equivalences in/of $(\infty,1)$-categories
The $(\infty,1)$-Grothendieck construction is a generalization of the Grothendieck construction – which establishes an equivalence
and
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
in the case of $\infty$-groupoids by a Quillen equivalence between the model structure for right fibrations and the projective global model structure on simplicial presheaves
in the case of $(\infty,1)$-categories by a Quillen equivalence between the model structure for Cartesian fibrations and the global model structure on functors with values in the model structure on marked simplicial sets.
The generalization of a category fibered in groupoids to quasi-category theory is a right fibration of quasi-categories.
($(\infty,0)$-Grothendieck construction)
Let $C$ be an (∞,1)-category. There is an equivalence of (∞,1)-categories
where
on the left we have the (∞,1)-category of right fibrations $RFib(C)$ – incarnated as the full SSet-subcategory of the overcategory $SSet/C$ on right fibrations;
and on the right the (∞,1)-category of (∞,1)-functors from the opposite category $C^{op}$ to ∞Grpd, i.e. the (∞,1)-category of (∞,1)-presheaves on $C$.
In the next section we discuss how this statement is presented in terms of model categories.
We discuss a presentation of the $(\infty,0)$-Grothendieck construction by a simplicial Quillen adjunction between simplicial model categories. (HTT, section 2.2.1).
(extracting a simplicial presheaf from a fibration)
Let
$S$ be a simplicial set, $\tau_hc(S)$ the corresponding SSet-category (under the left adjoint $\tau_{hc} : SSet \to SSet Cat$ of the homotopy coherent nerve, denoted $\mathfrak{C}$ in HTT);
$C$ an SSet-category;
$\phi : \tau_{hc}(S) \to C$ a morphism of SSet-categories.
In particular we will be interested in the case that $\phi$ is the identity, or at least an equivalence, identifying $C$ with $\tau_{hc}(S)$.
For any object $(p : X\to S)$ in $sSet/S$ consider the sSet-category $K(\phi,p)$ obtained as the (ordinary) pushout in SSet Cat
where $X^{\triangleright} = X \star \{v\}$ is the join of simplicial sets of $X$ with a single vertex $v$.
Using this construction, define a functor, the straightening functor,
from the overcategory of sSet over $S$ to the enriched functor category of sSet-enriched functors from $C^{op}$ to $sSet$ by defining it on objects $(p : X \to S)$ to act as
The straightening functor effectively computes the fibers of a Cartesian fibration $(p : X \to C)$ over every point $x \in C$. As an illustration for how this is expressed in terms of morphisms in that pushout, consider the simple situation where
$C = *$ only has a single point;
$X = \left\{ a \to b \;\;\; c\right\}$ is a category with three objects, two of them connected by a morphism
$p : X\to C$ is the only possible functor, sending everything to the point.
Then
and
Therefore the category of morphisms in this pushout from $*$ to $v$ is indeed again the category $\{a \to b \;\;\; c\}$.
More on this is at Grothendieck construction in the section of adjoints to the Grothendieck construction.
With the definitions as above, let $\pi : C \to C'$ be an sSet-enriched functor between sSet-categories. Write
for the left sSet-Kan extension along $\pi$.
There is a natural isomorphism of the straightening functor for the composite $\pi \circ \phi$ and the original straightening functor for $\phi$ followed by left Kan extension along $\pi$:
This is HTT, prop. 2.2.1.1.. 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:
let
be an sSet-enriched functor. Define from this a new sSet-category $C_F$ by setting
$Obj(C_F) = Obj(C) \coprod \{\nu\}$
$C_F(c,d) = \left\{ \array{ C(c,d) & for c,d \in Obj(C) \\ F(c) & for c \in Obj(c) and d = \nu \\ \emptyset & for c = \nu and d \in Obj(C) \\ * & for c = d = \nu } \right.$
The composition operation is that induced from the composition in $C$ and the enriched functoriality of $F$.
Observe that the sSet-category $K(\phi,p)$ appearing in the definition of the straightening functor is
(because $K(\phi,p)$ is $C$ with a single object $\nu$ and some morphisms to $\nu$ adjoined, such that there are no non-degenerate morphisms originating at $\nu$, we have that $K(\phi,p)$ is of form $C_F$ for some $F$; and $St_\phi(X)$ is that $F$ by definition).
To prove the proposition, we need to compute the pushout
and show that indeed $Q \simeq C'_{\pi_! St_\phi(X)}$.
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 $C_F$, the pushout of the canonical inclusion $C\to C_F$ along any $sSet$-functor $\pi : C \to C'$ is $C'_{\pi_! F}$.
This follows by inspection of what a cocone
is like: by the nature of $C_F$ the functor $d$ is characterized by a functor $d|_C : C \to Q$, an object $d(\nu) \in Q$ together with a natural transformation
being the component $F_{c,\nu} : C_F(c,\nu) \to Q(d(c), d(\nu))$ of the $sSet$-functor.
We may write this natural transformation as
where $d^*$ etc. means precomposition with the functor $d$.
By commutativity of the diagram this is
Now by the definition of left Kan extension $\pi_!$ 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 $Q := C'_{\pi_! F}$ with $r : C' \to Q$ the canonical inclusion and $C_{F} \to C'_{\pi_! F}$ given by $\pi$ on the restriction to $C$ and by the unit $F \to \pi^* \pi_! F$ on $C_F(c,\nu)$. For this the adjunct transformation $\eta$ is the identity, which makes this universal among all cocones.
This functor has a right adjoint
that takes a simplicial presheaf on $C$ to a simplicial set over $S$ – this is the unstraightening functor.
One checks that $St_\phi$ preserves colimits. The claim then follows with the adjoint functor theorem.
(presentation of the $(\infty,0)$-Grothendieck construction)
The straightening and the unstraightening functor constitute a Quillen adjunction
between the model structure for right fibrations and the global projective model structure on simplicial presheaves on $S$.
If $\phi$ is a weak equivalence in the model structure on simplicial categories then this Quillen adjunction is a Quillen equivalence.
This is HTT, theorem 2.2.1.2.
This models the Grothendieck construction for ∞-groupoids in the following way:
the (∞,1)-category presented by $sSet/S$ is $RFib(S)$
the (∞,1)-category presented by the global model structure on simplicial presheaves $[C^{op}, SSet]$ is (∞,1)-category of (∞,1)-presheaves $PSh_{(\infty,1)}(N_{hc}(C))$
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.
Let $C$ itself be an $\infty$-groupoid. Then $RFib(C) \simeq \infty Grpd/C$ and hence
By the fact that there is the standard model structure on simplicial sets we have that every morphism of $\infty$-groupoids $X \to C$ factors as
where the top morphism is an equivalence and the right morphism a Kan fibration. Moreover, as discussed at right fibration, over an $\infty$-groupoid the notions of left/right fibrations and Kan fibrations coincide. This shows that the full sub-(∞,1)-category of $\infty Grpd/X$ on the right fibrations is equivalent to all of $\infty Grpd/X$.
The generalization of a fibered category to quasi-category theory is a Cartesian fibration of quasi-categories.
($(\infty,1)$-Grothendieck construction)
Let $C$ be an (∞,1)-category. There is an equivalence
where
on the left we have the $(\infty,1)$-category of Cartesian fibrations over $C$ – incarnated as the full sSet-subcategory of the overcategory $sSet/C$ on Cartesian fibrations;
and on the right the (∞,1)-category of (∞,1)-functors from $C^{op}$ to the (∞,1)-category of (∞,1)-categories.
In the next section we discuss how this statement is presented in terms of model categories.
Regard the (∞,1)-category $C$ in its incarnation as a simplicially enriched category.
Let $S$ be a simplicial set, $\tau_{hc}(S)$ the corresponding simplicially enriched category (where $\tau_{hc}$ is the left adjoint of the homotopy coherent nerve) and let $\phi : \tau_{hc}(S) \to C$ 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 $S$ to marked simplicial presheaves on $C^{op}$ 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 $f: d \rightarrow e$ of $X \in sSet/S$ gives rise to an edge $\tilde f \in St_\phi (X)(d) = K(\phi,p)(d,v)$: the join 2-simplex $f \star v$ of $X^{\triangleright}$
with image $\tilde f : \tilde d \to f^* \tilde e$ in the pushout $K(\phi,p)(d,v)=St_\phi X(d)$.
We define the straightening functor to assign that marking of edges which is the minimal one such that all such morphisms $\tilde f$ are marked in $St_\phi X(d)$, for all marked $f : d \to e$ in $X$: this means that this marking is being completed under the constraint that $St_\phi(X)$ be sSet-enriched functorial.
For that, recall that the hom simplicial sets of $sSet^+$ are the spaces $Map^\sharp(X,Y)$, which consist of those simplices of the internal hom $Map(X,Y) := Y^X$ whose edges are all marked:
So we need to find a marking on the $St_\phi(X)(-)$ such that for all $g : \Delta[1] \to C(c,d)$ 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 $\tilde f \times \{0 \to 1\}$ i.e. 1-cells $(\tilde f , Id) : \Delta[1] \to St_\phi(X)(d) \times \Delta[1]$ to marked edges
in $St_\phi(X)(c)$. So we need to ensure that the edges of this form are marked:
we define that the straightening functor marks an edge in $St_\phi(X)(c)$ iff it is of this form $g^* \tilde f$, for $f : d \to e$ a marked edge of $X$ and $g \in C(c,d)_1$.
As in the unmarked cae, the straightening functor has an sSet-right adjoint, the unstraightening functor
This functor $Un_\phi$ exhibits the $(\infty,1)$-Grothendieck-construction proper, in that it constructs a Cartesian fibration from a given $(\infty,1)$-functor:
(presentation of $(\infty,1)$-Grothendieck construction)
This induces a Quillen adjunction
between the model structure for Cartesian fibrations and the projective global model structure on functors with values in the model structure on marked simplicial sets.
If $\phi$ is an equivalence in the model structure on simplicial categories then this Quillen adjunction is a Quillen equivalence.
This is HTT, theorem 3.2.0.1.
In the case that $C$ happens to be an ordinary category, the $(\infty,1)$-Grothendieck construction can be simplified and given more explicitly.
This is HTT, section 3.2.5.
(relative nerve functor)
Let $C$ be a small category and let $f : C \to sSet$ be a functor. The simplicial set $N_f(C)$, the relative nerve of $C$ under $f$ is defined as follows:
an $n$-cell of $N_f(C)$ is
a functor $\sigma : [n] \to C$;
for every $[k] \subset [n]$ a morphism $\tau(k) : \Delta[k] \to f(\sigma(k))$;
such that for all $[j] \subset [k] \subset [n]$ the diagram
commutes.
There is a canonical morphism
to the ordinary nerve of $C$, obtained by forgetting the $\tau$s.
This is HTT, def. 3.2.5.2.
When $f$ is constant on the point, then $N_f(C) \to N(C)$ is an isomorphism of simplicial sets, so $N_f(C)$ this is the ordinary nerve of $C$.
The fiber of $N_f(C) \to N(C)$ over an object $c \in C$ is given by taking $\sigma$ to be constant on $C$. Then all the $\tau$s are fixed by the maximal $\tau(n) : \Delta[n] \to f(c)$. So the fiber of $N_f(C)$ over $c$ is $f(c)$.
(marked relative nerve functor)
Let $C$ be a small category. Define a functor
by
where $f : C^{op} \stackrel{F}{\to} sSet^+ \to sSet$ is $F$ with the marking forgotten, where $N_f(C)$ is the relative nerve of $C$ under $f$, and where the marking $E_F$ is given by
This is HTT, def. 3.2.5.12.
This functor has a left adjoint $\mathcal{F}^+$. The value of $\mathcal{F}^+(F)$ on some $c \in C$ is equivalent to the value of $St(F)$.
This is HTT, Lemma 3.2.5.17.
($(\infty,1)$-Grothendieck construction over a category)
The adjunction
is a Quillen equivalence between the model structure for coCartesian fibrations and the projective global model structure on functors.
This is HTT, prop. 3.2.5.18.
Let $S$ be a simplicial set.
There is a sequence of Quillen adjunctions
Where from left to right we have
the model structure on an overcategory for the Joyal model structure for quasi-categories;
some localizaton of the model structure for Cartesian fibrations;
the model structure on an overcategory for the Quillen model structure on simplicial sets;
The first and third Quillen adjunction here is a Quillen equivalence if $S$ is a Kan complex.
For the base category $S$ being the point $S = {*}$, the $(\infty,1)$-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 $sSet_{Quillen}$ with itself that plays a central role in the proof of the corresponding Quillen equivalence over general $S$.
Definition
Let $Q : \Delta \to sSet$ be the cosimplicial simplicial set given by
where
Then: nerve and realization associated to $Q$ yield a Quillen equivalence of $sSet_{Quillen}$ with itself.
…
A Cartesian fibration $p : K \to \Delta[1]$ over the 1-simplex corresponds to a morphism $\Delta[1]^{op} \to$ (∞,1)Cat, hence to an (∞,1)-functor $F : D \to C$.
