Subsection 9.7.3 (0155): Epimorphisms of Categories

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Table of Contents
Part 3: Category Theory
Chapter 9: Categories
Section 9.7: More Conditions on Functors
Subsection 9.7.3: Epimorphisms of Categories (cite)

9.7.3 Epimorphisms of Categories

Let $\mathcal{C}$ and $\mathcal{D}$ be categories.

Definition 9.7.3.1.1 ▶ Epimorphisms of Categories

A functor $F\colon \mathcal{C}\to \mathcal{D}$ is a epimorphism of categories if it is a epimorphism in $\mathsf{Cats}$ (see ).

Proposition 9.7.3.1.2 ▶ Properties of Epimorphisms of Categories

Let $F\colon \mathcal{C}\to \mathcal{D}$ be a functor.

Characterisations. The following conditions are equivalent:
¹
1. The functor $F$ is a epimorphism of categories.
2. For each morphism $f\colon A\to B$ of $\mathcal{D}$, we have a diagram
  in $\mathcal{D}$ satisfying the following conditions:
  1. We have $f=\alpha _{0}\circ \phi _{1}$.
  2. We have $f=\psi _{m}\circ \alpha _{2m}$.
  3. For each $0\leq i\leq 2m$, we have $\alpha _{i}\in \textup{Mor}\webleft (\mathrm{Im}\webleft (F\webright )\webright )$.
Surjectivity on Objects. If $F$ is an epimorphism of categories, then $F$ is surjective on objects.

¹Further Terminology: This statement is known as Isbell’s zigzag theorem.

[Proof of Proposition 9.7.3.1.2]

Item 1: Characterisations

See [Isbell, Epimorphisms and Dominions III].

Item 2: Surjectivity on Objects

Omitted.

Question 9.7.3.1.3. [Characterisations of Functors With Epic Pre/Postcomposition]
Is there a characterisation of functors $F\colon \mathcal{C}\to \mathcal{D}$ such that:

For each $\mathcal{X}\in \text{Obj}\webleft (\mathsf{Cats}\webright )$, the precomposition functor
\[ F^{*}\colon \mathsf{Fun}\webleft (\mathcal{D},\mathcal{X}\webright )\to \mathsf{Fun}\webleft (\mathcal{C},\mathcal{X}\webright ) \]

is an epimorphism of categories?
For each $\mathcal{X}\in \text{Obj}\webleft (\mathsf{Cats}\webright )$, the postcomposition functor
\[ F_{*}\colon \mathsf{Fun}\webleft (\mathcal{X},\mathcal{C}\webright )\to \mathsf{Fun}\webleft (\mathcal{X},\mathcal{D}\webright ) \]

is an epimorphism of categories?

This question also appears as [MO 468125].

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