This third volume of the Handbook of categorical algebra is neither a book on topos theory nor a book on sheaf theory. Our main concern is to study various approaches to the notion of a “set valued sheaf” and describe the structure and the properties of the corresponding categories of sheaves. Those categories are toposes, indeed, so that this book can serve also as a first introduction to the theory of toposes and, of course, to the theory of sheaves.

The crucial idea behind the notion of a sheaf is to work not just with a “plain” set of elements, but with a whole system of elements at various levels. Of course, reasonable rules are imposed concerning the interactions between the various levels: an element at some level can be restricted to all lower levels and, if a compatible family of elements is given at various individual levels, it is possible to “glue” the family into an element defined at the global level covered by the individual ones. The various notions of sheaf depend on the way the words “level”, “restriction” and “covering” are defined.

The easiest examples are borrowed from topology, where the various “levels” are the open subsets of a fixed space X: for example a continuous function on X may very well be defined “at the level of the open subset U ⊆ X”, without being the restriction of a continuous function defined on the whole of X.