unitary group

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## Definition

For a [[natural number]] $n \in \mathbb{N}$, the **unitary group** $U(n)$ is the [[group]] of [[isometry|isometries]] of the $n$-dimensional complex [[Hilbert space]] $\mathbb{C}^n$.   This is canonically identified with the group of $n \times n$ [[unitary matrices]].

More generally, for a Hilbert space $\mathcal{H}$, $U(\mathcal{H})$ is the group of [[unitary operator]]s on that Hilbert space. For the purposes of studying unitary representations of Lie groups, the topology is chosen to be the [[operator topology|strong operator topology]], although other topologies on $U(\mathcal{H})$ are frequently considered for other purposes.

## Properties

The unitary groups are naturally [[topological group]]s and [[Lie groups]] (infinite dimensional if $\mathcal{H}$ is infinite dimensional).

+-- {: .num_prop #UnitaryGroupIsCompact}
###### Proposition

The unitary group $U(n)$ is [[compact topological space]], hence in particular a [[compact Lie group]].

=--

### Homotopy groups
 {#HomotopyGroups}

+-- {: .num_prop #InclusionOfUnitaryGroupnIntoUnitaryGroupnPlusIneIsnMinus1Equivalence}
###### Proposition

For $n,N \in \mathbb{N}$, $n \leq N$, then the canonical inclusion of unitary groups

$$
  U(n) \hookrightarrow U(N)
$$

is a [[n-equivalence|2n-equivalence]], hence induces an [[isomorphism]] on [[homotopy groups]] in degrees $\lt 2n$ and a [[surjection]] in degree $2n$.

=--

+-- {: .proof}
###### Proof

Consider the [[coset]] [[quotient]] [[projection]]

$$
  U(n)
  \longrightarrow
  U(n+1)
  \longrightarrow
  U(n+1)/U(n)
  \,.
$$

By prop. \ref{UnitaryGroupIsCompact} and by [this corollary](coset#QuotientProjectionForCompactLieSubgroupIsPrincipal), the projection $U(n+1)\to U(n+1)/U(n)$ is a [[Serre fibration]]. Furthermore, example \ref{nSphereAsUnitaryCosetSpace} identifies the [[coset]] with the [[n-sphere|(2n+1)-sphere]]

$$
  S^{2n+1}\simeq U(n+1)/U(n)
  \,.
$$

Therefore the [[long exact sequence of homotopy groups]] of the [[fiber sequence]] $U(n)\to U(n+1) \to S^{2n+1}$ is of the form

$$
  \cdots
    \to
  \pi_{\bullet+1}(S^{2n+1})
    \longrightarrow
  \pi_\bullet(U(n))
    \longrightarrow
  \pi_\bullet(U(n+1))
    \longrightarrow
  \pi_\bullet(S^{2n+1})
   \to
  \cdots
$$

Since $\pi_{\leq 2n}(S^{2n+1}) = 0$, this implies that

$$
  \pi_{\lt 2n}(U(n))
    \overset{\simeq}{\longrightarrow}
  \pi_{\lt 2n}(U(n+1))
$$

is an isomorphism and that

$$
  \pi_{2n}(U(n))
    \longrightarrow
  \pi_{2n}(U(n+1))
$$

is surjective. Hence now the statement follows by induction over $N-n$.

=--

It follows that the [[homotopy groups]] $\pi_k(U(n))$ are independent of $n$ for $n \gt \frac{k}{2}$ (the "stable range"). So if $U = \underset{\longrightarrow}{\lim}_n U(n)$, then $\pi_k(U(n)) = \pi_k(U)$. By [[Bott periodicity]] we have

$$
\array{
\pi_{2k+0}(U) &= 0\\
\pi_{2k+1}(U) &= \mathbb{Z}.
}
$$

In the unstable range for low $n$ they instead start out as follows

| $G$ | $\pi_1$ | $\pi_2$ | $\pi_3$ | $\pi_4$ | $\pi_5$ | $\pi_6$ | $\pi_7$ | $\pi_8$ | $\pi_9$ | $\pi_10$ | $\pi_11$ | $\pi_12$ | $\pi_13$ | $\pi_14$ | $\pi_15$ |
|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|--|
| $U(1)$ | $\mathbb{Z}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| $U(2)$ | " | 0 | $\mathbb{Z}$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ | $\mathbb{Z}_{12}$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2$ | $\mathbb{Z}_3$ | $\mathbb{Z}_{15}$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2^{\oplus 2}$ | $\mathbb{Z}_3\oplus\mathbb{Z}_{12}$ | $\mathbb{Z}_2^{\oplus 2}\oplus\mathbb{Z}_{84}$ | $\mathbb{Z}_2\oplus\mathbb{Z}_2$ |
| $U(3)$ | " | " | " | 0 | $\mathbb{Z}$ | $\mathbb{Z}_6$ | 0 | $\mathbb{Z}_{12}$ | $\mathbb{Z}_3$ | $\mathbb{Z}_{30}$ | $\mathbb{Z}_4$ | $\mathbb{Z}_{60}$ | $\mathbb{Z}_6$ | $\mathbb{Z}_2\oplus\mathbb{Z}_{84}$ | $\mathbb{Z}_{36}$ |
| $U(4)$ | " | " | " | " | " | 0 | $\mathbb{Z}$ | $\mathbb{Z}_{24}$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2\oplus\mathbb{Z}_{120}$ | $\mathbb{Z}_4$ | $\mathbb{Z}_{60}$ | $\mathbb{Z}_4$ | $\mathbb{Z}_2\oplus\mathbb{Z}_{1680}$ | $\mathbb{Z}_2\oplus\mathbb{Z}_{72}$ |
| $U(5)$ | " | " | " | " | " | " | " | 0 | $\mathbb{Z}$ | $\mathbb{Z}_{120}$ | 0 | $\mathbb{Z}_{360}$ | $\mathbb{Z}_4$ | $\mathbb{Z}_{1680}$ | $\mathbb{Z}_6$ |
| $U(6)$ | " | " | " | " | " | " | " | " | " | 0 | $\mathbb{Z}$ | $\mathbb{Z}_{720}$ | $\mathbb{Z}_2$ | $\mathbb{Z}_2\oplus\mathbb{Z}_{5040}$ | $\mathbb{Z}_6$ |
| $U(7)$ | " | " | " | " | " | " | " | " | " | " | " | 0 | $\mathbb{Z}$ | $\mathbb{Z}_{5040}$ | 0 |
| $U(8)$ | " | " | " | " | " | " | " | " | " | " | " | " | " | 0 | $\mathbb{Z}$ |

Due to the [[unitary group#SUn | relation to the special unitary group]], the higher homotopy groups of $U(n)$ and $SU(n)$ agree. The $U(2)$ row can be found using the fact that $SU(2)$ is [[diffeomorphism|diffeomorphic]] to the [[3-sphere]] $S^3$. The $U(3)$ row can be found using [Mimura-Toda 63](#MimuraToda63). Otherwise the table is given in columns $\pi_k$, $k= 6,\ldots, 15$, and in rows $U(n)$, $n=4,\ldots,8$, by the [[Encyclopedic Dictionary of Mathematics]], Table 6.VII in Appendix A.

### In infinite dimension

A good discussion about the various topologies one might place on $U(\mathcal{H})$ and how they all agree and make $U(\mathcal{H})$ a [[Polish space|Polish]] group is in ([Espinoza-Uribe](#EspinozaUribe)).

+-- {: .num_prop}
###### Proposition

For $\mathcal{H}$ a [[Hilbert space]], which can be either finite or infinite dimensional, the unitary group $U(\mathcal{H})$ and the [[general linear group]] $GL(\mathcal{H})$, regarded as [[topological group]]s, have the same [[homotopy type]].

Additionally, $U(\mathcal{H})$ is a [[maximal compact subgroup]] of $GL(\mathcal{H})$ for finite-dimensional $\mathcal{H}$.

=--

+-- {: .proof}
###### Proof

By the [[Gram-Schmidt process]].

=--

+-- {: .num_theorem}
###### Theorem
**(Kuiper's theorem)**

For a separable infinite-dimensional complex [[Hilbert space]] $\mathcal{H}$, the unitary group $U(\mathcal{H})$ is [[contractible]] in the [[norm topology]].

=--

See also [[Kuiper's theorem]]. Note that $U(\mathcal{H})$ is also contractible in the [[strong operator topology]] (due to Dixmier and Douady).

+-- {: .num_remark}
###### Remark

This in contrast to the finite dimensional situation. For $n \in \mathbb{N}$ ($n \ge 1$), $U(n)$ is not contractible.

=--

Write $B U(n)$ for the [[classifying space]] of the [[topological group]] $U(n)$. Inclusion of matrices into larger matrices gives a canonical sequence of inclusions

$$
  \cdots \to B U(n) \hookrightarrow B U(n+1) \hookrightarrow B U(n+2) \to \cdots
  \,.
$$

The [[homotopy limit|homotopy]] [[direct limit]] over this is written

$$
  B U := {\lim_\to}_n B U(n)
$$

or sometimes $B U(\infty)$. Notice that this is very different from $B U(\mathcal{H})$ for $\mathcal{H}$ an infinite-dimensional Hilbert space. See [[topological K-theory]] for more on this.

### Relation to special unitary group {#SUn}

+-- {: .num_prop}
###### Proposition

For all $n \in \mathbb{N}$, the [[unitary group]] $U(n)$ is a [[split exact sequence|split]] [[group extension]] of the [[circle group]] $U(1)$ by the [[special unitary group]] $SU(n)$

$$
  SU(n) \to U(n) \to U(1)
  \,.
$$

Hence it is a [[semidirect product group]]

$$
  U(n) \simeq SU(n) \rtimes U(1)
  \,.
$$

=--

### Relation to orthogonal, symplectic and general linear group
 {#RelationToOrthogonalSymplecticAndGeneralLinearGroup}

The unitary group $U(n)$ is equivalently the [[intersection]]
of the [[orthogonal group]] $O(2n)$, the [[symplectic group]] $Sp(2n,\mathbb{R})$ and the complex [[general linear group]] $GL(n,\mathbb{C})$ inside the real [[general linear group]] $GL(2n,\mathbb{R})$.

Actually it is already the intersection of any two of these three,
a fact also known as the "2 out of 3-property" of the unitary group.

This intersection property makes a [[G-structure]] for $G = U(n)$ (an [[almost Hermitian structure]]) precisely a joint [[orthogonal structure]], [[almost symplectic structure]] and [[almost complex structure]]. In the [[integrability of G-structure|first-order integrable case]] this is precisely a joint [[orthogonal structure]] ([[Riemannian manifold]] structure), [[symplectic structure]] and [[complex structure]].

## Examples

$U(1)$ is the [[circle group]].

### Coset spaces

+-- {: .num_example #nSphereAsUnitaryCosetSpace}
###### Example

The [[n-spheres|(2n+1)-spheres]] are [[coset spaces]] of unitary groups

$$
  S^{2n+1} \simeq U(n+1)/U(n)
  \,.
$$

=--

+-- {: .num_example #ComplexStiefelManifold}
###### Example

For $n \leq k$, the [[coset]]

$$
   V_n(\mathbb{C}^k) \coloneqq U(k)/U(k-n)
$$

is called the $n$th _complex [[Stiefel manifold]]_ of $\mathbb{C}^k$.

=--

+-- {: .num_prop }
###### Proposition

The complex [[Stiefel manifold]] $V_n(\mathbb{C}^k)$ (example \ref{ComplexStiefelManifold}) is [[n-connected topological space|2(k-n)-connected]].

=--

+-- {: .proof}
###### Proof

Consider the [[coset]] [[quotient]] [[projection]]

$$
  U(k-n)
    \longrightarrow
  U(k)
    \longrightarrow
  U(k)/U(k-n)
    =
  V_n(\mathbb{C}^k)
  \,.
$$

By prop. \ref{UnitaryGroupIsCompact} and by [this corollary](QuotientProjectionForCompactLieSubgroupIsPrincipal) the projection $U(k)\to U(k)/U(k-n)$ is a [[Serre fibration]]. Therefore there is induced the [[long exact sequence of homotopy groups]] of this [[fiber sequence]], and by prop. \ref{InclusionOfUnitaryGroupnIntoUnitaryGroupnPlusIneIsnMinus1Equivalence} it has the following form in degrees bounded by $n$:

$$
  \cdots
    \to
  \pi_{\bullet \leq 2(k-n)}(U(k-n))
    \overset{epi}{\longrightarrow}
  \pi_{\bullet \leq 2(k-n)}(U(k))
    \overset{0}{\longrightarrow}
  \pi_{\bullet \leq 2(k-n)}(V_n(\mathbb{C}^k))
    \overset{0}{\longrightarrow}
  \pi_{\bullet-1 \lt 2(k-n)}(U(k))
    \overset{\simeq}{\longrightarrow}
  \pi_{\bullet-1 \lt 2(k-n)}(U(k-n))
    \to
  \cdots
  \,.
$$

This implies the claim.

=--

## Related concepts

* [[stable unitary group]]

* [[U(ℋ)|$U(\mathcal{H})$]]

* The subgroup of unitary matrices with [[determinant]] equal to 1 is the [[special unitary group]]. The [[quotient]] by the [[center]] is the [[projective unitary group]].  The space of equivalence classes of unitary matrices under conjugation is the [[symmetric product of circles]].

* The analog of the unitary group for real metric spaces is the [[orthogonal group]].

* similarly: [[quaternionic unitary group]]

* The [[Lie algebra]] is the [[unitary Lie algebra]].

* [[unitary representation]]

* [[unitary operator]]

## References

(...)

* {#MimuraToda63} M. Mimura and H. Toda, _Homotopy Groups of $SU(3)$, $SU(4)$ and $Sp(2)$_, J. Math. Kyoto Univ. Volume 3, Number 2 (1963), 217-250. ([Euclid](http://projecteuclid.org/euclid.kjm/1250524818))

For [[U(ℋ)]]:

* {#EspinozaUribe} Jesus Espinoza, [[Bernardo Uribe]], _Topological properties of the unitary group_, JP Journal of Geometry and Topology
**16** (2014) Issue 1, pp 45-55 ([arXiv:1407.1869](https://arxiv.org/abs/1407.1869), [doi:10.18257/raccefyn.317](http://dx.doi.org/10.18257/raccefyn.317))

[[!redirects unitary group]]
[[!redirects unitary groups]]

[[!redirects infinite unitary group]]
[[!redirects infinite unitary groups]]

[[!redirects U(n)]]


unitary Lie algebra

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## Idea

The [[Lie algebra]] $\mathfrak{u}(n)$ of the [[unitary group]] $U(n)$. Canonically identified with [[matrices]] with [[complex number]] entries that are skew-hermitean.

## Related concepts

* [[special unitary Lie algebra]]

* [[special orthogonal Lie algebra]]


unitary matrix

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## Definition

An $n \times n$-[[matrix]] $U \in Mat(n, \mathbb{C})$ with entries in the [[complex numbers]] (for $n$ a [[natural number]]) is **unitary** if the following equivalent conditions hold

* it preserves the canonical [[inner product]] on $\mathbb{C}^n$;

* the operation $(-)^\dagger$ of [[transpose matrix|transposing]] it and then applying [[complex conjugation]] to all its entries takes it to its inverse:

  $$
    U^\dagger \;=\; U^{-1}
    \,.
  $$

  hence equivalently:

  $$
    U \cdot U^\dagger \;=\; \mathrm{I}
  $$

For fixed $n$, the unitary matrices under [[matrix product]] form a [[Lie group]]: the _[[unitary group]]_ $\mathrm{U}(n)$ (or other notations).

## Related concepts

* [[quaternion unitary group]]

[[!redirects unitary matrix]]
[[!redirects unitary matrices]]


unitary morphism

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## Definition

In a [[†-category]] $C$, a [[morphism]] $f : x \to y$ is said to be **unitary** if it is [[isomorphism|invertible]] and its [[inverse]] $f^{-1}$ is its dagger $f^{\dagger}$:

$$
  f^{-1} = f^\dagger\colon y \to x
  \,.
$$

For more details, see the entry [[†-category]].

## Examples

* The unitary morphisms in $C = $ [[Hilb]] are the ordinary [[unitary operator]]s between [[Hilbert space]]s

  In particular the unitary [[automorphism]]s of an object in $Hilb$ form the [[unitary group]].

* The unitary morphisms in $C = $ [[Rel]] are the ordinary [[bijections]]s between [[set]]s

  In particular the unitary [[automorphism]]s of an object in $Rel$ form the [[permutation group]].

## Related concepts

* [[self-adjoint morphism]]

* **unitary morphism**

* [[dagger monomorphism]]

* [[dagger epimorphism]]

[[!redirects unitary morphism]]
[[!redirects unitary morphisms]]
[[!redirects unitary isomorphism]]
[[!redirects unitary isomorphisms]]


unitary operator

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## Definition

A **unitary operator** is a [[unitary morphism]] in the [[†-category]] [[Hilb]].

Explicitly, this means the following.

### Unitary operators

A **unitary operator** $U$ on a [[Hilbert space]] $\mathcal{H}$ is a bounded linear operator $U: \mathcal{H} \to \mathcal{H}$  that satisfies

$U^{*}U=U U^{*}=I$

where $U^{*}$ is the [[Hilbert space adjoint]] of $U$ and $I$ is the [[identity]] operator.  This property is equivalent to saying that the range of $U$ is dense and that $U$ preserves the inner product $\langle \cdot,\cdot\rangle$ on the Hilbert space.  An operator is unitary if and only if $U^{-1}=U^{*}$.

Unitary operators are the [[isomorphism]]s of Hilbert spaces since they preserve the basic structure of the space, e.g. the topology.  The group of all unitary operators from a given Hilbert space to itself is sometimes called the [[unitary group]] $U(\mathcal{H})$ or  _Hilbert group_ of $H$ and is denoted Hilb($H$).

Sometimes operators may only obey the _[[linear isometry]]_ condition $U^{*}U=I$ or the _[[coisometry]]_ condition $U U^{*}=I$.

The generalization of a unitary operator is called a _unitary element_ of a [[unital algebra|unital]] [[star-algebra|*-algebra]].

### Unitary matrices

If a basis for a finite dimensional Hilbert space is chosen, the defnition of unitary operator reduces to that of unitary matrix.

A unitary matrix is an $n \times n$ matrix with complex entries that satisfies the condition

$U^{*}U=U U^{*}=I_{n}$.

This is equivalent to saying that both the rows and the columns of $U$ form an orthonormal basis in $\mathbb{C}^{n}$ with respect to the respective inner product.  $U$ is also a [[normal matrix]] whose [[eigenvalue]]s lie on the unit circle.

## Notation

The notation used here for the adjoint, $U^{*}$, is commonly used in linear algebraic circles (as is $U^{H}$).  In [[quantum mechanics]], $U^{\dagger}$ is exclusively used for the adjoint while $U^{*}$ is interpreted as the same thing as $\bar{U}$, i.e. the complex conjugate.

## Examples

* In [[harmonic analysis]], [[Parseval's theorem]] says that the [[Fourier transform]] is a unitary operator.

## Related concepts

* [[linear isometry]]

* [[unitary group]]

* [[special unitary group]]

* [[Hermitian operator]]

* [[antiunitary operator]]

* [[unitary quantum channel]]

* [[unitary fusion category]]

* [[unitary functorial field theory]]

## References

See also:

* Wikipedia, *[Unitary operator](https://en.wikipedia.org/wiki/Unitary_operator)*

[[!redirects unitary operators]]
[[!redirects unitary]]

[[!redirects unitary linear operator]]
[[!redirects unitary linear operators]]

[[!redirects unitary operator]]
[[!redirects unitary operators]]

[[!redirects unitary transformation]]
[[!redirects unitary transformations]]

[[!redirects unitary map]]
[[!redirects unitary maps]]

[[!redirects unitary linear map]]
[[!redirects unitary linear maps]]


unitary quantum channel

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## Idea

In [[quantum probability theory]]/[[open quantum systems]],
a *unitary quantum channel* is a [[quantum channel]] whose restriction to [[pure states]] acts by a [[unitary transformation]] just as a loss-less [[quantum gate]] does.

Concretely, in terms of [Kraus decomposition](quantum+channel#InTermsOfKrausDecompositions), a quantum channel

$$
  \array{
    \mathscr{H}_1
    \otimes
    \mathscr{H}_1^\ast
    &
    \longrightarrow
    &
    \mathscr{H}_2
    \otimes
    \mathscr{H}_2^\ast
    \\
    \rho
    &\mapsto&
    ch(\rho)
  }
$$

is unitary iff there exists a [[unitary operator]] $U \,\colon\, \mathscr{H}_1 \longrightarrow \mathscr{H}_2$ such that $ch$ is given by [[conjugation]] with this operator:

$$
  ch(\rho) \;=\; U \cdot \rho \cdot U^\dagger
  \,.
$$

## Properties

### General

\begin{proposition}
\label{UnitaryChannelsAreTheReversibleChannels}
**(unitary channels are the reversible channels)**
\linebreak
  Unitary channels are clearly [[invertible map|invertible]]. In fact, every invertible [[quantum channel]] must be a unitary quantum channel.
\end{proposition}
(eg. [Preskill](#PreskillRev04) [pp. 13](http://theory.caltech.edu/~preskill/ph219/chap3_15.pdf#page=13))

[[!include quantum channels and decoherence -- section]]

## Related concepts

* [[quantum channel]]

  * [[quantum measurement channel]]

  * [[partial trace quantum channel]]

## References

> (For more see the references at *[[quantum channel]]*.)

* [[Teiko Heinosaari]], [[Mário Ziman]], Exp. 4.6 in: *The Mathematical Language of Quantum Theory -- From Uncertainty to Entanglement*, Cambridge University Press  (2011) [[doi:10.1017/CBO9781139031103]( https://doi.org/10.1017/CBO9781139031103)]

* {#PreskillRev04} [[John Preskill]], *Reversibility*, Section 3.2.2 [[pdf](http://theory.caltech.edu/~preskill/ph219/chap3_15.pdf#page=13)] in: *Quantum Computation*, lecture notes, since 2004 [[web](http://theory.caltech.edu/~preskill/ph229/)]

* {#Attal} [[Stéphane Attal]], *Quantum Channels*, Lecture 6 in:  *Lectures on Quantum Noises* [[pdf](http://math.univ-lyon1.fr/~attal/Quantum_Channels.pdf), [[Attal-QuantumChannels.pdf:file]], [webpage](http://math.univ-lyon1.fr/~attal/chapters.html)]

[[!redirects unitary quantum channels]]


unitary representation

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A __unitary representation__ of a [[locally compact topological group]] $G$ on a [[Hilbert space]] $H$ is a representation of $G$ as an abstract group on $H$ by [[unitary linear operators]], i.e. the homomorphism of groups $\rho : G\to U(H)\subset End H$ which is continuous with respect to the topology on $U(H)$ [[induced topology|induced]] by the [[strong operator topology]] on $End H$. In other words, $\rho: G\to U(H)$ is a map satisfying $\rho(g h) = \rho(g)\rho(h)$, $\rho(g)^{-1} = \rho(g^{-1}) = \rho(g)^*$ for all $g,h\in G$ and for all $x\in H$, the function $g\mapsto \rho(g)(x)$ on $G$ is norm continuous.

## Examples

* [[unitary representation of the Poincaré group]]

## Related concepts

* [[super-unitary representation]]

## References

* [[Gerald B. Folland]], *A course in abstract harmonic analysis*, Studies in Advanced Mathematics. CRC Press (1995) [[pdf](https://sv.20file.org/up1/1415_0.pdf), [gBooks](http://books.google.com/books?hl=en&lr=&id=0VwYZI1DypUC)]

* Jeffrey Adams, Marc van Leeuwen, Peter Trapa, David A. Vogan Jr, _Unitary representations of real reductive groups_ ([arXiv:1212.2192](http://arxiv.org/abs/1212.2192))

[[!redirects unitary representations]]


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## Idea

The [[Poincaré group]] is the group of rigid [[spacetime]] symmetries of [[Minkowski spacetime]]. It is a [[topological group]] and as such has [[unitary representation]]s on infinite-dimensional [[Hilbert space]]s. For any [[quantum field theory]] in [[Minkowski space]] its [[space of states]] therefore decomposes into [[irreducible representation]]s of the [[Poincaré group]]. As was first observed by [[Hermann Weyl]], these irreducible representations encode the [[particle]] spectrum of the QFT.

We are interested in the **[[unitary representations]]** of a [[topological group]] $G$, i.e., the continuous [[group homomorphisms]]
$$U\colon G \to U(H)$$
into the [[unitary group]] of a [[Hilbert space]] $H$, especially those that are [[irreducible representation|irreducible]] in the usual sense of [[representation theory]]. The topology on $U(H)$ here is understood to be the [[operator topology|strong operator topology]].

In this and related articles, we study such representations in the case where

$$G = SL_2(\mathbb{C}) \ltimes \mathbb{R}^4$$

is the [[universal cover]] of the [[connected]] component of the identity of the [[Poincaré group]], which is important in the study of [[quantum field theory]]. A more physical name for such a representation is "[[elementary particle]]", and we will often use that term in this article. (NB: "elementary particle" will always refer to the formal mathematical representation it refers to.)

A full rounded account could become large; see the [blog discussion](http://golem.ph.utexas.edu/category/2009/03/unitary_representations_of_the.html), which despite its size was left in a still-nascent state. However, in a nutshell, the basic theorem is that (elementary) particles are classified up to isomorphism according to their [[mass]] and [[helicity]]; mass is a continuous parameter and helicity is a discrete parameter.

This theorem in commonly ascribed to [[Eugene Wigner]] and often referred to as the _[[Wigner classification]]_.

Wigner classified all irreducible unitary representations of the restricted Poincare group, including the unphysical ones. The latter cannot be used to define a [[free quantum field theory]] satisfying the [[Wightman axioms]]. Those that can are the physical ones and are characterized by a nonnegative real mass and a nonnegative half-integral spin; the zero component of the momentum has a nonnegative spectrum.
Many of these are realized by particles occurring in Nature, though not as 'elementary particles' but as bound states (in a suitable approximation, e.g., [[QCD]]). From the point of view of representation theory, the center of mass of a bound state behaves just like an elementary particle. Thus elementary is meant in this generalized sense.

## Related entries

Relevant topics in a full account will include

* [[spectral theorem|Spectral theorem]]

* [[rigged Hilbert space|Rigged Hilbert spaces]] and [[direct integral|direct integrals]]

* [[induced representations]]

* [[helicity]], [[spin]]

* [[Klein-Gordon equation]]

* [[Dirac equation]]

* [[spin-statistics theorem]]

* [[Maxwell equation]]

* [[Stone's theorem]]

* [[Mackey theory]]

## Tentative notes, to be expanded on...

In the first place, physicists tend to be a little carefree with the mathematics, so this account is written from the point of view of a 'stupid' mathematician (for the moment Todd Trimble) who wants to get details straight and precise.

For example, physicists tend to talk about "eigenstates" as if they were elements of the Hilbert space, and other states as linear combinations of eigenstates, whereas really we are dealing with some more complicated technology like rigged Hilbert spaces or direct integrals instead of direct sums. Failure to mention such details places hurdles of communication between physicists and mathematicians. In addition, there are stylistic differences in presentation, where a physicist will happily deal with formulas replete with lots of subscripts and superscripts, whereas many mathematicians prefer dealing with more conceptual, less notation-heavy explanations.

There seem to be at least three ways of dealing with spectral theory of (unbounded) self-adjoint operators on a Hilbert space:

* The usual Stone theory

* Direct integrals of Hilbert spaces

* Rigged Hilbert spaces

### Rigged Hilbert spaces

A rigged Hilbert space

## Induced representations

Simultaneous diagonalization.

Fact that Poincaré group is a [[semidirect product group]].

Let $\vec p \in \mathbb{R}^{d-1,1}$ be a given vector in [[Minkowski spacetime]]. Write

$$
  Stab_{Iso(\mathbb{R}^{d-1,1}))}(\vec p)
    \hookrightarrow
  Iso(\mathbb{R}^{d-1,1})
$$

for its [[stabilizer subgroup]] (often called _the little group_ in this context, going back to Wigner). Every unitary irrep of $Iso(\mathbb{R}^{d-1,1})$ of mass $p$ is an [[induced representation]] of a finite dimensional representation of the "little group" $Stab_{Iso(\mathbb{R}^{d-1,1}))}(\vec p)$. (recalled concisely e.g. in [Dragon 16, p. 2](#Dragon16)).

## Related concepts

* [[Wigner classification]]

* [[twistor]]

* [[unitary representation of the super Poincaré group]]

* [[Doplicher-Roberts reconstruction theorem]]

* [[finite-dimensional representation of the Poincaré group]]

* unitary representations of the [[anti de Sitter group]],

  [[singleton representation]]

## References

The observation that the irreps of the Poincaré group correspond to [[fundamental particles]] ([[Wigner classification]]) is due to

* [[Eugene Wigner]], _On unitary representations of the inhomogeneous Lorentz group_ , Ann. Math. **40:1** (1939), 149–204.

Review includes

* [[Rudolf Haag]], section I.3.1  of _[[Local Quantum Physics -- Fields, Particles, Algebras]]_

* {#Schroer96} [[Bert Schroer]], _Wigner Representation Theory of the Poincaré
Group, Localization, Statistics and the S-Matrix_, 1996  ([pdf](http://cds.cern.ch/record/308785/files/9608092.pdf))

* Eberhard Freitag, _Unitary representations of the Poincaré group_ ([pdf](http://www.rzuser.uni-heidelberg.de/~t91/skripten/unitdarst.pdf))

* [[Xavier Bekaert]], Nicolas Boulanger, _The unitary representations of the Poincare group in any spacetime dimension_ ([arXiv:hep-th/0611263](http://arxiv.org/abs/hep-th/0611263))

* {#Dragon16} [[Norbert Dragon]], _Currents for Arbitrary Helicity_ ([arXiv:1601.07825](http://arxiv.org/abs/1601.07825))

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[[!redirects irreducible representation of the Poincare group]]
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+-- {: .rightHandSide}
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### Context
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[super-unitary representation]] of the [[super Poincaré group]] ([[supersymmetry]])

The analog in [[superalgebra]]/[[supergeometry]] of a [[unitary representation of the Poincaré group]].

An [[irreducible representation]] of the [[super Poincaré Lie algebra]] is called  a **[[super multiplet]]**. This is in general itself a [[super vector space]] which contains an ordinary irreducible [[unitary representation of the Poincaré group]]. Via the [[Wigner classification]] of fundamental [[particles]] with Poincaré irreps, this may be identified with a [[boson|bosonic]] [[relativistic particle]] of some [[mass]] -- together with the images of these "bosonic" elements under the odd generators: the **superpartners** of the bosonic particles.

## Related concepts

* [[Deligne's theorem on tensor categories]]

* [[spin representation]]

* [[BPS state]]

* [[supersymmetry]]

* [[Wigner classification]]

  * [[unitary representation of the Poincaré group]]

* [[Doplicher-Roberts reconstruction theorem]]

* [[quantum field theory]]

## References

Reviews and lecture notes include

* {#CastellaniDAuriaFre91} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], section II.4.1 in _[[Supergravity and Superstrings - A Geometric Perspective]]_, World Scientific (1991)

* [[Veeravalli Varadarajan]], _Unitary representations of super Lie groups_ ([pdf](http://www.math.ucla.edu/~vsv/urtr.pdf))

* {#Freed01} [[Daniel Freed]], lecture 6 of _Classical field theory and Supersymmetry_, IAS/Park City Mathematics Series Volume 11 (2001) ([pdf](https://www.ma.utexas.edu/users/dafr/pcmi.pdf))

* [[Daniel Freed]], _Lecture 3 of [[Five lectures on supersymmetry]]_

* [[Antoine Van Proeyen]], _Tools for supersymmetry_ ([arXiv:hep-th/9910030](http://arxiv.org/abs/hep-th/9910030))

The classification is due to

* [[Werner Nahm]], _Supersymmetries and their representations_, Nucl. Phys. B135 (1978) 149

The analogous discussion generalized to [[parasupersymmetry]] is in

* A. Nikitin, V. Tretynyk, _Irreducible representations of the Poincaré parasuperalgebra_ ([pdf](http://www.imath.kiev.ua/~nikitin/publ80.pdf))

Expositional slides:

* Valerie Domcke, _Supermultiplets_ ([pdf](http://www.aphn.de/dateien/supermultiplets.pdf)) (in German)

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[[!redirects unitary representation of the super Poincare group]]
[[!redirects unitary representations of the super Poincare group]]

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unitization of a C-star-algebra

+-- {: .rightHandSide}
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### Context
#### Operator algebra
+-- {: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### Noncommutative geometry
+-- {: .hide}
[[!include noncommutative geometry - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

The generalization of the process of [[one-point compactification]] of possibly non-[[compact topological space|compact]] [[topological spaces]] from [[topology]] to [[non-commutative topology]] is the process of adding units to possibly [[non-unital ring|non-unital]] [[C*-algebras]], thought of as formal duals of non-commutative spaces in [[noncommutative topology]].

## Definition

+-- {: .num_defn}
###### Definition

For $ (A, {\Vert \cdot \Vert_{A}}) $ a non-[[unital algebra|unital]] [[C*-algebra]], its **[[unitisation]]** is the $ C^{\ast} $-algebra whose underlying vector space is the [[direct sum]]

$$
A^{+} \coloneqq A \oplus \mathbb{C}
$$

of $ A $ with the field of [[complex numbers]], equipped with the multiplication law

$$
          (a_{1} \oplus z_{1}) \cdot (a_{2} \oplus z_{2})
\coloneqq (a_{1} a_{2} + z_{2} a_{1} + z_{1} a_{2}) \oplus z_{1} z_{2},
$$

and the [[involution]]

$$
(a \oplus z)^\ast \coloneqq a^{\ast} \oplus \overline{z}
$$

([[complex conjugation]] is taking place on the right). This really is a $C^\ast$-algebra. The [[norm]] can be characterised as

$$
          {\Vert a \oplus z \Vert}_{A^{+}}
= {\Vert L_{a} + z \cdot \operatorname{Id}_{A} \Vert}_{\mathcal{B}(A)}
\coloneqq \sup_{b \in A, \Vert b \Vert_{A} \leq 1} \Vert a b + z b \Vert_{A}.
$$

if $A$ does not have a multiplicative unit. (Note that the operatornorm vanishes on $1_A\oplus -1_{\mathbb{C}}$ if $A$ does have a multiplicative unit). For $C^\ast$-algebras with a multiplicative unit one can show that the norm equals
$$\Vert a\oplus z \Vert_{A^+} = \max\{\|a+z1_A\|,|z|\}$$

=--

## Examples

+-- {: .num_example #UnitisationGivesOnePointCompactification}
###### Example

For $X$ a [[locally compact topological space|locally compact]] [[Hausdorff topological space]] and $C(X)$ its possibly non-unital [[C*-algebra]] [[algebra of functions|of functions]], then

$$
  (C_0(X))^+ \simeq C_0(X) \oplus \mathbb{C} \simeq C(X^+)
$$

this is equivalently the unital $C^\ast$-algebra of functions on the [[one-point compactification]] of $X$.

There is hence a canonical projection

$$
  i^\ast \colon C(X^+) \to \mathbb{C}
$$

The [[topological K-theory]] of $X$ is the [[kernel]] of the induced map in [[operator K-theory]]

$$
  K(X) \simeq ker(i^\ast) \simeq K(C_0(X))
  \,.
$$

=--

## Properties

### The point at infinity

+-- {: .num_remark #DualPointInclusion}
###### Remark

The unitisation of $A$ comes with a canonical [[projection]] [[homomorphism]] of [[C*-algebras]]

$$
  i^\ast_A \colon A^+ \to \mathbb{C}
$$

given by

$$
 (a + z) \mapsto z
  \,.
$$

Dually this corresponds to the inclusion of the "point at infinity".

