The -Grothendieck construction is a generalization of the Grothendieck construction – which establishes an equivalence
between fibered categories/categories fibered in groupoids and pseudofunctors to Cat/to Grpd – from category theory to (∞,1)-category-theory.
The Grothendieck construction for ∞-groupoids constitutes an equivalence of (∞,1)-categories
between right fibrations of quasi-categories and (∞,1)-functors to ∞ Grpd, while the full Grothendieck construction for (∞,1)-categories constitutes an equivalence
between Cartesian fibrations of quasi-categories and (∞,1)-functors to (∞,1)Cat.
This correspondence may be modeled
For fibrations in -groupoids
The generalization of a category fibered in groupoids to quasi-category theory is a right fibration of quasi-categories.
Let be an (∞,1)-category. There is an equivalence of (∞,1)-categories
In the next section we discuss how this statement is presented in terms of model categories.
Model category presentation
We discuss a presentation of the -Grothendieck construction by a simplicial Quillen adjunction between simplicial model categories. (HTT, section 2.2.1).
(extracting a simplicial presheaf from a fibration)
In particular we will be interested in the case that is the identity, or at least an equivalence, identifying with .
For any object in consider the sSet-category obtained as the (ordinary) pushout in SSet Cat
where is the join of simplicial sets of with a single vertex .
Using this construction, define a functor, the straightening functor,
from the overcategory of sSet over to the enriched functor category of sSet-enriched functors from to by defining it on objects to act as
The straightening functor effectively computes the fibers of a Cartesian fibration over every point . As an illustration for how this is expressed in terms of morphisms in that pushout, consider the simple situation where
only has a single point;
is a category with three objects, two of them connected by a morphism
is the only possible functor, sending everything to the point.
Therefore the category of morphisms in this pushout from to is indeed again the category .
More on this is at Grothendieck construction in the section of adjoints to the Grothendieck construction.
With the definitions as above, let be an sSet-enriched functor between sSet-categories. Write
for the left sSet-Kan extension along .
There is a natural isomorphism of the straightening functor for the composite and the original straightening functor for followed by left Kan extension along :
This is HTT, prop. 220.127.116.11.. The following proof has kindly been spelled out by Harry Gindi.
We unwind what the sSet-categories with a single object adjoined to them look like:
be an sSet-enriched functor. Define from this a new sSet-category by setting
The composition operation is that induced from the composition in and the enriched functoriality of .
Observe that the sSet-category appearing in the definition of the straightening functor is
(because is with a single object and some morphisms to adjoined, such that there are no non-degenerate morphisms originating at , we have that is of form for some ; and is that by definition).
To prove the proposition, we need to compute the pushout
and show that indeed .
Using the pasting law for pushouts (see pullback) we just have to compute the lower square pushout. Here the statement is a special case of the following statement: for every sSet-category of the form , the pushout of the canonical inclusion along any -functor is .
This follows by inspection of what a cocone
is like: by the nature of the functor is characterized by a functor , an object together with a natural transformation
being the component of the -functor.
We may write this natural transformation as
where etc. means precomposition with the functor .
By commutativity of the diagram this is
Now by the definition of left Kan extension as the left adjoint to prescomposition with a functor, this is bijectively a transformation
Using this we see that we may find a universal cocone by setting with the canonical inclusion and given by on the restriction to and by the unit on . For this the adjunct transformation is the identity, which makes this universal among all cocones.
This functor has a right adjoint
that takes a simplicial presheaf on to a simplicial set over – this is the unstraightening functor.
This is HTT, theorem 18.104.22.168.
This models the Grothendieck construction for ∞-groupoids in the following way:
Hence the unstraightening functor is what models the Grothendieck construction proper, in the sense of a construction that generalizes the construction of a fibered category from a pseudofunctor.
Remark: -fibrations over an -groupoid
Let itself be an -groupoid. Then and hence
By the fact that there is the standard model structure on simplicial sets we have that every morphism of -groupoids factors as
where the top morphism is an equivalence and the right morphism a Kan fibration. Moreover, as discussed at right fibration, over an -groupoid the notions of left/right fibrations and Kan fibrations coincide. This shows that the full sub-(∞,1)-category of on the right fibrations is equivalent to all of .
For general fibered -categories
The generalization of a fibered category to quasi-category theory is a Cartesian fibration of quasi-categories.
Let be an (∞,1)-category. There is an equivalence
In the next section we discuss how this statement is presented in terms of model categories.
Model category presentation
Regard the (∞,1)-category in its incarnation as a simplicially enriched category.
Let be a simplicial set, the corresponding simplicially enriched category (where is the left adjoint of the homotopy coherent nerve) and let be an sSet-enriched functor.
(extracting a marked simplicial presheaf from a marked fibration) (HTT, section 3.2.1)
The straightening functor
from marked simplicial sets over to marked simplicial presheaves on is on the underlying simplicial sets (forgetting the marking) the same straightening functor as above.
On the markings the functor acts as follows.
Each edge of gives rise to an edge : the join 2-simplex of
with image in the pushout .
We define the straightening functor to assign that marking of edges which is the minimal one such that all such morphisms are marked in , for all marked in : this means that this marking is being completed under the constraint that be sSet-enriched functorial.
For that, recall that the hom simplicial sets of are the spaces , which consist of those simplices of the internal hom whose edges are all marked:
So we need to find a marking on the such that for all the composite
is a marked edge of the mapping complex. By the internal hom-adjunction this edge corresponds to a morphism
and to be marked needs to carry edges of the form i.e. 1-cells to marked edges
in . So we need to ensure that the edges of this form are marked:
we define that the straightening functor marks an edge in iff it is of this form , for a marked edge of and .
As in the unmarked cae, the straightening functor has an sSet-right adjoint, the unstraightening functor
This functor exhibits the -Grothendieck-construction proper, in that it constructs a Cartesian fibration from a given -functor:
Over an ordinary category
In the case that happens to be an ordinary category, the -Grothendieck construction can be simplified and given more explicitly.
This is HTT, section 3.2.5.
(relative nerve functor)
Let be a small category and let be a functor. The simplicial set , the relative nerve of under is defined as follows:
an -cell of is
a functor ;
for every a morphism ;
such that for all the diagram
There is a canonical morphism
to the ordinary nerve of , obtained by forgetting the s.
This is HTT, def. 22.214.171.124.
(marked relative nerve functor)
Let be a small category. Define a functor
where is with the marking forgotten, where is the relative nerve of under , and where the marking is given by …
This is HTT, def. 126.96.36.199.
This functor has a left adjoint . The value of on some is equivalent to the value of .
This is HTT, Lemma 188.8.131.52.
Relation beween the model structures
Cartesian fibrations over the point
For the base category being the point , the -Grothendieck construction naturally becomes essentially trivial. However, its model by the Quillen functor
is not entirely trivial and in fact produces a Quillen auto-equivalence of with itself that plays a central role in the proof of the corresponding Quillen equivalence over general .
Let be the cosimplicial simplicial set given by
Then: nerve and realization associated to yield a Quillen equivalence of with itself.
HTT, section 2.2.2.
Cartesian fibrations over the interval
A Cartesian fibration over the 1-simplex corresponds to a morphism (∞,1)Cat, hence to an (∞,1)-functor .
By the above procedure we can express as the image of under the straightening functor. However, there is a more immediate way to extract this functor, which we now describe.
First recall the situation for the ordinary Grothendieck construction: given a Grothendieck fibration , we obtain a functor between the fibers, by choosing for each object a Cartesian morphism . Then the universal property of Cartesian morphism yields for every morphism in the unique left vertical filler in
And again by universality, this assignment is functorial: .
Diagrammatically, the choice of Cartesian morphisms here is a lift in the diagram
This diagrammatic way of encoding the functor associated to a Grothendieck fibration over the interval generalizes straightforwardly to the quasi-category context.
Given a Cartesian fibration with fibers the quasi-categories and , an -functor associated to the Cartesian fibration is a functor such that there exists a commuting diagram in sSet
More generally, if we also specify possibly nontrivial equivalences of quasi-categories and , then a functor is associated to and this choice of equivalences if the first twoo conditions above are generalized to
This is HTT, def. 184.108.40.206.
For a Cartesian fibration, the associated functor exists and is unique up to equivalence in the (∞,1)-category of (∞,1)-functors .
This is HTT, prop 220.127.116.11.
Set and .
With the notation described at model structure for Cartesian fibrations, consider the commuting diagram
in the category of marked simplicial sets.
Here the left vertical morphism is marked anodyne: it is the smash product of the marked cofibration (monomorphism) with the marked anodyne morphism . By the stability properties discussed at Marked anodyne morphisms, this implies that the morphism itself is marked anodyne.
As discussed there, this means that a lift against the Cartesian fibration in
exists. This exhibits an associated functor .
Suppose now that another associated functor is given. It will correspondingly come with its diagram
Together this may be arranged to a diagram
where the top horizontal morphism picks the 2-horn in whose two edges are labeled by and , respectively.
Now, the left vertical morphism is still marked anodyne, and hence the lift exists, as indicated. Being a morphism of marked simplicial sets, it must map for each the edge to a Cartesian morphism in , and due to the commutativity of the diagram this morphism must be in , sitting over . But as discussed there, a Cartesian morphism over a point is an equivalence. This means that the restriction
is an invertible natural transformation between and , hence these are equivalent in the functor category.
Conversely, every functor gives rise to a Cartesian fibration that it is associated to, in the above sense.
Every -functor is associated to some Cartesian fibration , and this is unique up to equivalence.
This is HTT, prop 18.104.22.168.
The idea is that we obtain from first forming the cylinder and the identifying the left boundary of that with , using .
Formally this means that we form the pushout
in , where and are and with precisely the equivalences marked. This comes canonically with a morphism
and does have the property that , and that is associated to it in that the restriction of the canonical morphism to the 0-fiber is . But it may fail to be a Cartesian fibration.
To fix this, use the small object argument to factor as
where the first morphism is marked anodyne and the second has the right lifting property with respect to all marked anodyne morphisms and is hence (since every morphism in is marked) a Cartesian fibration.
It then remains to check that is still associated to this . This is done by observing that in the small object argument is built succesively from pushouts of the form
where the morphisms on the left are the generators of marked anodyne morphisms (see here). from this one checks that if the fiber is equivalent to , then so is and similarly for . By induction, it follows that is indeed associated to .
To see that the obtained this way is unique up to equivalence, consider…
Cartesian fibrations over simplices
… for the moment see HTT, section 3.2.2 …
The universal Cartesian fibration
for the moment see
The construction for -groupoid fibrations i.e. left/right fibrations is the content of section 2.2.1, that of quasi-category fibrations i.e. Cartesian fibrations that
More on model-category theoretic construction of the -Grothendieck construction with values in -groupoids is in