Tag Archives: sheaves

We Shall Come Rejoicing

Trying to get my head around the interplay between the locality and gluing axioms in a sheaf. In brief, and given a metaphorical association of the “more global” with the “above” and of the “more local” with the “below”:

The locality axiom means that “the below determines the (identity of the) above”: whenever two sections over an open set U are indistinguishable based on their restrictions to the sections over any open cover of U, they are the same. There is no way for data that are more-locally the same to correspond to data that are more-globally different. Our view can be enriched as we move from the global to the local, but not the other way around.

The gluing axiom means that “the above determines the (coherence of the) below”: each compatible-in-pairs way of gluing together the sections over an open cover of U has a representative among the sections over U, of which the sections in the glued assemblage are the restrictions. There is no coherent more-local assemblage that does not have such a more-global representation. The global provides the local with its law, indexing its coherence.

A theme of postmodernism, and particularly of Lyotard’s treatment of the postmodern, was “incommensurability”. Between distinct local practices – language games – there is no common measure, no universal metalanguage into which, and by means of which, every local language can be translated. The image of thought given by sheaves does not contradict this, but it complicates it. The passage from the local to the global draws out transcendental structure; the passage from the global to the local is one of torsion, enrichment, discrimination. The logics of ascent and descent are linked: we cannot “go down” into the local without spinning a web of coherence along the way, and we cannot “come up” into the global without obeying a strict rule of material entailment.

Sheaves for n00bs

What itinerary would a gentle introduction to sheaves have to take? I would suggest the following:

  • A basic tour of the category Set, introducing objects, arrows, arrow composition, unique arrows and limits. (OK, that’s actually quite a lot to start with).
  • Introduction to bundles and sections, with a nicely-motivated example.
  • Enough topology to know what open sets are and what a continuous map is, topologically speaking.
  • Now we can talk about product spaces and fiber bundles.
  • Now we can talk about the sheaf of sections on a fiber bundle.
  • Now we back up and talk about order structures – posets, lattices, Heyting algebras, and their relationship to the lattice of open sets in a topological space. We note in passing that a poset can be seen as a kind of category.
  • Functors, covariant and contravariant.
  • That’s probably enough to get us to a categorial description of first presheaves (contravariant functor from a poset category to e.g. Set) and sheaves (presheaves plus gluing axiom, etc). Show how this captures the fundamental characteristics of the sheaf of sections we met earlier.
  • Then to applications outside of the sheaf of sections; sheaf homomorphisms and sheaf categories; applications in logic and so on. This is actually where my understanding of the topic falls of the edge of the cliff, but I think that rehearsing all of the material up to this point might help to make some of it more accessible.

Anything really essential that I’ve missed? Anything I’ve included that’s actually not that important?