Subsection 8.6.3 (0155): Epimorphisms of Categories

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

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

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.

Proof of Proposition 8.6.3.1.2.

Item 1: Characterisations

Item 2: Surjectivity on Objects

Omitted.

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].

Footnotes

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

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