Subsection 11.2.2 (01BC): Corepresentably Full Morphisms

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Table of Contents
Part 5: Bicategories
Chapter 11: Types of Morphisms in Bicategories
Section 11.2: Epimorphisms in Bicategories
Subsection 11.2.2: Corepresentably Full Morphisms (cite)

11.2.2 Corepresentably Full Morphisms

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

Definition 11.2.2.1.1 ▶ Corepresentably Full Morphisms

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

Remark 11.2.2.1.2 ▶ Unwinding Definition 11.2.2.1.1

In detail, $f$ is corepresentably 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

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

Example 11.2.2.1.3 ▶ Examples of Corepresentably Full Morphisms

Here are some examples of corepresentably full morphisms.

Corepresentably Full Morphisms in $\mathsf{Cats}_{\mathsf{2}}$. The corepresentably full morphisms in $\mathsf{Cats}_{\mathsf{2}}$ are characterised in Chapter 9: Preorders, Item 7 of Proposition 9.6.2.1.2.
Corepresentably Full Morphisms in $\textbf{Rel}$. The corepresentably full morphisms in $\textbf{Rel}$ are characterised in Chapter 6: Relations, Item 2 of Proposition 6.3.10.1.1.

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