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A Finitary Adjoint Functor Theorem

Jirí Adámek, Lurdes Sousa

Abstract

Graduated locally finitely presentable categories are introduced, examples include categories of sets, vector spaces, posets, presheaves and Boolean algebras. A finitary functor between graduated locally finitely presentable categories is proved to be a right adjoint if and only if it preserves countable limits. For endofunctors on vector spaces or pointed sets even countable products are sufficient. Surprisingly, for set functors there is a single exception of a (trivial) finitary functor preserving countable products but not countable limits.

A Finitary Adjoint Functor Theorem

Abstract

Graduated locally finitely presentable categories are introduced, examples include categories of sets, vector spaces, posets, presheaves and Boolean algebras. A finitary functor between graduated locally finitely presentable categories is proved to be a right adjoint if and only if it preserves countable limits. For endofunctors on vector spaces or pointed sets even countable products are sufficient. Surprisingly, for set functors there is a single exception of a (trivial) finitary functor preserving countable products but not countable limits.
Paper Structure (5 sections, 8 theorems, 40 equations)

This paper contains 5 sections, 8 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Graduated categories
  3. The finitary adjoint functor theorem
  4. Absolute intersections
  5. Set functors preserving countable products

Key Result

Lemma 3.1

Every object $K$ of a graduated locally finitely presentable category is the directed colimit of the diagram of all its finitely presentable subobjects.

Theorems & Definitions (26)

  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Theorem 3.3
  • proof
  • Example 3.4
  • Remark 3.5
  • ...and 16 more