=--

### K-theory with compact support on non-unital $C^\ast$-algebras

The map $i^\ast_A \colon A^+ \to \mathbb{C}$ of remark \ref{DualPointInclusion} induces a [[homomorphism]] of [[operator K-theory]] [[groups]] of the form

$$
  K(i^\ast_A) \colon K(A^+) \to K(\mathbb{C}) \simeq \mathbb{Z}
  \,.
$$

+-- {: .num_defn}
###### Definition

The [[kernel]] of this map is the operator K-theory of the original possibly non-unital $C^\ast$-algebra $A$:

$$
  K(A) \coloneqq ker(K(i^\ast_A))
 \,.
$$

=--

Heuristically it is clear that this is the "compactly suppported" K-theory of the possibly non-compact non-commutative space given by the algebra $A$. This statement has been made precise for instance in ([Emerson 07](#Emerson07), theorem 3.8)):

+-- {: .num_prop}
###### Proposition

For $X$ a [[locally compact topological space|locally compact]] [[Hausdorff topological space]],  there is a [[natural isomorphism]]

$$
  K(X) \simeq [X,Fred]_{cs}
  \,,
$$

where on the right we [[homotopy classes]] of maps of [[compact support]] into the [[classifying space]] for K-theory (space of Fredholm operators).

=--

## References

* [[Heath Emerson]], _Equivariant representable K-theory_ ([arXiv:0710.1410](http://arxiv.org/abs/0710.1410))
 {#Emerson07}

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unitoid

A __unitoid__ is a [[magma]] with an [[identity element]].

The term was used by [[Postnikov]] in his 1980s course on [[homotopy theory]]; an [[H-space]] is a homotopy unitoid.  But the terminology is not standard, and 'unitoid' has been applied to other concepts.


unitor

+-- {: .rightHandSide}
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###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--

* [[associator]], **unitor**, [[Jacobiator]]

* [[pentagonator]], [[hexagonator]]

***

# Contents
* table of contents
{:toc}

## Idea

A _unitor_ in [[category theory]] and [[higher category theory]] is an [[isomorphism]] that relaxes the ordinary [[uniticity]] _equality_ of a binary operation.

### In bicategories

In a [[bicategory]] the [[composition]] of [[1-morphism]]s does not satisfy [[uniticity]] as an equation, but there are natural _unitor_ 2-morphisms

$$
  Id \circ f \stackrel{\simeq}{\Rightarrow} f
$$

$$
  f \circ Id \stackrel{\simeq}{\Rightarrow} f
$$

that satisfy a [[coherence law]] among themselves.

### In monoidal categories

By the [[periodic table of higher categories]] a [[monoidal category]] is a pointed [[bicategory]] with a single object, its [[object]]s are the 1-morphisms of the bicategory.

Accordingly, a [[monoidal category]] with [[tensor product]] $\otimes$ and [[tensor unit]] "$1$" is equipped with a [[natural isomorphism]] of the form

$$ \ell_x : 1 \otimes x \to x \,, $$

called the **left unitor**, and a natural isomorphism

$$ r_x : x \otimes 1 \to x \,, $$

called the **right unitor**.

## Related concepts

* [[unit]]

* [[unitality]]

* [[tensor unit]]

[[!redirects unitor]]
[[!redirects unitors]]
[[!redirects left unitor]]
[[!redirects left unitors]]
[[!redirects right unitor]]
[[!redirects right unitors]]


This page collects links related to the article

* [[William Lawvere]]:

  \linebreak

  **Unity and Identity of Opposites in Calculus and Physics**

  \linebreak

  Proceedings of ECCT 1994 Tours Conference,

  Applied Categorical Structures **4** 167-174 (1996)

  Kluwer Academic Publishers

  [doi:10.1007/BF00122250](https://doi.org/10.1007/BF00122250)

  [github](https://github.com/mattearnshaw/lawvere/blob/master/pdfs/1996-unity-and-identity-of-opposites-in-calculus-and-physics.pdf)

on the formalization of [[unity of opposites]] in [[calculus]] and [[physics]].

{#WhatItIsAbout} Section II re-casts the statement of the *[[Hadamard lemma]]* via the three homomorphisms of [[rings]] of [[smooth functions]] on a [[smooth manifold]] $X$ and its [[product manifold]] $X \times X$ (the article generalizes this to [[fiber products]] but that's not necessary to appreciate the idea):

\begin{tikzcd}[column sep=20pt]
  C^\infty(X \times X)
  \ar[
    from=rr,
    shift left=14pt,
    "{
      \mathrm{pr}_1^\ast
    }"{description}
  ]
  \ar[
    rr,
    "{
      \mathrm{diag}^\ast
    }"{description}
  ]
  \ar[
    from=rr,
    shift right=14pt,
    "{
      \mathrm{pr}_2^\ast
    }"{description}
  ]
  &&
  C^\infty(X)
  \mathrlap{\,,}
\end{tikzcd}

where $pr_i \,\colon\, X \times X \xrightarrow{\;} X$ are the two [[projections]] out of the [[Cartesian product]] and $diag \,\colon\, X \xrightarrow{\;} X \times X$ denotes the [[diagonal map]]. So:

* for $f \,\in\, C^\infty(X)$ we have

  $$
    pr_1^\ast(f) \,\colon\, (x_1, x_2) \,\mapsto\, f(x_1)
    ,\,
    \;\;\;\;
    pr_2^\ast(f) \,\colon\, (x_1, x_2) \,\mapsto\, f(x_2)
  $$

* for $g \,\in\, C^\infty(X \times X)$ we have

  $$
    diag^\ast(g) \,\colon\, x \,\mapsto\, g(x,x)
    \,.
  $$

These are the expressions that enter the formulation of the [[Hadamard lemma]] ([here](Hadamard+lemma#HadamardLemma)) and its corollary ([here](Hadamard+lemma#AndItsCorollary)).

Compare this to the discussion at *[[adjoint cylinder]]* and then to *[[objective and subjective logic]]* and maybe also *[[Aufhebung]]*.

### Abstract

> A significant fraction of dialectical philosophy can be modeled mathematically through the use of "[[adjoint cylinder|cylinders]]" (diagrams of shape $\Delta_1$) in a category, wherein the two identical [[subobjects]] (united by the third map in the diagram) are "opposite". In a [[bicategory]], oppositeness can be very effectively characterized in terms of [[adjunction|adjointness]], but even in an ordinary category it may sometimes be given a useful definition. For example, an effective basis for teaching calculus is a ringed category satisfying the Hadamard-Marx property. The description in engineering mechanics of continuous bodies that can undergo cracking is clarified by an example involving lattices, raising a new questions about the foundations of topology.

> ($\Delta_1$ is the category of order-preserving maps between totally-ordered sets of one and two elements, respectively.)

category: reference


univalence axiom

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### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Equality and Equivalence
+--{: .hide}
[[!include equality and equivalence - contents]]
=--
#### Universes
+-- {: .hide}
[[!include universe - contents]]
=--
#### Homotopy theory
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

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 equivalence]]s 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.

Extensionality principles like [[function extensionality]], [[propositional extensionality]] (where $X$ and $Y$ are [[h-propositions]], and univalence ("typal extensionality") are naturally regarded as a stronger form of _[[identity of indiscernibles]]_. In particular, the [[consistency]] of univalence means that in [[Martin-Löf type theory]] without univalence, one cannot define any [[predicate]] that provably distinguishes isomorphic [[types]]; thus isomorphic types are "externally indiscernible", and univalence incarnates that principle internally by making them identical.

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).  In the language of [[(∞,1)-category]] theory, a univalent bundle is an [[object classifier]].

The univalence axiom does not *literally* say that anything is univalent in this sense.  However, it is *equivalent* to saying that the canonical fibration over $Type$ is univalent: 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 $Type_1$).  For a description of this equivalence, see section 4.8 of the [[HoTT Book]] (syntactically) and [Gepner-Kock](#GepnerKock12) (semantically).

Univalence is a commonly assumed [[axiom]] in [[homotopy type theory]], and is central to the proposal ([Voevodsky](#Voevodsky)) that this provides a natively [[homotopy theory|homotopy theoretic]] [[foundation]] of [[mathematics]] (see at _[[univalent foundations for mathematics]]_).

## Definition

We work in a [[dependent type theory]] with [[identity types]], [[function types]], [[dependent product types]], [[product types]], and [[dependent sum types]].

There are multiple notions of [[equivalence types]] in [[dependent type theory]], which can be used for a definition of univalence for a [[type universe]] $U$; these include

* [[judgmentally strict equivalence types]]
* [[propositionally strict equivalence types]]
* various notions of [[weak equivalence types]]
  * the type of [[functions]] with [[contractible type|contractible]] [[fiber type|fibers]]
  * the type of [[spans]] with contractible fibers
  * the type of [[multivalued partial functions]] which are single-valued and [[total function|total]] and have contractible fibers
  * the type of [[one-to-one correspondences]]
* type of $U$-small equivalences, given a type universe $U$ and a definition of equivalence above

Let us assume an arbitrary notion of equivalence type $\simeq_0$. Every [[Russell universe]] $U$ is a [[reflexive graph]] with the graph type family $R(A, B)$ defined as $R(A, B) \coloneqq A \simeq_0 B$ and the function $\mathrm{idtofam}(A, B)$ is defined as

$$\mathrm{idtofam}(A, B) \coloneqq \mathrm{idtoequiv}(A, B)$$

Similarly, every [[Tarski universe]] $U$ with universal type family $T$ is a [[reflexive graph]] with the graph type family $R(A, B)$ defined as $R(A, B) \coloneqq T(A) \simeq_0 T(B)$ and the function $\mathrm{idtofam}(A, B)$ is defined as

$$\mathrm{idtofam}(A, B) \coloneqq \mathrm{transport}^T(A, B)$$

And finally, every [[Tarski universe]] $U$ with type of terms $T$ and function $\mathrm{typeof}:T \to U$ is a reflexive graph with the graph type family $R(A, B)$ defined as

$$R(A, B) \coloneqq \left(\sum_{t:T} \mathrm{typeOf}(t) =_U A\right) \simeq \left(\sum_{t:T} \mathrm{typeOf}(t) =_U B\right)$$

and the function $\mathrm{idtofam}(A, B)$ is defined as

$$\mathrm{idtofam}(A, B) \coloneqq \mathrm{transport}^{\sum_{t:T} \mathrm{typeOf}(t) =_U (-)}(A, B)$$

Now, let us assume an arbitrary notion of equivalence type $\simeq$. A Russell or Tarski universe is **univalent** if it is [[univalent reflexive graph|univalent]] as a [[reflexive graph]], or equivalently, if one of the following equivalent conditions by the [[fundamental theorem of identity types]] hold:

1. That for each $x:A$ the type of elements $y:A$ such that $R(x, y)$ is a [[contractible type]].
$$x:A \vdash \mathrm{ua}(x):\mathrm{isContr}\left(\sum_{y:A} R(x, y)\right)$$

1. That there is a family of equivalences
$$x:A, y:A \vdash \mathrm{ua}(x, y):(x =_A y) \simeq R(x, y)$$

1. That $R(x, y)$ is an [[identity system]].

1. That for each $x:A$ and $y:A$, the function $\mathrm{idtofam}(x, y)$ is an [[equivalence of types]]
$$x:A, y:A, \vdash \mathrm{ua}(x, y):\mathrm{isEquiv}(\mathrm{idtofam}(x, y))$$

1. That $\mathrm{idtofam}(x, y)$ is a [[retraction]] (This is due to [[Daniel Licata]] in [Licata 16](#Licata16))
$$x:A, y:A \vdash \mathrm{ua}(x, y):R(x, y) \to (x =_A y)$$
$$x:A, y:A, r:R(x, y) \vdash G(x, y):\mathrm{idtofam}(x, y, \mathrm{ua}(x, y, r)) =_{R(x, y)} r$$

1. That $R(x, y)$ with the function $\mathrm{idtofam}(x, y)$ satisfies the [[universal property]] of the [[unary sum]] of $x =_A y$.

See [[fundamental theorem of identity types]] for proofs that these definitions are the same.

Traditional [[homotopy type theory]] uses the type of functions with contractible fibers for both $\simeq$ and $R$, while [[Mike Shulman]]'s model of [[higher observational type theory]] uses the type of $U$-small one-to-one correspondences for $R$ and the type of judgmentally strict equivalences for $\simeq$.

### Stricter versions of univalence

There are stricter versions of univalence, where we replace the [[equivalence of types]] between the identity type $A =_U B$ and the type of equivalences of the universe $A \simeq B$ in the univalence axioms with some notion of [[equality]], such as [[judgmental equality]], [[propositional equality]], and [[typal equality]].

1. In any [[dependent type theory]] with [[judgmental equality]], given a [[type universe]] $U$, one could replace the equivalence of types in the definition of univalence with a [[judgmental equality]] of types. This results in **judgmental univalence**, which states that for all small types $A:U$ and $B:U$, one could judge that $(A =_U B) \equiv (A \simeq B) \; \mathrm{type}$.

2. Similarly, in the context of any [[logic over type theory]] with [[propositional equality]], given a [[type universe]] $U$, one could replace the equivalence of types in the definition of univalence with a [[propositional equality]] of types. This results in **propositional univalence**, which states that for all small types $A:U$ and $B:U$, $(A =_U B) \equiv (A \simeq B) \; \mathrm{true}$.

3. Finally, if we are working inside a [[Tarski universe]] $(\mathcal{V}, \mathcal{T})$, then given an internal [[Tarski universe]] $U:\mathcal{V}$ with $T:\mathcal{T}(U) \to \mathcal{V}$, one could replace the equivalence of types in the definition of univalence with a [[typal equality]] of types. This results in **typal univalence**, which states that for all small types $A:\mathcal{T}(U)$ and $B:\mathcal{T}(U)$, there is an [[identification]] $\mathrm{ua}(A, B):(T(A) =_U T(B)) =_{\mathcal{V}} (T(A) \simeq_{\mathcal{V}} T(B))$.

Each of these imply the usual versions of univalence either through the structural rules for [[judgmental equality]] and [[propositional equality]], or through [[identification elimination]], [[transport]], and [[action on identifications]] for [[typal equality]].

### Rules for judgmentally univalent universes

With judgmentally univalent Tarski universes $(U, T)$, identities $p:A =_U B$ are [[equivalences of types]] $A$ and $B$. Namely, given [[function types]] and the [[isEquiv]] type family, one could add rules to the type theory which says that $A =_U B$ behaves as an [[equivalence type]]:

Introduction rules:
$$\frac{\Gamma \vdash A:U \quad \Gamma \vdash B:U \quad \Gamma, x:T[A/X] \vdash f:T[B/X] \quad \Gamma \vdash y:\mathrm{isEquiv}(f)}{\Gamma \vdash \mathrm{equiv}(f, y):A =_U B}$$

Elimination rules:
$$\frac{\Gamma \vdash A:U \quad \Gamma \vdash B:U \quad \Gamma, x:T[A/X] \vdash f:T[B/X] \quad \Gamma, z:A =_U B \vdash C \; \mathrm{type} \quad \Gamma, x:T[A/X], f:T[B/X], y:\mathrm{isEquiv}(f) \vdash c:C[\mathrm{equiv}(f, y)/z]}{\Gamma, z:A =_U B \vdash \mathrm{ind}_{A =_U B}^C(c):C}$$

Computation rules:
$$\frac{\Gamma \vdash A:U \quad \Gamma \vdash B:U \quad \Gamma, x:T[A/X] \vdash f:T[B/X] \quad \Gamma, z:A =_U B \vdash C \; \mathrm{type} \quad \Gamma, x:T[A/X], f:T[B/X], y:\mathrm{isEquiv}(f) \vdash c:C[\mathrm{equiv}(f, y)/z]}{\Gamma, x:T[A/X], f:T[B/X], y:\mathrm{isEquiv}(f) \vdash \beta_{A =_U B}^C(c):\mathrm{ind}_{A =_U B}^C(c)[\mathrm{equiv}(f, y)/z] =_{C[\mathrm{equiv}(f, y)/z]} c}$$

Uniqueness rules:
$$\frac{\Gamma \vdash A:U \quad \Gamma \vdash B:U \quad \Gamma, x:T[A/X] \vdash f:T[B/X] \quad \Gamma, z:A =_U B \vdash C \; \mathrm{type} \quad \Gamma, x:T[A/X], f:T[B/X], y:\mathrm{isEquiv}(f) \vdash c:C[\mathrm{equiv}(f, y)/z] \quad \Gamma, z:A =_U B \vdash u:C \quad \Gamma, x:T[A/X], f:T[B/X], y:\mathrm{isEquiv}(f) \vdash i_\mathrm{in}(u):u[\mathrm{equiv}(f, y)/z] =_{C[\mathrm{in}(x, y)/z]} c}{\Gamma, e:A =_U B \vdash \eta_{A =_U B}^C(c):u[e/z] =_{C[e/z]} \mathrm{ind}_{A =_U B}^C(c)[e/z]}$$

### Weaker versions of univalence

In the context of [[function extensionality]], Ian Orton and Andrew Pitts showed [here](http://types2017.elte.hu/proc.pdf#page=93) that the univalence axiom can be simplified to the following special cases:

   * $unit : A = \sum_{a:A} 1$
   * $flip : (\sum_{a:A} \sum_{b:B} C(a,b)) = (\sum_{b:B} \sum_{a:A} C(a,b))$
   * $contract: IsContr(A) \to (A=1)$
   * $unit_\beta : coe(unit(a)) = (a,\star)$
   * $flip_\beta : coe(flip(a,b,c)) = (b,a,c)$.

   The proof constructs $ua(f): A=B$ (for $f:A\simeq B$) as the composite
   $$ A \overset{unit}{=} \sum_{a:A} 1 \overset{contract}{=} \sum_{a:A} \sum_{b:B} f a=b \overset{flip}{=} \sum_{b:B} \sum_{a:A} f a = b \overset{contract}{=} \sum_{b:B} 1 \overset{unit}{=} B $$
   and uses $unit_\beta$ and $flip_\beta$ to compute that $coe(ua(f))(a) = f(a)$, hence by function extensionality $coe(ua(f)) = f$.

This is weaker than univalence because in the absense of [[function extensionality]], it is no longer possible to prove univalence from this set of axioms.

In addition, unlike the definitions above, this definition of univalence cannot be generalized to [[reflexive graphs]].

### Internal univalence

Suppose the [[Tarski universe]] $U$ has

* internal identity types
$$\mathrm{Id}^U:\prod_{A:U} T(A) \times T(A) \to U$$
* internal dependent product types
$$\Pi_U (x:(-)).(-)(x):\left(U \times \prod_{A:U} T(A) \to U\right) \to U$$
* internal dependent sum types
$$\Sigma_U (x:(-)).(-)(x):\left(U \times \prod_{A:U} T(A) \to U\right) \to U$$

We assume the weakest notion of Tarski universe, where the type reflection $T(A)$ of each $A:U$ is only equivalent to the external type, since (judgmentally, propositionally, typally) strict Tarski universes are weakly Tarski universes, because the [[judgmental equality]] and [[propositional equality]] imply equivalence of types by the structural rules for judgmental and propostional equality, and [[typal equality]] in a [[type universe]] of types imply equivalence of types by [[identification elimination]], [[transport]], and [[action on identifications]].

This allows us to define the internal type of equivalences $A \simeq_U^\mathrm{in} B$, internal to the universe $U$, which comes with a canonical equivalence of types
$$\mathrm{canonical}_\simeq(A, B):T(A \simeq_U^\mathrm{in} B) \simeq (T(A) \simeq T(B))$$
This implies that the equivalence $T(A) \simeq T(B)$ is $U$-small, and [[transport]] being an [[equivalence]] then implies that in any universe $U$ which is closed under [[function types]], [[dependent product types]], and [[dependent sum types]], for all $A:U$ and $B:U$, the [[identity type]] $A =_U B$ is $U$-small.

The internal univalence axiom states that the canonical function
$$\mathrm{idtointernalequiv}(A, B):(A =_U B) \to T(A \simeq_U^\mathrm{in} B)$$
is an equivalence of types
$$\mathrm{idtoequiv}(A, B):(A =_U B) \simeq T(A \simeq_U^\mathrm{in} B)$$

This is not definable for [[strongly predicative]] [[type universes]], since [[strongly predicative]] [[type universes]] are by definition not closed under [[dependent product types]].

In addition, the internal and external versions of univalence imply each other. In order to show that the two axioms imply each other, we need to show that there is an [[identification]]
$$i(p):\mathrm{canonical}_\simeq(A, B)(\mathrm{idtoequiv}(A, B)(p)) =_{T(A) \simeq T(B)}  \mathrm{trans}^{T}(A, B)(p)$$
for all identifications $p:A =_U B$. By the [[J rule]] it is enough to show that $\mathrm{canonical}_\simeq(A, A)$ maps the identity equivalence of $T(A \simeq_U^\mathrm{in} A)$ to the identity equivalence in $T(A) \simeq T(A)$. Since the identity equivalence in $T(A \simeq_U^\mathrm{in} A)$ is just the [[identity function]] $\mathrm{canonical}_\simeq^{-1}(A, A)(\lambda x.x)$ the above statement is always true. Thus, if the universe is closed under [[identity types]], [[dependent product types]], and [[dependent sum types]] the two univalence axioms imply each other and both define the same notion of univalent universe.

## In categorical semantics
 {#InCategoricalSemantics}

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]]

$$
  b : B \vdash E(b) : Type
$$

corresponds to a [[morphism]] $E \to B$ in $\mathcal{C}$ that is a [[fibration]] between fibrant objects.

Then the [[dependent type|dependent]] [[function type]]

$$
  b_1, b_2 : B \vdash ( E(b_1) \to E(b_2)) : Type
$$

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

$$
  [p_1^* E \, , \, p_2^* E]_{\mathcal{C}/_{B \times B}}
  \simeq
  [E \times B \, , \, B \times E]_{\mathcal{C}/_{B \times B}}
  \in
  \mathcal{C}/_{B \times B}
  \,.
$$

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
$$ q \colon \Delta_B \to [E \times B , B \times E]_{\mathcal{C}/_{B \times B}} \,.$$
in $\mathcal{C}/_{B \times B}$.  By the defining ([[product]] $\dashv$ [[internal hom]])-[[adjunction]], it suffices to define a morphism
$$ \Delta_B \times_{\mathcal{C}/_{B \times B}} E \times B \to B \times E $$
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
$$ \Delta_B \times_{\mathcal{C}/_{B \times B}} E \times B \to \Delta_B \times_{\mathcal{C}/_{B \times B}} B \times E\,. $$
And since the composite pullback along either composite
$$ B \xrightarrow{\Delta_B} B\times B \xrightarrow{\pi_1} B$$
$$ B \xrightarrow{\Delta_B} B\times B \xrightarrow{\pi_2} B$$
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 $\mathcal{C}$

$$
  \array{
     B &&\stackrel{\simeq}{\hookrightarrow}&& B^I
     \\
     & {}_{\mathllap{\Delta_B}}\searrow
     && \swarrow_{\mathrlap{}}
     \\
     && B \times B
  }
$$

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 $\mathcal{C}/_{B \times B}$

$$
  \array{
    B &\stackrel{q}{\to}& [E \times B, B \times E]_{\mathcal{C}/_{B \times B}}
   \\
   \downarrow &{}^{\mathllap{\hat q}}\nearrow& \downarrow
   \\
   B^I &\to& B \times B = *_{\mathcal{C}/_{B \times B}}
  }
  \,.
$$

This lift is the interpretation of the [[identity type|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

$$
  \hat q : B^I \to Eq(E)
  \,.
$$

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.

(...)

### In simplicial sets
 {#InSimplicialSets}

We specialize the [general discussion](#InCategoricalSemantics) above to the realization in $\mathcal{C} = $ [[sSet]], equipped with the
standard [[model structure on simplicial sets]].

For $E \to B$ any fibration ([[Kan fibration]])
between fibrant objects ([[Kan complexes]]),
consider first the simplicial set

$$
  [E \times B , B \times E]_{B \times B}
  \in
  sSet/_{B \times B}
$$

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 category|slice]] and using that [[products]] in $sSet/_{B \times B}$ are [[pullbacks]] in $sSet$ -- the horizontal morphisms of simplcial sets in

$$
  \array{
    * &&\to&& [E \times B, B \times E]_{B \times B}
    \\
    & {}_{\mathllap{(b_1,b_2)}}\searrow && \swarrow
    \\
    && B \times B
  }
$$

correspond bijectively to the horizontal morphisms in

$$
  \array{
    E_{b_1} \times \{b_2\} &&\to&& \{b_1\} \times E_{b_2}
    \\
    & \searrow && \swarrow
    \\
    && B \times B
  }
$$

in $sSet$, which are precisely morphisms $E_{b_1} \to E_{b_2}$.

Let then

$$
  Eq(E) \hookrightarrow [E \times B, B \times E]_{B \times B}
  \in sSet/_{B \times B}
$$

be the full sub-simplicial set on those vertices that correspond to
[[weak equivalences]]
(([[weak homotopy equivalence|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

$$
  B \to Eq(E)
  \,.
$$

The fibration $E \to B$ is univalent, precisely when this morphism is  a weak equivalence.

This appears originally as [Voevodsky, def. 3.4](#UnivalentFoundationsProject)

### In simplicial presheaves
  {#InSimplicialPresheaves}

(...)

See ([Shulman 12](#Shulman12), [UF 13](UF13))

(...)

## Properties

### Relation to function extensionality

The univalence axiom implies [[function extensionality]].

A commented version of a formal proof of this fact can be found in ([Bauer-Lumsdaine](#BauerLumsdaine)).

### Univalence and axiom K

In this section we assume that the universe is a Tarski universe. [[Axiom K]] states that for all $A:U$ the type $T(A)$ is a set. This means the type reflection of the internal equivalences $T(A \simeq_U B)$ is an [[h-set]], and the univalence axiom then implies that $U$ is an [[h-groupoid]].

It is frequently stated that the univalence axiom and axiom K are inconsistent with each other. However, this is only true if the Tarski universe $U$ has internal univalent Tarski universes $V:U$ where the type $T(V)$ has terms $A:T(V)$ such that $T_V(A)$ is an [[h-set]] which is not an [[h-proposition]], such as the [[booleans type]]. As a result, $T(V)$ can be proven to not be a set, causing axiom K for $U$ to be inconsistent with the existence of $V:U$ and univalence.

### Canonicity and homotopy canonicity
 {#Canonicity}

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]]), [[homotopy canonicity]] has been shown in ([Shulman 12, section 13](#Shulman12).  Thus, in univalent homotopy 1-type theory with two universes, every [[term]] of [[type]] of the [[natural numbers]] is [[propositional equality|propositionally equal]] to a [[numeral]].

The construction in ([Shulman 12, section 13](#Shulman12)) uses [[Artin gluing]] of a suitable [[type-theoretic model category|type-theoretic fibration category]] with the [[category]] [[Set]] and [[Grpd]], respectively, effectively inducing canonicity from these categories. By ([Shulman 12, remark 13.13](#Shulman12)) 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]]. A proof of the full result has been announced by [[Christian Sattler]] and [[Krzysztof Kapulkin]] ([Sattler 19](#Sattler19)).

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 mathematics|constructive]] version of the [[proof]], which would imply the existence of an algorithm) have not succeeded.

It is also a *[[propositional equality|propositional]]* canonicity, as opposed to the [[judgmental equality|judgmental]] canonicity which many traditional [[type theories]] enjoy.  Another approach to canonicity for 1-truncated univalence can be found in ([Harper-Licata](#HarperLicata)), 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.

Canonicity has been proved for [[cubical type theory]].

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 [[predicative mathematics|impredicative]] [[universe]] of [[h-sets]].

### A univalent universe inside a non-univalent universe

In a post to the [Homotopy Type Theory Google Group](https://groups.google.com/forum/#!forum/homotopytypetheory), Peter LeFanu Lumsdaine [wrote](https://groups.google.com/d/msg/homotopytypetheory/3wp32xV7OX0/2HfAnBt44XoJ):

> Let $(x_0:X)$ be any [[pointed type]], and $(\mathcal{U}, El)$ be a universe (with rules as I set out a couple of emails ago). Then $X \times \mathcal{U}$ is again a universe, admitting all the same constructors as $\mathcal{U}$: take

> $El(x,A) = El(A)$,
> $(x,A) +_\mathcal{U} (y,B) = (x_0, A +_\mathcal{U} B)$,

> and so on; that is, constructor operations on $(X \times \mathcal{U})$ are constantly $x_0$ on the first component, and mirror those of $\mathcal{U}$ on the second component.

> Now if $\mathcal{U}$ is univalent, and $X$ has non-trivial $\pi_0$ (e.g. $X=S^1$), then $\mathcal{U} \rightarrow (X \times \mathcal{U}$) gives a univalent universe sitting inside a non-univalent one (again, with the rules as I set out earlier).

> Slightly more generally, given any cumulative pair of universes $\mathcal{U}_0 \rightarrow \mathcal{U}_1$, we can consider $\mathcal{U}_0 \rightarrow A \times \mathcal{U}_1$; this shows we can additionally have the smaller universe represented by an element of the larger one.

[[Mike Shulman]] [added](https://groups.google.com/d/msg/homotopytypetheory/3wp32xV7OX0/kIapR0lo1UEJ):

> [N]ot only is $X \times \mathcal{U}$ not univalent, it's not even "univalent on the image of $\mathcal{U}$", as was the case for the example in the groupoid model that I mentioned.

## Related concepts

* [[univalent foundations]]

* [[univalent type theory]]

* [[extensionality]]

  * 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]]_.

  * [[propositional extensionality]]

* contrary to univalence is the [[axiom UIP]]

* [[directed univalence axiom]]

## References
 {#References}

The earliest occurrence of the univalence axiom is due -- under the name "universe extensionality" -- to:

* {#HofmannStreicher98} [[Martin Hofmann]], [[Thomas Streicher]], Section 5.4 of:  _The groupoid interpretation of type theory_, in: [[Giovanni Sambin]] et al. (eds.), *Twenty-five years of constructive type theory*, Proceedings of a congress, Venice, Italy, October 19-21, 1995. Oxford: Clarendon Press. Oxf. Logic Guides. **36** (1998) 83-111   [[ISBN:9780198501275](https://global.oup.com/academic/product/twenty-five-years-of-constructive-type-theory-9780198501275), [ps](http://www.mathematik.tu-darmstadt.de/~streicher/venedig.ps.gz), [[HofmannStreicherGroupoidInterpretation.pdf:file]]]

But these authors refer to a subtly incorrect type of [[equivalences in homotopy type theory]] (see [there](equivalence+in+type+theory#TheIssueWithQuasiInverses) for details).

For comments on the early history see also:

* {#Altenkirch21} [[Thorsten Altenkirch]], *Martin Hofmann’s contributions to type theory: Groupoids and univalence*, Mathematical Structures in Computer Science **31** 9 (2021) 953-957 [[doi:10.1017/S0960129520000316](https://doi.org/10.1017/S0960129520000316)]

It is this notion of [[equivalence in homotopy type theory]] which was fixed in

* {#Voevodsky10} [[Vladimir Voevodsky]], p. 8, 10-11 of: *Univalent Foundations Project* (2010) [[pdf](http://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/univalent_foundations_project.pdf), [[Voevodsky-UFP2010.pdf:file]]]

ever since the univalence axiom is widely attributed to Voevodsky. Earlier documentation of the univalence axiom in modern form is hard to come by:

The first technical understanding of (the semantics of) univalence in simplicial sets seems to be due to:

* {#Bousfield06} [[Aldridge Bousfield]], [email to VV from 01 May 2006 10:10:30 CDT](https://groups.google.com/g/homotopytypetheory/c/K_4bAZEDRvE/m/YSQz-jJ_AAAJ) $[$[[BousfieldOnUnivalence.jpg:file]]$]$

(which 6 years later came to be written up as [Kapulkin, Lumsdaine & Voevodsky12](#KapulkinLumsdaineVoevodsky12) and another 10 years later was published as [Kapulkin & Lumsdaine 2021](#KapulkinLumsdaine21)).

The first mentioning by Voevodsky of the term "univalence" by email is 3.5 years later, from [Dec. 30 2009](https://groups.google.com/g/homotopytypetheory/c/K_4bAZEDRvE/m/3N2xxxfSAAAJ) (according to [Grayson, Oct. 2017](https://groups.google.com/g/homotopytypetheory/c/K_4bAZEDRvE/m/-Lp5dxTTAAAJ)).

The earliest recorded statement of the univalence axiom by Voevodsky's hand date may be [Voevodsky (2010), p. 11](#Voevodsky10)

see also:

* [[Vladimir Voevodsky]], *The equivalence axiom and univalent models of type theory* (Talk at CMU on February 4, 2010) ([arXiv:1402.5556](https://arxiv.org/abs/1402.5556))

Later in:

* [[Vladimir Voevodsky]],  *[The Origins and Motivations of Univalent Foundations](https://www.ias.edu/ideas/2014/voevodsky-origins)*, IAS Institute Letter Summer 2014

appears the claim that:

> have been working on the ideas that led to the discovery of univalent models since 2005 and gave the first public presentation on this subject at Ludwig-Maximilians-Universität München in November 2009.

A comprehensive discussion finally appears in the textbook:

* [[UF-IAS-2012|Univalent Foundations Project]], p. 4 & Sec. 2.10 in: *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]* (2013)  [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)]

Voevodsky's ([Bousfield's](#Bousfield06)) original idea for the [[universal Kan fibration]] as a model for a univalent universe in [[simplicial sets]]/[[infinity-groupoids|$\infty$-groupoids]] was eventually published as:

* {#KapulkinLumsdaine21} [[Chris Kapulkin]], [[Peter LeFanu Lumsdaine]], *The Simplicial Model of Univalent Foundations (after Voevodsky)*, Journal of the European Mathematical Society **23** (2021) 2071–2126  $[$[arXiv:1211.2851](https://arxiv.org/abs/1211.2851), [doi:10.4171/jems/1050](https://doi.org/10.4171/jems/1050)$]$

Exposition and survey

* [[Peter Aczel]], *On Voevodsky’s Univalence Axiom*, talk at Third European Set Theory Conference (2011) [[pdf](http://www.cs.man.ac.uk/~petera/Recent-Slides/Edinburgh-2011-slides_pap.pdf), [[Aczel-Univalence.pdf:file]]]

  > (in view of the [[structure identity principle]])

* [[Mike Shulman]], _Homotopy type theory, IV_ ([blog post](http://golem.ph.utexas.edu/category/2011/04/homotopy_type_theory_iv.html))

Additional definition of univalent universes appeared in section 17.1 of

* {#Rijke22} [[Egbert Rijke]], *[[Introduction to Homotopy Type Theory]]*, Cambridge Studies in Advanced Mathematics, Cambridge University Press ([arXiv:2212.11082](https://arxiv.org/abs/2212.11082))

as well as in this Google Groups thread:

* {#Licata16} [[Dan Licata]], *weak univalence with "beta" implies full univalence* ([web](https://groups.google.com/forum/#!msg/homotopytypetheory/j2KBIvDw53s/YTDK4D0NFQAJ))

