Presheaves, Sheaves, Cosheaves, and Čech Descent

Local-to-global reasoning begins with a variance choice. A presheaf restricts data from a larger open set to a smaller one, whereas a precosheaf assembles data from a smaller open set into a larger one. A sheaf is a presheaf whose compatible local sections glue uniquely. A cosheaf is a precosheaf whose local contributions present the global object by a colimit. For ordinary open covers these conditions can be tested by an equalizer and a coequalizer, respectively.

The Čech construction records the same cover through all of its finite intersections. In an abelian target it produces a cochain complex for a presheaf and a chain complex for a precosheaf. The ordinary sheaf and cosheaf axioms determine degree zero only; higher Čech groups, comparisons with sheaf cohomology, and homotopical descent require additional hypotheses. Keeping these levels separate is essential in applications to local observables.

Required background. Categories, Functors, Natural Transformations, and Universal Properties supplies variance, functoriality, limits, and colimits.

Helpful background. Chains, Homology, Cohomology, and Exact Sequences supplies the chain and cochain language used in the later discussion of Čech complexes; it is not needed for the basic gluing definitions.

Open sets as a category

Let $X$ be a topological space. Write $\operatorname{Open}(X)$ for the poset category whose objects are open subsets of $X$ and whose unique morphism $U\to V$ exists exactly when $U\subseteq V$. Composition is inclusion. Fix a target category $\mathcal C$.

A presheaf on $X$ with values in $\mathcal C$ is a contravariant functor

$$\mathcal F:\operatorname{Open}(X)^{\mathrm{op}} \longrightarrow\mathcal C.$$

For $U\subseteq V$, write its restriction morphism as

$$\rho_U^V:\mathcal F(V)\longrightarrow\mathcal F(U).$$

Functoriality says

$$\rho_U^U=\operatorname{id}_{\mathcal F(U)}, \qquad \rho_W^V=\rho_W^U\circ\rho_U^V \quad (W\subseteq U\subseteq V). \tag{1}$$

A precosheaf is instead a covariant functor

$$\mathcal G:\operatorname{Open}(X)\longrightarrow\mathcal C.$$

For $U\subseteq V$, denote its structure morphism by

$$\iota_U^V:\mathcal G(U)\longrightarrow\mathcal G(V).$$

These assembly maps obey

$$\iota_U^U=\operatorname{id}_{\mathcal G(U)}, \qquad \iota_U^W=\iota_V^W\circ\iota_U^V \quad (U\subseteq V\subseteq W). \tag{2}$$

Thus “reverse the arrows” is not an informal slogan: the presheaf really is a functor on the opposite category, while the precosheaf is a functor on the original category. Neither functor is automatically a sheaf or cosheaf. The functorial definitions, equalizer and coequalizer forms, and Čech sign conventions used below are developed in Curry 2014, §§2.1–2.3.

Sheaf gluing is an equalizer

Let $U=\bigcup_{i\in I}U_i$ be an open cover. First suppose that $\mathcal F$ is set-valued. A family

$$s_i\in\mathcal F(U_i)$$

is compatible when every pair has the same restriction to its overlap:

$$\rho_{U_i\cap U_j}^{U_i}(s_i) = \rho_{U_i\cap U_j}^{U_j}(s_j). \tag{3}$$

The presheaf is a sheaf if every compatible family has a unique section $s\in\mathcal F(U)$ with $\rho_{U_i}^{U}(s)=s_i$ for all $i$. Existence is the gluing property; uniqueness says that local agreement detects equality of global sections. Both clauses are part of the definition.

For a category $\mathcal C$ with the required products and equalizers, this condition has a coordinate-free form. Define two arrows by restricting the $i$th or the $j$th component to $U_i\cap U_j$. Then $\mathcal F$ is a sheaf when, for every open cover, the diagram

$$\mathcal F(U) \longrightarrow \prod_{i\in I}\mathcal F(U_i) \mathrel{\substack{\longrightarrow\\[-0.6ex]\longrightarrow}} \prod_{(i,j)\in I^2}\mathcal F(U_i\cap U_j) \tag{4}$$

is an equalizer. Repeated ordered pairs in (4) are harmless; one may use a smaller equivalent diagram after ordering the cover. For compatible local sections and unique gluing, see Stacks Project Authors 2026, §6.7, tag 006S; for the equalizer formulation, see Stacks Project Authors 2026, Definition 6.9.1, tag 0072.

For a two-open cover $U=U_1\cup U_2$, equation (4) says that

$$\mathcal F(U) \cong \mathcal F(U_1) \times_{\mathcal F(U_1\cap U_2)} \mathcal F(U_2). \tag{5}$$

If $\mathcal C$ is abelian, this is equivalently exactness of

$$0\longrightarrow\mathcal F(U) \longrightarrow \mathcal F(U_1)\oplus\mathcal F(U_2) \xrightarrow{\ d\ } \mathcal F(U_1\cap U_2), \tag{6}$$

where

$$d(s_1,s_2) = \rho_{U_1\cap U_2}^{U_2}(s_2) - \rho_{U_1\cap U_2}^{U_1}(s_1).$$

There is no terminal $0$ in (6): the sheaf axiom identifies the kernel of $d$, but does not say that every section on the overlap is a difference of restrictions. The empty cover also matters. It forces $\mathcal F(\varnothing)$ to be terminal in $\mathcal C$.

A genuine sheaf and a presheaf that fails

For a topological space $Y$, let

$$\mathcal F(W)=C^0(W,Y)$$

with restriction of functions. Compatible continuous functions on an open cover define one set-theoretic function on the union; continuity is local, so that function is continuous, and its values are forced by the local functions. Hence $\mathcal F$ is a sheaf.

By contrast, fix a field $k$ and define a presheaf $\mathcal P$ by

$$\mathcal P(W)= \begin{cases} k,&W\ne\varnothing,\\ 0,&W=\varnothing, \end{cases}$$

with identity restrictions between nonempty opens and the unique map to $0$ otherwise. On the discrete space $X=\{a,b\}$, cover $X$ by its two singletons. The sections $0\in\mathcal P(\{a\})$ and $1\in\mathcal P(\{b\})$ agree after restriction to the empty overlap, but no single element of $\mathcal P(X)=k$ restricts to both. This is the presheaf of globally constant functions, with value $0$ on the empty open. Its sheafification is the sheaf of locally constant $k$-valued functions.

Cosheaf assembly is a coequalizer

Now let $\mathcal G$ be a precosheaf and suppose that $\mathcal C$ has the coproducts and coequalizers required by the cover. The precosheaf is a cosheaf if, for every $U=\bigcup_iU_i$, the diagram

$$\coprod_{(i,j)\in I^2}\mathcal G(U_i\cap U_j) \mathrel{\substack{\longrightarrow\\[-0.6ex]\longrightarrow}} \coprod_{i\in I}\mathcal G(U_i) \longrightarrow \mathcal G(U) \tag{7}$$

is a coequalizer. The parallel arrows assemble an overlap contribution first through $U_i$ or first through $U_j$. Thus $\mathcal G(U)$ is generated by the local objects with precisely the identifications forced on overlaps, in the universal sense appropriate to $\mathcal C$.

For a two-open cover, equation (7) becomes the pushout

$$\mathcal G(U_1) \coprod_{\mathcal G(U_1\cap U_2)} \mathcal G(U_2) \cong \mathcal G(U). \tag{8}$$

In an abelian category, the corresponding right-exact sequence is

$$\mathcal G(U_1\cap U_2) \xrightarrow{\ x\mapsto (-\iota_{U_1\cap U_2}^{U_1}x, \ \iota_{U_1\cap U_2}^{U_2}x)\ } \mathcal G(U_1)\oplus\mathcal G(U_2) \longrightarrow \mathcal G(U) \longrightarrow0. \tag{9}$$

Unlike sheaf gluing, cosheaf assembly does not assign a unique decomposition of a global element into local pieces. It says that all such presentations give the same morphism out of the coequalizer. The empty cover forces $\mathcal G(\varnothing)$ to be initial in $\mathcal C$.

The same object assignment used in the failed presheaf example can also fail covariantly. Give $\mathcal Q(W)=k$ for nonempty $W$ and $\mathcal Q(\varnothing)=0$ covariant identity maps. For the two-singleton cover, the canonical assembly map is the fold $k\oplus k\to k$, $(x,y)\mapsto x+y$, rather than an isomorphism. Thus $\mathcal Q$ is a precosheaf but not a cosheaf.

