A $1$-morphism $f\colon A\to B$ of $\mathcal{C}$ is corepresentably fully faithful1 if the following equivalent conditions are satisfied:
-
1.
The $1$-morphism $f$ is corepresentably full (Definition 13.2.2.1.1) and corepresentably faithful (Definition 13.2.1.1.1).
-
2.
For each $X\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$, 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.
- 1Further Terminology: Corepresentably fully faithful morphisms have also been called lax epimorphisms in the literature (e.g. in [AESV, On Functors Which Are Lax Epimorphisms]), though we will always use the name “corepresentably fully faithful morphism” instead in this work.
-
1.
For all diagrams in $\mathcal{C}$ of the form
if we have
\[ \alpha \mathbin {\star }\operatorname {\mathrm{id}}_{f}=\beta \mathbin {\star }\operatorname {\mathrm{id}}_{f}, \]then $\alpha =\beta $.
-
2.
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
\[ \beta =\alpha \mathbin {\star }\operatorname {\mathrm{id}}_{f}. \] -
1.
Corepresentably Fully Faithful Morphisms in $\mathsf{Cats}_{\mathsf{2}}$. The fully faithful epimorphisms in $\mathsf{Cats}_{\mathsf{2}}$ are characterised in Chapter 11: Categories, Item 10 of Proposition 11.6.3.1.2.
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2.
Corepresentably Fully Faithful Morphisms in $\boldsymbol {\mathsf{Rel}}$. The corepresentably fully faithful morphisms of $\boldsymbol {\mathsf{Rel}}$ coincide (Chapter 8: Relations, Item 3 of Proposition 8.4.10.1.1) with the corepresentably full morphisms in $\boldsymbol {\mathsf{Rel}}$, which are characterised in Chapter 8: Relations, Item 2 of Proposition 8.4.10.1.1.
13.2.3 Corepresentably Fully Faithful Morphisms
Let $\mathcal{C}$ be a bicategory.
In detail, $f$ is corepresentably fully faithful if the conditions in Remark 13.2.1.1.2 and Remark 13.2.2.1.2 hold:
Here are some examples of corepresentably fully faithful morphisms.
