fibration in a 2-category

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#### 2-category theory
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## Idea

Recall the notion of a [[Grothendieck fibration]]: a [[functor]] $p \colon E \to B$ whose [[fibres]] $E_b$ are ([[contravariant functor|contravariantly]]) [[functor|functorial]] in $b \in B$.

This idea may be generalized to work in any suitable [[2-category]] $K$ besides [[Cat]], although if the 2-category is not [[strict 2-category|strict]], then one has to generalize instead the non-strict notion of *[[Street fibrations]]*.  The generalized definition can be given in several equivalent ways, such that:

+-- {: .standout #ElemFibSpecializesToCat }

When $K$ is [[Cat]], strict fibrations in this sense are precisely [[Grothendieck fibrations]], while non-strict ones are precisely [[Street fibrations]].

=--

## Definition

Fix a (non-strict) [[2-category]] $K$ with all [[finite limit|finite]] [[2-limits]].

For a [[pair]] of [[1-morphisms]] $f\colon A \to C$ and $g \colon B \to C$ in $K$ write

* $f/g$ for their [[comma object]]

* $f/_{\cong} g$ for their [[2-pullback]].

A [[1-morphism]] $p \colon E \to B$ in $K$ is called a (non-strict) **fibration** when the following equivalent conditions hold:

* $p_* = K(X,p) \colon K(X,E) \to K(X,B)$ is a [[Street fibration]] in $Cat$ for each $X \in K$, and for all $f \colon Y \to X$ in $K$
$$
\array{
  K(X,E) & \overset{f^*}{\to} & K(Y,E) \\
  \mathllap{p_*} \downarrow & & \downarrow \mathrlap{p_*} \\
  K(X,B) & \overset{f^*}{\to} & K(Y,B)
}
$$
is a [[morphism of fibrations]].

* For every morphism $f: X \to B$, the canonical map $i: f/_{\cong} p \to f/p$ has a [[right adjoint]] in the [[slice 2-category]] $K / X$.

* The canonical map $i \colon p \to B/p$ has a right adjoint in $K / B$.

* $p \colon E \to B$ is an algebra for the [[2-monad]] $L$ on $K/B$ given by $L p = B/p$.

* The induced [[F-functor]] $\Sigma_p : K\swarrow E \to K\swarrow B$ between the [[lax slice 2-category|oplax slice]] [[F-categories]] has a [[lax F-adjunction|right semi-lax F-adjoint]].  (See [Johnstone](#JohnstonePP).)

If $K$ is a [[strict 2-category]] with finite strict 2-limits, then we say that $p \colon E \to B$ is a **strict fibration** when the corresponding conditions hold where "Street fibration" is replaced by "Grothendieck fibration", slice 2-categories are replaced by strict-slice 2-categories, comma objects are replaced by strict comma objects, and 2-pullbacks are replaced by strict 2-pullbacks.  In this case the 2-monad $L$ is in fact a strict 2-monad, but we do *not* require $p$ to be a *strict* algebra, only a [[pseudoalgebra]]; strict algebras for $L$ would instead be [[split fibrations]].

Note also that the *first* definition makes perfect sense regardless of whether $K$ has any limits, although this is not true of the others.

## Details

By way of spelling out the first definition, we may define a 2-cell $\eta \colon e \to e' \colon X \to E$ to be **$p$-cartesian** if $f^*\eta = \eta f$ is $p_*$-cartesian for every $f \colon Y \to X$.  Since this definition already incorporates stability under pullback, we can then say that $p$ is a fibration in $K$ if for every 2-cell $\beta \colon b \to p e$ there is a cartesian $\hat\beta \colon e' \to e$ such that $p e' = b$ and $p \hat\beta = \beta$.

The third definition is perhaps the simplest.  Of course, it is implied by the second (take $f = 1$), but the converse is also true by the pasting lemma for [[comma object|comma]] and [[pullback]] squares.

We can show that the first and third definitions are equivalent by using the representability of fibrations and adjunctions, plus the following lemma, whose proof can be found at [[Street fibration]].

+--{: .un_lemma}
###### Lemma
A functor $p \colon E \to B$ is a [[cleavage|cloven]] Street fibration if and only if the canonical functor $i \colon B \to E/p$ has a right adjoint $r$ in $Cat / B$.
=--

It follows that a morphism $p \colon E \to B$ in any 2-category $K$ is representably a fibration (i.e. satisfies the first definition) if and only if the adjunction $i \dashv r$ exists in $K/B$ (i.e. it satisfies the third condition).

Now we connect the first three conditions with the fourth.  Because $K$ is finitely complete, we may form the [[tricategory]] $Span(K)$ of [[spans]] in $K$.  In particular, $K/B$ is equivalent to the hom-2-category $Span(K)(B,1)$.  Now [[comma object|recall]] that $B/p$ can be expressed as a pullback or span composite
$$
\array{
  & & & & B/p & & & & \\
  & & & \swarrow & & \searrow & & & \\
  & & B^{\mathbf{2}} & & & & E & & \\
  & \swarrow & & \searrow &
  & \swarrow & & \searrow & \\
  B & & & & B & & & & 1
}
$$
Write $\Phi B = B^{\mathbf{2}}\rightrightarrows B$.  The functor $L \colon p \mapsto B/p$ is then given by composition: $L p =\Phi B \circ p$.  To show that $p$ is a fibration iff it is an $L$-algebra, it suffices to show

**Lemma.** _$\Phi B$ is a [[colax-idempotent monad]] in $Span(K)$ with unit $i \colon B \to B^{\mathbf{2}} = B/B$._

+-- {: .proof}
_Proof_.  Write $\mathbf{\Delta}$ for the monoidal 2-category whose underlying 1-category is the [[augmented simplex category]] $\Delta_a$. See there for more information, but here it suffices to say this is the full sub-$2$-category of $Cat$ (or $Poset$) on the finite linear orders. Recall that for $n \geq 1$ each $[n] \in \mathbf{\Delta}$ is given by the $(n-1)$-fold composite of the cospan $[1] \to [2] \leftarrow [1]$ with itself, where we take a 0-fold composite to denote the identity $[1] \to [1] \leftarrow [1]$. Recall also that the monoidal structure $[n] \oplus [m]$ on $\mathbf{\Delta}$ is generated by composing the cospans $[n],[m] \colon [1] ⇸ [1]$ (together with the fact that $[0],[1]$ are the initial and terminal objects).

Thus for each $n \geq 1$, there is a span $B^{[n]} \colon B ⇸ B$, which is canonically equivalent to the $(n-1)$-fold composite $B^{[2]} \circ B^{[2]} \circ \cdots \circ B^{[2]} \colon B ⇸ B$, because cotensor preserves finite limits.  For the same reason, $B^- \colon Cat^{op}_{fp} \to K$ almost restricts to a monoidal 2-functor $\mathbf{\Delta}^{op} \to Span(K)(B,B)$, except that $[0]$ does not yield a span from $B$ to $B$.  Precomposing $B^-$ with the functor $\mathbf{\Delta}^{co} \to \mathbf{\Delta}^{op}$ that sends $[n]$ to $[n+1]$, while $\delta_i \mapsto \sigma_i$ and $\sigma_i \mapsto \delta_{i+1}$ does yield a monoidal 2-functor $\mathbf{\Delta}^{co} \to Span(K)(B,B)$, which by the universal property of $\mathbf{\Delta}$ corresponds to a unique colax-idempotent monad in $Span(K)(B,B)$.

In detail, the monoid object $[0] \to [1] \leftarrow [2]$ in $\mathbf{\Delta}$ is sent to the monad $\Phi B = B^{[2]}$ in $Span(K)$, with structure maps $i = (\sigma^1_0)^* \colon B \to \Phi B$ and $c = (\delta^2_1)^* \colon \Phi B \circ \Phi B \to \Phi B$.  Moreover, because of the adjunction $\delta^2_1 \dashv \sigma^2_0 = \sigma^1_0 \oplus [1]$ in $\mathbf{\Delta}$, we have $i \circ \Phi B \dashv c$ in $Span(K)(B,B)$ with identity unit.
=--

It follows that $L = \Phi B \circ -$ is a monad, and is colax-idempotent because, for a span $H \colon B ⇸ A$, we have $\eta_H = i \circ H$, $\mu_H = c \circ H$, and
$$
  \eta_{L H} = i \circ \Phi B \circ H \dashv c \circ H
$$
with identity unit.  It is clear, too, that the morphism $i \colon p \to B/p$ as above is given by $\eta_{p^\circ} = i \circ p^\circ$, where $p^\circ$ denotes the obvious span $B ⇸ 1$.  Thus, by the definition of a [[colax-idempotent monad]], a morphism $p \colon E \to B$ carries the structure of an $L$-algebra if and only if the unit $i = \eta_{p^\circ} \colon p \to B/p$ has a right adjoint.

