we can construct
"set-based" representations of 1-types, as long as they have "braided" loop
spaces. The main result with one of its proofs and the application have been
formalised in Agda.Comment: 15 pages; to appear at CSL'1
A Rewriting Coherence Theorem with Applications in Homotopy Type Theory
Higher-dimensional rewriting systems are tools to analyse the structure of
formally reducing terms to normal forms, as well as comparing the different
reduction paths that lead to those normal forms. This higher structure can be
captured by finding a homotopy basis for the rewriting system. We show that the
basic notions of confluence and wellfoundedness are sufficient to recursively
build such a homotopy basis, with a construction reminiscent of an argument by
Craig C. Squier. We then go on to translate this construction to the setting of
homotopy type theory, where managing equalities between paths is important in
order to construct functions which are coherent with respect to higher
dimensions. Eventually, we apply the result to approximate a series of open
questions in homotopy type theory, such as the characterisation of the homotopy
groups of the free group on a set and the pushout of 1-types.
This paper expands on our previous conference contribution "Coherence via
Wellfoundedness" (arXiv:2001.07655) by laying out the construction in the
language of higher-dimensional rewriting.Comment: 30 pages. arXiv admin note: text overlap with arXiv:2001.0765
A rewriting coherence theorem with applications in homotopy type theory
Higher-dimensional rewriting systems are tools to analyse the structure of formally reducing terms to normal forms, as well as comparing the different reduction paths that lead to those normal forms. This higher structure can be captured by finding a homotopy basis for the rewriting system. We show that the basic notions of confluence and wellfoundedness are sufficient to recursively build such a homotopy basis, with a construction reminiscent of an argument by Craig C. Squier. We then go on to translate this construction to the setting of homotopy type theory, where managing equalities between paths is important in order to construct functions which are coherent with respect to higher dimensions. Eventually, we apply the result to approximate a series of open questions in homotopy type theory, such as the characterisation of the homotopy groups of the free group on a set and the pushout of 1-types. This paper expands on our previous conference contribution Coherence via Wellfoundedness by laying out the construction in the language of higher-dimensional rewriting
The General Universal Property of the Propositional Truncation
In a type-theoretic fibration category in the sense of Shulman (representing a dependent type theory with at least 1, Sigma, Pi, and identity types), we define the type of constant functions from A to B. This involves an infinite tower of coherence conditions, and we therefore need the category to have Reedy limits of diagrams over omega. Our main result is that, if the category further has propositional truncations and satisfies function extensionality, the type of constant function is equivalent to the type ||A|| -> B. If B is an n-type for a given finite n, the tower of coherence conditions becomes finite and the requirement of nontrivial Reedy limits vanishes. The whole construction can then be carried out in Homotopy Type Theory and generalises the universal property of the truncation. This provides a way to define functions ||A|| -> B if B is not known to be propositional, and it streamlines the common approach of finding a proposition Q with A -> Q and Q -> B
A type theory for synthetic -categories
We propose foundations for a synthetic theory of -categories
within homotopy type theory. We axiomatize a directed interval type, then
define higher simplices from it and use them to probe the internal categorical
structures of arbitrary types. We define Segal types, in which binary
composites exist uniquely up to homotopy; this automatically ensures
composition is coherently associative and unital at all dimensions. We define
Rezk types, in which the categorical isomorphisms are additionally equivalent
to the type-theoretic identities - a "local univalence" condition. And we
define covariant fibrations, which are type families varying functorially over
a Segal type, and prove a "dependent Yoneda lemma" that can be viewed as a
directed form of the usual elimination rule for identity types. We conclude by
studying homotopically correct adjunctions between Segal types, and showing
that for a functor between Rezk types to have an adjoint is a mere proposition.
To make the bookkeeping in such proofs manageable, we use a three-layered
type theory with shapes, whose contexts are extended by polytopes within
directed cubes, which can be abstracted over using "extension types" that
generalize the path-types of cubical type theory. In an appendix, we describe
the motivating semantics in the Reedy model structure on bisimplicial sets, in
which our Segal and Rezk types correspond to Segal spaces and complete Segal
spaces.Comment: 78 pages; v2 has minor updates inspired by discussions at the
Mathematics Research Community on Homotopy Type Theory; v3 incorporates many
expository improvements suggested by the referee; v4 is the final journal
version to appear in Higher Structures with a more precise syntax for our
type theory with shape