A finite calculation in both variances

Let $X=\{a,b,c\}$ be discrete, let $U=\{a,b\}$, and let $V=\{b,c\}$. Their intersection is $\{b\}$ and their union is $X$.

For the sheaf $\mathcal F(W)=\operatorname{Map}(W,k)$,

$$\mathcal F(U)\times\mathcal F(V) \cong k^2\times k^2.$$

Write a pair as $((\alpha,\beta),(\beta',\gamma))$. The equalizer condition is $\beta=\beta'$, so restriction identifies $\mathcal F(X)$ with

$$\left\{ ((\alpha,\beta),(\beta',\gamma)): \beta=\beta' \right\} \cong k^3.$$

For instance, $(1,2)$ on $U$ and $(2,5)$ on $V$ glue uniquely to $(1,2,5)$ on $X$.

For the covariant assignment $\mathcal G(W)=k[W]$, the free vector space on the points of $W$, inclusions of sets induce inclusions of basis vectors. The cosheaf sequence is

$$k\langle b\rangle \longrightarrow k\langle a,b_U\rangle\oplus k\langle b_V,c\rangle \longrightarrow k\langle a,b,c\rangle \longrightarrow0,$$

where the first map sends

$$b\longmapsto(-b_U,b_V).$$

Its cokernel identifies the two copies of $b$ and leaves $a,b,c$ as a basis. This is more than a dimension count: a linear map from the cokernel to a vector space $L$ is exactly a pair of maps from $k[U]$ and $k[V]$ to $L$ that agree on $k[U\cap V]$. That universal property is the cosheaf assertion.

The Čech nerve remembers finite overlaps

Choose an ordering of an open cover $\mathcal U=\{U_i\}_{i\in I}$ of $U$ and write

$$U_{i_0\cdots i_p} = U_{i_0}\cap\cdots\cap U_{i_p}.$$

The Čech nerve is the simplicial diagram whose $p$-simplices are

$$\check U_p = \coprod_{(i_0,\ldots,i_p)\in I^{p+1}} U_{i_0\cdots i_p}. \tag{10}$$

A face map omits one index; a degeneracy repeats one. Formula (10) is the full nerve, including repeated indices. For the additive calculations below, we adopt the conventional alternating Čech complex indexed by strictly increasing tuples. This is an ordered-cover convention, not a claim that normalizing (10) simply deletes every non-increasing tuple.

For an abelian-valued presheaf $\mathcal F$, define Čech cochains

$$\check C^p(\mathcal U;\mathcal F) = \prod_{i_0<\cdots<i_p} \mathcal F(U_{i_0\cdots i_p}). \tag{11}$$

The differential is the alternating sum of restrictions:

$$(\delta c)_{i_0\cdots i_{p+1}} = \sum_{r=0}^{p+1}(-1)^r \rho_{U_{i_0\cdots i_{p+1}}}^{ U_{i_0\cdots\widehat{i_r}\cdots i_{p+1}}} \left(c_{i_0\cdots\widehat{i_r}\cdots i_{p+1}}\right). \tag{12}$$

In degree zero, for $i<j$,

$$(\delta s)_{ij}=s_j|_{U_i\cap U_j}-s_i|_{U_i\cap U_j}. \tag{13}$$

For an abelian-valued precosheaf $\mathcal G$, use direct sums and assembly maps instead:

$$\check C_p(\mathcal U;\mathcal G) = \bigoplus_{i_0<\cdots<i_p} \mathcal G(U_{i_0\cdots i_p}), \tag{14}$$ $$\partial|_{\mathcal G(U_{i_0\cdots i_p})} = \sum_{r=0}^{p}(-1)^r \iota_{U_{i_0\cdots i_p}}^{ U_{i_0\cdots\widehat{i_r}\cdots i_p}}. \tag{15}$$

An edge $i<j$ therefore has boundary “vertex $j$ minus vertex $i$,” matching the signs in (9). Functoriality supplies identical composites for the two ways of omitting a pair of indices, and the alternating signs cancel them. Hence $\delta^2=0$ and $\partial^2=0$.

The sheaf condition says

$$\mathcal F(U)\cong H^0\bigl(\check C^\bullet(\mathcal U;\mathcal F)\bigr), \tag{16}$$

and the cosheaf condition says

$$H_0\bigl(\check C_\bullet(\mathcal U;\mathcal G)\bigr) \cong\mathcal G(U). \tag{17}$$

These are degree-zero statements. They do not force the higher Čech groups to vanish.

A sheaf with nonzero higher Čech cohomology

Let $\underline{k}$ be the sheaf of locally constant $k$-valued functions on $S^1$. Choose three connected arcs whose pairwise intersections are nonempty and connected but whose triple intersection is empty. The nerve is the boundary of a triangle. Every nonempty intersection carries the vector space $k$, so

$$\check C^0\cong k^3, \qquad \check C^1\cong k^3, \qquad \check C^2=0,$$

and, with the vertices ordered $0<1<2$,

$$\delta(a_0,a_1,a_2) = (a_1-a_0,\ a_2-a_0,\ a_2-a_1).$$

The image has codimension one, so

$$\check H^1(\mathcal U;\underline{k})\cong k.$$

Thus an honest sheaf need not have an augmented Čech complex that is exact in every positive degree. The sheaf axiom itself governs the kernel in degree zero, not higher acyclicity.

Related but distinct descent notions

Several related constructions are often compressed into the word “descent.” They should be distinguished rather than arranged as a single hierarchy.

1. Ordinary sheaf or cosheaf descent is the strict equalizer or coequalizer condition (4) or (7) for an ordinary open cover.

2. Čech cohomology or homology uses the full intersection pattern of one chosen cover. Higher groups can depend on that cover. A comparison with sheaf cohomology exists, but equality requires acyclicity of the relevant finite intersections. A Leray cover for $\mathcal F$ supplies that condition. For constant or locally constant coefficients on manifolds, a good cover is a standard sufficient setting. The cochain construction and its comparison with sheaf cohomology under acyclicity hypotheses are given in Stacks Project Authors 2026, §20.9, tag 01ED and Stacks Project Authors 2026, §20.11, tag 01EO.

3. Groupoid-valued descent assigns local objects together with specified isomorphisms on double overlaps and cocycle coherence on triple overlaps. This is the stack-like extension of gluing, not the ordinary equalizer axiom for a sheaf of nonabelian groups. This descent datum and its effectiveness condition are Stacks Project Authors 2026, Definition 8.3.1, tag 026B and Stacks Project Authors 2026, §8.5, tag 02ZH.

4. Derived or homotopical descent evaluates a full simplicial Čech object and uses a totalization or homotopy (co)limit, with equivalence interpreted by the relevant weak equivalences. Presheaf coefficients give a cosimplicial diagram and precosheaf coefficients give a simplicial one; their full repeated-index levels are

   $$B_{\mathcal F}^p = \prod_{(i_0,\ldots,i_p)\in I^{p+1}} \mathcal F(U_{i_0\cdots i_p}), \qquad B^{\mathcal G}_p = \coprod_{(i_0,\ldots,i_p)\in I^{p+1}} \mathcal G(U_{i_0\cdots i_p}).$$

   The coface and codegeneracy maps on $B_{\mathcal F}^\bullet$, and the face and degeneracy maps on $B^{\mathcal G}_\bullet$, come from restriction or assembly. Schematically, the descent comparison maps have types

   $$\mathcal F(U) \longrightarrow \operatorname*{holim}_{[p]\in\Delta}B_{\mathcal F}^p, \qquad \operatorname*{hocolim}_{[p]\in\Delta^{\mathrm{op}}} B^{\mathcal G}_p \longrightarrow \mathcal G(U).$$

   These arrows indicate variance and direction only; their construction depends on the homotopical target. Čech descent and hyperdescent are not interchangeable without further conditions. For the distinction between Čech descent and descent for general hypercovers, see Dugger, Hollander, and Isaksen 2004, Appendix A.

The target category controls which line is meaningful. Sets support products and equalizers but no subtraction, so formulas such as (12) belong to an additive target. Abelian groups, modules, and vector spaces support kernels, cokernels, and alternating complexes. Coproducts of algebras are generally not direct sums of underlying vector spaces. In topological or completed categories, the correct products, coproducts, quotients, and completions must be used. Infinite covers also require the corresponding infinite products or coproducts. A calculation in one target cannot silently be transferred to another.

A controlled field-and-probe example

Let $M$ be a smooth paracompact manifold and $E\to M$ a finite-rank vector bundle. Smooth fields form a sheaf

$$\mathcal E(U)=\Gamma(U,E),$$

because restriction is contravariant and smooth sections that agree on overlaps glue uniquely.

Let

$$E^!=E^\vee\otimes\operatorname{Dens}_M$$

be the dual-density bundle, and set

$$\mathcal D_E(U)=\Gamma_c(U,E^!).$$

For an inclusion $j:U\hookrightarrow V$, a compactly supported section has support contained away from the boundary of $U$ and therefore extends smoothly by zero:

$$j_!: \mathcal D_E(U)\longrightarrow\mathcal D_E(V).$$

These maps make $\mathcal D_E$ covariant. Partitions of unity subordinate to open covers show that it is a vector-space-valued cosheaf. For $W=U\cup V$, one obtains

$$\Gamma_c(U\cap V,E^!) \longrightarrow \Gamma_c(U,E^!)\oplus\Gamma_c(V,E^!) \longrightarrow \Gamma_c(W,E^!) \longrightarrow0. \tag{18}$$

The first arrow has the signs of (9), and the second adds the two zero-extensions. Given a partition of unity $\chi_U+\chi_V=1$ subordinate to the cover, a compactly supported $\alpha$ on $W$ is the sum of $\chi_U\alpha$ and $\chi_V\alpha$. This proves existence of a local presentation; the coequalizer relation controls changes of presentation. The chosen partition is not canonical.

Each $\alpha\in\mathcal D_E(U)$ defines a linear probe of fields,

$$\mathcal O_\alpha(\phi) = \int_U\langle\alpha,\phi\rangle, \qquad \phi\in\mathcal E(U). \tag{19}$$

Restriction and extension by zero are paired:

$$\left\langle j_!\alpha,\phi\right\rangle_V = \left\langle\alpha,j^*\phi\right\rangle_U. \tag{20}$$

This example explains a basic variance pattern: fields restrict, while compactly supported probes assemble. It does not by itself construct an algebra of observables or prove equations of motion, additivity, causality, the time-slice property, factorization, positivity, unitarity, or equivalence of quantum field theories.

Factorization algebras use a different cover condition. A Weiss cover of $U$ is a family of opens such that every finite subset of $U$ lies in one member. This stronger condition makes Weiss covers a distinguished subclass of ordinary covers: singleton subsets show that every Weiss cover is ordinary, but the converse fails. If an ordinary two-open cover has a point exclusive to each member, that pair of points lies in neither member. Factorization algebras use Weiss descent together with additional structure for disjoint opens. The designated QFT continuation is Weiss Descent and Local-to-Global Observables. The present page stops at the ordinary sheaf, cosheaf, and Čech vocabulary. For that boundary beyond ordinary cosheaves, see Costello and Gwilliam 2017, Chapter 6 and Gwilliam and Rejzner 2020, §2.4.1.

Common pitfalls

Confusing a functor with a descent condition. A presheaf need not be a sheaf, and a precosheaf need not be a cosheaf. Descent is an additional equalizer or coequalizer requirement for every relevant cover.

Forgetting uniqueness in sheaf gluing. Compatible local sections must have exactly one global extension. Existence alone does not make a sheaf.

Using the same arrow direction twice. Restrictions have type $\mathcal F(V)\to\mathcal F(U)$ for $U\subseteq V$; assembly maps have type $\mathcal G(U)\to\mathcal G(V)$. Always type-check before writing a Čech differential.

Turning an overlap map into a surjection. Exactness of (6) stops at the overlap term. A sheaf does not make every overlap section a difference of sections from the two larger opens.

Replacing a categorical coproduct by a vector-space sum. Equation (9) uses direct sums only because its target is abelian. In algebras, topological vector spaces, and completed categories, the relevant colimit may be a very different object.

Equating all Čech and derived constructions. Higher Čech groups depend on a cover, their comparison with sheaf cohomology needs hypotheses, and homotopical descent uses homotopy (co)limits. Ordinary covers, Weiss covers, Čech descent, and hyperdescent are not synonyms.

Calling every cosheaf a factorization algebra. A factorization algebra also has compatible multi-input operations for disjoint opens and uses Weiss descent, often in a homotopical target. A covariant observable assignment alone supplies none of those consequences.

Check your understanding

Variance check. For $W\subseteq U\subseteq V$, write the two functoriality equations with their arrows in the correct order.

