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(0,1)-category

(0,1)-topos

# Contents

## Idea

In logic, the false proposition, called falsehood or falsity, is the proposition which is always false.

The faleshood is commonly denoted $false$, $F$, $\bot$, or $0$. These may be pronounced ‘false’ even where it would be ungrammatical for an adjective to appear in ordinary English.

## Definitions

### In classical logic

In classical logic, there are two truth values: false and true. Classical logic is perfectly symmetric between falsehood and truth; see de Morgan duality.

### In constructive logic

In constructive logic, $false$ is the bottom element in the poset of truth values.

Constructive logic is still two-valued in the sense that any truth value is false if it is not true.

### In linear logic

In linear logic, there is both additive falsity, denoted $0$, and multiplicative falsity, denoted $\bot$. Despite the notation, it is $0$ that is the bottom element of the lattice of linear truth values. (In particular, $0 \vdash \bot$ but $\bot \nvdash 0$.)

### In a topos

In terms of the internal logic of a topos (or other category), $false$ is the bottom element in the poset of subobjects of any given object (where each object corresponds to a context in the internal language).

However, not every topos is two-valued, so there may be other truth values besides $false$ and $true$.

### In type theory

In type theory with propositions as types, falsehood is represented by the empty type.

### In homotopy type theory

In homotopy type theory, falsehood is represented by the empty space.

## Examples

### In the topos $Set$

In the archetypical topos Set, the terminal object is the singleton set $*$ (the point) and the poset of subobjects of that is classically $\{\emptyset \hookrightarrow *\}$. Then falsehood is the empty set $\emptyset$, seen as the empty subset of the point. (See Internal logic of Set for more details).

The same is true in the archetypical (∞,1)-topos ∞Grpd. From that perspective it makes good sense to think of

• a set as a 0-truncated $\infty$-groupoid: a 0-groupoid;

• a subsingleton set as a $(-1)$-truncated $\infty$-groupoid: a (?1)-groupoid?.

In this sense, the object $false$ in Set or ∞Grpd may canonically be thought of as being the unique empty groupoid.

Last revised on December 10, 2017 at 18:52:04. See the history of this page for a list of all contributions to it.