Subsection 9.2.8 (01C6): Corepresentably Conservative Morphisms

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Table of Contents
Part 4: Bicategories
Chapter 9: Types of Morphisms in Bicategories
Section 9.2: Epimorphisms in Bicategories
Subsection 9.2.8: Corepresentably Conservative Morphisms (cite)

9.2.8 Corepresentably Conservative Morphisms

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

Definition 9.2.8.1.1 ▶ Corepresentably Conservative Morphisms

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

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

given by precomposition by $f$ is conservative.

Remark 9.2.8.1.2 ▶ Unwinding Definition 9.2.8.1.1

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

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

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

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