Subsection 9.1.7 (01AG): Representably Essentially Injective Morphisms

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Table of Contents
Part 4: Bicategories
Chapter 9: Types of Morphisms in Bicategories
Section 9.1: Monomorphisms in Bicategories
Subsection 9.1.7: Representably Essentially Injective Morphisms (cite)

9.1.7 Representably Essentially Injective Morphisms

Let $\mathcal{C}$ be a bicategory.

Definition 9.1.7.1.1 ▶ Representably Essentially Injective Morphisms

A $1$-morphism $f\colon A\to B$ of $\mathcal{C}$ is representably essentially injective if, for each $X\in \text{Obj}\webleft (\mathcal{C}\webright )$, the functor

\[ f_{*}\colon \mathsf{Hom}_{\mathcal{C}}\webleft (X,A\webright )\to \mathsf{Hom}_{\mathcal{C}}\webleft (X,B\webright ) \]

given by postcomposition by $f$ is essentially injective.

Remark 9.1.7.1.2 ▶ Unwinding Definition 9.1.7.1.1

In detail, $f$ is representably essentially injective if, for each pair of morphisms $\phi ,\psi \colon X\rightrightarrows A$ of $\mathcal{C}$, the following condition is satisfied:

If $f\circ \phi \cong f\circ \psi $, then $\phi \cong \psi $.

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