Pursuing Classifying Topoi

26 Mar 2016

(This post will outline the haze of thoughts and visions as we pursue Classifying Topoi. All facts stated are true modulo small lies . Allegories to be taken with a liberal pinch of bath salts. It will be wordy to boot, TeXing reserved for better days and better keyboards.)

1. We want to “classify” certain kinds of “structures” over a space/category/topos. Classifying means the following:
   • For every kind of structure we are interested in, there exists a special space, called (and denoted here) the Classifying Spåce for that structure.

   • Q: What kind of structures can be classified?
     A: Something something representable.

   • Probing the Classifying Space with a map from a Test Space corresponds to/yields a structure over the Test Space.
   • examples of structures: cohomology operations, nth cohomology, principal G-bundles, vector bundles etc. Each of these has a glorified Classifying Space.

A galois connection

24 Dec 2015

Let $P,S$ be posets, with maps going back and forth $\varphi: P\to S$ and $\psi:S\to P$. Viewing the posets as categories and the maps as functors, an adjunction $\varphi \dashv \psi$ simply states that

$$\varphi (p) \leq s \Leftrightarrow p \leq \psi (s).$$

This follows from the fact that in this setup, a hom-set consists of at most one element, expressing an inequality. The naturality condition of the adjunction is trivial, a direct consequence of monotonicity of poset maps and transitivity of order.

An adjunction between posets goes by another name: a Galois connection (the fundamental theorem of Galois theory relates the lattices of certain subgroups and field extensions in this way). Let’s see a cute example from set theory.

Consider a set map $f:X\to Y$. It induces poset maps on the powersets: direct image $f_*: 2^X\to 2^Y$ and inverse image $f^*:2^Y\to 2^X$. Recall that for any subset $A\subseteq X$, we have the inclusion

$$A\subseteq f^*f_*A,$$

which I try to remember by the slogan «push-pull inflates» (decorated with suitable hand gestures). Dually, «pull-push deflates»:

$$f_*f^*B \subseteq B$$

for any $B\subseteq Y$.

These observations show that there is a Galois connection $\ f_* \dashv f^*$:

$f_* A \subseteq B \Leftrightarrow A \subseteq f^* B$.

One consequence of this is that direct image, as a left adjoint, preserves coproducts, AKA unions (but not necessarily intersection), while inverse image preserves intersections. This however does not explain why inverse image also preserves union… could it be that is is also left-adjoint?

What’s a lattice anyway?

22 Dec 2015

In common language, a lattice is something like a regular grid. This picture agrees with some mathematical terminology, where e.g. the product $\mathbb{Z}\times \mathbb{Z}$ is (reasonably enough) called a lattice. This is not the kind of lattice we’ll be talking about here.

Let $P$ be a poset. An element $v\in P$ is called the join of the pair $a,b\in P$ if it is the least upper bound of $a$ and $b$. In this case, we write $v=a\vee b$. Similarly, we could consider the join of any subset $S\subseteq P$.

Examples. The powerset of a given set $X$ is naturally a poset under inclusion of subsets. The join of two subsets $A,B\subseteq X$ is then simply their union $A\cup B$. Another example is to consider the natural numbers ordered by divisibility, i.e. we say $a\leq b$ whenever $a\vert b$ in $\mathbb{N}$. The join $a\vee b$ is then the least common multiple $lcm(a,b)$.

From a categorical perspective, the join is precisely the coproduct (when viewing the poset as a category). In general, a coproduct is ‘only’ unique up to unique isomorphism, but note that in our case this is precisely equality. Dually, the product is then called the meet, which is the greatest lower bound in poset terms. In the above examples, the meet correspond to the intersection and greatest common divisor, respectively. It is (obviously) denoted $a\wedge b$.

The way I remember the terminology is that the meet corresponds to the intersection of sets, i.e. where the sets « meet », while the union contains both, so it « joins » them. This makes sense, I assure you.

Equipped with these concepts, we can now give the following

Definition. A lattice is a poset admitting all finite meets and joins.

Removing the finiteness condition in favor of arbitrary meets and joins, we get the notion of a complete lattice. Both examples above are complete lattices, but not every lattice is complete: under the same notions of meet and join as in the powerset example, the lattice of open subsets of a topological space is not complete, since the arbitrary intersection of opens need not be open.

Incohesive thoughts

• Does the powerset of a poset inherit some kind of order?
• Right-adjoint functors preserve limits, hence meets. What does an adjunction between posets look like?