Sheaf (mathematics)

In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain a bigger one. A presheaf is similar to a sheaf, but it may not be possible to glue. Sheaves, it turns out, enable one to discuss in a refined way what is a local property, as applied to a function.

References

• Topologie algébrique et théorie des faisceaux, Roger Godement
• The Theory of Sheaves (University of Chicago Press,1964) R. G. Swan (concise lecture notes)
• Sheaf Theory (London Math. Soc.Lecture Note Series 20, Cambridge University Press, 1975) B. R. Tennison (pedagogic treatment)
• Sheaf Theory, 2nd Edition (1997) Glen E. Bredon (oriented towards conventional topological applications)
• Sheaves in Geometry and Logic (Springer-Verlag, 1992) S. Mac Lane and I. Moerdijk (category theory and toposes emphasised)
• Topological methods in algebraic geometry (Springer-Verlag, Berlin, 1995) F. Hirzebruch (updated edition of a classic using enough sheaf theory to show its power)
• Sheaves on Manifolds (1990) M. Kashiwara and P. Schapira (advanced techniques such as the derived category and vanishing cycles on the most reasonable spaces)