Sheaves are indispensable tools in algebraic geometry and algebraic analysis , in which we study geometric objects algebraically. In algebraic topology, we have chosen to perform algebraic operations directly to spaces by introducing new concepts such as fibrations .

There are still some usages of the notion of sheaves in algebraic topology. For example, Mark Johnson found a description of iterated loop spaces as presheaves on the topological category of spaces .

And concepts related to sheaves, such as topoi , gerbes , stacks , and torsors are gradually gaining their popularity, partly because of the requirements from . Moerdijk’s [ Moe ] serves as a good introduction to these concepts.

[Moe] Ieke Moerdijk. Introduction to the language of stacks and gerbes, arXiv:math.AT/0212266 .