Subsection 9.2.9 (01C9): Strict Epimorphisms

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Table of Contents
Part 4: Bicategories
Chapter 9: Types of Morphisms in Bicategories
Section 9.2: Epimorphisms in Bicategories
Subsection 9.2.9: Strict Epimorphisms (cite)

9.2.9 Strict Epimorphisms

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

Definition 9.2.9.1.1 ▶ Strict Epimorphisms

A $1$-morphism $f\colon A\to B$ is a strict epimorphism in $\mathcal{C}$ 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 injective on objects, i.e. its action on objects

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

is injective.

Remark 9.2.9.1.2 ▶ Unwinding Definition 9.2.9.1.1

In detail, $f$ is a strict epimorphism if, for each diagram in $\mathcal{C}$ of the form

if $\phi \circ f=\psi \circ f$, then $\phi =\psi $.

Example 9.2.9.1.3 ▶ Examples of Strict Epimorphisms

Here are some examples of strict epimorphisms.

Strict Epimorphisms in $\mathsf{Cats}_{\mathsf{2}}$. The strict epimorphisms in $\mathsf{Cats}_{\mathsf{2}}$ are characterised in Chapter 8: Categories, Item 1 of Proposition 8.6.3.1.2.
Strict Epimorphisms in $\textbf{Rel}$. The strict epimorphisms in $\textbf{Rel}$ are characterised in Chapter 5: Relations, Proposition 5.3.9.1.1.

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