Sterling’s Chapter 2 (update)

As discussed last week– and mentioned in the first paragraph of Chapter 2–, Sterling’s chapter on topoi is “greatly influenced by [the notes] of Anel and Joyal”. This reference chapter is hard-going, so I thought I’d try to concurrently march forward and read Topos-Logie, which at first glance seems a little more approachable for me.

  • At this point, it seems I need to understand some of the sheaf theoretic ideas
    • Certainly I don’t intend to actually learn sheaf theory right now, but I need a mental picture of what Kripke-Joyal semantics actually is, and what the notion of “sheaf logic” refers to

Misc

  • The logic seminar this semester will be a learning seminar on first-order categorical logic, with a rough goal of describing completeness due to Makkai(?)
    • Ronnie thought there wouldn’t be enough time in a semester for a treatment of higher order/sorted categorical logic (ie topoi). “Too much machinery”
    • But it should help me understand some of these ideas

Bundles

I found myself going on a Wikipedia search Sheaf > Grothendieck Topology > Sieve > Fiber-product > Fiber bundle, trying to learn some of the spatial(/geometric?) intuition for topoi, as seems to be the point of Anel and Joyal, and Sterling’s Chapter 2.

I went over some basic material on fiber bundles, from Husemoller’s book, referenced by Wikipedia.

Chapter 2 - (simple) Bundles

A bundle is defined as just the triple \((E, p : E → B, B)\), which could be compressed to just \(p : E → B\).

\(E\) is called the total space, \(p\) the projection, and \(B\) the base space. The namesake comes from thinking of the bundle “as a union of fibers \(p^{-1}(b)\) for \(b ∈ B\), parameterized by \(B\) and ‘glued together’ by the topology of the space \(E\).”

A simple example that is often referenced is the product bundle, which is a trivial bundle where the fibers are constant. That is, the bundle \(π_1 : B × F → B\).

The notion of a subbundle is defined in the obvious way, but is a special case of a bundle morphism where the morphisms are inclusions (described below).

  • This is essentially the same as the standard practice of identifying the domain of an embedding with its image, and thought of as an inclusion.

A cross section of a bundle is defined, but it is just a section of the bundle, in the categorical sense.

A bundle morphism is just a morphism in the arrow category. That is, the category of bundles \(\mathsf{Bun}\) is some subcategory of the arrow category. (As is, it should just be the arrow category.)

A \(B\)-morphism is a morphism of bundles with equal codomain. That is, the category of bundles over \(B\), \(\mathsf{Bun}_B\) is just the slice category over \(B\).

The book remarks that a (cross) section of a bundle \(p : E → B\) can be identified with a \(B\)-morphism \(s : 1_B → p\). (I had written that this seemed like a slight error, but I realize the key point is \(B\)-morphism, not just (bundle) morphism.)

A space \(F\) is called the fiber of a bundle if it is isomorphic to each fibre. In other words, the bundle is isomorphic to the product bundle.

What about the product of bundles? Suppose we have bundles \(p : E → B\) and \(p' : E' → B'\). The product is simply \(p × p'\). (ie \(\mathsf{Bun}\) has products if the category of spaces does.)

But what about in \(\mathsf{Bun}_B\)? Well, it’s the pullback, also called the fiber product.

  • I find it weird that the product in \(\mathsf{Bun}\) is just the product, but the product in \(\mathsf{Bun}_B\) is the pullback
  • Maybe this is one way of understanding what a pullback is: The pullback is the total space of the product (\(B\)-)bundle.
    • An alternative characterization of it, is as the union of the products of the fibers.
    • That is, for each \(b ∈ B\), take the fibers \(p^{-1}(b) ⊆ E\) and \(p'^{-1}(b) ⊆ E'\), and take their product as subspaces, to get a subspace of \(E × E'\). And aggregate in \(E × E'\) the rest of the fibers.
    • We then get a (\(B\)-)bundle where each fiber is well, the product of fibers
    • The universality of the pullback is that there’s nothing extra in this total space
    • (ie the data of both bundles is minimally captured as such)
  • From this upwards view, it’s actually the product in \(\mathsf{Bun}\) that’s a little weird
  • The product in \(\mathsf{Bun}_B\) can be viewed as the product in \(\mathsf{Bun}\) where we additionally require that everything commutes with \(1_B\) between the base spaces.
  • So the product in \(\mathsf{Bun}\) can be seen as a case where “no such space saving technique” can be employed

A restriction of a bundle is a special case (as before, in the same way) of an induced bundle, whereby a bundle \(p : E → B\) and \(f : B' → B\), the bundle \(f^*(p) : B' ×_{B', E} E → B\) is induced by the pullback.

This other sense of pullback, in which we actually pull back the fibers along the graph \(f\), is more fitting of the name in some sense. Although of course, this is literally just a tilted version of the other pullback we just described (viewed from a different angle).

  • So the moral is that pullbacks are both a notion of (fibered) product, and renaming, simultaneously?
  • A much better description of the above is from the wikipedia page for pullbacks which says that the fibered product is just the product \(p × p' : E × E' → B × B\), pulled back across the diagonal \(Δ : B → B × B\)

Moreover, each \(f : B' → B\) induces a functor \(f^* : \mathsf{Bun}_B → \mathsf{Bun_{B'}}\).

Chapter 4 - Fiber Bundles

  • I don’t understand why we’re talking about topological groups, but they seem ubiquitous, so we’ll roll with it

I’m having trouble with the definition of fiber bundle, so let’s go over some simple examples:

  • The trivial bundle \(π_1 : B × F → B\) is a fiber bundle of \(F\) over \(B\). The total space is said to be “globally [a product”, not “just locally”, but I don’t know what that means at the moment
  • The Möbius strip is the total space \(E\) of the following bundle:
    • \(B = S^1\) and \(F = \mathbb{R}(?)\).
    • “The corresponding trivial bundle \(B × F\) would be a cylinder”
    • I guess we take it for granted that we have already constructed \(E\), so this isn’t a construction
    • I still don’t get why the Möbius bundle has no non-vanishing (global) section, so I still have to push on this textbook

A right \(G\)-space \((X, G)\) is a space \(X\) with an action \(X × G → X\) for topological group \(G\).

An example of a topological group is \((\mathbb{R}, +)\). I suppose trivially every topological group is also a \(G\)-space relative to itself. That is, \(\mathbb{R}\), forgetting the group structure, is compatible with the group action \((\mathbb{R}, +)\).

A \(G\)-morphism is a map between \(G\)-spaces that is compatible with the group action.

Two elements \(x, x' ∈ (X, G)\) are \(G\)-equivalent if \(∃s∈G. xs=x'\). That is, if if they are in the same orbit, as standard. Then let \(X \mod G\) denote \(\{xG | x ∈ X\}\), the class of orbits, with the quotient topology.

Given a \(G\)-space \(X\) Let \(α(X)\) be the bundle

A bundle \(p : X → B\) is a \(G\)-bundle if \(p\) and \(α(X)\)

…

Given a bundle \(p :\)