infinity-cohesive site

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohesive $\infty$-Toposes
+--{: .hide}
[[!include cohesive infinity-toposes - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An _$(\infty,1)$-cohesive site_ is a [[site]] such that the [[(∞,1)-category of (∞,1)-sheaves]] over it is a [[cohesive (∞,1)-topos]].

## Definition

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

A [[site]] $C$ is **$\infty$-cohesive** over [[∞Grpd]] if it is

* a [[strongly ∞-connected site]]

* and an [[∞-local site]].

In detail this means that $C$ is

* a [[site]] -- a [[small category]] $C$ equipped with a [[coverage]];

* with the property that

  * it has a [[terminal object]] $*$;

  * it is a [[cosifted category]]

    (for instance in that it has all [[finite products]], see at _[[categories with finite products are cosifted]]_);

  * for every [[covering]] family $\{U_i \to U\}$ in $C$

    * the [[Cech nerve]] $C(U) \in [C^{op}, sSet]$ is degreewise
      a [[coproduct]] of [[representable functor|representables]];

    * the [[simplicial set]] obtained by replacing each copy of a representable by a point is [[contractible]] (weakly equivalent to the
      point in the [[classical model structure on simplicial sets]])

      $$
        \underset{\longrightarrow}{\lim} C(U) \stackrel{\simeq}{\to} *
      $$

    * the simplicial set of points in $C(U)$ is weakly equivalent to the set of points of $U$:

      $$
        \underset{\longleftarrow}{\lim}C(U)
        = Hom_C(*, C(U)) \stackrel{\simeq}{\to} Hom_C(*,U)
        \,.
      $$

=--

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

These conditions are stronger than for a [[cohesive site]], as the latter only guarantees cohesiveness of the 1-topos over it.

This definition is supposed to model the following ideas:

* every object $U$ has an underlying set of points $Hom_C(*,U)$. We may think of each $U$  as specifying one way in which there can be cohesion on this underlying set of points;

* in view of the [[nerve theorem]] the condition that $\lim_\to C(U)$ is contractible means that $U$ itself is contractible, as seen by the [[Grothendieck topology]] on $C$. This reflects the _local_ aspect of cohesion: we only specify cohesive structure on contractible lumps of points;

* in view of this, the remaining condition that $Hom_C(*,C(U))$ is contractible is the $\infty$-analog of the condition on a [[concrete site]] that $Hom_C(*,\coprod_i U_i) \to Hom_C(*, U)$ is surjective.  This expresses that the notion of topology on $C$ and its concreteness over [[Set]] are consistent.

=--

## Examples

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

The site for a [[presheaf topos]], hence with trivial topology, is $\infty$-cohesive, def. \ref{InfinityCohesiveSite}, if it has [[finite products]].

=--

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

All covers $\{U_i \to U\}$ consist of only the [[identity]] morphism $\{U \stackrel{Id}{\to} U\}$. The [[Cech nerve]] $C\{U\}$ is then the [[simplicial object]] constant on $U$ and hence satisfies its two conditions above trivially.

=--

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

The following [[sites]] are $\infty$-cohesive, def. \ref{InfinityCohesiveSite}:

* the category [[CartSp]] with covering families given by open covers $\{U_i \hookrightarrow U\}$ by [[geodesically convex|convex]] subsets $U_i$;

  we can take the morphisms $\mathbb{R}^k \to \mathbb{R}^l$ in $CartSp$ to be

  * [[continuous maps]]

    -- in which case the sheaf topos over it models generalized [[topological space]]s, the [[2-sheaf]] [[2-topos]] contains for instance [[topological stack]]s;

  * or [[smooth maps]]

    -- in which case the sheaf topos models generalized [[smooth space]]s such as [[diffeological space]]s, the [[(∞,1)-sheaf (∞,1)-topos]] is that of [[∞-Lie groupoid]]s;

* the site [[ThCartSp]] $ \subset \mathbb{L}$ of [[smooth loci]] consisting of smooth loci of the form $R^n \times D^l_{(k)}$ with the second factor infinitesimal, where covering families are those of the form $\{U_i \times D^l_{(k)} \to U \times D^l_{(k)}\}$ with $\{U_i \to U\}$ a covering family in $CartSp$ as above.

  This is a site of definition for the [[Cahiers topos]].

More discussion of these two examples is at [[∞-Lie groupoid]] and [[∞-Lie algebroid]].

=--

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

Since every [[star-shaped]] region in $\mathbb{R}^n$ is [[diffeomorphic]] to an [[open ball]] (see there for details) we have that the covers $\{U_i \to U\}$ on [[CartSp]] by convex subsets are [[good open covers]] in the strong sense that any finite non-empty intersection is [[diffeomorphic]] to an [[open ball]] and hence diffeomorphic to a [[Cartesian space]]. Therefore these are [[good open cover]]s in the strong sense of the term and their [[Cech nerve]]s $C(U)$ are degreewise coproducts of representables.

The fact that $\lim_\to C(U) \simeq *$ follows from the [[nerve theorem]], using that a [[Cartesian space]] regarded as a [[topological space]] is [[contractible]].

=--

## Properties {#Properties}

### $\infty$-Sheaves on $\infty$-Cohesive sites

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

Let $C$ be an $\infty$-cohesive site. Then the [[(∞,1)-sheaf (∞,1)-topos]] $Sh_{(\infty,1)}(C)$ over $C$ is a [[cohesive (∞,1)-topos]] that satisfies the axiom "discrete objects are concrete" .

If moreover for all objects $U$ of $C$ we have that $C(*,U)$ is [[inhabited set|inhabited]], then the axiom "pieces have points" also holds.

=--

Since the [[(n,1)-topos]] over a site for any $n \in \mathbb{N}$ arises as the full [[sub-(∞,1)-category]] of the $(\infty,1)$-topos on the $n$-[[truncated]] objects and since the definition of cohesive $(n,1)$-topos is compatible with such truncation, it follows that

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

Let $C$ be an $\infty$-cohesive site. Then for all $n \in \mathbb{N}$ the [[(n,1)-topos]] $Sh_{(n,1)}(C)$ is cohesive.

=--

To prove this, we need to show that

1. $Sh_{(\infty,1)}(C)$ is a [[locally ∞-connected (∞,1)-topos]] and a [[∞-connected (∞,1)-topos]].

   This follows with the discussion at [[∞-connected site]].

1. $Sh_{(\infty,1)}(C)$ is a [[local (∞,1)-topos]].

   This follows with the discussion at [[∞-local site]].

1. The [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]] $\Pi : Sh_{(\infty,1)}(C) \to \infty Grpd$ preserves finite [[(∞,1)-products]].

1. If $\Gamma(U)$ is not empty for all $U \in C$, then _pieces have points_ in $Sh_{(\infty,1)}(C)$.

The last two conditions we demonstrate now.

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

The functor $\Pi : Sh_{(\infty,1)}(C) \to \infty Grpd$ whose existence is guaranteed by the above proposition preserves [[product]]s:

$$
  \Pi(A \times B) \simeq \Pi(A) \times \Pi(B)
  \,.
$$

=--

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

By the discussion at [[∞-connected site]] we have that $\Pi$ is given by the [[(∞,1)-colimit]] $\lim_\to : PSh_{(\infty,1)}(C) \to \infty Grpd$. By the assumption that $C$ is a [[cosifted (∞,1)-category]], it follows that this operation preserves finite products.

=--

Finally we prove that _pieces have points_ in $Sh_{(\infty,1)}(C)$ if all objects of $C$ have points.

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

By the above discussion both $\Gamma$ and $\Pi$ are presented by left Quillen functors on the projective model structure $[C^{op}, sSet]_{proj,loc}$. By Dugger's cofibrant replacement theorem (see [[model structure on simplicial presheaves]]) we have for $X$ any simplicial presheaf that a cofibrant replacement is given by an object that in the lowest two degrees is

$$
  \cdots
  \stackrel{\to}{\stackrel{\to}{\to}}
  \coprod_{U_0 \to U_1 \to X_1} U
   \stackrel{\to}{\to}
   \coprod_{U \to X_0} U
  \,,
$$

where the coproduct is over all morphisms out of representable presheaves $U_i$ as indicated.

The model for $\Gamma$ sends this to

$$
  \cdots
  \stackrel{\to}{\stackrel{\to}{\to}}\coprod_{U_0 \to U_1 \to X_0} C(*,U_0)
   \stackrel{\to}{\to}
   \coprod_{U \to X_0} C(*,U)
  \,,
$$

whereas the model for $\Pi$ sends this to

$$
  \cdots
  \stackrel{\to}{\stackrel{\to}{\to}}\coprod_{U_0 \to U_1 \to X_0} *
   \stackrel{\to}{\to}
   \coprod_{U \to X_0} *
  \,.
$$

The morphism from the first to the latter is the evident one that componentwise sends $C(*,U)$ to the point. Since by assumption each $C(*,U)$ is nonempty, this is componentwise an epi. Hence the whole morphism is an epi on $\pi_0$.

=--

### Aufhebung

A [[cohesive (∞,1)-topos]] over an $\infty$-cohesive site satisfies [[Aufhebung]] of the [[unity of opposites|moments]] of [[becoming]]. See at _[[Aufhebung]]_ the section _[Aufhebung of becoming -- Over cohesive sites](http://ncatlab.org/nlab/show/Aufhebung#ExamplesBecomingFormalization)_.

## Related concepts

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

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

  * [[strongly connected topos]] / [[strongly ∞-connected (∞,1)-topos]]

  * [[totally connected topos]] / [[totally ∞-connected (∞,1)-topos]]

* [[local topos]] / [[local (∞,1)-topos]].

* [[cohesive topos]] / [[cohesive (∞,1)-topos]]

and

* [[locally connected site]] / [[locally ∞-connected site]]

  * [[connected site]] / [[∞-connected site]]

  * [[strongly connected site]] / [[strongly ∞-connected site]]

  * [[totally connected site]] / [[totally ∞-connected site]]

* [[local site]] / [[∞-local site]]

* [[cohesive site]], **∞-cohesive site**

[[!redirects ∞-cohesive site]]
[[!redirects ∞-cohesive sites]]

[[!redirects (infinity,1)-cohesive site]]

[[!redirects (∞,1)-cohesive site]]

[[!redirects (infinity,1)-cohesive sites]]


infinity-connected (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

A [[site]] being _locally and globally $\infty$-connected_ means that it satisfies sufficient conditions such that the [[(∞,1)-category of (∞,1)-sheaves]] over it is a [[locally ∞-connected (∞,1)-topos]] and a [[∞-connected (∞,1)-topos]].

## Definition

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

A a [[site]] is **locally and globally $\infty$-connected** over [[∞Grpd]]
if

* it has a [[terminal object]] *

* for every [[covering]] family $\{U_i \to U\}$ in $C$

  1. the [[Cech nerve]] $C(\{U_i\}) \in [C^{op}, sSet]$ is degreewise
       a [[coproduct]] of [[representable functor|representables]];

  1. the [[simplicial set]] obtained by replacing each copy of
     a representable by a point is [[contractible]], equivalently
     the [[colimit]] $\lim_\to : [C^{op}, sSet] \to sSet$
     of $C(\{U_i\})$ has a [[weak homotopy equivalence]] to the point

     $$
       \lim_\to C(\{U\}) \stackrel{\simeq}{\to} *
       \,.
     $$

=--

## Properties

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

The [[(∞,1)-sheaf (∞,1)-topos]] $Sh_{(\infty,1)}(C)$ over locally and globally $\infty$-conneted site $C$, regarded as an [[(∞,1)-site]], is a ([[n-localic (∞,1)-topos|1-localic]]) [[locally ∞-connected (∞,1)-topos]] and [[∞-connected (∞,1)-topos]], in that it comes with a triple of [[adjoint (∞,1)-functor]]s

$$
  (\Pi \dashv \Delta \dashv \Gamma) :
  Sh_{(\infty,1)}(C)
  \stackrel{\stackrel{\Pi}{\to}}{\stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\to}}}
  \infty Grpd
$$

such that $\Pi$ preserves the [[terminal object]].

=--

To prove this, we we use the [[model structure on simplicial presheaves]] to [[locally presentable (∞,1)-category|present]] $Sh_{(\infty,1)}(C)$.

Write $[C^{op}, sSet]_{proj}$ for the projective global model structure and $[C^{op}, sSet]_{proj,loc}$ for its [[Bousfield localization of model categories|left Bousfield localization]] at the set of morphisms $C(\{U_i\}) \to U$ out of the [[Cech nerve]] for each covering family $\{U_i \to U\}$, and $[C^{op}, sSet]_{proj,loc}^\circ$ for the [[Kan complex]]-[[enriched category]] on the fibrant-cofibrant objects. By the discussion at [[model structure on simplicial presheaves]] we have

$$
  Sh_{(\infty,1)}(C) \simeq [C^{op}, sSet]_{proj, loc}^\circ
$$

and the [[adjoint (∞,1)-functor]]s on the left are presented by [[simplicial Quillen adjunction]]s on the right.

To establish these, we proceed by a sequence of lemmas.

+-- {: .un_lemma}
###### Standard fact

The model categories

* standard [[model structure on simplicial sets]] $sSet_{Quillen}$;

*  global [[model structure on simplicial presheaves]]  $[C^{op}, sSet]_{proj,loc}$;

*  local [[model structure on simplicial presheaves]]  $[C^{op}, sSet]_{proj,loc}$

are all  [[left proper model categories]].

=--

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

The first since all objects are cofibrant. The second by general statements about the global [[model structure on functors]], the third because [[Bousfield localization of model categories|left Bousfield localization]] preserves left propernes.

=--

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

For $\{U_i \to U\}$ a [[covering]] family in the $\infty$-connected site $C$, the [[Cech nerve]]
$C(\{U_i\}) \in [C^{op}, sSet]$ is a cofibrant [[resolution]] of $U$ both in the projective model structure $[C^{op}, sSet]_{proj}$ as well as in the Cech local model structure $[C^{op}, sSet]_{proj,loc}$.

=--

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

By assumption on $C$ we have that $C(\{U_i\})$ is a [[split hypercover]]. By Dugger's theorem on cofibrant objects in the projective model structure this implies that $C(U)$ is cofibrant in the global model structure. By general properties of left [[Bousfield localization of model categories|Bousfield localization]] we have that the cofibrations in the local model structure as the same as in the global one. Finally that $C(\{U_i\}) \to U$ is a weak equivalence in the local model structure holds effectively by definition (since we are localizing at these morphisms).

=--

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

On a locally and globally $\infty$-connected site $C$
the [[global section]] [[(∞,1)-geometric morphism]]

$$
  (\Delta \dashv \Gamma) : Sh_{(\infty,1)}(C)
  \stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\to}}
  \infty Grpd
$$

is [[presentable (∞,1)-category|presented]] by the
[[simplicial Quillen adjunction]]

$$
  (Const \dashv \Gamma) :  [C^{op}, sSet]_{proj,loc}
  \stackrel{\leftarrow}{\to}
  sSet_{Quillen}
  \,,
$$

where $\Gamma$ is the functor that evaluates on the point, $\Gamma X = X(*)$, and $Const$ is the functor that sends a simplicial set $S$ to the presheaf constant on that value, $Const S : U \mapsto S$.

=--

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

We use (as described there) that [[adjoint (∞,1)-functor]]s are modeled by [[simplicial Quillen adjunction]]s between the [[simplicial model categories]] that model the $(\infty,1)$-categories in question.

That we have an [[adjunction]] $(Const \dashv \Gamma)$ follows for instance by observing that since $C$ has a [[terminal object]] we may think of $\Gamma$ as being the functor $\Gamma = \lim_\leftarrow$ that takes the [[limit]].

To see that we have a [[Quillen adjunction]] first notice that we have a Quillen adjunction

$$
  (Const \dashv \Gamma) :
  [C^{op}, sSet]_{proj}
  \stackrel{\leftarrow}{\to}
  sSet_{Quillen}
$$

on the global model structure, since $\Gamma$ manifestly preserves fibrations and acyclic fibrations there. Since $[C^{op}, sSet]_{proj,loc}$ is [[proper model category|left proper]] and has the same cofibrations as the global model structure, it follows with [[Higher Topos Theory|HTT, corollary A.3.7.2]] (see the discussion of sSet-Quillen adjunctions) that for
this to descend to a Quillen adjunction on the local model structure it is sufficient that $\Gamma$ preserves fibrant objects. But every fibrant object in the local structure is in particular fibrant in the global structure, hence in particular fibrant over the terminal object of $C$.

The left [[derived functor]] of $Const : sSet_{Quillen} \to [C^{op},sSet]_{proj}$ preserves [[homotopy limit]]s (because [[(∞,1)-limit]]s in an [[(∞,1)-category of (∞,1)-presheaves]] are computed objectwise), and [[∞-stackification]], the left derived functor of $Id : [C^{op}, sSet]_{proj} \to [C^{op}, sSet]_{proj,loc}$ is a left [[exact (∞,1)-functor]], therefore the left derived functor of $Const : sSet_{Quillen} \to [C^{op}, sSet]_{proj,loc}$ preserves finite homotopy limits.

This means that our Quillen adjunction does model a [[(∞,1)-geometric morphism]] $Sh_{(\infty,1)}(C) \to \infty Grpd$. By the discussion at [[global section]] the space of these geometric morphisms to [[∞Grpd]] is [[contractible]], hence this is indeed a representative of the terminal geometric morphism as claimed.

=--

+-- {: .proof}
###### Proof of the theorem

By general abstract facts the [[sSet]]-functor $Const : sSet \to [C^{op}, sSet]$ given on $S \in sSet$ by $Const_S : U \mapsto S$ for all $U \in C$ has an  [[sSet]]-[[left adjoint]]

$$
  \Pi : X \mapsto \int^U X(U) = \lim_\to X
$$

naturally in $X$ and $S$, given by the [[colimit]] operation. Notice that since [[sSet]] is itself a [[category of presheaves]] (on the [[simplex category]]), these colimits are degreewise colimits in [[Set]]. Also notice that the colimit over a [[representable functor]] is the point (by a simple [[Yoneda lemma]]-style argument).

Regarded as a functor  $sSet_{Quillen} \to [C^{op}, sSet]_{proj}$ the functor $Const$ manifestly preserves fibrations and acyclic fibrations and hence

$$
  (\Pi \dashv Const) : [C^{op}, sSet]_{proj}
      \stackrel{\overset{\lim_\to}{\to}}{\underset{Const}{\leftarrow}}
    sSet_{Quillen}
$$

is a [[Quillen adjunction]], in particular $\Pi : [C^{op},sSet]_{proj} \to sSet_{Quillen}$ preserves cofibrations. Since by general properties of left [[Bousfield localization of model categories]] the cofibrations of $[C^{op},sSet]_{proj,loc}$ are the same, also $\Pi : [C^{op}, sSet]_{proj,loc} \to sSet_{Quillen}$ preserves cofibrations.

Since $sSet_{Quillen}$ is a [[left proper model category]] it follows as before with [[Higher Topos Theory|HTT, corollary A.3.7.2]] (see the discussion of sSet-Quillen adjunctions) that for

$$
  (\Pi \dashv Const) : [C^{op}, sSet]_{proj,loc}
   \stackrel{\overset{\lim_\to}{\to}}{\underset{Const}{\leftarrow}}
   sSet_{Quillen}
$$

to be a [[Quillen adjunction]], it suffices to show that $Const$ preserves fibrant objects. That means that constant simplicial presheaves satisfy [[descent]] along [[covering]] families in the $\infty$-cohesive site $C$: for every covering family $\{U_i \to U\}$ in $C$ and every simplicial set $S$ it must be true that

$$
  [C^{op}, sSet](U, Const S) \to [C^{op}, sSet](C(U), Const S)
$$

is a [[homotopy equivalence]] of [[Kan complexes]]. (Here we use that $U$, being a [[representable functor|representable]], is cofibrant, that $C(U)$ is cofibrant by the above lemma and that $Const S$ is fibrant in the projective structure by the assumption that $S$ is fibrant. So the simplicial hom-complexes in the above equaltion really are the correct [[derived hom-space]]s.)

But that this is the case follows by the condition on the $\infty$-cohesive site $C$ by which $\lim_\to C(U) \simeq *$: using this it follows that

$$
  [C^{op}, sSet](C(U), Const S) = sSet(\lim_\to C(U), S) \simeq sSet(*, S) = S
  \,.
$$

So we have established that also

$$
  (\Pi \dashv Const) : [C^{op}, sSet]_{proj,loc} \stackrel{\overset{\lim_\to}{\to}}{\underset{Const}{\leftarrow}}
  sSet_{Quillen}
$$

is a [[Quillen adjunction]].

It is clear that the left [[derived functor]] of $\Pi$ preserves the terminal object: since that is representable by assumption on $C$, it is cofibrant in $[C^{op}, sSet]_{proj,loc}$, hence $\mathbb{L} \lim_\to * = \lim_\to * = *$.

=--

## Examples

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

The sites

* [[CartSp]], [[ThCartSp]].

are locally and globally $\infty$-connected and in fact [[∞-cohesive site|∞-cohesive]].

=--

This implies that [[?LieGrpd]] is a [[cohesive (∞,1)-topos]]. See there for details.

## Related concepts

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

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

  * [[strongly connected topos]] / [[strongly ∞-connected (∞,1)-topos]]

  * [[totally connected topos]] / [[totally ∞-connected (∞,1)-topos]]

* [[local topos]] / [[local (∞,1)-topos]].

* [[cohesive topos]] / [[cohesive (∞,1)-topos]]

and

* [[locally connected site]] / [[locally ∞-connected (∞,1)-site]]

  * [[connected site]] / **∞-connected (∞,1)-site**

  * [[strongly connected site]] / [[strongly ∞-connected site]]

  * [[totally connected site]] / [[totally ∞-connected site]]

* [[local site]] / [[∞-local site]]

* [[cohesive site]], [[∞-cohesive site]]

[[!redirects ∞-connected site]]
[[!redirects infinity-connected site]]

[[!redirects ∞-connected sites]]
[[!redirects infinity-connected sites]]

[[!redirects ∞-connected (∞,1)-site]]
[[!redirects ∞-connected (∞,1)-sites]]


infinity-connected (infinity,1)-topos

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos theory
+--{: .hide}
=--
#### Discrete and concrete objects
+-- {: .hide}
[[!include discrete and concrete objects - contents]]
=--
=--
=--

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

## Idea

If we think of an [[(∞,1)-topos]] as a generalized [[topological space]], then it being **∞-connected** is the analogue of a topological space being (weakly) [[contractible]], i.e. [[weak homotopy equivalence|weak-homotopy equivalent]] to a [[point]].

It is an (∞,1)-[[categorification]] of the notion of a [[topos]] being [[connected topos|connected]].

## Definition

Let $\mathbf{H}$ be a ([[(∞,1)-sheaf (∞,1)-topos|(∞,1)-sheaf]]-)$(\infty,1)$-topos.  It therefore admits a unique [[geometric morphism]] $(L\Const\dashv\Gamma)\colon \mathbf{H}\xrightarrow{\Gamma} $ [[∞Grpd]] given by [[global section]]s.  We say that $\mathbf{H}$ is **$\infty$-connected** if $LConst$ is [[fully faithful (∞,1)-functor|fully faithful]].

More generally, we call a [[geometric morphism]] between $(\infty,1)$-toposes **connected** if its [[inverse image functor]] is [[full and faithful (∞,1)-functor|fully faithful]].

## Properties

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

An $\infty$-connected $(\infty,1)$-topos has the _shape_ of the point, in the sense of [[shape of an (∞,1)-topos]].

=--

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

By a basic property of [[adjoint (∞,1)-functor]]s, $LConst$ being a [[full and faithful (∞,1)-functor]]  is equivalent to the unit of $(LConst \dashv \Gamma)$ being an [[equivalence in a quasi-category|equivalence]]

$$
 Id_{\infty Grpd} \stackrel{\simeq}{\to} \Gamma LConst
 \,.
$$

By definition of [[shape of an (∞,1)-topos]] this means that $\mathbf{H}$ has the same shape as [[∞Grpd]], which is to say that it shape is [[representable functor|represented]], as a functor $\infty Grpd \to \infty Grpd$, by the terminal object $*$.  Hence it has the "shape of the point".

=--

## Locally ∞-connected and ∞-connected

As in the case of [[connected topos|connected 1-topoi]], we have the following.

+--{: .un_prop}
###### Proposition
If an $(\infty,1)$-topos $\mathbf{H}$ is [[locally ∞-connected (∞,1)-topos|locally ∞-connected]] (i.e. $LConst$ has a left adjoint $\Pi$), then $\mathbf{H}$ is connected if and only if $\Pi$ preserves the terminal object.
=--
+--{: .proof}
###### Proof
This is just like the 1-categorical proof.  On the one hand, if $\mathbf{H}$ is ∞-connected, so that $LConst$ is fully faithful, then by properties of [[adjoint (∞,1)-functors]] this implies that the counit $\Pi \circ LConst \to \Id$ is an equivalence.  But $LConst$ preserves the terminal object, since it is [[left exact]], so $\Pi(*) \simeq \Pi(LConst(*)) \simeq *$.

Conversely, suppose $\Pi(*)\simeq *$.  Then any $\infty$-groupoid $A$ can be written as $A = \colim^A *$, the [[(∞,1)-colimit]] over $A$ itself of the constant diagram at the terminal object (see the details here).  Since $LConst$ and $\Pi$ are both left adjoints, both preserve colimits, so we have
$$
\Pi(LConst(A)) \simeq \Pi(LConst(\colim^A *)) \simeq \colim^A \Pi(LConst(*)) \simeq \colim^A * \simeq A.
$$
Therefore, the counit $\Pi \circ LConst \to \Id$ is an equivalence, so $LConst$ is fully faithful, and $\mathbf{H}$ is ∞-connected.
=--

## Related concepts

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

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

* [[local topos]] / [[local (∞,1)-topos]].

* [[cohesive topos]] / [[cohesive (∞,1)-topos]]

[[!redirects ∞-connected (∞,1)-topos]]
[[!redirects ∞-connected (∞,1)-toposes]]
[[!redirects ∞-connected (∞,1)-topoi]]
[[!redirects ∞-connected (∞,1)-geometric morphism]]
[[!redirects ∞-connected (∞,1)-geometric morphisms]]

[[!redirects infinity-connected (infinity,1)-toposes]]
[[!redirects infinity-connected (infinity,1)-topoi]]

[[!redirects globally ∞-connected (∞,1)-topos]]
[[!redirects globally ∞-connected (∞,1)-toposes]]
[[!redirects globally ∞-connected (∞,1)-topoi]]
[[!redirects globally ∞-connected (∞,1)-geometric morphism]]
[[!redirects globally ∞-connected (∞,1)-geometric morphisms]]

[[!redirects globally infinity-connected (∞,1)-topos]]
[[!redirects globally infinity-connected (∞,1)-toposes]]
[[!redirects globally infinity-connected (∞,1)-topoi]]
[[!redirects globally infinity-connected (∞,1)-geometric morphism]]
[[!redirects globally infinity-connected (∞,1)-geometric morphisms]]


infinity-cosmos

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

#Contents#
* table of contents
{:toc}

## Idea

An $\infty$-**cosmos** is a "good place in which to do [[higher category theory]],"  for [[1-categories]], [[(∞,1)-categories]] , [[(∞,n)-categories]], or [[fibered category]]-versions of any of the above.

The word is chosen by analogy with the notion of _[[cosmos]]_ in [[enriched category theory]], which is similarly "a good place to do (ordinary) category theory." The notion is more similar to Street's "fibrational cosmoi" than to Bénabou's cosmoi.

## A source of examples

Roughly speaking, an $\infty$-**cosmos** is a simplicially enriched category of fibrations and fibrant objects.

Let $\mathcal{M}$ be a [[model category]] that is enriched over the [[Joyal model structure]] on simplicial sets. Then the full subcategory of fibrant objects defines an **$\infty$-cosmos**. If, as is frequently the case in practice, every fibrant object is cofibrant, then this subcategory defines an **$\infty$-cosmos with all objects cofibrant,** which admits a simpler axiomatization.

## Definition

For simplicity, we define only an $\infty$-cosmos with all objects cofibrant. See the references below for the general definition.

An **$\infty$-cosmos** (with all objects cofibrant) is a simplicially enriched category $\mathcal{K}$ whose homs $Fun(A,B)$ are quasi-categories equipped with a specified class of [[isofibration|isofibrations]] so that

1. As a simplicially enriched category, $\mathcal{K}$ admits products, cotensors with simplicial sets, pullbacks of isofibrations, splittings of idempotents, and limits of towers of isofibrations.

2. The class of isofibrations is closed under product, pullback, retract, limits of towers, and Leibniz cotensors with monomorphisms of simplicial sets. Furthermore, if $A \twoheadrightarrow B$ is a fibration in $\mathcal{K}$, then for any $X$, $\Fun(X,A) \twoheadrightarrow \Fun(X,B)$ is an isofibration of quasi-categories.

A map $A \to B$ is an **equivalence** if for any $X$, $\Fun(X,A) \rightarrow \Fun(X,B)$ is an equivalence of quasi-categories, and a **trivial fibration** if it is both an isofibration and an equivalence.

3. For any trivial fibration $p\colon E \twoheadrightarrow B$ and any map $f \colon A \to B$ there exists a lift of $f$ along $p$.

## The prototypical example

The category of quasi-categories and isofibrations, inner fibrations that lift also against the inclusion of either endpoint into the nerve of the walking isomorphism, defines an $\infty$-cosmos.

## Related concepts

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

## References

For an overview:

* {#RiehlVerity16} [[Emily Riehl]], [[Dominic Verity]], _$\infty$-Category theory from scratch_, ([arXiv:1608.05314](https://arxiv.org/abs/1608.05314))

For the most general notion of $\infty$-cosmos and constructions of many examples:

* {#RiehlVerity15} [[Emily Riehl]], [[Dominic Verity]], _Fibrations and Yoneda's lemma in an $\infty$-comos_ ([arXiv:1506.05500](https://arxiv.org/abs/1506.05500))

For a simplified notion with "all objects cofibrant"

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

Textbook account:

* {#RiehlVerity22} [[Emily Riehl]], [[Dominic Verity]], _[[Elements of ∞-Category Theory]]_, Cambridge studies in advanced mathematics **194**, Cambridge University Press (2022) $[$[doi:10.1017/9781108936880](https://doi.org/10.1017/9781108936880), ISBN:978-1-108-83798-9, [pdf](https://emilyriehl.github.io/files/elements.pdf)$]$

For an exploration of atypical examples of ∞-cosmoi whose objects are 2-categories or bicategories

* [[Emily Riehl]], Mira Wattal, _On ∞-cosmoi of bicategories_ ([arXiv:2108.11786](https://arxiv.org/abs/2108.11786))

[[!redirects infinity-cosmoi]]

[[!redirects ∞-cosmos]]
[[!redirects ∞-cosmoi]]


infinity-CS theory for binary non-degenerate invariant polynomial - table

**[[schreiber:∞-Chern-Simons theory]] from binary and non-degenerate [[invariant polynomial]]**

| $n \in \mathbb{N}$ | [[symplectic Lie n-algebroid]] |  [[Lie integration|Lie integrated]] [[smooth ∞-groupoid]] = [[moduli ∞-stack]] of [[field (physics)|fields]] of $(n+1)$-d [[sigma-model]] | [[higher symplectic geometry]] |  $(n+1)$d [[sigma-model]] | [[dg-manifold|dg-]][[Lagrangian submanifold]]/ [[real polarization]] [[leaf]] | =  [[brane]]  | [[n-module|(n+1)-module]] of [[quantum states]] in [[codimension]] $(n+1)$ | discussed in: |
|--|--|--|--|--|--|--|--|--|
| 0 | [[symplectic manifold]] | [[symplectic manifold]] | [[symplectic geometry]] |   | [[Lagrangian submanifold]] | -- | ordinary [[space of states (in geometric quantization)]] | _[[geometric quantization]]_ |
| 1 | [[Poisson Lie algebroid]] | [[symplectic groupoid]] | [[2-plectic geometry]]  | [[Poisson sigma-model]] |  [[coisotropic submanifold]] (of underlying [[Poisson manifold]]) |  [brane of Poisson sigma-model](Poisson+sigma-model#Branes)  | [[2-module]] = [[category of modules]] over [[strict deformation quantization|strict deformation quantiized]] [[algebra of observables]] | _[[extended geometric quantization of 2d Chern-Simons theory]]_ |
| 2 | [[Courant Lie 2-algebroid]] | [[symplectic 2-groupoid]] | [[3-plectic geometry]] |[[Courant sigma-model]]  | [[Dirac structure]]  | [[D-brane]] in [[type II geometry]] |  |  |
| $n$ | [[symplectic Lie n-algebroid]] |  [[symplectic infinity-groupoid|symplectic n-groupoid]] | [[n-plectic geometry|(n+1)-plectic geometry]] | $d = n+1$ [[AKSZ sigma-model]] |   |  |  |  |

(adapted from [[Some title containing the words "homotopy" and "symplectic", e.g. this one|Ševera 00]])


infinity-Dijkgraaf-Witten theory

#Contents#
* table of contents
{:toc}

## Idea

[[schreiber:infinity-Chern-Simons theory]] for [[discrete infinity-group]].

See at _[[local prequantum field theory]]_ for more.

## Examples

* [[Dijkgraaf-Witten theory]]

* [[Yetter model]]

## References

A sketch of a formalization is in section 3, section 8 of

* FHLT, _[[Topological Quantum Field Theories from Compact Lie Groups]]_

[[!redirects ∞-Dijkgraaf-Witten theory]]

[[!redirects ∞-Dijkgraaf-Witten theories]]
[[!redirects infinity-Dijkgraaf-Witten theories]]

[[!redirects n-dimensional Dijkgraaf-Witten theory]]


infinity-Dold-Kan correspondence

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

#Contents#
* table of contents
{:toc}

## Statement

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

Let $\mathcal{C}$ be a [[stable (∞,1)-category]]. Then the [[(∞,1)-categories]] of non-negatively graded [[sequential diagram|sequences]] in $C$ is [[equivalence of (∞,1)-categories|equivalent]] to the [[(∞,1)-category]] of [[simplicial objects in an (∞,1)-category]] in $\mathcal{C}$

$$
  Fun(N(\mathbb{Z}_{\geq 0}), C)
  \simeq
  Fun(N(\Delta)^{op}, C)
  \,.
$$

Under this equivalence, a [[simplicial object]] $X_\bullet$ is sent to the [[sequential diagram|sequence]] of [[geometric realizations]] ([[(∞,1)-colimits]]) of its [[simplicial skeleta]]

$$
  {\vert sk_0 X_\bullet \vert} \to {\vert sk_1 X_\bullet \vert} \to {\vert sk_2 X_\bullet \vert} \to \cdots
  \,.
$$

This constitutes a [[filtered object in an (infinity,1)-category|filtering]] on the [[geometric realization]] of $X_\bullet$ itself

$$
  {\vert X_\bullet \vert}
  \simeq
  \underset{\longrightarrow}{\lim}_n {\vert sk_n X_\bullet \vert}
  \,.
$$

=--

([[Higher Algebra|Higher Algebra, theorem 1.2.4.1]])

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

Given a [[simplicial objects in an (∞,1)-category|simplicial object]] $X_\bullet$ in a [[stable (∞,1)-category]] $\mathcal{C}$, its image in the [[triangulated category|triangulated]] [[homotopy category of an (∞,1)-category|homotopy category]] $Ho(\mathcal{C})$ is identified by the
ordinary [[Dold-Kan correspondence]] with a [[chain complex]]. On the other hand, by the discussion at [[spectral sequence of a filtered stable homotopy type]] in the section _[Filtered objects and their chain complexes](spectral+sequence+of+a+filtered+stable+homotopy+type#FilteredObjectsAndTheirChainComplexes)_, the skelton sequence $sk_\bullet X_\bullet$ also induces a chain complex. These are [[natural isomorphism|naturally isomorphic]].

In particular therefore first page of the _[[spectral sequence of a filtered stable homotopy type]]_ associated with the [[simplicial skeleton]] filtration consists of the [[Moore complexes]] of the [[simplicial objects]] $\pi_q(X_\bullet) \in \mathcal{A}^{\Delta^{op}}$

$$
  E_1^{\bullet,q} \simeq N(\pi_q X_\bullet)
  \,.
$$

=--

([[Higher Algebra|Higher Algebra, remark 1.2.4.3, 1.2.4.4]])

## Applications

* [[J-homomorphism and chromatic homotopy]]

## Related concepts

* [[spectral sequence of a simplicial stable homotopy type]]

* [[stable Dold-Kan correspondence]]

## References

  This _infinity-Dold-Kan correspondence_ is [theorem 12.8, p. 50](http://www.math.harvard.edu/~lurie/topoibook/DAGI.pdf) of

* [[Jacob Lurie]], _[[Stable ∞-Categories]]_

later absorbed in

* [[Jacob Lurie]], section 1.2.4 of _[[Higher Algebra]]_

[[!redirects ∞-Dold-Kan correspondence]]


infinity-field

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

#Contents#
* table of contents
{:toc}

## Idea

The notion of _$A_\infty$-field_ is to the notion of [[A-∞ algebra]] as that of _[[field]]_ is to [[associative algebra]]/[[ring]].

## Definition

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

An [[A-∞ ring]] or in fact just an [[H-space]] $A$ is a **field** if $\pi_\bullet A$ is a [[graded field]].

=--

For instance ([Lurie, lecture 24, def. 3](#LurieLecture)).

## Properties

For $E$ an $\infty$-field, def. \ref{InfinityField}, then it carries the structure of an [[∞-module]] over the $n$th [[Morava K-theory]] spectrum $K(n)$, for some $n$.

([Lurie 10, lect 25, cor 10](#LurieLect25))

This follows with the [[nilpotence theorem]].

## Examples

* For $k$ an ordinary [[field]], the [[Eilenberg-Mac Lane spectrum]] of $k$, $H k$, is an $A_\infty$-field.
* The [[Morava K-theory]] [[A-∞ rings]] $K(n)$ are the basic $A_\infty$-fields. See at _[Morava K-theory  -- As infinity-Fields](Morava+K-theory#AsInfinityFields)_, where $K(0) \simeq H \mathbb{Q}$ and we define $K(\infty)$ as $H \mathbb{F}_p$.

## Related concepts

* [[chromatic homotopy theory]]

## References

Definition 3 in

* [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, Lecture series 2010, Lecture 24 _Uniqueness of Morava K-theory_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture24.pdf))
 {#LurieLecture}

* [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, Lecture series 2010,
Lecture 25 _The Nilpotence lemma_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture25.pdf))
 {#LurieLect25}

[[!redirects infinity-fields]]

[[!redirects ∞-field]]
[[!redirects ∞-fields]]


infinity-gerbe

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

#Contents#
* table of contents
{: toc}

## Idea

In the language of [[(∞,1)-topos theory]] the ordinary definition of [[gerbe]] has the following simple re-formulation:

In a given [[(2,1)-topos]], a _[[gerbe]]_ is an object which is

1. [[1-connective]];

1. [[1-truncated]].

There are various evident generalizations of this, where one allows the degree in either of the two clauses to vary. In the literature one finds higher gerbes defined in either way, and so there are more general and more specific definitions.

## Definition

### General

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

Let $\mathcal{X}$ be an [[(∞,1)-topos]].

For $n \in \overline{\mathbb{N}}$ an [[extended natural number]],

* a **$n$-gerbe** is an object $\mathcal{G} \in \mathbf{G}$ that is

  1. [[1-connective]];

  1. [[n-truncated]].

* an **EM $n$-gerbe** is an an $n$-gerbe that is

  1. [[n-connective]]

  1. [[n-truncated]].

=--

The definition of "EM $n$-gerbes" appears as [[Higher Topos Theory|HTT, def. 7.2.2.20]]. For $n = 2$ the definition of "general $n$-gerbe" (called a _nonabelian 2-gerbe_ at [[2-gerbe]]) appears for instance in ([Breen](#Breen)).

In particular we have then the following.

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

An **$\infty$-gerbe** in $\mathcal{X}$ is a [[1-connective|connected]] object.

Write

$$
  \infty Gerbe \subset \mathcal{X}
$$

for the [[core]] of the full-[[sub-(∞,1)-category]] on the $\infty$-gerbes.

=--

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

The notion of [[morphisms]] between EM $n$-gerbes in [[Higher Topos Theory|HTT, pages 576, 577]] is more restrictive than this. This is the reason why the classification found there is a restriction of the general classification. This is discussed [below](#Properties).

=--

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

If one adds to the definition of "EM $n$-gerbe" the condition that $\mathcal{G}$ has a "global section", a [[global element]] $* \to \mathcal{G}$, then it becomes the definition of an [[Eilenberg-MacLane object]] in degree $n$. In other words, EM $n$-gerbes are just like Eilenberg-MacLane objects but without necessarily a global section.

By the main theorem at [[looping and delooping]] the connected _and_ [[pointed object|pointed]] objects in an [[(∞,1)-topos]] $\mathcal{X}$ are equivalently (the [[delooping]]s of) [[∞-group]] objects: we have an [[equivalence of (∞,1)-categories]]

$$
  (\Omega \dashv \mathbf{B})
  :
  \infty Grp(\mathcal{X})
   \stackrel{\overset{\Omega}{\leftarrow}}{\underset{\mathbf{B}}{\to}}
  \mathcal{X}_{*, \geq 1}
  \,.
$$

Therefore an $\infty$-gerbe is much like the [[delooping]] of an $\infty$-group, only that a global section may be missing.

But notice that a section always exists locally: for

$$
  (x^* \dashv x_*) : \infty Grpd
   \stackrel{\overset{x^*}{\leftarrow}}{\underset{x_*}{\to}}
  \mathcal{X}
$$

any [[point of a topos|topos point]] and $P \in \mathcal{X}$ an $\infty$-gerbe, the [[stalk]] $x^* P \in $ [[∞Grpd]] is [[connected]], because the [[inverse image]] $x^*$ preserves the [[finite limit]]s that define [[categorical homotopy groups in an (∞,1)-topos]] and preserves the [[terminal object]] $*$:

$$
  x^* P \simeq B G_x
  \,.
$$

Therefore it is natural to consider the notion of an _$G$-$\infty$-gerbe_ for a fixed $G \in \infty Grp(\mathcal{X})$. This is done [below](#GInfinityGerbes).

=--

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

A "restricted" $n$-gerbe $E$ has, by definition, a single non-trivial [[categorical homotopy group in an (infinity,1)-topos|homotopy sheaf]]. For $n \geq 2$ this is an sheaf of abelian groups $A$ in the underlying [[topos]]

$$
  \pi_n E \simeq A
  \,.
$$

This $A$ is called the **band** of $E$ and that $E$ is **banded by** $A$.

=--

In the case that $n = 1$ or that we have a "general" $n$-gerbe the notion of band is refined by [[nonabelian cohomology]]-information. See [below].

### $G$-$\infty$-Gerbes
 {#GInfinityGerbes}

Let $\mathcal{X}$ be an [[(∞,1)-topos]].

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

For $G \in \infty Grp(\mathcal{X})$ an [[∞-group]], a **$G$-$\infty$-gerbe** $P$ is

* an $\infty$-gerbe, def. \ref{InfinityGerbe};

* such that there exists an [[effective epimorphism in an (∞,1)-category|effective epimorphism]] $U \to *$ in $\mathcal{X}$ and an [[equivalence in an (∞,1)-category|equivalence]]

  $$
    P|_U \simeq \mathbf{B}G|_U
    \,.
  $$

=--

## Properties
 {#Properties}

### Classification of EM $n$-gerbes.

Let $A \in Grp(\mathcal{X}) \subset \infty Grpd(\mathcal{X})$ be an [[abelian group]] object and fix $n \in \mathbb{N}$, $n \geq 2$.

Recall the notion of $A$-banded $n$-gerbes from def. \ref{AbelianBand}.

Write

$$
  H^{n+1}_{\mathcal{X}}(X, A)
   :=
  \pi_0 \mathcal{X}(*, \mathbf{B}^{n+1}A)
$$

for the [[cohomology]] in $\mathcal{X}$ of the [[terminal object]] with coefficients in $A$ in degree $n+1$.

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

There is a canonical bijection

$$
  \pi_0 EM n Gerbe_A
   \simeq
  H^{n+1}_{\mathcal{X}}(X, A)
  \,,
$$
where $EM n Gerbe_A$ is the category of all $EM$ $n$-gerbes banded over $A$.
=--

This appears as [[Higher Topos Theory|HTT, cor. 7.2.2.27]].

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

The morphisms of EM $n$-gerbes in [[Higher Topos Theory|HTT, p. 576-577]] are more restrictive than the morphisms of these objects regarded as general $n$-gerbes.

By the discussion below, if EM $n$-gerbes are regarded as special $n$-gerbes, then their classification is by a [[nonabelian cohomology]]-extension of the above result.

=--

### Classification of $G$-$\infty$-gerbes

We discuss partial generalizations of the above result to nonabelian $\infty$-gerbes

Comparing with the discussion at [[associated ∞-bundle]] one finds that def. \ref{GInfinityGerbe} of $G$-$\infty$-gerbes defines "locally trivial $\mathbf{B}G$-fibrations". By the main theorem there, these are classified by [[cohomology]] in $\mathcal{X}$ with coefficients in the [[internalization|internal]] [[automorphism ∞-group]]

$$
  AUT(G) := \underline{Aut}(\mathbf{B}G)
$$

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

At least for $\mathcal{X}$ a [[n-localic (∞,1)-topos|1-localic]] [[(∞,1)-topos]] we have a canonical bijection

$$
  \pi_0 (G Gerbe)
  \simeq
  H^1(\mathcal{X}, AUT(G))
  \,.
$$

=--

This follows as a special case of the result by ([Wendt](#Wendt)) discussed at [[associated infinity-bundle]].

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

The right hand side classifies als $AUT(G)$-[[principal ∞-bundles]] and this equivalence identifies $G$-$\infty$-gerbes as the canonical $AUT(G)$-[[associated ∞-bundles]].

=--

(Compare to the analogous discussion in the special case of [[gerbe]]s.)

### Nonabelian banded $\infty$-gerbes

For $G \in \infty Grpd(\mathcal{X})$ an $n$-group (with $n \geq 1$) write

$$
  Out(G) := \tau_{n-1} AUT(G) = \tau_{n-1}\underline{Aut}(\mathbf{B}G)
  \,.
$$

Call this the [[outer automorphism infinity-group]] of $G$. By definition there is a canonical morphism

$$
  \mathbf{B} AUT(G) \to \mathbf{B} Out(G)
  \,.
$$

By the above classification, this induces a morphism

$$
  Band : \pi_0 G Gerbe \to H^1_{\mathcal{X}}(X, Out(G))
$$

from $G$-$n$-gerbes to [[nonabelian cohomology]] in degree 1 with coefficients in $Out(G)$. For $E \in G Gerbe$ the pair

$$
  (\pi_n, Band(E))
$$

is called the **band** of $E$. For $[K] \in H^1_{\mathcal{X}}(X, Out(G))$ the [[n-groupoid|(n+1)-groupoid]] $G Gerbe_K$ of $K$-banded $G$-$n$-gerbes is the [[homotopy pullback]]

$$
  \array{
    G Gerbe_K &\to& *
    \\
    \downarrow && \downarrow^{\mathrlap{K}}
    \\
    \mathcal{X}(X, \mathbf{B}AUT(G))
    &\to&
    \mathcal{X}(X, \mathbf{B}Out(G))
  }
  \,.
$$

Write $\mathbf{B}^2 Z(G)$ for the [[homotopy fiber]] of $\mathbf{B}AUT(G) \to \mathbf{B}Out(G)$, producing a [[fiber sequence]]

$$
  \mathbf{B}^2 Z(G) \to \mathbf{B} AUT(G) \to \mathbf{B}Out(G)
  \,.
$$

We call $Z(G)$ the [[center of an infinity-group|center of the infinity-group]].

In terms of this the above $G Gerbe_K$ is the  [[cocycle]] $(n+1)$-groupoid of the [[twisted cohomology|K-twisted Z(G)-cohomology]] in degree 2:

$$
  \pi_0 G Gerbe_K \simeq H^2_K(X,Z(G))
  \,.
$$

Notice that if $Z(G)$ itself is [[n-connected|higher connected]] then $H^2_K(X, Z(G))$ is accordingly cohomology in higher degree.

## Examples

### Classification of abelian 2-gerbes
  {#ClassificationOf2Gerbes}

For $X$ a [[topological space]], let $\mathcal{X} = Sh_{(\infty,1)}(X)$ be its [[(∞,1)-category of (∞,1)-sheaves]]. Write

$$
  U(1) \in Grp(\mathcal{X}) \subset \infty Grp(\mathcal{X})
$$

for the [[sheaf]] of [[circle group]]-valued functions. And $\mathbf{B}U(1)$ for its [[delooping]]

Then

$$
  AUT(\mathbf{B}U(1))
  :=
  \underline{Aut}(\mathbf{B}^2 U(1))
  \simeq
  [U(1) \stackrel{0}{\to} U(1) \stackrel{0}{\to} \mathbb{Z}_2]
  \,,
$$

where on the right we display the [[crossed complex]] corresponding to this 3-group (the morphisms are all constant on the unit element, the action of $\mathbb{Z}_2$ on either of the $U(1)$s is the canonical one given by $\mathb{Z}_2 \simeq Aut(U(1))$ ). This is seen as follows: every invertible [[2-functor]] $F : \mathbf{B}^2 U(1) \to \mathbf{B}^2 U(1)$ comes from a group auotmorphism of $U(1)$, of which there are $\mathbb{Z}_2$. A [[pseudonatural transformation]] necessarily goes from any such $F$ to itself, and there are $U(1)$ of them Similarly, any [[modification]] of these necessarily is an endo, and there are also $U(1)$ of them.

Therefore $\mathbf{B}U(1)$-2-gerbes are classified by the [[nonabelian cohomology]]

$$
  H^1(X,[U(1) \to 1 \to \mathbb{Z}_2])
  \,.
$$

This has a subgroup coming from the canonical inclusion

$$
  \mathbf{B}^3 U(1) = [U(1)\to 1 \to 1] \hookrightarrow [U(1) \to 1 \to \mathbb{Z}_2]
$$

on the trivial automorphism of $U(1)$. The classification of $U(1)$ EM-2-gerbes in [[Higher Topos Theory|HTT]] gives only this subgroup

$$
  H^3(X, U(1))
    \hookrightarrow
  H^1(X, [U(1) \to 1 \to \mathbb{Z}_2])
  \,.
$$

In the terminology of [[orientifold]]/[[Jandl gerbe]]s the more general objects on the right are the "Jandl 2-gerbes" or "orientifold 2-gerbes".

## Related entries

* [[principal bundle]] / [[torsor]] / [[associated bundle]]

* [[principal 2-bundle]] / [[gerbe]] / [[bundle gerbe]]

* [[principal 3-bundle]] / [[2-gerbe]] / [[bundle 2-gerbe]]

* [[principal ∞-bundle]] / [[associated ∞-bundle]]  / **$\infty$-gerbe**.

  * [[Super Gerbes]]

Notice that the notion of [[bundle gerbe]] and [[bundle 2-gerbe]] etc. is not unrelated, but is _a priori_ a rather different notion.

## References

The notion of "EM" $\infty$-gerbes in an $(\infty,1)$-topos appears in section 7.2.2 of

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

The general notion (but without a concept of ambient $(3,1)$-topos made explicit) for $n = 2$ (see [[2-gerbe]]) appears for instance in

* [[Lawrence Breen]], _On the classification of 2-gerbes and 2-stacks_ , Astérisque 225 (1994).

* [[Lawrence Breen]], _Notes on 1- and 2-gerbes_ in [[John Baez]], [[Peter May]] (eds.) _[[Towards Higher Categories]]_ ([arXiv:math/0611317](http://arxiv.org/abs/math/0611317)).
  {#Breen}

The general notion for arbitrary $n$ in an $(\infty,1)$-topos context is discussed in section 2.3 of

* [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_ .

Discussion of the classification of $G$-$\infty$-gerbes and more general fiber bundles in 1-localic $(\infty,1)$-toposes  is in

* [[Matthias Wendt]], _Classifying spaces and fibrations of simplicial sheaves_ , Journal of  Homotopy and Related Structures 6(1), 2011, pp. 1--38.  ([arXiv](http://arxiv.org/abs/1009.2930)) ([published version](http://tcms.org.ge/Journals/JHRS/volumes/2011/volume6-1.htm))
{#Wendt}

[[!redirects ∞-gerbe]]
[[!redirects ∞-gerbes]]
[[!redirects infinity-gerbe]]
[[!redirects infinity-gerbes]]

[[!redirects n-gerbe]]
[[!redirects n-gerbes]]


infinity-group

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

#Contents#
* table of contents
{:toc}

## Definition

An **∞-group** is a [[groupoid object in an (∞,1)-category|group object]] in [[∞Grpd]].

Equivalently (by the [[delooping hypothesis]]) it is a [[pointed object|pointed]] [[connected]] $\infty$-[[infinity-groupoid|groupoid]].

Under the [[homotopy hypothesis|identification]] of [[∞Grpd]] with [[Top]] this is known as a grouplike $A_\infty$-[[A-infinity-space|space]], for instance.

An **$\infty$-Lie group** is accordingly a group object in [[∞-Lie groupoid]]s. And so on.

## Properties

For details see [[groupoid object in an (∞,1)-category]].

## Models

By

* [[model structure on simplicial groups]]

* [[model structure on reduced simplicial sets]]

## Related concepts and Examples

* [[group]]

* [[2-group]], [[braided 2-group]]

* [[n-group]], [[k-tuply groupal n-groupoid]]

* [[pointed connected groupoid]]

* **$\infty$-group**, [[braided ∞-group]]

  * [[looping and delooping]]

  * [[automorphism ∞-group]]

    * [[∞-group of bisections]]

  * [[∞-group of units]]

    * [[Picard ∞-group]]

      * [[Brauer ∞-group]]

  * [[p-compact group]]

  * [[center of an ∞-group]]

  * [[normal morphism of ∞-groups]]

  * [[stabilizer ∞-group]]

  * [[outer automorphism ∞-group]]

  * [[Eilenberg-MacLane object]], [[∞-gerbe]]

  * [[∞-group cohomology]]

  * [[∞-group extension]]

  * [[augmented ∞-group]]

  * [[∞-group completion]]

  * [[finite ∞-group]]

* [[free infinity-group type]]

* [[∞-action]]

[[!include k-monoidal table]]

## References
 {#References}

(For more see also the references at _[[infinity-action]]_.)

A standard textbook reference on $\infty$-groups in the [[classical model structure on simplicial sets]] is

* {#GoerssJardine} [[Paul Goerss]], [[Rick Jardine]], chapter V of _[[Simplicial homotopy theory]]_ [chapter V](http://www.maths.abdn.ac.uk/~bensondj/papers/g/goerss-jardine/ch-5.dvi).

[[groupoid object in an (infinity,1)-category|Group objects in (infinity,1)-categories]] are the topic of

* {#Lurie} [[Jacob Lurie]], section 6.1.2 in  _[[Higher Topos Theory]]_

Model category presentations of group(oid) objects in $\infty Grpd$ by groupoidal [[complete Segal spaces]] are discussed in

* [[Julia Bergner]],

  _Adding inverses to diagrams encoding algebraic structures_, Homology, Homotopy and Applications 10 (2008), no. 2, 149–174. ([arXiv:0610291](http://arxiv.org/abs/math/0610291))

  _Adding inverses to diagrams II: Invertible homotopy theories are spaces_, Homology, Homotopy and Applications, Vol. 10 (2008), No. 2, pp.175-193. ([web](http://www.intlpress.com/hha/v10/n2/a9/), [arXiv:0710.2254](http://arxiv.org/abs/0710.2254))

Discussion from the point of view of [[category objects in an (∞,1)-category]] is in

* {#Lurie2} [[Jacob Lurie]], _[[(∞,2)-Categories and the Goodwillie Calculus]]_ ([arXiv:0905.0462](http://arxiv.org/abs/0905.0462))

The [[homotopy theory]] of $\infty$-groups that are [[n-connected]] and [[n-truncated|r-truncated]] for $n \leq r$ is discussed in

* [[A.R. Garzón]], J.G. Miranda, _Serre homotopy theory in subcategories of simplicial groups_, Journal of Pure and Applied Algebra Volume 147, Issue 2, 24 March 2000, Pages 107-123 (doi:10.1016/S0022-4049(98)00143-1)

Discussion of aspects of ordinary [[group theory]] in relation to $\infty$-group theory:

* [[Roman Mikhailov]], *Homotopy patterns in group theory*,  Proceedings of the [ICM 2022](https://icm2022.org) ([arXiv:2111.00737](https://arxiv.org/abs/2111.00737))

Discussion of [[infinity-groups|$\infty$-groups]] in [[homotopy type theory]]:

* {#BuchholtzDoornRijke18} [[Ulrik Buchholtz]], [[Floris van Doorn]], [[Egbert Rijke]], *[[Higher Groups in Homotopy Type Theory]]*, LICS '18: Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science (2018) 205-214 [[arXiv:1802.04315](https://arxiv.org/abs/1802.04315), [doi:10.1145/3209108.3209150](https://doi.org/10.1145/3209108.3209150)]

[[!redirects infinity-groups]]

[[!redirects infinity-group]]
[[!redirects infinity-groups]]

[[!redirects ∞-group]]
[[!redirects ∞-groups]]

[[!redirects pointed connected type]]
[[!redirects pointed connected types]]

category:∞-groupoid


infinity-group extension

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of _$\infty$-group extension_ generalizes the notion of [[group extension]] to [[homotopy theory]]/[[(∞,1)-category theory]] and from [[groups]] to [[∞-groups]]. It is also a generalization to [[nonabelian cohomology]] of the shifted group extensions that are classified by [[Ext]]-groups.

Under forming [[loop space objects]], $\infty$-group extensions are the special case of [[principal ∞-bundles]] whose base space is the [[moduli ∞-stack]] of the group being extended.

## Definition

Let $\mathcal{H}$ an [[(∞,1)-topos]] and $G, A, \hat G \in Grp(\mathbf{H})$ be [[∞-groups]] with [[deloopings]] $\mathbf{B}G$, $\mathbf{B}A$ and $\mathbf{B}\hat G$, respectively.

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

An **extension** $\hat G$ of $G$ by $A$ is a [[fiber sequence]] of the form

$$
  \mathbf{B}A \stackrel{i}{\to} \mathbf{B}\hat G \stackrel{p}{\to} \mathbf{B}G
  \,.
$$

=--

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

Equivalently this says that $A \to \hat G$ is a [normal morphism of ∞-groups](normal%20subgroup#NormalMorphismOfInfinityGroups) and that $G \simeq \hat G \to G$ is its quotient.

=--

Let moreover $A$ be a [[braided ∞-group]], with second [[delooping]] $\mathbf{B}^2 A \in \mathbf{H}$.

+-- {: .num_defn #E2Extensions}
###### Definition

A **[[higher central extension]]** $\hat G$ of $G$ by $A$ is a [[fiber sequence]] in $\mathbf{H}$ of the form

$$
  \array{
     \mathbf{B} A
     &
     \stackrel{\Omega \mathbf{c}}{\to}
     &
     \mathbf{B}\hat G
     \\
     && \downarrow
     \\
     && \mathbf{B}G
     &\stackrel{\mathbf{c}}{\to}&
     \mathbf{B}^2 A
     \,.
  }
  \,.
$$

=--

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

Def. \ref{E2Extensions} equivalently says that

* $\mathbf{B}\hat G \to \mathbf{B}G$ is an $\mathbf{B} A$-[[principal ∞-bundle]] over $\mathbf{B}G$;

* the extension is classified by the [[group cohomology]] class

  $$
    [\mathbf{c}] \in \pi_0 \mathbf{H}(\mathbf{B}G, \mathbf{B}^2 A)
    =
    H^2_{Grp}(G,A)
    \,.
  $$

=--

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

If here $A$ is an [[Eilenberg-MacLane object]] $A = \mathbf{B}^{n}\mathcal{A}$, then the above says that extension of $G$ by the $n$-fold [[delooping]]/[[suspension]] $\mathbf{B}^n\mathcal{A}$ is classified by degree-$n$ [[group cohomology]]

$$
  H_{Grp}(G, \mathbf{B}^n\mathcal{A})
  =
  \pi_0 \mathbf{H}(\mathbf{B}G, \mathbf{B}^{n+1}\mathcal{A})
  =
  H_{Grp}^{n+1}(G, \mathcal{A})
  \,.
$$

In particular if $G$ here is [[n-truncated object in an (infinity,1)-category|0-truncated]] (hence a plain [[group object]] in the underlying 1-[[topos]]) then this reproduces the traditional theory of [[group extensions]] of 1-groups by 1-groups.

=--

## Properties

Notably for abelian $A$, by the main classification result at _[[principal ∞-bundles]]_, the [[∞-groupoid]] of $\infty$-group extensions is equivalent to

$$
  \mathbf{Ext}^n(G, A)
  \coloneqq
  \mathbf{H}(\mathbf{B}G, \mathbf{B}^{n+1}A)
  \,.
$$

In particular they are classified by the intrinsic $n+1$st $A$-[[cohomology]] of $\mathbf{B}G$.

## Examples

* For $\mathbf{H} = Sh_\infty(C)$ the [[(∞,1)-category of (∞,1)-sheaves]] on some [[site]] $C$ the [[Dold-Kan correspondence]] embeds [[chain complexes]] of [[abelian sheaves]] over $C$ into $\mathbf{H}$. Under this embedding ordinary [[Ext]]-groups and the shifted extensions that they classify (see [here](http://ncatlab.org/nlab/show/Ext#RelationToGroupExtensions)) identify with $\infty$-group extensions in the above sense.

* The [[string 2-group]] is an extension in $\mathbf{H} = $ [[Smooth∞Grpd]] of the [[spin group]] by the [[circle 2-group]].

* The [[fivebrane 6-group]] is an extension in $\mathbf{H} = $ [[Smooth∞Grpd]] of the [[string 2-group]] by the [[circle n-group|circle 6-group]].

## Related concepts

* [[∞-group cohomology]]

* [[central extension]], [[higher central extension]]

* [[centrally extended groupoid]]

## References

The general concept is discussed in section 4.3 of

* [[Thomas Nikolaus]], [[Danny Stevenson]], [[Urs Schreiber]], _[[schreiber:Principal ∞-bundles -- theory, presentations and applications]]_

Extensions by [[braided 2-groups]] are discussed in

* Evan Jenkins, _Extensions of groups by braided 2-groups_ ([arXiv:1106.0772](http://arxiv.org/abs/1106.0772))

[[!redirects ∞-group extension]]
[[!redirects infinity-group extensions]]
[[!redirects ∞-group extensions]]

[[!redirects extension of ∞-groups]]
[[!redirects extension of infinity-groups]]
[[!redirects extensions of ∞-groups]]
[[!redirects extensions of infinity-groups]]

[[!redirects central ∞-group extension]]
[[!redirects central infinity-group extensions]]
[[!redirects central ∞-group extensions]]

[[!redirects central extension of ∞-groups]]
[[!redirects central extension of infinity-groups]]
[[!redirects central extensions of ∞-groups]]
[[!redirects central extensions of infinity-groups]]

[[!redirects higher central ∞-group extension]]
[[!redirects higher central infinity-group extensions]]
[[!redirects higher central ∞-group extensions]]

[[!redirects higher central extension of ∞-groups]]
[[!redirects higher central extension of infinity-groups]]
[[!redirects higher central extensions of ∞-groups]]
[[!redirects higher central extensions of infinity-groups]]


infinity-group of units

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The generalization in [[(∞,1)-category theory]] of the notion of _[[group of units]]_ in ordinary category theory.

## Definition

### Unaugmented definition

+-- {: .num_defn #PlainInfinityGroupOfUnits}
###### 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, write $(\pi_0 \Omega^\infty A)^\times$ for its [[group of units]].

Then the [[∞-group of units]]

$$
  A^\times \coloneqq GL_1(A)
$$

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

\[
  \label{GL1AsPullback}
  \array{
    GL_1(A) &\to& \Omega^\infty A
    \\
    \downarrow &(pb)& \downarrow
    \\
    (\pi_0 \Omega^\infty A)^\times &\to& \pi_0 \Omega^\infty A
  }
  \,.
\]

=--

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

In terms of [[derived algebraic geometry]] one has that

$$
  GL_1(A) \simeq \mathbb{G}_m(A) = Hom(Spec A, \mathbb{G}_m)
$$

is the [[mapping space]] from $Spec A$ into the [[multiplicative group]].
This point of view is adopted for instance in ([Lurie, p. 20](#Lurie)).

=--

### Augmented definition
 {#AugmentedDefinition}

There is slight refinement of the above definition, which essentially adds one 0-th "grading" homotopy group to $B gl_1(E)$ and thereby makes the $\infty$-group of units of [[E-∞ rings]] be canonically [[augmented ∞-group|augmented]] over the [[sphere spectrum]] ([Sagave 11](#Sagave11)).

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

There is a functor

$$
  gl_1^J \colon CRing_\infty \to AbGrp_\infty/\mathbb{S}
  \,,
$$

given by ...

=--

This is ([Sagave 11, def. 3.14 in view of example 3.8](#Sagave11), [Sagave-Schlichtkrull 11, above theorem 1.8](#SagaveSchlichtkrull11)). See also ([Sagave 11, section 1.4](#Sagave11)) for comments on how this yields an $\infty$-version of $\mathbb{Z}$-grading on an abelian group.

In fact this grading extends form the group of units to the full $\infty$-ring ([Sagave-Schlichtkrull 11, theorem 1.7- 1.8](#SagaveSchlichtkrull11)).

+-- {: .num_prop #FiberSequenceForExtendedGl1}
###### Proposition

For $E$ an [[E-∞ ring]], there is a [[homotopy fiber sequence]] of [[abelian ∞-groups]]

$$
  gl_1(E) \to gl_1^J(E) \to \mathbb{S}
  \,,
$$

where on the left we have the ordinary $\infty$-group of units of def. \ref{PlainInfinityGroupOfUnits} and on the right we have the [[sphere spectrum]], regarded (being a [[connective spectrum]]) as an [[abelian ∞-group]].

=--

Here the existence of the map $gl_1(E) \to gl_1^J(E)$ is ([Sagave 11, lemma 2.12 + lemma 3.16 ](#Sagave11)). The fact that the resulting sequence is a homotopy fiber sequence is ([Sagave 11, prop. 4.1](#Sagave11)).

Using this, there is now a modified delooping of the ordinary $\infty$-group of units:

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

Write $bgl_1^\ast(E)$ for the [[homotopy cofiber]] of $gl_1^J(E) \to \mathbb{S}$ to yield

$$
  gl_1(E) \to gl_1^J(E) \to \mathbb{S} \to bgl_1^\ast(E)
  \,.
$$

=--

([Sagave 11, prop. 4.3](#Sagave11))

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

It ought to be true that the non-connective delooping $bgl_1^\ast(E)$ sits inside the full [[Picard ∞-group]] of $E Mod$. ([Sagave 11, remark 4.11](#Sagave11)). (Apparently it's the full inclusion on those degree-0 twists which are grading twists, i.e. on the elements $(-)\wedge\Sigma^n E$.)

See also at _[twisted cohomology -- by R-module bundles](twisted+cohomology#InfSections)_.

=--

## Properties
 {#Properties}

### Adjointness to $\infty$-group $\infty$-ring
 {#AdjointnessToGroupRing}

####  Unaugmented case

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

Write

$$
  gl_1 \; \colon \;  CRing_\infty \to AbGrp_\infty
$$

for the [[(∞,1)-functor]] which sends a [[E-∞ ring|commutative ∞-ring]] to its [[∞-group of units]].

=--

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

The [[∞-group of units]] [[(∞,1)-functor]] of def. \ref{GroupOfUnitsFunctor} is a  right-[[adjoint (∞,1)-functor]]

$$
  CRing_\infty
  \stackrel{\overset{\mathbb{S}[-]}{\leftarrow}}{\underset{gl_1}{\to}}
  AbGrp_\infty
  \,.
$$

=--

This is ([ABGHR 08, theorem 2.1/3.2, remark 3.4](#ABGHR08)).

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

The [[left adjoint]]

$$
  \mathbb{S}[-] \colon AbGrp_\infty \to CRing_\infty
$$

is a higher analog of forming the [[group ring]] of an ordinary [[abelian group]] over the [[integers]]

$$
  \mathbb{Z}[-] \colon Ab \to CRing
  \,,
$$

which is indeed [[left adjoint]] to forming the ordinary [[group of units]] of a ring.

We might call $\mathbb{S}[A]$ the **[[∞-group ∞-ring]]** of $A$ over the [[sphere spectrum]].

=--

####  Augmented case

Also the augmented $\infty$-group of units functor of def. \ref{AugmentedGroupOfUnits} is a homotopy right adjoint. ([Sagave 11, theorem 1.7](#Sagave11)).

### Homotopy groups

The [[homotopy groups]] of $GL_1(E)$ are

$$
  \pi_n(GL_1(E))
  =
  \left\{
    \array{
       \pi_0(E)^\times & |\, n = 0
       \\
       \pi_n(E) & | \, n \geq 1
    }
  \right.
$$

### Cohomology and logarithm
 {#CohomologyAndLogarithm}

Given $E$ an [[E-∞ ring]], then write $gl_1(E)$ for its $\infty$-group of units regarded as a [[connective spectrum]]. For $X$ the [[homotopy type]] of a  [[topological space]], then the [[cohomology]] [[Brown representability theorem|represented]] by $gl_1(E)$ in degree 0 is the ordinary [[group of units]] in the [[cohomology ring]] of $E$:

$$
  H^0(X, gl_1(E))  \simeq (E^0(X))^\times
  \,.

$$

In positive degree the canonical map of pointed homotopy types $GL_1(E) = \Omega^\infty gl_1(E) \to \Omega^\infty E$ is in fact an [[isomorphism]] on all [[homotopy groups]]

$$
  \pi_{\bullet \geq 1} GL_1(E) \simeq \pi_{\bullet \geq 1} \Omega^\infty E
  \,.
$$

On cohomology elements this map

$$
  \pi_q(gl_1(E)) \simeq \tilde H^0(S^q, gl_1(E)) \simeq  (1+ \tilde R^0(S^q))^\times \subset (R^0(S^q))^\times
$$

is [[logarithm]]-like, in that it sends $1 + x \mapsto x$.

But there is not a homomorphism of [[spectra]] of this form. This only exists after [[K(n)-local stable homotopy theory|K(n)-localization]], where it is called then the [[logarithmic cohomology operation]], see there for more.

([Rezk 06](#Rezk06))

### Relation to Picard $\infty$-group and Brauer $\infty$-group
 {#RelationToPicardInfinityGroups}

Given an [[E-∞ ring]] $E$, the [[looping]] of the Brauer $\infty$-group is the [[Picard ∞-group]] ([Szymik 11, theorem 5.7](#Szymik11)).

$$
  \Omega Br(E) \simeq Pic(E).
$$

The [[looping]] of that is the ∞-group of units ([Sagave 11, theorem 1.2](#Sagave11)).

$$
  \Omega^2 Br(E) \simeq \Omega Pic(E) \simeq GL_1(E)
  \,.
$$

## Examples

### Snaith's theorem and the units of K-theory and complex cobordism
 {#SnatihTheorem}

[[Snaith's theorem]] asserts that

1. the [[K-theory spectrum]] for [[complex K-theory]] is the [[∞-group ∞-ring]] of the [[circle 2-group]] localized away from the [[Bott element]] $\beta$:

   $$
     KU \simeq (\mathbb{S}[B U(1)])[\beta^{-1}]
     \,;
   $$

1. the [[periodic complex cobordism spectrum]] is the [[∞-group ∞-ring]] of the [[classifying space]] for stable [[complex vector bundles]] (the classifying space for [[topological K-theory]]) localized away from the [[Bott element]] $\beta$:

   $$
     MU \simeq (\mathbb{S}[B U])[\beta^{-1}]
     \,.
   $$

### Units of topological modular forms

Analysis of the $\infty$-group of units of [[tmf]] is in ([Ando-Hopkins-Rezk 10, from section 12 on](#AndoHopkinsRezk10)).

### Inclusion of circle $n$-bundles into higher chromatic cohomology

By [Snaith's theorem](#SnatihTheorem) above there is a canonical map

$$
  B U(1) \to \mathbb{S}[B U(1)] \to  KU
$$

that sends [[circle bundles]] to cocycles in [[topological K-theory]].

At the next level there is a canonical map

$$
  B^2 U(1) \to \mathbb{S}[B^2 U(1)] \to tmf
$$

that sends [[circle 2-bundles]] to [[tmf]]. See at _[tmf -- Inclusion of circle 2-bundles](tmf#InclusionOfCircle2Bundles)_.

Write $gl_1(K(n))$ for the [[∞-group of units]] of the (a)
[[Morava K-theory]] spectrum.

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

For $p = 2$ and all $n \in \mathbb{N}$, there is an [[equivalence in an (infinity,1)-category|equivalence]]

$$
  Maps(B^{n+1}U(1), B gl_1(K(n)))
  \simeq
  \mathbb{Z}/(2)
$$

between the [[mapping space]] from the [[classifying space]] for [[circle n-bundle|circle (n+1)-bundles]] to the [[delooping]] of the [[∞-group of units]] of $K(n)$.

=--

([Sati-Westerland 11, theorem 1](#SatiWesterland11))

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

By the discussion at [[(∞,1)-vector bundle]] this means that for each such map there is a type of [[twisted cohomology|twist]] of Morava K-theory (at $p = 2$).

=--

## Related concepts

* [[(∞,1)-vector bundle]]

## References

A notion of spectrum of units of an $E_\infty$-ring was originally described in

* [[Peter May]], p. 54 of: _$E_\infty$ ring spaces and $E_\infty$ ring spectra_,  Lecture Notes in Mathematics, Vol. 577. Springer-Verlag, Berlin, 1977. With contributions by [[Frank Quinn]], Nigel Ray, and Jørgen Tornehave ([pdf](https://www.math.uchicago.edu/~may/BOOKS/e_infty.pdf))

One explicit model was given in

* {#Schlichtkrull04} [[Christian Schlichtkrull]], _Units of ring spectra and their traces in algebraic K-theory_, Geom. Topol. **8** (2004) 645-673 ([arXiv:math/0405079](http://arxiv.org/abs/math/0405079), [euclid:gt/1513883412](https://projecteuclid.org/euclid.gt/1513883412))

Further discussion in

* [[Peter May]], [[Johann Sigurdsson]], Section 22.2 of: _[[Parametrized Homotopy Theory]]_,  Mathematical Surveys and Monographs, vol. 132, AMS 2006 ([ISBN:978-0-8218-3922-5](https://bookstore.ams.org/surv-132), [arXiv:math/0411656](https://arxiv.org/abs/math/0411656), [pdf](http://www.math.uchicago.edu/~may/EXTHEORY/MaySig.pdf))

A general abstract discussion in [[stable (∞,1)-category]] theory is in

* {#Rezk06} [[Charles Rezk]], section II of _The units of a ring spectrum and a logarithmic cohomology operation_, J. Amer. Math. Soc. 19 (2006), 969-1014 ([arXiv:math/0407022](http://arxiv.org/abs/math/0407022))

* {#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)]

* [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], [[Michael Hopkins]], [[Charles Rezk]], _Units of ring spectra, orientations, and Thom spectra via rigid infinite loop space theory_, Journal of Topology **7** 4 (2014) [[arXiv:1403.4320](https://arxiv.org/abs/1403.4320), [doi:10.1112/jtopol/jtu009](https://doi.org/10.1112/jtopol/jtu009)]

* {#LurieI} [[Jacob Lurie]], construction 3.9.4 of _Elliptic Cohomology I: Spectral Abelian Varieties_ ([pdf](http://www.math.harvard.edu/~lurie/papers/Elliptic-I.pdf))

Remarks alluding to this are also on p. 20 of

* {#Lurie} [[Jacob Lurie]], _[[A Survey of Elliptic Cohomology]]_

Theorem 3.2 there is proven using classical results which are collected in

* [[Peter May]], _What precisely are $E_\infty$-ring spaces and $E_\infty$-ring spectra?_, Geometry and Topology Monographs 16 (2009) 215–282 ([pdf](http://www.math.uchicago.edu/~may/PAPERS/Final1.pdf))

A survey of the situation in [[(∞,1)-category theory]] is also in section 3.1 of

* [[Jacob Lurie]], _[[A Survey of Elliptic Cohomology]]_

A construction in terms of a [[model structure on spectra]] is in

* [[John Lind]], _Diagram spaces, diagram spectra, and spectra of units_ ([arXiv:0908.1092](http://arxiv.org/abs/0908.1092))

A refinement (using [[I-space|$\mathcal{I}$-spaces]]) of the construction of $\infty$-groups of units to [[augmented ∞-groups]] over the [[sphere spectrum]], such as to distinguish $gl_1$ of a [[periodic E-∞ ring]] from its [[connective cover]]:

* {#Sagave11} [[Steffen Sagave]], _Spectra of units for periodic ring spectra_, Algebr. Geom. Topol. 16 (2016) 1203-1251 ([arXiv:1111.6731](http://arxiv.org/abs/1111.6731))

based on ([Schlichtkrull 04](#Schlichtkrull04)). See also

* {#SagaveSchlichtkrull11} [[Steffen Sagave]], [[Christian Schlichtkrull]], *Diagram spaces and symmetric spectra*, Advances in Mathematics **231** 3-4 (2012) 2116-2193 [[arXiv:1103.2764](https://arxiv.org/abs/1103.2764), [doi:10.1016/j.aim.2012.07.013](https://doi.org/10.1016/j.aim.2012.07.013)]

* {#Szymik11} [[Markus Szymik]], _Brauer spaces for commutative rings and structured ring spectra_ ([arXiv:1110.2956](http://arxiv.org/abs/1110.2956))

* {#BakerRichterSzymik12} [[Andrew Baker]], [[Birgit Richter]], [[Markus Szymik]], _Brauer groups for commutative $\mathbb{S}$-algebras_, J. Pure Appl. Algebra 216 (2012) 2361–2376 ([arXiv:1005.5370](http://arxiv.org/abs/1005.5370))

The $\infty$-group of units of [[Morava K-theory]] is discussed in

* [[Hisham Sati]], [[Craig Westerland]], _Twisted Morava K-theory and E-theory_ ([arXiv:1109.3867](http://arxiv.org/abs/1109.3867))
 {#SatiWesterland11}

The [[cohomology]] with coefficients in $gl_1(E)$ and the corresponding [[logarithmic cohomology operations]] are discussed in

* {#Rezk06} [[Charles Rezk]], _The units of a ring spectrum and a logarithmic cohomology operation_, J. Amer. Math. Soc. 19 (2006), 969-1014 ([arXiv:math/0407022](http://arxiv.org/abs/math/0407022))

The group of units of [[tmf]] is analyzed from section 12 on in

* {#AndoHopkinsRezk10} [[Matthew Ando]], [[Mike 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))

[[!redirects ∞-group of units]]

[[!redirects ∞-groups of units]]
[[!redirects infinity-groups of units]]


infinity-groupoid

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

# Contents
* table of contents
{: toc}

## Idea

The notion of $\infty$-groupoid is the generalization of that of [[group]] and [[groupoid]]s to [[higher category theory]]:

an $\infty$-groupoid -- equivalently an [[(∞,0)-category]] -- is an [[∞-category]] in which all [[k-morphism]]s for all $k$ are [[equivalences]].

The collection of all $\infty$-groupoids forms the [[(∞,1)-category]] [[∞Grpd]].

Special cases of $\infty$-groupoids include [[groupoid]]s, [[2-groupoid]]s, [[3-groupoid]]s, [[n-groupoid]]s, [[delooping]]s of [[group]]s, [[2-group]]s, [[∞-group]]s.

## Properties

### Presentations

There are many ways to [[presentable (∞,1)-category|present]] the [[(∞,1)-category]] [[∞Grpd]] of all $\infty$-groupoids, or at least obtain its [[homotopy category]].

A simple and very useful incarnation of $\infty$-groupoids is available using a [[geometric definition of higher categories]] in the form of [[simplicial sets]] that are [[Kan complex]]es: the $k$-cells of the underlying simplicial set are the [[k-morphism]]s of the $\infty$-groupoid, and the Kan [[horn]]-[[filler]] conditions encode the fact that adjacent $k$-morphisms have a (non-unique) composite $k$-morphism and that every $k$-morphism is invertible with respect to this composition.
See [[Kan complex]] for a detailed discussion of how these incarnate $\infty$-groupoids.

The [[(∞,1)-category]] of all $\infty$-groupoids is [[presentable (∞,1)-category|presented]] along these lines by the Quillen [[model structure on simplicial sets]], whose fibrant-cofibrant objects are precisely the Kan complexes:
$$
  \infty Grpd \simeq (sSet_{Quillen})^\circ.
$$

One may turn this geometric definition into an [[algebraic definition of higher category|algebraic definition of ∞-groupoids]] by _choosing [[horn]]-fillers_ . The resulting notion is that of an [[algebraic Kan complex]] that has been shown by [[Thomas Nikolaus]] to yield an equivalent [[(∞,1)-category]] of $\infty$-groupoids.

There are various model categories which are [[Quillen equivalence|Quillen equivalent]] to $sSet_{Quillen}$. For instance the standard [[model structure on topological spaces]], a [[model structure on marked simplicial sets]] and many more. All these therefore present [[∞Grpd]].

Moreover, the corresponding  [[homotopy category of an (∞,1)-category]] $Ho(\infty Grpd)$, hence a category whose objects are [[homotopy types]] of $\infty$-groupoids, is given by the [[homotopy category]] of the [[category of presheaves]] over any _[[test category]]_. See there for more details.

Every other algebraic definition of [[omega-categories]] is supposed to yield an equivalent notion of $\infty$-groupoid when restricted to $\omega$-categories all whose [[k-morphism]]s are invertible. This is the statement of the [[homotopy hypothesis]], which is for Kan complexes and algebraic Kan complexes a theorem and for other definitions regarded as a consistency condition.

Notably in [[Pursuing Stacks]] and the earlier letter to [[Larry Breen]], [[Alexander Grothendieck]] initiated the study of $\infty$-groupoids and the homotopy hypothesis with his original definition of [[Grothendieck weak infinity-groupoid]]s, that has recently attracted renewed attention.

### Strict $\infty$-groupoids

One may also consider entirely strict $\infty$-groupoids, usually called $\omega$-groupoids or [[strict ∞-groupoids]]. These are equivalent to [[crossed complexes]] of [[groups]] and [[groupoids]].

### Relation to $\infty$-groups

[[pointed object|Pointed]], [[0-connected]] $\infty$-groupoids  are the [[delooping]] $\mathbf{B}G$ of [[∞-group]]s (see _[[looping and delooping]]_).

These are presented by [[simplicial group]]s. Notably, abelian simplicial groups are therefore a model for abelian $\infty$-groupoids (more precisely, [[HA|H$\mathbb{Z}$]]-[[(∞,1)-module|modules]]). Under the [[Dold-Kan correspondence]] these are equivalent to non-negatively graded [[chain complex]]es, which therefore also are a model for abelian $\infty$-groupoids. This way much of [[homological algebra]] is secretly the study of special (structured) $\infty$-groupoids.

## Related concepts

* [[discrete ∞-groupoid]], [[∞-stack]], [[cohesive ∞-groupoid]]

* [[homotopy theory]], [[(∞,1)-category]]

* [[model structure for ∞-groupoids]]

* [[∞Grpd]]

* [[homotopy hypothesis]], [[test category]], [[modelizer]]

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

## Terminology
 {#Terminology}

The term ∞-groupoid is sometimes considered to be too unwieldy,
and some alternatives have been suggested or used, but none has gained wide acceptance.

Historically, the word “space” is often (ab)used to mean ∞-groupoid, due to the traditional presentation of $\infty Gpd$ by the [[model structure on topological spaces]].  Some have [condemned](https://nforum.ncatlab.org/discussion/4781/against-spaces-in-homotopy-theory/) this usage, but others argue that a "homotopy space" is a valid notion of "space".

The term “space” is also often used to refer to [[simplicial sets]].
In particular, [[Bousfield]] and [[Kan]] in their book “[[Homotopy Limits,  Completions and Localizations]]” write:

> “These notes are written _simplicially_, i.e. whenever we say _space_ we mean _simplicial set_.

More recently, [[Jacob Lurie]]'s work continues this usage.

The term “homotopy type” is also quite close in meaning to “∞-groupoid”.  Historically, it differed in that morphisms of homotopy types were mere homotopy classes of maps of ∞-groupoids, but more recently (especially with the advent of [[homotopy type theory]] some have used "homotopy type" synonymously with "$\infty$-groupoid".

{#Anima} More radically, [[Dustin Clausen]] and [[Peter Scholze]] use the term “anima” (plural: anima) as a synonym for "$\infty$-groupoid", in particular in relation to [[condensed mathematics]].  Reference: [nCafe](https://golem.ph.utexas.edu/category/2020/03/pyknoticity_versus_cohesivenes.html#c057623).  In
_Purity for flat cohomology_ by
Kęstutis Česnavičius and Peter Scholze they write
“If the objects of $C$ are called widgets, then we call those of $Ani(C)$ animated widgets, except that we abbreviate $Ani(Set)$ to $Ani$ and the term ‘animated set’ to anima (plural: anima).”

## References

* [[Jacob Lurie]], Section 1.1.2 in: *[[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))

Formulations in/from [[homotopy type theory]]:

* [[Benno van den Berg]], [[Richard Garner]], *Types are weak omega-groupoids*, Proceedings of the London Mathematical Society **102** 2 (2011) 370-394 [[arXiv:0812.0298](https://arxiv.org/abs/0812.0298), [doi:10.1112/plms/pdq026](https://doi.org/10.1112/plms/pdq026)]

* [[Thorsten Altenkirch]], Ondrej Rypácek, _A Syntactical Approach to Weak $\omega$-Groupoids_, Computer Science Logic (CSL'12) **16** (2012) 16-30 [[pdf](http://www.cs.nott.ac.uk/~txa/publ/weakomega2.pdf), [doi:10.4230/LIPIcs.CSL.2012.16](https://doi.org/10.4230/LIPIcs.CSL.2012.16)]

See also at _[[category object in an (infinity,1)-category]]_ for more along these lines.

[[!redirects ∞-groupoid]]
[[!redirects ∞-groupoids]]
[[!redirects infinity-groupoid]]
[[!redirects infinity-groupoids]]
[[!redirects infinity groupoid]]
[[!redirects $\infty$-groupoid]]

[[!redirects weak omega-groupoid]]
[[!redirects weak ∞-groupoid]]
[[!redirects weak omega-groupoids]]
[[!redirects weak ∞-groupoids]]

[[!redirects weak ∞-groupoid]]
[[!redirects weak ∞-groupoids]]
[[!redirects weak infinity-groupoid]]
[[!redirects weak infinity-groupoids]]

[[!redirects anima]]

category:∞-groupoid


Infinity-Grpd

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### [[categories of categories - contents|categories of categories]]
+-- {: .hide}
[[!include categories of categories - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

**$\infty Grpd$** is the [[(∞,1)-category]] of [[∞-groupoids]], i.e. of [[(∞,0)-categories]]. This is the archetypical [[(∞,1)-topos]], the home of classical [[homotopy theory]].

[[equivalence of (∞,1)-categories|Equivalently]] this means all of the following:

1. $\infty Grpd$ is the [[simplicial localization]] of the [[category]] [[Top]]${}_k$ of ([[weakly Hausdorff topological space|weakly Hausdorff]]) [[locally compact topological spaces]] at the [[weak homotopy equivalences]]. As such it is the [[∞-category]]-enhancement of the [[classical homotopy category]]: $\tau_0(\infty Grpd) \simeq$ [[Ho(Top)]], itself [[presentable (∞,1)-category|presented]] by the [[classical model structure on topological spaces]]: $\infty Grpd \simeq L_{whe} Top_k$.

1. $\infty Grpd$ is the [[simplicial localization]] of the [[category]] [[sSet]] of [[simplicial sets]] at the simplicial [[weak homotopy equivalences]]. As such it is the [[∞-category]]-enhancement of the [[classical homotopy category]]: $\tau_0(\infty Grpd) \simeq$ [[Ho(sSet)]], itself [[presentable (∞,1)-category|presented]] by the [[classical model structure on simplicial sets]]: $\infty Grpd \simeq L_{whe} sSet$.

   Hence, as a [[simplicially enriched category|Kan-complex enriched category]] (a [[fibrant object]] in the [[model structure on sSet-categories]]) $\infty Grpd$ is the [[full subcategory|full]] [[simplicially enriched category|sSet enriched]]-[[subcategory]] of [[sSet]] on the [[Kan complexes]].

1. $\infty Grpd$ is the [[full sub-(∞,1)-category]] of [[(∞,1)Cat]] on those [[(∞,1)-categories]] that are [[∞-groupoids]].

## Properties

### As an $(\infty,1)$-topos

As an [[(∞,1)-topos]] $\infty Grpd$ is the [[terminal object|terminal]] $(\infty,1)$-topos: for every other [[(∞,1)-category of (∞,1)-sheaves|(∞,1)-sheaf]] [[(∞,1)-topos]] $\mathbf{H}$ there is up to a [[contractible]] space of  choices a unique [[geometric morphism]] $(LConst \dashv \Gamma) : \mathbf{H}\stackrel{\leftarrow}{\to} \infty Grpd$ -- the [[global section]] geometric morphism. See there for more details.

### Limits and colimits in $\infty Grpd$

[[limit in a quasi-category|Limits and colimits]] over a [[(∞,1)-functor]] with values in $\infty Grpd$ may be reformulation in terms of the  [[universal fibration of (infinity,1)-categories]].

Let the [[(∞,1)-functor]] $Z|_{Grpd} \to \infty Grpd^{op}$ be the [[universal fibration of (infinity,1)-categories|universal ∞-groupoid fibration]] whose fiber over the object denoting some $\infty$-groupoid is that very $\infty$-groupoid.

Then let $X$ be any [[∞-groupoid]] and

$$
  F : X \to \infty Grpd
$$

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:

$$
  \array{
    E_F &\to& Z|_{Grpd}
    \\
    \downarrow && \downarrow
    \\
    X &\stackrel{F}{\to}& \infty Grpd
  }
$$

+-- {: .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|(∞,1)-groupoid of sections]] of $E_F \to X$

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

=--

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

The statement for the colimit is corollary 3.3.4.6 in [[Higher Topos Theory|HTT]]. The statement for the limit is corollary 3.3.3.4.

=--

##Subcategories

The [[n-truncated object of an (infinity,1)-category|n-truncated objects]] of $\infty Grpd$ are the [[n-groupoids]] (including [[(-1)-groupoid]]s and the [[(-2)-groupoid]]).

## Related categories

* [[Set]]

* [[Grpd]],

  **$\infty Grpd$**

  [[Top]], [[Ho(Top)]]

* [[Cat]], [[(∞,1)Cat]]

* [[(∞,n)Cat]]

category: category

[[!redirects ∞Groupoid]]
[[!redirects ∞Groupoids]]

[[!redirects ∞Grpd]]
[[!redirects ∞ Grpd]]
[[!redirects ∞-Grpd]]
[[!redirects infinity-Grpd]]
[[!redirects ∞Gpd]]
[[!redirects ∞ Gpd]]
[[!redirects ∞-Gpd]]
[[!redirects infinity-Gpd]]
[[!redirects Infinity-Gpd]]

[[!redirects (∞,1)-category of spaces]]
[[!redirects (infinity,1)-category of spaces]]