nLab Tarski universe

Idea

In intensional type theory and homotopy type theory, there are typically two ways to construct a “type of small types”. One way is by Russell universes, where an element $A:U$ of a type universe $U$ is literally a type $A \; \mathrm{type}$. The alternative is by Tarski universes, where elements $A:U$ are not literally types, but rather the universe type $U$ comes with the additional structure of a type family $T$, such that the dependent type $T(A)$ represents the actual type. Usually universes are defined in intensional type theory and homotopy type theory using derivations and rules?, such as the type reflection rule

$$\frac{\Gamma \vdash A:U}{\Gamma \vdash T(A) \; \mathrm{type}}$$

for Tarski universes. Furthermore, universes are usually defined as being closed under all type formers inside the type theory, regardless of the type theory.

However, this approach of defining Tarski universes means that the notion of “Tarski universe” differs from type theory to type theory. By this approach, a Tarski universe in the bare intensional Martin-Löf type theory, which has identity types, dependent product types, dependent sum types, an empty type, a unit type, a booleans type, and a natural numbers type, would not be considered a Tarski universe in homotopy type theory, since it is missing important type formers present in homotopy type theory, such as internal pushout types, a torus type?, propositional truncations, W-types, and localizations. On the other hand, one could consider an even more spartan type theory which doesn’t even have the unit type, let alone the empty type, the natural numbers type, and the booleans type, whose Tarski universes are thus consisting of only internal identity types, dependent product types, dependent sum types.

All three notions of Tarski universe described above are in fact definable in all three versions of intensional type theory mentioned above, but Tarski universes are usually described as having, apart from addiitonal internal Tarski universes, exactly the internal type formers that the external type theory has. One usually doesn’t consider Tarski universes with an internal James construction type in bare intensional Martin-Löf type theory, even though a Tarski universe closed under James constructions is definable in Martin-Löf type theory. On the other hand, in homotopy type theory the name “Tarski universe” isn’t usually used for a universe only closed under identity types, dependent product types, dependent sum types.

From a global perspective, however, all of these types should be considered Tarski universes, since they model some notion of type of small types, even though the small types in the universe do not behave as the external types in the type theory do. And that is how we approach Tarski universes in this article: Tarski universes are a very general notion encompassing all of the above notions of Tarski universe and more, and one could describe more specific versions of Tarski universes by adding additional structure and axioms to Tarski universes, in the same way one adds additional structure and axioms to an abelian group to get a ring, a commutative ring, a $R$-module, a $R$-algebra, and an $R$-commutative algebra. We could then study in general Tarski universes and their properties as we do for abelian groups and rings. This requires a shift in the point of view of what Tarski universes are: Tarski universes are actual mathematical structure in intensional type theory, rather than being something in the foundations of mathematics to merely address size? issues or do mathematics in.

Definition

With a family of types

A Tarski universe or universe à la Tarski is simply a type $U$ with a type family $T$ whose dependent types $T(A)$ are indexed by elements $A:U$. The elements $A:U$ are usually called $U$-small types, or small types for short, and $T(A)$ is the type of elements of $A$.

With a type and a function

A Tarski universe or universe à la Tarski is a type $U$ of all $U$-small types with a type $T'$ of all elements, and a function $\mathrm{typeOf}:T' \to U$ which gets the associated type for every element $a:T'$.

Equivalence of definitions

From one direction, the individual dependent types $T(A)$ could be defined as the fiber type of $\mathrm{typeOf}$ at $A$

$$T(A) \coloneqq \mathrm{fiber}_{T', U}(\mathrm{typeOf},A) \equiv \sum_{t:T'} \mathrm{typeOf}(t) =_U A$$

From the other direction, the entire type of elements $T'$ is just the dependent sum type of all the type reflections of small types

$$T' \coloneqq \sum_{A:U} T(A)$$

We shall be using the first definition throughout this article, but everything could be translated into the second definition of a Tarski universe.

Tarski universes as models of type theory

The definitions above are already enough to build an internal model of dependent type theory inside the Tarski universe:

• The type $A \; \mathrm{type}$ is represented by the small type $A:U$
• The element $a:A$ is represented by the element $a:T(A)$ for small type $A:U$.
• The type family $B$ indexed by the type $A \; \mathrm{type}$ is represented by the function $B:T(A) \to U$ for small type $A:U$
• The dependent type $a:A \vdash B \; \mathrm{type}$ is represented by the section $B(a):U$ for element $a:T(A)$ and small type $A:U$.
• The section $a:A \vdash b:B$ is represented by the element $b:T(B(a))$ for element $a:T(A)$ and small type $A:U$.

