A functor $F\colon \mathcal{C}\to \mathcal{D}$ is pseudoepic if it satisfies the following conditions:
-
1.
For all diagrams of the form
if we have
\[ \alpha \mathbin {\star }\operatorname {\mathrm{id}}_{F}=\beta \mathbin {\star }\operatorname {\mathrm{id}}_{F}, \]then $\alpha =\beta $.
-
2.
For each $X\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$ and each $2$-isomorphism
of $\mathcal{C}$, there exists a $2$-isomorphism
of $\mathcal{C}$ such that we have an equality
of pasting diagrams in $\mathcal{C}$, i.e. such that we have
\[ \beta =\alpha \mathbin {\star }\operatorname {\mathrm{id}}_{F}. \] -
1.
Characterisations. The following conditions are equivalent:
-
(a)
The functor $F$ is pseudoepic.
-
(b)
For each $\mathcal{X}\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Cats}\webright )$, the functor
\[ F^{*}\colon \mathsf{Fun}\webleft (\mathcal{D},\mathcal{X}\webright )\to \mathsf{Fun}\webleft (\mathcal{C},\mathcal{X}\webright ) \]given by precomposition by $F$ is pseudomonic.
-
(c)
We have an isococomma square of the form
in $\mathsf{Cats}_{\mathsf{2}}$ up to equivalence.
-
(a)
-
2.
Dominance. If $F$ is pseudoepic, then $F$ is dominant (Definition 11.7.1.1.1).
-
1.
For each $\mathcal{X}\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Cats}\webright )$, the precomposition functor
\[ F^{*}\colon \mathsf{Fun}\webleft (\mathcal{D},\mathcal{X}\webright )\to \mathsf{Fun}\webleft (\mathcal{C},\mathcal{X}\webright ) \]is pseudoepic?
-
2.
For each $\mathcal{X}\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Cats}\webright )$, the postcomposition functor
\[ F_{*}\colon \mathsf{Fun}\webleft (\mathcal{X},\mathcal{C}\webright )\to \mathsf{Fun}\webleft (\mathcal{X},\mathcal{D}\webright ) \]is pseudoepic?
11.7.5 Pseudoepic Functors
Let $\mathcal{C}$ and $\mathcal{D}$ be categories.
Let $F\colon \mathcal{C}\to \mathcal{D}$ be a functor.
Proof of Proposition 11.7.5.1.2.
Item 1: Characterisations
Omitted.
Item 2: Dominance
If $F$ is pseudoepic, then
\[ F^{*}\colon \mathsf{Fun}\webleft (\mathcal{D},\mathcal{X}\webright )\to \mathsf{Fun}\webleft (\mathcal{C},\mathcal{X}\webright ) \]
is pseudomonic for all $\mathcal{X}\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Cats}\webright )$, and thus in particular faithful. By Item 5g of Item 5 of Proposition 11.6.1.1.2, this is equivalent to requiring $F$ to be dominant.
Is there a nice characterisation of the pseudoepic functors, similarly to the characterisaiton of pseudomonic functors given in Item 1b of Item 1 of Proposition 11.7.4.1.2?
This question also appears as [Liberti, Characterization of pseudo monomorphisms and pseudo epimorphisms in Cat].
A pseudomonic and pseudoepic functor is dominant, faithful, essentially injective, and full on isomorphisms. Is it necessarily an equivalence of categories? If not, how bad can this fail, i.e. how far can a pseudomonic and pseudoepic functor be from an equivalence of categories?
This question also appears as [Emily, Is a pseudomonic and pseudoepic functor necessarily an equivalence of categories?].
Is there a characterisation of functors $F\colon \mathcal{C}\to \mathcal{D}$ such that:
This question also appears as [Emily, Characterisations of functors $F$ such that $F^*$ or $F_*$ is [property], e.g. faithful, conservative, etc].
