9.7.5 Pseudoepic Functors

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

Definition 9.7.5.1.1 ▶ Pseudoepic Functors

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

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

    then $\alpha =\beta $.

  2. For each $X\in \text{Obj}\webleft (\mathcal{C}\webright )$ 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
    \[ \beta =\alpha \mathbin {\star }\text{id}_{F}. \]

Proposition 9.7.5.1.2 ▶ Properties of Pseudoepic Functors

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

  1. Characterisations. The following conditions are equivalent:
    1. The functor $F$ is pseudoepic.
    2. For each $\mathcal{X}\in \text{Obj}\webleft (\mathsf{Cats}\webright )$, 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.

    3. We have an isococomma square of the form
      in $\mathsf{Cats}_{\mathsf{2}}$ up to equivalence.
  2. Dominance. If $F$ is pseudoepic, then $F$ is dominant (Definition 9.7.1.1.1).

Proof of Proposition 9.7.5.1.2.
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 $\mathcal{X}\in \text{Obj}\webleft (\mathsf{Cats}\webright )$, and thus in particular faithful. By Item (g) of Item 5 of Proposition 9.6.1.1.2, this is equivalent to requiring $F$ to be dominant.

Question 9.7.5.1.3 ▶ Characterisations of Pseudoepic Functors

Question 9.7.5.1.4 ▶ Must a Pseudomonic and Pseudoepic Functor Be an Equivalence of Categories

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 [MO 468334].

Question 9.7.5.1.5 ▶ Characterisations of Functors With Pseudoepic Pre/Postcomposition

Is there a characterisation of functors $F\colon \mathcal{C}\to \mathcal{D}$ such that:

  1. For each $\mathcal{X}\in \text{Obj}\webleft (\mathsf{Cats}\webright )$, 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 $\mathcal{X}\in \text{Obj}\webleft (\mathsf{Cats}\webright )$, the postcomposition functor
    \[ F_{*}\colon \mathsf{Fun}\webleft (\mathcal{X},\mathcal{C}\webright )\to \mathsf{Fun}\webleft (\mathcal{X},\mathcal{D}\webright ) \]

    is pseudoepic?

This question also appears as [MO 468125].

Previous

View Subsection 9.7.5 as PDF

Statistics Cite

Noticed something off, or have any comments? Feel free to reach out!

You can also use the contact form below: