Definition 13.1.1.1.1 (019L): Representably Faithful Morphisms

Table of Contents
Part 6: Bicategories
Chapter 13: Types of Morphisms in Bicategories
Section 13.1: Monomorphisms in Bicategories
Subsection 13.1.1: Representably Faithful Morphisms
Definition 13.1.1.1.1: Representably Faithful Morphisms (cite)

13.1.1 Representably Faithful Morphisms

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

Definition 13.1.1.1.1 ▶ Representably Faithful Morphisms

A $1$-morphism $f\colon A\to B$ of $\mathcal{C}$ is representably faithful¹ if, for each $X\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$, 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 13.1.1.1.3.

Remark 13.1.1.1.2 ▶ Unwinding Definition 13.1.1.1.1

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

if we have

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

then $\alpha =\beta $.

Example 13.1.1.1.3 ▶ Examples of Representably Faithful Morphisms

Here are some examples of representably faithful morphisms.

1.
Representably Faithful Morphisms in $\mathsf{Cats}_{\mathsf{2}}$. The representably faithful morphisms in $\mathsf{Cats}_{\mathsf{2}}$ are precisely the faithful functors; see Chapter 11: Categories, Item 2 of Proposition 11.6.1.1.2.
2.
Representably Faithful Morphisms in $\boldsymbol {\mathsf{Rel}}$. Every morphism of $\boldsymbol {\mathsf{Rel}}$ is representably faithful; see Chapter 8: Relations, Item 1 of Proposition 8.4.8.1.1.

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