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
\[ \beta =\operatorname {\mathrm{id}}_{F}\mathbin {\star }\alpha . \] -
1.
Characterisations. The following conditions are equivalent:
-
(a)
The functor $F$ is pseudomonic.
-
(b)
The functor $F$ satisfies the following conditions:
-
(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.
-
(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.
-
(i)
-
(c)
We have an isocomma square of the form
in $\mathsf{Cats}_{\mathsf{2}}$ up to equivalence.
-
(d)
We have an isocomma square of the form
in $\mathsf{Cats}_{\mathsf{2}}$ up to equivalence.
-
(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.
-
(a)
-
2.
Conservativity. If $F$ is pseudomonic, then $F$ is conservative.
-
3.
Essential Injectivity. If $F$ is pseudomonic, then $F$ is essentially injective.
-
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
Let $\mathcal{C}$ and $\mathcal{D}$ be categories.
Let $F\colon \mathcal{C}\to \mathcal{D}$ be a functor.
Proof of Proposition 11.7.4.1.2.
Item 1: Characterisations
Omitted.
Item 2: Conservativity
Omitted.
Item 3: Essential Injectivity
Omitted.
