Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Categories by Kan extension

David I. Spivak

TL;DR

This work develops a unifying framework for producing categories from existing data via left Kan extensions, anchored by the density comonad that identifies categories with polynomial comonads on Set. It generalizes density constructions to settings with monads, comonads, and distributive laws, yielding broad families of categories including Lawvere theories, product completions, and the simplicial indexing category $\Delta^{op}$, as well as introducing selection categories as a dual, robust construction. The approach leverages pra-functors between copresheaf categories, colax monoidal structures, and enrichment in Prof to establish functoriality and coherence across these constructions. By connecting Kan extensions, density comonads, and selection categories, the paper unifies classical category-theoretic constructions and opens new avenues for structured category generation. Overall, the results provide a versatile toolkit for constructing and analyzing categories from polynomial, Kan-extension-based data with broad implications for algebraic theories and categorical modeling.

Abstract

Categories can be identified -- up to isomorphism -- with polynomial comonads on Set. The left Kan extension of a functor along itself is always a comonad -- called the density comonad -- so it defines a category when its carrier is polynomial. We provide a number of generalizations of this to produce new categories from old, as well as from distributive laws of monads over comonads. For example, all Lawvere theories, all product completions of small categories, and the simplicial indexing category $Δ^{op}$ arise in this way. Another, seemingly much less well-known, example constructs a so-called selection category from a polynomial comonad in a way that's somehow dual to the construction of a Lawvere theory category from a monad; we'll discuss this in more detail. Along the way, we will see various constructions of non-polynomial comonads as well.

Categories by Kan extension

TL;DR

This work develops a unifying framework for producing categories from existing data via left Kan extensions, anchored by the density comonad that identifies categories with polynomial comonads on Set. It generalizes density constructions to settings with monads, comonads, and distributive laws, yielding broad families of categories including Lawvere theories, product completions, and the simplicial indexing category , as well as introducing selection categories as a dual, robust construction. The approach leverages pra-functors between copresheaf categories, colax monoidal structures, and enrichment in Prof to establish functoriality and coherence across these constructions. By connecting Kan extensions, density comonads, and selection categories, the paper unifies classical category-theoretic constructions and opens new avenues for structured category generation. Overall, the results provide a versatile toolkit for constructing and analyzing categories from polynomial, Kan-extension-based data with broad implications for algebraic theories and categorical modeling.

Abstract

Categories can be identified -- up to isomorphism -- with polynomial comonads on Set. The left Kan extension of a functor along itself is always a comonad -- called the density comonad -- so it defines a category when its carrier is polynomial. We provide a number of generalizations of this to produce new categories from old, as well as from distributive laws of monads over comonads. For example, all Lawvere theories, all product completions of small categories, and the simplicial indexing category arise in this way. Another, seemingly much less well-known, example constructs a so-called selection category from a polynomial comonad in a way that's somehow dual to the construction of a Lawvere theory category from a monad; we'll discuss this in more detail. Along the way, we will see various constructions of non-polynomial comonads as well.

Paper Structure

This paper contains 6 sections, 20 theorems, 58 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Left Kan extensions
  3. Pra-functors between copresheaf categories
  4. Generalizing density comonads
  5. Categories by Kan extension
  6. Basic theory of selection categories

Key Result

Proposition 2.2

The symbol $$ is covariantly functorial in $p\colon\mathcal{C}\to\mathcal{D}$ and contravariantly functorial in $q\colon\mathcal{C}\to\mathcal{C'}$. That is, given $\varphi\colon p\Rightarrow p'$ and $\psi\colon q'\Rightarrow q$ we have

Figures (2)

  • Figure 1: Proof of coassociativity for the colax monoidal functor ${\newline p}{\newline p\circ -}$ associated to any accessible $p\colon\mathcal{C}\to\mathcal{D}$ and cocomplete $\mathcal{D}$. See \ref{['thm.monad_comonad_dist']}.
  • Figure 2: Proof of coassociativity for the comonad ${\newline p\circ t}{\newline p\circ k}$ associated to any accessible $p\colon\mathcal{C}\to\mathcal{D}$, cocomplete $\mathcal{D}$, and (co)monad $k,t\colon\mathcal{C}\to\mathcal{C}$. See \ref{['thm.monad_comonad_dist']}.

Theorems & Definitions (61)

  • Remark 2.1
  • Proposition 2.2
  • proof
  • Example 2.3
  • Remark 2.4
  • Theorem 2.5
  • proof
  • Proposition 2.6
  • proof
  • Lemma 3.1
  • ...and 51 more