Item 3 (015Y) — The Clowder Project

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

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.
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.
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.

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.

11.7.4 Pseudomonic Functors