In any context it is of interest to ask which kind of morphisms
arise as pullbacks along a classifying morphism to some universal object of some universal morphism
The Grothendieck construction describes this in the context of Cat: a morphism of categories – i.e. a functor – is called a fibered category or Grothendieck fibration if it is encoded in a pseudofunctor/2-functor .
The reconstruction of from the pseudofunctor is the Grothendieck construction
between 2-functors and Grothendieck fibrations over .
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 ).
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.
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 . Here denotes the (2-)category of “lax-pointed” categories, also known as the “lax slice” of under the terminal category . Its objects are pointed categories, i.e. pairs where is a category and is an object of , and its morphisms are pairs where is a functor and is a morphism in . The projection is just the forgetful functor.
This means that
the objects of are pairs , where and
and morphisms in are given by pairs . This may be visualized as
This extends to a 2-functor between bicategories
The more commonly described version of this construction works instead on contravariant pseudofunctors, i.e. pseudofunctors . In this case we use instead the “universal -cobundle” , where is the colax slice, whose objects are again pointed categories , but whose morphisms are pairs where and . Now the 2-pullback
describes a 2-functor
In this case,
the objects of are again pairs , where and , but
the morphisms in from to are pairs .
that is natural in , where is the constant functor with value . (See oplax colimit for an explanation of why lax natural transformations appear in the definition of an oplax colimit.)
A lax natural transformation from to is given by
We want to show that to each such lax transformation there corresponds an essentially unique functor . So firstly, given as above, let be the functor that sends to , and acts on arrows as
That is a functor follows from the coherence properties of with respect to identities and composition in .
Conversely, if is a functor, we get a lax transformation as follows:
As one might expect, the coherence conditions on the resulting follow from the functoriality of .
It is then easy to check that these two mappings form a bijection between the objects of and .
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 is a pseudonatural transformation if and only if the functor inverts (i.e. sends to an isomorphism) each member of the class of morphisms of whose second component is an identity. (These are in fact the opcartesian morphisms with respect to the projection .) The localization is therefore the (weak) 2-colimit of :
This last result appears in SGA4 Exposé VI, Section 6.
One can characterize the image of the Grothendieck construction as consisting precisely of those objects in that are Grothendieck fibrations.
objects are pseudofunctors ;
morphism are pseudonatural transformations;
2-morphism are modifications.
A morphism of Grothendieck fibrations is
a 2-morphism between morphism is a natural transformation of the underlying functors, that also makes the obvious diagram 2-commute, i.e. such that is trivial.
Compositions are those induced from the underlying functors and natural transformations.
This defines the 2-category of Grothendieck fibrations
Cartesian lifts are not required to be unique, but are automatically unique up to a unique vertical isomorphism connecting their domains.
The Grothendieck construction factors through Grothendieck fibrations over
and establishes an equivalence of bicategories
and establishes a similar equivalence
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.
For the case of pseudofunctors with values in groupoids, there is a model category version of the Grothendieck construction discussed in
This model category incarnation of the Grothendieck construction generalizes to a model category presentation of the (∞,1)-Grothendieck construction.
The Grothendieck construction 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 is the functor
that assigns to a functor the presheaf which sends to the comma category with objects given by pairs and morphisms by commutative triangles
This functor may equivalently be expressed as follows.
For given consider the (3,1)-pushout
of (2,1)-categories , where is with one terminal object adjoined (a join of categories). (Here , and are 1-catgeories regarded trivially as -categories and where will in general be a (2,1)-category with nontrivial 2-morphisms).
And hence the left adjoint to the Grothendieck construction may be realized as the assignment that sends to the pseudofunctor
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
By the general yoga of homotopy colimits (see there for details) we know that this -pushout here may be computed as an ordinary pushout in the 1-category sSet if the pushout diagram 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 .
But this is trivially verified since the cofibrations in are just the monomorphisms in sSet: the degreewise injective maps of simplicial sets. So every object in is cofibrant and the inclusion is a cofibration.
(The same conclusion would hold for the same simple reasons in the standard model structure on simplicial sets .)
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 -pushouts).
So we are reduced to computing the ordinary pushout
As recalled at limits and colimits by example in the section limits in presheaf categories, colimits (and hence pushouts) in the presheaf-category sSet are computed for each object as ordinary colimits in Set.
For we see that is the collection of objects of and one additional vertex :
For similarly we find that consists of the 1-cells in in and in addition of one 1-cell for each with (this 1-cell is really the terminal 1-cell in but with its source re-interpreted as being according to the identification of as above). In the fibrant replacement of the composite of original 1-cells and the new 1-cells will be freely added, so that the general 1-morphism will consist of a 1-morphism in together with a lift of to . This is just as in the comma category .
For we have in the 2-cells in as well as one 2-cell
for each 1-cell in with = .
In particular this means that if is a morphism in and is a morphism in , then the composite in is homotopic to any compatible direct morphism in .
This means that forming the fibrant replacement of in will not throw in superfluous 1-morphisms on top of those we already discussed in the previous paragraph…
This formulation of the Grothendieck construction as an adjunction
For a functor, let
be its postcomposition with geometric realization of categories
Then we have a weak homotopy equivalence
This is due to (Thomason 79).
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 is a left adjoint, it implements the homotopy colimit using the diagonal of a bisimplicial set, and the second functor is a right adjoint, it uses the codiagonal (also known as the totalization) of a bisimplicial set. Both functors fit into adjunctions and , where the other two adjoints can be seen as rectification functors: the right adjoint generalizes the cleavage construction, whereas the left adjoint generalizes the comma category construction above.
The two functors and become naturally weakly equivalent once we derive them, but they are not isomorphic. The functor restricted to the full subcategory of presheaves of groupoids recovers the nerve of the classical Grothendieck construction described above. The functor 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.
The correspondence between -categorical cartesian fibrations and (∞,1)-presheaves 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
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.
Standard references are in
The geometric realization of Grothendieck constructions has been analyzed in
The left adjoint to the Grothendieck construction is discussed in §3.1.1 of
The analog for simplicial sets instead of groupoids is discussed in
A model category presentation of the Grothendieck construction is given in
Discussion of the Grothendieck construction as a lax colimit includes (see also at (infinity,1)-Grothendieck construction)