# nLab cohesive homotopy type theory with two kinds of types

Contents

under construction

# Contents

## Introduction

In Shulman18, the authors write

One might argue that it would be better to include a separate kind of type representing objects in the base category, and type constructors represent-ing the functors $p_*$, $p^*$, etc. rather than the monads and comonads they induce. This would be in line with the “judgmental deconstruction” of Reed (2009) that decomposes $\Box$ and $\bigcirc$ into pairs of adjoint functors. It could be that this is indeed better; in fact, the current rules for $\flat$ and $\sharp$ were deduced from a generalization of Reed (2009) developed by Licata and Shulman (2016).

This is an attempt at developing such a cohesive homotopy type theory.

## Overview

This presentation of cohesive homotopy type theory is a dependent type theory consisting of two different kinds of types, spaces and homotopy types. There exist judgments

• for spaces

$\frac{\Gamma}{\Gamma \vdash S\ space}$
• for homotopy types

$\frac{\Gamma}{\Gamma \vdash T\ homotopy\ type}$
• for points

$\frac{\Gamma \vdash S\ space}{\Gamma \vdash s:S}$
• for terms

$\frac{\Gamma \vdash T\ homotopy\ type}{\Gamma \vdash t:T}$
• for fibrations

$\frac{\Gamma \vdash S\ space}{\Gamma, s:S \vdash A(s)\ space}$
• for dependent types

$\frac{\Gamma \vdash T\ homotopy\ type}{\Gamma, t:T \vdash B(t)\ homotopy\ type}$
• for sections

$\frac{\Gamma \vdash S\ space}{\Gamma, s:S \vdash a(s):A(s)}$
• for dependent terms

$\frac{\Gamma \vdash T\ homotopy\ type}{\Gamma, t:T \vdash b(t):B(t)}$

Cohesive homotopy type theory has the following additional judgments, 2 for turning spaces into homotopy types and the other two for turning homotopy types into spaces:

• Every space has an underlying homotopy type

$\frac{\Gamma \vdash S\ space}{\Gamma \vdash p_*(S)\ homotopy\ type}$
• Every space has a fundamental homotopy type

$\frac{\Gamma \vdash S\ space}{\Gamma \vdash p_!(S)\ homotopy\ type}$
• Every homotopy type has a discrete space

$\frac{\Gamma \vdash T\ homotopy\ type}{\Gamma \vdash p^*(T)\ space}$
• Every homotopy type has an indiscrete space

$\frac{\Gamma \vdash T\ homotopy\ type}{\Gamma \vdash p^!(T)\ space}$

The underlying homotopy type and fundamental homotopy type are sometimes represented by the greek letters $\Gamma$ and $\Pi$ respectively, but both are already used in dependent type theory to represent the context and the dependent product type.

## Modalities

From these judgements one could construct the sharp modality as

$\sharp(S) \coloneqq p^!(p_*(S))$

the flat modality as

$\flat(S) \coloneqq p^*(p_*(S))$

and the shape modality as

$\esh(S) \coloneqq p^*(p_!(S))$

for a space $S$.

## Other types

### Path spaces

$\frac{\Gamma, a:S, b:S}{\Gamma \vdash a =_S b\ space}$

### Identity types

$\frac{\Gamma, a:T, b:T}{\Gamma \vdash a =_T b\ homotopy\ type}$

### Mapping spaces

$\frac{\Gamma, A\ space, B\ space}{\Gamma \vdash A \to B\ space}$

### Function types

$\frac{\Gamma, A\ homotopy\ type, B\ homotopy\ type}{\Gamma \vdash A \to B\ homotopy\ type}$

### Section spaces

$\frac{\Gamma, s:S \vdash A(s)\ space}{\Gamma \vdash \prod_{s:S} A(s)\ space}$

### Dependent function types

$\frac{\Gamma, t:T \vdash A(t)\ homotopy\ type}{\Gamma \vdash \prod_{t:T} A(t)\ homotopy\ type}$

### Product spaces

$\frac{\Gamma, A\ space, B\ space}{\Gamma \vdash A \times B\ space}$

### Pair types

$\frac{\Gamma, A\ homotopy\ type, B\ homotopy\ type}{\Gamma \vdash A \times B\ homotopy\ type}$

### Total spaces

$\frac{\Gamma, s:S \vdash A(s)\ space}{\Gamma \vdash \sum_{s:S} A(s)\ space}$

### Dependent pair types

$\frac{\Gamma, t:T \vdash A(t)\ homotopy\ type}{\Gamma \vdash \sum_{t:T} A(t)\ homotopy\ type}$

### Unit space

$\frac{\Gamma}{\Gamma \vdash \mathbb{1}\ space}$

### Unit type

$\frac{\Gamma}{\Gamma \vdash \mathbb{1}\ homotopy\ type}$

## Other properties

The fundamental homotopy type of the unit space is equivalent to the unit type.

$p_!(\mathbb{1}) \cong \mathbb{1}$

The product of the fundamental homotopy types of spaces $A$ and $B$ is equivalent to the fundamental homotopy type of the product of spaces $A$ and $B$:

$p_!(A \times B) \cong p_!(A) \times p_!(B)$

## Sources

Created on June 19, 2022 at 19:30:27. See the history of this page for a list of all contributions to it.