By the above procedure we can express $F$ as the image of $p$ 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 $K \to \{0 \to 1\}$, we obtain a functor $f : K_1 \to K_0$ between the fibers, by choosing for each object $d \in K_1$ a Cartesian morphism $e_d \to d$. Then the universal property of Cartesian morphism yields for every morphism $d_1 \to d_2$ in $K_1$ the unique left vertical filler in
And again by universality, this assignment is functorial: $K_1 \to K_0$.
Diagrammatically, the choice of Cartesian morphisms here is a lift $e$ 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 $p : K \to \Delta[1]$ with fibers the quasi-categories $C := K_{0}$ and $D := K_{1}$, an $(\infty,1)$-functor associated to the Cartesian fibration $p$ is a functor $f : D \to C$ such that there exists a commuting diagram in sSet
such that
$F|_{1} = Id_D$;
$F|_{0} = f$;
and for all $d \in D$, $F(\{d\}\times \{0 \to 1\})$ is a Cartesian morphism in $K$.
More generally, if we also specify possibly nontrivial equivalences of quasi-categories $h_0 : C \stackrel{\simeq}{\to} K_{0}$ and $h_1 : D \stackrel{\simeq}{\to} K_{1}$, then a functor is associated to $K$ and this choice of equivalences if the first twoo conditions above are generalized to
$F|_{1} = h_1$;
$F|_{0} = h_0 \circ f$;
This is HTT, def. 5.2.1.1.
For $p : K \to \Delta[1]$ a Cartesian fibration, the associated functor exists and is unique up to equivalence in the (∞,1)-category of (∞,1)-functors $Func(K_{0}, K_{1})$.
This is HTT, prop 5.2.1.5.
Set $C := K_{0}$ and $D := K_{1}$.
With the notation described at model structure for Cartesian fibrations, consider the commuting diagram
in the category $sSet^+$ of marked simplicial sets.
Here the left vertical morphism is marked anodyne: it is the smash product of the marked cofibration (monomorphism) $Id : D^\flat \to D^\flat$ with the marked anodyne morphism $\Delta[1]^# \to \Delta[0]$. 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 $d : D^\flat \times \Delta[1]^# \to K^{\sharp}$ against the Cartesian fibration in
exists. This exhibits an associated functor $f := s_0$.
Suppose now that another associated functor $f'$ 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 $K$ whose two edges are labeled by $s$ and $s'$, respectively.
Now, the left vertical morphism is still marked anodyne, and hence the lift $k$ exists, as indicated. Being a morphism of marked simplicial sets, it must map for each $d \in D$ the edge $\{d\}\times \{0\to 1\}$ to a Cartesian morphism in $K$, and due to the commutativity of the diagram this morphism must be in $K_0$, sitting over $\{0\}$. But as discussed there, a Cartesian morphism over a point is an equivalence. This means that the restriction
is an invertible natural transformation between $f$ and $f'$, hence these are equivalent in the functor category.
Conversely, every functor $f : D \to C$ gives rise to a Cartesian fibration that it is associated to, in the above sense.
Every $(\infty,1)$-functor $f : D \to C$ is associated to some Cartesian fibration $p : K \to \Delta[1]$, and this is unique up to equivalence.
This is HTT, prop 5.2.1.3.
The idea is that we obtain $K$ from first forming the cylinder $D \times \Delta[1]$ and the identifying the left boundary of that with $C$, using $f$.
Formally this means that we form the pushout
in $sSet^+$, where $C^\sharp$ and $D^\sharp$ are $C$ and $D$ with precisely the equivalences marked. This comes canonically with a morphism
and does have the property that $N_0 = C$, $N_1 = D$ and that $f$ is associated to it in that the restriction of the canonical morphism $D \times \Delta[1] \to K$ to the 0-fiber is $f$. But it may fail to be a Cartesian fibration.
To fix this, use the small object argument to factor $N \to \Delta[1]$ 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 $\Delta[1]^#$ is marked) a Cartesian fibration.
It then remains to check that $f$ is still associated to this $K \to \Delta[1]^#$. This is done by observing that in the small object argument $K$ 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 $N_\alpha \times_{\Delta[1]} \{0\}$ is equivalent to $C$, then so is $N_{\alpha +1} \times_{\Delta[1]} \{0\}$ and similarly for $D$. By induction, it follows that $f$ is indeed associated to $K \to \Delta[1]$.
To see that the $K$ obtained this way is unique up to equivalence, consider
… for the moment see HTT, section 3.2.2
for the moment see
The construction for $\infty$-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 $\infty$-Grothendieck construction with values in $\infty$-groupoids is in