(infinity,1)-Grothendieck construction

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#### $(\infty,1)$-Category theory
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[[!include quasi-category theory contents]]
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#Contents#
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## Idea

The **$(\infty,1)$-Grothendieck construction** is the generalization of the [[Grothendieck construction]] from [[category theory]] to [[(∞,1)-category theory]].

Recall that the [[1-category]]-theoretic [[Grothendieck construction]] establishes  [[equivalence of categories|equivalences of categories]],

$$
  Fib(C) \simeq 2Func(C^{op}, Cat)
  {\phantom{AAAA}}
  and
  {\phantom{AAAA}}
  Fib_{Grpd}(C) \simeq 2Func(C^{op}, Grpd)
$$

between (a) [[fibered category|fibered categories]] (or just [[categories fibered in groupoids]]) and (b) [[pseudofunctors]] to [[Cat]] (or just to [[Grpd]]).

Analogously,
the full Grothendieck construction for [[(∞,1)-categories]] constitutes an [[equivalence of (∞,1)-categories]]

$$
  CartFib(C) \,\simeq\, \infty Func\big( C^{op}, (\infty,1)Cat \big)
$$

between (a) [[Cartesian fibrations]] [[fibrations of quasi-categories|of quasi-categories]] and (b) [[indexed (∞,1)-categories]], that is, [[(∞,1)-functors]] to [[(∞,1)Cat]];

while its restriction to the Grothendieck construction for [[∞-groupoids]] constitutes an [[equivalence of (∞,1)-categories]]

$$
  RFib(C) \,\simeq\, \infty Func\big(C^{op}, \infty Grpd\big)
$$

between [[right fibrations]] [[fibrations of quasi-categories|of quasi-categories]] and [[(∞,1)-functors]] to [[∞Grpd]].

This correspondence may be [[model category|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 over-sets|model structure on marked simplicial sets]].

## For fibrations in $\infty$-groupoids

The generalization of a [[category fibered in groupoids]] to [[quasi-category]] theory is a [[right fibration|right]] [[fibrations of quasi-categories|fibration of quasi-categories]].

\begin{prop}\label{Infinity0GrothendieckConstruction}
**($(\infty,0)$-Grothendieck construction)**

Let $\mathcal{C}$ be a small [[(∞,1)-category]]. The operation of [[homotopy pullback]] of the [[universal fibration of infinity-groupoids|universal fibration of $\infty$-groupoids]] constitutes an [[equivalence of (∞,1)-categories]]

$$
  Func\big(\mathcal{C}^{op}, \infty Grpd\big)
    \;\simeq\;
  RFib(C)
$$

* from the [[(∞,1)-category of (∞,1)-functors]] from the [[opposite category]] $\mathcal{C}^{op}$ to [[∞Grpd]], i.e. the [[(∞,1)-category of (∞,1)-presheaves]] on $C$;

* to the [[right fibration|(∞,1)-category of right fibrations]] $RFib(C)$ -- incarnated as the full [[SSet]]-[[subcategory]] of the [[overcategory]] $SSet/C$ on [[right fibrations]];

\end{prop}

Cisinski19

In the next section we discuss how this statement is presented in terms of [[model categories]].

\begin{prop}\label{GrpdFibsOverGrpds}
**($(\infty,0)$-fibrations over an $\infty$-groupoid)**
\linebreak
If $C$ itself be an [[infinity-groupoid|$\infty$-groupoid]], then $RFib(C) \simeq \infty Grpd/C$ is [[equivalence of (infinity,1)-categories|equivalently]] the [[slice (infinity,1)-category|slice $\infty$-category]] of [[∞Grpd]] over $C$, and hence the above Prop. \ref{Infinity0GrothendieckConstruction} reduces to

$$
  \infty Grpd/C \;\simeq\; Func\big(C^{op}, \infty Grpd\big)
  \,.
$$
\end{prop}
\begin{proof}
By the fact that there is the [[classical model structure on simplicial sets]] we have that every morphism of $\infty$-groupoids $X \to C$ factors as

$$
  \array{
     X &&\stackrel{\simeq}{\to}&& \hat X
     \\
     & \searrow && \swarrow_{\mathrlap{fib}}
     \\
     && C
  }
  \,,
$$

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 $\infty Grpd/X$.
\end{proof}

### Model category presentation {#GrpdModelCatVersion}

We discuss a [[presentable (∞,1)-category|presentation]] of the $(\infty,0)$-Grothendieck construction by a [[simplicial Quillen adjunction]] between [[simplicial model categories]]. ([[Higher Topos Theory|HTT, section 2.2.1]]).

+-- {: .un_def #UnmarkedStraighteningFunctor}
###### Definition
**(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 [[Higher Topos Theory|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]]

$$
  \array{
    \tau_{hc}(X) &\stackrel{}{\to}& \tau_{hc}(X^{\triangleright})
    \\
    {}^{\mathllap{\phi(p)}}\downarrow && \downarrow
    \\
    C &\to& K(\phi,p)
  }
  \,,
$$

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]]**,

$$
  St_\phi : sSet/S \to [C^{op}, sSet]
$$

from the [[overcategory]] of [[sSet]] over $S$ to the [[enriched functor category]] of [[sSet]]-[[enriched functor]]s from $C^{op}$ to $sSet$ by defining it on objects $(p : X \to S)$ to act as

$$
  St_\phi(X) := K(\phi,p)(-,v) : C^{op} \to SSet
  \,.
$$

=--

+-- {: .un_example}
###### Example

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

* $p \colon X \to C$ is the only possible functor, sending everything to the point.

Then

* $X^{\triangleright} =
    \left\{
      \array{
         a &\to& b && c
         \\
         & \searrow \Leftarrow& \downarrow & \swarrow
         \\
         && v
      }
    \right\}
  $

and

* $X^{\triangleright} \coprod_{X} C =
    \left\{
      \array{
         && \bullet
         \\
         & \swarrow & \downarrow & \searrow
         \\
         \downarrow& \Leftarrow & \downarrow
         \\
         & \searrow & \downarrow & \swarrow
         \\
         && v
      }
    \right\}
  $

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.

=--

+-- {: .un_prop}
###### Proposition

With the definitions [as above](#UnmarkedStraighteningFunctor), let $\pi : C \to C'$ be an [[sSet]]-[[enriched functor]] between [[sSet-categories]]. Write

$$
  \pi_! : [C^{op}, sSet] \to [{C'}^{op}, sSet]
$$

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 $\pi$:

$$
  St_{\pi \circ \phi} \simeq \pi_! \circ St_\phi
  \,.
$$

=--

This is [[Higher Topos Theory|HTT, prop. 2.2.1.1.]]. The following proof has kindly been spelled out by [[Harry Gindi]].

+-- {: .proof}
###### Proof

We unwind what the [[sSet-categories]] with a single object adjoined to them look like:

let

$$
  F : C^{op} \to sSet
$$

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](#UnmarkedStraighteningFunctor)  is

$$
  K(\phi,p) \simeq C_{St_\phi(X)}
$$

(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

$$
  \array{
    \tau_{hc}(X) &\to& \tau_{hc}(X^{\triangleright})
    \\
    \downarrow && \downarrow
    \\
    C &\to& K(\phi,p) = C_{St_\phi(X)}
    \\
   {}^{\mathllap{\pi}}\downarrow && \downarrow
   \\
   C' &\to& Q
  }
$$

and show that indeed $Q \simeq C'_{\pi_! St_\phi(X)}$.

Using the pasting law for [[pushout]]s (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 $\pi \colon C \to C'$ is $C'_{\pi_! F}$.

This follows by inspection of what a cocone

$$
  \array{
     C &\stackrel{\iota}{\to}& C_F
     \\
     {}^{\mathllap{\pi}}\downarrow && \downarrow^{\mathrlap{d}}
     \\
     C' &\underset{r}{\to}& Q
  }
$$

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]]

$$
  F(c) \to  Q(d|_C(c), d(\nu))
$$

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

$$
  F \to (d|_C)^* Q(-,d(\nu)) = \iota^* d^* \nu Q(-,d(\nu))
  \,,
$$

where $d^*$ etc. means precomposition with the functor $d$.

By commutativity of the diagram this is

$$
  \cdots \simeq \pi^* r^* Q(-,d(\nu))
  \,.
$$

Now by the definition of left [[Kan extension]] $\pi_!$ as the [[left adjoint]] to prescomposition with a functor, this is bijectively a transformation

$$
  \eta : \pi_! F \to r^* Q(-,d(\nu))
  \,.
$$

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 of an adjunction|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.

=--

+-- {: .un_prop}
###### Proposition

This functor has a [[right adjoint]]

$$
  Un_\phi : [C^{op}, sSet] \to sSet/S
  \,,
$$

that takes a [[simplicial presheaf]] on $C$ to a simplicial set over $S$ -- this is the **unstraightening functor**.

=--

+-- {: .proof}
###### Proof

One checks that $St_\phi$ preserves [[colimit]]s. The claim then
follows with the [[adjoint functor theorem]].

=--

\begin{prop}
**(presentation of the $(\infty,0)$-Grothendieck construction)**
\linebreak

The straightening and the unstraightening functor constitute a [[Quillen adjunction]]

$$
  (St_\phi \dashv Un_\phi)
  \,\colon\,
   sSet/S
  \underoverset
    {\underset{St_\phi}{\longrightarrow}}
    {\overset{Un_{\phi}}{\longleftarrow}}
    {}
  [C^{op}, sSet]
$$

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]].
\end{prop}

This is [[Higher Topos Theory|HTT, theorem 2.2.1.2]].

This models the Grothendieck construction for [[∞-groupoids]] in the following way:

* the [[presentable (∞,1)-category|(∞,1)-category presented by]] $sSet/S$ is $RFib(S)$

  ([[Higher Topos Theory|HTT, lemma 2.2.3.9]])

* the [[presentable (∞,1)-category|(∞,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]].

## For general fibered $(\infty,1)$-categories

The generalization of a [[fibered category]] to [[quasi-category]] theory is a [[Cartesian fibration|Cartesian]] [[fibrations of quasi-categories|fibration of quasi-categories]].

+-- {: .un_theorem}
###### Theorem
**($(\infty,1)$-Grothendieck construction)**

Let $C$ be a small [[(∞,1)-category]]. There is an equivalence

$$
  Cart(C) \simeq Func(C^{op}, (\infty,1) Cat)
$$

* on the left we have the $(\infty,1)$-category of [[Cartesian fibration]]s over $C$ with small fibers and cartesian functors between them -- incarnated as a [[sSet]]-[[subcategory]]
of the [[overcategory]] $sSet/C$ on [[Cartesian fibration]]s;

* 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](#GepnerHaugsengNikolaus15). The dual statement is made in remark 1.13 of [Mazel-Gee](#MazelGee).

In the next section we discuss how this statement is presented in terms of [[model categories]].

### Model category presentation {#ModCatCart}

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]].

+-- {: .un_def}
###### Definition
**(extracting a marked simplicial presheaf from a marked fibration)**
(HTT, section 3.2.1)

The **straightening functor**

$$
  St_\phi : sSet^+/S \to [C^{op}, sSet^+]
$$

from [[marked simplicial set]]s over $S$ to marked [[simplicial presheaves]] on $C^{op}$ is on the underlying simplicial sets (forgetting the marking) the same straightening functor as [above](#GrpdModelCatVersion).

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 of simplicial sets|join]] 2-simplex $f \star v$ of $X^{\triangleright}$

$$
  \array{
    d && \stackrel{f}{\to} && e
    \\
    & {}_{\mathllap{\tilde d}}\searrow
   & \stackrel{\tilde f}{\Rightarrow}
   & \swarrow_{\mathrlap{\tilde e}}
    \\
    && v
  }
$$

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 functor|enriched functorial]].

