Subsection 9.1.8 (01AK): Representably Conservative 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 9.1.8: Representably Conservative Morphisms (cite)

9.1.8 Representably Conservative Morphisms

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

Definition 9.1.8.1.1 ▶ Representably Conservative Morphisms

A $1$-morphism $f\colon A\to B$ of $\mathcal{C}$ is representably conservative 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 conservative.

Remark 9.1.8.1.2 ▶ Unwinding Definition 9.1.8.1.1

In detail, $f$ is representably conservative if, for each pair of morphisms $\phi ,\psi \colon X\rightrightarrows A$ and each $2$-morphism

of $\mathcal{C}$, if the $2$-morphism

is a $2$-isomorphism, then so is $\alpha $.

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