nLab
Grothendieck construction

Idea

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

$$\begin{array}{c}E\\ {\downarrow }^p\\ C\end{array}$$\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

$$\begin{array}{c}\hat{U}\\ {\downarrow }^{p_{\mathrm{univ}}}\\ U\end{array}.$$\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 categories – i.e. a functor – is called a fibered category or Grothendieck fibration if it is encoded in a pseudofunctor/2-functor $S_p:C^{\mathrm{op}}\to \mathrm{Cat}$.

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

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

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

The essential image of this functor consists of Grothendieck fibrations and this establishes an equivalence of 2-categories

$$\int :\mathrm{Func}(C^{\mathrm{op}},\mathrm{Cat})\stackrel{\simeq }{\to }\mathrm{Fib}(C)$$\int : Func(C^{op}, Cat) \stackrel{\simeq}{\to} Fib(C)

between 2-functors $C^{\mathrm{op}}\to \mathrm{Cat}$ and Grothendieck fibrations over $C$.

When restricted to pseudofunctors with values in Grpd $\subset$ Cat this identifies the Grothendieck fibrations in groupoids

$$\int :\mathrm{Func}(C^{\mathrm{op}},\mathrm{Grpd})\stackrel{\simeq }{\to }\mathrm{FibGrpd}(C).$$\int \;\;:\;\; Func(C^{op}, Grpd) \stackrel{\simeq}{\to} FibGrpd(C) \,.

This equivalence notably allows to discuss stacks equivalently as pseudofunctors or as groupoid fibrations (in each case satisfying a descent condition with respect to a Grothendieck topology on $D$).

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.

Definition

Let Cat be the 2-category of categories, functors and natural transformations.,

Recall from generalized universal bundle that the “universal Cat-bundle” is ${\mathrm{Cat}}_{*}\to \mathrm{Cat}$, where ${\mathrm{Cat}}_{*}$ is the category of pointed categories.

Then for $F:C\to \mathrm{Cat}$ a functor, the Grothendieck construction for $F$ is the (strict) pullback $\int F\to C$ of ${\mathrm{Cat}}_{*}\to \mathrm{Cat}$ along $F$:

$$\begin{array}{ccc}\int F& \to & {\mathrm{Cat}}_{*}\\ \downarrow & & \downarrow \\ C& \stackrel{F}{\to }& \mathrm{Cat}\end{array}.$$\array{ \int F &\to& Cat_* \\ \downarrow && \downarrow \\ C &\stackrel{F}{\to}& Cat } \,.

This means that the objects of $\int F$ are pairs $(c,a)$, where $c\in \mathrm{Obj}(C)$ and $a\in \mathrm{Obj}(F(c))$ and morphisms in $\int F$ are given by pairs $(c\stackrel{f}{\to }c\prime ,\alpha :F(f)(a)\to a\prime )$. This may be visualized as

$$\int F=\left\{\begin{array}{ccc}& & *\\ & {}^a\swarrow & {\Downarrow }^{\alpha }& {\searrow }^{a\prime }\\ F(c)& & \stackrel{F(f)}{\to }& & F(c\prime )\\ \\ c& & \stackrel{f}{\to }& & c\prime \end{array}\right\}.$$\int F = \left\{ \array{ && {*} \\ & {}^a\swarrow &\Downarrow^{\alpha}& \searrow^{a'} \\ F(c) && \stackrel{F(f)}{\to} && F(c') \\ \\ c &&\stackrel{f}{\to}&& c' } \right\} \,.

Properties

Definition

The Grothendieck construction factors through Grothendieck fibrations over $C$

$$\int :[C^{\mathrm{op}},\mathrm{Cat}]\to \mathrm{Fib}(C)\hookrightarrow \mathrm{Cat}/C$$\int : [C^{op}, Cat] \to Fib(C) \hookrightarrow Cat/C

and establishes an equivalence of 2-categories

$$[C^{\mathrm{op}},\mathrm{Cat}]\simeq \mathrm{Fib}(C).$$[C^{op}, Cat] \simeq Fib(C) \,.

When restricted to pseudofunctors that factor through Grpd $\hookrightarrow \mathrm{Cat}$ it factors through fibrations in groupoids

$$\int :[C^{\mathrm{op}},\mathrm{Grpd}]\to {\mathrm{Fib}}_{\mathrm{Grpd}}(C)\hookrightarrow \mathrm{Cat}/C$$\int : [C^{op}, Grpd] \to Fib_{Grpd}(C)\hookrightarrow Cat/C

and establishes an equivance of 2-categories

$$[C^{\mathrm{op}},\mathrm{Grpd}]sime{\mathrm{Fib}}_{\mathrm{Grpd}}(C).$$[C^{op}, Grpd] \sime Fib_{Grpd}(C) \,.

Proof

This can be verified by straightforward albeit somewhatt tedious checking. Details are spelled out in section 1.2 of

• Peter Johnstone, Sketches of an Elephant .

The statement itself is theorem 1.3.6 there, all definitions and lemmas are on the pages before that.

For the case of pseudofunctors with values in groupoids, there is a model category version of the Grothendieck construction discussed in

• Sharon Hollander, A homotopy theory for stacks (arXiv:math.AT/0110247).

There the statement of the above equivalence is the statement that the Grothendieck equivalence exhibits a Quillen equivalence between suitable model category structures on functors from and to $C$.

Adjoints to the Grothendieck construction

The Grothendieck construction functor

$$\int :[C^{\mathrm{op}},\mathrm{Cat}]\to \mathrm{Cat}/C$$\int : [C^{op}, Cat] \to Cat/C

has a left and a right adjoint functor.

Restricted to Grothendieck fibrations and fibrations in groupoids, these exhibit the above equivalences as an adjoint equivalences.

The left adjoint

The left adjoint is the functor

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

$$c/p=\left\{\begin{array}{ccc}& & c\\ & \swarrow & & \searrow \\ p(e_1)& & \to & & p(e_2)\end{array}\right\},$$c/p = \left\{ \array{ && c \\ & \swarrow && \searrow \\ p(e_1) &&\to&& p(e_2) } \right\} \,,

i.e.

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

This functor may equivalently be expressed as follows:

for given $(E\stackrel{p}{\to }C)$ consider the weak pushout

$$\begin{array}{ccc}E& \hookrightarrow & E^{▹}\\ {\downarrow }^p& ⇙& \downarrow \\ C& \to & K(p)\end{array}$$\array{ E &\hookrightarrow& E^{\triangleright} \\ \downarrow^{\mathrlap{p}} &\swArrow& \downarrow \\ C &\to& K(p) }

of 2-categories (so that $K(p)$ is a 2-category) where $K^{▹}$ is $K$ with one terminal object $v$ adjoined (a join of categories).

In terms of a cone construction

Claim

We have

$$c/p\simeq {\mathrm{Hom}}_{K(p)}(c,v).$$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 $p:E\to C$ to the pseudofunctor

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

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 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

$$\begin{array}{ccc}N(E)& \hookrightarrow & N(E{)}^{▹}\\ {\downarrow }^{N(p)}& ⇙& \downarrow \\ N(C)& \to & N(K(p))\end{array}$$\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 $N(E{)}^{▹}=N(E)\star *$ is the join of simplicial sets of $N(E)$ with the point.

By the general yoga of homotopy colimits (see there for details) we know that this $\mathrm{\infty }$-pushout here may be computed as an ordinary pushout in the 1-category sSet if the pushout diagram $N(C)\leftarrow N(E)\to N(E{)}^{▹}$ 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 ${\mathrm{sSet}}_{\mathrm{Joyal}}$.

But this is trivially verified since the cofibrations in ${\mathrm{sSet}}_{\mathrm{Joyal}}$ are just the monomorphisms in sSet: the degreewise injective maps of simplicial sets. So every object in ${\mathrm{sSet}}_{\mathrm{Joyal}}$ is cofibrant and the inclusion $N(E)\hookrightarrow N(E{)}^{▹}$ is a cofibration.

(The same conclusion would hold for the same simple reasons in the standard model structure on simplicial sets ${\mathrm{sSet}}_{\mathrm{Quillen}}$.)

From this it follows that simply because we passed from categories to their nerves, 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-morphisms is “freed”, the nerve is a suitably “puffed up” version of a category that is suitable for computing $\mathrm{\infty }$-pushouts).

So we are reduced to computing the ordinary pushout

$$\begin{array}{ccc}N(E)& \hookrightarrow & N(E{)}^{▹}\\ {\downarrow }^{N(p)}& & \downarrow \\ N(C)& \to & Q\end{array}$$\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 the bicategory $K(p)$ that we are after.

As recalled at limits and colimits by example in the section limits in presheaf categories, colimits (and hence pushouts) in the presheaf-category sSet $=\mathrm{Func}({\Delta }^{\mathrm{op}},\mathrm{Set})$ 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 $v$:

$$Q_0=N(C{)}_0\coprod \{v\}=p(\mathrm{Obj}(E))\coprod \{v\}$$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 $e\in \mathrm{Obj}(E)$ with $p(e)=c$ (this 1-cell is really the terminal 1-cell $e\to v$ in $E^{▹}$ 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

$$\begin{array}{ccccc}{c}_1& & \to & & c_2\\ & \searrow & {}^{(e_1\to e_2)}⇙& \swarrow \\ & & v\end{array}$$\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 ${\mathrm{sSet}}_{\mathrm{Joyal}}$ 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 an adjunction

$$(L⊣\int ):\mathrm{Fib}(C)\overleftarrow{\to }[C^{\mathrm{op}},\mathrm{Cat}]$$(L \dashv \int) : Fib(C) \stackrel{\leftarrow}{\to} [C^{op}, Cat]

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

Generalizations

$n=0$

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

$n=(\mathrm{\infty }\mathrm{,1})$

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

The correspondence between $(\mathrm{\infty }\mathrm{,1})$-categorical cartesian fibrations $E\to C$ and (∞,1)-presheaves $C\to (\mathrm{\infty }\mathrm{,1}){\mathrm{Cat}}^{\mathrm{op}}$ is 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

Warning on terminology

The term ‘Grothendieck Construction’ is applied in the literature to at least two very different constructions (and as 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 referes 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.

References

• Grothendieck construction (Wikipedia)
• The Grothendieck Construction (UCSB ITP Seminar)
• The Homotopy Theory of n-Fold Categories
• Inference System Integration via Logic Morphisms
• Category Theory for Computing Science

A model category presentation of the Grothendieck construction is given in

• Sharon Hollander, A homotopy theory for stacks (arXiv:math.AT/0110247)