13.1.2 Representably Full Morphisms

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

    Definition 13.1.2.1.1 ▶ Representably Full Morphisms

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

    1. 1Further Terminology: Also called simply a full morphism, based on Item 1 of Example 13.1.2.1.3.

    Remark 13.1.2.1.2 ▶ Unwinding Definition 13.1.2.1.1

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

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

    Example 13.1.2.1.3 ▶ Examples of Representably Full Morphisms

    Here are some examples of representably full morphisms.

  • 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.

    Representably Full Morphisms in $\boldsymbol {\mathsf{Rel}}$. The representably full morphisms in $\boldsymbol {\mathsf{Rel}}$ are characterised in Chapter 8: Relations, Item 2 of Proposition 8.4.8.1.1.

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