For that, [[marked simplicial set|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:

$$
  Map(X,Y)_n = Hom_{sSet^+}(X \times \Delta[n]^#, Y)
  \,.
$$

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

$$
  \Delta[1]
  \stackrel{g}{\to}
  C(c,d)
  \stackrel{St_\phi(X)(c,d)}{\to}
  Map(St_\phi(X)(d), St_\phi(X)(c))
$$

is a marked edge of the mapping complex. By the internal hom-adjunction this edge corresponds to a morphism

$$
  St_\phi(X)(g)
  :
  St_\phi(X)(d) \times \Delta[1] \rightarrow St_\phi(X)(c)
$$

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

$$
  g^* \tilde f : \Delta[1] \stackrel{(\tilde f,Id)}{\to}
  St_\phi(X)(d)\times \Delta[1] \stackrel{St_\phi(X)(g)}{\to}
  St_{\phi}(X)(c)
$$

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**

$$
  n_\phi :  [C^{op}, sSet^+] \to sSet^+/S
  \,.
$$

=--

This functor $Un_\phi$ exhibits the $(\infty,1)$-Grothendieck-construction proper, in that it constructs a [[Cartesian fibration]] from a given $(\infty,1)$-functor:

+-- {: .un_theorem}
###### Theorem
**(presentation of $(\infty,1)$-Grothendieck construction)**

This induces a [[Quillen adjunction]]

$$
  (St_\phi \dashv Un_\phi) : SSet^+/S
  \stackrel{\overset{Un_{\phi}}{\leftarrow}}{\underset{St_\phi}{\to}}
  [C^{op}, SSet^+]
$$

between the [[model structure for Cartesian fibrations]] and the projective [[global model structure on functors]] with values in the [[model structure on marked simplicial over-sets|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]].

=--

+-- {: .proof}
###### Proof

This is [[Higher Topos Theory|HTT, theorem 3.2.0.1]].

=--

#### Over an ordinary category {#OverCat}

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 [[Higher Topos Theory|HTT, section 3.2.5]].

+-- {: .un_def}
###### Definition
**(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

1. a functor $\sigma : [n] \to C$;

1. for every $[k] \subset [n]$ a morphism $\tau(k) : \Delta[k] \to f(\sigma(k))$;

1. such that for all $[j] \subset [k] \subset [n]$ the diagram

   $$
     \array{
       \Delta[j] &\stackrel{\tau(j)}{\to}& f(\sigma(j))
       \\
       \downarrow && \downarrow^{\mathrlap{f(\sigma(j\to k))}}
       \\
       \Delta[k] &\stackrel{\tau(k)}{\to}& f(\sigma(k))
     }
   $$

   commutes.

There is a canonical morphism

$$
  N_f(C) \to N(C)
$$

to the ordinary [[nerve]] of $C$, obtained by forgetting the $\tau$s.

=--

This is [[Higher Topos Theory|HTT, def. 3.2.5.2]].

+-- {: .un_remark}
###### Remark

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 $f(c)$.

=--

+-- {: .un_def}
###### Definition
**(marked relative nerve functor)**

Let $C$ be a small [[category]]. Define a functor

$$
  sSet^+/N(C) \leftarrow [C, sSet^+] : N^+
$$

by

$$
  (C \stackrel{F}{\to} sSet^+)
  \mapsto
  (N_f(C), E_F)
  \,,
$$

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 [[Higher Topos Theory|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 [[Higher Topos Theory|HTT, Lemma 3.2.5.17]].

+-- {: .un_prop}
###### Proposition
**($(\infty,1)$-Grothendieck construction over a category)**

The adjunction

$$
  (\mathcal{F}^+ \dashv N^+)
  :
  sSet^+_{/N(C)} \stackrel{\overset{\mathcal{F}^+}{\to}}{\underset{N^+}{\leftarrow}}
  [C,sSet^+]
  \,.
$$

is a [[Quillen equivalence]] between the [[model structure for Cartesian fibrations|model structure for coCartesian fibrations]] and the projective [[global model structure on functors]].

=--

+-- {: .proof}
###### Proof

This is [[Higher Topos Theory|HTT, prop. 3.2.5.18]].

=--

### Relation beween the model structures

\begin{prop}
Let $S$ be a [[simplicial set]].

There is a sequence of [[Quillen adjunction]]s

$$
  (sSet/S)_{Joyal}
  \stackrel{\overset{}{\to}}{\overset{}{\leftarrow}}
  sSet^+/S
  \stackrel{\overset{}{\to}}{\overset{}{\leftarrow}}
  (sSet^+/S)^{loc}
  \stackrel{\overset{}{\to}}{\overset{}{\leftarrow}}
  (sSet/S)_{rfib}
  \stackrel{\overset{}{\to}}{\overset{}{\leftarrow}}
  (sSet/S)_{Quillen}
  \,.
$$

Where from left to right we have

1. the [[model structure on an overcategory]] for the Joyal [[model structure for quasi-categories]];

1. the [[model structure for Cartesian fibrations]];

1. some localizaton of the model structure for Cartesian fibrations;

1. the [[model structure for right fibrations]]

1. 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]].

\end{prop}

([[Higher Topos Theory|HTT, section 3.1.5]])

## Properties

### As an (op)lax $\infty$-colimit

The $(\infty,1)$-Grothendieck construction on an $\infty$-functor is equivalently its [[lax (infinity,1)-colimit]] ([Gepner-Haugseng-Nikolaus 15](#GepnerHaugsengNikolaus15)).

See also at _[[Grothendieck construction]]_  _[as a lax colimit](Grothendieck%20construction#AsALaxColimit)_.

## Examples

### Cartesian fibrations over the point

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

$$
  St_\phi : sSet/* \simeq sSet \to [*,sSet] \simeq sSet
$$

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 object|cosimplicial]] [[simplicial set]] given by

$$
  Q[n] := |J^n|(x,v)
  \,,
$$

where

$$
  J^n = C^{\triangleleft}(\Delta[n] \to \{x\})
  \,.
$$

Then: [[nerve and realization]] associated to $Q$ yield a [[Quillen equivalence]] of $sSet_{Quillen}$ with itself.

[[Higher Topos Theory|HTT, section 2.2.2]].

...

### Cartesian fibrations over the interval {#FibsOverInterval}

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

$$
  \array{
    e_{d_1} &\to& d_1
    \\
    \downarrow && \downarrow
    \\
    e_{d_2} &\to& d_2
  }
  \,.
$$

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

$$
  \array{
    K_1 &\hookrightarrow& K
    \\
    \downarrow &\nearrow_e& \downarrow
    \\
    K_1 \times \{0 \to 1\} &\to& \{0 \to 1\}
  }
  \,.
$$

This diagrammatic way of encoding the functor associated to a Grothendieck fibration over the interval generalizes straightforwardly to the [[quasi-category]] context.

+-- {: .un_defn}
###### Definition

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]]

$$
  \array{
    D \times \Delta[1] &&\stackrel{F}{\to}&& K
    \\
    & \searrow && \swarrow_{\mathrlap{p}}
    \\
    && \Delta[1]
  }
$$

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 [[equivalence of quasi-categories|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 [[Higher Topos Theory|HTT, def. 5.2.1.1]].

+-- {: .un_prop}
###### Proposition

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]] $Func(K_{0}, K_{1})$.

=--

+-- {: .proof}
###### Proof

This is [[Higher Topos Theory|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

$$
  \array{
    D^\flat \times \{1\} &\hookrightarrow& K^{\sharp}
    \\
    \downarrow && \downarrow^{\mathrlap{p}}
    \\
    D^{\flat} \times \Delta[1]^{#} &\to&
    \Delta[1]^#
  }
$$

in the category $sSet^+$ of marked simplicial sets.

Here the left vertical morphism is [marked anodyne](http://ncatlab.org/nlab/show/model+structure+for+Cartesian+fibrations#MarkedAnodyne): 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](http://ncatlab.org/nlab/show/model+structure+for+Cartesian+fibrations#MarkedAnodyne), 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

$$
  \array{
    D^\flat \times \{1\} &\hookrightarrow& K^{\sharp}
    \\
    \downarrow &\nearrow_{s}& \downarrow^{\mathrlap{p}}
    \\
    D^{\flat} \times \Delta[1]^{#} &\to&
    \Delta[1]^#
  }
$$

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

$$
  \array{
    D^\flat \times \{1\} &\hookrightarrow& K^{\sharp}
    \\
    \downarrow &\nearrow_{s'}& \downarrow^{\mathrlap{p}}
    \\
    D^{\flat} \times \Delta[1]^{#} &\to&
    \Delta[1]^#
  }
  \,.
$$

Together this may be arranged to a diagram

$$
  \array{
    D^\flat \times \Lambda[2]_2
     &\stackrel{(s,s')}{\to}& K^{\sharp}
    \\
    \downarrow &\nearrow_{q}& \downarrow^{\mathrlap{p}}
    \\
    D^{\flat} \times \Delta[2]^{#} &\to&
    \Delta[1]^#
  }
  \,,
$$

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

$$
  k|_{D \times \{0 \to 1\}} \to K_0
$$

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.

+-- {: .un_prop}
###### Proposition

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.

=--

+-- {: .proof}
###### Proof

This is [[Higher Topos Theory|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]]

$$
  N := (D^\natural \times \Delta[1]^#) \coprod_{D^\natural \times \{0\}^#} C^\natural
$$

in $sSet^+$, where $C^\natural$ and $D^\natural$ are $C$ and $D$ with precisely the [[equivalence in a quasi-category|equivalences]] marked. This comes canonically with a morphism

$$
  N \to \Delta[1]^{\sharp}
$$

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

$$
  N \to K \to \Delta[1]^#
  \,,
$$

where the first morphism is [marked anodyne](http://ncatlab.org/nlab/show/model+structure+for+Cartesian+fibrations#MarkedAnodyne) 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 [[pushout]]s of the form

$$
  \array{
    A &\to& N_\alpha
    \\
    \downarrow && \downarrow & \searrow
    \\
    B &\to& N_{\alpha+1} &\to& \Delta[1]
  }
  \,,
$$

where the morphisms on the left are the generators of marked anodyne morphisms (see [here](http://ncatlab.org/nlab/show/model+structure+for+Cartesian+fibrations#MarkedAnodyne)). 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...

=--

### Cartesian fibrations over simplices {#CartOverSimplex}

... for the moment see [[Higher Topos Theory|HTT, section 3.2.2]] ...

### The universal Cartesian fibration

for the moment see

* [[universal fibration of (∞,1)-categories]].

## Related concepts

* [[operadic (∞,1)-Grothendieck construction]]

* [[Grothendieck construction for model categories]]

* [[Joyal locus]], [[tangent (infinity,1)-topos]]

## References

Textbook accounts:

* {#Lurie09} [[Jacob Lurie]], Section 3.2 of: _[[Higher Topos Theory]]_, Annals of Mathematics Studies **170**, Princeton University Press 2009 ([pup:8957](https://press.princeton.edu/titles/8957.html), [pdf](https://www.math.ias.edu/~lurie/papers/HTT.pdf))

* {#Cisinski19} [[Denis-Charles Cisinski]], Section 5.2 of: _[[Higher Categories and Homotopical Algebra]]_,  Cambridge University Press 2019 ([doi:10.1017/9781108588737](https://doi.org/10.1017/9781108588737), [pdf](http://www.mathematik.uni-regensburg.de/cisinski/CatLR.pdf))

  > (for $\infty$-groupoids)

More on model-category theoretic construction of the $\infty$-Grothendieck construction with values in $\infty$-groupoids is in

* {#HeutsMoerdijk13} [[Gijs Heuts]], [[Ieke Moerdijk]], _Left fibrations and homotopy colimits_ ([arXiv:1308.0704](http://arxiv.org/abs/1308.0704))

* [[Gijs Heuts]], [[Ieke Moerdijk]], *Left fibrations and homotopy colimits II* [[arXiv:1602.01274](https://arxiv.org/abs/1602.01274)]

Discussion in terms of [[lax (infinity,1)-colimits]] is in

* {#GepnerHaugsengNikolaus15} [[David Gepner]], [[Rune Haugseng]], [[Thomas Nikolaus]], _Lax colimits and free fibrations in $\infty$-categories_ ([arXiv:1501.02161](http://arxiv.org/abs/1501.02161))

Section 1 of this paper reviews properties of the Grothendieck construction for quasicategories:

* {#MazelGee} [[Aaron Mazel-Gee]], _All about the Grothendieck construction_ ([arXiv:1510.03525](https://arxiv.org/abs/1510.03525))

Another review is

* Anders Jess Pedersen, Magnus Baunsgaard Kristensen, _Straightening and Unstraightening_ [(Dropbox)](https://www.dropbox.com/s/alys3wg59rir3gt/Grothendieck_construction.pdf?dl=0)

Discussion of the $\infty$-Grothendieck construction for [[(infinity,1)-functors|$\infty$-functors]] to [[(infinity,1)-toposes|$\infty$-toposes]] from [[inverse images]] of [[infinity-groupoid|$\infty$-groupoids]]:

* {#BeardsleyPéroux22} [[Jonathan Beardsley]], [[Maximilien Péroux]], Lemma 3.10 in: *Koszul Duality in Higher Topoi*, Homol. Homot. Appl. [(2022)](https://www.intlpress.com/site/pub/pages/journals/items/hha/_home/acceptedpapers/index.php), [[arXiv:1909.11724](https://arxiv.org/abs/1909.11724)]

Discussion entirely within the context of [[quasi-categories]]:

* [[Denis-Charles Cisinski]], [[Hoang Kim Nguyen]], *The universal coCartesian fibration* [[arXiv:2210.08945](https://arxiv.org/abs/2210.08945)]

which lends itself to understanding the [[universal coCartesian fibration]] as [[categorical semantics]] for the [[type universe]] in [[directed homotopy type theory]]:

* {#CisinskiEtAl23} [[Denis-Charles Cisinski]], [[Hoang Kim Nguyen]], Tashi Walde: *Univalent Directed Type Theory*, lecture series in the *[CMU Homotopy Type Theory Seminar](https://www.cmu.edu/dietrich/philosophy/hott/seminars/index.html)* (13, 20, 27 Mar 2023) [[web](https://www.cmu.edu/dietrich/philosophy/hott/seminars/index.html#230313), video 1:[YT](https://www.youtube.com/watch?v=5YOltuTcBK8), 2:[YT](https://www.youtube.com/watch?v=xWmELBvHMPo), 3:[YT](https://www.youtube.com/watch?v=P0Cfb4eUJo4); slides 0:[pdf](https://www.cmu.edu/dietrich/philosophy/hott/seminars/slides/cisinski-nguyen-walde-intro_talk1.pdf), 1:[pdf](https://www.cmu.edu/dietrich/philosophy/hott/seminars/slides/cisinski-nguyen-walde-talk1.pdf), 2:[pdf](https://www.cmu.edu/dietrich/philosophy/hott/seminars/slides/cisinski-nguyen-walde-talk2.pdf), 3:[pdf](https://www.cmu.edu/dietrich/philosophy/hott/seminars/slides/cisinski-nguyen-walde-talk3.pdf)]

(see [video 3 at 1:16:58](https://youtu.be/P0Cfb4eUJo4?t=4618) and [slides 3.33](https://www.cmu.edu/dietrich/philosophy/hott/seminars/slides/cisinski-nguyen-walde-talk3.pdf#page=33))

[[!redirects (∞,1)-Grothendieck construction]]

[[!redirects infinity1-Grothendieck construction]]


(infinity,1)-Kan extension

> under construction

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of $(\infty,1)$-Kan extension is the generalization of the notion of [[Kan extension]] from [[category theory]] to [[(∞,1)-category theory]].

## Definition

### General

Independent of any models or concrete realizations chosen, the notion of $(\infty,1)$-Kan extension is intrinsically determined from just the notions of

* [[(∞,1)-functor]],

* [[(∞,1)-category of (∞,1)-functors]]

*  [[adjoint (∞,1)-functor]].

In terms of these, for $f : C \to C'$ any [[(∞,1)-functor]] and any [[(∞,1)-category]] $A$, there is an induced $(\infty,1)$-functor $f^* : Func_{(\infty,1)}(C',A) \to Func_{(\infty,1)}(C,A)$.

The  **left $(\infty,1)$-Kan extension functor** is the left [[adjoint (∞,1)-functor]] to $f^*$.

The  **right $(\infty,1)$-Kan extension functor** is the right [[adjoint (∞,1)-functor]] to $f^*$.

Given different incarnations of or models for the notion of [[(∞,1)-category]], there are accordingly different incarnations and models of this general abstract prescription.

#### In terms of quasi-categories

([LurieHTT, def. 4.3.2.2, 4.3.3.2](#LurieHTT))

#### In terms of Kan-complex enriched categories

see [[homotopy Kan extension]]

#### In terms of simplicial model categories

see [[homotopy Kan extension]]

## Properties

### Pointwise (strong)
 {#Pointwise}

$\infty$-Kan extensions as above are [pointwise/strong](Kan+extension#Pointwise). That is in fact the very content of ([LurieHTT, def. 4.3.2.2, 4.3.3.2](#LurieHTT)).

### As adjoints to pullbacks

left/right $\infty$-Kan extension is left/right [[adjoint (∞,1)-functor]] to restriction. ([LurieHTT, prop. 4.3.3.7](#LurieHTT))

## Related concepts

* [[Kan extension]]

* **$(\infty,1)$-Kan extension**

## References

A general concept of $(\infty,1)$-Kan extensions in terms of quasi-categories are discussed in section 4.3 of

* {#LurieHTT} [[Jacob Lurie]], _[[Higher Topos Theory]]_ .

For [[simplicially enriched categories]] and [[model categories]] a discussion is in section A.3.3 there.

Coinciding left/right ([[ambidextrous adjunction|ambidextrous]]) $\infty$-Kan extensions along maps of [[∞-groupoids]] are discussed in

* {#HopkinsLurie14} [[Michael Hopkins]], [[Jacob Lurie]], section 4 of _[[Ambidexterity in K(n)-Local Stable Homotopy Theory]]_

Pointwise [[homotopy Kan extensions]] are discussed in

* {#Radulescu-Banu06} Andrei Radulescu-Banu, _Cofibrations in Homotopy Theory_ ([arXiv:0610009](http://arxiv.org/abs/math/0610009))

* {#Cisinski09} [[Denis-Charles Cisinski]], _Locally constant functors_, Math. Proc. Camb. Phil. Soc. (2009), 147, 593 ([pdf](http://www.math.univ-toulouse.fr/~dcisinsk/lcmodcat3.pdf))

* {#Gonzales11} Beatriz Rodriguez Gonzalez, section 4 of _Realizable homotopy colimits_ ([arXiv:1104.0646](http://arxiv.org/abs/1104.0646))

See also

* [[Samuel Isaacson]], _A note on unenriched homotopy coends_ ([pdf](http://www.math.uwo.ca/~sisaacso/PDFs/coends.pdf))

[[!redirects (∞,1)-Kan extension]]
[[!redirects (∞,1)-Kan extensions]]
[[!redirects (infinity,1)-Kan extensions]]

[[!redirects pointwise (∞,1)-Kan extension]]
[[!redirects pointwise (∞,1)-Kan extensions]]

[[!redirects pointwise (infinity,1)-Kan extension]]
[[!redirects pointwise (infinity,1)-Kan extensions]]


(infinity,1)-module

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Higher linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of _$(\infty,1)$-module_ over an [[monoid object in an (∞,1)-category]] (for instance an [[A-∞ ring]] or [[E-∞ ring]]) is the generalization to ([[stable homotopy theory|stable]])[[homotopy theory]] of the notion of [[module]] over a [[ring]].

## Definition

See at _[[module over an algebra over an (∞,1)-operad]]_.

## Examples

* in the [[stable (∞,1)-category of spectra]]: [[module spectrum]] over an [[ring spectrum]]

## Properties

### Compact modules

+-- {: .num_prop}
###### Propositon

Let $R$ be an [[A-∞ ring]]. The (∞,1)-category of ∞-modules $R Mod$ is a [[compactly generated (∞,1)-category]] and the [[compact object in an (∞,1)-category|compact objects]] coincide with the [[perfect modules]]

=--

([[Higher Algebra|HA, prop. 8.2.5.2]])

### Relation to fiberwise stabilization

By the discussion an [[tangent (∞,1)-category]] we may realize $E_\infty$-modules over $R$ as objects in the [[stabilization]] of the [[over-(∞,1)-category]] over $R$:

+-- {: .num_prop}
###### Proposition

Let $E_\infty := Alg^{Comm}(\infty Grpd)$ be the [[(∞,1)-category]] of [[E-∞ ring]]s and let $R \in E_\infty$. Then the [[stabilization]] of the [[over-(∞,1)-category]] over $A$

$$
  Stab(E_\infty/R) \simeq A Mod(Spec)
$$

is equivalentl to the category of $R$-module spectra.

=--

This is ([Lurie, cor. 1.5.15](#Lurie)).

### Stable Dold-Kan correspondence
 {#StableDoldKanCorrespondence}

For $R$ an ordinary [[ring]], write $H R$ for the corresponding [[Eilenberg-MacLane spectrum]].

+-- {: .num_theorem #StableDoldKan}
###### Theorem

For $R$ any [[ring]] (or [[ringoid]], even) there is a [[Quillen equivalence]]

$$
  H R Mod \simeq Ch_\bullet(R Mod)
$$

between model structure on  $H R$-module spectra and the [[model structure on chain complexes]] (unbounded) of ordinary $R$-[[module]]s.

This presents a corresponding [[equivalence of (∞,1)-categories]]. If $R$ is a commutative ring, then this is an equivalence of [[symmetric monoidal (∞,1)-categories]].

=--

This equivalence on the level of [[homotopy categories]] is due to ([Robinson](#Robinson)). The refinement to a Quillen equivalence is ([SchwedeShipley, theorem 5.1.6](#SchwedeShipley)). See also the discussion at _[[stable model categories]]_. A direct description as an equivalence of $(\infty,1)$-categories appears as ([Lurie, theorem 7.1.2.13](#Lurie)).

+-- {: .num_remark}
###### Remark

This is a stable version of the [[Dold-Kan correspondence]].

=--

See at __[[algebra spectrum]]_ for the corresponding statement for $H R$-algebra spectra and [[dg-algebras]].

+-- {: .num_example}
###### Example

For $X$ a [[topological space]] and $R$ a ring, let $C_\bullet(X, R)$ be the standard [[chain complex]] for [[singular homology]] $H_\bullet(X, R)$ of $X$ with coefficients in $R$.

Under the stable Dold-Kan correspondence, prop. \ref{StableDoldKan}, this ought to be identified with the [[smash product]] $(\Sigma^\infty_+ X) \wedge H R$ of the [[suspension spectrum]] of $X$ with the [[Eilenberg-MacLane spectrum]]. Notice that by the general theory of [[generalized homology]] the [[homotopy groups]] of the latter are again singular homology

$$
  \pi_\bullet( (\Sigma^\infty_+ X) \wedge H R) \simeq H_\bullet(X, R)
  \,.
$$

While the correspondence $(\Sigma^\infty_+ X) \wedge H R \sim C_\bullet(X,R)$ under the above equivalence is suggestive, maybe nobody has really checked it in detail. It is sort of stated as true for instance on <a href="http://arxiv.org/PS_cache/arxiv/pdf/0906/0906.5198v1.pdf#page=15">p. 15 </a> of ([BCT](#BCT)).

=--

## Related concepts

* [[action]], [[∞-action]]

* [[module]], **$(\infty,1)$-module**

  * [[(∞,1)-category of (∞,1)-modules]]

  * [[tensor product of ∞-modules]]

  * [[(∞,1)-module bundle]]

  * [[perfect module]]

* [[2-module]]

* [[bimodule]]

  [[(∞,1)-bimodule]]

* [[(∞,n)-module]]

* [[representation]], [[∞-representation]]

## References

Discussion in terms of [[module objects]] in [[symmetric smash product of spectra|symmetric monoidal]] [[model categories of spectra]] includes

* [[Anthony Elmendorf]], [[Igor Kriz]], [[Peter May]], section 5 of _[[Modern foundations for stable homotopy theory]]_ ([pdf](http://hopf.math.purdue.edu/Elmendorf-Kriz-May/modern_foundations.pdf))

Modules over algebras over an arbitrary [[(∞,1)-operad]] are discussed in  section 3.3 of

* [[Jacob Lurie]], _[[Higher Algebra]]_

Modules specifically over [[A-∞ algebras]] are discussed in section 4.2 there.

Further discussion of [[(infinity,n)-bimodules]] is in

* {#Haugseng14} [[Rune Haugseng]], _The higher Morita category of $E_n$-algebras_ ([arXiv:1412.8459](http://arxiv.org/abs/1412.8459))

The equivalence between the [[homotopy categories]] of  $H R$-module spectra and $Ch_\bullet(R Mod)$ is due to

*{#Robinson}  Alan Robinson, _The extraordinary derived category_ , Math. Z. 196 (2) (1987) 231–238.

The refinement of this statement to a [[Quillen equivalence]] is due to

* {#SchwedeShipley} [[Stefan Schwede]], [[Brooke Shipley]], _Stable model categories are categories of modules_ , Topology 42 (2003), 103-153 ([pdf](http://www.math.uic.edu/~bshipley/classTopFinal.pdf))

Applications to [[string topology]] are discussed in

* {#BCT} [[Andrew Blumberg]], [[Ralph Cohen]], [[Constantin Teleman]], _Open-closed field theories, string topology and Hochschild homology_ ([arXiv:0906.5198](http://arxiv.org/abs/0906.5198))

See the section on string topology at [[sigma model]] for more on this.

[[!redirects ∞-module]]
[[!redirects infinity-modules]]
[[!redirects ∞-modules]]

[[!redirects infinity-module]]

[[!redirects (∞,1)-module]]
[[!redirects (∞,1)-modules]]
[[!redirects (infinity,1)-modules]]


(infinity,1)-module bundle

[[!redirects (infinity,1)-vector bundle]]

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of an _$(\infty,1)$-module bundle_ is a [[categorification]]/[[homotopy theory|homotopification]] of the notion of a [[module bundle]]/[[vector bundle]], where [[fields]] and [[rings]] are replaced by [[∞-rings]] and [[modules]] by [[∞-modules]]; a central notion in [[parameterized stable homotopy theory]].

Recall that for $k$ a [[field]], a [[vector space]] is a $k$-[[module]], and a [[vector bundle]] over a [[space]] $X$ is classified by a morphism
$\alpha : X \to k$[[Mod]] with $k$[[Mod]] regarded as an object in the relevant [[topos]]. For instance for _discrete_ or _flat_ vector bundles $k Mod$ is the [[category]] [[Vect]] of vector spaces.
There is the subcategory $k Line \hookrightarrow k Mod$ of 1-[[dimension]]al $k$-vector bundles, and morphisms that factor as $\alpha : X \to k Line \hookrightarrow k Mod$ are $k$-[[line bundle]]s.
In the discrete case the vector space of [[section]]s of the vector bundle classified by $\alpha$ is the [[colimit]] $\lim_\to \alpha$.

These statements [[categorification|categorify]] in a straightforward manner to the case where $k$ is generalized to a commutative [[∞-ring]]: an _[[E-∞ ring]]_ or _[[ring spectrum]]_ . Modules are replaced by [[module spectrum|module spectra]] and colimits by [[homotopy colimit]]s.

The resulting notion of $(\infty,1)$-vector bundles plays a central role in many constructions in [[orientation in generalized cohomology]], [[twisted cohomology]] and [[Thom isomorphism]]s.

Further generalization of the concept leads to [[(∞,n)-vector bundle]]s: an $(\infty,n)$-module over an [[E-∞-ring]] $K$ is an object of the [[(∞,n)-category]] $(\cdots (K Mod) Mod ) \cdots Mod$, where we are iteratively forming module $(\infty,k)$-categories over the monoidal $(\infty,k-1)$-category of $(\infty,k-1)$-modules, $n$ times.

## Discrete $(\infty,1)$-vector bundles

We discuss $(\infty,1)$-vector bundles internal to the [[(∞,1)-topos]] [[∞Grpd]] $\simeq$ [[Top]]. Since we are discussing objects with geometric interpretation, we are to think of this as the $(\infty,1)$-topos of _[[discrete ∞-groupoids]]_.

Discussion of $\infty$-vector  bundles internal to _structured_ (non-discrete) $\infty$-groupoids is [below](#Structured).

### $\infty$-Modules and $\infty$-Module bundles

Assume in the following choices

* $K$ -- an [[E-∞ ring]]

* $A$ -- a $K$-[[algebra spectrum|algebra]],

  hence an [[A-∞ algebra]] in [[Spec]] equipped with a $\infty$-algebra homomorphism $K \to A$.

Denote

* $A Mod$ -- the [[(∞,1)-category]] of $A$-[[module spectrum|module spectra]].

+-- {: .num_defn}
###### Definition

For $X$ a [[discrete ∞-groupoid]] (often presented as a [[topological space]]), the [[(∞,1)-category]] of **$A$-module $\infty$-bundles** over $X$ is the [[(∞,1)-functor (∞,1)-category]]

$$
  A Mod(X) := Func(X, A Mod)
  \,.
$$

=--

In this form this appears as ([ABG def. 3.7](#ABG)). Compare this to the analogous definition at _[[principal ∞-bundle]]_.

+-- {: .num_remark}
###### Remark

If $X$ is regarded as a [[topological space]] then the corresponding [[discrete ∞-groupoid]] is $\Pi X$, the [[fundamental ∞-groupoid]] of $X$ and the morphism encoding an $K$-module bundle over $X$ is reads

$$
  \alpha : \Pi(X) \to A Mod
  \,.
$$

This assignment of $A$-modules to points in $X$, of $A$-module morphism to paths in $X$ etc. may be regarded as the [[higher parallel transport]] of the (unique and flat, due to [[discrete ∞-groupoid|discreteness]]) [[connection on an ∞-bundle]] on $\alpha$.

Equivalently, this morphism may be regarded as an [[∞-representation]] of $\Pi(X)$. Notaby if $X = B G$ is the [[classifying space]] of a [[discrete group]] or [[discrete ∞-group]], a $K$-module $\infty$-bundle over $X$ is the same as an [[∞-representation]] of $G$ on $A Mod$.

=--

### $\infty$-Lines and $\infty$-line bundles

+-- {: .num_defn}
###### Definition

Write

$$
  A Line \hookrightarrow A Mod
$$

for the [[full sub-(∞,1)-category]] on the _$A$-lines_ : on those $A$-modules that are equivalent to $A$ as an $A$-module. The full subcatgeory of $A Mod(X)$ on morphisms factoring through this inclusion we call the $(\infty,1)$-catgeory of **$A$-line $\infty$-bundles**.

=--

This appears as ([ABG def. 3.12](#ABG)), ([ABGHR 08, 7.5](#ABGHR08)).

+-- {: .num_defn}
###### Definition

Let $A$ be an [[A-∞ algebra|A-∞]] [[ring spectrum]].

For $\Omega^\infty A$ the underlying [[A-∞ space]] and $\pi_0 \Omega^\infty A$ the ordinary [[ring]] of connected components, writ $(\pi_0 \Omega^\infty A)^\times$ for its [[group of units]].

Then the [[∞-group of units]] of $A$ is the [[(∞,1)-pullback]] $GL_1(A)$ in

$$
  \array{
    GL_1(A) &\to& \Omega^\infty A
    \\
    \downarrow && \downarrow
    \\
    (\pi_0 \Omega^\infty A)^\times &\to& \pi_0 \Omega^\infty A
  }
  \,.
$$

=--

+-- {: .num_prop}
###### Proposition

There is an [[equivalence in an (∞,1)-category|equivalence]] of [[∞-groups]]

$$
  GL_1(A) \simeq Aut_{A Line}(A)
$$

of the [[∞-group of units]] of $A$ with the [[automorphism ∞-group]] of $A$, regarded canonically as a module over itself.

Since every $A$-line is by definition equivalent to $A$ as an $A$-module, there is accordingly, an [[equivalence of (∞,1)-categories]], in fact of [[∞-groupoids]]:

$$
 A Line \simeq B GL_1(A) \simeq B Aut(A)
$$

that identifies $A Line$ as the [[delooping]] [[∞-groupoid]] of either of these two [[∞-groups]].

=--

This appears in ([ABG, 3.6](#ABG)) ([p. 10](http://arxiv.org/pdf/1002.3004v2.pdf#page=10)). See also ([ABGHR 08, section 6](#ABGHR08)).

+-- {: .num_remark}
###### Remark

This means that every $A$-line $\infty$-bundle is canonically [[associated ∞-bundle|associated]] to a $GL_1(A)$-[[principal ∞-bundle]] over $X$ which is [[moduli space|modulated]] by a map $X \to B GL_1(A)$.

=--

+-- {: .num_defn}
###### Definition

A $GL_1(A)$-[[principal ∞-bundle]] on $X$ is also called a **[[twisted cohomology|twist]]** -- or better: a **[[local coefficient ∞-bundle]]** -- for
$A$-[[generalized (Eilenberg-Steenrod) cohomology|cohomology]] on $X$.

=--

For the moment see _[[twisted cohomology]]_ for more on this.

### Sections and twisted cohomology
 {#SectionsAndTwistedCohomology}

+-- {: .num_defn}
###### Definition

The $A$-module of (dual) [[sections]] of an $(\infty,1)$-module bundle $f : X \to A Mod$ is the [[(∞,1)-colimit]] over this functor

$$
  X^f := \lim_\to (X \stackrel{\alpha}{\to} A Mod)
  \,.
$$

The corresponding _spectrum of sections_ is the $A$-dual

$$
  \Gamma(f) := Mod_A(X^f, A)
  \,.
$$

=--

This is ([ABG, def. 4.1](#ABG)) and ([ABG, p. 15](#ABG)), ([ABG11, remark 10.16](#ABGII)).

+-- {: .num_remark}
###### Remark

For $f$ an $A$-line bundle  $\Gamma(f)$ is called in ([ABGHR 08, def. 7.27, remark 7.28](#ABGHR08)) the **[[Thom spectrum|Thom A-module]]** of $f$ and written $M f$.

=--

Because for $A = S$ the [[sphere spectrum]], $M f$ is indeed the classical [[Thom spectrum]] of the spherical fibration given by $f$:

+-- {: .num_prop}
###### Proposition

For $K = S$ the [[sphere spectrum]], $f : X \to K Line = S Line$ an $S$-line bundle -- hence a spherical fibration, and $A$ any other $\infty$-ring with canonical inclusion $S \to A$, the Thom $A$-module of the composite $X \stackrel{f}{\to} S Mod \to A Mod$ is the classical [[Thom spectrum]] of $f$ tensored with $A$:

$$
  \Gamma(X \stackrel{f}{\to} S Line \to A Line \to A Mod)
  \simeq
   X^f \wedge_S A
  \,.
$$

=--

This is ([ABGHR 08, theorem 4.5](#ABGHR08)).

### Trivializations and orientations

+-- {: .num_defn}
###### Definition

For $f : X \to A Line$ an $A$-line $\infty$-bundle, its [[∞-groupoid]] of **trivializations** is the $\infty$-groupoid of lifts

$$
  \array{
     && *
     \\
     & \nearrow & \downarrow
    \\
    X &\stackrel{f}{\to}& A Line
  }
  \,.
$$

For $K \to A$ the canonical inclusion and $f : X \to K Line$ a $K$-line bundle, we say that an **$A$-orientation** of $f$ is a trivialization of the associated $A$-line bundle $X \stackrel{f}{\to} K Line \to A Line$.

=--

That this encodes the notion of [[orientation in generalized cohomology|orientation in A-cohomology]] is around
 ([ABGHR 08, 7.32](#ABGHR08)).

+-- {: .num_cor #ThomModuleInOrientedCase}
###### Corollary

Every trivialization/orientation of an $A$-line $\infty$-bundle $f : X \to A Line$ induces an equivalence

$$
  \Gamma(f) \simeq (\Sigma^\infty X )\wedge A
$$

of the $A$-module of sections of $f$ / the [[Thom spectrum|Thom A-module]] of $f$ with the [[homology|generalized A-homology]]-spectrum of $X$:

$$
  \pi_\bullet \Gamma(f) \simeq H_\bullet(X,A)
  \,.
$$

=--

This appears as ([ABGHR 08, cor. 7.34](#ABGHR08)).

Therefore if $f$ is _not_ trivializable, we may regard its $A$-module of sections as encoding $f$-[[twisted cohomology|twisted A-cohomology]]:

+-- {: .num_defn}
###### Definition

For $f : X \to A Line$ an $A$-line $\infty$-bundle, the $f$-**[[twisted cohomology|twisted A-homology]] of $A$ is

$$
  H_\bullet^f(X, A) := \pi_\bullet(\Gamma(f)) := \pi_\bullet(M f)
  \,.
$$

The **$f$-[[twisted cohomology|twisted A-cohomology]] is**

$$
  H^\bullet_f(X,A) := \pi_0 A Mod(M f, \Sigma^\bullet A)
  \,.
$$

=--

## Structured $(\infty,1)$-vector bundles
 {#Structured}

We discuss now $(\infty,1)$-vector bundles in more general [[(∞,1)-toposes]].

(...)

## Applications

* The [[string topology]] operations on a compact [[smooth manifold]] $X$ may be understood as arising from a [[sigma-model]] [[quantum field theory]] with [[target space]] $X$ whose [[background gauge field]] is a flat $A$-line $\infty$-bundle $(P,\nabla)$ which is $A$-oriented over $X$, hence trivializabe over $X$ (for instance for $A = H \mathbb{Q}$ the [[Eilenberg-MacLane spectrum]] this may be the sphereical fibration of [[Thom spectrum|Thom spaces]] induced from the [[tangent bundle]] if the manifold is [[oriented]] in the ordinary sense).

  By prop. \ref{ThomModuleInOrientedCase} this implies that the space of [[state]]s of the $\sigma$-model is the $A$-homology spectrum $\Gamma(P) \simeq X \edge A$ of $X$, and that for every suitable [[surface]] $\Sigma$ with incoming and outgoing boundary components $\partial_{in} \Gamma \stackrel{in}{\to} \Gamma \stackrel{out}{\leftarrow} \partial_{out} \Gamma$ the [[mapping space]] [[span]]

  $$
    X^{\partial_{in} \Gamma}
     \stackrel{X^{in}}{\leftarrow}
    X^{\Gamma}
     \stackrel{X^{out}}{\rightarrow}
    X^{\partial_{out} \Gamma}
  $$

  acts by [[path integral as a pull-push transform]] on these spaces of states

  $$
    (X^{out})_* (X^{in})^! : H_\bullet(X^{\partial_{in} \Gamma},A) \to
     H_\bullet(X^{\partial_{out} \Gamma}, A)
     \,.
  $$

## Related concepts

* [[∞-representation]]

* [[twisted cohomology]]

* [[Picard ∞-stack]]

## References
 {#References}

A systematic discussion of discrete $(\infty,1)$-module bundles has a precursor in

* {#AndoHopkinsRezk10} [[Matthew Ando]], [[Michael Hopkins]], [[Charles Rezk]], _Multiplicative orientations of KO-theory and the spectrum of topological modular forms_, 2010 ([pdf](http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf))

(discussing the [[string orientation of tmf]]) and is then discussed in more detail in the triple of articles

* {#ABGHR08} [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], [[Michael Hopkins]], [[Charles Rezk]], _Units of ring spectra and Thom spectra_ ([arXiv:0810.4535](http://arxiv.org/abs/0810.4535))

* {#ABG} [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], _Twists of K-theory and TMF_, in Robert S. Doran, Greg Friedman, [[Jonathan Rosenberg]], _Superstrings, Geometry, Topology, and $C^*$-algebras_, Proceedings of Symposia in Pure Mathematics [vol 81](http://www.ams.org/bookstore-getitem/item=PSPUM-81), American Mathematical Society ([arXiv:1002.3004](http://arxiv.org/abs/1002.3004))

* {#ABGII} [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], _Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map_ ([arXiv:1112.2203](http://arxiv.org/abs/1112.2203))

The last of these explains the relation to

* [[Peter May|May]], Sigurdsson, _[[Parametrized Homotopy Theory]]_

A streamlined version of ([ABGHR 08](#ABGHR08)) appears as

* {#ABGHR14} [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], [[Michael Hopkins]], [[Charles Rezk]], _An $\infty$-categorical approach to $R$-line bundles, $R$-module Thom spectra, and twisted $R$-homology_ ([arXiv:1403.4325](http://arxiv.org/abs/1403.4325))

Lecture notes on these articles are in

* Ben Knudsen, Scott Slinker, Paul VanKoughnett, Brian Williams, and [[Dylan Wilson]], _Thom spectra reading course_ ([web](http://www.math.northwestern.edu/~bwill/thom/))

Interpretation of the  [[algebraic K-theory]] $K(R)$ of a [[ring spectrum]] $R$ (see at _[[iterated algebraic K-theory]]_) as the [[Grothendieck group]] of [[(∞,1)-module bundles]] over $R$:

* [[John Lind]], _Bundles of spectra and algebraic K-theory_, Pacific J. Math. 285 (2016) 427-452 ([arXiv:1304.5676](https://arxiv.org/abs/1304.5676), [doi:10.2140/pjm.2016.285.427](https://doi.org/10.2140/pjm.2016.285.427))

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(infinity,1)-monad

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of $(\infty,1)$-monad is the [[vertical categorification]] of that of [[monad]] from the context of [[category|categories]] to that of [[(∞,1)-category|(∞,1)-categories]].

They relate to [[(∞,1)-adjunctions]] as [[monads]] relate to [[adjunctions]].

## Properties

### Barr-Beck monadicity theorem
 {#MonadicityTheorem}

+-- {: .num_prop #CanonicalMonadicAdjunction}
###### Proposition

Given an [[(∞,1)-monad]] $T$ on an [[(∞,1)-category]]
$\mathcal{C}$, there is an [[(∞,1)-adjunction]]

$$
  (F \dashv U)
  \;\colon\;
  Alg_{\mathcal{C}}(T)
  \stackrel{\overset{F}{\leftrightarrow}}{\underset{U}{\longrightarrow}}
  \mathcal{C}
  \,,
$$

where $Alg_{\mathcal{C}}(T)$ is the (Eilenberg-Moore) [[(∞,1)-category of algebras over an (∞,1)-monad|(∞,1)-category of algebras over the (∞,1)-monad]] and where $U$ is the [[forgetful functor]] that remembers the underlying [[object]] of $\mathcal{C}$.

=--

This appears in ([Riehl-Verity 13, def. 6.1.14](#RiehlVerity13)).

The following is the refinement to [[(∞,1)-category theory]] of the classical [[Barr-Beck monadicity theorem]] which states sufficient conditions for recognizing an [[(∞,1)-adjunction]] as being canonically [[equivalence|equivalent]] to the one in prop. \ref{CanonicalMonadicAdjunction}, hence to be a _[[monadic adjunction]]_.

+-- {: .num_theorem #InfinityBarrBeckTheorem}
###### Theorem

Let $(L \dashv R)$ a pair of [[adjoint (∞,1)-functors]] such that

1. $R$ is a [[conservative (∞,1)-functor]];

1. the [[domain]] [[(∞,1)-category]] of $R$ admits [[geometric realization]] ([[(∞,1)-colimit]]) of [[simplicial objects in an (∞,1)-category|simplicial objects]];

1. and $R$ preserves these

then for $T \coloneqq R \circ L$ the essentially unique $(\infty,1)$-monad structure on the composite endofunctor, there is an [[equivalence of (∞,1)-categories]] identifying the [[domain]] of $R$ with the [[(∞,1)-category of algebras over an (∞,1)-monad]] $Alg_{\mathcal{C}}(T)$ over $T$ and $R$ itself as the canonical [[forgetful functor]] $U$ from prop. \ref{CanonicalMonadicAdjunction}.

=--

This appears as ([[Higher Algebra|Higher Algebra, theorem 6.2.0.6, theorem 6.2.2.5]], [Riehl-Verity 13, section 7](#RiehlVerity13))

### Homotopy coherence
 {#HomotopyCoherence}

+-- {: .num_remark}
###### Remark

An [[(∞,1)-adjunction]] $(L \dashv R) \colon \mathcal{C} \leftrightarrow \mathcal{D}$ is uniquely determined already by its image in the [[homotopy 2-category]] ([Riehl-Verity 13, theorem 5.4.14](#RiehlVerity13)). This is not in general true for $(\infty,1)$-monads $T \colon \mathcal{C} \to \mathcal{C}$. As these are [[monoids in an (∞,1)-category]] of [[endomorphisms]], they in general have relevant [[coherence]] data all the way up in degree. However, by the previous statement and the monadicity theorem \ref{InfinityBarrBeckTheorem}, for $(\infty,1)$-monads given via specified [[(∞,1)-adjunctions]] as $T \simeq R \circ L$ are determined by less (further) coherence data ([[Higher Algebra|Higher Algebra, remark 6.2.0.7, prop. 6.2.2.3]], [Riehl-Verity 13, page 6](#RiehlVerity13)). (Of course there is, instead, extra data/information carried by the choice of $\mathcal{D}$.) This should justify the [[simplicial model category]]-theoretic discussion in ([Hess 10](#Hess10)) in [[(∞,1)-category theory]].

=--

## Related concepts

* [[higher monadic descent]]

* [[algebraic theory]] / [[Lawvere theory]] /  [[(∞,1)-algebraic theory]]

* [[monad]] / [[2-monad]]/ [[doctrine]] / **$(\infty,1)$-monad**

  * [[idempotent (∞,1)-monad]]

  * [[modal type theory]]

* [[operad]] / [[(∞,1)-operad]]

## References

A general treatment of $(\infty,1)$-monads in [[(∞,1)-category theory]] is in

* [[Jacob Lurie]], section 3 of _Noncommutative algebra_ ([math.CT/0702299](http://arxiv.org/abs/math/0702299))

later absorbed as

* [[Jacob Lurie]], section 6.2 of _[[Higher Algebra]]_

More explict discussion in terms of [[quasi-categories]] and [[simplicial sets]]:

* {#RiehlVerity13} {#RiehlVerity16} [[Emily Riehl]], [[Dominic Verity]], _Homotopy coherent adjunctions and the formal theory of monads_, Advances in Mathematics, Volume 286, 2 January 2016, Pages 802-888 ([arXiv:1310.8279](http://arxiv.org/abs/1310.8279), [doi:10.1016/j.aim.2015.09.011](https://doi.org/10.1016/j.aim.2015.09.011))

Some homotopy theory of ([[enriched functor|enriched]]) monads on ([[simplicial model category|simplicial]]) [[model categories]] is discussed (with an eye towards [[higher monadic descent]]) in

* {#Hess10} [[Kathryn Hess]], _A general framework for homotopic descent and codescent_ ([arXiv/1001.1556](http://arxiv.org/abs/1001.1556))

[[!redirects (∞,1)-monad]]

[[!redirects (infinity,1)-monads]]
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(infinity,1)-operad

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of _$(\infty,1)$-operad_ is to that of [[(∞,1)-category]] as [[operad]] is to [[category]].

So, roughly, an $(\infty,1)$-operad is an algebraic structure that has for each given type of input and one type of output an [[∞-groupoid]] of operations that take these inputs to that output.

There is a fairly evident notion of [[∞-algebra over an (∞,1)-operad]]s. Examples include

* [[E-∞ algebra]]s

* [[L-∞ algebra]]s;

* [[A-∞ algebra]]s.

$(\infty,1)$-Operads form an [[(∞,2)-category]] [[(∞,1)Operad]].

## Definitions

Two models for $(\infty,1)$-operads exist to date, one by [[Denis-Charles Cisinski|Cisinski]]--[[Ieke Moerdijk|Moerdijk]]--[[Ittay Weiss|Weiss]], the other by [[Jacob Lurie|Lurie]]. It is expected though not yet entirely proved that the two are equivalent ([[Higher Algebra]], draft, Remark 2.0.0.8).

The first one models $(\infty,1)$-operads as [[dendroidal set]]s in close analogy to (in fact as a generalization of how) [[simplicial set]]s model [[(∞,1)-category|(∞,1)-categories]].

The second models the [[(∞,1)-category]] version of a [[category of operators]] of an operad.

