The Clowder Project

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. 2.

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

3. 3.

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