Subsection 11.1.1 (019K): Representably Faithful Morphisms

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Table of Contents
Part 5: Bicategories
Chapter 11: Types of Morphisms in Bicategories
Section 11.1: Monomorphisms in Bicategories
Subsection 11.1.1: Representably Faithful Morphisms (cite)

11.1.1 Representably Faithful Morphisms

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

Definition 11.1.1.1.1 ▶ Representably Faithful Morphisms

A $1$-morphism $f\colon A\to B$ of $\mathcal{C}$ is representably faithful¹ 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 faithful.

¹Further Terminology: Also called simply a faithful morphism, based on Item 1 of Example 11.1.1.1.3.

Remark 11.1.1.1.2 ▶ Unwinding Definition 11.1.1.1.1

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

if we have

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

then $\alpha =\beta $.

Example 11.1.1.1.3 ▶ Examples of Representably Faithful Morphisms

Here are some examples of representably faithful morphisms.

Representably Faithful Morphisms in $\mathsf{Cats}_{\mathsf{2}}$. The representably faithful morphisms in $\mathsf{Cats}_{\mathsf{2}}$ are precisely the faithful functors; see Chapter 9: Preorders, Item 2 of Proposition 9.6.1.1.2.
Representably Faithful Morphisms in $\textbf{Rel}$. Every morphism of $\textbf{Rel}$ is representably faithful; see Chapter 6: Relations, Item 1 of Proposition 6.3.8.1.1.

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