(infinity,1)-profunctor

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## Idea

The concept of _$(\infty, 1)$-profunctor_ is the [[categorification]] of that of [[profunctors]] from [[category theory]] to [[(∞,1)-category theory]].

## Definition ##

If $C$ and $D$ are [[(∞,1)-categories]], then a **profunctor** from $C$ to $D$ is a [[(∞,1)-functor]] of the form

$$
  F \colon D^{op}\times C \to \infty Grpd
  \,.
$$

Such a  profunctor is usually written as $F \,\colon\,  C ⇸ D$. Composition of (∞,1)-profunctors in [[(∞,1)Prof]] is by the "tensor product of (∞,1)-functors" [[homotopy coend]] construction: if $H\colon A ⇸  B$ and $K\colon B ⇸  C$, their composite is given as a functor $C^{op}\times A \to \infty Grpd$ by
$$(c,a)\mapsto \int^{b\in B} H(b,a)\times K(c,b).$$

Every [[(∞,1)-functor]] $f\colon C\to D$ induces two (∞,1)-profunctors $D(1,f)\colon C ⇸ D$ and $D(f,1)\colon D ⇸ C$, defined by $D(1,f)(d,c) = D(d,f(c))$ and $D(f,1)(c,d) = D(f(c),d)$. (Here $D(-,-)$ denotes the [[hom functor]] of $D$ and $1$ denotes the identity (∞,1)-functor on the respective category.)

Since a profunctor is also known as a (bi)module or a distributor or a correspondence, we should expect other names to be used for $(\infty, 1)$-profunctor. In [[Higher Topos Theory]] (Definition 2.3.1.3), Lurie speaks of a **correspondence**.

## Properties

### Relation to presentable $\infty$-categories
 {#RelationToPresentableInfinityCategories}

In analogy to the situation for [[profunctors]] between [[1-categories]] (see [there](profunctor#FuncsOnPresheaves)), $\infty$-profunctors

$$
  \mathcal{C} ⇸ \mathcal{D}
$$

are equivalently plain but [[(infinity,1)-colimit|$\infty$-colimit]]-[[preserved limit|preserving]] [[(infinity,1)-functors|$\infty$-functors]]

$$
  PSh_\infty(\mathcal{C})
  \xrightarrow{\;}
  PSh_\infty(\mathcal{D})
$$

between the corresponding [[(infinity,1)-categories of (infinity,1)-presheaves|$\infty$-categories of $\infty$-presheaves]].

In this way [[small (infinity,1)-categories|small $\infty$-categories]] with $\infty$-profunctors between them is a [[full sub-(infinity,1)-category|full sub-$\infty$-category]] of of the $\infty$-category [[Pr(∞,1)Cat]] that of [[presentable (infinity,1)-categories|presentable $\infty$-categories]] with [[cocontinuous functor|cocontinuous]] [[(infinity,1)-functor|$\infty$-functors]] between them.

## Related pages

* [cograph of an (∞,1)-profunctor](cograph+of+a+profunctor#in_category_theory_2)

* [[(∞,1)Prof]]

## References

* [[Emily Riehl]], [[Dominic Verity]], _Kan extensions and the calculus of modules for ∞-categories_, ([arXiv:1507.01460](https://arxiv.org/abs/1507.01460))

* [[Rune Haugseng]], _Bimodules and natural transformations for enriched ∞-categories_, ([arXiv:1506.07341](https://arxiv.org/abs/1506.07341))

[[!redirects (infinity,1)-profunctors]]

[[!redirects (∞,1)-profunctor]]
[[!redirects (∞,1)-profunctors]]

[[!redirects (infinity, 1)-profunctor]]
[[!redirects (infinity, 1)-profunctor]]
[[!redirects (\infty, 1)-profunctor]]

[[!redirects infinity1-profunctor]]


(infinity,1)-pullback

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#### $(\infty,1)$-Category theory
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#Contents#
* table of contents
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## Idea

An **$(\infty,1)$-pullback**  is a [[limit in an (∞,1)-category]] $\mathcal{C}$ over a [[diagram]] of the shape

$$
  \{a \to c \leftarrow b\} \to \mathcal{C}
  \,.
$$

In other words it is a [[cone]]

$$
  \array{
    A \times_C B &\to& B
    \\
    \downarrow &\cong\swArrow& \downarrow
    \\
    A &\to& C
  }
$$

which is [[universal property|universal]] among all such cones in the $(\infty,1)$-categorical sense.

This is the analog in [[(∞,1)-category theory]] of the notion of [[pullback]] in [[category theory]].

## Incarnations

### In quasi-categories

Let $\mathcal{C}$ be a [[quasi-category]]. Recall the notion of [[limit in a quasi-category]].

The non-degenerate cells of the [[simplicial set]] $\Delta[1] \times \Delta[1]$ obtained as the [[cartesian product]] of the simplicial 1-[[simplex]] with itself look like

$$
  \array{
    (0,0) &\to& (1,0)
    \\
    \downarrow &\searrow& \downarrow
    \\
    (0,1) &\to& (1,1)
  }
$$

A **square** in a [[quasi-category]] $C$ is an image of this in $C$, i.e. a morphism

$$
  s : \Delta[1] \times \Delta[1] \to C
  \,.
$$

The simplicial square $\Delta[1]^{\times 2}$ is [[isomorphism|isomorphic]], as a [[simplicial set]], to the [[join of simplicial sets]] of a 2-[[horn]] with the point:

$$
  \Delta[1] \times \Delta[1]
  \simeq
  \{v\} \star \Lambda[2]_2
  =
  \left(
  \array{
    v &\to& 1
    \\
    \downarrow &\searrow& \downarrow
    \\
    0 &\to& 2
  }
  \right)
$$

and

$$
  \Delta[1] \times \Delta[1]
  \simeq
  \Lambda[2]_0 \star \{v\}
  =
  \left(
  \array{
    0&\to& 1
    \\
    \downarrow &\searrow& \downarrow
    \\
    2 &\to& v
  }
  \right)
  \,.
$$

If a square $\Delta[1] \times \Delta[1] \simeq \Lambda[2]_0 \star \{v\} \to C$ exhibits $\{v\} \to C$ as a [[limit in a quasi-category|quasi-categorical limit]] over $F : \Lambda[2]_0 \to C$, we say the limit

$$
  v := \lim_\leftarrow F := F(1) \prod_{F(0)} F(2)
$$

is [[generalized the|the]] **quasi-categorical pullback** of the diagram $F$.

#### Pasting law {#QuasiCatPastingLaw}

We have the following quasi-categorical analog of the familiar [pasting law of pullbacks](http://ncatlab.org/nlab/show/pullback#Pasting) in ordinary [[category theory]]:

A [[pasting]] diagram of two squares is a morphism

$$
  \sigma : \Delta[2] \times \Delta[1] \to \mathcal{c}
  \,.
$$

Schematically this looks like

$$
  \array{
     a &\to& b &\to& c
     \\
     \downarrow && \downarrow && \downarrow
     \\
     d &\to& e &\to& f
  }
$$

in $\mathcal{C}$.

+-- {: .un_prop}
###### Proposition
**(pasting law for quasi-categorical pullbacks)**

If the right square is a pullback diagram in $\mathcal{C}$, then the left square is precisely if the outer square is.

=--

This is [[Higher Topos Theory|HTT, lemma 4.4.2.1]]

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

Consider the diagram inclusions

$$
  \left(
    \array{
      && && c
      \\
      && && \downarrow
      \\
      d &\to& e &\to& f
    }
  \right)
  \;\;\to\;\;
  \left(
    \array{
      && b &\to& c
      \\
      && \downarrow && \downarrow
      \\
      d &\to& e &\to& f
    }
  \right)
  \;\;\leftarrow\;\;
  \left(
    \array{
      && b
      \\
      && \downarrow
      \\
      d &\to& e
    }
  \right)
$$

and the induced diagram of [[over quasi-categories]]

$$
  \mathcal{C}_{/\sigma(c,d,f)}
  \stackrel{\phi}{\leftarrow}
  \mathcal{C}_{/\sigma(b,c,d,e,f)}
  \stackrel{\psi}{\to}
  \mathcal{C}_{/\sigma(b,d,e)}
  \,.
$$

Notice that by definition of [[limit in a quasi-category]] the quasi-categorical pullback $\sigma(c) \times_{\sigma(f)} \sigma(d)$ is [[generalized the|the]] [[terminal object in a quasi-category|terminal object]] in $\mathcal{C}_{/\sigma(c,d,f)}$, while $\sigma(d) \times_{\sigma(e)} \sigma(b)$ is the terminal object in $\mathcal{C}_{/\sigma(b,d,e)}$.

The strategy now is to show that both these morphisms $\phi$ and $\psi$ are acyclic [[Kan fibration]]s. That will imply that these terminal objects coincide as objects of $\mathcal{C}$.

First notice that the inclusion

$$
  \left(
    \array{
      && b
      \\
      && \downarrow
      \\
      d &\to& e
    }
  \right)
   \;\; \to \;\;
  \left(
    \array{
      && b &\to& c
      \\
      && \downarrow && \downarrow
      \\
      d &\to& e &\to& f
    }
  \right)
$$

is a [[left anodyne morphism]], being the composite of [[pushout]]s of left [[horn]] inclusions

$$
  \begin{aligned}
  \left(
    \array{
      && b
      \\
      && \downarrow
      \\
      d &\to& e
    }
  \right)
  & \to
  \left(
    \array{
      && b &\to& c
      \\
      && \downarrow &&
      \\
      d &\to& e
    }
  \right)
  \\
  & \to
  \left(
    \array{
      && b &\to& c
      \\
      && \downarrow && \downarrow
      \\
      d &\to& e && f
    }
  \right)
  \\
  & \to
  \left(
    \array{
      && b &\to& c
      \\
      && \downarrow &\searrow& \downarrow
      \\
      d &\to& e && f
    }
  \right)
  \\
  & \to
  \left(
    \array{
      && b &\to& c
      \\
      && \downarrow &\searrow& \downarrow
      \\
      d &\to& e &\to& f
    }
  \right)
 \end{aligned}
  \,.
$$

We could also prove this by showing that this functor is [[homotopy final functor|homotopy initial]] using the characterization in terms of slice categories, and then invoking the theorem of [[HTT]] 4.1.1.3(4) which says (in dual form) that an inclusion of simplicial sets is homotopy initial if and only if it is left anodyne.

One of the properties of left anodyne morphisms is that restriction of [[over quasi-categories]] along left anodyne morphisms produces an acyclic [[Kan fibration]]. This shows the desired statement for $\psi$.

To see that $\phi$ is also an acyclic fibration observe that
$\phi$ can be factored as

$$
\mathcal{C}_{/\sigma(c,d,f)} \leftarrow
\mathcal{C}_{/\sigma(c,d,e,f)} \leftarrow
\mathcal{C}_{/\sigma(b,c,d,e,f)}
$$

Observe that $\mathcal{C}_{/\sigma(c,d,e,f)}\leftarrow\mathcal{C}_{/\sigma(b,c,d,e,f)}$
fits into a pullback diagram

$$
   \array{
      \mathcal{C}_{/\sigma(c,d,e,f)}
      & \leftarrow &
      \mathcal{C}_{/\sigma(b,c,d,e,f)} \\
      \downarrow & & \downarrow \\
     \mathcal{C}_{/\sigma(c,e,f)}
     & \leftarrow &
     \mathcal{C}_{/\sigma(b,c,e,f)}
    }
$$

and hence is an acyclic Kan fibration since
$\mathcal{C}_{/\sigma(c,e,f)} \leftarrow \mathcal{C}_{/\sigma(b,c,e,f)}$
is one, on account of the fact that the square

$$
   \array{
        \sigma(b) & \to & \sigma(c) \\
        \downarrow & & \downarrow \\
        \sigma(e) & \to & \sigma(f)
   }
$$

is a pullback in $\mathcal{C}$.  Finally,
$\mathcal{C}_{/\sigma(c,d,f)} \leftarrow \mathcal{C}_{/\sigma(c,d,e,f)}$
is a trivial fibration since

$$
   \array{
      \left(
          \array{
              & & c \\
              & & \downarrow \\
           d & \to & f
          }
     \right) & \to &
     \left(
        \array{
           & & & & c \\
           & & & & \downarrow \\
         d & \to & e & \to & f
         }
     \right)
   }
$$

is left anodyne; clearly this is a pushout of
$(d\to f)\to (d\to e\to f)$ and so it suffices to show
that $\Delta^{\{0,2\}}\to \Delta^{\{0,1,2\}}$ is left
anodyne.  But this map factors as
$\Delta^{\{0,2\}}\to \Lambda^2_0 \to \Delta^{\{0,1,2\}}$
and clearly $\Delta^{\{0,2\}}\to \Lambda^2_0$ is left
anodyne since it is a pushout of $\Delta^{\{0\}}\to
\Delta^{\{0,1\}}$.

=--

### In model categories

* [[homotopy pullback]]

### In derivators

* [[pullback in a derivator]]

## Examples

### Fiber sequence

If $\mathcal{C}$ has a [[terminal object]] and $* \to C \in \mathcal{C}$ is a [[pointed object]], then the **fiber** or **$(\infty,1)$-kernel** of a morphisms $f : B \to C$ is the $(\infty,1)$-pullback

$$
  \array{
    ker(f) &\to& *
    \\
    \downarrow && \downarrow
    \\
    B &\stackrel{f}{\to}& C
  }
  \,.
$$

For more on this see [[fiber sequence]].

## Related concepts

[[!include notions of pullback -- contents]]

## References

### In homotopy type theory

A formalization of homotopy pullbacks in [[homotopy type theory]] is [[Coq]]-coded in

* [[Guillaume Brunerie]], _[Hott/Coq/Limits/Pullbacks.v](https://github.com/guillaumebrunerie/HoTT/blob/master/Coq/Limits/Pullbacks.v)_

[[!redirects (∞,1)-pullback]]
[[!redirects (infinity,1)-pullbacks]]
[[!redirects (∞,1)-pullbacks]]

[[!redirects (∞,1)-fiber product]]
[[!redirects (∞,1)-fiber products]]

[[!redirects (infinity,1)-fiber product]]
[[!redirects (infinity,1)-fiber products]]


(infinity,1)-pushout

The dual notion to _[[(infinity,1)-pullback]]_.

[[!redirects (infinity,1)-pushouts]]


(infinity,1)-quantity

See [[schreiber:∞-quantity]] for now.

[[!redirects (∞,1)-quantity]]


(infinity,1)-quasitopos

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#Contents#
* table of contents
{:toc}

## Idea

The notion of _$(\infty,1)$-quasitopos is the [[(∞,1)-topos]]-analog of the notion of [[quasitopos]].

## Definition

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

An **(∞,1)-bisite** is an [[(∞,1)-category]] $C$ together with two [[(∞,1)-Grothendieck topologies]], $J$ and $K$ such that $J \subseteq K$.

=--

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

Let $C$ be an [[(∞,1)-bisite]]. Say an [[(∞,1)-presheaf]] $F \in (\infty,1)PSh(C)$ is $\left(J,K\right)$-**biseparated** if it is an [[(∞,1)-sheaf]] for $J$ and for every $K$-covering [[sieve]] $U \to X$ in $C$ we have that the induced morphism

$$
  (\infty,1)PSh_C(X,F) \hookrightarrow (\infty,1)PSh_C(U,F)
$$

in [[∞Grpd]] is a [[full and faithful (∞,1)-functor]].

We say it is $n-\left(J,K\right)$-**biseparated** if

the induced morphism

$$
  (\infty,1)PSh_C(X,F) \hookrightarrow (\infty,1)PSh_C(U,F)
$$

is an [[n-truncated object in an (∞,1)-category|(n-1)-truncated object]] in the [[over quasi-category|(∞,1)-overcategory]] $\left(\infty-Gpd\right)/(\infty,1)PSh_C(U,F)$.

=--

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

A **(Grothendieck) $(\infty,1)$-quasitopos** is an [[(∞,1)-category]] that is [[equivalence of quasi-categories|equivalent]] to the full [[sub-(∞,1)-category]] of some $(\infty,1)PSh_C$ on the $n-\left(J,K\right)$-**biseparated** $(\infty,1)$-presheaves, on some (∞,1)-bisite $\left(C,J,K\right)$.

=--

## Examples

For $\mathbf{H}$ a [[local (∞,1)-topos]]

$$
  \mathbf{H} \stackrel{\stackrel{\overset{Disc}{\leftarrow}}{\underset{\Gamma}{\to}}}{\underset{Codisc}{\leftarrow}}
  \infty Grpd
$$

and $C$ be a site of definition for $\mathbf{H}$, the $(\infty,1)$-quasitopos on $C$ that factors the [[geometric embedding]] $Codisc \infty Grpd \hookrightarrow \mathbf{H}$

$$
  \infty Grpd
   \stackrel{\overset{\Gamma}{\leftarrow}}{\underset{Codisc}{\hookrightarrow}}
  Conc(\mathbf{H})
   \stackrel{\overset{concretization}{\leftarrow}}{\underset{}{\hookrightarrow}}
  \mathbf{H}
$$

is that of **concrete objects** in $\mathbf{H}$, the analog of [[concrete sheaves]].

## Related concepts

* [[quasitopos]]

* **$(\infty,1)$-quasitopos**

## References

The definition as it stands, originated out of a discussion between [[Urs Schreiber]] and [[David Carchedi]]. The suggestion to rephrase the definition in terms of bisites came from [[Mike Shulman]].

[[!redirects (∞,1)-quasitopos]]
[[!redirects (∞,1)-quasitoposes]]
[[!redirects (∞,1)-quasitopoi]]
[[!redirects Grothendieck (∞,1)-quasitopos]]
[[!redirects Grothendieck (∞,1)-quasitoposes]]
[[!redirects Grothendieck (∞,1)-quasitopoi]]
[[!redirects (infinity,1)-quasitoposes]]
[[!redirects (infinity,1)-quasitopoi]]
[[!redirects Grothendieck (infinity,1)-quasitopos]]
[[!redirects Grothendieck (infinity,1)-quasitoposes]]
[[!redirects Grothendieck (infinity,1)-quasitopoi]]
[[!redirects (∞,1)-bisite]]


(infinity,1)-semitopos

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#### $(\infty,1)$-Category theory
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=--
#### $(\infty,1)$-Topos Theory
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[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

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

An [[(∞,1)-category]] $C$ is an **$(\infty,1)$-semitopos if

1. it is [[presentable (∞,1)-category]];

1. it has [[universal colimits]];

1. for every [[morphism]] the corresponding [[Cech nerve]] [[groupoid object in an (∞,1)-category|groupoid object]] is effective.

=--

This appears as [[Higher Topos Theory|HTT, def. 6.2.3.1]].

## Properties

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

If $C$ is an $(\infty,1)$-semitopos and $X \in C$ is any object, then also the [[over-(∞,1)-category]] is an $(\infty,1)$-semitopos.

=--

This appears as ([[Higher Topos Theory|HTT, remark 6.2.3.3]]).

## Examples

* Of course every [[(∞,1)-topos]] is an $(\infty,1)$-semitopos.

## Related concepts

* [[semitopos]]

* [[(∞,1)-topos]]

[[!redirects (∞,1)-semitopos]]
[[!redirects (∞,1)-semitoposes]]

[[!redirects (infinity,1)-semitoposes]]


(infinity,1)-sheaf

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[[!include (infinity,1)-topos - contents]]
=--
#### Locality and descent
+--{: .hide}
[[!include descent and locality - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of **$(\infty,1)$-sheaf** (or _[[∞-stack]]_ or _[[geometric homotopy type]]_) is the analog in [[(∞,1)-category]] theory of the notion of [[sheaf]] ([[geometric type]]) in ordinary [[category theory]].

See [[(∞,1)-category of (∞,1)-sheaves]] for more.

## Definition

Given an [[(∞,1)-site]] $C$, let $S$ be the class of [[monomorphism in an (∞,1)-category|monomorphisms]] in the [[(∞,1)-category of (∞,1)-presheaves]] $PSh_{(\infty,1)}(C)$ that correspond to [[covering]] [[(∞,1)-sieve]]s

$$
  \eta : U \hookrightarrow j(c)
$$

on objects $c \in C$, where $j$ is the [[(∞,1)-Yoneda embedding]].

Then an [[(∞,1)-presheaf]] $A \in PSh_{(\infty,1)}(C)$ is an **$(\infty,1)$-sheaf** if it is an $S$-[[local object]]. That is, if for all such $\eta$ the morphism

$$
  A(c) \simeq PSh_C(j(c),A) \stackrel{PSh_C(\eta,A)}{\to}
  PSh(U,A)
$$

is an [[equivalence in a quasi-category|equivalence]]. For a presheaf $A : C^{\op} \to E$ with values in an arbitrary ∞-category, we say it is a sheaf iff $E(e, A(-))$ is a sheaf for every object $e$ of $E$.

This is the analog of the ordinary sheaf condition for covering sieves. The [[∞-groupoid]] $PSh_C(U,A)$ is also called the [[descent]]-[[∞-groupoid]] of $A$ relative to the covering encoded by $U$.

As in the 1-categorial case, the sheaf condition for a covering sieve can be translated into a condition on a covering family that generates it:

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

Let $\{ u_i \to c \}$ be a family of morphisms of $C$ that generate the sieve corresponding to $\eta : U \hookrightarrow j(c)$, and let $r_\bullet : \mathbf{\Delta}^{\op} \to PSh_C$ be the [[groupoid object in an (infinity,1)-category#cech_nerves|Čech nerve]]
of $\amalg_i j(u_i) \to j(c)$.
Then a presheaf $A$ is local with respect to $\eta$ iff the induced map $A(c) \to \lim A(r_\bullet)$ is an equivalence.
=--

Thus, a presheaf $A$ is a sheaf iff every covering sieve contains
a generating family satisfying this condition. Spelling out the description of the Čech nerve, the condition is that we have
$$
  A(c) \simeq \lim\left(
  \prod_i A(u_i)
  \stackrel{\to}{\to}
  \prod_{i,j} PSh_C(j(u_i) \times_{j(c)} j(u_j), A)
  \stackrel{\to}{\stackrel{\to}{\to}}
  \cdots
  \right)
$$
If $C$ has pullbacks, this simplifies to
$$
  A(c) \simeq \lim\left(
  \prod_i A(u_i)
  \stackrel{\to}{\to}
  \prod_{i,j} A(u_i \times_c u_j)
  \stackrel{\to}{\stackrel{\to}{\to}}
  \cdots
  \right)
$$
and furthermore this formulation applies to presheaves with values
in an arbitrary ∞-category.

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

Taking colimits of Čech nerve computes $(-1)$-truncations in $(PSh_C)/j(X)$,
so $\colim(r_\bullet)$ is the subobject of $j(c)$ corresponding
to the sieve $\eta$. We have
$$
  PSh_C(\colim(r_\bullet), A) \simeq \lim PSh_C(r_\bullet, A)
$$
and so the theorem follows.

=--

## Terminology

An **($\infty$,1)-sheaf** is also called an [[∞-stack]] with values in [[∞-groupoid]]s.

The practice of writing "$\infty$-sheaf" instead of [[∞-stack]] is a rather reasonable one, since a [[stack]] is nothing but a _2-sheaf_.

Notice however that there is ambiguity in what precisely one may mean by an $\infty$-stack: it can be an $(\infty,1)$-sheaf or more specifically a [[hypercomplete]] $(\infty,1)$-sheaf. This is a distinction that only appears in [[(∞,1)-topos]] theory, not in [[(n,1)-topos]] theory for finite $n$.

## Related concepts

* [[sheaf]]

* [[2-sheaf]] / [[stack]]

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

  * [[sheaf of spectra]]

  * [[sheaf of L-∞ algebras]]

* [[(∞,2)-sheaf]]

* [[(∞,n)-sheaf]]

[[!include homotopy n-types - table]]

## References

Section 6.2.2 in

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

[[!redirects (infinity,1)-sheaves]]
[[!redirects infinity-sheaf]]
[[!redirects infinity-sheaves]]
[[!redirects ∞-sheaf]]
[[!redirects ∞-sheaves]]
[[!redirects (∞,1)-sheaf]]
[[!redirects (∞,1)-sheaves]]


(infinity,1)-sheafification

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#### $(\infty,1)$-Topos theory
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[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

As for ordinary [[sheafification]], _$(\infty,1)$-sheafification_ is the [[exact (∞,1)-functor|left exact]] [[left adjoint|left]] [[adjoint (∞,1)-functor]] to the inclusion of [[(∞,1)-sheaves]] into [[(∞,1)-presheaves]].

For more details see [[(∞,1)-category of (∞,1)-sheaves]].

## Properties

An $(n,1)$-presheaf can be $(n,1)$-sheafified by

* applying the [[plus-construction on presheaves]] $n+1$ times (see [Lurie, section 6.5.3](#Lurie))

* applying a single "[[hypercover]]-plus-construction" (see [DHI, theorem 7.6](#DHI))

## Related concepts

* [[sheafification]], [[plus-construction on presheaves]]

* **(∞,1)-sheafification** / [[∞-stackification]]

## References

The iterated plus-construction is mentioned in section 6.5.3 of

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

The "hypercover-plus construction" is discussed around theorem 7.6 of

* [[Dan Dugger]], [[Sharon Hollander]], [[Dan Isaksen]], _Hypercovers and simplicial presheaves_ ([pdf](http://www.math.uiuc.edu/K-theory/0563/spre.pdf))
 {#DHI}

[[!redirects (infinity,1)-sheafifications]]
[[!redirects (∞,1)-sheafification]]
[[!redirects (∞,1)-sheafifications]]


(infinity,1)-site

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

#Contents#
* table of contents
{:toc}

## Idea

The structure of an $(\infty,1)$-site on an [[(∞,1)-category]] $C$ is precisely the data encoding an [[(∞,1)-category of (∞,1)-sheaves]]

$$
  Sh(C) \hookrightarrow PSh(C)
$$

inside the [[(∞,1)-category of (∞,1)-presheaves]] on $C$.

The notion is the analog in [[(∞,1)-category]] theory of the notion of a [[site]] in 1-[[category theory]].

## Definition

The definition of $(\infty,1)$-sites parallels that of 1-categorical [[site]]s closely. In fact the structure of an $(\infty,1)$-site on an $(\infty,1)$-category is equivalent to that of a 1-categorical site on its [[homotopy category of an (infinity,1)-category|homotopy category]] (see below).

+-- {: .un_defn}
###### Definition
**($(\infty,1)$-Grothendieck topology)**

A **[[sieve]] in** an [[(∞,1)-category]] $C$ is a full [[sub-(∞,1)-category]] $D \subset C$ which is closed under precomposition with morphisms in $C$.

A **sieve on** an [[object]] $c \in C$ is a sieve in the [[over quasi-category|overcategory]] $C_{/c}$.

Equivalently, a sieve on $c$ is an [[equivalence class]] of [[monomorphism in an (infinity,1)-category|monomorphisms]]  $U \to j(c)$ in the [[(∞,1)-category of (∞,1)-presheaves]] $PSh(C)$, with $j : C \to PSh(C)$ the [[(∞,1)-Yoneda embedding]]. (See below for the proof of this equivalence).

For $S$ a sieve on $c$ and  $f : d \to c$ a [[morphism]] into $c$, we take the **pullback sieve** $f^* S$ on $d$ to be that spanned by all those morphisms into $d$ that become equivalent to a morphism in $S$ after postcomposition with $f$.

A **[[Grothendieck topology]]** on the $(\infty,1)$-category $C$ is the specification of a collection of sieves on each object of $C$ -- called the **covering sieves** , subject to the following conditions:

1. _the trivial sieve covers_ -- For each object $c \in C$ the overcategory $C_{/c}$ regarded as a maximal subcategory of itself is a covering sieve on $c$. Equivalently: the monomorphism $Id : j(c) \to j(c)$ covers.

1. _the pullback of a sieve covers_ -- If $S$ is a covering sieve on $c$ and $f : d \to c$ a morphism, then the pullback sieve $f^* S$ is a covering sieve on $d$. Equivalently, the [[pullback]]

   $$
     \array{
        f^* U &\to& U
        \\
        \downarrow && \downarrow
        \\
        d &\stackrel{f}{\to}& c
     }
   $$

   in $PSh(C)$ is covering.

1. _a sieve covers if its pullbacks cover_ -- For $S$ a covering sieve on $c$ and $T$ any sieve on $c$, if the pullback sieve $f^* T$ for every $f \in S$ is covering, then $T$ itself is covering.

An $(\infty,1)$-category equipped with a Grothendieck topology is an **$(\infty,1)$-site**.

=--

## Properties

### Of sieves

+-- {: .un_lemma}
###### Lemma

A sieve $S'$ on $c$ that contains a covering sieve $S \subset S'$ is itself covering.

=--

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

For every $f : d \to c$ an object of $S \subset C_{/c}$, the pullback sieve
$f^* S'$ equals the pullback sieve $f^* S$. So it covers $d$ by the second axiom on sieves. So by the third axiom $S'$ itself is covering.

=--

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

There is a natural bijection between sieves on $c$ in $C$ and equivalence class of [[monomorphism in an (infinity,1)-category|monomorphisms]] $U \to j(C)$ in $PSh(C)$.

=--

This is [[Higher Topos Theory|HTT, prop. 6.2.2.5]].

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

First observe that equivalence classes of $(-1)$-[[truncated]] object of $PSh(C_{/c})$ are in bijection with sieves on $c$:

An $(\infty,1)$-presheaf $F$ is $(-1)$-truncated if its value on any object is either the empty [[∞-groupoid]] $\emptyset$ or a [[contractible]] $\infty$-groupoid. The full subcategory of $C_{/c}$ on those objects on which $F$ takes a contractible value is evidently a sieve (because there is no morphism from a contractible to the empty $\infty$-groupoid). Conversely, given a sieve $S$ on $c$ we obtain a (-1)-truncated presheaf fixed by the demand that it takes the value $* = \Delta[0] \in \infty Grpd$ on those objects that are in $S$, and $\emptyset$ otherwise.

Now, as described at Interaction of presheaves and overcategories we have an equivalence

$$
  PSh(C_{/c})
  \simeq
  PSh(C)_{/j(c)}
  \,.
$$

Under this equivalence our bijection above maps to the statement that there is a bijection between sieves on $c$ and equivalence class of $(-1)$-[[truncated]] objects in $PSh(C)_{/j (c)}$. But such a (-1)-truncated object is precisely a [[monomorphism in an (infinity,1)-category|monomorphism]] $U \to j(c)$.

=--

### Of coverages

\begin{lemma}
The set of Grothendieck topologies on an $(\infty,1)$-category $C$ is in natural bijection with the set of Grothendieck topologies on its [[homotopy category of an (infinity,1)-category|homotopy category]].
\end{lemma}

This is [[Higher Topos Theory|HTT, remark 6.2.2.3]].

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

Because picking full sub-1-categories as well as full sub-$(\infty,1)$-categories amounts to picking sub-sets/sub-classes of the set of equivalence classes of objects.

=--

+-- {: .un_corollary}
###### Corollary

If the $(\infty,1)$-category $C$ happens to be an ordinary [[category]] (for instance in its incarnation as a [[quasi-category]] it is the [[nerve]] of an ordinary [[category]]), then the structure of an $(\infty,1)$-site on it is the same as the 1-categorical structure of a [[site]] on it.

=--

### Of sites

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

Structures of $(\infty,1)$-sites on an [[(∞,1)-category]] $C$ correspond bijectively to [[topological localization]]s of the [[(∞,1)-category of (∞,1)-presheaves]] to a [[(∞,1)-category of (∞,1)-sheaves]]. See there for more details.

=--

## Incarnations and models

If [[(∞,1)-categories]] are incarnated as [[simplicially enriched categories]], then an $(\infty,1)$-site appears as an

* [[sSet-site]]

If $(\infty,1)$-categories are [[presentable (∞,1)-category|presented]] by [[model categories]], then the notion of $(\infty,1)$-site appears as that of

* [[model site]].

## Examples

* The **trivial Grothendieck-topology** on an $(\infty,1)$-category is that where the only covering sieve on each object $c$ is $C_{/c}$ itself. Equivalently, where the only covering monomorphisms $U \to j(c)$ in $PSh(C)$ are the equivalences.

  The [[(∞,1)-category of (∞,1)-sheaves]] on this site is just the [[(∞,1)-category of (∞,1)-presheaves]] itself. The localization is an equivalence.

* [[étale (∞,1)-site]]

## Related concepts

* [[site]]

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

* **$(\infty,1)$-site**

  * [[model site]], [[simplicial site]]

  *[[infinity-cohesive site]]

* [[internal site]] / [[internal (infinity,1)-site]]

## References

* [[Jacob Lurie]], Section 6.2.2 of: _[[Higher Topos Theory]]_ (2009)

* [[Raffael Stenzel]], *Notions of $(\infty,1)$-sites and related formal structures* [[arXiv:2306.06619](https://arxiv.org/abs/2306.06619)]

* [[Raffael Stenzel]], *Higher sites and their higher categorical logic*, talk at *[[Homotopy Type Theory Electronic Seminar Talks|HoTT Electronic Seminar]]* (18 November 2021) [video:[YT](https://www.youtube.com/watch?v=hUJlK8XgZxQ), slides:[pdf](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Stenzel-2021-11-18-HoTTEST.pdf)]

[[!redirects (∞,1)-site]]

[[!redirects (infinity,1)-sites]]
[[!redirects (∞,1)-sites]]
[[!redirects (∞,1)-Grothendieck topology]]
[[!redirects (∞,1)-Grothendieck topologies]]


(infinity,1)-topos

+-- {: .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}

## Idea

Recall the following familiar 1-categorical statement:

* Working in the 1-[[category]] [[Set]] of [[0-category|0-categories]] amounts to doing [[set theory]]. The point of [[sheaf topos|sheaf]] [[toposes]] is to pass to _parameterized_  [[0-category|0-categories]], namely  [[presheaf]] categories. Although these [[topos|topoi]] behave much like the 1-topos [[Set]], their objects are generalized [[spaces]] that may carry more structure. For instance, a (pre)[[sheaf]] on [[Diff]] is a [[generalized smooth space]].

The idea of $(\infty,1)$-toposes is to generalize the above situation from $1$ to $(\infty,1)$  (recall the notion of [[(n,r)-category]] and see the general discussion at [[∞-topos]]):

* Working in the [[(∞,1)-category]] [[∞Grpd]] of [[infinity-groupoid|(∞,0)-categories]] amounts to doing [[homotopy theory]]. The point of [[(∞,1)-sheaves]] is to pass to _parameterized_ [[(∞,0)-categories]], namely  [[(∞,1)-presheaf]] categories. Although these $(\infty,1)$-topoi behave much like the $(\infty,1)$-topos [[∞Grpd]], their objects are generalized [[spaces]] with higher [[homotopies]] that may carry more structure. More generally we have topoi of [[sheaves]], and $(\infty,1)$-topoi of [[(∞,1)-sheaves]].  For instance, an [[∞-Lie groupoid]] is an [[(∞,1)-sheaf]] on [[CartSp]].

## Definition {#Definition}

### As a geometric embedding into a $(\infty,1)$-presheaf category
 {#AsAGeometricEmbedding}

Recall that [[sheaf toposes are equivalently the left exact reflective subcategories of presheaf toposes]] and that the inclusion functor is necessarily an [[accessible functor]]. This characterization has the following immediate generalization to a definition in [[(∞,1)-category theory]], where the only subtlety is that accessibility needs to be explicitly required:

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

A [[Alexander Grothendieck|Grothendieck]]--[[Charles Rezk|Rezk]]--[[Jacob Lurie|Lurie]] **$(\infty,1)$-topos** $\mathbf{H}$ is an [accessible](reflective%20sub-%28infinity,1%29-category#AccessibleReflectiveSubcategory) [left exact](reflective+sub-%28infinity%2C1%29-category#ExactLocalizations) [[reflective sub-(∞,1)-category]] of an [[(∞,1)-category of (∞,1)-presheaves]]

$$
  \mathbf{H}
  \stackrel{\overset{lex}{\leftarrow}}{\hookrightarrow}
  PSh_{(\infty,1)}(C)
  \,.
$$

If the above localization is a [[topological localization]] then $\mathbf{H}$ is an **[[(∞,1)-category of (∞,1)-sheaves]]**.

=--

### By Giraud-Rezk-Lurie axioms
 {#GiraudAxioms}

Equivalently:

\begin{proposition}
**(Giraud-Rezk-Lurie axioms)**
\linebreak
An $(\infty,1)$-topos $\mathbf{H}$ is

an [[(∞,1)-category]] that satisfies the $(\infty,1)$-category-theoretic analogs of [[Giraud's axioms]]:

* $\mathbf{H}$ is [[presentable (infinity,1)-category|presentable]];

* [[limit in quasi-categories|(∞,1)-colimits]] in $\mathbf{H}$ [[universal colimits|are universal]];

* [[coproducts]] in $\mathbf{H}$ are [[disjoint coproduct|disjoint]];

* every [[groupoid object in an (infinity,1)-category|groupoid object]] in $\mathbf{H}$ is [[groupoid objects in an (∞,1)-topos are effective|effective]] (i.e. has a [[delooping]]).

\end{proposition}

This is  part of the statement of [[Higher Topos Theory|HTT, theorem 6.1.0.6]].

This is derived from the following equivalent one:

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

An [[(∞,1)-topos]] is

* a [[presentable (∞,1)-category]] with [[universal colimits]]

* that has [[object classifiers]].

=--

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

An [[object classifier]] is a (small) _self-reflection_ of the $\infty$-topos inside itself ([[type of types]], internal [[universe]]). See also ([WdL, book 2, section 1](Science+of+Logic#WesenAlsReflexionInIhmSelbst)).

=--

A further equivalent one (essentially by an invocation of the [[adjoint functor theorem]]) is:

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

An [[(∞,1)-topos]] is

* a [[presentable (∞,1)-category]]

* in which all colimits are [[van Kampen colimits]].

=--

### Morphisms

A [[morphism]] between $(\infty,1)$-toposes is an [[(∞,1)-geometric morphism]].

The [[(∞,1)-category]] of all $(\infty,1)$-topos is [[(∞,1)Toposes]].

## Types of $(\infty,1)$-toposes

### Topological localizations / $(\infty,1)$-sheaf toposes

for the moment see

* [[topological localization]]

### Hypercomplete $(\infty,1)$-toposes

for the moment see

* [[hypercomplete (∞,1)-topos]]

### Cubical type theory

The Cartesian cubical model of [[cubical type theory]] and [[homotopy type theory]] is [conjectured](https://groups.google.com/d/msg/homotopytypetheory/RQkLWZ_83kQ/s6iazlFdBgAJ) to be an (∞,1)-topos not equivalent to (∞,1)-groupoids.

## Models

Another main theorem about $(\infty,1)$-toposes is that
[[models for ∞-stack (∞,1)-toposes]] are given by the [[model structure on simplicial presheaves]]. See there for details

## Properties

### Global sections geometric morphism
 {#GlobalSectionsGeometricMorphism}

Every [[∞-stack]] $(\infty,1)$-topos $\mathbf{H}$ has a canonical [[(∞,1)-geometric morphism]] to the terminal $\infty$-stack $(\infty,1)$-topos [[∞Grpd]]: the [[direct image]] is the [[global section]]s [[(∞,1)-functor]] $\Gamma$, the [[inverse image]] is the [[constant ∞-stack]] functor

$$
  (LConst \dashv \Gamma)
  \;\colon\;
  \mathbf{H}
  \underoverset
    {\underset{\Gamma}{\longrightarrow}}
    {\overset{LConst}{\longleftarrow}}
    {\;\;\; \bot \;\;\;}
  \infty Grpd
  \,.
$$

In fact, this is unique, up to [[equivalence in an (infinity,1)-category|equivalence]]: Since every $\infty$-groupoid is an [[(infinity,1)-colimit|$(\infty,1)$-colimit]] (namely over itself, by [this Prop.](infinity1-limit#EveryInfinityGroupoidIsHomotopyColimitOfConstantFunctorOverItself)) of the [[point]] (hence of the [[terminal object]]), and since the [[inverse image]] $\infty$-functor $LConst$ needs to preserve these $\infty$-colimits (being a [[left adjoint]]) as well as the point (being a [[lex functor]]).

### Closed monoidal structure {#ClosedMonoidalStructure}

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

Every $(\infty,1)$-topos is a [[cartesian closed (∞,1)-category]].

=--

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

By the fact that every $(\infty,1)$-topos $\mathbf{H}$ has [[universal colimits]] it follows that for every object $X$ the [[(∞,1)-functor]]

$$
  X \times (-) : \mathbf{H} \to \mathbf{H}
$$

preserves all [[(∞,1)-colimit]]s. Since every $(\infty,1)$-topos is a [[locally presentable (∞,1)-category]] it follows with the [[adjoint (∞,1)-functor theorem]] that there is a [[right adjoint|right]] [[adjoint (∞,1)-functor]]

$$
  (X \times (-) \dashv [X,-]) : \mathbf{H} \stackrel{\overset{X \times (-)}{\leftarrow}}{\underset{[X,-]}{\to}} \mathbf{H}
  \,.
$$

=--

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

For $C$ an [[(∞,1)-site]] for $\mathbf{H}$ we have that the [[internal hom]] ([[mapping stack]]) $[X,-]$ is given on $A \in \mathbf{H}$ by the [[(∞,1)-sheaf]]

$$
  [X,A] \,\colon\, U \mapsto \mathbf{H}(X \times L y(U), A)
  \,,
$$

where $y : C \to \mathbf{H}$ is the [[(∞,1)-Yoneda embedding]] and $L : PSh_C \to \mathbf{H}$ denotes [[∞-stackification]].

=--

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

The argument is entirely analogous to that of the [[closed monoidal structure on sheaves]].

We use the [[full and faithful (∞,1)-functor|full and faithful]] [[geometric embedding]] $(L \dashv i) : \mathbf{H} \hookrightarrow PSh_C$ and the [[(∞,1)-Yoneda lemma]] to find for all $U \in C$ the value

$$
  [X,A](U) \simeq PSh_C(y U, [X,A])
$$

and then the fact that [[∞-stackification]] $L$ is [[left adjoint]] to inclusion to get

$$
  \cdots \simeq \mathbf{H}(L y(U), [X,A])
  \,.
$$

Then the defining adjunction $(X \times (-) \dashv [X,-])$ gives

$$
  \cdots \simeq \mathbf{H}(X \times L y(U) , A)
  \,.
$$

=--

[[!include powering of ∞-toposes over ∞-groupoids -- section]]

### Slice-$(\infty,1)$-toposes

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

For $\mathbf{H}$ an $(\infty,1)$-topos and $X \in \mathbf{H}$ an object, the [[slice-(∞,1)-category]] $\mathbf{H}_{/X}$ is itself an $(\infty,1)$-topos -- an **[[over-(∞,1)-topos]]**. The projection $\pi_! : \mathbf{H}_{/X} \to \mathbf{H}$ part of an  [[essential geometric morphism]]

$$
  \pi : \mathbf{H}_{/X}
  \stackrel{\overset{\pi_!}{\to}}{\stackrel{\overset{\pi^*}{\leftarrow}}{\underset{\pi_*}{\to}}}
  \mathbf{H}
  \,.
$$

=--

This is [[Higher Topos Theory|HTT, prop. 6.3.5.1]].

The $(\infty,1)$-topos $\mathbf{H}_{/X}$ could be called the [[gros topos]] of $X$. A [[geometric morphism]] $\mathbf{K} \to \mathbf{H}$ that factors as $\mathbf{K} \xrightarrow{\simeq} \mathbf{H}_{/X} \stackrel{\pi}{\to} \mathbf{H}$ is called an [[etale geometric morphism]].

### Syntax in univalent homotopy type theory

$(\infty,1)$-Toposes provide [[categorical semantics]] for [[homotopy type theory]] with a [[univalence|univalent]] Tarskian [[type of types]] (which inteprets as the [[object classifier]]).

For more on this see at

* _[[homotopytypetheory:model of type theory in an (infinity,1)-topos]]_

* _[relation between type theory and category theory -- Univalent homotopy type theory and infinity-toposes](relation+between+type+theory+and+category+theory#HomotopyWithUnivalence)_

## $(\infty,1)$-Topos theory {#ToposTheory}

Most of the standard constructions in [[topos theory]] have or should have  immediate generalizations to the context of $(\infty,1)$-toposes, since all notions of [[category theory]] exist for [[(∞,1)-categories]].

For instance there are evident notions of

* [[geometric morphism]]s between $(\infty,1)$-toposes, such as the [[global section]] geometric morphism to the [[terminal object|terminal]] [[(∞,1)-category of (∞,1)-sheaves|(∞,1)-sheaf]] $(\infty,1)$-topos [[∞Grpd]].

Moreover, it turns out that $(\infty,1)$-toposes come with plenty of internal structures, more than canonically present in an ordinary topos.
Every $(\infty,1)$-topos comes with its intrinsic notion of

* [[cohomology|cohomology in an (∞,1)-topos]]

and with an intrinsic notion of

* [[homotopy groups in an (∞,1)-topos|homotopy in an (∞,1)-topos]].

In classical topos theory, cohomology and homotopy of a topos $E$ are defined in terms of [[simplicial object]]s in $C$. If $E$ is a [[sheaf topos]] with [[site]] $C$ and [[point of a topos|enough point]]s, then  this classical construction is secretly really a model for the intrinsic cohomology and homotopy in the above sense of the  [[hypercomplete (∞,1)-topos]] of [[∞-stack]]s on $C$.

The beginning of a list of all the structures that exist intrinsically in a big $(\infty,1)$-topos is at

* [[cohesive (∞,1)-topos]].

But **$(\infty,1)$-topos theory** in the style of an $\infty$-analog of the [[Elephant]] is only barely beginning to be conceived.

There are some indications as to what the

* [[internal logic of an (∞,1)-topos]]

should be.

## Related concepts

* [[elementary (∞,1)-topos]], [[(∞,1)-pretopos]]

* [[model topos]]

* [[structured (∞,1)-topos]]

* [[compact topos]], [[coherent (∞,1)-topos]]

* [[category object in an (∞,1)-topos]]

* [[tangent (∞,1)-topos]]

* [[doubly monoidal (∞,1)-topos]]

[[!include flavors of higher toposes -- list]]

[[!include locally presentable categories - table]]

## References

### General
 {#ReferencesGeneral}

In retrospect, at least the [[homotopy category of an (∞,1)-category|homotopy categories]] of [[(∞,1)-toposes]] have been known since

* [[Kenneth Brown]], *[[BrownAHT|Abstract homotopy theory and generalized sheaf cohomology]]*,  Transactions of the American Mathematical Society,  Vol. 186 (1973), 419-458  ([jstor:1996573](http://www.jstor.org/stable/1996573)).

presented there via [[categories of fibrant objects]] among [[simplicial presheaves]]. The enhancement of this to [[model categories]] [[model structure on simplicial presheaves|of simplicial presheaves]] originates wit:h

* [[André Joyal]], Letter to [[Alexander Grothendieck]], 11. 4. 1984, ([pdf scan](http://webusers.imj-prg.fr/~georges.maltsiniotis/ps/lettreJoyal.pdf)).

* {#JardineLecture} [[John F. Jardine]], _Simplicial presheaves_, Journal of Pure and Applied Algebra 47 (1987), 35-87 ([pdf](https://core.ac.uk/download/pdf/82485559.pdf))

A more intrinsic characterization of these "[[model toposes]]" ([Rezk 2010](#Rezk10)) as $\infty$-toposes (the term seems to first appear here in [Simpson 1999](#Simpson99)) is due to:

* {#Simpson99} [[Carlos Simpson]],  *A Giraud-type characterization of the simplicial categories associated to closed model categories as $\infty$-pretopoi* ([arXiv:math/9903167](http://arxiv.org/abs/math/9903167))

The generalization of these [[model toposes]] from 1-sites to [[simplicial site|simplicial]] [[model sites]] is due to

* {#ToenVezzosi05} [[Bertrand Toën]], [[Gabriele Vezzosi]], _Homotopical Algebraic Geometry I: Topos theory_, Advances in Mathematics **193** 2 (2005)  257-372  [[arXiv:math.AG/0207028](http://arxiv.org/abs/math.AG/0207028), [doi:10.1016/j.aim.2004.05.004](https://doi.org/10.1016/j.aim.2004.05.004)]

The term *$\infty$-topos* is due to

* {#Lurie03} [[Jacob Lurie]], *On $\infty$-Topoi* ([arXiv:math/0306109](https://arxiv.org/abs/math/0306109))

The term *[[model topos]]* was later coined in:

* {#Rezk10} [[Charles Rezk]], _Toposes and homotopy toposes_, 2010 ([pdf](http://www.math.uiuc.edu/~rezk/homotopy-topos-sketch.pdf), [[Rezk_HomotopyToposes.pdf:file]])

A comprehensive conceptualization and discussion of [[(∞,1)-toposes]] is then due to

* [[Jacob Lurie]], section 6.1 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))

building on [Rezk 2010](#Rezk10). There is is also proven that the Brown-Joyal-Jardine-Toën-Vezzosi models indeed precisely model $\infty$-stack $(\infty,1)$-toposes. Details on this relation are at [[models for ∞-stack (∞,1)-toposes]].

Overview:

* [[Denis-Charles Cisinski]], _Catégories supérieures et théorie des topos_, Séminaire Bourbaki, 21.3.2015, [pdf](http://www.math.univ-toulouse.fr/~dcisinsk/1097.pdf).

* {#Rezk19} [[Charles Rezk]], _Lectures on Higher Topos Theory_, Leeds (2019) [[pdf](https://rezk.web.illinois.edu/leeds-lectures-2019.pdf), [[RezkHigherToposTheory2019.pdf:file]]]

A useful collection of facts of [[simplicial homotopy theory]] and [[(infinity,1)-topos theory]] is in

* [[Zhen Lin Low]], _[[Notes on homotopical algebra]]_

A quick introduction to the topic is in

* [[André Joyal]], _[[A crash course in topos theory -- The big picture]]_, lecture series at [Topos à l'IHES](https://indico.math.cnrs.fr/event/747/), November 2015, Paris

On $\infty$-toposes [[category object in an (infinity,1)-category|internal to]] other $\infty$-toposes;

* [[Louis Martini]], [[Sebastian Wolf]], *Internal higher topos theory* [[arXiv:2303.06437](https://arxiv.org/abs/2303.06437)]

### Giraud-Rezk-Lurie axioms

A discussion of the $(\infty,1)$-[[universal colimits]] in terms of [[model category]] presentations is due to

* [[Charles Rezk]], _Fibrations and homotopy colimits of simplicial sheaves_ ([pdf](http://www.math.uiuc.edu/~rezk/rezk-sharp-maps.pdf))

More on this with an eye on [[associated ∞-bundles]] is in

* [[Matthias Wendt]], _Classifying spaces and fibrations of simplicial sheaves_ ([arXiv](http://arxiv.org/abs/1009.2930))

### Homotopy type theory

Proof that all [[∞-stack]] [[(∞,1)-topos]] have [[presentable (∞,1)-category|presentations]] by [[model categories]] which interpret (provide [[categorical semantics]]) for [[homotopy type theory]] with [[univalence|univalent]] [[type universes]]:

* {#Shulman19} [[Michael Shulman]], _All $(\infty,1)$-toposes have strict univalent universes_ ([arXiv:1904.07004](https://arxiv.org/abs/1904.07004)).

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(infinity,1)-topos - contents

**[[(∞,1)-topos theory]]**

## Background {#sidebar_background}

* [[sheaf and topos theory]]

* [[(∞,1)-category]]

* [[(∞,1)-functor]]

* [[(∞,1)-presheaf]]

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

## Definitions {#sidebar_definitions}

* [[elementary (∞,1)-topos]]

* [[(∞,1)-site]]

* [[reflective sub-(∞,1)-category]]

  * [[localization of an (∞,1)-category]]

  * [[topological localization]]

  * [[hypercompletion]]

* [[(∞,1)-category of (∞,1)-sheaves]]

  * [[(∞,1)-sheaf]]/[[∞-stack]]/[[derived stack]]

* [[(∞,1)-topos]]

* [[(n,1)-topos]], [[n-topos]]

  * [[truncated|n-truncated object]]

  * [[connected|n-connected object]]

  * [[topos|(1,1)-topos]]

    * [[presheaf]]

    * [[sheaf]]

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

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

* [[(∞,1)-quasitopos]]

  * [[separated (∞,1)-presheaf]]

  * [[quasitopos]]

    * [[separated presheaf]]

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

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

* [[(∞,2)-topos]]

* [[(∞,n)-topos]]

## Characterization {#sidebar_characterization}

* [[universal colimits]]

* [[object classifier]]

* [[groupoid object in an (∞,1)-category|groupoid object in an (∞,1)-topos]]

  * [[effective epimorphism]]

## Morphisms {#sidebar_morphisms}

* [[(∞,1)-geometric morphism]]

* [[(∞,1)Topos]]

* [[Lawvere distribution]]

## Extra stuff, structure and property {#sidebar_extra}

* [[hypercomplete (∞,1)-topos]]

  * [[hypercomplete object]]

  * [[Whitehead theorem]]

* [[over-(∞,1)-topos]]

* [[n-localic (∞,1)-topos]]

* [[locally n-connected (n,1)-topos]]

* [[structured (∞,1)-topos]]

  * [[geometry (for structured (∞,1)-toposes)]]

* [[locally ∞-connected (∞,1)-topos]], [[∞-connected (∞,1)-topos]]

* [[local (∞,1)-topos]]

  * [[concrete (∞,1)-sheaf]]

* [[cohesive (∞,1)-topos]]

## Models {#sidebar_models}

* [[models for ∞-stack (∞,1)-toposes]]

  * [[model category]]

  * [[model structure on functors]]

  * [[model site]]/[[sSet-site]]

  * [[model structure on simplicial presheaves]]

  * [[descent for simplicial presheaves]]

  * [[Verity on descent for strict omega-groupoid valued presheaves|descent for presheaves with values in strict ∞-groupoids]]

## Constructions {#sidebar_constructions}

**structures in a [[cohesive (∞,1)-topos]]**

* [[shape of an (∞,1)-topos|shape]] / [[coshape of an (∞,1)-topos|coshape]]

* [[cohomology]]

* [[homotopy groups in an (∞,1)-topos|homotopy]]

  * [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]]/[[fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos|of a locally ∞-connected (∞,1)-topos]]

  * [[categorical homotopy groups in an (∞,1)-topos|categorical]]/[[geometric homotopy groups in an (∞,1)-topos|geometric]] homotopy groups

  * [[Postnikov tower in an (∞,1)-category|Postnikov tower]]

  * [[Whitehead tower in an (∞,1)-topos|Whitehead tower]]

* [[rational homotopy theory in an (∞,1)-topos|rational homotopy]]

* [[dimension]]

  * [[homotopy dimension]]

  * [[cohomological dimension]]

  * [[covering dimension]]

  * [[Heyting dimension]]

[Edit this sidebar](/nlab/edit/%28infinity%2C1%29-topos+-+contents)

[[!redirects infinity,1-topos - contents]]


(infinity,1)-topos theory

[[!redirects (∞,1)-topos theory]]

+-- {: .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}

## Idea

The theory of [[(∞,1)-toposes]], generalizing [[topos theory]] from [[category theory]] to [[(∞,1)-category theory]]: "[[geometric homotopy theory]]".

## Related concepts

* [[topos theory]]

* [[2-topos theory]]

* **$(\infty,1)$-topos theory**

* [[higher topos theory]]

* [[Awodey's conjecture]]

## References

An quick introduction is in part 3, 4 of

* [[André Joyal]], _[[A crash course in topos theory -- The big picture]]_, lecture series at [Topos à l'IHES](https://indico.math.cnrs.fr/event/747/), November 2015, Paris

For origins of the notion of $(\infty,1)$-topos itself see the references at [[(∞,1)-topos]].

Early frameworks for [[Grothendieck topos|Grothendieck]] (as opposed to "elementary") $(\infty,1)$-topoi are due [[Charles Rezk]] via [[model categories]]

* {#Rezk10} [[Charles Rezk]], _Toposes and homotopy toposes_, 2010 ([pdf](http://www.math.uiuc.edu/~rezk/homotopy-topos-sketch.pdf))

and due to [[Bertrand Toen|Toën]]--[[Gabriele Vezzosi|Vezzosi]] in two versions (preprints 2002), via [[simplicially enriched category|simplically enriched categories]] and via [[Segal categories]]:

* [[Bertrand Toën]], [[Gabriele Vezzosi]], _Homotopical algebraic geometry. I. Topos theory_, Adv. Math. 193 (2005) no. 2, 257--372 [doi](https://doi.org/10.1016/j.aim.2004.05.004) [arXiv:math.AT/0207028](http://arxiv.org/abs/math/0207028)

* [[Bertrand Toën]] , [[Gabriele Vezzosi]], _Segal topoi and stacks over Segal categories_, [arXiv:math.AG/0212330](http://arXiv.org/abs/math/0212330)

A general abstract conception of $(\infty,1)$-topos theory in terms of [[(∞,1)-category theory]]  was given in

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

The analog of the [[Elephant]] for $(\infty,1)$-topos theory is still to be written.


(infinity,1)-Yoneda extension

#Contents#
* automatic table of contents goes here
{:toc}

## Idea ##

In ordinary [[category theory]] the [[Yoneda extension]] of a [[functor]] $F : C \to D$ is its left [[Kan extension]] through the [[Yoneda embedding]] of its domain to a functor $\hat F : PSh(C) \to D$.

In [[higher category theory]] there should be a corresponding version of this construction.

In particular with categories replaced by [[(∞,1)-category|(∞,1)-catgeories]] there should be a version with the category of [[presheaf|presheaves]] replaced by a [[(∞,1)-category of (∞,1)-presheaves]], corresponding to the [[Yoneda lemma for (∞,1)-categories]].

This in turn should have a [[presentable (infinity,1)-category|presentation]] in terms of the global [[model structure on simplicial presheaves]].

## Statement ##

+-- {: .standout}

  [[Urs Schreiber]]: this here is something I thought about. Check. Even to the extent that this is right, it is clearly not yet a full answer, but at best a step in the right direction.

=--

Let $C$ be a [[category]] and write $[C^{op}, SSet]_{proj} = SPSh(C)_{proj}$ for the projective [[nLab:model structure on simplicial presheaves|model structure on simplicial presheaves]] on $C$. Let $\mathbf{D}$ be  any [[combinatorial simplicial model category]].

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

If $F$ takes values in cofibrant objects of $\mathbf{D}$ then the [[SSet]]-[[enriched category theory|enriched]] [[Yoneda extension]] $\hat F$ of $F$ is the [[nLab:left adjoint|left adjoint]] part of an [[SSet]]-[[Quillen adjunction]]

$$
   \hat F : SPSh(C)_{proj}
   \stackrel{\leftarrow}{\to}
   \mathbf{D}
   :
   R
   \,.
$$

=--

Accordingly, if $F$ does not take values in cofibrant objects but where a cofibrant replacement functor $Q : \mathbf{D} \to \mathbf{D}$ is given, the [[Yoneda extension]] $\widehat{Q F}$ of $Q F$ is an $(\infty,1)$-extension up to weak equivalence of $F$.

### Proof ###

We prove this in two steps.

+-- {: .num_lemma}
###### Lemma

The [[Yoneda extension]] $F : SPSh(C)_{proj} \to \mathbf{D}$ preserves cofibrations and acyclic cofibrations.

=--

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

Recall that the [[Yoneda extension]] of $F : SPSh(C)_{proj} \to \mathbf{D}$ is given by the [[nLab:coend|coend]] formula

$$
  \hat F : X \mapsto \int^{U \in C} F(U) \cdot X(U)
  \,,
$$

where in the integrand we have the [[copower|tensoring]] of the object $F(U) \in \mathbf{D}$ by the [[simplicial set]] $X(U)$.

The lemma now rests on the fact that this [[coend]] over the [[copower|tensor]]

$$
  \int (-)\cdot (-)
   :
     [C,\mathbf{D}]_{inj}
     \cdot
     [C^{op}, SSet]_{proj}
   \to
   \mathbf{D}
$$

is a [[Quillen bifunctor]] using the injective and projective [[global model structure on functors]] as indicated. This is [[Higher Topos Theory|HTT prop. A.2.9.26 & rmk. A.2.9.27]] and recalled at [[Quillen bifunctor]].

Since by assumption $F(U)$ is cofibrant for all $U$ we have that $\hat F$ itself is cofibrant regarded as an object of $[C,\mathbf{D}]_{inj}$. From the definition of [[Quillen bifunctor]]s it follows that

$$
  \hat F = \int^U F(U) \cdot (-)(U) : SPSh(C)_{proj} \to \mathbf{D}
$$

preserves cofibrations and acyclic cofibrations.

=--

+-- {: .un_lemma }
###### Lemma

The functor $\hat F$ has an [[enriched functor|enriched]]  [[right adjoint]]

$$
  R : \mathbf{D} \to \mathrm{SPSh}(C)
$$

given by

$$
  R(A)
  =
  \mathbf{D}(F(-), A)
  \,.
$$

=--

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

This is a standard argument.

We demonstrate the Hom-isomorphism that characterizes the [[adjunction]]:

Start with the above [[coend]] description of $\hat F$

$$
  \mathbf{D}({\hat F}(X), A)
  \simeq
  \mathbf{D}(
    \int^{U \in S}
     F(U) \cdot X(U)
     ,
     A
  )
  \,.
$$

Then use the continuity of the enriched Hom-functor to pass it through the coend and obtain the following [[end]]:

$$
  \cdots \simeq
  \int_{U \in S}
    \mathbf{D}({\hat F}(U) \cdot X(U), A)
  \,.
$$

The defining property of the tensoring operation
implies that this is equivalent to

$$
  \simeq
  \int_{U \in S}
  SSet( X(U), \mathbf{D}(F(U),A))
  \,.
$$

But this is the [[enriched functor category|end-formula]] for the $SSet$-object of
[[natural transformation]]s between [[simplicial presheaf|simplicial presheaves]]:

$$
  \cdots
  \simeq
  [C^{op},SSet](X, \mathbf{D}(\Pi(-), A))
  \,.
$$

By definition this is the desired right hand of the hom isomorphism

$$
  \cdots
  =  [C^{op}, SSet](X, R(A))
  \,.
$$

=--

These two lemmas together constitute the proof of the proposition.

[[!redirects (∞,1)-Yoneda extension]]


(infinity,1)Cat

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

**$(\infty,1)Cat$** is the [[(∞,2)-category]] of all small [[(∞,1)-categories]].

Its full [[subcategory]] on [[∞-groupoid]]s is [[∞Grpd]].

#Contents#
* table of contents
{:toc}

## The $(\infty,2)$-category

### As an $SSet$-category

One incarnation of [[(∞,2)-categories]] is given by [[quasi-category]]-enriched categories (see there for details). As such $(\infty,1)Cat$ is the full [[SSet]]-[[enriched category|enriched]] [[subcategory]] of [[SSet]] on those [[simplicial set]]s that are [[quasi-categories]]. By the fact described at [[(∞,1)-category of (∞,1)-functors]] this is indeed a [[quasi-category]]-enriched category.

### As an enriched model category

The [[model category]] presenting this [[(∞,2)-category]] is the Joyal [[model structure for quasi-categories]] $sSet_{Joyal}$. Its full [[sSet]]-[[subcategory]] is the [[quasi-category]] enriched category of quasi-categories from above.

## The $(\infty,1)$-category

Sometimes it is useful to consider inside the full $(\infty,2)$-catgeory of $(\infty,1)$-categories just the maximal $(\infty,1)$-category and discarding all non-invertible [[k-morphism|2-morphisms]]. This is the [[(∞,1)-category of (∞,1)-categories]].

### As an $SSet$-category

As an [[SSet]]-[[enriched category]] the [[(∞,1)-category of (∞,1)-categories]] is obtained from the quasi-category-enriched version by picking in each [[hom-object]] simplicial set of $(\infty,1)Cat$ the maximal [[Kan complex]].

### As an enriched model category

One [[model category]] structure presenting this is the [[model structure on marked simplicial over-sets|model structure on marked simplicial sets]]. As a plain [[model category]] this is [[Quillen equivalence|Quillen equivalent]] to $sSet_{Joyal}$, but as an [[enriched model category]] it is $sSet_{Quillen}$ enriched, so that its full [[SSet]]-subcategory on fibrant-cofibrant objects presents the $(\infty,1)$-category of $(\infty,1)$-categories.

### Properties

#### Limits and colimits in $(\infty,1)$Cat
 {#LimitsAndColimits}

[[limit in a quasi-category|Limits and colimits]] over a [[(∞,1)-functor]] with values in $(\infty,1)Cat$ may be reformulation in terms of the  [[universal fibration of (infinity,1)-categories]] $Z \to (\infty,1)Cat^{op}$

Then let $X$ be any [[(∞,1)-category]] and

$$
  F : X \to (\infty,1)Cat
$$

an [[(∞,1)-functor]]. Recall that the [[Cartesian fibration|coCartesian fibration]] $E_F \to X$ classified by $F$ is the pullback of the [[universal fibration of (∞,1)-categories]] $Z$ along F:

$$
  \array{
    E_F &\to& Z
    \\
    \downarrow && \downarrow
    \\
    X &\stackrel{F}{\to}& (\infty,1)Cat
  }
$$

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

Let the assumptions be as above. Then:

* The colimit of $F$ is equivalent to $E_F$:

  $$
    E_F \simeq colim F
  $$

* The limit of $F$ is equivalent to the [[(infinity,1)-category of cartesian section]] of $E_F \to X$

  $$
    \Gamma_X(E_F) \simeq lim F
    \,.
  $$

=--

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

This is [[Higher Topos Theory|HTT, section 3.3]].

=--

#### Automorphisms
 {#Automorphisms}

+-- {: .un_theorem }
###### Theorem

The full subcategory of the [[(∞,1)-category of (∞,1)-categories]] $Func((\infty,1)Cat, (\infty,1)Cat)$ on those [[(∞,1)-functor]]s that are equivalences is equivalent to $\{Id, op\}$: it contains only the identity functor and the one that sends an $(\infty,1)$-category to its [[opposite (infinity,1)-category]].

=--

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

This is due to

* [[Bertrand Toen]],  _Vers une axiomatisation de la théorie des catégories supérieures_ , K-theory 34 (2005), no. 3,
233-263.

It appears as [[Higher Topos Theory|HTT, theorem 5.2.9.1]] ([arxiv v4+](http://arxiv.org/abs/math.CT/0608040) only)

First of all the statement is true for the ordinary category of [[poset]]s. This is [prop. 5.2.9.14](http://arxiv.org/PS_cache/math/pdf/0608/0608040v4.pdf#page=311).

From this the statement is deduced for $(\infty,1)$
-categories by observing that posets are characterized by the fact that two parallel functors into them that are objectwise equivalent are already equivalent, [prop. 5.2.9.11](http://arxiv.org/PS_cache/math/pdf/0608/0608040v4.pdf#page=310), which means that posets $C$ are characterized by the fact that

$$
  \pi_0 (\infty,1)Cat(D,C)
  \to
  Hom_{Set}(
   \pi_0 (\infty,1)Cat(*,D)
   ,
   \pi_0 (\infty,1)Cat(*,C)
  )
$$

is an injection for all $D \in (\infty,1)Cat$.

This is preserved under automorphisms of $(\infty,1)Cat$, hence any such automorphism preserves posets, hence restricts to an automorphism of the category of posets, hence must be either the identity or $(-)^{op}$ there, by the above statement for posets.

Now finally the main point of the proof is to see that the linear posets $\Delta \subset (\infty,1)Cat$ are [[dense functor|dense]] in $(\infty,1)Cat$, i.e. that the identity transformation of the inclusion functor $\Delta \hookrightarrow (\infty,1)Cat$ exhibits $Id_{(\infty,1)Cat}$ as the left [[Kan extension]]

$$
  \array{
    \Delta &\hookrightarrow& (\infty,1)Cat
    \\
    \downarrow & \nearrow_{Lan = \mathrlap{Id}}
    \\
    (\infty,1)Cat
  }
  \,.
$$

=--

## Presentations

[[!include table - models for (infinity,1)-operads]]

## Related concepts

* [[formal (infinity,1)-category theory|formal $(\infty,1)$-category theory]]

* [[Pos]]

* [[Set]]

* [[Grpd]], [[∞Grpd]]

* [[Cat]], [[Operad]]

* [[2Cat]]

* **$(\infty,1)$Cat**, [[(∞,1)Operad]]

  * [[homotopy 2-category of (∞,1)-categories]]

* [[(∞,n)Cat]]

* [[(infinity, 1)Prof]]

category: category

[[!redirects (∞,1)Cat]]

[[!redirects (∞,1)Categories]]
[[!redirects (infinity,1)Categories]]

[[!redirects (infinity,2)-category of (infinity,1)-categories]]


(infinity,1)Prof

[[!redirects (infinity, 1)Prof]]
#Contents#
* table of contents
{:toc}

## Definition

$\mathbf{(\infty, 1)Prof}$ is the [[(∞,2)-category]] of [[(∞,1)-categories]], [[(\infty, 1)-profunctor|(∞,1)-profunctors]], and [[natural transformations]].

Recall that a (∞,1)-profunctor from $A$ to $B$ is a (∞,1)-functor $B^{op}\times A\to \infty Grpd$.  Composition of (∞,1)-profunctors in $(\infty, 1)Prof$ is by the "tensor product of (∞,1)-functors" [[homotopy coend]] construction: if $H\colon A ⇸  B$ and $K\colon B ⇸  C$, their composite is given as a functor $C^{op}\times A \to \infty Grpd$ by
$$(c,a)\mapsto \int^{b\in B} H(b,a)\times K(c,b).$$
The identity on an (∞,1)-category $A$ is its hom-functor $Hom_A(-,-)$.

## Properties

Note that every (∞,1)-functor $f\colon A\to B$ gives two [[representable functor|representable]] (∞,1)-profunctors $B(f-,-)$ and $B(-,f-)$.  This defines two [[(∞,2)-functors]] $(\infty,1)Cat \to (\infty,1)Prof$ that are the identity on objects.  The relationship between [[(∞,1)Cat]] and $(\infty,1)Prof$ encoded in this way makes them into an $(\infty, 1)$-version of an [[equipment]], see ([Haugseng 15](#Haugseng15)).

$(\infty, 1)Prof$ can act as a classifying object for kinds of (∞,1)-functors; see [higher exponentiable functor](Conduché+functor#highercategorical_versions).

$(\infty,1)Prof$ is equivalent to the full subcategory of [[Pr(∞,1)Cat]] whose objects are presheaf categories.

## Related concepts

* [[Prof]]
* [[Pr(∞,1)Cat]]

##References

* {#Haugseng15} [[Rune Haugseng]], _Bimodules and natural transformations for enriched ∞-categories_, ([arXiv:1506.07341](https://arxiv.org/abs/1506.07341))

[[!redirects (∞,1)Prof]]


(infinity,1)Topos

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

#Contents#
* table of contents
{:toc}

## Idea

By **$(\infty,1)Topos$** is denoted the collection of all [[(∞,1)-topos]]es. This is the [[(∞,1)-category]]-theory analog of [[Topos]].

## Definition

**$(\infty,1)\,Topos$**  is the (non-full) [[sub-(∞,1)-category]] of [[(∞,1)Cat]] on [[(∞,1)-toposes]] and [[(∞,1)-geometric morphisms]] between them.

Morally, this should actually be an [[(∞,2)-category]], just as [[Topos]] is a [[2-category]], but since the technology of $(\infty,2)$-categories is not well developed, this point of view is not often taken yet.

## Properties

In the following, $(\infty,1)Cat$ will refer to the (superlarge)
$(\infty,1)$ category of _large_ $(\infty, 1)$-categories.

### Existence of sites of definition
  {#SitesOfDefinition}

(...)

### Existence of limits and colimits
 {#ExistenceOfLimitsAndColimits}

We discuss existence of [[(∞,1)-limit]]s and [[(∞,1)-colimit]]s in $(\infty,1)Topos$.

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

The $(\infty,1)$-category $(\infty,1)Topos$ has all small $(\infty,1)$-colimits and functor

$$
  (\infty,1)Topos^{op} \to (\infty,1)Cat
$$

preserves small limits.

=--

This is [[Higher Topos Theory|HTT, prop. 6.3.2.3]]. In the notation there, $LTop$ is the $(\infty,1)$-category of toposes whose arrows are the inverse image morphisms, and thus opposite to $(\infty,1)Topos$.

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

The $(\infty,1)$-category $(\infty,1)Topos$ has [[filtered (infinity,1)-category|filtered]] [[(∞,1)-limit]]s and the inclusion

$$
  (\infty,1)Topos \to (\infty,1)Cat
$$

preserves these.

=--

This is [[Higher Topos Theory|HTT, prop. 6.3.3.1]].

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

The $(\infty,1)$-category $(\infty,1)Topos$ has all small [[(∞,1)-limit]]s.

=--

This is [[Higher Topos Theory|HTT, prop. 6.3.4.7]].

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

The $(\infty,1)$-limits in $(\infty,1)Topos$ coincide actually with the proper $(\infty,2)$-limits.

=--

This is [[Higher Topos Theory|HTT, remark 6.3.4.10]].

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

The [[terminal object in an (infinity,1)-category|terminal object]] in
[[(∞,1)Topos]] is [[∞Groupoids]].

=--

This is [[Higher Topos Theory|HTT, Prop. 6.3.4.1]], for details see at *[[terminal geometric morphism]]*.

### Computation of limits and colimits
 {#ComputationOfLimitsAndColimits}

We discuss more or less explicit descriptions of
[[(∞,1)-limits]] and [[(∞,1)-colimits]] in $(\infty,1)Topos$.

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

Let

$$
  \array{
    && \mathcal{X}
    \\
    && \downarrow^{\mathrlap{(g^* \dashv g_*)}}
    \\
    \mathcal{Y}
    &\stackrel{(f^* \dashv f_*)}{\to}&
    \mathcal{Z}
  }
$$

be a [[diagram]] of [[(∞,1)-toposes]]. Then its [[(∞,1)-pullback]] is computed, roughly, by the [[(∞,1)-pushout]] of their [[(∞,1)-sites]] of definition (see [above](#SitesOfDefinition)).

More in detail: there exist [[(∞,1)-sites]] $\tilde \mathcal{D}$, $\mathcal{D}$, and $\mathcal{C}$ with [[finite limit|finite]] [[(∞,1)-limit]] and [[morphisms of sites]]

$$
  \array{
    && \mathcal{D}
    \\
    && \uparrow^{\mathrlap{g}}
    \\
    \tilde \mathcal{D}
    &\stackrel{f}{\leftarrow}&
    \mathcal{C}
  }
$$

such that

$$
  \left(
  \array{
    && \mathcal{X}
    \\
    && \downarrow^{\mathrlap{(g^* \dashv g_*)}}
    \\
    \mathcal{Y}
    &\stackrel{(f^* \dashv f_*)}{\to}&
    \mathcal{Z}
  }
  \right)
  \,\,\,
   \simeq
  \,\,\,
  \left(
  \array{
    && Sh_{(\infty,1)}(\mathcal{D})
    \\
    && \downarrow^{\mathrlap{(Lan_g \dashv (-)\circ g)}}
    \\
    Sh_{(\infty,1)}(\tilde \mathcal{D})
    &\stackrel{(Lan_f \dashv (-)\circ f)}{\to}&
    Sh_{(\infty,1)}(\mathcal{C})
  }
  \right)
  \,.
$$

Let then

$$
  \array{
    \tilde \mathcal{D} \coprod_{\mathcal{C}} \mathcal{D} &\stackrel{f'}{\leftarrow}& \mathcal{D}
    \\
    {}^{\mathllap{g'}}\uparrow &\swArrow_{\simeq}& \uparrow^{\mathrlap{g}}
    \\
    \tilde \mathcal{D}
    &\stackrel{f}{\leftarrow}&
    \mathcal{C}
  }
  \,\,\,\,\,
  \in
  (\infty,1)Cat^{lex}
$$

be the [[(∞,1)-pushout]] of the underlying [[(∞,1)-categories]] in the [[full sub-(∞,1)-category]] [[(∞,1)Cat]]${}^{lex} \subset (\infty,1)Cat$ of $(\infty,1)$-categories with finite $(\infty,1)$-limits.

Let moreover

$$
  Sh_{(\infty,1)}(\tilde \mathcal{D} \coprod_{\mathcal{C}} \mathcal{D})
  \hookrightarrow
  PSh_{(\infty,1)}(\tilde \mathcal{D} \coprod_{\mathcal{C}} \mathcal{D})
$$

be the [[reflective sub-(∞,1)-category]] obtained by [[localization]] at the class of morphisms generated by the inverse image $Lan_{f'}(-)$ of the [[covering]]s of $\mathcal{D}$ and the inverse image $Lan_{g'}(-)$ of the coverings of $\tilde \mathcal{D}$.

Then

$$
  \array{
    Sh_{(\infty,1)}(\tilde \mathcal{D} \coprod_{\mathcal{C}} \mathcal{D})
    &\to& \mathcal{X}
    \\
    \downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{(g^* \dashv g_*)}}
    \\
    \mathcal{Y}
    &\stackrel{(f^* \dashv f_*)}{\to}&
    \mathcal{Z}
  }
$$

is an [[(∞,1)-pullback]] square.

=--

This is [[Higher Topos Theory|HTT, prop. 6.3.4.6]].

### Colimits of over-toposes
 {#ColimitsOfOverToposes}

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

For any $(\infty,1)$ topos $\mathbf{H}$, there is a colimit preserving functor $\mathbf{H} \to (\infty,1)Topos$ sending an object $X$ to its over-topos $\mathbf{H}_{/X}$, and sending an arrow $f : X \to Y$ to the essential geometric morphism $f_* : \mathbf{H}_{/X} \to \mathbf{H}_{/Y}$.

=--

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

[[Higher Topos Theory|HTT, Prop. 6.1.3.9]] implies the "terminal vertex" Cartesian fibration $\mathbf{H}^{[1]} \to \mathbf{H}$ is classified by a limit preserving functor $\mathbf{H}^{op} \to Pr^L$, the $(\infty,1)$-category of locally presentable $(\infty,1)$-categories and colimit-preserving functors between them.

This functor factors through the subcategory $(\infty,1)Topos^{op} \subseteq Pr^L$ that sends a geometric morphism to its inverse image part. By [HTT, Prop. 6.3.2.3] and [HTT, Prop. 5.5.1.13], it follows that this is also a limit preserving functor.

The opposite category is then formed by taking right adjoints.

=--

## Related concepts

* [[Topos]]

* **$(\infty,1)$Topos**

## References

section 6.3 in

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

category: category

[[!redirects (infinity,1)Topos]]
[[!redirects (infinity,1)Topoi]]
[[!redirects (infinity,1)Toposes]]
[[!redirects (infinity,1)-Topos]]
[[!redirects (infinity,1)-Topoi]]
[[!redirects (infinity,1)-Toposes]]
[[!redirects (∞,1)Topos]]
[[!redirects (∞,1)Topoi]]
[[!redirects (∞,1)Toposes]]
[[!redirects (∞,1)-Topos]]
[[!redirects (∞,1)-Topoi]]
[[!redirects (∞,1)-Toposes]]

[[!redirects (infinity,1)Toposes]]


(infinity,2)-Categories and the Goodwillie Calculus

[[!redirects (∞,2)-Categories and the Goodwillie Calculus]]

this entry is about the article

* [[Jacob Lurie]], _$(\infty,2)$-categories and the Goodwillie calculus_  ([pdf](http://www.math.harvard.edu/~lurie/papers/GoodwillieI.pdf), [arXiv:0905.0462](http://arxiv.org/abs/0905.0462))

on

1. [[(∞,1)-category|(∞,1)-categories]] of [[(∞,n)-categories]] via [[category objects in an (∞,1)-category]] in general and

1. of [[(∞,2)-categories]] in particular.

1. and about [[Goodwillie calculus]].

category: reference

[[!redirects (∞,2)-Categories and the Goodwillie Calculus]]


(infinity,2)-category

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

#Contents#
* table of contents
{:toc}

## Idea

An __$(\infty,2)$-category__ is the special case of $(\infty,n)$-[[(infinity,n)-category|category]] for $n=2$.

It is best known now through a [[geometric definition of higher category]].

Models include:

* the definition by [[Carlos Simpson]] and Tamsamani;

* the definition in terms of [[n-fold Segal spaces]];

* a definition in terms of scaled simplicial sets, following Verity's [[simplicial model for weak omega-categories]] by Jacob Lurie (see reference below)

See also the list of all definitions of higher categories at [[(∞,n)-category]].

## Properties

### Models for the $(\infty,1)$-category of $(\infty,2)$-categories {#TotMod}

In _[[(∞,2)-Categories and the Goodwillie Calculus]]_ [[Jacob Lurie]] discusses a variety of [[model category]] structures, all [[Quillen equivalence|Quillen equivalent]], that all model the [[(∞,2)-category]] of $(\infty,1)$-categories, in generalization of the standard model category models for [[(∞,1)-category|(∞,1)-categories]] themselves (see there for details).

Recall that

* the standard [[model structure on simplicial sets]] models [[∞-groupoid]]s.

A [[simplicial model category|simplicially]] [[enriched model category]] with respect to the standard model structure on simplicial sets hence models [[∞Grpd]]-enriched categories, hence [[(∞,1)-category|(∞,1)-categories]].

Along this pattern $(\infty,2)$-categories should be modeled by [[enriched category|categories enriched]] in the [[model structure on simplicial sets|Joyal model structure]] that models the [[(∞,1)-category of (∞,1)-categories]].

Write $SSet^J$ for [[SSet]] equipped with the Joyal model structure. Then, indeed, there is a diagram of [[Quillen equivalence]]s of [[model category]] structures

$$
  SSet^J Cat \to SSet SegSp \to [\Delta^{op}, SSet^J]
$$

between Joyal-$SSet$-enriched categories, Joyal-$SSet$-enriched [[complete Segal space]]s and simplicial Joyal-simplicial sets.

This is [remark 0.0.4, page 5](http://arxiv.org/PS_cache/arxiv/pdf/0905/0905.0462v2.pdf#page=5) of the article.
There are many more models. See there for more.

## Examples

Classes of examples include

* [[(∞,2)-toposes]]

* For $\mathcal{C}$ a suitable [[monoidal (∞,1)-category]] there is the $(\infty,2)$-category $Mod(\mathcal{C})$ of $\infty$-algebras and $\infty$-bimodules in $\mathcal{C}$. See at _[bimodule - Properties - The (∞,2)-category of ∞-algebras and ∞-bimodules](bimodule#Infinity2CategoryOfInfinityAlgebrasAndBimodules)_.

## Related concepts

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

* [[category]]

* [[2-category]]

* [[3-category]]

* [[n-category]]

* [[(∞,0)-category]]

* [[(∞,1)-category]]

* **(∞,2)-category**

* [[(∞,n)-category]]

* [[(n,r)-category]]

##References

* [[Jacob Lurie]], _[[(∞,2)-Categories and the Goodwillie Calculus]]_

[[!redirects (infinity,2)-categories]]
[[!redirects (∞,2)-category]]
[[!redirects (∞,2)-categories]]


(infinity,2)-sheaf

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### Higher topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An _$(\infty,2)$-sheaf_ or _$(\infty,2)$-stack_ is the higher analog of an [[(∞,1)-sheaf]] / [[∞-stack]].

For $\mathcal{C}$ an [[(∞,1)-category]] equipped with the structure of an [[(∞,1)-site]], an $(\infty,2)$-sheaf on $\mathcal{C}$ is an [[(∞,1)-functor]]

$$
  X : \mathcal{C}^{op} \to Cat_{(\infty,1)}
$$

to [[(∞,1)Cat]], that satisfies [[descent]]: hence which is a [[local object]] with respect to the [[covering]] [[sieve]] inclusions in $Func(\mathcal{C}^{op}, Cat_{(\infty,1)})$.

The [[(∞,2)-category]] of $(\infty,2)$-sheaves

$$
  Sh_{(\infty,2)}(\mathcal{C})
$$

is an [[(∞,2)-topos]], the [[homotopy theory]]-generalization of a [[2-topos]] of [[2-sheaves]].

## Examples

### Codomain fibration / canonical $(\infty,2)$-sheaf
 {#CodomainFibration}

Let $\mathcal{X}$ be an [[(∞,1)-topos]], regarded as a ([[large site|large]]) [[(∞,1)-site]] equipped with the [[canonical topology]]. Then an [[(∞,1)-functor]]

$$
  A : \mathcal{X}^{op} \to CAT_{(\infty,1)}
$$

is an $(\infty,2)$-sheaf precisely if it preserves [[(∞,1)-limits]] (takes [[(∞,1)-colimits]] in $\mathcal{X}$ to [[(∞,1)-limits]] in [[(∞,1)Cat]]).

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

For $\mathcal{X}$ an $(\infty,1)$-topos, the functor

$$
  Cod : \mathcal{X}^{op} \to CAT_{(\infty,1)}
$$

$$
  Cod : A \mapsto \mathcal{X}_{/A}
$$

is a ([[universe enlargement|large]]) $(\infty,2)$-sheaf on $\mathcal{X}$ , regarded as a [[(∞,1)-site]] equipped with the [[canonical topology]]. Here $\mathcal{X}_{/A}$ is the [[slice (∞,1)-topos]] over $A$.

=--

This is a special case of ([Lurie, lemma 6.1.3.7](#Lurie)).

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

The functor $Cod$ [[(∞,1)-Grothendieck construction|classifies]] the [[codomain fibration]]. Its fiberwise [[stabilization]] to the [[tangent (∞,1)-category]] is the $(\infty,2)$-sheaf of [[quasicoherent sheaves]] on $\mathcal{X}$.

=--

## Related concepts

* [[sheaf]]

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

* [[(∞,1)-sheaf]] / [[∞-stack]]

* **$(\infty,2)$-sheaf**

* [[(∞,n)-sheaf]]

## References

* {#Lurie} [[Jacob Lurie]], section 6.1.3 of _[[Higher Topos Theory]]_

Discussion of a local [[model structure on simplicial presheaves]] $[S^op, sSet_{Joyal}]_{loc}$ with respect to the [[Joyal model structure]] $sSet_{Joyal}$ for [[quasicategories]] is in

* {#Meadows15} [[Nicholas Meadows]], _The Local Joyal Model Structure_ ([arXiv:1507.08723](http://arxiv.org/abs/1507.08723))

and with respect to the [[model structure for complete Segal spaces]] in

* [[Nicholas Meadows]], _Local Complete Segal Spaces_ ([arXiv:1607.05794](http://arxiv.org/abs/1607.05794))

[[!redirects (∞,2)-sheaf]]
[[!redirects (∞,2)-sheaves]]
[[!redirects (infinity,2)-sheaves]]


(infinity,2)-site

[[!redirects (∞,2)-site]]

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,2)$-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### $(\infty,2)$-Topos theory
+--{: .hide}
[[!include (infinity,2)-topos theory - contents]]
=--
=--
=--

An [[(n,r)-site]] for $n = \infty$, $r = 2$.

[[!redirects (infinity,2)-sites]]


(infinity,2)-topos

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### Higher topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An [[(∞,2)-category]] of [[(∞,2)-sheaves]]. The joint higher generalization of the notion of [[(∞,1)-topos]] and [[2-topos]].

The archetypical example is the [[(infinity,2)-category of (infinity,1)-categories]] $Cat_{(\infty,1)}$. See also at *[[formal category theory]]*.

## Related concepts

* [[formal (infinity,1)-category theory|formal $(\infty,1)$-category theory]]

* [[directed homotopy type theory]]

[[!include flavors of higher toposes -- list]]

## References

Discussion of, potentially, the [[internal language]] of $(\infty,2)$-toposes as a form of [[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)]

  > **Abstract:** We will introduce a version of [[dependent type theory]] that is suitable to develop a [[synthetic mathematics|synthetic]] theory of [[1-categories]]. The [[axioms]] are both a [[fragment]] and an extension of ordinary [[dependent type theory]]. The [[axioms]] are chosen so that [[(∞,1)-category theory]] (in the form of [[quasi-categories]] or [[complete Segal spaces]]) gives a [[categorical semantics|semantic interpretation]], in a way which extends [[Vladimir Voevodsky|Voevodsky]]'s interpretation of [[univalence axiom|univalent]] [[dependent type theory]] in the [[homotopy theory]] of [[Kan complexes]]. More generally, using a slight generalization of [Shulman's methods](relation+between+type+theory+and+category+theory#Shulman19), we should be able to see that the theory of [[internal (∞,1)-category|(∞,1)‑categories internally]] in any [[(∞,1)-topos|∞‑topos]] (as [developed by Martini and Wolf](internal+infinity1-category#ReferencesInternalToInfinityTopos)) is a [[categorical semantics|semantic interpretation]] as well (hence so is [[Parametrized Higher Category Theory and Higher Algebra|parametrized higher category theory]] introduced by Barwick, Dotto, Glasman, Nardin and Shah). There are of course strong links with [[∞-cosmoi]] of Riehl and Verity as well as with [[cubical type theory|cubical HoTT]] (as strongly suggested by the work of Licata and Weaver), or [[simplicial type theory|simplicial HoTT]] (as in the work of Buchholtz and Weinberger). We will explain the axioms in detail and have a glimpse at basic theorems and constructions in this context ([[(∞,1)-Yoneda lemma|Yoneda Lemma]], [[(∞,1)-Kan extension|Kan extensions]], [[localization of an (∞,1)-category|Localizations]]). We will also discuss the perspective of reflexivity: since the theory speaks of itself (through directed [[univalence axiom|univalence]]), we can use it to justify new deduction rules that express the idea of working up to [[equivalence]] natively (e.g. we can produce a logic by rectifying the idea of having a locally cartesian type). In particular, this logic can be used to produce and study [[categorical semantics|semantic interpretations]] of [[HoTT]].

[[!redirects (infinity,2)-toposes]]

[[!redirects (∞,2)-topos]]
[[!redirects (∞,2)-toposes]]


(infinity,2)-topos theory - contents

**[[(∞,2)-topos]]**

* [[(∞,2)-presheaf]], [[(∞,2)-site]],

**[[(∞,2)-sheaf]]**

* [[codomain fibration]], [[tangent (∞,1)-category]], [[quasicoherent ∞-stack]]

## Truncations

**[[2-topos]]**

* [[2-pretopos]]

* [[2-presheaf]], [[2-site]], [[2-sheaf]], [[stack]]

* [[n-localic 2-topos]]

* [[2-Giraud theorem]]

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

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

**[[(∞,1)-topos]]**

* [[(∞,1)-presheaf]], [[(∞,1)-site]], [[(∞,1)-sheaf]]

**[[1-topos]]**

* [[presheaf]], [[site]], [[sheaf]]