Properties

Univalence

There are two notions of equivalence in a Tarski universe $(U, T)$, equality $A =_U B$ and equivalence $T(A) \simeq T(B)$. By the properties of the identity type, there is a canonical transport dependent function

$$\mathrm{transport}^T:\prod_{A:U} \prod_{B:U} (A =_U B) \to T(A) \simeq T(B)$$

A Tarski universe is a univalent Tarski universe if for all elements $A:U$ and $B:U$ the function $\mathrm{transport}^T(A, B)$ is an equivalence of types

$$\prod_{A:U} \prod_{B:U} \mathrm{isEquiv}(\mathrm{transport}^T(A, B))$$

This is the extensionality principle for any Tarski universe $(U, T)$.

Equivalently, by the fundamental theorem of identity types, a Tarski universe is a univalent Tarski universe if it comes with a dependent function

$$t:\prod_{X:U} C(X, X, \mathrm{id}_X) \vdash J(t):\prod_{A:U} \prod_{B:U} \prod_{R:T(A) \simeq T(B)} C(A, B, R)$$

and a dependent function

$$t:\prod_{X:U} C(X, X, \mathrm{id}_X) \vdash \beta(t):\prod_{A:U} J(t, A, A, \mathrm{id}_A) =_{C(A, A, \mathrm{id}_A)} t(A)$$

When the Tarski universe is closed under identity types, dependent sum types, and dependent product types (see below for more on closure of Tarski universes under type formers), one is able to internalize the type of equivalences in the universe as $A \simeq_U B$. It can be proven that $T(A \simeq_U B)$ is equivalent to $T(A) \simeq T(B)$, and that the definition of univalence using transport is equivalent to the usual definition given by the internal type of equivalences and the canonical function $\mathrm{idtoequiv}$. See at Univalence axiom#For Tarski universes for more details on this.

Construction of univalent Tarski universes via a higher inductive type

Let $(U, T)$ be a Tarski universe. Then one could construct a univalent Tarski universe as the higher inductive type $(U', T')$ generated by

• a function $F:U \to U'$
• a function $G(A):T(A) \to T'(A)$ for all elements $a:U$
• a dependent function
  $$\mathrm{ua}_{U'}:\prod_{A:U} \prod_{B:U} \mathrm{isEquiv}(\mathrm{transport}^{T'}(A, B))$$

Regularity and product-regularity

A univalent Tarski universe $(U, T)$ is regular if it is closed under dependent sum types: namely, for all $U$-small types $A:U$ and $U$-small type families $B:T(A) \to U$, the dependent sum type $\sum_{x:T(A)} T(B(x))$ is essentially $U$-small.

$$\mathrm{ax}_\Sigma:\sum_{\Sigma:\prod_{A:U} (T(A) \to U) \to U} \prod_{A:U} \prod_{B:T(A) \to U} T(\Sigma(A, B)) \simeq \sum_{x:T(A)} T(B(x))$$

Regular univalent Tarski universes are the type theoretic equivalent of regular cardinals in set theory.

A univalent Tarski universe $(U, T)$ is product-regular if it is closed under dependent product types: namely, for all $U$-small types $A:U$ and $U$-small type families $B:T(A) \to U$, the dependent product type $\sum_{x:T(A)} T(B(x))$ is essentially $U$-small.

$$\mathrm{ax}_\Pi:\sum_{\Pi:\prod_{A:U} (T(A) \to U) \to U} \prod_{A:U} \prod_{B:T(A) \to U} T(\Pi(A, B)) \simeq \prod_{x:T(A)} T(B(x))$$

Product-regular univalent Tarski universes are the type theoretic equivalent of product-regular cardinals in set theory.

Since product-regularity implies that function types between $U$-small types are essentially $U$-small, and regularity implies that subtypes of essentially $U$-small types are essentially $U$-small, this implies that in a regular and product-regular universe, equivalence types between $U$-small types are essentially $U$-small, because equivalence types are subtypes of function types. In combination with univalence, regularity and product-regularity implies that the identity types between $U$-small types are also essentially $U$-small.

Closure under identity types

A univalent Tarski universe $(U, T)$ is closed under identity types if for all $U$-small types $A:U$ and elements $x:T(A)$ and $y:T(A)$, the identity type $x =_{T(A)} y$ is essentially $U$-small.

$$\mathrm{ax}_\mathrm{Id}:\sum_{\mathrm{Id}:\prod_{A:U} (T(A) \times T(A)) \to U} \prod_{A:U} \prod_{x:T(A)} \prod_{y:T(A)} T(\mathrm{Id}(A, x, y)) \simeq (x =_{T(A)} y)$$

Regularity and closure under identity types implies that the universe is closed under pullbacks.

Singletons

A Tarski universe $(U, T)$ satisfies the axiom of singletons if it is closed under the type of all $U$-small contractible types: namely, the dependent sum type $\sum_{A:U} \mathrm{isContr}(T(A))$ is essentially $U$-small:

$$\mathrm{axSingl}_U:\sum_{\mathbb{1}:U} T(\mathbb{1}) \simeq \sum_{A:U} \mathrm{isContr}(T(A))$$

Empty type

A univalent Tarski universe satisfies the axiom of empty type if it is closed under the (weak) empty type: if the empty type $\mathbb{0}$ is essentially $U$-small. The axiom of empty type comes from the formation rule and the dependent universal property of the empty type, which are represented by the following elements:

• formation rules: $\mathbb{0}:U$

• dependent universal property:

  $$\mathrm{up}_\mathbb{0}^U:\prod_{C:T(\mathbb{0}) \to U} \mathrm{isContr}\left(\prod_{x:T(\mathbb{0})} T(C(x))\right)$$

By using dependent sum types, these can be combined into a single element of the following type:

$$\mathrm{axempty}_U:\sum_{\mathbb{0}:U} \prod_{C:T(\mathbb{0}) \to U} \mathrm{isContr}\left(\prod_{x:T(\mathbb{0})} T(C(x))\right)$$

Product regularity and the axiom of empty type imply the axiom of singletons, because the dependent universal property of the empty set states that for every type family $C:T(\mathbb{0}) \to U$ the dependent function type $\prod_{x:T(\mathbb{0})} T(C(x))$ is a singleton, and product regularity implies that $\prod_{x:T(\mathbb{0})} T(C(x))$ is essentially $U$-small.

Propositional resizing

A Tarski universe $(U, T)$ satisfies propositional resizing if it is closed under the type of all $U$-small propositions: namely, the dependent sum type $\sum_{A:U} \mathrm{isProp}(T(A))$ is essentially $U$-small:

$$\mathrm{propresize}_U:\sum_{\Omega:U} T(\Omega) \simeq \sum_{A:U} \mathrm{isProp}(T(A))$$

This is a version internal to the universe $U$ of having a type of all propositions in the type theory itself. While propositional resizing implies that the universe is impredicative, it does not imply that the type theory as a whole is impredicative; the latter requires an actual type of all propositions in the type theory.

Infinity

A univalent Tarski universe satisfies the axiom of infinity if it is closed under the (weak) natural numbers type: if the natural numbers type $\mathbb{N}$ is essentially $U$-small. The axiom of infinity comes from the formation rule, the introduction rules, and the dependent universal property of the natural numbers, which are represented by the following elements:

• formation rules: $\mathbb{N}:U$

• introduction rules: $0:T(\mathbb{N})$ and $s:T(\mathbb{N}) \to T(\mathbb{N})$

• dependent universal property:

  $$\mathrm{up}_\mathbb{N}^U:\prod_{C:\mathbb{N} \to U} \prod_{c_0:T(C(0))} \prod_{c_s:\prod_{x:\mathbb{N}} T(C(x)) \to T(C(s(x)))} \exists!c:\prod_{x:\mathbb{N}} T(C(x)).(c(0) =_{T(C(0))} c_0) \times \prod_{x:\mathbb{N}} (c(s(x)) =_{T(C(s(x)))} c_s(c(x)))$$

  where $\exists!x:A.B(x)$ is the uniqueness quantifier for the type family $x:A \vdash B(x)$.

By using dependent sum types, these can be combined into a single element of the following type:

$$\mathrm{axinf}_U:\sum_{\mathbb{N}:U} \sum_{0:T(\mathbb{N})} \sum_{s:T(\mathbb{N}) \to T(\mathbb{N})} \prod_{C:\mathbb{N} \to U} \prod_{c_0:T(C(0))} \prod_{c_s:\prod_{x:\mathbb{N}} T(C(x)) \to T(C(s(x)))} \exists!c:\prod_{x:\mathbb{N}} T(C(x)).(c(0) =_{T(C(0))} c_0) \times \prod_{x:\mathbb{N}} (c(s(x)) =_{T(C(s(x)))} c_s(c(x)))$$

Replacement

A univalent Tarski universe satisfies the axiom of replacement if for every essentially $U$-small type $A$, every locally $U$-small type $B$, and every function $f:A \to B$, the image of $f$, $\mathrm{im}(f)$, is essentially $U$-small.

Equivalently, given types $A$ and $B$, there is a function

$$\mathrm{axrepl}_U^{A, B}:\left(\sum_{C:U} T(C) \simeq A\right) \times \left(\sum_{R:B \times B \to U} \prod_{a:B} \prod_{b:B} T(R(a, b)) \simeq (a =_B b)\right) \to \left(\sum_{I:(A \to B) \to U} \prod_{f:A \to B} T(I(f)) \simeq \mathrm{im}(f)\right)$$

Choice operators and existence

A choice operator on a type is a function from its propositional truncation to the type itself, and represents the concept that if there exists an element of the set (i.e. the propositional truncation has an element), then the set itself has an element chosen by the choice operator. A Tarski universe satisfies the type-theoretic axiom of existence if every $U$-small type in the Tarski universe has a choice operator, represented by the following dependent function

$$\varepsilon_U:\prod_{A:U} \vert T(A) \vert \to T(A)$$

$U$ satisfying the type-theoretic axiom of existence implies that $U$ satisfies axiom K or UIP. If $U$ is also univalent, then it is an h-groupoid.

There is also a set-theoretic axiom of existence, which only requires the sets in the universe to have choice operators. This coincides with the axiom of existence in set theory and allows for non-set types in the Tarski universe like the circle type as well. This is represented by the following dependent function

$$\varepsilon_U:\prod_{A:U} \mathrm{isSet}(T(A)) \times \vert T(A) \vert \to T(A)$$

Closure of Tarski universes under other type formers

There are many ways of defining type formers internally in a universe:

• by an equivalence or definitional equality with an existing global type former for the entire type theory.

• by translating the rules for the type former into internal structure on the universe - the formation rules, the introduction rules, and the dependent universal property of the structure.

• by using the universal properties corresponding to the categorical semantics of the types

Using a definitional equality with an existing global type former for each type former results in a strict Tarski universe, while using equivalences for each type former results in a weak Tarski universe. There are also various Tarski universes where some type formers are strictly closed, and some type formers are only weakly closed, resulting in Tarski universes which are intermediate between strict Tarski universes and weak Tarski universes.

Tarski universes with all propositions

A Tarski universe $(U, T)$ has all propositions if given a type $A$ with a family of identities $x:A, y:A \vdash \tau_{-1}(x, y):x =_A y$, the universe $U$ comes with the structure of an element $A_U:U$ and an equivalence $\delta_A:T(A_U) \simeq A$.

The rules for this condition on Tarski universes is as follows:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A, y:A \vdash \tau_{-1}(x, y):x =_A y}{\Gamma \vdash A_U:U} \qquad \frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A, y:A \vdash \tau_{-1}(x, y):x =_A y}{\Gamma \vdash \delta_A:T(A_U) \simeq A}$$

Given a Tarski universe which has all propositions, the type $\sum_{A:U} \mathrm{isProp}(T(A))$ is the type of all propositions. This is a form of strong impredicativity in the type theory, as given a type $A$, the type $A \to \sum_{A:U} \mathrm{isProp}(T(A))$ is the power set on $A$.

Tarski subuniverses

Let $(U, T_U)$ be a Tarski universe. A Tarski universe $(V, T_V)$ is a Tarski subuniverse of $(U, T_U)$ if it comes with an embedding $t:V \hookrightarrow U$ and a dependent function

$$p:\prod_{a:V} T_U(t(a)) \simeq T_V(a)$$

This is equivalent to a family of Tarski universes $(U(i), T(i))$ indexed by elements $i:\mathbb{2}$ of the interval preorder $(\mathbb{2}, \leq_\mathbb{2})$, which comes with

• a dependent function

  $$t:\prod_{a:\mathbb{2}} \prod_{b:\mathbb{2}} (a \leq_\mathbb{2} b) \to (U(a) \hookrightarrow U(b))$$

• a dependent function

  $$q:\prod_{a:\mathbb{2}} \prod_{b:\mathbb{2}} \prod_{p:a \leq_\mathbb{2} b} \prod_{A:U(a)} T(a)(A) \simeq T(b)(t(a, b)(p)(A))$$

  indicating that lifting each small type results in a type equivalent to the original small type.

Modalities and comodalities

A modal operator on a Tarski universe $(U, T)$ is just an endofunction $L:U \to U$. Given a Tarski universe, the modal unit family is a family of functions

$$\eta:\prod_{L:U \to U}\prod_{A:U} T(A) \to T(L(A))$$

and the comodal counit family is a family of functions

$$\epsilon:\prod_{L:U \to U}\prod_{A:U} T(L(A)) \to T(A)$$

Given a modal operator $L:U \to U$, $\eta(L)$ is called the modal unit of $L$ and $\epsilon(L)$ is called the comodal counit of $L$. A small type $A:U$ is $L$-modal if the function

$$\eta(L)(A):T(A) \to T(L(A))$$

is an equivalence of types, and $L$-comodal if the function

$$\epsilon(L)(A):T(L(A)) \to T(A)$$

is an equivalence of types.

A modality is…

…and a comodality is…

Reflective and coreflective Tarski subuniverses

(Section under construction see reflective subuniverse for the Russell universe variant for the time being)

Essentially small Tarski universes

A Tarski universe $(V, T_V)$ inside of a Tarski universe $(U, T_U)$ consists of an element $V:U$ and a function $T_V:T_U(V) \to U$.

Given a Tarski universe $(U, T_U)$, a Tarski universe $(V, T_V)$ is essentially $U$-small if it comes with

• an element $V':U$

• a function $T_{V'}:T_U(V') \to U$

• an equivalence

  $$\mathrm{smallType}:V \simeq T_U(V')$$

• a dependent function

  $$\mathrm{smallFamily}:\prod_{A:V} T_V(A) \simeq T_U(T_{V'}(\mathrm{smallType}(A)))$$

Equivalently, the entire structure could be regarded as a family of Tarski universes $(U(a), T(a))$ indexed by elements $a:\mathbb{2}$ of the interval preorder $(\mathbb{2}, \leq_\mathbb{2})$, which comes with

• a dependent function

  $$U':\prod_{a:\mathbb{2}} \prod_{b:\mathbb{2}} \prod_{p:a \leq_\mathbb{2} b} \left((b \leq_\mathbb{2} a) \to \emptyset\right) \to U(b)$$

• a dependent function

  $$T':\prod_{a:\mathbb{2}} \prod_{b:\mathbb{2}} \prod_{p:a \leq_\mathbb{2} b} \left((b \leq_\mathbb{2} a) \to \emptyset\right) \to \left(T(b)(U'(a, b, p)) \to U(b)\right)$$

• an dependent function

  $$\mathrm{smallType}:\prod_{a:\mathbb{2}} \prod_{b:\mathbb{2}} \prod_{p:a \leq_\mathbb{2} b} \left((b \leq_\mathbb{2} a) \to \emptyset\right) \to \left(U(a) \simeq T(b)(U'(a, b, p))\right)$$

• a dependent function

  $$\mathrm{smallFamily}:\prod_{a:\mathbb{2}} \prod_{b:\mathbb{2}} \prod_{p:a \leq_\mathbb{2} b} \left((b \leq_\mathbb{2} a) \to \emptyset\right) \to \prod_{A:U(a)} T(a)(A) \simeq T(b)(T'(a, b, p)(\mathrm{smallType}(a, b, p)(A)))$$

Hierarchy of Tarski universes

Let $P$ be a preorder. Then, a $P$-indexed hierarchy of Tarski universes is a type family $U$ such that for all indices $a:P$, there is a Tarski universe $(U(a), T(a))$, such that for all indices $a:P$ and $b:P$ and witnesses $p:a \leq_P b$, the Tarski universe $(U(a), T(a))$ is an essentially $U(b)$-small Tarski subuniverse of the Tarski universe $(U(b), T(b))$.

