Why Do Rings Represent Spaces?

This article is currently unfinished.

It is commonly known that many notions of "spaces" are dual to categories of rings. This is often derived from two well-known cases: the first is the case of locally compact Hausdorff spaces being dually emebedded into the category of C*-algebras as exactly the commutative C*-algebras, and the second is the case of algebraic varieties over an algebraically closed field 𝔽 being dually embedded into the category of commutative 𝔽-algebras. This correspondence can also be extended to other cases, such as the dual embedding of smooth manifolds into the category of commutative ℝ-algebras. There is also the Stone duality between Stone spaces and Boolean algebras, though Boolean algebras aren't rings, but they are close enough for our purposes. Besides, the theory Boolean algebras is equivalent to the theory of Boolean rings, so rings once again appear.

Of course, this type of correspondence can easily be explained abstractly by Isbell duality, the formal duality between algebraic and geometric objects. Yet, if this is the case, why is it that categories of spaces that are actually interesting are dual to categories of vaguely ring-like objects? I have wondered for some time if this is because we are merely short-sighted, or if there is a more fundamental reason for why this must be true. It was recently that I had a sudden realization that gave this correspondence some amount of clarity.

When working with categories, two important kinds of objects are initial and terminal objects. To recap, an initial object βˆ… is such that, given any other object c in the category, there is a unique morphism from βˆ… to c. Similarly, a terminal object * is such that, given any other object c in the category, there is a unique morphism from c to *. In many categories that occur "in practice," at least one of these is present, and often, both are present. In fact, it turns out that demanding certain properties of these two kinds of objects to hold when they "interact with" each other is very powerful.

It was everyone's favorite "alleged" daughter-fucker, Peter Freyd (my intuition tells me that it is advisable to add "alleged" to this article), who realized that pretopoi and Abelian categories can be described by a common set of axioms about exactness, producing what are known as AT categories. In this setup, what differentiates pretopoi from Abelian categories is precisely how initial objects interact with products. Given an object X, one has a product βˆ…Γ—X, with projections Ο€1, onto βˆ…, and Ο€2, onto X. An object is of "type T" if Ο€1 is an isomorphism, and is of "type A" if Ο€2 is an isomorphism. If every object is of type T, then the AT category is a pretopos, and if every object is of type A, then it is an Abelian category. These two cases are essentially the only interesting ones, as all AT categories are equivalent to a product of a pretopos and an Abelian category.

The property of βˆ… being an "absorbing element" of products is a distinct feature of many categories, often considered some form of "spaces." For instance, this holds in the category Set, whose objects can somewhat trivially be considered spaces. It also holds, as mentioned earlier, in any pretopos, and in particular in any topos. Generalizing in a different direction, this also holds in Top, as well as similar categories like PreOrd. What should also be of note is that all of the previously-mentioned categories have further nice interactions between products and more general colimits. In fact, various topoi, and PreOrd, are Cartesian closed categories, meaning that functors formed by taking products always have right adjoints. This means that products not only preserve initial objects, but also all colimits that exist in these categories, including some large ones.

The other case, of βˆ… being another "identity element" of products, can be reduced to stating that the AT category's βˆ… equals its *. This means that we are working in a category containing a zero object. There are some well-known cases of this occurring. Perhaps the simplest to construct is the category of pointed sets, where a pointed set containing only the distinguished point is a zero object. In fact, any category with a zero object is automatically enriched over pointed sets with the smash product, so in some sense, this case is fundamental. Extrapolating from this case, categories with zero objects are often called "pointed categories." Of note is that, unlike the previously-discussed categories, the opposite category of a pointed category is again a pointed category.

A natural extension of the concept of initial and terminal objects collapsing into zero objects is that of products and coproducts coinciding. If one has finite products and finite coproducts, and there are natural isomorphisms between the two, one says that the category has biproducts. Perhaps the most well-known case of this is the category of vector spaces over some field. This property also holds for the broader class of Abelian groups, and more generally, left (or right) modules over some ring R. However, what is the most natural case of a category with biproducts, analogous to the pointed case from before, is the category of commutative monoids. This is because any category with finite biproducts is automatically enriched over commutative monoids with the standard tensor product. Another way to state this is that every category with finite biproducts is automatically a semiadditive category. Given the exactness required to define an AT category, it can be said that a sufficiently exact semiadditive category is automatically Abelian. In fact, Abelian categories are precisely the pointed AT categories, so more generally, sufficently exact pointed categories are automatically Abelian.

Continuing the pattern from earlier, the opposite category of a semiadditive category is also semiadditive. In fact, it's possible for a nontrivial semiadditive category to be dual to itself. Take the category of locally compact Hausdorff topological groups. For any object in this category, we can form its dual, which will also be a locally compact Hausdorff topological group. If we perform this operation again, we get a group that is isomorphic to the original. There is also a good notion of dualization in the category of suplattices. As an aside, demanding that a category has all small biproducts automatically enriches it over the category of suplattices with the standard tensor product.

One important question, then, is how commutative monoids are related to bare sets. The answer lies in tensor products, and why we care about them. Of course, the role of a tensor product has been sufficiently generalized to the context of monoidal categories, but this setting is somewhat abstract. In general, a good "tensor product" should allow one to encode "multilinear operations" in a meaningful way. Meaning, we have a function, not necessarily a morphism, that takes multiple inputs, such that, separately in each argument, it's a morphism of the category. We then want to be able to interpret these "multilinear" morphisms as "linear" morphisms from a sufficiently constructed object. In the case of Abelian groups, and more generally commutative monoids, this notion gives us the classical tensor product. In this vein, the case of pointed sets gives us the smash product as our equivalent of the tensor product. Finally, the case of bare sets turns out to be such that the standard Cartesian product is the analogue of the tensor product.

Once we have an appropriately-defined tensor product, we are now working within a monoidal category, meaning we can form monoid objects. Further, we are working within a symmetric monoidal category, meaning we can form commutative monoid objects. Finally, we are working within a closed symmetric monoidal category, meaning that functors derived from tensor products always have right adjoints. This means that the tensor product interacts nicely with all possible colimits, including some large ones. We can now ask ourselves: what are the categories of commutative monoids over these categories like? Over Set, the category of commutative monoids is just that: the category of commutative monoids. Yet, these have their own tensor product. What if we were to "double" the operation of constructing categories of commutative monoid objects? These give us some structure with two commutative monoid structures, such that the second one produces monoid morphisms of the first one. It doesn't take long to see that this condition can be written down concretely as the condition that the second operation distributes over the first. This means that "doubling" the operation of constructing commutative monoids, starting with Set, gives us the category of commutative semirings! Given that Abelian categories are the sufficiently exact semiadditive categories, we can extrapolate that rings are (perhaps obviously) the "nice" semirings.

Now, given our previous emphasis on properties of initial and terminal objects, what sorts of important properties do semirings have? It turns out, they satisfy the dual property to that of categories of "spaces," meaning that * is an "absorbing element" of coproducts. While it is somewhat by fiat, it turns out that, in our definition, the category of (semi)rings is dual to a category of "spaces," like in practice. In fact, declaring sufficiently Set-like categories to be categories of "spaces" lines up nicely with another aspect of algebraic geometry: When one defines notions such as schemes and their close cousins, one equips CRingop with a coverage, then takes sheaves on this site. While this is naΓ―vely a large site, it turns out that it is also permissible to merely give CRingfinop a coverage. Because this category is essentially small, it turns out that the category of sheaves on this site then forms a Grothendieck topos. Grothendieck topoi are at the perfect insersection of "sufficiently Set-like" and "sufficiently spatial," so this all comes full circle.

At this point, perhaps some of you are wondering if the set -> commutative monoid -> commutative semiring sequence has a next step. It unfortunately doesn't. To see this, notice that we defined our notion of a decent tensor product in terms of "bilinearity," being "linear" in each variable separately. On the category of commutative semirings, this must preserve both addition and 0 in each variable, but it also must preserve multiplication and 1 in each. It doesn't take long to see that, in the codomain of any such "bilinear" map, 0 and 1 must be equal to each other, so the codomain must be the zero ring. This means that attempting to construct a next step in out sequence will only produce essentially trivial examples. Perhaps this was evident from the key property of "repeating" the commutative monoid construction: Categories whose initial objects absorb products are taken to pointed categories, which get taken to categories whose terminal objects abosrb coproducts. There is nowhere else to go after this last step, so the collapse into essential triviality, in truth, should be expected.

This article is currently unfinished.