Subsection 11.1.2 (019R): Representably Full 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 11.1.2: Representably Full Morphisms (cite)

11.1.2 Representably Full Morphisms

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

Definition 11.1.2.1.1 ▶ Representably Full Morphisms

A $1$-morphism $f\colon A\to B$ of $\mathcal{C}$ is representably full¹ 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 full.

¹Further Terminology: Also called simply a full morphism, based on Item 1 of Example 11.1.2.1.3.

Remark 11.1.2.1.2 ▶ Unwinding Definition 11.1.2.1.1

In detail, $f$ is representably full if, for each $X\in \text{Obj}\webleft (\mathcal{C}\webright )$ and each $2$-morphism

of $\mathcal{C}$, there exists a $2$-morphism

of $\mathcal{C}$ such that we have an equality

of pasting diagrams in $\mathcal{C}$, i.e. such that we have

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

Example 11.1.2.1.3 ▶ Examples of Representably Full Morphisms

Here are some examples of representably full morphisms.

Representably Full Morphisms in $\mathsf{Cats}_{\mathsf{2}}$. The representably full morphisms in $\mathsf{Cats}_{\mathsf{2}}$ are precisely the full functors; see Chapter 9: Categories, of Proposition 9.6.2.1.2.
Representably Full Morphisms in $\textbf{Rel}$. The representably full morphisms in $\textbf{Rel}$ are characterised in Chapter 6: Relations, Item 2 of Proposition 6.3.8.1.1.

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