Sheaves on my mind

So 5am rolls around, and I’m still not asleep.  Of course.  It’s not like I have to be up in 2.5 hours or anything.  My brain is conspiring against me.

Whilst rolling around in bed NOT SLEEPING, my thoughts turned to sheaves: just what are they?  I posted a bit ago (longer than I like) about sheaves, but it was kind of a lame post, caught up in the definition of the objects.  This is a way of remedying that, and a bit more of some dry definitions (you really can’t escape them, unfortunately).

Put simply, I think of a sheaf (of sets) as a means of “observing” or “exploring” a space X.  A topological space is a mathematical object, independent of our puny reality and ordinary means of observation.  Think of a scientist walking around “in/ on” X.  Let’s call him monsieur Faisceau.  Monsieur Faisceau walks around X, recording what he sees in every open set U \subseteq X.  He starts noticing patterns in his observations…like X happens to look red at every open set.  Being a good scientist and having dutifully recorded all of his observations, he sees that the observation of X looking red in every open set agrees “on overlaps”.  That is, whenever he compares the results of his observations, they agree.  He then deduces (and maybe even publishes a paper) that “X looks red”.

This is obviously a vast simplification of what scientists can do (i.e. serve us better than acting as locally constant functions).  Monsieur Faisceau could be taking temperature readings on every open set of X, i.e. compiling a family of continuous functions (or smooth, depending on what he wants/ if X is a smooth manifold) T_i : U \to \mathbb{R} that give the give the temperature of U at every point.  Suppose, for the sake of the argument, that \cup U_i = X.  Again, he notices that his results agree on overlaps.  In mathspeak, T_i |_{U_i \cap U_j} = T_j |_{U_i \cap U_j}, for any i,j (in whatever indexing set we’re dealing with).  Being a good lazy intellectual, he doesn’t like keeping the data of each function T_i, so he instead defines a function T : X \to \mathbb{R} via T(x) = T_i(x) if x \in U_i.  This is well defined, since T_i(x) = T_j(x) if x \in U_i and x \in U_j.  He then, of course, publishes a paper about his findings.

There’s then a theorem that says specifying a family of sheaves on an open cover \{U_i\}_{i \in I} (i.e. F_i is a sheaf on U_i for each i) subject to some nice “gluing” conditions amounts to specifying a unique sheaf defined on all of X (see Hartshorne’s Algebraic Geometry, ex. 22 in section 2.1).  This can then be thought of as a collection of scientists who work together observing X.  They each work on their own open set, and talk to each other whenever they work on overlapping interests.  That is, scientist F_i specializes in studying the open set U_i, and scientist F_j specializes in studying U_j.  Thus they both specialize in studying U_i \cap U_j as well, and any observation made by one scientist on U_i \cap U_j is communicated “isomorphically” to the other scientist about that area.  Obviously, any scientist communicates with himself about his work without having to do any extra work.  Papers published about global results then amount to “joint work” by the scientists.

Another fundamental process to understand is “sheafification” (an AWESOME word, btw).  To understand this, we must understand why presheaves are so lacking.  These can be thought of as people with really poor memory who are walking around in X.  They’re just as smart as a scientist, and thus see all the same things when they look at each open set U \subseteq X.  They just have a hard time remembering their results/ observations, and can’t infer global data by “gluing” local data.  This puts the “forgetful functor \textbf{Sh}(X) \to \textbf{Psh}(X) (now aptly named) in a new perspective: it takes a scientist and hits him in the head, rendering him unable to remember anything he sees.

Hopefully this makes some sense to other people.  I’ll return with sheafification soon.

Author: brianhepler

I'm a second-year math postdoc at the University of Wisconsin-Madison, and I think math is pretty neat. Especially the more abstract stuff. It's really hard to communicate that love with the general population, but I'm going to do my best.

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