2.6: Summary and further reading

In this chapter we thought of elements of preorders as describing resources, with the or- der detailing whether one resource could be obtained from another. This naturally led to the question of how to describe what could be built from a pair of resources, which led us to consider monoid structures on preorders. More abstractly, these monoidal preorders were seen to be examples of enriched categories, or V-categories, over the symmetric monoidal preorder Bool. Changing Bool to the symmetric monoidal pre- order Cost, we arrived upon Lawvere metric spaces, a slight generalization of the usual notion of metric space. In terms of resources, Cost-categories tell us the cost of obtaining one resource from another.

At this point, we sought to get a better feel for V-categories in two ways. First, we introduced various important constructions: base change, functors, products. Second, we looked at how to present V-categories using labelled graphs; here, perhaps surpris- ingly, we saw that matrix multiplication gives an algorithm to compute the hom-objects from a labelled graph.

Resource theories are discussed in much more detail in [CFS16; Fri17]. The authors provide many more examples of resource theories in science, including in thermodynamics, Shannon’s theory of communication channels, and quantum entanglement. They also discuss more of the numerical theory than we did, including calculating the asymptotic rate of conversion from one resource into another.

Enrichment is a fundamental notion in category theory, and we will we return to it in Chapter 4, generalizing the definition so that categories, rather than mere preorders, can serve as bases of enrichment. In this more general setting we can still perform the constructions we introduced in Section 2.4—base change, functors, products—and many others; the authoratitive, but by no means easy, reference on this is the book by Kelly [Kel05].

While preorders were familiar before category theory came along, Lawvere metric spaces are a beautiful generalization of the previous notion of (symmetric) metric space, that is due to, well, Lawvere. A deeper exploration than the taste we gave here can be found in his classic paper [Law73], where he also discusses ideas like Cauchy completeness in category-theoretic terms, and which hence generalize to other categorical settings.

We observed that while any symmetric monoidal preorder can serve as a base for enrichment, certain preorders—quantales—are better than others. Quantales are well known for links to other parts of mathematics too. The word quantale is in fact a portmanteau of ‘quantum locale’, where quantum refers to quantum physics, and locale is a fundamental structure in topology. For a book-length introduction of quantales and their applications, one might check [Ros90]. The notion of cartesian closed categories, later generalized to monoidal closed categories, is due to Ronnie Brown [Bro61].

Note that while we have only considered commutative quantales, the noncommu- tative variety also arise naturally. For example, the power set of any monoid forms a quantale that is commutative iff the monoid is. Another example is the set of all binary relations on a set X, where multiplication is relational composition; this is non-commutative. Such noncommutative quantales have application to concurrency theory, and in particular process semantics and automata; see [AV93] for details.