9.2.3 Corepresentably Fully Faithful Morphisms

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

Definition 9.2.3.1.1 ▶ Corepresentably Fully Faithful Morphisms

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

  1. The $1$-morphism $f$ is corepresentably full (Definition 9.2.2.1.1) and corepresentably faithful (Definition 9.2.1.1.1).
  2. 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 fully faithful.

Remark 9.2.3.1.2 ▶ Unwinding Definition 9.2.3.1.1

In detail, $f$ is corepresentably fully faithful if the conditions in Remark 9.2.1.1.2 and Remark 9.2.2.1.2 hold:

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

    if we have

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

    then $\alpha =\beta $.

  2. 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 9.2.3.1.3 ▶ Examples of Corepresentably Fully Faithful Morphisms

Here are some examples of corepresentably fully faithful morphisms.

  1. Corepresentably Fully Faithful Morphisms in $\mathsf{Cats}_{\mathsf{2}}$. The fully faithful epimorphisms in $\mathsf{Cats}_{\mathsf{2}}$ are characterised in Chapter 8: Categories, Item 9 of Proposition 8.5.3.1.2.
  2. Corepresentably Fully Faithful Morphisms in $\textbf{Rel}$. The corepresentably fully faithful morphisms of $\textbf{Rel}$ coincide (Chapter 5: Relations, Item 3 of Proposition 5.3.10.1.1) with the corepresentably full morphisms in $\textbf{Rel}$, which are characterised in Chapter 5: Relations, Item 2 of Proposition 5.3.10.1.1.

Footnotes

[1] Further Terminology: Corepresentably fully faithful morphisms have also been called lax epimorphisms in the literature (e.g. in [Adámek–Bashir–Sobral–Velebil, On Functors Which Are Lax Epimorphisms]), though we will always use the name “corepresentably fully faithful morphism” instead in this work.
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