sheaf

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

#Contents#
* table of contents
{:toc}

## Idea

A [[presheaf]] on a [[site]] is a _sheaf_ if its value on any object of the site is given by its compatible values on any [[covering]] of that object.

See also

* [[motivation for sheaves, cohomology and higher stacks]].

A competing, though related, definition which one sometimes sees uses "sheaf" as a synonym for [[étale space]]; see [discussion below](#AsEtaleSpace).

## Definition

There are several equivalent ways to characterize sheaves. We start with the general but explicit componentwise definition and then discuss more [[category theory|general abstract]] equivalent reformulations. Finally we give special discussion applicable in various common special cases.

### General definition in components
 {#GeneralDefinitionInComponents}

The following is an explicit component-wise definition of sheaves that is fully general (for instance not assuming that the [[site]] has [[pullbacks]]).

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

Let $(C,J)$ be a [[site]] in the form of a [[small category]] $C$ equipped with a [[coverage]] $J$.

A [[presheaf]] $A \in PSh(C)$ is a **sheaf** with respect to $J$ if

* for every [[covering]] family $\{p_i : U_i \to U\}_{i \in I}$ in $J$

* and for every _[[matching family|compatible family]] of elements_, given by tuples $(s_i \in A(U_i))_{i \in I}$ such that for all $j,k \in I$ and all [[morphisms]] $U_j \stackrel{f}{\leftarrow} K \stackrel{g}{\to} U_k$ in $C$ with $p_j \circ f = p_k \circ g$ we have $A(f)(s_j) = A(g)(s_k) \in A(K)$

then

* there is a _unique_ element $s \in A(U)$ such that $A(p_i)(s) = s_i$ for all $i \in I$.

=--

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

If in the above definition there is _at most_ one such $s$, we say that $A$ is a [[separated presheaf]] with respect to $J$.

=--

In this form the definition appears for instance in  ([Johnstone, def. C2.1.2](#Johnstone)).-

### General definition abstractly
 {#GeneralDefinitionAbstractly}

We now reformulate the [above component-wise definition](#GeneralComponentwiseDefinition) in [[category theory|general abstract]] terms.

Write

$$
  j : C \hookrightarrow PSh(C)
$$

for the [[Yoneda embedding]].

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

Given a [[covering]] family $\{f_i : U_i \to U\}$ in $J$, its **[[sieve]]** is the presheaf $S(\{U_i\})$ defined as the [[coequalizer]]

$$
   \coprod_{j,k} j(U_j) \times_{j(U)} j(U_k)
     \stackrel{\overset{}{\to}}{\to}
   \coprod_i j(U_i) \to S(\{U_i\})
$$

in $PSh(C)$.

=--

Here the [[coproduct]] on the left is over the [[pullbacks]]

$$
  \array{
    j(U_j) \times_{j(U)} j(U_k) &\stackrel{p_j}{\to}& j(U_j)
    \\
    {}^{\mathllap{p_k}}\downarrow && \downarrow^{\mathrlap{j(f_j)}}
    \\
    j(U_k) &\stackrel{j(f_k)}{\to}& j(U)
  }
$$

in $PSh(C)$, and the two morphisms between the coproducts are those induced componentwise by the two projections $p_j, p_k$ in this pullback diagram.

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

Using that [[limits]] and [[colimits]] in a [[category of presheaves]] are computed objectwise, we find that the [[sieve]] $S(\{U_i\})$ defined this way is the presheaf that sends any $K \in C$ to the set of [[morphisms]] $K \to U$ in $C$ that factor through one of the $f_i$.

=--

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

For every [[covering]] family there is a canonical morphism

$$
  i_{\{U_i\}} :  S(\{U_i\}) \to j(U)
$$

that is induced by the [[universal property]] of the [[coequalizer]] from the morphisms $j(f_i) : j(U_i) \to j(U)$ and $j(U_j) \times_{j(U)} j(U_k) \to j(U)$.

=--

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

A **sheaf** on $(C,J)$ is a [[presheaf]] $A \in PSh(C)$ that is a [[local object]] with respect to  all $i_{\{U_i\}}$: an object such that for all [[covering]] families $\{f_i : U_i \to U\}$ in $J$ we have that the [[hom-functor]] $PSh_C(-,A)$ sends the canonical morphisms $i_{\{U_i\}} : S(\{U_i\}) \to j(U)$ to [[isomorphisms]].

$$
  PSh_C(i_{\{U_i\}}, A) :
  PSh_C(j(U), A) \stackrel{\simeq}{\to} PSh_C(S(\{U_i\}), A)
  \,.
$$

=--

Equivalently, using the [[Yoneda lemma]] and the fact that the [[hom-functor]] $PSh_C(-,A)$ sends [[colimits]] to [[limits]], this says that the diagram

$$
  A(U)
   \to
  \prod_i A(U_i)
   \stackrel{\to}{\to}
  \prod_{j,k} PSh_C(j(U_j) \times_{j(U)} j(U_k), A)
$$

is an [[equalizer]] diagram for each covering family.

This is also called the **[[descent]] condition** for descent along the covering family.

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

For many examples of sites that appear in practice -- but by far not for all -- it happens that the pullback presheaves $j(U_j) \times_{j(U)} \times j(U_k)$ are themselves again representable, hence that the [[pullback]] $U_j \times_U U_k$ exists already in $C$, even before passing to the [[Yoneda embedding]].

In this special case we may apply the [[Yoneda lemma]] once more to deduce

$$
  PSh_C(j(U_j) \times_{j(U)} j(U_k), A)
  \simeq
  A(U_j \times_U U_k)
  \,.
$$

Then the sheaf condition is that all diagrams

$$
  A(U)
   \to
  \prod_i A(U_i)
   \stackrel{\to}{\to}
  \prod_{j,k} A(U_j \times_U U_k)
$$

are [[equalizer]] [[diagram]]s.

=--

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

The condition that $PSh_C(S(\{U_i\}), A)$ is an [[isomorphism]] is equivalent to the condition that the set $A(U)$ is isomorphic to the set of [[matching families]] $(s_i \in A(U_i))$ as it appears in the [above component-wise definition](#GeneralComponentwiseDefinition).

=--

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

We may express the set of [[natural transformation]]s $PSh_C(j(U_j) \times_{j(U)} j(U_k), A)$ (as described there) by the [[end]]

$$
  PSh_C(j(U_j) \times_{j(U)} j(U_k), A)
  \simeq
  \int_{K \in C}
   Set( C(K,U_j) \times_{C(K,U)} C(K,U_k) , A(K))
  \,.
$$

Using this in the expression of the [[equalizer]]

$$
  \prod_i A(U_i)
  \simeq
  \prod_i
  \int_{K \in C}
   Set( C(K,U_i), A(K))
  \stackrel{\to}{\to}
   \prod_{j,k}
  \int_{K \in C}
   Set( C(K,U_j) \times_{C(K,U)} C(K,U_k) , A(K))
$$

as a [[subset]] of the product set on the left manifestly yields the componenwise definition above.

=--

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

A **morphism of sheaves** is just a morphism of the underlying presheaves.
So the [[category of sheaves]] $Sh_J(C)$ is the [[full subcategory]] of the [[category of presheaves]] on the sheaves:

$$
  Sh_J(C) \hookrightarrow PSh(C)
$$

=--

### Characterizations over special sites

We discuss equivalent characterizations of sheaves that are applicable if the underlying [[site]] enjoys certain special properties.

#### Characterizations over sites of opens
 {#CharacterizationsOverSitesOfOpens}

An important special case of sheaves is those over a [[(0,1)-site]] such as a [[category of open subsets]] $Op(X)$ of a [[topological space]] $X$.
We consider some equivalent ways of characterizing sheaves among presheaves in such a situation.

(The following was mentioned in Peter LeFanu Lumsdaine's comment [here](http://mathoverflow.net/questions/23268/geometric-intuition-for-limits/23276#23276)).

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

Suppose $Op = Op(X)$ is the [[category of open subsets]] of some [[topological space]] regarded as a [[site]] with the canonical [[coverage]] where $\{U_i \hookrightarrow U\}$ is [[covering]] if the [[union]] $\cup_i U_i \simeq U$ in $Op$.

Then a [[presheaf]] $\mathcal{F}$ on $Op$  is a **sheaf** precisely if for every [[complete category|complete]] [[subcategory|full subcategory]] $\mathcal{U} \hookrightarrow Op$, $\mathcal{F}$ takes the [[colimit]] in $Op$ over $\mathcal{U} \hookrightarrow Op$ to a [[limit]]:

$$
  \mathcal{F}(\underset{\to}{lim} \mathcal{U}) \simeq \underset{\leftarrow}{lim} \mathcal{F}(\mathcal{U})
  \,.
$$

=--

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

A complete full subcategory $\mathcal{U} \hookrightarrow Op$ is a collection $\{U_i \hookrightarrow X\}$ of [[open subsets]] that is closed under forming [[intersections]] of subsets. The [[colimit]]

$$
  \underset{\to}{\lim} (\mathcal{U} \hookrightarrow Op)
  \simeq
  \cup_{i \in I} U_i
$$

is the [[union]] $U \coloneqq \cup_{i \in I} U_i$ of all these open subsets. Notice that by construction the component maps $\{U_i \hookrightarrow U\}$ of the colimit are a [[covering]] family of $U$.

Inspection then shows that the [[limit]] $\underset{\leftarrow}{\lim}_{i \in I} \mathcal{F}(U_i)$ is the corresponding set of [[matching families]] (use the description of [limits in terms of products and equalizers](http://ncatlab.org/nlab/show/limit#ConstructionFromProductsAndEqualizers) ). Hence the statement follows with def. \ref{GeneralComponentwiseDefinition}.

=--

#### As étale spaces
 {#AsEtaleSpace}

Further in the case where the [[site]] is the [[category of open subsets]] of a [[topological space]] $B$.

Some authors (e.g., [[Robert Goldblatt|Goldblatt]] in _Topoi: The Categorial Analysis of Logic_, §4.5, p. 96) use _sheaf_ to mean what we call an [[étale space]]: a topological [[bundle]] where the projection map is a [[local homeomorphism]].

As discussed at *[[étale space#RelationToSheaves]]*, there is an [[equivalence of categories]] between the "sheaves" in this sense over a given base space $B$ (i.e., the étale spaces over $B$), and the sheaves as defined above over $B$.

#### Characterization over canonical topologies
 {#CharacterizationOverCanonicalTopologies}

The above prop. \ref{AsContinuousFunctorOverOpenSubsets} shows that often sheaves are characterized as contravariant functors that take some [[colimits]] to [[limits]]. This is true in full generality for the following case

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

Let $\mathcal{T}$ be be a [[topos]], regarded as a [[large site]] when equipped with the [[canonical topology]]. Then a [[presheaf]] (with values in [[small sets]]) on $\mathcal{T}$ is a sheaf precisely if it sends all [[colimits]] to [[limits]].

=--

## Sheaves and localization

We now describe the derivation and the detailed description of various aspects of sheaves, the [[descent]] condition for sheaves and [[sheafification]], relating it to all the related notions

* [[geometric embedding]]

  * [[localization]]

  * [[homotopy category]]

* [[coverage]]

  * [[Grothendieck topology]]

  * [[Lawvere-Tierney topology]]

* [[local isomorphism]]

  * [[sieve]]

    * [[cover]]

    * [[hypercover]]

  * [[dense monomorphism]]

  * [[local epimorphism]]

We start by assuming that a [[geometric embedding]] into a [[presheaf]] category is given and derive the consequences.

So let $S$ be a [[small category]] and write $PSh(S) = PSh_S = [S^{op}, Set]$ for the corresponding [[topos]] of [[presheaf|presheaves]].

Assume then that another topos $Sh(S) = Sh_S$ is given together with a [[geometric embedding]]

$$
  f : Sh(S) \to PSh(S)
$$

i.e. with a [[full and faithful functor]]

$$
  f_* : Sh(S) \to PSh(S)
$$

and a left [[exact functor]]

$$
  f^* : PSh(S) \to Sh(S)
$$

Such that both form a pair of [[adjoint functor]]s

$$
  f^* \dashv f_*
$$

with $f^*$ [[left adjoint]] to $f_*$.

Write $W$ for the category

$$
  Core(PSh(S)) \hookrightarrow W \hookrightarrow PSh(S)
$$

consisting of all those morphisms in $PSh(S)$ that are sent to [[isomorphism]]s under $f^*$.

$$
  W = (f^*)^{-1}(Core(Sh_S))
  \,.
$$

From the discussion at [[geometric embedding]] we know that $Sh(S)$ is equivalent to the full [[subcategory]] of $PSh(S)$ on all $W$-[[local object]]s.

Recall that an object $A \in PSh(S)$ is called a $W$-[[local object]] if for all $p : Y \to X$ in $W$ the morphism

$$
  p^* : PSh_S(X,A) \to PSh_S(Y,A)
$$

is an [[isomorphism]]. This we call the [[descent]] condition on presheaves (saying that a presheaf "descends" along $p$ from $Y$ "down to" $X$). Our task is therefore to identify the category $W$, show how it determines and is determed by a [[Grothendieck topology]] on $S$ -- equipping $S$ with the structure of a  [[site]] -- and characterize the $W$-[[local object]]s. These are (up to equivalence of categories) the objects of $Sh$, i.e. the sheaves with respect to the given [[Grothendieck topology]].

