Subsection 9.1.6 (01AB): Morphisms Representably Fully Faithful on Cores

Preferences

Font:

PDF Style:

Here's a breakdown of the differences between each PDF style:

Style	Class	Font	Theorem Environments
Style 1	`book`	Alegreya Sans	`tcbthm`
Style 2	`book`	Alegreya Sans	`amsthm`
Style 3	`book`	Arno*	`amsthm`
Style 4	`book`	Computer Modern	`amsthm`

*To be replaced with Linus Romer's Elemaints when it is released.

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

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

The $1$-morphism $f$ is representably faithful on cores (Definition 9.1.5.1.1) and representably full on cores (Definition 9.1.4.1.1).
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.

In detail, $f$ is representably fully faithful on cores if the conditions in Remark 9.1.4.1.2 and Remark 9.1.5.1.2 hold:

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

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

You can also use the contact form below: