Categories and Modules

Modules (also known as profunctors or distributors) and morphisms among them subsume categories and functors and provide more general and abstract framework to explore the theory of structures. In this book we generalize and redevelop the basic notions and results of category theory using this framework of modules. Topics Chapter 1 introduces modules and cells among them. A moduleM ∶ X⇢A from a category X to A is a functor of the formM ∶ Xop×A→ Set, assigning a set of arrows to each pair of objects x ∈ X and a ∈ A. A cell from a moduleM to N sends each arrow ofM to an arrow of N . The hom-functor of a category C forms an endomodule ⟨C⟩ ∶ C⇢C called the hom of C, and the arrow function of a functor F ∶ C →D forms a cell ⟨F⟩ ∶ ⟨C⟩ ⇢ ⟨D⟩ from the hom of C to the hom of D. Modules and cells thus subsume categories and functors in this way and set up a more general and conceptual framework to explore the structure of mathematics. Presheaves and copresheaves are called right and left modules respectively in this book and studied as special instances of modules. Chapter 2 discusses the action of a module on its domain and codomain, the operation that yields the Yoneda embedding functor in the case of a hom endomodule. The chapter introduces an important class of modules called representable, which each functor produces by the composition with the hom of its codomain. Chapter 3 presents two variants of modules, namely collages and commas, which are special sorts of cospans and spans between two categories. We will establish an isomorphism between the category of modules and the category of collages, and, later in Chapter 9, construct an adjoint equivalence between the category of commas and the category of collages. Two forgetful functors fromMOD (the category of modules and cells) to CAT are de ned through construction of collages and commas, and it is shown that they form left and right adjoints of the embedding CAT↪MOD given by the hom operation C↦ ⟨C⟩. Chapter 4 introduces the notion of frames of a module. A cylindrical frame of an endomodule abstracts a natural transformation between two functors, and a conical frame of a right (resp. left) module abstracts a cone between an object and a functor. Inner and outer cylinders, which are, so to speak, natural and extranatural transformations spanning a module, are de ned using cylindrical frames. Likewise, cones are de ned along a module using conical frames. Chapter 5 succeeds Chapter 2 and discusses the actions of the domain and codomain of a module on its arrows and frames. It is shown, as a generalization of the Yoneda embedding, that these actions embed a module X⇢A in (the hom of) the category of right modules (i.e. presheaves) over X and the category of left module (i.e. copresheaves) over A. The Yoneda lemma is presented in a general form to state that the morphisms from the representable module of a functor F ∶ X → A to an arbitrary module M ∶ X ⇢ A correspond one-to-one with the cylinders de ned between F andM. Using the lemma, we show a variety of bijective correspondences between frames and cells. Chapters 6, 7, 8, and 10 explore the fundamental concepts of category theory in the framework of modules. Universals are de ned along a module and we study if they are preserved by a cell. The embedding of a module in the category of presheaves (resp. copresheaves) makes the de nition of a universal simple: an arrow of a module is universal if its image under the embedding is an isomorphism. Lifts, of which Kan lifts are special instances, are de ned as universal cylinders, and extensions, of which Kan extensions are special instances, are de ned as universal cells. We introduce the notion of pointwise lifts and show that the