The usefulness and richness of 2-polygraphs is confirmed by the large number and variety of categories they present. In order to show that a given polygraph is a presentation of a given category, one can either tackle the issue directly, by using rewriting tools, or take a modular approach, by combining already known presentations: this is the route taken in the present chapter. Three significant applications are given. First addressed is the presentation of limits and colimits by means of given presentations of the base categories, and precisely shown is how to systematically build presentations of products, coproducts, and pushouts. Next, it is shown how to add formal inverses to some morphisms of a category at the level of presentations. Finally, distributive laws are investigated in relation to factorization systems on categories. A notion of composition along a distributive law between two small categories sharing the same set of objects is introduced, and it is shown how to derive a presentation of this composite from presentations of the components.

We define and study new categories of ‘pre $C^\infty$-rings with corners’ and ‘ $C^\infty$-rings with corners’. These are pairs (C,C_{ex}) of sets with many algebraic operations. If X is a manifold with corners then taking C to be the set of smooth maps X -> R, and C_{ex} the set of smooth maps X -> [0,infinity), gives a $C^\infty$-ring with corners. It helps to consider C a $C^\infty$-ring or R-algebra, and C_{ex} a commutative monoid which adds ‘corner structure’ to C.

A (pre) $C^\infty$-ring with corners can equivalently defined as a product-preserving functor from the category of Euclidean spaces with corners [0,infinity)^k x R^{n-k} to the category of sets. This is an example of an Algebraic Theory, so theorems from Algebraic Theories apply.

We define interesting subcategories of $C^\infty$-rings with corners with special properties - interior, firm, toric, ... and study their categorical properties, e.g. existence of (co)limits.

Much of the theory of commutative algebras and $C^\infty$-rings used in scheme theory extends to $C^\infty$-rings with corners, e.g. local $C^\infty$-rings with corners and localization, free $C^\infty$-rings with corners, generators, and relations.

“Mutual Coercion, Mutually Agreed Upon” (the phrase comes from Garrett Hardin's classic essay "Tragedy of the Commons") sees the democratic reforms and social reorganization of Attica by the Athenian statesman Cleisthenes in 508 BCE as a case study in systems leveraging. Cleisthenes’s reforms are situated in a nexus of Presocratic (Pythagorean) thinking about limit (peras) and in the context of ideas that circulated at the time under the banners of isonomy (isonomia) and harmony (harmonia). The ancient Athenians, newly freed from political tyranny and the social upheaval of 508, recognized the intrinsic value of limits and restraint and built them into the structures of democratic life. Their example, I argue further, stands as a challenge to environmental and social problems faced by democratic regimes today.

The basis of Aristotle’s strategy for bringing together time and distance covered in an account of motion is his understanding time, distance, and motion as continua. Against the background of Parmenides’ notion of continuity and the atomistic understanding of magnitudes, this chapter investigates Aristotle’s notion of continuity. It demonstrates the extent to which Aristotle adopts a mathematical notion of continuity according to which divisibility without end is crucial, and it expounds his appropriation of this mathematical notion for the physical realm. The main characteristics of Aristotle’s account of continuity turn out to be a new part-whole relation (in which the whole is prior to the parts since it is only by dividing the whole that we gain parts), a new account of limits, a new understanding of infinity, and a careful distinction between potential divisibility and actual division. We see how this new understanding of continuity helps counter the mereological problems of Zeno’s motion paradoxes by making it illegitimate to define time, space, and motion in terms of an additive part-whole-relation.

This chapter argues for both a broad similarity and a crucial distinction between the ways in which Lucretius (De rerum natura) and Seneca (Naturales quaestiones) employ the category of the sublime in their natural scientific (including cosmological) writings. Both authors, as previous scholars such as Gian Biago Conte and Gareth Williams have observed, use the sublime as part and parcel of their didactic and consolatory projects. Moreover, in both authors the sublime first causes the pupil-reader to "take fright" in the face of nature and then builds him up to conquer nature with his own knowledge. Yet in Lucretius and Seneca the knowledge that allows such conquest is quite distinct: for the Epicurean, scientific knowledge is limited, while for the Stoic, it approaches omniscience.