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 indexed (∞,1)-categories, that is, (∞,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 a small (∞,1)-category. There is an equivalence
on the left we have the $(\infty,1)$-category of Cartesian fibrations over $C$ with small fibers and cartesian functors between them – incarnated as a 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.
Furthermore, this is equivalence is natural for $C$, where $Cart(-)$ acts by taking pullbacks and $Func(-^{op}, (\infty,1) Cat)$ acts by composition.
A reference for the naturality in small $C$ is corollary A.32 of Gepner-Haugseng-Nikolaus 15. The dual statement is made in remark 1.13 of Mazel-Gee.
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.
The $(\infty,1)$-Grothendieck construction on an $\infty$-functor is equivalently its lax (infinity,1)-colimit (Gepner-Haugseng-Nikolaus 15).
See also at Grothendieck construction as a lax colimit.
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. The characterization via lax colimits leads to describing $K$ as the mapping cylinder $(\Delta^1 \times B) \amalg_{\Delta^{\{0\}} \times B} A$.
However, there is a more immediate way to extract this functor, which we now describe. This construction also provides additional strictness properties in the quasicategory model.
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
Discussion in terms of lax (infinity,1)-colimits is in
Section 1 of this paper reviews properties of the Grothendieck construction for quasicategories:
Another review is
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