The Waves of a Total Category

For any total category K , with defining adjunction ∨ a Y : K // set op , the expression W (A)(K) = set K op (K (A, ∨ −), [K,−]), where [K,−] is evaluation at K, provides a well-defined functor W : K // K̂ = set op . Also, there are natural transformations β : W ∨ // 1 K̂ and γ : ∨ W // 1K satisfying ∨ β = γ ∨ and βW = Wγ. A total K is totally distributive if ∨ has a left adjoint. We show that K is totally distributive iff γ is invertible iff W a ∨ . The elements of W (A)(K) are called waves from K to A. Write K̃ (K,A) for W (A)(K). For any total K there is an associative composition of waves. Composition becomes an arrow • : K̃ ◦K K̃ // K̃ . Also, there is an augmentation ̄ (−) : K̃ (−,−) // K (−,−) corresponding to a natural δ :W // Y constructed via β. We show that if K is totally distributive then • : K̃ ◦K K̃ // K̃ is invertible and then K̃ supports an idempotent comonad structure. In fact, K̃ ◦K K̃ = K̃ ◦K̃ K̃ so that • is the coequalizer of •K and K •, making K̃ a taxon in the sense of Koslowski [KOS]. For a small taxon T , the category of interpolative modules iMod(1,T ) is totally distributive [MRW]. Here we show, for any totally distributive K , that there is an equivalence K // iMod(1, K̃ ). 1. Preliminaries 1.1. Most of our notations and conventions are carried over from [MRW]. For a category K , we write |K | for the set of objects of K . If K and A are objects of K , K (K,A) denotes the set of arrows from K to A. We assume the existence of full categories of sets called set and SET, both toposes, with set a (full) subcategory of SET, and |set| an object of SET. The sets in set are called small sets. We assume that set has all sums indexed by objects of set and SET has all sums indexed by objects of SET. We will write i : set // SET for the inclusion. We write CAT for the 2-category of category objects in SET. If a category K in CAT has all its hom-sets K (K,A) in set, we say that K is locally small. For K and A in CAT, we write PRO(K ,A ) for the category The author gratefully acknowledges financial support from the Canadian NSERC. Diagrams typeset using M. Barr’s diagxy package Received by the editors 2015-07-14 and, in revised form, 2015-11-09. Transmitted by Ross Street. Published on 2015-11-30. 2010 Mathematics Subject Classification: 18A30.