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#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
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#Contents#
* table of contents
{:toc}

## Idea

For $X_\bullet$ a [[simplicial object]] in [[Top]] --  a [[simplicial topological space]] -- its _[[geometric realization]]_ is a plain [[topological space]] ${|X_\bullet|} \in Top$ obtained by gluing all topological space $X_n$ together, as determined by the face and degeneracy maps.

The construction of ${|X_\bullet|}$ is a direct analog of the ordinary notion of [[geometric realization]] of a [[simplicial set]], but taking into account the [[topology]] on the spaces of $n$-simplices $X_n$.

The [[duality|dual]] concept is that of [[totalization]] of [[cosimplicial topological spaces]].

## Definition

Let [[Top]] in the following denote either

* the [[category]] of [[compactly generated space|compactly generated]] [[weakly Hausdorff space]]s, or

* the category of [[k-space]]s.

Let $\Delta$ denote the [[simplex category]], and write $\Delta_{Top} \colon \Delta \to Top \colon  [n] \mapsto \Delta^n_{Top}$ for the standard [[cosimplicial object|cosimplicial]] [[topological space]] of topological [[simplices]].

Write equivalently

$$
  Top^{\Delta^{op}} \coloneqq  sTop \coloneqq  [\Delta^{op}, Top]
$$

for the category of [[simplicial topological spaces]].

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

For $X_\bullet \colon  \Delta^{op} \to Top$ a [[simplicial topological space]], its **geometric realization** is the [[coend]]

$$
  {|X_\bullet|} \coloneqq  \int^{n \in \Delta} X_n \times \Delta^n_{Top}
$$

formed in [[Top]].
=--

This operation naturally extends to a [[functor]]

$$
  {|-|} \colon sTop \to Top
  \,.
$$

More explicitly, $\vert X_\bullet \vert$ is the [[topological space]] given by the [[quotient]]

$$
  {\vert X_\bullet \vert} = \coprod_{n} X_n \times \Delta^n_{Top} /\sim
$$

where the [[equivalence relation]] "$\sim$" identifies, for every morphism $[k] \to [l]$ in $\Delta$, the points $(x,f_* p) \in X_l \times \Delta^l_{Top}$ and $(f^* x,p) \in X_k \times \Delta^k_{Top}$.

This form of geometric realization of simplicial topological spaces goes back to ([Segal 1968](#Segal68)). An early reference that realizes this construction as a [[coend]] is ([MacLane](#MacLane)).

One also considers geometric realization after [restricting](semi-simplicial+set#AdjunctionWithSimplicialSets) to the subcategory $\Delta_+ \overset{j^{op}}{\hookrightarrow} \Delta$ of the [[simplex category]] on the strictly increasing maps (that is, the coface maps only---no codegeneracies). (This yields the underlying *[[semi-simplicial set]]*).

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

The corresponding [[coend]] in [[Top]] is called the **fat geometric realization**

$$
  {\Vert X_\bullet\Vert} \coloneqq   \int^{n \in \Delta_+} X_n \times \Delta^n_{Top}
  \,.
$$

=--

This is called _fat_ ,  because it does not [[quotient]] out the [[relation]]s induced by the degeneracy maps and hence is "bigger" than ordinary geometric realization.

Explicitly, this is the [[topological space]] given by the [[quotient]]

$$
  {\Vert X_\bullet \Vert} = \coprod_{n} X_n\times \Delta^n_{Top} /{\sim_+}
$$

where the [[equivalence relation]] "$\sim_+$" identifies $(x,f_* p) \in X_l\times \Delta^l_{Top}$ with $(f^* x,p) \in X_k\times \Delta^k_{Top}$ only when $[k] \to [l]$ is a coface map.

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

The geometric realization of the point --- the [[simplicial topological space]] that is in each degree the 1-point topological space --- is [[homeomorphic]] to the point, but the fat geometric realization of the point is an "infinite dimensional topological ball": the [[terminal object|terminal]] morphism

$$
  {\vert * \vert} \stackrel{\simeq_{iso}}{\longrightarrow} *
$$

is an [[isomorphism]], but the morphism

$$
  {\Vert * \Vert} \stackrel{\simeq_{h.e.}}{\longrightarrow} *
$$

is just a [[homotopy equivalence]].

=--

+-- {: .num_remark}
###### Remark
The ordinary geometric realization can be described as the [[tensor product of functors]] ${|X_\bullet|}=\Delta^\bullet \otimes_{\Delta} X_\bullet$, and the fat geometric realization can likewise be described as ${\Vert X_\bullet\Vert} = i^* \Delta^\bullet \otimes_{\Delta_+} i^* X_\bullet$, where $i\colon \Delta_+ \hookrightarrow \Delta$ is the inclusion.  By general facts about tensor product of functors (essentially, the associativity of composition of [[profunctors]]), it follows that we can also write ${\Vert X_\bullet\Vert} \cong \Delta^\bullet \otimes_{\Delta} \Lan_i i^* X_\bullet \cong {| Lan_i i^* X_\bullet |} $.  In other words, the fat geometric realization of $X_\bullet$ is the ordinary geometric realization of a "fattened up" version of $X_\bullet$, which is obtained by forgetting the degeneracy maps of $X_\bullet$ and then "freely throwing in new ones."
=--

## Reminder on nice simplicial topological spaces

Simplicial topological spaces are in [[homotopy theory]] presentations for certain [[topological ∞-groupoid]]s . In this context what matters is not the operation of geometric realization itself, but its [[derived functor]]. This is obtained by evaluating ordinary geometric realization on "sufficiently nice" [[resolution]]s of simplicial topological spaces. These we discuss now.

Recall the following definitions and facts from [[nice simplicial topological space]].

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

Let $X \colon  \Delta^{op} \to Top$ be a [[simplicial topological space]].

Such $X$ is called

* **good** ([Segal 1973, App. 2](#Segal73); [Segal 1974, Def. A.4](#Segal74))

  if all the degeneracy maps $X_{n-1} \hookrightarrow X_n$ are [[closed cofibrations]];

* **proper** ([May 1972](#May72))

  if the inclusion $s X_n \hookrightarrow X_n$ of the degenerate simplices is a [[closed cofibration]], where $s X_n = \bigcup_i s_i(X_{n-1})$.

Noticing that the union of degenerate simplices appearing here is a [[Reedy model structure|latching object]] and that closed cofibrations are cofibrations in the [[Strøm model structure]] on [[Top]], the last condition equivalently says:

* $X_\bullet$ is **proper** if it is cofibrant in the [[Reedy model structure]] $[\Delta^{op}, Top_{Strom}]_{Reedy}$ on [[simplicial object]]s in [[Top]] with respect to the [[Strøm model structure]] on [[Top]].

=--

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

A good simplicial topological space is proper:

$$
  (X_\bullet \; good) \Rightarrow (X_\bullet\; proper)
  \,.
$$

=--

([Roberts & Stevenson 2012, App. A](#RobertsStevenson), following hints in [Gaunce Lewis 1982](#GaunceLewis82))

### Bisimplicial sets and good resolutions

We now discuss the [[resolution]] of any simplicial topological space by a good one.
Write

$$
  ({\vert- \vert} \dashv Sing)
   \colon
  Top
    \stackrel{\overset{{|-|}}{\longleftarrow}}{\underset{Sing}{\longrightarrow}}
  sSet
$$

for the ordinary [[geometric realization]]/[[singular simplicial complex]] [[adjunction]] (see [[homotopy hypothesis]]).  Recall that the composite ${|Sing(-)|}\colon Top\to Top$ is a cofibrant replacement functor: for any space $X$, the space ${|Sing(X)|}$ is a CW complex and comes with a natural map ${|Sing X|} \to X$ (the [[unit of an adjunction|counit]] of the adjunction $({|-|} \dashv Sing)$) which is a [[weak homotopy equivalence]].

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

Let $X_\bullet$ be a simplicial topological space.
Then the simplicial topological space

$$
  {|Sing(X_\bullet)|} \in [\Delta^{op}, Top] \,,
$$

obtained by applying ${|Sing(-)|} \colon  Top \to Top$ degreewise, is good and hence proper.  Moreover, we have a natural morphism
$$
{|Sing X_\bullet|} \to X_\bullet
$$
which is degreewise a weak homotopy equivalence.

=--

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

Each space $|Sing X_n|$ is a [[CW-complex]], hence in particular a [[locally equi-connected space]]. By ([Gaunce Lewis 1982, p. 153](#GaunceLewis82)) inclusions of [[retract]]s of locally equi-connected spaces are [[closed cofibration]]s, and since degeneracy maps are retracts, this means that the degeneracy maps in $|Sing X_\bullet|$ are closed cofibrations.

The second sentence follows directly by the remarks above.

=--

{#OtherGoodResolutions} Note that there is nothing special about ${|Sing(-)|}$ in the proof; any functorial CW replacement would do just as well (such as that obtained by the [[small object argument]], or the constructions denoted $\tau(-)$ and $simp(-)$ in [Segal 1974, p. 308-309](#Segal74)).

However, ${|Sing(-)|}$ has the advantage that its geometric realization can be computed alternately in terms of diagonals of bisimplicial sets, as we now show.

If $S_{\bullet,\bullet} \colon  \Delta^{op} \times \Delta^{op} \to Set$ is a [[bisimplicial set]], we write $d S$ for its [[diagonal]], which is the composite
$$\Delta^{op} \to \Delta^{op} \times \Delta^{op} \stackrel{S}{\longrightarrow} Set.$$
On the other hand, we can also consider a bisimplicial set as a simplicial object in $sSet$ and take its "geometric realization":
$$ {|S_{\bullet,\bullet}|} \coloneqq \int^{[n]\in\Delta^{op}} S_{n,\bullet} \times \Delta^n_{sSet} $$
where $\Delta^n_{sSet}$ denotes the $n$-simplex as a simplicial set, i.e. the representable functor $\Delta(-,[n])\colon \Delta^{op}\to Set$.

+-- {: .num_lemma #DiagonalIsRealization}
###### Lemma
For a bisimplicial set $S_{\bullet,\bullet}$, there is a natural isomorphism of simplicial sets $d S \cong {|S|}$.
=--
+-- {: .proof}
###### Proof
Both functors $d$ and ${|-|}$ are [[cocontinuous functor|cocontinuous]] functors between [[presheaf categories]] $Set^{\Delta^{op}\times\Delta^{op}} \to Set^{\Delta^{op}}$, so it suffices to verify that they agree on representables.  The representable $(\Delta\times\Delta)(-,([n],[m]))$ is called a *bisimplex* $\Delta^{n,m}$; its diagonal is
$$(d \Delta^{n,m})(-) = \Delta(-,n) \times \Delta(-,m) = \Delta^n_{sSet} \times \Delta^m_{sSet}.$$
Now note that geometric realization of a bisimplicial set is left adjoint to the "singular complex" $Sing\colon sSet \to sSet^{\Delta^{op}}$ defined by
$$Sing(X_\bullet)(n,m) = \underline{Map}(\Delta^n_{sSet}, X_\bullet)_m = sSet(\Delta^n_{sSet} \times \Delta^m_{sSet}, X_\bullet) = sSet(d \Delta^{n,m}, X_\bullet)$$
where $\underline{Map}(-,-)$ denotes the simplicial mapping space between two simplicial sets.  But by the Yoneda lemma, $Sing(X_\bullet)(n,m) = sSet^{\Delta^{op}}(\Delta^{n,m},Sing(X_\bullet))$, so this shows that $d \Delta^{n,m}$ has the same universal property as ${|\Delta^{n,m}|}$; hence they are isomorphic.
=--

In particular, the lemma implies that for the two different ways of considering a bisimplicial set as a simplicial simplicial set ("vertically" or "horizontally"), the resulting "geometric realizations" as simplicial sets are isomorphic (since both are isomorphic to the diagonal, which is symmetrically defined).

Note that Lemma \ref{DiagonalIsRealization} can be interpreted as an isomorphism between two [[profunctors]] $\Delta ⇸ \Delta\times\Delta$, of which the first is [[representable functor|representable]] by the diagonal functor $\Delta \to \Delta\times\Delta$.  It follows that if $S_{\bullet,\bullet}$ is a bisimplicial object in any [[cocomplete category]], we also have
$$d S_{\bullet,\bullet} \cong \int^{[n]\in\Delta} S_{n,\bullet}\,\times \,(\Delta^n_{sSet})_{\bullet}$$
where the right-hand side is a "realization" functor from bisimplicial objects to simplicial objects in any cocomplete category.  On the other hand, if $S$ is a bisimplicial *space*, then we also have the *levelwise* realization
$$ \int^{[n]\in\Delta} S_{n,\bullet} \,\times\, \Delta^n_{Top} $$
which will not, in general, agree with the diagonal and the abstract realization considered above.  It does agree, however, after we pass to a further geometric realization as a single topological space.

+-- {: .num_prop #RealizationOfDiagonalIsRealizationOfRealization}
###### Proposition
For $S_{\bullet,\bullet}$ any bisimplicial space, there is a [[homeomorphism]] between the geometric realizations of the following two simplicial spaces: (1) the diagonal $d S_{\bullet,\bullet}$ of $S$, and (2) the levelwise realization $|S_{\bullet,\bullet}|$ of $S$.
=--
+-- {: .proof}
###### Proof
Applying Lemma \ref{DiagonalIsRealization} as above, we have
$$ d S_{\bullet,\bullet} \cong \int^{[n]\in \Delta} S_{n,\bullet} \times \Delta^n_{sSet}$$
If we then take the realization of these simplicial spaces, we find
$$
{| d S_{\bullet,\bullet} |}
\cong \left| \int^{[n]\in\Delta} S_{n,\bullet} \times \Delta^n_{sSet} \right|
\cong \int^{[n]\in\Delta} {|S_{\bullet,\bullet}|} \times \Delta^n_{Top}
\cong {\vert({\vert S_{\bullet,\bullet} \vert})\vert}
$$
using the fact that geometric realization of simplicial sets preserves colimits and products, and ${|\Delta^n_{sSet}|} = \Delta^n_{Top}$.
=--

This fact is attributed to Tornehave by Quillen on page 94 of his 'Higher Algebraic K-theory I'.

Finally, for any simplicial space $X_\bullet$, we have a bisimplicial set $Sing(X_\bullet)_\bullet$.  Applying the previous proposition to this bisimplicial set, regarded as a discrete bisimplicial space, we find a homeomorphism
$$
  {\vert({\vert Sing(X_\bullet) \vert})\vert}  \cong {| d Sing(X_\bullet)_\bullet |}
  \,.
$$
This also follows from results of ([Gaunce Lewis 1982](#GaunceLewis82)).  Thus, as a resolution of $X_\bullet$, the levelwise realization of the levelwise singular complex ${\vert Sing(X_\bullet) \vert}$ has the pleasant property that its geometric realization, as a simplicial *space*, can be calculated as the realization of a single simplicial *set* (the diagonal of $Sing(X_\bullet)_\bullet$).

## Properties

### General

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

We have the following degenerate case of geometric realization of simplicial topological spaces.

* If $X_\bullet \colon  \Delta^{op} \to Set \hookrightarrow Top$ is a degreewise [[discrete space]], hence just a [[simplicial set]], the notion of geometric realization above coincides with the notion of [[geometric realization|geometric realization of simplicial sets]].

* If $X_\bullet$ is simplicially constant on a topological space $X_0$, then its geometric realization is [[homeomorphic]] to that space:

  $$
    {\vert X_\bullet \vert} \simeq X_0
    \,.
  $$

  (This is not true for the fat geometric realization, only the ordinary one.  The fat geometric realization will be homotopy equivalent to $X$.)

=--

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

Geometric realization of simplicial topological spaces has a [[right adjoint|right]] [[adjoint functor]] $\underline{Sing}$:

$$
  ({\vert - \vert} \dashv \underline{Sing})  \colon
   sTop
   \stackrel{\overset{{\vert - \vert}}{\longrightarrow}}{\underset{\underline{Sing}} {\longleftarrow}}
   Top
  \,,
$$

For $X \in $ [[Top]] a [[topological space]], we have by definition

$$
  \underline{Sing}(X) \colon  [n] \mapsto [\Delta^n_{Top}, X]
  \,,
$$

where on the right we have the [[internal hom]], or [[exponential law for spaces|exponential]], space from the standard topological $n$-simplex to $X$.

=--

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

For every $X \in Top$ there is a [[weak homotopy equivalence]]

$$
  {|\underline{Sing}(X)|} \to X
  \,.
$$

=--

{#StricklandSeymour} The proof can be found in [[Neil Strickland]]'s answer to this [mathoverflow question](http://mathoverflow.net/q/171662/381). (An incorrect argument appears as ([Seymour, prop. 3.1](#Seymour)) where it is claimed that $\underline{Sing}(X)$ is proper. In fact, the first degeneracy map $X \to [\Delta^1_{Top}, X]$ is not in general a cofibration as explained in the linked question.)

Ordinary geometric realization has the following two disadvantages:

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

* If $X_\bullet$ has in each degree the [[homotopy type]] of a [[CW-complex]], its realization ${\vert X_\bullet \vert}$ in general need not.

* If a morphism $f \colon  X_\bullet \to Y_\bullet$ is degreewise a [[homotopy equivalence]], its geometric realization ${\vert f \vert}$ need not be a homotopy equivalence.

=--

See ([Segal74, appendix A](#Segal74))

This is different for the fat geometric realization.

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

* If $X_\bullet$ is degreewise of the [[homotopy type]] of a [[CW-complex]] (i.e. is degreewise [[m-cofibrant space|m-cofibrant]]), then so is ${\Vert X_\bullet \Vert}$.

* If $f \colon  X_\bullet \to Y_\bullet$ is degreewise a [[homotopy equivalence]], then also ${\Vert f \Vert}$ is a homotopy equivalence.

=--

This appears as ([Segal74, prop. A.1](#Segal)).

### Relation between fat and ordinary geometric realization

\begin{proposition}\label{FatRealizationOfGoodSimplicialSpaces}

If the simplicial topological space $X_\bullet$ is [good](#GoodAndProper) then the natural morphism from its [fat geometric realization](#FatGeometricRealization) to its [ordinary geometric realization](#GeometricRealization) is a [[homotopy equivalence]]

$$
  {\Vert X_\bullet \Vert} \stackrel{\simeq}{\longrightarrow} {|X_\bullet|}
  \,.
$$

\end{proposition}

A direct proof of this (not using that [good implies proper](#GoodImpliesProper)) appears as ([Segal74, prop. A.1 (iv)](#Segal74)) and a more detailed proof, using properness, is due to [tom Dieck 1974, Prop. 1](#tomDieck74).

A recent paper treating the special case where $X_\bullet$ is the nerve of a [[topological category]], and each $X_n$ is of the [[homotopy type]] of a [[CW-complex]], is [Wang 2017](#Wang17), [Wang 18](#Wang18).

### Compatibility with limits
 {#CompatibilityWithLimits}

We discuss how geometric realization interacts with [[limits]] of simplicial topological spaces.

#### Ordinary geometric realization

+-- {: .num_prop #RealizationPreservesPullbacks}
###### Proposition
**(geometric realization prerserves pullbacks)** \linebreak

In the category [[Top]] of [[compactly generated topological spaces]],
geometric realization of simplicial spaces [[preserved limit|preserves]] [[pullbacks]]: for $X_\bullet \to Y_\bullet \leftarrow Z_\bullet$ a [[diagram]] in $Top^{\Delta^{op}}$ there are [[natural transformation|natural]] [[homeomorphisms]]

$$
  {|X_\bullet|} \,\times_{|Y_\bullet|}\, {|Z_\bullet|}
  \simeq
  {|X_\bullet \,\times_{Y_\bullet}\, Z_\bullet|}
  \,.
$$

=--

This appears for instance as ([May 1972, Cor. 11.6](#May72)).  See also the proof that [[geometric realization]] of [[simplicial sets]] [[preserved limit|preserves]] [[pullbacks]], at [[geometric realization]] ([here](geometric+realization#GeometricRealizationIsLeftExact)).

\begin{remark}
It is essential in Prop. \ref{RealizationPreservesPullbacks} to work in a category $Top$ such as [[compactly generated topological spaces]] (also "k-spaces"): in the category of *all* topological spaces the proposition would not be true.  It works in these cases because product and/or quotient topologies in these categories are slightly different from in the category of all topological spaces.
\end{remark}

It follows that:

\begin{prop}\label{GeometricRealizationPreservesFiniteLimits}
  Geometric realization of simplicial compactly-generated topological spaces preserves all [[finite limits]].
\end{prop}
\begin{proof}

  By [this Prop.](finite+limit#FiniteLimitsFromPullbacksAndTerminalObject)
  finite limits are preserved as soon as

  1. the [[terminal object]] is preserved, which here is evidently the case;

  1. [[pullbacks]] are preserved, which is the case by Prop. \ref{RealizationPreservesPullbacks}.

\end{proof}

#### Fat geometric realization

The operation of [fat geometric realization](#FatGeometricRealization) does not preserve [[fiber product]]s on the nose, in general, but it does preserve all [[finite limit]]s up to [[homotopy]].

Write ${\Vert * \Vert}$ for the fat geometric realization of the point. Notice that due to the identification of $sTop$ with its [[overcategory]] over the point (the simplicial topological space constant on the point), $sTop\simeq sTop/*$, we may regard fat geometric gealization as a functor with values in the [[overcategory]] $Top/{\Vert*\Vert}$ over the
[fat geometric realization of the point](#GeometricRealizationOfThePoint).

\begin{prop}\label{FatRealizationPreservesConnectedFiniteLimits}
The functor

$$
  {\Vert - \Vert} \colon  Top^{\Delta^{op}} \to Top/{\Vert*\Vert}
$$

preserves all [[finite limits]].
\end{prop}

This is claimed in [Gepner-Henriques 07, Remark 2.23](#GepnerHenriques07).

### Preservation of homotopy limits
 {#PreservationOfHomotopyLimits}

On [[homotopy limits]]:

\begin{proposition}
\label{SufficientConditionsForRealizationToPreserveHomotopyPullback}
**([[homotopy pullbacks]] [[preserved limit|preserved]] by geometric realization)**
\linebreak
For a morphism of simplicial spaces
\[
  \label{AMorphismOfSimplicialTopologicalSpacesToBeHomotopyPulledBack}
  f_\bullet \,\colon\, X_\bullet \xrightarrow{\;} Y_\bullet
  \,,
\]
the following are sufficient conditions for the geometric realization of any [[homotopy pullback]]-square of $f_\bullet$ to be a [[homotopy pullback]]-square in topological spaces (assuming throughout that all simplicial spaces are good or that the realization is taken to be fat, so that realization computes the [[homotopy colimit]]):

1. {#BousfieldFriedlanderPiStarKanCondition} [Bousfield & Friedlander 1978, Thm. B.4](#BousfieldFriedlander78):

   1. on [[simplicial sets]] of [[connected components]], $\pi_0(f)_\bullet$ is a [[Kan fibration]],

   1. the simplicial spaces $X_\bullet$ and $Y_\bullet$ satisfy the [[π-Kan condition]];

1. [Anderson 1978, p. 2](#Anderson78):

   1. on [[simplicial sets]] of [[connected components]], $\pi_0(f)_\bullet$ is a [[Kan fibration]],

   1. the component spaces $X_n$, $Y_n$ ($n \in \mathbb{N}$) are each [[connected topological space|connected]] or [[discrete topological space|discrete]];

1. {#RezkConditionForPreservationOfHomotopyPullbacks} [Rezk 2014, Prop. 5.4](#Rezk14):

   * the [[simplicial set]] of [[connected components]] of the base is [[constant functor|constant]]: $\pi_0(Y)_\bullet \simeq const\big(\pi_0(Y_0)\big)_\bullet$.

   This is also essentially [Lurie HA, Lem. 5.5.6.17](#LurieHigherAlgebra) (using that simplicial $\infty$-colimits are [[sifted infinity-colimit|sifted]], by [this Prop.](sifted+infinity-colimit#SimplicialInfinityColimitsAreSifted)), which generalizes the statement to [[(infinity,1)-toposes|$(\infty,1)$-toposes]] other than $L_W Top$.

