The Univalence Principle

02/11/2021
by   Benedikt Ahrens, et al.
0

The Univalence Principle is the statement that equivalent mathematical structures are indistinguishable. We prove a general version of this principle that applies to all set-based, categorical, and higher-categorical structures defined in a non-algebraic and space-based style, as well as models of higher-order theories such as topological spaces. In particular, we formulate a general definition of indiscernibility for objects of any such structure, and a corresponding univalence condition that generalizes Rezk's completeness condition for Segal spaces and ensures that all equivalences of structures are levelwise equivalences. Our work builds on Makkai's First-Order Logic with Dependent Sorts, but is expressed in Voevodsky's Univalent Foundations (UF), extending previous work on the Structure Identity Principle and univalent categories in UF. This enables indistinguishability to be expressed simply as identification, and yields a formal theory that is interpretable in classical homotopy theory, but also in other higher topos models. It follows that Univalent Foundations is a fully equivalence-invariant foundation for higher-categorical mathematics, as intended by Voevodsky.

READ FULL TEXT
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

04/14/2020

A Higher Structure Identity Principle

The ordinary Structure Identity Principle states that any property of se...
05/20/2020

Comprehension and quotient structures in the language of 2-categories

Lawvere observed in his celebrated work on hyperdoctrines that the set-t...
02/03/2022

Univalent foundations and the equivalence principle

In this paper, we explore the 'equivalence principle' (EP): roughly, sta...
04/28/2022

On Quantitative Algebraic Higher-Order Theories

We explore the possibility of extending Mardare et al. quantitative alge...
04/17/2019

A 2-Categorical Study of Graded and Indexed Monads

In the study of computational effects, it is important to consider the n...
10/23/2020

An Intuitionistic Set-theoretical Model of Fully Dependent CCω

Werner's set-theoretical model is one of the simplest models of CIC. It ...
10/14/2020

Representable Markov Categories and Comparison of Statistical Experiments in Categorical Probability

Markov categories are a recent categorical approach to the mathematical ...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.