### In terms of dendroidal sets

Here [[simplicial set]]s are generalized to [[dendroidal set]]s. The theory of $(\infty,1)$-operads is then formulated in terms of dendroidal sets in close analogy to how the theory of [[(∞,1)-category|(∞,1)-categories]] is formulated in terms of [[simplicial set]]s.

There is a [[model  structure on dendroidal sets]] whose fibrant objest are the **quasi-operad**s in direct analogy to the notion of [[quasi-category]].

So the [[model structure on dendroidal sets]] is a [[presentable (∞,1)-category|presention]] of the [[(∞,1)-category]] of $(\infty,1)$-operads. It is [[Quillen equivalence|Quillen equivalent]] to the standard [[model structure on operads]] enriched over [[Top]] or [[sSet]]. Therefore, conversely, the traditional homotopy-theoretic constructions on topological and chain operads (such as cofibrant [[resolution]]s in order to present homtopy algebras such as [[A-∞ algebra]]s, [[L-∞ algebra]]s, [[homotopy BV-algebra]]s and the like) are also indeed presentations of $(\infty,1)$-operads.

### In terms of $(\infty,1)$-categories of operators
 {#InTermsOfInfinityCategoriesOfOperators}

Every [[operad]] $A$ encodes and is encoded by its [[category of operators]] $C_A$. In the approach to $(\infty,1)$-operators described below, the notion of category of operators is generalized to an [[(∞,1)-category]] of operators.

In this approach an $(\infty,1)$-operad $C^\otimes$ is regarded as an [[(∞,1)-category]] $C$ -- the unary part of the $(\infty,1)$-operad to be described-- with extra structure that determines [[(∞,1)-functor]]s $C^{\times n} \to C$.

This and the conditions on these are encoded in requiring that $C^\otimes$ is an $(\infty,1)$-functor $C^\otimes \to \Gamma$ over [[Segal Gamma-category|Segal's category]] $\Gamma$ of pointed finite sets, satisfying some conditions.

In particular, any [[symmetric monoidal (∞,1)-category]] yields an example of an $(\infty,1)$-operad in this sense.  In fact, symmetric monoidal $(\infty,1)$-categories can be *defined* as $(\infty,1)$-operads such that the functor $C^\otimes \to \Gamma$ is a [[coCartesian fibration]].  (For the moment, see [[monoidal (infinity,1)-category]] for more comments and references on higher operads in this context.)

This is the approach described in ([LurieCommutative](#LurieCommutative))

#### Basic definitions
 {#BasicDefinitions}

We are to generalize the following construction from [[categories]] to [[(∞,1)-categories]].

+-- {: .num_defn}
###### Definition

For $\mathcal{O}$ a [[symmetric multicategory]], write $\mathcal{O}^\otimes \to FinSet^{*/}$ for its [[category of operators]].

Here $\mathcal{O}^\otimes$ is the [[category]] whose

* [[objects]] are finite sequences ([[tuples]]) of objects of $\mathcal{O}$;

* [[morphisms]] $(X_1, \cdots, X_{n_1}) \to (Y_1, \cdots, Y_{n_2})$ are given by a morphism $\alpha \colon \langle n_1\rangle \to \langle n_2\rangle$ in $FinSet_*$ together with a collection of [[multimorphisms]]

  $$
    \left\{
      \phi_j \in \mathcal{O}\left( \left\{ X_i\right\}_{i \in \alpha^{-1}\left\{j\right\}} , Y_j \right)
    \right\}_{1 \leq j \leq n_2}
    \,.
  $$

The [[functor]] $p \colon \mathcal{O}^\otimes \to FinSet^{*/}$ is the evident [[forgetful functor]].

=--

In ([Lurie](#Lurie)) this is construction 2.1.1.7.

This motivates the following definition of the generalization of this situation to [[(∞,1)-category theory]].

+-- {: .num_defn #FinSetPointed}
###### Definition

Write $FinSet^{*/}$ for the category of [[pointed object|pointed]] [[finite set]] ([[Segal Gamma-category|Segal's Gamma-category]]).

For $n \in \mathbb{N}$ we write

$$
  \langle n\rangle
  \coloneqq
  {*} \coprod [n]
  \in
  FinSet^{*/}
$$

for the [[pointed set]] with $n+1$ elements.

A morphism in $FinSet^{*/}$

* is called an  **[[inert morphism]]** if it is a surjection, and an [[injection]] on those elements that are not sent to the base point. That is, the preimage of every non-base point is a singleton.

* called an **[[active morphism]]** if only the basepoint goes to the basepoint.

For $n \in \mathbb{N}$ and $1 \leq i \leq n$ write

$$
  \rho^i \colon \langle n\rangle \to \langle 1\rangle
$$

for the inert morphism that sends all but the $i$th element to the basepoint.

Notice that for each $n \in \mathbb{N}$ there is a unique [[active morphism]] $\langle n\rangle \to \langle 1\rangle$.

=--

([Lurie, def. 2.1.1.8](#Lurie))

+-- {: .num_defn #InfinityOperadByCategoryOfOperators}
###### Definition

The **$(\infty,1)$-category of operators of an $(\infty,1)$-operad** is a morphism

$$
  p \colon \mathcal{O}^\otimes \to FinSet^{*/}
$$

of [[quasi-categories]] such that the following conditions hold:

1. For every [[inert morphism]] in $FinSet^{*/}$ and every [[object]] over it, there is a lift to a $p$-[[coCartesian morphism]] in $\mathcal{O}^\otimes$. In particular, for $f \colon \langle n_1\rangle \to \langle n_2\rangle$ inert, there is an induced [[(∞,1)-functor]]

   $$
     f_!
       \colon
     \mathcal{O}^\otimes_{\langle n_1\rangle}
      \to
     \mathcal{O}^\otimes_{\langle n_2\rangle}
     \,.
   $$

1. The coCartesian lifts of the inert projection morphisms induce an equivalence of [[derived hom-spaces]] in $\mathcal{O}^{\otimes}$ between maps into multiple objects and the products of the maps into the separete objects:

   For $f \colon \langle n_1 \rangle \to \langle n_2 \rangle$ write $\mathcal{O}^\otimes_f(-,-) \hookrightarrow \mathcal{O}^\otimes(-,-)$ for the components of the [[derived hom-space]] covering $f$, then the $(\infty,1)$-functor

   $$
     \mathcal{O}^\otimes_f(C_1,C_2)
     \to
     \underset{1 \leq k \leq n_2}{\prod}
     \mathcal{O}^\otimes_{\rho^i\circ f}(C_1,(C_2)_i)
   $$

   induced as above is an [[equivalence of infinity-groupoids|equivalence]].

1. For every finite collection of objects $C_1, \cdots c_n \in \mathcal{O}^\otimes_{\langle 1\rangle}$ there exists a multiobject $C \in \mathcal{O}^\otimes_{\langle n\rangle}$ and a collection of $p$-[[coCartesian morphisms]] $\{C \to C_i\}$ covering $\rho^i$.

   Equivalently (given the first two conditions): for all $n \in \mathbb{N}$ the $(\infty,1)$-functors $\{(\rho^i)_!\}_{1 \leq i \leq n}$ induce an [[equivalence of (∞,1)-categories]]

   $$
     \mathcal{O}^\otimes_{\langle n\rangle}
     \to
     (\mathcal{O}^\otimes_{\langle 1\rangle})^{\times^n}
   $$

=--

([Lurie, def. 2.1.1.10, remark 2.1.1.14](#Lurie))

+-- {: .num_remark }
###### Remark

The conditions in def. \ref{InfinityOperadByCategoryOfOperators} mean that $p \colon \mathcal{O}^\otimes \to FinSet^{*/}$ encodes

1. an [[(∞,1)-category]] $\mathcal{O} \coloneqq \mathcal{O}^\otimes_{\langle 1\rangle}$;

1. for each $n \in \mathbb{N}$ an $n$-ary operation given by the $(\infty,1)$-functor

   $$
     \mathcal{O}^{n}
     =
     (\mathcal{O}^\otimes_{\langle 1\rangle})^{\times n}
     \simeq
     \mathcal{O}^{\otimes}_{\langle n\rangle}
     \to
     \mathcal{O}^\otimes_{\langle 1\rangle}
     =
     \mathcal{O}
   $$

   induced by the unique [[active morphism]] $\langle n\rangle \to \langle 1\rangle$

1. a coherently associative multicomposition of these operations.

=--

+-- {: .num_remark }
###### Remark

The notion of def. \ref{InfinityOperadByCategoryOfOperators} may also be called a **symmetric $(\infty,1)$-multicategory** or **colored $(\infty,1)$-operad**. The _colors_ are the [[objects]] of $\mathcal{O}$.

=--

We now turn to the definition of [[homomorphisms]] of $(\infty,1)$-operads.

+-- {: .num_defn #InertMorphismsInInfinityOperad}
###### Definition

Given an $(\infty,1)$-operad $p \colon \mathcal{O}^\otimes \to FinSet^{*/}$ as in def. \ref{InfinityOperadByCategoryOfOperators}, a [[morphism]] $f$ in $\mathcal{O}^\otimes$ is called an **inert morphism** if

1. $p(f)$ is an [[inert morphism]] in $FinSet^{*/}$ by def. \ref{FinSetPointed};

1. $f$ is a $p$-[[coCartesian morphism]].

=--

+-- {: .num_defn #MorphismOfInfinityOperads}
###### Definition

A **[[morphism of (∞,1)-operads]]** is a map between their [[(∞,1)-categories of operators]] over $FinSet^{*/}$ that preserves the inert morphisms of def. \ref{InertMorphismsInInfinityOperad}.

=--

Morphisms of operads $\mathcal{O}_1 \to \to \mathcal{O}_2$ can be understood equivalently as exhibiting an $\mathcal{O}_1$-[[algebra over an operad|algebra]] in $\mathcal{O}_2$. Therefore:

+-- {: .num_defn }
###### Definition

For $\mathcal{O}_1, \mathcal{O}_2$ to $(\infty,1)$-operads, write

$$
  Alg_{\mathcal{O}_1}(\mathcal{O}_2)
  \hookrightarrow
  qCat_{/FinSet^{*/}}(\mathcal{O}_1, \mathcal{O}_2)
$$

for the full [[sub-(∞,1)-category]] of the [[(∞,1)-functor (∞,1)-category]] on those that are [[morphisms of (∞,1)-operads]] by def. \ref{MorphismOfInfinityOperads}.

=--

([Lurie, def. 2.1.2.7](#Lurie)).

We also have the notion of

* _[[fibration of (∞,1)-operads]]_;

* _[[coCartesian fibration of (∞,1)-operads]]_.

See there for more details.

#### Model for $(\infty,1)$-categories of operators {#ModelForinfOpera}

There is a [[model category]] that [[presentable (infinity,1)-category|presents]] the [[(∞,1)-category]] $(\infty,1)Cat_{Oper}$ of $(\infty,1)$-categories of operations.

+-- {: .num_prop}
###### Proposition

There exists a

* [[proper model category|left proper]]

* [[combinatorial model category|combinatorial]]

[[model category]] $\mathcal{P} Op_{(\infty,1)}$

* whose underlying [[category]] has

  * [[object]]s are [[marked simplicial set]] $S$ equipped with a morphism $S \to N(FinSet_*)$ such that marked edges map to inert morphisms in $FinSet_*$ (those for which the preimage of the marked point contains just the marked point)

  * [[morphism]]s are morphisms of [[marked simplicial set]]s $S \to T$ such that the triangle

    $$
      \array{
        S &&\to&& T
        \\
        & \searrow && \swarrow
        \\
        && N(FinSet_*)
      }
    $$

    commutes;

* which is canonically an [[SSet]]-[[enriched category]];

* and whose [[model category|model structure]] is given by

  * cofibrations are those morphisms whose underlying morphisms of [[simplicial set]]s ate cofibrations, hence [[monomorphism]]s

  * weak equivalences are those morphisms $S \to T$ such that for all $A \to N(FinSet_*)$ that are $(\infty,1)$-categories of operations by the above definition, the morphism of [[SSet]]-[[hom object]]s

    $$
      \mathcal{P}Op_\infty(T,A) \to
      \mathcal{P}Op_\infty(S,A)
    $$

    is a homotopy equivalence of simplicial sets.

  * an object is fibrant if and only if it is an $(\infty,1)$-category of operations, by the above definition.

=--

This is prop 1.8 4 in

* [[Jacob Lurie]], _Commutative algebra_ ([pdf](http://www.math.harvard.edu/~lurie/papers/DAG-III.pdf))

## Examples

We list some examples of $(\infty,1)$-operads incarnated as their [[(∞,1)-categories of operators]] by def. \ref{InfinityOperadByCategoryOfOperators}.

The first basic examples to follow are in fact all given by [[1-categories]] [[category of operators|of operators]].

+-- {: .num_example}
###### Example

The [[identity]] functor on $FinSet^{*/}$ exhibits an $(\infty.1)$-operad. This is the **[[commutative operad]]**

$$
  Comm^\otimes = FinSet^{*/} \stackrel{id}{\to} FinSet^{*/}
  \,.
$$

The [[(∞,1)-algebras over an (∞,1)-operad]] over this $(\infty,1)$-operad are [[E-∞ algebras]].

=--

+-- {: .num_example}
###### Example

The **[[associative operad]]** has $Assoc^\otimes$ the category whose objects are the natural numbers, whose $n$-ary operations
are labeled by the [[total orders]] on $n$ elements, equivalently the elements of the [[symmetric group]] $\Sigma_n$, and whose composition is given by forming consecutive total orders in the obvious way.

The [[(∞,1)-algebras over an (∞,1)-operad]] over this $(\infty,1)$-operad are [[A-∞ algebras]]

=--

In ([Lurie](#Lurie)) this is remark 4.1.1.4.

+-- {: .num_example}
###### Example

The **[[operad for modules over an algebra]]** $LM$ is the [[colored operad|colored]] [[symmetric operad]] whose

* [[objects]] are two elements, to be denoted $\mathfrak{a}$ and $\mathfrak{n}$;

* [[multimorphisms]] $(X_i)_{i = 1}^n \to Y$ form

  * if $Y = \mathfrak{a}$ and $X_i = \mathfrak{a}$ for all $i$ then: the set of [[linear orders]] on $n$ elements, equivalently the elements of the [[symmetric group]] $\Sigma_n$;

  * if $Y = \mathfrak{n}$ and exactly one of the $X_i = \mathfrak{n}$ then: the set of linear order $\{i_1 \lt \cdots \lt i_n\}$ such that $X_{i_n} = \mathfrak{n}$

  * otherwise: the empty set;

* [[composition]] is given by composition of linear orders as for the [[associative operad]].

The [[(∞,1)-algebras over an (∞,1)-operad]] over this $(\infty,1)$-operad are pairs consisting of [[A-∞ algebras]] with [[(∞,1)-modules]] over them.

=--

In ([Lurie](#Lurie)) this appears as def. 4.2.1.1.

+-- {: .num_defn}
###### Definition

The **[[operad for bimodules over algebras]]** $BMod$ is the [[colored operad|colored]] [[symmetric operad]] whose

* [[objects]] are three elements, to be denoted $\mathfrak{a}_-, \mathfrak{a}_+$ and $\mathfrak{n}$;

* [[multimorphisms]] $(X_i)_{i = 1}^n \to Y$ form

  * if $Y = \mathfrak{a}_-$ and all $X_i = \mathfrak{a}_-$ then: the set of [[linear orders]] of $n$ elements;

  * if $Y = \mathfrak{a}_*$ and all $X_i = \mathfrak{a}_*$ then again: the set of [[linear orders]] of $n$ elements;

  * if $Y = \mathfrak{n}$: the set of linear orders $\{i_1 \lt \cdots \lt i_n\}$ such that there is exactly one index $i_k$ with $X_{i_k} = \mathfrak{n}$ and $X_{i_j} = \mathfrak{a}_-$ for all $j \lt k$ and $X_{i_j} = \mathfrak{a}_+$ for all $k \gt k$.

* [[composition]] is given by the composition of linear orders as for the [[associative operad]].

The [[(∞,1)-algebras over an (∞,1)-operad]] over this $(\infty,1)$-operad are pairs consisting of two [[A-∞ algebras]] with an [[(∞,1)-bimodule]] over them.

=--

## Properties

### Relation between the two definitions
  {#relation}

+-- {: .standout}

At the time of this writing there is no discussion in "the literature" of the relation between the definition of $(\infty,1)$-operads in terms of dendroidal sets (Cisinski, Moerdijk, Weiss) and $(\infty,1)$-categories of operators (Lurie). The following are some tentative observations. - [[Urs Schreiber|Urs]]

update: meanwhile this has been worked out by some people. Results should appear in preprint form soon.

=--

There is an obvious way to regard a [[tree]] as an $(\infty,1)$-category of operators:

+-- {: .num_defn}
###### Definition
**(dendroidal $(\infty,1)$-category of operators)**

Let

$$
  \omega : \Omega \hookrightarrow Op \stackrel{C_{(-)}}{\to}
  Cat/FinSet_* \stackrel{N}{\to}
  \mathcal{P}Op_{(\infty,1)}
$$

be the dendroidal object given by the following composition:

* $\Omega \hookrightarrow Op$ is the functor from the [[tree category]] $\Omega$ to the category of symmetric colored [[operad]]s (over [[Set]]) that sends a tree to the operad freely generated from it;

* $Op \stackrel{C_{(-)}}{\to} Cat/FinSet_*$ sends an [[operad]] to its [[category of operators]];

* $Cat/FinSet_*
  \stackrel{N}{\to} \mathcal{P}Op_{(\infty,1)}$ takes
  the [[nerve]] of this category, regarded as
  a  [[marked simplicial set]] over $N(FinSet_*)$,
  whose marked edges are the inert morphisms in
  the category of operations.

=--

Following the general pattern of [[nerve and realization]], we get:

+-- {: .num_defn}
###### Definition
**(dendroidal nerve of Lurie-$\infty$-operad)**

The functor

$$
  N_d :=
   Hom_{\mathcal{P}Op_{(\infty,1)}}(\omega(-), -):
   \mathcal{P}Op_{(\infty,1)} \to dSet
$$

that sends a [[marked simplicial set]] $A \to N(FinSet_*)$ to the [[dendroidal set]] which sends a [[tree]] $T$ to the set of morphisms of $\omega(T)$ into $A$

$$
  N_d(A) : T \mapsto Hom_{\mathcal{P}Op_{(\infty,1)}}(\omega(T), A)
$$

is the **dendroidal nerve** of $A$.

=--

+-- {: .standout}

One expects that $N_d$ induces a [[Quillen adjunction]] and indeed a [[Quillen equivalence]] between the above model category structure on $\mathcal{P}Op_{(\infty,1)}$ and the [[model structure on dendroidal sets]]. The following is as far as I think I can prove aspects of this. -[[Urs Schreiber|Urs]].

=--

+-- {: .num_prop}
###### Proposition

The dendroidal nerve functor has the following properties:

* it is the [[right adjoint]] of a [[SSet]]-[[enriched functor|enriched]] [[adjunction]]

  $$
    C_{(-)} : dSet \stackrel{\leftarrow}{\to} \mathcal{P}Op_{(\infty,1)}
   : N_d
  $$

* it sends fibrant objects to fibrant objects

  i.e. it sends $(\infty,1)$-categories of
  operations to $(\infty,1)$-operads
  in their incarnation as "quasi-operads";

* it sends objects $\pi : A \to N(FinSet_*)$ that come
  from grouplike [[symmetric monoidal (infinity,1)-category|symmetric monoidal]] [[∞-groupoid]]s to fully Kan dendroidal sets (that have the extension property with respect to all horns)

* it sends objects $\pi : A \to N(FinSet_*)$ that come
  from
  [[symmetric monoidal (infinity,1)-category|symmetric monoidal (∞,1)-categories]] to dendroidal sets that have the extension property with respect to at least one outer horn $\Lambda_{v} T$ for $v \in T$ an $n$-corolla, for all $n \in \mathbb{N}$.

* its [[left adjoint]] sends cofibrations to cofibrations and acyclic cofibrations with cofibrant domain to acyclic cofibrations.

=--

+-- {: .proof}
###### Proof

**respect for fibrant objects**. If $A \to N(FinSet_*)$ is fibrant, then in particual $A$ is a [[weak Kan complex]] hence has the extension property with respect to all inner [[horn]] inclusions of [[simplex|simplices]]. We need to show that this implies that $N_d(A)$ has the extension property with respect to all inner horn inclusions of [[tree]]s.

By an (at the moment unpublished) result by [[Ieke Moerdijk|Moerdijk]], right [[lifting property]] with respect to inner horn inclusions of trees is equivalend to right lifting property with respect to inclusions of [[spine]]s of trees: the union over all the corollas in a tree.

For this the extension property means that if we find a collection $\{C_{k_i} \to N_d(A)\} = Sp(T)$ of corollas in $N_d(A)$ that match at some inputs and output, then these can be composed to an image $T \to N_d(A)$ of the corresponding tree $T$ in $N_d(A)$.

An image of $T$ in $N_d(A)$ is an image of $\omega(T)$ in $A$. In the [[category of operators]] $\omega(A)$ every tree may be represented as the composite of a sequence of morphisms each of which consists of precisely one of the corollas $C_{k_i}$ in parallel to identity morphisms. This way gluing the tree from the corollas is a matter of composing a sequence of edges in $A$. But this is guaranteed to be possible if $A$ is a [[weak Kan complex]].

**symmetric monoidal product and outer horn lifting**

As described at [[cartesian morphism]], an edge $f : \Delta^1 \to A$ in $A$ is coCartesian if for all diagrams

$$
  \array{
    \Delta^{0,1}
    \\
    \downarrow & \searrow^f
    \\
    \Lambda^n_0 &\to & A
    \\
    \downarrow && \downarrow
    \\
    \Delta^n &\to& N(FinSet_*)
  }
$$

of 0-horn lifting problems where the first edge of the horn is $f$ itself, there exists a lift

$$
  \array{
    \Delta^{0,1}
    \\
    \downarrow & \searrow^f
    \\
    \Lambda^n_0 &\to & A
    \\
    \downarrow &\nearrow & \downarrow
    \\
    \Delta^n &\to& N(FinSet_*)
  }
  \,.
$$

For $f$ the parallel application of an $n$-corolla with a collection of identity morphisms this implies that any outer horn $\Lambda_v T \to N_d(A)$ for which the vertex $v : C_n \to N_d(A)$ maps to $f$, the dendroidal set $N_d(A)$ has the extension property with respect to the inclusion $\Lambda_d T \hookrightarrow T$.

**the left adjoint and its respect for cofibrations**

By general nonsense the [[left adjoint]] to $N_d$ is given by the [[coend]]

$$
  C_{(-)} : dSet \to \mathcal{P}Op_{(\infty,1)}
$$

$$
  C_P = \int^{T \in \Omega} \omega(T) \cdot P(T)
  \,,
$$

where in the integrand we have the tautological [[copower|tensoring]] of $\mathcal{P}Op_{(\infty,1)}$ over [[Set]].

Notice that $\omega : \Omega \to \mathcal{P}Op_{(\infty,1)}$ is an [[SSet]]-[[enriched functor]] for the ordinary category $\Omega$ regarded as a simplicially enriched category by the canonical embedding $Set \hookrightarrow SSet$. Therefore this adjunction $F \dashv N_d$ is defined entirely in [[SSet]]-[[enriched category theory]] and hence is a simplicial adjunction.

The [[model structure on dendroidal sets]] has a set of [[cofibrantly generated model category|generating cofibrations]] given by the boundary inclusions of trees. $\partial \Omega[T] \hookrightarrow \Omega[T]$. Tese evidenly map to monomorphisms of underlying simplicial sets under $F$, hence to cofibrations.

For $f : P \hookrightarrow Q$ an acyclic cofibration with cofibrant domain, we need to check that $C_f : C_X \to C_Y$ is a weak equivalence in $\mathcal{P}Op_{(\infty,1)}$. This is by definition the case if for every fibrant object $A$ the morphism

$$
  \mathcal{P}Op_{(\infty,1)}(C_Y,A) \to
  \mathcal{P}Op_{(\infty,1)}(C_X,A)
$$

is a weak equivalence in the standard [[model structure on simplicial sets]]. By the simplicial adjunction $F \dashv N_d$ this is equivalent to

$$
  dSet(f,N_d(A))
  :
  dSet(Y,N_d(A)) \to
  dSet(X,N_d(A))
$$

being a weak equivalence. By the above $N_d(A)$ is fibrant. By section 8.4 of the lecture notes on dendroidal sets cited at [[model structure on dendroidal sets]] a morphism between cofibrant dendroidal sets is a weak equivalence precisely if homming it into any fibrant dendroidal set produces an equivalence of homotopy categories.

Since $f$ is a weak equivalence between cofibrant objects by assumption, it follows that indeed $dSet(f,N_d(A))$ is a weak equivalence for all fibrant $A$.

> (AHM, or does it? there is a prob here, but I need to run now...)

Hence $C_f$ is a weak equivalence.

=--

## Related concepts

* [[table - models for (∞,1)-operads]]

* [[algebraic theory]] / [[Lawvere theory]] /  [[(∞,1)-algebraic theory]]

* [[monad]] / [[(∞,1)-monad]]

* [[operad]] / **$(\infty,1)$-operad**, [[model structure on operads]]

  * [[morphism of (∞,1)-operads]], [[fibration of (∞,1)-operads]]

  * [[generalized (∞,1)-operad]], [[family of (∞,1)-operads]]

  * [[algebra over an (∞,1)-operad]], [[model structure on algebras over an operad]]

    * [[module over an algebra over an (∞,1)-operad]], [[model structure on modules over an algebra over an operad]]

  * [[coherent (∞,1)-operad]]

* [[operadic (∞,1)-Grothendieck construction]]

* [[cohomology of operads]]

## References

The formulation in terms of [[dendroidal sets]] is due to

* [[Ieke Moerdijk]], [[Ittay Weiss]], _Dendroidal sets_ ([web](http://cat.inist.fr/?aModele=afficheN&cpsidt=20087314))

* [[Denis-Charles Cisinski]], [[Ieke Moerdijk]], _Dendroidal sets as models for homotopy operads_ ([arXiv](http://arxiv.org/abs/0902.1954)) .

Here are two blog entries on talks on this stuff:

* [Dendroidal Sets and Infinity-Operads](http://golem.ph.utexas.edu/category/2009/02/dendroidal_sets.html)

* [Moerdijk on Infinity Operads](http://golem.ph.utexas.edu/category/2009/02/moerdijk_on_infinityoperads.html)

The formulation in terms of an $(\infty,1)$-version of the [[category of operators]] is introduced in

* [[Jacob Lurie]], _[[Commutative Algebra]]_ ([pdf](http://www.math.harvard.edu/~lurie/papers/DAG-III.pdf))
{#LurieCommutative}

and further discussed in

* _[[Ek-Algebras]]_ .

Now in section 2 of the textbook

* _[[Higher Algebra]]_
 {#Lurie}

The equivalence between the [[dendroidal set]]-formulation and the one in terms of $(\infty,1)$-categories of operators is shown in

* [[Gijs Heuts]], [[Vladimir Hinich]], [[Ieke Moerdijk]], _The equivalence between Lurie's model and the dendroidal model for infinity-operads_ ([arXiv:1305.3658](http://arxiv.org/abs/1305.3658))

Further equivalence to Barwick's complete Segal operads is discussed in

* Hongyi Chu, [[Rune Haugseng]], [[Gijs Heuts]], _Two models for the homotopy theory of $\infty$-operads_ ([arXiv:1606.03826](https://arxiv.org/abs/1606.03826))

For an account in terms of _analytic_ [[monads]], that is, monads that are cartesian (multiplication and unit transformations are cartesian) and the underlying endofunctor preserves sifted colimits and wide pullbacks (or equivalently all weakly contractible limits), see

* [[David Gepner]], [[Rune Haugseng]], [[Joachim Kock]], _∞-Operads as Analytic Monads_, ([arXiv:1712.06469](https://arxiv.org/abs/1712.06469))

On the [[Eckmann-Hilton argument]] for (∞,1)-operads:

* [[Tomer Schlank]], [[Lior Yanovski]], *The ∞-Categorical Eckmann-Hilton Argument*, Algebr. Geom. Topol. 19 (2019) 3119-3170 ([arXiv:1808.06006](https://arxiv.org/abs/1808.06006), [doi:10.2140/agt.2019.19.3119](https://doi.org/10.2140/agt.2019.19.3119))

[[!redirects (∞,1)-operad]]
[[!redirects (∞,1)-operads]]
[[!redirects (infinity,1)-operads]]


(infinity,1)-presheaf

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition ##

Write $(\infty,0)Cat$ for [[generalized the|the]] category [[∞Grpd]] of $\infty$-[[infinity-groupoid|groupoids]] regarded as an [[(∞,1)-category]].

Let $S$ be a [[simplicial set]] (which in particular may be a [[quasi-category]]).

An **$(\infty,1)$-presheaf** on $S$ is an [[(∞,1)-functor]]

$$
  F : S^{op} \to (\infty,0)Cat
  \,.
$$

The [[(infinity,1)-category of (infinity,1)-presheaves|(∞,1)-category of $(\infty,1)$-presheaves]] is the corresponding [[(∞,1)-category of (∞,1)-functors]]

$$
 PSh(S) := Fun(S^{op}, (\infty,0)Cat)
 \,.
$$

## Remarks ##

* $(\infty,1)$-presheaves are the basic building block for the definition of [[(infinity,1)-category of (infinity,1)-sheaves|(∞,1)-categories of (∞,1)-sheaves]].

## Model structures

$(\infty,1)$-presheaves can be presented by many different [[model categories]], corresponding to several of the model structures for [[(∞,1)-categories]].  These include:

* [[global model structures on simplicial presheaves]] (injective and projective)
* [[model structure for right fibrations]] of [[quasicategories]]
* [[model structure on internal simplicial presheaves]]
* [[model structure for right fibrations of Segal spaces]]
* [[model structure on simplicial presheaves over simplices]]

Various Quillen equivalences between these model structures are constructed in the references.  For special cases of the domain $S$ there exist other model structures that are also Quillen equivalent to these, such as:

* The [[Reedy model structure]] on simplicial presheaves, when $S$ is a [[Reedy category]], or more generally a [[generalized Reedy category]]
* The analogous [[model structure on internal inverse diagrams]], when $S$ is an [[inverse EI (∞,1)-category]].

## Related concepts

* [[(0,1)-presheaf]]

* [[presheaf]]

* [[(2,1)-presheaf]]

* **$(\infty,1)$-presheaf**, [[(∞,1)-sheaf]], [[∞-stack]] ,

  * [[simplicial presheaf]]

  * [[model structure on functors]]

* [[(∞,n)-presheaf]]

## References ##

### $(\infty,1)$-categorical definition

This is in Section 5.1 of

* [[Jacob Lurie]], _[[Higher Topos Theory]]_ .

### Model structures

The various model structures, and their Quillen equivalences, can be found in the following references.

The [[global model structures on simplicial presheaves]] are a standard special case of the [[global model structures on functors]].  The fact that the injective model structure exists is a bit less classical; see [[injective model structure]] for references.

The [[model structure for right fibrations]] of [[quasicategories]] is constructed in [[Higher Topos Theory]].  It is shown there to be Quillen equivalent to the global model structure on simplicial presheaves by a [[straightening functor]].  Alternative proofs of such an equivalence can be found in

* [[Gijs Heuts]], [[Ieke Moerdijk]], *Left fibrations and homotopy colimits II*, [arXiv](http://arxiv.org/abs/1602.01274)

* [[Danny Stevenson]], *Covariant Model Structures and Simplicial Localization*, [arXiv](https://arxiv.org/abs/1512.04815)

The latter also constructs the [[model structure on simplicial presheaves over simplices]] and links it with Quillen equivalences to the other two.

The [[model structure on internal simplicial presheaves]] is constructed in

* [[Geoffroy Horel]], *A model structure on internal categories in simplicial sets*, [TAC](http://tac.mta.ca/tac/volumes/30/20/30-20abs.html)

The [[model structure for right fibrations of Segal spaces]] is constructed in

* Pedro Boavida de Brito, *Segal objects and the Grothendieck construction*, [arXiv](http://arxiv.org/abs/1605.00706)

and shown to be Quillen equivalent to both the [[model structure on internal simplicial presheaves]] and the [[model structure for right fibrations]] of quasicategories.

Finally, the [[model structure on internal inverse diagrams]] is constructed, and shown to be Quillen equivalent to the model structure for internal simplicial presheaves (hence all the others) in

* [[Mike Shulman]], *Univalence for inverse EI diagrams*, [arXiv](http://arxiv.org/abs/1508.02410).

[[!redirects (∞,1)-presheaf]]
[[!redirects (∞,1)-presheaves]]
[[!redirects (infinity,1)-presheaves]]
[[!redirects ∞-presheaf]]
[[!redirects ∞-presheaves]]

[[!redirects ∞-prestack]]
[[!redirects ∞-prestacks]]
[[!redirects (∞,1)-prestack]]
[[!redirects (∞,1)-prestacks]]

[[!redirects infinity-presheaf]]
[[!redirects infinity-presheaves]]

[[!redirects infinity-prestack]]
[[!redirects infinity-prestacks]]
[[!redirects (infinity,1)-prestack]]
[[!redirects (infinity,1)-prestacks]]


(infinity,1)-pretopos

[[!redirects (∞,1)-pretopos]]

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### Context
#### $(\infty,1)$-Topos Theory
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[[!include (infinity,1)-topos - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The concept of _$(\infty,1)$-pretopos_ ([Lurie, appendix A](#Lurie)) is a version of the concept of _[[pretopos]]_ as one passes from [[toposes]] to [[(∞,1)-toposes]]. The definition is a variant of the characterization of [[Grothendieck (∞,1)-toposes]], via the [Giraud-Rezk-Lurie axioms](infinity-topos#GiraudAxioms), asking only for [[finite (∞,1)-limits]] and [[finite (∞,1)-colimits]] with some exactness properties relating them.

## Definition

+-- {: .num_defn #InfinityPreTopos}
###### Definition

Let $\mathcal{C}$ be an [[(∞,1)-category]]. This is called an _$(\infty,1)$-pretopos_ if

1. $\mathcal{C}$ has a [[terminal object in an (∞,1)-category|terminal object]] and [[homotopy fiber products]];

1. $\mathcal{C}$ has [[finite (∞,1)-colimits]];

1. finite [[coproducts]] in $\mathcal{C}$ are [[universal colimits|universal]] and [[disjoint coproducts|disjoint]];

1. [[groupoid object in an (infinity,1)-category|groupoid objects]] in $\mathcal{C}$ are effective:

1. realization of groupoid objects is [[universal colimit|universal]].

If these conditions hold except possibly for the existence of a [[terminal object in an (∞,1)-category|terminal object]], then $\mathcal{C}$ is a _local $(\infty,1)$-pretopos._

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[Lurie, def. A:6.1.1](#Lurie)

## Examples

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###### Example

Every [[Grothendieck (∞,1)-topos]] is an $(\infty,1)$-pretopos (def. \ref{InfinityPreTopos}).

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([Lurie, example A.6.1.5](#Lurie))

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###### Example

Let $\mathbf{H}$ be a [[Grothendieck (∞,1)-topos]] then the [[full sub-(∞,1)-category]]

$$
  \mathbf{H}_{coh} \hookrightarrow \mathbf{H}
$$

on the [[coherent objects]] is a local $(\infty,1)$-pretopos (def. \ref{InfinityPreTopos}).

If moreover $\mathbf{H}$ is an [[coherent (∞,1)-topos]], then $\mathbf{H}_{coh}$ is an $(\infty,1)$-pretopos.

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([Lurie, prop. A.6.1.6, cor. 6.1.7](#Lurie))

## Related concepts

* [[elementary (∞,1)-topos]]

## References

* {#Lurie} [[Jacob Lurie]], appendix A of _[[Spectral Algebraic Geometry]]_ ([pdf]( http://math.harvard.edu/~lurie/papers/SAG-rootfile.pdf))