Čech cohomology

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

## Idea ##

### General idea
 {#GeneralIdea}

Čech cohomology is a tool, or an algorithm, which, when it applies, computes [[abelian sheaf cohomology]] (of some $X$ with [[coefficients]] in some $A$) by use of [[coverings]] and systems of [[coefficients]] on the covering and all its non-empty finite intersections. More generally, it applies also to [[nonabelian cohomology]] and may hence for instance (in degree 1) be used to compute classes of [[principal bundles]] and generally (in higher degree) those of [[principal ∞-bundles]]. Quite generally Čech cohomology is the way to express the intrinsic [[cohomology]] of [[(∞,1)-sheaf]] [[(∞,1)-toposes]] by mapping out of [[Čech nerves]] of [[covers]].

Hence Čech cohomology is more an algorithm for computing [[cohomology]] (see also at _[[Čech methods]]_) than a cohomology theory in itself. In particular it applies (when it applies) to all sorts of [[coefficient]] [[sheaves]]/[[stacks]]/[[∞-stacks]] $A$.

When no further qualification of the coefficients $A$ is given then one usually has in mind by default either that $A = \mathbb{Z}$ is the group of [[integers]] (or a [[delooping]] thereof, in which case one computes [[ordinary cohomology]] by Čech methods) or, if working on a [[ringed space]]/[[ringed topos]]/[[structured (∞,1)-topos]] that $A = \mathcal{O}^\times$ is the [[group of units]] in the [[structure sheaf]].

Notice that there are technical conditions on a [[site]] and a [[cover]] to ensure that the result of the Čech cohomology algorithm really coincides with the more abstractly defined [[abelian sheaf cohomology]] -- and generally with the [[hom-spaces]] in the given [[(∞,1)-topos]]. (Technically the condition is that the [[Čech nerve]] is a [[cofibrant resolution]] in a [[model structure on simplicial presheaves]] for which the coefficient sheaf is a [[fibrant object]]). If these conditions do not apply, then the definition of Čech cohomology groups still works as such, but typically no longer qualifies as a "cohomology theory", axiomatically. They are just some groups. Strictly speaking one should maybe not use the term "Čech cohomology" in such cases, but beware that some authors do.

### More technical idea

From the discussion at [[abelian sheaf cohomology]] we know that the [[right derived functor]] definition computes the hom-set in the [[homotopy category of an (infinity,1)-category|homotopy category]] of an [[(infinity,1)-topos]] $\mathbf{H}$ that may alternatively be computed as a [[colimit]] over [[resolutions]] of the [[domain]] object

$$
  H(X,A)
  \coloneqq
  \pi_0 \mathbf{H}(X,A)
  = colim_{Y \stackrel{\in W \cap F}{\longrightarrow} X}
  C_H(Y,A)/_{homotopy}
$$

where the colimit is over all acyclic fibrations $Y \stackrel{\in W \cap F}{\longrightarrow} X$ in an appropriate [[model category]] $C_H$ that [[presentable (infinity,1)-category|presents]] $\mathbf{H}$. For $\mathbf{H}$ an [[infinity-stack]] [[(infinity,1)-topos]] this $C_H$ is a [[model structure on simplicial presheaves]] and the acyclic fibrations $Y \stackrel{\in W \cap F}{\to} X$ for $X$ an ordinary space are the [[hypercovers]].

Now, for some coefficient objects $A$ it is sufficient to take the colimit here not over all [[hypercovers]], but just over [[Čech covers]]. The resulting formula

$$
  H(X,A)

  = colim_{Y \stackrel{\mathop{Čech} cover}{\to} X}
  C_H(Y,A)/_{homotopy}
$$

is then called the formula for **Čech cohomology** on $X$ with values in $A$.

Here a [[Čech cover]] is a simplicial presheaf that arises from an ordinary covering map $U \to X$ of $X$ by another space $U$ as the corresponding [[Čech nerve]] simplicial presheaf

$$
 \begin{aligned}
  Y := \left(
    \cdots
    U \times_X U \times_X U
    \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}
    U \times_X U \stackrel{\longrightarrow}{\longrightarrow} U
  \right)
  \simeq
  hocolim_{[k] \in \Delta} U^{\times_X k}
 \end{aligned}
  \,.
$$

See [[descent for simplicial presheaves]] for more on the manipulations involved here.

To amplify, a general [[hypercover]] would start in degree 0 with a $U$ as above, but then in the next degree would have a cover $V \to U \times_X U$ of the fiber product, and so on, each fiber product in turn being covered by another space.

If $Y$ is not simply a [[Čech cover]] but also not the most general [[hypercover]] in that this iterative choice of further covering stops in degree $n$, then one also speaks of **Čech cover of level $n$** and of the corresponding cohomology formula as **higher Čech cohomology**. See for instance the reference by Tibor Beke below.

## General ("nonabelian") Čech cohomology
 {#NonabelianČechCohomology}

We start with describing the general " _nonabelian_ " Čech cohomology (compare the terminology and remarks at [[cohomology]] and [[nonabelian cohomology]]), i.e. the plain unwrapping of the above definition, before assuming that our coefficient object is [[abelian sheaf cohomology|abelian]] and before applying the [[Moore complex]] functor that sends the following simplicial computation to the maybe more familiar one in [[chain complexes]].

The reasoning parallels that described at [[descent]] to some extent, but is maybe still worthwhile repeating here.

So let $C$ be some [[site]] and let $SSh(C)$ the category of [[simplicial presheaf|simplicial (pre)sheaves]] on $C$. Let $C(U)$ be the Čech nerve for a cover $\pi : U \to X$ of some simplicial presheaf $X$ and let $A$ be any other simplicial presheaf that serves as the coefficient object.

Using [[end]] and [[coend]] notation we compute the required hom

$$
 \begin{aligned}
    SSh(C(U), A)
    &
    \simeq
    SSh( \int^{[k] \in \Delta} \Delta^k \cdot C(U)_k, A )
    \\
    &
    \simeq
    \int_{[k] \in \Delta} SSh(\Delta^k \cdot C(U)_k, A )
    \\
    &
    \simeq
    \int_{[k] \in \Delta} SSet(\Delta^k , SSh(C(U)_k,A) )
    \\
    &
    \simeq
    \int_{[k] \in \Delta} SSet(\Delta^k , A(U \times_X \cdots \times_X U) )
    \\
    &
    \simeq
    \int_{[k] \in \Delta} SSet(\Delta^k , \prod_{i_0, \cdots, i_k} A(U_{i_0 i_1, \cdots ,i_k}) )
 \end{aligned}
  \,.
$$

Here in the last line we have assumed that $U = \sqcup_i U_i$ for $\{U_i \to X\}$ an open cover of some space $X$ and we abbreviate as usual with
$
  U_{i_0, i_1, \cdots, i_k}
$ the $(k+1)$-fold intersections
$\cap_{0 \leq r \leq k} U_{i_r}$.

An object in this last expression -- an $A$-valued cocycle on $X$ relative to $U$ -- is a collection of morphisms in [[SSet]] of the form

$$
  \left(
  \array{
    \vdots && \vdots
    \\
    \Delta^2
    &\stackrel{f}{\longrightarrow}&
    \prod_{i j k} A(U_{i j k})
    \\
    \Delta^1
    &\stackrel{g}{\longrightarrow}&
    \prod_{i j}A(U_{ i j})
    \\
    \Delta^0
    &\stackrel{a}{\longrightarrow}&
    \prod_i A(U_i)
  }
  \right)
$$

that make all diagrams of the form

$$
  \array{
    \vdots && \vdots
    \\
    \uparrow
    &&
    \uparrow
    \\
    \Delta^2
    &\stackrel{f}{\longrightarrow}&
    \prod_{i j k} A(U_{i j k})
    \\
    \uparrow
    &&
    \uparrow
    \\
    \Delta^1
    &\stackrel{g}{\to}&
    \prod_{i j}A(U_{ i j})
    \\
    \uparrow^{\delta_{\cdot}}
    &&
    \uparrow^{A(d_{C(U)})_\cdot}
    \\
    \Delta^0
    &\stackrel{a}{\to}&
    \prod_i A(U_i)
  }
$$

commute. Here the vertical morphism on the right, when one traces their origin through the derivation above, are the various inclusion and restriction maps of elements of $A$ evaluated on a $k$-fold intersection to some other intersection.

For instance the three possible vertical maps at the
bottom have components on the cartesian factors given by

$$
  A(d_1)|_{U_i} := (-)|_{U_{i j}}
  :
  A(U_i)
  \to
  A(U_{i j})
$$

and

$$
  A(d_0)|_{U_i} := (-)|_{U_{i j}}
  :
  A(U_j)
  \to
  A(U_{i j})
$$

and

$$
  A(s_0)|_{U_{i i}}
  :
  A(U_{i i})
  \to
  A(U_{i })
  \,.
$$

This picks

* an object $a \in A(U)$

* a morphism $g : \pi_1 ^* a \to \pi_2^* a in A(U \times_X U)$

* a 2-morphism
  $$

    \array{
      && \pi_{2}^* a
      \\
      & {}^{\pi_{12}^* g}\nearrow &\Downarrow^{f}& \searrow^{\pi_{23}^* g}
      \\
      \pi_{1}^* a
      &&\stackrel{\pi_{13}^* g}{\to}&&
      \pi_3^* a
    }
  $$
  in $A(U \times_X U \times_X U)$

* etc.

If we follow tradition and write

* $U = \sqcup_i U_i$;

* and $U \times_X U = \sqcup_{i j} U_{i j} := \sqcup_{i j} U_i \cap U_j$

* etc

this is

* an collection of objects $(a_i \in A(U_i))$

* a collection of morphisms
  $(g_{i j} : a_i|_{U_{ij}} \to a_{j}|_{U_{i j}}
  in A(U_{i j}))$

* a collection of 2-morphisms
  $$
   \left(
    \array{
      && a_{j}|_{U_{i j k}}
      \\
      & {}^{g_{ij}|_{U_{i j k}}}\nearrow
      &\Downarrow^{f_{i j k}}&
       \searrow^{g_{j k}|_{U_{i j k}}}
      \\
      a_i|_{U_{i j k}}
       &&\stackrel{g_{i k}|_{U_{i j k}}}{\to}&&
      a_k|_{U_{i j k}}
    }
   \right)
  $$
  in $A(U_{i j k})$

* etc.

This is a **Čech-cocycle** on $X$ with values in $A$ relative to $U$.

A **transformation/homotopy/coboundary** between two such cocycles is a cylinder over these diagrams, i.e.

* a collection of morphism
  $ h_i : a_i  \to a'_i  in A(U_i)$;

* a collection of 2-morphism
  $$
    \array{
      a_i|_{U_{i j}}
       &\stackrel{g_{i j}}{\to}&
      a_j|_{U_{i j}}
      \\
      \;\;\;\downarrow^{h_i|_{U_{i j}}}
      &\Downarrow^{a_{i j}}&
      \;\;\;\downarrow^{h_j|_{U_{i j}}}
      \\
      a'_i|_{U_{i j}}
       &\stackrel{g_{i j}}{\to}&
      a'_j|_{U_{i j}}
    }
  $$
  on $U_{i j}$

* a collection of 3-morphisms
  $$
    \array{
      && a_j|_{U_{i j k}}
      \\
      & {}^{g_{ij}|_{U_{i j k}}}\nearrow
      &\Downarrow^{f_{i j k}}&
      \searrow^{g_{j k}|_{U_{i j k}}}
      \\
      a_i|_{U_{i j k}}
       &&\stackrel{g_{i k}}{\to}&&
      a_k|_{U_{i j k}}
      \\
      \;\;\;\downarrow^{h_i|_{U_{i j}}}
      &&\Downarrow^{a_{i k}}&&
      \;\;\;\downarrow^{h_j|_{U_{i j}}}
      \\
      a'_i|_{U_{i j k}}
       &&\stackrel{g_{i k}}{\to}&&
      a'_k|_{U_{i j k}}
    }
    \;\;
    \;\;
    \;\;
    \;\;
    \;\;
    \stackrel{\rho_{i j k}}{\Rightarrow}
    \;\;
    \;\;
    \;\;
    \;\;
    \;\;
    \array{
      && a_j|_{U_{i j k}}
      \\
      &
      {}^{g_{ij}|_{U_{i j k}}}\nearrow
      &\Downarrow^{a_{i j}\cdot a_{j k}|_{U_{i j k}}}&
      \searrow^{g_{j k}|_{U_{i j k}}}
      \\
      a_i|_{U_{i j k}}
       && a'_j|_{U_{i j k}} &&
      a_k|_{U_{i j k}}
      \\
      \downarrow^{h_{i}|_{U_{i j k}}}
      &
        {}^{g'_{i j}|_{U_{i j k}}}\nearrow
      &
       \Downarrow^{f'_{ijk}}
      &
        \searrow^{g'_{j k}|_{U_{i j k}}}
      &
      \downarrow^{h_{k}|_{U_{i j k}}}
      \\
      a'_i|_{U_{i j k}}
      &&\stackrel{g'_{i k}|_{U_{i j k}}}{\to}&&
      a'_k|_{U_{i j k}}
    }
  $$

We now plug in some concrete coefficient object
$n$-types for low $n$ and reproduce some concrete formulas
from this.

