Subsection 11.2.9 (01C9): Strict Epimorphisms

Preferences





Font:

PDF Style:

Here's a breakdown of the differences between each PDF style:

Style	Class	Font	Theorem Environments
Style 1	`book`	Alegreya Sans	`tcbthm`
Style 2	`book`	Alegreya Sans	`amsthm`
Style 3	`book`	Arno*	`amsthm`
Style 4	`book`	Computer Modern	`amsthm`

*To be replaced with Linus Romer's Elemaints when it is released.

Table of Contents
Part 5: Bicategories
Chapter 11: Types of Morphisms in Bicategories
Section 11.2: Epimorphisms in Bicategories
Subsection 11.2.9: Strict Epimorphisms (cite)

11.2.9 Strict Epimorphisms

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

Definition 11.2.9.1.1 ▶ 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 11.2.9.1.2 ▶ Unwinding Definition 11.2.9.1.1

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 11.2.9.1.3 ▶ 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 9: Preorders, Item 1 of Proposition 9.7.3.1.2.
Strict Epimorphisms in $\textbf{Rel}$. The strict epimorphisms in $\textbf{Rel}$ are characterised in Chapter 6: Relations, Proposition 6.3.9.1.1.

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

You can also use the contact form below: