Grothendieck construction

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###Context###
#### Category theory
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[[!include category theory - contents]]
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#Contents#
* table of contents
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## Idea

In any context it is of interest to ask which kind of [[morphisms]]

$$
  \array{
    E
    \\
    \downarrow^{\mathrlap{p}}
    \\
    C
  }
$$

arise as [[pullbacks]] along a classifying morphism $S_p : C \to U$ to
some universal object $U$ of some universal morphism

$$
  \array{
    \hat U
    \\
    \downarrow^{\mathrlap{p_{univ}}}
    \\
    U
  }
  \,.
$$

The **Grothendieck construction** describes this in the context of [[Cat]]: a morphism $p : E \to C$ of [[category|categories]] -- i.e. a [[functor]] -- is called a [[fibered category]] or [[Grothendieck fibration]] if it is encoded in a [[pseudofunctor]]/[[2-functor]] $S_p : C^{op} \to Cat$.

The reconstruction of $p$ from the pseudofunctor $S_p$ is the **Grothendieck construction**

$$
  \textstyle{\int} \;\; : \;\; Func(C^{op}, Cat) \to Cat/C
$$

which is a [[2-functor]] from the [[2-category]] of pseudofunctors $C^{op} \to Cat$ to the [[overcategory]] of [[Cat]] over $C$.

The essential image of this functor consists of [[Grothendieck fibration]]s and this establishes an [[equivalence of 2-categories]]

$$
  \textstyle{\int}
  \,\colon\,
  Func(C^{op}, Cat)
    \overset{\simeq}{\longrightarrow}
  Fib(C)
$$

between 2-functors $C^{op} \to Cat$ and [[Grothendieck fibration]]s over $C$.

When restricted to pseudofunctors with values in [[Grpd]] $\subset$ [[Cat]] this identifies the [[fibration fibered in groupoids|Grothendieck fibrations in groupoids]]

$$
  \textstyle{\int}
  \;\colon\;
  Func(C^{op}, Grpd)
    \overset{\simeq}{\longrightarrow}
  FibGrpd(C)
  \,.
$$

This equivalence notably allows one to discuss [[stack]]s equivalently as pseudofunctors or as groupoid fibrations (in each case satisfying a [[descent]] condition with respect to a [[Grothendieck topology]] on $C$).

The Grothendieck construction is one of the central aspects of [[category theory]], together with the notions of universal constructions such as [[limit]], [[adjunction]] and [[Kan extension]]. It is expected to have suitable analogs in all sufficiently good contexts of [[higher category theory]]. Notably there is an [[(∞,1)-Grothendieck construction]] in [[(∞,1)-category]] theory.

The Grothendieck construction can also be generalized beyond fibrations, to the correspondence between [[displayed categories]] and arbitrary categories over $C$.

## Definition
 {#Definition}

Let [[Cat]] denote the [[2-category]] of [[categories]], [[functors]] and [[natural transformations]].

In line with the philosophy of [[generalized universal bundles]], consider the "universal [[Cat]]-[[bundle]]"

$$
  \begin{array}{c}
    Cat_{*,\ell}
    \\
    \big\downarrow
    \\
    Cat
    \mathrlap{\,,}
  \end{array}
$$

namely the [[2-category]] of "lax-[[pointed object|pointed]]" [[categories]], also known as the "[[lax slice 2-category|lax slice]]" of [[Cat]] under the [[terminal category]] $\ast \,\in\, Cat$:

* Its objects are [[pointed categories]] (i.e. [[pairs]] $(A,a)$ where $A$ is a category and $a$ is an [[object]] of $A$)

* and its [[morphisms]] $(A,a) \to (B,b)$ are pairs $(f,\gamma)$, where $f \colon A\to B$ is a [[functor]] and $\gamma\colon f(a)\to b$ is a [[morphism]] in $B$.

* The projection $Cat_{*,\ell} \to Cat$ is the evident [[forgetful functor]].

Now if $F \colon C \to Cat$ is a [[pseudofunctor]] from a category $C$ to $Cat$, then its **Grothendieck construction** is the (strict) [[2-pullback]] $ p \colon \int F \to C$ of $Cat_{*,\ell} \to Cat$ along $F$:

$$
  \array{
    \int F &\longrightarrow& Cat_{*,\ell}
    \\
    \mathllap{{}^{p}}\big\downarrow
    &&
    \big\downarrow
    \\
    C &\underset{F}{\longrightarrow}& Cat
    \mathrlap{\,.}
  }
$$

This means that:

* the [[objects]] of $\int F$ are pairs $(c,a)$, where $c \in Obj(C)$ and $a \in Obj(F(c))$,

* {#TheMorphisms} and morphisms in $\int F$ are given by pairs $\big(c \overset{f}{\to} c',\, F(f)(a) \overset{\phi}{\to} a'\big)$. As systems of [[diagrams]] in [[Cat]] this looks as follows:

This construction extends to a [[2-functor]] between [[bicategories]]

$$
  \textstyle{\int}
  \;\colon\;
  [C, Cat]
    \longrightarrow
  Cat/C
$$

from [[pseudofunctors]] on $C$ to the [[overcategory]] of [[Cat]] over $C$.

The more commonly described version of this construction works instead on [[contravariant functor|contravariant]] pseudofunctors, i.e. pseudofunctors $C^{op}\to Cat$ from an [[opposite category]].  In this case we use instead the "universal $Cat$-cobundle" $(Cat_{*,c})^{op} \to Cat^{op}$, where $(Cat_{*,c})$ is the colax slice, whose objects are  again pointed categories $(A,a)$, but whose morphisms $(A,a) \to (B,b)$ are pairs $(f,\gamma)$ where $f\colon A\to B$ and $\gamma\colon b \to f(a)$.  Now the 2-pullback

$$
  \array{
    \int F &\to& (Cat_{*,c})^{op}
    \\
    {}^{\mathllap{p}}\downarrow && \downarrow
    \\
    C &\stackrel{F}{\to}& Cat^{op}
  }
  \,.
$$

constitutes a [[2-functor]]

$$
  \textstyle{\int}
  \;\colon\;
  [C^{op},Cat]
    \longrightarrow
  Cat/C
  \,.
$$

In this case,

* the objects of $\int F$ are again pairs $(c,a)$, where $c \in Obj(C)$ and $a \in Obj(F(c))$, but

* {#TheMorphismsContra} the morphisms in $\int F$ from $(c,a)$ to $(c',a')$ are pairs $\big(c \overset{f}{\to} c', a \overset{\phi}{\to} F(f)(a')\big)$:

Note that this is not the same as the first (covariant) Grothendieck construction applied to $C^{op} \to Cat$, since, in that case, the morphisms between the objects of $C$ go in the opposite direction to one another.

## Properties

### (Co)Limits in a Grothendieck construction
 {#CoLimitsInAGrothendieckConstruction}

We discuss existence and characterization of ([[colimit|co]])[[limits]] *in* a Grothendieck construction.

\begin{proposition}
Given a [[pseudofunctor]] $F \colon C^{op} \to Cat$.

If

1. $C$ is [[complete category|complete]].

1. $F(J)$ is complete for all $J \in C$.

1. $F(f) : F(J) \to F(K)$ [[preserved limit|preserves limits]] for all $f \colon K \to
J$ in $C$.

then $\int F$ is complete.

Dually, if

1. $C$ is [[cocomplete category|cocomplete]].

1. $F(J)$ is cocomplete for all $J \in C$.

1. $F(f) : F(J) \to F(K)$ has a [[left adjoint]] for all $f : K \to J$ in $C$.

Then $\int F$ is cocomplete.
\end{proposition}
This proven in [Tarlecki, Burstall & Goguen (1991), §3.1, 3.2](#TBG91).

The case of colimits is also described in [Harpaz & Prasma (2015), Prop. 2.4.4](Grothendieck+construction+for+model+categories#HarpazPrasma15):

Given a [[pseudofunctor]]
\[
  \label{ThePseudofunctor}
  \array{
    \mathllap{
      \mathbf{C} \,\colon\,
      \;
    }
    Base &\longrightarrow& Cat
    \\
    \mathcal{X} &\mapsto& \mathbf{C}_{\mathcal{X}}
    \\
    \Big\downarrow\mathrlap{{}^{f}}
    &&
    \mathllap{^{f_!}}\Big\downarrow
    \dashv
    \Big\uparrow\mathrlap{{}^{f^\ast}}
    \\
    \mathcal{Y} &\mapsto& \mathbf{C}_{\mathcal{Y}}
  }
\]
such that

1. $Base$ is cocomplete

2. $\mathbf{C}_{\mathcal{X}}$ is [[cocomplete category|cocomplete]] for each $\mathcal{X} \in Base$

then also the [[Grothendieck construction]] $\int_{\mathcal{X}} \mathbf{C}_{\mathcal{X}} \,\in\, Cat$ is cocomplete.

Explicitly, [[colimits]] in $\int_{\mathcal{X}} \mathbf{C}_{\mathcal{X}}$ are computed as follows:

Given a diagram in the Grothendieck construction

$$
  \mathscr{V}_{\mathcal{X}}
  \;\colon\;
  I
  \longrightarrow
  \textstyle{\int} \mathbf{C}_{(-)}
$$

its [[underlying]] diagram in $Base$

$$
  \mathcal{X}
  \;\colon\;
  I
  \overset{ \mathscr{V}_{\mathcal{X}} }{\longrightarrow}
  \int \mathbf{C}_{(-)}
  \overset{\pi}{\longrightarrow}
  Base
$$

has a colimit by assumption on $Base$, with [[coprojection]] morphisms to be denoted like this:

$$
  \mathcal{X}(i)
   \overset{\;\; q(i) \;\;}{\longrightarrow}
  \underset{\underset{j \in I}{\longrightarrow}}{\lim}
  \mathcal{X}(j)
  \,.
$$

Now the idea is that the full colimit in $\int \mathbf{C}_{(-)}$ is obtained by

1. first pushing all morphisms $\phi \colon f_!\mathscr{V}(i) \to \mathscr{V}(j)$ in the diagram forward along the respective $q_j$

1. to hence obtain a diagram $q_! \mathscr{V}$ in $\mathbf{C}_{\underset{\longrightarrow}{\lim} \mathcal{X}}$

1. whose colimit $\underset{\longrightarrow}{\lim} q_! \mathscr{V}$ exists by assumption on $\mathbf{C}$

and then $\big(\underset{\longrightarrow}{\lim} q_! \mathscr{V}\big)_{\underset{\longrightarrow}{\lim}\mathcal{X}}$ is the desired colimit in $\int \mathbf{C}$

\begin{example}\label{CartesianProductInGrothendieckConstruction}
**([[Cartesian product]] in Grothendick construction is [[external tensor product|external product]] on fiber categories )**
\linebreak

  Given a [[contravariant functor|contravariant]] [[pseudofunctor]]
  $$
    \array{
       X
       \\
       \Big\downarrow\mathrlap{{}^{f}}
       \\
       Y
    }
    \;\;\;\;\;\mapsto\;\;\;
    \array{
       \mathcal{C}_X
       \\
       \Big\uparrow\mathrlap{{}^{f^\ast}}
       \\
       \mathcal{C}_Y
    }
  $$
  where the base category and all fiber categories $\mathcal{C}_{(-)}$ have [[Cartesian products]] and all [[base change]] maps $f^\ast$ [[preserved limit|preserve]] these products, then the Grothendieck construction $\int_X \mathcal{C}_X$ has cartesian products given on objects
$$
  \mathscr{V}_X \,\equiv\,
  \big(
    \mathscr{V} \,\in\, \mathcal{C}_X
  \big)
$$
by the formula
\[
  \label{CartesianProductInGrothendieckConstruction}
  \mathscr{V}_X
  \times
  \mathscr{W}_Y
  \;\;
  \simeq
  \;\;
  \Big(
    \big(pr_X^\ast \mathscr{V}\big)
    \,\times\,
    \big(pr_Y^\ast \mathscr{W}\big)
  \Big)_{X \times Y}
  \,,
\]
where we are denoting by
$$
  \array{
    && X \times Y
    \\
    &
    \mathllap{{}^{pr_X}}\swarrow
    &&
    \searrow\mathrlap{{}^{pr_Y}}
    \\
    X && && Y
  }
$$
the product [[projection]] maps in the base category.

A product of the form (eq:CartesianProductInGrothendieckConstruction) is known as an *[[external tensor product]]*, here the "external Cartesian product" on the fiber categories; see also [this Proposition](free+coproduct+completion#FreeDistributiveCategoryMonad) at *[[free coproduct completion]]*.
\end{example}

### As an oplax colimit
 {#AsALaxColimit}

The Grothendieck construction on $F : C \to Cat$ is equivalently the [[oplax colimit]] of $F$ (e.g [Gepner-Haugseng-Nikolaus 15](#GepnerHaugsengNikolaus15)).  That means that for each category $X$ there is an [[equivalence|equivalence of categories]]

$$
  Lax(F, \Delta X) \simeq [{\textstyle \int} F, X]
$$

that is natural in $X$, where $\Delta X$ is the constant functor with value $X$.  (See [[oplax colimit]] for an explanation of why *lax* natural transformations appear in the definition of an *oplax* colimit.)

A [[lax natural transformation]] $\alpha$ from $F$ to $\Delta X$ is given by

* for each object $c$ of $C$, a functor $\alpha_c \colon F c \to X$, and
* for each morphism $m \colon c \to d$ in $C$, a [[natural transformation]] $\alpha_m \colon \alpha_c \Rightarrow \alpha_d \circ m_*$ (writing $m_* = F m$),

such that $\alpha_{1_c}$ is the isomorphism $F 1_c \cong 1_{F c}$ given by [[pseudofunctor|pseudofunctoriality]] of $F$, and that if $m \colon c \to d$, $n \colon d \to e$ is a composable pair in $C$, then $\alpha_{n m}$ is equal to the obvious [[pasting]] of $\alpha_m$ and $\alpha_n$.

We want to show that to each such lax transformation there corresponds an essentially unique functor $\int F \to X$.  So firstly, given $\alpha$ as above, let $A$ be the functor that sends $x \in F c$ to $\alpha_c x$, and acts on arrows as

$$
  (m \colon c \to d, f \colon m_* x \to y)
  \quad \mapsto \quad
  \alpha_c x \overset{\alpha_m x}{\to}
  \alpha_d m_* x \overset{\alpha_d f}{\to} \alpha_d y
$$

That $A$ is a functor follows from the coherence properties of $\alpha$ with respect to identities and composition in $C$.

Conversely, if $A \colon \int F \to X$ is a functor, we get a lax transformation $\alpha$ as follows:

* For each $c \in C$, $\alpha_c$ is the restriction of $A$ to the category $F c$, which is the subcategory of $\int F$ whose objects are those of $F c$ and whose morphisms are those with first component an identity morphism.  This clearly makes $\alpha_c$ a functor.
* For each $m \colon c \to d$ in $C$, $\alpha_m$ has components $\alpha_c x \to \alpha_d m_* x$ given by $A$'s value at the morphism $(m,1_{m_* x})$.  This is a natural transformation because, if $k \colon x \to x'$ is a morphism in $F c$, then both sides of the naturality square are the value of $A$ at the morphism $(m, m_*k)$.

As one might expect, the coherence conditions on the resulting $\alpha$ follow from the functoriality of $A$.

It is then easy to check that these two mappings form a bijection between the objects of $Lax(F, \Delta X)$ and $[\int F, X]$.

As for the morphisms involved, the [[modifications]] between lax transformations and the [[natural transformations]] between functors, it is straightforward to show that these are in bijective correspondence too.  Hence we have shown that the above equivalence holds.

By inspecting the above proof, it is easy to see that the lax transformation associated to a functor $\int F \to X$ is a [[pseudonatural transformation]] if and only if the functor inverts (i.e. sends to an isomorphism) each member of the class $S$ of morphisms of $\int F$ whose second component is an identity.  (These are in fact the [[cartesian morphism|opcartesian]] morphisms with respect to the projection $\int F \to C$.)  The [[localization]] $\int F[S^{-1}]$ is therefore the (weak) [[2-colimit]] of $F$:

$$
  Ps(F, \Delta X) \simeq [{\textstyle \int} F, X]_{S^{-1}}
   \simeq [{\textstyle \int} F[S^{-1}], X]
$$

This last result appears in [[SGA4]] Exposé VI, Section 6.

### Local smallness

In general, a [[weighted colimit|(weighted)]] colimit of a [[large category|large]] diagram of [[locally small categories]] need no longer be locally small.  However, in the case of the oplax colimit, i.e. the Grothendieck construction, we have:

+-- {: .un_theorem}
###### Theorem
If $F:C^{op}\to Cat$ is a pseudofunctor, $C$ is locally small, and each category $F(c)$ is locally small, then the Grothendieck construction $\int F$ is also locally small.
=--
+-- {: .proof}
###### Proof
Recall that the morphisms in $\int F$ from $(c,a)$ to $(c',a')$ are pairs $(c \overset{f}{\to} c', a \overset{\alpha}{\to} F(f)(a'))$.  Local smallness of $C$ means that there is only a set of such $f$'s, and local smallness of $F(c)$ means that for each $f$ there is only a set of such $\alpha$'s.
=--

For example, consider the [[canonical indexing]] of a locally small category $A$, i.e. the pseudofunctor $Set^{op}\to Cat$ sending each set $X$ to the category $A^X$.  This satisfies the conditions of the above theorem, so its Grothendieck construction, which is the category of [[families]] of objects of $A$, is locally small.

### The equivalence between fibrations and pseudofunctors
 {#EquivalenceBetweenFiberedCategoriesAndPsuedofunctors}

One can characterize the _image_ of the Grothendieck construction
as consisting precisely of those objects in $Cat/C$ that are
[[Grothendieck fibration]]s.

We recall the definition of the [[bicategory]] of Grothendieck fibrations
and [[pseudofunctor]]s and then state the main equivalence theorem.

#### The bicategory of pseudofunctors

A [[pseudofunctor]] from a 1-[[category]] $C$ to a [[2-category]] ([[bicategory]]) $A$ is nothing but a (non-strict) [[2-functor]] between bicategories, with the ordinary category regarded as a special bicategory.

We write $[C^{op}, A]$ for the [[2-functor]] 2-category from the
[[opposite category]] of $C$ to $A$ (the $op$ here is just convention):

* objects are pseudofunctors $F : C^{op} \to A$;

* morphisms are [[lax natural transformation|pseudonatural transformations]];

* 2-morphism are [[modification]]s.

#### The bicategory of fibrations

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

A [[functor]] $p : E \to C$ is a **[[Grothendieck fibration]]**
if for every object $e \in E$ and every morphism $f : c \to p(e)$
in $C$ there is a morphism $\hat f : \hat c \to e$ in $E$ that lifts
$f$ in that $p(\hat f) = f$ and which is a [[Cartesian morphism]].

A **morphism of Grothendieck fibrations $F : (p : E \to C) \to (p' : E' \to C)$ is

* a functor $F : E \to E'$

* such that

  * $F$ sends [[Cartesian morphism]]s to Cartesian morphisms;

  * the diagram

    $$
      \array{
        E &&\stackrel{F}{\to}&& E'
        \\
        & {}_{\mathllap{p}}\searrow && \swarrow_{\mathrlap{p'}}
        \\
        && C
      }
    $$

    in [[Cat]] commutes (strictly).

* a **2-morphism** between morphism $\eta : F \to F'$ is a
  [[natural transformation]] of the underlying functors, that also makes
  the obvious diagram 2-commute, i.e. such that
  $p' \cdot \eta$ is trivial.

Compositions are those induced from the underlying functors and natural transformations.

This defines the **[[2-category]] of Grothendieck fibrations**

$$
  Fib(C) \hookrightarrow Cat/C
$$

over $C$, being a 2-[[subcategory]] of the [[overcategory]] of [[Cat]] over $C$.

=--

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

Cartesian lifts are not required to be unique, but are automatically unique up to a unique vertical [[isomorphism]] connecting their domains.

=--

#### Statement of the equivalence

\begin{proposition}

The Grothendieck construction factors through [[Grothendieck fibrations]] over $C$

$$
  \textstyle{\int}
  \;\colon\;
  [C^{op}, Cat]
    \longrightarrow
  Fib(C)
   \hookrightarrow
  Cat/C
$$

and establishes an [[equivalence of bicategories]]

$$
  \textstyle{\int}
  \;\colon\;
  [C^{op}, Cat]
    \overset{\simeq}{\longrightarrow}
  Fib(C)
  \,.
$$

In fact, it is more than that: it is an equivalence of [[strict 2-categories]], in the sense of strict 2-category theory, i.e. an equivalence of $Cat$-[[enriched categories]].

When restricted to pseudofunctors that factor through [[Grpd]] $\hookrightarrow Cat$ it factors through [[fibrations in groupoids]]

$$
  \textstyle{\int}
  \;\colon\;
  [C^{op}, Grpd]
    \longrightarrow
  Fib_{Grpd}(C)
    \hookrightarrow
  Cat/C
$$

and establishes a similar equivalence

$$
  [C^{op}, Grpd] \simeq Fib_{Grpd}(C)
  \,.
$$

\end{proposition}

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

This can be verified by straightforward albeit somewhat tedious checking. Details are spelled out in [Johnstone (2002), B1.3](#Johnstone). (The statement itself is theorem B1.3.6 there, all definitions and lemmas are on the pages before that.)

=--

#### Model category version

There is refinement of the Grothendieck construction to [[model categories]].

See at _[[Grothendieck construction for model categories]]_.

This model category incarnation of the Grothendieck construction
generalizes to a model category presentation of the
[[(∞,1)-Grothendieck construction]].

### Adjoints to the Grothendieck construction
  {#adjunction}

The Grothendieck construction functor

$$
  \textstyle{\int}
  \;\colon\;
  [C^{op}, Cat]
    \longrightarrow
  Cat/C
$$

has a [[left adjoint|left]] and a [[right adjoint|right]] [[adjoint functor]].

Restricted to [[groupoid]]-valued functors and [[fibrations in groupoids]], both of these exhibit the above equivalences as [[adjoint equivalence]]s. The intuition for this is that all of the categorical structure of a groupoid is contained in the automorphism groups, so one does not need to look at functors $[C^{op}, GpdProf]$ in order to get all of the objects of $Gpd/C$.

Notice that much of the traditional literature discusses (just) the right adjoint.

#### The left adjoint

The [[left adjoint]] is the functor

$$
  L : (p : E \to C) \mapsto ( (-)/p : C^{op} \to Cat)
$$

that assigns to a functor $p$ the presheaf which sends $c \in C$ to the [[comma category]] $c/p$ with objects given by pairs $(e, c \to p(e))$ and morphisms by commutative triangles

$$
  \array{
      && c &&
      \\
      & \swarrow && \searrow &
      \\
      p(e_1) &&\to&& p(e_2)
  }
$$

i.e.

$$
  L(E \stackrel{p}{\to}C) : c \mapsto c/p
  \,.
$$

This functor may equivalently be expressed as follows.

##### In terms of a cone construction {#Cone}

For given $(E \stackrel{p}{\to} C)$ consider the [[(infinity,1)-limit|(3,1)-pushout]]

$$
  \array{
    E &\hookrightarrow& E^{\triangleright}
    \\
    \downarrow^{\mathrlap{p}} &\swArrow& \downarrow
    \\
    C &\to& K(p)
  }
$$

of _[[(2,1)-categories]]_ , where $K^{\triangleright}$ is $K$ with one [[terminal object]] $v$ adjoined (a [[join of quasi-categories|join]] of categories). (Here $E$, $C$ and $E^{\triangleright}$ are 1-catgeories regarded trivially as $(2,1)$-categories and where $K(p)$ will in general be a [[(2,1)-category]] with nontrivial [[2-morphism]]s).

\begin{proposition}
We have

$$
  c_{/p}
    \,\simeq\,
  Hom_{K(p)}(c,v)
  \,.
$$

And hence the [[left adjoint]] to the Grothendieck construction may be realized as the assignment that sends $p \colon E \to C$ to the pseudofunctor

$$
  L(p) := Hom_{K(p)}(-, v) : C^{op} \to Cat
  \,.
$$
\end{proposition}

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

It is convenient to compute the weak pushout by embedding the situation from [[Cat]] into the bigger context of [[(∞,1)-categories]] and using the [[model category|model]] of that provided by [[sSet]]: the [[model structure for quasi-categories]]. This also facilitates the generalization of the argument from 1-categories to higher categories.

So consider equivalently the weak pushout diagram

$$
  \array{
    N(E) &\hookrightarrow& N(E)^{\triangleright}
    \\
    \downarrow^{\mathrlap{N(p)}} &\swArrow& \downarrow
    \\
    N(C) &\to& N(K(p))
  }
$$

of [[quasi-categories]], where $N(-)$ is the [[nerve]] operation and where $N(E)^{\triangleright} = N(E) \star *$ is the [[join of simplicial sets]] of $N(E)$ with the [[point]].

By the general yoga of [[homotopy colimit]]s (see there for details) we know that this $\infty$-pushout here may be computed as an ordinary [[pushout]] in the 1-category [[sSet]] if the pushout diagram $N(C) \leftarrow N(E) \to N(E)^{\triangleright} $ has the property that

* all three objects are cofibrant;

* at least one of the two morphisms is a cofibration

in the [[model structure for quasi-categories]] $sSet_{Joyal}$.

But this is trivially verified since the cofibrations in $sSet_{Joyal}$ are just the [[monomorphism]]s in [[sSet]]: the degreewise injective maps of simplicial sets. So every object in $sSet_{Joyal}$ is cofibrant and the inclusion $N(E) \hookrightarrow N(E)^{\triangleright}$ is a cofibration.

(The same conclusion would hold for the same simple reasons in the standard [[model structure on simplicial sets]] $sSet_{Quillen}$.)

From this it follows that simply because we passed from categories to their [[nerve]]s, the computation of the weak pushout reduces to the computation of an ordinary pushout (one may think of passing to nerves as providing a cofibrant replacement: since in the nerve all composition of [[k-morphism]]s is "freed", the nerve is a suitably "puffed up" version of a category that is suitable for computing $\infty$-pushouts).

So we are reduced to computing the ordinary [[pushout]]

$$
  \array{
    N(E) &\hookrightarrow& N(E)^{\triangleright}
    \\
    \downarrow^{\mathrlap{N(p)}} && \downarrow
    \\
    N(C) &\to& Q
  }
$$

in [[sSet]]. The fibrant replacement of $Q$ is then the [[nerve]] of [[generalized the|the]] [[bicategory]] $K(p)$ that we are after.

As recalled at [[limits and colimits by example]] in the section [limits in presheaf categories](http://ncatlab.org/nlab/show/limits+and+colimits+by+example#limitsinpresheafcat), [[colimit]]s (and hence pushouts) in the [[presheaf]]-category [[sSet]] $= Func(\Delta^{op}, Set)$ are computed for each object $[n] \in \Delta$ as ordinary colimits in [[Set]].

For **$n=0$** we see that $Q_0$ is the collection of objects of $C$ and one additional vertex $v$:

$$
 Q_0 = N(C)_0  \coprod \{ v\} = p(Obj(E)) \coprod \{v \}
$$

For **$n=1$**  similarly we find that $Q_1$ consists of the 1-cells in in $C$ and in addition of one 1-cell $e : c \to v$ for each $e \in Obj(E)$ with $p(e) = c$ (this 1-cell is really the terminal 1-cell $e \to v$ in $E^{\triangleright}$ but with its source re-interpreted as being $p(e) = c$ according to the identification of $Q_0$ as above). In the fibrant replacement of $Q$ the composite of original 1-cells $c_1 \to c_2$ and the new 1-cells $e : c_2 \to v$ will be freely added, so that the general 1-morphism $c_1 \to v$ will consist of a 1-morphism $c_1 \to c_2$ in $C$ together with a lift of $c_2$ to $E$. This is just as in the [[comma category]] $c/p$.

For **$n=2$** we have in $Q_2$ the 2-cells in $C$ as well as one 2-cell

$$
  \array{
    c_1 &&\to&& c_2
    \\
    & \searrow &{}^{(e_1 \to e_2)}\swArrow& \swarrow
    \\
    && v
  }
$$

for each 1-cell $(e_1 \to e_2)$ in $N(E)$ with $p(e_1 \to e_2)$ = $(c_1 \to c_2)$.

In particular this means that if $e_2: c_2 \to v$ is a morphism in $Q$ and $c_1 \to c_2$ is a morphism in $C$, then the composite $c_1 \to c_2 \to v$ in $Q$ is homotopic to any compatible direct morphism $c_1 \to v$ in $Q$.

This means that forming the fibrant replacement of $Q$ in $sSet_{Joyal}$ will not throw in superfluous 1-morphisms on top of those we already discussed in the previous paragraph...

Now furthermore...

=--

This formulation of the Grothendieck construction as part of an [[adjoint functors|adjunction]]

$$
  (L \dashv \textstyle{\int})
  \;\colon\;
  Fib(C)
  \rightleftarrows
  [C^{op}, Cat]
$$

with the [[left adjoint]] given by hom-objects in a pushout object as above is the starting point for its [[vertical categorification]] described at *[[(∞,1)-Grothendieck construction]]*.

#### The right adjoint

We also have an adjunction
$$
  (R \vdash \textstyle{\int})
  \;\colon\;
  Fib(C)
  \rightleftarrows
  [C^{op}, Cat]
  \,,
$$
where the [[right adjoint]] $R$ sends a [[Grothendieck fibration]] $F$ over $C$
to the [[presheaf]]
$$
  c
   \;\mapsto\;
  Hom(\textstyle{\int} y(c), F)
  \,,
$$
where $\int y(c)$ is the Grothendieck construction applied to the
representable presheaf of sets (hence [[discrete categories]]) on $c$
and $Hom$ denotes the category of morphisms between two Grothendieck fibrations.

Informally, an object of $R(F)(c)$ is given by a choice of a pullback
along each morphism with codomain $c$,
and these pullbacks must be functorial.

### Behaviour under simplicial nerve

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

For $F \colon \mathcal{D} \to Cat$ a [[functor]], let

$$
  {\vert N(F(-))\vert}
   \;\colon\;
  \mathcal{D}
    \overset{F}{\longrightarrow}
  Cat
   \stackrel{\vert N(-) \vert}{\to}
  Top
$$

be its postcomposition with [[geometric realization of categories]]

Then we have a [[weak homotopy equivalence]]

$$
  {\left\vert N\left(\textstyle{\int} F \right) \right\vert}
    \simeq
  hocolim {\vert N(F(-)) \vert}
$$

exhibiting the [[homotopy colimit]] in [[Top]] over $\vert N(F (-)) \vert$ as the geometric realization of the [[Grothendieck construction]] $\int F$ of $F$.

=--

This is due to ([Thomason 79](#Thomason79)).

### Further properties

\begin{proposition}
\label{PrecompositionOfPseudofunctorWithAnAdjunction}
Given a [[contravariant functor|contravariant]] [[pseudofunctor]]
$$
  \array{
    \mathbf{D}^{op}
    &\overset{C_{(-)}}{\longrightarrow}&
    Cat
    \\
    \mathbf{x}
      &\mapsto&
    C_{\mathbf{x}}
    \\
    \Big\downarrow\mathrlap{{}^{\mathbf{f}}}
    &&
    \Big\uparrow\mathrlap{{}^{\mathbf{f}^\ast}}
    \\
    \mathbf{x}'
      &\mapsto&
    C_{\mathbf{x}'}
  }
$$
and a pair of [[adjoint functors]] of the form
$$
  \mathcal{D}
  \underoverset
    {\underset{R}{\longleftarrow}}
    {\overset{L}{\longrightarrow}}
    {\;\; \bot \;\;}
  \mathbf{D}
$$
then there is an induced [[adjoint functor|adjunction]] between the [[Grothendieck constructions]] on $C_{(-)}$ and on $C_{L(-)}$ covering the given adjunction
$$
\array{
  \Bigg\{
    \mathscr{V}_{x}
      \xrightarrow{ \phi_{f} }
    \mathscr{V}'_{x'}
    \;\displaystyle{\Bigg\vert}\;
    \array{
      \mathscr{V}
        &\xrightarrow{ \phi }&
      L(f)^\ast \mathscr{V}'
      & \in C_{L(x)}
      \\
      x
        &\xrightarrow{ f }&
      x'
      & \in \mathcal{D}
    }
  \Bigg\}
  \,\equiv
  &
  \Big(
    \underset
      {x \in \mathcal{D}}
      {\textstyle{\int}}
    C_{L(x)}
  \Big)
  &
  \underoverset
    {\underset{\widehat R}{\longleftarrow}}
    {\overset{\widehat L}{\longrightarrow}}
    {\;\; \bot \;\;}
  &
  \Big(
    \underset
      {\mathbf{x} \in \mathbf{D}}
      {\textstyle{\int}}
    C_{\mathbf{x}}
  \Big)
  &
  \equiv
  \,
  \Bigg\{
    \mathscr{V}_{\mathbf{x}}
      \xrightarrow{ \phi_{\mathbf{f}} }
    \mathscr{V}'_{\mathbf{x}'}
    \;\displaystyle{\Bigg\vert}\;
    \array{
      \mathscr{V}
        &\xrightarrow{ \phi }&
      \mathbf{f}^\ast \mathscr{V}'
      & \in C_{\mathbf{x}}
      \\
      \mathbf{x}
        &\xrightarrow{ \mathbf{f} }&
      \mathbf{x}'
      & \in \mathbf{D}
    }
  \Bigg\}
  \\
  &
  \Big\downarrow && \Big\downarrow
  \\
  &
  \mathcal{D}
  &
  \underoverset
    {\underset{R}{\longleftarrow}}
    {\overset{L}{\longrightarrow}}
    {\;\; \bot \;\;}
  &
  \mathbf{D}
 }
$$
where $\widehat L$ acts as $L$ on [[underlying]] morphisms and as the [[identity functor|identity]] on components:
$$
  \array{
  \mathscr{V}_{x}
  \\
  \Big\downarrow\mathrlap{{}^{\phi_{f}}}
  \\
  \mathscr{V}'_{x'}
  }
  \;\;\;\;\;\;\;
  \overset{\widehat L}{\mapsto}
  \;\;\;\;\;
  \array{
  \mathscr{V}_{L(x)}
  \\
  \Big\downarrow\mathrlap{{}^{\phi_{L(f)}}}
  \\
  \mathscr{V}'_{L(x')}
  \mathrlap{\,,}
  }
$$
while $\widehat R$ acts as $R$ on [[underlying]] morphisms and on components by [[base change]] along the [[adjunction unit]] $\epsilon_{\mathbf{x}} \colon L \circ R(\mathbf{x}) \to \mathbf{x}$:
$$
  \array{
  \mathscr{V}_{\mathbf{x}}
  \\
  \Big\downarrow\mathrlap{{}^{\phi_{\mathbf{f}}}}
  \\
  \mathscr{V}'_{\mathbf{x}'}
  }
  \;\;\;\;\;\;\;
  \overset{\widehat R}{\mapsto}
  \;\;\;\;\;
  \array{
    \big(\epsilon_{\mathbf{x}}^\ast
      \mathscr{V}\big)_{R(\mathbf{x})}
    \\
    \Big\downarrow
    \mathrlap{{}^{ (\epsilon_\mathbf{x}^\ast \phi)_{ R(\mathbf{f}) } }}
    \\
    \big(\epsilon_{\mathbf{x}'}^\ast \mathscr{V}\big)_{ R(\mathbf{x}')}
  }
$$
and the [[adjunction counit]] is the [[identity morphism]] on components covering the [[underlying]] [[adjunction counit]]:
\[
  \label{AdjunctionCounitCoveringUnderlyingAdjunctionCounit}
  \epsilon^{
    \widehat{L} \dashv \widehat{R}
  }_{
    \mathscr{V}_{\mathbf{x}}
  }
  \;\colon\;
  \widehat{L}
  \widehat{R}
  \big(\mathscr{V}_{\mathbf{x}}\big)
  =
  \Big(
    \big(\epsilon^{L \dashv R}_{\mathbf{x}}\big)^\ast
    \mathscr{V}
  \Big)_{ L R(\mathbf{x}) }
  \overset{
    (id)_{\epsilon^{L \dashv R}_{\mathbf{x}}}
  }{\longrightarrow}
  \mathscr{V}_{\mathbf{x}}
\]
\end{proposition}
\begin{proof}
First notice that $\widehat R$ is indeed well-defined in that we have a [[natural isomorphism]] on the right of
$$
  \epsilon_{\mathbf{x}}^\ast \mathscr{V}
  \xrightarrow{ \epsilon_{\mathbf{x}}^\ast \phi }
  \epsilon_{\mathbf{x}}^\ast \mathbf{f}^\ast \mathscr{V}'
  \simeq
  \big(L R(\mathbf{f})\big)^\ast
  \epsilon_{\mathbf{x}'}^\ast \mathscr{V}'
$$
given by the [[contravariant functor|contravariant]] [[pseudo-functor|pseudo-functoriality]] of $C_{(-)}$ applied to the [[commuting square|commutativity]] of the [[naturality square]] of the [[adjunction counit]]:
$$
  \array{
    L R(\mathbf{x})
    &\overset{ L R(\mathbf{f}) }{\longrightarrow}&
    L R(\mathbf{x}')
    \\
    \mathllap{{}^{ \epsilon_{\mathbf{x}} }}
    \Big\downarrow
    &&
    \Big\downarrow
    \mathrlap{{}^{ \epsilon_{\mathbf{x}'} }}
    \\
    \mathbf{x}
      &\underset{\;\; \mathbf{f} \;\;}{\longrightarrow}&
    \mathbf{x}'
    \mathrlap{\,.}
  }
$$

Now
if we write $f \colon x \to R(\mathbf{x}')$ for the $(L \dashv R)$-[[adjunct]] of a given $\mathbf{f} \colon L(x) \to \mathbf{x}'$ then we have natural bijections
$$
\begin{array}{ll}
  \Big\{
    \widehat L\big( \mathscr{V}_x \big)
    \xrightarrow{ \phi_{\mathbf{f}} }
    \mathscr{V}'_{\mathbf{x}'}
  \Big\}
  \\
  \;\simeq\;
  \Big\{
    \mathscr{V}_{L(x)}
    \xrightarrow{ \phi_{\mathbf{f}} }
    \mathscr{V}'_{\mathbf{x}'}
  \Big\}
  &
  \text{by def.}
  \\
  \;\simeq\;
  \Big\{
    \mathscr{V}_{x}
    \xrightarrow{ \phi_{f} }
    \big(
      \epsilon^\ast_{\mathbf{x}'}
      \mathscr{V}'
    \big)_{R(\mathbf{x}')}
  \Big\}
  &
  \text{see below}
  \\
  \;\simeq\;
  \Big\{
    \mathscr{V}_{x}
    \xrightarrow{ \phi_{f} }
    \widehat R \big(
      \mathscr{V}'_{\mathbf{x}'}
    \big)
  \Big\}
  &
  \text{by def.,}
\end{array}
$$
where the one step that is not a definition is on [[underlying]] morphisms the $(L \dashv R)$-[hom-isomorphism](adjoint+functor#eq:HomIsomorphismForAdjointFunctors)
$$
\begin{array}{ll}
  \Big\{
    L(x) \xrightarrow{ \mathbf{f} } \mathbf{x}'
  \Big\}
  \;\simeq\;
  \Big\{
    x \xrightarrow{ f } R(\mathbf{x}')
  \Big\}
\end{array}
$$
and on components the following natural bijection
$$
\begin{array}{ll}
  \Big\{
    \mathscr{V}
    \xrightarrow{ \phi }
    \mathbf{f}^\ast \mathscr{V}'
  \Big\}
  \\
  \;\simeq\;
  \Big\{
    \mathscr{V}
    \xrightarrow{ \phi }
    \big( \epsilon_{\mathbf{x}'} \,\circ\, L(f) \big)^\ast
    \mathscr{V}'
  \Big\}
  &
  \text{ formula for adjuncts }
  \\
  \;\simeq\;
  \Big\{
    \mathscr{V}
    \xrightarrow{ \phi }
    L(f)^\ast
    \big(
    \epsilon_{\mathbf{x}'}^\ast
    \mathscr{V}'
    \big)
  \Big\}
  &
  \text{ pseudo-functoriality, }
\end{array}
$$
where the first step uses the general formula $\mathbf{f} \,=\, \epsilon_{\mathbf{x}'} \circ L(f)$ ([here](adjoint+functor#GeneralAdjunctsInTermsOfAdjunctionUnitCounit)) that expresses [[adjuncts]] in terms of the [[counit of an adjunction|counit]] and the second step is the [[contravariant functor|contravariant]] [[pseudo-functor|pseudo-functoriality]] of $C_{(-)}$.

This establishes a [hom-isomorphism](adjoint+functor#eq:HomIsomorphismForAdjointFunctors) exhibiting adjoint functors $\widehat L \dashv \widehat R$. Moreover, the image of an [[identity morphism]] on $\widehat{R}(-)$ under this hom-isomorphism is the claimed counit (eq:AdjunctionCounitCoveringUnderlyingAdjunctionCounit).
\end{proof}

## Generalizations

### $n = 0$

The analog of the Grothendieck construction one categorical dimension down is the [[category of elements]] of a [[presheaf]].

### $n = (\infty,0)$
 {#ForInfinityGroupoids}

The analog of the Grothendieck construction for [[∞-groupoids]] is examined in detail in [Heuts-Moerdijk 13](#HeutsMoerdijk13).

The category of presheaves in groupoids is replaced by the [[model category of simplicial presheaves]] equipped with the [[projective model structure]]
and the category of Grothendieck fibrations in groupoids is replaced by the model category of simplicial sets over the [[nerve of a category|nerve]] of the source category, equipped with the [[contravariant model structure]].

In this case there is not one, but two different functors that generalize the Grothendieck construction.

The first functor $h_!$ is a [[left adjoint]], it implements the [[homotopy colimit]] using the [[diagonal of a bisimplicial set]],
and the second functor $r^*$ is a [[right adjoint]], it uses the [[codiagonal]] (also known as the [[totalization]]) of a bisimplicial set.
Both functors fit into [[adjunctions]] $h_!\dashv h^*$ and $r_!\dashv r^*$,
where the other two adjoints can be seen as rectification functors:
the right adjoint $h^*$ generalizes the cleavage construction,
whereas the left adjoint $r_!$ generalizes the comma category construction above.

The two functors $h_!$ and $r^*$ become naturally weakly equivalent once we [[derived functor|derive]] them, but they are not isomorphic.
The functor $r^*$ restricted to the full subcategory of presheaves of groupoids recovers the nerve of the classical Grothendieck construction described above.
The functor $h_!$ restricted to the same full subcategory
does not even land in quasicategories, so it doesn't give
rise to a new construction in the classical case.

### $n = (\infty,1)$  {#(oo1)case}

The analog of the Grothendieck construction for [[(infinity,1)-category|(∞,1)-categories]] is described at [[Cartesian fibration]] and at [[universal fibration of (∞,1)-categories]].

The correspondence between $(\infty,1)$-categorical [[cartesian fibrations]] $E \to C$ and [[(infinity,1)-presheaf|(∞,1)-presheaves]] $C \to (\infty,1)Cat^{op}$ is [[model category|modeled]] by the [[Quillen equivalence]] between the [[model structure on marked simplicial over-sets]] and the projective [[global model structure on simplicial presheaves]].

For more details see

* [[(∞,1)-Grothendieck construction]]

### Normal lax functors into $Prof$

The Grothendieck construction can be generalized from pseudofunctors into $Cat$ to normal lax functors into [[Prof]].  Instead of fibrations over $C$, such normal lax functors correspond to arbitrary functors into $C$.  See [[displayed category]] for more.

## Warning on terminology ##

The term 'Grothendieck Construction' is applied in the literature to at least two very different constructions (and as [[Alexander Grothendieck|Grothendieck]] introduced so many new ideas and constructions to mathematics, perhaps there are others!). One concerns the construction of a [[fibered category]] from a [[pseudofunctor]] and will be treated in more detail in the entry on [[Grothendieck fibration]].  The other refers to constructing the [[Grothendieck group]] is  in the context of [[K-theory]] from isomorphism classes of vector bundles on a space by the introduction of formal inverses, 'virtual bundles'.  This constructs an Abelian group from the semi-group of isomorphism classes.

## Examples

\begin{example}
**(representable functor and slice categories)**
\linebreak
A [[representable functor]] $C(-,X) \colon  C^{op} \to Set \hookrightarrow Cat$ maps under the Grothendieck construction to the [[slice category]] $C/X$. The corresponding fibrations $C/X \to C$ are also called *[[representable fibered categories]]*.
\end{example}

\begin{example}\label{SliceCategoriesAndArrowCategory}
**(slice categories and arrow category)**
\linebreak
For $\mathcal{S}$ any category, let
$$
  \mathcal{S}_{(-)}
  \,\colon\,
  \mathcal{S}
  \longrightarrow
  Cat
$$
be the [[pseudofunctor]] which sends

* an [[object]] $B \,\in\, \mathcal{S}$ to the [[slice category]] $\mathcal{S}_{/B}$,

* a [[morphism]] $f \colon B \to B'$ to the left [[base change]] [[functor]] $f_! \,\colon\, \mathcal{C}_{B} \to \mathcal{C}_{/B'}$ given by post-[[composition]] in $\mathcal{C}$.

The [[Grothendieck construction]] on this functor is the [[arrow category]] $\mathcal{S}^{I}$ of $\mathcal{S}$:

$$
  \mathcal{S}^{I}
  \;\;\;
    \simeq
  \;\;\;
  \textstyle{\int}_{B \in \mathcal{S}}
  \mathcal{S}_{/B}
  \mathrlap{\,.}
$$

This follows readily by unwinding the definitions.
In the refinement to the [[Grothendieck construction for model categories]] (here: [[slice model categories]] and [[model structures on functors]]) this equivalence is also considered for instance in [Harpaz & Prasma (2015), above Cor. 6.1.2](Grothendieck+construction+for+model+categories#HarpazPrasma15).

The correponding [[Grothendieck fibration]] is also known as the *[[codomain fibration]]*, a priori an [[opfibration]].

But if $\mathcal{S}$ has all [[pullbacks]], then there is also the [[contravariant functor|contravariant]] [[pseudofunctor]]
$$
  \mathcal{S}_{(-)}
  \,\colon\,
  \mathcal{S}^{op}
  \longrightarrow
  Cat
$$
which sends

* an [[object]] $B \,\in\, \mathcal{S}$ to the [[slice category]] $\mathcal{S}_{/B}$,

* a [[morphism]] $f \colon B \to B'$ to the [[base change]] [[functor]] $f^\ast \,\colon\, \mathcal{C}_{B'} \to \mathcal{C}_{/B}$ given by [[pullback]] in $\mathcal{C}$.

The corresponding (contravariant) Grothendieck construction is still the [[arrow category]] of $\mathcal{S}$, but now exhibited as a [[Grothendieck fibration]] (instead of or rather: in addition to being an [[opfibration]]) over $\mathcal{S}$. This is often the default meaning of the term *[[codomain fibration]]*.
\end{example}

In slight variation of Exp. \ref{SliceCategoriesAndArrowCategory}:

\begin{example}\label{PointedSliceCategoriesAndRetractiveSpaces}
**(pointed slice categories and retractive spaces)**
\linebreak
Let $\mathcal{S}$ be a category with all [[pushouts]] and consider the [[pseudofunctor]]

$$
  \big(\mathcal{S}_{(-)}\big)^{\ast/}
  \,\colon\,
  \mathcal{S}
  \longrightarrow
  Cat
$$
which sends

* an [[object]] $B \,\in\, \mathcal{S}$ to the category of [[pointed objects]] in the [[slice category]] $\mathcal{S}_{/B}$,

* a [[morphism]] $f \colon B \to B'$ to the [[functor]] $f_! \;\colon\; \big(\mathcal{S}_{/B}\big)^{\ast/} \to \big(\mathcal{S}_{/B'}\big)^{\ast/}$ which forms the [[pushout]] of [[retraction]] [[diagrams]]:

\begin{tikzcd}[
    row sep=30pt
  ]
    B
    \ar[d, "{i}"{description}]
    \ar[rr, "{f}"{description}]
    \ar[
      drr,
      phantom,
      "{
        \scalebox{.6}{
          (po)
        }
      }"{pos=.7}
    ]
    &&
    B'
    \ar[d, "{ f_!(i) }"{description}]
    \\
    E
    \ar[d, "{p}"{description}]
    \ar[rr, "{\widehat f}"{description}]
    \ar[
      drr,
      phantom,
      "{
        \scalebox{.6}{
          (po)
        }
      }"{pos=.7}
    ]
    &&
    f_! E
    \ar[d, "{f_!(p)}"{description}]
    \\
    B
    \ar[
      rr,
      "{f}"{description}
    ]
    &&
    B'
\end{tikzcd}

The corresponding [[Grothendieck construction]] is also known (at least when $\mathcal{S}$ is regarded as a category of "spaces") as the category $\mathcal{S}_{\mathcal{R}}$ of "[[retractive spaces]]" in $\mathcal{S}$:

$$
  \mathcal{S}_{\mathcal{R}}
  \;\;\;
    \simeq
  \;\;\;
  \int_{B \in \mathcal{S}}
  \big(
    \mathcal{S}_{/\mathcal{B}}
  \big)^{\ast/}
  \,.
$$

This follows readily from the definitions, but see also [Braunack-Mayer (2021), Rem. 1.15](retractive+space#BraunackMayer19); [Hebestreit, Sagave & Schlichtkrull (2020), Lem. 2.14](retractive+space#HebestreitSagaveSchlichtkrull20), where this is the basis of a [[model structure for parameterized spectra|model category-presentation]] of the [[tangent (infinity,1)-category|tangent $\infty$-category]] of (the [[simplicial localization]] of) $\mathcal{S}$.
\end{example}

In alternative slight variation of Exp. \ref{SliceCategoriesAndArrowCategory}:

\begin{example}\label{AbelianizedSlicesAndTangentCategory}
**(abelianized slice categories and tangent category)**
For $\mathcal{S}$ a category with [[finite limits]], let

$$
  Ab\big(
    \mathcal{S}_{/(-)}
  \big)
  \;\;
  \colon
  \;\;
  \mathcal{S}^{op}
  \longrightarrow
  Cat
$$
be the [[contravariant functor|contravariant]] [[pseudofunctor]] which sends

* any [[object]] $B \in \mathcal{S}$ to the category $Ab\big(\mathcal{S}_{/B}\big)$ of [[abelian group|abelian]] [[group objects]] [[internalization|internal]] to the [[slice category]] $\mathcal{S}_{/B}$

* any [[morphism]] $f \colon B \to B'$ to the [[base change]] [[functor]] $f^\ast \colon Ab\big(\mathcal{S}_{/B'}\big) \to Ab\big(\mathcal{S}_{/B}\big)$ given by [[pullback]] in $\mathcal{C}$ (which [[preserved limit|preserves]] [[group objects]]).

The [[Grothendieck construction]] on this functor may be called the *[[tangent category]]* of $\mathcal{S}$.
\end{example}

## Related concepts

* [[category of elements]]

* [[two-sided fibration]]

* [[indexed Grothendieck construction]]

* [[2-Grothendieck construction]]

* [[(infinity,1)-Grothendieck construction]]

* [[Grothendieck construction for monoidal categories]]

## References

The Grothendieck construction originates in:

* [[Alexander Grothendieck]], §VI.8 of: _Revêtements Étales 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)]

Review:

* [[Angelo Vistoli]], §3.1.3 in: *Grothendieck topologies, fibered categories and descent theory*, in: *[[Fundamental algebraic geometry -- Grothendieck's FGA explained]]*, Mathematical Surveys and Monographs **123**, Amer. Math. Soc. (2005) 1-104 [[ISBN:978-0-8218-4245-4](https://bookstore.ams.org/surv-123-s), [math.AG/0412512](http://arxiv.org/abs/math/0412512)]

Further textbook accounts:

* [[Francis Borceux]], Section 8.3 of: *[[Handbook of Categorical Algebra]]*, Vol. 2: *Categories and Structures* Encyclopedia of Mathematics and its Applications **50**, Cambridge University Press (1994) ([doi:10.1017/CBO9780511525865](https://doi.org/10.1017/CBO9780511525865))

* {#Johnstone} [[Peter Johnstone]], sections A1.1.7, B1.3.1 of:  _[[Sketches of an Elephant]]_ (2002)

* {#JohnsonYau20} [[Niles Johnson]], [[Donald Yau]], Chapter 10 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))

Survey in the generality of [[enriched category theory|enriched-]], [[internal category|internal-]] and [[(infinity,1)-category theory|$\infty$-category theory]] (see also the *[[enriched Grothendieck construction|enriched-]] and [[(infinity,1)-Grothendieck construction|$\infty$-Grothendieck construction]]):

* [[Liang Ze Wong]], *The Grothendieck Construction in Enriched, Internal and ∞-Category Theory*, PhD thesis, Univ. Washington (2019) [[pdf](http://sheaves.github.io/slides/thesis.pdf), [[Wong-GrothendieckConstruction.pdf:file]]]

The [[geometric realization of categories|geometric realization]] of Grothendieck constructions has been analyzed in

* {#Thomason79} [[R. W. Thomason]], _Homotopy colimits in the category of small categories_ , Math. Proc. Cambridge Philos. Soc. 85 (1979), no. 1, 91109.

The left adjoint to the Grothendieck construction is discussed in §3.1.1 of

* [[Georges Maltsiniotis]], _La théorie de l'homotopie de Grothendieck_

On limits and colimits in Grothendieck constructions:

* {#TBG91} [[Andrzej Tarlecki]], [[Rod M. Burstall]], [[Joseph A. Goguen]], *Some fundamental algebraic tools for the semantics of computation: Part 3. Indexed categories*, Theoretical Computer Science **91** 2 (1991) 239-264 [doi:10.1016/0304-3975(91)90085-G]

The analog for simplicial sets instead of groupoids is discussed in

* {#HeutsMoerdijk13}  [[Gijs Heuts]], [[Ieke Moerdijk]], _Left fibrations and homotopy colimits_ ([arXiv:1308.0704](https://arxiv.org/abs/1308.0704)).

See also

* [Grothendieck construction](http://en.wikipedia.org/wiki/Grothendieck_construction) (Wikipedia)
* [The Homotopy Theory of n-Fold Categories](http://www-personal.umd.umich.edu/~tmfiore/1/ThomasonNFoldBeamerSlides.pdf)
* [Inference System Integration via Logic Morphisms](http://www.kestrel.edu/home/projects/logicware/slides-9806/sld001.htm)
* [Category Theory for Computing Science](http://www.math.mcgill.ca/triples/Barr-Wells-ctcs.pdf)

A [[model category]] presentation of the Grothendieck construction see at _[[Grothendieck construction for model categories]]_.

Discussion of the Grothendieck construction as a [[lax colimit]] includes (see also at [[(infinity,1)-Grothendieck construction]])

* {#GepnerHaugsengNikolaus15} [[David Gepner]], [[Rune Haugseng]], [[Thomas Nikolaus]], _Lax colimits and free fibrations in $\infty$-categories_ ([arXiv:1501.02161](http://arxiv.org/abs/1501.02161))

On the [[enriched Grothendieck construction]]:

* {#BeardsleyWong18} [[Jonathan Beardsley]], [[Liang Ze Wong]], _The Enriched Grothendieck Construction_, Advances in Mathematics, **344** (2019) 234-261 [[arXiv:1804.03829](https://arxiv.org/abs/1804.03829), [doi:10.1016/j.aim.2018.12.009](https://doi.org/10.1016/j.aim.2018.12.009)]

* {#BeardsleyWong19} [[Jonathan Beardsley]], [[Liang Ze Wong]], *The operadic nerve, relative nerve and the Grothendieck construction*, Theory and Applications of Categories, **34** 13 (2019) 349-374 [[tac:34-13](http://www.tac.mta.ca/tac/volumes/34/13/34-13abs.html)]

A [[monoidal category|monoidal]] version:

* {#MoellerVasilakopoulou19} [[Joe Moeller]], [[Christina Vasilakopoulou]], *Monoidal Grothendieck Construction*,  2019 ([arXiv:1809.00727](https://arxiv.org/abs/1809.00727))

[[!redirects Grothendieck constructions]]

[[!redirects enriched Grothendieck construction]]
[[!redirects enriched Grothendieck constructions]]


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

#Contents#
* table of contents
{:toc}

## Idea

The [[Grothendieck construction]] may be lifted from [[categories]] to [[model categories]], where it serves as a presentation for the [[(infinity,1)-Grothendieck construction|$\infty$-Grothendieck construction]].

## Definition
 {#Definition}

Let $\mathcal{M}$ be a [[model category]] and

\[
  \label{ThePseudofunctor}
  \array{
    \mathcal{M} &\longrightarrow& Mod Cat
    \\
    X &\mapsto& F(X)
    \\
    \Big\downarrow\mathrlap{{}^{f}}
    &&
    \mathllap{^{f_!}}\Big\downarrow
    {}^{\dashv}
    \Big\uparrow\mathrlap{{}^{f^\ast}}
    \\
    Y &\mapsto& F(Y)
  }
\]

a [[pseudofunctor]], where [[ModCat|$ModelCat$]] is the [[2-category]] of [[model categories]], [[Quillen adjunctions]] (pointing in the direction of their [[left adjoints]] $f_!$), and [[conjugate transformations of adjoints]] ([[mate]]-pairs of [[natural isomorphisms]]).

We say that:

\begin{definition}\label{RelativePseudoFunctor}
$F$ is *[[relative functor]]* if for any [[weak equivalence]] $f$ in $M$, the associated Quillen adjunction $f_! \dashv f^*$ is a [[Quillen equivalence]].
\end{definition}

\begin{definition}\label{ProperPseudofunctor}
$F$ is "*proper*", if when $f \colon A\to B$ is an [[acyclic cofibration]] (resp. an [[acyclic fibration]]) in $\mathcal{M}$ then $f_!$ (resp. $f^*$) preserves all [[weak equivalences]].
\end{definition}

\begin{definition}\label{IntegralModelStructure}
**(integral model structure)**
\linebreak
Given a [[pseudofunctor]] as in (eq:ThePseudofunctor),
we say that a morphism
$$
  (f,\phi)
   \;\colon\;
  (X,A) \longrightarrow (Y,B)
$$
in its [[Grothendieck construction]] $\int_{X \in \mathcal{M}} F(X)$
(where $f \colon X\to Y$ in $\mathcal{M}$ and $\phi \colon f_!(A) \to B$ in $F(Y)$) is:

* an *integral equivalence* iff

  1. $f \colon X \to Y$ is a [[weak equivalence]] in $\mathcal{M}$

  2. $f_!(Q A) \to f_!(A) \xrightarrow{\phi} B$ is a [[weak equivalence]] in $F(Y)$, where $Q(-)$ denotes [[cofibrant replacement]].

     (Since $f_!\dashv f^*$ is a Quillen equivalence by relativeness, this is equivalent to the [[adjunct]] condition that $A \xrightarrow{\widetilde{\phi}} f^*(B) \to f^*(R B)$ is a weak equivalence in $F(X)$ for [[fibrant replacement]] $R(-)$.)

* an *integral cofibration* iff

  1. $f$ is a [[cofibration]] in $\mathcal{M}$

  1. $\phi \colon f_!(A) \to B$ is a [[cofibration]] in $F(Y)$.

* an *integral fibration* iff

  1. $f$ is a [[fibration]] in $\mathcal{M}$

  1. the [[adjunct]] $\widetilde{\phi} \colon A \to f^*(B)$ is a [[fibration]] in $F(X)$.

\end{definition}

\begin{proposition}\label{ExistenceStatement}
If the pseudofunctor (eq:ThePseudofunctor) is relative (Def. \ref{RelativePseudoFunctor}) and proper (Def. \ref{ProperPseudofunctor}) then the classes of maps in Def. \ref{IntegralModelStructure}
make $\int_{X \in \mathcal{M}} F(X)$ a [[model category]].
\end{proposition}
This is [Harpaz & Prasma (2015), Theorem 3.0.12](#HarpazPrasma15).

## Properties

\begin{proposition}
Given a proper relative $F \colon \mathcal{M} \to ModelCat$, we can compose with the underlying [[(infinity,1)-functor|$(\infty,1)$-functor]] $Ho \colon ModelCat \to QCat$ with values in (say) [[quasicategories]].
Since $F$ is relative, this map takes weak equivalences in $\mathcal{M}$ to [[equivalences of quasicategories]], so it induces a functor of quasicategories $Ho(M) \to Ho(QCat) = (\infty,1)Cat$.  The [[(∞,1)-Grothendieck construction]] of this functor is then equivalent, over $Ho(M)$, to the underlying $(\infty,1)$-category of the Grothendieck-construction model structure on $\int F$
\end{proposition}
([Harpaz & Prasma (2015), Proposition 3.1.2](#HarpazPrasma15))

## Examples
 {#Examples}

### Free coproduct completions
 {#BasicExamples}

\begin{example}\label{ModelStructureOnIndexedSetsOfObjects}
**(model structure on indexed sets of objects)**
\linebreak
  For $\mathcal{C}$ any [[model category]], consider the [[pseudofunctor]] on [[Set]] which assigns categories of [[indexed sets]] of objects of $\mathcal{C}$, equipped with the [product model structure](model+category#ProductModelStructure):
$$
  \array{
    Set
    &\longrightarrow&
    ModCat
    \\
    S
      &\mapsto&
    \underset{s \in S}{\prod} \mathcal{C}
    \\
    \Big\downarrow{{}^\mathrlap{f}}
    &&
    \mathllap{{}^{f_!}}\Big\downarrow
    {}^{\dashv}
    \Big\uparrow\mathrlap{{}^{f^*}}
    \\
    T
      &\mapsto&
    \underset{s \in S}{\prod} \mathcal{C}
  }
$$
Here $f_!$ forms [[coproducts]] of objects in the same [[fibers]] of $f$. The corresponding [[Grothendieck construction]] is the [[free coproduct completion]] of the [[underlying]] [[category]] $\mathcal{C}$ (see [there](free+coproduct+completion#AsAGrothendieckConstruction)).

If we regard [[Set]] as equipped with its [[trivial model structure]] (whose [[weak equivalences]] are the [[isomorphisms]] and all morphisms are both [[fibrations]] as well as [[cofibrations]]) then this is evidently a relative and proper functor in the sense of Def. \ref{RelativePseudoFunctor},  Def. \ref{ProperPseudofunctor} (since with weak equivalences $f$ in $Set$ being isomorphisms, the associated functors $f_!, f^\ast$ are certainly [[Quillen equivalences]] but in fact are plain [[equivalences of categories]] compatible with the model structure and hence also preserve all weak equivalences).

Therefore with Prop. \ref{ExistenceStatement} the integral model structure on the [[Grothendieck construction]] exists.
If $\mathcal{C}$ is an [[extensive category]], then
the [[objects]] in this Grothendieck construction may be thought of as [[bundles]] of $\mathcal{C}$-objects over sets (under the unique [[coproduct]]-[[preserved colimit|preserving]] embedding $Set \hookrightarrow \mathcal{C}$) and the [[morphisms]] as morphisms of such bundles covering possibly non-trivial maps of base sets:
\[
  \label{GrothConstForObjectsParameterizedOverSets}
  \int_{S \in Set}
    \underset{s \in S}{\prod} \mathcal{C}
  \;\;\;\;
  =
  \;\;\;\;
  \left\{
  \begin{array}{ccc}
    \underset{s \in S}{\coprod} X_s
    &\overset{\phi}{\longrightarrow}&
    \underset{t \in T}{\coprod} Y_t
    \\
    \big\downarrow
    &&
    \big\downarrow
    \\
    S &\underset{f}{\longrightarrow}& T
  \end{array}
  \right\}
\]

Unwinding the definition \ref{IntegralModelStructure} of the integral model structure in this case, gives that such a morphism $\phi_f$ in this Grothendieck construction is:

* a fibration iff all the components $X_s \longrightarrow Y_{f(s)}$ are fibrations in $\mathcal{C}$ for all $s \in S$,

* a cofibration iff all the co-components $\underset{s \in f^{-1}(\{t\})}{\coprod}  X_s \longrightarrow Y_t$ are cofibrations in $\mathcal{C}$, for all $t \in T$,

* a weak equivalence iff $f$ is a [[bijection]] of index sets and all the (co)components maps --- which in this case are all of the form $X_s \longrightarrow X_{f(s)}$ --- are weak equivalences in $\mathcal{C}$

  (here the composition with (co)fibrant replacements can be omitted, since, as above, $f_!, f^\ast$ preserve such resolution weak equivalences).

\end{example}

\begin{remark}
\label{CoFibrantObjectsInFreeCoproductCompletion}
The [[cofibrant objects]] in the integral model structure on a [[free coproduct completion]] (Exp. \ref{ModelStructureOnIndexedSetsOfObjects}) are exactly the [[tuples]] of [[cofibrant objects]] in the coefficient model structure $\mathcal{C}$; analogously for the [[fibrant objects]].

To see this, observe that the [[initial object]] in the [[Grothendieck construction]] of Exp. \ref{ModelStructureOnIndexedSetsOfObjects} is

$$
  0_{\varnothing}
  \;\coloneqq\;
  \Big(
    \varnothing \,\in\, Set
    ,\;
    0 \,\in\, \underset{\varnothing}{\prod} \mathcal{C}
  \Big)
  \,,
$$
where

* $\varnothing$ (the [[empty set]]) denotes the [[initial object]] in $Set$

* $0$ denotes the [[initial object]] in the [[empty diagram|empty]]-[[product category]], [which is](terminal+object#TerminalObjectIsEmptyLimit) the [[terminal category]], so that $0$ is in fact a [[zero object]], indeed it is the only object in that category

  $$
    \underset{\varnothing}{\prod} \mathcal{C}
    \;\simeq\;
    \{0\}
     \,.
  $$

Now given any object $\mathscr{V}_S \,\in\, \int_{S \in Set}\;\underset{s \in S}{\prod} \, \mathcal{C}$ observe that there is a unique morphism

$$
  0_{\varnothing} \longrightarrow \mathscr{V}_S
  \,.
$$
Namely, unwinding the definitions, such a morphism is a [[dependent pair]] consisting of

1. a morphism $f \,\colon\,\varnothing \longrightarrow S$ in [[Set]],

1. a morphism $\widetilde{\phi} \,\colon\, f_! 0 \longrightarrow  \mathscr{V}$ in $\underset{s \in S}{\prod} \, \mathcal{C}$

But both of these exist uniquely, by the fact that their respective [[domains]] are [[initial objects]], using here that $f_!$, [[left adjoints preserve colimits|being a left adjoint]], [[preserved colimit|preserves]] [[colimits]] and hence [[initial objects]]

This shows that also $0_{\varnothing}$ is the initial object in $\int_S \,\prod_{s \in S}\, \nathcal{C}$. (Of course this also follows by the general formula for [colimits in a Grothendieck construction](Grothendieck+construction#CoLimitsInAGrothendieckConstruction) specialized to the [[empty diagram|empty]] [[colimit]].) And it shows (since all morphisms in the [[trivial model structure]] on $Set$ are cofibrations while the cofibrations in a [product model category](model+category#ProductModelStructure) are the [[tuples]] of cofibrations) that $\mathcal{V}_S$ is cofibant precisely if $\mathcal{V}_s \,\in\, \mathcal{C}$ is so for all $s \in S$.
\end{remark}

\begin{example}
**(model structure on skeletal simplicial groupoids)**
As a special case of Exp. \ref{ModelStructureOnIndexedSetsOfObjects}, consider $\mathcal{C} \coloneqq sGrpd$ the [[model structure on simplicial groups]].

Notice that forming simplicial [[delooping groupoids]] is a [[fully faithful functor]] from [[sGrp]] to the *[[1-category]]* of [[sSet]]-[[enriched groupoids]] (Dwyer-Kan's"[[simplicial groupoids]]")
$$
  \array{
    sGrp &\longrightarrow& sSet\text{-}Grpd
    \\
    \mathcal{G} &\mapsto& \mathbf{B}\mathcal{G}
  }
$$
which extends to identify the [[Grothendieck construction]] (eq:GrothConstForObjectsParameterizedOverSets) in the present case with the [[full subcategory]] of [[disjoint unions]] of simplicial [[delooping groupoids]], hence with that of simplicial [[skeletal groupoids]]:
\[
  \label{InclusionOfSkeletalSimplicialGroupoids}
  \array{
    \int_S \underset{s \in S}{\prod} sGrp
    &\;\simeq\;&
    sSet\text{-}Grpd_{skl}
    &\xhookrightarrow{\phantom{--}}&
    sSet\text{-}Grpd
    \\
    \big(\mathcal{G}_s\big)_{s \in S}
      &\mapsto&
    \underset{s \in S}{\coprod}
      \mathbf{B}\mathcal{G}_s
  }
\]

Since (assuming the [[axiom of choice]] in the [[underlying]] [[Sets]]) every [[sSet]]-[[enriched groupoid]] is [[Dwyer-Kan equivalence|DK-equivalent]] to a skeletal simplicial groupoid, the [[full subcategory]] inclusion (eq:InclusionOfSkeletalSimplicialGroupoids) is also a [[Dwyer-Kan equivalence]] of [[sSet-enriched categories]].

Moreover, if we consider on $sSet\text{-}Grpd$ the usual [[model structure on simplicial groupoids]], then the inclusion functor
$$
  sSet\text{-}Grpd_{skl}
  \hookrightarrow
  sSet\text{-}Grpd
$$
preserves [[weak equivalences]] and [[fibrations]]. However, since there is no [[left adjoint]] to this functor (due to the choices involved in passing to a [[skeleton]] such an adjoint exists only as a [[pseudofunctor]], hence on the level of [[2-category theory]]) this is *not* a [[right Quillen functor]], and in particular *not* a [[Quillen equivalence]].

\end{example}

\linebreak

## Related pages

* The [[formal duality|dual]] notion is a [[model structure on sections]]

* [[tangent (infinity,1)-topos]]

* [[Joyal locus]]

* [[parameterized homotopy theory]]

* [[parameterized stable homotopy theory]]

## References

The first [[model category]] version of the [[Grothendieck construction]] was given in

* {#Roig94} [[Agustí Roig]], _Model category structures in bifibred categories_, Journal of Pure and Applied Algebra **95** 2 (1994) 203–223 [doi:10.1016/0022-4049(94)90074-4]

This article ([Roig 94](#Roig94)) had a mistake, which was fixed in

* {#Stanculesu12} [[Alexandru E. Stanculescu]], *Bifibrations and weak factorisation systems*, Applied Categorical Structures **20** 1 (2012) 19–30 [doi:10.1007/s10485-009-9214-3]

The construction was then generalized in

* {#HarpazPrasma15} [[Yonatan Harpaz]], [[Matan Prasma]], _The Grothendieck construction for model categories_, Advances in Mathematics **281** (2015) 1306-1363 [[arXiv:1404.1852](https://arxiv.org/abs/1404.1852), [10.1016/j.aim.2015.03.031](https://doi.org/10.1016/j.aim.2015.03.031)]

and further in

* [[Pierre Cagne]], [[Paul-André Melliès]], *On bifibrations of model categories*, Advances in Mathematics **370** (2020) 107205 [[arXiv:1709.10484](https://arxiv.org/abs/1709.10484), [doi:10.1016/j.aim.2020.107205](https://doi.org/10.1016/j.aim.2020.107205)]

For the special case of pseudofunctors with values in [[groupoids]], a [[model category]] version of the Grothendieck construction was discussed in

* [[Sharon Hollander]], _A homotopy theory for stacks_ ([arXiv:math.AT/0110247](http://arxiv.org/abs/math.AT/0110247)).

Dicussion of further examples:

Integral model structure for graupal Segal-space actions:

* [[Yonatan Harpaz]], [[Matan Prasma]], *An integral model structure and truncation theory for coherent group actions*,  Isr. J. Math. **221** (2017) 511–561 [[arXiv:1506.04117](https://arxiv.org/abs/1506.04117), [doi:10.1007/s11856-017-1551-6](https://doi.org/10.1007/s11856-017-1551-6)]

[[!redirects Grothendieck construction of model categories]]
[[!redirects model category-theoretic Grothendieck construction]]

[[!redirects integral model structure]]
[[!redirects integral model structures]]
[[!redirects lax colimit model structure]]
[[!redirects lax colimit model structures]]
[[!redirects model structure on a lax colimit]]
[[!redirects model structures on a lax colimit]]
[[!redirects model structures on lax colimits]]

[[!redirects model structure on Grothendieck fibration]]
[[!redirects model structures on Grothendieck fibrations]]

[[!redirects model structure on a Grothendieck construction]]
[[!redirects model structure on the Grothendieck construction]]
[[!redirects model structure on Grothendieck constructions]]
[[!redirects model structures on Grothendieck constructions]]


Grothendieck construction for monoidal categories

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#### Monoidal categories
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#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[Grothendieck construction]] gives a way of gluing together the constituent categories $F(x)$ of an [[indexed category]] $F \colon X^{op} \to \mathsf{Cat}$ to get a category $\int F$ which admits an obvious [[Grothendieck fibration|fibration]] over the base $\int F \to X$. This gives an [[equivalence of 2-categories|equivalence]] between the 2-category of indexed categories and the 2-category of fibrations.

Two ways [[monoidal category|monoidal structures]] can join this story is on the total category $\int F$, or on the fibres $F(x)$. The 2-equivalence then lifts to two monoidal variants, one where the fibres are equipped with a monoidal structure, and one where the total category is equipped with a monoidal structure. Under certain conditions on the base category, these two settings are equivalent to each other as well. That is to say, under the right conditions, one can glue together the monoidal structures on the fibres to get a monoidal structure on the total category.

## Related Concepts

* [[Grothendieck construction]]

* [[monoidal category]]

* [[monoidal fibration]]

* [[indexed monoidal category]]

## References

* [[Mike Shulman]], _Framed bicategories and monoidal fibrations_, Theory and Applications of Categories, Vol. 20, No. 18, 2008, pp. 650–738. [TAC:20-18](http://www.tac.mta.ca/tac/volumes/20/18/20-18abs.html)

* {#Moeller_Vasilakopoulou19} [[Joe Moeller]], [[Christina Vasilakopoulou]], _Monoidal Grothendieck Construction_, Theory and Applications of Categories, **35** 31 (2020) 1159-1207 $[$[arXiv:1809.00727](http://arxiv.org/abs/1809.00727), [TAC:35-31](http://www.tac.mta.ca/tac/volumes/35/31/35-31abs.html)$]$

[[!redirects monoidal Grothendieck construction]]
[[!redirects monoidal Grothendieck constructions]]


Grothendieck context

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#### Geometry
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[[!include higher geometry - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Duality
+-- {: .hide}
[[!include duality - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _Grothendieck context_ is a pair of two [[symmetric monoidal categories]] $(\mathcal{X}, \otimes_X, 1_{X})$, $(\mathcal{Y}, \otimes_Y, 1_Y)$ which are connected by an [[adjoint triple]] of [[functors]] such that the leftmost one is a [[closed monoidal functor]].

This is the variant/special case of the [[yoga of six operations]] with two [[adjoint pairs]] $(f_! \dashv f^!)$ and $(f^\ast \dashv f_\ast)$ for $f_! \simeq f_\ast$.

$$
  f^\ast \dashv (f_\ast = f_!) \dashv f^!
  \;\colon\;
  \mathcal{X}
   \;
    \array{
     \overset{f^\ast}{\longleftarrow}
     \\
     \overset{f_\ast = f_! }{\longrightarrow}
     \\
     \overset{f^!}{\longleftarrow}
    }
    \;
  \mathcal{Y}
  \,.
$$

(The other specialization of the [[six operations]] where $f^\ast \simeq f^!$ is called the _[[Wirthmüller context]]_).

The existence of the (derived) right adjoint $f^!$ to $f_\ast$ is what is called _[[Grothendieck duality]]_.

## Examples

### Quasicoherent sheaves on schemes
 {#QuasicoeherntSheavesOnSchemes}

A [[homomorphism]] of [[schemes]] $f \;\colon\; X \longrightarrow Y$ induces an [[inverse image]] $\dashv$ [[direct image]] [[adjunction]] on the [[derived categories]] $QCoh(-)$ of [[quasicoherent sheaves]]

$$
  (f^\ast \dashv f_\ast)
   \;\colon\;
   QCoh(X)
    \underoverset
     \overset{f^\ast}{\longleftarrow}
     \overset{f_\ast}{\longrightarrow}
     {\bot}
   QCoh(Y)
  \,.
$$

(all [[derived functors]]) If $f$ is a [[proper morphism of schemes]] then under mild further conditions there is a further [[right adjoint]] $f^!$

$$
  (f^\ast \dashv f_\ast \dashv f^!)
   \;\colon\;
   QCoh(X)
    \;
    \array{
      \overset{f^\ast}{\longleftarrow}
      \\
      \overset{f_\ast}{\longrightarrow}
      \\
      \overset{f^!}{\longleftarrow}
     }
   \;
   QCoh(Y)
  \,.
$$

This is originally due to [[Grothendieck]], whence the name. Refined accounts are in  ([Deligne 66](#Deligne66), [Verdier 68](Verdier68), [Neeman 96](#Neeman96)).

### Quasicoherent sheaves in $E_\infty$-geometry

Generalization of the pull-push [[adjoint triple]] to [[E-∞ geometry]] is in ([LurieQC, prop. 2.5.12](#LurieQC)) and the [[projection formula]] for this is in ([LurieProp, remark 1.3.14](#LurieProper)).

## Related concepts

* [[Wirthmüller context]]

* [[Verdier-Grothendieck context]]

## References

The original construction for [quasicoherent sheaves on schemes](#QuasicoeherntSheavesOnSchemes) is due to [[Alexander Grothendieck]], whence the name "Grothendieck context".

Further stream-lined accounts then appeared in

* [[Pierre Deligne]], _Cohomology à support propre en construction du foncteur $f^!$_, Appendix to: _Residues and Duality_, Lecture Notes in Math., vol. 20, Springer-Verlag, Heidelberg, 1966, pp. 404{421. MR 36:5145
 {#Deligne66}

* [[Jean-Louis Verdier]], _Base change for twisted inverse images of coherent sheaves_, Collection: Algebraic Geometry (Internat. Colloq.), Tata Inst. Fund. Res., Bombay, 1968, pp. 393-408. MR 43:227
 {#Verdier68}

Further refinement and highlighting of the close relation to the [categorical Brown representability theorem](Brown+representability+theorem#CategoricalBrownRepresentability) is in

* [[Amnon Neeman]], _The Grothendieck duality theorem via Bousfield's techniques and Brown representability_, J. Amer. Math. Soc. 9 (1996), 205-236 ([web](http://www.ams.org/journals/jams/1996-9-01/S0894-0347-96-00174-9/))
 {#Neeman96}

Discussion of [[integral transforms]] in Grothendieck contexts is in

* [[Alexander Polishchuk]], _Kernel algebras and generalized Fourier-Mukai transforms_ ([arXiv:0810.1542](http://arxiv.org/abs/0810.1542))
 {#Polishchuk}

Generalization of the pull-push [[adjoint triple]] to [[E-∞ geometry]] is in

* {#LurieQC} [[Jacob Lurie]], section 2.5 of _[[Quasi-Coherent Sheaves and Tannaka Duality Theorems]]_

and the [[projection formula]] for this triple appears as remark 1.3.14 of

* {#LurieProper} [[Jacob Lurie]], _[[Proper Morphisms, Completions, and the Grothendieck Existence Theorem]]_

A clear discussion of axioms of [[six operations]], their specialization to Grothendieck context and [[Wirthmüller context]] and their consequences is in

* {#May05} [[Halvard Fausk]], [[Po Hu]], [[Peter May]],  *Isomorphisms between left and right adjoints*, Theory and Applications of Categories, **11** 4 (2003) 107-131 [[tac:11-04](http://www.tac.mta.ca/tac/volumes/11/4/11-04abs.html), [pdf](http://www.math.uiuc.edu/K-theory

* {#BalmerDellAmbrogioSanders15} [[Paul Balmer]], [[Ivo Dell'Ambrogio]], [[Beren Sanders]], _Grothendieck-Neeman duality and the Wirthmüller isomorphism_, [arXiv:1501.01999](http://arxiv.org/abs/1501.01999).

[[!redirects Grothendieck contexts]]


Grothendieck duality

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#### Duality
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[[!include duality - contents]]
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=--

# Grothendieck duality
* table of contents
{: toc}

## Idea

Given a [[homomorphism]] $f$ of [[schemes]], one says that it satisfies
_Grothendieck duality_  if the ([[derived functor|derived]]) [[direct image]] functor $f_\ast$ on [[quasicoherent sheaves]] has a (derived) [[right adjoint]] $f^!$. This is [[Verdier duality]] in a "[[Grothendieck context]]" of [[six operations]].

Grothendieck duality is intimately connected to **dualizing complexes**. This was the original approach of Grothendieck in the book _Residues and Duality_.

## Statement

Suppose $f\colon X \to Y$ is a [[quasi-compact map|quasi-compact]] and [[quasi-separated map|quasi-separated]] [[morphism]] of [[schemes]]; then the [[triangulated functor]] $\mathbf{R}f_*\colon D_{qc}(X)\to D(Y)$ has a bounded below [[right adjoint]]. In other words, $\mathbf{R}Hom_X(\mathcal{F}, f^\times \mathcal{G})\stackrel{\sim}{\to} \mathbf{R}Hom_Y(\mathbf{R}f_*\mathcal{F}, \mathcal{G})$ is a [[natural isomorphism]].

## Dualizing Complexes

Let $X$ be a noetherian scheme. A dualizing complex on $X$ is a complex
$\mathcal{R} \in \mathsf{D}(\mathsf{Mod} X)$ that has these three properties:

* $\mathcal{R} \in \mathsf{D}^{\mathrm{b}}_{\mathrm{c}}(\mathsf{Mod} X)$ (i.e. $\mathcal{R}$ has bounded coherent cohomology sheaves).

* $\mathcal{R}$ has finite injective dimension.

* The canonical morphism
$\mathcal{O}_X \to \mathrm{R} \mathcal{Hom}_{X}(\mathcal{R}, \mathcal{R})$
in $\mathsf{D}(\mathsf{Mod} X)$ is an isomorphism.

The following two structures are basically equivalent to each other, for a given category of noetherian schemes $\mathsf{S}$:

* A psudofunctor $f \mapsto f^!$, called the **twisted inverse image**,
that assigns a functor
\[ f^! : \mathsf{D}^{+}_{\mathrm{c}}(\mathsf{Mod} Y)
\to \mathsf{D}^{+}_{\mathrm{c}}(\mathsf{Mod} X) \]
to each map of schemes $f : X \to Y$ in $\mathsf{S}$,
and has several known properties.

* A dualizing complex $\mathcal{R}_X$ for every scheme $X$ in the category
$\mathsf{S}$, with several known functorial properties.

The relation between these two structures is demonstrated in the following Example.

**Example**.
Suppose $K$ is a regular finite dimensional noetherian ring, and let $\mathsf{S}$ be the category of finite type $K$-schemes.
Given a twisted inverse image psudofunctor $f \mapsto f^!$, we define dualizing complexes as follows: on any $X \in \mathsf{S}$ with structural morphism
$\pi_X : X \to \operatorname{Spec} K$, we let
$\mathcal{R}_X := \pi_X^!(K)$.

Conversely, suppose we are given a dualizing complex $\mathcal{R}_X$ on each
$X \in \mathsf{S}$. This gives rise to a duality (contrvariant equivalence)
$D_X$ of $\mathsf{D}^{}_{\mathrm{c}}(\mathsf{Mod} X)$,
exchanging $\mathsf{D}^{+}_{\mathrm{c}}(\mathsf{Mod} X)$ with
$\mathsf{D}^{-}_{\mathrm{c}}(\mathsf{Mod} X)$, with formula
\[ D_X(\mathcal{M}) := \mathrm{R} \mathcal{Hom}_{X}(\mathcal{M}, \mathcal{R}_X) . \]
We then define
\[ f^!(\mathcal{M}) := D_Y( \mathrm{L} f^* ( D_X(\mathcal{M}) )) . \]

### Rigid Dualizing Complexes

The notion of _rigid dualizing complex_ was introduced by Van den Bergh in 1997, for a _noncommutative ring_ $A$ over a base field $K$.

Yekutieli and Zhang have shown how to define a rigid dualizing complex $R_{A/K}$, when $K$ is a regular finite dimensional noetherian ring, and $A$ is an essentially finite type $K$-ring (both commutative). A refined variant of the rigid dualizing complex, namely the _rigid residue complex_ $\mathcal{K}_{A/K}$, was shown to exist, and to be unique (up to a unique isomorphism of complexes).

These rigid residue complexes have all the good functorial properties alluded to above, and even more. Specifically, they are covariant for essentially etale ring homomorphisms $A \to A'$ (via the rigid localization homomorphism), and contravariant
(as graded modules) for all ring homomorphisms $A \to B$ (via the ind-rigid trace homomorphism).

The rigid localization homomorphism permits the gluing of the rigid residue complexes
$\mathcal{K}_{A/K}$ on affine open sets $U = \operatorname{Spec} A$ of a scheme $X$ into a rigid residue complex $\mathcal{K}_{X/K}$ on $X$. In this way one obtains a collection of dualizing complexes $\mathcal{K}_{X/K}$ on all essentially finite type $K$-schemes $X$, consisting of quasi-coherent injective sheaves. For any map of scheme $f : X \to Y$ there is the ind-rigid trace homomoprhism
\[ \mathrm{Tr}_f : f_*(\mathcal{K}_{Y/K}) \to \mathcal{K}_{X/K} , \]
which is a homomorphism of graded quasi-coherent sheaves.
The _Residue Theorem_ says that when $f$ is proper,
$\mathrm{Tr}_f$ is a homomorphism of complexes.
The _Duality Theorem_ says that when $f$ is proper, $\mathrm{Tr}_f$ induces global duality. As explained above, there is a corresponding functor $f^!$;
and the Duality Theorem says that $\mathrm{R} f_*$ and $f^!$ are adjoint functors.

Moreover, the rigidity method works also for finite type **Deligne-Mumford stacks** over $K$. The key observation is that the rigid residue complexes are complexes of quasi-coherent sheaves in the etale site over $K$. Details of this extension of the theory are still under preparation.

## Noncommutative Grothendieck Duality

(To be added later)

## Related concepts

* [[six operations]]

* [[Poincare duality]], [[Verdier duality]]

* [[Fourier-Mukai transform]]

## References

* [[Robin Hartshorne]], _Residues and duality_ (Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne.) Springer LNM 20, 1966 [MR222093](http://www.ams.org/mathscinet-getitem?mr=222093)
* Domingo Toledo, Yue Lin L. Tong,  _Duality and intersection theory in complex manifolds. I._, Math. Ann. __237__ (1978), no. 1, 41–77, [MR80d:32008](http://www.ams.org/mathscinet-getitem?mr=506654), [doi](http://dx.doi.org/10.1007/BF01351557)
* Mitya Boyarchenko, [[Vladimir Drinfeld]], _A duality formalism in the spirit of Grothendieck and Verdier_, [arxiv/1108.6020](http://arxiv.org/abs/1108.6020)
* Z. Mebkhout, _Le formalisme des six opérations de Grothendieck pour les $\mathcal{D}_X$-modules cohérents_, Travaux en Cours __35__. Hermann, Paris, 1989. x+254 pp. [MR90m:32026](http://www.ams.org/mathscinet-getitem?mr=1008245)
* [[Amnon Neeman]], _Derived categories and Grothendieck duality_, in:  Triangulated categories, 290–350, London Math. Soc. Lecture Note Ser. __375__, Cambridge Univ. Press 2010
* Amnon Neeman, _The Grothendieck duality theorem via Bousfield's techniques and Brown representability_, J. Amer. Math. Soc. __9__ (1996), no. 1, 205–236, [MR96c:18006](http://www.ams.org/mathscinet-getitem?mr=1308405), [doi](http://dx.doi.org/10.1090/S0894-0347-96-00174-9)
* Brian Conrad, _Grothendieck duality and base change_, Springer Lec. Notes Math. __1750__ (2000) vi+296 pp.
* [[Joseph Lipman]], _Notes on derived functors and Grothendieck duality_, in: Foundations of Grothendieck duality for diagrams of schemes, 1–259, Lecture Notes in Math. __1960__, Springer 2009, [doi](http://dx.doi.org/10.1007/978-3-540-85420-3), [draft pdf](http://www.math.purdue.edu/~lipman/Duality.pdf)
* J. Lipman, _Grothendieck operations and coherence in categories_, conference slides, 2009, [pdf](http://www.math.purdue.edu/~lipman/Madrid.pdf)
* Alonso Tarrío, Leovigildo; Jeremías López, Ana; Joseph Lipman, _Studies in duality on Noetherian formal schemes and non-Noetherian ordinary schemes_, Contemporary Mathematics __244__ Amer. Math. Soc. 1999. x+126L. [MR2000h:14017](http://www.ams.org/mathscinet-getitem?mr=1716706); _Duality and flat base change on formal schemes_, Contemporary Math. __244__ (1999), pp. 3–90.
* J. Ayoub, _Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique. I._, Astérisque No. 314 (2007), x+466 pp. (2008) [MR2009h:14032](http://www.ams.org/mathscinet-getitem?mr=2423375); _II._ Astérisque No. 315 (2007), vi+364 pp. (2008) [MR2009m:14007](http://www.ams.org/mathscinet-getitem?mr=2438151);
also a file at K-theory archive [THESE.pdf](http://www.math.uiuc.edu/K-theory/0761/THESE.pdf)

* [[Amnon Yekutieli]], James Zhang,
Rings with Auslander Dualizing Complexes, J. Algebra 213 (1999), 1-51;
_Rigid dualizing complexes over commutative rings_, Algebr. Represent. Theory __12__ (2009), no. 1, 19–52, [doi](http://dx.doi.org/10.1007/s10468-008-9102-9);
Dualizing Complexes and Perverse Sheaves on Noncommutative Ringed Schemes,
Selecta Math. 12 (2006), 137-177;
Dualizing Complexes and Perverse Modules over Differential Algebras,
Compositio Mathematica 141 (2005), 620-654.

* [[Amnon Yekutieli]],
An Explicit Construction of the Grothendieck Residue Complex, Astérisque **208** (1992);
The residue complex of a noncommutative graded algebra, J. Algebra __186__ (1996), no. 2, 522–543; _Smooth formal embeddings and the residue complex_, Canad. J. Math. __50__ (1998), no. 4, 863–896, [MR99i:14004](http://www.ams.org/mathscinet-getitem?mr=1638635); _Rigid dualizing complexes via differential graded algebras (survey)_, in: Triangulated categories, 452–463, London Math. Soc. Lecture Note Ser. __375__, Cambridge Univ. Press 2010, [MR2011h:18015](http://www.ams.org/mathscinet-getitem?mr=2681716);
 Residues and Differential Operators on Schemes,
Duke Math. J. 95 (1998), 305-341;
Duality and Tilting for Commutative DG Rings, [arXiv:1312.6411](http://arxiv.org/abs/1312.6411);
Residues and Duality for Schemes and Stacks
([lecture notes](http://www.math.bgu.ac.il/~amyekut/lectures/resid-stacks/handout.pdf)).

* [[Roy Joshua]], _Grothendieck-Verdier duality in enriched symmetric monoidal $t$-categories_ ([[JoshuaDuality.pdf:file]])

* {#Belmans} [[Pieter Belmans]], section 2.2 of _Grothendieck duality: lecture 3_, 2014 ([[BelmansDuality.pdf:file]])

* [[Amnon Neeman]], _An improvement on the base-change theorem and the functor $f^!$_, [arXiv](http://arxiv.org/abs/1406.7599).


Grothendieck existence

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#Contents#
* table of contents
{:toc}

## Idea

A **Grothendieck's existence problem** is a question of [[representable functor|representability]] of a [[presheaf]] of sets (or sometimes of groups) on the [[category]] of [[affine schemes]] (or some variant of it) as a [[scheme]] (or as an [[algebraic space]], [[Deligne-Mumford stack]] or some other algebraic variant of a space making the representability problem nontrivial...). In early days of [[scheme]] theory, at the end of 1950-s Grothendieck proved several major positive results of this sort, like the constructions of Hilbert, [[Quot scheme|Quot]] and Picard schemes, and the **Grothendieck's existence theorem** in [[formal geometry]] with major applications in [[deformation theory]]. Another major result of that sort is the [[Artin's representability theorem]] (and its refined version in [[higher geometry]], the [[Artin-Lurie representability theorem]]).

## Related concepts

* [[Artin representability theorem]]

* [[Artin-Lurie representability theorem]]

## References

Articles on the original theorem include

* [[Luc Illusie]], _Grothendieck's existence theorem in formal geometry_, [[FGA explained]], 179--233, MR2223409; (draft version [pdf](http://cdsagenda5.ictp.it//askArchive.php?categ=a0255&id=a0255s3t3&ifd=15021&down=1&type=lecture_notes))

* [[Michael Artin]], _Algebraization of formal moduli. I_, Global Analysis (Papers in Honor of K. Kodaira), Princeton Univ. Press, Princeton, N.J., 1969, pp. 21-71. [MR0260746](http://www.ams.org/mathscinet-getitem?mr=0260746) (41:5369)

* Siegmund Kosarew, _Grothendieck's existence theorem in [[analytic geometry]] and related results_,  Trans. Amer. Math. Soc. 328 (1991), 259-306, [pdf](http://www.ams.org/journals/tran/1991-328-01/S0002-9947-1991-1014252-X/S0002-9947-1991-1014252-X.pdf)

Discussion in [[E-∞ geometry]] is in

* [[Jacob Lurie]], section 5.3 of _[[Proper Morphisms, Completions, and the Grothendieck Existence Theorem]]_

* [[Jacob Lurie]], _[[Representability theorems]]_

[[!redirects Grothendieck's existence theorem]]
[[!redirects Grothendieck's existence problem]]
[[!redirects Grothendieck existence theorem]]
[[!redirects Grothendieck existence problem]]
[[!redirects Grothendieck's existence]]


Grothendieck Festschrift

__The Grothendieck Festschrift__ is a 3-volume collection of articles written in honor of the 60th birthday (1988) of [[Alexander Grothendieck]], published by Birkhäuser in series Progress in Mathematics in 1990 as volumes 86, 87, 88. More recently a paperback edition appeared in series Modern Birkhäuser Classics (French edition).

Vol. 1

* Foreword

* Bibliographie d'Alexander Grothendieck ([pdf](http://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/GrothBiblio.pdf))

* [[Jean Dieudonne]], De l'Analyse fonctionnelle aux fondements de la géométrie algébrique

* Allen B. Altman and Steven L. Kleiman, The presentation functor and the compactified Jacobian

* M. Artin, J. Tate, [[M. Van den Bergh]], Some algebras associated to automorphisms of elliptic curves

* V. Balaji, C.S. Seshadri, Cohomology of a moduli space of vector bundles

* Arnaud Beauville, Sur les hypersurfaces dont les sections hyperplanes sont a module constant

* A.A. Beilinson, A.B. Goncharov, V.V. Schechtman and A.N. Varchenko, Aomoto dilogarithms, mixed Hodge structures and motivic cohomology of pairs of triangles on the plane

* Pierre Berthelot, William Messing, Théorie de Dieudonné cristalline. III. Théorèmes d'équivalence et de pleine fidélité

* Jean-Michel Bismut, Henri Gillet and Christophe Soulé, Complex immersions and Arakelov geometry

* Spencer Bloch, Kazuya Kato, L-Functions and Tamagawa numbers of motives

* [[Lawrence Breen]], Bitorseurs et cohomologie non abélienne (pp. 401--476);

* [[Jean-Luc Brylinski]], Noncommutative Ruelle-Sullivan type currents (pp. 477--498).

Vol. 2

* [[Pierre Cartier]], André Voros, Une nouvelle interprétation de la formule des traces de Selberg (pp. 1--67)
* [[Carlos Contou-Carrere]], Jacobiennes généralisées globales relatives (pp. 69--109)
* [[Pierre Deligne]], [[Catégories Tannakiennes]] (pp. 111--195)
* Torsten Ekedahl, On the adic formalism (pp. 197--218)
* [[Gerd Faltings]], $F$-isocrystals on open varieties: results and conjectures (pp. 219--248)
* Jean-Marc Fontaine, Représentations $p$-adiques des corps locaux. I (pp. 249--309)
* Helmut A. Hamm, LęDung Tráng, Rectified homotopical depth and Grothendieck conjectures

Vol 3

* Alain Lascoux, Anneau de Grothendieck de la variété de drapeaux 1
* S. Lichtenbaum, New results on weight-two motivic cohomology 35
* George Lusztig, Symmetric spaces over a finite field 57
* Zoghman Mebkhout, Le théorème de positivité de irrégularité pour les $D_X$-modules 83
* Arthur Ogus, The convergent topos in characteristic p 133
* A.N. Parshin, Finiteness theorems and hyperbolic manifolds 163
* Michel Raynaud, p-groupes et réduction semi-stable des courbes 179
* G.B. Shabat, V.A. Voevodsky, Drawing curves over number fields 199
* L. Szpiro, Sur les propriétés numériques du dualisant relatif d'une surface arithmetique 229
* R.W. Thomason, Thomas Trobaugh, Higher algebraic K-theory of schemes and of derived categories
* A. Treibich, J.-L. Verdier (with an Appendix by J. Oesterle), Solitons elliptiques 437
* Yu. G. Zarhin, Linear simple Lie algebras and ranks of operators 481

category: reference