I skimmed through books 1, 4, and 5 of my new batch and am currently skimming through 3; it seems I don’t have the mathematical prerequisites to get much out of 2. It will take me a long time to digest all of the interesting things I’ve learned, but I thought I’d discuss an interesting idea coming from Lawvere and Schanuel.
An important idea in mathematics is to reduce an object into its “connected components.” This has various meanings depending on context; it is perhaps clearest in the categories 
Let
be a concrete category with a forgetful functor
. If it exists, let
be the left adjoint to
. Then
describes the “discrete” (i.e. “totally disconnected”) objects of
describing the “connected components” of an object in
I think this is a nice illustration of a construction that is illuminated by the abstract approach, so I’ll briefly describe how this works for a few of my favorite categories.
-algebras, we have seen that 
is a functor which sends homomorphisms to continuous functions. However, this is not true for general commutative rings. What we want is for a ring homomorphism 
to be sent to a continuous function 
is just a surjective homomorphism 
where 
. However, this homomorphism need no longer be surjective, so it may land in a subring of 
. The ideal 
is maximal in 
, but its contraction is the ideal 
, which is prime but not maximal. 
where 
is some integral domain, and the contracted ideal of a prime ideal is always prime because a subring of an integral domain is still an integral domain. Now, therefore, is an appropriate time to replace 
, the space of all prime ideals equipped with the Zariski topology, and this time 
. ![\text{MaxSpec } k[x_1, ... x_n] \simeq \mathbb{A}^n(k)](https://s0.wp.com/latex.php?latex=%5Ctext%7BMaxSpec+%7D+k%5Bx_1%2C+...+x_n%5D+%5Csimeq+%5Cmathbb%7BA%7D%5En%28k%29&bg=ffffff&fg=333333&s=0&c=20201002)
, affine 
-space over an algebraically closed field 
. A generic Zariski-closed subset is usually referred to instead as an algebraic set (although some authors also call these varieties), and the terminology does not apply to non-algebraically closed fields. The additional difficulty that arises in the non-algebraically-closed case is that it’s harder to think about points. For example, ![\text{MaxSpec } \mathbb{R}[x]](https://s0.wp.com/latex.php?latex=%5Ctext%7BMaxSpec+%7D+%5Cmathbb%7BR%7D%5Bx%5D&bg=ffffff&fg=333333&s=0&c=20201002)
has two types of points corresponding to the two types of irreducible polynomials: the usual points 
on the real line and additional points 
. These points can be thought of as orbits of the action of 
on ![\text{MaxSpec } \mathbb{C}[x]](https://s0.wp.com/latex.php?latex=%5Ctext%7BMaxSpec+%7D+%5Cmathbb%7BC%7D%5Bx%5D&bg=ffffff&fg=333333&s=0&c=20201002)
by this group action. This picture generalizes.
. In this case, and more generally in the algebraically closed case, there is a reasonably simple description of what the category of affine varieties looks like, but first we have to describe what the morphisms look like and then we have to take the strong Nullstellensatz on faith, since we haven’t proven it yet. 
is a compact Hausdorff space and 
is its ring of functions. From this definition it follows that 
of compact Hausdorff spaces to the category 
of 
-algebras (which we are assuming have identities). Explicitly, a continuous function
is sent to a continuous function 
. The contravariance may look weird if you’re not used to it, but it’s perfectly natural in the case that 
is an 
with the restriction of 
to the image of 
whose kernel is the set of functions which are zero on 
, so it exhibits 
?
Annoying Precision 