9.2.5 Morphisms Corepresentably Full on Cores

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

Definition 9.2.5.1.1 ▶ Morphisms Corepresentably Full on Cores

A $1$-morphism $f\colon A\to B$ of $\mathcal{C}$ is corepresentably full on cores if, for each $X\in \text{Obj}\webleft (\mathcal{C}\webright )$, the functor

\[ f^{*}\colon \mathsf{Core}\webleft (\mathsf{Hom}_{\mathcal{C}}\webleft (B,X\webright )\webright )\to \mathsf{Core}\webleft (\mathsf{Hom}_{\mathcal{C}}\webleft (A,X\webright )\webright ) \]

given by precomposition by $f$ is full.

Remark 9.2.5.1.2 ▶ Unwinding Definition 9.2.5.1.1

In detail, $f$ is corepresentably full on cores if, 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

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

Previous
Next

View Subsection 9.2.5 as PDF

Statistics Cite

Noticed something off, or have any comments? Feel free to reach out!

You can also use the contact form below: