Homotopy Type Theory type > history (Rev #7)

“A type is defined as the range of significance of a propositional function, i.e., as the collection of arguments for which the said function has values.” – Bertrand Russell, 1908

Idea

A type theory is a formal system in which every term has a ‘type’, and operations in the system are restricted to acting on specific types.

A number of type theories have been used or proposed for doing homotopy type theory.

Here we will describe the types present in “Book HoTT”, the type theory given in the HoTT Book.

Universes

Russell-style universe

With Russell-style universes, all types are seen as elements of a type called $\mathcal{U}$. This is the universe type. Due to Russel-like paradoxes, we cannot have $\mathcal{U} : \mathcal{U}$, therefore we have an infinite hierarchy of universes

for every $n$. For most cases it will not matter what universe we are in.

Tarski-style universe

With Tarski-style universes, all types are seen as dependent types of a universal type family $\mathcal{T}$ given a type $\mathcal{U}$ called the universe type and a term $X:\mathcal{U}$, i.e. $X:\mathcal{U} \vdash \mathcal{T}(X) type$. The existence of Russell-like paradoxes means that we similarly cannot have a type $\mathcal{U}^\prime:\mathcal{U}$ such that $\mathcal{T}(\mathcal{U}^\prime) \equiv \mathcal{U}$.

Thus, we likewise have an infinite hierarchy of universes

for every $n$. For most cases it will not matter what universe we are in.

Function types

Given a type $A$ and $B$, there is a type $A \to B$ called the function type representing the type of functions from $A$ to $B$. A function can be defined explicitly using lambda notation $\lambda x . y$. There is a computation rule saying there is a reduction $(\lambda x . y ) a \equiv y[a / x]$ for some $a : A$. The notation $y[a / x]$ means to replace all occurances of $x$ with $a$ in $y$, giving us a term of $B$.

Universes are closed under function types, i.e.

and

Pi types

(…)

Pair types

Given a type $A$ and $B$, there is a type $A \times B$ called the pair type? or product type? representing the type of pairs $(a, b)$ for $a:A$ and $b:B$.

Universes are closed under pair types, i.e.

and

Sigma types

(…)

Empty type

(…)

Unit type

(…)

Identity types

(…)

Homotopy pushout types

Given types $A$, $B$, and $C$, and functions $f:C \to A$ and $g:C \to A$, there is a type $A \sqcup_C B$ called the homotopy pushout type? representing the homotopy pushout of $f$ and $g$.

Special cases of homotopy pushout types include the type of booleans, which is defined as

the coproduct type, which is defined as

and the circle type, which is defined as

References

• HoTT Book
• On the homotopy groups of spheres in homotopy type theory