Grothendieck fibration

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## Idea

A *Grothendieck fibration* (named after [[Alexander Grothendieck]], also called a *fibered category* or just a *fibration*) is a [[functor]] $p \colon E\to B$ such that the [[fibers]] $E_b = p^{-1}(b)$ depend (contravariantly) [[pseudofunctor|pseudofunctorially]] on $b\in B$.

One also says that $E$ is a *fibered category* over $B$.
One calls $E$ the *total category* of the fibration (not to be confused with the independent notion of *[[total category]]*) and $B$ its *base category*.

[[formal duality|Dually]], in a (Grothendieck) *opfibration* the dependence is [[covariant functor|covariant]].

There is an [[equivalence of categories|equivalence]] of [[strict 2-category|2-categories]]

$$
  Fib(B)
   \overset
    {\simeq}
    {\leftrightarrow}
  [B^{op}, Cat]
  \;\colon\;
  \textstyle{\int}
$$

between the 2-category of fibrations, cartesian functors, and vertical natural transformations over $B$, and the 2-category $[B^{op},Cat]$ of contravariant [[pseudofunctor]]s from $B$ to [[Cat]], also called $B$-[[indexed categories]].

The "[[Grothendieck construction]]" $\int \colon [B^{op}, Cat] \to Fib(B) \colon F \mapsto \int F$ of a fibration from a pseudofunctor.

Those fibrations corresponding to pseudofunctors that factor through [[Grpd]] are called **[[categories fibered in groupoids]]**.

## Definition
 {#Definition}

\begin{definition}\label{CartesianMorphism}
**([[cartesian morphism]])**
\linebreak
Given a [[functor]] $P \,\colon\, \mathcal{E} \longrightarrow \mathcal{B}$,
a [[morphism]] $f \colon x \to y$ in its [[domain]] [[category]] $\mathcal{E}$ is called *[[cartesian morphism|cartesian]]* if for any morphism $g \colon z\to y$ in $\mathcal{E}$ and $w \colon P(z)\to P(x)$ in $\mathcal{B}$ such that $P(f)\circ w = P(g)$, there exists a *unique* $\widehat{w} \,\colon\, z\to x$ such that $g = f \circ \widehat{w}$ and $P(\widehat{w}) = w$:

\begin{tikzcd}[
  sep=20pt
]
  &
  z
  \ar[
    drrr,
    bend left=20,
   "{ \forall \, g }"{description}
  ]
  \ar[
    dr,
    dashed,
    "{
       \exists! \, \widehat{w}
    }"{description}
  ]
  \\
  \mathcal{E}
  \ar[dd, "{ P }"]
  &&
  x
  \ar[
     rr,
     "{ f }"{description},
     "{ \mathrm{cartesian} }"{swap, yshift=-1pt}
  ]
  &&
  y
  \\
  &
  P(z)
  \ar[drrr, bend left=20, "{ P(g) }"{description}]
  \ar[
    dr,
    "{ \forall \, w }"{description}
  ]
  \\
  \mathcal{B}
  &&
  P(x)
  \ar[rr, "{ P(f) }"{description}]
  &&
  P(y)
\end{tikzcd}

\end{definition}

\begin{definition}\label{GrothendieckFibration}
**(Grothendieck fibration)**
\linebreak
A [[functor]] $p \colon \mathcal{E} \to \mathcal{B}$ is a *fibration* if for all [[objects]] $y \in \mathcal{E}$ and [[morphisms]] $Pf \colon Px \to P(y)$, there is a cartesian morphism $f \colon x \to y$ (Def. \ref{CartesianMorphism}) such that $P(f) = Pf$.
\end{definition}

Such $f$ is also called a "cartesian lifting" of $Pf$ to $y$, and a choice of cartesian lifting for every $y$ and $Pf$ is called a *[[cleavage]]*.  Thus, assuming the [[axiom of choice]], a functor is a fibration iff it admits some cleavage.

\begin{remark}
**(weakly cartesian morphisms)**
\linebreak
We say that a morphism $f$ is [[prefibered category|weakly cartesian]] if it has the property of a Cartesian morphism (Def. \ref{CartesianMorphism}) only when $w$ is an [[identity morphism]].

One can prove that $P$ is a fibration (Def. \ref{GrothendieckFibration}) if and only if firstly, it has the above property with "Cartesian" replaced by "weakly cartesian," and secondly, the composite of weakly cartesian arrows is weakly cartesian.

In a fibration, every weakly cartesian lifting $f$ of $Pf$ to $y$ is in fact cartesian (as one can show by combining the [[universal properties]] of $f$ and of a given cartesian lifting to $e$), but this is not true in general.

Some sources say "cartesian" and "strongly cartesian" instead of "weakly cartesian" and "cartesian," respectively.  If weakly cartesian liftings exist but weakly cartesian arrows are *not* necessarily closed under composition, one sometimes speaks of a [[prefibered category]].
\end{remark}

\begin{definition}\label{MorphismOfFibrations}
**(morphism of fibrations -- [[fibered functor]])**
\linebreak
For $P$ and $P'$ fibrations (Def. \ref{GrothendieckFibration}), a [[commuting square]] of [[functors]] of the form
\[
  \label{DiagramForFiberedFunctor}
  \array{
    \mathcal{E}'
     &
    \longrightarrow
    &
    \mathcal{E}
    \\
    \mathllap{{}^{P'}}\big\downarrow
    &&
    \big\downarrow
    \mathrlap{{}^{P}}
    \\
    \mathcal{B}'
      & \longrightarrow &
    \mathcal{B}
  }
\]
is a *morphism of fibrations* (also called a *cartesian square* or *[[fibred functor]]*) if the top functor preserves Cartesian morphisms (Def. \ref{CartesianMorphism}).
\end{definition}

\begin{definition}\label{TwoMorphismOfFibrations}
**([[2-morphisms]] of fibrations -- [[fibered natural transformation]])**
\linebreak
A [[2-morphism]] (or *[[fibered natural transformation]]*) between morphisms of fibrations (Def. \ref{MorphismOfFibrations}) is a pair of [[natural transformations]], one lying over the other in (eq:DiagramForFiberedFunctor).
\end{definition}

\begin{remark}
If the functor on base categories is an [[identity functor|identity]], then a 2-morphism (Def. \ref{TwoMorphismOfFibrations}) is a [[natural transformations]] between total categories which is "vertical" in that its component morphisms lie in [[fibers]].
\end{remark}

## Fibrations versus pseudofunctors

Given a fibration $p:E\to B$, we obtain a pseudofunctor $B^{op}\to Cat$ by sending each $b\in B$ to the category $E_b = p^{-1}(b)$ of objects mapping onto $b$ and morphisms mapping onto $1_b$.  To obtain the action on morphisms, given an $f:a\to b$ in $B$ and an object $e\in E_b$, we choose a cartesian arrow $\phi:e'\to e$ over $f$ and call its [[source]] $f^*(e)$.  The universal factorization property of cartesian arrows then makes $f^*$ into a functor $E_b \to E_a$, and it is easy to verify that it is a pseudofunctor.  The functor in the other direction is called the [[Grothendieck construction]].  This yields a [[strict 2-equivalence of 2-categories]] between

* fibrations over $B$, morphisms of fibrations over $Id_B$, and 2-cells over $id_{Id_B}$, and

* [[pseudofunctors]] of the form $B^{op}\to Cat$, [[pseudonatural transformations]], and [[modifications]].

In fact, this is an instance of the general theory of representability for [[generalized multicategories]].  There is a monad $T$ whose pseudoalgebras are pseudofunctors $B^{op}\to Cat$, and whose "generalized multicategories" are functors $E\to B$.  Such a multicategory is "representable" precisely when it is a fibration, and moreover there is an induced monad $\hat{T}$ on $Cat/B$ which is [[lax-idempotent 2-monad|colax-idempotent]] and whose pseudoalgebras are precisely the fibrations.

This correspondence also generalizes to the correspondence between arbitrary functors over $B$ and [[displayed categories]] over $B$, i.e. normal lax functors $B\to Prof$.

## Fibrations versus presheaves of categories

The [[Grothendieck construction]] is an equivalence of [[bicategories]]
from the bicategory of presheaves of categories to the bicategory of Grothendieck fibrations.

As explained there, this equivalence, interpreted as a functor between 1-categories,
has both a left and a right adjoint (which are equivalent in the bicategorical context).
Roughly speaking, the left adjoint strictifies a Grothendieck fibration
by adding formal pullbacks of objects, which themselves pullback strictly,
whereas the right adjoint strictifies a Grothendieck fibration
by equipping object with a functorial choice of a pullback for each possible morphism.
See [[Grothendieck construction]] for more details.

These two adjunctions can be turned into [[Quillen equivalences]] of [[model categories]].
This can be deduced, for example, from two [[Quillen equivalences]]
between [[cartesian fibrations]] and [[presheaves]] of [[marked simplicial sets]]
on the [[quasicategory]] given by the [[nerve]] of $C$.
See [[straightening functor]] for more details.

## Remarks

* Fibrations are a "nonalgebraic" structure, since the base change functors $f^*$ are determined by universal properties, hence uniquely up to isomorphism.  By contrast, pseudofunctors are an "algebraic" structure, since the functors $f^*$ are specified, together with the necessary coherence data and axioms; the latter come for free in a fibration because of the universal property.

* A [[stack]], being a particular type of pseudofunctor, can also be described as a particular sort of fibration.  This was the original application for which Grothendieck introduced the notion.

## Examples

\begin{example}
Let [[VectBund|$\VBun$]] denote the category whose [[objects]] are [[pairs]] $(M,V)\in\Man\times\Man$ of [[manifolds]] where $q:V\longrightarrow M$ is a [[vector bundle|vector bundle]] over $M$. A morphism $(M_{1},V_{1})\longrightarrow (M_{2},V_{2})$ is a [[commutative square]]:
\begin{center}
\begin{tikzcd}
V_{1} \arrow[r, "g"] \arrow[d, "q_{1}"'] & V_{2} \arrow[d, "q_{2}"] \\
M_{1} \arrow[r, "f"']                    & M_{2}
\end{tikzcd}
\end{center}
where $f$ is a smooth map, and $g$ is a smooth map such that $g\mid_{q_{1}^{-1}(m)}:q_{1}^{-1}(m)\longrightarrow q_{2}^{-1}(f(m))$ is linear for every $m\in M_{1}$. Composition is given as in the [[arrow category|arrow category]]. There is a forgetful functor $\mathcal{P}:\VBun\longrightarrow\Man$ which projects onto the base data. To see that this functor is a Grothendieck fibration, first consider a morphism of the form:
\begin{center}
\begin{tikzcd}
f^{*}M_{2} \arrow[r] \arrow[d] & V_{2} \arrow[d, "q_{2}"] \\
M_{1} \arrow[r, "f"']                      & M_{2}
\end{tikzcd}
\end{center}
given by the [[pullback bundle|pullback bundle]] of a vector bundle $V_{2}\longrightarrow M_{2}$ along a smooth map $M_{1}\longrightarrow M_{2}$. Certainly, the [[universal construction|universal property]] of the [[pullback|pullback]] will supply the conditions for this square to define a [[Cartesian morphism|cartesian morphism]] in $\VBun$ with respect to the the functor $\mathcal{P}$. We show that these are in fact all the cartesian morphisms in $\VBun$.
Let $e' = (V_{1},M_{1})$, $e = (V_{2},M_{2})$, and $e'' = (V,M)$, then by applying the definition from above directly, we see that a morphism $\phi:e'\longrightarrow e$ in $\VBun$, i.e., a square:
\begin{center}
\begin{tikzcd}
V_{1} \arrow[r] \arrow[d, "e'"'] & V_{2} \arrow[d, "e"] \\
M_{1} \arrow[r]                    & M_{2}
\end{tikzcd}
\end{center}
is cartesian, if given any morphism $\psi:e''\longrightarrow e$ in $\VBun$, i.e., a square (which we draw around the first):
\begin{center}
\begin{tikzcd}
V \arrow[rrd, bend left] \arrow[dd, "e''"']   &                           &                 \\
                                      & V_{1} \arrow[r] \arrow[d, "e'"'] & V_{2} \arrow[d, "e"] \\
M \arrow[rr, bend right, shift right] & M_{1} \arrow[r]           & M_{2}
\end{tikzcd}
\end{center}
and any morphism $g:\mathcal{P}(e'')\longrightarrow \mathcal{P}(e')$ in $\Man$ such that $\mathcal{P}(\phi)\circ g = \mathcal{P}(\psi)$, i.e. a smooth map $g:M\longrightarrow M_{1}$ that fills in the above diagram to a commuting diagram:
\begin{center}
\begin{tikzcd}
V \arrow[rrd, bend left] \arrow[dd, "e''"']             &                           &                 \\
                                                & V_{1} \arrow[r] \arrow[d, "e'"'] & V_{2} \arrow[d, "e"] \\
M \arrow[rr, bend right, shift right] \arrow[r, "g"] & M_{1} \arrow[r]           & M_{2}
\end{tikzcd}
\end{center}
then there exists a unique morphism $\chi:e''\longrightarrow e'$ in $\VBun$ such that $\psi = \phi\circ\chi$ and $\mathcal{P}(\chi) = g$, i.e., there exists a unique morphism $V\longrightarrow V_{1}$ that further fills the diagram to a commuting diagram:
\begin{center}
\begin{tikzcd}
V \arrow[rrd, bend left] \arrow[dd, "e''"'] \arrow[rd, dashed] &                           &                 \\
                                                       & V_{1} \arrow[r] \arrow[d, "e'"'] & V_{2} \arrow[d, "e"] \\
M \arrow[rr, bend right, shift right] \arrow[r, "g"]        & M_{1} \arrow[r]           & M_{2}
\end{tikzcd}
\end{center}
It immediately follows from the above unfolding of the definition that $\VBun(e'',e')\cong\VBun(e'',p)$ naturally in $e''$, where $p$ is the pullback bundle of $V_{2}\longrightarrow M_{2}$ along $M_{1}\longrightarrow M_{2}$, and thus by the [[Yoneda lemma#YonedaCorollaries|Yoneda lemma]] $e'\cong p$, and thus $V_{1}\cong\mathcal{P}(e')^{*}V_{2}$. Note that the isomorphisms $e'\cong p$ preserve the commutativity of the diagrams we have drawn above by the universal property of the pullback, and by the definition of a cartesian morphism. Now that we have identified the cartesian morphisms, the statement that $\mathcal{P}:\VBun\longrightarrow\Man$ is a Grothendieck fibration is equivalent to the statement that vector bundles can be pulled back along smooth maps. This is of course true, and thus $\mathcal{P}:\VBun\longrightarrow\Man$ is a Grothendieck fibration.
The associated functors $f^{*}$ are the [[pullback#PullbackFunctor|pullback psuedofunctors]] $f^{*}:\VBun_{M_{2}}\longrightarrow \VBun_{M_{1}}$ that pullback vector bundles over a manifold $M_{2}$ to vector bundles over a manifold $M_{1}$ along a smooth map $f:M_{1}\longrightarrow M_{2}$.
\end{example}

\begin{example}
Let [[Ring]] be the category of [[ring|rings]], and [[Mod]] the category of pairs $(R,M)$ where $R$ is a ring and $M$ is a (left) $R$-module.  Then the evident forgetful functor $Mod\to Ring$ is a fibration and an opfibration.  The functors $f^*$ are given by restriction of scalars, $f_!$ is extension of scalars, and the right adjoint $f_*$ is coextension of scalars.
\end{example}

\begin{example}
Let $C$ be any category with [[pullbacks]] and $\mathbf{2}$ the [[free-living]] arrow, so that $C^{\mathbf{2}}$ is the category of arrows and commutative squares in $C$.  Then the "codomain" functor $C^{\mathbf{2}} \to C$ is a fibration and opfibration.  The cartesian arrows are precisely the pullback squares, and the functors $f_!$ are just given by composition.  The right adjoints $f_*$ exist iff $C$ is [[locally cartesian closed category|locally cartesian closed]].  The term "cartesian" is motivated by this example, which is usually called the **[[codomain fibration]]** over $C$.
\end{example}

\begin{example}
\label{SimpleFibration}
Let $C$ be a category with binary [[cartesian products]]. Then for each object $\Gamma$, we can construct an indexed comonad $C \to Comonad$, the [[coreader comonad|environment comonad]] $\Gamma \times -$ on $C$, and taking [[Kleisli categories]] we get a functor $C^{op} \to Cat$. The corresponding fibration is called the [[simple fibration]] over $C$ and can be seen as a full subfibration of the [[codomain fibration]], with objects being the [[projections]] $\Gamma \times A \to \Gamma$.
\end{example}

\begin{example}\label{FreeCoproductCompletion}
**([[families]], [[free coproduct completion]])**
\linebreak
Let $C$ be any category and let $Fam(C)$ be the category of set-indexed [[families]] of objects of $C$ (the [[free coproduct completion]] of $C$, which is a [[Grothendieck construction]] by the discussion [there](free+coproduct+completion#AsAGrothendieckConstruction)).  The forgetful functor $Fam(C)\to Set$ taking a family to its indexing set is a fibration; the functors $f^*$ are given by reindexing.  They have left adjoints iff $C$ has small coproducts, and right adjoints iff $C$ has small products.
\end{example}

## Properties

* It is easy to verify that fibrations are closed under   [[pullback]] in [[Cat]], and that the composite of fibrations is a fibration.  This latter property is notably difficult to even express in the language of pseudofunctors.

* Every fibration (or opfibration) is a [[Conduché functor]], and therefore an [[exponentiable morphism]] in [[Cat]].

* Every fibration or opfibration is an [[isofibration]].  In particular, [[strict 2-limit|strict 2-pullbacks]] of fibrations are also [[2-pullbacks]].

* {#FiberedLimit} Suppose that $p\colon A\to B$ is a fibration, and $B$ has $I$-indexed [[limits]]. Then $A$ has $I$-indexed limits and $p$ strictly preserves them if and only if $p$ has $I$-indexed [[fibered limit|fibred limits]]. This is in [Gray 66](#Gray); for a general perspective, see [Hermida 99, Corollary 4.9](#Hermida). Concretely, for instance, from right to left, given a [[diagram]] $f\colon I\to A$, let $L$ be the limit of $p f\colon I\to B$, with projections $\phi_i\colon L \to p(f(i))$.  Then for each $i\in I$, let $g(i) = \phi_i^*(f(i)) \in p^{-1}(L)$; these objects form a diagram $g\colon I\to p^{-1}(L)$ whose limit is the limit of $f$.

* Dually, if $p\colon A\to B$ is an opfibration, then [[colimits]] in $A$ can be constructed out of those in $B$ and in the fiber categories.

* Suppose that $p\colon A\to B$ is a fibred [[cartesian closed category]] with [[simple products]]. Then if $B$ is cartesian closed, so is $A$ and $p$ preserves the structure. (See [Hermida 99, Corollary 4.12](#Hermida).) But the converse does not hold, see Section 4.3.2 of [Hermida's thesis](#HermidaThesis).

* If $p\colon A\to B$ is a fibration and $B$ admits an [[orthogonal factorization system]] $(E,M)$, then there is a factorization system $(E',M')$ on $A$, where $M'$ is the class of cartesian arrows whose image in $B$ lies in $M$, and $E'$ is the class of all arrows whose image in $B$ lies in $E$.  A dual construction is possible if $p$ is an opfibration.  If it is a bifibration (or more generally, an [[ambifibration]] over $(E,M)$), then these together form a [[ternary factorization system]].

* Under suitable hypotheses, versions of the preceding fact can be shown  [[weak factorization systems]] and [[model structures]] as well.

* Generalizing in a different direction, if $p\colon A\to B$ is a fibration and $(E,M)$ is a [[factorization structure for sinks]] on $B$, then $A$ admits a factorization structure for sinks $(E',M')$, where $M'$ is the class of cartesian arrows whose image in $B$ lies in $M$, and $E'$ is the class of all arrows whose image in $B$ lies in $E$.  Similarly, we can lift factorization structures for *cosinks* along an opfibration.  To lift in the "opposite" way we require more of $p$; see [[topological concrete category]].

## Variations

### Discrete and groupoidal fibrations

One important special case of a fibration is when each fiber is a [[groupoid]]; these correspond to pseudofunctors $B^{op}\to Grpd$.  These are also called *categories fibered in groupoids*.  A fibration $E\to B$ is fibered in groupoids precisely when *every* morphism in $E$ is cartesian.

Another important special case is when each fiber is a [[discrete category]]; these correspond to functors $B^{op}\to Set$.  These are also called *[[discrete fibrations]]*.  A functor $p\colon E\to B$ is a discrete fibration precisely when for every $e\in E$ and $f\colon b\to p(e)$ there is a *unique* lifting of $f$ to a morphism $e'\to e$.

### Opfibrations and bifibrations

We say that $p\colon E\to B$ is an **opfibration** if $p^{op}:E^{op}\to B^{op}$ is a fibration.  These correspond to covariant pseudofunctors $B\to Cat$.  A functor that is both a fibration and an opfibration is called a **[[bifibration]]**.  It is not hard to see that a fibration is a bifibration iff each functor $f^*$ has a left adjoint, written $f_!$ or $\Sigma_f$.  In many cases $f^*$ also has a right adjoint, written $f_*$ or $\Pi_f$, but this is not as easily expressible in fibrational language.

Grothendieck originally called an opfibration a *cofibered category*, and if the fibers are groupoids a [[category cofibered in groupoids]] (cf. [[SGA I]], [[Higher Topos Theory]]).  However, that term has fallen out of favor in the homotopy-theory and category-theory communities (though still used sometimes in algebraic geometry), because an opfibration still has a _lifting_ property, as is characteristic of other notions of [[fibration]], as opposed to the _extension_ property exhibited by [[cofibrations]] in [[homotopy theory]].

Note that an opfibration is the same as an internal fibration in the 2-category $Cat^{co}$, while it is the fibrations in the 2-category $Cat^{op}$ which are more deserving of the name "cofibration."

Note also that a given pseudofunctor $B^{op}\to Cat$ can be represented both by a fibration $E_1\to B$ and by an opfibration $E_2\to B^{op}$.  However $E_2$ is _not_ the opposite category of $E_1$.

### Version respecting the Principle of equivalence {#StreetFibration}

There is something apparently in violation of the [[principle of equivalence]] about the notion of fibration, namely the requirement that for every $f:a\to b$ and $e\in E_b$ there exists a $\phi:e'\to e$ such that $p(e')$ is _[[equality|equal]]_, rather than merely [[isomorphism|isomorphic]], to $a$.  This is connected with the fact that we use strict fibers, rather than [[essential fiber]]s, and that fibrations and pseudofunctors can be recovered from each other up to isomorphism rather than merely equivalence.

Note that almost any fibration between "concrete" categories that arises in practice does satisfy this strict property.  However, even stating the strict property requires our categories to be [[strict categories]] (i.e. to have a notion of equality of objects), so when working in a context where not all categories are strict (such as [[internal categories in homotopy type theory]], or with objects in a [[bicategory]]) it is problematic.  Sometimes (such as in type theory) this can be avoided by working with [[displayed categories]] instead, but in some cases (such as internally in a bicategory) one does not have classifying objects either, so it is sometimes useful to have a version which manifestly accords to the [[principle of equivalence]].

The correct modification, first given by [[Ross Street]], is simply to require that for any $f:a\to b$ and $e\in E_b$ there exists a cartesian $\phi:e'\to e$ and an _isomorphism_ $h:p(e') \cong a$ such that $f\circ h = p(\phi)$; the definition of "cartesian" is unchanged; this gives the notion of [[Street fibration]].  Every [[equivalence of categories]] is a Street fibration, which is not true for the concept of Grothendieck fibrations according to the [[principle of equivalence]], but every Street fibration can be factored as an equivalence of categories followed by a Grothendieck fibration.

We might also think that it violates the [[principle of equivalence]] to say that the target of the cartesian arrow $\phi$ is equal to the given object $e$, akin to the topological distinction between [[Serre fibrations]] and [[Dold fibrations]], where the initial point of a lifted path can only be specified up to homotopy.  However, this part of the definition is really better regarded as a typing assertion, akin to saying, in the definition of a [[product]] of two objects $A$ and $B$, that the target of the two projections $A\times B\to A$ and $A\times B \to B$ are *equal* to $A$ and $B$.  Moreover, any weakening along these lines would actually be equivalent to the version above: if we demand only that for any $f\colon a\to b$ in $B$ and $e\in E_b$, there exists a cartesian $\phi\colon e' \to \hat{e}$ with $p(\phi)=f$ and an isomorphism $\hat{e}\cong e$ in the fiber $E_b$, then you can just compose $\phi$ with the isomorphism $\hat{e}\cong e$ to get a cartesian arrow $\hat{\phi}\colon e'\to e$ with $p(\phi)=f$ still.  The reason this doesn't work in topology is that paths come with parametrizations, and requiring the lower triangle (in the square drawn at [[Dold fibration]]) to commute strictly prevents the reparametrization necessary to compose with a vertical homotopy.

The idea of [[proto-fibration]] is closely related to this.

### Internal version

In a [[strict 2-category]] $K$, a morphism $p:E\to B$ is called a fibration if for every object $X$, $K(X,E)\to K(X,B)$ is a fibration of categories, and for every morphism $f:Y\to X$, the square
$$\array{ K(X,E) & \to & K(Y,E) \\ \downarrow && \downarrow \\ K(X,B) & \to & K(Y,B)}$$
is a morphism of fibrations.  There is an alternate characterization in terms of [[comma object]]s and adjoints, see [[fibration in a 2-category]].  The same definition works in a [[bicategory]], as long as we use the version in accord with the [[principle of equivalence]] above.  Interpreted in [[Cat]] we obtain the explicit notion we started with.

### Two-sided version

A **two-sided fibration**, or a **fibration from $B$ to $A$**, is a span $A \leftarrow E \rightarrow B$ such that $E\to A$ is a fibration, $E\to B$ is an opfibration, and the structure "commutes" in a natural way.  Such two-sided fibrations correspond to pseudofunctors $A^{op}\times B \to Cat$.  If they are discrete, they correspond to functors $A^{op}\times B\to Set$, i.e. to [[profunctors]] from $B$ to $A$.

See [[two-sided fibration]] for more details.

Note that a pseudofunctor $A^{op}\times B \to Cat$ can also be represented by an opfibration $E_1\to A^{op}\times B$ and by a fibration $E_2\to A\times B^{op}$, but there is no simple relationship between the three categories $E$, $E_1$, and $E_2$.

### Higher categorical versions

There is also a notion of fibration for 2-categories that has been studied by Hermida.  See [[n-fibration]] for a general version.

For [[(∞,1)-category|(∞,1)-categories]] the notion of fibered category is modeled by the notion of [[Cartesian fibration]] of simplicial sets. The corresponding analog of the Grothendieck construction is discussed at [[(∞,1)-Grothendieck construction]].

### Versions for other categorical structures

* There is an analog for [[multicategories]]. See [[fibration of multicategories]].

* For higher multicategories, see [[Cartesian fibration of dendroidal sets]].

* For [[double categories]], see [[double fibration]].

## Alternate definitions

There are several alternate characterizations of when a functor is a fibration, some of which are more convenient for [[fibration in a 2-category|internalization]].  Here we mention a few.

### In terms of adjoints

Since Grothendieck fibrations are a [[strict category|strict]] notion, in what follows we denote by $B\downarrow p$ the [[strict category|strict]] [[comma category]] (i.e. determined up to [[isomorphism]], not merely up to [[equivalence of categories|equivalence]]) and by $Cat/B$ the [[strict slice 2-category]].

+--{: .un_lemma}
###### Lemma
A functor $p \colon E \to B$ is a cloven fibration if and only if the canonical functor $i \colon E \to B\downarrow p$ has a right adjoint $r$ in $Cat / B$.
=--
+-- {: .proof }
###### Proof
First, recall that the strict slice 2-category $Cat/X$ has objects the functors $C \to X$, as morphisms the commuting triangles
$$\array{C & \stackrel{h}{\to} & C' \\ & f \searrow \swarrow g &  \\  & X,  & }$$
and as 2-cells the natural transformations $\alpha : h_1 \to h_2$ such that $g\alpha = id_f$.

Next, recall that the [[comma category]] $B\downarrow p$ has objects the triples $(x, e, k)$, with $k \colon x \to p e$.  Let $\pi \colon B\downarrow p \to B$ denote the projection $(x, e, k) \mapsto x$.

The canonical morphism $i:E \to B\downarrow p$ is simply the inclusion functor of identity maps $i e  = 1_{p e} \colon p e \to p e$.

Somewhat imprecisely, seeing both categories $E$ and $B\downarrow p$ as sitting over $B$ means that functors between those should be the identity on the $b$ component, and natural transformations should have the identity as their $b$ component.

To give an adjunction $i \dashv r$ it suffices to give, for each $k \colon x \to p e$ in $B\downarrow p$, an object $r k$ in $E$ such that $p r k = x$ and an arrow $i r k = 1_x \to k$ in $B\downarrow p$ that is [[universal arrow|universal]] from $i$ to $k$.  For the adjunction to live in $Cat / B$ we must have that $\pi \circ i r k = 1_{p r k} = 1_x$, so the universal arrow must be of the form
$$
\array{
  x & \overset{1}{\to} & x \\
  \mathllap{1} \downarrow & & \downarrow \mathrlap{p \epsilon_k} \\
  x & \underset{k}{\to} & p e
}
$$
and thus amounts to a choice of $\epsilon_k \colon r k \to e$ in $E$ such that $p \epsilon_k = k$.

The universal property of $\epsilon_k$ tells us that for any other morphism in $B\downarrow p$ from some $i y$ to $k$, i.e., for any $y$ and any pair $(f,g)$ making the square
$$
\array{
  p y & \stackrel{1}{\to} & p y \\
  \mathllap{f} \downarrow & & \downarrow \mathrlap{p g} \\
  x & \underset{k}{\to} & p e
}
$$
commute, there is a unique map $h \colon y \to r k$ in $B$ such that the above square factors in $B\downarrow p$ as
$$
\array{
  p y & \stackrel{1}{\to} & p y \\
  \mathllap{p h} \downarrow &  & \downarrow \mathrlap{p h} \\
  \mathllap{p r k =} x & \stackrel{1}{\to} & x \mathrlap{= p r k}\\
  \mathllap{1} \downarrow & & \downarrow \mathrlap{p \epsilon_k} \\
  x & \underset{k}{\to} & p e.
}
$$

In other words, the universal property provides a unique $h$ such that $\epsilon_k h = g$ and $p h = f$, which exactly asserts that $\epsilon_k$ is a cartesian lift of $k$.

So the existence of a right adjoint to $i$ means precisely that for each morphism $k \colon x \to p e$ a choice is given of a cartesian lift of $k$, which means in turn that $p$ is a cloven fibration.
=--

### In terms of pseudoalgebras

Fibrations are pseudoalgebras for a [[lax-idempotent pseudomonad]] (see there for more details).

## Discussions

The following discussion brings out some interesting points about the equivalence between fibrations and pseudofunctors.

+--{: .query}
[[Sridhar Ramesh]]: I have a (possibly stupid) question about the nature of this equivalence. I assume the idea here is that moving from a cloven fibration to the corresponding pseudofunctor is in some sense "inverse" to carrying out the Grothendieck construction in the other direction. But, in trying to get a good intuition for the nuances of non-splittable fibrations, I seem to be stumbling upon just in what sense this is so. For example, consider the nontrivial group homomorphism from Z (integer addition) to Z_2 (integer addition modulo 2); this gives us a non-splittable fibration (and, for that matter, an opfibration), for which a cleavage can be readily selected. No matter what cleavage is selected, the corresponding (contravariant) pseudofunctor from Z_2 to Cat, it would appear to me, is the one which sends the unique object in Z_2 to the subcategory of Z containing only even integers (let us call this 2Z), and which sends both of Z_2's morphisms to identity; thus, it is actually a genuine functor, and indeed, a "constant" functor. Applying the Grothendieck construction now, I would seem to get back the projection from Z_2 X 2Z onto Z_2. But can this really be equivalent to the fibration I started with? After all, Z and Z_2 X 2Z are very different groups. So either "equivalence" means something trickier here than I realize, or I keep making a mistake somewhere along the line. Either way, it'd be great if someone could help me see the light.

[[Mike Shulman]]: Good question!  I think the missing subtlety is that a pseudofunctor is not uniquely determined by its action on objects and morphisms, even if its domain is a mere category (or a mere group); there are also natural coherence isomorphisms $g^* f^* \cong (g f)^*$ to take into account.  For instance, if $g$ is the nonidentity element of $\mathbb{Z}/2$, then $g g = 1$, so even if $g$ acts by the identity on $2\mathbb{Z}$, a pseudofunctor also contains the additional data of a natural automorphism of the identity of $2\mathbb{Z}$, i.e. a (central) element of $2\mathbb{Z}$.  If you start from $\mathbb{Z}$, then depending on your cleavage your element can be anything that's 2 mod 4, while if you start from $\mathbb{Z}/2\times 2\mathbb{Z}$, your element can be anything that's 0 mod 4.  Finally, there is a pseudonatural equivalence between two such pseudofunctors just when their corresponding elements differ by a multiple of 4, so you get exactly two equivalence classes of pseudofunctors, corresponding to the two groups $\mathbb{Z}$ and $\mathbb{Z}/2\times 2\mathbb{Z}$.  Of course we are reproducing the classification of group extensions via group cohomology.

By the way, this sort of thing (by which I mean, the cohomology class that classifies some categorical structure arising as the trace of a coherence isomorphism) happens in lots of other places too.  For instance, a [[2-group]] is classified by a group $G$, an abelian group $H$, an action of $G$ on $H$, and an element in $H^3(G;H)$.  If you replace a 2-group by a [[skeleton|skeletal]] one, then $G$ is the group of objects (which is strictly associative and unital, by skeletality), $H$ is the group of endomorphisms of the unit, and the action is defined by "whiskering".  The cohomology class comes from the [[associator]] isomorphism, which can (and often must) still be nontrivial even though the multiplication is "strictly associative" at the level of objects (by skeletality).

_Toby_:  So the multiplication is strictly associative, but the $2$-group itself is not a strict $2$-group, since it uses a different associator than the identity.  As in the example of the pseudofunctor from $Z_2$ to $Cat$, there is some additional structure here which is not trivial, even though it seems like it could be.

[[Sridhar Ramesh]]: Ah, of course, that's what I was missing. Thanks, both of you; that clears it all up.
=--

## Related concepts

* [[opfibration]], [[bifibration]]

  [[two-sided fibration]]

* [[monoidal fibration]]

* [[fibration of 2-categories]]

* [[n-fibration]]

* [[Cartesian fibration]]

* [[foliated category]]

## References
 {#References}

The concept was introduced in the context of [[descent theory]] and in the guise now known as [[indexed categories]], in

* {#Grothendieck60} [[Alexander Grothendieck]], p. 300 in: *Technique de descente et théorèmes d’existence en géométrie algébrique. I. Généralités. Descente par morphismes fidèlement plats*, Séminaire [[N. Bourbaki]] exp. no190 (1960) 299-327 [[numdam:SB_1958-1960__5__299_0](http://www.numdam.org/item/?id=SB_1958-1960__5__299_0)]

and then elaborated on in

* [[Alexander Grothendieck]], exposé VI of: _Revêtements Etales et Groupe Fondamental - Séminaire de Géometrie Algébrique du Bois Marie 1960/61_ ([[SGA 1]]) , LNM **224** Springer (1971) [updated version with comments by M. Raynaud: [arxiv.0206203](http://arxiv.org/abs/math/0206203)]

Another important early reference:

* {#Gray} [[John W. Gray]], *Fibred and Cofibred Categories*, pp. 21-83 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)]

Early development of the theory:

* {#Bénabou1975} [[Jean Bénabou]], _Fibrations petites et localement petites_, C. R. Acad. Sci. Paris **281** Série A (1975) 897-900 [[gallica](http://gallica.bnf.fr/ark:/12148/bpt6k6228235m/f171.image#)]

* {#Bénabou1985} [[Jean Bénabou]], *Fibered Categories and the Foundations of Naive Category Theory*,  The Journal of Symbolic Logic, Vol. **50** 1 (1985) 10-37 [[doi:10.2307/2273784](http://dx.doi.org/10.2307/2273784)]

reviewed in:

* {#Streicher18} [[Thomas Streicher]], _Fibred Categories à la Jean Bénabou_,  [[arXiv:1801.02927](https://arxiv.org/abs/1801.02927)]

Further review with emphasis on [[descent]] and Grothendieck fibrations as incarnations of [[stacks]]:

*  [[Angelo Vistoli]], _Grothendieck topologies, fibered categories and descent theory_ [[math.AG/0412512](http://arxiv.org/abs/math/0412512), [MR2223406](http://www.ams.org/mathscinet-getitem?mr=2223406)]  in: Fantechi et al. (eds.), _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), [MR2007f:14001](http://www.ams.org/mathscinet-getitem?mr=2007f:14001)]

* [[Barbara Fantechi]], [[Lothar Göttsche]], [[Luc Illusie]], [[Steven L. Kleiman]], [[Nitin Nitsure]], [[Angelo Vistoli]], Chapter 3 of: _Fundamental algebraic geometry. Grothendieck's [[FGA explained]]_, Mathematical Surveys and Monographs __123__, Amer. Math. Soc. (2005) [[MR2007f:14001](http://www.ams.org/mathscinet-getitem?mr=2007f:14001), [ISBN:978-0-8218-4245-4](https://bookstore.ams.org/surv-123-s), [lecture notes](http://indico.ictp.it/event/a0255/other-view?view=ictptimetable)]

Textbook accounts:

* [[Michael Barr]], [[Charles Wells]], Chapter 12 of: *Category Theory for Computing Science* , Prentice Hall 1995³. ([TAC reprints no.22 (2012)](http://www.tac.mta.ca/tac/reprints/articles/22/tr22abs.html))

* [[Francis Borceux]], Chapter 8 of: *[[Handbook of Categorical Algebra]] vol. 2*, Cambridge UP 1994.

* [[Bart Jacobs]], Chapters 1 and 9 in: *Categorical Logic and Type Theory*, Studies in Logic and the Foundations of Mathematics **141**, Elsevier (1998)  [[ISBN:978-0-444-50170-7](https://www.sciencedirect.com/bookseries/studies-in-logic-and-the-foundations-of-mathematics/vol/141), [pdf](https://people.mpi-sws.org/~dreyer/courses/catlogic/jacobs.pdf)]

  > (in the context of [[categorical semantics of dependent types]])

* [[Peter Johnstone]], _[[Sketches of an Elephant]] vol.1, Oxford UP 2002. (Part B)

  > (beware that this uses the non-standard terms "prone" and "supine" for "cartesian" and "opcartesian" morphisms)

* {#JohnsonYau20} [[Niles Johnson]], [[Donald Yau]], Chapter 9 of: _2-Dimensional Categories_, Oxford University Press 2021 ([arXiv:2002.06055](http://arxiv.org/abs/2002.06055), [doi:10.1093/oso/9780198871378.001.0001](https://oxford.universitypressscholarship.com/view/10.1093/oso/9780198871378.001.0001/oso-9780198871378))

See also:

* André Joyal, *[[joyalscatlab:Grothendieck fibrations]]*

* Wikipedia, *[Fibered category](http://en.wikipedia.org/wiki/Fibred_category)*

On the [[2-category]] of fibrations and application to [[computer science]]:

* {#Hermida} [[Claudio Hermida]], *Some properties of $Fib$ as a fibred 2-category*, Journal of Pure and Applied Algebra **134** (1999) 83-109 [<a href="https://doi.org/10.1016/S0022-4049(97)00129-1">doi:10.1016/S0022-4049(97)00129-1</a>]

* {#HermidaThesis} [[Claudio Hermida]], [PhD thesis, University of Edinburgh](https://era.ed.ac.uk/bitstream/handle/1842/14057/Hermida1993.Pdf)

A [[2-comonad]] characterizing Grothendieck fibrations:

* [[Jacopo Emmenegger]], [[Luca Mesiti]], [[Giuseppe Rosolini]], [[Thomas Streicher]], *A comonad for Grothendieck fibrations* [[arXiv:2305.01474](https://arxiv.org/abs/2305.01474)]

On the connection between Grothendieck fibrations and [[factorisation systems]] (see also [[stable factorisation system]]):

* J. Hughes and [[Bart Jacobs]], _Factorization systems and fibrations: Toward a fibred Birkhoff variety
theorem_, Electr. Notes in Theor. Comp. Sci., 69 (2002)

* [[Jiří Rosický ]], [[Walter Tholen]], _Factorization, fibration and torsion_, [arxiv/0801.0063](http://arxiv.org/abs/0801.0063), Journal of homotopy and Related Structures

* Miloslav Štěpán, _Factorization systems and double categories_, [arXiv:2305.06714](https://arxiv.org/abs/2305.06714).

See also:

* [[Ronnie Brown|R. Brown]], R. Sivera, _Algebraic colimit calculations in homotopy theory using fibred and cofibred categories_, TAC **22** (2009) pp.222-251. ([pdf](http://www.tac.mta.ca/tac/volumes/22/8/22-08.pdf))

[[!redirects Grothendieck fibrations]]
[[!redirects Grothendieck opfibration]]
[[!redirects Grothendieck cofibration]]
[[!redirects Grothendieck opfibrations]]
[[!redirects Grothendieck cofibrations]]
[[!redirects opfibration]]
[[!redirects opfibrations]]
[[!redirects bifibration]]
[[!redirects bifibrations]]

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

[[!redirects cofibered category]]
[[!redirects cofibered categories]]

[[!redirects cofibred category]]
[[!redirects cofibred categories]]

[[!redirects (Grothendieck) fibration]]
[[!redirects (Grothendieck) fibrations]]


Grothendieck group

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

#Contents#
* table of contents
{:toc}

## Idea

The term _Grothendieck group_ has two closely related meanings:

* [Grothendieck group of commutative monoids](#OfCommutativeMonoids)

* [Grothendieck group of stable infinity-categories](#OfStableCategories)

In its restricted sense the Grothedieck group of a commutative monoid (i.e. of a commutative [[semi-group]] with [[unitality|unit]]) $A$ is a specific presentation of its [[group completion]], given as a certain [[group]] structure on a [[quotient]] of the [[Cartesian product]] $A \times A$. This is such that applied to the additive monoid of [[natural numbers]] $\mathbb{N}$ it produces the additive group of [[integers]] $\mathbb{Z}$ represented by pairs of natural numbers $(n_+, n_-)$ subject to an [[equivalence relation]] which identifies them with their [[difference]] $n_+ - n_-$.

A vaguely similar procedure applied to [[decategorification|isomorphism classes]] of certain [[Quillen exact category|Quillen exact categories]] happens to compute a group that is called the [[algebraic K-theory]] of these categories. See [[K-theory]] for some general abstract nonsense behind this.

Notably, the Grothendieck group completion of the [[decategorification]] of the category of [[topological vector bundles]] on some ([[compact Hausdorff space|compact Hausdorff]]) [[topological space]] $X$ produces the group known as the [[topological K-theory]] of $X$.

Given this one speaks more generally of the ([[algebraic K-theory|algebraic]]) [[K-theory]] group of a suitable category (one presenting a [[stable (∞,1)-category]] in some way) as its _Grothendieck group_ .

In that sense, the Grothendieck group of a ($\infty$-)category $C$ with a notion of [[fibration sequence|cofibration sequence]]s is the [[decategorification]] set $K(C)$ equipped with a notion of addition that is encoded in these homotopy exact sequences.

We now first state the definition of "Grothendieck group completion" -- which is really just the _free group completion of an abelian [[monoid]]_ --  and then the definition of Grothendieck group in the sense of [[algebraic K-theory]]. Notice that a priori both concepts are entirely independent constructions on different entities. But in various special case both can be applied to specific objects so as to produce the same result.

## Of commutative monoids
 {#OfCommutativeMonoids}

Every [[abelian  group]] is in particular a [[commutative monoid]]. The [[forgetful functor]] $U \colon Ab \to CMon$ from the [[category]] [[Ab]] to the category [[CMon]] has a [[left adjoint]] $F$ (the corresponding [[free functor]]), the _[[group completion]]_ functor

$$
  Ab
    \underoverset
      {\underset{U}{\longrightarrow}}
      {\overset{F}{\longleftarrow}}
      {\bot}
  CMon
$$

The _[[Grothendieck group construction on commutative monoids]]_ (def. \ref{GrothendieckGroupViaQuotientOfCartesianProduct} below) is an explict presentation of this [[group completion]] functor.

For more details see at _[[Grothendieck group of a commutative monoid]]_.

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

The idea of the free group on an abelian monoid is a very simple algebraic idea that, at least for a [[cancellative monoid]] (so that the unit is monic and one can reasonably use the term 'completion') certainly predates [[Grothendieck]]. That the [[integers]] $\mathbb{Z}$ is the group completion of the [[natural numbers]] $\mathbb{N}$ goes back at least to [[Kronecker]].

=--

+-- {: .num_defn #GrothendieckGroupViaQuotientOfCartesianProduct}
###### Definition
**([[Grothendieck group of a commutative monoid]])**

Let $(A,+)$ be a [[commutative monoid]] (i.e. a commutative [[semi-group]]).

On the [[Cartesian product]] of underlying sets $A \times A$ (the set of ordered [[pairs]] of elements in $A$), consider the [[equivalence relation]]

$$
  \big(
    (a_+, a_-) \sim_1 (b_+, b_-)
  \big)
    \;\Leftrightarrow\;
  \left(
    \underset{k \in A}{\exists}
    \left(
      a_+ + b_- + k = b_+ + a_- + k
    \right)
  \right)
$$

or equivalently the equivalence relation

$$
  \big(
    (a_+, a_-) \sim_2 (b_+, b_-)
  \big)
    \;\Leftrightarrow\;
  \left(
    \underset{k_1, k_2 \in A}{\exists}
    \left(
      (a_+ + k_1, a_- + k_1)
      =
      (b_+ + k_2, b_- + k_2)
    \right)
  \right)
  \,.
$$

Write

$$
  G(A) \coloneqq (A \times A)/\sim
$$

for the set of [[equivalence classes]] under this equivalence relation. This inherits a binary operation

$$
  + \;\colon\; G(A) \times G(A) \longrightarrow G(A)
$$

by applying the addition in $A$ on representatives:

$$
  [a_+,a_-] + [b_+,b_-] \coloneqq [ a_+ + b_+ , a_- + b_- ]
  \,.
$$

This defines the structure of an [[abelian group]]

$$
  (G(A),+)
$$

and this is the _Grothendieck group_ of $A$.

This comes with a canonical [[homomorphism]] of [[semigroups]]

$$
  \array{
    A &\overset{\phantom{A} \eta_A \phantom{A} }{\longrightarrow}& G(A)
    \\
    a &\overset{\phantom{AAA}}{\mapsto}& [a,0]
  }
  \,.
$$

=--

## Of stable $\infty$-categories
 {#OfStableCategories}

In this sense, a Grothendieck group is fundamentally something assigned to a [[stable (∞,1)-category]].  We start with the naive [[decategorification]] of $C$, i.e. the set of [[equivalence]] classes of objects, which inherits the structure of an abelian monoid.  Then in addition to group-completing it as above, we add additional [[generators and relations|relations]] by the rule that for every [[fibration sequence]]

$$
  A \to X \to B
$$

in $C$, the equivalence classes $[A]$, $[B]$ and $[X]$ must satisfy

$$
  [X] = [A] + [B]
  \,.
$$

The result is also called the [[K-theory]] of $C$.

In particular, the additive inverse $-[A]$ of an element $[A]$ is the class of its [[loop space object]] $\Omega A$, or equivalently of its [[delooping]] $\mathbf{B} A$ called the _suspension_ $\Sigma A$, since by definition the sequences

$$
  \Omega A \to 0 \to A
$$

and

$$
  A \to 0 \to \Sigma A
$$

are [[fibration sequence]], so that

$$
  [A] + [\Omega A] = 0
$$

and

$$
 [A] + [\Sigma A] = 0
 \,.
$$

But there are many ways to _model_ a [[stable (∞,1)-category]] by an ordinary category. Essentially for each of these ways there is a seperate prescription for how to model the above general construction in terms of concrete 1-categorical constructions.

In particular from an

* [[abelian category]]

and a

* [[Quillen exact category]]

one obtains the corresponding [[category of chain complexes|categories of chain complexes]]. These are [[stable (infinity,1)-category|stable (∞,1)-categories]]. Below we list presciptions for how to compute the Grothendieck/K-theory groups of these in terms of the underlying 1-categories.

Apart from the case of abelian categories, this requires some handle on the fibration sequences. A tool developed to handle exactly this for the purpose of computing Grothendieck/K-theory groups is the notion of a [[Waldhausen category]]. That provides the sufficient extra information to get a hand on the homotopy exact sequences.

### of an abelian category ###

Let $C$ be an [[abelian category]].  The **Grothendieck group** or [[algebraic K-theory]] group of $C$, denoted $K(C)$, is the [[abelian group]] generated by [[decategorification|isomorphism class]]es of objects of $C$, with relations of the form

$$[X] = [A] + [B] $$

whenever there is a [[short exact sequence]]

$$ 0 \to A \to X \to B \to 0 $$

### of a Quillen exact category ###

An exact category $C$ in the present sense is a full [[subcategory]]
of an [[abelian category]] $\hat C$ such that the collection
of all sequences $0 \to A \to X \to B \to 0$
in $C$ that are [[exact sequence]]s in $\hat C$ has the property that for every [[exact sequence]] $A \to X \to B$ in $\hat C$ with $A$ and $B \in C$ also their "sum" $X$ is in $C$.

Given an exact category $C$ with the inherited notion of exact sequences this way, the definition of its Grothendieck group is as above.

### of a Waldhausen category ###

A [[Waldhausen category]] is a [[category with weak equivalences]] with an [[initial object]] -- called $0$ -- and equipped with the notion of auxiliary morphism called _cofibrations_. These satisfy some axioms which are such that the ordinary 1-categorical [[cokernel]] of a cofibration $A \hookrightarrow X $, i.e. the ordinary [[pushout]]

$$
  \array{
    A &\hookrightarrow& X
    \\
    \downarrow && \downarrow
    \\
    0 &\to &  B
  }
$$

computes the desired [[homotopy colimit|homotopy]] [[pushout]]. (This is exactly dual to the reasoning by which one computes [[homotopy pullback]]s in a [[category of fibrant objects]]. See there for details.)

Therefore in a Waldhausen category a [[fibration sequence|cofibration sequence]] is a [[pushout]] sequence

$$
 A \hookrightarrow X \to B
$$

where the first morphism is a cofibration.

The Grothendieck/[[K-theory]]-group of the [[Waldhausen category]] $C$ is then, as before, on the [[decategorification]] $K(C)$ the abelian group structure given by

$$
  [X] = [A] + [B]
$$

for all cofibration sequences as above.

### Examples

* The Grothendieck group of the [[Quillen exact category]] of [[vector bundle]]s on a space $X$ is called the [[topological K-theory]] of $X$.

  Notice that vector bundles do not form an [[abelian category]].

* The Grothendieck group of the category of finite-dimensional complex-linear [[representations]] of a [[group]] is called its [[representation ring]].

These two examples illustrate a general fact: the Grothendieck group of a [[monoidal category|monoidal]] [[abelian category]] inherits a ring structure from the tensor product in this category, and thus becomes a ring, called the [[Grothendieck ring]]. See also the general discussion at [[decategorification]].

* Every [[Quillen exact category]] $C$ is canonically equipped with the structure of a [[Waldhausen category]]. The two different prescriptions for forming the Grothendieck group $K(C)$ of $C$ do coincide.

## Related concepts

* [[group completion]]

* [[K-theory]]

* [[Eilenberg swindle]]

## References

* [[Charles Weibel]], _The K-book: An introduction to algebraic K-theory_ ([web](http://www.math.rutgers.edu/~weibel/Kbook.html))

  section 2: _The Grothendieck group $K_0$_ ([pdf](http://www.math.rutgers.edu/~weibel/Kbook/Kbook.II.pdf))

* [[Mikhail Khovanov]] (notes by [[You Qi]]), §4 in: *Introduction to categorification*, lecture notes, Columbia University (2010, 2020) [[web](https://www.math.columbia.edu/~khovanov/cat2020/), [web](https://you-qi2121.github.io/mypage/categorificationnotes.html), full:[pdf](https://www.dropbox.com/scl/fi/wdesax1c8il6tgwbi20t4/KhovanovYouQi-Categorification.pdf?rlkey=l5cm3khnzu604ijdnl06o89od&dl=0)]

See also

* [The Grothendieck Construction](http://online.itp.ucsb.edu/online/ktheory/wu/) (UCSB ITP Seminar)

* Wikipedia, _[Grothendieck group](https://en.wikipedia.org/wiki/Grothendieck_group)_

[[!redirects Grothendieck groups]]

[[!redirects Grothendieck group completion]]
[[!redirects Grothendieck group completions]]


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

#Contents#
* table of contents
{:toc}

## Idea

The _Grothendieck group construction_ is an explicit presentation of the [[group completion]] of a [[commutative monoid]] to an [[abelian group]]. For a [[cancellative monoid]] it reduces to the age-old construction that turns the additive monoid of [[natural numbers]] into the additive group of [[integers]], or the multiplicative monoid of non-zero [[integers]] into the multiplicative group of non-zero [[rational numbers]]. But the construction also applies to non-[[cancellative monoids]]. The archetypical application of the construction is to monoids of [[topological vector bundles]] over some [[topological space]] $X$ under [[direct sum of vector bundles]], in which case it yields the _[[topological K-theory]] group_ $K(X)$ of $X$.

Motivated by the example of [[topological K-theory]] there is a vaguely related construction of [[algebraic K-theory]] groups from [[Quillen exact categories]]. Applied to the category of [[topological vector bundles]] this coincides with the Grothendieck group of the monoid of vector bundles, and hence is also called _Grotheniek group construction_. For more on this [[category theory|category theoretic]] operation see at _[[Grothendieck group]]_.

## Definition

+-- {: .num_defn #GrothendieckGroupViaQuotientOfCartesianProduct}
###### Definition
**([[Grothendieck group of a commutative monoid]])**

Let $(A,+)$ be a [[commutative monoid]] (i.e. a commutative [[semi-group]]).

On the [[Cartesian product]] of underlying sets $A \times A$ (the set of ordered [[pairs]] of elements in $A$), consider the [[equivalence relation]]

$$
  \big(
    (a_+, a_-) \sim_1 (b_+, b_-)
  \big)
    \;\Leftrightarrow\;
  \left(
    \underset{k \in A}{\exists}
    \left(
      a_+ + b_- + k = b_+ + a_- + k
    \right)
  \right)
$$

or equivalently the equivalence relation

$$
  \big(
    (a_+, a_-) \sim_2 (b_+, b_-)
  \big)
    \;\Leftrightarrow\;
  \left(
    \underset{k_1, k_2 \in A}{\exists}
    \left(
      (a_+ + k_1, a_- + k_1)
      =
      (b_+ + k_2, b_- + k_2)
    \right)
  \right)
  \,.
$$

Write

$$
  G(A) \coloneqq (A \times A)/\sim
$$

for the set of [[equivalence classes]] under this equivalence relation. This inherits a binary operation

$$
  + \;\colon\; G(A) \times G(A) \longrightarrow G(A)
$$

by applying the addition in $A$ on representatives:

$$
  [a_+,a_-] + [b_+,b_-] \coloneqq [ a_+ + b_+ , a_- + b_- ]
  \,.
$$

This defines the structure of an [[abelian group]]

$$
  (G(A),+)
$$

and this is the _Grothendieck group_ of $A$.

This comes with a canonical [[homomorphism]] of [[monoids]] ([[semigroups]] with [[unitality|unit]]):

$$
  \array{
    A &\overset{\phantom{A} \eta_A \phantom{A} }{\longrightarrow}& G(A)
    \\
    a &\overset{\phantom{AAA}}{\mapsto}& [a,0]
  }
  \,.
$$

=--

+-- {: .num_prop #UniversalProperty}
###### Proposition
**([[universal property]] of Grothendieck group)**

The Grothendieck group in def. \ref{GrothendieckGroupViaQuotientOfCartesianProduct} is well defined, and the homomorphism $A \to G(A)$ satisfies the [[universal property]] of the [[group completion]] of $A$:

Given an [[abelian group]] $B$ and a [[homomorphism]] of commutative [[semi-groups]] ([[commutative monoids]]) $f \colon A \longrightarrow B$ then there is a unique homomorphism of [[abelian groups]] $\tilde A \;\colon\; G(A) \longrightarrow B$ such that $f = \tilde f\circ \eta_A$:

$$
  \array{
    A
    \\
    {}^{\mathllap{\eta_A}}\downarrow & \searrow^{\mathrlap{f}}
    \\
    G(A) &\overset{\exists ! \tilde f}{\longrightarrow}& B
  }
$$

=--

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

First to see that the two equivalence relations in def. \ref{GrothendieckGroupViaQuotientOfCartesianProduct} are indeed the same:

If $a_+ + b_- + k = b_+ + a_- + k$ then take $k_1 \coloneqq b_- + k$ and $k_2 \coloneqq a_- + k$ to find that

$$
  \begin{aligned}
    (a_+ + k_1 , a_- + k_1)
    & =
    ( a_+ + b_- + k, a_- + b_- + k)
    \\
    & =
    (b_+ + a_- + k, a_- + b_- + k)
    \\
    & =
    (b_+ + k_2, b_- + k_2)
  \end{aligned}
  \,.
$$

Conversely, if $(a_+ + k_1 , a_- + k_1) = (b_+ + k_2, b_- + k_2)$ then take $k \coloneqq k_1 + k_2$ to find that

$$
  \begin{aligned}
    a_+ + b_- + k
      & =
    a_+ + k_1 + b_- + k_2
    \\
      & =
    b_+ + k_2 + a_- + k_1
    \\
      & =
    b_+ + a_- + k
  \end{aligned}
  \,.
$$

Now to see that $(G(A),+)$ is indeed an abelian group:

1. the second equivalence relation also makes it immediate that the [[neutral element]] is the class

   $$
     [0,0] =  [a,a]
   $$

   for all $a \in A$.

1. with this the second equivalence relation makes it immediate that the [[inverse element]] to any $[a_+, a_-]$ is

   $$
     -[a_+, a_-] = [a_-, a_+]
     \,,
   $$

That this group is abelian is immediate from the fact that $A$ is assumed to be abelian.

Regarding the universal property: let $B$ be any abelian group and let

$$
  \tilde f \colon G(A) \longrightarrow B
$$

be a homomorphism of abelian groups. Observe from the above that then

$$
  \begin{aligned}
    \tilde f([a_+,a_-])
    & =
    \tilde f( [a_+,0]  - [a_-, 0] )
    \\
    & =
    \tilde f([a_+,0]) - \tilde f([a_-,0])
    \\
    & =
    \tilde f(\eta_A(a_+)) - \tilde f(\eta_A(a_-))
    \\
    & =
    f(a_+) - f(a_-)
  \end{aligned}
$$

by the linearity of $f$ and the definition of $\eta_A \colon A \to G(A)$.

Conversely, given $f \colon A \to B$ then this equation uniquely defines $\tilde f$ with $f = \tilde f \circ \eta_A$.

=--

+-- {: .num_remark #GrothendieckGroupForCancellativeMonoids}
###### Remark
**(Grothendieck group for [[cancellative monoids]])**

If $(A,+)$ is a [[cancellative monoid]], in that

$$
  \underset{a,b,z \in A}{\forall}
  \left(
    \left(
      a + z = b + z
   \right)
     \Rightarrow
   \left(
     a = b
   \right)
  \right)
$$

then, as is immediate from the first of the two equivalence relations in def. \ref{GrothendieckGroupViaQuotientOfCartesianProduct}, the definition of the Grothendieck group $G(A)$ simplifies to

$$
  G(A) = (A \times A)/ \sim
$$

with

$$
  \big(
    (a_+,a_-)
      \sim
    (b_+,b_-)
  \big)
     \;\Leftrightarrow\;
  \big(
    a_+ + b_- = b_+ + a_-
  \big)
  \,.
$$

=--

## Examples

+-- {: .num_example #GrothendieckGroupOfNaturalNumbersUnderAdditionIsTheIntegers}
###### Example
**(Grothendieck group of the [[natural numbers]] is the [[integers]])

Let $(\mathbb{N}, +)$ be the [[commutative monoid]] of [[natural numbers]] under [[addition]]. By def. \ref{GrothendieckGroupViaQuotientOfCartesianProduct} its Grothendieck group consists of pairs $(n_+, n_-) \in \mathbb{N} \times \mathbb{N}$ subject to some equivalence relation, and since $(\mathbb{N}, +)$ is [[cancellative monoid|cancellative]], remark  \ref{GrothendieckGroupForCancellativeMonoids} says that this equivalence relation is simply

$$
  \big(
    (a_+,a_-)
      \sim
    (b_+,b_-)
  \big)
     \;\Leftrightarrow\;
  \big(
    a_+ + b_- = b_+ + a_-
  \big)
  \,.
$$

Let

$$
  \array{
    (G(\mathbb{N}),+) &\longrightarrow& (\mathbb{Z},+)
    \\
    (n_+, n_-) &\mapsto&  n_+ - n_-
  }
$$

be the evident [[homomorphism]] of abelian groups to the additive group of [[integers]].

This is manifestly [[surjective function|surjective]]. For it to be injective we need that

$$
  (a_+, a_-) \sim (b_+,b_-)
$$

precisely if

$$
  a_+ - a_- = b_+ - b_- \;\; \in \mathbb{Z}
  \,.
$$

The last condition holds precisely if

$$
  a_+ + b_- = b_+ + a_- \;\; \in \mathbb{Z}
$$

which is precisely the above equivalence relation. Therefore the above homomorphism is a [[bijection]] and hence the Grothendieck group of the natural numbers is the [[integers]]:

$$
  (G(\mathbb{N}), +) \simeq \mathbb{Z}
  \,.
$$

=--

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

Consider the commutative monoid $(\mathbb{Z}^\times, \cdot)$ of non-zero [[integers]] under [[multiplication]].

Consider the homomorphism

$$
  \array{
     (G(\mathbb{Z}), \cdots)
       &\longrightarrow&
     (\mathbb{Q}^\times, \cdot)
     \\
     ( n_+ ,n_- )
        &\mapsto&
     n_+/n_-
  }
$$

to the non-zero [[rational numbers]] under multiplication.

It is immediate that this is surjective. For it to be injective we need that

$$
  (a_+, a_-)
    \sim
  (b_+, b_-)
$$

precisely if

$$
  a_+/ a_- = b_+ / b_- \;\; \in \mathbb{Q}
$$

which is the case precisely if

$$
  a_+ \cdot b_- = b_+ \cdot a_-
  \,.
$$

Since $(\mathbb{Z}^\times, \cdot)$ is a [[cancellative monoid]], this is indeed the equivalence relation on $G(\mathbb{Z}^\times)$, according to remark \ref{GrothendieckGroupForCancellativeMonoids}.

=--

+-- {: .num_example}
###### Example
**([[topological K-theory]])**

Let $X$ be a [[topological space]] and let $(Vect(X)_{/\sim}, \oplus)$ be the monoid of [[isomorphism classes]] of [[topological vector bundles]] on $X$ with addition induced from the [[direct sum of vector bundles]]. (This is in general not a [[cancellative monoid]]). Then the Grothendieck group

$$
  K(X) \coloneqq (G(Vect(X)_{/\sim}), +)
$$

is called the _[[topological K-theory]]_ group of $X$.

=--

## References

See also

* Wikipedia, _[Grothendieck group](https://en.wikipedia.org/wiki/Grothendieck_group)_

[[!redirects Grothendieck groups of commutative monoids]]

[[!redirects Grothendieck group construction on commutative monoids]]


Grothendieck inequality

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### Context
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
=--
=--

# Grothendieck inequality
* table of contents
{: toc}

## Statement

Let $B$ be the [[unit ball]] of a [[separable space|separable]] [[Hilbert space]] over the [[real number|real]] or [[complex numbers]]. Then the [[inner product|scalar product]], $\langle\cdot,\cdot\rangle : B \times B \to \mathbb{C}$ has the following special property:

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

There exist [[infinite sequence|sequences]] $f_n,g_n: B \to \mathbb{C}$ of [[norm topology|norm]]-[[continuous functions]], such that

* $\langle x, y \rangle = \sum_n f_n(x) g_n(y)$ for all $x, y \in B$

* $\sum_n \, \sup_B \left| f_n \right| \, \sup_B \left| g_n \right| \, < \, \infty$
=--

In other words, $\langle\cdot,\cdot\rangle$, as a [[function of two variables]], is an element of the [[projective tensor product]] $C(B) {\displaystyle\hat{\otimes}} C(B)$. Its [[projective tensor norm]] is known as __Grothendieck's constant__. The precise value of this constant is different in the real and complex case, and neither one is known exactly.

## Related concepts

* [[inequality]]

* [[Cauchy-Schwarz inequality]]

* [[Bell's inequality]]

## References

### General

Due to:

* [[Alexander Grothendieck]], *Résumé des résultats essentiels dans la théorie des produits tensoriels topologiques et des espaces nucléaires*, Annales de l'Institut Fourier, **4** (1952) 73-112 [[numdam:AIF_1952__4__73_0](http://www.numdam.org/item/?id=AIF_1952__4__73_0)]

Review:

* Leqi Zhu, *Grothendieck's inequality* (2018) [[pdf](http://www.cs.toronto.edu/~toni/Courses/Proofs-SOS-2018/Lectures/grothendieck.pdf)]

* Ron Blei, _Analysis in integer and fractional dimensions_, Cambridge University Press (2009) [[doi:10.1017/CBO9780511543012](https://doi.org/10.1017/CBO9780511543012)]

See also:

* Wikipedia, *[Grothendieck inequality](https://en.m.wikipedia.org/wiki/Grothendieck_inequality)*

### In quantum physics
 {#ReferencesInQuantumPhysics}

Discussion of Grothendieck's inequality in [[quantum physics]], in relation to [[Bell's inequality]], originates with:

* [[Boris S. Tsirelson]], *Quantum analogues of the Bell inequalities. The case of two spatially separated domains*, Journal of Soviet Mathematics **36** (1987) 557–570 [[doi:10.1007/BF01663472](https://doi.org/10.1007/BF01663472)]

reviewed in

* [[Boris S. Tsirelson]], *Some results and problems on quantum Bell-type inequalities* Hadronic Journal Supplement **8** 4 (1993) 329-345 [[pdf](https://www.tau.ac.il/~tsirel/download/hadron.pdf), [[Tsirelson-QuantumBellType.pdf:file]] [web](https://www.tau.ac.il/~tsirel/download/hadron.html)]

  > (but see the erratum [here](https://www.tau.ac.il/~tsirel/Research/bellopalg/main.html))

* Wikipedia, *[Tsirelson's bound](https://en.wikipedia.org/wiki/Tsirelson%27s_bound)*

Further discussion:

* Antonio Acín, Nicolas Gisin, and Benjamin Toner, *Grothendieck’s constant and local models for noisy entangled quantum states*, Phys. Rev. A **73** (2006) 062105 [[doi:10.1103/PhysRevA.73.062105](https://doi.org/10.1103/PhysRevA.73.062105), [arXiv:quant-ph/0606138](https://arxiv.org/abs/quant-ph/0606138)]

* Hoshang Heydari, *Quantum Violation: Beyond Clauser-Horne-Shimony-Holt Inequality*, J. Phys. A: Math. Gen. **39** (2006) 11869-11875 [[arXiv:quant-ph/0603050](https://arxiv.org/abs/quant-ph/0603050), [doi:10.1088/0305-4470/39/38/012](https://iopscience.iop.org/article/10.1088/0305-4470/39/38/012)]

* Jop Briët, Harry Buhrman & Ben Toner, *A Generalized Grothendieck Inequality and Nonlocal Correlations that Require High Entanglement*, Comm. Math. Phys. **305** (2011) 827–843 [[doi:10.1007/s00220-011-1280-3](https://doi.org/10.1007/s00220-011-1280-3)]

* Flavien Hirsch, Marco Túlio Quintino, Tamás Vértesi, Miguel Navascués, Nicolas Brunner, Quantum **1** (2017) 3 *Better local hidden variable models for two-qubit Werner states and an upper bound on the Grothendieck constant $K_G(3)$* [[doi:10.22331/q-2017-04-25-3](https://doi.org/10.22331/q-2017-04-25-3), [arXiv:1609.06114](https://arxiv.org/abs/1609.06114)]

* A. Vourdas, *Grothendieck bound in a single quantum system*, J. Phys. A: Math. Theor. **55** (2022) 435206 [[arXiv:2212.11663](https://arxiv.org/abs/2212.11663), [doi:10.1088/1751-8121/ac9dcf](https://iopscience.iop.org/article/10.1088/1751-8121/ac9dcf)]

[[!redirects Grothendieck inequality]]
[[!redirects Grothendieck's inequality]]

[[!redirects Grothendieck constant]]
[[!redirects Grothendieck's constant]]
[[!redirects Grothendieck's constant]]
[[!redirects Grothendieck\'s constant]]


Grothendieck pretopology

> This entry is about pretopologies on [[sites]]. For a similarly-named generalization of [[topological space]]s based on neighborhoods, see [[pretopological space]].

***

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[[!include topos theory - contents]]
=--
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=--

# Contents
* automatic table of contents goes here
{:toc}

## Idea

A **Grothendieck pretopology** or **basis for a Grothendieck topology** is a collection of families of [[morphisms]] in a [[category]] which can be considered as [[cover|covers]].

Every Grothendieck pretopology generates a genuine _[[Grothendieck topology]]_. Different pretopologies may give rise to the same topology.

An even weaker notion than a Grothendieck pretopology, which also generates a Grothendieck topology, is a [[coverage]].  A Grothendieck pretopology can be defined as a coverage that also satisfies a couple of extra saturation conditions.  (Note that it is coverages, not pretopologies, that most directly corresponds to [[bases of topological spaces]].

## Definition

See Definition II.1.3 in [[SGA]] 4.

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

Let $C$ be a [[category]].
A **Grothendieck pretopology** or **basis (for a [[Grothendieck topology]])** on $C$ is an assignment to each [[object]] $U$ of $C$ of a collection of families $\{U_i \to U\}$ of morphisms, called **[[covering]] families** such that

1. (Stability under [[base changes]].)
The collection of covering families is stable under [[pullback]]: if $\{U_i \to U\}$ is a covering family and $f : V \to U$ is any morphism in $C$, then $\{f^* U_i \to V\}$ exists and is a covering family;

1. (Stability under composition.)
If $\{U_i \to U\}_{i \in I}$ is a covering family and for each $i$ also $\{U_{i,j} \to U_i\}_{j \in J_i}$ is a covering family, then also the family of composites $\{U_{i,j} \to U_i \to U\}_{i\in I, j \in J_i}$ is a covering family.

1. (Isomorphisms cover.)
Every family consisting of a single [[isomorphism]] $\{V \stackrel{\cong}{\to}U\}$ is a covering family;

=--

If we drop the second and third conditions, we obtain something a bit stronger than a [[coverage]]; at the page [[coverage]] this notion is called a *cartesian coverage*.  Conversely every coverage on a category with pullbacks generates a Grothendieck pretopology by an evident closure process.  However, many coverages that arise in practice are actually already Grothendieck pretopologies. On the other hand, for some analogues in noncommutative algebraic geometry, rather the stability axiom fails.

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

The [[Grothendieck topology]] on $C$ _generated_ from a basis of covering families is that for which a [[sieve]] $\{S_i \to U\}$ is covering precisely if it contains a covering family of morphisms.

=--

## Properties

Given any [[Grothendieck topology]] on $C$, there is a **maximal basis** which generates it: this has as covering families precisely thoses families of morphisms that generate a covering sieve under completion under precomposition.

## Examples

The prototype is the pretopology on the [[category of open subsets]] $Op(X)$ of a [[topological space]] $X$, consisting of [[open covers]] of $X$.

Given a [[base for the topology]] on $X$,
we can construct a pretopology on $Op(X)$
by declaring that a family $\{U_i\to V\}_{i\in I}$ is a covering family
if it is an open cover and for every $i\in I$ the open set $U_i$
is the intersection of $V$ and an element of the base.

If instead we impose a stronger requirement that $U_i$ belongs to the base,
then the resulting [[coverage]] is not a pretopology,
since the intersection of such a covering family
with an arbitrary open subset of $X$ is an open cover
whose elements need not belong to the base.

Grothendieck pretopologies on [[Top]] include:

* [[local section|Local-section]]-admitting surjections
* Surjective [[open map|open maps]]
* Surjective [[topological submersion|topological submersions]]
* Surjective [[local homeomorphism|local homeomorphisms]]

An example for the category [[Diff]] of manifolds is the pretopology of [[surjective submersion]]s.  All of these have covering families consisting of single morphisms. Such a pretopology is called a *singleton* pretopology (and, in particular, it is a *singleton coverage*).

An example of a [[coverage]] that is not a pretopology is the coverage of [[good open covers]], say on [[Diff]]. In general the pullback of a good open cover is just an [[open cover]], not necessarily still one where all finite non-empty intersections are contractible.

## References

The definition appears for instance as definition 2 on page 111 of

* [[Saunders MacLane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_
* MO questions:
  * [example-of-a-grothendieck-pretopology-satisfying-a-weak-saturation-condition](http://mathoverflow.net/questions/42437/example-of-a-grothendieck-pretopology-satisfying-a-weak-saturation-condition);
  * [colimits-of-covers](http://mathoverflow.net/questions/44893/colimits-of-covers);
  * [When is a basis of a topological space a Grothendieck pretopology?](https://mathoverflow.net/questions/416691/when-is-a-basis-of-a-topological-space-a-grothendieck-pretopology/416706#416706)

[[!redirects pretopology]]
[[!redirects pretopologies]]

[[!redirects Grothendieck pretopology]]
[[!redirects Grothendieck pretopologies]]

[[!redirects base for a Grothendieck topology]]
[[!redirects basis for a Grothendieck topology]]
[[!redirects bases for a Grothendieck topology]]
[[!redirects bases for Grothendieck topologies]]
[[!redirects basis for the Grothendieck topology]]
[[!redirects bases for the Grothendieck topology]]
[[!redirects base of a Grothendieck topology]]
[[!redirects basis of a Grothendieck topology]]
[[!redirects bases of a Grothendieck topology]]
[[!redirects bases of Grothendieck topologies]]


Grothendieck ring

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[[!include homological algebra - contents]]
=--
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=--

#Contents#
* table of contents
{:toc}

##Idea

Any [[abelian category]] $C$ gives rise to an abelian group $K(C)$ called its [[Grothendieck group]] (see there for more).  If we apply this construction to a [[monoidal category|monoidal]] abelian category or generally to a [[rig category]], $K(C)$ is a [[ring]], called the **Grothendieck ring**.

If $C$ is a [[braided monoidal category]], $K(C)$ becomes a [[commutative ring]].

If $C$ is a [[symmetric monoidal category]], $K(C)$ becomes a $\Lambda$-[[Lambda-ring|ring]].

## Definition

Let $C$ be an [[abelian category]]. The [[Grothendieck group]] $K(C)$ is the abelian group generated by objects of $C$, quotiented by the relation that $[A]+[B]=[C]$ if there is a short exact sequence

$$0\to A\to C\to B.$$

In the presence of a [[monoidal category | monoidal]] structure $\otimes$, we can turn $C$ into a ring by defining

$$[A]\cdot [B]=[A\otimes B].$$

This endows $K(C)$ with the structure of a ring because

$$A\otimes(B\oplus C)\cong (A\otimes B)\oplus (A\otimes C),$$

and every monoidal category has a tensor unit.

## Examples

* The Grothendieck ring of the monoidal category of finite [[G-sets]] is called the [[Burnside ring]] of $G$.

## References

* [[Peter May]], _Picard groups, Grothendieck rings,and Burnside rings of categories_, Advances in Mathematics 163, 1–16 (2001), ([pdf](http://www.math.uchicago.edu/~may/PAPERS/PicJan01.pdf)).

[[!redirects Grothendieck rings]]


Grothendieck ring of varieties

## Idea

Grothendieck group of varieties is the free Abelian group generated by isomorphism classes of [[quasiprojective varieties]] modulo all relations of the form $[X]\sim [U] + [X\backslash U]$ where $U$ is open in $X$. Product of varieties induces a multipication on this Abelian group, making it into a [[ring]], the Grothendieck ring of varieties.

## Literature

Chapter 2 of

*  Antoine Chambert-Loir, Johannes Nicaise, Julien Sebag, _Motivic integration_, Progress in Mathematics __325__ (2018) [doi](https://doi.org/10.1007/978-1-4939-7887-8)

* Bjorn Poonen, _The Grothendieck ring of varieties is not a domain_, Math. Res. Lett. __9__:4 (2002) 493-497
[doi](https://doi.org/10.4310/MRL.2002.v9.n4.a8) arXiv:[math.AG/0204306](https://arxiv.org/abs/math/0204306)

* Michael Larsen, Valery A. Lunts, _Motivic measures and stable birational geometry_,  Mosc. Math. J. 3 (2003), no. 1, 85-95, 259  [MR1996804](https://mathscinet.ams.org/mathscinet-getitem?mr=1996804) [doi](https://doi.org/10.17323/1609-4514-2003-3-1-85-95)

> We study the motivic Grothendieck group of algebraic varieties from the point of view of stable [[birational geometry]]. In particular, we obtain a counter-example to a conjecture of M. Kapranov on the rationality of motivic zeta-function.

An analogue in the setup of [[pretriangulated categories]] is introduced in

* [[Alexei Bondal|A. I. Bondal]], [[Michael Larsen|M. Larsen]], [[V. A. Lunts]], _Grothendieck ring of pretriangulated categories_,  Intern. Math. Res. Notices __29__ (2004) 1461--1495 [doi](https://doi.org/10.1155/S1073792804140385)
[math.AG/0401009](https://arxiv.org/abs/math/0401009)

category: algebraic geometry
[[!redirects Grothendieck group of varieties]]


Grothendieck spectral sequence

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#Contents#
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## Idea

A _Grothendieck spectral sequence_ is a [[spectral sequence]] that computes the [[cochain cohomology]] of the [[composition|composite]] of two [[derived functor in homological algebra|derived functors]] on [[categories of chain complexes]].

## Statement
 {#Statement}

Let $\mathcal{A},\mathcal{B},\mathcal{C}$ be [[abelian category|abelian categories]] and let $F \colon \mathcal{A}\to \mathcal{B}$ and $G \colon \mathcal{B}\to \mathcal{C}$  be [[exact functor|left exact]] [[additive functor|additive]] [[functors]]. Assume that $\mathcal{A}, \mathcal{B}$ have [[injective object|enough injectives]].

+-- {: .num_theorem}
###### Theorem ([[Tohoku]])

Write $R_F\subset \mathrm{Ob} A$ and $R_G\subset\mathrm{Ob} B$ for the [[class of adapted objects|classes of objects adapted to]] $F$ and $G$ respectively, and let furthermore $F(R_A)\subset R_B$. Then the derived functors $R F:D^+(A)\to D^+(B)$, $R G:D^+(B)\to D^+(C)$ and $R(G\circ  F):D^+(A)\to D^+(C)$ are defined and the natural morphism $R(G\circ F)\to R G\circ R F$ is an isomorphism.

=--

+-- {: .num_theorem}
###### Theorem ([[Tohoku]])

In the above situation, assume that for every [[injective object]] $I \in \mathcal{A}$ the object $F(I) \in \mathcal{B}$ is a $G$-[[acyclic object]].

Then for every object $A \in \mathcal{A}$ there is a [[spectral sequence]] $\{E^r_{p,q}(A)\}_{r,p,q}$ called the **Grothendieck spectral sequence** whose $E_2$-page is the composite

$$
  E^{p,q}_2(A) = R^p G \circ R^q F (A)
$$

of the [[derived functor in homological algebra|right derived functors]] of $F$ and $G$ in degrees $q$ and $p$, respectively
and which is converging to to the derived functors $R^n(G\circ F)$ of the composite of $F$ and $G$:

$$
  E^{p,q}_\infty(A) \simeq R^{p+q}(G \circ F)(A)
  \,.
$$

Moreover, this is [[natural isomorphism|natural]] in $A \in \mathcal{A}$.

=--

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

By assumption of enough injectives, we may find an _[[injective resolution]]_

$$
  A \stackrel{\simeq_{qi}}{\to} C^\bullet
$$

of $A$. Next, by the discussion at _[injective resolution -- Existence and construction](projective+resolution#ExistenceAndConstructionOfResolutionsOfComplexes)_ we may find a _fully injective_ resolution of the chain complex $F(C^\bullet)$:

$$
  0 \to F(C^\bullet) \to I^{\bullet, \bullet}
  \,,
$$

where hence $I^{\bullet, \bullet}$ is a [[double complex]] of [[injective objects]] such that for each $n \in \mathbb{N}$ the component $0 \to F(C^n) \to I^{n,\bullet}$ is an ordinary injective resolution of $F(C^n) \in \mathcal{B}$.

Thus we have the corresponding double complex $G(I^{\bullet,\bullet})$ in $\mathcal{C}$. The claim is that the Grothendieck spectral sequence is the [[spectral sequence of a double complex]] for $G(I^{\bullet, \bullet})$ equipped with the vertical-degree [[filtered chain complex|filtration]] $\{{}^{vert}E^r_{p,q}(A)\}$:

$$
  {}^{vert} E^2_{p,q}(A) \simeq R^p G (R^q F(A))
  \,.
$$

To see this, notice that by the assumption that $I^{\bullet,\bullet}$ is a _fully_ injective projective resolution, the short exact sequences

$$
  0 \to B^{q,p}(I) \to Z^{q,p}(I) \to H^{q,p}(I) \to 0
$$

are [[split exact sequence|split]] (by the discussion there) and hence so is their image under any functor and hence in particular under $G$. Accordingly we have

$$
  \begin{aligned}
    {}^{vert}E^{p,q}_1
    & \simeq
    H^q(G(I^{\bullet,p}))
    \\
    & \simeq (G(Z^{q,p})) / (G(B^{q,p}))
    \\
    & \simeq G H^{q,p}
  \end{aligned}
$$

(the first two equivalences by general properties of the filtration spectral sequence, the last by the above splitness). Hence it follows that

$$
  \begin{aligned}
    {}^{vert}E^{p,q}_2
    & \simeq
    H^p(G(H^{q,\bullet}))
    \\
    & \simeq
    R^p G (R^q F (A))
  \end{aligned}
  \,,
$$

where in the last step we used that $H^{q,\bullet}$ is be construction an injective resolution of $H^q(F(C^\bullet)) \simeq R^q F(A)$ (using the $G$-acyclicity of $F(C^\bullet)$).

This establishes the spectral sequence and its second page as claimed. It remains to determine its convergence.

To that end, consider dually, the spectral sequence $\{{}^{hor}E^{p,q}_r\}$ coming from the horizontal filtration on the double complex $G(I^{\bullet, \bullet})$. By the general properties of [[spectral sequence of a double complex]] this converges to the same value as the previous one. But for this latter spectral sequence we find

$$
  \begin{aligned}
    {}^{hor}E^{p,q}_1 & \simeq H^q(G I^{p,\bullet})
    \\
    & \simeq R^q G(F(C^p))
  \end{aligned}
  \,,
$$

the first equivalence by the general properties of filtration spectral sequences, the second then by the definition of [[derived functor in homological algebra|right derived functors]]. But by assumption $F(C^p)$ is $G$-[[acyclic object|acyclic]] and hence all these derived functors vanish in positive degree, so that

$$
  {}^{hor}E^{p,q}_1 \simeq
  \left\{
    \array{
       G(F(C^p)) & if\; q = 0
       \\
       0 & otherwise
    }
  \right.
  \,.
$$

Next, the $E_2$-page then contains just horizontal homology of $G(F(C^\bullet))$ and this is by definition now the derived functor of the composite of $F$ with $G$:

$$
  {}^{hor}E^{p,q}_2 \simeq
  \left\{
    \array{
      R^p(G \circ F) & if \; q  = 0
      \\
      0 & otherwise
    }
  \right.
  \,.
$$

Since this is concentrated in the $(q = 0)$-row the spectral sequence of the horizontal filtration collapses here and hence

$$
  \begin{aligned}
    H^n(Tot(G(I^{\bullet,\bullet})))
    &\simeq
    G^n H^{n+0}(Tot(G(I^{\bullet,\bullet})))
    \\
    & \simeq
    E^{n,0}_\infty
 \end{aligned}
$$

So in conclusion we have

$$
  \begin{aligned}
    R^p G(R^q F(A))
    & \simeq
    {}^{vert}E^{p,q}_2
    \\
    & \Rightarrow
    {}^{vert} E^{p,q}_\infty
    \\
    & \simeq
    G^p_{vert} H^{p+q}(Tot(G(I^{\bullet, \bullet})))
    \\
    & \simeq
    H^{p+q}(Tot(G(I^{\bullet, \bullet})))
    \\
    & \simeq
    G^{p+q}_{hor} H^{p+q}(Tot(G(I^{\bullet, \bullet})))
    \\
    & \simeq
    {}^{hor} E^{p+q,0}_\infty(A)
    \\
    & \simeq
    R^{p+q}(G \circ F)(A)
  \end{aligned}
$$

=--

## Examples

Many other classes of spectral sequences are special cases of the Grothendieck spectral sequence, for instance the

* [[Hochschild-Serre spectral sequence]]

* [[Leray spectral sequence]]

* [[base change spectral sequence]]

## References

Leture notes include

* [[James Milne]], section 10 of _[[Lectures on Étale Cohomology]]_

* Jinhyun Park, _Personal notes on Grothendieck spectral sequence_ ([pdf](http://mathsci.kaist.ac.kr/~jinhyun/note/g_s_sequence/sequence.pdf))


Grothendieck topology

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

#Contents#
* table of contents
{:toc}

## Idea

A _Grothendieck topology_ on a category is a choice of morphisms in that category which are regarded as [[cover|covers]].

A category equipped with a Grothendieck topology is a [[site]].  Sometimes all sites are required to be [[small category|small]].

Probably the main point of having a site is so that one can define [[sheaf|sheaves]], or more generally [[stack|stacks]], on it.  In particular, the category of sheaves on a (small) site is a [[Grothendieck topos]].

## Original definition

In the original definition ([[Michael Artin]]'s seminar notes “Grothendieck topologies”), a Grothendieck topology on a [[category]] $C$ is defined as a [[set]] $T$ of [[coverings]] satisfying certain closure properties.

More precisely (to cite from Artin's notes):

\begin{definition}
A __Grothendieck topology__ on a [[category]] $C$ is a set $T$ of families of maps $\{\phi_i\colon U_i\to U\}_{i\in I}$
(known as __coverings__)
such that

* for any [[isomorphism]] $\phi$ we have $\{\phi\}\in T$;

* if $\{U_i\to U\}\in T$ and $\{V_{i,j}\to U_i\}\in T$ for each $i$, then $\{V_{i,j}\to U\}\in T$;

* if $\{U_i\to U\}\in T$ and $V\to U$ is a morphism, then $U_i\times_U V$ exist and $\{U_i\times_U V\to V\}\in T$.

\end{definition}

This is almost identical to the current definition of [[Grothendieck pretopology]],
except that in Artin's definition only the relevant pullbacks are required to exist.

## Definition using sieves

If $g:d\to c$ is a morphism in a category $C$ and $F\subset C(-,c)$ a [[sieve]] on $c$ then
$$
g^* F = \{ h: dom(h)\to d | g\circ h \in F\}\subset C(-,d)
$$
is a sieve on $d$, the pullback sieve of $F$ along $g$.

The following definition can be found in [[SGA]] 4, Exposé II.

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

A **Grothendieck topology** $J$ on a [[category]] $C$ is an assignment to each object $c \in C$ of a collection of [[sieves]] on $c$ which are called _covering sieves_, satisfying the following axioms:

1. (Stability under [[base change]].)
If $F$ is a [[sieve]] that covers $c$ and $g: d \to c$ is any morphism, then the pullback sieve $g^* F$ covers $d$.

1. (Local character condition.)
If $F$ is a sieve on $c$ such that the sieve $\bigcup_d \{g: d \to c| g^* F \; covers \; d\}$ contains a covering sieve of $c$, then $F$ itself covers $c$.

1. The [[maximal sieve]] $id: \hom(-, c) \hookrightarrow \hom(-, c)$ is always a covering sieve;

=--

The set of covering sieves of an object $c$ is denoted $J(c)$.

A category equipped with a Grothendieck topology is called a **[[site]]** .

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

The axiom of stability under [[base change]] guarantees that we have a functor $J: C^{op} \to Set$.  Thus $J$ itself can be regarded as an object of the [[presheaf topos]] $[C^{op},Set]$; in this way Grothendieck topologies on $C$ are identified with [[Lawvere-Tierney topologies]] on $[C^{op},Set]$.

=--

Given a Grothendieck topology $J$ on a [[small category]] $C$, one can define the category $Sh(C,J)$ of [[sheaves]] on $C$ relative to $J$, which is a [[reflective subcategory]] of the category $[C^{op},Set]$ of [[presheaves]] on $C$.  Thus we have a functor $C\to Sh(C,J)$ given by the composite of the [[Yoneda embedding]] with the reflection (or "sheafification").  This composite functor is [[full and faithful functor|fully faithful]] if and only if all representable presheaves are sheaves for $J$; a topology with this property is called [[subcanonical site|subcanonical]].

## Saturation {#Saturation}

Grothendieck topologies may be and in practice quite often are obtained as closures of collections of morphisms that are not yet closed under the operations above (that are not yet sieves, not yet pullback stable, etc.).

Two notions of such unsaturated collections of morphisms inducing Grothendieck topologies are

*  [[Grothendieck pretopology]],

*  [[coverage]].

## Examples {#Examples}

### Topology on open subsets of a topological space

The archetypical example of a Grothendieck topology is that on a [[category of open subsets]] $Op(X)$ of a [[topological space]] $X$. A covering family of an open subset $U \subset X$ is a collection of open subsets $V_i \subset U$ that cover $U$ in the ordinary sense of the word, i.e. which are such that every point $x \in U$ is in at least one of the $V_i$.

### Regular topology

Any [[regular category]] $C$ admits a [[subcanonical site|subcanonical]] Grothendieck topology whose covering families are generated by single [[regular epimorphisms]].  If $C$ is [[exact category|exact]] or has pullback-stable [[reflexive coequalizer]]s, then its [[codomain fibration]] is a [[stack]] for this topology (the necessary and sufficient condition is that any pullback of a kernel pair is again a kernel pair).

### Extensive topology

Any [[extensive category]] admits a [[Grothendieck topology]] whose [[cover|covering families]] are (generated by) the families of inclusions into a [[coproduct]] (finite or small, as appropriate).  We call this the **extensive [[coverage]]** or **extensive topology**.  The [[codomain fibration]] of any extensive category is a [[stack]] for its extensive topology.

### Coherent topology

Any [[coherent category]] $C$ admits a [[subcanonical site|subcanonical]] Grothendieck topology in which the covering families are generated by finite, jointly [[regular epimorphism|regular-epimorphic]] families.  Equivalently, they are generated by single [[regular epimorphisms]] and by finite unions of [[subobject]]s.  If $C$ is [[extensive category|extensive]], then its coherent topology is generated by the regular topology together with the extensive topology.  (In fact, the coherent topology is [[superextensive site|superextensive]].)

### Canonical topology

On any category there is a largest subcanonical topology.  This is called the *[[canonical topology]]*, with "subcanonical" a back-formation from this (since a topology is subcanonical iff it is contained in the canonical topology).  On a [[Grothendieck topos]], the covering families in the canonical topology are those which are jointly epimorphic.

## Historical note on terminology

The term "topology" for this concept is hallowed by tradition and the name of Grothendieck, but some people consider it to be somewhat unfortunate. While "a category equipped with a (Grothendieck) topology" is, in fact, sort of a generalization of "a set equipped with an (ordinary) topology," the relationship between a category and its topology is quite different from the relationship between a set and its topology.  In particular, when we construct a site from a topological space, the objects of the category are the _open sets_, not the points, of the space.

This use of "topology" can also lead to confusion since for a topologist, there is a completely different and very natural meaning of "topology on a category:" namely, topologies on its sets of objects and morphisms making it into an [[internal category]] in [[Top]].  The topologist's definition is, of course, a conservative extension of the classical notions of "topology on a set" and even "topology on a group," while there are no nontrivial Grothendieck topologies on a group considered as a 1-object category.

Furthermore, the use of "topology" for a category with a system of covers also leads to the use of "continuous" for a functor which preserves covers.  This is (in some people's opinion) doubly unfortunate, since "continuous functor" is used not only by topologists to mean an internal or [[enriched category|enriched]] functor over Top, but also by many category-theorists to mean a functor that preserves all small [[limit|limits]].  This is especially confusing since covers are more akin to colimits than limits. Moreover, while a continuous function between topological spaces does induce a a cover-preserving functor between categories of open sets, the function and the functor go in opposite directions.

In [[Elephant|Sketches of an Elephant]], Peter Johnstone introduced the term __coverage__ used above for a system of covers on a category, with "Grothendieck coverage" as a proposed replacement for "Grothendieck topology."  See [[coverage]] for his definitions.  Proposed replacements for "Lawvere-Tierney topology" include:

* _Local operator_ (used by Johnstone).

* _Local modality_ or _geometric modality_, since in the [[internal logic]] of the topos, it represents a [[modal operator]] with the intutive meaning of "it is locally the case that...".

* _Lawvere-Tierney operator_ or _Lawvere-Tierney modality_ to avoid possible conflict with other uses of "local" or "geometric" in modal logic.

## Related notions

A more general notion is simply a collection of "covering families," not necessarily sieves, satisfying only pullback-stability; this suffices to define an equivalent notion of sheaf.  Following the [[Elephant]], we call such a system a [[coverage]].  A Grothendieck topology may then be defined as a coverage that consists of sieves (which the Elephant calls "sifted") and satisfies certain extra saturation conditions; see [[coverage]] for details.

An intermediate notion is that of a [[Grothendieck pretopology]], which consists of covering families that satisfy some, but not all, of the closure conditions for a Grothendieck topology.  Many examples are "naturally" pretopologies, but must be "saturated" under the remaining closure conditions to produce Grothendieck topologies.

As remarked above, Grothendieck topologies on a small category $C$ are also in bijective correspondence with [[Lawvere-Tierney topology|Lawvere-Tierney topologies]] on the [[presheaf]] [[topos]] $[C^{op},Set]$.  See [[Lawvere-Tierney topology]] for a description of the correspondence.

See also

* [[coverage]], [[site]], [[Lawvere-Tierney topology]], [[Grothendieck pretopology]], [[Q-category]], [[cd-structure]]

* [[Weiss topology]]

* [[(∞,1)-Grothendieck topology]]

## References

The original definition is in

* [[Michael Artin]], _Grothendieck Topologies.  Notes on a Seminar by M. Artin.  Spring, 1962_, Harvard University, Department of Mathematics.  [PDF](http://www.math.nagoya-u.ac.jp/~larsh/teaching/S2013_AG/grothendiecktopologies.pdf)

The definition in terms of [[sieves]] is found in

* [[SGA]] 4: Théorie des topos et cohomologie étale des schémas.

Standard books include

* [[Saunders MacLane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_

* [[Francis Borceux]], _[[Handbook of Categorical Algebra]]_, Volume 3.

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_

[[!redirects Grothendieck topologies]]