### Nonabelian 1-cocycles: principal bundles

For $G$ a [[group]] let
$\mathbf{B} G$ by abuse of notation denote

* first of all the corresponding
[[delooping|delooped]] [[groupoid]],

* then the corresponding [[nerve]]

* then finally the corresponding simplicial sheaf that for each object $U$ assigns the [[nerve]] for the group $Hom(U,G)$:

  $$
    \mathbf{B}G : U \mapsto
    \left\{
      \array{
        \mathbf{B}G(U)_0 = Hom(U,{*}) = {*}
        \\
        \mathbf{B}G(U)_1 = Hom(U,G)
        \\
        \mathbf{B}G(U)_2 = Hom(U,G)\times Hom(U,G)
        \\
        \vdots
      }
    \right.
  $$

The above then says that

* a $\mathbf{B}G$-Čech cocycle is

  * a collection of $G$-valued functions

     $$
       (g_{ij} \in Hom(U_{i j}, G))
     $$

  * a collection of identities between $G$-valued functions

    $$
      g_{ij}|_{U_{i j k }}
       \cdot
      g_{j k}|_{U_{i j k }}
      =
      g_{i k}|_{U_{i j k }}
    $$

* a $\mathbf{B}G$-Čech coboundary is

  * a collection of $G$-valued functions

    $$
      (h_i \in Hom(U_i , G))
    $$

  * a collection of identites between $G$-valued functions

    $$
      g_{ij} \cdot h_j|_{U_{i j}}
      =
      h_i|_{U_{i j}} \cdot g'_{j k}
      \,.
    $$

Remembering that the Čech cohomology is the colimit over refinement of covers over cohomology classes defined this way, one sees the standard

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

Čech cohomology with coefficients in $\mathbf{B}G$
(as above) classies $G$-[[principal bundles]]

$$

  colim_{U \stackrel{\mathop{Čech} cover}{\to} X}
  SSh(C(U), \mathbf{B}G)/_{homtopy}
  \simeq
  G Bund(X)/_\sim
$$

=--

### Nonabelian 2-cocycles: gerbes and principal 2-bundles

In one degree higher the general [[homotopy n-type|homotopy 2-type]] coefficient object is modeled using a strict [[2-group]] $H$ coming from a [[crossed module]]
$G_2 \stackrel{\delta}{\to} G_1$.

The [[nerve]] of the corresponding [[delooping|delooped]]
[[2-groupoid]], regarded immediately as a simplicial sheaf starts out like

$$
  \mathbf{B}(G_2 \to G_1)
  :
  U
  \mapsto
  \left\{
    \array{
       \mathbf{B}(G_2 \to G_1)(U)_{0}
       =
       Hom(U, G_1)
       \\
       \mathbf{B}(G_2 \to G_1)(U)_{1}
       =
       Hom(U, G_1) \times Hom(U, G_1)
       \times Hom(U, G_2)
       \\

       \vdots
    }
  \right.
  \,.
$$

One finds that a Čech-cocycle now is
(see also the diagrams at [[group cohomology]] for this)

* a collection of functions

  $$
    (g_{i j} \in Hom(U_{i j}, G_1))
  $$

* a collection of functions

  $$
    (f_{i j k }\in Hom(U_{i j k}, G_2) )
  $$

* a collection of identities

  $$
    \delta(f_{i j k}|_{U_{i j k}})
    g_{i j}|_{U_{i j k}}
    g_{j k}|_{U_{i j k}}
    =
    g_{i k}|_{U_{i j k}}
  $$

* and a collection of identities

  $$
    f_{i k l}|_{U_{i j k l}}
    \cdot
    f_{i j k}|_{U_{i j k l}}
    =
    f_{i j l}|_{U_{i j k l}}
    \cdot
    \alpha(g_{i j}|_{U_{i j k l}})(f_{j k l}|_{U_{i j k l}}),
    $$

where $\alpha:G_1\to Aut(G_2)$ is the homomorphism associated with the crossed-module description of the 2-group.

This is a **nonabelian Čech 2-cocycle**.

Reading off the formulas for the coboundaries is
left as an excercise for the reader.

For the specical case the

$$
  (G_2 \to G_1) = (G \to Aut(G))
$$

this is the nonabelian cohomology classifying

[[gerbes]].

## Abelian Čech cohomology
 {#AbelianČechCohomology}

In much of the literature _Čech cohomology_ denotes exclusively the abelian case, which we now describe.

In the special case that the coefficient [[simplicial presheaf]] $A$ is in the image of the [[nerve]] operation

$$
  N : Ch_+ \to sAb \to SSet
$$

on [[chain complexes]] with values in [[simplicial sets]] that happen to be abelian [[simplicial groups]], the computation of Čech cocycles may be entirely pulled back to the world of [[homological algebra]] by making use of the [[Dold-Kan correspondence]] that provides an [[adjoint equivalence]] between simplicial sets with values in abelian groups and non-negatively graded chain complexes.

### Definition
 {#AbelianČechCohomologyDefinition}

We discuss the [[double complex]] which gives the traditional definition of Čech cohomology with coefficients in [[sheaves of abelian groups]].

Let

* $X$ be a [[topological space]];

* $A_\bullet = (\cdots \stackrel{\partial}{\to} A_2 \stackrel{\partial}{\to} A_1 \stackrel{\partial}{\to} A_0)$ be a [[chain complex]] of [[abelian sheaves]] on $X$ (i.e. on the [[category of open subsets]] $S \colon Op(X)$ of $X$, regarded as a [[site]] with the usal [[covering]] [[Grothendieck topology]]);

* $\{U_i \to X\}$ be an [[open cover]] of $X$.

We write

$$
  U_{i_0,\cdots i_k}
  \coloneqq
  U_{i_0}
  \underset{X}{\times}
  U_{i_1}
  \underset{X}{\times}
  \cdots
  \underset{X}{\times}
  U_{i_k}
$$

for the $k$-fold intersections of these open subsets. Given an inclusion of open subsets $U_1 \to U_2$ and given any group element $a\in A_\bullet(U_2)$ we write $a|_{U_1}$ for its restriction along this map.

More generally, we may allow $S$ to be any [[site]]. For simplicity of the following formulas assume that $S$ has [[finite products]] (which in the case that $S$ is a [[category of open subsets]] are the [[fiber products]] above.) Then $A_\bullet$ is a chain complex of [[abelian sheaves]] on that site.

+-- {: .num_defn #CechComplex}
###### Definition
**(Čech complex)**

The **Čech cochain complex** $C^\bullet((X,\{U_i\}),A_\bullet)$
of $X$ with respect to the cover $\{U_i \to X\}$ and with [[coefficients]] in $A_\bullet$ is in degree $k \in \mathbb{N}$ given by the [[abelian group]]

$$
  C^k((X,\{U_i\}),A_\bullet)
  \coloneqq
  \oplus_{{l,n} \atop {k = n-l}}
  \oplus_{i_0, i_1, \cdots, i_n}
  A_l(U_{i_0, \cdots, i_n})
$$

which is the [[direct sum]] of the values of $A_\bullet$ on the given intersections as indicated; and whose [[differential]]

$$
  d
  \colon
  C^{k}((X,\{U_i\}),A_\bullet)
  \longrightarrow
  C^{k+1}((X,\{U_i\}),A_\bullet)
$$

is defined componentwise (see at [[matrix calculus]] for conventions on maps between [[direct sums]]) by

$$
  \begin{aligned}
    (d a)_{i_0, \cdots, i_{k+1}}
    & \coloneqq
    (\partial_A a + (-1)^k \delta a)_{i_0, \cdots, i_{k+1}}
    \\
    & \coloneqq
    \partial_A a_{i_0, \cdots, i_{k+1}}
    +
    (-1)^k
    \sum_{0 \leq j \leq k+1} (-1)^{j}
    a_{i_0, \cdots, i_{j-1}, i_{j+1}, \cdots, i_{k+1}}
    |_{U_{i_0, \cdots, i_{k+1}}}
  \end{aligned}
$$

where on the right the sum is over all components of $a$ obtained via the canonical restrictions obtained by discarding one of the original $(k+1)$ subscripts.

The **Čech cohomology** groups of $X$ with coefficients in $A_\bullet$ __relative to the given cover__ are the [[chain homology]] groups of the Čech complex

$$
  H_{Cech}^k((X,\{U_i\}), A_\bullet)
  \coloneqq
  H^k(C^\bullet((X,\{U_i\}),A_\bullet))
  \,.
$$

The **Čech cohomology** groups as such are the [[colimit]] ("[[directed limit]]") of these groups over refinements of covers

$$
  H^k_{Cech}(X, A_\bullet)
  \coloneqq
  \underset{\longrightarrow}{\lim}_{\{U_i \to X\}}
  H_{Cech}^k((X,\{U_i\}), A_\bullet)
  \,.
$$

=--

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

Often Čech cohomology is considered for the case that $A_\bullet$ is concentrated in a single degree, in which case the first term in the sum defining the differential in def. \ref{CechComplex} disappears.
When $A_\bullet$ is not concentrated in a single degree, then for emphasis and following terminology of _[[hypercohomology]]_
one may speak of the **Čech hypercomplex** computing **Čech hypercohomology**.

=--

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

The Čech chain complex in def. \ref{CechComplex} is the [[total complex]] of the [[double complex]] whose vertical differential is that of $A_\bullet$ and whose horizontal differential is the _Čech differential_ $\delta$ given by alternating sums over restrictions along patch inclusions

$$
  \array{
     \vdots && \vdots
     \\
     \downarrow^{\mathrlap{\partial_A}}
     &&
     \downarrow^{\mathrlap{\partial_A}}
     \\
     \oplus_i A_1(U_i)
     &\stackrel{\delta}{\longrightarrow}&
     \oplus_{i_1, i_2} A_1(U_{i_1, i_2})
     &\stackrel{\delta}{\longrightarrow}&
     \cdots
     \\
     \downarrow^{\mathrlap{\partial_A}}
     &&
     \downarrow^{\mathrlap{\partial_A}}
     \\
     \oplus_i A_0(U_i)
     &\stackrel{\delta}{\longrightarrow}&
     \oplus_{i_1, i_2} A_0(U_{i_1, i_2})
     &\stackrel{\delta}{\longrightarrow}&
     \cdots
  }
$$

=--

### Relation to nonabelian Čech cohomology
 {#RelationAbelianNonabelian}

We discuss how the abelian Čech complex of def. \ref{CechComplex} arises as a special case of the simplicial nonabelian cocycle complex of general nonabelian Čech cohomology [above](#NonabelianČechCohomology), under the [[Dold-Kan correspondence]].

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

It may be helpful to keep the following pictures in mind to match the signs to the orientations of simplices.

In dimension one we have:

$$
  (0 \to 1)
  \;\;\;

  \mapsto
  \;\;\;
  (a_0 \stackrel{a_{0 1}}{\to} a_1)
  \;\;\;
  \leftrightarrow
  \;\;\;
  (a_1 = a_0 + d_A a_{01})
$$

In dimension 2 we have:

$$
  \array{
    && 1
    \\
    & \nearrow &\Downarrow& \searrow
    \\
    0 &&\to&& 2
  }
  \;\;\;
  \mapsto
  \array{
    && a_1
    \\
    & {}^{a_{0 1}}\nearrow &\Downarrow^{a_{0 1 2}}
    & \searrow^{a_{1 2}}
    \\
    a_0 &&\stackrel{a_{0 2}}{\to}&& a_2
  }
  \;\;\;
  \leftrightarrow
  (a_{0 2} = a_{0 1} + a_{1 2} + d a_{0 1 2})
  \,.
$$

Here the relation on the right is the
[[Dold-Kan correspondence]] relating coboundaries in the
complex $A_\bullet$ to simplices .

=--

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

The following derivation of abelian Čech cohomology from nonabelian Čech cohomology restricted to nerves of chain complexes needs of the [[Dold-Kan correspondence]] only that it is an [[adjunction]], not that it is an [[adjoint equivalence]].

=--

+-- {: .num_prop }
###### Proposition (recovering abelian Čech cohomology)

Let $A_\bullet$ be a sheaf with values in $Ch_+$
and write $N(A_\bullet)$ for the corresponding
simplicial sheaf under the [[nerve]] operation
of the [[Dold-Kan correspondence]].

Then the general (nonabelian) Čech cohomology
of $N(A_\bullet)$ as defined above coincides
with the cohomology of the Čech complex of
$A_\bullet$:

$$
  H(X, N(A_\bullet))
  \simeq
  colim_U H_0(C(U,A_\bullet))
  \,.
$$

=--

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

The underlying idea is to use the
adjunction of the [[Dold-Kan correspondence]]
to move the nerve operation $N(A_\bullet)$
on the right to the
free abelian chain complex operation
$C_bullet(F(C(U)))$
on the
[[Čech cover]] simplicial sheaf $C(U)$ and then
use the [[Yoneda lemma]] to evaluate $A_\bullet$
on $N_\bullet(F(C(U)))$. The result is the
Čech complex of $A_\bullet$.
Cocycles and homotopies/coboundaries
are in bijection on both sides.

Spelled out in full detail this looks a bit more lengthy,
but is nothing but this simple idea.

Before starting the computation notice the following
observation on the image of the [[Čech cover]]
$C(U)$ under the [[Dold-Kan correspondence]]:

For $Y = C(U)$ a [[Čech cover]] on a sieve
$\{U_i \to X\}_{i \in I}$, and for $W$ any test object,
the non-degenerate $n$-simplices in $C(U)(W)$ are the

$$
  U_{i_0, i_1, \cdots, i_n}(W)
$$

with all indices pairwise different, $i_k \neq i_l$.

Accordingly the free abelian simplicial group-valued
sheaf $F(C(U))$ is for each $W$ and $n$ the
free abelian group generated from these
$U_{i_0, i_1, \cdots, i_n}(W)$ with pairwise distinct elements, i.e. that given by
formal interger combinations of these element.

In turn, the normalized [[Moore complex]] sheaf

$$
  N_\bullet(F(C(U)))
$$

assigns to a test domain $W$ the complex that in
each degree is the free abelian group on these
elements.

With this in hand,
we compute the set of $N(A_\bullet)$-valued cocycles on $U$ as follows:

$$
  \begin{aligned}
    Hom_{SPSh}(C(U), N(A_\bullet))
    &\simeq
    \int_W Hom_{SSet}( C(U)(W) , N(A_\bullet)(W) )
    \\
    & \simeq
    \int_W Hom_{SSet}( C(U)(W) , N(A(W)_\bullet) )
    \\
    & \simeq
    \int_W Hom_{SAb}( F(C(U)(W)) , N(A(W)_\bullet) )
    \\
    & \simeq
    \int_W Hom_{Ch_+}( N_\bullet(F(C(U)(W))) ,
         A(W)_\bullet )
    \\
    & \simeq
    \int_W \int_{n}
       Hom_{Ab}( N_n(F(C(U)(W)), A(W)_n )
    \\
    & \simeq
       \int_{n}
       \int_W
       Hom_{Ab}( N_n(F(C(U)(W)), A(W)_n )
    \\
    & \simeq
       \int_{n}
       \int_W
       Hom_{Set}( \coprod_{i_0, i_1, \cdots, i_k}
       U_{i_0,i_1, \dots, i_n}(W) , A(W)_n )  )
    \\
    & \simeq
    \int_{n}
       \prod_{i_0, i_1, \cdots, i_n}
       A(U_{i_0,i_1, \dots, i_n})_n
    \\
    & =
    ker d_{C(U, A_\bullet)} \subset C(U,A_\bullet)_0
  \end{aligned}
  \,.
$$

Here

* The first step is the definition of morphism of [[presheaf|presheaves]] using the [[end]] notation;

* the second step is the definition of the [[nerve]] of of [[chain complexes]] applied to chain complex valued sheaves;

* the third step uses that the free simplicial abelian group functor is [[left adjoint]] to the forgetful one that remembers the underlying simplicial set;

* the fourth step then uses the [[Dold-Kan correspondence]], or actually just that the normalized [[Moore complex]] functor is [[left adjoint]] to the [[nerve]] of [[chain complexes]];

* the fifth step expresses the set of morphisms of [[chain complexes]] as an [[end]] (being itself a [[natural transformation]]);

* the sixth step uses the [[Fubini theorem]] of [[enriched category theory]] to commute the two [[ends]];

* the seventh step uses that the chain complex in the left argument is generated freely on elements of a set to rewrite the hom of abelian groups into one of sets;

* this finally allows to apply the [[Yoneda lemma]] in step eight

* and in step nine, by inspection, one notices that the result thus obtained is the set of 0-cycles in the Čech complex $C(U,A_\bullet)$ as previously defined.

An entirely analogous argument shows that dividing out homotopies is respected.

One starts by observing that the cohomology coboundaries are given by homotopies in the hom-simplicial set, i.e. by

$$
  Hom_{SPSh}(C(U) \times \Delta^1, N(A_\bullet))
  \,.
$$

With this one goes in the above computation.
After applying the
[[Dold-Kan correspondence|Dold-Kan adjunction]]
we now have on the left in the integrand the term

$$
  N_\bullet(F(C(U)(W) \times \Delta^1))
  \,.
$$

Using first that the free simplicial group functor is monoidal and then that the normalized [[Moore complex]]
functor is lax monoidal (as described at
[[Dold-Kan correspondence]])

> what to do about laxness versus pseudoness??

we get a morphism to that from

$$
  N_\bullet(F(C(U)(W))) \otimes  N_\bullet(F(\Delta^1))
  \,.
$$

Using that the normalized Moore complex is
isomorphic to the Moore complex divided by the part
generated by degenerate cells,
on the right we identify the [[interval object]] complex

$$
   (  \cdots \to 0 \to I  \stackrel{1 \oplus -1}{\to} I \oplus I)
  \,.
$$

To see that one just needs to observe that the normalized [[Moore complex]] of the 1-[[simplex]] serves as an [[interval object]] in chain complexes.

=--

### Relation to abelian sheaf cohomology

For $A$ a complex of sheaves, there is a canonical morphism

$$
  \check{H}(X,A) \to H(X,A)
$$

from the Čech cohomology to the full (hypercompleted) cohomology, which is [[abelian sheaf cohomology]] in the case that $A$ is in the image of the [[Dold-Kan correspondence|Dold-Kan map]] from [[chain complexes]]. Using the description of abelian sheaf cohomology in terms of morphisms out of hypercovers described at the beginning of this entry, this morphism is the obvious one coming from the inclusion of [[Čech covers]] into all [[hypercovers]].

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

If $X$ is a [[paracompact space]] the canonical morphism

$$
  \check{H}(X,A) \to H(X,A)
$$

from Čech cohomology to [[abelian sheaf cohomology]] is an isomorphism from every $A \in Sh(X, Ch_+)$.

=--

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

Recalled as theorem 1.3.13 in

* Brylinski, _Loop spaces, characteristic classes and geometric quantization_

=--

When $X$ is not paracompact, we still have the following condition under which Čech cohomology computes [[abelian sheaf cohomology]]-.

For $A$ a complex of sheaves, there is a canonical morphism

$$
  H_0(C(U,A)_\bullet) \to H(X,A)
$$

from the cohomology of the Čech complex with respect to a cover $U$ with coefficients in $A$ to the [[abelian sheaf cohomology]] of $X$ with values in $A$. Using the description of abelian sheaf cohomology in terms of morphisms out of hypercovers described at the beginning of this entry, this morphism is the obvious one coming from the inclusion of [[Čech covers]] into all [[hypercovers]].

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

Let $A$ be a complex of sheaves on $X$ concentrated
in a single degree $p \geq 0$,
and let $\{U_i \to X\}$ be a [[cover]]
such that the $A$-cohomology of all intersections vanishes,

$$
 \forall U_{i_0 , \cdots, i_k} H(U_{i_0, \cdots, i_k},A) = 0
$$

Then the canonical morphism

$$
  H_0(C(U,A)_\bullet) \to H(X,A)
$$

is an [[isomorphism]].

=--

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

One considers the spectral sequence associated with the Čech double complex. Details are on page 28 of

* Brylinski, _Loop spaces, characteristic classes and geometric quantization_

=--

### Examples ###

We now list examples for abelian Čech cohomology expressed in terms of the Čech complex described above.

#### Line bundles ####

Let $G = U(1)$ be the circle group, an abelian
group. The [[nerve]] $N(\mathbf{B}G)$
of its [[delooping]] $\mathbf{B}G$ is the
[[bar construction]] of $G$. This is equivalently
the nerve of the [[chain complex]]

$$
  U(1)[1]
  := ( \cdots \to 0 \to U(1) \to 0 )
  \,.
$$

Equivalently we get the corresponding simplicial sheaves
and complexes of sheaves, eg

$$
  U(1)[1] : W \mapsto
  ( \cdots \to 0 \to Hom(W,U(1)) \to 0 )
  \,.
$$

So given a cover $\{U_i \to X\}$ a Čech cocycle is a collection

$$
  c = ( \{g_{i j} \in U(1)[1](U_{i j})\}|_{i,j})
$$

such that the Čech differential evaluated on it

$$
  \begin{aligned}
    (d_{C(U,U(1)[1])} c)_{i j k}
    &=
    g_{j k}|_{U_{i j k}} - g_{i k}|_{U_{i j k}}
    + g_{i j}|_{U_{i j k}} )
  \end{aligned}
$$

vanishes:

$$
  g_{i j}|_{U_{i j k}} + g_{j k}|_{U_{i j k}} = g_{i k}|_{U_{i j k}}
$$

for all $i, j, k$.

Here now we write plus signs for the operation in the abelian group $U(1)$ which above we have written by juxtaposition or using a dot. So up to notation for the group operation this is

$$
  g_{i j} g_{j k} = g_{i k}
$$

on $U_{i j k}$ as before in the nonabelian case of
Čech cocycles for
$U(1)$-[[principal bundles]].

Here and from now on we shall notationally suppress the restriction maps $(-)|_{U_{i_0, i_1, \cdots, i_n}}$
as they are unambiguously obviuous in every case.

#### Line bundle gerbes

Similarly by shifting $U(1)$ ever higher in chain degree,
one finds Čech cocycles for [[bundle gerbes]], bundle 2-gerbes, etc.

A Čech cocycle for

$$
  U(1)[2]
  := ( \cdots \to 0 \to U(1) \to 0 \to 0 )
  \,.
$$

is

$$
  c = ( \{g_{i j k} \in U(1)[2](U_{i j k})\}|_{i,j, k})
$$

such that on $U_{i,j,k,l}$ we have

$$
  g_{i j k} - g_{i j l} + g_{i k l} - g_{j k l} = 0
$$

which in the nonabelian context would be written as

$$
  g_{i j k} g_{i k l} = g_{i j l} g_{j k l}
  \,.
$$

#### Čech-Deligne cohomology

When refining the complexes of sheaves $U(1)[n]$
to the [[Deligne cohomology|Deligne complex]]

$$
  U(1)[n] \hookrightarrow \mathbb{Z}(n+1)_D^\infty
$$

and then evaluating Čech cohomology with coefficients
in the Deligne complex, we obtain the formulas for
Čech-Deligne cohomology.

* For $n = 1$ a cocycle is a collection
  $$
    ( A_i \in \Omega^1(U_i),
      g_{i j} \in C^\infty(U_{i j}, U(1)) )
  $$
  such that for all $i, j$
  $$
    d log g_{i j} + A_i - A_j -  = 0
  $$
  and for all $i,j ,k$
  $$
    g_{i j} g_{j k} = g_{i k}
    \,.
  $$
  Such cocycles classify $U(1)$-[[principal bundles]]
  with [[connection on a bundle|connection]].

  These $n=1$ Čech-Deligne cocycles appear naturally in the study of the [[electromagnetic field]].

* For $n = 2$ a cocycle is a collection
  $$
    ( B_i \in \Omega^2(U_i),
      A_{i j} \in \Omega^1(U_{i j}),
      g_{i j k} \in C^\infty(U_{i j k}, U(1)) )
  $$
  such that for all $i, j$
  $$
    d A_{i j} + B_i - B_j = 0
  $$
  and for all $i,j ,k$
  $$
    A_{i j} - A_{i k} + A_{j k} + d log g_{i j k} = 0
  $$
  and for all $i, j, k ,l$

  $$
    g_{i j k} g_{i k l} = g_{i j l} g_{j k l}
    \,.
  $$
  Such cocycles classify $U(1)$-[[bundle gerbes]]
  with connection.

## Related entries

* For discussion of the isomorphism between isomorphism classes of [[topological vector bundles]] and Čech cohomology on [[topological spaces]] with coefficients in continuous functions with values in the [[general linear group]] see [there](topological+vector+bundle#TransitionFunctionsAndCechCohomology).

## References

Classical accounts (in the generality of [[nonabelian cohomology]]):

* [[Alexander Grothendieck]], *A General Theory of Fibre Spaces With Structure Sheaf*, University of Kansas, Report No. 4 (1955, 1958) [[pdf](https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Kansasnotes.pdf), [[Grothendieck-FibreSpaces.pdf:file]]]

* [[Jean Frenkel]], *Cohomologie à valeurs dans un faisceau non abélien*, C. R. Acad. Se., t. 240 (1955) 2368-2370

* [[Jean Frenkel]], *Cohomologie non abélienne et espaces fibrés*, Bulletin de la Société Mathématique de France, **85** (1957) 135-220 [[numdam:BSMF_1957__85__135_0](http://www.numdam.org/item/BSMF_1957__85__135_0)]

* {#Godement58} [[Roger Godement]], *Topologie algébrique et theorie des faisceaux*, Actualités Sci. Ind. **1252**, Hermann, Paris (1958) [[webpage](https://www.editions-hermann.fr/livre/topologie-algebrique-et-theorie-des-faisceaux-roger-godement), [[Godement-TopologieAlgebrique.pdf:file]]]

Textbook account:

* [[Torsten Wedhorn]], *Torsors and Non-abelian Čech Cohomology*, chapter 7 in: *Manifolds, Sheaves, and Cohomology*, Springer (2016) [[doi:10.1007/978-3-658-10633-1](https://doi.org/10.1007/978-3-658-10633-1)]

A historical survey (of some aspects) is in

* [[David Edwards]], Harold M. Hastings, _Cech Theory: Its past, present, and future_, Rocky Mountain J. Math. Volume 10, Number 3 (1980), 429-468. ([Euclid](http://projecteuclid.org/euclid.rmjm/1250128825))

A motivational introduction from within [[complex analytic geometry]] is in

* Michael Weiss, _The Search for a Global Primitive -- Cech Cohomology with Coefficients in a Sheaf_ ([pdf](http://files.meetup.com/4699592/cech-ipad.pdf))

A discussion of Čech cohomology in the wider context of [[cohomology]] particularly realized in terms of the [[model structure on simplicial presheaves]] and with an emphasis on the shades of notions between [[Čech cover]] and [[hypercover]] is

* [[Tibor Beke]], _Higher Čech theory_ ([journal](http://www.math.uiuc.edu/K-theory/0646/), [pdf](http://www.math.uiuc.edu/K-theory/0646/cech.pdf))

Abelian Čech cohomology is discussed in some detail in section I.3 of

* [[Jean-Luc Brylinski]], _Loop spaces, characteristic classes and geometric quantization_

A discussion of the [[model structure on simplicial presheaves]] with an eye towards the distinction between Čech and hyperlocalization is in

* [[Daniel Dugger]], _Sheaves and Homotopy theory_ ([web](http://math.uoregon.edu/~ddugger/cech.html), [pdf](http://ncatlab.org/nlab/files/cech.pdf))

(But beware that, while providing useful insights, these are unfinished abandoned notes that seem to have some gaps right at this point.)

A long list of reasons why higher Čech cohomology might
is after all better behaved than its [[hypercompletion]] where a cycle is with respect to an arbitrary [[hypercover]] is in section 6.5.4, **Descent versus Hyperdescent** of

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

The relation between smooth and continuous Čech cohomology is discussed in

* Christoph Müller, [[Christoph Wockel]], _Equivalences of smooth and continuous principal bundles with infinite-dimensional structure groups_, Advances in Geometry. Volume 9, Issue 4, Pages 605–626 (2009)

[[!redirects {?}ech cohomology]]
[[!redirects ?ech cohomology]]
[[!redirects ?ech cohomology]]
[[!redirects Cech cohomology]]


Čech cohomology > history

< [[Čech cohomology]]


Čech cover

# Čech covers
* table of contents
{: toc}

## Idea

A _Čech cover_ is a [[Čech nerve]] $C(U)$ that comes from a [[cover]] $U \to X$.

## Definition

Let $C$ be a [[site]] and $\{U_i \to X\}$ a covering [[sieve]]. Write $U = \sqcup_i Y(U_i)$ for the [[coproduct]] of the patches in the [[presheaf]] category $PSh(C)$ ($Y$ is the [[Yoneda embedding]]).

Write $X$ for $Y(X)$, for short.

Then the [[Čech nerve]] $C(U)$ of $U \to X$ in $PSh(C)$, i.e. the [[simplicial presheaf]]

$$
  \cdots
  U \times_X U \times_X U
  \stackrel{\stackrel{\to}{\to}}{\to}
  U \times_X U \stackrel{\to}{\to} U
$$

is called a **Čech cover**.

## Properties

Consider the [[local model structure on simplicial presheaves]] on $C$. If the [[sheaf]] [[topos]] $Sh(C)$ has [[point of a topos|enough points]], then the weak equivalences (called local weak equivalences for emphasis) are the [[stalk]]-wise weak equivalences of [[simplicial set ]] (with respect to the standard [[model structure on simplicial sets]]).

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

Write

$$
  \pi_0(C(U)) = colim_{[n] \in \Delta^{op}} C(U)_n
$$

for the presheaf of connected components (see [[simplicial homotopy group]]) of $C(U)$. Regard this as a simplicially constant [[simplicial presheaf]].

=--

**Remark**. By the discussion in the section "Interpretation in terms of descent and codescent" at [[sieve]] this $\pi_0(C(U))$, regarded as an ordinary presheaf, is precisely the subfunctor of $Y(X)$ that corresponds to the [[sieve]] $\{U_i \to X\}$.

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

For every Čech cover $C(U)$ the morphism of simplicial presheaves

$$
  C(U) \to \pi_0(C(U))
$$

is a local weak equivalence.

=--

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

Being a simplicially discrete simplicial sheaf, for every test object $V$ $\pi_0(C(U))(V)$ has all [[simplicial homotopy group]]s trivial except possibly the set of connected components.
But by the very definition of $\pi_0(C(U))$ the morphism
$C(U) \to \pi_0(C(U))$ is a bijection on $\pi_0$.

Over each test domain $V$ the [[simplicial set]]
$C(U)(V)$ is just the [[nerve]] of the [[Cech groupoid|Čech groupoid]]
$$
   \left(
     C(V,U)\times_{C(V,U \times_X U)}
     \stackrel{\to}{\to}
     C(V,U)
   \right)
  \,.
$$
The nerve of that groupoid is readily seen to have vanishing first [[simplicial homotopy group]]. Being the [[nerve]] of a 1-[[groupoid]], also all higher [[simplicial homotopy group]]s vanish.

So $C(U) \to \pi_0(C(U))$ induces for each object $V \in C$ an isomorphism of [[simplicial homotopy group]]s. It therefore is an objectwise weak equivalence of simplicial sets.

See also for instance lemma 3.3.5 in

* Daniel Dugger, _Sheaves and homotopy theory_ ([web](http://www.uoregon.edu/~ddugger/cech.html), [pdf](http://ncatlab.org/nlab/files/cech.pdf))

=--

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

Every Čech cover

$$
  C(U) \to X
$$

is a stalkwise weak equivalence.

=--

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

From the above we know that $C(U) \to X$ factors as

$$
  C(U) \to \pi_0(C(U)) \to X
$$

and that the first morphism is an objectwise, hence also a [[stalk]]wise weak equivalence. It therefore suffices to show that $\pi_0(C(U)) \to X$ is a stalkwise weak equivalence.

But by the remark above, $\pi_0(C(U)) \to X$ is actually the [[local isomorphism]] corresponding to the [[cover]] $U$. It is therefore even a [[stalk]]wise [[isomorphism]].

See also for instance lemma 3.4.9 in

* Daniel Dugger, _Sheaves and homotopy theory_ ([web](http://www.uoregon.edu/~ddugger/cech.html), [pdf](http://ncatlab.org/nlab/files/cech.pdf))

=--

+-- {: .un_remark}
###### Remark
So this says that every Čech cover is a [[hypercover]]. But not conversely. [[localization|Localization]] of [[simplicial presheaf|simplicial presheaves]] at Čech covers yields [[Čech cohomology]].
=--

[[!redirects Čech cover]]
[[!redirects Čech covers]]
[[!redirects Cech cover]]
[[!redirects Cech covers]]


Čech descent

## Idea

The **Čech descent** condition is used to refer to a specific [[∞-descent]] condition for [[∞-presheaves]].

Concretely, Čech descent can be implemented by performing the [[left Bousfield localization]] of the [[model category]] of [[simplicial presheaves]] equipped with the [[projective model structure]] (say) at [[Čech nerves]] of [[covering families]].
Then [[fibrant objects]] in the resulting [[model structure]] will be precisely [[presheaves]] of [[Kan complexes]] that satisfy the Čech descent condition.

Čech descent should be contrasted with [[hyperdescent]].
The [[hyperdescent]] condition is stronger than the Čech descent condition, since it uses arbitrary [[hypercovers]] instead of [[Čech nerves]] of [[covering families]].
However, for [[hypercomplete sites]] such as [[Zariski site]], [[Nisnevich site]], [[CartSp]], [[Stein site]], and most of the other sites used in differential and complex geometry, the [[hyperdescent]] condition coincides with the Čech descent condition.

## Related concepts

* [[hyperdescent]]

* [[∞-presheaf]], [[∞-sheaf]], [[simplicial presheaf]]

* [[Čech nerve]]


Čech groupoid

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For $\{U_i \to X\}$ a [[cover]] of a [[space]] $X$, the corresponding **Čech groupoid** is the [[internal groupoid]]

$$
  C(\{U_i\}) = (\coprod_{i j} U_i \cap U_j \rightrightarrows \coprod_i U_i)
$$

whose set of objects is the [[disjoint union]] $\coprod_i U_i$ of the covering patches, and the set of morphisms is the disjoint union of the [[intersections]] $U_i \cap U_j$ of these patches.

This is the $2$-[[coskeleton]] of the full [[Čech nerve]]. See there for more details.

If we speak about [[generalized element|generalized points]] of the $U_i$ (which are often just ordinary points, in applications), then

* an [[object]] of $C(\{U_i\})$ is a pair $(x,i)$ where $x$ is a point in $U_i$;

* there is a unique [[morphism]] $(x,i) \to (x,j)$ for all pairs of objects labeled by the same $x$ such that $x \in U_i \cap U_j$;

* hence the composition of morphism is of the form

  $$
    \array{
       && (x,j)
       \\
       & \nearrow &=& \searrow
       \\
      (x,i) &&\to&& (x,k)
    }
    \,.
  $$

## Definition
 {#Definition}

+-- {: .num_defn #CechGroupoid}
###### Definition
**(Cech groupoid)**

Let $\mathcal{C}$ be a [[site]], and $X \in \mathcal{C}$ an [[object]] of that site. For each [[covering]] family $\{ U_i \overset{\iota_i}{\to} X\}$ of $X$ in the given [[coverage]], the _[[Cech groupoid]]_ is the [[presheaf of groupoids]]

$$
  C(\{U_i\})
    \;\in\;
  [\mathcal{C}^{op}, Grpd]
  \;\simeq\;
  Grpd\left( [\mathcal{C}^{op}, Set]  \right)
$$

which, regarded as an [[internal category]] in the [[category of presheaves]] over $\mathcal{C}$, has as [[presheaf]] of [[objects]] the [[coproduct]]

$$
  Obj_{C(\{U_i\})} \;\coloneqq\; \underset{i}{\coprod}  y(U_i)
$$

of the [[representable presheaf|presheaves represented]] (under the [[Yoneda embedding]]) by the [[covering]] objects $U_i$, and as [[presheaf]] of [[morphisms]] the [[coproduct]] over all [[fiber products]] of these:

$$
  Mor_{C(\{U_i\})}
    \;\coloneqq\;
  \underset{i,j}{\coprod}  y(U_i) \times_{y(X)} y(U_j)
  \,.
$$

This means that for any $V \in \mathcal{C}$ the [[groupoid]] assigned by $C(\{U_i\})$ has as set of objects [[pairs]] consisting of an index $i$ and a morphism $V \overset{\kappa_i}{\to} U_i$ in $\mathcal{C}$, and there is a unique morphism between two such objects

$$
  \kappa_i \longrightarrow \kappa_j
$$

precisely if

\[
  \label{CechMatchingCondition}
  \iota_i \circ \kappa_i
  \;=\;
  \iota_j \circ \kappa_j
  \phantom{AAAAAAAA}
  \array{
    && V
    \\
    & {}^{\mathllap{\kappa_i}}\swarrow && \searrow^{\mathrlap{\kappa_j}}
    \\
    U_i && && U_j
    \\
    & {}_{\mathllap{\iota_i}}\searrow && \swarrow_{\mathrlap{\iota_j}}
    \\
    && X
  }
\]

=--

## Properties

### Codescent
 {#Codescent}

We discuss (Prop. \ref{CechGroupoidCoRepresents} below) how the Cech groupoid co-represents _[[matching families]]_ as they appear in the definition of [[sheaves]].

For reference, we first recall that definition:

+-- {: .num_defn #CompatibleElements}
###### Definition
**([[matching family]] -- [[descent object]])**

Let $\mathcal{C}$ be a [[small category]] equipped with a [[coverage]], hence a [[site]] and consider a [[presheaf]] $\mathbf{Y} \in [\mathcal{C}^{op}, Set]$ (Example \ref{CategoryOfPresheaves}) over $\mathcal{C}$.

Given an [[object]] $X \in \mathcal{C}$ and a [[covering]] $\left\{ U_i \overset{\iota_i}{\to} X \right\}_{i \in I}$ of it  we say that a _[[matching family]]_ (of probes of $\mathbf{Y}$) is a [[tuple]] $(\phi_i \in \mathbf{Y}(U_i))_{i \in I}$ such that for all $i,j \in I$ and [[pairs]] of [[morphisms]] $U_i \overset{\kappa_i}{\leftarrow} V \overset{\kappa_j}{\to} U_j$ satisfying

\[
  \label{MatchingCondition}
  \iota_i \circ \kappa_i
  \;=\;
  \iota_j \circ \kappa_j
  \phantom{AAAAAAAA}
  \array{
    && V
    \\
    & {}^{\mathllap{\kappa_i}}\swarrow && \searrow^{\mathrlap{\kappa_j}}
    \\
    U_i && && U_j
    \\
    & {}_{\mathllap{\iota_i}}\searrow && \swarrow_{\mathrlap{\iota_j}}
    \\
    && X
  }
\]

we have

\[
  \label{GluingCondition}
  \mathbf{Y}(\kappa_i)(\phi_i)
  \;=\;
  \mathbf{Y}(\kappa_j)(\phi_j)
  \,.
\]

We write

\[
  \label{SetOfMatching}
  Match\big(
    \{U_i\}_{i \in I}
    \,,\,
    \mathbf{Y}
  \big)
  \subset
  \underset{i}{\prod} \mathbf{Y}(U_i)
  \;\in\;
  Set
\]

for the set of [[matching families]] for the given presheaf and covering.

This is also called the _[[descent object]]_ of $\mathbf{Y}$ for _[[descent]]_ along the [[covering]] $\{U_i \overset{\iota_i}{\to}X\}$.

