9.7.2 Monomorphisms of Categories

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

Definition 9.7.2.1.1 ▶ Monomorphisms of Categories

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

Proposition 9.7.2.1.2 ▶ Properties of Monomorphisms of Categories

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

  1. Characterisations. The following conditions are equivalent:
    1. The functor $F$ is a monomorphism of categories.
    2. The functor $F$ is injective on objects and morphisms, i.e. $F$ is injective on objects and the map
      \[ F\colon \textup{Mor}\webleft (\mathcal{C}\webright )\to \textup{Mor}\webleft (\mathcal{D}\webright ) \]

      is injective.

Proof of Proposition 9.7.2.1.2.

Item 1: Characterisations
Omitted.

Question 9.7.2.1.3 ▶ Characterisations of Functors With Monic Pre/Postcomposition

Is there a characterisation of functors $F\colon \mathcal{C}\to \mathcal{D}$ such that:

  1. 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 a monomorphism of categories?

  2. 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 a monomorphism of categories?

This question also appears as [MO 468125].

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