§ Sheaves in geometry and logic 1.3: Characteristic functions of subobjects

    !
S --> 1
v     |
|     | true
v     v
X---->2
phi(S)

• true: 1 -> 2 is the unique monic such that true(1) = 1 (where 2 = {0, 1})
• For all monic m: S -> X , there must be a unique phi(S): X -> 2 such that the diagram is a pullback.
• Then 1 -true-> 2 is called as the subobject classifier. See that 1 is also determined (it is terminal object). So the only "choice" is in what 2 is and what the morphism 1 -true-> 2 is.
• The definition says that every monic is the pullback of some universal monic true.

§ Subobject category

• Define an equivalence relation between two monics m, m': S, S' -> X where m ~ m' iff there is an iso i: S -> S'' such that the triangle commutes:
  S --i--> S'
\      /
m\    /m'
v  v
X

• $Sub_C(X)$ is the set of all subobjects of $X$.
• to make the idea more concrete, let C = Set and let X = {1, 2}. This has subobjects [{}], [{1}], [{2}], [{1, 2}].
• To be clear, these are given by the map m0: {} -> {1, 2} (trivial), m1: {*} -> {1, 2} where m1(*) = 1, m2: {*} -> {1, 2} where m2(*) = 2, and finally m3: {1, 2} -> {1, 2} given by id.
• The category $C$ is well powered when $Sub_C(X)$ is a small set for all $X$. That is, the class of subobjects for all $X$ is set-sized.
• Now given any arrow $f: Y \to X$, then pulllback of a monic $m: S -> X$ along $f$ is another monic $m': S' \to Y$. (recall that pullback of monic along any arrow is monic).
• This means that we can contemplate a functor $Sub_C: C^{op} \to \texttt{Set}$ which sends an object $C$to its set of subobjects, and a morphism $f: Y \to X$ to the pullback of the subobjects of $X$ along $f$.
• If this functor is representable, that is,

§ $G$ bundles

• If $E \to X$ is a bundle, it is a $G$-bundle if $E$ has a $G$ action such that $\pi(e) = \pi(e')$ iff there is a unique $g$ such that $ge = e'$. That is, the base space is the quotient of $E$ under the group, and the group is "just enough" to quotient --- we don't have redundancy, so we get a unique $g$.
• Now define the space $GBund(X)$ to be the set of all $G$ bundles over $X$.
• See that if we have a morphism $f: Y \to X$, we can pull back a $G$ bundle $E \to X$ to get a new bundle $E' \to Y$.
• Thus we can contemplate the functor GBund: Space^op -> Set which sends a space to the set of bundles over it.
• A bundle V -> B is said to be a classifying bundle if any bundle E -> X can be obtained as a pullback of the universal bundle V -> B along a unique morphism X -> B.
• in the case of the orthogonal group Ok, let V be the steifel manifold. Consider the quotient V/Ok, which is the grassmanian. So the bundle V -> Gr is a G bundle. Now, some alg. topology tells us that is in fact the universal bundle for Ok.
• The key idea is that this universal bundle V -> B represents the functor that sends a space to its set of bundles, This is because any bundle E -> X is uniquely determined by a pullback X -> B! So the base space B determines every bundle. We can recover the bundle V -> B by seeing what we get along the identity B -> B.

§ Sieves / Subobject classifiers of presheaves

• Let P: C^op -> Set be a functor.
• Q: C^op -> Set is a subfunctor of P iff Q(c) ⊂ P(c) for all c ∈ Cand that Qf is a restriction of Pf.
• The inclusion Q -> P is a monic arrow in [C^op, Set]. So each subfunctor is a subobject.
• Conversely, all subobjects are given by subfunctors. If θ: R -> P is a monic natural transformation (ie, monic arrow) in the functor category [C^op, Set], then each θC: RC -> PC is an injection (remember that RC, PClive in Set, so it's alright to call it an injection)
• For each C, let QC be the image of θC. So (QC = θC(RC)) ⊂ PC.
• This Q is manifestly a subfunctor.
• For an arbitrary presheaf C^ = [C^op, Set], suppose there is a subobject classifier O.
• Then this O must at the very least classify yonedas (ie, must classify yC = Hom(-, C): [C^op, Set].
• Recall that Sub_C(X) was the functor that sent X ∈ C to the set of subobjects of X, and that the category C had a subobject classifier O iff Sub_C(X) is represented by the subobject classifier O. Thus we must have that Sub_C(X) ~= Hom(X, O).
• Let y(C) = Hom(-, C). Thus we have the isos Sub_C^(yC) = Hom_C^(yC, O) =[yoneda] O(C).
• This means that the subobject classifier O: C^op -> Set, if it exists, must be defined on objects as O(C) = Sub_C^(yC). This means we need to build the set of all subfunctors of Hom(-, C).

§ Sieves

• for an object c, a sieve on c is a set S of arrows with codomain c such that f ∈ S and for all arrows fh which can be defined, we have fh ∈ S.
• If we think of paths f as things allowed to get through c, this means that some path to some other b (via a h) followed by an allowed path to c (via f is allowed). So if b -f-> c is allowed, so is a -h-> b -f-> c.
• If C is a monoid, then a sieve is just a right ideal
• For a partial order, a sieve on c is a set of elements that is downward closed/smaller closed. If b is in the sieve, then so too is any element a such that a .
•  So a sieve is a smaller closed subset: if a small object passes the sieve, then so does anything smaller!
•  Let Q ⊂ Hom(-, c) = yc be a subfunctor. Then define the set S_Q = { f | f: a -> c and f ∈ Q(a) }.
•  Another way of writing it maybe to say that we take S_q = { f ∈ Hom(a, c) | f ∈ Q(a) }.
•  This is a sieve because fh is pulling back f: a -> c along h: z -> a, and the action on the hom functor will pull back the set Hom(a, c) to Hom(z, c), which will maintain sieveiness, as if f ∈ Hom(a, c) then fh ∈ Hom(z, c).
•  This means that a sieve on c is the same as a subfunctor on yc = Hom(c, -).
•  this makes us propose a subobject classifier on [C^op, Set] to be defined as O(c) = set of sieves of Hom(c, -).