-
  • Sterling’s Chapter 2
    • +
  • Sterling’s Chapter 2 (update)
  • Misc
  • Bundles -

    Sterling’s Chapter 2

    +

    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 @@ -85,9 +85,10 @@

    Sterling’s Chapter 2

    which at first glance seems a little more approachable for me.

      -
    • At this point, it seems I need to roll up the sleeves and understand some of the sheaf theoretic ideas +
    • 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
        • +
      • 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 +<!–
      • Some concepts which seem related to learn the “101” of:
        • Sheafs @@ -96,10 +97,11 @@

          Sterling’s Chapter 2

        • I’m hoping Kripke-Joyal semantics as motivation itself will be enough for me, so I’ll punt on other examples
        • -
      • The canonical example of the cylinder v. mobius strip fiber bundles
        • +
      • The canonical example of the cylinder v. Möbius strip fiber bundles
      • Boolean valued models
          -
        • There’re many books that go much further into this, but let’s table that
          • +
        • There’re many books that go much further into this, but let’s table that +–>
      @@ -129,17 +131,17 @@

      Bundles

      from Husemoller’s book, referenced by Wikipedia.

      Chapter 2 - (simple) Bundles

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

      +\((E, p : E → B, B)\), 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 \(b \in B\), +“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 \(\pi_1 : B \times F \to B\).

      +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 @@ -160,31 +162,17 @@

      Chapter 2 - (simple) Bundles

      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 \(p : E \to B\) can be identified with -a \(B\)-morphism \(s : 1_B \to p\).

      - -
        -
      • Although this view is a little weird. -Certainly, given a section, we can construct the \(B\)-morphism -where the base-morphism is the identity, -and the top-morphism is the section. -But given such a \(B\)-morphism \(: 1_B \to p\), -you don’t necessarily get a total section?
        • -
      • (Consider the trivial \(B\)-morphism that maps every \(b\) in the base to a chosen one, -and every \(b\) in the total space to a chose element of the fiber over the chosen \(b\). -This hardly gives a total section.
        • -
      - -

      I think this is actually a small error in the book? -Every section is such a \(B\)-morphism, so every general property of morphisms applies to sections, but the “precisely” is not warranted.

      +

      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 \(p : E \to B\) and \(p' : E' \to B'\). -The product is simply \(p \times p'\). +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 \(\mathsf{Bun}_B\)? @@ -199,10 +187,10 @@

      Chapter 2 - (simple) Bundles

      • An alternative characterization of it, is as the union of the products of the fibers.
        • -
      • That is, for each \(b \in B\), - take the fibers \(p^{-1}(b) \subseteq E\) and \(p'^{-1}(b) \subseteq E'\), - and take their product as subspaces, to get a subspace of \(E \times E'\). - And aggregate in \(E \times E'\) the rest 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)
        • @@ -217,8 +205,8 @@

        Chapter 2 - (simple) Bundles

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

        +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 \(f\), @@ -230,22 +218,63 @@

        Chapter 2 - (simple) Bundles

        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 \(p \times p' : E \times E' \to B \times B\), pulled back across the diagonal \(\Delta : B \to B \times B\)
        • +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 \(f : B' \to B\) -induces a functor \(f^* : \mathsf{Bun}_B \to \mathsf{Bun_{B'}}\).

      +

      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
        • +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 \(X\) is st there is a action \(X \times G \to X\) +

      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 \(p :\)

      @@ -286,7 +315,7 @@

      related

    • (the answer is yes, you should.)
    -

    site version: 2024-01-28

    +

    site version: 2024-01-29