Axiomatic Sheaf Theory : Some Constructions and Applications

Abstract

After having thought for some time about what it was that I wanted to say in these lectures, I finally decided they should serve two purposes, and two purposes only. First, I wanted to provide a leisurely, and reasonably complete, introduction to the axiomatic theory of sheaves developed recently by F. W. Lawvere and myself. Grothendieck, and those around him, have long maintained (see [11] ) that in sheaf theory it is the topos itself - i.e., the whole category of sheaves - that is important, and not the site, or small category, from which it is derived. He himself, however, had never consequently developed this point of view. Thus, this was our first goal, and it is the one I would like to concentrate on here. Later, we began to think of the notion of topos as a kind of set theory useful for dealing with many kinds of “sets” other than just “abstact” sets. Though this is perhaps the most interesting aspect of topos, I shall hardly mention it here; I hope the interested reader will consult [4] or [9] for more information. After developing the basic general theory, we will turn to the more specialized topics whose exposition was my second aim. Here, among several possible ways of constructing topos, I would like to describe thesis from the topos point of view. Thus, this section, as well as the preceding, may be considered background material Tor [10]. The Continuum Hypothesis forms the subject of the last section, so these lectures are independent of [10], though the reader might want to consult this for more details.