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

A morphism $Y \to X$ is in $W$ if and only if for every [[representable functor|representable presheaf]] $U$ and every morphism $U\to X$ the pullback $Y \times_X U \to U$ is in $W$
$$
  \array{
     Y \times_X U &\to& Y
     \\
     \downarrow^{\in W} && \downarrow^{\Leftrightarrow \in W}
     \\
     U &\to& X
  }
  \,.
$$
=--

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

Since $W$ is stable under [[pullback]] (as described at [[geometric embedding]]: simply because $f^*$ preserves finite limits) it is clear that $Y \times_X U \to U$ is in $W$ if $Y \to X$ is.

To get the other direction, use the [[co-Yoneda lemma]] to write $X$ as a [[colimit]] of [[representable functor|representables]] over the [[comma category]] $(Y/const_X)$ (with $Y$ the [[Yoneda embedding]]):

$$
  X \simeq colim_{U_i \to X} U_i
  \,.
$$

Then pull back $Y \to colim_{U_i \to X} U$ over the entire colimiting cone, so that over each component we have

$$
  \array{
    Y \times_X U_i &\to& Y
    \\
    \downarrow && \downarrow
    \\
    U_i &\to& X
  }
  \,.
$$

Using that in $PSh(S)$ [[commutativity of limits and colimits|colimits are stable under base change]] we get

$$
  colim_i (Y \times_X U_i) \simeq (colim_i U_i) \times_X Y
  \,.
$$

But since $X \simeq colim_i U_i$ the right hand is $X \times_X Y$, which is just $Y$. So $Y = colim_i (Y \times_X U_i)$ and we find that $Y \to X$ is a morphism of colimits. But under $f^*$ the two respective diagrams become isomorphic, since $Y \times_X U_i \to U_i$ is in $W$. That means that the corresponding morphism of colimits $f^*(Y \to X)$ (since $f^*$ preserves colimits) is an isomorphism, which finally means that $Y \to X$ is in $W$.
=--

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

A presheaf $A \in PSh(S)$ is a [[local object]] with respect to all of $W$ already if it is local with respect to those morphisms in $W$ whose codomain is [[representable functor|representable]]
=--

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

Rewriting the morphism $Y \to X$ in $W$ in terms of
colimits as in the above proof

$$
  \array{
     colim_{U \to X} U_i \times_X Y
      &\stackrel{\simeq}{\to}& Y
     \\
     \downarrow && \downarrow
     \\
     colim_{U \to X} U &\stackrel{\simeq}{\to}& X
  }
$$

we find that $A(X) \to A(Y)$ equals

$$
  lim_{U \to X} (A(U) \to A(U \times_X Y))
  \,.
$$

If $A$ is local with respect to morphisms $W$ with representable codomain, then by the above if $Y \to X$ is in $W$ all the morphisms in the limit here are isomorphisms, hence

$$
  \cdots = Id_{A(X)}
  \,.
$$
=--

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

Every morphism $Y \to X$ in $W \subset PSh(S)$ factors as an epimorphism followed by a monomorphism in $PSh(S)$ with both being morphisms in $W$.
=--

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

Use factorization through [[image]] and [[coimage]], use exactness of $f^*$ to deduce that the factorization exists not only in $PSh(S)$ but even in $W$.

More in detail, given $Y \to X$ we get the diagram

$$
  \array{
    Y \times_X Y &&\to&& Y
    \\
     &&& \swarrow
    \\
    \downarrow &&Y \sqcup_{Y \times_X Y} Y && \downarrow
    \\
    & \nearrow && \searrow
    \\
    Y && \to && X
  }
  \,.
$$

Because $f^*$ is exact, the pullbacks and pushouts in this diagram remain such under $f^*$. But since $f^*(Y \to X)$ is an isomorphism by assumption, the all these are pullbacks and pushouts along isomorphisms in $Sh(S)$, so all morphisms in the above diagram map to isomorphisms in $Sh(S)$, hence the entire diagram in $PSh(S)$ is in $W$.

Since the morphism $Y \sqcup_{Y \times_X Y} Y  \to X$ out of the [[coimage]] is at the same time the [[equalizer|equalizing]] morphism into the [[image]] $lim(X \stackrel{\to}{\to} X \sqcup_Y X)$, it is a [[monomorphism]].
=--

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

The monomorphisms in $PSh(S)$ which are in $W$ are
called [[dense monomorphism]]s.

=--

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

Every [[monomorphism]] $Y \to X$ with $X$ [[representable functor|representable]] is of the form

$$
  Y = colim ( U \times_X U \to U  )
$$

for $U = \sqcup_{\alpha} U_\alpha$ a disjoint union of representables
=--

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

This is a direct consequence of the standard fact that
subfunctors are in bijection with [[sieve]]s.
=--

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

If a presheaf $A$ is [[local object|local]] with
respect to all [[dense monomorphism]]s, then it is already local with respect to all morphisms $Y \to X$ of the form

$$
  \array{
    Y
    \\
    \downarrow
    \\
    X
  }
  =
  colim
  \left(
  \array{
    W &\stackrel{\to}{\to}& U
    \\
    \;\;\downarrow^{dense mono} && \downarrow^{Id}
    \\
    U \times_X U & \stackrel{\to}{\to}&  U
  }
  \right)
$$

with the left vertical morphism a [[dense monomorphism]]

(and with $U = \sqcup_\alpha U_\alpha$ the disjoint union (of representable presheaves) over a [[cover]]ing family of objects.)
=--

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

The morphisms in $W$ with representable codomain

* of the form $colim (U \times_X U \stackrel{\to}{\to} U) \to X$ as above are [[cover]]s:

* of the form $colim (W  \stackrel{\to}{\to} U) \to X$ (with $W$ a cover of $U \times_X U$) as above are [[hypercover]]s

of the representable $X$.
=--

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

A presheaf $A$ is $W$-local, i.e. a sheaf, already if it is local (satisfies [[descent]]) with respect to all [[cover]]s, i.e. all [[dense monomorphism]]s with codomain a [[representable functor|representable]].
=--

>[[Urs Schreiber|Urs]]: the above shows this almost. I am not sure yet how to see the remaining bit directly, without making recourse to the full machinery leading up to section VII, 4, corollary 7 in [[Sheaves in Geometry and Logic]].

So we finally conclude:

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

We have:

* Systems $W$ of weak equivalences defined by choice of [[geometric embedding]] $f : Sh(S) \to PSh(S)$ are in canonical bijection with choice of [[Grothendieck topology]].

* A presheaf $A$ is $W$-local, i.e. local with respect to all [[local isomorphism]]s, if and only if it is local already with respect to all [[dense monomorphism]], i.e. if and only if it satisfies sheaf condition for all covering [[sieve]]s.
=--

From the _assumption_ that $f : Sh(S) \to PSh(S)$ is a [[geometric embedding]] follows at once the following explicit description of the [[sheafification]] functor
$f^* : PSh(S) \to Sh(S)$.

+-- {: .un_lemma}
###### Lemma (Sheafification)

For $A \in PSh(S)$ a presheaf, its [[sheafification]] $\bar A := f_* f^* A$ is the presheaf given by

$$
  \bar A : U \mapsto colim_{(Y \to U) \in W} A(Y)
$$
=--

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

By the discussion at [[geometric embedding]] the category $Sh(S)$ is equivalent to the [[localization]] $PSh(S)[W^{-1}]$, which in turn is the category with the same objects as $PSh(S)$ and with morphisms given by spans out of hypercovers in $W$

$$
  PSh(S)[W^{-1}](X,A) =
  colim_{(Y \to X) \in W} A(Y)
  \,.
$$

So we have

$$ \array {
  Sh(S) &&\stackrel{\stackrel{f_*}{\to}}{\stackrel{f^*}{\leftarrow}}&
  PSh(S)
  \\
  & \searrow_{\simeq}&\Downarrow^{\simeq}&  \downarrow
  \\
  &&
  PSh(S)[W^{-1}]
  \,.
} $$

and deduce

* by [[Yoneda lemma|Yoneda]] that $\bar A(U) = PSh_S(U, \bar A)$;

* by the [[adjoint functor|hom-adjunction]] this is $\cdots \simeq Sh_S(\bar U, \bar A)$;

*  by the equivalence just mentioned this is
$\cdots \simeq PSh_S[W^{-1}](U,A)$.
=--

+-- {: .un_remark}
###### Remark: covers versus hypercovers

For checking the sheaf condition the [[dense monomorphism]]s, i.e. the ordinary [[cover]]s are already sufficient. But for [[sheafification]] one really needs the [[local isomorphism]]s, i.e. the [[hypercover]]s. If one takes the colimit in the sheafification prescription above only over [[cover]]s, one obtains instead of sheafification the [[plus construction on presheaves|plus-construction]].
=--

+-- {: .un_definition}
###### Definition: plus-construction

For $A \in PSh(S)$ a presheaf, the **[[plus construction on presheaves|plus-construction]]**
on $A$ is the presheaf

$$
  A^+ : X \mapsto  colim_{(Y \hookrightarrow X) \in W } A(Y)
$$

where the colimit is over all [[dense monomorphism]]s (instead of over all [[local isomorphism]]s as for [[sheafification]] $\bar A$).
=--

+-- {: .un_remark}
###### Remark: plus-construction versus sheafification

In general $A^+$ is not yet a sheaf. It is however in general closer to being a sheaf than $A$ is, namely it is a [[separated presheaf]].

But applying the plus-construction twice yields the desired sheaf

$$
  (A^+)^+ = \bar A
  \,.
$$

This is essentially due to the fact that in the context of ordinary sheaves discussed here, all [[hypercover]]s are already of the form

$$
  colim(W \stackrel{\to}{\to} U)
$$

for $W \to U \times_X U$ a cover. For higher [[stack]]s the hypercover is in general a longer simplicial object of covers and accordingly if one restricts to covers instead of using hypercovers one will need to use the plus-construction more and more often.  Specifically, for stacks of $n$-groupoids one needs to apply the plus-construction $n+2$ times; see [[plus construction on presheaves]].

When $n=\infty$, even a countable sequence of applications does not suffice in general, but a sufficiently long transfinite sequence does.  In this case, using hypercovers instead actually produces a different answer, namely the reflection into the [[hypercompletion]] of the sheaf $\infty$-topos.
=--

## Examples

The archetypical example of sheaves are _sheaves of [[function]]s_:

* for $X$ a topological space, $\mathbb{C}$ a topological space and $O(X)$ the [[site]] of open subsets of $X$, the assignment $U \mapsto C(U,\mathbb{C})$ of continuous functions from $U$ to $\mathbb{C}$ for every open subset $U \subset X$ is a sheaf on $O(X)$.

* for $X$ a complex manifold and $\mathbb{C}$ a complex manifold, the assignment $U \mapsto C_{hol}{X,\mathbb{C}}$ of holomorphic functions in a sheaf.

## Related concepts

* [[(0,1)-sheaf]] / [[ideal]]

* [[presheaf]] / [[separated presheaf]] /  **sheaf** / [[cosheaf]]

  * [[sheafification]]

  * [[abelian sheaf]], [[sheaf of abelian groups]], [[sheaf of modules]], [[quasicoherent sheaf]], [[sheaf of meromorphic functions]]

  * [[locally constant sheaf]], [[constructible sheaf]]

  * [[sheaf with transfer]]

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

* [[(∞,1)-sheaf]] / [[∞-stack]]

  * [[sheaf of spectra]]

* [[(∞,2)-sheaf]]

* [[(∞,n)-sheaf]]

* [[abelian sheaf cohomology]]

  * [[soft sheaf]]

  * [[fine sheaf]]

  * [[flabby sheaf]]

* [[descent]]

  * [[cover]]

  * [[cohomological descent]]

  * [[descent morphism]]

  * [[monadic descent]],

    * [[Sweedler coring]]

    * [[higher monadic descent]]

    * [[descent in noncommutative algebraic geometry]]

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

## References

The original definition is in

* [[Jean Leray]], _L'anneau d'homologie d'une représentation_.  Comptes rendus hebdomadaires des séances de l'Académie des Sciences 222 (1946), 1366–1368.  [[Leray-0.pdf:file]]

Subsequent development by Leray, incorporating ideas of [[Henri Cartan]]:

* [[Jean Leray]], _L'anneau spectral et l'anneau filtré d'homologie d'un espace localement compact et d'une application continue_, Journal de Mathématiques Pures et Appliquées. Neuvième Série, 29 (1950), 1–80, 81–139.

[[Henri Cartan]]'s account of the theory:

* [[Henri Cartan]], _Faisceaux sur un espace topologique. I, II_, Séminaire Henri Cartan, Exposés 14, 15.
numdam: [I](http://www.numdam.org/item/SHC_1950-1951__3__A14_0/), [II](http://www.numdam.org/item/SHC_1950-1951__3__A15_0/).

It refers to a previous exposition of the theory
in Exposés 12–17 of the first year (1948/1949),
which apparently are not scanned, unlike Exposés 1–11.

Further references:

Section C2 in

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_
{#Johnstone}

* [[Saunders MacLane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_
* [[Yuri Manin]], _Methods of homological algebra_

A concise and contemporary overview can be found in

* {#CentazzoVitale04}C. Centazzo, [[Enrico Vitale|E. M. Vitale]], _Sheaf theory_ , pp.311-358 in Pedicchio, Tholen (eds.), _Categorical Foundations_ , Cambridge UP 2004. ([draft](https://perso.uclouvain.be/enrico.vitale/chapter7.pdf))

The book by Kashiwara and Schapira discusses sheaves with motivation from [[homological algebra]], [[abelian sheaf cohomology]] and [[homotopy theory]], leading over in the last chapter to the notion of [[stack]].

* [[Masaki Kashiwara]], [[Pierre Schapira]], _[[Categories and Sheaves]]_, Grundlehren der Mathematischen Wissenschaften __332__, Springer (2006)

A quick pedagogical introduction with an eye towards the generalization to [[(∞,1)-sheaves]] is in

* [[Dan Dugger]], _Sheaves and homotopy theory_, [pdf](http://ncatlab.org/nlab/files/cech.pdf)

Classics of sheaf theory on topological spaces are

* Roger Godement, _Topologie algébrique et théorie des faisceaux_, Hermann, 1958, 283 p. [gBooks](http://books.google.fr/books/about/Topologie_alg%C3%A9brique_et_th%C3%A9orie_des_fa.html?id=JVrvAAAAMAAJ)

* [[A. Grothendieck]], [[Tohoku]]

Recently, an improvement in understanding the interplay of derived functors (inverse image and proper direct image) in sheaf theory on topological spaces has been exhibited in

* Olaf M. Schnuerer, Wolfgang Sergel, _Proper base change for separated locally proper maps_, [arxiv/1404.7630](http://arxiv.org/abs/1404.7630)

[[!redirects sheaves]]

[[!redirects sheaf of sets]]
[[!redirects sheaves of sets]]

[[!redirects 1-sheaf]]
[[!redirects (1,1)-sheaf]]

[[!redirects 1-sheaves]]
[[!redirects (1,1)-sheaves]]


sheaf and topos theory

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

#Contents#
* table of contents
{:toc}

## Idea

_Topos theory_ is the part of [[category theory]] that studies [[categories]] which are [[topos]]es. This includes in particular [[Grothendieck toposes]], i.e. [[categories of sheaves]].

There are always two ways to think of topos theory: as being

* about [[logic]]

* about [[higher geometry|geometry]].

## Related concepts

* **topos theory**

  * [[Giraud's theorem]]

  * [[fundamental theorem of topos theory]]

* [[2-topos theory]]

* [[(∞,1)-topos theory]]

* [[higher topos theory]]

* [[microlocal sheaf theory]]

## References

### Original texts

Discussion in [[algebraic topology]]:

* {#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]]]

Discussion in [[algebraic geometry]]:

* {#SGA4} [[Michael Artin]], [[Alexander Grothendieck]], [[Jean-Louis Verdier]] (eds.), _Théorie des Topos et Cohomologie Etale des Schémas - SGA 4_ , LNM **269** Springer Heidelberg 1972.

* [[Monique Hakim]], _Topos annelés et schémas relatifs_, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 64, Springer, Berlin, New York (1972) ([doi:10.1007/978-3-662-59155-0](https://www.springer.com/gp/book/9783540055730))

### Introductions
 {#Introductions}

Brief expositions:

* [[Michael Barr]], [[Colin McLarty]], [[Charles Wells]],  *Variable set theory*, prepared for *Scientific American* but unpublished (~1985) [[pdf](https://www.math.mcgill.ca/barr/papers/vst.pdf), [[BarrMcLartyWells-VariableSetTheory.pdf:file]]]

* [[Ieke Moerdijk]], *Background in topos theory*, chapter I in: _Classifying Spaces and Classifying Topoi_, Lecture Notes in Mathematics **1616**, Springer (1995) [[doi:10.1007/BFb0094441](https://doi.org/10.1007/BFb0094441)]

* [[Luc Illusie]], *What is... a Topos?*, Notices of the AMS **51** 9 (2004) [[pdf](https://www.ams.org/notices/200409/what-is-illusie.pdf), full volume:[pdf](https://www.ams.org/notices/200409/200409FullIssue.pdf)]

* [[Tom Leinster]], _[[Leinster2010|An informal introduction to topos theory]]_ (2010) [[arXiv:1012.5647](https://arxiv.org/abs/1012.5647)]

* [[John Baez]], *Topos Theory in a Nutshell* (2021) [[web](http://math.ucr.edu/home/baez/topos.html)]

* MathProofsable, Category Theory - Toposes [video playlist] (https://www.youtube.com/watch?v=gKYpvyQPhZo&list=PL4FD0wu2mjWM3ZSxXBj4LRNsNKWZYaT7k)

Lecture notes:

* [[Ross Street]], _A survey of topos theory_, lecture notes (1978) [[pdf](http://www.math.mq.edu.au/~street/ToposSurvey.pdf), [[Street-SurveyToposTheory.pdf:file]]]

* {#Wyler} [[Oswald Wyler]], *Lecture Notes on Topoi and Quasitopoi*, World Scientific (1991) [[doi:10.1142/1047](https://doi.org/10.1142/1047)]

* {#Joyal15} [[André Joyal]], _[[A crash course in topos theory -- The big picture]]_, lecture series at [Topos à l'IHES](https://indico.math.cnrs.fr/event/747/), Paris (November 2015)

* [[André Joyal]], _Geometric aspects of topos theory in relation with logical doctrines_, talk at _[[New Spaces for Mathematics and Physics]]_, IHP Paris 2015 ([video recording](https://www.youtube.com/watch?v=kaZpOEOAUzE))

* [[Ieke Moerdijk]], [[Jaap van Oosten]], _Topos Theory_, Master class notes (2007) [[pdf](http://www.staff.science.uu.nl/~ooste110/syllabi/toposmoeder.pdf)]

* [[Jaap van Oosten]], *Topos Theory* (2018) [[pdf](https://webspace.science.uu.nl/~ooste110/syllabi/topostheory.pdf), [[vanOosten-ToposTheory.pdf:file]]]

* [[Francis Borceux]], *Some glances at topos theory*, lecture notes, Como (2018) [[pdf](https://tcsc.lakecomoschool.org/files/2018/01/Borceux.pdf), [[Borceux-ToposTheory.pdf:file]], [video playlist](https://www.youtube.com/watch?v=s_fN9euuVAY&list=PLh_3Q6ZRqWs0LBptMGClJ8OArR0fBT6Pp&index=11)]

* [[Pierre Schapira]], *An Introduction to Categories and Sheaves*, lecture notes (2023) [[pdf](https://webusers.imj-prg.fr/~pierre.schapira/LectNotes/CatShv.pdf)]

A monograph that aims to be more expository, focusing on [[presheaf toposes]]:

* [[Marie La Palme Reyes]], [[Gonzalo E. Reyes]], [[Houman Zolfaghari]], _Generic figures and their glueings.  A constructive approach to functor categories_, Polimetrica (2008) [[pdf](https://marieetgonzalo.files.wordpress.com/2004/06/generic-figures.pdf), ISBN:8876990046]

### Textbooks
 {#Textbooks}

* {#Johnstone77} [[Peter Johnstone]], _Topos theory_, London Math. Soc. Monographs __10__, Acad. Press 1977, xxiii+367 pp. (Available as Dover Reprint, Mineola 2014)

* [[Michael Barr]], [[Charles Wells]], _[[Toposes, Triples, and Theories]]_ , Springer Heidelberg 1985. (Available as [TAC reprint no.12](http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html) 2005)

* [[Colin McLarty]], *[[Elementary Categories, Elementary Toposes]]*, Oxford University Press 1992 ([ISBN:9780198514732](https://global.oup.com/academic/product/elementary-categories-elementary-toposes-9780198514732?cc=ae&lang=en&))

* [[Saunders MacLane]], [[Ieke Moerdijk]], *[[Sheaves in Geometry and Logic]]*, Springer (1992) [[doi:10.1007/978-1-4612-0927-0](https://link.springer.com/book/10.1007/978-1-4612-0927-0)]

* [[Francis Borceux]], _[[Handbook of Categorical Algebra]] 3 - Categories of Sheaves_, Cambridge UP 1994 ([ISBN:9780521061247](https://www.cambridge.org/de/academic/subjects/mathematics/logic-categories-and-sets/handbook-categorical-algebra-volume-3?format=PB))

* [[Peter Johnstone]], *[[Elephant|Sketches of an elephant: a topos theory compendium]]*, 2002

* [[Alexandru Dimca]], *Sheaves in Topology*, Universitext, Springer (2004) $[$[doi:10.1007/978-3-642-18868-8](https://doi.org/10.1007/978-3-642-18868-8)$]$

* [[Masaki Kashiwara]], [[Pierre Schapira]], *[[Categories and Sheaves]]*, Grundlehren der Mathematischen Wissenschaften **332** Springer (2006)  $[$[doi:10.1007/3-540-27950-4](https://link.springer.com/book/10.1007/3-540-27950-4),  [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/kashiwara2.pdf)$]$

Introducing even category theory from the scratch while still managing to cover some ground, the following textbook is the _royal road to topos theory_ for people with some background in [[first-order logic]]:

* {#Goldblatt84} [[Robert Goldblatt]], _Topoi: The Categorial Analysis of Logic_, 2nd ed. North-Holland Amsterdam 1984. ([Dover reprint](https://store.doverpublications.com/0486450260.html) New York 2006; free online at [Project Euclid](http://projecteuclid.org/euclid.bia/1403013939).)

See also

* [[Olivia Caramello]], _Theories, Sites, Toposes_, Oxford UP 2017.

### Course notes

A survey is in

* [[Ross Street]], _A survey of topos theory_ (notes for students, 1978) [pdf](http://www.math.mq.edu.au/~street/ToposSurvey.pdf)

A nice and concise introduction is available in

* [[Francis Borceux]], _Some glances at topos theory_ , lecture notes Como 2018. ([pdf](http://tcsc.lakecomoschool.org/files/2018/06/Como2018.pdf), [video playlist](https://www.youtube.com/watch?v=s_fN9euuVAY&list=PLh_3Q6ZRqWs0LBptMGClJ8OArR0fBT6Pp&index=11))

* [[Ieke Moerdijk]], [[Jaap van Oosten]], _Topos theory_ Master Class notes (2007) ([pdf](http://www.staff.science.uu.nl/~ooste110/syllabi/toposmoeder.pdf))

### History

* {#Lawvere00}[[F. William Lawvere]], *Comments on the development of topos theory*, pp.715-734 in Pier (ed.),  _Development of Mathematics 1950 - 2000_ , Birkhäuser Basel 2000. ([tac reprint](http://www.tac.mta.ca/tac/reprints/articles/24/tr24abs.html))

* [[Colin McLarty]], _The Uses and Abuses of the History of Topos Theory_ , Brit. J. Phil. Sci., 41 (1990) ([JSTOR](http://www.jstor.org/stable/687825)) [PDF](http://bjps.oxfordjournals.org/content/41/3/351.full.pdf)

A historical analysis of Grothendieck's 1973 Buffalo lecture series on toposes and their precedents is in

* {#McLarty18} [[Colin McLarty]], _Grothendieck's 1973 topos lectures_, Séminaire Lectures grothendieckiennes, 3 May (2018) ([YouTube video](https://www.youtube.com/watch?v=5AR55ZsHmKI))

* _[The peripatetic seminar on sheaves and logic 1976-1999](http://www.wra1th.plus.com/gcw/math/PSSL/index.html)_

[[!redirects Sheaf and Topos Theory]]
[[!redirects sheaf theory]]
[[!redirects topos theory]]


Sheaf and Topos Theory > history

< [[Sheaf and Topos Theory]]
[[!redirects Sheaf and Topos Theory -- history]]


sheaf cohomology

#entries related to sheaf cohomology#

* [[motivation for sheaves, cohomology and higher stacks]]

* [[sheaf]]

* [[cohomology]]

  * [[abelian sheaf cohomology]]

  * [[nonabelian sheaf cohomology]]

    * [[nonabelian cocycle]]

    * [[nonabelian cosheaf homotopy]]

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

  * [[category of sheaves]]

  * [[(infinity,1)-category of (infinity,1)-sheaves]]

  * [[infinity-stack]]

    * [[stack]]

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

category: sheaf theory


sheaf of abelian groups

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

#Contents#
* table of contents
{:toc}

## Idea

For $C$ a [[site]], a _sheaf of abelian groups_ on $C$ is an [[abelian group|abelian]] [[group object]] in the [[sheaf topos]] $Sh(C)$.

The [[category]] $Ab(Sh(C))$ of sheaves of abelian groups is an [[abelian category]] and hence serves as a context for [[homological algebra]] "parameterized over $C$". For the case that $C = *$ is the point, this is just [[Ab]] itself.

More generally, for $\mathcal{A}$ an [[abelian category]] one can consider $\mathcal{A}$-valued sheaves $Sh(C,\mathcal{A})$: **[[abelian sheaves]]**. For this to have good properties $\mathcal{A}$ has to be a [[Grothendieck category]].

## References

A basic textbook introduction begins for instance around Definition 1.6.5 of

* [[Charles Weibel]], _[[An Introduction to Homological Algebra]]_

A detailed textbook discussion is in section 18 of

* [[Masaki Kashiwara]], [[Pierre Schapira]], _Categories and Sheaves_, Grundlehren der Mathematischen Wissenschaften __332__, Springer (2006)

category: sheaf theory
[[!redirects sheaves of abelian groups]]


sheaf of ideals

An (left, right, two-sided) [[ideal]] of ring (or $k$-algebra) $R$ is a (left, right, bi) $R$-submodule of $R$ itself. One can consider similarly the notion of a [[presheaf of modules]] over a presheaf of rings: and the presheaf of ideals is then the presheaf of submodules of a presheaf of rings.

In algebraic geometry, [[quasicoherent sheaves]] of modules are of special importance; thus also quasicoherent ideals as those ideals which are quasicoherent as sheaves of modules.  An important example of a quasicoherent sheaf of ideals is the [[defining sheaf of ideals]] of a [[closed subscheme]].

Wikipedia uses less preferrable term [ideal sheaf](http://en.wikipedia.org/wiki/Ideal_sheaf).

Having a [[finitary monad]] $T$ in the category of sets, the [[sheafification functor]] from the category of presheaves of sets to the category of sheaves of sets, can be strictly lifted to the category of presheaves of $T$-modules (or, if you like, $T$-algebras) to the category of sheaves of $T$-modules. More generally, one can consider sheaves of finitary monads and corresponding categories of sheaves of modules and of bimodules. The discussions of sheaves of ideals extends easily to this setting.

category: algebraic geometry, sheaf theory
[[!redirects ideal sheaf]]


sheaf of L-∞ algebras

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

#Contents#
* table of contents
{:toc}

## Idea

A _sheaf of $L_\infty$-algebras_ is a suitable [[sheaf]]/[[stack]]/[[∞-stack]] of [[L-∞ algebras]]/[[dg-Lie algebras]].

These structures appear in the [[deformation theory]] of the given base [[space]]/[[site]]: where a single [[L-∞ algebra]] is equivalently a [[formal moduli problem]] (hence "over the point), so a sheaf of these over some base $X$ may be thought of as an $X$-parameterized [[formal moduli problem]] (eg. [Hinich 03](#Hinich03)).

## Local $L_\infty$-algebras {#LocalLieAlgebra}

\begin{definition}
\label{LocalLieAlgebra}

A __local $L_\infty$-algebra__ on a [[smooth manifold]] $X$ is defined by the following data:

1. A graded vector bundle $L\rightarrow X$,

1. A square $0$ [[differential operator]] $\mathrm{d}:\Gamma(X,L)\rightarrow\Gamma(X,L)$ of cohomological degree $1$,

1. A collection $\{ l_n \}_{n \geq 2}$ of poly-differential operators
$$l_n \,:\; \Gamma(X,L)^{\otimes n} \rightarrow \Gamma(X,L),$$
which are alternating, of cohomological degree $2-n$ and such that they endow $\Gamma(X,L)$ with an $L_\infty$ structure.

\end{definition}

One usually denotes by $\mathcal{L}$ the sheaf of sections of the bundle $L$.

\begin{definition}

The __Chevalley-Eilenberg cochain complex__ $\mathrm{CE}(\mathcal{L})$ of the local $L_\infty$-algebra $\mathcal{L}$ is defined by

$$ \mathrm{CE}\left(\mathcal{L}(U)\right) \;=\; \prod_{n\geq 0} \mathrm{Hom}\left( (\mathcal{L}(U)[1])^{\hat{\otimes} n}, \,\mathbb{R}  \right)_{S_n} $$

equipped with the usual [[Chevalley-Eilenberg algebra|Chevalley-Eilenberg]] differential; where $\hat{\otimes}$ denotes the completed projective tensor product, $\mathrm{Hom}$ denotes the dg-vector space of continuous linear maps and the subscript $S_n$ denotes the [[coinvariants]] respect to the action of the [[symmetric group]] $S_n$.

\end{definition}

\begin{definition}

The __reduced Chevalley-Eilenberg cochain complex__ $\mathrm{CE}_{\mathrm{red}}(\mathcal{L})$ of the local $L_\infty$-algebra $\mathcal{L}$ is defined by the [[kernel]]

$$ \mathrm{CE}_{\mathrm{red}}(\mathcal{L}) \subset \mathrm{CE}(\mathcal{L}) $$

of the natural augmentation map $\mathrm{CE}\left(\mathcal{L}(U)\right) \rightarrow \mathbb{R}$.

\end{definition}

Both $\mathrm{CE}(\mathcal{L})$ and $\mathrm{CE}_{\mathrm{red}}(\mathcal{L})$ are differentiable pro-cochain complexes.

\begin{definition}

The __local Chevalley-Eilenberg cochain complex__ $\mathrm{CE}_{\mathrm{loc}}(\mathcal{L})$ of the local $L_\infty$-algebra $\mathcal{L}$ is defined by

$$ \mathrm{CE}_{\mathrm{loc}}(\mathcal{L}) \;\coloneqq\;  \mathscr{Dens}_X \otimes_{D_X} \mathrm{CE}_{\mathrm{red}}(\mathscr{Jet}(L)) $$

where $\mathscr{Jet}(L)$ is the local $L_\infty$-algebra, in the category of $D_X$-modules, of sections of the [[jet bundle]] of $L$ and $mathscr{Dens}_X$ is the right $D_X$-module of densities on $X$.

\end{definition}

The cochain complex $\mathrm{CE}_{\mathrm{loc}}(\mathcal{L})$ can be thought as the sheaf of [[Lagrangian density|Lagrangian densities]] on the graded vector bundle $L$.

## Related concepts

* [[descent for L-∞ algebras]]

* [[factorization algebras]]

* [[formal moduli problems]]

* [[L-∞ algebroid]]

* [[homotopy theory of L-∞ algebras]]

* [[sheaf of spectra]]

## References

Discussion in the context of [[deformation theory]]/parameterized [[formal moduli problems]] is in

* [[Vladimir Hinich]], _Deformations of sheaves of algebras_ ([arXiv:math/0310116](http://arxiv.org/abs/math/0310116))
 {#Hinich03}

[[quasi-coherent sheaf|Quasi-coherent sheaves]] of dg-Lie algebras appear in

* [[Amnon Yekutieli]], appendix A of _Twisted Deformation Quantization of Algebraic Varieties_ ([arXiv:0905.0488](http://arxiv.org/abs/0905.0488))

* I. Ciocan-Fontanine, [[Mikhail Kapranov]], _Derived Quot schemes_ ([arXiv:math/9905174](http://arxiv.org/abs/math/9905174))

Discussion in [[physics]], of sheaves of $L_\infty$-algebras of [[physical fields]] in [[prequantum field theory]] includes

* [[Anton Zeitlin]], _Conformal Field Theory and Algebraic Structure of Gauge Theory_ ([arXiv:0812.1840](http://arxiv.org/abs/0812.1840))

Local $L_\infty$-algebras are defined and discussed by

* [[Kevin Costello]], [[Owen Gwilliam]], _[[Factorization algebras in perturbative quantum field theory]]_

[[!redirects sheaf L-∞ algebras]]
[[!redirects sheaves L-∞ algebras]]

[[!redirects sheaves of L-∞ algebras]]

[[!redirects sheaf of L-infinity algebras]]
[[!redirects sheaves of L-infinity algebras]]

[[!redirects sheaf of dg-Lie algebras]]
[[!redirects sheaves of dg-Lie algebras]]

[[!redirects local L-∞ algebras]]
[[!redirects local L-∞ algebra]]
[[!redirects local L-infinity algebras]]
[[!redirects local L-infinity algebra]]


sheaf of meromorphic functions

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

# Contents
* table of contents
{: toc}

## Definition ##

Let $(X, \mathcal{O}_X)$ be a [[ringed space]].  Consider the [[sheaf|subsheaf]] of [[sets]] $\mathcal{S} \subset \mathcal{O}_X$ of the [[structure sheaf]] such that for each [[open subset]] $U \subset X$, $\Gamma(U, \mathcal{S})$ consists of only the regular sections of $\mathcal{O}_X$ over $U$, i.e. those elements $s\in\Gamma(U, \mathcal{O}_X)$ for which $s|_V f=0$ implies $f=0$ for all $f\in\Gamma(V,\mathcal{O}_X)$ for all opens $V\subseteq X$.  Consider the [[presheaf]] of rings on $X$
  \[ U \quad\mapsto\quad \Gamma(U, \mathcal{O}_X)[\Gamma(U, \mathcal{S})^{-1}] \]
which assigns to $U$ the [[ring of fractions]] of $\Gamma(U, \mathcal{O}_X)$ with denominators in $\Gamma(U, \mathcal{S})$; its [[sheafification]] $\mathcal{M}_X$ is called the **sheaf of ([[germs]] of) [[meromorphic functions]] on $X$**.  The sections of $\mathcal{M}_X$ over $X$ are called the **meromorphic functions on X** and we denote this ring $M(X) = \Gamma(X, \mathcal{M}_X)$.

## Properties ##

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

For every open subset $U \subset X$ there is a canonical [[isomorphism]] between $\mathcal{M}_U$ and the restriction of $\mathcal{M}_X$ to $U$.

=--

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

For every point $x \in X$ there is a canonical isomorphism between the stalk $\mathcal{M}_{X,x}$ and $\mathcal{O}_{X,x}[\mathcal{S}_x^{-1}]$.

=--

## References ##

* [[Alexander Grothendieck]], [[Jean Dieudonné]], _[[EGA]]_ (IV, 20.1).

* [[Stacks Project]], [sheaf of meromorphic functions and sections](http://stacks.math.columbia.edu/tag/01X1), tag 01X1.

* Steven L. Kleiman, Misconceptions about $K_X$, *L’Enseignement Mathématique* **25** (1979), 203–206 ([doi:10.5169/seals-50379](http://doi.org/10.5169/seals-50379))

category: sheaf theory, algebraic geometry
[[!redirects sheaf of germs of meromorphic functions]]


sheaf of modules

A [[sheaf of abelian groups]] which is a [[module]] over a [[sheaf of rings]].

[[!redirects sheaves of modules]]


sheaf of n-types

The term _sheaf of $n$-types_ is another word for an [[∞-stack]] that has trivial [[homotopy group]]s above degree $n$.

Under the [[homotopy hypothesis]] a [[homotopy n-type]] is just an [[∞-groupoid]] that has trivial [[homotopy group]]s above degree $n \in \mathbb{N}$. Hence a "sheaf of $n$-types" is just a special case of a "sheaf of $\infty$-groupoid"s which is usually called an _$n$-stack_ .

[[!redirects sheaves of n-types]]


sheaf of rational functions

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

# Contents
* table of contents
{: toc}

## Idea

Given a [[ringed space]]/[[ringed topos]] $(X, \mathcal{O}_X)$, with (commutative) [[structure sheaf]] $\mathcal{O}_X$, then the _sheaf of rational functions_ $K_X$ associated with this data is the [[sheafification]] of the [[presheaf]] obtained from the [[structure sheaf]] by [[localization of a ring|localizing]] its values $\mathcal{O}_X(U)$ at the multiplicative set of all $f \in \mathcal{O}_X(U)$ which are [[stalk|stalkwise]] not [[zero-divisors]].

On an [[integral scheme]] this amounts to forming [[fields of fractions]] of the [[regular function]]s.

## Properties

### Function field analogy

[[!include function field analogy -- table]]

## Related concepts

* [[sheaf of meromorphic functions]]

* [[function field of a variety]]

## References

* Wikipedia, _[Function field (scheme theory)](http://en.wikipedia.org/wiki/Function_field_(scheme_theory))_

[[!redirects sheaf of rational functions]]
[[!redirects sheaves of rational functions]]


sheaf of spectra

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

#Contents#
* table of contents
{:toc}

## Idea

The [[stabilization]] of an [[(∞,1)-topos]] $\mathbf{H}$

$$
  (\Sigma^\infty \dashv \Omega^\infty) :
  \mathbf{H}
   \stackrel{\overset{\Omega^\infty}{\leftarrow}}{\underset{\Sigma^\infty}{\to}}
  Stab(\mathbf{H})
$$

consist of [[spectrum objects]] in $\mathbf{H}$. By the "[stable Giraud theorem](http://ncatlab.org/nlab/show/stable+%28infinity%2C1%29-category#StabGiraud)" this is the [[localization of an (∞,1)-category|localization]] of an [[(∞,1)-category of (∞,1)-functors]] with values in the [[stable (∞,1)-category of spectra]]: $\infty$-sheaves of
spectra.

This may be [[presentable (infinity,1)-category|presented]] by a [[model structure on presheaves of spectra]].

## Properties

### Relation to sheaves of chain complexes and abelian sheaf cohomology

The [[stable Dold-Kan correspondence]]

$$
  DK \;\colon\; Ch_\bullet(R) \stackrel{\simeq}{\longrightarrow} H R Mod(Spectra) \stackrel{U}{\longrightarrow} Spectra
$$

is an [[(∞,1)-limits]]-preserving [[(∞,1)-functor]] from the [[(∞,1)-category of chain complexes]] (over a given [[commutative ring]]) to the [[(∞,1)-category of spectra]]. Hence every [[(∞,1)-sheaf]]/[[∞-stack]] of chain complexes (as it appears (maybe implicitly) in [[abelian sheaf cohomology]]/[[hypercohomology]] canonically incarnates as an $(\infty,1)$-sheaf of spectra).

### Symmetric monoidal structure

The [[smash product|smash]] [[tensor product]] of the [[symmetric monoidal (∞,1)-category of spectra]] induced also a smash tensor product on any [[(∞,1)-category]] of sheaves of spectra

([[Spectral Schemes|Lurie, "Spectral Schemes", prop 1.5]]).

## Examples

* [[generalized étale cohomology]]

* [[tangent cohesion]]

  * [[smooth spectrum]]

  * [[differential cohomology]],

* [[differential algebraic K-theory]]

## Related concepts

* [[sheaf of L-∞ algebras]]

* [[parameterized spectrum]]

## References

### General

The [[homotopy categories]] of sheaves of [[combinatorial spectra]] were discussed in

* {#Brown73} [[Kenneth Brown]], part II of _[[Abstract homotopy theory and generalized sheaf cohomology]]_ (1973)

A [[model category]] structure of presheaves of spectra akin to the [[model structure on simplicial presheaves]] is discussed in

* [[Rick Jardine]], _Stable homotopy theory of simplicial presheaves_, Canad. J. Math. 39(1987), 733-747 ([pdf](http://cms.math.ca/cjm/v39/cjm1987v39.0733-0747.pdf))

Plenty of further discussion in terms of [[model category]] theory is in

* {#Jardine97} [[Rick Jardine]], _Generalized Étale cohomology theories_, 1997 Progress in mathematics volume 146

Discussion in terms of [[(∞,1)-category]]/[[(∞,1)-topos]]-theory is in

* [[Jacob Lurie]], _Sheaves of spectra_, section 1 of _[[Spectral Schemes]]_

### Application to K-theory

* [[Bertrand Toën]], section 1.2 of _K-theory and cohomology of algebraic stacks: Riemann-Roch theorems, D-modules and GAGA theorems_ ([arXiv:math/9908097](http://arxiv.org/abs/math/9908097))

* Michael Paluch, _Algebraic K-theory and topological spaces_ ([pdf](http://www.math.uiuc.edu/K-theory/0471/alg-top.pdf))

### Application to differential cohomology
 {#ReferencesApplicationToDifferentialCohomology}

Discussion of sheaves of spectra on the [[infinity-cohesive site|$\infty$-cohesive site]] of [[SmoothManifolds]] (or, equivalently, its [[dense subsite]] of [[Cartesian spaces]]) as a model for [[Whitehead-generalized cohomology theory|Whitehead-generalized]] [[differential cohomology]]

* [[Urs Schreiber]], Section 4.1.2 in: *[[schreiber:differential cohomology in a cohesive topos|Differential cohomology in a cohesive $\infty$-topos]]* ([arXiv:1310.7930](https://arxiv.org/abs/1310.7930))

and observing that the [[differential cohomology hexagon]] emerges in this context:

* {#BunkeNikolausVoelkl13} [[Ulrich Bunke]], [[Thomas Nikolaus]], [[Michael Völkl]], _Differential cohomology theories as sheaves of spectra_,  J. Homotopy Relat. Struct. 11, 1–66 (2016) ([arXiv:1311.3188](http://arxiv.org/abs/1311.3188))

Discussion within the broader context of differential [[non-abelian cohomology]]:

* {#FSS20} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], Section 4.3 in: _[[schreiber:The Character Map in Twisted Non-Abelian Cohomology|The character map in (twisted differential) non-abelian cohomology]]_ ([arXiv:2009.11909](https://arxiv.org/abs/2009.11909))

A book-length overview:

* {#AmabelDebrayHaine21} [[Araminta Amabel]], [[Arun Debray]], [[Peter J. Haine]] (eds.), _Differential Cohomology: Categories, Characteristic Classes, and Connections_. Based on [Fall 2019 talks at MIT's Juvitop seminar](https://math.mit.edu/juvitop/pastseminars/2019_Fall.html) by: A. Amabel, D. Chua, A. Debray, S. Devalapurkar, D. Freed, P. Haine, M. Hopkins, G. Parker, C. Reid, and A. Zhang. ([arXiv:2109.12250](https://arxiv.org/abs/2109.12250))

[[!redirects sheaves of spectra]]
[[!redirects stack of spectra]]
[[!redirects stacks of spectra]]

[[!redirects presheaf of spectra]]
[[!redirects presheaves of spectra]]

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

[[!redirects ∞-stack of spectra]]
[[!redirects ∞-stacks of spectra]]


sheaf of vertex operator algebras

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

#Contents#
* table of contents
{:toc}

## Idea

A [[sheaf]] of [[vertex operator algebras]].

This arises naturally in the descriptions of [[2d CFTs]] such as the [[2d (2,0)-superconformal QFT]] which patchwise on [[target space]] is a [[free field theory]], but not globally so.

If the degree-0 part of a sheaf of vertex operator algebras is the [[structure sheaf]] of the base space, then one one may think of the whole sheaf as defining a "[[horizontal categorification]]" of a vertex operator algebra and hence one also speaks of a _[[vertex operator algebroid]]_.

## Examples

* [[chiral de Rham complex]]

## References

* [[Fyodor Malikov]], _Lagrangian approach to sheaves of vertex operator algebras_, ESI preprint 1828 (2006) ([pdf](http://www.mat.univie.ac.at/~esiprpr/esi1828.pdf))

Formalization of the [[Witten genus]] in terms of sheaves of vertex operator algebras is in

* Pokman Cheung, _The Witten genus and vertex algebras_ ([arXiv:0811.1418](http://arxiv.org/abs/0811.1418))

[[!redirects sheaves of vertex operator algebras]]

[[!redirects sheaf of vertex algebras]]
[[!redirects sheaves of vertex algebras]]


sheaf on a noncommutative space

In [[noncommutative algebraic geometry]] the analogues of collection of open sets, and more generally, the analogues of [[site]]s usually fail to be stable under pullback: a pullback of an open cover is not an open cover in general. This requires an extension of the usual concepts of Grothendieck topology and Grothendieck pretopology. Consequently the notion of a sheaf and of a stack need to be adapted to this formalism.

The point of view that localizations are analogues of Zariski open sets, and the appropriate notion of descent for quasicoherent sheaves for covers by noncommutative localizations is implied already in Gabriel's thesis [[Des Categories Abeliennes]] and later explicitly studied in a number of works, including

* Freddy M. J. Van Oystaeyen, Alain H. M. J. Verschoren, _Noncommutative algebraic geometry. An introduction_, Lec. Notes in Math. __887__, Springer 1981. vi+404 pp.
* [[A. L. Rosenberg]], _Non-commutative affine semischemes and schemes_, Seminar on supermanifolds __26__, Dept. Math., U. Stockholm (1988)
* [[Fred Van Oystaeyen]], Luc Willaert, _Grothendieck topology, coherent sheaves and Serre's theorem for schematic algebras_, J. Pure Appl. Algebra __104__ (1995), no. 1, 109–122, [MR97a:16086](http://www.ams.org/mathscinet-getitem?mr=1359695), doi
* [[F. van Oystaeyen]], _Algebraic geometry for associative algebras_, Marcel Dekker 2000. vi+287 pp.

* [[M. Kontsevich]], A. L. Rosenberg, _Noncommutative spaces and flat descent_, MPI-2004-36 [dvi](http://www.mpim-bonn.mpg.de/preblob/2304),[ps](http://www.mpim-bonn.mpg.de/preblob/2332),

More general point of view closer to the formalism of topologies/[[sieve]]s than to [[Grothendieck pretopologies]] is also a notion of [[Q-category]] due Rosenberg and his work on sheaves and later work with Kontsevich on stacks on Q-categories. For example, noncommutative analogues of smooth, fppf and fpqc topologies can be formalized in this framework.

* [[A. L. Rosenberg]], _Almost quotient categories, sheaves and localizations_, 181 p. Seminar on supermanifolds __25__, University of Stockholm, D. Leites editor, 1988 (in Russian; partial remake in English exists)

* [[M. Kontsevich]], A. Rosenberg, _Noncommutative spaces_, preprint MPI-2004-35 ([dvi](http://www.mpim-bonn.mpg.de/preblob/2303),[ps](http://www.mpim-bonn.mpg.de/preblob/2331)),  _Noncommutative stacks_, MPI-2004-37 [dvi](http://www.mpim-bonn.mpg.de/preblob/2305),[ps](http://www.mpim-bonn.mpg.de/preblob/2333)

category: noncommutative geometry
[[!redirects sheaves on a noncommutative space]]
[[!redirects sheaves on a noncommutative spaces]]
[[!redirects sheaf on noncommutative space]]
[[!redirects sheaf over a noncommutative space]]


sheaf on a topological space

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

#Contents#
* table of contents
{:toc}

## Idea

In general, the concept of _[[sheaf]]_ is relative to a choice of _[[site]]_, hence to a choice of [[small category]] $\mathcal{C}$ equipped with a [[coverage]]. Given this, the [[category of sheaves]] on $\mathcal{C}$ is the [[full subcategory]] of the [[category of presheaves]] $[\mathcal{C}^{op}, Set]$ on those presheaves which satisfy the sheaf condition:

$$
  Sh(\mathcal{C})
  \overset{\phantom{AAA}}{\hookrightarrow}
  [\mathcal{C}^{op}, Set]
  \,.
$$

Often sheaves are introduced and discussed in the more restrictive sense of _sheaves on a topological space_. This is indeed a special case: A sheaf on a [[topological space]] $X$ is a sheaf on the [[site]] $Op(X)$ whose underlying [[category]] is the [[category of open subsets]] of $X$, and whose [[coverage]] consists of the [[open covers]] of topological spaces. It is usual to write $Sh(X)$ as shorthand for the resulting category of sheaves, but the more systematic name is "$Sh(Op(X))$":

$$
  Sh(X) \;\coloneqq\; Sh(Op(X))
    \overset{\phantom{AAA}}{\hookrightarrow}
  [Op(X)^{op}, Set]
  \,.
$$

A [[topos]] which is [[equivalence of categories|equivalent]] to a category of sheaves on a topological spaces, this way, is called a _spatial topos_.

## Properties

### Localic reflection

Similarly, if $X$ is a [[locale]], there is its [[frame of opens]] $Op(X)$ and one obtains the [[sheaf topos]] $Sh(X) \coloneqq Sh(Op(X))$ as above. A [[topos]] in the [[essential image]] of this construction is called a _[[localic topos]]_.  A _[[sober topological space]]_ is canonically identified as a [[locale]], and hence the category of sheaves over a sober topological space is a [[localic topos]].

Restricted to [[sober topological spaces]], the construction of forming [[categories of sheaves]] is a [[fully faithful functor]] to the category [[Topos]] of [[toposes]] with [[geometric morphisms]] between them:

$$
  Sh(-)
  \;\colon\;
  SoberTopologicalSpace
  \overset{ \phantom{AA} }{\hookrightarrow}
  Topos
  \,.
$$

([MacLaneMoerdijk, theorem IX.3 1](#MacLaneMoerdijk)) or as ([Johnstone, lemma C.1.2.2](#Johnstone))

For more see at _[[locale]]_ the section _[Localic reflection](locale#RelationToToposes)_.

## Examples

Examples of spatial toposes which are not _manifestly_ spatial, by usual their definition, include the following:

* The [[Bohr topos]] of a [[C*-algebra]]

* The category of [[hypergraphs]]

* The [[Sierpinski topos]]

## Related concepts

* [[étendue]]

* [[localic topos]], [[spatial locale]]

* [[abelian sheaf]], [[sheaf of abelian groups]]

* [[sheaves on a simplicial topological space]]

## References

* {#Johnstone} [[Peter Johnstone]], part C (volume 2) _[[Elephant|Sketches of an elephant: a topos theory compendium]]_.

* {#MacLaneMoerdijk} [[Saunders MacLane]], [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_

[[!redirects sheaves on a topological space]]
[[!redirects sheaves on topological spaces]]

[[!redirects category of sheaves on a locale]]
[[!redirects categories of sheaves on a locale]]

[[!redirects category of sheaves on a topological space]]
[[!redirects categories of sheaves on a topological space]]
[[!redirects categories of sheaves on topological spaces]]

[[!redirects spatial topos]]
[[!redirects spatial toposes]]
[[!redirects spatial topoi]]


sheaf semantics of concurrent interacting objects

[[!redirects Sheaf semantics of concurrent interacting objects]]
[_Sheaf Semantics for Concurrent Interacting Objects_](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.52.4296) is a paper by [[Joseph Goguen]], which formulates the behaviour of
systems of concurrent interacting objects categorically. Goguen uses [[sheaf|sheaves]] to represent the behaviours of individual objects, and [[limits]] to combine the behaviours of objects into that of the system. The approach is very general, and is not restricted to "objects" in the sense of object-oriented programming, or indeed to computation.

I intend this page to refer to Goguen's formulation and its relatives, rather than just his paper. I've given it this long title, rather than something shorter such as "Sheaf semantics", to avoid ambiguity because I suspect that sheaves have been used to describe other semantics. E.g. in topos theory?

### Relevance to computing and simulation ###

In [[A Categorical Manifesto]], Goguen writes that:
> Given a species of structure, say widgets, then the result of
> interconnecting a system of widgets to form a super-widget corresponds to
> taking the
> colimit of the diagram of widgets in which the morphisms show how
> they are interconnected.
He continues by saying that, at least for him, this dogma arose in the context of [[General Systems Theory]]. I don't know how widely recognised the use of [[colimit|colimits]] for "putting together" is, but to the extent that it is, it this paper and earlier ones on which it's based may have helped promote the idea, and thereby be of historical interest.

Moreover, if one is having trouble designing a system to describe or run simulations — whether on a computer or not — Goguen's formulation seems, to me (JNIP) at least, to offer a clean elegant starting place. Certainly cleaner than the semantics of most object-oriented programming languages.

### Abstract ###

> This paper uses concepts from sheaf theory to explain phenomena in
> concurrent systems, including object, inheritance, deadlock, and
> non-interference, as used in computer security. The approach is very
> general, and applies not only to concurrent object oriented systems, but
> also to systems of differential equations, electrical circuits, hardware
> description languages, and much more. Time can be discrete or continuous,
> linear or branching, and distribution is allowed over space as well as
> time. Concepts from category theory help to achieve this generality:
> objects are modelled by sheaves; inheritance by sheaf morphisms; systems by
> diagrams; and interconnection by diagrams of diagrams. In addition,
> behaviour is given by limit, and the result of interconnection by colimit.
> The approach is illustrated with many examples, including a semantics for a
> simple concurrent object-based programming language.

### Reference ###

Joseph A. Goguen, _Sheaf Semantics for Concurrent Interacting Objects_, in _Mathematical Structures in Computing Science_, volume 2, pages 159-91, 1992.

### Link ###

[_Sheaf Semantics for Concurrent Interacting Objects_](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.52.4296). CiteSeerX page.


sheaf toposes are equivalently the left exact reflective subcategories of presheaf toposes

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

#Contents#
* table of contents
{:toc}

## Statement

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

Let $(\mathcal{C}, \tau)$ be a [[small site|small]] [[site]].
Then the [[full subcategory]] inclusion $i \colon Sh(\mathcal{C},\tau) \hookrightarrow PSh(\mathcal{C})$ of its [[category of sheaves]] ([[Grothendieck topos]]) into its [[category of presheaves]] is a [[reflective subcategory]] inclusion

$$
  Sh(\mathcal{C},\tau)
   \underoverset
     {\underset{i}{\hookrightarrow}}
     {\overset{L}{\longleftarrow}}
     {\bot}
  PSh(\mathcal{C})
$$

such that the reflector $L \colon PSh(\mathcal{C})  \to Sh(\mathcal{C})$ [[preserved limit|preserves]] [[finite limits]] (the reflector being [[sheafification]]).

Moreover, up to [[equivalence of categories|equivalence]], every [[Grothendieck topos]] arises this way: Given a [[small category]] $\mathcal{C}$ there is a [[bijection]] between

1. the [[equivalence classes]] of left exact reflective subcategories $\mathcal{E} \hookrightarrow PSh(\mathcal{C})$ of the [[category of presheaves]]

1. [[Grothendieck topologies]]  $\tau$ on $\mathcal{C}$,

which is such that $\mathcal{E} \simeq Sh(\mathcal{C}, \tau)$.

=--

(e.g. [Borceux 94, prop. 3.5.4, cor. 3.5.5](#Borceux94), [Johnstone, C.2.1.11](#Johnstone), or [Kelly](#Kelly)).

+-- {: .num_prop #AccessibleReflection}
###### Proposition
**(accessible embedding is implied)**

In the situation of prop. \ref{GrothendieckToposesAreLexReflectionsOfPresheafToposes}
it follows that the inclusion $i \colon Sh(\mathcal{C},\tau) \hookrightarrow PSh(\mathcal{C})$ is an [[accessible functor]], hence an [accessible reflective subcategory inclusion](reflective+subcategory#AccessibleReflectiveSubcategories).

=--

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

Every  [[Grothendieck topos]] like $Sh(\mathcal{C}, \tau)$ and $PSh(\mathcal{C})$ is a [[locally presentable category]] ([this prop.](Grothendieck+topos#LocallyPresentable), [Borceux 94, vol 3, prop. 3.4.16](#Borceux94)). Therefore with prop. \ref{GrothendieckToposesAreLexReflectionsOfPresheafToposes} the statement follows by the [[adjoint functor theorem]] for locally presentable categories ([this prop.](adjoint+functor+theorem#AdjFuncTheoremForLocallyPresentableCats)).

=--

+-- {: .num_remark #GeneralizationToInfinityToposes}
###### Remark
**(generalization to [[(∞,1)-toposes]])**

For [[(∞,1)-toposes]] the accessibility of the reflection is no longer implied in general, contrary to prop. \ref{AccessibleReflection} above, but needs to be required. It is however still implied for [[topological localizations]] ([Lurie, Cor. 6.2.1.6](#Lurie)).

=--

## References

* {#Borceux94} [[Francis Borceux]], vol. 3, section 3.5 of _[[Handbook of Categorical Algebra]]_, Cambridge University Press (1994)

* {#Johnstone} [[Peter Johnstone]], _[[Sketches of an Elephant]]_

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

* {#Kelly} [[Shane Kelly]], _What is the relationship between Grothendieck pretopologies and Grothendieck topologies?_, 2022 ([web] (https://www.ms.u-tokyo.ac.jp/~kelly/pdfs/grothendieckTop.pdf)) .

[[!redirects sheafification preserves finite limits]]


sheaf with transfer

[[!redirects Nisnevich sheaves with transfer]]

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Motivic cohomology
+--{: .hide}
[[!include motivic cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given some [[category]] ([[site]]) $S$ of test spaces, suppose one fixes some  category $Corr_p(S)$ of [[correspondences]] in $S$ equipped with certain cohomological data on their correspondence space. Then a _sheaf with transfer_ on $S$ is a contravariant functor on $Corr_p(S)$ such that the restriction along the canonical embedding $S \to Corr_p(S)$ makes the resulting [[presheaf]] a [[sheaf]].

Traditionally this is considered for $S$ the [[Nisnevich site]] and $Corr_p(S)$ constructed from correspondences equipped with algebraic cycles as discussed at _[[pure motive]]_, (e.g. [Voevodsky 00, 2.1 and def. 3.1.1](#Voevodsky00)).

The idea is that, looking at it the other way around, the extension of a sheaf to a sheaf with transfer defines a kind of [[Umkehr map]]/[[fiber integration]] by which the sheaf is not only pulled back along maps, but also pushed forward, hence "transferred" (this concept of course makes sense rather generally in [[cohomology]], see e.g. [Piacenza 84, 1.1](#Piacenza84)).

The [[derived categories]] of those [[abelian sheaves]] with transfers for the [[Nisnevich site]] with are [[A1-homotopy theory|A1-homotopy invariant]] provides a model for [[motive|motives]] known as _[Voevodsky motives](motive#VoevodskyMotives)_ or similar ([Voevodsky 00, p. 20](#Voevodsky00)).

## Definition

On this page we construct the category of Nisnevich sheaves with transfer over a base scheme $S$ which is assumed to be [[regular scheme|regular]] and [[noetherian scheme|noetherian]], following [Déglise](#Deglise).  Nisnevich sheaves with transfer play an important role in the theory of [[mixed motives]].

1.  Let $S$ be a base [[scheme]] which is [[regular scheme|regular]] and [[noetherian scheme|noetherian]].  Let $Sm_S$ denote the category of [[schemes]] that are [[smooth]] and of [[finite type]] over $S$.  For $X,Y \in \Sm_S$, one defines the group of **finite $S$-[[correspondences]]** $C_S(X, Y)$ as the [[free abelian group]] generated by closed integral subschemes $Z \subset X \times_S Y$ such that the projection $Z \to X$ is [[finite]] and [[equidimensional morphism|equidimensional]].  This defines an [[additive category]] $Sm_S^{cor}$ whose objects are the same as $Sm_S$ and whose morphisms are finite $S$-correspondences.

2.  Note that there is a canonical functor $\gamma_S : \Sm_S \to \Sm_S^cor$ which acts on a [[morphism]] by taking the finite correspondence induced by its [[graph]].

3.  Recall that the [[fibred product]] over $S$ defines a [[symmetric monoidal structure]] on $\Sm_S$.  One can check that this also induces a [[symmetric monoidal structure]] on $\Sm_S^{cor}$.  The functor $\gamma_S$ respects these structures.

4.  Let $P_S$ (resp. $P_S^{tr}$) denote the category of [[abelian presheaves]] on $\Sm_S$ (resp. on $\Sm_S^{cor}$).  Objects of $P_S^{tr}$ are called **presheaves with transfer** on $S$.  We will write $L_S[X]$ for the [[presheaf]] with transfers [[representable|represented]] by an object $X \in \Sm_S$.

5.  Let $N_S$ denote the category of [[sheaves]] on $\Sm_S$ with respect to the [[Nisnevich topology]].  Let $N_S^{tr} \subset P_S^{tr}$ be the [[full subcategory]] spanned by [[presheaves]] $F : \Sm_S^{cor} \to Ab$ whose [[composition]] with $\gamma_S : \Sm_S \to \Sm_S^{cor}$ is a [[sheaf]] on $Sm_S$ (with respect to the [[Nisnevich topology]]).  Objects of $N_S^{tr}$ are called **Nisnevich sheaves with transfer**.

6.  **Proposition.** The category $N_S^{tr}$ is [[cocomplete]] and [[Grothendieck abelian category|Grothendieck abelian]] with an [[essentially small]] set of [[generators]] given by the objects $L_S[X]$ for $X \in \Sm_S$.

7.  This allows one to define a [[symmetric monoidal structure]] on $N_S^{tr}$, which is in fact the unique one for which the functor $\Sm_S \to \N_S^{tr}$ is [[symmetric monoidal]] and $- \otimes_S^{tr} -$ preserves [[colimits]].  Given $F$ and $G$ in $N_S^{tr}$, we can write
  $$ F = \colim_{X \to F} L_S[X] ; \qquad G = \colim_{Y \to G} L_S[Y] $$
where the [[colimits]] are taken over the family of morphisms $L_S[X] \to F$ (resp. $L_S[Y] \to G$.  Then we define
  $$ F \otimes_S^{tr} G = \colim_{X \to F ; Y \to G} L_S[X \times_S Y]. $$

8.  This [[symmetric monoidal structure]] is further [[closed monoidal structure|closed]]: one defines
  $$ \underline{\Hom}_{N_S^{tr}} (F,G) (X) = \Hom_{N_S^{tr}} (F \otimes_S^{tr} L_S[X], G) $$
for all $X \in \Sm_S$.

## Related concepts

* [[Becker-Gottlieb transfer]], [[transfer context]]

* [[Mackey functor]]

## References

* {#Deglise} [[Frédéric Déglise]], _Finite correspondences and transfers over a regular base_, [pdf](http://www.math.uiuc.edu/K-theory/0765/regular_base.pdf).

* {#Voevodsky00} [[Vladimir Voevodsky]], section 3.1 of _Triangulated categories of motives over a field_, K-Theory, 74 (2000) ([pdf](http://math.illinois.edu/K-theory/0368/s5.pdf), [web](http://www.math.uiuc.edu/K-theory/0074/))

* {#Levine06} [[Marc Levine]], section 3.2 of _Six lectures on motives_, ICTP lectures 2006 ([pdf](http://users.ictp.it/~pub_off/lectures/lns023/Levine/Levine.pdf))

Relation to [[Mackey functors]]:

* {#KahnYamazaki11} [[Bruno Kahn]], Takao Yamazaki, _Voevodsky's motives and Weil reciprocity_, Duke Mathematical Journal 162, 14 (2013) 2751-2796 ([arXiv:1108.2764](http://arxiv.org/abs/1108.2764))

In the more general context of [[abelian sheaf cohomology]] a kind of "transfer" is discussed in

* {#Piacenza84} [[Robert Piacenza]], _Transfer in generalized sheaf cohomology_, Proceedings of the AMS, Volume 90, Number 4 (1984)

[[!redirects sheaves with transfer]]
[[!redirects sheaf with transfer]]

[[!redirects sheaves with transfers]]
[[!redirects sheaf with transfers]]

[[!redirects presheaves with transfer]]
[[!redirects presheaf with transfer]]

[[!redirects presheaves with transfers]]
[[!redirects presheaf with transfers]]

[[!redirects Nisnevich sheaf with transfer]]


sheafification

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

#Contents#
* table of contents
{:toc}

## Idea

Given a [[site]] $C$, the _sheafification_ [[functor]] universally turns [[presheaves]] on $C$ into [[sheaves]].

It is characterized as being the [[left adjoint|left]] [[adjoint functor]] $L : PSh(C) \to Sh(C)$ to the inclusion $Sh(C) \hookrightarrow PSh(C)$ of sheaves into all presheaves, exhibiting this as a [[reflective subcategory]].

Therefore sheafification is a special case of a very general phenonemon of [[localization|localizations]] of categories. See [[category of sheaves]] for more.

## Definition

Let $(C,J)$ be a [[site]] in the sense of: [[small category]] equipped with a [[coverage]]. Write $PSh(C)$ for the [[category of presheaves]] on $C$ and

$$
  j : C \to PSh(C)
$$

for the [[Yoneda embedding]]. Write

$$
  i :  Sh_J(C) \hookrightarrow PSh(C)
$$

for the [[category of sheaves]]: the [[full subcategory]] on those presheaves that are $J$-[[sheaves]].

### Existence

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

The inclusion of sheaves into presheaves admits a [[left adjoint|left]] [[adjoint functor]], which hence exhibits $Sh(C)$ as a [[reflective subcategory]] or reflective [[localization]] of $PSh(C)$:

$$
  (L \dashv i) : Sh_J(C) \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow}
   PSh(C)
  \,.
$$

=--

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

This follows by general properties discussed at [[reflective subcategory]]. We spell out the argument using the theory of [[localization]] at a set of morphisms satisfying a [[calculus of fractions]].

Recall from the discussion at [[sheaf]] that $Sh_J(C)$ is by definition the [[full subcategory]] of $PSh(C)$ on the [[local object|local objects]] with respect to the morphisms

$$
  W =
  \left\{
  S(\{U_i\}) := \lim_\to\left(
    \coprod_{i,j} j(U_i) \times_{j(U)} j(U_j) \stackrel{\to}{\to} \coprod_i j(U_i) \right) \to j(U)
   |
   \;\;
  \{U_i \to U\}_{i \in I} \in J
  \right\}
$$

of [[sieve]] inclusions for all [[covering]] families of the [[coverage]] $J$. (Here the [[colimit]] $\lim_\to$ is the [[coequalizer]] of the two injection maps, as indicated. This is spelled out in more detail at [[sheaf]] and at [[sieve]].)

Now we invoke the following results:

1. The localization proposition says that every [[full subcategory]] of a [[locally presentable category]] on the $W$-[[local object|local objects]] for a [[small set]] $W$ of morphisms is a [[reflective subcategory]], given by the [[localization]] at these morphisms;

1. By Gabriel-Zisman's theorem every reflective subcategory is the [[localization]] at the collection of morphisms inverted by the left adjoint (which by the localization proposition is the saturation of the original set of morphisms).

1. If $W$ satisfies the axioms of a [[calculus of fractions]] then, by the discussion there, this localization is equivalently given by the category $PSh(C)[W^{-1}]$ whose objects are those of $PSh(C)$ and whose morphisms are given by $PSh(C)[W^{-1}](X,A) \simeq {\lim_{\to}}_{\hat X  \stackrel{w \in W}{\to} X} PSh_C(\hat X,A)$.

Notice that an object is a [[local object]] with respect to the above set of  morphisms $W$ precisely if it is local with respect to the set of all [[small set|small]] [[colimit|colimits]] (in the [[arrow category]] $Arr(PSh(C))$ ) of such morphims (since the [[hom-functor]] $PSh_C(-,A)$ sends colimits in the first argument to [[limit|limits]], and a limit of [[isomorphism|isomorphisms]] is an isomorphism).

Let hence $\bar W$ be the completion of $A$ under forming small colimits in $Arr(PSh(C))$.

We claim that the morphisms in $\bar W$ form a [[calculus of fractions]]. The first condition to check is that for all morphisms of presheaves $X \to j(U)$ and every covering family $\{U_i \to U\}$ there is a morphism $Y \to X$ in $\bar W$ and a [[commuting diagram]]

$$
  \array{
    Y &\to& S(\{U_i\})
    \\
    \downarrow && \downarrow^{\mathrlap{s}}
    \\
    X &\to& j(U)
  }
$$

in $PSh(C)$. (It is sufficient to demand this for $s \in W \subset \bar W$ to deduce the stability conditions for all morphisms in $\bar W$, since by [[universal colimits]] in the [[presheaf topos]] $PSh(C)$ the pullback of a colimit is a colimit of pullbacks.)

Similarly, to see that we can find $Y \to X$ we use the [[co-Yoneda lemma]] to decompose $X$ as a [[colimit]] of [[representable functor|representables]] $X \simeq {\lim_{\to}_j} K_j$ and then use [[universal colimits]] to deduce that we are looking at a diagram of the form

$$
  \array{
    {\lim_{\to}}_r s^* K_r &\to& S(\{U_i\})
    \\
    \downarrow && \downarrow^{\mathrlap{s}}
    \\
    {\lim_\to}_r j(K_r) &\to& j(U)
  }
  \,.
$$

Since $\bar W$ is closed under colimits, it is hence sufficient that we show the stability condition for $X$ any representable. So we need to fill diagrams of the form

$$
  \array{
    Y &\to& S(\{U_i\})
    \\
    \downarrow && \downarrow^{\mathrlap{s}}
    \\
    j(K) &\to& j(U)
  }
  \,.
$$

For this now use the single condition on a [[coverage]]: that for $\{U_i \to U\}$ a covering family in the [[site]] $(C,J)$ we can find a covering family $\{K_j \to K\}$ such that every $K_j \to K$ factors through one of the $U_i \to U$. But this means that also the [[sieve|sieves]] factor, and we have a commuting diagram

$$
  \array{
    S(\{K_j\}) &\to& S(\{U_i\})
    \\
    \downarrow && \downarrow
    \\
    j(K) &\to& j(U)
  }
  \,.
$$

This shows that $\bar W$ satisfies the first condition at [[factorization system]].

The second condition at [[calculus of fractions]] demands that if two composites of the form

$$
   X \stackrel{\to}{\to} S(\{U_i\}) \to j(U)
$$

are equal in $PSh(C)$, then there is a morphism $Y \to X$ in $PSh(C)$ such that the two composites

$$
  Y \to X \stackrel{\to}{\to} S(\{U_i\})
$$

are equal. But the [[sieve]] inclusions are [[monomorphism|monomorphisms]], hence this condition is trivially satisfied (choose $Y \to X$ to be the [[identity]] on $X$). Again by decomposing into colimits by the [[co-Yoneda lemma]] and using [[universal colimits]] and the pasting law for [[pullback|pullbacks]], the same follows for general morphisms in $\bar W$

$$
  \array{
    {\lim_\to}_k X_k &\stackrel{\simeq}{\to}& X
    \\
    \downarrow \downarrow && \downarrow \downarrow
    \\
    {\lim_\to}_k S(U_{k,i}) &\to& Y
    \\
    \downarrow && \downarrow
    \\
    {\lim_\to}_k j(U_k) &\stackrel{\simeq}{\to}& j(U)
  }
  \,.
$$

by applying the above on each component $k$ of the colimit.

This gives us the [[localization]] $L : PSh(C) \to Sh_J(C)$ as described at [[calculus of fractions]]. By the discussion there
we have that $Sh_J(C)$ is equivalently given by the category $PSh(C)[W^{-1}]$ with the same objects as $PSh(C)$ and [[hom-set|hom-sets]] given by

$$
  PSh(C)[W^{-1}](X,A)
   \simeq
 {\lim_\to}_{\{\hat X \stackrel{w \in \hat W}{\to} X\}}
  PSh_C(\hat X,A)
  \,.
$$

So we have that for $X \in PSh(C)$ a presheaf and $A \in Sh_J(C)$ the [[hom-set]] $Sh_C(L(X),A)$ is given by

$$
  Sh_C(L(X), A) \simeq
  {\lim_{\to}}_{\hat X \stackrel{w \in \bar W}{\to} X} PSh_C(\hat X,A)
  \,.
$$

But if $A$ is a sheaf, it is a $\bar W$-[[local object]] and hence $PSh_C(\hat X \stackrel{w}{\to} X, A)$ is an [[isomorphism]] for all $w \in \bar W$. Hence the above colimit is over a diagram constant on its value at $w = Id : X \to X$ therefore we have a [[natural isomorphism]]

$$
  \cdots \simeq
  PSh_C(X,A)
  \,.
$$

This demonstrates the [[adjunction]] $(L \dashv i)$.

=--

### Construction

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

For $X \in PSh(C)$ a [[presheaf]] on the [[site]] $(C,J)$, its **sheafification** $L(X)$ is the presheaf given on any $U \in C$ by

$$
  L(X) : U \mapsto {\lim_\to}_{\{\hat U \stackrel{w}{\to} j(U)\}} PSh_C(\hat U, X)
  \,,
$$

where the [[colimit]] on the right is over all $w \in \bar W$.

=--

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

By the [[Yoneda lemma]] we have

$$
  L(X)(U) \simeq PSh_C(j(U), L(X))
  \,.
$$

By the [above proposition](#ExistenceOfLeftAdjoint) this is

$$
  \cdots \simeq Sh_C(L(j(U)), L(X))
  \,.
$$

By the proof of the above proposition, using the formulas discussed at [[calculus of fractions]], this hom-set is given by

$$
  \cdots \simeq {\lim_\to}_{\{ \hat U \stackrel{w \in \bar W}{\to} j(U)\}} PSh_C(\hat U, X)
  \,.
$$

=--

\begin{remark}

By the definition of $\bar W$, the morphisms $\hat U \to j(U)$ in $\bar W$ are colimits of diagrams of [[covering]] [[sieve|sieves]]

$$
  \hat U \simeq
    {\lim_\to}_{\{K \to U\}} S(\{K_i\}) \to {\lim_\to}_{\{K \to U\}} j(K)
  \,.
$$

This means (...) that the above colimit may be computed as two consecutive colimits of the form

$$
  {\lim_{\to}}_{\{S(\{U_i\}) \to j(U)\}} PSh_C(S(\{U_i\}), X)
  \,.
$$

One such application is called the [[plus construction]].

\end{remark}

\begin{proposition}

A morphism $S(\{U_i\}) \to X$ out of a [[sieve]] into any presheaf is in components precisely a [[matching family]] of the presheaf $X$ on the [[covering]] $\{U_i \to U\}$.

\end{proposition}

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

Use that the sieve is the [[coequalizer]]

$$
  S(\{U_i\}) \to \coprod_i j(U_i) \stackrel{\to}{\to}
   \coprod_{i,j} j(U_i) \times_{j(U)} j(U_j)
$$

and that the [[hom-functor]] $PSh_C(-,X)$ sends [[colimit|colimits]] to [[limit|limits]]. More details on this computation are at [[sheaf]].

=--

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

The [[unit of an adjunction|unit]] of the $(L \dashv i)$-[[adjunction]] has as components natural morphisms

$$
  X \to L X
$$

in $PSh(C)$, from any presheaf into its sheafification. By general properties of [[reflective subcategories]] these morphisms are mapped to [[isomorphism|isomorphisms]] by $L : PSh(C) \to Sh(C)$. Therefore these are [[local isomorphism|local isomorphisms]].

So every presheaf is related by a [[local isomorphism]] to its sheafification.

=--

## For sheaves with values in categories other than $Set$

For presheaves with values in categories other than
[[Set]], sheafification may be a difficult problem, unless one has some extra assumptions.

### With values in models for finite-limit theories ###

Consider a type of structure $T$ defined in terms of an [[essentially algebraic theory]] finite limits (such as [[group|groups]], [[algebra|algebras]], [[module|modules]], etc.), then [[internal logic|internal]] $T$-[[algebra over an algebraic theory|models]] are preserved by both [[direct image|direct images]] and [[inverse image|inverse images]] of [[geometric morphism|geometric morphisms]].  Therefore, the [[adjunction]] $(L \dashv i) : Sh_J(C) \to PSh(C)$  directly induces an adjunction between $T$-models in sheaves and presheaves.  And since finite limits of sheaves and presheaves are computed pointwise, $T$-models in the category of (pre)sheaves are the same as (pre)sheaves of $T$-models-in-$Set$.

### Using the IPC-property

If a [[category]] $A$ satisfies the following assumptions, sheafification of presheaves in $[S^{op}, A]$ exists and is constructed analogously as for [[Set]]-valued sheaves.

* $A$ admits small [[limit|limits]];
* $A$ admits small [[colimit|colimits]];
* small [[filtered category|filtered colimits]] in $A$ are exact;
* $A$ satisfies the [[IPC-property]] .

This is true for instance for

* the category [[Set]] of sets;

* the category [[Grp]] of groups;

* the category $k Alg$ of $k$-algebras;

* the category $Mod(R)$ of [[module|modules]],

(but all of these are also $T$-models for finite-limit theories $T$).

+-- {.query}
One should say more: there are so many applications
and fairly difficult theorems there; for example van
Osdol's  work.

[[Mike Shulman|Mike]]: Another way to think about this is: if you have a type of structure $T$ defined in terms of finite limits (such as groups, algebras, modules, etc.), then [[internal logic|internal]] $T$-models are preserved by both direct and inverse images of geometric morphisms.  Therefore, the adjunction between sheaves and presheaves of sets directly induces an adjunction between $T$-models in sheaves and presheaves.  And since finite limits of sheaves and presheaves are computed pointwise, $T$-models in the category of (pre)sheaves are the same as (pre)sheaves of $T$-models-in-$Set$.

If $T$ is _not_ defined in terms of finite limits, then internal $T$-models in sheaves need not be the same as sheaves of $T$-models-in-$Set$.  My intuition would be that the former, rather than the latter, is the more interesting and important notion.  For instance, a local ring in a topos of sheaves is a sheaf of rings whose _stalks_ are local, rather than a sheaf taking values in the category of local rings, and this is usually what people care about.  But since people have studied the other version, there must be important examples of it as well?

[[Urs Schreiber|Urs]]: okay, I have added this
to the above now -- so is the IPC-property business
really unnecessary for the examples above?

[[Mike Shulman|Mike]]: I'm pretty sure it is not.  Does anyone have any examples where the IPC-property business is important?
=--

## Related concepts

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

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

## References

The description of sheafification in terms of [[local isomorphism|local isomorphisms]] is in section 16.3 (for [[Set]]-valued presheaves) and section 17.4 (for more general presheaves) of

* Kashiwara--Schapira, _[[Categories and Sheaves]]_

The description in terms of [[dense monomorphism|dense monomorphisms]] using [[Lawvere-Tierney topology]] is in section V.3 of

* [[Saunders Mac Lane]] and [[Ieke Moerdijk]], _[[Sheaves in Geometry and Logic]]_

Extension of sheafification of presheaves with values in other categories has been advanced in

* [[Alex Heller]], K. A. Rowe, _On the category of sheaves_, Amer. J. Math. 84, 1962, 205-216

* Barr, Grillet, and Van Osdol, _Exact categories and categories of sheaves_, Lecture Notes in Math., Vol. 236, Springer, Berlin, 1971

* Friedrich Ulmer, _On the existence and exactness of the associated sheaf functor_, J. Pure Appl. Algebra 3, 1971, 295-306

* [[Alexander Rosenberg]], _Almost quotient categories, sheaves and localizations_, 181 p. Seminar on supermanifolds 25, University of Stockholm, D. Leites editor, 1988 (in Russian; partial remake in English exists)

Discussion in [[homotopy type theory]] is in

* {#QuirinTabareau16} Kevin Quirin, Nicolas Tabareau, _Lawvere-Tierney sheafification in Homotopy Type Theory_, Journal of Formalized Reasoning, Vol 9, No 2, (2016) ([web](https://jfr.unibo.it/article/view/6232))

category: sheaf theory
[[!redirects sheafification functor]]
[[!redirects associated sheaf]]
[[!redirects associated sheaves]]
[[!redirects associated sheaf functor]]


sheafification in a Lawvere-Tierney topos > history

< [[sheafification in a Lawvere-Tierney topos]]
[[!redirects sheafification in a Lawvere-Tierney topos -- history]]


shear map

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

#Contents#
* table of contents
{:toc}

## Definition

Given a [[group]] $G$ (generally: a [[group object]] in some ambient [[category]] $\mathcal{C}$, and a [[quasigroup]]-object suffices) and a [[group action]] (generally: an [[action object]] in $\mathcal{C}$)

\[
  \label{ActionMorphism}
  G \times A \overset{\rho}{\longrightarrow} A
\]

the *shear map* is the [[morphism]]

$$
  \array{
    G \times A
    &\overset{ (\rho,pr_2) }{\longrightarrow}&
    A \times A
    \\
    (g,a) &\mapsto& \big( \rho(g)(a), a \big)
    \,.
  }
$$

form the [[Cartesian product]] of (the [[objects]] underlying) $G$ and $A$ to that of $A$ with itself, whose first component is the [[action]] morphism (eq:ActionMorphism) and whose second component is the [[projection]] onto the second factor (or the other way around, equivalently).

The action $(A,\rho)$ is called:

* a *[[free action]]* if the shear map is a [[monomorphism]],

* a *[[transitive action]]* if the shear map is an [[epimorphism]],

* a *[[regular action]]* or *$G$-[[torsor]]* if the shear map is an [[isomorphism]] (or rather a *[[pseudo-torsor]]* if $A$ is not required to be [[inhabited]]).

Often this is considered in the case that:

1. $\mathcal{C}$ is a [[slice category]] over an object $X$,

1. $G$ is a [[trivial bundle]] of groups over $X$, then still denoted $G$

in which case

1. $A = (P \overset{p}{\longrightarrow} X)$ is a [[bundle]] over $X$,

1. $A \times A \,=\, P \times_X P$ is the [[fiber product]] over $X$,

1. $\rho$ is a [[fiber]]-wise [[action]],

and so in which case the shear map, seen as a morphism in $\mathcal{C}$, reads as follows:

\[
  \label{ShearMapForBundles}
  \array{
    G \times P
    &\overset{ (\rho, pr_2) }{\longrightarrow}&
    P \times_X P
    \\
    (g,p) &\mapsto& \big( \rho(g)(p), p \big)
    \,.
  }
\]

Here $P$ with this action is called a *$G$-[[principal bundle]]* (not necessarily [[locally trivial]]) if the shear map is an [[isomorphism]], or rather a *[[formally principal bundle]]* if $P$ is allowed to be an [[empty bundle]].

Notice that this condition (eq:ShearMapForBundles) is equivalent to the condition that we have a [[pullback square]] as follows:

$$
  \array{
     G \times P
     &\overset{\rho}{\longrightarrow}&
     P
     \\
     {}^{\mathllap{pr_2}}
     \big\downarrow
     &{}^{{}_{(pb)}}&
     \big\downarrow {}^{\mathrlap{p}}
     \\
     P
     &\underset{p}{\longrightarrow}&
     X
     \mathrlap{\,,}
  }
$$

because the shear map (eq:ShearMapForBundles) is the universal comparison morphism induced from the [[commuting square|commutativity]] of this square to the manifest [[fiber product]] pullback.

## Related concepts

* [[group action]]

* [[torsor]], [[pseudo torsor]]

* [[principal bundle]]

## References

Early explicit appearance of the shear map, alongside discussion of its [[isomorphism|isomorphy]] ([[pseudo-torsor]]-condition):

* {#Grothendieck60} [[Alexander Grothendieck]], p. 312 (15 of 30) in: [[FGA]] *Technique de descente et théorèmes d’existence engéométrie algébrique. I. Généralités. Descente par morphismes fidèlement plats*, Séminaire N. Bourbaki, 1960, exp. no190, p. 299-327 ([numdam:SB_1958-1960__5__299_0](http://www.numdam.org/item/?id=SB_1958-1960__5__299_0))

* {#Grothendieck61} [[Alexander Grothendieck]], p. 105 (8 of 21) in: [[FGA ]] _Techniques de construction et théorèmes d'existence en géométrie algébrique III : préschémas quotients_, Séminaire Bourbaki : années 1960/61, exposés 205-222, Séminaire Bourbaki, no. 6 (1961), Exposé no. 212,   ([numdam:SB_1960-1961__6__99_0](http://www.numdam.org/item/?id=SB_1960-1961__6__99_0), [pdf](http://www.numdam.org/item/SB_1960-1961__6__99_0.pdf), [English transl. pdf](https://labs.thosgood.com/builds/fga-3-III.pdf))

* {#Grothendieck67} [[Alexander Grothendieck]], 16.5.15 in: *[[Éléments de géométrie algébrique]]* : IV. *Étude locale des schémas et des morphismes de schémas (Quatrième partie)*, Publications mathématiques de l’I.H.É.S., tome  32 (1967), p. 5-361 ([numdam:PMIHES_1967__32__5_0](http://www.numdam.org/item/PMIHES_1967__32__5_0))

* {#Grothendieck71} [[Alexander Grothendieck]], p. 9 of _Exemples et Complements_ ([doi:10.1007/BFb0058666](https://link.springer.com/chapter/10.1007/BFb0058666), [pdf](https://link.springer.com/content/pdf/10.1007%2FBFb0058666.pdf)), which is p. 293 in:

  [[Alexander Grothendieck]], [[Michèle Raynaud]], *Revêtements étales et groupe fondamental* ([[SGA 1]]),  Lecture Notes in Mathematics **224**, Springer 1971 ([arXiv:math/0206203](https://arxiv.org/abs/math/0206203), [doi:10.1007/BFb0058656](https://link.springer.com/book/10.1007/BFb0058656))

[[!redirects shear maps]]

[[!redirects shearing map]]
[[!redirects shearing maps]]


Sheaves in Geometry and Logic

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

This entry collects hyperlinks related to the textbook

* [[Saunders Mac Lane]], [[Ieke Moerdijk]]:

  \linebreak

  **Sheaves in Geometry and Logic**

  *A first introduction to topos theory*

  \linebreak

  Springer (1992)

   [doi:10.1007/978-1-4612-0927-0](https://link.springer.com/book/10.1007/978-1-4612-0927-0)

on [[sheaf and topos theory]] and its application in [[categorical logic]].

For a similar link lists see also

* _[[Categories and Sheaves]]_

* _[[Sketches of an Elephant]]_

# Contents #
* table of contents
{:toc}

## Categorical Preliminaries

## I Categories of Functors

* [[presheaf]]

* [[category of presheaves]]

## II Sheaves of Sets

### 1. Sheaves

### 2. Sieves and Sheaves

* [[category of open subsets]]

* [[sieve]]

* [[sheaf]]

* [[global section]]/[[constant sheaf]]

### 3. Sheaves and Manifolds

### 4. Bundles

* [[bundle]]

### 5. Sheaves and Cross-Sections

### 6. Sheaves as Étale spaces

* [[etale space]]

### 7. Sheaves with algebraic structure

* [[internalization]]

* [[group object]]

### 8. Sheaves are Typical

### 9. Inverse Image Sheaf

##III Grothendieck Topologies and Sheaves

* [[presheaf]]

* [[site]]

  * [[coverage]]

  * [[basis for a topology]]

  * [[Grothendieck topology]]

  * [[sheaf]]

    * [[sheafification]]

    * [[direct image]]

    * [[inverse image]]

    * [[restriction and extension of sheaves]]

    * [[stalk]]

  * [[category of sheaves]]

* [[Grothendieck topos]]

##IV First Properties of Elementary Topoi

### 1. Definition of a topos

* [[subobject classifier]]

  * [[subobject]]

* [[topos]]

  * [[Grothendieck topos]]

  * [[well-pointed topos]]

  * [[Boolean topos]]

  * [[pretopos]]

  * [[universe in a topos]]

* [[point of a topos]]

* [[fundamental group of a topos]]

* [[dependent product]]

### 2. The construction of exponentials

* [[exponential object]]

### 3. Direct image

* [[direct image]]

### 4. Monads and Beck's theorem

* [[monad]]

* [[Eilenberg-Moore category]]

* [[monadic functor]]

* [[monadicity theorem]]

### 5. The construction of colimits

* [[quotient type]]

### 6. Factorization and images

* [[image]]

### 7. The slice category as a topos

* [[over-topos]]

### 8. Lattice and Heyting algebra objects in a topos

* [[lattice]]

* [[Heyting algebra]]

### 9. The Beck-Chevalley condition

* [[Beck-Chevalley condition]]

### 10. Injective objects

* [[injective objects]]

## V Basic Constructions of Topoi

### 1. Lawvere-Tierney topologies

* [[Lawvere-Tierney topology]]

### 2. Sheaves

* [[dense monomorphism]]

* [[sheaf]]

### 3. The associated sheaf functor

* [[sheafification]]

### 4. Lawvere-Tierney subsumes Grothendieck

* [[Grothendieck topology]]

### 5. Internal versus external

### 6. Group actions

* [[group]], [[action]]

* [[group object]], [[action object]]

* [[permutation representation]]

### 7. Category actions

### 8. The topos of coalgebras

* [[comonad]]

* [[topos of coalgebras]]

### 9. The filter-quotient construction

##VI Topoi and Logic

* [[ETCS]]

* [[continuum hypothesis]]

* [[logic]]

* [[foundations and logic]]

  * [[internal logic]]

    * [[intuitionistic logic]]

    * [[classical logic]]

    * [[paraconsistent logic]]

    * [[coherent logic]]

* [[Boolean algebra]]

* [[Heyting algebra]]

## VII Geometric Morphisms

### VII 1. Geometric Morphisms and Basic Examples

* [[geometric morphism]]

  * [[direct image]], [[inverse image]]

Examples:

* [[continuous function]]

* [[base change geometric morphism]]

* [[sheafification]], [[geometric embedding]]

* [[global section geometric morphism]]

### VII 2. Tensor products

* [[Kan extension]]

### VII 3. Group actions

* [[group object]]

### VII 4. Embeddings and surjections

* [[geometric embedding]]

* [[geometric surjection]]

* [[geometric surjection/embedding factorization]]

### VII 5. Points

* [[point of a topos]]

* [[stalk]]

### VII 6. Filtering functors

### VII 7. Morphisms into Grothendieck Topoi

### VII 8. Filtering functors into a topos

### VII 9. Geometric morphisms as filtering functors

### VII 10. Morphisms between sites

* [[morphism of sites]]

* [[covering lifting property]]

##VIII Classifying Topoi

* [[classifying topos]]

##IX Localic Topoi

* [[locale]]

* [[localic topos]]

##Geometric Logic and Classifying Topoi

* [[classifying topos]]

##Appendix: Sites for Topoi

* [[Giraud's theorem]]

category: reference

[[!redirects Sheaves in geometry and logic]]
[[!redirects sheaves in geometry and logic]]
[[!redirects Mac Lane-Moerdijk]]


sheaves on a simplicial topological space

> under construction

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

## Definition

For $X_\bullet : \Delta^{op} \to $ [[Top]] a [[simplicial object]] [[topological space]], write $Sh(X_n)$ for the [[category of sheaves]] on (the [[category of open subsets]]) $X_n$.

The [[category]] $Sh(X_\bullet)$ of sheaves on the simplicial space is defined to be the category whose

* objects are

  * collections $(S_n \in Sh(X_n))_n$

  * equipped for each $\alpha : [n] \to [m]$ with morphisms

    $$
      S(\alpha) : X(\alpha)^* S_n \to S_m
    $$

  * such that

    * $S(Id_{[n]}) = Id_{S_n}$;

    * for every $\alpha : [n] \to [m]$ and $\beta : [m] \to [k]$
      the diagram

      $$
        \array{
          X(\beta)^* X(\alpha)^* S_n
          &\stackrel{X(\beta)^* S(\alpha)}{\to}&
          X(\beta)^* S_m
          \\
          {\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{S(\beta)}}
          \\
          X(\beta \alpha)^* S_m &\underset{S(\beta \alpha)}{\to}&
          S_k
        }
      $$

      [[commuting diagram|commutes]];

* morphisms are collections $(S_n \to T_n)$ of morphisms of sheaves,
  compatible with all structure maps.

## Examples

For $C$ a [[topological category]] and $N C : \Delta^{op} \to Top$ its [[nerve]], $Sh(N_\bullet C)$ is the [[classifying topos]] for $C$-torsors. see [[classifying topos of a localic groupoid]].


sheaves on large sites > history

< [[sheaves on large sites]]
[[!redirects sheaves on large sites -- history]]