=--

+-- {: .num_example #CechGroupoidCoRepresents}
###### Proposition
**([[Cech groupoid]] co-represents [[matching families]] -- [[codescent]])**

For [[Grpd]] regarded as a [[cosmos]] for [[enriched category theory]], via its [[cartesian closed category]]-structure, and $\mathcal{C}$ a [[site]], let

$$
  \mathbf{Y} \in [\mathcal{C}^{op}, Set] \hookrightarrow [\mathcal{C}^{op}, Grpd]
$$

be a [[presheaf]] on $\mathcal{C}$, regarded as a [[Grpd]]-[[enriched presheaf]], let $X \in \mathcal{C}$ be any [[object]] and $\{U_i \overset{\iota_i}{\to} X\}_i$ a [[covering]] family with induced [[Cech groupoid]] $C(\{U_i\}_i)$ (Def.\ref{CechGroupoid}).

Then there is an [[isomorphism]]

$$
  [\mathcal{C}^{op},Grpd]
  \left(
    C\left(\{U_i\}_i\right), \, \mathbf{Y}
  \right)
  \;\simeq\;
  Match\left( \{U_i\}_i, \, \mathbf{Y} \right)
$$

between the [[hom-object|hom-groupoid]] of [[Grpd]]-[[enriched presheaves]]  and the set of [[matching families]] (Def. \ref{CompatibleElements}).

Since therefore the Cech-groupoid co-represents the [[descent object]], it is sometimes called the _[[codescent object]]_ along the given covering.

Moreover, under this identification the canonical morphism $C\left( \{U_i\}_i \right) \overset{p_{\{U_i\}_i}}{\longrightarrow} X$ induces the comparison morphism

$$
  \array{
    [\mathcal{C}^{op}, Grpd]\left( X, \, \mathbf{Y} \right)
    & \simeq &
    \mathbf{Y}(X)
    \\
    {}^{
      \mathllap{
        [\mathcal{C}^{op}, Grpd](p_{\{U_i\}_i}, \mathbf{Y})
      }
    }\downarrow && \downarrow
    \\
    [\mathcal{C}^{op},Grpd]
    \left(
      C\left(\{U_i\}_i\right), \, \mathbf{Y}
    \right)
    &\simeq&
    Match\left( \{U_i\}_i, \, \mathbf{Y} \right)
  }
  \,.
$$

In conclusion, this means that the [[presheaf]] $\mathbf{Y}$ is a [[sheaf]] (Def. \ref{Sheaf}) precisely if homming Cech groupoid projections into it produces an isomorphism.

=--

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

The hom-groupoid is computed as the [[end]]

$$
  [\mathcal{C}^{op},Grpd]
  \left(
    C\left(\{U_i\}_i\right), \, \mathbf{Y}
  \right)
  \;=\;
  \int_{V \in \mathcal{C}}
  \left[
    C\left(\{U_i\}_i\right)(V), \, \mathbf{Y}(V)
  \right]
  \,,
$$

where the "integrand" is the [[functor category]] (here: a [[groupoid]]) from the [[Cech groupoid]] at a given $V$ to the set (regarded as a groupoid) assigned by $\mathbf{Y}$ to $V$.

Since $\mathbf{Y}(V)$ is just a set, that functor groupoid, too, is just a set, regarded as a groupoid. Its elements are the [[functors]] $C\left(\{U_i\}_i\right)(V) \longrightarrow \mathbf{Y}(V)$, which are equivalently those  [[functions]] on sets of objects

$$
  \underset{i}{\coprod} y(U_i)(V)
  =
  Obj_{C\left(\{U_i\}_i\right)(V)}
    \longrightarrow
  Obj_{\mathbf{Y}(V)}
  =
  \mathbf{Y}(V)
$$

which respect the [[equivalence relation]] induced by the morphisms in the Cech groupoid at $V$.

Hence the hom-groupoid is a subset of the [[end]] of these [[function sets]]:

$$
  \begin{aligned}
    \int_{V \in \mathcal{C}}
    \left[
      C\left(\{U_i\}_i\right)(V), \, \mathbf{Y}(V)
    \right]
    & \hookrightarrow
    \int_{V \in \mathcal{C}}
    \left[
      \underset{i}{\coprod} y(U_i)(V), \, \mathbf{Y}(V)
    \right]
    \\
    & \simeq
    \int_{V \in \mathcal{C}}
    \underset{i}{\prod}
    \left[
       y(U_i)(V), \, \mathbf{Y}(V)
    \right]
    \\
    & \simeq
    \underset{i}{\prod}
    \int_{V \in \mathcal{C}}
    \left[
       y(U_i)(V), \, \mathbf{Y}(V)
    \right]
    \\
    & \simeq
    \underset{i}{\prod}
    \mathbf{Y}(U_i)
  \end{aligned}
$$

Here we used: first that the [[internal hom]]-functor turns colimits in its first argument into limits (see at _[[internal hom-functor preserves limits]]_), then that [[limits commute with limits]], hence that in particular [[ends]] commute with [[products]] , and finally the [[enriched Yoneda lemma]], which here comes down to just the plain [[Yoneda lemma]] . The end result is hence the same [[Cartesian product]] set that also the set of matching families is defined to be a subset of, in (eq:SetOfMatching).

This shows that an element in
$ \int_{V \in \mathcal{C}}
    \left[
      C\left(\{U_i\}_i\right)(V), \, \mathbf{Y}(V)
    \right]
$ is a [[tuple]] $(\phi_i \in \mathbf{Y}(U_i))_i$, subject to some condition. This condition is that for each $V \in \mathcal{C}$ the assignment

$$
  \array{
    C\left(\{U_i\}_i\right)(V)
      & \longrightarrow &
    \mathbf{Y}(V)
    \\
    (V \overset{\kappa_i}{\to} U_i)
    &\mapsto&
    \kappa_i^\ast \phi_i
    =
    \mathbf{Y}(\kappa_i)(\phi_i)
  }
$$

constitutes a [[functor]] of [[groupoids]].

By definition of the [[Cech groupoid]], and since the [[codomain]] is a just [[set]] regarded as a [[groupoid]], this is the case precisely if

$$
  \mathbf{Y}(\kappa_i)(\phi_i)
  \;=\;
  \mathbf{Y}(\kappa_j)(\phi_j)
  \phantom{AAAA}
  \text{for all}\, i,j
  \,,
$$

which is exactly the condition (eq:GluingCondition) that makes $(\phi_i)_i$ a matching family.

=--

## Examples

For $X$ a [[smooth manifold]] and $\{U_i \to X\}$ an [[atlas]] by [[coordinate chart]]s, the Čech groupoid is a [[Lie groupoid]] which is equivalent to $X$ as a Lie groupoid:
$C(\{U_i\}) \stackrel{\simeq}{\to} X$

For $\mathbf{B}G$ the Lie groupoid with one object coming from a [[Lie group]] $G$ morphisms of Lie groupoids of the form

$$
  \array{
     C(\{U_i\}) &\stackrel{g}{\to}& \mathbf{B}G
     \\
     \downarrow^{\mathrlap{\simeq}}
     \\
     X
  }
$$

are also called [[anafunctor]]s from $X$ to $\mathbf{B}G$. They correspond to smooth $G$-[[principal bundle]]s on $X$.

## Related concepts

* [[Cech nerve]]

## References

For instance

* {#Jardine06} [[J. F. Jardine]], Example 5 in: _Homotopy classification of gerbes_ ([arXiv:math/0605200](https://arxiv.org/abs/math/0605200))

[[!redirects Cech groupoid]]
[[!redirects Cech groupoids]]
[[!redirects Čech groupoids]]


Čech homology

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic topology
+--{: .hide}
[[!include algebraic topology - contents]]
=--
=--
=--

#Čech homology#
* table of contents
{:toc}

## Idea

Given a [[topological space]] $X$, its *Čech homology* is the [[limit]] of [[simplicial homology]]-[[homology group|groups]] of the [[Čech nerves]] of its [[open covers]], under [[refinement of open covers]].

Compare to the [[singular homology]] of $X$, which is the [[simplicial homology]] of its [[singular simplicial complex]]. For well behaved topological spaces the two notions agree and are jointly known as computing the *[[ordinary homology]]* of $X$.

## Definition

Given a [[topological space]] $X$ with an [[open cover]],

$$
  \mathcal{U}
  \;=\;
  \Big\{
    U_i \xhookrightarrow{open} X
  \Big\}_{i \in I}
$$

we write

\[
  \label{CechNerve}
  C(X,\mathcal{U})
  \;\coloneqq\;
  \pi_0
  \big(
    U^{\times^\bullet_X}
  \big)
  \;\in\;
  sSet
\]

for the [[simplicial set]] which is its [Čech+nerve](Čech+nerve#FromACover):
whose $n$-[[simplex|simplices]] are the [[inhabited set|inhabited]] $(n + 1)$-fold [[intersections]] of the [[open subsets]] $U_i$ in $\mathcal{U}$.

\begin{remark}
For sufficiently well-behaved topological spaces $X$ ([[paracompact spaces]]) and *[[good open cover|good]]* open covers, the Čech nerve (eq:CechNerve) is [[simplicial homotopy equivalence|simplicially homotopy equivalent]] to the [[singular simplicial complex]] of $X$ -- this is the statement of the *[[nerve theorem]]*.

However, in general $C(X,\mathcal{U})$ may differ from the [[weak homotopy type]] of $X$ (even in the [[limit]] below) in which case Čech homology may *differ* from the [[singular homology]] of $X$. (See for instance the discussion at *[[well group]]*.)
\end{remark}

If $\mathcal{U}$ is a [[refinement of an open cover|refinement]] open cover, i.e. such that for each $U' \in \mathcal{U}$, there is a $U \in \mathcal{U}$ with $U' \subseteq U$, then these inclusions induce a [[morphism]] of [[simplicial sets]]

$$
  C(X,\mathcal{U}) \longrightarrow C(X,\mathcal{U}')
$$

This yields an [[inverse system]] of [[simplicial sets]].

\begin{definition}
The $n$th **Čech homology group** of the space $X$ is the [[limit]]
$$
  \check{H}_n(X)
  \;\coloneqq\;
  \underset{
    \underset{\mathcal{U}}{\longleftarrow}
  }{lim}
  H_n(X,\,\mathcal{U})
$$
over the [[inverse system]] of [[open covers]] $\alpha$, of the [[simplicial homology]] [[homology group|groups]] of the [Čech nerve](Čech+nerve#FromACover) $C(X,\alpha)$:
$$
  H_n(X,\,\mathcal{U})
  \;\coloneqq\;
  H_n\big(C(X,\,\mathcal{U})\big)
  \,.
$$
\end{definition}

\begin{remark}
It is to be noted that these groups do not constitute a [[homology theory]] in the sense of the [[Eilenberg-Steenrod axioms]] as the _exactness axiom_ fails in general. There is a "corrected" theory known under the name [[strong homology]].
\end{remark}

## Related concepts

* [[Čech methods]], [[nerve theorem]]

* [[singular homology]], [[cellular homology]]

* [[ordinary homology]]

* [[étale homotopy type]]

* [[well group]]

## References

Exposition:

* [[David H. Fremlin]], Section 2 of: _Singular homology for amateurs_ (2016) $[$[pdf](https://www1.essex.ac.uk/maths/people/fremlin/n16703.pdf), [[FremlinSIngularHomology.pdf:file]]$]$

On [[closed covers]] in Čech homology:

* [[Edwin E. Floyd]], *Closed coverings in Čech homology theory*, Trans. Amer. Math. Soc. **84** (1957) 319-337  [[doi:10.1090/S0002-9947-1957-0087100-2](https://doi.org/10.1090/S0002-9947-1957-0087100-2), [pdf](http://www.ams.org/journals/tran/1957-084-02/S0002-9947-1957-0087100-2/S0002-9947-1957-0087100-2.pdf)]

[[!redirects Cech homology]]
[[!redirects Čech homology]]

[[!redirects Čech homology theory]]
[[!redirects Cech homology]]


Čech homotopy

# Čech homotopy
* table of contents
{: toc}

##Idea and discussion##

In Čech cohomology, in its *traditional* form, you resolve the space by an inverse system  or [[pro-object]], $\check{C}(X,-)$, of nerves of open covers (or if you prefer the nerves of the corresponding sheaf  of groupoids). For constant and Abelian coefficients (and we will keep to those for the moment), $A$, we throw $H^n(-,A)$ at this pro-object.  As cohomology switches 'variance' we get a _direct system_ or _ind-object_ of Abelian groups, and usually we then take its (direct) limit (i.e. its colimit) to get $\check{H}^n(X,A)$.

Now dualise ... .  Replace cohomology by homotopy $[A,-]$, but although there will be some very useful long exact (Puppe) sequences available in homotopy, we cannot go this way to get them!  In dualising, things can go slightly 'wrong'. We have $[A,\check{C}(X,-)]$ is a pro-object and so we might want to take limits not colimits to get the thing we want, but $lim$ is not an exact functor on the usual categories available here, such as groups, sets, modules, etc.

There is another problem, coherence!  $\check{C}(X,-)$ is a pro-object in the homotopy category of simplicial sets, not in the category of simplicial sets itself.  It is a homotopy coherent system, but the choices of covering refinement maps cause the problem.

We are very near the topics discussed in [[shape theory]], [[Čech methods]] and [[strong shape theory]] (when that has been written!).
One interpretation of Čech homotopy is [[shape theory]], but that does not usually handle the 'homotopy is dual to cohomology' idea so this will need more work here.

##Handling the problem of coherence##

###Giving homotopy coherence directly:###

As explained in [[Čech methods]], given an open cover $\beta$ of a space $X$ and a refinement $\alpha$ of $U$
for a given set in $V$ there may be many different sets in $U$ containing it, so the seemingly natural way to get a simplicial map

$$\check{C}(X,\alpha)\to \check{C}(X,\beta)$$

involves a choice. We choose a 'refinement mapping' $\varphi : \alpha \to \beta$ so that
for each $V\in \alpha$, $V\subseteq \varphi(V)$. There is an obvious extension to $n$-simplices.  We send $\langle V_0, \ldots, V_n\rangle$ to $\langle \varphi(V)_0, \ldots, \varphi(V)_n\rangle$. This still does not get us a functor from $Cov(X)$ to $SSets$. The fact that we had to choose make it unlikely in the extreme that given three (or more) covers, each a refinement of the next, that the resulting diagram of simplicial sets will turn out (by some miracle) to be commutative. However once we choose a refinement maps for each pair $(\alpha,\beta)$, we _do_ get a [[homotopy coherent diagram]]

$$\check{C}(X,-) : Cov(X)\to SSets.$$

with the coherence being able to be spelled out completely, (see _Coverings, Hypercoverings and Homotopy Coherence_, thesis (1999) from Bangor, by Alinor Abdul Kadir, available [here](http://www.maths.bangor.ac.uk/research/ftp/theses/kadir.pdf)).

###Using the Vietoris complex: ###

The [[Vietoris complex]] is an alternative way of obtaining a simplicial complex (or simplicial set) from a space $X$ and an open cover $\alpha$. The disadvantage of it is that whilst the nerve of the open cover would be a finite simplicial set if the open cover is finite (for instance this is useful when handling compact spaces as we can replace the indexing set of _all_ open covers by that of _finite open covers_ without changing the result up to isomorphism), the Vietoris complex has as many vertices as there are points in $X$. It has the advantage that the induced maps between the simplicial complexes are independent of the choice of refinement maps so we do get a pro-simplicial set, $V(X,-) : Cov(X)\to SSet$. We therefore get a well defined [[pro-homotopy type]] $V(X)$ in the homotopy category $Ho(pro-SSet)$.  (The category $pro-SSet$ has various [[model category]] structures and [[cofibration category]] structures, with related basic notions of homotopy equivalence. These are discussed in the entry on [[pro-homotopy theory]].)

How are the two ways around the coherence problem related?

There is a result [[Dowker's Theorem]] which shows that the geometric realisations of $V(X,\alpha)$ and $\check{C}(X,\alpha)$ are homotopy equivalent. (This explains why Čech and Vietoris homology are isomorphic.) Given a commutative diagram of spaces (or simplicial sets), if you change some of the
spaces to homotopy equivalent spaces then you can build from those new spaces, a homotopy coherent diagram equivalent to the original one. Using Dowker's theorem we get that $\check{C}(X,-)$ is a homotopy coherent diagram, (by a second route).

## Homotopy groups or homotopy progroups?##

One obvious case to investigate of the above is to take a pointed space $X$, apply the $n^th$ homotopy group functor $\pi_n = [S^n, -]$ to the Čech or Vietoris complex to get a progroup, $\pi_n(\check{C}(X))$. This will lead to long exact sequences for pairs, fibrations etc., but these will be of progroups so are less amenable to study. If, on the other hand, you take the limit of these progroups, as mentioned above, you destroy the exactness of the sequences and thus, to a large extent, the simplicity of their use.

### Definition

The **$n^{th}$ Čech homotopy group** of a pointed space $X$ is defined to be $\check{\pi_n}(\check{C}(X)) = lim \pi_n(\check{C}(X))$.

### Examples

If $X$ is compact metric then it has countable cofinal subcategories of covers, so the progroups
$\pi_n(\check{C}(X))$ can be replaced up to isomorphism by inverse sequences, indexed by the natural numbers.  (These are sometimes called *towers*.)

Here are two specific examples:

1. The [[Warsaw circle]], $S_W$:  The progroup $\pi_1(\check{C}(S_W))$ is isomorphic to the  constant tower with value $C_\infty$,  the bonding maps being the identity maps.

1. The [[dyadic solenoid]]: This time the progroup is isomorphic in $Pro(Groups)$ to the inverse sequence

$$C_\infty\stackrel{a}{\leftarrow} C_\infty\stackrel{a}{\leftarrow}C_\infty\stackrel{a}{\leftarrow}C_\infty\stackrel{a}{\leftarrow}C_\infty\stackrel{a}{\leftarrow}\ldots $$
where if $c$ is the generator of $C_\infty$, $a(c) = c^2$.

This second example is certainly not isomorphic to a constant one. Note that it has trivial limit group, so is it possible to get some 'useful' limiting information from the progroup by taking some other type of 'limiting' approach?  Yes.  There are two related ways.

* **The [[derived functors of lim]]**.  Applied to the above progroup you get a definite measure of the non-triviality of the progroup. (This example is in that entry.)

* (A second method will be put here but will need additional work over on [[proper homotopy theory]].)

+--{: .query}
That is a start on it anyhow!
=--

[[!redirects Čech homotopy]]
[[!redirects Cech homotopy]]


Čech methods

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic topology
+--{: .hide}
[[!include algebraic topology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea ##

The construction of Čech homology used coverings of the space by families of open sets. The way the open sets overlap gives 'combinatorial' information on the space.

This has been abstracted and extended to [[homotopy]] and also in the direction of algebraic geometry. This article aims to introduce some of the main ways this idea has interacted with some of the many themes in the nLab.

##History (Čech 1930s)##

The term '[[Čech nerve|nerve of a covering]]' has been used from the late
1920s for a construction that starts with a space and an open cover of it and builds a [[simplicial complex]].
Historically these data were organized as a [[simplicial complex]], rather than a [[simplicial set]]. The idea had been independently worked out by [[Eduard Cech|Čech]] and [[P. S. Aleksandrov]]
so one often talks about Aleksandrov--Čech (co)homology. There was an alternative approach described by Vietoris (1927), which led to what is known as [[Vietoris homology]]  There are separate entries in nlab for [[Cech homology|Čech homology]]
and [[Čech cohomology]]. The first is not exact, and so is  not really
a homology theory in the sense of Steenrod--Eilenberg axioms. It is also not the (Alexander--Spanier) dual of Čech cohomology.  The two problems  can be avoided at the same time using coherent homotopy theory. The resulting homology is one which is exact.  It also is one that satisfies the wedge axiom, and then it is the unique such theory, nowadays sometimes called [[strong homology]] and in special cases it reduces to what was called [[Steenrod-Sitnikov homology]] (but which was originally constructed in a different way).

##Nerve of an open cover##
+--{: .un_defn}
######Definition######
([[Pavel Sergeevič Aleksandrov]], 1926)
Given a (compact) space $X$ and a (finite) open cover, $\alpha$, of $X$, we can
form a [[simplicial set]], $C(X,\alpha)$, called the *nerve of the cover* whose $n$-[[simplex|simplices]] are $(n + 1)$-strings
of open sets  from $\alpha$, i.e. $\langle U_0, \ldots, U_n\rangle$, each $U_i \in \alpha$,
satisfying $\cap U_i \neq \emptyset$.

If $\beta$ is another cover such that for each $V\in \beta$, there is a $U\in
\alpha$ with $V\subseteq U$, then the assignment $V\to U$ in this case defines
a map

$$C(X,\alpha)\to C(X,\beta)$$

dependent on the choice of $U$ for each
$V$, but independent 'up to homotopy' (see below).
=--

##Application of Čech methods##

Applying one's favourite homotopy
functor, $F : \mathbf{S}\to \mathbf{A}$, to each $C(X,\alpha)$ and to these homotopy
classes of induced transition maps, yields an inverse  system of objects in
$\mathbf{A}$, i.e.,  a
[[pro-object]]  in $\mathbf{A}$, but the $C(X,\alpha)$ by themselves do _not_ define a pro-object in the category of simplicial sets. They do form a pro-object in $Ho(sSets)$, or alternative, a 'coherent pro-object' in $sSets$, (see [[Čech homotopy]] for more discussion on this point.)

### Čech homology and cohomology ###

Aleksandrov and Čech in the 1930s, applied
homology and cohomology in this situation to extend simplicially-based
homology to a much wider class of spaces.  Lefschetz, without the language of
category theory, studied, again in the 1930s, the formal properties of inverse
systems of polyhedra and maps between them and his student Christie looked at the homotopy groups in this setting.

* see [[Čech cohomology]] and [[Čech homotopy]] for more on this.

### Shape theory as a Čech homotopy theory ###

Borsuk (1968) developed [[shape theory]], which although initially very geometric in flavour turned out also to be described in terms of Christie's
theory of the "Čech extensions" of homotopy theory.

The first papers on this approach were by [[Tim Porter]]. They used the [[Vietoris complex]] of a space relative to an open cover as well as the [[Čech nerve]] itself.  (By [[Dowker's Theorem]] the two complexes give the same information up to homotopy, but the [[Vietoris complex]] is a functor on the category of covers, having values in the category of simplicial sets, whilst the Čech complex does not give so nice a structure as most naturally it takes values in the homotopy category of simplicial sets. In fact, that functor can be rigidified but this requires a certain amount of work.)

Porter also gave a partial solution to the problem of the stability of a space, that is whether it has the same Čech homotopy as a polyhedron. The methods developed for that problem then suggested the existence of a homotopy theory for pro-simplicial sets related to homotopy coherence.  That observation was also made by Edwards and Hastings, who proposed a strong shape theory based on what they termed [[Steenrod homotopy]]. This relates to [[Alexander-Čech duality|Steenrod-Sitnikov homology]].

(See [[shape theory]] and [[Čech homotopy]] for references and more on this area).

### Čech-like methods in algebraic geometry ###

The use of modified Čech methods in algebraic geometry was well
established when Grothendieck and his collaborators in Paris started adapting
it to work with a [[Grothendieck topology]].  Verdier in SGA4,
introduced hypercoverings and in Artin and Mazur (SLN 100), you can find their use
in homotopy theory.  (The work of Lubkin (1967) should also be
mentioned here as it contains much that is parallel to the development by
Verdier, Artin and Mazur and is sometimes much easier to decipher for the
non-specialist algebraic geometer. A summary of his construction is given under [[Lubkin's construction]].)

Given that a [[Grothendieck topology]] is essentially about abstracting a notion
of '[[cover|covering]]', it is not surprising that
modified Čech methods can be applied. Artin and Mazur
used Verdier's idea of a [[hypercover|hypercovering]] to get, for each Grothendieck topos,
$\mathbb{E}$, a pro-object in $Ho(\mathbf{S})$ (i.e. an inverse system of
simplicial sets), which they call the étale [[homotopy type]] of the topos
$\mathbb{E}$ (which for them is 'sheaves for the étale topology on a
variety').

Applying homotopy group functors gives pro-groups $\pi_i(\mathbb{E})$
such that $\pi_1(\mathbb{E})$ is essentially the same as Grothendieck's [[algebraic fundamental group]],
$\pi_1(\mathbb{E})$.  (Here you need to know that the category of [[profinite
group]]s in the topological sense, is equivalent to the category of [[pro-object]]s in the category of
finite groups (as explained in [[profinite group]]).  (If you remove
'finite' the result does not hold, but you can recover it in part by
working with "pro-discrete [[localic groups]]" instead of topological groups,
i.e. take limits of finite groups within the category of '[[locale|localic]]' rather than
'[[topological group|topological]]' groups, remembering that 'locales' are almost 'spaces without
points'.

Grothendieck's nice $\pi_1$ has thus an interpretation as a formal limit of a Čech
type, or shape theoretic, system of $\pi_1$s of 'hypercoverings'.

Can [[shape
theory]] (or its more powerfully structured 'strong' or 'homotopy coherent' version,
cf. Lisica and Mardešić, Edwards and Hastings, Porter
(papers 1976--78) be useful for studying étale homotopy
type?  Not without extra work, since the Artin--Mazur--Verdier approach leads
one  to look at inverse systems in $proHo(\mathbf{S})$, i.e. inverse systems
(diagrams) in a homotopy category not a homotopy category of inverse systems
as in [[strong shape theory]].  Attempts to 'rigidify' the hypercovering approach,
so as to get into $Hopro(\mathbf{S})$ have been made (e.g. by Lubkin, (1967)
or using simplicial schemes in Friedlander, (1982),
but it is not completely clear if one of them is
the definitive method.
+-- {: .query}
What is the consensus on this here?  I have sometimes seen talks which claim to have THE method but none has been clearly THE ONE. Perhaps I did not go to the right conferences! (Some is needed here, e.g. for more ways of getting around the difficulty.)
=--

One of the difficulties with this hypercovering approach is that
'hypercovering' is a difficult concept and to the 'non-expert' seem
non-geometric and lacking in intuition.  Thankfully for us, there is an
alternative approach put forward by Ken Brown (1973), (see [[BrownAHT]]).

As the Grothendieck topos $\mathbb{E}$ 'pretends to be' the category of
$Sets$, but with a strange logic, we can 'do' simplicial set theory in
$Simp(\mathbb{E})$ as long as we take care of the arguments we use.  To see a
bit of this in action we can note that the object $[0]$ in $Simp(\mathbb{E})$
will be the constant simplicial sheaf with value the ordinary $[0]$,
"constant" here taking on two meanings at the same time,

(a) constant sheaf, i.e. not varying 'over $X$' if $\mathbb{E}$ is thought of as $Sh(X)$, and

(b)
constant simplicial object, i.e. each $K_n$ is the same and all face and
degeneracy maps are identities.  Thus $[0]$ interpreted as an étale space
is the identity map $X\to X$ as a space over $X$.  Of course not all
simplicial objects are constant and so $Simp(\mathbb{E})$ can store a lot of
information about the space (or site) $X$.

One can look at the homotopy
structure of $Simp(\mathbb{E})$.  Ken Brown in [[BrownAHT]] showed it had a
fibration category structure and if we look at
those fibrant objects $K$ in which the natural map

$$p : K \to [0]$$

 is a weak
equivalence, we find that these $K$ are _exactly_ the [[hypercover|hypercoverings]].  Global
sections of $p$ give a simplicial set, $\Gamma(K)$ and varying $K$ amongst the
hypercoverings gives a pro-simplicial set (still in $proHo(\mathbf{S})$ not in
$Hopro(\mathbf{S})$ unfortunately) which determines the Artin--Mazur
[[etale homotopy type|pro-homotopy type]] of $\mathbb{E}$

###Links with derived category theory###

This makes it clear there is a link between Čech methods and [[derived category]]
theory.  In the first, the 'space' is resolved using '[[cover]]ings'
and these, in a [[sheaf]] theoretic setting, lead to [[simplicial object]]s in $Sh(X)$
that are weakly equivalent to $[0]$; in the second, to evaluate the
[[derived functor]] of some functor $F : \mathbf{C} \to \mathbf{A}$, say, on an object $C$,
one takes the 'average' of the values of $F$ on objects weakly equivalent to
$G$, i.e. one works with the functor

$$F' : \mathbf{W}(C) \to \mathbf{A}$$

(where $ \mathbf{W}(C)$ has objects, $\alpha : C \to C'$, $\alpha$ a
weak equivalence, and maps, the commuting 'triangles', and this has a 'domain'
functor $\delta :\mathbf{W}(C) \to \mathbf{C}$, $\delta(\alpha) = C'$
and $F'$ is the composite $F\delta$).

This is in many cases a pro-object
in $\mathbf{A}$ -- unfortunately standard derived functor theory interprets
'commuting triangles' in too weak a sense and thus corresponds to shape rather
than strong shape theory -- one thus, in some sense, arrives in $pro
Ho(\mathbf{A})$ instead of in $Ho(pro \mathbf{A})$.  This problem has been in part resolved with Grothendieck's theory of _Derivateurs_ (see the volume by Maltsiniotis.)

* see also [[abelian sheaf cohomology]]

## Related entries

* [[Čech nerve]]

* [[Čech homology]], [[Čech cohomology]]

* [[nerve theorem]]

*(...)

[[!redirects Čech methods]]

[[!redirects Čech method]]

[[!redirects Cech method]]
[[!redirects Cech methods]]


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

#Contents#
* table of contents
{:toc}

## Idea

The **Čech model structure on simplicial presheaves** on a [[site]] $C$ is a model for the [[topological localization]] of an [[(∞,1)-category of (∞,1)-presheaves]] on $C$ to the [[(∞,1)-category of (∞,1)-sheaves]].

It is obtained from the the [[global model structure on simplicial presheaves]] on $C$ by [[Bousfield localization of model categories|left Bousfield localization]]s at [[Cech cover]]s:  its fibrant objects are [[∞-stack]]s that satisfy [[descent]] over [[Cech cover]]s but not necessarily over [[hypercover]]s.

Further [[Bousfield localization of model categories|left Bousfield localization]] at [[hypercover]]s leads from the Čech model structure to the Joyal-Jardine [[local model structure on simplicial presheaves]] that presents the [[hypercomplete (∞,1)-topos]] which is the [[hypercompletion]] of that presented by the Čech model structure.

## Definition

Let $C$ be a small [[site]] and write $[C^{op}, sSet]_{proj}$ and $[C^{op}, sSet]_{inj}$ for the projective and injective [[global model structure on simplicial presheaves]], respectively.

For $\{U_i \to V\}_i$ a covering family in the [[site]] $C$, let

$$
  C(\{U_i\})
  :=
  \left(
    \cdots\stackrel{\to}{\stackrel{\to}{\to}}\coprod_{i j} U_{i j}\stackrel{\to}{\to}\coprod_i U_i
  \right)
$$

be the corresponding [[Cech nerve]], regarded as a [[simplicial presheaf]] on $C$. This comes canonically with a morphism

$$
  C(\{U_i\}) \to V
$$

of simplicial presheaves, the corresponding _Čech cover morphism_ .

Notice that by the discussion at [model structure on simplicial presheaves - fibrant and cofibrant objects](http://ncatlab.org/nlab/show/model+structure+on+simplicial+presheaves#FibAndCofibObjects) this is a morphism between cofibrant objects.

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

The injective (projective) **Čech model structure on simplicial presheaves** $[C^{op},sSet]_{Cech}$ on $C$ is the [[Bousfield localization of model categories|left Bousfield localization]] of $[C^{op}, sSet]_{inj}$ ($[C^{op}, sSet]_{proj}$) at the set of Čech cover morphisms.

=--

By the general poperties of [[Bousfield localization of model categories|Bousfield localization]] this means that the fibrant-cofibrant objects $A$ of $[C^{op},sSet]_{Cech}$ are precisely those that are fibrant-cofibrant in the global model structure and in addition satisfy the [[descent]] condition that for all covers $\{U_i \to V\}$ the morphism of simplicial sets

$$
  A(U) \simeq [C^op,sSet](V,U) \to [C^{op},sSet](C(\{U_i\}), A)
$$

is a weak equivalence in the standard [[model structure on simplicial sets]].

This is the model for the $\infty$-analog of the [[sheaf]] condition, modelling the [[topological localization]] of an $(\infty,1)$-presheaf $(\infty,1)$-topos.

+--{: .query}
[[Mike Shulman]]: Two questions, one (hopefully) easy and one (perhaps) hard:

1. Is there a Quillen equivalent Čech model structure on simplicial *sheaves*?  Can we just lift the model structure for simplicial presheaves along the sheafification adjunction?

1. Is there a characterization of the weak equivalences in either Čech model structure?

I am particularly interested in this for the following reason.  According to Beke in *Sheafifiable homotopy model categories*, the weak equivalences in the [[local model structure on simplicial sheaves]] are precisely those maps $f\colon X\to Y$ of simplicial objects in the corresponding 1-topos of sheaves of sets such that the statement "$f$ is a weak equivalence of simplicial sets" is true in the [[internal logic]] of the topos (at least, interperiting "$f$ is a weak equivalence of simplicial sets" by one particular set of geometric sentences whose interpretation in $Set$ is equivalent to saying that a simplicial map is a weak equivalence).  But if, as [[HTT]] teaches us, Čech descent is often to be preferred to hyperdescent, then we should be interested in Čech weak equivalences instead.  So I would really like to know what it means for a map of simplicial sheaves to be a Čech weak equivalence, in the *internal logic* of the 1-topos of sheaves of sets.  If nothing else, I think such a characterization would help me understand the real meaning of [[hypercompletion]].  But any sort of characterization of them would be better than none.

[[Urs Schreiber]]: below is a reply to the first question.

[[Mike Shulman]]: Thanks for attacking this.  I thought I should also mention, for anyone listening in, that this question is evidently also relevant to what the correct notion of [[internal ∞-groupoid]] may be.
=--

> check

We may form the [[transferred model structure]] on simplicial _[[sheaf|sheaves]]_ by transferring along the degreewise [[sheafification]] [[adjunction]]

$$
  Sh(C) \stackrel{\overset{sh}{\leftarrow}}{\underset{}{\hookrightarrow}}
  PSh(C)
  \,.
$$

This defines fibrations and weak equivalences in $sSh(C)$ to be those morphisms that are fibrations or weak equivalences, respectively, as morphism in  $sPSh(C)_{Cech} = [C^{op},sSet]_{Cech}$.

As discussed there, sufficient conditions for this to be a model structure is that

* the inclusion $Sh(C) \hookrightarrow PSh(C)$ preserves [[filtered colimit]]s;

* $sSh(C)$ has functorial fibrant replacement and functorial [[path object]]s for fibrant objects.

Since [[sheafification]] does preserve [[filtered colimit]]s the first condition is satisfied degreewise and hence is satisfied.

+--{: .query}
[[Mike Shulman]]: I believe that sheafification preserves $\kappa$-filtered colimits for some sufficiently large $\kappa$, but if the site has covers of infinite cardinality, I don't see why sheafification would preserve $\omega$-filtered colimits.  But I think this is enough for the proof to work.
=--

Since the [[small object argument]] holds in $sSh(C)$ for generating acyclic cofibrations we have functorial fibrant replacement. And a path object is obtained just by forming objectwise the standard path object in [[sSet]], as in $[C^{op}, sSet]$.

+--{: .query}
[[Mike Shulman]]: The small object argument doesn't automatically produce functorial fibrant replacements in this context... isn't the whole question whether the map to the "fibrant replacement" is still a weak equivalence (in the underlying category)?  I.e. whether $F(J)$-cell complexes are still weak equivalences.
=--

## References

A detailed though unfinished account of the Čech model structure is given in

* Daniel Dugger, _Sheaves and homotopy theory_ ([web](http://www.uoregon.edu/~ddugger/cech.html), [pdf](http://ncatlab.org/nlab/files/cech.pdf))

But beware of this document is unfinished. Some aspects of this appeared in

* [[Daniel Dugger]], _[[DuggerUniv.pdf:file]]_

[[!redirects ?ech model structure on simplicial presheaves]]
[[!redirects Cech model structure on simplicial presheaves]]
[[!redirects ?ech model structure on simplicial presheaves]]


+-- {: .rightHandSide}
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###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The **Čech model structure on simplicial sheaves** on a [[site]] $C$ is a model by [[simplicial sheaves]] for the [[topological localization]] of an [[(∞,1)-category of (∞,1)-presheaves]] on $C$ to the [[(∞,1)-category of (∞,1)-sheaves]].

It is obtained from the the [[Čech model structure on simplicial presheaves]] on $C$ by [[transferred model structure|transfer]] along the [[sheafification]] [[adjunction]].

Further [[Bousfield localization of model categories|left Bousfield localization]] at "internal" weak homotopy equivalences leads from the Čech model structure to the [[model structure on simplicial sheaves]] that presents the [[hypercomplete (∞,1)-topos]] which is the [[hypercompletion]] of that presented by the Čech model structure.

## Definition

Let $C$ be a small [[site]], let $sSh (C)$ be the [[category]] of [[simplicial sheaves]] on $C$, and write $[C^{op}, sSet]_{\check{C},proj}$ and $[C^{op}, sSet]_{\check{C},inj}$ for the projective and injective [[Čech model structure on simplicial presheaves]], respectively.

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

The injective **Čech model structure on simplicial sheaves** on $C$ is the unique model structure on $sSh (C)$ with the following properties:

* The weak equivalences are the morphisms in $sSh (C)$ that are weak equivalences in $[C^{op}, sSet]_{\check{C},inj}$.
* The cofibrations are the monomorphisms.
* The fibrations are the morphisms in $sSh (C)$ that are fibrations in $[C^{op}, sSet]_{\check{C},inj}$.

=--

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

The projective **Čech model structure on simplicial sheaves** on $C$ is the unique [[model category]] structure on $sSh (C)$ with the following properties:

* The weak equivalences are the morphisms in $sSh (C)$ that are weak equivalences in $[C^{op}, sSet]_{\check{C},proj}$.
* The trivial fibrations are the morphisms in $sSh (C)$ that are trivial fibrations in the projective Čech model structure on $[C^{op}, sSet]_{\check{C},proj}$, i.e. the componentwise trivial Kan fibrations.

=--

## Constructions

To construct the injective Čech model structure on $sSh (C)$, we use the following facts:

* The inclusion $sSh (C) \hookrightarrow [C^{op}, sSet]$ is fully faithful.
* The left adjoint (i.e. sheafification) $[C^{op}, sSet] \to sSh (C)$ preserves monomorphisms, and the class of monomorphisms in $sSh (C)$ is closed under pushouts, transfinite composition, and retracts.
* The adjunction unit is a natural weak equivalence with respect to the (injective) Čech model structure on $[C^{op}, sSet]$: see Theorem A.2 in [DHI](#DuggerHollanderIsaksen02).

We may then apply Kan's recognition principle for [[cofibrantly generated model category|cofibrantly generated model structures]] to transfer the injective Čech model structure from $[C^{op}, sSet]$ to $sSh (C)$. By construction, the sheafification adjunction becomes a Quillen equivalence.

On the other hand, to construct the projective Čech model structure on $sSh (C)$, we use Smith's recognition principle for [[combinatorial model category|combinatorial model structures]] and build it like a [[mixed model structure]].

## References

* {#DuggerHollanderIsaksen02} [[Daniel Dugger]], [[Sharon Hollander]], [[Daniel Isaksen]], [_Hypercovers and simplicial presheaves_](http://dx.doi.org/10.1017/S0305004103007175) ([arXiv:math/020502](http://arxiv.org/abs/math/0205027)).

