nLab
Grothendieck construction

Contents

Idea

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

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

arise as pullbacks along a classifying morphism S p:CUS_p : C \to U to some universal object UU of some universal morphism

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

The Grothendieck construction describes this in the context of Cat: a morphism p:ECp : 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 opCatS_p : C^{op} \to Cat.

The reconstruction of pp from the pseudofunctor S pS_p is the Grothendieck construction

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

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

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

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

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

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

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

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

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. In line with the philosophy of generalized universal bundles, the “universal Cat-bundle” is Cat *,CatCat_{*,\ell} \to Cat. Here Cat *,Cat_{*,\ell} denotes the (2-)category of “lax-pointedcategories, also known as the “lax slice” of CatCat under the terminal category *Cat*\in Cat. Its objects are pointed categories, i.e. pairs (A,a)(A,a) where AA is a category and aa is an object of AA, and its morphisms (A,a)(B,b)(A,a) \to (B,b) are pairs (f,γ)(f,\gamma) where f:ABf\colon A\to B is a functor and γ:f(a)b\gamma\colon f(a)\to b is a morphism in BB. The projection Cat *,CatCat_{*,\ell} \to Cat is just the forgetful functor.

Then if F:CCatF\colon C \to Cat is a pseudofunctor from a category CC to CatCat, the Grothendieck construction for FF is the (strict) 2-pullback p:FC p : \int F \to C of Cat *,CatCat_{*,\ell} \to Cat along FF:

F Cat *, p C F Cat. \array{ \int F &\to& Cat_{*,\ell} \\ {}^{p}\downarrow && \downarrow \\ C &\overset{F}{\to}& Cat } \,.

This means that

  • the objects of F\int F are pairs (c,a)(c,a), where cObj(C)c \in Obj(C) and aObj(F(c))a \in Obj(F(c))

  • and morphisms in F\int F are given by pairs (cfc,F(f)(a)αa)(c \overset{f}{\to} c', F(f)(a) \overset{\alpha}{\to} a'). This may be visualized as

F={ * a α a F(c) F(f) F(c) c f c}. \int F = \left\{ \array{ && {*} \\ & {}^a\swarrow &\seArrow^{\alpha}& \searrow^{a'} \\ F(c) && \stackrel{F(f)}{\to} && F(c') \\ \\ c &&\stackrel{f}{\to}&& c' } \right\} \,.

This extends to a 2-functor between bicategories

:[C,Cat]Cat/C \int \;\; : \;\; [C, Cat] \to Cat/C

from pseudofunctors on CC to the overcategory of Cat over CC.

The more commonly described version of this construction works instead on contravariant pseudofunctors, i.e. pseudofunctors C opCatC^{op}\to Cat. In this case we use instead the “universal CatCat-cobundle” (Cat *,c) opCat op(Cat_{*,c})^{op} \to Cat^{op}, where (Cat *,c)(Cat_{*,c}) is the colax slice, whose objects are again pointed categories (A,a)(A,a), but whose morphisms (A,a)(B,b)(A,a) \to (B,b) are pairs (f,γ)(f,\gamma) where f:ABf\colon A\to B and γ:bf(a)\gamma\colon b \to f(a). Now the 2-pullback

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

describes a 2-functor

:[C op,Cat]Cat/C. \int \quad\colon\quad [C^{op},Cat] \to Cat/C.

In this case,

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

  • the morphisms in F\int F from (c,a)(c,a) to (c,a)(c',a') are pairs (cfc,aαF(f)(a))(c \overset{f}{\to} c', a \overset{\alpha}{\to} F(f)(a')).

Properties

As an oplax colimit

The Grothendieck construction on F:CCatF : C \to Cat is equivalently the oplax colimit of FF. That means that for each category XX there is an equivalence of categories

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

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

A lax natural transformation α\alpha from FF to ΔX\Delta X is given by

  • for each object cc of CC, a functor α c:FcX\alpha_c \colon F c \to X, and
  • for each morphism m:cdm \colon c \to d in CC, a natural transformation α m:α cα dm *\alpha_m \colon \alpha_c \Rightarrow \alpha_d \circ m_* (writing m *=Fmm_* = F m),

such that α 1 c\alpha_{1_c} is the isomorphism F1 c1 FcF 1_c \cong 1_{F c} given by pseudofunctoriality of FF, and that if m:cdm \colon c \to d, n:den \colon d \to e is a composable pair in CC, then α nm\alpha_{n m} is equal to the obvious pasting of α m\alpha_m and α n\alpha_n.

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

(m:cd,f:m *xy)α cxα mxα dm *xα dfα dy (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 AA is a functor follows from the coherence properties of α\alpha with respect to identities and composition in CC.

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

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

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

It is then easy to check that these two mappings form a bijection between the objects of Lax(F,ΔX)Lax(F, \Delta X) and [F,X][\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 FX\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 SS of morphisms of F\int F whose second component is an identity. (These are in fact the opcartesian morphisms with respect to the projection FC\int F \to C.) The localization F[S 1]\int F[S^{-1}] is therefore the (weak) 2-colimit of FF:

Ps(F,ΔX)[F,X] S 1[F[S 1],X] 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.

The equivalence between fibrations and pseudofunctors

One can characterize the image of the Grothendieck construction as consisting precisely of those objects in Cat/CCat/C that are Grothendieck fibrations.

We recall the definition of the bicategory of Grothendieck fibrations and pseudofunctors and and then state the main equivalence theorem.

The bicategory of pseudofunctors.

