natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism = propositions as types +programs as proofs +relation type theory/category theory
equality (definitional, propositional, computational, judgemental, extensional, intensional, decidable)
isomorphism, weak equivalence, homotopy equivalence, weak homotopy equivalence, equivalence in an (∞,1)-category
Examples.
(in category theory/type theory/computer science)
of all homotopy types
of (-1)-truncated types/h-propositions
In intensional type theory, identity types behave like path space objects; this viewpoint is called homotopy type theory. This induces furthermore a notion of homotopy fibers, hence of homotopy equivalences between types.
On the other hand, if type theory contains a universe Type, so that types can be considered as points of $Type$, then between two types we also have an identity type $Paths_{Type}(X,Y)$. The univalence axiom says that these two notions of “sameness” for types are the same. Morally, this says that Type behaves like an object classifier.
The name univalence (due to Voevodsky) comes from the following reasoning. A fibration or bundle $p\colon E\to B$ of some sort is commonly said to be universal if every other bundle of the same sort is a pullback of $p$ in a unique way (up to homotopy). Less commonly, a bundle is said to be versal if every other bundle is a pullback of it in some way, not necessarily unique. By contrast, a bundle is said to be univalent if every other bundle is a pullback of it in at most one way (up to homotopy). The univalence axiom then says that the canonical fibration over $Type$ is univalent, for every fibration with small fibers is an essentially unique pullback of this one (while those with large fibers are not, they are pullbacks of the next higher $Type_1$).
We state univalence first in (intensional) type theory and then in its categorical semantics.
Let $X$ and $Y$ be types. There is a canonically defined map from the identity type $(X = Y)$ of paths (in Type) between them to the function type $(X \stackrel{\simeq}{\to} Y)$ of equivalences in homotopy type theory between them. It can be defined by path induction, i.e. the eliminator for the identity types, by specifying that it takes the identity path $1_X \colon (X=X)$ to the identity equivalence of $X$.
Univalence: For any two types $X,Y$, this map $(X=Y)\to (X\simeq Y)$ is an equivalence.
When $X$ and $Y$ are propositions, univalence corresponds to propositional extensionality.
Univalence is a commonly assumed axiom in homotopy type theory, and is central to the proposal (Voevodsky) that this provides a natively homotopy theoretic foundation of mathematics (the Univalent Foundations Project.)
Let $\mathcal{C}$ be a locally cartesian closed model category in which all objects are cofibrant.
By the categorical semantics of homotopy type theory, a dependent type
corresponds to a morphism $E \to B$ in $\mathcal{C}$ that is a fibration between fibrant objects.
Then the dependent function type
is interpreted as the internal hom $[-,-]_{\mathcal{C}/_{B \times B}}$ in the slice category $\mathcal{C}/_{B \times B}$ after extending $E$ to the context $B \times B$ by pulling back along the two projections $p_1, p_2 : B \times B \to B$, respectively. Hence this is interpreted as
Consider then the diagonal morphism $\Delta_B : B \to B \times B$ in $\mathcal{C}$ as an object of $\mathcal{C}/_{B \times B}$. We would like to define a morphism
in $\mathcal{C}/_{B \times B}$. By the defining (product $\dashv$ internal hom)-adjunction, it suffices to define a morphism
in $\mathcal{C}/_{B \times B}$. But now by the universal property of pullback, it suffices to define just in $\mathcal{C}_{/B}$ a morphism
And since the composite pullback along either composite
is the identity, both $\Delta_B \times_{\mathcal{C}/_{B \times B}} E \times B$ and $\Delta_B \times_{\mathcal{C}/_{B \times B}} B \times E$ are isomorphic to $E$; thus here we can take the identity morphism.
Now, using the path object factorization in $\mathcal{C}$
by an acyclic cofibration followed by a fibration, we obtain a fibrant replacement of $\Delta_B$ in the slice model category $\mathcal{C}_{B \times B}$.
Since also $[E \times B, B \times E]_{\mathcal{C}/_{B \times B}}$ is fibrant by the axioms on the locally cartesian closed model category $\mathcal{C}$, we have a lift $\hat q$ in the diagram in $\mathcal{C}/_{B \times B}$
This lift is the interpretation of the path induction that deduces a map on all paths $\gamma \in B^I$ from one on just the identity paths $id_b \in B \hookrightarrow B^I$.
Finally, let $Eq(E) \hookrightarrow [E \times B , B \times E]_{\mathcal{C}/_{B \times B}}$ be the subobject on the weak equivalences (…), and observe that $q$ and $\hat q$ factor through this to give a morphism
The fibration $E \to B$ is univalent in $\mathcal{C}$ if this morphism is a weak equivalence. By the 2-out-of-3 property, of course, it is equivalent to ask that $q\colon B\to Eq(E)$ be a weak equivalence.
(…)
We specialize the general discussion above to the realization in $\mathcal{C} =$ sSet, equipped with the standard model structure on simplicial sets.
For $E \to X$ any fibration (Kan fibration) between fibrant objects (Kan complexes), consider first the simplicial set
defined as the internal hom in the slice category $sSet/_{B \times B}$.
Notice that the vertices of this simplicial set over a fixed pair $(b_1, b_2) : * \to B \times B$ of vertices in $B$ form the set of morphisms $E_{b_1} \to E_{b_2}$ between the fibers in $sSet$.
This is because – by the defining property of the internal hom in the slice and using that products in $sSet/_{B \times B}$ are pullbacks in $sSet$ – the horizontal morphisms of simplcial sets in
correspond bijectively to the horizontal morphisms in
in $sSet$, which are precisely morphisms $E_{b_1} \to E_{b_2}$.
Let then
be the full sub-simplicial set on those vertices that correspond to weak equivalences ((weak) homotopy equivalences).
By a similar consideration, one sees that the diagonal morphism $\Delta_B : B \to B \times B$ in $sSet$, regarded as an object $B \in sSet/_{B \times B}$, comes with a canonical morphism
The fibration $E \to B$ is univalent, precisely when this morphism is a weak equivalence.
This appears originally as Voevodsky, def. 3.4
(…)
See (Shulman 12, UF 13)
(…)
The univalence axiom implies function extensionality.
A commented version of a formal proof of this fact can be found in (Bauer-Lumsdaine).
It is currently open whether the univalence axiom enjoys canonicity in general, but for the special case of 1-truncated homotopy types (groupoids) (and two nested univalent universes and function extensionality), a “homotopical” sort of canonicity has been shown in (Shulman 12, section 13. Thus, in univalent homotopy 1-type theory with two universes, every term of type of the natural numbers is propositionally equal to a numeral.
The construction in (Shulman 12, section 13) uses Artin gluing of a suitable type-theoretic fibration category with the category Set and Grpd, respectively, effectively inducing canonicity from these categories. By (Shulman 12, remark 13.13) for this construction to generalize to untruncated univalent type theory, one seems to need a sufficiently strict global sections functor with values in some model for infinity-groupoids.
Notice that this sort of canonicity does not yet imply computational effectiveness?, which would require also an algorithm to extract that numeral from the given term. There may be such an algorithm, but so far attempts to extract one from the proof (or to give a constructive version of the proof, which would imply the existence of an algorithm) have not succeeded.
It is also a propositional canonicity, as opposed to the judgmental canonicity which many traditional type theories enjoy. Another approach to canonicity for 1-truncated univalence can be found in (Harper-Licata), which involves modifying the type theory by adding more judgmental equalities, resulting in a judgmental canonicity. However, no algorithm for computing canonical forms has yet been given for this approach either.
Another approach to solving canonicity is to build a model for homotopy type theory inside a theory which is known to be constructive, such as ordinary (univalence-free) intensional type theory. Then one could induce canonicity, with an algorithm, from the latter to the former. This is the direction of the work of (Coquand-Huber 13), which models univalent homotopy type theory in constructive cubical sets, or the truncated form in constructive truncated simplicial sets. One might also try to construct the Hoffman-Streicher groupoid model in a constructive framework; Awodey and Bauer have done some work in this direction with an impredicative universe of h-sets.
Univalence is closely related to the “completeness” condition in the theory of Segal spaces/semi-Segal spaces. See complete Segal space/_complete semi-Segal space_.
Perhaps the earliest occurrence of the univalence axiom is in section 5.4 of
under the name “universe extensionality”. They formulate almost the modern univalence axiom; the only difference is the lack of a coherent definition of equivalence. The univalence axiom in its modern form was introduced and promoted by Vladimir Voevodsky around 2005. (?)
A quick survey is for instance in
An exposition is at
An accessible account of Voevodsky’s proof that the universal Kan fibration in simplicial sets is univalent is at
A quick elegant proof of the object classifier/universal associated infinity-bundle in simplicial sets/$\infty$-groupoids is in
See also
A guided walk through the formal proof that univalence implies functional extensionality is at
A discussion of univalence in categories of diagrams over an inverse category with values in a category for which univalence is already established is discussed in
This discusses canonicity of univalence in its section 13. Another approach to showing canonicity is (via cubical sets) in
On the issue of strict pullback of the univalent universe see
The computational interpretation of univalence / canonicity is discussed in
Dan Licata, Robert Harper, Computing with Univalence (2012) (pdf)
Robert Harper, Daniel Licata, Canonicity for 2-dimensional type theory (2011) (pdf)
Daniel Licata The computational interpretation of HoTT (in 2D), talk at UF-IAS-2012 (video)
Simon Huber (with Thierry Coquand), Towards a computational justification of the Axiom of Univalence , talk at TYPES 2011 (pdf)
Bruno Barras, Thierry Coquand, Simon Huber, A Generalization of Takeuti-Gandy Interpretation (pdf)
Thierry Coquand (with Marc Bezem and Simon Huber), Computational content of the Axiom of Univalence, September 2013 (pdf)
A study of the semantic side of univalence in (infinity,1)-toposes, as well as further cases of locally cartesian closed (infinity,1)-categories is in
This does not yet show that the univalence axiom in its usual form holds in the internal type theory of (infinity,1)-toposes, however, due to the lack of a (known) sufficiently strict model for the object classifier. (But it works with Tarskian type universes, see there). Constructions of such a model in some very special cases are in Shulman12 above, and also in
Michael Shulman, The univalence axiom for elegant Reedy presheaves, arXiv:1307.6248.
Denis-Charles Cisinski, Univalent universes for elegant models of homotopy types (arXiv:1406.0058)
For more references see homotopy type theory.