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How to Represent Non-Representable Functors

Paolo Perrone

Abstract

The arrows of a category are elements of particular sets, the hom-sets. These sets are functorial, and their functoriality specifies how to compose the arrows with other arrows of the same category. In particular, it allows to form diagrams, making many abstract concepts graphically visible. Presheaves and set-valued functors, in general, are not representable, and so their elements are not arrows in the usual sense. They can however still be seen as "arrow-like structures", which can be post-composed but not pre-composed (for the case of set functors), or pre-composed but not post-composed (for the case of presheaves). Therefore, we can still represent their structure graphically. In this exposition we show how to draw and interpret these generalized diagrams, and how to use them to prove theorems. We will then study in detail, and represent graphically, a few concepts of category theory which are often considered hard to visualize: representability, weighted limits and colimits, and Cauchy completion (for unenriched categories). We also sketch how to interpret the more general case of profunctors, and the Day convolution of presheaves.

How to Represent Non-Representable Functors

Abstract

The arrows of a category are elements of particular sets, the hom-sets. These sets are functorial, and their functoriality specifies how to compose the arrows with other arrows of the same category. In particular, it allows to form diagrams, making many abstract concepts graphically visible. Presheaves and set-valued functors, in general, are not representable, and so their elements are not arrows in the usual sense. They can however still be seen as "arrow-like structures", which can be post-composed but not pre-composed (for the case of set functors), or pre-composed but not post-composed (for the case of presheaves). Therefore, we can still represent their structure graphically. In this exposition we show how to draw and interpret these generalized diagrams, and how to use them to prove theorems. We will then study in detail, and represent graphically, a few concepts of category theory which are often considered hard to visualize: representability, weighted limits and colimits, and Cauchy completion (for unenriched categories). We also sketch how to interpret the more general case of profunctors, and the Day convolution of presheaves.

Paper Structure

This paper contains 20 sections, 38 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Where these ideas have appeared before.
  3. Prerequisites.
  4. Acknowledgements.
  5. Diagrams for set functors and presheaves
  6. Virtual objects and virtual arrows
  7. Representable functors and universal properties
  8. Natural transformations and the Yoneda embedding
  9. Weighted limits and colimits
  10. Weighted cones and cocones
  11. Basic examples
  12. Core results
  13. Weighted (co)limits as (co)ends
  14. Pointwise Kan extensions
  15. Functor pairing
  16. ...and 5 more sections

Key Result

Theorem 2.7

A functor $F:{\mathbf{C}}\boldsymbol{\rightarrow}{\mathbf{Set}}$ is represented by an object $R$ if and only if the inclusion functor $I:{\mathbf{C}}\boldsymbol{\hookrightarrow}{\mathbf{C}}^{+F}$ admits a retraction functor $\Pi:{\mathbf{C}}^{+F}\boldsymbol{\rightarrow}{\mathbf{C}}$ mapping ${\mathb

Theorems & Definitions (110)

  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.7
  • Corollary 2.8
  • proof
  • Remark 2.9
  • ...and 100 more