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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 $j(U_j) \times_{j(U)} j(U_k)$ 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)) 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) Lecture notes: * {#Warner12} [[Garth Warner]]: *Fibrations and Sheaves*, EPrint Collection, University of Washington (2012) &lbrack;[hdl:1773/20977](http://hdl.handle.net/1773/20977), [pdf](https://sites.math.washington.edu//~warner/Warner_FIBRATIONS%20AND%20SHEAVES.pdf), [[Warner-FibrationsAndSheaves.pdf:file]]&rbrack; * {#Schapira23} [[Pierre Schapira]], *An Introduction to Categories and Sheaves*, lecture notes (2023) &lbrack;[pdf](https://webusers.imj-prg.fr/~pierre.schapira/LectNotes/CatShv.pdf), [[Schapira-Sheaves.pdf:file]]&rbrack; 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 alg&#233;brique et th&#233;orie 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]] </textarea> <p> <input id="alter_title" name="alter_title" onchange="toggleVisibility();" type="checkbox" value="1" /> <label for="alter_title">Change page name.</label><br/> <span id="title_change" style="display:none"><label for="new_name">New name:</label> <input id="new_name" name="new_name" onblur="addRedirect();" type="text" value="sheaf" /></span> </p> <div> <p style="font-size: 0.8em; width: 70%;"> For non-trivial edits, please briefly describe your changes below. 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