Subsection 9.8.1 (016F): Injective on Objects Functors

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

A functor $F\colon \mathcal{C}\to \mathcal{D}$ is injective on objects if the action on objects

\[ F\colon \text{Obj}\webleft (\mathcal{C}\webright )\to \text{Obj}\webleft (\mathcal{D}\webright ) \]

of $F$ is injective.

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

Characterisations. The following conditions are equivalent:
1. The functor $F$ is injective on objects.
2. The functor $F$ is an isocofibration in $\mathsf{Cats}_{\mathsf{2}}$.

Proof of Proposition 9.8.1.1.2.

Item 1: Characterisations

Omitted.

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