11.8.9 Functors Corepresentably Fully Faithful on Cores

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

    Definition 11.8.9.1.1 ▶ Functors Corepresentably Fully Faithful on Cores

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

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

    is fully faithful.

    Remark 11.8.9.1.2 ▶ Unwinding Definition 11.8.9.1.1

    In detail, a functor $F\colon \mathcal{C}\to \mathcal{D}$ is corepresentably fully faithful on cores if it satisfies the conditions in Remark 11.8.7.1.2 and Remark 11.8.8.1.2, i.e.:

  • 1.

    For all diagrams of the form

    if $\alpha $ and $\beta $ are natural isomorphisms and we have

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

    then $\alpha =\beta $.

  • 2.

    For each $\mathcal{X}\in \operatorname {\mathrm{Obj}}(\mathsf{Cats})$ and each natural isomorphism

    there exists a natural isomorphism
    such that we have an equality
    of pasting diagrams in $\mathsf{Cats}_{\mathsf{2}}$, i.e. such that we have

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

    • Question 11.8.9.1.3 ▶ Characterisation of Functors Corepresentably Fully Faithful on Cores

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

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