Sheaf

2020 Mathematics Subject Classification: Primary: 14-XX [MSN][ZBL]

A sheaf is a pre-sheaf $F$ (cf. also Sheaf theory) on a topological space $X$ such that for every union $\def\l{\lambda} U=\bigcup_\l U_\l$ of open subsets $U_\l$ of $X$ the following conditions are satisfied:

a) if on every $U_\l$ the restrictions of two elements $s$ and $s'$ in $F(U)$ coincide, then $s'=s$;

b) if $s_\l\in F(U_\l)$ are such that for any pair of indices $\l$ and $\mu$ the restrictions of $s_\l$ and $s_\mu$ to $U_\l\cap U_\mu$ coincide, then there exists an element $s\in F(U)$ which on each $U_\l$ has restriction coinciding with $s_\l$.

Every sheaf on $X$ is isomorphic to the sheaf of continuous sections of a certain covering space $p:E\to X$ over $X$, which is determined uniquely up to an isomorphism (by a covering space one means a continuous mapping from $E$ onto $X$ which is a local homeomorphism), therefore a sheaf is also commonly understood to be the covering space $p:E\to X$ itself (see Sheaf theory).

Comments

Generalizing the above notion of a sheaf on a topological space, it is also possible to define sheaves on an arbitrary site. Cf. also Topos.

For a more detailed treatment of sheaves, and additional references, see Sheaf theory.

References

[Br]	G.E. Bredon, "Sheaf theory", McGraw-Hill (1967) MR0221500 Zbl 0158.20505
[Go]	R. Godement, "Topologie algébrique et théorie des faisceaux", Hermann (1958) MR0102797 Zbl 0080.16201