+-- {: .rightHandSide}
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###Context###
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
#### Integration theory
+--{: .hide}
[[!include integration theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The special case of [[fiber integration in differential cohomology]] for _[[ordinary differential cohomology]]_
is the partial [[higher parallel transport|higher]] [[holonomy]] operation for [[circle n-bundles with connection]]:

for $Y \to X$ a [[bundle]] of [[compact space|compact]] [[smooth manifolds]] $S$ of [[dimension]] $k$ and $[\nabla] \in H_{diff}^n(Y)$ a class in [[ordinary differential cohomology]] of degree $n$ on $Y$, its [[fiber integration]]

$$
  \left[\exp(i \int_{Y/X} \nabla)\right] \in H^{n-k}_{diff}(X)
$$

is a differential cohomology class on $X$ of degree $k$ less.

In the particular case that $X = *$ is the [[point]] and $dim Y = k = n-1$ the element

$$
  \exp(i \int_{Y} \nabla) \in H^{1}_{diff}(*) \simeq U(1)
$$

is the [[higher parallel transport|higher]] [[holonomy]] of $\nabla$ over $Y$.

## Definition

### Differential orientation

The operation of [[fiber integration]] in [[generalized (Eilenberg-Steenrod) cohomology]] requires a choice of [[orientation in generalized cohomology]]. For fiber integration in [[differential cohomology]] this is to be refined to a _differential orientation_ .

Accordingly, instead of a [[Thom class]] there is a _differential Thom class_ .

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

For $X$ a [[compact space|compact]] [[smooth manifold]] and $V \to X$ a smooth real [[vector bundle]] of [[rank]] $k$ a **differential Thom cocycle** on $V$ is

* a [[compact support|compactly supported]] [[cocycle]] $\hat \omega$ in the [[ordinary differential cohomology]] of degree $k$ of $V$;

* such that for each $x \in X$ we have

  $$
    \int_{V_x} \omega = \pm 1
    \,.
  $$

=--

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

The underlying class $[\hat \omega] \in H^{k}_{compact}(V, \mathbb{Z})$ in [[compact support|compactly supported]] [[integral cohomology]] is an ordinary [[Thom class]] for $V$.

=--

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

Let $p : X \to Y$ be a [[smooth function]] of [[smooth manifold]]s.

An **$H \mathbb{Z}_{diff}$-orientation** on $p$ is

1. A factorization through an [[embedding of smooth manifolds]]

   $$
     p : X \hookrightarrow Y \times \mathbb{R}^N \stackrel{}{\to} Y
   $$

   for some $N \in \mathbb{N}$;

1. a [[tubular neighbourhood]] $W \hookrightarrow Y \times \mathbb{R}^N$ of $X$;

1. a differential Thom cocycle, def. \ref{DifferentialThomCocycle}, $U$ on $W \to X$.

=--

This appears as ([HopkinsSinger, def. 2.9](#HopkinsSinger)).

### Via differential Thom cocycles
 {#FiberIntegration}

Write $H^n_{diff}(-)$ for [[ordinary differential cohomology]]. For any choice of presentation, there is a fairly evident fiber integration of [[compact support|compactly supported]] [[cocycles]] along trivial [[Cartesian space]] bundles $Y \times \mathbb{R}^N \to Y$ over a [[compact space|compact]] $Y$:

$$
  \int_{\mathbb{R}^N}
   : H^{n+N}_{diff,cpt}(Y \times \mathbb{R}^n)
  \to
  H^n_{diff}(Y)
  \,.
$$

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

Let $X \to Y$ be a [[smooth function]] equipped with differential $H\mathbb{Z}$-orientation $U$, def. \ref{DifferentialOrientation}. Then the corresponding **fiber integration** of ordinary differential cohomology is the composite

$$
  \int_{X/Y} :
   H_{diff}^{n+k}(X)
    \stackrel{(-)\cup U}{\to}
   H_{diff, cpt}^{n+N}(X \times \mathbb{R}^N)
    \stackrel{\int_{\mathbb{R}^N}}{\to}
   H_{diff}^n(Y)
  \,.
$$

=--

This appears as ([HopkinsSinger, def. 3.11](#HopkinsSinger)).

### In terms of Deligne cocycles
 {#ViaDeligneComplex}

We discuss an explicit formula for fiber integration along [[product]]-bundles with [[compact topological space|compact]] [[fibers]] in terms of [[Deligne complex]], following ([Gomi-Terashima 00](#GomiTerashima00)).

For $X$ a [[smooth manifold]], write $\mathbf{H}(X, \mathbf{B}^n U(1)_{conn})$ for the [[Deligne complex]] in degree $(n+1)$ over $X$.

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

Let $X$ be a [[paracompact topological space|paracompact]] [[smooth manifold]] and let $F$ be a [[compact topological space|compact]] [[smooth manifold]] of [[dimension]] $k$ without [[boundary]]. Then there is a morphism

$$
  \int_F
    \;\colon\;
  \mathbf{H}(X, \mathbf{B}^n U(1)_{conn})
    \to
  \mathbf{H}(X, \mathbf{B}^{n-k} U(1)_{conn})
$$

given by (...)

=--

([Gomi-Terashima 00, section 2, corollary 3.2](#GomiTerashima00))

### In terms of smooth homotopy types
 {#ViaSmoothHomotopyType}

The [above](#ViaDeligneComplex) formulation of fiber integration in ordinary differential cohomology serves as a presentation for a more abstract construction in [[smooth infinity-groupoid|smooth homotopy theory]].

Let $\mathbf{H} \coloneqq $ [[Smooth∞Grpd]] be the ambient [[cohesive (∞,1)-topos]] of [[smooth ∞-groupoids]]/smooth [[∞-stacks]]. As discussed there, the [[Deligne complex]], being a [[sheaf]] of [[chain complexes]] of [[abelian groups]], presents under the [[Dold-Kan correspondence]] a [[simplicial presheaf]] on the [[site]] [[CartSp]], which in turn presents an object

$$
  \mathbf{B}^n U(1)_{conn} \in \mathbf{H}
  \,,
$$

discussed _[here](http://nlab.mathforge.org/nlab/show/cohesive+%28infinity%2C1%29-topos+--+structures#DifferentialCohomology)_: the smooth [[moduli ∞-stack]] of [[circle n-bundles with connection]].

Let now $\Sigma_k$ be a [[compact topological space|compact]] [[smooth manifold]] of [[dimension]] $k \in \mathbb{N}$ without [[boundary]]. There is the [internal hom in an (infinity,1)-topos](infinity,1%29-topos#ClosedMonoidalStructure)

$$
  [\Sigma_k, \mathbf{B}^n U(1)_{conn}] \in \mathbf{H}
  \,,
$$

which is the smooth moduli $n$-stack of circle $n$-connections _on_ $\Sigma_k$.

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

For all $k \leq n$ there is a natural morphism

$$
  \exp(2\pi i\int_\Sigma(-))
  \; \colon \;
  [\Sigma_k, \mathbf{B}^n U(1)_{conn}]
  \to
  \mathbf{B}^{n-k} U(1)_{conn}
  \;\;\;
  \in
  \mathbf{H}
  \,.
$$

which for $U \in $ [[SmthMfd]] a smooth test manifold sends $n$-connections on $\Sigma_k$ on $U \times \Sigma_k$ to the $(n-k)$-connection on $U$ which is their fiber integration over $\Sigma_k$.

=--

([Fiorenza-Sati-Schreiber 12](#FiorenzaSatiSchreiber12))

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

To see this, observe that

1. by definition $\mathbf{H}(U, [\Sigma_k, \mathbf{B}^n U(1)_{conn}]) \simeq \mathbf{H}(U \times \Sigma_k, \mathbf{B}^n U(1)_{conn})$;

1. if $\{U_i \to \Sigma_k\}$ is a fixed [[good open cover]] of $\Sigma_k$, then $\{U \times U_i \to U \times \Sigma_k\}$ is also a good open cover, for every $U \in $ [[CartSp]];

1. hence the [[Cech nerve]] $C(\{U \times U_i\})$ is a natural (functorial in $U \in CartSp$) [[cofibrant object]] [[resolution]] of $U \times \Sigma_k$ in the projective local [[model structure on simplicial presheaves]] $[CartSp^{op}, sSet]_{proj,loc}$ which [[presentable (infinity,1)-category|presents]] $\mathbf{H} = $[[Smooth∞Grpd]] (as discussed there);

1. the (image under the [[Dold-Kan correspondence]]) of the [[Deligne complex]] $\mathbb{Z}(n+1)^\infty_D$ is a is fibrant in this model structure (since every circle $n$-bundle is trivializable over a [[contractible topological space|contractible space]] $U \in $ [[CartSp]]).

This means that a presentation of $[\Sigma_k, \mathbf{B}^n U(1)_{conn}]$ by an object of $[CartSp^{op}, sSet]_{proj,loc}$ is given by the [[simplicial presheaf]]

$$
  U \mapsto DK \mathbb{Z}(n+1)^\infty_D(C(\{U \times U_i\}))
$$

that sends $U$ to the [[Cech cohomology|Cech]]-[[Deligne complex|Deligne]] [[hypercohomology]] [[chain complex]] with respect to the cover $\{U \times U_i \to U \times \Sigma_k\}$.

On this def. \ref{GomiFiberIntegration} provides a morphism of simplicial sets

$$
  DK \mathbb{Z}(n+1)^\infty_D(C(\{U \times U_i\}))
  \to
  DK \mathbb{Z}(n+1)^\infty_D(U)
$$

which one directly sees is natural in $U$, hence extends to a morphism of simplicial presheaves, which in turn presents the desired morphism in $\mathbf{H}$.

=--

Applications are to

* [[transgression]]

* [[double dimensional reduction]]

## Properties

### General

(...)

### Abstract formulation

At least the fiber integration all the way to the point exists on general grounds for the intrinsic differential cohomology in any [[cohesive (∞,1)-topos]]: the general abstract formulation is in the section _Higher holonomy and Chern-Simons functional_ and the implementation in [[smooth ∞-groupoid]]s is in the section _Smooth higher holonomy and Chern-Simons functional_ .

## Examples

### $\infty$-Chern-Simons functionals in higher codimension
 {#CSInHigherCodimension}

(...)

Differential universal characteristic class / extended $\infty$-Chern-Simons Lagrangian:

$$
  \hat \mathbf{c} : \mathbf{B}G_{conn}  \to \mathbf{B}^{n}U(1)_{conn}
$$

moduli $\infty$-stack of [[higher gauge fields]] on a given $\Sigma_k$:

$$
  [\Sigma_k, \mathbf{B}G_{conn}] \in \mathbf{H}
$$

Lagrangian of $\hat \mathbf{c}$-Chern-Simons theory:

$$
  [\Sigma_k, \hat \mathbf{c}]
   :
  [\Sigma_k, \mathbf{B}G_{conn}]
   \to
  [\Sigma_k, \mathbf{B}^n U(1)_{conn}]
$$

extended action functional of $\hat \mathbf{c}$-Chern-Simons theory in codimension $(n-k)$

$$
  \exp(2 \pi i \int_{\Sigma_k}
  [\Sigma_k, \hat \mathbf{c}]
  )
   :
  [\Sigma_k, \mathbf{B}G_{conn}]
    \stackrel{[\Sigma_k, \hat \mathbf{c}]}{\to}
  [\Sigma_k, \mathbf{B}^n U(1)_{conn}]
   \stackrel{\exp(2 \pi i\int_{\Sigma_k} (-))}{\to}
  \mathbf{B}^{n-k} U(1)_{conn}
  \,.
$$

(...)

## Related concepts

* [[higher parallel transport]], [[higher holonomy]]

* [[fiber integration in differential K-theory]]

## References
 {#References}

A discussion in the general sense of [[fiber integration]] in [[generalized (Eilenberg-Steenrod) cohomology]] is in section 3.4 of

* {#HopkinsSinger} [[Mike Hopkins]], [[Isadore Singer]], _[[Quadratic Functions in Geometry, Topology,and M-Theory]]_

and around prop. 2.1 (in the context of [[Chern-Simons theory]]) in

* [[Daniel Freed]], _Classical Chern-Simons theory_ II. Special issue for S. S. Chern. Houston J. Math. 28 (2002), no. 2, 293–310.

Explicit formulas for fiber integration of _cocycles_ in [[Cech cohomology|Cech]]-[[Deligne cohomology]] are given in

* {#GomiTerashima00} [[Kiyonori Gomi]] and Yuji Terashima, _A Fiber Integration Formula for the Smooth Deligne Cohomology_ International Mathematics Research Notices 2000, No. 13 ([pdf](http://imrn.oxfordjournals.org/content/2000/13/699.full.pdf), [pdf](http://numr.wdfiles.com/local--files/differential-cohomology/gomi-terashima.pdf))

and their generalization from [[higher holonomy]] to [[higher parallel transport]] in

* [[Kiyonori Gomi]] and Yuji Terashima, _Higher dimensional parallel transport_ Mathematical Research Letters 8, 25–33 (2001) ([pdf](http://intlpress.com/_newsite/site/pub/files/_fulltext/journals/mrl/2001/0008/0001/MRL-2001-0008-0001-A-004.pdf))

and

* David Lipsky, _Cocycle constructions for topological field theories_ (2010) ([[LipskyThesis.pdf:file]])

See also

* [[Johan Dupont]], Rune Ljungmann, _Integration of simplicial forms and Deligne cohomology_ Math. Scand. 97 (2005), 11–39 ([pdf](http://www.mscand.dk/article.php?id=897))

The observation that the construction in [Gomi-Terashima 00](#GomiTerashima00) induces refines to [[smooth infinity-groupoid|smooth]] [[moduli infinity-stack|higher moduli stacks]] is discussed in

* {#FiorenzaSatiSchreiber12} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Extended higher cup-product Chern-Simons theories]]_, Journal of Geometry and Physics Volume 74, 2013, Pages 130–163 ([arXiv:1207.5449](http://arxiv.org/abs/1207.5449))

for the case without boundary and for the general case in

* [[Urs Schreiber]] et al. _[[schreiber:Local prequantum field theory]]_.

[[!redirects differential Thom class]]

[[!redirects integration in ordinary differential cohomology]]


fiber sequence

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###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _homotopy fiber sequence_ is a "long left-[[exact sequence]]" in an [[(∞,1)-category]]. (The dual concept is that of _[[cofiber sequence]]_.)

Traditionally fiber sequences have been considered in the context of [[homotopical category|homotopical categories]] such as [[model category|model categories]] and [[category of fibrant objects|Brown category of fibrant objects]] which [[presentable (infinity,1)-category|present]] the [[(∞,1)-category]] in question. In particular, classically this was considered for [[Top]] itself. In these cases they are obtained in terms of [[homotopy pullbacks]]. Since, as discussed there, the homotopy fiber of a morphism may be computed as the ordinary 1-categorical [[fiber]] of any [[fibration]] [[resolution]] of this morphism, one often also speaks of _fibration sequences_.

## Definition
 {#Definition}

### In $(\infty,1)$-category theory

Let $C$ be an [[(∞,1)-category]] with small [[limits]] and consider [[pointed objects]] of $C$, i.e. morphisms ${*} \to A$ from the [[terminal object]] ${*}$ ([[generalized the|the]] [[point]]) to some object $A$. All unlabeled morphisms from the point in the following are these chosen ones and all other morphisms are taken with respect to these points.

Notice that in the case that $C$ happens to be a [[stable (∞,1)-category]] for which ${*} = 0$ all objects are canonically pointed and the notions of left- and right-exact fibration sequences coincide.

But for the notion of fibration sequence to make sense, we do not need to assume that $C$ is a stable $(\infty,1)$-category. In particular, in the context of [[nonabelian cohomology]] (see [[gerbe]] and [[principal 2-bundle]]) one considers fibration sequences in non-stable $(\infty,1)$-categories.

Now let $f : A \to B$ be a morphism in $C$.

The **homotopy fiber** or **homotopy kernel** or **[[mapping cocone]]** of $f$ is [[generalized the|the]] [[pullback]] (which in our $(\infty,1)$-categorical context means [[homotopy limit|homotopy pullback]]) of the point along $f$:

$$
  \array{
     ker(f) &\to& {*}
     \\
     \downarrow && \downarrow
     \\
     A &\stackrel{f}{\to}& B
  }
  \,.
$$

### In categories of fibrant objects

> under construction

In ([Quillen 67, section I.3](#Quillen67)) it was shown how the theory of fiber sequences and [[cofiber sequences]] arises in the abstract homotopy theory of [[model categories]]. Focusing on the fiber sequences, this perspective depends only on the [[category of fibrant objects]] inside the model category, and in fact makes sense generally in this context. This was spelled out in ([Brown 73, section 4](#Brown73)), which we review here.

+-- {: .num_lemma #FiberOfFibrationIsCompatibleWithWeakEquivalences}
###### Lemma

In [[pointed objects]] $\mathcal{C}_f^{\ast/}$ of a [[category of fibrant objects]] $\mathcal{C}_f$, def. \ref{FullSubcategoriesOfFibrantCofibrantObjects}, consider a morphism of [[fiber]]-diagrams, hence a [[commuting diagram]] of the form

$$
  \array{
    fib(f_1) &\longrightarrow& X_1 &\underoverset{\in Fib}{f_1}{\longrightarrow}& Y_1
    \\
    \downarrow^{\mathrlap{}} && \downarrow && \downarrow
    \\
    fib(f_2) &\longrightarrow& X_2 &\underoverset{\in Fib}{f_2}{\longrightarrow}& Y_2
  }
  \,.
$$

If the two vertical morphisms on the right are weak equivalences, then so is the vertical morphism in the left

=--

([[BrownAHT|Brown 73, section 4, lemma 3]])

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

Factor the diagram in question

$$
  \array{
    fib(f_1) &\longrightarrow& X_1 &\underoverset{\in Fib}{f_1}{\longrightarrow}& Y_1
    \\
    \downarrow^{\mathrlap{}} && \downarrow && \downarrow^{\mathrlap{f}}
    \\
    fib(f_2) &\longrightarrow& X_2 &\underoverset{\in Fib}{f_2}{\longrightarrow}& Y_2
  }
$$

through the pullback of the bottom horizontal line:

$$
  \array{
    fib(f_1) &\longrightarrow& X_1 &\underoverset{\in Fib}{f_1}{\longrightarrow}& Y_1
    \\
    \downarrow^{\mathrlap{}} && \downarrow^{\mathrlap{\in W}} && \downarrow^{\mathrlap{id}}
    \\
    fib(\phi) &\longrightarrow& f^\ast X_2 &\underoverset{\in Fib}{\phi}{\longrightarrow}& Y_1
    \\
    \downarrow^{\mathrlap{}} && \downarrow^{\mathrlap{\in W}} && \downarrow^{\mathrlap{f}}_{\mathrlap{\in W}}
    \\
    fib(f_2) &\longrightarrow& X_2 &\underoverset{\in Fib}{f_2}{\longrightarrow}& Y_2
  }
$$

Here $f^\ast X_2 \to X_2$ is a weak equivalence by lemma \ref{InCfPullbackAlongFibrationPreservesWeakEquivalences} and with this $X_1 \to f^\ast X_2$ is a weak equivalence by assumption and [[two-out-of-three]].

Moreover, this diagram exhibits $fib(f_1)\to fib(\phi)$ as the base change (along $\ast \to Y_2$) of $X_1 \to f^\ast X_2$.

Hence it is now sufficient to observe that in category of fibrant objects, base change preserves weak equivalences (...).

=--

Hence we say:

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

Let $\mathcal{C}$ be a [[model category]]. For $f \colon X \longrightarrow Y$ any morphism, then its **[[homotopy fiber]]**

$$
  hofib(f)\longrightarrow X
$$

is the morphism in the [[homotopy category of a model category|homotopy category]] $Ho(\mathcal{C})$, def. \ref{HomotopyCategoryOfAModelCategory}, which is represented by the [[fiber]], def. \ref{FiberAndCofiberInPointedObjects}, of any fibration resolution of $f$.

=--

We may now state the abstract version of the statement of prop. \ref{SerreFibrationGivesExactSequenceOfHomotopyGroups}:

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

Let $\mathcal{C}$ be a [[model category]]. For $f \colon X \to Y$ any morphism of [[pointed objects]], and for $A$ a [[pointed object]], def. \ref{CategoryOfPointedObjects}, then the sequence

$$
  [A,hofib(f)]_\ast \overset{i_\ast}{\longrightarrow} [A,X]_\ast \overset{f_\ast}{\longrightarrow} [A,Y]_{\ast}
$$

is [[exact sequence|exact]] (the sequence being the image of the [[homotopy fiber]] sequence of def. \ref{HomotopyFiber} under the hom-functor of the pointed [[homotopy category of a model category]]

$$
  [A,-]_\ast \;\colon\; Ho(\mathcal{C}^{/\ast}) \longrightarrow Set^{\ast/}
  \,.
$$

=--

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

We may choose representatives such that $A$ is cofibrant, and $f$ is a fibration. Then we are faced with an ordinary pullback diagram

$$
  \array{
    hofib(f) &\overset{i}{\longrightarrow}& X
    \\
    \downarrow && \downarrow^{\mathrlap{p}}
    \\
    \ast &\longrightarrow& Y
  }
$$

and the hom-classes are represented by genuine morphisms in $\mathcal{C}$. From this it follows immediately that $ker(p_\ast)$ includes $im(i_\ast)$. Hence it remains to show that every element in $ker(p_\ast)$ indeed comes from $im(i_\ast)$.

But an element in $ker(p_\ast)$ is represented by a morphism $\alpha \colon A \to X$ such that there is a left homotopy as in the following diagram

$$
  \array{
     && A &\overset{\alpha}{\longrightarrow}& X
     \\
     && {}^{\mathllap{i_0}}\downarrow &{}^{\tilde \eta}\nearrow& \downarrow^{\mathrlap{p}}
     \\
     A &\overset{i_1}{\longrightarrow} & Cyl(A) &\overset{\eta}{\longrightarrow}& Y
     \\
     \downarrow && && \downarrow^{\mathrlap{=}}
     \\
     \ast && \longrightarrow && Y
  }
  \,.
$$

Now by lemma \ref{ComponentMapsOfCylinderAndPathSpaceInGoodSituation} the square here has a lift $\tilde \eta$, as shown. This means that $i_1 \circ\tilde \eta$ is left homotopic to $\alpha$. But by the universal property of the fiber, $i_1 \circ \tilde \eta$ factors through $i \colon hofib(f) \to X$.

=--

### In homotopy type theory

In [[homotopy type theory]] the homotopy fiber of a function term $f : A \to B$ over a function term $pt_B : * \to B$ is the type

$$
  \sum_{a : A} (f(a) = pt_B)
  \,,
$$

hence the [[dependent sum]] over $A$ of the [[identity type]] on $B$ with $f(a)$ and $pt_B$ [[substitution|substituted]]. (A special case of the discussion at [[homotopy pullback]])

For the corresponding [[Coq]] code  see

* [[Vladimir Voevodsky]], _[Foundations/Generalities/uuo.v](https://github.com/vladimirias/Foundations/blob/master/Generalities/uu0.v)_

See also at [[fiber type]].

## Long fibration sequences

A crucial difference between $\infty$-categorical fibration sequences
and ordinary 1-categorical sequences is that the former are always _long_ : in contrast to the ordinary kernel of a kernel, which is necessarily trivial, the homotopy kernel of a homotopy kernel is typically far from trivial, but is a [[loop space object]]. Due to that, each fibration sequence extend to the left by as many steps (times 3) as the objects involved have nontrivial [[homotopy group]]s.

### Kernel of a kernel: loop objects

In particular the homotopy fiber of the point ${*} \to B$ is the [[loop space object]] $\Omega B$ of $B$ (by definition):

$$
  \array{
     \Omega B &\to& {*}
     \\
     \downarrow && \downarrow
     \\
     {*} &\stackrel{}{\to}& B
  }
  \,.
$$

Notice that the ordinary 1-categorical [[pullback]] of a point to itself is necessarily just the point again. Much of what makes [[(∞,1)-category]]-theory richer than ordinary category theory is this fact that the kernel of the point is not trivial, but loops. This implies in particular that **the kernel of the kernel is in general nontrivial**.

Namely the homotopy kernel of the morphism $ker(f) \to A$ constructed above is by definition the homotopy limit in the diagram

$$
  \array{
    ker(ker(f)) &\to& ker(f)
    \\
    \downarrow && \downarrow
    \\
    {*} &\to & A
  }
$$

This is the same kind of diagram as before, just depicted after taking its mirror image along a diagonal. The point of drawing it this way is that this suggests to form the pasting diagram with the one that defines $ker(f)$

$$
  \array{
    ker(ker(f)) &\to& ker(f) &\to& {*}
    \\
    \downarrow && \downarrow && \downarrow
    \\
    {*} &\to& A &\stackrel{f}{\to}& B
  }
  \,.
$$

Since the $(\infty,1)$-categorical pullback [satisfies the pasting law](http://ncatlab.org/nlab/show/limit+in+a+quasi-category#PushoutPasting) just as ordinary [[pullback]] diagrams do, it follows that the total outer square obtained this way is itself a [[homotopy pullback]]. But by definition of the [[loop space object]] $\Omega B$ this means that the kernel of the kernel is loops:

$$
  ker(ker(f)) \simeq \Omega B
  \,.
$$

I.e. all three squares in

$$
  \array{
    \Omega B &\to& ker(f) &\to& {*}
    \\
    \downarrow && \downarrow && \downarrow
    \\
    {*} &\to& A &\stackrel{f}{\to}& B
  }
$$

are (homotopy) pullback squares.

### Long fibration sequences
 {#ContinuingToLongFiberSequences}

Continuing this way to the left with the [[pasting law]], we obtain a long fiber sequence of morphisms to the left of the form

$$
  \cdots
    \to
  \Omega ker(f)
    \longrightarrow
  \Omega A
    \stackrel{\Omega f}{\longrightarrow}
  \Omega B
    \longrightarrow
  ker(f)
    \stackrel{g}{\longrightarrow}
  A
    \stackrel{f}{\longrightarrow} B
  \,.
$$

A subtlety to be aware of here is that $\Omega B$ is not quite $ker(ker(f))$, but the latter instead is $\bar \Omega f$, where $\bar \Omega$ denotes loops with reversed orientation.

A classical discussion of this in terms of computing homotopy fibers via [[path object]] fibrant replacements is e.g. in
([Switzer 75, around 2.57](#Switzer75)). But let's see it just diagrammatically:

First observe that it is indeed $\Omega f$ and not $\bar \Omega f$ that appears in the above: by "bending around" the bottom left "$\ast \to $" we get

$$
  \array{
    \Omega A &\longrightarrow& \ast
    \\
    \downarrow^{\mathrlap{\Omega f}} && \downarrow
    \\
    \Omega B & \longrightarrow & ker(f) &\longrightarrow& \ast
    \\
    \downarrow && \downarrow^{\mathrlap{g}} && \downarrow
    \\
    \ast &\longrightarrow & A &\stackrel{f}{\longrightarrow}& B
  }
  \;\;\;\;\;
  \simeq
  \;\;\;\;\;
  \left(
  \array{
    \ast &\longrightarrow& \ast
    \\
    \downarrow && \downarrow
    \\
    ker(f) &\longrightarrow& \ast
    \\
    \downarrow^{\mathrlap{g}} && \downarrow
    \\
    A &\stackrel{f}{\longrightarrow}& B
    \\
    \uparrow && \uparrow
    \\
    \ast &\longrightarrow& \ast
  }
  \right)
  \stackrel{\underset{\longleftarrow}{\lim}}{\mapsto}
  \left(
    \Omega A \stackrel{\Omega f}{\longrightarrow} \Omega B
  \right)
  \,.
$$

On the other hand, if we define the homotopy fiber of any morphism $\phi$ by the diagram

$$
  \array{
    ker(\phi) &\longrightarrow& \ast
    \\
    \downarrow &\swArrow& \downarrow
    \\
    & \stackrel{\phi}{\longrightarrow} &
  }
$$

then $ker(g)$ is given by the diagram

$$
  \array{
    ker(g) &\longrightarrow& \ast
    \\
    \downarrow &\swArrow& \downarrow
    \\
    ker(f) &\stackrel{g}{\longrightarrow}& A
  }
$$

but what appears in the above pasting diagram is instead this diagram "reflected at the diagonal axis"

$$
  \array{
    ker(g) &\longrightarrow& \ast
    \\
    \downarrow &\swArrow& \downarrow
    \\
    ker(f) &\stackrel{g}{\longrightarrow}& A
  }
  \;\;\;
  \simeq
  \;\;\;
  \array{
    \Omega B &\longrightarrow& \ast
    \\
    \downarrow &\swArrow& \downarrow
    \\
    ker(f) &\stackrel{g}{\longrightarrow}& A
  }
  \;\;\;\;
  \simeq
  \;\;\;\;
  \left(
  \array{
    \Omega B &\longrightarrow& ker(f)
    \\
    \downarrow &\swArrow& \downarrow
    \\
    \ast &\longrightarrow& A
  }
  \right)^{-1}
$$

Here "$(-)^{-1}$" denotes the inverse of the [[2-morphism]] ([[homotopies]]). Since it is these 2-morphisms/homotopies that become the loops in the loop space, it is here that loop reversal appears in translating between the naive iterated homotopy fiber to the construction that actually appears in the above pasting composite.

### Long exact sequences in cohomology {#LongSequCoh}

Usually, when looking at fibration sequences in 1-categorical contexts of the [[homotopy category of an (∞,1)-category]], one doesn't see these long fibration squences directly, but only "in cohomology".

This can be usefully understood as follows:

recall from [[cohomology]] that for $X$ and $A$ objects in an [[(∞,1)-category]] $C$ that is an [[(∞,1)-topos]], the $\infty$-groupoid of $A$-valued cocycle on $X$ is just $Hom_C(X,A)$, so that the corresponding cohomology classes are

$$
  H(X,A) = \Pi_0 Hom_C(X,A) = Ho_C(X,A)
  \,,
$$

where $Ho_C$ is the corresponding [[homotopy category of an (∞,1)-category]].

The upshot being that in the right $(\infty,1)$-context cohomology is just the [[hom-object]].

But the [[hom-functor]] has the crucial property that it is an [[exact functor]] in both arguments. This holds for $(\infty,1)$-categories just as well as for ordinary categories. For our context this means in particular that for

$$
  \array{
    A \times_K B &\to& B
    \\
    \downarrow && \downarrow
    \\
    A &\to& K
  }
$$

a homotopy pullback in $C$, for every $X \in K$ the induced diagram

$$
  \array{
     Hom_C(X,A \times_K B) &\to& Hom_C(X,B)
    \\
    \downarrow && \downarrow
    \\
    Hom_C(X,A) &\to& Hom_C(X,K)
  }
$$

is again homotopy pullback diagram (of [[∞-groupoids]]); in particular the morphism $Hom_C(X,A \times_K B)\to Hom_C(X,A) \times_{Hom_{C}(X,K)} Hom_C(X,B)$ induced by the universal property of homotopy pullback is an equivalence.

So in particular for

$$
  \cdots
  \to
  \Omega \Omega B
  \to
  \Omega ker(f)
  \to
  \Omega A \stackrel{\bar \Omega f}{\to}
  \Omega B \to ker(f) \to A \stackrel{f}{\to} B
$$

a fibration sequence and for $X$ any object, there is a fibration sequence

$$
  \cdots
  \to
  Hom_C(X,\Omega \Omega B)
  \to
  Hom_C(X,\Omega ker(f))
  \to
  Hom_C(X,\Omega A) \stackrel{Hom_C(X,\bar \Omega f)}{\to}
  Hom_C(X,\Omega B) \to Hom_C(X,ker(f)) \to Hom_C(X,A) \stackrel{Hom_C(X,f)}{\to} Hom_C(X,B)
$$

is again a fibration sequence, now of $\infty$-groupoids. By projecting everything to connected components with $\Pi_0$ this then yields an ordinary long exact sequence of pointed sets

$$
  \cdots
  \to
  Ho_C(X,\Omega \Omega B)
  \to
  Ho_C(X,\Omega ker(f))
  \to
  Ho_C(X,\Omega A) \stackrel{Ho_C(X,\bar \Omega f)}{\to}
  Ho_C(X,\Omega B) \to Ho_C(X,ker(f)) \to Ho_C(X,A) \stackrel{Ho_C(X,f)}{\to} Ho_C(X,B)
  \,.
$$

Due to the identitfication of [[cohomology]] with these homotopy hom-sets via $Ho_C(X,A) =: H(X,A)$, this is a "long exact sequence in cohomology"

$$
  \cdots
  \to
  H(X,\Omega \Omega B)
  \to
  H(X,\Omega ker(f))
  \to
  H(X,\Omega A) \stackrel{H(X,\bar \Omega f)}{\to}
  H(X,\Omega B) \to H(X,ker(f)) \to H(X,A) \stackrel{H(X,f)}{\to} H(X,B)
  \,.
$$

### Long exact sequences of homotopy groups

See [[long exact sequence of homotopy groups]].

## Examples

### Characterization of equivalences {#CharOfEquivalences}

We may read off from the non-triviality of the homotopy fiber $A$ of a morphism $f : B \to C$ to which extent $f$ fails to be an [[equivalence in a quasi-category|equivalence]].

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

A morphism $f : B \to C$ in [[∞Grpd]] is an equivalence precisely if all its homotopy fibers over every point of $C$ are [[contractible]], i.e. are equivalent to $*$.

More generally, a morphism $f : B \to C$ in any [[(∞,1)-category]] $\mathcal{C}$ is an equivalence if for all objects $X \in \mathcal{C}$ all homotopy fibers of the morphism $\mathcal{C}(X,f) : \mathcal{C}(X,B) \to \mathcal{C}(X,C)$ are contractible.

=--

For more on this see [[n-truncated object of an (∞,1)-category]]. Also [[Higher Topos Theory|HTT, around example 5.5.6.13]]. For more on homotopy fibers of hom-spaces see [the section below](#OfFuncCats).

### Fiber sequences of $(\infty,1)$-functor categories {#OfFuncCats}

We have seen that for $A \to B \stackrel{f}{\to} C$ a fiber sequence in
an $(\infty,1)$-category $\mathcal{C}$,
then for any other object $X$ we obtain
a fiber sequence

$$
  Hom_{\mathcal{C}}(X,A)
  \to
  Hom_{\mathcal{C}}(X,B)
  \stackrel{f_*}{\to}
  Hom_{\mathcal{C}}(X,C)
$$

in [[∞Grpd]], where the point of $Hom_{\mathcal{C}}(X,C)$ is $X \to * \to C$ with $* \to C$ the point of $C$, so that $Hom_{\mathcal{C}}(X,A)$ is the homotopy fiber over this point of the morphism given by postcomposition with $B \to C$.

Often it is important to know the homotopy fibers of $f_*$ also over other objects in $Hom_{\mathcal{C}}(X,C)$. This is notably the case when considering [[twisted cohomology]] with coefficients in $A$.

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

The homotopy fiber of  $Hom_{\mathcal{C}}(X,B) \stackrel{f_*}{\to} Hom_{\mathcal{C}}(X,C)$ over a morphism $c : X \to C$ may be identified with the [[hom-object in a quasi-category|hom-object]]

$$
  Hom_{\mathcal{C}_{/C}}(c,f)
$$

in the [[over quasi-category|over (∞,1)-category]] $\mathcal{C}_{/C}$.

=--

This is [[Higher Topos Theory|HTT, prop. 5.5.5.12]].

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

Model all [[(∞,1)-categories]] as [[quasi-categories]].

Using the discussion at [[hom-object in a quasi-category]], we observe that

$$
  Hom_{\mathcal{C}}(X,C) \simeq \mathcal{C}_{/C} \times_{\mathcal{C}} \{X\}
$$

and

$$
  Hom_{\mathcal{C}}(X,B) \simeq \mathcal{C}_{/B} \times_{\mathcal{C}} \{X\}
  \simeq \mathcal{C}_{/f} \times_{\mathcal{C}} \{X\}
$$

under which identification the map $f_*$ is induced by the canonical

$$
  \phi''  : \mathcal{C}_{/f} \to \mathcal{C}_{/C}
  \,.
$$

So we are done if we can show that the ordinary [[pullback]] [[diagram]]

$$
  \array{
     \mathcal{C}_{/f} \times_{\mathcal{C}_{/C}} \{c\}
     &\to& \mathcal{C}_{/f} \times_{\mathcal{C}} \{X\}
     \\
     \downarrow^{\mathrlap{\phi}} && \downarrow^{\mathrlap{\phi'}}
     \\
     \{c\} &\to& \mathcal{C}_{/C} \times_{\mathcal{C}} \{X\}
  }
$$

is a [[homotopy pullback]] square, because we have an isomorphism of simplicial sets

$$
  \mathcal{C}_{/f} \times_{\mathcal{C}_{/C}} \{c\}
  \simeq
  Hom^R_{\mathcal{C}_{/X}}(c,f)
  \,,
$$

as one checks.

Since in the above diagram all objects are [[Kan complex]]es, for the diagram to be a homotopy pullback it is sufficient that $\phi'$ is a [[Kan fibration]] for which in turn it is sufficient that it is a [[left fibration]].

That follows by noticing that the right vertical morphism fits into the pullback diagram

$$
  \array{
     \mathcal{C}_{/f} \times_{\mathcal{C}} \{X\} &\to& \mathcal{C}_{/f}
     \\
     \downarrow^{\mathrlap{\phi'}} && \downarrow^{\mathrlap{\phi''}}
     \\
     \mathcal{C}_{/C} \times_{\mathcal{C}} \{X\}
     &\to&
     \mathcal{C}_{/C}
  }
  \,.
$$

But by general properties of [[left fibration]]s, the right vertical map is a left fibration. And since these are stable under pullbacks, so is $\phi'$.

=--

### Principal $\infty$-bundles

Fibration sequences are familiar from the context of [[principal bundles]].

Let $G$ be a [[group]] and let $\mathbf{B}G$ denote the corresponding one-object [[groupoid]] (in the world of [[∞-groupoids]]) or else the [[classifying space]] $\mathcal{B}G$.

Notice that

$$
  G \simeq \Omega \mathbf{B} G
  \,.
$$

Then that a $G$-[[principal bundle]] $P \to X$ is classified by morphism $X \to \mathbf{B}G$ means that it is the homotopy fiber of this morphism.

Indeed, as indicated at [[generalized universal bundle]] and at [[homotopy limit]], we may compute the homotopy pullback

$$
  \array{
  P &\to& {*}
  \\
  \downarrow && \downarrow
  \\
  X &\to& \mathbf{B}G
 }
$$

by first forming the standard _fibrant replacement_ of the diagram $X \to \mathbf{B}G \leftarrow {*}$. That is given by
the diagram

$$
  X \to \mathbf{B}G \leftarrow \mathbf{E}G
  \,,
$$

where $\mathbf{E}G \simeq {*}$ is the total "space" (or 2-groupoid) of the universal $G$-bundle. Once we have done this weakly equivalent replacement, the [[homotopy pullback]] may be computed as the _ordinary_ pullback

$$
  \array{
   P &\to & \mathbf{E}G
   \\
   \downarrow && \downarrow
   \\
   \hat X &\stackrel{g}{\to}& \mathbf{B}G
  }
  \,,
$$

in the ordinary 1-category of $n$-groupoids or spaces, using a replacement $\hat X \stackrel{\simeq}{\twoheadrightarrow} X$ of $X$ by an acyclic fibration (called "[[hypercover]]" in this context) (for instance the Čech groupoid associated with an open cover of $X$).

One recognizes the usual statement that principal $G$-bundles all arise as pullbacks of the universal $G$-principal bundle.

The fact that such pullbacks really are bundles whose fiber is $G$ is the statement of the long fibration sequence induced by $g$ which says that picking any point ${*} \to X$ of $X$ and then pulling back $P$ to that point (i.e. restricting it to that point) yields $\Omega \mathbf{B}G = G$:

$$
  \array{
   G \simeq
   \Omega \mathbf{B}G
   &\to&
   P &\to & \mathbf{E}G
   \\
   \downarrow && \downarrow && \downarrow
   \\
   {*}&\stackrel{x}{\to}& \hat X &\stackrel{g}{\to}& \mathbf{B}G
  }
  \,,
$$

The same logic -- even the same diagrams -- work for [[principal 2-bundles]] and generally for [[principal ∞-bundles]].

### Action groupoids {#ActionGroupoid}

Let $G$ be an [[∞-group]] in that $\mathbf{B}G$ is an [[∞-groupoid]] with a single object. An action of $G$ on an [[(∞,1)-category]] is an [[(∞,1)-functor]]

$$
  \rho : \mathbf{B}G \to (\infty,1)Cat
$$

to [[(∞,1)Cat]]. This takes the single object of $\mathbf{B}G$ to some $(\infty,1)$-category  $V$.

The _[[action groupoid]]_ $V//G$ is the [[limit in a quasi-category|(∞,1)-categorical colimit]] over the action:

$$
  C//G := \lim_\to \rho
  \,.
$$

By the result [described here](http://ncatlab.org/nlab/show/limit+in+a+quasi-category#WithValInooGrpd) this is, equivalent to the pullback of the "universal $(\infty,1)Cat$-bundle" $Z \to (\infty,1)Cat$, namely to the [[coCartesian fibration]]

$$
  \array{
    V//G &\to& Z
    \\
    \downarrow && \downarrow
    \\
    \mathbf{B}G &\stackrel{\rho}{\to}& (\infty,1)Grpd
  }
$$

classified by $\rho$ under the [[(∞,1)-Grothendieck construction]]. We obtain a [[fiber sequence]] to the left by adjoining the $(\infty,1)$-categorical pullback along the point inclusion $* \to \mathbf{B}G$

$$
  \array{
    V&\to& V//G &\to& Z
    \\
    \downarrow && \downarrow && \downarrow
    \\
    {*} &\to& \mathbf{B}G &\stackrel{\rho}{\to}& (\infty,1)Grpd
  }
  \,.
$$

The resulting total $(\infty,1)$-pullback rectangle is the fiber of $Z \to (\infty,1)Cat$ over the $(\infty,1)$-category $C$, which is $V$ itself, as indicated.

Notice that every fibration sequence $V \to V//G \to \mathbf{B}G$ with $V//G \to \mathbf{B}G$ a [[coCartesian fibration]] arises this way, up to equivalence.

### Integral versus real cohomology {#Integralreal}

One of the most basic fibration sequences that appears all over the place in practice is the sequence of [[Eilenberg-MacLane object]]s

$$
  \cdots
  \to
  \mathbf{B}^{n-1} \mathbb{R}/\mathbb{Z}
  \to
  \mathbf{B}^n \mathbb{Z}
  \to
  \mathbf{B}^n \mathbb{R}
  \to
  \mathbf{B}^n \mathbb{R}/\mathbb{Z}
  \to
  \mathbf{B}^{n+1} \mathbb{Z}
  \to
  \cdots
$$

in $\mathbf{H} = $ [[∞Grpd]]/[[Top]], where $\mathbb{R}$ is the abelian group (under addition) of [[real number]], $\mathbb{Z}$ the abelian group of [[integer]]s, and where $\mathbb{R}/\mathbb{Z} = S^1 = U(1)$ is their [[quotient]] group, the circle group or [[unitary group]] $U(n)$ for $n=1$.

A quick way to see that this is indeed a fibration sequence is to realize that $\mathbb{R}/\mathbb{Z}$ is equivalent to the weak quotient [[action groupoid]] $\mathbb{R}//\mathbb{Z}$. Since everything here is in the image of the [[Dold-Kan correspondence]]

$$
  \infty Grpd \simeq
  KanCplx
  \hookrightarrow
  sSet
   \stackrel{\overset{U}{\leftarrow}}{\underset{F}{\to}}
  sAb
  \stackrel{\overset{\Xi}{\leftarrow}}{\underset{N^\bullet}{\to}}
  Ch^+
$$

it is useful to model this fiber sequence as a sequence of [[chain complex]]es

$$
  (
  \mathbf{B}^n\mathbb{R}
  \to
  \mathbf{B}^n \mathbb{R}//\mathbb{Z}
  \to
  \mathbf{B}^{n+1} \mathbb{Z}
  )
  \;\;\;
  =
  \;\;\;
  U \Xi
  \left(
     \left[
     \array{
       \vdots
       \\
       \downarrow
       \\
       0
       \\
       \downarrow
       \\
       \mathbb{R}
       \\
       \downarrow
       \\
       \vdots
     }
     \right]
     \;\;\;
     \to
     \;\;\;
     \left[
     \array{
       \vdots
       \\
       \downarrow
       \\
       \mathbb{Z}
       \\
       \downarrow
       \\
       \mathbb{R}
       \\
       \downarrow
       \\
       \vdots
     }
     \right]
     \;\;\;
     \to
     \;\;\;
     \left[
     \array{
       \vdots
       \\
       \downarrow
       \\
       \mathbb{Z}
       \\
       \downarrow
       \\
       0
       \\
       \downarrow
       \\
       \vdots
     }
     \right]
  \right)
  \,.
$$

Written this way the second morphism is evidently a degreewise surjection, hence is a fibration in the [[model structure on chain complexes]]. Therefore this being a fibration sequence is equivalent (as described at [[homotopy pullback]]) to the first morphism being equivalent to the ordinary [[kernel]] of the second, which clearly it is.

From this fiber sequence we obtain long exact sequences in cohomology, for instance in [[singular cohomology]]: let $X$ be a [[topological space]] and for $A$ an abelian group let

$$
  H^n(X,A) := \pi_0 \infty Grpd( \Pi(X), \mathbf{B}^n A)
$$

be its cohomology with coefficients in $A$, computed in [[∞Grpd]] which we may [[presentable (∞,1)-category|present]] by [[sSet]], where the [[fundamental ∞-groupoid]] $\Pi(X)$ is the singular simplicial complex $Sing X$ and we have

$$
  H^n(X,A) = \pi_0 sSet(Sing X, U \Xi A[n])
  \,.
$$

Then, as discussed [above](LongSequCoh), the fiber sequence of coefficients  $
  \cdots
  \to
  \mathbf{B}^{n-1} \mathbb{R}/\mathbb{Z}
  \to
  \mathbf{B}^n \mathbb{Z}
  \to
  \mathbf{B}^n \mathbb{R}
  \to
  \mathbf{B}^n \mathbb{R}/\mathbb{Z}
  \to
  \mathbf{B}^{n+1} \mathbb{Z}
  \to
  \cdots
$ yields the long exact sequence in cohomology

$$
  \cdots
  H^{n-1}(X,\mathbb{R}/\mathbb{Z})
  \to H^n(X,\mathbb{Z}) \to H^n(X,\mathbb{R}) \to
  H^n(X,\mathbb{R}/\mathbb{Z})
  \to
  H^{n+1}(X,\mathbb{Z})
  \to
  \cdots
  \,.
$$

In applications it often happens that one has a situation where the _real_ cohomology is trivial, i.e $H^n(X,\mathbb{R}) = 0$ in some degree $n$. In that case the exactness of the sequence

$$
  \cdots
  \to H^n(X,\mathbb{Z})
  \to 0 \to
  H^n(X,\mathbb{R}/\mathbb{Z})
  \to
  H^{n+1}(\mathbb{Z})
  \to
  0
  \to
  \cdots
$$

implies that in this case we have an [[isomorphism]]

$$
  H^n(X,\mathbb{R}/\mathbb{Z})
  \stackrel{\simeq}{\to}
  H^{n+1}(\mathbb{Z})
  \,.
$$

### Mayer-Vietoris sequences
 {#MayerVietoris}

A **[[Mayer-Vietoris sequence]]**  is a fiber sequence obtained from an $(\infty,1)$-pullback diagram of [[pointed objects]]:

if

$$
  \array{
    A \times_C B &\to& B
    \\
    \downarrow && \downarrow^{\mathrlap{g}}
    \\
    A &\stackrel{f}{\to}& C
  }
$$

is an [[limit in a quasi-category|infinity-pullback]] diagram in $\mathbf{H}$, then it naturally induces a fiber sequence that starts out as

$$
  \Omega C \to A \times_C B \to A \times B
  \,.
$$

This -- or its associated [[long exact sequence of homotopy groups]] -- is called the _[[Mayer-Vietoris sequence]]_ of the pullback. See there for details.

#### Of a cover

The original example of Mayer-Vietoris sequences is obtained from the situtation where a [[homotopy pushout]] diagram

$$
  \array{
    U \cap V &\hookrightarrow& U
    \\
    \downarrow && \downarrow
    \\
    V &\to& X
  }
$$

in $\mathbf{H} = $ [[∞Grpd]]/[[Top]] is given (which modeled in [[sSet]] or in terms of [[CW-complex]]es in [[Top]] may be modeled by an ordinary pushout), and where $A \in \infty Grpd$ is some coefficient group object. Then applying the $(\infty,1)$-categorical hom $\mathbf{H}(-, A) : \mathbf{H}^{op} \to \infty Grpd$ yields the $\infty$-pullback diagram

$$
  \array{
    \mathbf{H}(X, A) &\to& \mathbf{H}(U,A)
    \\
    \downarrow && \downarrow
    \\
    \mathbf{H}(V,A) &\to& \mathbf{H}(U \cap V, A)
  }
  \,.
$$

By the above this is equivalent to

$$
  \mathbf{H}(X,A) \to \mathbf{H}(U,A) \times \mathbf{H}(V,A)
  \to \mathbf{H}(U \cap V, A)
$$

being a fibration sequence. The corresponding long exact sequence in cohomology (as discussed above) is what is traditionally called the Mayer-Vietoris sequence of the cover of $X$ by $U$ and $V$ in $A$-cohomology.

## Related concepts

* [[fiber]], [[essential fiber]]

* [[tower of homotopy fibers]]

* [triangulated category -- long fiber-cofiber sequences](triangulated+category#LongExactSequences)

## References

The observation of [[long exact sequences of homotopy groups]] for homotopy fiber sequences originates (according to [Switzer 75, p. 35](#Switzer75)) in

* M. G. Barratt, _Track groups I, II_. Proc. London Math. Soc. 5, 71-106, 285-329 (1955).

The first exhaustive study of these is due to

* {#Puppe58} [[Dieter Puppe]], _Homotopiemengen und ihre induzierten Abbildungen I_, Math. Z. 69,  299-344 (1958).

whence the terminology _Puppe sequences_.

Classical textbook accounts include

* {#Switzer75} [[Robert Switzer]], around 2.57 of _Algebraic Topology - Homotopy and Homology_, Die  Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.

* {#Kochmann96} [[Stanley Kochmann]], prop. 3.2.6 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

Lecture notes:

* [[Michael Hopkins]] (notes by [[Akhil Mathew]]), Lecture 1 of: _Spectra and stable homotopy theory_, 2012 ([pdf](http://math.uchicago.edu/~amathew/256y.pdf), [[HopkinsMathewStableHomotopyTheory.pdf:file]])

The construction in the axiomatic homotopy theory of [[model categories]] is due to

* {#Quillen67} [[Daniel Quillen]], chapter I.3 of  _Axiomatic homotopy theory_ in  _Homotopical algebra_, Lecture Notes in Mathematics, No. 43 43, Berlin (1967)

and in the context of [[categories of fibrant objects]] due to

* {#Brown73} [[Kenneth Brown]], section 4 of _[[BrownAHT|Abstract Homotopy Theory and Generalized sheaf Cohomology]]_, 1973 .

A discussion of fiber sequences in terms of [[associated ∞-bundles]] is in

* [[Matthias Wendt]], _Classifying spaces and fibrations of simplicial sheaves_ ([arXiv](http://arxiv.org/abs/1009.2930))

Related discussion on the behaviour of [[fiber sequence]]s under left [[Bousfield localization of model categories]] is in

* [[Matthias Wendt]], _Fibre sequences and localization of simplicial sheaves_,
Alg. Geom. Topol. 13(3), 2013, pp. 1779--1813. ([preprint pdf](http://home.mathematik.uni-freiburg.de/arithmetische-geometrie/preprints/wendt-flocal.pdf) and [ArXiv](http://arxiv.org/abs/1011.4784).)

[[!redirects fiber sequences]]

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[[!redirects fibration sequences]]

[[!redirects Puppe sequence]]
[[!redirects Puppe sequences]]

[[!redirects homotopy fiber]]
[[!redirects homotopy fibers]]
[[!redirects homotopy fibre]]
[[!redirects homotopy fibres]]
[[!redirects homotopy kernel]]
[[!redirects homotopy kernels]]

[[!redirects homotopy exact sequence]]
[[!redirects homotopy exact sequences]]

[[!redirects homotopy fiber sequence]]
[[!redirects homotopy fiber sequences]]
[[!redirects homotopy fibre sequence]]
[[!redirects homotopy fibre sequences]]

[[!redirects (∞,1)-fiber]]
[[!redirects (∞,1)-fibers]]
[[!redirects (∞,1)-fibre]]
[[!redirects (∞,1)-fibres]]

[[!redirects long homotopy fiber sequence]]
[[!redirects long homotopy fiber sequences]]


fiber type

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

\tableofcontents

## Definition

In a [[dependent type theory]] with [[function types]], [[dependent sum types]], and [[identity types]], the **fiber** (or "fiber type") of a function $f:A \to B$ over a term $b:B$ is defined to be the type

$$
  \mathrm{fiber}_{A, B}(f, b)
  \;\coloneqq\;
  \sum_{a : A}
  \big(
    f(a) = b
  \big)
$$

hence the [[dependent sum]] over $A$ of the [[identity type]] on $B$ with $f(a)$ and $b$ [[substitution|substituted]].

### Rules for fiber types

If the dependent type theory does not have [[dependent sum types]], it is still possible to define fiber types by adding the formation, introduction, elimination, computation, and uniqueness rules for fiber types

Formation rules for fiber types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, f:A \to B, y:B, x:A \vdash f(x) =_B y \; \mathrm{type}}{\Gamma, f:A \to B, y:B \vdash \mathrm{fiber}_{A, B}(f, y) \; \mathrm{type}}$$

Introduction rules for fiber types:
$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B \; \mathrm{type} \quad \Gamma, f:A \to B, y:B, x:A \vdash b(x):f(x) =_B y \quad \Gamma \vdash a:A \quad \Gamma, f:A \to B, y:B \vdash p:f[a/x] =_B y}{\Gamma, f:A \to B, y:B \vdash (a, b):\mathrm{fiber}_{A, B}(f, y)}$$

Elimination rules for fiber types:
$$\frac{\Gamma, f:A \to B, y:B \vdash z:\mathrm{fiber}_{A, B}(f, y)}{\Gamma \vdash \pi_1(z):A} \qquad \frac{\Gamma, f:A \to B, y:B \vdash z:\mathrm{fiber}_{A, B}(f, y)}{\Gamma, f:A \to B, y:B \vdash \pi_2(z):f(\pi_1(z)) =_B y}$$

Computation rules for fiber types:
$$\frac{\Gamma, f:A \to B, y:B, x:A \vdash b(x):f(x) =_B b \quad \Gamma \vdash a:A}{\Gamma \vdash \beta_{\mathrm{fiber}_{A, B} 1}:\pi_1(a, b) =_A a} \qquad \frac{\Gamma, f:A \to B, y:B, x:A \vdash b(x):f(x) =_B y \quad \Gamma \vdash a:A}{\Gamma, f:A \to B, y:B \vdash \beta_{\mathrm{fiber}_{A, B} 2}:\pi_2(a, b) =_{f(\pi_1(a, b)) =_B y} b}$$

Uniqueness rules for fiber types:
$$\frac{\Gamma, f:A \to B, y:B \vdash z:\mathrm{fiber}_{A, B}(f, y)}{\Gamma, f:A \to B, y:B \vdash \eta_{\mathrm{fiber}_{A, B}}:z =_{\mathrm{fiber}_{A, B}(f, y)} (\pi_1(z), \pi_2(z))}$$

## See also

* [[equivalence of types]]

* [[fiber]]

* [[homotopy fiber]]

* [[cofiber type]]

## References

The notion of fiber types (and with it the modern notion of [[equivalence in homotopy type theory]]) originates around:

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

Textbook account:

* {#UFP13} [[Univalent Foundations Project]], Def. 4.2.4 in: *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]* (2013) [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)]

For the corresponding [[Coq]] code  see

* [[Vladimir Voevodsky]], _[Foundations/Generalities/uuo.v](https://github.com/vladimirias/Foundations/blob/master/Generalities/uu0.v)_

[[!redirects fiber type]]
[[!redirects fiber types]]

[[!redirects homotopy fiber type]]
[[!redirects homotopy fiber types]]

[[!redirects type of fibers]]
[[!redirects types of fibers]]

[[!redirects type of homotopy fibers]]
[[!redirects types of homotopy fibers]]


fibered category > history

< [[fibered category]]


fibered heterogeneous identity type

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

\tableofcontents

## Idea

A form of [[heterogeneous equality]] in [[dependent type theory]].

## Definition
 {#Definition}

Given a [[type]] $A$ and an $A$-[[dependent type]] $a \colon A \;\vdash\; B(a)$, the *fibered heterogeneous identity type*  is the [[dependent type]] indexed by

1. a pair of parameters

   $$
     a ,\, a' \;\colon\;  A
   $$

   indexing a [[pair]] of [[fibers]] of $B$,

1. a pair of terms of these fiber types

   $$
     b \;\colon\; B(a)
     ,\;\;\;\;
     b' \;\colon\; B(a')
   $$

   whose "[[heterogeneous identity type|heterogeneous]]" [[identification]] is to be witnessed,

and defined equivalently as follows:

* [[Winterhalter 2023](#Winterhalter23), [p. 30](https://dutchcats.github.io/Weak_type_theories/slides_winterhalter.pdf#page=30)] as the [[dependent sum type]]

  $$
    fhId(a, b; a', b')
    \;\coloneqq\;
    \sum_{p \colon \mathrm{Id}_A(a, a')}
    \mathrm{hId}(b, p, b')
  $$

  of the [[heterogeneous identity types]]

  $$
   hId(b, p, b')
   \;\equiv\;
   \mathrm{Id}_{B(b)}\big(
     p_\ast(b)
     ,\,
     b'
   \big)
   \,,
  $$

  over all [[identifications]] $p \colon \mathrm{Id}_A(a, b)$, where

  $$
    p_\ast \,\colon\, B(a) \overset{\sim}{\to} B(a')
  $$

  denotes the [[type transport]] along $p$;

* as the [[identification type]] of [[dependent pairs]] in the [[dependent sum type]] $\sum_{x:A} B(x)$

  $$
    fhId(a, b; a', b')
    \;\coloneqq\;
    \mathrm{Id}_{\sum_{x:A} B(x)}
    \big(
      (a, b)
      ,\,
      (a', b')
    \big)
    \mathrlap{\,.}
  $$

The two definitions are [[equivalence of types|equivalent]]
due to the [extensionality property](dependent+sum+type#ExtensionalityPrinciple) of [[dependent sum types]].

### Inference rules with UIP
 {#InferenceRulesWithUIP}

> Beware that the following text writes "$\mathrm{JMEq}(a, b, x, y)$" for what [above](#Definition) is denoted $fhId(a,x; b,y)$.

When the type theory has a [[set truncation axiom]] such as [[UIP]] or [[axiom K]], one could use one of the following [[inference rules]] to define John Major equality:

#### McBride's axiomatization

The axioms given in [McBride 1999](#McBride99) are in [[OLEG]] [[pseudocode]], use [[Russell universes]], and interpret relations impredicatively as functions into a [[type of all propositions]]. Here, the axioms are written as [[inference rules]] in a [[deduction system]], modified slightly to use [[Tarski universes]] instead of [[Russell universes]], as well as interpreting [[relations]] [[predicative mathematics|predicatively]] as prop-valued type families instead of functions into a [[type of all propositions]]. The axioms are as follows:

$$\frac{\Gamma \vdash A:U \quad \Gamma \vdash B:U \quad \Gamma \vdash u:\mathrm{El}(A) \quad \Gamma \vdash v:\mathrm{El}(B)}{\Gamma \vdash \mathrm{JMEq}(A, B, u, v) \; \mathrm{type}}$$

$$\frac{\Gamma \vdash A:U \quad \Gamma \vdash B:U \quad \Gamma \vdash u:\mathrm{El}(A) \quad \Gamma \vdash v:\mathrm{El}(B)}{\Gamma, p:\mathrm{JMEq}(A, B, u, v), q:\mathrm{JMEq}(A, B, u, v) \vdash \mathrm{propTruncJMEq}(A, B, u, v, p, q):\mathrm{Id}_{\mathrm{JMEq}(A, B, u, v)}(p, q)}$$

$$\frac{\Gamma \vdash A:U \quad \Gamma \vdash u:\mathrm{El}(A)}{\Gamma \vdash \mathrm{refl}(A, u):\mathrm{JMEq}(A, A, u, u)}$$

$$\frac{\Gamma, A:U, B:U, u:\mathrm{El}(A), v:\mathrm{El}(B), e:\mathrm{JMEq}(A, B, u, v) \vdash \Phi(A, B, u, v, e) \; \mathrm{type}}{\Gamma, t:\prod_{X:U} \prod_{x:\mathrm{El}(X)} \Phi(X, X, x, x, \mathrm{refl}(X, x)), A:U, B:U, u:\mathrm{El}(A), v:\mathrm{El}(B), e:\mathrm{JMEq}(A, B, u, v) \vdash \mathrm{eqIndElim}(t, A, B, u, v, e):\Phi(A, B, u, v, e)}$$

$$\frac{\Gamma, A:U, B:U, u:\mathrm{El}(A), v:\mathrm{El}(B), e:\mathrm{JMEq}(A, B, u, v) \vdash \Phi(A, B, u, v, e) \; \mathrm{type}}{\Gamma, t:\prod_{X:U} \prod_{x:\mathrm{El}(X)} \Phi(X, X, x, x, \mathrm{refl}(X, x)), A:U, u:\mathrm{El}(A) \vdash \mathrm{eqIndElim}(t, A, A, u, u, \mathrm{refl}(A, u)) \equiv t(a, u):\Phi(A, A, u, u, \mathrm{refl}(A, u))}$$

The use of Tarski universes instead of Russell universes makes it clear that John Major equality could be defined using inference rules for any type family $x:A \vdash B(x)$ instead of just the Tarski universe type family $X:U \vdash \mathrm{El}(X)$:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash u:B(a) \quad \Gamma \vdash v:B(b)}{\Gamma \vdash \mathrm{JMEq}(a, b, u, v) \; \mathrm{type}}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash u:B(a) \quad \Gamma \vdash v:B(b)}{\Gamma, p:\mathrm{JMEq}(a, b, u, v), q:\mathrm{JMEq}(a, b, u, v) \vdash \mathrm{propTruncJMEq}(a, b, u, v, p, q):\mathrm{Id}_{\mathrm{JMEq}(a, b, u, v)}(p, q)}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma \vdash a:A \quad \Gamma \vdash u:B(a)}{\Gamma \vdash \mathrm{refl}(a, u):\mathrm{JMEq}(a, a, u, u)}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, a:A, b:A, u:B(a), v:B(b), e:\mathrm{JMEq}(a, b, u, v) \vdash \Phi(a, b, u, v, e) \; \mathrm{type}}{\Gamma, t:\prod_{x:A} \prod_{y:B(a)} \Phi(x, x, y, y, \mathrm{refl}(x, y)), a:A, b:A, u:B(a), v:B(b), e:\mathrm{JMEq}(a, b, u, v) \vdash \mathrm{eqIndElim}(t, a, b, u, v, e):\Phi(a, b, u, v, e)}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma, a:A, b:A, u:B(a), v:B(b), e:\mathrm{JMEq}(a, b, u, v) \vdash \Phi(a, b, u, v, e) \; \mathrm{type}}{\Gamma, t:\prod_{x:A} \prod_{y:B(a)} \Phi(x, x, y, y, \mathrm{refl}(x, y)), a:A, u:B(a) \vdash \mathrm{eqIndElim}(t, a, a, u, u, \mathrm{refl}(a, u)) \equiv t(a, u):\Phi(a, a, u, u, \mathrm{refl}(a, u))}$$

#### Winterhalter's axiomatization

The axioms given in [Winterhalter 2023](#Winterhalter23) are in [[Coq]] [[pseudocode]] and use [[Russell universes]]. Here, the axioms are written as [[inference rules]] in a [[deduction system]] and modified slightly to use [[Tarski universes]] instead of [[Russell universes]]. The axioms are as follows:

$$\frac{\Gamma \vdash A:U \quad \Gamma \vdash B:U \quad \Gamma \vdash u:\mathrm{El}(A) \quad \Gamma \vdash v:\mathrm{El}(B)}{\Gamma \vdash \mathrm{JMEq}(A, B, u, v) \; \mathrm{type}}$$

$$\frac{\Gamma \vdash A:U \quad \Gamma \vdash u:\mathrm{El}(A)}{\Gamma \vdash \mathrm{hrefl}(A, u):\mathrm{JMEq}(A, B, u, v)}$$

$$\frac{\Gamma \vdash A:U \quad \Gamma \vdash u:\mathrm{El}(A) \quad \Gamma \vdash v:\mathrm{El}(A) \quad \Gamma \vdash p:\mathrm{Id}_{\mathrm{El}(A)}(u, v)}{\Gamma \vdash \mathrm{eqToHeq}(A, u, v, p):\mathrm{JMEq}(A, A, u, v) \; \mathrm{type}}$$

$$\frac{\Gamma \vdash A:U \quad \Gamma \vdash u:\mathrm{El}(A) \quad \Gamma \vdash v:\mathrm{El}(A) \quad \Gamma \vdash p:\mathrm{JMEq}(A, A, u, v)}{\Gamma \vdash \mathrm{heqToEq}(A, u, v, p):\mathrm{Id}_{\mathrm{El}(A)}(u, v) \; \mathrm{type}}$$

$$\frac{\Gamma \vdash A:U \quad \Gamma \vdash B:U \quad \Gamma \vdash p:\mathrm{Id}_{U}(A, B) \quad \Gamma \vdash t:\mathrm{El}(A)}{\Gamma \vdash \mathrm{heqTransport}(A, B, p, t):\mathrm{JMEq}(A, B, t, \mathrm{transport}_{X:U.\mathrm{El}(X)}(A, B, p, t))}$$

The use of Tarski universes instead of Russell universes makes it clear that John Major equality could be defined using inference rules for any type family $x:A \vdash B(x)$ instead of just the Tarski universe type family $X:U \vdash \mathrm{El}(X)$:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash u:B(a) \quad \Gamma \vdash v:B(b)}{\Gamma \vdash \mathrm{JMEq}(a, b, u, v) \; \mathrm{type}}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma \vdash a:A \quad \Gamma \vdash u:B(a)}{\Gamma \vdash \mathrm{hrefl}(a, u):\mathrm{JMEq}(a, b, u, v)}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma \vdash a:A \quad \Gamma \vdash u:B(a) \quad \Gamma \vdash v:B(a) \quad \Gamma \vdash p:\mathrm{Id}_{B(a)}(u, v)}{\Gamma \vdash \mathrm{eqToHeq}(a, u, v, p):\mathrm{JMEq}(a, a, u, v) \; \mathrm{type}}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma \vdash a:A \quad \Gamma \vdash u:B(a) \quad \Gamma \vdash v:B(a) \quad \Gamma \vdash p:\mathrm{JMEq}(a, a, u, v)}{\Gamma \vdash \mathrm{heqToEq}(a, u, v, p):\mathrm{Id}_{B(a)}(u, v) \; \mathrm{type}}$$

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \quad \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash p:\mathrm{Id}_{A}(a, b) \quad \Gamma \vdash t:B(a)}{\Gamma \vdash \mathrm{heqTransport}(a, b, p, t):\mathrm{JMEq}(a, b, t, \mathrm{transport}_{x:A.B(x)}(a, b, p, t))}$$

### Inference rules without UIP

It should be possible to take McBride's axiomatization of John Major equality above and simply remove the requirement that John Major equality be a proposition-valued type family.

Then, the inference rules for forming John Major equality and terms are as follows.  First the inference rule that defines the John Major equality itself, as a [[dependent type]], in some [[context]] $\Gamma$.

Formation rule for John Major equality:
$$\frac{
    \begin{array}{l}
      \Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash B(x) \; \mathrm{type} \\
      \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash y:B(a) \quad \Gamma \vdash z:B(b)
    \end{array}
  }{\Gamma \vdash \mathrm{JMEq}_{x:A.B(x)}(a, b, y, z) \; \mathrm{type}}$$

Now the basic "introduction" rule, which says that everything is equal to itself in a canonical way.

Introduction rule for John Major equality:
$$\frac{
    \begin{array}{l}
      \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \\
      \Gamma \vdash a:A \quad \Gamma \vdash y:B(a)
    \end{array}
  }{\Gamma \vdash \mathrm{refl}_{x:A.B(x)}(a, y):\mathrm{JMEq}_{x:A.B(x)}(a, a, y, y)}$$

Next we have the "elimination" rule:

Elimination rule for John Major equality:
$$\frac{
    \begin{array}{l}
      \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \\
      \Gamma, a:A, b:A, y:B(a), z:B(b), p:\mathrm{JMEq}_{x:A.B(x)}(a, b, y, z) \vdash C(a, b, y, z, q) \; \mathrm{type} \\
      \Gamma \vdash t:\prod_{a:A} \prod_{y:B(a)} C(a, a, y, y, \mathrm{refl}_{x:A.B(x)}(a, y)) \\
      \Gamma \vdash a:A \quad \Gamma \vdash b:A \quad \Gamma \vdash y:B(a) \quad \Gamma \vdash z:B(b) \quad \Gamma \vdash p:\mathrm{JMEq}_{x:A.B(x)}(a, b, y, z)
    \end{array}
  }{\Gamma \vdash \mathrm{ind}_{\mathrm{JMEq}_{x:A.B(x)}}(t, a, b, y, z, p):C(a, b, y, z, p)}$$

The elimination rule then says that if:

1. for any elements $a:A$ and $b:A$, any elements $y:B(a)$ and $z:B(b)$, and any John Major equality $p:\mathrm{JMEq}_{x:A.B(x)}(a, b, y, z)$ we have a type $C(a, b, y, z, p)$, and
1. we have a specified dependent function
$$t:\prod_{a:A} \prod_{y:B(a)} C(a,a,y,y,\mathrm{refl}_{x:A.B(x)}(a, y))$$

then we can construct a canonically defined term called the *eliminator*
$$\mathrm{ind}_{\mathrm{hId}_{x:A.B(x)}}(t,a,b,y,z,q):C(a,b,y,z,q)$$
for *any* $a$, $b$, $y$, $z$, and $q$.

Then, we have the [[computation rule]] or [[beta-reduction]] rule. This says that for all elements $a:A$ and $y:B(a)$, substituting the dependent function $t:\prod_{a:A} \prod_{y:B(a)} C(a,a,y,y,\mathrm{refl}_{x:A.B(x)}(a, y))$ into the eliminator along [[reflexive relation|reflexivity]] for $a$ and $y$ yields an element equal to $t(a, y)$ itself.  Normally "equal" here means [[judgmental equality]] (a.k.a. definitional equality), but it is also possible for it to mean [[propositional equality]] (a.k.a. typal equality), so there are two possible computation rules.

Computation rules for John Major equality:

* Judgmental computational rule
$$\frac{
    \begin{array}{l}
      \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \\
      \Gamma, a:A, b:A, y:B(a), z:B(b), p:\mathrm{JMEq}_{x:A.B(x)}(a, b, y, z) \vdash C(a, b, y, z, p) \; \mathrm{type} \\
      \Gamma \vdash t:\prod_{a:A} \prod_{y:B(a)} C(a, a, y, y, \mathrm{refl}_{x:A.B(x)}(a, y)) \quad \Gamma \vdash a:A \quad \Gamma \vdash y:B(a)
    \end{array}
  }{\Gamma \vdash \mathrm{ind}_{\mathrm{JMEq}_{x:A.B(x)}}(t, a, a, y, y, \mathrm{refl}_{x:A.B(x)}(a, y)) \equiv t(a, y):C(a, a, y, y, \mathrm{refl}_{x:A.B(x)}(a, y))}$$

* Propositional computational rule
$$\frac{
    \begin{array}{l}
      \Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B(x) \; \mathrm{type} \\
      \Gamma, a:A, b:A, y:B(a), z:B(b), p:\mathrm{JMEq}_{x:A.B(x)}(a, b, y, z) \vdash C(a, b, y, z, p) \; \mathrm{type} \\
      \Gamma \vdash t:\prod_{a:A} \prod_{y:B(a)} C(a, a, y, y, \mathrm{refl}_{x:A.B(x)}(a, y)) \quad \Gamma \vdash a:A \quad \Gamma \vdash y:B(a)
    \end{array}
  }{\Gamma \vdash \beta_{\mathrm{JMEq}_{x:A.B(x)}}(t, a, y):\mathrm{Id}_{C(a, a, y, y, \mathrm{hrefl}_{x:A.B(x)}(a, y))}(\mathrm{ind}_{\mathrm{JMEq}_{x:A.B(x)}}(t, a, a, y, y, \mathrm{refl}_{x:A.B(x)}(a, y)), t(a, y))}$$

Finally, one might consider a [[uniqueness rule]] or [[eta-conversion]] rule.  But similar to the case for [[identity types]], a judgmental uniqueness rule for John Major equality implies that the type theory is an [[extensional type theory]], in which case there is not much need for John Major equality, so such a rule is almost never written down.  And as for identity types and other inductive types, the propositional/typal uniqueness rule is provable from the other four inference rules, so we don't write it out explicitly.

## Related concepts

* [[heterogeneous identity type]]

## References

* {#McBride99} [[Conor McBride]], §5.1.3 in: *Dependently Typed Functional Programs and their Proofs*, (1999) [[pdf](https://www.lfcs.inf.ed.ac.uk/reports/00/ECS-LFCS-00-419/ECS-LFCS-00-419.pdf)]

  > (who speaks of "John Major equality" [McBride 1999 ](#McBride99) with reference to British political discussion of that times)

* {#Winterhalter23} [[Théo Winterhalter]], *A conservative and constructive translation from extensional type theory to weak type theory*, Strength of Weak Type Theory, [[DutchCATS]], 11 May 2023. ([slides](https://dutchcats.github.io/Weak_type_theories/slides_winterhalter.pdf))

[[!redirects John Major equality]]

[[!redirects fibered heterogeneous identity types]]


fibered limit

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

#Contents#
* table of contents
{:toc}

## Idea

An appropiate notion of [[limit]] in [[fibered category theory]].

## Related concepts

* [[Grothendieck fibration]]

## References

The notion may be due to:

* {#Gray} [[John W. Gray]], *Fibred and Cofibred Categories*, in: [[Samuel Eilenberg|S. Eilenberg]], [[D. K. Harrison]], [[S. MacLane]], [[H. Röhrl]] (eds.): *[[Proceedings of the Conference on Categorical Algebra - La Jolla 1965]]*, Springer (1966) [[doi:10.1007/978-3-642-99902-4](https://doi.org/10.1007/978-3-642-99902-4)]

The general perspective is in:

* {#Hermida} [[Claudio Hermida]], Def. 4.7 and Cor. 4.9 in: *Some properties of $Fib$ as a fibred 2-category*, Journal of Pure and Applied Algebra **134** (1999) 83-109 [doi:10.1016/S0022-4049(97)00129-1]

[[!redirects fibered limits]]

[[!redirects fibred limit]]
[[!redirects fibred limits]]

[[!redirects fibered colimit]]
[[!redirects fibered colimits]]

[[!redirects fibred colimit]]
[[!redirects fibred colimits]]

[[!redirects fibered terminal object]]
[[!redirects fibered terminal objects]]

[[!redirects fibred terminal object]]
[[!redirects fibred terminal objects]]


fibered n category > history

< [[fibered n category]]
[[!redirects fibered n category -- history]]


Fibonacci numbers

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

#Contents#
* table of contents
{:toc}

## Definition

The Fibonacci numbers are a [[subset]] of the [[natural numbers]] [[inductive definition|defined inductively]] by the equation

$$f(x + 2) \coloneqq f(x + 1) + f(x)$$

with [[initial conditions]] $f(0) = 0$ and $f(1) = 1$.

## See also

* [[natural numbers]]


fibrant object

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

#Contents#
* table of contents
{:toc}

## Idea

A [[model category]] is a [[homotopical category]] equipped with especially nice control over the [[weak equivalences]]. In particular every [[object]] of the category is weakly equivalent to an object that is particularly well-behaved for forming [[derived hom-spaces]] _into_ it -- these are the fibrant objects, as well as weakly equivalent to a object that is particularly well-behaved for forming [[derived hom-spaces]] _out of_ it -- these are the cofibrant objects.

Hence fibrant and cofibrant objects are particularly good representatives of objects, which are the "same" as the given objects up to [[weak equivalence]].

These concepts exists also in [[homotopical categories]] with less [[extra structure]] than that of a full [[model category]]. For instance a [[category of fibrant objects]] implements roughly half of the model category axioms, namely those for [[fibrations]] and, as the name indicates, it has a concept of weakly equivalent replacement by fibrant objects, but in general not by cofibrant object. And [[formal dual|dually]], in a [[cofibration category]] there is a notion of cofibrant objects but not necessarily of fibrant objects.

## Definition

In a [[model category]], an [[object]] $X$ is said to be **fibrant** if the unique morphism $X\to 1$ to the [[terminal object]] is a [[fibration]].

$$
  (X \;\; \text{fibrant})
   \;\;\Leftrightarrow\;\;
  ( X \overset{\in Fib}{\longrightarrow} 1)
$$

[[formal dual|Dually]], $X$ is said to be **cofibrant** if the unique morphism $0\to X$ from the [[initial object]] is a [[cofibration]].

$$
  (X \;\; \text{cofibrant})
   \;\;\Leftrightarrow\;\;
  ( 0 \overset{\in Cof}{\longrightarrow} X)
$$

Hence the axiom that every morphism in a model category factors

1. as an [[acyclic cofibration]] followed by a [[fibration]]

1. as a [[cofibration]] followed by an [[acyclic fibration]]

implies [[fibrant resolution]] and [[cofibrant resolution]] of objects:

For $X$ any object then

1. the factorization of the terminal morphism as an acyclic cofibration followed by a fibration yields a fibrant object $X_{fib}$ weakly equivalent to $X$

   $$
      X
        \overset{\in Cof \cap W}{\longrightarrow}
      X_{fib}
        \overset{\in Fib}{\longrightarrow}
      1
   $$

1. the factorization of the initial morphism as a cofibration followed by an acyclic fibration yields a cofibrant object $X_{cof}$ weakly equivalent to $X$

   $$
     0
       \overset{\in Cof}{\longrightarrow}
     X_{cof}
       \overset{\in Fib \cap W}{\longrightarrow}
     X
   $$

## Examples

The standard examples appear

1. in the [[classical model structure on topological spaces]], here every object $X$ is fibrant (namely the [[continuous function]] $X \to \ast$ to the [[point space]] is a [[Serre fibration]]), and the cofibrant objects are the [[retracts]] of [[cell complexes]], in particular the [[CW-complexes]];

1. in the [[classical model structure on simplicial sets]], here every object is cofibrant, and the fibrant objects are the [[Kan complexes]].

Other examples include:

* In the projective [[model structure on dgc-algebras]] in non-negative degree, the cofibrant objects are the [[Sullivan algebras]] (see [there](model+structure+on+dg-algebras#SullivanAlgebras)). This plays a key role in [[rational homotopy theory]]

* ...

## Related concepts

* [[cofibrant object]], [[bifibrant object]]

* [[fibration]], [[cofibration]]

* [[fibrant replacement]]

* [[fibrant type]]

* [[functorial factorization]]

[[!redirects fibrant object]]
[[!redirects fibrant objects]]

[[!redirects cofibrant object]]
[[!redirects cofibrant objects]]

[[!redirects fibrant and cofibrant object]]
[[!redirects fibrant and cofibrant objects]]

[[!redirects fibrancy]]
[[!redirects cofibrancy]]

[[!redirects fibrant]]
[[!redirects cofibrant]]


fibrant replacement

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###Context###
#### Model category theory
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[[!include model category theory - contents]]
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#Contents#
* table of contents
{:toc}

## Definition

In a [[model category]] every [[morphism]] may be factored as a [[weak equivalence]] followed by a [[fibration]]. Specifically if the morphism is that to the [[terminal object]], this process finds a weakly equivalent [[fibrant object]].
This is a **fibrant replacement** or [[resolution]] of the original object.

The dual concept is called **cofibrant replacement**.

If the factorization is functorial, then it yields a **fibrant replacement functor**. See at _[[functorial factorization]]_.

## Related concepts

* [[functorial factorization]]
* [[algebraic weak factorization system]]
* [[contextual fibrancy]]

[[!redirects fibrant replacement]]
[[!redirects cofibrant replacement]]
[[!redirects fibrant replacement functor]]
[[!redirects cofibrant replacement functor]]
[[!redirects fibrant resolution]]
[[!redirects cofibrant resolution]]
[[!redirects fibrant resolution functor]]
[[!redirects cofibrant resolution functor]]

[[!redirects fibrant replacements]]
[[!redirects cofibrant replacements]]
[[!redirects fibrant replacement functors]]
[[!redirects cofibrant replacement functors]]
[[!redirects fibrant resolutions]]
[[!redirects cofibrant resolutions]]


fibrant type

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### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### Homotopy theory
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In the [[categorical semantics]] of [[dependent type theory]] in some suitable [[category]] $\mathcal{C}$ (see at _[[relation between category theory and type theory]]_) a [[type]] $X$ in the empty [[context]] is usually interpreted as an [[object]] $[X]$ in the given target [[category]], such that the unique [[morphism]] $[X]\to \ast$ to the [[terminal object]] (which interprets the [[unit type]]) is a special morphism called a [[display map]]. Specifically in the context of [[homotopy type theory]] display maps are typically [[fibrations]] with respect to some [[homotopical category]] structure on $\mathcal{C}$ (for instance the structure of a [[category of fibrant objects]] or even that of a [[model category]], in particular that of a [[type-theoretic model category]]). Since in $\mathcal{C}$ objects $[X]$ for which $[X] \to \ast$ is a [[fibration]] are called _[[fibrant objects]]_, one may then call a closed type which is to be interpreted this way as a _fibrant type_.
More generally, an open type $T$ in context $\Gamma$ is called _fibrant_ if $[\Gamma.T] \to [\Gamma]$ is a fibration.

Usually _all_ types are required to be interpreted this way, and hence often the term "fibrant type" is not used, since often non-fibrant types are never considered.

However in general one may want to consider flavors of [[dependent type theory]] with [[categorical semantics]] in suitable [[homotopical categories]] which explicitly distinguishes between fibrant and non-fibrant interpretation. One example of this is what has come to be called _[[Homotopy Type System]]_ (HTS). Default [[homotopy type theory]] (HoTT) has interpretation in [[type-theoretic model categories]] which is such that, roughly, [[Quillen equivalence]] is respected in that the interpretation in a way only depends on the [[(infinity,1)-category]] which is [[presentable (infinity,1)-category|presented]] by the model category. Since up to [[weak equivalence]] every object in the model category is fibrant, this means that in this sense there is no sensible distinction between fibrant and non-fibrant objects/types. On the other hand, [[HTS]] is more a language for the [[type-theoretic model category]] itself (or whatever homotopical category plays its role), not just for the [[(infinity,1)-category]] that it is going to present. In its interpretation one explicitly wants to be able to distinguish between any object and its fibrant replacement. Hence in this context one distinguishes between "fibrant types" that are required to be interpreted as [[fibrant objects]] and "non-fibrant types" which may be interpreted as more general objects.

## Related concepts

* [[fibrant replacement]]
* [[contextual fibrancy]]

[[!redirects fibrant types]]


fibration

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###Context###
#### Homotopy theory
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[[!include homotopy - contents]]
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#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

There are a number of different types of [[morphism]] bearing the name **fibration**, which are all connected to each other at least by a [[zigzag]] of relationships.

#Contents#
* table of contents
{:toc}

## Classical homotopy theory

In classical [[homotopy theory]], a fibration $p:E\to B$ is a [[continuous function]] between [[topological spaces]] that has a certain [[lifting property]].  The most basic property is that given a point $e\in E$ and a path $[0,1] \to B$ in $B$ starting at $p(e)$, the path can be lifted to a path in $E$ starting at $e$.

One generally also assumes the lifting of additional structures (including "higher homotopies") in $B$ which, in particular, imply that the path lifting is unique up to [[homotopy]]. Different choices of what can be lifted give rise to different notions of fibration, for example:

* In a [[Hurewicz fibration]], all sorts of homotopies can be lifted.

* In a [[Serre fibration]], topological $n$-cells can be lifted for all $n$.

* In a [[Dold fibration]] or "halb-fibration," all homotopies can be lifted, but the lifting only has to agree with the given initial map up to vertical homotopy.  A Hurewicz fibration is a Dold fibration where the vertical homotopy is stationary.

All three of these definitions give rise to a
[[long exact sequence of homotopy groups]].  In fact, the [[exact sequence]] would follow from only requiring up-to-homotopy lifting for [[cubes]].  There doesn't seem to be a name for this sort of map, but there is the following:

* A [[quasifibration]] (not to be confused with the completely different notion related to [[quasi-categories]] below) is a map which induces a [[long exact sequence of homotopy groups]].  Equivalently, it is a map each of whose [[fibers]] is [[weak homotopy equivalence|weakly homotopy equivalent]] to the corresponding [[homotopy fiber]].

## Abstract homotopy theory

Inspired by the role of fibrations in
[[algebraic topology]], part of the structure of a [[model category]] or a [[category of fibrant objects]] is a class of maps called "fibrations," which also possess a lifting property relating them to the rest of the structure ([[cofibrations]] and [[weak equivalences]]).  Examples include:

* [[Kan fibrations]] between [[simplicial sets]] (whose study predates the definition of a [[model category]]), which allow lifting of $n$-[[simplices]] for every $n$.

* [[isofibration]]s between [[categories]], which allow lifting of [[isomorphisms]].

* fibrations in the [[model structure for quasi-categories]] between [[simplicial sets]] are a common  generalization of [[Kan fibrations]] and [[isofibrations]], just as [[quasi-category|quasicategories]] are a common generalization of [[Kan complex]]es and categories.

Fibrations have many good properties in [[homotopy theory]].  For example, under some extra assumptions [[pullback]] of a fibration is already a [[homotopy pullback]] (see there for details). Sufficient conditions are that in addition all three [[objects]] involved are [[fibrant objects]] or that the model category is a [[right proper model category]].

Generally, every morphism can be replaced by a
[[weak equivalence|weakly equivalent]] fibration, which gives one way to compute a homotopy pullback by comuting an ordinary pullback of a fibrant enough weakly quivalent [[cospan]] [[diagram]].

## Transports and classifying spaces

There is a classical theorem that [[covering spaces]] $p:E\to B$ of a [[locally connected space]] $B$ (which have _unique_ path lifting) are equivalent to functors $\Pi(B)\to Set$ from the [[fundamental groupoid]] of $B$ to [[Set]] (hence to [[permutation representations]] of the [[fundamental group]] ).

The functor corresponding to $p:E\to B$ takes a point $b\in B$ to its fiber $p^{-1}(b)$, and a path $\alpha$ from $b$ to $b'$ to the function $p^{-1}(b) \to p^{-1}(b')$ defined by "the endpoint of the lift of $\alpha$."

Generalizing this massively, arbitrary topological fibrations $p:E\to B$ correspond to functors from the fundamental $\infty$-groupoid of $B$ to the $(\infty,1)$-category of spaces, in an analogous way.  Points are sent to fibers, paths to the endpoints of liftings, homotopies between paths to the result of lifting such homotopies, and so on.  This functor is sometimes called the **[[parallel transport]]** corresponding to the fibration.  There are a number of ways to make this precise, some of which make sense in classical homotopy theory (e.g. actions by a loop space) and some of which require higher categorical machinery. A discussion is at [[higher parallel transport]] in the section [Flat ∞-parallel transport in Top](http://ncatlab.org/nlab/show/higher+parallel+transport#FlatInTop).

More generally, one can consider fibrations in which the fibers are equipped with some extra structure, such as being a [[vector space]] or having an [[action]] by a [[group]].  This corresponds to restricting the [[codomain]] of the transport to some category of spaces with structure and structure-preserving maps.  The most common version of this is a [[bundle]] with some structure [[group]] $G$, in which case the transport lands in $\mathbf{B}G$, the [[delooping]] [[groupoid]] of $G$ with a singleobject (thought of as a generic $G$-[[torsor]]) and $G$ as its endomorphisms.  In such cases the word "parallel" is often added in front of "transport."

If we replace the [[topological groupoid]] $\mathbf{B}G$  its  [[classifying space]], $\mathcal{B}G$ -- which is the [[geometric realization of simplicial topological spaces]] of the [[nerve]] of $\mathbf{B}G$ -- then passing to $\pi_0$ recovers the classical fact that "classifying spaces classify": there is a bijection between $G$-bundles over a space $X$ and homotopy classes of maps $X\to \mathcal{B}G$.  This bijection is realized by pulling back to $X$ the "universal $G$-bundle" $\mathcal{E}G \to \mathcal{B}G$ over the space $\mathcal{B}G$.  There are also classifying spaces for more general types of fibrations, constructed from the relevant subcategories of $Top$.

## Fibrations in category theory

It is common in [[category theory]] to consider the objects of a [[over category|slice category]] $C/X$ as "objects of $C$ varying over $X$."  For example, an object $A\to X$ of $Set/X$ can be identified with an $X$-indexed family $\{A_x\}_{x\in X}$ of sets, where $A_x$ is the [[fiber]] of $A\to X$ over $x\in X$.  Likewise, if $X$ is a [[topological space]], we can regard an object of $Top/X$ as a family of spaces (the fibers) "varying continuously" over $X$.  But as we have seen, in the topological case, in order to make this varying into a "functor" $X\to Top$ we need the map to be a fibration.

In category theory, there is analogous notion of when a functor $p:E\to B$ is a __fibration__ or __[[fibered category]]__ or __[[Grothendieck fibration]]__ or __Cartesian fibration__, which is exactly what is needed to make the assignment $b \mapsto p^{-1}(b)$ into a [[pseudo-functor]] $B\to Cat$.  (Actually, there are two such notions, since a functor on a non-groupoid can be either [[covariant functor|covariant]] or [[contravariant functor|contravariant]].)  Thus, in this case [[Cat]] is the analogue of the "[[classifying space]]", and there is a "[[universal Cartesian fibration]]" $Cat_* \to Cat$ of which every other fibration is a [[pullback]] (modulo size issues).

Categorical fibrations also have a "lifting" property, but the liftings must satisfy an extra "universality" condition.  For this reason, they are not the fibrations in any [[model structure]] on [[Cat]].  However, fibrations of this sort between [[groupoids]] are the same as [[isofibrations]], and thus are the fibrations in the [[folk model structure]] on [[Grpd]].  See [[Grothendieck fibration]] for more details.

A __[[discrete fibration]]__ is one in which we use [[Set]] instead of [[Cat]] as the classifying space.

## Fibrations in higher category theory

* [[fibrations of quasi-categories]]

  * [[Cartesian fibration]]

  * [[left fibration]], [[right fibration]]

  * [[right/left fibration of simplicial spaces]]

  * [[inner Kan fibration]]

  * [[minimal Joyal fibration]]

* [[homotopy Kan fibration]]

## Fibrations in type theory

Fibrations are employed in [[type theory]] as the [[categorical models of dependent types]].

## Related concepts

Both forms of [[opposite 2-category|2-categorical dualization]] are commonly encountered in the context of fibrations. Moreover, the distinction between the two is appreciable. [[Cofibrations]] play a role dual to that of fibrations in homotopy theory, notably in the axioms for a [[model category]]. In this context, cofibrations have an entirely different geometric flavor from fibrations. On the other hand, [[opfibrations]] are the same as fibrations homotopically because of the invertibility of 2-cells. The duality between fibrations and opfibrations is more visible in [[category theory]], where it's the same duality as one finds between limits and colimits, for example. Confusingly, some (much?) of the older categorical (and algebro-geometric?) literature uses "cofibration" to mean "opfibration".

An [[object]] such that the unique morphism to the [[terminal object]] is a fibration (in abstract [[homotopy theory]]) is called a _[[fibrant object]]_.

A replacement of a morphism by a weakly equivalent fibration is also called a _[[resolution]]_.

[[!redirects fibrations]]
[[!redirects Fibrations]]


fibration category

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###Context###
#### Model category theory
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[[!include model category theory - contents]]
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#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
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#Contents#
* table of contents
{:toc}

## Idea

Something like a

* [[category of fibrant objects]]

or similar

## Related entries

* [[category of fibrant objects]]

* [[categorical semantics of dependent types]]

  * [[clan]], [[tribe]]

  * [[type-theoretic model category]]

* [[cofibration category]]

[[!redirects fibration categories]]


fibration fibered in groupoids

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### Context
#### Category theory
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[[!include category theory - contents]]
=--
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#Contents#
* table of contents
{:toc}

## Idea

A **[[Grothendieck]] fibration fibered in groupoids** -- usually called a **category fibered in groupoids** -- is a [[Grothendieck fibration]] $p : E \to B$ all whose [[fiber]]s are [[groupoid]]s.

## Definition

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

A **fibration fibered in groupoids** is a [[functor]] $p : E \to B$ such that the corresponding (strict) functor $B^{op} \to $ [[Cat]] classifying $p$ under the [[Grothendieck construction]] factors through the inclusion [[Grpd]] $\hookrightarrow$ [[Cat]].

=--

Under forming [[opposite categories]] we obtain the notion of an **op-fibration fibered in groupoids**. In old literature this is sometimes called a "cofibration in groupoids" but that terminology collides badly with the notion of *[[cofibration]]* in [[homotopy theory]] and [[model category]] theory.

## Properties

Fibrations in groupoids have a simple characterization in terms of their [[nerve]]s. Let $N : Cat \to sSet$ be the [[nerve]] functor and for $p : E \to B$ a morphism in [[Cat]], let $N(p) : N(E) \to N(B)$ be the corresponding morphism in [[sSet]].

Then

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

The functor $p : E \to B$ is an  op-fibration in groupoids precisely if the morphism $N(p) : N(E) \to N(B)$ is a [[left Kan fibration]] of simplicial sets, i.e. precisely if for all [[horn]] inclusion

$$
  \Lambda[n]_i \hookrightarrow \Delta[n]
$$

for all $n \in \mathbb{N}$ and all $i$ _smaller_ than $n$ -- $0 \leq i \lt n$, we have that every commuting diagram

$$
  \array{
    \Lambda[n]_i &\to& N(E)
    \\
    \downarrow && \downarrow^{\mathrlap{N(p)}}
    \\
    \Delta[n] &\to& N(B)
  }
$$

has a lift

$$
  \array{
    \Lambda[n]_i &\to& N(E)
    \\
    \downarrow &\nearrow& \downarrow^{\mathrlap{N(p)}}
    \\
    \Delta[n] &\to& N(B)
  }
  \,.
$$

=--

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

For instance [[Higher Topos Theory|HTT, prop. 2.1.1.3]].

=--

## Related concepts

* [[Grothendieck fibration]], [[Grothendieck construction]]

* [[discrete fibration]]

* [[left fibration]], [[right fibration]], [[Cartesian fibration]], [[Kan fibration]]

## References

Review:

* [[Angelo Vistoli]], §3.3 in: *Grothendieck topologies, fibered categories and descent theory*, in: *[[Fundamental algebraic geometry -- Grothendieck's FGA explained]]*, Mathematical Surveys and Monographs **123**, Amer. Math. Soc. (2005) 1-104 [[ISBN:978-0-8218-4245-4](https://bookstore.ams.org/surv-123-s), [math.AG/0412512](http://arxiv.org/abs/math/0412512)]

[[!redirects category fibered in groupoids]]
[[!redirects category fibred in groupoids]]
[[!redirects categories fibered in groupoids]]
[[!redirects categories fibred in groupoids]]

[[!redirects category fibered in groupoids]]
[[!redirects category fibred in groupoids]]
[[!redirects categories fibered in groupoids]]
[[!redirects categories fibred in groupoids]]
[[!redirects category cofibered in groupoids]]

[[!redirects category cofibred in groupoids]]
[[!redirects categories cofibered in groupoids]]
[[!redirects categories cofibred in groupoids]]

[[!redirects fibration in groupoids]]
[[!redirects opfibration in groupoids]]
[[!redirects fibrations in groupoids]]
[[!redirects opfibrations in groupoids]]

[[!redirects fibrations fibered in groupoids]]