(0,1)-category

(0,1)-topos

# Contents

## Idea

In logic, the true proposition, or truth, is the proposition which is always true.

The truth is commonly denoted $\mathrm{true}$, $T$, $\top$, or $1$.

## Definitions

### In classical logic

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

### In constructive logic

In constructive logic, $\mathrm{true}$ is the top element in the poset of truth values.

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

### In a topos

In terms of the internal logic of a topos (or other category), $\mathrm{true}$ is the top 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 $\mathrm{true}$ and $\mathrm{false}$.

### In homotopy type theory

In homotopy type theory the true is represented by any contractible type.

## Examples

### In the topos $\mathrm{Set}$

In the archetypical topos Set, the terminal object is the singleton set $\left\{*\right\}$ (the point) and the poset of subobjects of that is classically $\left\{\varnothing ↪*\right\}$. Then truth is the singleton set $\left\{*\right\}$, seen as the improper subset of itself. (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 $\left(-1\right)$-truncated $\infty$-groupoid: a (−1)-groupoid;

• the singleton set as the $\left(-2\right)$-truncated $\infty$-groupoid: the unique (up to equivalence) (−2)-groupoid.

In this sense, the object $\mathrm{true}$ in Set or ∞Grpd may canonically be thought of as being the unique (−2)-groupoid.

homotopy leveln-truncationhomotopy theoryhigher category theoryhigher topos theoryhomotopy type theory
h-level 0(-2)-truncatedcontractible space(-2)-groupoidtrue/unit type/contractible type
h-level 1(-1)-truncated(-1)-groupoid/truth valueh-proposition
h-level 20-truncateddiscrete space0-groupoid/setsheafh-set
h-level 31-truncatedhomotopy 1-type1-groupoid/groupoid(2,1)-sheaf/stackh-groupoid
h-level 42-truncatedhomotopy 2-type2-groupoidh-2-groupoid
h-level 53-truncatedhomotopy 3-type3-groupoidh-3-groupoid
h-level $n+2$$n$-truncatedhomotopy n-typen-groupoidh-$n$-groupoid
h-level $\infty$untruncatedhomotopy type∞-groupoid(∞,1)-sheaf/∞-stackh-$\infty$-groupoid

Revised on September 10, 2012 20:22:08 by Urs Schreiber (131.174.188.17)