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

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

The bicategory of fibrations

Definition

A functor p:ECp : E \to C is a Grothendieck fibration if for every object eEe \in E and every morphism f:cp(e)f : c \to p(e) in CC there is a morphism f^:c^e\hat f : \hat c \to e in EE that lifts ff in that p(f^)=fp(\hat f) = f and which is a Cartesian morphism.

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

  • a functor F:EEF : E \to E'

  • such that

    • FF sends Cartesian morphisms to Cartesian morphisms;

    • the diagram

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

      in Cat commutes (strictly).

  • a 2-morphism between morphism η:FF\eta : F \to F' is a natural transformation of the underlying functors, that also makes the obvious diagram 2-commute, i.e. such that pηp' \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)Cat/C Fib(C) \hookrightarrow Cat/C

over CC, being a 2-subcategory of the overcategory of Cat over CC.

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

Definition

The Grothendieck construction factors through Grothendieck fibrations over CC

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

and establishes an equivalence of bicategories

:[C op,Cat]Fib(C). \int : [C^{op}, Cat] \stackrel{\simeq}{\to} 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 CatCat-enriched categories.

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

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

and establishes a similar equivalence

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

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

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

Model category version

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

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 CC.

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

Adjoints to the Grothendieck construction

The Grothendieck construction functor

:[C op,Cat]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, both of these exhibit the above equivalences as adjoint equivalences. Notice that much of the traditional literature discusses (just) the right adjoint.

The left adjoint

The left adjoint is the functor

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

that assigns to a functor pp the presheaf which sends cCc \in C to the comma category

c/p={ c p(e 1) p(e 2)}, c/p = \left\{ \array{ && c \\ & \swarrow && \searrow \\ p(e_1) &&\to&& p(e_2) } \right\} \,,

i.e.

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

This functor may equivalently be expressed as follows.

In terms of a cone construction

For given (EpC)(E \stackrel{p}{\to} C) consider the (3,1)-pushout

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

of (2,1)-categories , where K K^{\triangleright} is KK with one terminal object vv adjoined (a join of categories). (Here EE, CC and E E^{\triangleright} are 1-catgeories regarded trivially as (2,1)(2,1)-categories and where K(p)K(p) will in general be a (2,1)-category with nontrivial 2-morphisms).

Claim

We have

c/pHom 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:ECp : E \to C to the pseudofunctor

L(p):=Hom K(p)(,v):C opCat. 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

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

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

By the general yoga of homotopy colimits (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)N(E)N(E) 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 JoyalsSet_{Joyal}.

But this is trivially verified since the cofibrations in sSet JoyalsSet_{Joyal} are just the monomorphisms in sSet: the degreewise injective maps of simplicial sets. So every object in sSet JoyalsSet_{Joyal} is cofibrant and the inclusion N(E)N(E) 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 QuillensSet_{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 \infty-pushouts).

So we are reduced to computing the ordinary pushout

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

in sSet. The fibrant replacement of QQ is then the nerve of the bicategory K(p)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 =Func(Δ op,Set)= Func(\Delta^{op}, Set) are computed for each object [n]Δ[n] \in \Delta as ordinary colimits in Set.

For n=0n=0 we see that Q 0Q_0 is the collection of objects of CC and one additional vertex vv:

Q 0=N(C) 0{v}=p(Obj(E)){v} Q_0 = N(C)_0 \coprod \{ v\} = p(Obj(E)) \coprod \{v \}

For n=1n=1 similarly we find that Q 1Q_1 consists of the 1-cells in in CC and in addition of one 1-cell e:cve : c \to v for each eObj(E)e \in Obj(E) with p(e)=cp(e) = c (this 1-cell is really the terminal 1-cell eve \to v in E E^{\triangleright} but with its source re-interpreted as being p(e)=cp(e) = c according to the identification of Q 0Q_0 as above). In the fibrant replacement of QQ the composite of original 1-cells c 1c 2c_1 \to c_2 and the new 1-cells e:c 2ve : c_2 \to v will be freely added, so that the general 1-morphism c 1vc_1 \to v will consist of a 1-morphism c 1c 2c_1 \to c_2 in CC together with a lift of c 2c_2 to EE. This is just as in the comma category c/pc/p.

For n=2n=2 we have in Q 2Q_2 the 2-cells in CC as well as one 2-cell

c 1 c 2 (e 1e 2) v \array{ c_1 &&\to&& c_2 \\ & \searrow &{}^{(e_1 \to e_2)}\swArrow& \swarrow \\ && v }

for each 1-cell (e 1e 2)(e_1 \to e_2) in N(E)N(E) with p(e 1e 2)p(e_1 \to e_2) = (c 1c 2)(c_1 \to c_2).

In particular this means that if e 2:c 2ve_2: c_2 \to v is a morphism in QQ and c 1c 2c_1 \to c_2 is a morphism in CC, then the composite c 1c 2vc_1 \to c_2 \to v in QQ is homotopic to any compatible direct morphism c 1vc_1 \to v in QQ.

This means that forming the fibrant replacement of QQ in sSet JoyalsSet_{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):Fib(C)[C op,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=0n = 0

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

n=(,1)n = (\infty,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 (,1)(\infty,1)-categorical cartesian fibrations ECE \to C and (∞,1)-presheaves C(,1)Cat opC \to (\infty,1)Cat^{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

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 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

A representable functor C(,X):C opSetCatC(-,X) : C^{op} \to Set \hookrightarrow Cat maps under the Grothendieck construction to the slice category C/XC/X. The corresponding fibrations C/XCC/X \to C are also called representable fibered categories.

References

Standard references are in

See also

A model category presentation of the Grothendieck construction is given in

Revised on April 8, 2014 09:07:35 by Tim Porter (2.26.40.160)