11.1.6 Morphisms Representably Fully Faithful on Cores

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

Definition 11.1.6.1.1 ▶ Morphisms Representably Fully Faithful on Cores

A $1$-morphism $f\colon A\to B$ of $\mathcal{C}$ is representably fully faithful on cores if the following equivalent conditions are satisfied:

  1. The $1$-morphism $f$ is representably faithful on cores (Definition 11.1.5.1.1) and representably full on cores (Definition 11.1.4.1.1).
  2. For each $X\in \text{Obj}\webleft (\mathcal{C}\webright )$, the functor
    \[ f_{*}\colon \mathsf{Core}\webleft (\mathsf{Hom}_{\mathcal{C}}\webleft (X,A\webright )\webright )\to \mathsf{Core}\webleft (\mathsf{Hom}_{\mathcal{C}}\webleft (X,B\webright )\webright ) \]

    given by postcomposition by $f$ is fully faithful.

Remark 11.1.6.1.2 ▶ Unwinding Definition 11.1.6.1.1

In detail, $f$ is representably fully faithful on cores if the conditions in Remark 11.1.4.1.2 and Remark 11.1.5.1.2 hold:

  1. For all diagrams in $\mathcal{C}$ of the form

    if $\alpha $ and $\beta $ are $2$-isomorphisms and we have

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

    then $\alpha =\beta $.

  2. For each $X\in \text{Obj}\webleft (\mathcal{C}\webright )$ and each $2$-isomorphism
    of $\mathcal{C}$, there exists a $2$-isomorphism
    of $\mathcal{C}$ such that we have an equality
    of pasting diagrams in $\mathcal{C}$, i.e. such that we have
    \[ \beta =\text{id}_{f}\mathbin {\star }\alpha . \]

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