## Properties

### The 2-fibration of fibrations

For a [[strict 2-category]] $K$, let $Fib_K$ be the 2-category of strict fibrations in $K$, morphisms of fibrations, and 2-cells between them.  Then the codomain functor $cod : Fib_K \to K$ is a strict [[2-fibration]].  (This is arguably an instance of the [[microcosm principle]].)

Similarly, for any [[bicategory]] $K$, the bicategory $Fib_K$ of weak fibrations in $K$, morphisms of fibrations, and 2-cells admits a weak 2-fibration $cod : Fib_K \to K$.

### Iterated fibrations

It is easy to show that a composite of fibrations is a fibration.  Moreover, if $Fib(X)= Fib_K(X)$ denotes the 2-category of fibrations over $X\in K$, then we have:

+--{: .num_theorem}
###### Theorem
A morphism in $Fib(X)$ is a fibration in the 2-category $Fib(X)$ iff its underlying morphism in $K$ is a fibration.
=--

This is a standard result, at least in the case $K=Cat$, and is apparently due to Benabou.  References include ([Bénabou 1985](#Benabou85)), ([Hermida 1999, Section 4.3](#Hermida99)) and ([Jacobs 1999, Chapter 9](#Jacobs)). Therefore, for any fibration $A\to X$ in $K$ we have $Fib_K(A) \simeq Fib_{Fib_K(X)}(A\to X)$, and similarly for opfibrations.  This is a fibrational 2-categorical analogue of the standard equivalence $K/A \simeq (K/X)/(A\to X)$ for ordinary [[slice categories]].

## Related concepts

* [[Grothendieck fibration]], [[Street fibration]], [[discrete fibration]], [[two-sided fibration]]

* [[Cartesian fibration]]

## References

* [[Ross Street]], _Fibrations and Yoneda's lemma in a 2-category_, Lecture Notes in Mathematics, Vol. 420, 1974, pp. 104–133. [[doi:10.1007/BFb0063102](https://doi.org/10.1007/BFb0063102)]

* [[Syméon Bozapalides]]. _Fibrations et faisceaux dans une 2-catégorie_, Archivum Mathematicum 12.4 (1976) 199-208. [[10338.dmlcz/106944](https://dml.cz/handle/10338.dmlcz/106944), [pdf](https://dml.cz/bitstream/handle/10338.dmlcz/106944/ArchMath_012-1976-4_4.pdf)]

* [[Ross Street]], _Fibrations in bicategories_, [[Cahiers de Topologie et Géométrie Différentielle Catégoriques]], 21 no. 2 (1980), p. 111--160 ([numdam](http://www.numdam.org/numdam-bin/fitem?id=CTGDC_1980__21_2_111_0)).

*  [[Mark Weber]], _Yoneda structure from 2-toposes_,  Applied Categorical Structures **15** (2007) pp259-323. doi:[10.1007/s10485-007-9079-2](https://doi.org/10.1007/s10485-007-9079-2) ([author's page](https://sites.google.com/site/markwebersmaths/home/yoneda-structures-from-2-toposes))

* {#JohnstonePP} [[Peter Johnstone]], _Fibrations and partial products in a 2-category_, Applied Categorical Structures **1** (1993) pp141-179 doi:[10.1007/BF00880041](https://doi.org/10.1007/BF00880041)

* {#Benabou85} [[Jean Bénabou]], _Fibered categories and the foundations of naive category theory_, Journal of Symbolic Logic, **50**(1) (1985) pp10-37. doi:[10.2307/2273784](https://doi.org/10.2307/2273784), ([JSTOR](http://www.jstor.org/stable/2273784))

* {#Jacobs} [[Bart Jacobs]], _Categorical Logic and Type Theory_, Studies in Logic and the Foundations of Mathematics, 141. North-Holland Publishing Co., Amsterdam, 1999. xviii+760 pp. ISBN: 0-444-50170-3 ([pdf](https://people.mpi-sws.org/~dreyer/courses/catlogic/jacobs.pdf)) ([contents](http://www.cs.ru.nl/B.Jacobs/CLT/bookinfo.html))

* {#Hermida99} [[Claudio Hermida]], _Some properties of Fib as a fibred 2-category_, Journal of Pure and Applied Algebra **134** Issue 1 (1999) pp83-109 doi:[10.1016/S0022-4049(97)00129-1](https://doi.org/10.1016/S0022-4049%2897%2900129-1), ([Core](https://core.ac.uk/display/82656524))

[[!redirects fibrations in a 2-category]]
[[!redirects fibrations in 2-categories]]
[[!redirects internal fibration]]
[[!redirects internal fibrations]]
[[!redirects opfibration in a 2-category]]
[[!redirects opfibrations in a 2-category]]
[[!redirects opfibrations in 2-categories]]
[[!redirects internal opfibration]]
[[!redirects internal opfibrations]]


fibration of (∞,1)-operads

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#Contents#
* table of contents
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## Definition

In the sense of [Lurie, def. 2.1.2.10](#Lurie):

a [[morphism of (∞,1)-operads]] such that the underlying map of [[(∞,1)-categories of operators]] is a [[fibration]] in the [[model structure for quasi-categories]]

## Properties

### Relation to $(\infty,1)$-Algebras over an $(\infty,1)$-operad

For $\mathcal{C}^\otimes \to \mathcal{O}^\otimes$ a fibration of $(\infty,1)$-operads, then for $\mathcal{P}^\otimes \to \mathcal{O}^\otimes$ any other [[morphism of (∞,1)-operads|homomorphism]], an [[(∞,1)-algebra over an (∞,1)-operad|(∞,1)-algebra over]] $\mathcal{P}^\otimes$ in $\mathcal{C}^\otimes$ is a homomorphism of [[(∞,1)-operads]] from $\mathcal{P}$ to $\mathcal{C}$ over $\mathcal{O}$

([Lurie, def. 2.1.3.1](#Lurie))

## Related concepts

* [[morphism of (∞,1)-operads]]

  * **fibration of (∞,1)-operads**

  * [[coCartesian fibration of (∞,1)-operads]]

* [[Kan fibration]], [[anodyne morphism]]

* [[right/left Kan fibration]], [[right/left anodyne map]]

* [[inner fibration]]

* [[Cartesian fibration]]

## References

Def. 2.1.2.10

* [[Jacob Lurie]], _[[Higher Algebra]]_
 {#Lurie}

[[!redirects fibrations of (∞,1)-operads]]

[[!redirects fibration of (infinity,1)-operads]]
[[!redirects fibrations of (infinity,1)-operads]]


fibration of bigroupoids

This page is primarily to explore the notion of [[fibration]] of weak [[2-groupoid]]s. There is (well, will be) original research here, so beware.

Preliminary content soon...

Applicable references:

* I. Bakovic _Grothendieck construction for bicategories_ [pdf](http://www.irb.hr/korisnici/ibakovic/sgc.pdf) (preliminary version)

* C. Hermida _Descent on 2-fibrations and 2-regular 2-categories_ [preprint](http://maggie.cs.queensu.ca/chermida/papers/2-descent-ACS.pdf)  Applied Categorical Structures, 12(5-6), 427--459, 2004

* Stuff by [[Mike Shulman]]


fibration of multicategories

#Contents#
* table of contents
{:toc}

## Idea

The analogue of the notion of _[[Grothendieck fibration]]_ generalized from [[categories]] to [[multicategories]].

## Properties

### Relation to algebras over an operad

For a multicategory regarded as a ([[planar operad|non-symmetric]]) [[operad]], discrete fibrations over it are equivalent to [[algebra over an operad|algebras over that operad]] ([Hermida, proposition 5.1](#Hermida)).

For [[symmetric multicategories]] we have the following.
Let $P$ be a [[symmetric operad]] over [[Set]]

+-- {: .num_theorem}
###### Theorem

The [[operadic Grothendieck construction]] induces an [[equivalence of 2-categories]]

$$
  Alg_P(Cat)
  \simeq
  opFib_P
$$

between the [[∞-algebra over an (∞,1)-operad|weak]] [[algebra over an operad|algebras over]] $P$ and op-fibrations over $P$.

=--

This is ([Heuts, theorem 1.6](#Heuts)).

### Relation to representable multicategories

Fibrations over the [[terminal object|terminal]] multicategory are equivalently [[representable multicategories]] ([Hermida, corollary 4.3](#Hermida)).

## Related concepts

The generalization to the context of [[(∞,1)-operads]] is given by the notion of [[Cartesian fibration of dendroidal sets]].

## References

Fibrations of planar [[multicategories]] are discussed in

* [[Claudio Hermida]], _Fibrations for abstract multicategories_, Fields Institute Communications ([pdf](http://sqig.math.ist.utl.pt/pub/HermidaC/fib-mul.pdf))
 {#Hermida}

For [[symmetric multicategories]] a discussion of (op)fibrations and of the operadic [[Grothendieck construction]] is in section 1 of

* [[Gijs Heuts]], _Algebras over infinity-operads_ ([arXiv:1110.1776](http://arxiv.org/abs/1110.1776))
  {#Heuts}

[[!redirects fibrations of multicategories]]


fibration of points

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#### Category theory
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#### Universes
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[[!include universe - contents]]
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\tableofcontents

## Idea

Given a [[category]] $C$ and an [[object]] $I\in Ob(C)$, the __category of points over $I$__ is the category $Pt_I(C)$ of [[pointed objects]] of the [[slice category]] $C/I$; or alternatively the [[under category|coslice]]-slice category

$$
  Pt_I(C)
   \;=\;
  (I,id_I)\backslash(C/I)
$$

This can be unpacked the following way:

* An [[object]] of $Pt_I(C)$ is a [[pair]] $(p:X\leftrightarrow I:s)$ of morphisms of $C$ such that $s$ is a [[section]] of $p$; in other words it is a [[split epimorphism]] $p$ to $I$ with a fixed choice of splitting $s$.

* A morphism $f:(q:Y\leftrightarrow I:t)\to(p:X\leftrightarrow I:s)$ is any morphism $f:Y\to X$ such that $p\circ f= q$ and $f\circ t=s$.

By the construction, the category $Pt_I(C)$ of points over $I$ is [[pointed category|pointed]] and if the category $C$ has finite limits, then [[finitely complete category|finitely complete]]; moreover the __[[inverse image]] functor__ $v^*:Pt_I(C)\to Pt_J(C)$ induced by $v:J\to I$ is a [[left exact functor]].

The __category $Pt(C)$ of points of $C$__ has objects $Ob(Pt(C))=\coprod_I Ob(Pt_I(C))$. In other words, the objects are the split epimorphisms $p:X\to I$ of $C$ with a choice of a splitting, or equivalently the [[retracts]] in $C$. At the level of morphisms it is just a bit more complex than the morphisms in each $Pt_I(C)$, namely the slice and coslice triangles become squares. More precisely, a morphism $(u,v): (q:Y\leftrightarrow J:t)\to (p:X\leftrightarrow I:s)$ is a pair of morphisms $u:Y\to X$, $v:J\to I$ in $C$ such that $u\circ t = s\circ v$ and $v\circ q =p\circ u$.

$$\array{
Y &\stackrel{u}\to & X \\
q\downarrow\uparrow t & & p\downarrow\uparrow s\\
J&\stackrel{v}\to & I
}$$

The __fibration of points__ is the [[codomain]]-assigning functor $\pi \,\colon\, Pt(C)\to C$, $\pi:(p:X\leftrightarrow I:s)\to I$, $(u,v)\mapsto v$. It is a [[fibered category]] in the sense of Grothendieck (cf. *[[codomain fibration]]*). Its [[fibers]] are the $Pt_I(C)$ which are (as mentioned above) pointed and finitely complete.
A morphism $(u,v)$ (in notation as above) is [[cartesian morphism|cartesian]] iff $q,u,p,v$ are the sides of a [[pullback square]] in $C$ (i.e. $q$ is a pullback of $p$ along $v$ and $u$ a pullback of $v$ along $p$). The inverse image functor for this fibration is exactly described by the rule $v\mapsto v^*$ above.

## Related concepts

* [[codomain fibration]]

## References

The notion appears in Section 3 of:

* [[Dominique Bourn]], *Normalization equivalence, kernel equivalence and affine categories*, Lecture Notes in Mathematics **1488** (1991) [[doi:10.1007/BFb0084212](https://doi.org/10.1007/BFb0084212)]

[[!redirects category of points]]


fibration of quasi-categories

[[!redirects fibrations of quasi-categories]]

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#Contents#
* table of contents
{:toc}

## Idea

For ordinary [[categories]] there is the notion of

* [[Grothendieck fibration]] between two categories.

* and the special case of a [[category fibered in groupoids]].

The analog of this for [[quasi-categories]] are

* [[Cartesian fibrations]]

* [[left fibrations]]/[[right fibrations]] of quasi-categories.

There are more types of fibrations between the [[simplicial sets]] underlying the [[quasi-category]]

* [[inner fibrations]] -- these correspond to **bundles of [[quasi-categories]]** : an inner fibration $E \to \Delta[1]$ over the interval is the quasi-categorical analog of a [[cograph of a profunctor]]: it characterizes the fibers $C, D$ over the endpoints $0,1 \in \Delta[1]$ as [[quasi-categories]]. Notably having an inner fibration $C \to \Delta[0]$ over the point says precisely that $C$ is a [[quasi-category]].

* categorical fibrations -- these appear as the fibrations in the sense of [[model category]] theory in the Joyal [[model structure for quasi-categories]] $sSet_{Joyal}$ . But they have no particular intrinsic meaning in [[higher category theory]]. In fact, there is also the [[model structure on marked simplicial over-sets|model structure on marked simplicial sets]] which is [[Quillen equivalence|Quillen equivalent]] to $sSet_{Joyal}$ and in which the model-theoretic fibrations coincide precisely with the [[Cartesian fibration]]s that do have an intrinsic category theoretic meaning.

* [[minimal Joyal fibrations]]

## Definition

We list the different definitions in the order of their generality. The examples of each definition are also examples of the following definitions.

All morphisms in the following are morphisms of [[simplicial set]]s.

### Trivial fibration

A **trivial fibration** (trivial [[Kan fibration]]) is a morphism that has the [[right lifting property]] with respect to the boundary inclusions $\partial \Delta[n] \hookrightarrow \Delta[n], n \geq 1$.

### Kan fibration

* for the moment see [[Kan fibration]]

A morphism with [[left lifting property]] against all Kan fibrations is called **anodyne**.

### (Left/)Right fibration

A [[morphism]] of [[simplicial set]]s $f : X \to S$ is a **left fibration** or **[[left Kan fibration]]** if it has the [[right lifting property]] with respect to all [[horn]] inclusions except the right outer ones. It is a **right fibration** or **right Kan fibration** if its extends against all horns except the left outer ones.

$$
  \array{
    \Lambda[n]_{k \gt 0} &\to& X
    \\
    \downarrow &{}^{\exists}\nearrow& \downarrow
    \\
    \Delta[n] &\to& S
  }
  \,.
$$

Morphisms with the [[left lifting property]] against all left/right fibrations are called **left/right [[anodyne morphism]]** maps.

Write

$$
  RFib(S) \subset sSet/S
$$

for the full [[SSet]]-[[subcategory]] of the [[overcategory]] of [[SSet]] over $S$ on those morphisms that are right fibrations.

This is a [[Kan complex]]-enriched category and as such an incarnation of the **[[(∞,1)-category]] of right fibrations**.
It is modeled by the [[model structure for left fibrations|model structure for right fibrations]]. For details on this see the discussion at [[(∞,1)-Grothendieck construction]].

### (co)Cartesian fibration

A [[Cartesian fibration]] is an [[inner fibration]] $p : X \to S$ such that

* for every edge $f : X \to Y$ of $S$

* and every lift $\tilde{y}$ of $y$ (that is, $p(\tilde{y})=y$),

there is a [[cartesian morphism|Cartesian edge]] $\tilde{f} : \tilde{x} \to \tilde{y}$ in $X$ lifting $f$.

(HTT, def 2.4.2.1)

see also

* [[model structure for Cartesian fibrations]]

* [[(∞,1)-Grothendieck construction]]

### Categorical fibration

A **categorical fibration** is a fibration in the [[model structure for quasi-categories]]: morphism $f : X \to S$ with the [[right lifting property]] against all monomorphic _categorical equivalences_ .

(HTT, p. 81).

### Inner fibration

A [[morphism]] of [[simplicial set]]s $f : X \to S$ is an **inner fibration** or **[[inner Kan fibration]]** if its has the [[right lifting property]] with respect to all inner [[horn]] inclusions.

The morphisms with the [[left lifting property]] against all inner fibrations are called **inner anodyne**.

#### Minimal fibration

* [[minimal Kan fibration]]

* [[minimal Joyal fibration]]

## Properties

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

By the [[small object argument]] we have that every morphism $f : X \to Y$ of simplicial sets may be factored as

$$
  f : X \to Z \to Y
$$

with $X \to Z$ a left/right/inner anodyne cofibration and $Z \to Y$ accordingly a left/right/inner Kan fibration.

=--

### Trivial fibration

...

### Kan fibration

...

### (Left/)Right fibration

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

Under the operation of forming the [[opposite quasi-category]], left fibrations turn into right fibrations, and vice versa: if $p : C \to D$ is a left fibration then $p^{op} : C^{op} \to D^{op}$ is a right fibration.

Therefore it is sufficient to list properties of only one type of these fibrations, that for the other follows.

=--

#### Homotopy lifting property

In classical [[homotopy theory]], a continuous map $p : E \to B$ of [[topological spaces]] is said to have the [[homotopy lifting property]] if it has the [[right lifting property]] with respect to all morphisms $Y \stackrel{(Id, 0)}{\to} Y \times I$ for $I = [0,1]$ the standard [[interval]] and every commuting diagram

$$
  \array{
    Y &\to& E
    \\
    \downarrow && \downarrow
    \\
    Y \times I &\to& B
  }
$$

there exists a lift $\sigma : Y \times I \to E$ making the two triangles

$$
  \array{
    Y &\to& E
    \\
    \downarrow &{}^\sigma\nearrow& \downarrow
    \\
    Y \times I &\to& B
  }
$$

commute. For $Y = *$ the [[point]] this can be rephrased as saying that the
universal morphism $E^I \to B^I \times_B E$ induced by the commuting square
commuting square

$$
  \array{
    E^I &\to& E
    \\
    \downarrow && \downarrow
    \\
    B^I &\to& B
  }
$$

is an [[epimorphism]]. If it is even an [[isomorphism]] then the lift $\sigma$ exists _uniquely_ . This is the situation that the following proposition generalizes:

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

A morphism $p : X \to S$ of simplicial sets is a left fibration precisely if the canonical morphism

$$
  X^{\Delta[1]} \to X^{\{0\}} \times_{S^{\{0\}}} S^{\Delta^1}
$$

is a trivial Kan fibration.

=--

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

This is a corollary of the characterization of left anodyne morphisms in [Properties of left anodyne maps](#PropRightAnodyne) by [[Andre Joyal]], recalled in [[Higher Topos Theory|HTT, corollary 2.1.2.10]].

=--

#### As fibrations in $\infty$-groupoids

The notion of right fibration of quasi-categories generalizes the notion of [[category fibered in groupoids]]. This follows from the following properties.

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

For $C \to *$ a right (left) fibration over the [[point]], $C$ is a [[Kan complex]], i.e. an [[∞-groupoid]].

=--

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

Due to [[Andre Joyal]]. Recalled at [[Higher Topos Theory|HTT, prop. 1.2.5.1]].

=--

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

Right (left) fibrations are preserved by [[pullback]] in [[sSet]].

=--

+-- {: .num_corollary}
###### Corollary

It follows that the fiber $X_c$ of every right fibration $X \to C$ over every point $c \in C$, i.e. the [[pullback]]

$$
  \array{
    X_c &\to& X
    \\
    \downarrow && \downarrow
    \\
    \{c\} &\to& C
  }
$$

is a [[Kan complex]].

=--

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

For $C$ and $D$ quasi-categories that are ordinary [[categories]] (i.e. simplicial sets that are [[nerve]]s of ordinary categories), a morphism $C \to D$ is a right fibration precisely if the correspunding ordinary [[functor]] exhibits $C$ as a [[category fibered in groupoids]] over $D$.

=--

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

This is [[Higher Topos Theory|HTT, prop. 2.1.1.3]].

=--

A canonical class of examples of a [[fibered category]] is the [[codomain fibration]]. This is actually a [[bifibration]]. For an ordinary category, a bifiber of this is just a set. For an $(\infty,1)$-category it is an $\infty$-groupoid. Hence fixing only one fiber of the bifibration should yield a fibration in $\infty$-groupoids. This is asserted by the following statement.

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

Let $p : K \to C$ be an arbitrary morphism to a [[quasi-category]] $C$ and let $C_{p/}$ be the corresponding [[over quasi-category|under quasi-category]]. Then the canonical projection $C_{p/} \to C$ is a left fibration.

=--

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

Due to [[Andre Joyal]]. Recalled as [[Higher Topos Theory|HTT, prop 2.1.2.2]].

=--

#### (Left/)Right anodyne moprphisms {#PropRightAnodyne}

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

The collection of left anodyne morphisms (those with [[left lifting property]] against left fibrations) is equivalently $LAn = LLP(RLP(LAn_0))$ for the following choices of $LAn_0$:

$LAn_0 =$

* the collection of all left [[horn]] inclusions

  $\{ \Lambda[n]_{i} \to \Delta[n] | 0 \leq i \lt n \}$;

* blah-blah

* blah-blah

=--

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

This is due to [[Andre Joyal]], recalled as [[Higher Topos Theory|HTT, prop 2.1.2.6]].

=--

### (co)Cartesian fibration

### Categorical fibration

### Inner fibrations

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

A [[simplicial set]] $K$ is the [[nerve]] of an ordinary [[category]] $C$, $K \simeq_{iso} N(C)$ precisely if the terminal morphism $K \to \Delta[0]$ is an inner fibration with _unique_ inner [[horn]] fillers, i.e. precisely if for all morphisms

$$
   \Lambda[n]_i \to K
$$

with $n \in \mathbb{N}$ and $0 \lt i \lt n$ there is a _unique_ morphism $\Delta[n] \to K$ making the diagram

$$
  \array{
    \Lambda[n]_i &\to& K
    \\
    \downarrow & \nearrow
    \\
    \Delta[n]
  }
$$

commute.
=--

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

This is [[Higher Topos Theory|HTT, prop. 1.1.2.2]]

=--

+-- {: .num_corollary}
###### Corollary

It follows that under the [[nerve]] _every_ functor $f : C \to D$ between ordinary [[categories]] is an inner fibration.

=--

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

This is immediate, but let's spell it out:

In any commutative diagram

$$
  \array{
    \Lambda[n] &\to& N(C)
    \\
    \downarrow && \downarrow^{\mathrlap{N(f)}}
    \\
    \Delta[n] && \to& N(D)
  }
$$

by the above the bottom morphism is already uniquely specified by the remaining diagram.

$$
  \array{
    \Lambda[n] &\to& N(C)
    \\
    \downarrow && \downarrow^{\mathrlap{N(f)}}
    \\
    \Delta[n] && N(D)
  }
  \,.
$$

By the above there exists a unique lift into $N(C)$

$$
  \array{
    \Lambda[n] &\to& N(C)
    \\
    \downarrow &\nearrow& \downarrow^{\mathrlap{N(f)}}
    \\
    \Delta[n] && N(D)
  }
$$

and by uniqueness of lifts into $N(D)$ this must also make the lower square commute

$$
  \array{
    \Lambda[n] &\to& N(C)
    \\
    \downarrow &\nearrow& \downarrow^{\mathrlap{N(f)}}
    \\
    \Delta[n] &\to& N(D)
  }
  \,.
$$

=--

## Related concepts

* [[fibered category]]

* [[n-fibration|$n$-fibration]]

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

## References

* [[Jacob Lurie]], chapter 2 of:  _[[Higher Topos Theory]]_

* {#AyalaFrancis} [[David Ayala]] and [[John Francis]], _Fibrations of ∞-categories_, ([arxiv:1702.02681](https://arxiv.org/abs/1702.02681))

* [[Clark Barwick]], [[Jay Shah]], _Fibrations in ∞-category theory_ ([arXiv:1607.04343](https://arxiv.org/abs/1607.04343))

Discussion in [[homotopy type theory]]:

* [[Ulrik Buchholtz]], [[Jonathan Weinberger]], *Synthetic fibered (∞,1)-category theory* [[arXiv:2105.01724](https://arxiv.org/abs/2105.01724)]

[[!redirects fibered (infinity,1)-category]]
[[!redirects fibered (infinity,1)-categories]]

[[!redirects fibered (∞,1)-category]]
[[!redirects fibered (∞,1)-categories]]

[[!redirects fibration of quasi-categories]]

[[!redirects fibration of quasi-categories]]
[[!redirects fibrations of quasi-categories]]
[[!redirects fibration of simplicial sets]]
[[!redirects fibrations of simplicial sets]]

[[!redirects weak fibration]]
[[!redirects weak fibrations]]
[[!redirects Joyal fibration]]
[[!redirects Joyal fibrations]]


fibration of simplicial sets > history

< [[fibration of simplicial sets]]


fibration theory

The notion of fibration theory was created by James Wirth in his PhD thesis in 1965. It allows classification of what would now be called $\infty$-[[principal infinity-bundle|bundles]].

### The axioms ###

A *fibration theory* $E$ is an assignment of a [[category]] $E(B)$ to each [[topological space]] $B$ and a [[contravariant functor]] $f^*:E(C) \to E(B)$ to each continuous map $f:B \to C$ such that $id^*$ is the [[identity functor]]. $E$ is required to satisfy the following
+-- {: .query}
How can $id^*$ be an identity functor when that is not contravariant?  Maybe each $f^*$ is a covariant functor but the mapping $f \mapsto f^*$ is a contravariant functor?  But then it\'s automatic that $id^* = id$ (and furthermore that $(f ; g)^* = f^* \circ g^*$).  ---Toby

[[David Roberts]]: whoops! I didn't pick that up. I think you are partly right: it should be some sort of contravariant assignment $f\mapsto f^*$, but maybe not functorial (since I believe that *category* should be replaced as I said below). The protoypical example, AFAICS, is assigning the category of locally homotopy trival fibrations over the given space. It is not spelled out in detail in the paper.
=--

1. For a [[numerable open cover]] $U = \coprod U_i$ of a space $B$ and a system of objects (morphisms) $E_i$ over each $U_i$ such that $E_i$ and $E_j$ agree over $U_i\cap U_j$, then there is a unique common extension of the $E_i$ over $B$
   +-- {: .query}
   What makes an open cover 'numerable'?  ---Toby

   [[David Roberts]]: A cover is numerable if it admits a subordinate partition of unity. [[numerable open cover|Numerable open covers]] form a site. The axiom is there to link locally homotopically trivial fibrations and Dold fibrations (see theorem 2.3 in Wirth-Stasheff, due to Dold.)

   Also, the uniqueness should at least be demoted to unique-up-to-isomorphism.
   =--
1. If $\phi$ is a morphism in $E(B)$ such that each restriction $\phi|_{U_i}$ for a numerable open cover $U$ of $B$ is a [[homotopy equivalence]], then $\phi$ is a homotopy equivalence. If $H\in E(I\times B)$, then the restrictions $H|_{\{t\}\times B}$ are homotopy equivalent (for objects) or homotopic (for morphisms)
1. (Mapping cylinder axiom) If $\phi:F \to F' \in E(B)$ is a homotopy equivalence, then there is an object $M(\phi)\in E(I\times B)$ which serves as a *mapping cylinder* for $\phi$. That is, $M(\phi)$ restricts to $F$ at $t=0$ and to $F'$ at $t=1$ with a characterising homotopy equivalence $\psi_M:M(\phi) \to I\times F'$ which restricts to $\{0\}\times\phi$, respectively $\{1\}\times id$.

+--{: .query}
[[David Roberts]]: The axioms are just copied from Wirth--Stasheff JHRS 1(1) 2006, p 273. They need to be clarified a little, as the notion of homotopy and homotopic are undefined. We could ask that $E(B)$ is a category of fibrant objects or a Quillen model category or $(\infty,1)$-category or a category with an interval objects or something. One could even ask for a subcategory of $Top$ which is closed under some conditions. In that instance, something needs to be said about the compatibility of homotopies etc with the functors $f^*$.

_Toby_:  I know that you\'re just copying things, so maybe you don\'t know the answers to my questions, but so far I don\'t even understand the parts that I should be able to understand!
=--


fibred category theory

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

#Contents#
* table of contents
{:toc}

## Idea
**Fibred category theory** is that part of [[category theory]] concerned with the study of [[fibred categories]].

Fibred categories behave like (and, in a sense, are) [[categories]] parametrized by some other category or, put differently, like families of categories bundled together in a single object.
Hence the [[2-category]] of fibred categories over a fixed base has many formal analogies with the [[Cat|2-category of categories]], and this is one of the main focus of fibred category theory: to study the 'category theory of fibred categories'.

Lots of fibred category theory is done at a higher level of generality, dealing with [[Street fibrations]] in any 2-category instead of specializing to [[Cat]].

## See also

* [[fibred category]], [[fibred functor]], [[fibred transformation]]

* [[fibrations in a 2-category]]

* [[displayed category]]

## References

For the moment see the [references](Grothendieck+fibration#References) at *[[fibered category]]*.

[[!redirects fibered category theory]]


fibred category theory - contents

*[[fibred category theory]]*

## Concepts

* [[Grothendieck fibration]]

* [[indexed category]]

* [[Grothendieck construction]]

* [[fibered limit]]

## Theorems

* [[fibred Yoneda lemma]]
* [[iterated fibrations]]

## Generalizations

* [[fibration in a 2-category]]

## Applications

* [[categorical semantics of dependent type theory]]


fibred functor

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

#Contents#
* table of contents
{:toc}

## Idea
**Fibred functors** are morphisms between [[fibred categories]].
They are 1-cells in the [[2-category]] $\mathbf{Fib}$.

## Definition
A **fibred functor** between [[fibred categories]] $p:\mathcal{E} \to \mathcal{B}$ and $p':\mathcal{E}' \to \mathcal{B}'$ is a pair of functors $F_0:\mathcal{B} \to \mathcal{B}'$ and $F_1:\mathcal{E} \to \mathcal{E}'$ such that the following commutes:
\begin{tikzcd}
\mathcal{E} \arrow[swap]{d}{p} \arrow{r}{F_1} & \mathcal{E}' \arrow{d}{p'}\\
\mathcal{B} \arrow{r}{F_0} & \mathcal{B}',
\end{tikzcd}
and such that $F_1$ preserve [[cartesian morphisms]].

Symbolically, preserving cartesian morphisms means that for each $E:\mathcal{E}$ and $f:B \to p(E)$ morphism in the base, we have $F_0(f)^*E \cong F_1(f^*E)$, where $(-)^*$ denotes reindexing.

\begin{remark}
When $F_0$ is the identity we recover the notion of morphism between fibrations over the same base (which are the 1-cells in $\mathbf{Fib}(\mathcal{B})$).
In that case, $F_1$ induces a family of functors between fibers over the same object in the base.
\end{remark}

## For cloven fibrations
If $p$ and $p'$ are assumed to be [[cloven fibrations]], then a **(cloven) fibred functor** between them is assumed to preserve the cleavage, i.e. the choice of cartesian lifts.

## Properties
The total category of a fibration supports a [[factorization system]] whose left class is comprised of vertical maps (those projecting to an isomorphism) and whose right class are [[cartesian morphism|cartesian maps]].
Fibred functors, by definition, only support the latter, because vertical maps are supported by square of functors in general:

\begin{proposition}
Consider a [[commutative square]] of [[functors]]:
\begin{tikzcd}
\mathcal{E} \arrow[swap]{d}{p} \arrow{r}{F_1} & \mathcal{E}' \arrow{d}{p'}\\
\mathcal{B} \arrow{r}{F_0} & \mathcal{B}',
\end{tikzcd}
Then $F_1$ sends $p$-vertical maps to $p'$-vertical maps.
\end{proposition}
\begin{proof}
Let $\varphi:D \to E$ be a vertical map in $\mathcal{E}$, meaning $p(\varphi) = 1_{B}$, where $B=p(D)=p(E)$. Then $F_1(\varphi) : F_1(D) \to F_1(E)$ is a 1-cell over $p'(F_1\varphi) = F_0(p(\varphi)) = 1_{F_0B}$.
\end{proof}

Thus fibred functors preserve the vertical-cartesian factorization of maps.

## See also
* [[fibred category]], [[fibred natural transformation]]

[[!redirects morphism of fibrations]]
[[!redirects morphisms of fibrations]]
[[!redirects fibered functor]]
[[!redirects fibered functors]]
[[!redirects fibred functors]]


fibred natural transformation

[[!redirects fibred transformation]]

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

#Contents#
* table of contents
{:toc}

## Idea
**Fibred natural transformations** are natural transformations between [[fibred functors]].
They are 2-cells in the [[2-category]] $\mathbf{Fib}$.

## Definition
Given [[fibred category|fibrations]] $p$ and $p'$ and [[fibred functors]] $F,G:p \to p'$, a **fibred natural transformation** $\alpha=(\alpha_1,\alpha_0):F \Rightarrow G$ is a pair of 2-cells as follows:
\begin{tikzcd}
	{\mathcal{E}} && {\mathcal{E}'} \\
	{\mathcal{B}} && {\mathcal{B}'}
	\arrow[""{name=0, anchor=center, inner sep=0}, "{F_0}"{pos=0.45}, bend left=20, from=2-1, to=2-3]
	\arrow[""{name=1, anchor=center, inner sep=0}, "{F_1}"{pos=0.45}, bend left=20, from=1-1, to=1-3]
	\arrow[""{name=2, anchor=center, inner sep=0}, "{G_1}"'{pos=0.55}, bend right=20, from=1-1, to=1-3]
	\arrow[""{name=3, anchor=center, inner sep=0}, "{G_0}"'{pos=0.55}, bend right=20, from=2-1, to=2-3]
	\arrow["p"', from=1-1, to=2-1]
	\arrow["{p'}", from=1-3, to=2-3]
	\arrow[shorten <=2pt, shorten >=2pt, Rightarrow, from=1, to=2]
	\arrow[shorten <=2pt, shorten >=2pt, Rightarrow, from=0, to=3]
\end{tikzcd}
This means $\alpha_0:F_0 \Rightarrow G_0$, $\alpha_1:F_1 \Rightarrow G_1$ and $p'(\alpha_1) = \alpha_0 p$.

When looking at fibrations over a fixed base $\mathcal{B}$, then $F_0=G_0=1_{\mathcal{B}}$, and also $\alpha_0$ is the identity natural transformation.
In that case, $\alpha$ reduces to a natural transformation between the functor $F_1$ and $G_1$ whose components are vertical, i.e. stay in the fibers.

## See also
* [[fibred category]], [[fibred functor]]

* [[natural transformation]]

[[!redirects fibred transformation]]
[[!redirects fibred transformations]]
[[!redirects fibered transformations]]
[[!redirects fibered natural transformation]]
[[!redirects fibered natural transformations]]
[[!redirects fibred natural transformations]]

[[!redirects transformation of fibred functors]]
[[!redirects transformations of fibred functors]]
[[!redirects transformation of fibered functors]]
[[!redirects transformations of fibered functors]]

[[!redirects natural transformation of fibrations]]
[[!redirects natural transformations of fibrations]]

[[!redirects 2-cell of fibrations]]
[[!redirects 2-cells of fibrations]]


fictitious gauge field

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

#Contents#
* table of contents
{:toc}

## Idea

In general the term *fictitious gauge field* could refer to any [[auxiliary field|auxiliary]] [[gauge field]]. Specifically the term is used:

* in [[anyon statistics]] for a [[model (in theoretical physics)|model]] of anyons as [[fermions]] which each source and each couple to a flat [[electromagnetic field|U(1)-gauge field]] causing mutual [[Aharonov-Bohm effect|Aharonov-Bohm phases]];

* in [[solid state physics]] for effects analogous to [[Berry curvatures]] caused by [[crystal]] strain or dislocations.

### For anyon statistics
 {#ForAnyonStatistics}

A model for [[anyon statistics]] via otherwise [[free field|free]] [[fermions]] in 2d, mutually [[interaction|interacting]] through a flat "fictitious gauge field"  has been proposed in [Arovas, Schrieffer, Wilczek & Zee 1985](#ArovasSchriefferWilczekZee85), developed in
[Chen, Wilczek, Witten & Halperin 1989](#ChenWilczekWittenHalperin89). This model has been advertized in several early reviews (e.g. [Wilczek 1990, §I.3](#Wilczek90), [Wilczek 1991](#Wilczek91), but seems not to have been developed much since:

The model regards anyons as *a priori* [[free field|free]] [[fermions]], but equipped now with a non-local mutual [[interaction]] via a "fictitious gauge field" ([CWWH89, §2](#ChenWilczekWittenHalperin89)), in that
each of the particles is modeled as the singular source of a [[flat connection|flat]] [[circle bundle with connection|circle connection]] (a [[vector potential]] with vanishing [[field strength]]), which hence exerts no [[Lorentz force]] but has the effect that globally each particle is subject to the same [[Aharonov-Bohm effect]] as would be caused by a tuple of infinite [[solenoids]] piercing through each of the other particle's positions.

For emphasis, from [CWWH89, p. 359](#ChenWilczekWittenHalperin89):

> Here the particles are to be regarded (in the absence of
interactions) as fermions; the interaction then makes them anyons with statistical
parameter $\theta = \pi(1 - 1/n)$.

It follows that (quoting from [Fröhlich, Gabbiani & Marchetti 1990, p. 20](#FroehlichGabbianiMarchetti90)):

> If $\theta \in\!\!\!\!\!/ \frac{1}{2}\mathbb{Z}$ the [[space of quantum states|Hilbert space]] of [[anyon]] [[wave functions]] must be chosen to be a space of [[multi-valued functions]] with half-[[monodromies]] given by the [[complex phase|phase]] factors $exp(2 \pi \mathrm{i} \theta)$. Such wave functions can be viewed as single-valued functions on the [[universal cover]] $\widetilde M_n$ of $M_n$ $[$the [[configuration space of points]]$]$.

Incidentally, the [[quasi-particle]]-excitations *of* (or *in*) a gas of such Aharonov-Bohm phased anyons are argued to be [[vortices]] ([CWWH89, p. 457](#ChenWilczekWittenHalperin89)):

> we are led to conclude that *in anyon superconductivity, charged quasi-particles and vortices do not constitute two separate sorts of elementary excitations - they are one and the same.*

## References

### For anyon statistics

* {#ArovasSchriefferWilczekZee85} [[Daniel P. Arovas]], [[Robert Schrieffer]], [[Frank Wilczek]], [[Anthony Zee]], *Statistical mechanics of anyons*, Nuclear Physics B **251** (1985) 117-126 (reprinted in [Wilczek 1990, p. 173-182](#Wilczek90)) $[$doi:10.1016/0550-3213(85)90252-4$]$

* {#ChenWilczekWittenHalperin89} [[Yi-Hong Chen]], [[Frank Wilczek]], [[Edward Witten]], [[Bertrand Halperin]], *On Anyon Superconductivity*,  International Journal of Modern Physics B **03** 07 (1989) 1001-1067 (reprinted in [Wilczek 1990, p. 342-408](#Wilczek90))  $[$[doi:10.1142/S0217979289000725](https://doi.org/10.1142/S0217979289000725), [[CWWH-AnyonSuperfluidity.pdf:file]]$]$

Review in:

* {#FroehlichGabbianiMarchetti90} [[Jürg Fröhlich]], [[Fabrizio Gabbiani]], [[Pieralberto Marchetti]], *Braid statistics in three-dimensional local quantum field theory*, in: H.C. Lee (ed.) *Physics, Geometry and Topology* NATO ASI Series, **238** Springer (1990)  $[$[doi:10.1007/978-1-4615-3802-8_2](https://doi.org/10.1007/978-1-4615-3802-8_2), [[FroehlichGabbianiMarchetti-BraidStatistics.pdf:file]]$]$

* {#Wilczek90} [[Frank Wilczek]], *Fractional Statistics and Anyon Superconductivity*, World Scientific (1990) $[$[doi:10.1142/0961](https://doi.org/10.1142/0961)$]$

* {#Wilczek91} [[Frank Wilczek]], *States of Anyon Matter*,  International Journal of Modern Physics B **05** 09 (1991) 1273-1312 $[$[doi:10.1142/S0217979291000626](https://doi.org/10.1142/S0217979291000626)$]$

### In solid state physics

Fictitious gauge fields:

* Mircea Trif, Pramey Upadhyaya, Yaroslav Tserkovnyak, *Theory of electromechanical coupling in dynamical graphene* $[$[arXiv:1210.7384](https://arxiv.org/abs/1210.7384), [doi:10.1103/PhysRevB.88.245423](https://doi.org/10.1103/PhysRevB.88.245423)$]$


Fida El Chami

* [Maths Genelogy page](http://www.genealogy.math.ndsu.nodak.edu/id.php?id=56653)

## related $n$Lab entries

* [[super-Cartan geometry]]

* [[torsion constraints of supergravity]]

category: people


Field

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

#Contents#
* table of contents
{:toc}

## Definition

__$Field$__ is the [[category]] with [[field|fields]] as [[objects]] and [[field homomorphism|field homomorphisms]] as [[morphisms]]. It is a [[full subcategory]] of [[CRing]], the more general category of [[commutative rings]], which is more often considered. This is due to $Field$ not being a very well-behaved category.

