Item 2 (0173) — The Clowder Project

Table of Contents
Part 4: Categories
Chapter 11: Categories
Section 11.8: Even More Conditions on Functors
Subsection 11.8.6: Functors Representably Fully Faithful on Cores
Item 2 (cite)

11.8.6 Functors Representably Fully Faithful on Cores

Let $\mathcal{C}$ and $\mathcal{D}$ be categories.

Definition 11.8.6.1.1 ▶ Functors Representably Fully Faithful on Cores

A functor $F\colon \mathcal{C}\to \mathcal{D}$ is representably fully faithful on cores if, for each $X\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Cats}\webright )$, the postcomposition by $F$ functor

\[ F_{*}\colon \mathsf{Core}\webleft (\mathsf{Fun}\webleft (\mathcal{X},\mathcal{C}\webright )\webright )\to \mathsf{Core}\webleft (\mathsf{Fun}\webleft (\mathcal{X},\mathcal{D}\webright )\webright ) \]

is fully faithful.

Remark 11.8.6.1.2 ▶ Unwinding Definition 11.8.6.1.1

In detail, a functor $F\colon \mathcal{C}\to \mathcal{D}$ is representably fully faithful on cores if it satisfies the conditions in Remark 11.8.4.1.2 and Remark 11.8.5.1.2, i.e.:

1.
For all diagrams of the form
with $\alpha $ and $\beta $ natural isomorphisms, if we have $\operatorname {\mathrm{id}}_{F}\mathbin {\star }\alpha =\operatorname {\mathrm{id}}_{F}\mathbin {\star }\beta $, then $\alpha =\beta $.

2.
For each $\mathcal{X}\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Cats}\webright )$ and each natural isomorphism
of $\mathcal{C}$, there exists a natural isomorphism
of $\mathcal{C}$ such that we have an equality
of pasting diagrams in $\mathsf{Cats}_{\mathsf{2}}$, i.e. such that we have

\[ \beta =\operatorname {\mathrm{id}}_{F}\mathbin {\star }\alpha . \]