1. {#KanLikeFibrationConditionForPreservationOfHomotopyPullbacks} [Mazel-Gee 2014, Cor. 6.7](#MazelGee14), [Lurie 2011, Prop. 10](#Lurie11):

   * the morphism (eq:AMorphismOfSimplicialTopologicalSpacesToBeHomotopyPulledBack) is a [[homotopy Kan fibration]], in that for all [[positive numbers]] $n \in \mathbb{N}_+$ and all $0 \leq k \leq n$ the induced map

     \[
       \label{HomotopyTheoreticKanFibration}
       X(\Delta^n)
       \xrightarrow{\;\;}
       X(\Lambda^n_k)
         \underset
           { Y(\Lambda^n_k) }
           {\times^h}
       Y(\Delta^n)
     \]

     (into the [[homotopy fiber product]] of the space of space of [[horns|$(n,k)$-horns]] in $X$ with that of [[simplex|$n$-simplices]] in $Y$)

     is [[surjective]] on [[connected components]],

     hence in that for all solid homotopy-commutative squares as follows, there exists a dashed lift up to homotopy (with the left morphism being the [[horn|$(n,k)$-horn]]-inclusion in [[simplicial sets]] regarded as degree-wise [[discrete topological spaces]]):

\begin{tikzcd}
        \Lambda^n_k
        \ar[
          rr,
          "{\ }"{below, name=s1, pos=.7}
        ]
        \ar[d]
        &&
        \mathrm{X}_\bullet
        \ar[
          d,
          "{f_\bullet}"{right}
        ]
        \\
        \Delta^n
        \ar[
          rr,
          "{\ }"{above, name=t2, pos=.3}
        ]
        \ar[
          urr,
          dashed,
          "{\ }"{above, name=t1},
          "{\ }"{below, name=s2}
        ]
        &&
        \mathrm{Y}_{\bullet}
        \mathrlap{\,,}
        %
        \ar[
          from=s1,
          to=t1,
          Rightarrow,
          dashed
        ]
        \ar[
          from=s2,
          to=t2,
          Rightarrow,
          dashed
        ]
\end{tikzcd}

\end{proposition}

### Relation to the homotopy colimit
  {#RelationToHomotopyColimit}

#### For proper simplicial spaces

Recall that a simplicial topological space is proper if it is Reedy cofibrant relative to the [[Strøm model structure]] on [[Top]], in which the weak equivalences are the honest [[homotopy equivalences]].  Nevertheless, in certain cases geometric realisation computes the [[homotopy colimit]] of the [[diagram]] $X_\bullet \colon  \Delta^{op} \to Top$ given by the simplicial space, with respect to the standard [[Quillen model structure on topological spaces]] in which the weak equivalences are the [[weak homotopy equivalences]].

+-- {: .num_lemma #RealizPresWHE}
###### Lemma
If $X_\bullet \to Y_\bullet$ is an objectwise weak homotopy equivalence between proper simplicial spaces, then the induced map ${|X_\bullet|} \to {|Y_\bullet|}$ is a weak homotopy equivalence.
=--
+-- {: .proof}
###### Proof
The following proof is essentially from ([May 1974, A.4](#May_Einf)); see also ([Dugger, prop. 17.4, example 18.2](#Dugger)).  It relies on two facts relating [[Hurewicz cofibration]]s to [[weak homotopy equivalence]]s:

1. [[pushout|Pushouts]] along [[Hurewicz cofibration]]s preserve [[weak homotopy equivalence]]s, and

1. [[colimit|Colimits]] of sequences of Hurewicz cofibrations preserve weak homotopy equivalences.

Let $i_n \colon \Delta_{\le n} \hookrightarrow \Delta$ denote the inclusion of the objects $\le n$, and write ${|X_\bullet|}_n = i_n^* \Delta \otimes_{\Delta_{\le n}} i_n^* X_\bullet$.  Writing $L_n X$ for the $n$th [[latching object]] (the subspace of degeneracies in $X_n$), we have pushouts
$$\array{ (L_n X \times \Delta^n) \sqcup_{L_n X \times \partial\Delta^n} (X_n \times \partial\Delta^n) & \to &  {|X_\bullet|}_{n-1}\\
  \downarrow & & \downarrow\\
  X_n \times \Delta^n & \to &  {|X_\bullet|}_n }
$$
Since $X$ is proper, $L_n X \to X_n$ is a cofibration, and of course $\partial\Delta^n \to \Delta^n$ is a cofibration.  Thus, by the [[pushout-product axiom]] for the [[Strøm model structure]], the left-hand vertical map is a cofibration; hence so is the right-hand vertical map.

Now ${|X_\bullet|}$ is the [[colimit]] of the sequence of cofibrations
$$  {|X_\bullet|}_0 \to {|X_\bullet|}_1 \to {|X_\bullet|}_2 \to \cdots$$
and likewise for ${|Y_\bullet|}$.  In other words, the geometric realization is [[filtration|filtered]] by simplicial degree.  Thus, by point (2) above, it suffices to show that each map ${|X_\bullet|}_n \to {|Y_\bullet|}_n$ is a weak homotopy equivalence.

Since ${|X_\bullet|}_0 = X_0$, this is true for $n=0$.  Moreover, by the above pushout square, ${|X_\bullet|}_n$ is a pushout of ${|X_\bullet|}_{n-1}$ along a cofibration.  Thus, by point (1) above, since $X_n \times \Delta^n \to Y_n \times \Delta^n$ is certainly a weak homotopy equivalence, it will suffice for an induction step to prove that
$$(L_n X \times \Delta^n) \sqcup_{L_n X \times \partial\Delta^n} (X_n \times \partial\Delta^n) \to
(L_n Y \times \Delta^n) \sqcup_{L_n Y \times \partial\Delta^n} (Y_n \times \partial\Delta^n)
$$
is a weak homotopy equivalence.  However, by definition we have a pushout
$$ \array{
  L_n X \times \partial\Delta^n & \to & X_n \times \partial\Delta^n \\
  \downarrow & & \downarrow\\
  L_n X \times \Delta^n & \to &
  (L_n X \times \Delta^n) \sqcup_{L_n X \times \partial\Delta^n} (X_n \times \partial\Delta^n) }
$$
This is also a pushout along a Hurewicz cofibration, and cartesian product preserves weak homotopy equivalences, so it will suffice to show that $L_n X \to L_n Y$ is a weak homotopy equivalence.

Recall that $L_n X$ can be written as $\colim^{[n]\to [k]} X_{[k]}$, where the colimit is over all codegeneracy maps $[n]\to [k]$ in $\Delta$ *except* the identity $[n] \to [n]$.  For $0\le m \le n$, write $L^m_n X$ for the corresponding colimit over all codegeneracies which factor through $\sigma_{p}\colon [n]\to [n-1]$ for some $0\le p\le m$.  Then $L^0_n X = X_{n-1}$ and $L^n_n X = L_n X$, and for $0\lt m \le n$ we have a pushout square
$$ \array{ L^{m-1}_{n-1} X & \to & L^{m-1}_n X \\
  \downarrow & & \downarrow \\
  X_{n-1} & \to & L^m_n X. }
$$
We claim that $L^{m-1}_n X \to L^m_n X$ is a cofibration for all $0\le m \le n$, and we prove it by induction on $n$.  For $n=0$ it is obvious.  If it holds for $(n-1)$ (and all $m$ with $0\le m \le n-1$), then by composition and properness of $X$, each map $L^{m-1}_{n-1} X \to X_{n-1}$ is a cofibration.  Hence, by the above pushout square, so is $L^{m-1}_n X \to L^m_n X$.  This proves the claim.

Now, using the above pushout square again and point (1) above, we can prove by induction on $n$, and for fixed $n$, by induction on $m$, that each map $L^m_n X \to L^m_n Y$ is a weak homotopy equivalence.  In particular, taking $m=n$, we find that $L_n X \to L_n Y$ is a weak homotopy equivalence, as desired.
=--

Recall that one way to compute the homotopy colimit of a diagram $X\colon D^{op}\to Top$, with respect to the standard (Quillen) model structure, is as the tensor product
$$hocolim X \coloneqq N(D/ -) \otimes_D Q X,$$
where $(D/ -)\colon D \to Cat$ sends each object of $D$ to its [[overcategory]], $N$ denotes the [[nerve]] of a small category, and $Q$ denotes a functorial cofibrant replacement in the [[Quillen model structure on topological spaces]] (e.g. [[CW complex|CW]] replacement via singular [[nerve and realization]]).  When $D=\Delta$, there is a canonically defined map $N(\Delta / -) \to \Delta$ (where the second $\Delta$ denotes the canonical cosimplicial simplicial set) called the [[Bousfield-Kan map]].  This map induces, for each simplicial space $X$, a map
$$ N(D/ -) \otimes_D X_\bullet \to \Delta \otimes_{\Delta} X_\bullet = {|X_\bullet|} $$
which is also called the **Bousfield-Kan map**.

Since the Strøm model structure is a [[simplicial model category]], standard arguments involving Reedy model structures imply that the [[Bousfield-Kan map]] is a Strøm [[weak equivalence]] (i.e. a [[weak homotopy equivalence]]) whenever $X$ is Strøm Reedy cofibrant (i.e. proper).  Thus we have (see also [Arkhipov & Ørsted 2018, Exp. 6.4](#ArkhipovOrsted18)):

+-- {: .num_prop #RealizationOfGoodSimplicialSpacesIsHomotopyColimit}
###### Proposition
Let $X_\bullet$ be a proper simplicial space.  Then the composite
$$ \hocolim X_\bullet = N(D/ -) \otimes_D Q X_\bullet \xrightarrow{Bousfield-Kan} {|Q X_\bullet|} \longrightarrow {|X_\bullet|} $$
is a weak homotopy equivalence.
=--
+-- {: .proof}
###### Proof
Since the definition of $\hocolim$ doesn't depend on the choice of an objectwise-cofibrant replacement of $X$, we may as well take $Q$ to be, instead of the composite with a functorial cofibrant replacement in $Top_{Quillen}$, rather a cofibrant replacement in the Reedy model structure on $Top^{\Delta^{op}}$ with respect to the Quillen model structure on $Top$.  (Any Reedy cofibrant diagram is in particular objectwise cofibrant.)  Then $Q X_\bullet$ is Quillen-Reedy cofibrant, hence also proper, and so the Bousfield-Kan map is a homotopy equivalence.  On the other hand, $Q X_\bullet \to X_\bullet$ is a levelwise weak homotopy equivalence, while $Q X_\bullet$ and $X_\bullet$ are proper, so by Lemma \ref{RealizPresWHE} its realization is a weak homotopy equivalence.
=--

By naturality, the above composite is also equal to the composite
$$ \hocolim X_\bullet = N(D/ -) \otimes_D Q X_\bullet \to N(D/ -) \otimes_D X_\bullet \xrightarrow{Bousfield-Kan} {|X_\bullet|}; $$
hence this composite is also a weak homotopy equivalence.

\linebreak

#### For degreewise cofibrant simplicial spaces
 {#RelationToHomotopyColimitForDegreewiseCofibrationSimplicialSpaces}

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

Let $X_\bullet$ be a [[simplicial topological space]] which is degreewise a [[cofibrant object]] in the [[classical model structure on topological spaces]], hence which is degreewise a [[retract]] of a [[cell complex]] (for instance: degreewise a [[CW-complex]]).

Then its [[fat geometric realization]] models the [[homotopy colimit]] over $X_\bullet \;\colon\; \Delta^{op} \longrightarrow Top_{Quillen}$ in the [[classical model structure on topological spaces]]:

$$
  \left\Vert X_\bullet \right\Vert
  \;\simeq_{whe}\;
  hocolim X_\bullet
  \,.
$$

=--

This is claimed in [Wang 18, Theorem 4.3, Remark 4.4](#Wang18).

## Examples and Applications

### Topological principal $\infty$-bundles
 {#TopologicalPrincipalInfinityBundle}

We discuss aspects of [[principal ∞-bundle]]s equipped with topological [[cohesive (∞,1)-topos|cohesion]] and their geometric realization to [[principal bundle]]s in [[Top]].

For $X,Y \in sTop$, write

$$
  sTop(X,Y) \coloneqq \int_{[k] \in \Delta} [X_k, Y_k] \;\; \in Top
$$

for the [[Top]]-[[hom-object]], where in the integrand of the [[end]] $[-,-] \colon  Top^{op} \times Top^{op} \to Top$ is the [[internal hom]] of topological spaces.

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

We say a morphism $f \colon  X \to Y$ of [[simplicial topological space]]s is a **global Kan fibration** if for all $n \in \mathbb{N}$ and $0 \leq i \leq n$ the canonical morphism

$$
  X_n \to Y_n \;\times_{sTop(\Lambda[n]_i, Y)}\; sTop(\Lambda[n]_i, X)
$$

in [[Top]] has a [[section]], where  $\Lambda[n]_i \in $ [[sSet]] $\hookrightarrow Top^{\Delta^{op}}$ is the $i$th $n$-[[horn]] regarded as a [[discrete space|discrete]] [[simplicial topological space]].

We say a [[simplicial topological space]] $X_\bullet \in Top^{\Delta^{op}}$ is **(global) Kan simplicial space** if the unique morphism $X_\bullet \to *$ is a global Kan fibration, hence if for all $n \in \mathbb{N}$ and all $0 \leq i \leq n$ the canonical [[continuous function]]

$$
  X_n \to sTop(\Lambda[n]_i, X)
$$

into the [[topological space]] of $i$th $n$-[[horn]]s admits a [[section]] (in [[Top]], hence a global, continuous section).

=--

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

This global notion of topological Kan fibration is considered in ([Brown & Szczarba, def. 2.1, def. 6.1](#BrownSzczarba)). In fact there a stronger condition is imposed: a [[Kan complex]] in [[Set]] automatically has the lifting property not only against all full [[horn]] inclusions but also against sub-horns; and in ([Brown & Szczarba](#BrownSzczarba)) all these fillers are required to be given by global sections. This ensures that with $X$ globally Kan also the [[internal hom]] $[Y,X] \in sTop$ is globally Kan, for any simplicial topological space $Y$. This is more than we need and want to impose here. For our purposes it is sufficient to observe that if $f$ is globally Kan in the sense of ([Brown & Szczarba, def. 6.1](#BrownSzczarba)), then it is so also in the [above sense](#GloballyKanSimplicialTopologicalSpace).

=--

Recall from the discussion at [[universal principal ∞-bundle]] that for $G$ a [[simplicial topological group]] the [[universal simplicial principal bundle]] $\mathbf{E}G \to \mathbf{B}G$ is presented by the morphism of [[simplicial topological spaces]] traditionally denoted
$W G \to \overline{W} G$ (see also at *[[universal principal simplicial complex]]*).

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

Let $G$ be a [[simplicial topological group]]. Then:

1. $G$ is a globally Kan simplicial topological space;

1. $\overline{W} G$ is a globally Kan simplicial topological space;

1. $W G \to \overline{W} G$ is a global Kan fibration.

=--

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

The first statement appears as ([Brown & Szczarba, theorem 3.8](#BrownSzczarba)), the second is noted in ([Roberts & Stevenson](#RobertsStevenson)), the third appears as ([Brown & Szczarba, lemma 6.7](#BrownSzczarba)).

=--

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

If $G$ is a [[well-pointed simplicial topological group]] then both $W G$ and $\overline{W} G$ are [good simplicial topological spaces](#GoodAndProper).

=--

For $\bar W G$  this is ([RobertsStevenson, prop. 19](#RobertsStevenson)). For $W G$ this follows with ([RobertsStevenson, lemma 10, lemma 11](#RobertsStevenson)) which says that $W G = Dec_0 \bar W G$ and the observations in the proof of ([RobertsStevenson, prop. 16](#RobertsStevenson)) that $Dec_0 X$ is good if $X$ is.

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

For $G$ a [[well-pointed simplicial topological group]], the geometric realization of the [[universal simplicial principal bundle]] $W G \to \bar W G$

$$
  {\vert W G \vert} \to {\vert \bar W G \vert}
$$

is a fibration [[resolution]] in $Top_{Quillen}$ of the point inclusion
$* \to B{|G|}$ into the [[classifying space]] of the geometric realization of $G$.

=--

This is ([RobertsStevenson, prop. 14](#RobertsStevenson)).

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

Let $X_\bullet$ be a [good](#GoodAndProper) [[simplicial topological space]] and $G$ a [[well-pointed simplicial topological group]]. Then for every morphism

$$
  \tau \colon  X \to \bar W G
$$

the corresponding topological [[simplicial principal bundle]] $P$ over $X$ is itself a good simplicial topological space.

=--

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

The bundle is the [[pullback]] $P = X \times_{\overline{W} G} W G$ in $sTop$

$$
  \array{
    P
      &\longrightarrow&
    W G
    \\
    \big\downarrow
     &&
    \big\downarrow
    \\
    X
      &\stackrel{\tau}{\longrightarrow}&
    \overline{W} G
    \mathrlap{\,.}
  }
$$

By assumption on $X$ and $G$ and using prop. \ref{BarWGIsGoodIfGIsWellSectioned} we have that $X$, $\bar W G$ and $W G$ are all good simplicial spaces.

This means that the degeneracy maps of $P_\bullet$ are induced degreewise by morphisms between pullbacks in [[Top]] that are degreewise [[closed cofibrations]], where one of the morphisms in each pullback is a fibration. By the properties discussed at [[closed cofibration]] ([here](Hurewicz+cofibration#InteractionWithPullbacks)), this implies that also these degeneracy maps of $P_\bullet$ are closed cofibrations.

=--

Let for the following $Top_s \hookrightarrow Top$ be any [[small category|small]] [[full subcategory]].

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

Under the degreewise [[Yoneda embedding]] $sTop_s \hookrightarrow [Top_s^{op}, sSet]$ simplicial topological spaces embed into the category of [[simplicial presheaves]] on $Top_s$.

We equip this with the projective [[model structure on simplicial presheaves]] $[Top_s^{op}, sSet]_{proj}$, and we speak of [[homotopy limit]]s in $sTop$ under this embedding.

=--

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

Under this embedding a [global Kan fibration](#GloballyKanSimplicialTopologicalSpace) $f \colon  X \to Y$ in $sTop_s$ maps to a fibration in $[Top_s^{op}, sSet]_{proj}$.

=--

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

By definition, a morphism $f \colon  X \to Y$ in $[Top_s^{op}, sSet]_{proj}$ is a fibration if for all $U \in Top$ and all $n \in \mathbb{N}$ and $0 \leq i \leq n$ diagrams of the form

$$
  \array{
    \Lambda[n]_i \cdot U &\to& X
    \\
    \downarrow && \downarrow^{\mathrlap{f}}
    \\
    \Delta[n] \cdot U &\to& Y
  }
$$

have a lift. This is equivalent to saying that the [[function]]

$$
  Hom(\Delta[n]\cdot U, X)
   \to
  Hom(\Delta[n]\cdot U,Y)
    \times_{Hom(\Lambda[n]_i \cdot U, Y)}
  Hom(\Lambda[n]_i \cdot U, X)
$$

is surjective. Notice that we have

$$
  \begin{aligned}
    Hom_{[Top_s^{op}, sSet]}(\Delta[n]\cdot U, X)
    & =
    Hom_{sTop}(\Delta[n]\cdot U, X)
    \\
     & = \int_{[k] \in \Delta} Hom_{Top}( \Delta[n]_k \times U, X_k)
   \\
    & = \int_{[k] \in \Delta} Hom_{Top}(U, [\Delta[n]_k, X_k])
   \\
    & = Hom_{Top}(U, \int_{[k] \in \Delta} [\Delta[n]_k, X_k])
   \\
    & = Hom_{Top}(U, sTop(\Delta[n], X))
   \\
    & = Hom_{Top}(U, X_n)
  \end{aligned}
$$

and analogously for the other factors in the above morphism. Therefore the lifting problem equivalently says that the function

$$
  Hom_{Top}(U, \; X_n \to Y_n \times_{sTop(\Lambda[n]_i, Y)} sTop(\Lambda[n]_i,X) \;)
$$

is surjective. But by the assumption that $f \colon  X \to Y$ is a global Kan fibration of simplicial topological spaces, def. \ref{GloballyKanSimplicialTopologicalSpace}, we have a section $\sigma \colon  Y_n \times_{sTop(\Lambda[n]_i), Y} sTop(\Lambda[n]_i,X) \to X_n$. Therefore $Hom_{Top}(U, \sigma)$ is a section of our function.

=--

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

The [[homotopy colimit]] operation

$$
  sTop \hookrightarrow [Top_s^{op}, sSet]_{proj} \stackrel{hocolim}{\to} Top_{Quillen}
$$

preserves [[homotopy fiber]]s of morphisms $\tau \colon  X \to \bar W G$ with $X$ good and globally Kan and $G$ well-pointed.

=--

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

By prop. \ref{SimplicialTopologicalUniversalBundle}
and prop. \ref{GlobalKanFibImpliesProjectiveFib} we have that $W G \to \bar W G$ is a fibration [[resolution]] of the point inclusion $* \to \bar W G$ in $[Top^{op}, sSet]_{proj}$.
By the general discussion at [[homotopy limit]] this means that the [[homotopy fiber]] of a morphism $\tau \colon  X \to \bar W G$ is computed as the ordinary [[pullback]] $P$ in

$$
  \array{
    P &\to& W G
    \\
    \downarrow && \downarrow
    \\
    X &\stackrel{\tau}{\to}& \bar W G
  }
$$

(since all objects $X$,  $\bar W G$ and $W G$ are fibrant and at least one of the two morphisms in the pullback diagram is a fibration) and hence

$$
  hofib(\tau) \simeq P
  \,.
$$

By prop. \ref{SimplicialTopologicalUniversalBundle} and prop. \ref{SimplicialTopolgicalBundleIsGood} it follows that all objects here
are good simplicial topological spaces. Therefore by prop. \ref{RealizationOfGoodSimplicialSpacesIsHomotopyColimit} we have

$$
  hocolim P_\bullet \simeq {|P_\bullet|}
$$

in [[Ho(Top)]]. By prop. \ref{RealizationPreservesPullbacks} we have that

$$
  \cdots = {|X_\bullet|} \times_{|\bar W G|} {|W G|}
  \,.
$$

But prop. \ref{RealizationSimplicialTopologicalUniversalBundle} says that this is again the presentation of a homotopy pullback/homotopy fiber by an ordinary pullback

$$
  \array{
    {|P|} &\to& {|W G|}
    \\
    \downarrow && \downarrow
    \\
    {|X|} &\stackrel{\tau}{\to}& {|\bar W G|}
  }
  \,,
$$

because $|W G| \to |\bar W G|$ is again a fibration resolution of the point inclusion. Therefore

$$
  hocolim P_\bullet \simeq hofib( {|\tau|} )
  \,.
$$

Finally by prop. \ref{RealizationOfGoodSimplicialSpacesIsHomotopyColimit} and using the assumption that $X$ and $\bar W G$ are both good, this is

$$
  \cdots \simeq hofib (hocolim \tau)
  \,.
$$

In total we have shown

$$
  hocolim (hofib \tau) \simeq hofib (hocolim \tau)
  \,.
$$

=--

### Classifying spaces
  {#ClassifyingSpaces}
For $G$ a [[topological group]], write $\mathbf{B}G$ for its [[delooping]] [[topological groupoid]]: the topological groupoid with a single object and $Mor_{\mathbf{B}G}(*,*) \coloneqq  G$, with composition given by the product on $G$.

The [[nerve]] $N \mathbf{B}G$ of this topological groupoid is naturally a [[simplicial topological space]], with

$$
  N \mathbf{B}G \colon  [n] \mapsto G^{\times_n}
  \,.
$$

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

The geometric realization of $N \mathbf{B}G$ is a model for the [[classifying space]] $B G$ of $G$-[[principal bundle]]s

$$
  {|N \mathbf{B}G|} \simeq B G
  \,.
$$

=--

An early reference for this classical fact is ([Segal68](#Segal68)).

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

Let $G$ be a [[well-pointed simplicial topological group|well-pointed]] [[topological group]], $B G$ its [[classifying space]] and $\Omega B G$ the [[loop space]] of the classifying space.

There is a [[weak homotopy equivalence]]

$$
  \Omega B G \stackrel{\simeq}{\to} G
  \,.
$$

=--

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

By prop. \ref{ClassifyingSpaceByGeometricRealization} we have
that $B G \simeq |\mathbf{B}G|$ (we suppress the [[nerve]] operation notationally, which injects [[groupoid]]s into [[∞-groupoid]]s).

By standard facts about the $\bar W$-functor (see [[simplicial principal bundle]]) we have a [[pullback]] square of [[simplicial topological space]]s

$$
  \array{
    G &\to& W G
    \\
    \downarrow && \downarrow
    \\
    * &\to& \bar W G
  }
$$

exhibiting the [[homotopy pullback]]

$$
  \array{
    G &\to& *
    \\
    \downarrow &\swArrow_{\simeq}& \downarrow
    \\
    * &\to& \mathbf{B}G
  }
  \,.
$$

Under geoemtric realization this maps to

$$
  \array{
    G &\to& *
    \\
    \downarrow &\swArrow_{\simeq}& \downarrow
    \\
    * &\to& B G
  }
$$

in [[Top]]. By prop \ref{HocolimPreservesHomotopyFibers} this is still a homotopy pullback, and hence exhibits $G$ as the [[loop space]] of $B G$.

=--

### Cech nerves
 {#CechNerves}

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

Let $X,Y$ be  [[topological space]]s and $\pi \colon  Y \to X$ a [[continuous function]] that admits local [[section]]s. Write $C(\pi) \in sTop$ for the [[Cech nerve]].

Then the canonical map

$$
  {|C(\pi)|} \to X
$$

from the geometric realization of $X$ back to $X$ is a [[weak homotopy equivalence]]. If $X$ is a [[paracompact topological space]] then it is even a [[homotopy equivalence]].

=--

For paracompact $X$ this goes back to ([Segal68](#Segal68)). The general case is discussed in ([DuggerIsaksen](#DuggerIsaksen)). A generalization to parameterized spaces is in ([RobertsStevenson, lemma 22](#RobertsStevenson)).

## Related concepts

* [[simplicial topological space]], [[nice simplicial topological space]]

* [[simplicial topological group]]

* [[geometric realization]]

  * [[geometric realization of categories|of categories]], **of simplicial topological spaces**, [[geometric realization of cohesive ∞-groupoids|of cohesive ∞-groupoids]]

## References

### General
 {#General}

The realization construction of simplicial topological spaces is implicit in classical discussion of [[classifying spaces]] $B G$, in particular

* {#Milgram67} [[R. James Milgram]], *The bar construction and abelian H-spaces*, Illinois J. Math. 11(2): 242-250 (1967) ([doi:10.1215/ijm/1256054662](https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-11/issue-2/The-bar-construction-and-abelian-H-spaces/10.1215/ijm/1256054662.full))

which may be understood as the realization of the [[nerves]] of their [[topological groupoid|topological]] [[delooping groupoids]] $G \rightrightarrows \ast$,

More explicit allusion to realization at least of [[semi-simplicial objects|semi-simplicial]] [[nerves]] of [[topological categories]] appeared, without details, in:

* {#Segal68} [[Graeme Segal]],  Section 2 of: _Classifying spaces and spectral sequences_, Publications Mathématiques de l'IHÉS, 34 (1968), p. 105-112  ([numdam:PMIHES_1968__34__105_0](http://www.numdam.org/item?id=PMIHES_1968__34__105_0))

The detailed [[coend]] formula for and the relevance of [[compactly generated topological spaces]] for [[preserved limit|preservation]] at least of [[finite products]] is due (with reference to [Milgram 1967](#Milgram67)) to:

* {#MacLane} [[Saunders MacLane]], Section 6 of: _The Milgram bar construction as a tensor product of functors_,  In: F.P. Peterson  (eds.) *The Steenrod Algebra and Its Applications: A Conference to Celebrate N.E. Steenrod's Sixtieth Birthday* Lecture Notes in Mathematics *168*,  Springer, 1970 ([doi:10.1007/BFb0058523](https://doi.org/10.1007/BFb0058523), [pdf](https://link.springer.com/content/pdf/10.1007/BFb0058523.pdf))

Further details, including proof of the preservation not just of binary products but also of pullbacks appears in:

* {#May72} [[Peter May]], Section 11 of: *The geometry of iterated loop spaces*, Springer 1972 ([pdf](https://www.math.uchicago.edu/~may/BOOKS/geom_iter.pdf), [doi:10.1007/BFb0067491](https://link.springer.com/book/10.1007/BFb0067491))

The notion of *good* simplicial spaces is due to

* {#Segal73} [[Graeme Segal]], Appendix 2 of: *Configuration-spaces and iterated loop-spaces*, Invent. Math. __21__ (1973), 213–221.  ([doi:10.1007/BF01390197](https://doi.org/10.1007/BF01390197), [pdf](http://dodo.pdmi.ras.ru/~topology/books/segal.pdf), MR 0331377)

and the notion of *fat* geometric realization together with its homotopy equivalence to ordinary realization for good simplicial spaces is due to:

* {#Segal74} [[Graeme Segal]], Appendix A of: _[[SegalCategoriesAndCohomologyTheories.pdf:file]]_, Topology, 13:293–312, 1974 (<a href="https://doi.org/10.1016/0040-9383(74)90022-6">doi:10.1016/0040-9383(74)90022-6</a>)

* {#tomDieck74} [[Tammo tom Dieck]], *On the homotopy type of classifying spaces*, Manuscripta Math 11, 41–49 (1974) ([doi:10.1007/BF01189090](https://doi.org/10.1007/BF01189090))

Discussion of fat geometric realization as representing the [[homotopy colimit]] (as a [[homotopy coend]]):

* {#ArkhipovOrsted18} [[Sergey Arkhipov]], [[Sebastian Ørsted]], Exp. 6.4 in: *Homotopy (co)limits via homotopy (co)ends in general combinatorial model categories* ([arXiv:1807.03266](https://arxiv.org/abs/1807.03266))

More discussion of the comparison of different realization functors is in:

* {#Wang17} Yi-Sheng Wang, _Fat realization and Segal's classifying space_ ([arXiv:1710.03796](https://arxiv.org/abs/1710.03796))

* {#Wang18} Yi-Sheng Wang, _Geometric realization and its variants_ ([arXiv:1804.00345](https://arxiv.org/abs/1804.00345))

A proof that good simplicial spaces are proper is implicit in the proof of   [Segal 1974, Lemma A.5](#Segal74). It appears explicitly in

* {#GaunceLewis82} [[L. Gaunce Lewis, Jr.]], _When is the natural map $X\to \Omega \Sigma X$ a cofibration?_ , Trans. Amer. Math. Soc. **273** (1982) no. 1, 147--155 ([jstor:1999197](https://www.jstor.org/stable/1999197))

A generalization of the statement that good implies proper to other [[topological concrete categories]] and a discussion of the geometric realization of $W G \to \bar W G$ for $G$ a [[simplicial topological group]] is in

* {#RobertsStevenson} [[David Roberts]], [[Danny Stevenson]], _Simplicial principal bundles in parametrized spaces_, New York Journal of Mathematics Volume 22 (2016) 405-440 ([arXiv:1203.2460](https://arxiv.org/abs/1203.2460), [nyjm:22-19](http://nyjm.albany.edu/j/2016/22-19.html))

* {#Stevenson} [[Danny Stevenson]], _Classifying theory for simplicial parametrized groups_ ([arXiv:1203.2461](http://arxiv.org/abs/1203.2461))

Comments on the relation between properness and cofibrancy in the [[Reedy model structure]] on $[\Delta^{op}, Set]$ are made in

* [[Paul Goerss]], [[Kristen Schemmerhorn]], _Model Categories and Simplicial Methods_ ([pdf](http://www.math.northwestern.edu/~pgoerss/papers/ucnotes.pdf)).

The relation between (fat) geometric realization and [[homotopy colimit]]s is considered as prop. 17.5 and example 18.2 of

* {#Dugger} [[Daniel Dugger]], _A primer on homotopy colimits_ ([pdf](http://pages.uoregon.edu/ddugger/hocolim.pdf))

The proof that geometric realization of proper simplicial spaces preserves weak equivalences is from

* {#May_Einf} [[Peter May]], *$E_\infty$-spaces, group completions, and permutative categories*,  London Math. Soc. Lecture Notes No. 11, 1974, 61-94 ([doi:10.1017/CBO9780511662607.008](https://doi.org/10.1017/CBO9780511662607.008), [pdf](http://www.math.uchicago.edu/~may/PAPERS/13.pdf))

A definition of the Bousfield-Kan map, and the Reedy model category theory necessary to show that it is a weak equivalence, can be found in

* Hirschhorn, *Model categories and their localizations*, AMS Mathematical Surveys and Monographs No. 99, 2003

On the (fat) geometric realization of ([[nerves]] of) [[topological groupoids]]:

* {#GepnerHenriques07} [[David Gepner]], [[André Henriques]], Section 2.3 in: _Homotopy theory of orbispaces_ ([arXiv:math/0701916](https://arxiv.org/abs/math/0701916))

See also

* [[David Carchedi]], section 3.2 of _On The Homotopy Type of Higher Orbifolds and Haefliger Classifying Spaces_ ([arXiv:1504.02394](http://arxiv.org/abs/1504.02394))

On globally Kan simplicial spaces:

* {#BrownSzczarba} [[Edgar H. Brown]], R. H. Szczarba, _Continuous cohomology and real homotopy type_ , Trans. Amer. Math. Soc. 311 (1989), no. 1, 57 ([pdf](http://www.ams.org/journals/tran/1989-311-01/S0002-9947-1989-0929667-6/S0002-9947-1989-0929667-6.pdf))

On the [[right adjoint]] to geometric realization of simplicial topological spaces

* {#Seymour} R. M. Seymour, _Kan fibrations in the category of simplicial spaces_ Fund. Math., 106(2):141-152, 1980.

On geometric relatization of [[semi-simplicial topological spaces]]:

* [[Johannes Ebert]], [[Oscar Randal-Williams]], *Semi-simplicial spaces*, Algebr. Geom. Topol. 19 (2019) 2099-2150 ([arXiv:1705.03774](https://arxiv.org/abs/1705.03774))

On geometric realization of general [[Cech nerves]]:

* {#DuggerIsaksen} [[Dan Dugger]], [[Dan Isaksen]], _Topological hypercovers and $\mathbb{A}^1$- realizations, Math. Z. 246 (2004) no. 4

On geometric realization of simplicial spaces as a [[left Quillen functor]] on the [[Reedy model category]] of [[Top]]:

* [[Lars Hesselholt]], *Reedy model structure on the category of simplicial spaces; geometric realization is a left Quillen functor*, Lecture 11 in [Algebraic Topology I](https://www.math.nagoya-u.ac.jp/~larsh/teaching/S2008/), 2008 ([pdf](https://www.math.nagoya-u.ac.jp/~larsh/teaching/S2008/lecture11.pdf), [[Hesselholt_RealSimpSpOnReedy.pdf:file]])

### (Non-)Compatibility with homotopy pullbacks
 {#ReferencesCompatibilityHomotopyPullback}

Discussion of sufficient conditions (cf. *[[π-Kan condition]]*) for geometric realization to be compatible with [[homotopy pullbacks]]:

* {#BousfieldFriedlander78} [[Aldridge Bousfield]], [[Eric Friedlander]], _Homotopy theory of $\Gamma$-spaces, spectra, and bisimplicial sets_, Springer Lecture Notes in Math., Vol. 658, Springer, Berlin, 1978, pp. 80-130. ([pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/bousfield-friedlander.pdf), [[BousfieldFriedlanderSpectra.pdf:file]])

* {#Anderson78} [[Donald Werner Anderson]], _Fibrations and geometric realization_,  Bull. Amer. Math. Soc. Volume 84, Number 5 (1978), 765-788. ([euclid:1183541139](http://projecteuclid.org/euclid.bams/1183541139))

* {#Lurie11} [[Jacob Lurie]], *Simplicial spaces*, Lecture 7 of: *[Algebraic L-theory and Surgery](https://www.math.ias.edu/~lurie/287x.html)*, 2011 ([pdf](https://www.math.ias.edu/~lurie/287xnotes/Lecture7.pdf))

* {#Rezk14} [[Charles Rezk]], _When are homotopy colimits compatible with homotopy base change?_, 2014 ([pdf](https://faculty.math.illinois.edu/~rezk/i-hate-the-pi-star-kan-condition.pdf), [[RezkHomotopyColimitsBaseChange.pdf:file]])

* Edoardo Lanari, _Compatibility of homotopy colimits and homotopy pullbacks of simplicial presheaves_ ([pdf](http://algant.eu/documents/theses/lanari.pdf), [[LanariHomotopyColimitsBaseChange.pdf:file]])

  > (expanded version of [Rezk 14](#Rezk14))

* {#MazelGee14} [[Aaron Mazel-Gee]], _Model $\infty$-categories I: some pleasant properties of the $\infty$-category of simplicial spaces_ ([arXiv:1412.8411](https://arxiv.org/abs/1412.8411))

* {#LurieHigherAlgebra} [[Jacob Lurie]], around Lem. 5.5.6.17 in: *[[Higher Algebra]]* (~2017)

* [[Jacob Lurie]], Thms. A.5.4.1, *[[Spectral Algebraic Geometry]]* (2018) [[pdf](https://math.harvard.edu/~lurie/papers/SAG-rootfile.pdf)]

* {#BEBP19} [[Daniel Berwick-Evans]], [[Pedro Boavida]], [[Dmitri Pavlov]], §3 of: _Classifying spaces of infinity-sheaves_, Algebraic & Geometric Topology  [[arXiv:1912.10544](https://arxiv.org/abs/1912.10544)]

### Realization of topological stacks

On geometric realizations of [[topological stacks]]:

* [[Johannes Ebert]], *The homotopy type of a topological stack* ([arXiv:0901.3295](https://arxiv.org/abs/0901.3295))

[[!redirects geometric realization of a simplicial space]]
[[!redirects geometric realization of a simplicial topological space]]

[[!redirects geometric realization of simplicial spaces]]

[[!redirects fat geometric realization of simplicial spaces]]
[[!redirects fat geometric realization of simplicial topological spaces]]
[[!redirects fat geometric realization of a simplicial space]]
[[!redirects fat geometric realization of a simplicial topological space]]
[[!redirects fat geometric realization]]
[[!redirects fat realization]]

[[!redirects topological realization of simplicial topological spaces]]
[[!redirects topological realizations of simplicial topological spaces]]

[[!redirects geometric realization of topological stacks]]
[[!redirects geometric realizations of topological stacks]]


geometric representation theory

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#### Representation theory
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=--
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#Contents#
* table of contents
{:toc}

## Idea

**Geometric representation theory** studies [[representation]]s (of various symmetry objects like [[algebraic group|algebraic groups]], [[Hecke algebra]]s, [[quantum groups]], [[quiver]]s etc.) realizing them by [[higher geometry|geometric means]], e.g. by geometrically defined [[actions]] on [[sections]] of various [[bundles]] or [[sheaves]] as in [[geometric quantization]] (see at _[[orbit method]]_), [[D-modules]], [[perverse sheaf|perverse sheaves]], [[deformation quantization]] [[modules]] and so on.

Typically the underlying spaces for the sheaves involved are [[Grassmannian]]s, [[flag variety|flag varieties]], [[configuration space]]s and the like. Another important tool are cohomological vanishing theorems in appropriate contexts. Some historical landmarks are the [[Borel-Weil theorem]], the [[Borel-Weil-Bott theorem]], [[Kazhdan-Lusztig conjecture]], the [[BBDG decomposition theorem]], the [[Beilinson-Bernstein localization theorem]] and the [[Lusztig conjectures]].

Quoting from ([MSRI 14](#MSRI14)):

> Representation theory is the study of the basic symmetries of mathematics and physics. Symmetry groups come in many different flavors: finite groups, Lie groups, p-adic groups, loop groups, adelic groups,.. A striking feature of representation theory is the persistence of fundamental structures and unifying themes throughout this great diversity of settings. One such theme is the Langlands philosophy, a vast nonabelian generalization of the Fourier transform of classical harmonic analysis, which serves as a visionary roadmap for the subject and places it at the heart of number theory.

> The fundamental aims of geometric representation theory are to uncover the deeper geometric and categorical structures underlying the familiar objects of representation theory and harmonic analysis, and to apply the resulting insights to the resolution of classical problems. A groundbreaking example of its success is Beilinson-Bernstein's generalization of the Borel-Weil-Bott theorem, giving a uniform construction of all representations of Lie groups via the geometric study of differential equations on flag varieties.

> The geometric study of representations often reveals deeper layers of structure in the form of categorification. Categorification typically replaces numbers (such as character values) by vector spaces (typically cohomology groups), and vector spaces (such as representation rings) by categories (typically of sheaves). It is a primary explanation for miraculous integrality and positivity properties in algebraic combinatorics. A recent triumph of geometric methods is Ngô's proof of the Fundamental Lemma, a key technical ingredient in the Langlands program. The proof relies on the cohomological interpretation of orbital integrals, which makes available the deep topological tools of algebraic geometry (such as Hodge theory and the Weil conjectures).

A similar description was given by [[David Ben-Zvi]] in an [MO answer](http://mathoverflow.net/questions/126474/new-geometric-methods-in-number-theory-and-automorphic-forms/126540#126540)

> Representation theory is the study of the basic symmetries of mathematics and physics. The primary aim of the subject is to understand concrete linear models for abstract symmetry groups. A signature triumph of the past century is our understanding of compact Lie groups. At the foundation, there is Cartan's classification of Lie algebras and Borel-Weil-Bott's uniform construction of all representations in the cohomology of line bundles on flag varieties. Thus we have a list of every compact Lie group we could ever encounter and every way in which it could appear concretely as a matrix group. Furthermore, there is a deep combinatorial theory of other key structures such as the tensor product of representations. The ideas and results of this subject are the basic input to diverse areas from number theory to quantum field theory.

> Though mysteries still remain, the theory of compact Lie groups is a representative model for what we would like to achieve with other symmetry groups. Because of their universal importance, symmetry groups come in many different flavors: finite groups, Lie groups, p-adic groups, loop groups, adelic groups,... and the list will only increase with the discovery of new important structures. For the above examples, our understanding is still very coarse though the last decades have witnessed breathtaking advances. The Langlands program along with its geometric spinoffs provide a visionary roadmap for where the subject could go in the coming years. In particular, current developments such as the recent proof of the Fundamental Lemma create great optimism that geometric techniques will have a deep impact.

> Geometric representation theory seeks to understand groups and representations as a consequence of more subtle but fundamental symmetries. A groundbreaking example of its success is Beilinson-Bernstein's uniform construction of all representations of Lie groups via the geometry of D-modules on flag varieties. The result is not difficult to state and prove but has the Borel-Weil-Bott theorem and the Kazhdan-Lusztig multiplicity conjectures as immediate consequences. A reasonable reaction is to wonder what allows one to prove deep results about representations of Lie groups with so little effort. One answer is that the true focus of the Beilinson-Bernstein theory is not representations but rather the symmetries of flag varieties. The geometric notion of D-module allows one to localize symmetries and to apply the sheaf-theoretic techniques of algebraic geometry. In this way, our understanding of representations of Lie groups follows from that of infinitesimal symmetries of algebraic varieties.

> There are now many instances where difficult questions about representations can be translated into more tractable questions about geometry. Other famous examples include the Deligne-Lusztig theory of representations of finite groups of Lie type, the Springer theory of representations of Weyl groups, the Kazhdan-Lusztig theory of modules for Hecke algebras, and Lusztig and Nakajima's theory of representations of Kac-Moody algebras and quantum groups via quiver varieties. In the theory of automorphic forms, one of the fundamental tools is the realization of representations of adelic groups in the cohomology of Shimura varieties (in the case of number fields) and Drinfeld modular varieties (in the case of function fields). The recent proof of the Fundamental Lemma exploits the fact that computing orbital integrals in p-adic groups can be reduced to calculating cohomology of fibers of Hitchin's integrable system.

> The twin modern goals of geometric representation theory are to explore the above dictionaries and to discover new unexpected ones. As an important consequence, the geometric realization of representations often reveals deeper layers of structure in the form of categorification. Categorification typically turns numbers (for example, the coefficients of Kazhdan-Lusztig polynomials) into the dimensions of vector spaces (in this case, the Ext groups of intersection cohomology sheaves). It is a primary explanation for miraculous integrality and positivity properties in algebraic combinatorics. At the center of geometric representation theory is Grothendieck's categorification of functions by ℓ-adic sheaves. An important example is Lusztig's theory of character sheaves: it provides a uniform geometric source for the characters of all finite groups of Lie type. Another important example is the theory of canonical bases: here categorification replaces representation spaces with linear categories equipped with canonical generating objects. Another broad example is the geometric Langlands program: it provides a categorification of the Langlands program in the setting of function fields, providing new insights into many classical constructions.

> Geometric representation theory has close and profound connections to many fields of mathematics, which we expect to play a significant role in the program. Perhaps the most significant are to number theory, via the theory of automorphic forms, L-functions and modularity. Much current activity in the field is motivated either directly by problems in number theory or by more tractable geometric analogues thereof. Another major influence on the subject comes from physics, in particular gauge theory, integrable systems, and recently topological string theory. There is a significant interaction with the theory of C∗-algebras through the Baum-Connes conjecture, an instance of which provides an organizing principle for representation theory of real and p-adic groups. Finally, geometric representation theory is closely entwined with very active areas in combinatorics such as Schubert calculus, its affine analogue, and the theory of Macdonald polynomials.

## References

* A. Beilinson, J. Bernstein, _Localisations de $\mathfrak{g}$–modules_, C. R. Acad. Sci. Paris 292 (1981), 15--18.

* N. Chriss, V. Ginzburg, _Representation theory and complex geometry_, book

* K. Vilonen, _Geometric methods of representation theory_, [math.AG/0410032](http://arxiv.org/abs/math/0410032)

* R. Hotta, K. Takeuchi, T. Tanisaki, _D-modules, perverse sheaves, and representation theory_, Progress in Mathematics __236__, Birkhäuser 2008

* M. Kashiwara, _Equivariant derived category and representation of real semisimple Lie groups_, in CIME Summer school _Representation Theory and Complex Analysis_, Cowling, Frenkel et al. eds. LNM __1931__ Springer ([pdf](http://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/sd.pdf)).

* A. Borel et al., _Algebraic D-modules_, Perspectives in Mathematics, Academic Press, 1987.

* M.Kashiwara, W.Schmid, _Quasi-equivariant D-modules, equivariant
derived category, and representations of reductive Lie groups_, Lie Theory
and Geometry, in Honor of Bertram Kostant, Progress in Mathematics,
Birkhäuser, 1994, pp. 457--488.

* [[Dennis Gaitsgory]], _Geometric representation theory_, 61 pp. 2005 course notes [cat0.pdf](http://www.math.harvard.edu/~gaitsgde/267y/catO.pdf)
* [[David Kazhdan]], _Algebraic geometry and representation theory_, Proc. ICM 1986, vol. I, 849-852, [djvu:201K](http://www.mathunion.org/ICM/ICM1986.1/Main/icm1986.1.0849.0852.ocr.djvu), [pdf:399K](http://www.mathunion.org/ICM/ICM1986.1/Main/icm1986.1.0849.0852.ocr.pdf)
* V. Ginzburg (Paper read by D. Vogan.) _Geometrical aspects of representation theory_, Proc. ICM 1986, vol. I, 840-848, [djvu:487K](http://www.mathunion.org/ICM/ICM1986.1/Main/icm1986.1.0840.0848.ocr.djvu), [pdf:940K](http://www.mathunion.org/ICM/ICM1986.1/Main/icm1986.1.0840.0848.ocr.pdf)

* [MSRI program Geometric Representation Theory](http://www.msri.org/web/msri/scientific/programs/show/-/event/Pm8951) (2014)
 {#MSRI14}

* [[David Ben-Zvi]], _[[Loop Groups, Characters and Elliptic Curves]]_, talk (2012)

See also the references at _[[orbit method]]_.


geometric Satake equivalence

# Contents
* table of contents
{: toc}

## Idea

The _geometric Satake equivalence_ identifies for a suitable [[algebraic group]] $G$, and suitable [[local field]] $K$ with [[ring of integers]] $\mathcal{O}_K$, suitable functions on the [[double coset]]/[[Grassmannian]] $G(\mathcal{O}_K)\backslash G(K)/G(\mathcal{O}_K)$ with the [[representation ring]] of the [[Langlands dual group]] ${}^L G$.

Notice that the double coset appearing here is akin to that which controls the [[Langlands correspondence]], whose geometric meaning is discussed for instance at _[[Weil uniformization]]_ and at _[[function field analogy]]_.

## Related concepts

* [[geometric Langlands correspondence]]

* [[categorification via groupoid schemes]] -- [section on geometric Satake](categorification+via+groupoid+schemes#GeometricSatake)

## References

* Wikipedia, _[Satake isomorphism](http://en.wikipedia.org/wiki/Satake_isomorphism)_

* Benedict Gross, _On the Satake isomorphism_ ([pdf](http://www.math.harvard.edu/~gross/preprints/sat.pdf))

* [[Dennis Gaitsgory]], [Graduate Seminar. Geometric Representation theory. Fall 2009--Spring 2010](http://www.math.harvard.edu/~gaitsgde/grad_2009/), _Feb. 23: Affine Grassmannian-II: the Satake equivalence (Ryan Reich)_ ([pdf](http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Feb23%28Gr-II%29.pdf))

* example 1.7 in these rough notes, apparently taken in some talk by [[Jacob Lurie]]: [[lurie.pdf:file]]

[[!redirects geometric Satake equivalence]]
[[!redirects geometric Satake equivalences]]

[[!redirects Satake isomorphism]]
[[!redirects Satake isomorphism]]

[[!redirects Satake equivalence]]
[[!redirects Satake equivalences]]

[[!redirects geometric Satake]]


geometric series

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* table of contents
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## Definition

The _geometric series_ is the [[series]]

$$
  \sum_{n = 0}^\infty r^n
  \,.
$$

## Properties

For $r \in \mathbb{R}$ with ${\Vert r \Vert} \lt 1$ this [[convergence of a sequence|sequence converges]]

$$
  \underset{n \to \infty}{\lim}
  \underoverset{k = 0}{n}{\sum} r^k
  \;=\;
  \frac{1}{1 - r}
  \,.
$$

For $r \in \mathbb{Z}_p$ with $r = p$ the above sequences converges in the p-adic metric.

## Related entries

* [[series]], [[power series]]

* [[Mercator series]]

## References

See also:

* Wikipedia, _[Geometric series](https://en.wikipedia.org/wiki/Geometric_series)_


geometric shape for higher structures

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## Overview

Handling higher structures such as [[higher category theory|higher categories]] usually involves conceiving them as  conglomerates of _cells_ of certain _shape_.

Examples for possible shapes used to model higher categories are

* [[globe|globes]];

* [[simplex|simplices]];

* [[prisms]];

* [[cube|cubes]];

* [[trees]];

* [[cell category|globular cells]];

* [[opetopes]], aka [[multitopes]].

There are corresponding categories whose

* objects are the "standard cellular shapes" of the given sort: [[globe|globes]], [[simplex|simplices]], [[cube|cubes]], respectively.  For globes, simplices, and cubes, there is one object for each natural number $n \in \mathbb{N}$, so they are usually denoted $[n]$.  For other sorts of shapes, there can be multiple shapes of each dimension.

* morphisms are generated from (some of) the ways of mapping these standard cellular shapes to each other such that their cellular structure is preserved;

* and composition of such morphisms is subject to the relations which are inherited from the geometric meaning of these maps, which says for instance that the left boundary of the top boundary of a [[cube]] is the same as the top boundary of its left boundary -- these are the  _globular identities_, the _simplicial identities_ and the _cubical identities_, respectively.

The resulting categories of basic cellular shapes are

* the _[[globe category]]_ $\mathbb{G}$

* the _[[simplex category]]_ $\Delta$

* the _[[cube category]]_ $\square$

* the _[[tree category]]_ $\Omega$

* the _[[opetope category]]_ $\mathbb{O}$

In general, there are two types of morphism in such categories, called *face inclusions* (or *cofaces*) and *degeneracies*.  Face inclusions exhibit a lower-dimensional cell as a "face" of a higher-dimensional one, while degeneracies collapse a higher-dimensional cell down to a lower-dimensional one.  All such categories have face inclusions (these are the really basic "structural" maps that determine the shapes of the cells), but some (such as $\Delta$, $\square$, and $\Omega$) include degeneracies while some (such as $\mathbb{G}$ and $\mathbb{O}$) do not usually.

A general notion which includes all of these categories is that of a [[generalized Reedy category]].  If degeneracies are excluded, then we obtain the simpler notion of a (generalized) [[direct category]].

A higher structure based on these geometric shapes is a [[presheaf]] on one of these categories. These are called

* [[globular sets]]

* [[simplicial sets]]

* [[cubical sets]]

* [[dendroidal sets]]

* [[opetopic sets]]

respectively.

There are also other notions of geometric shape which have been found useful in higher category theory, such as the shapes encapsulated in Joyal's [[disk category]] $\Theta$ (which include both globes and simplices as special cases).

Many definitions of [[higher category theory|higher categories]] are given by one of these presheaves

* either equipped with _extra properties_, in the [[geometric definition of higher category|geometric definition of higher categories]];

* or equipped with _extra structure_, in the [[algebraic definition of higher category|algebraic definition of higher categories]].

For instance, the usual definition of [[strict omega-category|strict ∞-categories]] is based on giving [[globular sets]] extra structure, as is the weak version of [[Batanin ∞-category]].  Similarly, [[n-fold category|n-fold categories]] give extra structure to [[cubical sets]].  On the other hand, the notion of [[quasi-category|quasi-categories]] is based on [[simplicial sets]], while other more "homotopical" notions of higher category are based on cubical or multisimplicial techniques.  In general, the "geometric" definitions can usually be thought of as a [[nerve]] of the "algebraic" ones.

Sometimes, two definitions that use different kinds of shapes nevertheless capture equivalent notions.  This is hoped to be true for all proposed definitions of [[n-category]], although they use many different kinds of shapes.  On the other hand, sometimes the use of different shapes indicates a fundamentally different structure being considered.  For instance, edge-symmetric $n$-fold categories with connections give the same notion as globular $n$-categories, while arbitrary $n$-fold categories are something quite different.  Moreover, even in the case where different kinds of shapes produce "equivalent" notions, it can sometimes be markedly easier to work with one kind of shape than another for some particular application.

## Related notions

See for instance

* [[pasting diagram]]

* [[parity complex]]

* [[directed complex]]

* [[test category]]

* [[direct category]]

## References ##

* [This Week's Finds in Mathematical Physics (Week 242)](http://golem.ph.utexas.edu/category/2006/12/this_weeks_finds_in_mathematic_4.html#c006655) (Discussion at the n-Cafe)

* On a type-theoretic approach to formalizing higher structures: [[Nicolai Kraus]]: [seminar slides, 2017](http://www.cs.nott.ac.uk/~psznk/docs/nicolai_highercatstruct_handout.pdf)

[[!redirects geometric shape for higher categories]]
[[!redirects geometric shapes for higher categories]]
[[!redirects geometric shape for higher structures]]
[[!redirects geometric shapes for higher structures]]


geometric stability theory

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#### Model theory
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#Contents#
* table of contents
{:toc}

## Idea

__Geometric stability theory__ is the principal part of the branch of [[model theory]] called __[[geometric model theory]]__. It was introduced in works of [[Boris Zilber]], Gregory Cherlin, [[Ehud Hrushovski]], Anand Pillay, and others.

Geometric stability theory has largely to do with the model-theoretic classification of [[structure in model theory|structures]] in terms of dimension-like quantities that can be axiomatized in terms of notions of [[combinatorial geometry]] such as [[matroid]]. Key guiding examples include [[vector spaces]] and [[algebraically closed fields]], and many of the guiding concepts have an [[algebraic geometry|algebro-geometric]] flavor (e.g., Morley rank as a generalization of Krull dimension).

Such a structural geometric approach can make geometric stability theory an attractive key of entry into modern model theory for those mathematicians who are not already logicians.

## Basic concepts

Perhaps the key axiomatic notions of geometric stability theory (which at the outset don't seem particularly wedded to logic) are those of _pregeometry_ and _geometry_.

+-- {: .num_defn}
###### Definition
Let $X$ be a set. A **pregeometry** on $X$ is a [[closure operator]] (i.e., a [[monad]] $cl \colon P X \to P X$ on the [[power set]]), satisfying the following two conditions:

* The monad $cl$ is [[finitary functor|finitary]], i.e., $A \in P X$ and $a \in cl(A)$, then there is a finite $A_0 \subseteq A$ such that $a \in cl(A_0)$.

* (Exchange condition) If $A \in P X$, $a,b \in X$, and $a \in cl(A\cup\{b\})$, then $a \in cl(A)$ or $b \in cl(A \cup \{\a\})$. (Cf. [[matroid]])

A **geometry** is a pregeometry such that $cl(\emptyset) = \emptyset$ and $cl(\{x\}) = \{x\}$ for all $x \in X$.
=--

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

* Let $X$ be a [[vector space]], and let $cl$ be the monad on $P X$ whose algebras are vector subspaces of $X$. Clearly $cl$ is finitary (any subspace is the set-theoretic union of finite-dimensional subspaces), and the exchange condition is a classical fact about vector spaces related to the notion of independence. Thus $cl$ is a pregeometry.

* Similarly, let $X$ be a [[projective space]] $\mathbb{P}V$, and let $cl$ be the monad on $P X$ whose algebras are projective subspaces. Then $cl$ is a geometry (the closure of a point is a point). Any pregeometry $cl$ gives rise to a geometry in a similar way, in the sense that a pregeometry $cl$ induces a geometry on the image of the function $X \to P X$, $x \mapsto cl(\{x\})$, as explained in Remark \ref{project}.

* Let $X$ be an algebraically closed field; let $cl$ be the monad on $P X$ whose algebras are algebraically closed subfields. Then $cl$ is a pregeometry. That the exchange condition is satisfied is a classical result credited to Steinitz[^fine].
=--

+-- {: .num_defn}
###### Definition
Given a pregeometry $(X, cl)$, a subset $A \in P X$ is **independent** if for all $a \in A$, $a \notin cl(A - \{a\})$. An independent set $A$ said to be a **basis** for $Y \in P X$ if $Y \subseteq cl(A)$. All bases of $Y$ have the same cardinality, called the **dimension** of $Y$.
=--

+-- {: .num_remark}
###### Remark
Given a pregeometry $(X, cl)$ and a subset $Y \subseteq X$, there is a restriction pregeometry $cl^Y: P(Y) \to P(Y)$ defined by the formula

$$cl^Y(S) = cl(S) \cap Y.$$

It is immediate that $A \subseteq Y$ is independent in $(X, cl)$ iff it is independent in $(Y, cl^Y)$, and that it is a basis for $Y$ in $(X, cl)$ iff it is a basis for $Y$ in $(Y, cl^Y)$. By this observation, to prove that any $Y \in P(X)$ has a well-defined dimension, we may assume without loss of generality that $Y = X$. That a pregeometry $X$ has well-defined dimension was proven [here](nlab/show/matroid#welldefined).
=--

+-- {: .num_remark #project}
###### Remark
There is a standard way of getting a geometry from a pregeometry $(X, cl)$. First, if $cl(\emptyset) \neq \emptyset$, then replace $X$ by $X' \coloneqq X - cl(\emptyset)$, equipped with the restriction pregeometry of the previous remark. Then define an equivalence relation on $X'$ by $x \sim y$ if $cl(\{x\}) = cl(\{y\})$, and define a pregeometry on the quotient set $X'/\sim$ by

$$\widehat{cl}(A) \coloneqq \{[b]: b \in cl(A)\}$$

where $[b]$ denotes the equivalence class of $b$. This abstracts the process of taking a projectivization of a vector space.
=--

### Minimal and strongly minimal sets

We'll suppose for now that $\mathbf{L}$ is a countable [[signature (in logic)|signature]], so that the [[first-order language|language]] it generates consists of countably many formulas. Let $\mathbf{M}$ be an $\mathbf{L}$-structure, with underlying set $M$.

+-- {: .num_defn}
###### Definition
For $A \subseteq M$, an element $b \in M$ is **algebraic** over $A$ if there is a formula $\phi(y, w_1, \ldots, w_n)$ and elements $a_1, \ldots, a_n \in A$ such that $\mathbf{M} \models \phi(b, a_1, \ldots, a_n)$ and

$$\{c \in M: \mathbf{M} \models \phi(c, a_1, \ldots, a_n)\}$$

is finite. The **algebraic closure** of $A$, denoted $acl(A)$, is the set of elements of $M$ that are algebraic over $A$.
=--

+-- {: .num_prop #finclos}
###### Proposition
The algebraic closure $A \mapsto acl(A)$ defines a finitary closure operator on $M$.
=--

+-- {: .proof}
###### Proof
That $acl$ is monotone (preserves order) is obvious. Also the fact that $acl$ is finitary is easy: given that $b \in acl(A)$ satisfies $\mathbf{M} \models \phi(b, a_1, \ldots, a_n)$ for $a_1, \ldots, a_n \in A$, then similarly $b \in acl(A_0)$ for $A_0 = \{a_1, \ldots, a_n\}$.

Taking $\phi(y, w)$ to be the equality predicate $y = w$, we see $A \subseteq acl(A)$.

For idempotence of $acl$, suppose $b_1, \ldots, b_k \in acl(A)$ and $\mathbf{M} \models \psi(c, b_1, \ldots, b_k)$ where $\{y \in M: \mathbf{M} \models \psi(y, b_1, \ldots, b_k)\}$ has exactly $m$ elements. Write down a formula $F_{m, \psi}(x_1, \ldots, x_k)$ that says $\{y \in M: \mathbf{M} \models \psi(y, x_1, \ldots, x_k)\}$ has at most $m$ elements (by adding some extra inequalities, we can make this "exactly $m$ elements"):

$$F_{m, \psi}(x_1, \ldots, x_k) \coloneqq \exists_{y_1, \ldots, y_m} \forall_y \psi(y, x_1, \ldots, x_k) \Leftrightarrow (y = y_1) \vee \ldots \vee (y = y_m).$$

Now for $i = 1, \ldots, k$, let $\phi_i$ be a formula witnessing $b_i \in acl(A)$, i.e., $\mathbf{M} \models \phi_i(b_i, a_{i, 1}, \ldots, a_{i, n_i})$ where $a_{i, k} \in A$ and $\{x \in M: \mathbf{M} \models \phi_i(x, a_{i, 1}, \ldots, a_{i, n_i})\}$ has finitely many elements. Then

$$\exists_{x_1, \ldots, x_k} F_{m, \psi}(x_1, \ldots, x_k) \wedge \psi(y, x_1, \ldots, x_k) \wedge \bigwedge_{i=1}^k \phi_i(x_i, a_{i, 1}, \ldots, a_{i, n_i})$$

is a formula with parameters in $A$ that witnesses $c \in acl(A)$. This proves the idempotence of $acl$.
=--

As before, this notion of algebraic closure $acl: P(M) \to P(M)$ can be restricted to a closure operator $acl^X: P(X) \to P(X)$ for a subset $X \subseteq M$, via the definition $acl^X(A) \coloneqq X \cap acl(A)$ for $A \subseteq X$. One often writes just $acl$ instead of $acl^X$, provided that $X$ is understood.

+-- {: .num_defn}
###### Definitions
For an $\mathbf{L}$-structure $\mathbf{M}$, let $D \subseteq M^n$ be definable. $D$ is **minimal** if the only definable subsets of $D$ are finite or cofinite in $D$. Slightly abusing language, if $\phi(x_1, \ldots, x_n, a_1, \ldots, a_k)$ is a formula with parameters that defines $D$, we also say $\phi$ is minimal. We say $D$ (or $\phi$) is **strongly minimal** if it is minimal in any elementary extension of $\mathbf{M}$.

A theory $\mathbf{T}$ is _strongly minimal_ if for any model $\mathbf{M}$ of $\mathbf{T}$, the underlying set $M$ (definable by the formula $x = x$) is strongly minimal.
=--

+-- {: .num_example}
###### Examples
1. Let $K$ be an algebraically closed field. (Elimination of quantifiers, Chevalley's theorem, etc.) Conclusion: ACF is a strongly minimal theory.

1. Divisible torsionfree abelian groups.

1. Non-example of dense linear orders.

=--

+-- {: .num_theorem}
###### Theorem
**(Baldwin, Lachlan)**
The algebraic closure operator on a minimal set $X$ is a pregeometry.
=--

+-- {: .proof}
###### Proof
Let $A \subseteq X$ and suppose $c \in acl(A \cup \{b\}) - acl(A)$ for $b, c \in X$; we want to show $b \in acl(A \cup \{c\})$. Thus, suppose $\phi(c, b)$ is a formula with parameters from $A$ such that $\mathbf{M} \models \phi(c, b)$ and $card(\{x \in X: \mathbf{M} \models \phi(x, b)\}) = n$, a finite number. As in the proof of Proposition \ref{finclos}, let $\psi(w)$ be a formula that says $card(\{x \in X: \mathbf{M} \models \phi(x, w)\}) = n$. This is a formula with parameters from $A$ and $\psi(b)$ is satisfied in $\mathbf{M}$.

If $\{y \in X: \mathbf{M} \models \phi(c, y) \wedge \psi(y)\}$ is finite, then since $b$ belongs to this set, $\phi(c, y) \wedge \psi(y)$ would witness $b \in acl(A \cup \{c\})$ and we would be done. So suppose otherwise. Then this set is cofinite in $X$, so that

$$card(X - \{y \in X: \mathbf{M} \models \phi(c, y) \wedge \psi(y)\}) = m$$

for some finite $m$, and we can again write down a formula $\chi(x)$ with parameters from $A$ that says

$$card(X - \{y \in X: \mathbf{M} \models \phi(x, y) \wedge \psi(y)\}) = m.$$

If $\chi(x)$ defines a finite subset of $X$, then since $\chi(c)$ is satisfied, we would reach the conclusion that $c \in acl(A)$, a contradiction. Hence $\chi(x)$ defines a subset cofinite in $X$. We may therefore choose $n+1$ elements $a_1, \ldots, a_{n+1} \in X$ such that $\chi(a_i)$ is satisfied. By our supposition,

$$B_i \coloneqq \{u \in X: \mathbf{M} \models \phi(a_i, u) \wedge \psi(u)\}$$

is cofinite for $i = 1, \ldots, n+1$. Hence $\bigcap_{i=1}^{n+1} B_i$ is inhabited, say by an element $b'$. We have at least $n+1$ elements $x \in X$ such that $\phi(x, b')$ is satisfied, namely $x = a_1, \ldots, a_{n+1}$. But now this contradicts the fact $\psi(b')$.
=--

=--

## Related concepts

* [[stability in model theory]]

* [[Zariski geometry]]

* [[matroid]]

## References

* Anand Pillay, _Geometric stability theory_, Oxford Logic Guides __32__
* slides from conference "[Geometric model theory](http://www.maths.ox.ac.uk/events/borisfest)", Oxford 2010: directory [html](http://people.maths.ox.ac.uk/bays/misc/borisfest/notes)
* Misha Gavrilovich, _Model theory of universal covering space of complex algebraic varieties_, thesis, [pdf](http://people.maths.ox.ac.uk/bays/misha-thesis.pdf)

* [[Boris Zilber]], _Elements of Geometric Stability Theory_, 2003 ([pdf](http://people.maths.ox.ac.uk/zilber/est.pdf))

[^fine]: Although oddly enough, as explained at the [MacTutor biography page](http://www-history.mcs.st-andrews.ac.uk/Biographies/Steinitz.html), what is called the Steinitz exchange condition was set out by Ernst Steinitz in a 1913 publication on convergent series and apparently not (?), as might be supposed, in his _Algebraische Theorie der Körper_ (Crelle's Journal, 1910). His 1913 lemma was for _vector spaces_.


geometric stack

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

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

## Idea

A [[stack]] $X$ on a [[site]] $C$ is **geometric** if, roughly, it is [[representable functor|represented]] by a suitably well-behaved [[groupoid]] object $\mathcal{G} = (\mathcal{G}_1 \stackrel{\to}{\to} \mathcal{G}_0)$ [[internal category|internal]] to $C$, i.e. if to an object $U \in C$ the stack assigns the (ordinary) [[groupoid]]

$$
  X : U \mapsto (C(U,\mathcal{G}_1) \stackrel{\longrightarrow}{\longrightarrow} C(U,\mathcal{G}_0))
  \,.
$$

A crucial difference between the groupoid object $\mathcal{G}$ in $C$ and the geometric stack $X$ is that the [[equivalence class]] of the stack in general contains _more_ (geometric) stacks than there are groupoid objects internally equivalent to $\mathcal{G}$: two groupoid objects with equivalent geometric stacks are called **Morita equivalent** groupoid objects.

## Special cases

Geometric stacks for the following choices of sites $C$ are called

* for $C = $ [[Top]] -- [[topological stack]];

* for $C = $ [[Diff]] -- [[differentiable stack]];

* for $C = $ [[CRing]]${}^{op}$ -- [[algebraic stack]];

* for $C = $ [[complex manifold|CplxMfd]] -- [[complex analytic stack]];

* for $C = $ [[Loc]] -- [[localic stack]];

## Definition

There are slight variations in the literature on what precisely is required of a [[stack]] $X$ on a [[site]] $C$  with [[subcanonical coverage|subcanonical topology]] in order that it qualifies as **geometric**.

A general requirement is that

1. the [[diagonal]] morphism $\Delta : X \to X \times X$ is a [[representable morphism of stacks]]

1. there exists an [[atlas]] for the stack, in that there is a [[representable functor|representable]] $U \in C$ and a surjective morphism

  $$
    p : U \to X
    \,.
  $$

  This is necessarily itself representable, precisely if $\Delta_X$ is.

Further conditions are the following

* for $C = Sch_{et}$ the [[site]] of [[scheme]]s with the [[etale topology]]

  * $\Delta_X$ is required to be [[quasicompact]] and [[separated]]

  * for [[Deligne-Mumford stack]]s $p$ is moreover required to be etale

  * for [[Artin stack]]s $p$ is required to be [[smooth morphism of schemes|smooth]].

## Relation to groupoid objects

The _groupoid object_ associated to a geometric stack $X$ with [[atlas]] $p : U \to X$ is the [[Cech groupoid]] of $p$ (this is simply the Cech groupoid of $p$ seen as a singleton cover) defined by $\mathcal{G}_0 := U$ and $\mathcal{G}_1 = U \times_X U$, where the latter is the 2-categorical [[pullback]]

$$
  \array{
   \mathcal{G}_1 &\stackrel{s}{\to}& U
   \\
   \downarrow^{\mathrlap{t}} &{}^{\simeq}\swArrow& \downarrow^{\mathrlap{p}}
   \\
   U &\to& X
  }
$$

## Related concepts

* **geometric stack**

* [[geometric ∞-stack]]

## References

A good discussion of topological and differentiable stacks is around definition 2.3 in

* [[Jochen Heinloth]], *Notes on differentiable stacks* (2004) [[pdf](https://math.nyu.edu/~tschinke/WS04/pdf/heinloth.pdf), [[Heinloth-DifferentiableStacks.pdf:file]]]

Differentiable stacks are discussed in

* [[Kai Behrend]], [[Ping Xu]], _Differentiable Stacks and Gerbes_ ([arXiv](http://front.math.ucdavis.edu/0605.5694)).

Specifically for the relation to groupoid objects see

3.1 and 3.3 in

* Metzler, _Topological and smooth stacks_ ([arXiv:math/0306176](http://arxiv.org/abs/math/0306176))

paragraphs 2.4.3, 3.4.3, 3.8, 4.3 in

* G. Laumon, L. Moret-Bailly, _Champs algébriques_ , Ergebn. der Mathematik und ihrer Grenzgebiete 39 , Springer-Verlag, Berlin, 2000

paragraph 4.4 in

* [[Eugene Lerman]], _Orbifolds as stacks?_ ([arXiv:0806.4160](http://arxiv.org/abs/0806.4160))

See also

* [[Bertrand Toen]], _[[Master course on algebraic stacks]]_.

* [[Aise Johan de Jong]],  _[[The Stacks Project]]_ ([pdf](http://www.math.columbia.edu/algebraic_geometry/stacks-git/book.pdf)) ([project website](http://www.math.columbia.edu/algebraic_geometry/stacks-git/))
{#deJong}

Geometric stacks over the site of schemes modeled on [[smooth loci]] is in section 8 of

* [[Dominic Joyce]], _Algebraic geometry over $C^\infty$-rings_ ([arXiv:1001.0023](http://arxiv.org/abs/1001.0023))

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[[!redirects geometric stacks]]
[[!redirects presentable stack]]
[[!redirects presentable stacks]]