An accessible account of Voevodsky's proof (following [Bousfield 06](#Bousfield06)) that the universal [[Kan fibration]] in [[simplicial sets]] is univalent:

* {#KapulkinLumsdaineVoevodsky12} [[Chris Kapulkin]], [[Peter LeFanu Lumsdaine]], [[Vladimir Voevodsky]], _Univalence in simplicial sets_ [[arXiv:1203.2553](http://arxiv.org/abs/1203.2553)]

A quick elegant proof of the [[object classifier]]/universal [[associated infinity-bundle]] in simplicial sets/$\infty$-groupoids is in

* {#Moerdijk} [[Ieke Moerdijk]] (notes by Chris Kapulkin), _Fiber bundles and univalence_ ([pdf](http://www-home.math.uwo.ca/~kkapulki/notes/fiber_bundles_univalence.pdf))

See also

* [[Colin McLarty]], _A univalent universe in finite order arithmetic_ ([arXiv:1412.6714](http://arxiv.org/abs/1412.6714))

Implementation of univalence in [[proof assistants]]:

in [[Agda]]:

* [[Martín Escardó]], *[Voevodsky’s univalence axiom](https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/HoTT-UF-Agda.html#univalence)*, §3.11 in: *Introduction to Univalent Foundations of Mathematics with Agda* [[arXiv:1911.00580](https://arxiv.org/abs/1911.00580),  [webpage](https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/HoTT-UF-Agda.html)]

[[cubical Agda]]:

* [[1lab]], *[Univalence](https://1lab.dev/1Lab.Univalence.html)*

in [[Coq]]:

* _[HoTT/HoTT theories/Types/Universe.v](https://github.com/HoTT/HoTT/blob/master/theories/Types/Universe.v)_

* _[HoTT/HoTT theories/UnivalenceAxiom.v](https://github.com/HoTT/HoTT/blob/master/theories/UnivalenceAxiom.v)_

A guided walk through the formal proof that univalence implies [[functional extensionality]] is at

* {#BauerLumsdaine} [[Andrej Bauer]], [[Peter LeFanu Lumsdaine]], _[[Oberwolfach HoTT-Coq tutorial]]_

Some details regarding the univalence axiom for [[weakly Tarski universes]] appeared on MathOverflow in:

* Madeleine Birchfield, Valery Isaev, *Univalence for weakly Tarski universes*, MathOverflow, ([web](https://mathoverflow.net/q/431723))

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

* [[Michael Shulman]], _Univalence for inverse diagrams, oplax limits, and gluing, and homotopy canonicity_ ([arXiv:1203.3253](http://arxiv.org/abs/1203.3253))
 {#Shulman12}

This discusses [[canonicity]] of univalence in its section 13. Another approach to showing canonicity is (via [[cubical sets]]) in

* [[Thierry Coquand]], Simon Huber, _A model of type theory in cubical sets_, 2013  ([pdf](http://www.cse.chalmers.se/~coquand/mod1.pdf), [Haskell code](https://github.com/simhu/cubical), [discussion](https://groups.google.com/forum/#!topic/homotopytypetheory/GmXKEArD3HY))
  {#CoquandHuber13}

A proof of canonicity is presented in the talk

* {#Sattler19} [[Christian Sattler]], _Homotopy Canonicity_, ([abstract](http://www.ii.uib.no/~bezem/abstracts/TYPES_2019_paper_110))

On the issue of strict pullback of the univalent universe see

* Univalent Foundations Mailing List, _[Quotients](https://groups.google.com/d/msg/univalent-foundations/Glo7NgNvhJA/4j9SewiFvQ0J)_, March 2013
  {#UF13}

The computational interpretation of univalence / [[canonicity]] is discussed in

* [[Dan Licata]], [[Robert Harper]], _Computing with Univalence_  (2012) ([pdf](http://4wft.fmf.uni-lj.si/wp-content/uploads/2012/04/Licata.pdf))

* [[Robert Harper]], [[Daniel Licata]], _Canonicity for 2-dimensional type theory_ (2011) ([pdf](http://www.cs.cmu.edu/~rwh/papers/2dtt-can/paper.pdf))
 {#HarperLicata}

* [[Daniel Licata]] _The computational interpretation of HoTT (in 2D)_, talk at [[UF-IAS-2012]]  ([video](http://video.ias.edu/stream&ref=1674))

* Simon Huber (with [[Thierry Coquand]]), _Towards a computational justification of the Axiom of Univalence_ , talk at _TYPES 2011_ ([pdf](http://www.cse.chalmers.se/~simonhu/slides/types11.pdf))

* Bruno Barras, [[Thierry Coquand]], Simon Huber, _A Generalization of Takeuti-Gandy Interpretation_ ([pdf](http://uf-ias-2012.wikispaces.com/file/view/semi.pdf))

and realized in [[cubical type theory]] in

* {#Coquand13} [[Thierry Coquand]] (with [[Marc Bezem]] and [[Simon Huber]]), _Computational content of the Axiom of Univalence_, September 2013 ([pdf](http://www.humboldt-kolleg.iam.unibe.ch/talks/Coquand.pdf))

* [[Cyril Cohen]], [[Thierry Coquand]], [[Simon Huber]], [[Anders Mörtberg]], _Cubical Type Theory: a constructive interpretation of the univalence axiom_ ([pdf](https://hal.inria.fr/hal-01378906/document))

* {#BezemCoquandHuber17} [[Marc Bezem]], [[Thierry Coquand]], [[Simon Huber]], _The univalence axiom in cubical sets_ ([arXiv:1710.10941](https://arxiv.org/abs/1710.10941))

A study of the [[semantics|semantic]] side of univalence in [[(infinity,1)-toposes]], as well as further cases of [[locally cartesian closed (infinity,1)-categories]] is in

* {#GepnerKock12} [[David Gepner]], [[Joachim Kock]], _Univalence in locally cartesian closed infinity-categories_ ([arXiv:1208.1749](http://arxiv.org/abs/1208.1749))

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 [[Tarski universes]], see there and [[type universes]]). Constructions of such a model in some very special cases are in [Shulman12](#Shulman12) above, and also in

* [[Michael Shulman]], _The univalence axiom for elegant Reedy presheaves_, [arXiv:1307.6248](http://arxiv.org/abs/1203.3253).

* {#Cisinski14} [[Denis-Charles Cisinski]], _Univalent universes for elegant models of homotopy types_ ([arXiv:1406.0058](http://arxiv.org/abs/1406.0058))

Finally, full proof that all [[∞-stack]] [[(∞,1)-topos]] have [[presentable (∞,1)-category|presentations]] by [[model categories]] which interpret (provide [[categorical semantics]]) for [[homotopy type theory]] with [[univalence|univalent]] [[type universes]]:

* {#Shulman19} [[Michael Shulman]], _All $(\infty,1)$-toposes have strict univalent universes_ ([arXiv:1904.07004](https://arxiv.org/abs/1904.07004)).

For more references see _[[homotopy type theory]]_.

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univalent bicategory

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## Definition

In an [[intensional type theory]] such as [[homotopy type theory]], a [[locally univalent bicategory]] is **univalent** if a Rezk completion condition is satisfied for the objects of the bicategory: equality of objects is [[adjoint equivalence]] of objects.

More specifically, it is a locally univalent bicategory $C$ for which the canonical function
$$idtoiso^{2,0}(a, b):(a = b) \to adjequiv(a, b)$$
is an equivalence of types for all objects $a:Ob(C)$, $b:Ob(C)$.

### In set-level foundations

A univalent bicategory in [[set-level foundations]] is the same thing as a [[locally posetal 2-category|locally posetal]] [[gaunt category]].

## See also

* [[bicategory]]

* [[locally univalent bicategory]]

## References

* [[Benedikt Ahrens]], [[Dan Frumin]], [[Marco Maggesi]], [[Niels van der Weide]], _Bicategories in Univalent Foundations_, Mathematical Structures in Computer Science **31** 10 (2022) 1232-1269 [[arXiv:1903.01152](https://arxiv.org/abs/1903.01152), [doi:10.1017/S0960129522000032](https://doi.org/10.1017/S0960129522000032)&rbrack

[[!redirects univalent bicategories]]


* [[Benedikt Ahrens]], [[Chris Kapulkin]], [[Michael Shulman]]:

  **Univalent categories and the Rezk completion**

  Math. Structures Comput. Sci. **25** 5 (2015) 1010-1039.

  [arXiv:1303.0584](http://arxiv.org/abs/1303.0584)

  [doi:10.1017/S0960129514000486](https://doi.org/10.1017/S0960129514000486)

on [[internal categories in homotopy type theory]] and their [[Rezk completion]] to [[univalent categories]].

> **Abstract.**  We develop [[category theory]] within [[Univalent Foundations]], which is a [[mathematical foundation|foundational system for mathematics]] based on a [[homotopy type theory|homotopical interpretation]] of [[dependent type theory]]. In this system, we propose a definition of "[[category]]" for which [[equality]] and [[equivalence of categories]] agree. Such categories satisfy a version of the [[univalence axiom|Univalence Axiom]], saying that the [[type]] of [[isomorphisms]] between any two [[objects]] is [[equivalence in homotopy type theory|equivalent]] to the [[identity type]] between these objects; we call them "saturated" or [[univalent categories|"univalent" categories]]. Moreover, we show that any category is [[weak equivalence|weakly equivalent]] to a univalent one in a [[universal construction|universal]] [[Rezk completion|way]]. In [[relation between category theory and type theory|homotopical and higher-categorical semantics]], this construction corresponds to a truncated version of the [[Rezk completion]] for [[Segal spaces]], and also to the [[stackification|stack completion]] of a [[prestack]].

category: reference


univalent category

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## Idea

In [[set-level foundations]], there are two ways to compare two [[objects]] of a [[category]] for sameness: by [[equality]] or by [[isomorphism]].  The [[principle of equivalence]] argues that isomorphism is the "correct" notion of "sameness" for objects of a category, but the notion of equality is still present to be ignored.

Note that even in a [[skeletal category]], where the mere *property* of "being isomorphic" coincides with the property of "being equal", one cannot collapse the notions of isomorphism and equality because an object can still be isomorphic to itself in more than one way.  This distinction only disappears in a [[gaunt category]], but not every category is equivalent to a gaunt one.

By contrast, in [[higher foundations]], it is possible to expand the notion of "equality" so that it coincides with isomorphism: just as two objects can be isomorphic in more than one way, they can also be equal in more than one way.  The notion of "category" in which equality and isomorphism coincide in this way is called *univalent* or *saturated*.  This is an analogue for 1-categories of imposing antisymmetry on a [[preordered set]] (a.k.a. a [[(0,1)-category]]) to obtain a [[partially ordered set]] (a.k.a. a skeletal or gaunt (0,1)-category).

Since we are talking about [[1-categories]] whose hom-types are [[h-sets]], in a univalent category the type of objects is a [[1-type]].  By contrast, a "[[precategory]]" in higher foundations maintains equality of objects (which can have arbitrary [[h-level]]) as separate from isomorphism, while a "[[strict category]]" requires that the type of objects form a set.

## Definition ##

We work in a [[dependent type theory]] with some notion of [[identity type]] or [[path type]], denoted $a=b$.  If $A$ is a [[precategory]] and $a,b:ob(A)$, we write $a\cong b$ for the type of isomorphisms from $a$ to $b$.

\begin{lemma}
There is a map
$$idtoiso : (a=b) \to (a \cong b)$$
\end{lemma}
\begin{proof}
By induction on identity, we may assume $a$ and $b$ are the same. But then we have $1_a:hom_A(a,a)$, which is clearly an isomorphism.
\end{proof}

\begin{definition}
A **univalent category** or **saturated category** is a precategory which is *Rezk complete*, meaning that the above canonical function
$$idtoiso : (a=b) \to (a \cong b)$$
an equivalence of types.
\end{definition}

## Comparing notions of category

In [[homotopy type theory]], univalent categories are arguably the best notion of "category".  Specifically, in HoTT/UF:

1. Most naturally occurring precategories are in fact univalent categories, such as the category [[Set]] and most categories of structured objects built from it such as [[Grp]], [[Ring]], [[Vect]], [[Mod]], [[Top]], [[Met]], [[StrCat]], etc.
2. Furthermore, *every* precategory is [[equivalence of categories|weakly equivalent]] to a univalent category (its [[Rezk completion]]).
3. The theory of univalent categories is abstractly nicer than that of either precategories or strict categories.  For instance, the statement "a fully faithful and essentially surjective functor is an equivalence" for strict categories is equivalent to the [[axiom of choice]], for precategories it is just false, and for univalent categories it is just true. (On the other hand it can be argued that one should be using [[split essentially surjective functors]] instead of [[essentially surjective functors]], since every equivalence of precategories is fully faithful and split essentially surjective.)
4. Under the semantics of HoTT/UF in [[(infinity,1)-toposes]], univalent categories are interpreted by the correct notion of [[stacks]] of 1-categories.  (See [[relation between type theory and category theory]] and [[category object in an (infinity,1)-category]].)  This is not true for either precategories or strict categories.  (Although in the "classical" model in [[∞Gpd]], univalent categories are interpreted by 1-truncated [[complete Segal spaces]], while strict categories are interpreted by 1-truncated [[Segal categories]] and precategories are interpreted by 1-truncated [[Segal spaces]].  Thus, in this model only, strict categories also give a correct notion of "category", although precategories still do not.)

For these reasons, some of the HoTT/UF literature (including the [[HoTT Book]]) refers to univalent categories as simply "categories".

The situation in plain [[intensional type theory]] is less clear, since without the [[univalence axiom]], naturally-occurring precategories cannot be shown to be univalent (or strict) and the Rezk completion need not exist.  Thus, in this case precategories seem unavoidable.  Fortunately, a surprising amount of category theory can be developed with precategories.

(To be completely precise, the notion of "univalent category" can also be defined in [[set-level foundations]].  In that case it turns out to be equivalent to the notion of [[gaunt category]], which again is not satisfactory as a general notion of "category".  More generally, a [[strict category]] is univalent if and only if it is gaunt.)

On this page we will not use the plain term "category" at all, but speak only of "precategories", "univalent categories", and "strict categories".

\begin{remark}
Most nLab pages about category theory are, and should remain, written in a fairly foundation-agnostic way.  If the reader is interpreting them in HoTT/UF, it is usually best to read "category" as meaning "univalent category" for the reasons above.  However, in many cases it is also possible to read it as meaning "precategory" or "strict category", as would be necessary if interpreting them in plain intensional type theory.

It is not usually necessary for pages about ordinary category theory to remark on this issue at all.  However, if there are subtleties regarding the choice of interpretation (e.g. if some construction, such as a [[Kleisli category]], may lead out of the world of univalent categories; or if some structure, such as a [[dagger category]], requires changing the notion of "univalence"), it is appropriate to include a remark.  Such a remark should generally be lower down on the page, not at the top, so that it doesn't confuse first-time readers.
\end{remark}

## Examples

Note: All categories given can become univalent via the [[Rezk completion]].

* As noted above, every [[gaunt category]] is a [[strict category|strict]] univalent category, or equivalently a [[skeletal category|skeletal]] univalent category.

* There is a precategory [[Set]], whose type of objects is $Set$, and with $hom_{Set}(A,B)\equiv (A \to B)$.  The [[univalence axiom]] implies that this is a univalent category.  One can also show from this that any precategory of set-level structures such as groups, rings topological spaces, etc. is also univalent.

* For any [[1-type]] $X$, there is a univalent category with $X$ as its type of objects and with $hom(x,y)\equiv(x=y)$. If $X$ is a set we call this the **discrete category** on $X$. In general, we call it a **(univalent) groupoid**.

* More generally, or any type $X$, there is a precategory with $X$ as its type of objects and with $hom(x,y)\equiv \| x= y \|_0$. The composition operation
$$\|y=z\|_0 \to \|x=y\|_0 \to \|y=z\|_0$$
is defined by induction on [[truncation]] from concatenation of the [[identity type]]. We call this the **fundamental pregroupoid** of $X$.  It is univalent if and only if $X$ is a 1-type; its Rezk completion is the **fundamental (univalent) groupoid** of $X$.

* There is a precategory whose type of objects is the [[type universe]] $\mathcal{U}$ and with $hom(X,Y)\equiv \| X \to Y\|_0$, and composition defined by induction on truncation from ordinary composition of functions.  We call this the **homotopy precategory of types**; it is not univalent (although it contains the univalent category [[Set]] as a sub-precategory).

## Properties ##

\begin{theorem}
Given two categories $A$ and $B$, if $B$ is univalent, then the functor category $B^A$ is univalent.
\end{theorem}

\begin{proof}
Let $F:A \to B$ and $G:A \to B$ be functors from $A$ to $B$.

Suppose that $\gamma:F \to G$ is a natural isomorphism. Then for any object $a : Ob(A)$, there is an isomorphism $\gamma_a: F_{ob}(a) \cong G_{ob}(a)$ in $B$. Since $B$ is univalent, this shows that $F_{ob}(a) = G_{ob}(a)$ for all objects $a$, and by function extensionality, that $F_{ob} = G_{ob}$.

Now, for any $a : Ob(A)$ and $b : Ob(B)$, the functions $F_{mor}(a,b):Mor_A(a,b) \to Mor_B(F_{ob}(a),F_{ob}(b))$ and $G_{mor}(a,b):Mor_A(a,b) \to Mor_B(G_{ob}(a),G_{ob}(b))$ are equal, because we established above that $F(a) = G(a)$ for all $a : Ob(a)$. Because the object functions and morphism functions of $F$ and $G$ are equal to each other, $F = G$.

Thus, if there is an isomorphism $\gamma:F \cong G$, then $F = G$.

this proof needs to be revised since it is missing the explicit identifications
\end{proof}

An [[equivalent-on-objects functor]] between two precategories $A$ and $B$ is a functor $F:A \to B$ where the object function $F_{ob}:Ob(A) \to Ob(B)$ is a equivalence of types.

A [[split essentially surjective functor]] between $A$ and $B$ is a functor $F:A \to B$ where for every object $b : Ob(B)$, there is a specified object $a : Ob(A)$ and an isomorphism $f:F_{ob}(a) \cong b$.

A [[fully faithful functor]] is a functor $F:A\to B$ such that each map on [[hom-sets]] $F_{a,b} : A(a_0,a_1) \to B(F(a_0),F(a_1))$ is an [[equivalence of types]].

These definitions are important for defining the two notions of equality for categories:

An **[[isomorphism]] of categories** is a fully faithful equivalent-on-objects functor.

An **[[equivalence of categories]]** is a fully faithful split essentially surjective functor. Equivalently, it is a functor $F:A \to B$ which has a left adjoint $G:B \to A$ such that the unit $\eta:1_A \to G F$ and counit $\epsilon:F G \to 1_B$ are isomorphisms.

\begin{theorem}
For univalent categories, isomorphism of categories and equivalence of categories coincide.
\end{theorem}

...

\begin{lemma}
In a univalent category, the type of objects is a 1-type.
\end{lemma}

\begin{proof}
It suffices to show that for any $a,b:A$, the type $a=b$ is a set. But $a=b$ is equivalent to $a \cong b$ which is a set.
\end{proof}

There is a canonical way to turn a [[category]] into a univalent category via the [[Rezk completion]].

## Related pages

* [[category]]

* [[gaunt category]]

* [[locally univalent bicategory]]

* [[type-theoretic definition of category]]

* [[Rezk completion]] (for [[displayed categories]])

* [[homotopy groups of spheres in homotopy type theory]]

* [[cohomology in homotopy type theory]]

* [[Hopf construction in homotopy type theory]]

* [[synthetic (infinity,1)-category theory|synthetic $(\infty,1)$-category theory]]

## References
 {#References}

### General
  {#ReferencesGeneral}

The relation between [[complete Segal space|Segal completeness]] (now often "[[Rezk completion|Rezk completeness]]") for internal categories in HoTT and the [[univalence axiom]] had been pointed out in

* [[Urs Schreiber]], _[Segal completenes and Univalence](https://golem.ph.utexas.edu/category/2012/05/segalcompleteness_and_univalen.html)_ (2012)

A comprehensive discussion for [[1-categories]] then appeared in:

* {#AhrensKapulkinShulman13} [[Benedikt Ahrens]], [[Chris Kapulkin]], [[Michael Shulman]], _Univalent categories and the Rezk completion_, Mathematical Structures in Computer Science **25** 5 (*From type theory and homotopy theory to Univalent Foundations of Mathematics*),  (2015) 1010-1039   $[$[arXiv:1303.0584](https://arxiv.org/abs/1303.0584), [doi:10.1007/978-3-319-21284-5_14](https://doi.org/10.1007/978-3-319-21284-5_14)$]$

Exposition:

* [[Mike Shulman]],  _[Category Theory in Homotopy Type Theory](https://golem.ph.utexas.edu/category/2013/03/category_theory_in_homotopy_ty.html)_, 2013

* [[Amélia Liao]], *Univalent Category Theory*, [[Homotopy Type Theory Electronic Seminar Talks]], (October 2022) [[video](https://www.youtube.com/watch?v=asY6dXkR2yg)]

Implementation in [[Coq]]:

* [[Jason Gross]], [[Adam Chlipala]], [[David Spivak]], *Experience Implementing a Performant Category-Theory Library in Coq*,  In:  *Interactive Theorem Proving. ITP 2014* Lecture Notes in Computer Science, **8558** Springer (2014)  $[$[arXiv:1401.7694](http://arxiv.org/abs/1401.7694), [doi:10.1007/978-3-319-08970-6_18](https://doi.org/10.1007/978-3-319-08970-6_18)$]$

[[Coq]] code formalizing this includes the following:

* _[github.com/benediktahrens/rezk_completion](https://github.com/benediktahrens/rezk_completion)_

and in [[cubical Agda]]:

* [[1lab]]: *[Cat.Base](https://1lab.dev/Cat.Base.html)*

Generalization to [[(n,1)-categories]] is discussed in

* {#CapriottiKraus17} [[Paolo Capriotti]], [[Nicolai Kraus]], _Univalent Higher Categories via Complete Semi-Segal Types_ ([arXiv:1707.03693](https://arxiv.org/abs/1707.03693))

and, by different means, in

* {#RiehlShulman17} [[Emily Riehl]], [[Michael Shulman]], *A type theory for synthetic $\infty$-categories* ([arXiv:1705.07442](https://arxiv.org/abs/1705.07442))

* {#Riehl18} [[Emily Riehl]], _The synthetic theory of ∞-categories vs the synthetic theory of ∞-categories_, talk at [Vladimir Voevodsky Memorial Conference 2018](http://www.math.ias.edu/vvmc2018) ([pdf](http://www.math.jhu.edu/~eriehl/Voevodsky.pdf))

Formalization of [[bicategories]]:

* [[Benedikt Ahrens]], Dan Frumin, Marco Maggesi, Niels van der Weide, _Bicategories in Univalent Foundations_ ([arXiv:1903.01152](https://arxiv.org/abs/1903.01152))

Lemmas and proofs above are taken from:

* [[Univalent Foundations Project]], *[[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]]* (2013)

On [[monoidal category|monoidal]] [[univalent categories]]:

* [[Kobe Wullaert]], [[Ralph Matthes]], [[Benedikt Ahrens]], *Univalent Monoidal Categories* [[arXiv:2212.03146](https://arxiv.org/abs/2212.03146)]

### By coinduction
  {#ReferencesCoinduction}

A formalization in [[HoTT]]-[[Agda]] of general [[(n,r)-categories]] for $n,r \in \mathbb{N} \sqcup \{\infty\}$, defined as [[coinductive types]] of infinity-graphs, with operations defined by induction-coinduction, is implemented in

* {#Morrison19} Darin Morrison, _[agda-nr-cats](https://github.com/5HT/agda-nr-cats)_,
_[agda-infinity-categories](https://github.com/freebroccolo/agda-infinity-categories/)

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