[[!redirects Cech model structure on simplicial sheaves]]


Čech nerve

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###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

# Čech nerve
* table of contents
{:toc}

## Definition

### In category theory

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

In a [[category]] $C$ with [[pullbacks]] (possibly [[homotopy pullbacks]]), given a [[morphism]] $U \to X$ in $C$ its corresponding **Čech nerve** $C(U)$ is the [[simplicial object]] in $C$ that in degree $k$ is given by the $(k+1)$-fold [[fiber product]] of $U$ over $X$ with itself :

$$
  C(U) \coloneqq
  \left(
    \cdots
    U \times_X U \times_X U
   \overset{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} U \times_X U \stackrel{\longrightarrow}{\longrightarrow} U
  \right)
  \,.
$$

=--

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

This is the [[internal nerve]] of the [[internal groupoid]] corresponding to the [[kernel pair]] of the morphism $U \to X$.

=--

### In $(\infty,1)$-category theory

The notion of Čech nerve makes sense in any [[(∞,1)-category]] with [[limit in a quasi-category|(∞,1)-pullbacks]].

Consider a morphism $f: A \rightarrow B$  in an $(\infty,1)$-category $M$. The Cech nerve of that morphism is, by definition, the [[augmented simplicial object]] in $M$ obtained by the [[right Kan extension]] of that morphism (regarded as a functor from the obvious category into $M$) along the inclusion of the first two objects into the augmented simplicial category. See Lurie HTT 6.1.2.11. One can show that this gives a [[groupoid object]] in $M$. The Cech nerve is essentially unique by uniqueness of adjoint $\infty-$ functors.

See [[groupoid object in an (∞,1)-category]].

## Applications and occurences

* The [[cohomology]] theory obtained by mapping out of [[Čech covers]] instead of general [[hypercovers]] is [[Čech cohomology]].

* A [[groupoid object in an (infinity,1)-category]] that is a Čech nerve $U \to X$ exhibits $X$ as a [[delooping]].

  * In an [[infinity-stack]] [[(infinity,1)-topos]] every [[groupoid object in an (infinity,1)-category]] is a Čech nerve.

## Examples

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

For $U = \coprod_i U_i$ the [[disjoint union]] of  a [[covering]] [[sieve]] $\{U_i \to X\}$ with respect to a [[coverage]], the objectwise [[simplicial homotopy group|connected components]] of the Čech nerve is the [[subfunctor]] corresponding to the [[sieve]]

$$
  \Pi_0 C(U) = \bigcup_i hom(-,U_i)
  \,.
$$

This is described in more detail in the section "Interpretation in terms of higher descent and codescent" at [[sieve]]. This example is important in understanding the construction of the [[etale homotopy type]] of a scheme or more generally of objects in certain types of topos.

=--

+-- {: .num_remark}
###### Remark (Historical and pedagogic note)

This example is more or less the way that [[Eduard Čech]] gave the original form of the construction that now carries his name. More on this can be found in the entry on [[Čech methods]], and the discussion there of the **nerve of an open cover**. For the case of triangulable spaces (polyhedra), for the cover by open stars of vertices of a triangulation, one retrieves the simplicial complex used to triangulate the space. This is one of the strong roots of the modern theory of cohomology as nerves of open covers can be seen as analogues of triangulations, and then [[Čech cohomology]] is seen to extend simplicial cohomology to spaces that are locally nice. That is one of the first steps on the long route to [[Grothendieck]]'s definition of topos as a generalisation of space, expressly so as to define a cohomology further extending Čech cohomology to the geometric objects studied in algebraic geometry.

=--

##Related entries

* [[Čech groupoid]]

* [[effective epimorphism]]

* [[nerve theorem]]

## References

An early appearance (in the form of the 0-coskeleton) is in Enlightenment 8.5(b) of

* [[Michael Artin]], [[Barry Mazur]], _Etale Homotopy_, Lecture Notes in Mathematics  100 (1969), [doi](http://dx.doi.org/10.1007/bfb0080957).

An early appearance of the term “Čech nerve” is in

* [[Eric M. Friedlander]], _Etale Homotopy of Simplicial Schemes_, Annals of Mathematics Studies 104 (1983), [doi](http://dx.doi.org/10.1515/9781400881499).

[[!redirects Cech nerve]]
[[!redirects Cech-nerve]]
[[!redirects Čech nerve]]
[[!redirects Čech-nerve]]
[[!redirects Čech nerve]]
[[!redirects Čech-nerve]]
[[!redirects Cech complex]]

[[!redirects Čech nerves]]
[[!redirects Cech nerves]]


Čedomir Crnković

[[!redirects Cedomir Crnković]]

* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=206648)

* [inspire page](https://inspirehep.net/authors/2295325)

## Selected writings

On the [[covariant phase space]] for [[field theories]]:

* [[Čedomir Crnković]], [[Edward Witten]], *Covariant Description of Canonical Formalism in Geometrical Theories*, chapter 16 in: S. W. Hawking and W. Israel (eds.) *Three Hundred Years of Gravitation*, Cambridge University Press (1987) 676-684 [[pdf](https://www.ias.edu/sites/default/files/sns/files/CovariantPaper-1987.pdf)]

* [[Čedomir Crnković]], *Symplectic Geometry of the Covariant Phase Space*, Class. Quant. Grav. **5** (1988) 1557-1575 [[doi:10.1088/0264-9381/5/12/008](https://doi.org/10.1088/0264-9381/5/12/008), [inspire:247290](https://inspirehep.net/literature/247290)]

category: people

[[!redirects Cedomir Crnkovic]]


Łoś ultraproduct theorem

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+-- {: .toc .clickDown tabindex="0"}
### Context
#### Model theory
+-- {: .hide}
[[!include model theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
The Łoś [[ultraproduct]] theorem characterizes the [[first-order theory]] of an ultraproduct of $\mathcal{L}$-structures (for $\mathcal{L}$ some [[signature (in logic)|signature]]). It generalizes the _transfer principle_ from [[nonstandard analysis]], by replacing the hyperreals $^*\mathbb{R}$ with an ultraproduct and formulas from the standard reals $\mathbb{R}$ with formulas which are true on an "ultrafilter-large" subset of factors of the ultraproduct.

## Definition
The theorem is usually given in this form:

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

let $(A_i)_{i \in I}$ be a family of nonempty $\mathcal{L}$-structures, and let $\mathcal{U}$ be an [[ultrafilter]] on $I$. Let $^* A$ be the ultraproduct of $A_i$ with respect to $\mathcal{U}$. Since each $A_i$ was nonempty, $^* A$ is the quotient of the product $\prod_{i \in I} A_i$ by the equivalence relation which identifies sequences which coincide on a subset of indices belonging to $\mathcal{U}$. Let $(a_i)_{i \in I}$ be such a sequence, and let $[(a_i)]$ denote its equivalence class. Then for each $\mathcal{L}$-formula $\varphi(x)$,

$$^* A \models \varphi([a_i]) \iff \{j \in I | A_j \models \varphi(a_j)\} \in \mathcal{U}.$$

 =--

The above follows from a more fundamental fact:
+-- {: .num_prop}
###### Proposition

Any functor

$$[\mathcal{U}] : \mathbf{Set}^I \to \mathbf{Set}$$

which specifies a choice of ultraproduct (and hence comparison maps, since ultraproducts are colimits) is elementary (logical in the terminology of Makkai-Reyes, i.e. a [[pretopos]] morphism.)

=--

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

Recall that the ultraproduct sends the family of sets $A_i$ to the colimit of $s \mapsto \prod_{i \in s} A_i$, indexed by $s \in U^op$. $U^op$ (the opposite category of the sub-poset $U \subseteq \mathcal{P}(I)$) is filtered because $U$ is a filter.

A functor is a pretopos morphism precisely when it preserves finite limits, initial objects, disjoint sums, and quotients by internal equivalence relations.

Finite limits: a $s$-indexed product of $J$-indexed limits is a $J$-indexed limit of $s$-indexed products, and finite limits commute with filtered colimits.

Initial objects: a $s$-indexed product of empty sets is empty (as each $s$ is nonempty by properness of $U$), and a colimit of empty sets is empty.

Disjoint sums: a $s$-indexed product of disjoint sums $\prod_{i:s} (A_{0i} + A_{1i})$ is a disjoint sum of products $\sum_{f : s \to 2} \prod_{i:s} A_{f(i)i}$. For any $(f,a)$ in this set, there is a $U$-large subset $t \subset s$ such that the restriction of $f$ to $t$ is constantly $b$ ($b \in \{0,1\}$). Therefore the restriction map identifies $(f,a)$ with $(const_b,a|_t)$, and the $U^op$-indexed colimit of the $\sum_{f : s \to 2} \prod_{i:s} A_{f(i)i}$ is equivalent to the $U^op$-indexed colimit of the $\sum_{b : 2} \prod_{i:s} A_{bi} = \prod_{i:s} A_{0i} + \prod_{i:s} A_{1i}$, and colimits commute with colimits.

Quotients: Given $X_i$ and equivalence relations $E_i \to X_i \times X_i$, a $s$-indexed product of quotients $\prod_{i:s} (X_i / E_i)$ is a quotient of products $(\prod_{i:s} X_i) / (\prod_{i:s} E_i)$ (because the natural map from the latter to the former is an iso, using the axiom of choice), and colimits commute with colimits.
 =--

Now we can show the theorem:

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

Since the ultraproduct functor is elementary, then the process of taking points $M(X)$ of a definable set $X$ inside models $M$ commutes with taking ultraproducts. In symbols,

$$\left(\prod_{i \in I} M_i / \mathcal{U}\right) (X) \simeq \left(\prod_{i \in I} M_i(X) / \mathcal{U}\right).$$

Since this is a filtered colimit, a sequence $\overline{x}$ satisfies that its germ $[\overline{x}]$ is in $\prod_{i \in I} M_i / \mathcal{U}(X)$ if and only if there is some $J \in \mathcal{U}$ such that the restriction of $\overline{x}$ to $J$ is in $\prod_{j \in J} M_j(X)$. i.e. if $x_j \in M_j(X)$ for each $j \in J$.
=--

## Examples

An immediate consequence of the Łoś theorem is the [[nonstandard analysis|transfer principle]] for the hyperreals.

The [[compactness theorem]] also follows quickly from the Łoś theorem, so anything that you [build](http://math.stackexchange.com/questions/413770/most-astonishing-applications-of-compactness-theorem-outside-logic) using compactness can be realized a bit more concretely as an ultraproduct.

## Remarks

The proposition should be interpreted as saying that [[Set]] simultaneously carries a [[pretopos]] structure and an [[ultracategory]] structure, and that these two structures commute. Indeed, one can view [[Set]] as a [[dualizing object]] for a generalized [[Stone duality]] between pretoposes and ultracategories; this is the starting point of [[Makkai duality]].

There is an analogous statement for ultraproducts of structures in [[continuous logic]].

## Related concepts

* [[ultrafilter]]

* [[ultrapower]]

* [[stalk]]

## References

* Hisashi Aratake, _Sheaves of Structures, Heyting-Valued Structures, and a Generalization of Łoś's Theorem_ , arXiv:2012.04317 (2020). ([abstract](https://arxiv.org/abs/2012.04317))

* [[Michael Makkai]], _Stone duality for first order logic_, Advances in Mathematics, 65(2):97–170, 1987.

[[!redirects Los theorem]]
[[!redirects Los ultraproduct theorem]]
[[!redirects transfer principle]]


Łukasiewicz logic

\tableofcontents

## Idea

(...)

## Related concepts

* [[fuzzy logic]]

* [[linear logic]]

* [[substructural logic]]

* [[MV-algebra]]

## References

* Wikipedia, [Łukasiewicz logic](https://en.wikipedia.org/wiki/%C5%81ukasiewicz_logic)

Łukasiewicz logic is discussed in section 5 of:

* [[Peter Freyd]], *Algebraic real analysis*, Theory and Applications of Categories, Vol. 20, 2008, No. 10, pp 215-306 ([tac:20-10](http://www.tac.mta.ca/tac/volumes/20/10/20-10abs.html))


δ-ring

\tableofcontents

## Definition

(Definition 2.1 in Bhatt–Scholze.)

Fix a prime $p$.
A __δ-ring__ is a pair $(R,\delta)$,
where $R$ is a [[commutative ring]]
and $\delta\colon R\to R$ is a map of underlying sets
such that $\delta(0)=0$, $\delta(1)=0$,
$$\delta(x y)=x^p \delta(y)+y^p \delta(x) + p\delta(x)\delta(y),$$
and
$$\delta(x+y)=\delta(x)+\delta(y)+(x^p+y^p-(x+y)^p)/p.$$

## Properties

If $(R,\delta)$ is a δ-ring,
then the map $\phi\colon R\to R$
given by $\phi(x)=x^p + p\delta(x)$
is a ring homomorphism
that lifts the [[Frobenius endomorphism]] on $R/p$.

For $p$-torsionfree rings,
the above correspondence between δ-structures
and lifts of the Frobenius endomorphism on $R/p$ to $R$
is bijective.
This motivates the identities in the definition of a δ-structure.

## Related entries

* [[perfectoid ring]]

* [[perfectoid space]]

* [[prism]]

* [[prismatic cohomology]]

## References

The original notion is due to [[André Joyal]]:

* [[André Joyal]], δ-anneaux et vecteurs de Witt, C. R. Math. Rep. Acad. Sci. Canada 7 (1985), no. 3, 177–182.

Recent developments can be found in

* [[Bhargav Bhatt]], [[Peter Scholze]], _Prisms and prismatic cohomology_, [arXiv](http://arxiv.org/abs/1905.08229v3).