Expanded out, the family of Tarski universes $(U(a), T(a))$ indexed by elements $a:P$ additionally come with

• a dependent function

  $$U':\prod_{a:P} \prod_{b:P} \prod_{p:a \leq_P b} \left((b \leq_P a) \to \emptyset\right) \to U(b)$$

• a dependent function

  $$T':\prod_{a:P} \prod_{b:P} \prod_{p:a \leq_P b} \left((b \leq_P a) \to \emptyset\right) \to \left(T(b)(U'(a, b, p)) \to U(b)\right)$$

• an dependent function

  $$\mathrm{smallType}:\prod_{a:P} \prod_{b:P} \prod_{p:a \leq_P b} \left((b \leq_P a) \to \emptyset\right) \to \left(U(a) \simeq T(b)(U'(a, b, p))\right)$$

• a dependent function

  $$\mathrm{smallFamily}:\prod_{a:P} \prod_{b:P} \prod_{p:a \leq_P b} \left((b \leq_P a) \to \emptyset\right) \to \prod_{A:U(a)} T(a)(A) \simeq T(b)(T'(a, b, p)(\mathrm{smallType}(a, b, p)(A)))$$

• a dependent function

  $$t:\prod_{a:P} \prod_{b:P} (a \leq_P b) \to (U(a) \hookrightarrow U(b))$$

• a dependent function

  $$q:\prod_{a:P} \prod_{b:P} \prod_{p:a \leq_P b} \prod_{A:U(a)} T(a)(A) \simeq T(b)(t(a, b)(p)(A))$$

Usually, hierarchies of Tarski universes are indexed by the type of natural numbers. A type theory may have multiple hierarchies of type universes.

Examples

The empty type, unit type, boolean domain, and FinSet are all regular univalent Tarski universes. The types of propositions in a type theory are univalent Tarski universes: they model a dependent type theoretic model of propositional logic with function types, product types, disjunction higher inductive types, empty type, and unit type.

Every concrete category and every concrete dagger category? is a Tarski universe, as given a concrete category or concrete dagger category $C$ by definition for each object $A:Ob(C)$ there is an h-set $El(A)$. This includes models of Mike Shulman‘s SEAR, which is a concrete power allegory, as well as ETCS with elements, which is a concrete elementary topos.

See also

For universes as a replacement for type judgments, see

• Russell universe

Regarding the general notion of types of small types, see

• type universe

Other mathematical structures and their univalent counterparts:

• preorder, partial order
• precategory, univalent category
• dagger precategory, univalent dagger category
• locally univalent bicategory, univalent bicategory

References

Historical review of Alfred Tarski‘s original notion in set theory:

• Ralf Krömer, §6.4.5 in: Tool and object: A history and philosophy of category theory, Science Networks. Historical Studies 32 (2007) [doi:10.1007/978-3-7643-7524-9]

The terminology “universe à la Tarski” — and now in the context of type universes for Martin-Löf dependent type theory — is due to:

• Per Martin-Löf (notes by Giovanni Sambin): p. 48 in: Intuitionistic type theory, Lecture notes Padua 1984, Bibliopolis, Napoli (1984) [pdf, pdf]

Further discussion:

• Martin Hofmann, Universes, section 2.3.5 in: Syntax and semantics of dependent types, Chapter 2 in: Extensional concepts in intensional type theory, Ph.D. thesis, University of Edinburgh (1995), Distinguished Dissertations, Springer (1997) [ECS-LFCS-95-327, pdf, doi:10.1007/978-1-4471-0963-1]

See also:

• Zhaohui Luo, Notes on Universes in Type Theory, 2012 (pdf)

• Cesare Gallozzi, Constructive Set Theory from a Weak Tarski Universe, MSc thesis (2014) (arXiv:1411.5591)

• Egbert Rijke, Introduction to Homotopy Type Theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press (arXiv:2212.11082)

• Homotopy Type Theory: Univalent Foundations of Mathematics, The Univalent Foundations Program, Institute for Advanced Study, 2013. (web, pdf)

• Egbert Rijke, Mike Shulman, Bas Spitters, Modalities in homotopy type theory (arXiv:1706.07526)