Subsection 9.7.3 (0155): Epimorphisms of Categories

Preferences





Font:

PDF Style:

Here's a breakdown of the differences between each PDF style:

Style	Class	Font	Theorem Environments
Style 1	`book`	Alegreya Sans	`tcbthm`
Style 2	`book`	Alegreya Sans	`amsthm`
Style 3	`book`	Arno*	`amsthm`
Style 4	`book`	Computer Modern	`amsthm`

*To be replaced with Linus Romer's Elemaints when it is released.

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.

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

Proof of Proposition 9.7.3.1.2.

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].

Noticed something off, or have any comments? Feel free to reach out!

You can also use the contact form below: