universal enveloping algebra

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

### For Lie algebras
 {#ForLieAlgebras}

Given a [[Lie algebra object]] $L$ [[internalization|interna]] to some [[symmetric monoidal category|symmetric monoidal]] $k$-[[linear category]] $C = (C,\otimes, \mathbf{1},\tau)$, an __enveloping monoid__ (or __enveloping algebra__) of $L$ in $C$ is any morphism $f \colon L\to Lie(A)$ of Lie algebras in $C$ where $A$ is a monoid (= algebra) in $C$, and $Lie(A)$ is the underlying object of $A$ equipped with the [[Lie bracket]] $[,]_{Lie(A)}=\mu-\mu\circ\tau_{A,A}$.

A morphism of enveloping algebras
$$
  \phi
    \;\colon\;
  \big(A, f \colon L\to Lie(A)\big)
  \longrightarrow
  \big(A, f' \colon L\to A'\big)
$$
is a morphism $g \colon A\to A'$ of [[monoid objects]] completing a commutative triangle of morphisms in $C$, i.e. $g\circ f = f'$. With an obvious composition of morphisms, the enveloping algebras of $L$ form a [[category]].

A __universal enveloping algebra__ of $L$ in $C$ is any universal [[initial object]] $i_L \colon L\to U(L)$ in the category of enveloping algebras of $L$, which implies that it is unique up to an isomorphism if it exists.

If it exists for all Lie algebras in $C$, then the rule $L\mapsto U(L)$ can be extended to a [[functor]] $U$ which is the [[left adjoint]] to the [[forgetful functor]] $Lie \colon A\mapsto Lie(A)$ defined above and the morphism $i_L:L\to U(L)$ is the [[unit of an adjunction|unit of the adjunction]].

\begin{remark}\label{ForSuperLieAlgebras}
**(for [[super Lie algebras]] as eg. in [[algebraic topology]])**
\linebreak
  The [[internalization|internal]] generality of the above definitions is used notably in [[algebraic topology]] where the relevant Lie algebras arise from [[Whitehead brackets]] which are [[super Lie algebras]] (see [there](super+Lie+algebra#AsLieAlgebrasInternalToSuperVectorSpaces)), i.e. [[Lie algebra objects]] [[internalization|internal]] to [[super vector spaces]] (cf. eg. [Milnor & Moore 1965, §5](#MilnorMoore65); [May & Ponto 2012, Def. 22.1.3](#MayPonto12)).
\end{remark}

### For $L_\infty$-algebras

In the more general context of [[higher algebra]] there is a notion of universal enveloping [[E-n algebra]] of an [[L-infinity algebra]] for all $n \in \mathbb{N}$ which generalizes the notion of universal associative algebra envelope of a Lie algebra. See at _[[universal enveloping E-n algebra]]_.

## Existence

The existence of the universal enveloping algebra is easy in many concrete symmetric monoidal categories, e.g. the symmetric monoidal category of bounded chain complexes (giving the universal enveloping [[differential graded algebra|dg-algebra]] of a [[differential graded Lie algebra|dg-Lie algebra]]), but not true in general.

First of all if $C$ admits countable coproducts, form the tensor algebra $TL=\coprod_{n=0}^\infty L^{\otimes n}$ on the object $L$; this is a monoid in $C$.
In most standard cases, one can also form the smallest 2-sided ideal (i.e. $A$-subbimodule) $I$ in monoid $A$ among those ideals whose inclusion into $A$ is factorizing the map $([,]-m_{TL}+m_{TL}\circ\tau)\circ \otimes :L\otimes L\to TL$; if the coequalizers exist in $C$ then we can form the quotient object $TL/I$ and there is an induced monoid structure in it. Under mild conditions on $C$, the natural morphism $i_L:L\to TL/I$ is an universal enveloping monoid of $L$ in $C$. If $C$ is an abelian tensor category and some flatness conditions on the tensor product are satisfied, then the enveloping monoid $i_L:L\to TL/I$ is a monic morphism in $C$ and $U(L\coprod L)\cong U(L)\otimes U(L)$.

## Properties

### Isomorphism problem
 {#IsomorphismProblem}

Notice that the enveloping algebras of two Lie algebras may be [[isomorphism|isomorphic]] as [[associative algebras]] even if the two Lie algebras they arise from are not isomorphic to each other.

This happens even in [[characteristic zero]], in which case, however, at least the [[dimension]] of the Lie algebra is encoded in its universal enveloping algebra, in the guise of the *Gelfand-Kirillov dimension* $GK\big(U(L)\big)$.

### Poisson algebra structure on $U(\mathfrak{g})$

The universal enveloping algebra $U(\mathfrak{g})$ of a [[Lie algebra]] is naturally a (non-commutative) [[Poisson algebra]] with the restriction of the Poisson bracket to generators being the original [[Lie bracket]]

### Hopf algebra structure on $U(\mathfrak{g})$

Suppose the universal enveloping algebras of Lie algebras exist in a $k$-linear symmetric monoidal category $C$ and the functorial choice $L\mapsto U(L)$ realizing the above construction with tensor products is fixed. For example, this is true in the category of $k$-[[modules]] where $k$ is a [[commutative ring]]. Then the projection $L\to 0$ (where $0$ is the trivial Lie algebra) induces the counit $\epsilon:U(L)\to U(0)=\mathbf{1}$. The coproduct $\Delta:U(L)\to U(L\times L)\cong U(L)\otimes U(L)$ is induced by the [[diagonal]] map $L\to L\times L$ whereas the antipode $S=U(-id):U(L)\to U(L)$. One checks that these morphisms make $U(L)$ into a [[Hopf algebra]] in $C$. (e.g [Milnor-Moore 65, section 5](#MilnorMoore65)) The _[[Milnor-Moore theorem]]_ states conditions under which the converse holds (hence under which a primitively generated Hopf algebra is a universal enveloping algebra of a Lie algebra).

If the category is simply the vector spaces over a field $k$, then for $l\in L$, after we identify $L$ with its image in $U(L)$, $\Delta(l) = l\otimes 1 + 1\otimes l$, i.e. the elements in $L$ are the [[primitive element]]s in $U(L)$.

### PBW theorem

The [[Poincaré–Birkhoff–Witt theorem]] states that the
[[associated graded]] algebra of an enveloping algebra $U(g)$ in [[characteristic zero]] is canonically isomorphic to a [[symmetric algebra]] $Sym(g)$, and $U(g)$ is isomorphic to $S(g)$ as a coalgebra, via the projection map $U(g)\to Gr U(g)$.

### Relation to formal deformation quantization

See at _[[formal deformation quantization]]_ the section _[Relation to universal enveloping algebras](formal+deformation+quantization#RelationToUniversalEnvelopingAlgebras)_.

### Relation to the group algebra
 {#RelationToTheGroupAlgebra}

> The universal enveloping algebra of a Lie algebra is the analogue of the usual [[group algebra]] of a group. It has the analogous function of exhibiting the [[category]] of [[Lie algebra modules]] as a category of [[modules]] for an [[associative algebra]]. This becomes more than an analogy when the universal enveloping algebra is viewed with its full [[Hopf algebra]] structure. By dualization, one obtains a commutative Hopf algebra which, in the case where the Lie algebra is that of an irreducible [[algebraic group]] over a [[field]] of [[characteristic 0]], contains the algebra of polynomial functions of that group as a sub Hopf algebra in a natural fashion.

(quoted from [Hochschild 1981, p. 221](#Hochschild81), see [ibid. Thm. 3.1 on p. 230](#Hochschild81); [Tjin 1992, Thm. 1](#Tjin92))

## Examples

\begin{example}\label{WeylAlgebraAndHeisenbergAlgebra}
Consider the standard [[symplectic form]] on the [[Cartesian space]] $\mathbb{R}^{2n}$, making a [[symplectic vector space]]. This gives rise to the corresponding [[Heisenberg Lie algebra]].

Depending on conventions, the [[universal enveloping algebra]] of the [[Heisenberg Lie algebra]] either already is the [[Weyl algebra]] on $2n$ generators or else it becomes so after after forming the [[quotient algebra]] in which the central generator is identified with the [[unit element]] of the [[ground field]] -- whereas in the former case (considered eg. in [Kravchenko 2000, Def. 2.1](#Kravchenko00); [Bekaert 2005, p. 9](#Bekaert05)) the central generator plays the role of the formal [[Planck constant]] $\hbar$ with the Weyl algebra regarded as a [[formal deformation quantization]] of the [[symplectic manifold]] $\mathbb{R}^{2m}$.

Accordingly, given a [[Heisenberg Lie n-algebra|Heisenberg Lie $n$-algebra]] it makes sense to call its [[universal enveloping E-n algebra|universal enveloping $E_n$-algebra]] a _[[Weyl n-algebra|Weyl $n$-algebra]]_.
\end{example}

\begin{example}
**(Universal enveloping of a tangent Lie algebra)**
\linebreak
The universal enveloping algebra of the [[tangent Lie algebra]] of a  finite-dimensional [[Lie group]] $G$ over the real or complex numbers is canonically isomorphic to the algebra of the left invariant  [[differential operators]] on $G$.
\end{example}

## Related concepts

* [[Hausdorff series]]

* [[Poincaré–Birkhoff–Witt theorem]]

* [[coexponential map]]

* [[Duflo isomorphism]]

* [[oidification]] to [[universal enveloping algebroid]]

* [[universal enveloping E-n algebra]]

## References

* {#Dixmier74} [[Jacques Dixmier]]; _[[Algèbres Enveloppantes]]_, Cahiers Scientifique (1974), Engl transl: _Enveloping Algebras_, Graduate Studies in Mathematics **11** American Mathematical Society (1996) [[ams:gsm-11](https://bookstore.ams.org/gsm-11)]

* [[Nicolas Bourbaki]], §I.2 in: _Lie groups and Lie algebras -- Chapters 1-3_, Springer (1975, 1989) [[ISBN:9783540642428](https://link.springer.com/book/9783540642428)]

* {#Hochschild81} [[Gerhard P. Hochschild]], *The Universal Enveloping Algebra*, Chapter XVI in: *Basic Theory of Algebraic Groups and Lie Algebras*, Graduate Texts in Mathematics **75**, Springer (1981) [[doi:10.1007/978-1-4613-8114-3_16](https://doi.org/10.1007/978-1-4613-8114-3_16)]

Further discussion in the context of [[quantum groups]]:

* {#Tjin92} [[Tjark Tjin]], *An introduction to quantized Lie groups and algebras*, Int. J. Mod. Phys. A **7** (1992) 6175-6213 [[arXiv:hep-th/9111043](https://arxiv.org/abs/hep-th/9111043), [doi:10.1142/S0217751X92002805](https://doi.org/10.1142/S0217751X92002805)]

Discussion in the generality of [[super Lie algebras]] such as notably of the [[Whitehead Lie algebras]] arising in [[algebraic topology]]:

* {#MilnorMoore65} [[John Milnor]], [[John Moore]], §5 in _On the structure of Hopf algebras_, Annals of Math. __81__ (1965) 211-264 [[doi:10.2307/1970615](https://doi.org/10.2307/1970615), [pdf](http://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/milnor-moore-ann-math-1965.pdf)]

* {#MayPonto12} [[Peter May]], [[Kate Ponto]], Def. 22.1.3 in: *More concise algebraic topology -- Localization, Completion, and Model Categories*, University of Chicago Press (2012) [[ISBN:9780226511795](https://press.uchicago.edu/ucp/books/book/chicago/M/bo12322308.html), [pdf](https://www.math.uchicago.edu/~may/TEAK/KateBookFinal.pdf)]

See also:

* Wikipedia: *[Universal enveloping algebra](http://en.wikipedia.org/wiki/Universal_enveloping_algebra)*, *[PBW theorem](http://en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Birkhoff%E2%80%93Witt_theorem)*

* F. A. Berezin, _Some remarks about the associated envelope of a Lie algebra_, Func. Analysis and Its Appl. 1967, 1:2, 91–102; Rus. original in Fun. Anal. Pril. 1:2 (1967) 1–14 [Rus. pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=faa&paperid=2813&volume=1&year=1967&issue=2&fpage=1&what=fullt&option_lang=eng) [MR219671](http://www.ams.org/mathscinet-getitem?mr=219671)

* MathOverflow question [What is the universal enveloping algebra](http://mathoverflow.net/questions/25020/what-is-the-universal-enveloping-algebra) which is looking for a rather general construction in a class of symmetric monoidal [[pseudoabelian category|pseudoabelian categories]].

[[!redirects universal enveloping algebras]]


universal enveloping algebroid

The construction of a [[universal enveloping algebra]] of a Lie algebra can be generalized in several ways. One of them is to generalize it to [[Lie algebroid]]s and, slightly more general, to the [[Lie-Rinehardt algebra]]s. This leads to the universal enveloping algebroids. Some standard theorems like the PBW theorem generalize to some extend to this context.

* George S. Rinehart, _Differential forms on general commutative algebras_, Trans. Amer. Math. Soc. 108 1963 195--222 (MR0154906 (27 #4850))

* Hinich, Schechtman

* Kapranov


universal enveloping E-n algebra

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For all $n \in \mathbb{N}$ there is supposed to be a pair of [[adjoint (∞,1)-functors]]

$$
  E_n Alg \stackrel{\overset{\mathcal{U}_n}{\leftarrow}}{\underset{}{\to}}
  L_\infty
$$

between [[E-n algebras]] and [[L-∞ algebras]], suitably factoring through [[Poisson n-algebras]].

The [[left adjoint]] $\mathcal{U}_n$ sends an [[L-∞ algebra]] to its _universal enveloping $E_n$-algebra_ in that for $n = 1$ and for $\mathfrak{g}$ an ordinary [[Lie algebra]], $\mathcal{U}_1(\mathfrak{g})$ is the [[associative algebra]] (an $E_1$=[[A-∞ algebra|A-∞]] algebra) which is the ordinary [[universal enveloping algebra]] of $\mathfrak{g}$.

## Examples and applications

* [[Weyl n-algebra]]

## References
 {#References}

Discussion for $n = 1$, hence universal [[A-∞ algebra|A-∞]]-enveloping algebras of [[L-∞ algebras]] is around theorem 3.1, 3.3 in

* [[Tom Lada]], [[Martin Markl]], _Strongly homotopy Lie algebras_ ([arXiv:hep-th/9406095](http://arxiv.org/abs/hep-th/9406095))

and more details have been worked out here:

* Vladimir Baranovsky, _A universal enveloping for L-infinity algebras_ ([https://arxiv.org/abs/0706.13960](https://arxiv.org/abs/0706.1396))

Aspects of general enveloping $E_n$-alebras are mentioned in the context of [[factorization homology]] in section 5 and in particular around the bottom of p. 18 in

* [[David Ayala]], [[John Francis]], _Factorization homology of topological manifolds_ ([arXiv:1206.5522](http://arxiv.org/abs/1206.5522))

and more specifically in the context of [[factorization algebras of observables]] around remark 4.5.5 of

* {#Gwilliam} [[Owen Gwilliam]], _Factorization algebras and free field theories_ PhD thesis ([pdf](http://math.berkeley.edu/~gwilliam/thesis.pdf))

The fact that $\mathcal{U}_1$ reproduces the traditional [[universal enveloping algebra]] of a Lie algebra is prop. 4.6.1 in ([Gwilliam](#Gwilliam)).

See also

* [[John Francis]], [[Dennis Gaitsgory]], _Chiral Koszul duality_ ([arXiv:1103.5803](http://arxiv.org/abs/1103.5803))

[[!redirects universal enveloping E-n algebras]]


universal epimorphism

A morphism $f\colon X \to Y$ is a __universal epimorphism__ if for every morphism $u\colon V \to Y$ there is a [[pullback]] $X \times_Y V$ and its projection $u^*(f)\colon X \times_Y V \to V$ is an [[epimorphism]].

In particular, setting $u = id_Y\colon Y \to Y$, we see that $f$ itself is an epimorphism.

A morphism $g\colon X\to Y$ is a __universal monomorphism__ if its opposite $g^\circ\colon Y^\circ \to X^\circ$ is a universal epimorphism in the [[opposite category]]. In particular, it is a [[monomorphism]].

[[!redirects universal epimorphism]]
[[!redirects universal epimorphisms]]

[[!redirects universal monomorphism]]
[[!redirects universal monomorphisms]]


universal equivariant PU(H)-bundle

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

> under construction

#Contents#
* table of contents
{:toc}

## Idea

In [[equivariant cohomology|equivariant]] generalization of how the [[universal principal bundle]] with [[structure group]] the [[projective unitary group]] [[PU(ℋ)]] on a [[separable Hilbert space]] $\mathcal{H}$ classifies the [[twisted cohomology|3-twist]] of [[twisted K-theory|twisted]] [[KU]]-[[cohomology theory]], so the the
[[universal equivariant principal bundle]] with this [[structure group]] [[PU(ℋ)]] serves to classify the [[twisted cohomology|3-twists]] of [[twisted equivariant K-theory|equivariant]] [[KU]]-[[cohomology theory]].

## Details

For [[finite group|finite]] [[equivariance group]] $G$ and
in specialization of the [Murayama-Shimakawa construction](equivariant+bundle#calBGammaForDiscreteG), the base space of the universal $G$-equivariant $PU(\mathcal{H})$-principal bundle is

\[
  \label{TheEquivariantClassifyingSpace}
  \mathcal{B} PU(\mathcal{H})
  \;\;
    =
  \;\;
  TopGrpds
  \big(
    G \times G \rightrightarrows G
    ,\,
    PU(\mathcal{H}) \rightrightarrows \ast
  \big)
  \;\;\;
  \in
  \;
  \in G Actions(TopSpaces)
  \,,
\]

where $G$ [[group action|acts]] by right multiplication on the arguments.

This space is considered in [BEJU 2014, Thm. 3.5](#BEJU14) (without reference to [Murayama & Shimakawa 1995](equivariant+bundle#MurayamaShimakawa95)) as the equivariant classifying space for the 3-twist of [[twisted equivariant K-theory]].

For [[subgroups]] $H \subset G$ the $H$-[[fixed locus]] of this space is the [[Borel construction]]

$$
  \big(
    \mathcal{B} PU(\mathcal{H})
  \big)^H
  \;\;
    =
  \;\;
  Grps\big(H, PU(\mathcal{H})\big) \sslash_{\!ad} PU(\mathcal{H})
$$

of the [[adjoint action]] of [[PU(ℋ)]] on the space of [[group homomorphisms]] from $H$.

## Properties

### Equivariant homotopy groups
 {#EquivariantHomotopyGroups}

\begin{proposition}\label{EquivariantHomotopyGroupsOfBaseSpace}
  For $H \subset G$ any [[subgroup]], the higher [[homotopy groups]]
  of the $H$-[[fixed locus]] of the equivariant classifying space (eq:TheEquivariantClassifyingSpace) are concentrated on the [[integers]] in degree 3 and the [[Pontrjagin dual]] of $K$ in degree 1:
  $$
    \pi_{k \gt 0}
    \Big(
      \big(
        \mathcal{B} PU(\mathcal{H})
      \big)^H
    \Big)
    \;\;
    =
    \;\;
    \left\{
    \begin{array}{cll}
      0 &\vert& k \geq 4
      \\
      \mathbb{Z} &\vert& k = 3
      \\
      0 &\vert& k = 2
      \\
      Grps(H,S^1) &\vert& k = 1
    \end{array}
    \right.
  $$
\end{proposition}
([BEJU 2014, around Cor. 1.11](#BEJU14))

## Related concepts

* [[U(ℋ)|$U(\mathcal{H})$]]

## References

* {#BEJU14} [[Noé Bárcenas Torres|Noé Bárcenas]], [[Jesús Francisco Espinoza Fierro|Jesús Espinoza]], [[Michael Joachim]], [[Bernardo Uribe]], _Universal twist in Equivariant K-theory for proper and discrete actions_, Proceedings of the London Mathematical Society, Volume 108, Issue 5 (2013) ([arXiv:1202.1880](https://arxiv.org/abs/1202.1880), [doi:10.1112/plms/pdt061](https://doi.org/10.1112/plms/pdt061))

* {#UribeLueck13} [[Bernardo Uribe]], [[Wolfgang Lück]], Section 15 of: _Equivariant principal bundles and their classifying spaces_, Algebr. Geom. Topol. 14 (2014) 1925-1995 ([arXiv:1304.4862](https://arxiv.org/abs/1304.4862), [doi:10.2140/agt.2014.14.1925](http://dx.doi.org/10.2140/agt.2014.14.1925))

* [[Noé Bárcenas Torres|Noé Bárcenas]], [[Jesús Francisco Espinoza Fierro|Jesús Espinoza]], [[Bernardo Uribe]], [[Mario Velasquez]], Section 3.2 of: _Segal's spectral sequence in twisted equivariant K-theory for proper and discrete actions_, Proceedings of the Edinburgh Mathematical Society **61** 1 (2018) ([arXiv:1307.1003](https://arxiv.org/abs/1307.1003), [doi:10.1017/S0013091517000281](https://doi.org/10.1017/S0013091517000281))

* [[Jesus Espinoza]], [[Bernardo Uribe]], *Topological properties of spaces of projective unitary representations*, Rev. Acad. Colombiana Cienc. Exact. Fís. Natur. 40 (2016), no. 155, 337-352 ([arXiv:1511.06785](https://arxiv.org/abs/1511.06785), [scielo:S0370-39082016000200013](http://www.scielo.org.co/scielo.php?script=sci_arttext&pid=S0370-39082016000200013), [doi:10.18257/raccefyn.317](https://doi.org/10.18257/raccefyn.317))

[[!redirects universal equivariant PU-bundle]]
[[!redirects universal equivariant PUH-bundle]]

[[!redirects universal equivariant PU(ℋ)-bundle]]


universal fibration of (infinity,1)-categories

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

##  Idea

The _universal fibration of [[(infinity,1)-category|(∞,1)-categories]] is the [[generalized universal bundle]] of $(\infty,1)$-categories in that it is [[Cartesian fibration]]

$$
  p \colon Z \to (\infty,1)Cat^{op}
$$

over the [[opposite category]] of the [[(∞,1)-category of (∞,1)-categories]] such that

* its fiber  $p^{-1}(C)$ over $C \in (\infty,1)Cat$ is just the $(\infty,1)$-category $C$ itself;

* every [[Cartesian fibration]] $p : C \to D$ arises as the [[pullback]] of the universal fibration along an [[(∞,1)-functor]] $S_p : D \to (\infty,1)Cat^{op}$.

Recall from the discussion at [[generalized universal bundle]] and at [[stuff, structure, property]] that for [[n-category|n-categories]] at least for low $n$ the corresponding universal object was the $n$-category $n Cat_*$ of [[pointed object|pointed]] $n$-categories. $Z$ should at least morally be $(\infty,1)Cat_*$.

## Definition

### For $(\infty,1)$-categories

One description of the universal cartesian fibration is given in section 3.3.2 of [[Higher Topos Theory|HTT]] as the contravariant [[(∞,1)-Grothendieck construction]] applied to the identity functor $((\infty,1)Cat^{op})^{op} \to (\infty,1)Cat$.

We can also give a more direct description:

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

$Z^{op}$ is equivalent to the full subcategory of $(\infty,1)Cat^{[1]}$ spanned by the morphisms of the form $[C_{x/} \to C]$ for small (∞,1)-categories $C$ and objects $x \in C$.

The universal fibration $Z \to (\infty,1)Cat^{op}$ is opposite to the target evaluation.

Dually, the universal cocartesian fibration is $Z' \to (\infty,1)Cat$ where $Z'$ is the (∞,1)-category of arrows of the form $[C_{/x} \to C]$.

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

This is the proof idea of [this mathoverflow post](#direct).

By proposition 5.5.3.3 of Higher Topos Theory, there are presentable fibrations $RFib \to (\infty,1)Cat$ and $(\infty,1)Cat^{[1]} \xrightarrow{tgt} (\infty,1)Cat$ classifying functors $C \mapsto \mathcal{P}(C)$ and $C \mapsto (\infty,1)Cat_{/C}$.

By proposition 5.3.6.2 of Higher Topos Theory, the yoneda embedding $j : C \to \mathcal{P}(C)$ is a natural transformation, and the covariant Grothendieck construction provides a cocartesian functor $Z' \to RFib$. Since it is fiberwise fully faithful and $(-)_!$ preserves representable presheaves, we can identify $Z'$ with the full subcategory of $RFib$ consisting of the representable presheaves.

The Grothendieck construction provides a fully faithful $\mathcal{P}(C) \to
(\infty,1)Cat_{/C}$ whose essential image is the right fibrations. The contravariant Grothendieck construction a cartesian functor $RFib \to (\infty,1)Cat^{[1]}$. Since it is fiberwise fully faithful and pullbacks preserve right fibrations, we can identify $RFib$ with the full subcategory of $( \infty,1)Cat^{[1]}$ spanned by right fibrations.

By the relationship between the covariant and contravariant Grothendieck constructions, the universal cartesian fibration is classified by $op : ((\infty,1)Cat^{op})^{op} \to (\infty,1)Cat$.

=--

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

A key point of this description is that for any small (∞,1)-category $C$, the functor $x \mapsto C_{/x}$ (where $x \to y$ acts by composition) is a fully faithful functor $C \to (\infty,1)Cat_{/C}$. Dually, $x \mapsto C_{x/}$ is a fully faithful functor $C^{op} \to (\infty,1)Cat_{/C}$

=--

The hom-spaces of the universal cocartesian fibration can be described as
$$
  Z'([C_{/x} \to C], [D_{/y} \to D]) \simeq Core(eval_x \downarrow y)
$$
where $eval_x : D^C \to D$. This should be compared with the lax [[slice 2-category]] construction. In fact, $Z'$ can be constructed by taking the underlying (∞,1) category of the lax coslice (or colax, depending on convention) over the point of the (∞,2)-category of (∞,1)-categories.

### For $\infty$-Groupoids
  {#RestInfGrpd}

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

The universal fibration of $(\infty,1)$-categories restricts to a [[Cartesian fibration]] $Z|_{\infty Grpd} \to \infty Grpd^{op}$ over [[∞Grpd]] by [[pullback]] along the inclusion morphism $\infty Grpd \hookrightarrow (\infty,1)Cat$

$$
  \array{
    Z|_{\infty Grpd} &\longrightarrow& Z
    \\
    \big\downarrow && \big\downarrow
    \\
    \infty Grpd^{op} &\hookrightarrow& (\infty,1)Cat^{op}
  }
  \,.
$$

=--

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

The [[∞-functor]] $Z|_{\infty Grpd} \to \infty Grpd^{op}$ is even a [[right fibration]] and it is the _universal right fibration_. In fact it is (when restricted to small objects) the [[object classifier]] in the [[(∞,1)-topos]] [[∞Grpd]], see at [object classifier -- In ∞Grpd](%28sub%29object+classifier+in+an+%28infinity%2C1%29-topos#ObjectClassifierInInfinityGroupoid).

=--

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

The universal left fibration is the forgetful functor
$\infty Grpd_* \to \infty Grpd$. Its opposite is the universal right fibration.

=--

([Lurie 2009, Prop, 3.3.2.7](#Lurie09), [Cisinski 2019, Sec. 5.2](#Cisinski19), for the further restriction to the [[universal Kan fibration]] see also [Kapulkin & Lumsdaine 2021](#KapulkinLumsdaine12))

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

The following are equivalent:

* An [[∞-functor]] $p : C \to D$ is a [[right Kan fibration]].

* Every functor $S_p : D \to (\infty,1)Cat$ that classifies $p$ as a [[Cartesian fibration]] factors through [[∞-Grpd]].

* There is a functor $G_p : D \to \infty Grpd$ that classifies $p$ as a [[right Kan fibration]].

=--

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

This is proposition 3.3.2.5 in [[Higher Topos Theory|HTT]].

=--

## Models

For concretely constructing the relation between [[Cartesian fibration]]s $p : E \to C$ of [[(infinity,1)-categories|(∞,1)-categories]] and [[(∞,1)-functor]]s $F_p : C \to (\infty,1)Cat$ one may use a [[Quillen equivalence]] between suitable [[model category|model categories]] of [[marked simplicial set]]s.

For $C$ an [[(∞,1)-category]] regarded as a [[quasi-category]] (i.e. as a [[simplicial set]] with certain properties), the two model categories in question are

* the projective [[global model structure on simplicial presheaves]] on $[C,SSet]$ -- this models the [[(∞,1)-category of (∞,1)-functors]] $(\infty,1)Func(C,(\infty,1)Cat)$.

* the _covariant model structure_ on the [[over category]] $SSet/C$ -- this models the $(\infty,1)$-category of [[Cartesian fibration]]s over $C$.

The [[Quillen equivalence]] between these is established by the [[relative nerve]] construction

$$
  N_{-}(C) : [C,SSet] \to SSet/C
  \,.
$$

By the [[adjoint functor theorem]] this functor has a [[left adjoint]]

$$
  F_{-}(C) : SSet/C \to [C,SSet]
  \,.
$$

For $p : E \to C$ a [[left Kan fibration]] the functor $F_p(C) : C \to SSet$ sends $c \in Obj(C)$ to the [[fiber]] $p^{-1}(c) := E \times_C \{c\}$

$$
  F_p(C) : c \mapsto p^{-1}(c)
  \,.
$$

(See remark 3.2.5.5 of [[Higher Topos Theory|HTT]]).

## Related concepts

* [[universal Kan fibration]]

* [[fibration of (∞,1)-categories]]

## References

Textbook accounts:

* {#Lurie09} [[Jacob Lurie]], Section 3.3.2 of: _[[Higher Topos Theory]]_, Annals of Mathematics Studies **170**, Princeton University Press 2009 ([pup:8957](https://press.princeton.edu/titles/8957.html), [pdf](https://www.math.ias.edu/~lurie/papers/HTT.pdf))

  > (the concrete description in terms of model theory on marked simplicial sets is in section 3.2. A simpler version of this is in section 2.2.1)

* {#Cisinski19} [[Denis-Charles Cisinski]], Section 5.2 of: _[[Higher Categories and Homotopical Algebra]]_,  Cambridge University Press 2019 ([doi:10.1017/9781108588737](https://doi.org/10.1017/9781108588737), [pdf](http://www.mathematik.uni-regensburg.de/cisinski/CatLR.pdf))

  > (the universal left fibration)

The direct description of the universal fibration is discussed at

* {#direct} MathOverflow: _[A construction of the universal cocartesian fibration](https://mathoverflow.net/questions/383275/)_

Discussion of the [[universal Kan fibration]] as a [[relation between type theory and category theory|categorical semantics]] for a [[univalence axiom|univalent]] [[type universe]] in [[homotopy type theory]]:

Discussion of fibrations via [[(∞,2)-category]] theory

* [[Jacob Lurie]], chapter 5 _[Fibrations of ∞-Categories](https://kerodon.net/tag/01J2)_ (2021) in: _[[Kerodon]]_

On the [[relation between type theory and category theory|categorical semantics]] of [[homotopy type theory]] in [[simplicial sets]]/[[infinity-groupoids|$\infty$-groupoids]]:

* {#KapulkinLumsdaine12} [[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)$]$

Discussion of [[straightening and unstraightening]] entirely within the context of [[quasi-categories]]:

* [[Denis-Charles Cisinski]], [[Hoang Kim Nguyen]], *The universal coCartesian fibration* [[arXiv:2210.08945](https://arxiv.org/abs/2210.08945)]

which (along the lines of the discussion of the universal left fibration from [Cisinski 2019](#Cisinski19)) allows to understand the [[universal coCartesian fibration]] as [[categorical semantics]] for the [[univalence axiom|univalent]] [[type universe]] in [[directed homotopy type theory]]:

* {#CisinskiEtAl23} [[Denis-Charles Cisinski]], [[Hoang Kim Nguyen]], Tashi Walde: *Univalent Directed Type Theory*, lecture series in the *[CMU Homotopy Type Theory Seminar](https://www.cmu.edu/dietrich/philosophy/hott/seminars/index.html)* (13, 20, 27 Mar 2023) [[web](https://www.cmu.edu/dietrich/philosophy/hott/seminars/index.html#230313), video 1:[YT](https://www.youtube.com/watch?v=5YOltuTcBK8), 2:[YT](https://www.youtube.com/watch?v=xWmELBvHMPo), 3:[YT](https://www.youtube.com/watch?v=P0Cfb4eUJo4); slides 0:[pdf](https://www.cmu.edu/dietrich/philosophy/hott/seminars/slides/cisinski-nguyen-walde-intro_talk1.pdf), 1:[pdf](https://www.cmu.edu/dietrich/philosophy/hott/seminars/slides/cisinski-nguyen-walde-talk1.pdf), 2:[pdf](https://www.cmu.edu/dietrich/philosophy/hott/seminars/slides/cisinski-nguyen-walde-talk2.pdf), 3:[pdf](https://www.cmu.edu/dietrich/philosophy/hott/seminars/slides/cisinski-nguyen-walde-talk3.pdf)]

(see [video 3 at 1:16:58](https://youtu.be/P0Cfb4eUJo4?t=4618) and [slide 3.33](https://www.cmu.edu/dietrich/philosophy/hott/seminars/slides/cisinski-nguyen-walde-talk3.pdf#page=33)).

[[!redirects universal fibration of (∞,1)-categories]]

[[!redirects universal left fibration]]
[[!redirects universal right fibration]]

[[!redirects universal Cartesian fibration]]
[[!redirects universal coCartesian fibration]]

[[!redirects universal fibration of ∞-groupoids]]


universal higher central extension

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In any context of [[nLab:higher algebra]] where [[higher central extension]] make sense, one may ask for the _universal_ such. Since an ordinary [[universal central extension]] of some algebraic object is characterized as the [[initial object]] in the [[category]] of all its ordinary [[central extensions]], an evident definition of universal higher central extension is as a suitable initial object in a [[higher category]] of [[higher central extensions]].

For higher central extensions in the context of [[discrete infinity-groupoid|discrete]] [[2-groups]] this is considered in [Epa-Ganter 16, Def. 2.4](#EpaGanter16)

## References

* {#EpaGanter16} [[Narthana Epa]], [[Nora Ganter]], _Platonic and alternating 2-groups_, Higher Structures 1(1):122-146, 2017. ([arXiv:1605.09192](http://arxiv.org/abs/1605.09192))

[[!redirects universal higher central extensions]]

[[!redirects higher universal central extension]]
[[!redirects higher universal central extensions]]


Universal Homotopy Theories

This entry is about the article

* [[Daniel Dugger]]:

  \linebreak

  **[[DuggerUniv.pdf:file]]**

  \linebreak

  Advances in Mathematics __164__

  (2001) 144-176

  [doi:10.1006/aima.2001.2014](https://doi.org/10.1006/aima.2001.2014)

  [arXiv:math/0007070](http://arxiv.org/abs/math/0007070)

on the projective local [[model structure on simplicial presheaves]].

\linebreak

  **Abstract** _Given a small category $C$, we show that there is a universal way of expanding $C$ into a model category, essentially by formally adjoining homotopy colimits. The technique of localization becomes a method for imposing "relations" into these universal gadgets. The paper develops this formalism and discusses various applications, for instance to the study of homotopy colimits, the Dwyer-Kan theory of framings, and to the homotopy theory of schemes._

The article discusses the _projective_ global and local [[model structure on simplicial presheaves]].

The general discussion is in parts is based on the unfinished but useful notes

* Dan Dugger, _Sheaves and homotopy theory_ ([web](http://www.uoregon.edu/~ddugger/cech.html), [dvi](http://www.uoregon.edu/~ddugger/cech.dvi), [pdf](http://ncatlab.org/nlab/files/cech.pdf))

In particular this

* shows that and how the projective model structure on simplicial presheaves on $C$ is something like the free completion of $C$ under [[homotopy colimit]]s. This is the model-category theoretic analog of the fact that the [[(∞,1)-category of (∞,1)-presheaves]] on $C$ is the
  [[free (∞,1)-cocompletion]] of $C$;

* announces the statement of **[Dugger's theorem](http://ncatlab.org/nlab/show/combinatorial+model+category#DuggerTheorem)** which states that every [[combinatorial model category]] arises as the [[Bousfield localization of model categories|left Bousfield localization]] of the projective [[global model structure on simplicial presheaves]]. The full proof of this is in the companion article

  * Daniel Dugger, _[[Combinatorial model categories have presentations]]_

* discusses [cofibrant replacement in the projective model structures](http://ncatlab.org/nlab/show/model+structure+on+simplicial+presheaves#FibAndCofibObjects) (necessarily both for the global as well as for the local structure)

* discusses _geometric realization_ of $\infty$-stacks on [[Diff]] essentially by using the geometric homotopy $\infty$-groupoid functor as disscussed at [[homotopy groups in an (∞,1)-topos]].

## Free $(\infty,1)$-cocompletion

The main theorem of the article is the following. For more details see [[(∞,1)-category of (∞,1)-presheaves]].

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

Let $A$ and $B$ be [[model categories]], $D$ a plain [[category]] and

$$
  \array{
    D &\stackrel{r}{\to}& A
    \\
    & \searrow_\gamma
    \\
    && B
  }
$$

two plain [[functor]]s. Say that a **model-category theoretic factorization** of $\gamma$ through $A$ is

1. a [[Quillen adjunction]] $(L \dashv R) : A \stackrel{\overset{L}{\to}}{\underset{R}{\leftarrow}} B$

1. a [[natural transformation|natural]] weak equivalence $\eta  : L \circ r \to \gamma$

   $$
     \array{
       D &&\stackrel{r}{\to}&& A
       \\
       & \searrow_\gamma &{}^\eta\swArrow& \swarrow_{L}
       \\
       && B
     }
     \,.
   $$

Let the [[category]] of such factorizations have morphisms $((L \dashv R), \eta ) \to ((L' \dashv R'), \eta' )$ given by [[natural transformation]]s $L \to L'$ such that for all all objects $d \in D$ the diagrams

$$
  \array{
     L\circ r(d) &&\to&& L'\circ r(d)
     \\
     & {}_{\eta_{d}}\searrow && \swarrow_{\eta'_{d}}
     \\
     &&
     \gamma()
  }
$$

commutes.

=--

Notice that the [[(∞,1)-category]] presented by a [[model category]] -- at least by a [[combinatorial model category]] -- has all [[limit in a quasi-category|(∞,1)-categorical colimits]], and that the Quillen left adjoint functor $L$ presents, via its [[derived functor]], a left [[adjoint (∞,1)-functor]] that preserves $(\infty,1)$-categorical colimits. So the notion of factorization as above is really about factorizations through colimit-preserving $(\infty,1)$-functors into $(\infty,1)$-categories that have all colimits.

+-- {: .un_theorem}
###### Theorem
**(model category presentation of free $(\infty,1)$-cocompletion)**

For $C$ a [[small category]], the projective global [[model structure on simplicial presheaves]] $[C^{op}, sSet]_{proj}$ on $C$ is universal with respect to such factorizations of functors out of $C$:

every functor $C \to B$ to any [[model category]] $B$ has a factorization through $[C^{op}, sSet]_{proj}$ as above, and the category of such factorizations is [[contractible]].

=--

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

This is theorem 1.1 in the article.

=--

category: reference

[[!redirects Universal homotopy theories]]


Universal homotopy theories > history

< [[Universal homotopy theories]]


universal Kan fibration

## Idea

A [[Kan fibration]] such that every other sufficiently small Kan fibration arises, up to equivalence, as a [[pullback]].

See at:

* _[[universal fibration of (infinity,1)-categories]]_ the section _[for infinity-groupoids](https://ncatlab.org/nlab/show/universal+fibration+of+%28infinity,1%29-categories#RestInfGrpd)_

* _[[object classifier]]_, the section _[in infinity-groupoids](https://ncatlab.org/nlab/show/%28sub%29object+classifier+in+an+%28infinity%2C1%29-topos#ObjectClassifierInInfinityGroupoid)_

* *[[univalent universe]]*

[[!redirects universal Kan fibrations]]
[[!redirects universal fibration of infinity-groupoids]]
[[!redirects universal fibration of ∞-groupoids]]


universal localization

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion _universal localization_ or _Cohn localization_ of a [[ring]] is a variant of the notion of [[localization of a ring]] which forces not just elements of the ring to become invertible (which one may think of as $1 \times 1$-[[matrices]]) but forces more general [[matrices]] with [[coefficients]] in the ring to become invertible.

## Definition

Let $\Sigma$ be a set of finite square [[matrices]] (of different sizes) over a (typically noncommutative) [[ring]] $R$. Without loss of generality, one assumes that $\Sigma$ is left or right multiplicative. It is left multiplicative if for any matrices $A,B,C$ of right sizes such that $A,B\in\Sigma$ and $C$ fits into matrix $New = \left(\array{ A & 0\\ C & B}\right)$, matrix $New$ is also in $\Sigma$.

We say that a [[homomorphism]] of rings $f: R\to S$ is _$\Sigma$-inverting_ if all matrices $f(A)$ over $S$ where $A\in \Sigma$ are invertible in $S$.
The Cohn localization of a ring $R$, is a homomorphism of rings $R\to \Sigma^{-1} R$ which is initial in the category of all $\Sigma$-inverting maps (which is the subcategory of coslice category $R/Ring$). In the left hand version, the elements in the localized ring are thought of as solutions of linear equations $A x = b$ where $b$ is a column vector with elements in $R$ and $A\in\Sigma$.

### More general definition

Given a [[ring]] $R$ and a family $S$ of morphisms in the [[category]] $R$[[Mod]] of (say left) [[finitely generated module|finitely generated]] [[projective module|projective]] $R$-[[modules]], we say that a morphism of rings $f:R\to T$ is **$S$-inverting** if the [[extension of scalars]] from $R$ to $T$ along $f$

$$
  T \otimes_R (-) \colon R Mod \to T Mod
$$

sends all morphisms of $S$ into [[isomorphism]] in the category of left $T$-modules.

P. M. Cohn has shown that there is a universal object $R\to Q_S R$ in the category of $S$-inverting morphisms. The ring $Q_S R$ (and more precisely the universal morphism itself) are called the **universal localization** or **Cohn localization** of the ring $R$ at $S$.

## Properties

Cohn localization induces a hereditary torsion theory, i.e. a localization endofunctor on the category of all modules, but it lacks good flatness properties at the level of full module category. However when restricted to the subcategory of finite-dimensional projectives it has all good properties -- it is not any worse than [[Ore localization]].

Universal localization is much used in algebraic K-theory, algebraic L-theory and surgery theory -- see [[Andrew Ranicki]]'s slides in the references at Cohn localization and his papers, specially the series with Amnon Neeman.

## Related concepts

* [[localization of a ring]]

* [[localization of a commutative ring]]

* [[noncommutative localization]]

## References

The original articles:

* [[Paul M. Cohn]], _Free rings and their relations_, Academic Press (1971) [[pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/cohnfree.pdf)]

* [[Paul M. Cohn]], _Inversive localization in noetherian rings_, Communications on Pure and Applied Mathematics __26__ 5-6 (1973 ) 679-691 [[doi:10.1002/cpa.3160260510](http://dx.doi.org/10.1002/cpa.3160260510)]

Reviews and lecture notes:

* [[V. Retakh]], R. Wilson, _Advanced course on quasideterminants and universal localization_ (2007) [[[RetakhWilson-Quasideterminants.pdf:file]]]

*  [[Andrew Ranicki]] (ed.),  _Noncommutative localization in algebra and topology_, (Proceedings of Conference at ICMS, Edinburgh, 29-30 April 2002) London Math. Soc. Lecture Notes Series **330** Cambridge University Press (2006) [[pdf](http://www.maths.ed.ac.uk/~aar/books/nlat.pdf)]

containing this article on applications to [[topology]]:

* [[Andrew Ranicki]], *Noncommutative localization in topology* [[arXiv:math/0303046](https://arxiv.org/abs/math/0303046)]

reviewed in:

* [[Andrew Ranicki]], _Noncommutative localization in algebra and topology_, talk at Knot theory meeting (2008) [slides [pdf](http://www.maths.ed.ac.uk/~aar/slides/heidel2.pdf)]

* [[Andrew Ranicki]], _Noncommutative localization_, [[Pierre Vogel]] 65th birthday conference, Paris, 27 October 2010 [[slides pdf](http://www.maths.ed.ac.uk/~aar/slides/vogel.pdf)]

See also:

* [[Z. Škoda]], _Noncommutative localization in noncommutative geometry_, in (NLOC, above) pp. 220--313, [math.QA/0403276](http://arxiv.org/abs/math.QA/0403276)

On localization with inverses just from one side:

* [[P. M. Cohn]], _One-sided localization in rings_, J. Pure Appl. Algebra __88__ (1993), no. 1-3, 37–42 []()

Universal localization of [[group rings]] (and connections to certain noncommutative rational function rings and [[Fox derivative]]s) is discussed in

* [[M. Farber]], [[Pierre Vogel]], _The Cohn localization of the free group ring_, Math. Proc.  Camb. Phil. Soc. (1992) 111, 433  ([pdf](http://www.maths.ed.ac.uk/~aar/papers/fv.pdf))

[[!redirects Cohn localization]]
[[!redirects Cohn universal localization]]


universal polynomial


universal principal bundle

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For $G$ a [[topological group]] there is a notion of $G$-[[principal bundles]] $P \to X$ over any [[topological space]] $X$. Under [[continuous maps]] $f : X \to Y$ there is a notion of [[pullback]] of principal bundles $f^* : G Bund(Y) \to G Bund(X)$.

A **universal $G$-principal bundle** is a $G$-principal bundle, which is usually written $E G \to B G$, such that for every [[CW-complex]] $X$ the map

$$
  [X, B G] \to G Bund(X)/_\sim
$$

from [[homotopy]] classes of [[continuous functions]] $X \to B G$ given by $[f] \mapsto f^* E G$, is an [[isomorphism]].

In this case one calls $B G$ a [[classifying space]] for $G$-principal bundles.

The universal principal bundle is characterized, up to equivalence, by its total space $E G$ being [[contractible]].

More generally, we can ask for a universal bundle for _numerable_ bundles, that is principal bundles which admit a trivialisation over a [[numerable]] [[open cover]]. Such a bundle exists, and classifies numerable bundles over _all_ topological spaces, not just [[paracompact]] spaces or CW-complexes.

## Details

See at _[[classifying space]]_.

## Related concepts

* [[classifying space]]

* [[Milnor construction]]

* [[universal vector bundle]], [[universal complex line bundle]]

* [[universal principal infinity-bundle]].

## References

### General

Among the earliest references that consider the notion of universal bundles is

* [[Shiing-shen Chern]], Sun, ... (1949)

A review is for instance in

* Stephen Mitchell, _Universal principal bundles and classifying spaces_ ([pdf](http://www-math.mit.edu/~mbehrens/18.906/prin.pdf))
* [[Dale Husemoeller]], _Fibre bundles_, McGraw-Hill 1966 (300 p.); Springer Graduate Texts in Math. __20__, 2nd ed. 1975 (327 p.), 3rd. ed. 1994 (353 p.) [gBooks](https://books.google.hr/books?id=DPr_BSH89cAC)  [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/husemoller)

For more see the references at _[[classifying space]]_.

### For equivariant bundles
 {#ReferencesForEquivariantBundles}

Discussion of universal [[equivariant principal bundles]]:

* {#tomDieck69} [[Tammo tom Dieck]], _Faserbündel mit Gruppenoperation_, Arch. Math 20, 136–143 (1969) ([doi:10.1007/BF01899003](https://doi.org/10.1007/BF01899003))

* {#Lashof82} [[Richard Lashof]], _Equivariant bundles_, Illinois J. Math. 26(2): 257-271, 1982 ([doi:10.1215/ijm/1256046796.full](https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-26/issue-2/Equivariant-bundles/10.1215/ijm/1256046796.full))

* {#May90} [[Peter May]], _Some remarks on equivariant bundles and classifying spaces_, Théorie de l'homotopie, Astérisque, no. 191 (1990), 15 p. ([numdam:AST_1990__191__239_0](http://www.numdam.org/item/?id=AST_1990__191__239_0))

* {#MurayamaShimakawa95} [[Mitutaka Murayama]], [[Kazuhisa Shimakawa]], _Universal equivariant bundles_, Proc. Amer. Math. Soc. 123 (1995), 1289-1295 ([doi:10.1090/S0002-9939-1995-1231040-9](https://doi.org/10.1090/S0002-9939-1995-1231040-9))

* {#UribeLueck14} [[Bernardo Uribe]], [[Wolfgang Lück]], _Equivariant principal bundles and their classifying spaces_, Algebr. Geom. Topol. 14 (2014) 1925-1995 ([arXiv:1304.4862](https://arxiv.org/abs/1304.4862), [doi:10.2140/agt.2014.14.1925](http://dx.doi.org/10.2140/agt.2014.14.1925))

* {#GuillouMayMerling17} [[Bertrand Guillou]], [[Peter May]], [[Mona Merling]] _Categorical models for equivariant classifying spaces_, Algebr. Geom. Topol. 17 (2017) 2565-2602 ([arXiv:1201.5178](https://arxiv.org/abs/1201.5178), [doi:10.2140/agt.2017.17.2565](https://doi.org/10.2140/agt.2017.17.2565))

[[!redirects universal principal bundles]]


universal principal infinity-bundle

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Yoneda lemma
+--{: .hide}
[[!include Yoneda lemma - contents]]
=--
#### $(\infty,1)$-Topos theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _universal principal ∞-bundle_ over an [[∞-group]]-object in an [[(∞,1)-topos]] $\mathbf{H}$ is a morphism $\mathbf{E}G \to \mathbf{B}G$ in a _1-categorical model_ $C$ for $\mathbf{H}$ (a [[homotopical category]]) such that every $G$-[[principal ∞-bundle]] $P \to X$ in $\mathbf{H}$ is modeled in $C$ by an (ordinary) [[pullback]] of $\mathbf{E}G \to \mathbf{B}G$.

Notice that in the proper [[nLab:(∞,1)-topos]]-context the _universal_ $G$-[[nLab:principal ∞-bundle]] for an [[nLab:∞-group]] $G$ is nothing but the point inclusion $* \to \mathbf{B}G$ into the [[nLab:delooping]] of $G$: every $G$-principal $\infty$-bundle $P \to X$ is the [[(∞,1)-pullback]]

$$
  \array{
    P &\to& *
    \\
    \downarrow &{}^{\mathllap{\simeq}}\swArrow& \downarrow
    \\
    X &\stackrel{}{\to}& \mathbf{B}G
  }
$$

of the point in $\mathbf{H}$, namely the [[nLab:homotopy kernel]] of its classifying map $g$. In other words, in a full $(\infty,1)$-categorical context the notion of universal bundle disappears. It is a notion genuinely associated with 1-categorical _models_ for $\mathbf{H}$.

## Standard models

Assume that we have a [[homotopical category]] model $C$ for $\mathbf{H}$ that has the structure of a [[category of fibrant objects]]. Notably this can be the [[full subcategory]] on fibrant objects of a [[model structure on simplicial presheaves]].

### By fibrations

By standard results on [[homotopy pullback]]s every morphism $\mathbf{E}G \to \mathbf{B}'G$ that

1. is a fibration

1. fits into a diagram

   $$
     \array{
       \mathbf{E}G &\stackrel{\simeq}{\to}& *
       \\
       \downarrow && \downarrow
       \\
       \mathbf{B}'G &\stackrel{\simeq}{\to}& \mathbf{B}G
     }
   $$

   with the horizontal morphisms being weak equivalences;

is a model for the universal $G$-principal $\infty$-bundle.

### By path fibrations

A standard construction of a fibration $\mathbf{E}G \to \mathbf{B}G$ is above is obtained as follows:

by standard results on [[homotopy pullback]]s, we have that the bundle $P \to X$ classified by a morphism $X \stackrel{\simeq}{\leftarrow} \hat X\to \mathbf{B}G$ is given by the [[limit]]

$$
  \array{
     P &\to& &\to& *
     \\
     \downarrow && && \downarrow
     \\
     && (\mathbf{B}G)^I &\stackrel{\simeq}{\to}& \mathbf{B}G
     \\
     \downarrow && \downarrow^{\mathrlap{\simeq}}
     \\
     * &\to& \mathbf{B}G
  }
  \,,
$$

where $(\mathbf{B}G)^I $ is a [[path space object]] for $\mathbf{B}G$.

This limit may be computed as two consecutive [[pullback]]s

$$
  \array{
     P &\to& \mathbf{E}G &\to& *
     \\
     \downarrow && && \downarrow
     \\
     && (\mathbf{B}G)^I &\to& \mathbf{B}G
     \\
     \downarrow && \downarrow
     \\
     * &\to& \mathbf{B}G
  }
  \,.
$$

The intermediate pullback

$$
  \mathbf{E}G := (\mathbf{B}G)^I \times_{\mathbf{B}G} *
$$

is the path fibration over $\mathbf{B}G$. By the [[factorization lemma]] we have that the projection $\mathbf{E}G \to \mathbf{B}G$ is indeed a fibration and by the fact that the acyclic fibration $(\mathbf{B}G)^I \to \mathbf{B}G$ is preserved under pullback that indeed $\mathbf{E}G \to *$ is a weak equivalence.

### By decalage

For $X$ a [[Kan complex]] with a single vertex, the [[decalage]] construction $Dec X \to X$ is a [[Kan fibration]] that fits into a diagram

$$
  \array{
    Dec X &\stackrel{\simeq}{\to}& *
    \\
    \downarrow && \downarrow
    \\
    X &\stackrel{=}{to}& X
  }
  \,.
$$

For $G$ a [[simplicial group]] the standard simplicial model for the [[delooping]] of $G$ in $\mathbf{H} = $[[∞Grpd]] is denoted $\bar W G$. This is a Kan complex with a single vertex and $Dec \bar W G$ is the standard model for the universal [[simplicial principal bundle]], traditionally written $W G$.

$$
  Dec \bar W G = W G \to \bar W G
  \,.
$$

These constructions are functorial and hence extend to models for [[(∞,1)-topos]]es by a [[model structure on simplicial presheaves]].

The model $W G$ for the universal $G$-principal bundle has the special property that it is a [[groupal model for universal principal ∞-bundles]].

## Related concepts

* [[universal principal bundle]]

* [[principal ∞-bundle]]

* **universal principal $\infty$-bundle** , [[groupal model for universal principal ∞-bundles]]

* [[universal connection]]

* [[classifying space]], [[classifying stack]]

* [[moduli space]], [[moduli stack]], [[derived moduli space]]

* [[classifying topos]]

* [[subobject classifier]]

* [[classifying morphism]]

[[!redirects universal principal ∞-bundle]]

[[!redirects universal principal ∞-bundles]]

[[!redirects universal principal infinity-bundles]]


universal property > history

< [[universal property]]


universal quantifier

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea
 {#Idea}

In [[first-order logic]], the _universal quantifier_ "$\forall$" is the [[quantifier]] used to express --- given a [[predicate]] $P$ with a [[free variable]] $t$ of [[type]] $T$ --- the [[proposition]] denoted

$$
  \underset{
    t \colon T
  }
  {\forall} P(t)
  \,,
$$

which is meant to be [[true]] if and only if $P(t)$ holds true for ALL possible [[elements]] $t$ of $T$ (all [[terms]] of [[type]] $T$) --- whence the notation "A" (even if upside-down) for this quantifier.

This notion is "dual" (in fact *[[adjoint functor|adjoint]]*, see [below](#InTermOfAdjunctions)) to the [[existential quantifier]] "$\exists$" which asserts that $P(t)$ holds true for *some* $t$.

But beware that is quite possible that $P(t)$ may be provable (in a given [[context]]) for every *[[term]]* $t$ of type $T$ that can actually be constructed in that context, and yet
$
  \underset{
    t \colon T
  }
  {\forall} \phi(t)
$
cannot be proved: The proper internal definition of universal quantification is as a [[right adjoint]] to [[context extension]] as brought out by the definition (eq:FirstDefinition) below and expanded on [further below](#InTermOfAdjunctions).

## Definition
 {#Definition}

We work in a [[logic]] in which we are concerned with which [[propositions]] entail which other propositions (in a given [[context]]); in particular, two propositions which entail each other are considered [[logical equivalence|logically equivalent]], denoted by "$\Leftrightarrow$".

Let

* $\Gamma$ be an arbitrary [[context]],

* $T$ a [[dependent type]] in $\Gamma$,

so that

* $\Delta \coloneqq \Gamma, T$ is the [[context extension]] of $\Gamma$ by a [[free variable]] $t$ of [[type]] $T$.

We assume that we have a [[weakening rule]] that allows us to interpret any proposition $Q$ in $\Gamma$ as

* $T^\ast Q$ being a proposition in $\Delta$.

Then for

* $P$ a [[proposition]] in $\Delta$, which we think of as a [[predicate]] in $\Gamma$ with the [[free variable]] of type $T$,

its __universal quantification__ is any proposition $\underset{T}{\forall} P$ in $\Gamma$ such that, *given any proposition $Q$ in $\Gamma$*, we have:

\[
  \label{FirstDefinition}
  Q \;\vdash_{\Gamma}\;
  \underset{
    T
  }{\forall}
  \;
  P
  \;\;\;\;\;\;\;
    \Leftrightarrow
  \;\;\;\;\;\;\;
  T^\ast Q \;\vdash_{\Gamma, T}\; P
  \,.
\]

It is then a theorem that universal quantification of $P$, if it exists, is unique.  The existence is typically asserted as an [[axiom]] in a particular logic, but this may be also be deduced from other principles (as in the topos-theoretic discussion [below](#InTermOfAdjunctions)).

Often one makes the appearance of the [[free variable]] in $P$ explicit by thinking of $P$ as a [[propositional function]] and writing $P(t)$ instead.  Then the rule appears as follows:

\[
  Q \;\vdash_{\Gamma}\;
  \underset{
    t \colon T
  }{\forall}  P(t)
  \;\;\;\;\;\;
    \Leftrightarrow
  \;\;\;\;\;\;
  T^\ast Q \;\vdash_{\Gamma, t \colon T}\; P(t)
  \,.
\]

In terms of [[semantics]] (as for example topos-theoretic semantics; see [below](#InTermOfAdjunctions)), the [[context extension]] from $Q$ to $T^\ast Q$ corresponds to [[pullback|pulling back]] along a [[product]] projection $\sigma(T) \times A \to A$, where $\sigma(T)$ is the interpretation of the type $T$, and $A$ is the interpretation of $\Gamma$. In other words, if a statement $Q$ read in context $\Gamma$ is interpreted as a [[subobject]] of $A$, then the statement $Q$ read in context $\Delta = \Gamma, t \colon T$ is interpreted by pulling back along the [[projection]], obtaining a subobject of $\sigma(T) \times A$.

As observed by [[Lawvere]], we are not particularly constrained to product projections; we can pull back along any map $f \colon B \to A$. (Often we have a class of [[display maps]] and require $f$ to be one of these.) Alternatively, any pullback functor $f^\ast\colon Set/A \to Set/B$ can be construed as pulling back along an object $X = (f \colon B \to A)$, i.e., along the unique map $!\colon X \to 1$ corresponding to an object $X$ in the slice $Set/A$, since we have the identification $Set/B \simeq (Set/A)/X$.

In [[natural deduction]] the [[inference rules]] for the universal quantifier are given as

$$\frac{\Gamma, x:A \vdash P(x) \; \mathrm{prop}}{\Gamma \vdash \forall x:A.P(x) \; \mathrm{prop}} \qquad \frac{\Gamma, x:A \vdash P(x) \; \mathrm{true}}{\Gamma \vdash \forall x:A.P(x) \; \mathrm{true}} \qquad \frac{\Gamma \vdash \forall x:A.P(x) \; \mathrm{true}}{\Gamma, x:A \vdash P(x) \; \mathrm{true}}$$

## Internal universal quantifier in set theory

In [[set theory]], recall that a [[predicate]] on a [[set]] $A$ in the [[internal logic of set theory]] is represented by the [[preimage]] $i^*(a)$ of an [[injection]] $i:B \hookrightarrow A$. Because $i$ is an [[injection]], each preimage $i^*(a)$ is a [[subsingleton]] for all $a \in A$, which represents the internal [[propositions]] of the [[set theory]]. The **internal universal quantifier** is represented by the [[Cartesian product]] of the family of sets $(i^*(a))_{a \in A}$:

$$\forall a \in A.i^*(a) \coloneqq \prod_{a \in A} i^*(a)$$

## In topos theory / in terms of adjunctions
 {#InTermOfAdjunctions}

In terms of the [[internal logic]] in some ambient [[topos]] $\mathcal{E}$,

* a [[type]] $X$ is given by an [[object]] $X \in \mathcal{E}$,

* the [[predicate]] $\phi$ is a [[(-1)-truncated]] object of the [[over-topos]] $\mathcal{E}/X$;

* a [[truth value]] is a [[(-1)-truncated]] object of $\mathcal{E}$ itself.

Universal quantification is essentially the restriction of the [[direct image]] [[right adjoint]] of a canonical [[étale geometric morphism]] $X_\ast$,

$$
    \mathcal{E}/X
    \stackrel{\overset{X_!}{\to}}{\stackrel{\overset{X^*}{\leftarrow}}{\underset{X_*}{\to}}}
    \mathcal{E}
  \,,
$$

where $X^\ast$ is the functor that takes an object $A$ to the [[product]] projection $\pi \colon X \times A \to X$, where $X_! = \Sigma_X$ is the [[dependent sum]] (i.e., forgetful functor taking $f \colon A \to X$ to $A$) that is left adjoint to $X^\ast$, and where $X_\ast = \Pi_X$ is the [[dependent product]] operation that is right adjoint to $X^\ast$.

The dependent product operation restricts to [[propositions]] by pre- and postcomposition with the [[truncated|truncation]] [[adjunctions]]

$$
  \tau_{\leq -1} \mathcal{E}
  \stackrel{\overset{\tau_{\leq -1}}{\leftarrow}}{\underset{}{\hookrightarrow}}
  \mathcal{E}
$$

to give universal quantification over the domain (or context) $X$:

$$
  \array{
    \mathcal{E}/X
    &
    \stackrel{\overset{X_!}{\to}}{\stackrel{\overset{X^*}{\leftarrow}}{\underset{X_*}{\to}}}
    &
    \mathcal{E}
    \\
    {}^\mathllap{\tau_{\leq_{-1}}}\downarrow \uparrow
    &&
    {}^\mathllap{\tau_{\leq_{-1}}}\downarrow \uparrow
    \\
    \tau_{\leq_{-1}}
    \mathcal{E}/X
    &
     \stackrel{\overset{\exists_X}{\to}}{\stackrel{\overset{}{\leftarrow}}{\underset{\forall_X}{\to}}}
    &
    \tau_{\leq_{-1}}\mathcal{E}
  }
  \,.
$$

Dually, the extra [[left adjoint]] $\exists_X$, obtained from the dependent sum $X_!$ by pre- and post-composition with the truncation adjunctions, expresses the [[existential quantifier]].  (The situation with the universal quantifier is somewhat simpler than for the existential one, since the dependent product automatically preserves $(-1)$-truncated objects (= subterminal objects), whereas the dependent sum does not.)

The same makes sense, verbatim, also in the [[(∞,1)-logic]] of any [[(∞,1)-topos]].

This interpretation of universal quantification as the right adjoint to context extension is also used in the notion of _[[hyperdoctrine]]_.

## In dependent type theory

In [[dependent type theory]], the universal quantifier is the [[propositional truncation]] of the [[dependent product type]] of a family of [[h-propositions]]:

$$\forall (x:A).B(x) \coloneqq \left[\prod_{x:A} B(x)\right]$$

The axiom of [[function extensionality]] or [[weak function extensionality]] implies that the dependent product type of a family of h-propositions is always an h-proposition.

One doesn't need all [[dependent product types]] to define universal quantifiers. The [[isProp]] types are definable without all dependent product types, by use of [[center of contraction]], which are also definable without all dependent product types.

Formation rules for the universal quantifier:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A \vdash p(x):\mathrm{isProp}(B(x))}{\Gamma \vdash \forall (x:A).B(x) \; \mathrm{type}}$$

Introduction rules for the universal quantifier:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A \vdash p(x):\mathrm{isProp}(B(x)) \quad \Gamma, x:A \vdash b(x):B(x)}{\Gamma \vdash \lambda(x:A).b(x):\forall (x:A).B(x)}$$

Elimination rules for the universal quantifier:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A \vdash p(x):\mathrm{isProp}(B(x)) \quad \Gamma \vdash f:\forall (x:A).B(x) \quad \Gamma \vdash a:A}{\Gamma \vdash f(a):B[a/x]}$$

Computation rules for the universal quantifier:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A \vdash p(x):\mathrm{isProp}(B(x)) \quad \Gamma, x:A \vdash b(x):B(x) \quad \Gamma \vdash a:A}{\Gamma \vdash \beta_\forall:\lambda(x:A).b(x)(a) =_{B[a/x]} b[a/x]}$$

Uniqueness rules for the universal quantifier:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A \vdash p(x):\mathrm{isProp}(B(x)) \quad \Gamma \vdash f:\forall (x:A).B(x)}{\Gamma \vdash \eta_\forall:f =_{\forall (x:A).B(x)} \lambda(x).f(x)}$$

Closure of universal quantifiers under h-propositions rule:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, x:A \vdash p(x):\mathrm{isProp}(B(x))}{\Gamma \vdash \mathrm{weakfunext}:\mathrm{isProp}(\forall (x:A).B(x))}$$

This means that one could define the type-theoretic internal logic of a [[Heyting category]] or [[Boolean category]] which are not [[locally cartesian closed]], for [[strongly predicative mathematics]].

## Examples

Let $\mathcal{E} = $ [[Set]], let $X = \mathbb{N}$ be the set of [[natural number]]s and let $\phi \coloneqq 2\mathbb{N} \hookrightarrow \mathbb{N}$ be the [[subset]] of even natural numbers. This expresses the [[proposition]] $\phi(x) \coloneqq IsEven(x)$.

Then the [[dependent product]]

$$
  (\phi \in Set/{\mathbb{N}})
  \mapsto
  (\prod_{x\colon X} \phi(x) \in Set)
$$

is the set of [[section]]s of the [[bundle]] $2 \mathbb{N} \hookrightarrow \mathbb{N}$. But since over odd numbers the [[fiber]] of this bundle is the [[empty set]], there is no possible value of such a section and hence the set of sections is itself the [[empty set]], so the statement "all natural numbers are even" is, indeed, [[false]].

## Related concepts

* [[necessity]], [[example]]

* [[existential quantifier]]

* [[generalized quantifier]]

[[!include logic symbols -- table]]

## References

The observation that [[substitution]] forms an [[adjoint pair]]/[[adjoint triple]] with the existential/universal quantifiers is due to

* {#Lawvere69} [[William Lawvere]], _Adjointness in Foundations_, ([tac:16](http://www.emis.de/journals/TAC/reprints/articles/16/tr16abs.html)), Dialectica 23 (1969), 281-296

* [[William Lawvere]], _Quantifiers and sheaves_, Actes, Congrès intern, math., 1970. Tome 1, p. 329 à 334 ([pdf](http://www.mathunion.org/ICM/ICM1970.1/Main/icm1970.1.0329.0334.ocr.pdf))

and further developed in

* [[William Lawvere]], _Equality in hyperdoctrines and
comprehension schema as an adjoint functor_, Proceedings of the AMS Symposium on Pure Mathematics XVII (1970), 1-14.

[[!redirects universal quantifier]]
[[!redirects universal quantifiers]]
[[!redirects universal quantification]]
[[!redirects universal quantifications]]

[[!redirects every]]


universal spacetime

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Universal spacetimes_ are [[spacetimes]] for which all [[conserved current|conserved]] symmetric [[tensor|rank-2 tensors]], constructed as contractions of [[polynomials]] from the [[metric]], the [[Riemann tensor]] and its [[covariant derivatives]] of arbitrary order, are multiples of the metric. Consequently, metrics of universal spacetimes solve [[vacuum]] [[equations of motion]] of all [[gravity|gravitational theories]], i.e. [[variational calculus|extremize]] the [[Einstein-Hilbert action]] with given [[cosmological constant]] and _arbitrary_ [[higher curvature correction]]. In the literature, universal metrics are also discussed as metrics with vanishing quantum corrections and as classical solutions to [[string theory]]. Widely known examples of universal metrics are certain [[Ricci flat]] [[pp-wave spacetimes]].

## References

The first examples are due to

* [[Daniele Amati]], [[Ctirad Klimčík]], _Nonperturbative computation of the Weyl anomaly for a class of nontrivial backgrounds_, Phys. Lett. B, 219:443–447, 1989 ([spire:269390](inspirehep.net/record/269390), 10.1016/0370-2693(89)91092-7)

* [[Gary Horowitz]], Alan R. Steif, _Spacetime singularities in string theory_, Phys. Rev. Lett. 64, 260 1990 ([doi:10.1103/PhysRevLett.64.260](https://doi.org/10.1103/PhysRevLett.64.260))

The terminology was introduced in

* A. A. Coley, [[Gary Gibbons]], S. Hervik, [[Christopher Pope]], _Metrics with vanishing quantum corrections_, Class. Quantum Grav., 25:145017, 2008 ([arXiv:0803.2438](https://arxiv.org/abs/0803.2438))

Classification results are discussed in

* {#HervikPravdaPravdova13} Sigbjørn Hervik, [[Vojtěch Pravda]], Alena Pravdová, _Type III and N universal spacetimes_, Class. Quantum Grav. 31: 215005, 2014 ([arXiv:1311.0234](https://arxiv.org/abs/1311.0234))

* Sigbjørn Hervik, Tomáš Málek, [[Vojtěch Pravda]], Alena Pravdová, _Type II universal spacetimes_, Class. Quantum Grav. 32: 245012, 2015 ([arXiv:1503.08448](https://arxiv.org/abs/1503.08448))

* Sigbjørn Hervik, [[Vojtěch Pravda]], Alena Pravdová, _Universal spacetimes in four dimensions_, JHEP 1710 (2017): 28 ([arXiv:1707.00264](https://arxiv.org/abs/1707.00264))

* Martin Kuchynka, Tomáš Málek, [[Vojtěch Pravda]], Alena Pravdová, _Almost universal spacetimes in higher-order gravity theories_ ([arXiv:1810.02178](https://arxiv.org/abs/1810.02178))

Discussion for [[black hole spacetimes]]:

* Sigbjørn Hervik, Marcello Ortaggio, _Universal Black Holes_, J. High Energ. Phys. 2020, 47 (2020) ([arXiv:1907.08788](https://arxiv.org/abs/1907.08788))

Discussion for [[Einstein-Yang-Mills theory]]:

* Martin Kuchynka, _Einstein-Yang-Mills fields immune to quantum corrections_ ([arXiv:2001.03768](https://arxiv.org/abs/2001.03768))

See also:

* Metin Gurses, Yaghoub Heydarzade, Cetin Senturk, *Kerr-Schild-Kundt Metrics in Generic Gravity Theories with Modified Horndeski Couplings* ([arXiv:2109.01244](https://arxiv.org/abs/2109.01244))

[[!redirects universal spacetimes]]


universal Vassiliev invariant

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Knot theory
+--{: .hide}
[[!include knot theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
  {#Idea}

A _universal Vassiliev invariant_ is a [[Vassiliev knot invariant]] with [[coefficients]] in [[formal power series]] of [[Jacobi diagrams]] in [[Planck's constant]] $\hbar$ whose leading term in $\hbar$ on a given [[singular knot]] is its [[chord diagram]].

By evaluating (framed) [[weight systems]] on the [[Jacobi diagram]] [[coefficients]] of a universal Vassiliev invariant it defines a [[function]] from framed [[weight systems]] on [[Jacobi diagrams]], [[chord diagrams modulo 4T are Jacobi diagrams modulo STU|equivalently]] on [[round chord diagrams]] to actual ([[ground field]]-valued) [[Vassiliev knot invariants]].

This construction in particular yields the proof that [[weight systems are the associated graded objects of Vassiliev invariants]].

In principle there is a vector space of universal Vassiliev invariants, but all that appear in the literature tend to agree ([BNGRT 97](#BNGRT97))
and are identified  ([AF 96](#AF96)) with the un-traced [[Wilson loop observable]] of [[perturbative quantization of 3d Chern-Simons theory|perturbative Chern-Simons theory]]:

> from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

The main other known universal Vassiliev iunvariant is the [[Kontsevich integral]] (see [Lescop 02](#Lescop02) for the comparison).

\linebreak

## Ingredients
 {#Ingredients}

Write

\[
  \label{CSFeynmanDiagrams}
  \mathcal{D}^{cs} \;\in\; Set^{\mathbb{N}}
\]

for the [[graded set]] of [[isomorphism classes]] of  trivalent framed knot graphs -- [[Feynman diagrams]] for [[Chern-Simons theory]] in the presence of [[Wilson loops]], called "Wilson graphs" [AF 96, Section 1](#AF96), slightly differing from the un-framed knot graphs in [CCRL 02](#CattaneoCottaRamusinoLongoni02).

By definition, a [[graph]] $\Gamma \in \mathcal{D}^{cs}$ must have [[even number]] of [[vertices]], and its degree is half that number ([AF 96, (2.9)](#AF96))

$$
  \Gamma \;\in\; \mathcal{D}^{cs}_{(\# Vert_\Gamma)/2}
  \,.
$$

For any $\Gamma \in \mathcal{D}^{cs}$ write

$$
  Aut(\Gamma)
  \;\in\;
  Grp
$$

for its [[automorphism group]], a [[finite group]] whose [[order of a group|order]] we denote by

\[
  \label{AutomorphismGroup}
  \left\vert Aut(\Gamma)\right\vert
  \;\in\;
  \mathbb{N}
  \,.
\]

Write

\[
  \label{KnotGraphComplex}
  \mathcal{G}^\bullet \;\in\; Ch^\bullet(\mathbb{R})
\]

for the framed [[knot graph complex]] and

$$
  H^\bullet(\mathcal{G})_3
  \subset
  H^\bullet(\mathcal{G})
$$

for the sub-vector space of its cohomology spanned by cocycles made of trivalent graphs.

Write

\[
  \label{JacobiDiagramsModuloSTU}
  \mathcal{A}_\bullet
  \;\coloneqq\;
  Span
  \left(
    \mathcal{D}^t
  \right)/(STU)
  \;
  \in
  \;
  Vect_\bullet(\mathbb{R})
\]

for the [[graded vector space]] of [[Jacobi diagrams]] [[quotient vector space|modulo]] the [[STU-relations]].

Also write

\[
  \label{LinearizationMaps}
  \array{
    && \mathcal{D}^{cs}
    \\
    &
      {}^{\mathllap{ [-]_{{}_{\mathcal{A}}} }}
      \swarrow
    &&
     \searrow^{\mathrlap{ [-]_{\mathcal{G}} }}
    \\
    \mathcal{A}_\bullet
    &&
    &&
    \mathcal{G}^\bullet
  }
\]

for the [[functions]] that send a [[graph]] to the defining [[linear basis|basis]] [[vector]] that it represents in these [[vector spaces]], respectively.

The space $\mathcal{A}_\bullet$ is the graded [[linear dual]] of the space of [[weight systems]]

$$
  \mathcal{W}^\bullet
  \;\coloneqq\;
  (\mathcal{A}_\bullet)^\ast
  \,.
$$

Hence if we regard

$$
  \mathcal{A}_\bullet
  \;=\;
  (\mathcal{A}^{-\bullet}, d= 0)
$$

as a [[cochain complex]] in non-positive degree with vanishing [[differential]], then its [[tensor product of cochain complexes]] with the knot graph complex is the [[cochain complex]] whose closed elements are the graded [[linear maps]] from $\mathcal{W}^\bullet$ to the [[cochain cohomology]] $H^\bullet(GraphCplx)$ of the [[knot graph complex]]:

\[
  \label{TensorCochainsAsMaps}
  H^0
  \big(
    \mathcal{A}_\bullet
    \otimes
    \mathcal{G}^\bullet
  \big)
  \;\simeq\;
  Hom
  \big(
    \mathcal{W}^\bullet
    ,
    H^\bullet(\mathcal{G})
  \big)
\]

## Statement
  {#Statement}

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

The element

\[
  \label{WisonLoopObservable}
    \left\langle
      Tr_{(-)}
      \text{P}\exp
      \left(
        \int_{(-)} A
      \right)
    \right\rangle
    \;\coloneqq\;
    \underset{
      n \in \mathbb{N}
    }{\sum}
      \hbar^n
    \underset{
     \Gamma \in \mathcal{D}^{cs}_n
    }{\sum}
    \left(
      \frac{1}{\left\vert \Gamma\right\vert}
      \,
      [\Gamma]_{\mathcal{A}} \otimes [\Gamma]_{\mathcal{G}}
    \right)
   \;
    \in
    \;\;
    \mathcal{A}_\bullet \otimes \mathcal{G}^\bullet
\]

(hence the [[sum]] over [[Feynman diagrams]] (eq:CSFeynmanDiagrams) of the [[tensor product of chain complexes|tensor product]] of their [[images]] (eq:LinearizationMaps) in [[Jacobi diagrams]] [[quotient vector space|modulo]] the [[STU-relations]] (eq:JacobiDiagramsModuloSTU) and in the [[knot graph complex]] (eq:KnotGraphComplex), respectively, weighted by the inverse [[order of a group|order]] of their [[automorphism group]] (eq:AutomorphismGroup) )

is closed

$$
  d_{\mathcal{G}}
  \left\langle
      Tr_{(-)}
    \text{P}\exp
      \left(
        \int_{(-)} A
      \right)
    \right\rangle
   \;=\;
   0
$$

=--

This is [AF 96, Theorem 1](#AF96).

Hence (eq:WisonLoopObservable) defines a [[cochain cohomology]]-[[cohomology  class|class]]

$$
  \begin{aligned}
    \left\langle
      Tr_{(-)}
      \text{P}\exp
      \left(
        \int_{(-)} A
      \right)
    \right\rangle
    \;\coloneqq\;
    H^0
    \big(
       \mathcal{A}_\bullet \otimes \mathcal{G}^\bullet
    \big)
    \\
    &
    \;\simeq\;
    Hom
    \big(
      \mathcal{W}^\bullet
      ,
      H^\bullet(\mathcal{G})
    \big)
  \end{aligned}
$$

and hence, by (eq:TensorCochainsAsMaps), it defines a graded [[linear function]]

$$
  \mathcal{W}^\bullet
  \overset{
    \left\langle
      Tr_{(-)}
      \text{P}\exp
      \left(
        \int_{(-)} A
      \right)
    \right\rangle
  }{
    \longrightarrow  }
  H^\bullet(\mathcal{G})_3
  \hookrightarrow
  H^\bullet(\mathcal{G})
$$

from [[weight systems]] on [[Jacobi diagrams]] ([[chord diagrams modulo 4T are Jacobi diagrams modulo STU|equivalently]] on [[round chord diagrams]]) to the [[cochain cohomology]] of the framed [[knot graph complex]] spanned by trivalent graphs.

According to [CCRL 02, Prop. 7.6](#CattaneoCottaRamusinoLongoni02) this map is a [[bijection]].

To see this, use 1) [AF 96, Theorem 5, Condition U2](#AF96) to find that the map is an [[injection]], and 2) the fact that [[weight systems are associated graded of Vassiliev invariants]].

## Related concepts

* [[weight systems are cohomology of knot graph complex]]

## References

### General

Textbook account

* {#Lescop20} [[Christine Lescop]], _Invariants of links and 3-manifolds from graph configurations_ ([arXiv:2001.09929](https://arxiv.org/abs/2001.09929))

Original articles:

* {#Kontsevich93} Theorem 2.3 of: [[Maxim Kontsevich]], _Vassiliev's knot invariants_, Advances in Soviet Mathematics, Volume 16, Part 2, 1993 ([pdf](http://pagesperso.ihes.fr/~maxim/TEXTS/VassilievKnot.pdf))

* [[Le Tu Quoc Thang]], [[Jun Murakami]], _The universal Vassiliev-Kontsevich invariant for framed oriented links_, Compositio Math. 102 (1996), 42–64 ([arXiv:hep-th/9401016](https://arxiv.org/abs/hep-th/9401016))

* {#BarNatan95} [[Dror Bar-Natan]], 4.4.2 of: _On the Vassiliev knot invariants_, Topology Volume 34, Issue 2, April 1995, Pages 423-472 (doi:10.1016/0040-9383(95)93237-2, [pdf](https://www.math.toronto.edu/drorbn/papers/OnVassiliev/OnVassiliev.pdf))

* [[Le Tu Quoc Thang]], [[Jun Murakami]], _Representation of the category of tangles by Kontsevich's iterated integral_,  Comm. Math. Phys., Volume 168, Number 3 (1995), 535-562. ([euclid.cmp/1104272488](https://projecteuclid.org/euclid.cmp/1104272488))

* [[Daniel Altschuler]], [[Laurent Freidel]], _On Universal Vassiliev Invariants_, Commun. Math. Phys. **170** (1995) 41-62 [[arXivhep-th/9403053](https://arxiv.org/abs/hep-th/9403053), [doi:10.1007/BF02099438](https://doi.org/10.1007/BF02099438)]

* {#AF96} [[Daniel Altschuler]], [[Laurent Freidel]], Theorem 5 of: _Vassiliev knot invariants and Chern-Simons perturbation theory to all orders_, Commun. Math. Phys. *187** (1997) 261-287 [[arXiv:q-alg/9603010](https://arxiv.org/abs/q-alg/9603010), [doi:10.1007/s002200050136](https://doi.org/10.1007/s002200050136)]

* [[Le Tu Quoc Thang]], [[Jun Murakami]], Tomotada Ohtsuki, _On a universal perturbative invariant of 3-manifolds_, Topology Volume 37, Issue 3, May 1998, Pages 539-574 (https://doi.org/10.1016/S0040-9383(97)00035-9)

Review:

* {#BarNatanStoimenow97} [[Dror Bar-Natan]], Alexander Stoimenow, _The Fundamental Theorem of Vassiliev Invariants_ ([arXiv:q-alg/9702009](https://arxiv.org/abs/q-alg/9702009))

* {#ChmutovDuzhinMostovoy11} [[Sergei Chmutov]], [[Sergei Duzhin]], [[Jacob Mostovoy]], Section 8.8 of: _Introduction to Vassiliev knot invariants_, Cambridge University Press, 2012 ([arxiv:1103.5628](http://arxiv.org/abs/1103.5628), [doi:10.1017/CBO9781139107846](https://doi.org/10.1017/CBO9781139107846))

* [[David Jackson]], [[Iain Moffat]], Section 18 of: _An Introduction to Quantum and Vassiliev Knot Invariants_, Springer 2019 ([doi:10.1007/978-3-030-05213-3](https://link.springer.com/book/10.1007/978-3-030-05213-3))

### Comparison

Comparison between the two main [[universal Vassiliev invariants]]: the [[Kontsevich integral]] and the [[Wilson loop observable]] of [[perturbative quantization of 3d Chern-Simons theory|perturbatively quantized]] [[Chern-Simons theory]]:

* {#Lescop02} [[Christine Lescop]], _On configuration space integrals for links_, Geom. Topol. Monogr. 4 (2002) 183-199 ([arXiv:math/0211062](https://arxiv.org/abs/math/0211062))

### Special values

Computation of the [[perturbative quantization of 3d Chern-Simons theory|perturbative Chern-Simons]]  [[Wilson loop observable]] ([[universal Vassiliev invariant]]) of the [[unknot]] ("[[Wheels theorem]]"):

* [[Dror Bar-Natan]], [[Le Tu Quoc Thang]], [[Dylan Thurston]], _Two applications of elementary knot theory to Lie algebras and Vassiliev invariants_, Geom. Topol. Volume 7, Number 1 (2003), 1-31 ([euclid.gt/1513883092](https://projecteuclid.org/euclid.gt/1513883092))

following

* {#BNGRT97} [[Dror Bar-Natan]], [[Stavros Garoufalidis]], [[Lev Rozansky]], [[Dylan Thurston]], _Wheels, wheeling, and the Kontsevich integral of  the unknot_ ([q-alg/9703025](http://arxiv.org/abs/q-alg/9703025))

[[!redirects universal Vassiliev invariants]]


universal vector bundle

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[topology]] a _universal vector bundle_ of some [[rank of a vector bundle|rank]] $n$ is a [[vector bundle]] $\zeta_n \to B GL(n)$ (over a base space to be called a _[[classifying space]]_) such that every other [[vector bundle]] $E \to X$ of rank $n$ over a suitably [[nice topological space]] ([[paracompact topological space]]) arises as the [[pullback bundle]] $E \simeq f^\ast \zeta_n$ of the universal bundle, along some morphisms ([[continuous function]]) $f \colon X \to B GL(n)$ which is unique up to [[homotopy]]:

$$
  \array{
    \zeta_n &\longrightarrow& E GL(n)
    \\
    \downarrow &(pb)& \downarrow
    \\
    X &\underset{f}{\longrightarrow}& B GL(n)
  }
  \,.
$$

The _universal real vector_ $\zeta_n$ of rank $n$ is the [[vector bundle]] which is [[associated bundle|associated]] to the [[universal principal bundle]] $E GL(n) \to B GL(n)$ (with [[structure group]] the [[general linear group]]) over the given [[classifying space]], equivalently to $E O(n) \to B O(n)$:

$$
  \zeta_n \coloneqq (E O(n))\underset{O(n)}{\times} \mathbb{R}^n
  \,.
$$

Similarly for [[complex vector bundles]]

$$
  \zeta^{\mathbb{C}}_n \coloneqq (E U(n))\underset{U(n)}{\times} \mathbb{R}^{2n}
  \,.
$$

etc.

## Constructions

### Via Grassmannians and Stiefel manifolds

For $n, k \in \mathbb{N}$, and $n \leq k$, there is the _[[Grassmannian manifold]]_ given as the [[coset]] [[topological space]]

$$
  Gr_n(k)
    \coloneqq
  O(k)/(O(n)\times O(k-n))
  \,.
$$

Similarly, the _[[Stiefel manifold]]_ is the [[coset]]

$$
  V_n(k) \coloneqq O(k)/O(n)
  \,.
$$

The [[quotient]] [[projection]]

$$
  V_{k-n}(k)\longrightarrow Gr_n(k)
$$

is an $O(n)$-[[principal bundle]], with [[associated bundle]] $V_n(k)\times_{O(n)} \mathbb{R}^n$ a [[vector bundle]] of [[rank]] $n$. In the limit ([[colimit]]) that $k \to \infty$ is this gives a presentation of the $O(n)$-[[universal principal bundle]] and of the [[universal vector bundle]] of rank $n$, respectively.. The base space $Gr_n(\infty)\simeq_{whe} B O(n)$ is the [[classifying space]] for $O(n)$-[[principal bundles]] and rank $n$ vector bundles.

## Examples

* [[universal complex line bundle]]

* [[tautological line bundle]]

## References

Textbook accounts include

* {#Kochmann96} [[Stanley Kochmann]], section 1.3 of of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* {#Hatcher} [[Allen Hatcher]], _Vector bundles and K-Theory_, (partly finished book) [web](http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html)

[[!redirects universal vector bundles]]

[[!redirects universal complex vector bundle]]
[[!redirects universal complex vector bundles]]


universality

## In physics

See [[universality class]].

## In mathematics, category theory

[[universal construction]] (having universal property), e.g. limits, representable object etc.

A [[terminal object]] is also called a universal terminal object, and an [[initial object]] is also called a universal initial object. There is also a notion of [[universal arrow]].

While every colimit has a universal property, there is also an additional  terminology. A colimit is a [[universal colimit]] if it is [[stability under pullback|stable under pullback]]s.

See also [[universe]], [[Grothendieck universe]], [[constructible universe]].


universality class

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

> This entry is about the phenomenon of [[universality]] in [[physics]]. See also [[universal property]] for a (different) concept of a similar name in [[mathematics]].

***

#Contents#
* table of contents
{:toc}

## Idea ##

In [[physics]], **universality** is a term which means the following:

+-- {: .standout}
Under the proper conditions, different systems can exhibit the same behaviour, as measured by quantitative indices, if they meet the same qualitative criteria.  Sets of systems which are equivalent in this manner are known as **universality classes**.
=--

If we have some complicated phenomenon we can't understand directly, and we figure out (or make a good stab at guessing) the universality class to which it belongs, we can make testable predictions about the complicated thing by working with a simpler member of that universality class.  Membership in a universality class depends on properties like how many spatial [[dimensions]] a system lives in, [[symmetry|symmetries]] and the like.  People have identified universality classes, with varying degrees of rigour.  Lots of them have names, sometimes even with dashes inside; the ones we understand less well and aren't so familiar with get abstruse symbols for labels instead.  A non-exhaustive tabulation of these labels might look something like this:

Stable Distributions | Equilibrium | Random Matrices | Non-Equilibrium | Extreme-Value Distributions | Dynamical Maps |
--------------|-------------------|-----------------|-----------------|--------------|-----------|
Gaussian      | 2D Ising          | [[unitary group|Unitary]]         | Directed percolation | Gumbel  | 1D Feigenbaum |
Cauchy        | 2D Potts          | [[orthogonal group|Orthogonal]]      | Dynamic percolation  | Fréchet | 2D Volume-preserving |
Lévy          | 2D Tricritical Ising    | [[symplectic group|Symplectic]] | CDP  | Weibull | $\vdots$ |
               | 2D Tricritical Potts | $U(N+M)/U(N)\times U(M)$ | TDP | | |
               | 2D Other RSOS | $ Sp(N+M)/Sp(N)\times Sp(M)$ | Manna | | |
               | 3D Ising | $ Sp(2N) $ | Edwards--Wilkinson | | |
               | 3D Potts | $ SO(2N) $ | KPZ | | |
               | $\vdots$ | $ O(N+M) / O(N) \times O(M) $ | $\vdots$ | | |
               |   | $ SO(2N) / U(N) $ |   | | |
               |   | $ Sp(2N) / U(N) $ |   | | |

The general concept of [[renormalization]] is really important here.  The universality classes we understand best correspond to fixed points of renormalization-group transforms.

The first column, "stable distributions", basically comes from the [[central limit theorem]] and the ways in which the conditions necessary for it to apply can fail to obtain.  The middle column with all the funny symbols comes from [[Élie Cartan]]'s classification of [[symmetric space|symmetric spaces]].  The 2D part of the [[equilibrium]] statistical systems can be given a taxonomy based on the [[ADE classification]] of [[Dynkin diagram|Dynkin diagrams]] and [[conformal field theories]].  As [Cardy (2010)](#Cardy2010) writes,

+-- {: .standout}
[T]he study of the [[representation theory]] of the [[Virasoro algebra]] gives a way of classifying all possible [[CFT]]s and thereby universality classes in 2d. The breakthrough in this direction came following the seminal 1984 paper of Belavin, Polyakov and Zamolodchikov (BPZ) in which they showed that, for certain special rational values of $c \lt 1$, the [[CFT]] closes with only a finite number of [[representation]]s of the Virasoro algebra, and, for, these cases, all the [[critical exponent]]s and multi-point [[correlation function]]s are calculable. Shortly thereafter Friedan, Qiu and Shenker showed that unitary CFTs (corresponding to local, positive definite Boltzmann weights) are a subset of this list, with $c = 1 - 6/m(m + 1)$ and $m$ an integer $\geq 3$. This gives rise to what might be termed the 'conformal periodic table'. The first few examples may be identified with well-known universality classes. The 'hydrogen atom' of CFT is the scaling limit of the critical Ising model, 'helium' is the tricritical Ising model, and so on. Note, however, that at the next value of $c = 4/5$ two possible 'isotopes' arise. In the second, corresponding to the critical 3-state Potts model, not all the scaling dimensions allowed by BPZ in fact occur, but some of those that do actually appear twice. In fact the constraint of unitarity is not sufficient to determine exactly which representations actually occur in a given CFT. The answer to this is provided by demanding consistency of the theory on a [[torus]], by interchanging the interpretations of space and imaginary time [...]. For the torus, this is a [[modular form|modular]] transformation, and the requirement of modular invariance has become another powerful tool in classifying CFTs, completely solved in the case $c \lt 1$ by Cappelli, Itzykson and Zuber.
=--

Some relationships between columns can be identified. For example, dynamical surface growth and [[random matrix theory]] are, unexpectedly, linked: the [[eigenvalue]] distributions of the [[Gaussian Unitary Ensemble|Gaussian Unitary]] and [[Gaussian Orthogonal Ensemble|Gaussian Orthogonal Ensembles]] show up as surface heights in Kardar-Parisi-Zhang phenomena. See ([Takeuchi _et al._](#Takeuchi)).

## Timeline ##

The modern terminology of this subject dates to the late 1960s and early '70s.  By 1968, the quantitative concurrence of [[critical exponent]]s experimentally observed in magnetic materials and fluid phase transitions was sufficiently well established that it was reasonable to ask, "To what extent are the observed values of the critical-point exponents universal?" ([Fisher 1968](#Fisher1968); for a more recent presentation of relevant empirical figures, see [Sethna 2006](#Sethna2006), p. 267).  At the 1970 Enrico Fermi summer school in Varenna, Italy, Leo P. Kadanoff summarized recent results with a "hypothesis of universality" which the conference proceedings record as follows:

+-- {: .standout}
All phase transition problems can be divided into a small number of different classes depending upon the dimensionality of the system and the symmetries of the order state. Within each class, all phase transitions have identical behaviour in the critical region, only the names of the variables are changed. ([Kadanoff 1971](#Kadanoff1971))
=--

The following is a capsule history tabulating some important events in the historical development of the modern understanding of universality.

|Year| |Contributors     | |Event|
|----|-|-----------------|-|-----|
|1733| |de Moivre| | Suggestion of what we now know as the [[central limit theorem]]|
|1822| |Cagniard de la Tour| | Observation of critical point at end of liquid-gas coexistence curve |
|1873--75| |van der Waals, Maxwell| | Equation of state for non-ideal gases, and its application to phase transitions |
|1893| | van der Waals | | Use of critical exponents to characterize a phase transition ([Levelt Sengers 1976](#LeveltSengers1976))|
|1895| |P. Curie| | Experimental observation of demagnetization by heat; pointing out similarity between liquid-gas and ferromagnetic transitions |
|1896--1900| | Verschaffelt | | Discovery that the critical exponents predicted from van der Waals theory do not match experiment ([Levelt Sengers 1976](#LeveltSengers1976))|
|1901| |Lyapunov| | Demonstration of why the central limit theorem works|
|1908| |Smoluchowski| | Discovery that critical opalescence is due to density fluctuations across many scales |
|1937| |Landau| | General mean-field theory of phase transitions|
|1944| |Onsager| | Exact solution of 2D [[Ising model]], illuminating the breakdown of analyticity and providing a valuable comparison point for later approximation techniques|
|1947--49| |Feynman, Schwinger, Tomonaga| | Workable [[renormalization]] techniques for "sweeping infinities under the rug" in [[quantum electrodynamics]] |
|1954--56| |Stueckelberg, Gell-Mann, Bogoliubov _et al._ | | Discovery that the "renormalization group" (RG) captured an important formal property of [[quantum field theory]] |
|1955| | Wigner | | Eigenvalue statistics of random matrices |
|1970--72| |Kadanoff, Wilson _et al._ | | Development of RG concept, discovery that it explains universal properties of phase transitions like critical exponents |
|1976 | | Doi| | Second-quantization formalism for classical stochastic many-particle dynamics |
|1976--79| |Feigenbaum| | Observation and use of RG to explain universal scaling exponents in 1D dynamical maps |
|1977| | Hohenberg, Halperin | | Classification of critical phenomena in dynamic continuum theories |
|1978--80| |Grassberger, Sundermeyer, de la Torre, Cardy, Sugar| | Connection of [[Reggeon field theory]] with directed percolation |
|1980| |Libchaber _et al._ | | Observation of Feigenbaum's constants in experimental systems, starting with convection of mercury |
| 1981--82 | | Janssen, Grassberger | | Formulation of DP conjecture, setting out the likely criteria for phase transitions to belong to the directed-percolation class |
| 1984 | | Belavin, Polyakov, Zamolodchikov | | [[conformal field theory]] in 2D |
| 1986 | | Zamolodchikov | | [[c-theorem]] for 2D theories, identifying RG fixed points with CFTs of specific [[Virasoro algebra|central charges]] |
|1986| |Kardar, Parisi, Zhang | | KPZ equation for dynamical surface growth by deposition, showing that RG is applicable to its far-from-equilibrium dynamics |
|1996--97| |Zirnbauer, Altland| | Symmetry classes (not _exactly_ the same thing as universality classes, but clearly part of the same picture) for random matrices, following Cartan's classification of symmetric spaces |
|1998--99 | | Freedman, _et al._ | | First indications that the [[AdS/CFT correspondence]] and related implementations of gauge/gravity duality could lead to a "holographic $c$-theorem" generalising Zamolodchikov's work to higher dimensions |

## Related concepts

* [[statistical physics]]

* [[thermodynamic limit]]

## References ##

General sources:

* [[Terence Tao|T. Tao]] (2010), "A second draft of a non-technical article on universality" ([web](http://terrytao.wordpress.com/2010/09/14/a-second-draft-of-a-non-technical-article-on-universality/))

Statistical distributions:

* Wikipedia, [stable distribution](http://en.wikipedia.org/wiki/Stable_distribution)
* Wikipedia, [Fisher–Tippett–Gnedenko theorem](http://en.wikipedia.org/wiki/Fisher%E2%80%93Tippett%E2%80%93Gnedenko_theorem)

Equilibrium critical phenomena:

* B. Berche _et al._ (2009), "Critical phenomena: 150 years since Cagniard de la Tour" _J. Phys. Studies_ **13**: 3201, [arXiv:0905.1886](http://arxiv.org/abs/0905.1886).
* J. P. Sethna (2006), _Statistical Physics: Entropy, Order Parameters, and Complexity_. Oxford University Press. [Available on the author's website](http://pages.physics.cornell.edu/sethna/StatMech/).  See in particular chapter 12.
  {#Sethna2006}
* Kardar's _Statistical Physics of Particles_ and _Statistical Physics of Fields_
* Henkel's _Conformal Invariance and Critical Phenomena_ (1999)
* P. A. Pearce (1994), "Recent progress in solving _A--D--E_ lattice models" _Physica A_ **205**: 15--30. ([pdf](http://mac0916.ms.unimelb.edu.au/~pap/Publications_pdf/1994PearceRecentProgressADE.pdf)) This one brings in [[Temperley-Lieb algebra]]s, two-colour braid monoid algebras and [[Reidemeister moves]].
* J. Cardy (2010), _The Ubiquitous 'c': From the Stefan--Boltzmann Law to Quantum Information_, J.Stat.Mech. **2010**: P10004, ([arXiv:1008.2331](http://arxiv.org/abs/1008.2331))
  {#Cardy2010}
* A. Cappelli and J.-B. Zuber (2010), _A-D-E Classification of Conformal Field Theories_ Scholarpedia, 5(4): 10314 ([web](http://www.scholarpedia.org/article/A-D-E_Classification_of_Conformal_Field_Theories)).  Also available as [arXiv:0911.3242](http://arxiv.org/abs/0911.3242).
* Michael E. Fisher (1968), _Renormalization of Critical Exponents by Hidden Variables_ Physical Review **176**: 257--72. ([web](http://link.aps.org/doi/10.1103/PhysRev.176.257))
  {#Fisher1968}
* L. P. Kadanoff (1971), _Critical Behavior. Universality and Scaling_ in Proceedings of the International School of Physics Enrico Fermi, Course LI (27 July -- 8 August 1970). Edited by M. S. Green.
  {#Kadanoff1971}
* J. M. H. Levelt Sengers (1976), _Critical Exponents at the Turn of the Century_ Physica A **82**: 319--51. ([web](http://www.sciencedirect.com/science/article/pii/0378437176900121))
  {#LeveltSengers1976}

Random matrix theory:

* Yan Fyodorov (2011) Random matrix theory. Scholarpedia, 6(3):9886 ([web](http://www.scholarpedia.org/article/Random_matrix_theory))
* this conversation at the n-Category Café on [the Tenfold Way](http://golem.ph.utexas.edu/category/2010/12/the_threefold_way_part_2.html#c035973).

Nonequilibrium critical phenomena:

* P. C. Hohenberg and B. I. Halperin (1977). "Theory of dynamic critical phenomena" _Rev. Mod. Phys._ **49**: 435. ([web](http://www.physics.mcgill.ca/~omid/Hohenberg_Halperin.pdf))
* _Nonequilibrium Phase Transitions_ (2008) by Henkel _et al._
* Vollmayr-Lee's talks on the "field theory approach to diffusion-limited reactions", [Boulder School for Condensed Matter and Materials Physics](http://boulderschool.yale.edu/2009/boulder-school-2009-lecture-notes) (2009), in particular lecture 4
* G. Odor (2004), "Universality classes in nonequilibrium lattice systems" _Rev. Mod. Phys._ **76**: 663. [arXiv:cond-mat/0205644v7](http://arxiv.org/abs/cond-mat/0205644)
* I. Corwin (2011), "The Kardar-Parisi-Zhang equation and universality class" [arXiv:1106.1596](http://arxiv.org/abs/1106.1596).
* Takeuchi _et al._ _Growing interfaces uncover universal fluctuations behind scale invariance_  Sci Rep. (Nature) **1** 34, ([arXiv:1108.2118](http://arxiv.org/abs/1108.2118))
  {#Takeuchi}

Dynamical maps and chaos theory:

* P. Cvitanovic, ed. _Universality in Chaos_ (1983).

[[!redirects universality classes]]


universally closed morphism

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _universally closed morphism_ is a [[closed morphism]] all whose [[pullback]]s are also closed.

## Definition

Let $C$ be a [[category]] with [[pullback]]s and with a notion of [[closed morphism]] which is stable under [[composition]] and contains all the [[isomorphisms]].

A morphism $f:X\to Y$ in $C$ is __universally closed__ if for every
$h: Z\to Y$ the [[pullback]] $h^*(f): Z\times_Y X\to Z$ is a [[closed morphism]].

In particular, for $h=id_Y$ we see that a universally closed morphism is itself closed.

## Examples

* The assumptions are satisfied in the category of [[scheme]]s. See [[valuative criterion of properness]] for a characterization of universally closed morphisms of schemes via valuation rings.

[[!redirects universally closed morphisms]]