11.7.4 Pseudomonic Functors

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

    A functor $F\colon \mathcal{C}\to \mathcal{D}$ is pseudomonic if it satisfies the following conditions:

    1. 1.

      For all diagrams of the form

      if we have

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

      then $\alpha =\beta $.

    2. 2.

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

      there exists a natural isomorphism
      such that we have an equality
      of pasting diagrams, i.e. such that we have

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

    Let $F\colon \mathcal{C}\to \mathcal{D}$ be a functor.

    1. 1.

      Characterisations. The following conditions are equivalent:

      1. (a)

        The functor $F$ is pseudomonic.

      2. (b)

        The functor $F$ satisfies the following conditions:

        1. (i)

          The functor $F$ is faithful, i.e. for each $A,B\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$, the action on morphisms

          \[ F_{A,B} \colon \operatorname {\mathrm{Hom}}_{\mathcal{C}}\webleft (A,B\webright ) \to \operatorname {\mathrm{Hom}}_{\mathcal{D}}\webleft (F_{A},F_{B}\webright ) \]

          of $F$ at $\webleft (A,B\webright )$ is injective.

        2. (ii)

          For each $A,B\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$, the restriction

          \[ F^{\operatorname {\mathrm{iso}}}_{A,B} \colon \operatorname {\mathrm{Iso}}_{\mathcal{C}}\webleft (A,B\webright ) \to \operatorname {\mathrm{Iso}}_{\mathcal{D}}\webleft (F_{A},F_{B}\webright ) \]

          of the action on morphisms of $F$ at $\webleft (A,B\webright )$ to isomorphisms is surjective.

      3. (c)

        We have an isocomma square of the form

        in $\mathsf{Cats}_{\mathsf{2}}$ up to equivalence.

      4. (d)

        We have an isocomma square of the form

        in $\mathsf{Cats}_{\mathsf{2}}$ up to equivalence.

      5. (e)

        For each $\mathcal{X}\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Cats}\webright )$, the postcomposition1 functor

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

        is pseudomonic.

  • 2.

    Conservativity. If $F$ is pseudomonic, then $F$ is conservative.

  • 3.

    Essential Injectivity. If $F$ is pseudomonic, then $F$ is essentially injective.


    1. 1Asking the precomposition functors
      \[ F^{*}\colon \mathsf{Fun}\webleft (\mathcal{D},\mathcal{X}\webright )\to \mathsf{Fun}\webleft (\mathcal{C},\mathcal{X}\webright ) \]
      to be pseudomonic leads to pseudoepic functors; see Item 1b of Item 1 of Proposition 11.7.5.1.2.

    Item 1: Characterisations
    Omitted.

    Item 2: Conservativity
    Omitted.

    Item 3: Essential Injectivity
    Omitted.


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