Answer check. They are $\rho_W^V=\rho_W^U\circ\rho_U^V$ and $\iota_W^U$ followed by $\iota_U^V$, namely $\iota_W^V=\iota_U^V\circ\iota_W^U$.

Gluing check. Why do $(1,2)$ on $\{a,b\}$ and $(2,5)$ on $\{b,c\}$ glue, while $(1,2)$ and $(3,5)$ do not?

Answer check. The two sections must have the same value at the overlap $\{b\}$. The first pair agrees there and gives $(1,2,5)$; the second has values $2$ and $3$ at $b$.

Sign check. For an ordered pair $i<j$, recover the first Čech coboundary and the first Čech boundary from (12) and (15).

Answer check. The cochain formula is $s_j|_{U_i\cap U_j}-s_i|_{U_i\cap U_j}$. The chain formula sends an overlap element to its image at vertex $j$ minus its image at vertex $i$.

Degree check. Does the sheaf axiom imply $\check H^1(\mathcal U;\mathcal F)=0$?

Answer check. No. It identifies global sections with the degree-zero kernel. The three-arc cover of $S^1$ gives a sheaf with $\check H^1\cong k$.

Cover check. Give an ordinary cover that is not a Weiss cover.

Answer check. In a two-open cover $U=A\cup B$, choose $x\in A\setminus B$ and $y\in B\setminus A$. The finite set $\{x,y\}$ is contained in neither member, so the cover is not Weiss.

What has been established

Presheaves and precosheaves encode opposite variance on $\operatorname{Open}(X)$. Sheaves impose unique local-to-global gluing by a limit, while cosheaves impose universal local-to-global assembly by a colimit. The Čech nerve organizes all finite intersections, and its additive complexes recover the ordinary axioms in degree zero without forcing higher acyclicity. Smooth fields and compactly supported dual-density probes give a controlled geometric example of the two variances.

The designated chapter-wide continuation is Derived, Higher, and Factorization Frameworks: a Boundary Map. It marks the comparison with the stronger structures deliberately left outside this page; the Mathematical QFT link above marks their first specialist application.

References

• Kevin Costello and Owen Gwilliam, Factorization Algebras in Quantum Field Theory, Volume 1, Cambridge University Press (2017), Chapter 6, DOI:10.1017/9781316678626.006. Weiss covers, strict and homotopical factorization-algebra descent, and the boundary beyond ordinary cosheaves.

• Justin Curry, Sheaves, Cosheaves and Applications, PhD dissertation, University of Pennsylvania (2014), arXiv:1303.3255v2, §§ 2.1–2.3. Functorial definitions, equalizer and coequalizer forms, compactly supported cosheaf examples, and Čech sign conventions.

• Daniel Dugger, Sharon Hollander, and Daniel C. Isaksen, “Hypercovers and Simplicial Presheaves,” Mathematical Proceedings of the Cambridge Philosophical Society 136 (2004), 9–51, arXiv:math/0205027, Appendix A. The distinction between Čech descent and descent for general hypercovers.

• Owen Gwilliam and Kasia Rejzner, “Relating Nets and Factorization Algebras of Observables: Free Field Theories,” Communications in Mathematical Physics 373 (2020), 107–174, DOI:10.1007/s00220-019-03652-9, arXiv:1711.06674, § 2.4.1. Prefactorization structure, Weiss covers, strict and homotopical cosheaf qualifications, and the distinction between ordinary and Weiss covers.

• The Stacks Project Authors, Cohomology of Sheaves, The Stacks Project (continuously updated; accessed August 11, 2026), Stacks Project Authors, § 20.9, tag 01ED and Stacks Project Authors, § 20.11, tag 01EO. Čech cochains and differentials, the degree-zero sheaf statement, and the comparison between Čech and sheaf cohomology under acyclicity hypotheses.

• The Stacks Project Authors, Sheaves on Spaces, The Stacks Project (continuously updated; accessed August 11, 2026), Stacks Project Authors, § 6.7, tag 006S and Definition 6.9.1, tag 0072. Compatible local sections, unique gluing, the empty cover, and the equalizer formulation of the sheaf condition.

• The Stacks Project Authors, Stacks, The Stacks Project (continuously updated; accessed August 11, 2026), Definition 8.3.1, tag 026B and Stacks Project Authors, § 8.5, tag 02ZH. Descent data as local objects, overlap isomorphisms, and cocycle coherence, and the effectiveness condition for stacks in groupoids.