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

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

    • Question 11.8.6.1.3 ▶ Characterisation of Functors Representably Fully Faithful on Cores

    Is there a characterisation of functors representably fully faithful on cores?

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