Item 2 (01AY) — The Clowder Project

Table of Contents
Part 6: Bicategories
Chapter 13: Types of Morphisms in Bicategories
Section 13.1: Monomorphisms in Bicategories
Subsection 13.1.10: Pseudomonic Morphisms
Item 2 (cite)

13.1.10 Pseudomonic Morphisms

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

Definition 13.1.10.1.1 ▶ Pseudomonic Morphisms

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

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

given by postcomposition by $f$ is pseudomonic.

Remark 13.1.10.1.2 ▶ Unwinding Definition 13.1.10.1.1

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

1.
For all diagrams in $\mathcal{C}$ 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 $X\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$ 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

\[ \beta =\operatorname {\mathrm{id}}_{f}\mathbin {\star }\alpha . \]

Proposition 13.1.10.1.3 ▶ Properties of Pseudomonic Morphisms

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

1.
Characterisations. The following conditions are equivalent:
1. (a)
  The morphism $f$ is pseudomonic.
2. (b)
  The morphism $f$ is representably full on cores and representably faithful.
3. (c)
  We have an isocomma square of the form
  in $\mathcal{C}$ up to equivalence.
2.
Interaction With Cotensors. If $\mathcal{C}$ has cotensors with $\mathbb {1}$, then the following conditions are equivalent:
1. (a)
  The morphism $f$ is pseudomonic.
2. (b)
  We have an isocomma square of the form
  in $\mathcal{C}$ up to equivalence.