##Properties

* $Field$ is neither [[finitely complete category|finitely complete]] nor [[finitely cocomplete category|finitely cocomplete]], meaning in particular, that it has neither [[products]] nor [[coproducts]]. ([Riehl 17, p. 126](#RiehlCTInContext)) This implies that it is not [[locally presentable]].
* However, $Field$ is a [[locally multipresentable category]], which means in particular that it has [[connected limits]] and [[multicolimits]].
* The [[forgetful functor]] $U\colon Field\to Set$ does not have a [[left adjoint]] (hence $Field$ is not a [[reflective subcategory]] of $Set$), hence there is no free field construction (contrary to many [[free functors]] for other [[algebraic categories]]). The same holds for weaker forgetful functors like $U\colon Field\to Ring$, $U\colon Field\to Ab$ or the [[group of units]] $-^\times\colon Field\to Ab$. ([Riehl 17, Example 4.1.11.](#RiehlCTInContext))
* Every morphism in $Field$ is a [[monomorphism]], hence $Field$ is a [[left-cancellative category]].
* The isomorphisms in $Field$ are the bijective homomorphisms. ([Riehl 17, Examples 1.1.6. (ii)](#RiehlCTInContext))

## Subcategories

$Field$ is not [[connected category|connected]] as there are no [[field homomorphism|field homomorphisms]] between fields of different characteristic. The connected component ([[full subcategory]] of $Field$) corresponding to [[characteristic]] $p$ (with $p=0$ or $p$ [[prime]]) is denoted $Field_p$.

The [[field]] of [[rational numbers]] $\mathbb{Q}$ is the [[initial object]] of $Field_0$ and the [[prime field]] $\mathbb{F}_p$ is the [[initial object]] of $Field_p$, but none are in $Field$, which has neither an initial nor [[terminal object]]. ([Riehl 17, Examples 1.6.18. (vi)](#RiehlCTInContext))

## References

* {#RiehlCTInContext} [[Emily Riehl]], _[[Category Theory in Context]]_, Dover Publications (2017) [[pdf](http://www.math.jhu.edu/~eriehl/context.pdf)]

[[!redirects category of fields]]
[[!redirects categories of fields]]


field

> This entry is about the notion in [[algebra]]. For the different concept of the same name in [[differential geometry]] see at _[[vector field]]_, and for that in [[physics]] see at _[[field (physics)]]_.

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### Context
#### Linear algebra
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[[!include homotopy - contents]]
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#### Algebra
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[[!include higher algebra - contents]]
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# Fields
* table of contents
{: toc}

## Idea

A field is a mathematical structure with the usual arithmetic operations of [[addition]], [[subtraction]], [[multiplication]], and [[division]] of non-zero elements.

The original concept of a field was developed by [[Gustav Lejeune Dirichlet]] and [[Richard Dedekind]] in 1871 under the name "body" (Körper in German), and referred solely to [[subfields]] of the [[complex numbers]] (cf. [Dirichlet & Dedekind 1871](#DirichletDedekind71)). Later in 1893 [[Heinrich Weber]] generalized the notion of field to today's definition of field as an arbitrary [[commutative ring|commutative]] [[division ring]] (cf. [Weber 1893](#Weber93)). The word "field" itself in the English language was coined by [[E. Hastings Moore]] in 1893 (cf. [Moore 1893](#Moore93)) to refer to the algebraic structure.

## Definitions

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

A **field** is a [[commutative ring]] in which every nonzero element has a multiplicative [[inverse]] and $0 \neq 1$ (which may be combined as: an element is invertible if and only if it is nonzero).
=--

Fields are studied in *field theory*, which is a branch of [[commutative algebra]].

If we omit the commutativity axiom, then the result is a [[skewfield]] or [[division ring]] (also in some contexts simply called a "field"). For example, the [[free field (algebra)|free field]] of Cohn and Amitsur is in fact noncommutative.

### Constructive notions
{#constructive}

Fields are (arguably) not a purely [[algebra|algebraic]] notion in that they don\'t form an [[algebraic category]] (see [discussion below](#category)).  For this reason, it should be unsurprising that in [[constructive mathematics]] (including the [[internal logic]] of a [[topos]]) there are different inequivalent ways to define a field.  In this case the classical definition is not usually the best one; for instance, the real numbers do not satisfy it.  There are several potential replacements with their own advantages and disadvantages. These include:

* [[discrete field]]
* [[Heyting field]]
* [[weak Heyting field]]
* [[Kock field]]
* [[local ring]]
* [[weak local ring]]

\begin{definition}\label{discrete}
If we replace "an element is invertible iff it is nonzero" in Definition \ref{classical} by "an element is invertible [[xor]] it equals zero" (which is equivalent in [[classical logic]] but stronger in [[constructive logic]]), then we obtain the notion of **discrete field**.  This condition means that every element is either $0$ or invertible, and it also implies that $0\neq 1$.
\end{definition}

Such a field $F$ is 'discrete' in that it decomposes as a coproduct $F = \{0\} \sqcup F^\times$ (where $F^\times$ is the subset of invertible elements).  An advantage is that this is a [[coherent logic|coherent theory]] and hence also a [[geometric theory]]; for this reason [Johnstone](#Johnstone77) calls such fields **geometric fields**.  A disadvantage is that this axiom is not satisfied (constructively) by the ring of [[real numbers]] (however these are defined), although it is satisfied by the ring of [[rational number|rational]] (or even [[algebraic number|algebraic]]) numbers and by the [[finite field]]s as usual.

\begin{definition}\label{heyting}
If we interpret 'nonzero' in Definition \ref{classical} as a reference to a [[tight apartness relation]], thus defining the apartness relation $\#$ by $x # y$ iff $x - y$ is invertible, then we obtain the notion of **[[Heyting field]]**. (As shown [here](/nlab/show/local+ring#internal), the ring operations become strongly extensional functions.)   In addition to $0\# 1$, the condition then means that every element apart from $0$ is invertible.
\end{definition}

This is how 'practising' constructive analysts of the Bishop school usually define the simple word 'field'.  An advantage is that the (located Dedekind) [[real numbers]] form a Heyting field, although (for example) the (less located) [[MacNeille real number]]s need not form a Heyting field; another disadvantage is that this is not a coherent axiom and so cannot be [[internalization|internalized]] in as many categories.

\begin{definition}
If we replace "an element is invertible iff it is nonzero" in Definition \ref{classical} by "an element is noninvertible iff it is zero" (which is equivalent in [[classical logic]] but incomparable in [[constructive logic]]), we obtain the notion of **[[weak Heyting field]]** (cf. [Richman (2020)](#Richman20)) or **[[Johnstone residue field]]** (cf. [Johnstone (1977)](#Johnstone77)), which is not quite the same as the [[residue fields]] of [[algebraic geometry]]). In addition to $0\neq 1$, this condition means that every noninvertible element (i.e. element $x$ such that $x y\neq 1$ for all $y$) is zero.
\end{definition}

An advantage is that even more versions of the [[real numbers]] (including the [[MacNeille real number]]s) form a weak Heyting field; disadvantages are that this axiom is not coherent either and that a residue field lacks an [[apartness relation]] (in particular, the MacNeille reals have no apartness). Another example of a weak Heyting field include the set of [[Laurent series]] of [[real numbers]].

Every discrete field is also a Heyting field, and every Heyting field is also a weak Heyting field. A Heyting field is a discrete field if and only if [[decidable equality|equality is decidable]]; it is in this sense that a discrete field is 'discrete'. It is not true that every weak Heyting field with decidable equality is Heyting. See [this proof](https://ncatlab.org/nlab/show/Heyting+field#Weak+Heyting+fields) for details.

A weak Heyting field is a Heyting field if and only if it is a [[local ring]].  Furthermore, the quotient ring (or 'residue ring') of any local ring by its ideal of noninvertible elements is a Heyting field; in particular, it is a [[residue field]].  On the other hand, not every weak Heyting field is even a local ring (the MacNeille reals are not), so not every residue field is the residue ring of any local ring. The name "residue field" as defined in [Johnstone (1977)](#Johnstone77) comes from the fact that these fields are precisely the residue rings of *weak local rings* (rings in which the noninvertible elements form an ideal).

Counterexamples were remarked above, but to be explicit: The (located Dedekind) [[real numbers]] form a Heyting field which need not be discrete. The [[MacNeille real number]]s form a weak Heyting field which need not be Heyting; see section D4.7 of _[[Elephant|Sketches of an Elephant]]_.

The three definitions above do not exhaust the possible constructive notions of field.  For instance, in [MRR87](#MRR87) the unadorned word **field** is defined like a Heyting field above, but with $\#$ being an arbitrary [[inequality relation]] rather than a tight apartness.  If the inequality is the [[denial inequality]], this reproduces the original classical definition, which in [Johnstone77](#Johnstone77) is called a **field of fractions** since they are precisely the fields of fractions of "weak [[integral domains]]" (defined as rings in which the product of two nonzero elements is nonzero).  In [MRR87](#MRR87) a **denial field** is defined to be a Heyting field with respect to the [[denial inequality]] in which additionally $(0)$ is a [[prime ideal]].

\begin{remark}\label{TrivialRingAsAField}
In [LombardiQuitté2010](#LombardiQuitté2010), the authors' definitions of discrete field and Heyting field do not include the non-equational axiom $1 \neq 0$ or $1 \# 0$ respectively. With such a definition, the [[trivial ring]] counts as a discrete field as well as a Heyting field and constitutes the [[terminal object]] in the [[category|categories]] of such fields.
\end{remark}

## Properties

### Category of fields
{#category}

Fields are not as well-behaved [[category theory|categorically]] as most other common algebraic structures ([[groups]], [[rings]], [[modules]], etc.).  In particular, [[Field]], the [[category]] of fields and field [[homomorphisms]] (a [[full subcategory]] of the category _[[Rings]]_ of [[rings]] and [[ring homomorphisms]]) is not [[complete category|complete]] or [[cocomplete category|cocomplete]], although it is [[accessible category|accessible]].

In particular, it lacks a [[terminal object]] and also lacks an [[initial object]] (though it has a [[weakly initial set]], namely the set of [[prime fields]], hence has a "[multi-initial object](multi-adjoint#MultiInitialObjectsInFields)").  In particular, it is therefore not [[algebraic category|algebraic]] or [[locally presentable category|locally presentable]].

### Accessibility and sketchability
{#AccSketch}

[[Field]] is [[accessible category|accessible]], even *finitely* accessible, and therefore can be presented as the category of models (in [[Set]]) of a mixed limit-colimit [[sketch]].  It is moreover straightforward to write down such a sketch.

We suppose as given to start with a [[limit sketch]] whose models are [[commutative rings]], with $F$ denoting the ring.  We can construct via limit constructions a subobject $I\hookrightarrow F$ consisting of the invertible elements, as the equalizer of the two maps
$$ F \times F \;\rightrightarrows\; F,$$
the first being given by multiplication and the second by the composite $F\times F \to * \overset{1}{\to} F$, where $*$ is terminal and the map labeled "1" picks out the element $1\in F$.  We now assert that if we take the pullback
$$\array{P & \overset{}{\to} & * \\
  \downarrow && \downarrow^0\\
  I& \hookrightarrow & F,}$$
where the map labeled "0" picks out the element $0\in F$, then the object $P$ is initial (i.e. $0$ is not invertible, or equivalently not equal to $1$), and moreover the pullback is also a pushout (i.e. every element of $F$ is either $0$ or invertible).  Of course, in making these last two assertions we use the fact that we are allowing ourselves a limit-colimit sketch instead of just a limit sketch.

Note that this gives us the notion of *discrete* field (see the [constructive definitions](#constructive) above).  The other constructive notions of field can also be described as models for different limit-colimit sketches.

## Examples
 {#Examples}

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

There are the fields of:

* [[rational numbers]],

* (all) [[algebraic numbers]], (Kontsevich-Zagier) [[period]]s

* [[real numbers]],

* [[complex numbers]],

* [[p-adic numbers]] (for $p$ a [[prime number]]).

Other examples

* [[finite field]]s $\mathbf{F}_{p^n}$ where $p$ is a prime

* algebraic [[number field]]s -- finite degree extensions of the field of rational numbers

* [[function field]]s

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

The canonical  [[ring object|local ring object]] of the [[Zariski site|gros Zariski topos]] of any [[scheme]] (given by $S \mapsto \Gamma(S, \mathcal{O}_S)$, that is to say, the affine line $\mathbb{A}^{1}_{S}$) is in fact moreover a field object, where the latter is defined by requiring that Definition \ref{classical} holds in the internal logic of this topos. For a proof, see Proposition 2.2 in the article [Universal projective geometry via topos theory](http://www.sciencedirect.com/science/article/pii/0022404976900025) of Anders Kock. The ring $\mathcal{O}_X$ (the [[structure sheaf]]) in the [[category of sheaves|sheaf topos]] (i.e. the petit Zariski topos) is a [[residue field]] if $X$ is a [[reduced scheme|reduced]] scheme.

=--

## Related concepts

* [[prefield ring]]

* [[reciprocal ring]]

* [[division ring]], [[division monoid]]

* [[subfield]]

* [[ring]]

* [[finite field]], [[field with one element]]

* [[possibly trivial field]]

* [[residue field]]

* [[global field]]

  * [[number field]]

  * [[function field]] (over a finite field)

* [[positive characteristic]]

* [[topological field]]

* [[graded field]]

* [[model theory of fields]]

* [[infinity-field]]

* [[Kock field]]

| [[commutative ring]] | [[reduced ring]] | [[integral domain]] |
---------------------|-----------------|-----------------|
| [[local ring]] | [[reduced local ring]] | [[local integral domain]] |
| [[Artinian ring]] | [[semisimple ring]] | [[field]] |
| [[Weil ring]] | [[field]] | [[field]] |

## References

* {#DirichletDedekind71} [[Gustav Lejeune Dirichlet]] (1871), [[Richard Dedekind]] (ed.), Vorlesungen über Zahlentheorie (Lectures on Number Theory) (in German), vol. 1 (2nd ed.), Braunschweig, Germany: Friedrich Vieweg und Sohn ([Google Books](https://books.google.com/books?id=SRJTAAAAcAAJ&pg=PA424))

* {#Weber93} [[Heinrich Weber]] (1893), "Die allgemeinen Grundlagen der Galois'schen Gleichungstheorie", Mathematische Annalen (in German), 43 (4): 521–549, $[$[doi:10.1007/BF01446451](https://doi.org/10.1007%2FBF01446451), [ISSN:0025-5831](https://www.worldcat.org/issn/0025-5831), [JFM:25.0137.01](https://zbmath.org/?format=complete&q=an:25.0137.01), [S2CID:120528969](https://api.semanticscholar.org/CorpusID:120528969)$]$

* {#Moore93} [[E. Hastings Moore]] (1893), "A doubly-infinite system of simple groups", Bulletin of the American Mathematical Society, 3 (3): 73–78, $[$[doi:10.1090/S0002-9904-1893-00178-X](https://doi.org/10.1090%2FS0002-9904-1893-00178-X), [MR:1557275](https://mathscinet.ams.org/mathscinet-getitem?mr=1557275)$]$

* {#MRR87} Ray Mines, [[Fred Richman]], Wim Ruitenburg,  _A course in constructive algebra_, Universitext, Springer, 1987.

* {#Elephant} [[Peter Johnstone]], [[Sketches of an Elephant]], Part D.  The [[classifying topos]] for fields is discussed in section D3.1.11(b).

* {#Johnstone77} [[Peter Johnstone]], *Rings, Fields, and Spectra*, Journal of Algebra **49** (1977) 238-260 [doi](https://doi.org/10.1016/0021-8693%2877%2990284-8)

* [[Olivia Caramello]], [[Peter Johnstone]], _De Morgan's law and the theory of fields_ ([arXiv:0808.1972](http://arxiv.org/abs/0808.1972))

* {#LombardiQuitté2010} [[Henri Lombardi]], [[Claude Quitté]] (2010): *Commutative algebra: Constructive methods (Finite projective modules)* Translated by Tania K. Roblo, Springer (2015) ([doi:10.1007/978-94-017-9944-7](https://link.springer.com/book/10.1007/978-94-017-9944-7), [pdf](http://hlombardi.free.fr/CACM.pdf))

* {#Richman20} [[Fred Richman]], *Laurent series over $\mathbb{R}$*. Communications in Algebra, Volume 48, Issue 5, 11 Jan 2020 Pages 1982-1984 [[doi:10.1080/00927872.2019.1710166](https://doi.org/10.1080/00927872.2019.1710166)]

Discussion in [[univalent foundations of mathematics]] ([[homotopy type theory]]):

* [[Marc Bezem]], [[Ulrik Buchholtz]], [[Pierre Cagne]], [[Bjørn Ian Dundas]], [[Daniel R. Grayson]]: Chapter 8 of: *[[Symmetry]]* (2021) $[$[pdf](https://unimath.github.io/SymmetryBook/book.pdf)$]$

[[!redirects field]]
[[!redirects fields]]

[[!redirects discrete field]]
[[!redirects discrete fields]]