Subsection 9.2.1 (01B6): Corepresentably Faithful 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 9.2.1: Corepresentably Faithful Morphisms (cite)

9.2.1 Corepresentably Faithful Morphisms

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

Definition 9.2.1.1.1 ▶ Corepresentably Faithful Morphisms

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

Remark 9.2.1.1.2 ▶ Unwinding Definition 9.2.1.1.1

In detail, $f$ is corepresentably faithful if, for all diagrams in $\mathcal{C}$ of the form

if we have

\[ \alpha \mathbin {\star }\text{id}_{f}=\beta \mathbin {\star }\text{id}_{f}, \]

then $\alpha =\beta $.

Example 9.2.1.1.3 ▶ Examples of Corepresentably Faithful Morphisms

Here are some examples of corepresentably faithful morphisms.

Corepresentably Faithful Morphisms in $\mathsf{Cats}_{\mathsf{2}}$. The corepresentably faithful morphisms in $\mathsf{Cats}_{\mathsf{2}}$ are characterised in Chapter 8: Categories, Item 4 of Proposition 8.5.1.1.2.
Corepresentably Faithful Morphisms in $\textbf{Rel}$. Every morphism of $\textbf{Rel}$ is corepresentably faithful; see Chapter 5: Relations, Item 1 of Proposition 5.3.10.1.1.

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