nLab Grothendieck construction




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

which is a fully faithful 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) \textstyle{\int} \,\colon\, Func(C^{op}, Cat) \overset{\simeq}{\longrightarrow} 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). \textstyle{\int} \;\colon\; Func(C^{op}, Grpd) \overset{\simeq}{\longrightarrow} 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 CC).

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 CC.


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

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

namely the 2-category of “lax-pointedcategories, also known as the “lax slice” of Cat under the terminal category *Cat\ast \,\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 the evident forgetful functor.

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

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

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)ϕa)\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

:[C,Cat]Cat/C \textstyle{\int} \;\colon\; [C, Cat] \longrightarrow 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 from an opposite category. 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} } \,.

constitutes a 2-functor

:[C op,Cat]Cat/C. \textstyle{\int} \;\colon\; [C^{op},Cat] \longrightarrow 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,aϕF(f)(a))\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 opCatC^{op} \to Cat, since, in that case, the morphisms between the objects of CC go in the opposite direction to one another.


(Co)Limits in a Grothendieck construction

We discuss existence and characterization of (co)limits in a Grothendieck construction.


Given a pseudofunctor F:C opCatF \colon C^{op} \to Cat.


  1. CC is complete.

  2. F(J)F(J) is complete for all JCJ \in C.

  3. F(f):F(J)F(K)F(f) : F(J) \to F(K) preserves limits for all f:KJf \colon K \to J in CC.

then F\int F is complete.

Dually, if

  1. CC is cocomplete.

  2. F(J)F(J) is cocomplete for all JCJ \in C.

  3. F(f):F(J)F(K)F(f) : F(J) \to F(K) has a left adjoint for all f:KJf : K \to J in CC.

Then F\int F is cocomplete.

This proven in Tarlecki, Burstall & Goguen (1991), §3.1, 3.2.

The case of colimits is also described in Harpaz & Prasma (2015), Prop. 2.4.4:

Given a pseudofunctor

(1)C:Base Cat 𝒳 C 𝒳 f f ! f * 𝒴 C 𝒴 \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. BaseBase is cocomplete

  2. C 𝒳\mathbf{C}_{\mathcal{X}} is cocomplete for each 𝒳Base\mathcal{X} \in Base

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

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

Given a diagram in the Grothendieck construction

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

its underlying diagram in BaseBase

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

has a colimit by assumption on BaseBase, with coprojection morphisms to be denoted like this:

𝒳(i)q(i)limjI𝒳(j). \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 C ()\int \mathbf{C}_{(-)} is obtained by

  1. first pushing all morphisms ϕ:f !𝒱(i)𝒱(j)\phi \colon f_!\mathscr{V}(i) \to \mathscr{V}(j) in the diagram forward along the respective q jq_j

  2. to hence obtain a diagram q !𝒱q_! \mathscr{V} in C lim𝒳\mathbf{C}_{\underset{\longrightarrow}{\lim} \mathcal{X}}

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

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


(Cartesian product in Grothendick construction is external product on fiber categories )

Given a contravariant pseudofunctor

X f Y𝒞 X f * 𝒞 Y \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 𝒞 ()\mathcal{C}_{(-)} have Cartesian products and all base change maps f *f^\ast preserve these products, then the Grothendieck construction X𝒞 X\int_X \mathcal{C}_X has cartesian products given on objects

𝒱 X(𝒱𝒞 X) \mathscr{V}_X \,\equiv\, \big( \mathscr{V} \,\in\, \mathcal{C}_X \big)

by the formula

(2)𝒱 X×𝒲 Y((pr X *𝒱)×(pr Y *𝒲)) X×Y, \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

X×Y pr X pr Y X Y \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 (2) is known as an external tensor product, here the “external Cartesian product” on the fiber categories; see also this Proposition at free coproduct completion.

As an oplax colimit

The Grothendieck construction on F:CCatF : C \to Cat is equivalently the oplax colimit of FF (e.g Gepner-Haugseng-Nikolaus 15). 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.

Local smallness

In general, a (weighted) colimit of a 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:


If F:C opCatF:C^{op}\to Cat is a pseudofunctor, CC is locally small, and each category F(c)F(c) is locally small, then the Grothendieck construction F\int F is also locally small.


Recall that 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')). Local smallness of CC means that there is only a set of such ff‘s, and local smallness of F(c)F(c) means that for each ff there is only a set of such α\alpha’s.

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

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 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


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.


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


The Grothendieck construction factors through Grothendieck fibrations over CC

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

and establishes an equivalence of bicategories

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

and establishes a similar equivalence

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


This can be verified by straightforward albeit somewhat tedious checking. Details are spelled out in Johnstone (2002), B1.3. (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

The Grothendieck construction functor

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

has a left and a right adjoint functor.

Restricted to groupoid-valued functors and fibrations in groupoids, both of these exhibit the above equivalences as adjoint equivalences. 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][C^{op}, GpdProf] in order to get all of the objects of Gpd/CGpd/C.

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/pc/p with objects given by pairs (e,cp(e))(e, c \to p(e)) and morphisms by commutative triangles

c p(e 1) p(e 2) \array{ && c && \\ & \swarrow && \searrow & \\ p(e_1) &&\to&& p(e_2) }


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).


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 \colon E \to C to the pseudofunctor

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


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 part of an adjunction

(L):Fib(C)[C op,Cat] (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):Fib(C)[C op,Cat], (R \vdash \textstyle{\int}) \;\colon\; Fib(C) \rightleftarrows [C^{op}, Cat] \,,

where the right adjoint RR sends a Grothendieck fibration FF over CC to the presheaf

cHom(y(c),F), c \;\mapsto\; Hom(\textstyle{\int} y(c), F) \,,

where y(c)\int y(c) is the Grothendieck construction applied to the representable presheaf of sets (hence discrete categories) on cc and HomHom denotes the category of morphisms between two Grothendieck fibrations.

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

Behaviour under simplicial nerve


For F:𝒟CatF \colon \mathcal{D} \to Cat a functor, let

|N(F())|:𝒟FCat|N()|Top {\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

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

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

This is due to (Thomason 79).

Further properties


Given a contravariant pseudofunctor

D op C () Cat x C x f f * x C x \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

𝒟RLD \mathcal{D} \underoverset {\underset{R}{\longleftarrow}} {\overset{L}{\longrightarrow}} {\;\; \bot \;\;} \mathbf{D}

then there is an induced adjunction between the Grothendieck constructions on C ()C_{(-)} and on C L()C_{L(-)} covering the given adjunction

{𝒱 xϕ f𝒱 x|𝒱 ϕ L(f) *𝒱 C L(x) x f x 𝒟} (x𝒟C L(x)) R^L^ (xDC x) {𝒱 xϕ f𝒱 x|𝒱 ϕ f *𝒱 C x x f x D} 𝒟 RL D \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 L^\widehat L acts as LL on underlying morphisms and as the identity on components:

𝒱 x ϕ f 𝒱 xL^𝒱 L(x) ϕ L(f) 𝒱 L(x), \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 R^\widehat R acts as RR on underlying morphisms and on components by base change along the adjunction unit ϵ x:LR(x)x\epsilon_{\mathbf{x}} \colon L \circ R(\mathbf{x}) \to \mathbf{x}:

𝒱 x ϕ f 𝒱 xR^(ϵ x *𝒱) R(x) (ϵ x *ϕ) R(f) (ϵ x *𝒱) R(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:

(3)ϵ 𝒱 x L^R^:L^R^(𝒱 x)=((ϵ x LR) *𝒱) LR(x)(id) ϵ x LR𝒱 x \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}}


First notice that R^\widehat R is indeed well-defined in that we have a natural isomorphism on the right of

ϵ x *𝒱ϵ x *ϕϵ x *f *𝒱(LR(f)) *ϵ x *𝒱 \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 pseudo-functoriality of C ()C_{(-)} applied to the commutativity of the naturality square of the adjunction counit:

LR(x) LR(f) LR(x) ϵ x ϵ x x f x. \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:xR(x)f \colon x \to R(\mathbf{x}') for the (LR)(L \dashv R)-adjunct of a given f:L(x)x\mathbf{f} \colon L(x) \to \mathbf{x}' then we have natural bijections

{L^(𝒱 x)ϕ f𝒱 x} {𝒱 L(x)ϕ f𝒱 x} by def. {𝒱 xϕ f(ϵ x *𝒱) R(x)} see below {𝒱 xϕ fR^(𝒱 x)} by def., \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 (LR)(L \dashv R)-hom-isomorphism

{L(x)fx}{xfR(x)} \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

{𝒱ϕf *𝒱} {𝒱ϕ(ϵ xL(f)) *𝒱} formula for adjuncts {𝒱ϕL(f) *(ϵ x *𝒱)} pseudo-functoriality, \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 f=ϵ xL(f)\mathbf{f} \,=\, \epsilon_{\mathbf{x}'} \circ L(f) (here) that expresses adjuncts in terms of the counit and the second step is the contravariant pseudo-functoriality of C ()C_{(-)}.

This establishes a hom-isomorphism exhibiting adjoint functors L^R^\widehat L \dashv \widehat R. Moreover, the image of an identity morphism on R^()\widehat{R}(-) under this hom-isomorphism is the claimed counit (3).


n=0n = 0

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

n=(,0)n = (\infty,0)

The analog of the Grothendieck construction for ∞-groupoids is examined in detail in Heuts-Moerdijk 13.

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

The two functors h !h_! and r *r^* become naturally weakly equivalent once we derive them, but they are not isomorphic. The functor r *r^* restricted to the full subcategory of presheaves of groupoids recovers the nerve of the classical Grothendieck construction described above. The functor h !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=(,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

Normal lax functors into ProfProf

The Grothendieck construction can be generalized from pseudofunctors into CatCat to normal lax functors into Prof. Instead of fibrations over CC, such normal lax functors correspond to arbitrary functors into CC. 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 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.



(representable functor and slice categories)
A representable functor C(,X):C opSetCatC(-,X) \colon 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.


(slice categories and arrow category)
For 𝒮\mathcal{S} any category, let

𝒮 ():𝒮Cat \mathcal{S}_{(-)} \,\colon\, \mathcal{S} \longrightarrow Cat

be the pseudofunctor which sends

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

𝒮 I B𝒮𝒮 /B. \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.

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

But if 𝒮\mathcal{S} has all pullbacks, then there is also the contravariant pseudofunctor

𝒮 ():𝒮 opCat \mathcal{S}_{(-)} \,\colon\, \mathcal{S}^{op} \longrightarrow Cat

which sends

  • an object B𝒮B \,\in\, \mathcal{S} to the slice category 𝒮 /B\mathcal{S}_{/B},

  • a morphism f:BBf \colon B \to B' to the base change functor f *:𝒞 B𝒞 /Bf^\ast \,\colon\, \mathcal{C}_{B'} \to \mathcal{C}_{/B} given by pullback in 𝒞\mathcal{C}.

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

In slight variation of Exp. :


(pointed slice categories and retractive spaces)
Let 𝒮\mathcal{S} be a category with all pushouts and consider the pseudofunctor

(𝒮 ()) */:𝒮Cat \big(\mathcal{S}_{(-)}\big)^{\ast/} \,\colon\, \mathcal{S} \longrightarrow Cat

which sends

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

𝒮 B𝒮(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; Hebestreit, Sagave & Schlichtkrull (2020), Lem. 2.14, where this is the basis of a model category-presentation of the tangent \infty -category of (the simplicial localization of) 𝒮\mathcal{S}.

In alternative slight variation of Exp. :


(abelianized slice categories and tangent category) For 𝒮\mathcal{S} a category with finite limits, let

Ab(𝒮 /()):𝒮 opCat Ab\big( \mathcal{S}_{/(-)} \big) \;\; \colon \;\; \mathcal{S}^{op} \longrightarrow Cat

be the contravariant pseudofunctor which sends

The Grothendieck construction on this functor may be called the tangent category of 𝒮\mathcal{S}.


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]


Further textbook accounts:

Survey in the generality of enriched-, internal- and \infty -category theory (see also the enriched- and \infty -Grothendieck construction):

  • Liang Ze Wong, The Grothendieck Construction in Enriched, Internal and ∞-Category Theory, PhD thesis, Univ. Washington (2019) [pdf, pdf]

The geometric realization of Grothendieck constructions has been analyzed in

  • 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

On limits and colimits in Grothendieck constructions:

The analog for simplicial sets instead of groupoids is discussed in

See also

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)

On the enriched Grothendieck construction:

A monoidal version:

Last revised on June 12, 2024 at 10:42:50. See the history of this page for a list of all contributions to it.