11.2.10 Pseudoepic Morphisms

Let $\mathcal{C}$ be a bicategory.

Definition 11.2.10.1.1 ▶ Pseudoepic Morphisms

A $1$-morphism $f\colon A\to B$ of $\mathcal{C}$ is pseudoepic if, for each $X\in \text{Obj}\webleft (\mathcal{C}\webright )$, the functor

\[ f^{*}\colon \mathsf{Hom}_{\mathcal{C}}\webleft (B,X\webright )\to \mathsf{Hom}_{\mathcal{C}}\webleft (A,X\webright ) \]

given by precomposition by $f$ is pseudomonic.

Remark 11.2.10.1.2 ▶ Unwinding Definition 11.2.10.1.1

In detail, a $1$-morphism $f\colon A\to B$ of $\mathcal{C}$ is pseudoepic if it satisfies the following conditions:

  1. For all diagrams in $\mathcal{C}$ 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 11.2.10.1.3 ▶ Properties of Pseudoepic Morphisms

Let $f\colon A\to B$ be a $1$-morphism of $\mathcal{C}$.

  1. Characterisations. The following conditions are equivalent:
    1. The morphism $f$ is pseudoepic.
    2. The morphism $f$ is corepresentably full on cores and corepresentably faithful.
    3. We have an isococomma square of the form
      in $\mathcal{C}$ up to equivalence.

Proof of Proposition 11.2.10.1.3.

Item 1: Characterisations
Omitted.

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