14.1.2 Representably Full Morphisms

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

Definition 14.1.2.1.1 ▶ Representably Full Morphisms

A $1$-morphism $f\colon A\to B$ of $\mathcal{C}$ is representably full1 if, for each $X\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, the functor

\[ f_{*}\colon \mathsf{Hom}_{\mathcal{C}}(X,A)\to \mathsf{Hom}_{\mathcal{C}}(X,B) \]

given by postcomposition by $f$ is full.

  1. 1Further Terminology: Also called simply a full morphism, based on Item 1 of Example 14.1.2.1.3.

Remark 14.1.2.1.2 ▶ Unwinding Definition 14.1.2.1.1

In detail, $f$ is representably full if, for each $X\in \operatorname {\mathrm{Obj}}(\mathcal{C})$ 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

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

Example 14.1.2.1.3 ▶ Examples of Representably Full Morphisms

Here are some examples of representably full morphisms.

  1. 1.

    Representably Full Morphisms in $\mathsf{Cats}_{\mathsf{2}}$. The representably full morphisms in $\mathsf{Cats}_{\mathsf{2}}$ are precisely the full functors; see Chapter 11: Categories, Unresolved reference of Proposition 11.6.2.1.2.

  2. 2.

    Representably Full Morphisms in $\boldsymbol {\mathsf{Rel}}$. The representably full morphisms in $\boldsymbol {\mathsf{Rel}}$ are characterised in Chapter 8: Relations, Unresolved reference of Unresolved reference.

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