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A Note on the Non-Existence of Functors

Emmanuel Dror Farjoun, Sergei O. Ivanov, Aleksandr Krasilnikov, Anatolii Zaikovskii

TL;DR

The work investigates when functors between large and small categories must be trivial, establishing general nonexistence results for functors from groups (and related categories) to small targets and proving a strong embedding property for nontrivial subfunctors of the identity. It develops a framework showing that, under broad conditions, such subfunctors force the presence of simple groups in their essential image and prohibit nontrivial subfunctors into many reflective subcategories; it also analyzes quotients of the identity functor, demonstrating strong restrictions in the group and abelian categories and highlighting nuanced behavior in abelian settings via divisible and $\mathbb{Q}$-vector-space targets. The paper connects these categorical nonexistence results to questions in homotopy theory and $\infty$-categories, proposing natural analogues and highlighting the limits of natural transformations to identities and augmented/coaugmented functors. Overall, it clarifies when natural subobjects or quotients of identity fail to exist and characterizes the images forced by any hypothetical such functors.

Abstract

We consider several types of non-existence theorems for functors. For example, there are no nontrivial functors from the category of groups (or the category of pointed sets, or vector spaces) to any small category. Another type of questions that we consider are questions about nonexistence of subfunctors and quotients of the identity functor on the category of groups (or abelian groups). For example, there is no a natural non-trivial way to define an abelian subgroup of a group, or a perfect quotient group of a group. As an auxiliary result we prove that, for any non-trivial subfunctor $F$ of the identity functor on the category of groups, any group can be embedded into a simple group that lies in the essential image of $F.$ The paper concludes with a few questions regarding the non-existence of certain (co-)augmented functors in the $\infty$-category of spaces.

A Note on the Non-Existence of Functors

TL;DR

The work investigates when functors between large and small categories must be trivial, establishing general nonexistence results for functors from groups (and related categories) to small targets and proving a strong embedding property for nontrivial subfunctors of the identity. It develops a framework showing that, under broad conditions, such subfunctors force the presence of simple groups in their essential image and prohibit nontrivial subfunctors into many reflective subcategories; it also analyzes quotients of the identity functor, demonstrating strong restrictions in the group and abelian categories and highlighting nuanced behavior in abelian settings via divisible and -vector-space targets. The paper connects these categorical nonexistence results to questions in homotopy theory and -categories, proposing natural analogues and highlighting the limits of natural transformations to identities and augmented/coaugmented functors. Overall, it clarifies when natural subobjects or quotients of identity fail to exist and characterizes the images forced by any hypothetical such functors.

Abstract

We consider several types of non-existence theorems for functors. For example, there are no nontrivial functors from the category of groups (or the category of pointed sets, or vector spaces) to any small category. Another type of questions that we consider are questions about nonexistence of subfunctors and quotients of the identity functor on the category of groups (or abelian groups). For example, there is no a natural non-trivial way to define an abelian subgroup of a group, or a perfect quotient group of a group. As an auxiliary result we prove that, for any non-trivial subfunctor of the identity functor on the category of groups, any group can be embedded into a simple group that lies in the essential image of The paper concludes with a few questions regarding the non-existence of certain (co-)augmented functors in the -category of spaces.
Paper Structure (6 sections, 26 theorems, 21 equations)

This paper contains 6 sections, 26 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Functors from large to small categories
  3. Subfunctors and quotients of the identity functor
  4. Subfunctors of the identity functor of the category of groups
  5. Quotients of the identity functor of the category of groups
  6. The category of abelian groups

Key Result

Theorem 1

If there is a cardinal $\kappa$ such that $\mathpzc{C}$ is a strongly connected category that has products indexed by sets of cardinality $\leq \kappa,$ and $\mathpzc{D}$ is an essentially small category whose hom-sets have cardinality $\leq \kappa,$ then any functor from $\mathpzc{C}$ to $\mathpzc{

Theorems & Definitions (54)

  • Theorem
  • Theorem
  • Theorem
  • Theorem
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • proof
  • ...and 44 more