Introduction to strong and weak maps, including issues around realizability and topological representations, is the topic of this chapter.

Introduction to the oriented matroid abstraction of convex polytopes is the goal of this chapter. The chapter shows that the existence of a polar of an oriented matroid polytope is closely related to the existence of an adjoint. The Lawrence construction is introduced and used to produce interesting examples.

Some basic properties and operations, including poset structure of covector sets and flats, are provided in this chapter.

The chapter begins with localizations, including a topological proof of Las Vergnas’s characterization of localizations. Adjoints and their relationship to extensions are discussed. The final part of the chapter discusses intersection properties, particularly on the Euclidean property and non-Euclidean oriented matroids.

Several analogs to fans and triangulations of point configurations are introduced and motivated as representability issues. The equivalence of some of these analogs is established, while others remain open. Results on the topology of triangulations are proved, most notably for Euclidean oriented matroids.

A proof of the Topological Representation Theorem, including an introduction to shelling, topological interpretation of oriented matroid concepts, and an application to counting topes, are provided in this chapter.

Corecursive algebras are algebras that admit unique solutions of recursive equation systems. We study these and a generalization: completely iterative algebras. The terminal coalgebra turns out to be the initial corecursive algebra as well as the initial completely iterative algebra. Dually, the initial algebra is the initial (parametrically) recursive coalgebra. These results explain the title of the chapter. We apply recursive coalgebras in order to obtain a new proof of the Initial Algebra Theorem from Chapter 6.

Well-founded coalgebras generalize well-foundedness for graphs, and they capture the induction principle for well-founded orders on an abstract level. Taylor’s General Recursion Theorem shows that, under hypotheses, every well-founded coalgebra is parametrically recursive. We give a new proof of this result, and we show that it holds for all set functors, and for all endofunctors preserving monomorphisms on a complete and well-powered category with smooth monomorphisms. The converse of the theorem holds for set functors preserving inverse images. We provide an iterative construction of the well-founded part of a given coalgebra: It is carried by the least fixed point of Jacobs’ next-time operator.

This chapter presents a number of sufficient conditions to guarantee that an endofunctor has an initial algebra or a terminal coalgebra. We generalize Kawahara and Mori’s notion of a bounded set functor and prove that for a cocomplete and co-well-powered category with a terminal object, every endofunctor bounded by a generating set has a terminal coalgebra. We use this to show that every accessible endofunctor on a locally presentable category has an initial algebra and a terminal coalgebra. We introduce pre-accessible functors and prove that on a cocomplete and co-well-powered category, the initial-algebra chain of a pre-accessible functor converges, and so the initial algebra exists. If the base category is locally presentable and the functor preserves monomorphisms, then the terminal coalgebra exists.