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The support of a section of a sheaf is the closure of the set of points in the space where the section is not equal to the zero or base element.

A Grothendieck topology is a structure on a category that allows the definition of sheaves for the category, generalizing the notion of open sets for topological spaces to more abstract contexts.

A section of a sheaf over an open set is an element of the sheaf associated with that open set, often representing a continuous function or another geometric object.

The stalk of a sheaf at a point is the direct limit of the sets or groups of the sheaf over all open sets containing that point.

The Godement resolution of a sheaf is a canonical way to construct an injective resolution of a sheaf, which is used in the computation of sheaf cohomology.

A flabby (or flasque) sheaf is a sheaf in which every section over an open set can be extended to a global section.

Given a continuous map $f: X \rightarrow Y$ and a sheaf $\mathcal{F}$ on $X$, the direct image sheaf $f_*\mathcal{F}$ on $Y$ assigns to each open set $V \subseteq Y$, the sections of $\mathcal{F}$ over the preimage $f^{-1}(V)$.

A monopresheaf is a presheaf where the restriction mappings are monomorphisms, meaning they are injective.

Given a continuous map $f: X \rightarrow Y$ and a sheaf $\mathcal{G}$ on $Y$, the inverse image sheaf $f^{-1}\mathcal{G}$ on $X$ consists of data on $X$ that is defined in terms of the direct limit of stalks of $\mathcal{G}$ at points in $Y$ lying over open sets of $X$.

Sheaf cohomology is a mathematical tool for computing topological invariants of a space by studying the global sections and relations among sections of a sheaf over that space.

A sheaf is a data structure that associates information to open sets of a topological space in a way that is consistent on overlaps.

The kernel of a sheaf morphism $\phi: \mathcal{F} \rightarrow \mathcal{G}$ is a sheaf that assigns to each open set the kernel of the morphism between sections of $\mathcal{F}$ and $\mathcal{G}$ over that set.

A presheaf is similar to a sheaf but does not necessarily satisfy the gluing axiom required for sheaves. It assigns data to open sets with restriction maps.

A morphism of sheaves is a natural transformation between two sheaves, giving a compatible system of maps between the data on corresponding open sets.